Superuniversality of Superdiffusion

Enej Ilievski,1 Jacopo De Nardis,2 Sarang Gopalakrishnan,3 Romain Vasseur,4 and Brayden Ware41Faculty for Mathematics and Physics, University of Ljubljana, Jadranska ulica 19,

1000 Ljubljana, Slovenia2Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium

3Department of Physics and Astronomy, CUNY College of Staten Island, Staten Island, New York 10314;Physics Program and Initiative for the Theoretical Sciences, The Graduate Center,

CUNY, New York, New York 10016, USA4Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA

(Received 24 October 2020; revised 4 May 2021; accepted 19 May 2021; published 28 July 2021)

Anomalous finite-temperature transport has recently been observed in numerical studies of variousintegrable models in one dimension; these models share the feature of being invariant under a continuousnon-Abelian global symmetry. This work offers a comprehensive group-theoretic account of this elusivephenomenon. For an integrable quantum model with local interactions, invariant under a global non-Abelian simple Lie group G, we find that finite-temperature transport of Noether charges associated withsymmetry G in thermal states that are invariant under G is universally superdiffusive and characterized bythe dynamical exponent z ¼ 3=2. This conclusion holds regardless of the Lie algebra symmetry, localdegrees of freedom (on-site representations), Lorentz invariance, or particular realization of microscopicinteractions: We accordingly dub it “superuniversal.” The anomalous transport behavior is attributed tolong-lived giant quasiparticles dressed by thermal fluctuations. We provide an algebraic viewpoint on thecorresponding dressing transformation and elucidate formal connections to fusion identities amongst thequantum-group characters. We identify giant quasiparticles with nonlinear soliton modes of classical fieldtheories that describe low-energy excitations above ferromagnetic vacua. Our analysis of these fieldtheories also provides a complete classification of the low-energy (i.e., Goldstone-mode) spectra ofquantum isotropic ferromagnetic chains.

DOI: 10.1103/PhysRevX.11.031023 Subject Areas: Statistical Physics

I. INTRODUCTION

A complete characterization and classification ofdynamical properties of isolated interacting many-bodysystems remains one of the central unsettled problems instatistical mechanics. Especially in one-dimensional sys-tems, a range of exotic dynamical phenomena have beendemonstrated, both theoretically and experimentally. Twoprominent examples are integrable and many-body local-ized quantum systems [1–3], which feature extensivelymany conserved quantities and therefore can persist innonthermal “generalized Gibbs states” that are measurablydifferent from the orthodox thermal ensemble [4–9].Because these extensive conservation laws lead to non-standard equilibrium states, and because hydrodynamicsbegins with an assumption of local thermal equilibrium, itfollows that hydrodynamics is also modified for integrable

systems. Thus, instead of normal diffusion, nondisorderedintegrable systems typically exhibit ballistic transport withfinite Drude weights [10,11], whereas in localized models,transport is entirely absent [2].Integrable systems feature coherent quasiparticle exci-

tations with an infinite lifetime [12,13] that propagatethrough the system in a ballistic manner while scatteringelastically off one another. The same picture remains validin thermal ensembles at finite temperature, where one canthink of quasiparticles being “dressed” due to interactionswith a macroscopic thermal environment [14]. Thermalfluctuations are responsible for screening, and thus con-served charges carried by quasiparticles can be appreciablydifferent from the bare values. This effect is captured by theversatile framework of generalized hydrodynamics (GHD)[15,16]. Among other results, GHD has enabled the explicitcharacterization of ballistic transport [17–20] and analytictreatments of various other nonequilibrium protocols[21–25]. Remarkably, despite the ballistic motion ofindividual excitations, there are situations in which certainintegrable models do not exhibit ballistic transport onmacroscopic scales but instead display normal diffusionor even anomalous diffusion; this happens specifically for a

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distinguished subset of conserved quantities linked withinternal degrees of freedom, whereas other conser-ved quantities (such as, e.g., energy) undergo ballistictransport [26].In this work, we address the phenomenon of super-

diffusive transport in integrable models. This unexpectedphenomenon was first found in Ref. [29] in the Heisenbergspin-1=2 chain, an archetypal example of a quantummany-body interacting system. Nowadays, there is numeri-cal evidence [30] that the dynamical exponent and asymp-totic scaling profiles of dynamical structure factors belongto the Kardar-Parisi-Zhang (KPZ) universality class [31].Specifically, the diagonal dynamical spin correlationsCjjðx; tÞ≡ hSjðx; tÞSjð0; 0Þi among the spin componentsSj (for all j ∈ fx; y; zg), evaluated in thermal equilibriumat half filling, have been found to comply with thescaling form

Cjjðx; tÞ ≃ Cjj

ðλKPZtÞ2=3fKPZ

�x

ðλKPZtÞ2=3�; ð1Þ

characterized by the scaling function fKPZ (tabulated inRef. [32]) and the KPZ nonlinearity coupling parameter λ,and normalized by diagonal static charge susceptibilitiesCjj ¼ R

dxCjjðx; tÞ. Kardar-Parisi-Zhang physics is a wide-spread phenomenon in stochastic growth processes anddynamical interfaces [33–36]. However, stochasticity isnot of fundamental importance, and there are knowncases of KPZ dynamics—most famously, the Fermi-Pasta-Ulam-Tsingou model of coupled anharmonic oscillators—also in deterministic Hamiltonian dynamical systems.Even though the microscopic mechanism responsiblefor superdiffusion in the quantum Heisenberg chain (along-side other integrable models with non-Abelian symmetryas discussed here) is not yet understood in full detail, itnevertheless seems to fall outside of the conventionalframework of nonlinear fluctuating hydrodynamics(NLFHD) that describes the underlying mode-couplingmechanism in nonintegrable models with a finite number oflocal conservation laws. The fact that spin (or charge)superdiffusion of the KPZ type has also been numericallyobserved in a number of other quantum chains—such asthe higher-spin SU(2) integrable chains and the SO(5)-symmetric spin ladder [37], the Fermi-Hubbard model [38],and even in the classical Landau-Lifshitz equation [39–41](and its higher-rank analogues that exhibit symmetry ofnon-Abelian unitary groups [42])—gives a strong indica-tion that there is a general principle behind superdiffusionin integrable systems with non-Abelian symmetries that isyet to be uncovered.GHD has already provided a number of invaluable

theoretical insights into this question. Specifically, forthe case of the quantum Heisenberg spin chains, boththe dynamical exponent z ¼ 3=2 [43] and the KPZ cou-pling constant λKPZ [44] (though not the scaling function)

have been inferred with the aid of a heuristic extension ofthe GHD framework, taking full advantage of the exactknowledge of the Bethe ansatz quasiparticles. Recently, aphenomenological explanation for the observed KPZ phe-nomenon has been given in Ref. [45], which invokes thenotion of hydrodynamic “soft gauge modes” coupled to aneffective noise stemming from thermal fluctuations of otherhydrodynamic modes. The current understanding is thatsuch soft models are a manifestation of the so-called giantquasiparticles in the long-wavelength regime, whose emer-gent dynamics is governed by a classical action [44]. Thisidea appears to suggest that the observed superdiffusivespin dynamics in classical rotationally symmetric [e.g.,SO(3)-invariant] spin chains has the same microscopicorigin as that of quantum spin chains. In spite of thistheoretical progress, however, a fundamental questionnonetheless remains unsettled: In integrable Hamiltoniandynamical systems, what are the necessary and sufficientconditions for the superdiffusive charge dynamics charac-terized by anomalous dynamical exponent z ¼ 3=2? Wededicate this paper to answering this question on generalgrounds, in both integrable lattice models and integrablequantum field theories.Before delving into mostly technical aspects, we offer a

broader perspective on the problem at hand. By invokingthe universal GHD formulas for the spin diffusion con-stants, it is evident from the outset that divergent chargediffusion constants are only possible in systems withinfinitely many types of quasiparticles in the spectrum.This necessary condition is quite generally fulfilled inintegrable lattice models, including, in particular, the classof models with non-Abelian symmetries, which is ourprimary focus. Given that such models do have infinitelymany quasiparticle types (owing to formation of boundstates), however, there are two scenarios that seem par-ticularly plausible:

(I) All integrable Hamiltonians that are symmetricunder a non-Abelian Lie group G display universalsuperdiffusive dynamics of the Noether charges inG-invariant (i.e., unpolarized) Gibbs states.

(II) Integrable models allow for a wider range ofdynamical exponents, depending, e.g., on the rankor type of Lie algebras and their representationsassigned to local degrees of freedom.

There are several recent studies, e.g., Refs. [37,38,46],which offer evidence in favor of scenario (I). Anotherstrong piece of evidence comes from a recent study [42] oftransport properties in classical integrable matrix models indiscrete space-time, representing discretizations of higher-rank analogues of SUðnÞ-symmetric Landau–Lifshitz fieldtheories on complex projective spaces and Grassmannianmanifolds. Besides providing convincing numerical evi-dence for the KPZ scaling profiles independently of rankr ¼ n − 1, Ref. [42] speculates that the same phenomenonwill persist in all classical (nonrelativistic) G-invariantintegrable field theories whose target manifolds belong

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to Hermitian symmetric spaces. Indeed, the class of modelsconsidered in Ref. [42] governs low-energy spectra ofintegrable SUðnÞ-invariant Lai–Sutherland ferromagneticspin chains [47,48] that are included as part of our survey.On this basis, it is reasonable to anticipate that coherentsemiclassical modes analogous to the aforementioned giantquasiparticles emerge as a general feature of integrablequantum chains. Nevertheless, their microscopic descrip-tion is not known at this moment.However, option (II) cannot be a priori rejected either.

We note that the phenomenological framework of nonlinearfluctuating hydrodynamics [49,50] allows for, in principleat least, an infinite tower of anomalous dynamical expo-nents [51]. Now, NLFHD does not directly apply tointegrable systems, and there are good reasons to doubtthat it can capture superdiffusion in the integrable case [45]:Notably, in NLFHD, superdiffusion arises as a correction toballistic transport, rather than as the leading dynamicalbehavior. Regardless, the existing numerical evidence ondynamical exponents in integrable systems is also ambigu-ous: For instance, a study of the integrable SU(4) spin chainfound a distinct exponent z ≈ 5=3 [52], suggesting thepossibility that different non-Abelian symmetries may, afterall, realize distinct universality classes of anomalous trans-port. A key result of our work will be to show, instead, thatz ¼ 3=2 is superuniversal and that there is no room for adifferent exponent such as z ¼ 5=3. We also provideimproved numerics showing that transport in SU(4) spinchains (and in quantum chains with other non-Abelian Liegroups) is in fact consistent with z ¼ 3=2. In addition to itsfundamental interest, this question is experimentally rel-evant because interacting quantum lattice systems possess-ing SUðNÞ symmetry can be implemented using ultracoldalkaline-earth atoms [53,54]. In the context of solid-statephysics, several strongly coupled ladder compounds alsorealize (approximately) the SU(4)-symmetric ladder in thepresence of fields [55–57].

A. Summary

In this work, we outline a systematic theoretical analysisof anomalous charge transport, specializing to the classof integrable quantum “spin chains” that exhibit globalsymmetries of non-Abelian simple Lie groups. Our analysispartially relies on a universal algebraic description linked torepresentation theory of quantum groups (and the associ-ated fusion relations). For the reader’s convenience, ourfocus will be mainly to collect and discuss all the relevantstatements while relating the largely technical material toRef. [58]. For clarity, we first summarize our main findings.Our central result is that an anomalous algebraic

dynamical exponent z ¼ 3=2 associated with transport ofthe Noether charges is a common feature of integrableHamiltonian systems invariant under a general non-AbelianLie groupG, including both quantum spin chains with localinteractions and quantum field theories. The key extra

requirement is that the equilibrium state does not breakthe global symmetry by the presence of finite chemicalpotentials. This statement holds independently of thetype of simple Lie algebra or the unitary representationsassociated with local Hilbert spaces (in quantum chains) orlocal degrees of freedom (in integrable QFTs): The onlyrequirement is that the charge Q whose correlation func-tions we study must transform nontrivially underG (unlike,e.g., the energy). We thereby establish superuniversality ofsuperdiffusive charge transport.Our analysis incorporates the following complementary

approaches:(i) We carry out a scaling analysis of the universal nested

Bethe ansatz dressing equations, concluding that thespectrum of “giant quasiparticles” exhibits the sametype of asymptotic scaling relations irrespectively ofthe non-Abelian symmetry algebra. This conclusionimplies that the kinetic theory argument outlined inRef. [43] carries through, in general. To solidify thisconclusion,wederive an explicit closed-form solutionof the dressing equations for the case of higher-rankunitary groups SUðNÞ [including the general depend-ence on the U(1) chemical potentials coupling to theCartan charges].

(ii) Second, we verify our predictions through tensor-network-based numerical simulations by computingdynamical charge correlation functions for a numberof representative cases, including unitary, orthogo-nal, and symplectic Lie groups (shown in Figs. 1 and2); evidently, all the cases we have studied yield theexponent z ¼ 3=2.

(iii) Lastly, we elucidate the physical nature of the giantquasiparticles. The latter govern the semiclassicallong-wavelength dynamics of the charge density inhighly excited eigenstates. From the spin chainviewpoint, they correspond to macroscopically largecoherent states made out of interacting, quadraticallydispersing (i.e., “magnonlike”), Goldstone modesabove a ferromagnetic vacuum. When viewed asclassical objects, they are none other than localizednonlinear modes called solitons. Although it mayseem counterintuitive (at first glance at least) thatlong-wavelength modes matter to high-temperaturephysics, we notice that it is precisely due to inte-grability that these large bound-state excitationsremain well-defined quasiparticles even away fromthe vacuum and also in thermal states in the presenceof other modes; however, their (bare) properties getdressed by thermal fluctuations. To corroborate this,we establish a one-to-one correspondence betweenthe Bethe ansatz quasiparticles and the spectrum ofGoldstone modes. This correspondence is subtle:There are many more distinct Goldstone modes for agiven symmetry-breaking pattern than there aremagnon species (flavors) in the Bethe ansatz spec-trum. Therefore, counting the Goldstone modes

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correctly requires the notion of composite quasipar-ticles called “stacks.” To our knowledge, thesemultiflavored stacks have not been explicitly clas-sified thus far.

While we mostly focus on integrable quantum chainswith ferromagnetic exchange, we also briefly discuss, in theAppendix C, how KPZ superuniversality emerges even inintegrable quantum field theories possessing internal iso-tropic degrees of freedom (which take values in compactsimple Lie groups G or coset spaces thereof), both forLorentz-invariant theories and for nonrelativistic interact-ing fermions. Such integrable QFTs are characterized bynondiagonal scattering, signifying that their elementaryquasiparticle excitations can exchange isotropic degrees offreedom upon elastic collisions. This process results indynamically produced massless pseudoparticles that areresponsible for charge transport at finite temperature. Theseso-called “auxiliary quasiparticles” are interacting magnonwaves and bound states thereof, which formally resemblemagnons of the quantum ferromagnetic spin chains [59].Divergent charge diffusion constants are thus, once again,attributed to the presence of interacting giant magnons.Outline. The paper is structured as follows. In Sec. II,

we succinctly review the formalism of generalized hydro-dynamics, provide closed formulas for the charge diffu-sion constants, and proceed by summarizing the generalphysical picture and scaling arguments that lead to

superuniversality. Technical derivations are provided inthe Supplemental Material [58], which contains detailedinformation about the nested Bethe ansatz diagonalizationtechnique and the algebraic structure of the quasiparticlespectra for a family of quantum spin chains, includingRefs. [59–109]. In Sec. III, we discuss our numerical resultson transport, while Sec. IV is devoted to the semiclassicallimit of integrable models and related concepts. In Sec. V,we summarize our results and propose areas for futureexploration.

II. SUPERDIFFUSION FROM GIANTQUASIPARTICLES

We aim to characterize charge dynamics in Hamiltoniansystems invariant under a non-Abelian continuous sym-metry groupG. We begin our presentation by first outliningthe general setting and introducing the key quantities ofthe linear transport theory. We provide compact spectralrepresentations for the charge diffusion constant in terms ofquasiparticle spectra, largely following previous works onthe subject [43,109–112].We specialize to simple Lie groups G of rank r,

generated by Lie algebra g. Owing to the Noether theorem,the system possesses local conserved (Noether) charges,

QðσÞ ¼Z

dxqðσÞðxÞ; ð2Þ

FIG. 1. Dynamical charge correlation functions computed withTEBD, showing asymptotic scaling with dynamical exponent z ¼3=2 for each of the integrable chains, measured by (top) the widthΔxðtÞ of the expanding charge profiles and (bottom) the returnprobability Cð0; tÞ. The dashed lines in the background showΔx ∝ t2=3 and Cð0; tÞ ∝ t−2=3.

FIG. 2. Asymptotic charge correlator profiles rescaled using thedynamical exponent z ¼ 3=2 collapse. The highlighted regionshows the KPZ scaling function for a range of widths fit to theseprofiles—the fit fails, as the tails of the scaling function appear tofall off faster than in KPZ scaling. The inset shows the same plotwith a linear axis.

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with local densities qðσÞðxÞ, one for each Hermitian gen-erator Xσ of Lie algebra g. For simplicity of notation, we donot make explicit distinctions between lattice and con-tinuum models; i.e., in lattice models, spatial integration inEq. (2) should be understood as a discrete summation.Our first objective is to obtain closed-form expressions

for transport coefficients in equilibrium. We specialize tothermal states characterized by grand-canonical Gibbsensembles (at inverse temperature β), corresponding todensity matrices of the form

ϱβ;h ¼ 1

Zβ;hexp

�−βH þ

Xri¼1

hiQðiÞ�: ð3Þ

Here, parameters h≡ fhi ∈ Rg are the U(1) chemicalpotentials that have been assigned to a maximal set of com-muting (Cartan) charges QðiÞ with i ∈ Ir ≡ f1; 2;…; rg,while normalization Zβ;h ≡ Trϱβ;h represents the partitionsum. Our results generalize to an arbitrary generalizedGibbs ensemble, including other conserved charges pre-serving the symmetry.The presence of finite chemical potentials h allows us to

study transport at generic values of the charge densities.When all hi ≠ 0, the ensembles (3) explicitly violate theglobal symmetry of G, and one is left with a residualsymmetry of the maximal Abelian subgroup T ≡ Uð1Þr ⊂G (called the maximal torus). The complete set of dimðgÞNoether charges can accordingly be separated into twosectors: the “longitudinal” charges QðiÞ assigned to the“unbroken” (Cartan) generators and the non-Abelian setof “transversal” charges QðσÞ for σ ∉ I r, satisfying½QðiÞ; Qðσ∉IÞ� ≠ 0, associated with the “broken” generators.In this work, we are exclusively interested in emergentanomalous charge transport that arises in the limit hi → 0where the distinction between longitudinal and transversalcorrelators disappears (since G invariance of ϱβ;h getsrestored); it will thus be sufficient to focus exclusivelyon the longitudinal sector (see, e.g., Ref. [113] for remarksabout the nature of transversal correlators).

A. Generalized hydrodynamics

We now formulate the transport theory for integrablesystems, known as generalized hydrodynamics [15,16](cf. Ref. [114] for an overview). As our starting point,we begin by an infinite tower of local (including quasilocal[9,104]) conservation laws

∂tqkðx; tÞ þ ∂xjkðx; tÞ ¼ 0: ð4Þ

For the time being, k ∈ N is just a formal discrete modeindex. The complete set of (quasi)local charges canbe constructed explicitly with the aid of the algebraicBethe ansatz techniques as in Ref. [104]. The associatedcurrents can be constructed in a similar spirit; see, e.g.,

Refs. [115,116]. We note that, for practical considerations,these steps can be entirely circumvented as long as oneoperates at the level of thermal averages.The GHD formalism provides an explicit prescription

to describe large-scale spatiotemporal variations of thedensities of conserved charges, as provided by the localcontinuity equations (4). The key ingredient is to expressexpectation values of the current densities as functionalsof the charge averages. While there exist various thermo-dynamic state functions to choose from, a particularly con-venient one is to use quasiparticle rapidity densities ρAðθÞ;we have introduced the “type” label A, running overthe entire model-specific (thermodynamic) quasiparticlecontent, and the rapidity variable θ, which parametrizestheir bare momenta, that is, pA ¼ pAðθÞ. Importantly, aninfinite collection of functions fρAðθÞg uniquely specifies amacrostate, physically representing unbiased microcanoni-cal ensembles of locally indistinguishable thermodynamiceigenstates. Furthermore, by exploiting a one-to-one corre-spondence between the expectationvalues of the (quasi)localconservation laws andquasiparticle content in an equilibriummacrostate [100,101], it proves useful to recast Eq. (4) as acontinuity equation for the quasiparticle densities

∂tρAðθ; x; tÞ þ ∂xjAðθ; x; tÞ ¼ 0: ð5Þ

The quasiparticle current densities take a simple factorizedform at the Euler scale [15,16],

jAðθÞ ¼ veffA ðθÞρAðθÞ: ð6Þ

Here, veffA are state-dependent effective group velocitiesdetermined from the dressed quasiparticle dispersions

veffA ðθÞ ¼ ∂θεAðθÞ∂θpAðθÞ

: ð7Þ

For a simple Lie group G of rank r, elementaryquasiparticle excitations (defined with respect to a refer-ence ferromagnetic order parameter) come in exactly rdifferent “flavors.” In addition, quasiparticles participate inthe formation of bound states; a quasiparticle that carries squanta of flavor a is accordingly assigned a pair of quantumnumbers A ¼ ða; sÞ. Therefore, the presence of infinitelymany bound states, where s takes values in an infinitecountable set (typically ranging from 1 to∞), is a commonfeature for all the models we consider subsequently.Expanding above a reference equilibrium state with

densities ρAðθÞ, that is, ρAðθ; x; tÞ ¼ ρAðθÞ þ δρAðθ; x; tÞ,the hydrodynamic evolution of density (or charge) fluctua-tions δρAðθ; x; tÞ on the Euler scale is encoded in the linearpropagator (flux Jacobian) A ¼ ∂j=∂ρ, reading

∂tδρðx; tÞ þ ∂xðAδρÞ ¼ 0; ð8Þ

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where (for compactness of presentation) we have employedthe tensor notation by flattening the quasiparticle andrapidity labels. Equation (8) can be diagonalized by per-forming a basis transformation

δϕ ¼ Ω½n�δρ; ð9Þdepending (nonlinearly) on Fermi functions n of theunderlying equilibrium state, where δϕ are the normalmodes of GHD (defined uniquely modulo normalization ofstatic covariances). In the basis of normal modes, thepropagator A acts diagonally, with eigenvalues given bythe effective velocities (7), that is,

ΩAΩ−1 ¼ veff : ð10ÞThe physical interpretation of Ω½n� becomes transparentin the formalism of the thermodynamic Bethe ansatz, whereit plays the role of a dressing operator for conservedquantities,

qdr ¼ Ω½n�q: ð11Þ

B. Diffusion constants

We now introduce the linear transport coefficients. Here,we define them based on the asymptotic behavior ofdynamical structure factors (or dynamical charge suscep-tibilities). For our purpose, it will be sufficient to onlyconsider the Cartan charges and introduce the associatedr-dimensional matrix of dynamical susceptibilities

Cijðx; tÞ≡ hqðiÞðx; tÞqðjÞð0; 0Þi; ð12Þ

where the brackets h•i designate the connected correlatorevaluated in a grand-canonical Gibbs ensemble given byEq. (3). Static susceptibilities are accordingly given by thetime-invariant sum rule Cij ¼ R

dxCijðx; tÞ.For generic values of the Cartan chemical potentials hi,

the variance of the structure factor in integrable modelsexperiences ballistic spreading,Z

dxx2Cijðx; tÞ ≃Dijt2=z; ð13Þ

signaled by the ballistic dynamical exponent z ¼ 1 andcharge Drude weights Dij [10,117,118]. The charge Drudeweights admit the following mode resolution [18,19]:

Dij ¼XA

ZdθρAðθÞnAðθÞ(veffA ðθÞ)2qðiÞdrA qðjÞdrA : ð14Þ

Here, nAðθÞ≡ 1 − nAðθÞ denote thermal Fermi occupationfunctions associated with vacancies (holes). All the thermo-dynamic quantities in the integrand of Eq. (14) depend ontemperature and chemical potentials.

Drude weights do not provide complete informationabout the late-time relaxation of Cijðx; tÞ. To deduce theasymptotic behavior of Cijðx; tÞ on a sub-ballistic scale, weadopt the kinetic theory framework of Refs. [43,111]. Onenormally envisions a thermodynamic system divided upinto large fluid cells of size l, with each cell beingapproximately in local thermal equilibrium. In a hydro-

dynamic description, both the dressed charges qðiÞdrA andlocal chemical potentials hi get promoted to dynamicalquantities, which, in a macroscopic fluid cell of length l,exhibit thermal fluctuations of the order Oðl−1=2Þ, whichwill, in turn, lead to diffusion.In generic local equilibrium states, diffusion is a sub-

leading correction to the ballistic transport characterized bythe Drude weight. However, in unpolarized thermal ensem-bles in systems with non-Abelian symmetries, the chargeDrude weight vanishes and the leading transport behavior issub-ballistic. Charge Drude weights are proportional to

dressed charges qðiÞdrA carried by quasiparticles, cf. Eq. (14).The latter are quite different from their bare (quantized)

values qðiÞA and depend nontrivially on chemical potentialsof the background equilibrium state, including the U(1)chemical potentials hi. In unpolarized thermal states thatexhibit full invariance under G, the dressed quasiparticlecharges vanish simply by symmetry under G (see, e.g.,Refs. [17,19,45]): One can see this as the screeningof the quasiparticle charge by the thermal environment.Therefore, the charge Drude weight (14) vanishes.The leading response therefore occurs at the diffusive

scale, where one treats the chemical potentials as dynami-cally fluctuating quantities, with fluctuations that aresuppressed by the hydrodynamic scale, Oðl−1=2Þ. Forsufficiently small hi, the quasiparticles carry dressed chargeslinearly proportional to hi; i.e., they behave paramagneti-cally. Fluctuations of chemical potentials induce fluctuationsof thermally dressed charges in accordance with

qðiÞdrA qðjÞdrA ¼ 1

2

Xk;l

∂2ðqðiÞdrA qðjÞdrA Þ∂hk∂hl

����h→0

hkhl þ…: ð15Þ

Notice that the chemical potentials can be simply related todensities of the Cartan charges via

hkhl ¼Xi;j

ðC−1ÞkiðC−1ÞljqðiÞqðjÞ: ð16Þ

Here, Cij ¼ ∂qðiÞ=∂hj ¼ ð∂2=∂hi∂hjÞ logZ are staticcharge susceptibilities, and it will be helpful below to expressthem in terms of a mode expansion [18,19] analogous toEq. (14):

Cij ¼XA

ZdθρAðθÞnAðθÞqðiÞdrA qðjÞdrA : ð17Þ

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The latter also determine the magnitude of charge fluctua-tions, hqðiÞqðjÞi ¼ Cij=l. By combining these two results,thermal fluctuations of dressed charges (or dressed suscep-tibilities) carried by individual quasiparticle modes can beexpressed in the form

hqðiÞdrA qðjÞdrA ih→0 ¼ϒij

A

l; ð18Þ

where

ϒijA ¼ 1

2

Xk;l

ðC−1Þkl ∂2ðqðiÞdrA qðjÞdrA Þ∂hk∂hl

����h→0

¼ CijϒA: ð19Þ

The quantities ϒijA can be physically interpreted as effective

paramagnetic moments, assuming nontrivial dependence onboth quasiparticle quantum numbers A ¼ ða; sÞ.In diffusive dynamics, the variance of the dynamical

structure factor Cij experiences linear growth at late times,ZdxCijðx; tÞ ≃ 2σijt; ð20Þ

characterized by the charge conductivity matrix

σ ¼ DC; ð21Þwith static susceptibility matrix C and charge diffusionmatrix D [119]. A full expression for the conductivity(Onsager) matrix σ has been derived in Refs. [110,112]using the form-factors approach, and its diagonal part inRef. [111] using a kinetic theory formulation. Here, weprovide a compact expression for D in models of higher-rank symmetry (restricted to the Cartan sector), specializingto the limit of vanishing chemical potentials, h → 0. To thisend, we substitute the fluctuation relation (18) into Eq. (13)with Eq. (14) on a characteristic scale l set by quasipar-ticles’ effective velocities, l ¼ jveffA ðθÞjt [43], yielding thefollowing spectral resolution of the conductivity matrix:

σijðh ¼ 0Þ ¼ 1

2

XA

ZdθρAðθÞnAðθÞjveffA ðθÞjϒij

A : ð22Þ

The diffusion matrix is thus proportional to the identity,corresponding to a single value of charge diffusion constant

D ¼ 1

2

XA

ZdθρAðθÞnAðθÞjveffA ðθÞjϒA: ð23Þ

The quantities ϒijA can be easily computed explicitly at

infinite temperature in various integrable spin chains(see Appendixes). For instance, in the An ≡ SUðnÞ ferro-magnetic integrable chains, we find ϒA1

s ¼ 49ðsþ 1Þ4,

ϒA2a;s ¼ 1

16ð1þ sÞ2ð2þ sÞ2, while for n ≥ 4, it depends

on the flavor label a ∈ f1; 2;…; n − 1g. Its precise form

will not matter for our purposes, but note the large-s scalingϒA ∼ s4.By adopting quasiparticle densities as variational objects

(see Appendix A for the derivation) and comparing theresult to Eq. (22), we deduce the following relation betweenthe dressed charge fluctuations and the dressed differentialscattering phases Kdr

AA0 ,

hqðiÞdra;s qðiÞdra;s ih→0 ¼Cii

l

�Xa0qðiÞa0 lims0→∞s0Kdr

as;a0s0

�2

: ð24Þ

Thus, we get the following nontrivial identity:

ϒa;s ¼�X

a0qðiÞa0 lims0→∞s0Kdr

as;a0s0

�2

: ð25Þ

We remark that this expression can be thought of asa generalization of the so-called “magic formula” ofRef. [109], which was originally introduced as a way torelate the different expressions of the diffusion constant ofthe XXZ spin chain with interaction anisotropy Δ > 1obtained from form factors [112] and from a kinetic theoryargument [43] similar to the one we used above.

C. Emergence of superdiffusion

From the considerations above, one might expect to findnormal diffusion with z ¼ 2, with the diffusion constantgiven by Eq. (22). Explicit computation shows, however,that this result diverges as hi → 0 [43,46]. This divergencehas been catalogued for various specific cases; here, weexplain why it holds, in general. To this end, we define theregularized diffusion constant by truncating the spectralsum (22) at some string index s�,

D� ¼Xr

a¼1

Xs�s¼1

Da;s; ð26Þ

where Da;s ≡DA corresponds to each summand inEq. (22). As a finite sum, Eq. (26) is manifestly finite.If D� converges as s� → ∞, then one has normal diffusion.First, we observe that the convergence properties of D�

are tightly linked to those of the static susceptibility.As h → 0, the static charge susceptibilities Cij, given byEq. (17), must approach a constant value. At finite h, theoccupation factor for a quasiparticle of type ða; sÞ(a bound state of s magnons of flavor a) is suppressedwith a factor of expð−shaÞ, and thus quasiparticles withsha ≫ 1 do not contribute to the susceptibility at chemicalpotential ha. The label “a” essentially plays no role in thefollowing argument, so we drop it. Uniformly writingha → h, truncating the sum (17) at s ∼ 1=h, and usingthe paramagnetic behavior of the dressed charges, we haveCij ∼ h2

Pra¼1

Ps<1=h C

ija;s. Requiring that the Cij have a

finite yet nonvanishing limit as h → 0, we see that the sum

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over s has to scale as about s2, so the summand Cija;s must

grow linearly in s. Crucially, however, the summands inEqs. (17) and (22) are the same, except for a factor of thevelocity. Suppose the characteristic velocity of a quasipar-ticle with labels s scales as jveffs j ∼ s−ν. Then, the diffusionconstant must scale as

D� ∼Xs≤s�

s1−ν ∼ s2−ν; ν < 2: ð27Þ

If ν < 2, the diffusion constant diverges, yielding super-diffusive transport, whereas if ν > 2, it converges and onehas diffusive transport. (In the marginal case ν ¼ 2, one hasa logarithmically divergent diffusion constant [120].)Next, we relate this divergence of D� to the time

dependence of transport. Recall that the dynamical criticalexponent z is defined by the relation x2 ∼ t2=z. Followingstandard notation, we define a time-dependent diffusionconstant via the relation x2 ∼DðtÞt, and (by equating thetwo relations above) we observe that

DðtÞ ∼ tð2=zÞ−1: ð28Þ

We now apply the kinetic argument [43] to relate theexponents ν and z. At time t, we can divide quasiparticlesinto “light” quasiparticles for which vst ∼ t=sν > xðtÞ ∼t1=z and “heavy” quasiparticles for which vst < t1=z. Thecrossover between light and heavy quasiparticles takesplace when vst ∼ t=sν ∼ t1=z. Thus, we have thats� ∼ tð1−1=zÞ=ν. Plugging this result into Eq. (27), we arriveat the conclusion

DðtÞ ∼D(s�ðtÞ) ∼ ½tð1−1=zÞ=ν�2−ν: ð29Þ

Equating the exponents in Eqs. (28) and (29), we find thefollowing relation between the exponents z and ν:

z ¼�1þ ν

2

�; ν < 2: ð30Þ

From the analytic structure of the Bethe ansatz solutions,one can infer that the exponent ν ≥ 0 is integer valued. Thisfact rules out all but three possible values for z. The threeremaining possibilities are ballistic transport (ν ¼ 0), whichis ruled out by the vanishing Drude weight discussed above;KPZ scaling (ν ¼ 1), which we will argue is generic; anddiffusion with possible logarithmic corrections (ν ≥ 2),which occurs in certain fine-tuned models [120] wherethe single-magnon dispersion is fine-tuned to scale as ω ∼k3 (or slower) at long wavelengths k → 0. The onlysuperdiffusive exponent that is allowed by the structureof these integrable models is z ¼ 3=2.In what follows, we will establish z ¼ 3=2 within a

universal algebraic description of the thermodynamicdressing equations and link them to the representation

theory of the underlying symmetry structures—quantumgroups called Yangians, in particular. We find, remarkably,that the Fermi functions assume universal algebraic scalingat large s,

na;s ∼1

s2; ð31Þ

which is a direct corollary of fusion identities amongst thequantum characters associated to the Yangian symmetry[67–69,121,122]. Similarly, we find that the total statedensities and the dressed velocities of giant magnons (whenmultiplied by a regular function and integrated over therapidity domain [58]) decay as

ρtota;s ¼ρa;sna;s

∼1

sand veffa;s ∼

1

s; ð32Þ

respectively, for all flavors a ¼ 1;…; r. Most remarkably,these large-s scaling properties hold irrespectively of thelocal on-site degrees of freedom (i.e., finite-dimensional,irreducible, unitary representations of g); they can beunderstood as kinematic constraints stemming from theunderlying quantum symmetry algebra. Finally, notice thatϒA ∼ s4 in conjunction with the above scaling relationsimplies υ ¼ 1. Thus, the upshot is that the anomalousfractional algebraic dynamical exponent z ¼ 3=2 is deeplyrooted in the fusion relations for the quantum characters.This case establishes the superuniversality of superdiffu-sion with the dynamical exponent z ¼ 3=2 and rules out thepossibility of different exponents such as z ¼ 5=3, whichwas suggested in Ref. [52] from a numerical analysis of SU(4) integrable chains. We will confirm this predictionnumerically in the next section.

III. NUMERICAL ANALYSIS

We verify our predictions for the superdiffusive chargetransport on a number of representative instances ofintegrable quantum spin chains. In our analysis, we includerepresentative examples of quantum chains invariant undersimple Lie groups from the classical series. Specifically, weconsider homogeneous integrable spin-chain Hamiltonianswhose local degrees of freedom transform in the funda-mental representation of various symmetry groups (seeSupplemental Material [58] for explicit expressions).Exceptional algebras are exempted from our numericalanalysis. Moreover, we leave out the B series, as thefundamental integrable B2 ≡ SOð5Þ chain has already beenstudied numerically in a recent paper [37].We employ numerical tensor-network-based computa-

tions of the dynamical correlation function Ciiðx; tÞ [asdefined in Eq. (1)] in the canonical Gibbs equilibrium atinfinite temperature. Owing to the non-Abelian symmetryG of both the Hamiltonian H and infinite-temperatureGibbs density matrix, all the components i ¼ 1; 2;…; r are

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identical, and we thus suppress the redundant index i. Weperform the time-evolving block-decimation (TEBD) algo-rithm in the Heisenberg picture [123–125], evolving thelocal charge density operator qð0; tÞ in a matrix productoperator (MPO) representation.Numerical methods. In the simulations, a fourth-order

Trotter decomposition is used to propagate the operatorforward in time steps of size δt, followed by truncations ateach bond keeping the largest χ Schmidt states. Thesimulations are accelerated by taking advantage of time-reversal symmetry and translation symmetry to computethe full correlator Cðx; tÞ ¼ hqðx; t=2Þqð0;−t=2Þi using asingle MPO evolution; additionally, the TEBD implementsthe maximal Abelian subgroup of the on-site symmetrygroup for efficiency. The computations are checked forconvergence in δt and bond dimension χ (up to χ ¼ 1024).Our TEBD scheme uses a Trotter step size of δt ¼ 0.4.

The operators are truncated initially with a constantdiscarded weight ε ¼ 10−8, allowing the bond dimensionχ to grow until it reaches a threshold χmax ¼ 512.Subsequent truncations keep only χmax states. The simu-lations are checked for convergence in the step size δt, thetruncation error ε, and the threshold bond dimension χmax.Additionally, the results shown here include rescaling of thecorrelations at each time to explicitly enforce the chargesum rule

RdxCðx; tÞ ¼ Cð0; 0Þ, which improves the con-

vergence significantly.Results. Our main results are succinctly summarized in

Fig. 1, where we display the dynamical width of the chargeprofiles

½ΔxðtÞ�2 ¼Rdxx2Cðx; tÞRdxCðx; tÞ : ð33Þ

We find a clear signature of an asymptotic power-law growthΔxðtÞ ∝ t1=z, and the return probability shows power-law decay Cð0; tÞ ∝ t−1=z, with a numerically estimateddynamical exponent z ¼ 3=2 with great accuracy. Thescaling collapse in Fig. 2 shows that the correlators obeythe asymptotic scaling form Cðx; tÞ ≃ t−1=zfscðxt−1=zÞ.Comparing fsc to the KPZ scaling function fKPZ given byEq. (1), we find some discernible deviations in the tails[including the basic SU(2) case studied previously inRef. [30] ]. On accessible timescales, we are unable todetermine whether this discrepancy persists at late timesor if it is merely due to transient effects.

IV. SEMICLASSICAL SPECTRUM

The basic picture that we have established thus far is thatthe anomalous charge transport comes from giant quasi-particles (i.e., bound states carrying large bare Noethercharge) immersed in a charge-neutral thermal background.It is natural to expect such giant quasiparticles to have asemiclassical description. In this section, we discuss such a

semiclassical description. It is helpful to first review someresults along these lines that were shown for SO(3) spinchains [44]. We then proceed by outlining how thesearguments generalize to other symmetry groups.Consider a one-dimensional classical SO(3) ferromagnet

in its ground state, i.e., all spins aligned in some direction.The elementary excitations of the ferromagnet are softGoldstone modes in which the spin orientation changesslowly across the lattice. One can regard any sufficientlyslowly varying spin texture as being composed ofGoldstone modes, as it locally consists of slow modulations(rotations) of the vacuum orientation. The long-wavelengthdynamics of these ferromagnetic, quadratically dispersing,Goldstone modes is governed by a linear PDE. Away fromthe ground state, the evolution of the order parameter atlarge scales is governed instead by the nonlinear Landau-Lifshitz equation. Remarkably, the latter is an integrablePDE [126] that features stable, nonlinear, ballisticallypropagating modes of any width w [127], known assolitons. The properties of these soliton solutions can beinferred by observing that they are wave packets ofGoldstone modes that are stabilized by nonlinearity. Forexample, a soliton of width w is made up of Goldstonemodes with characteristic momentum 1=w and thus char-acteristic energy density 1=w2; therefore, its characteristicenergy scales as 1=w and its velocity also as 1=w. On theother hand, its spin orientation is away from the vacuum byOð1Þ, so it carries a finite amount of magnetization, aboutw. These “bare” properties of classical solitons preciselymatch those of the large-s Bethe ansatz strings, suggestingthat there is some correspondence between them. Indeed,the exact correspondence between large-s strings and large-w solitons can be explicitly established, not only at the levelof individual spin-wave configurations as usual but even inhigh-temperature equilibrium ensembles [44]. (Note that,for this correspondence to hold, it is irrelevant whether themicroscopic model has ferromagnetic or antiferromagneticcouplings: for the considerations in this paper, the Bethevacuum always corresponds to the ferromagnetic groundstate, and elementary excitations like solitons are alwaysconstructed above this state.)This classical quasiparticle construction generalizes to

arbitrary quantum or classical ferromagnets. As discussedbelow, distinguished common features of such ferromag-nets are quadratically dispersing Goldstone modes. One cananalogously construct slowly varying wave packets of theseGoldstone modes; by construction, these have the samescaling as the large-s strings, so it is tempting to identify thetwo types of excitations in the general case also. Althoughwe have not yet attempted to analytically solve the classicalproblem in full generality, by noting that large-s stringsmust correspond to objects in the semiclassical spectrumand that Goldstone-like excitations exhaust this spectrum, itis tempting to infer the existence of such solitons from theintegrability of the quantum model.

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However, one major puzzle arises when one tries toperform such an identification: The number of magnonflavors (which always coincides with the rank r of thegroupG) can be far fewer than that of Goldstone modes; thelatter is given by one-half of the real dimension of a cosetmanifold G=H, where H is the residual symmetry group ofthe ferromagnetic state. There are apparently some “miss-ing” Goldstone bosons that must, in some fashion, emergeout of the Bethe ansatz spectrum; however, they cannot beidentified directly with elementary (physical and auxiliary)magnons of integrable quantum chains. Curiously, theyrather correspond to “stacks” of magnons, which are long-lived wave packets in a finite system, and only becomesharp eigenstates in the infinite system limit.The main result of this section is identifying the

correspondence between Goldstone modes in the semi-classical spectrum and stacks of magnons (and condensatesthereof, representing semiclassical Bethe s strings) in theBethe ansatz spectrum. Using this correspondence, onecan identify large-spin semiclassical excitations abovethe ferromagnetic vacuum precisely as stacked strings.These semiclassical excitations can then be identified assolitons that involve smooth maps fromR to the coset spaceG=H. While it would be interesting to construct suchsolitons directly in the classical theory, this task is left tofuture work.

A. Counting nonrelativistic Goldstone modes

Before detailing the spectra of classical continuousferromagnets, we begin by first characterizing the spec-trum of linear fluctuations of a ferromagnetic order param-eter. This approach leads to the notion of Goldstoneexcitations, which naturally arise in systems with sponta-neously broken, continuous internal symmetry. To thisend, we briefly revisit the Goldstone theorem in a generalcontext of nonrelativistic field theories. Our primary objec-tive here is to characterize the (nonrelativistic) Goldstonemodes that govern the low-energy spectrum for the class ofintegrable ferromagnetic quantum spin chains discussed inthis work.Let us first recapitulate the main result on the counting

of Goldstone modes in systems without Lorentz invariance,as independently worked out in Refs. [128–130] (improv-ing upon the earlier Nielsen-Chadha inequality [131]).The conventional setting concerns Hamiltonian systemsinvariant under a non-Abelian Lie group G, possessing adegenerate ferromagnetic ground-state manifold. Spon-taneous breaking of internal symmetry amounts to pickinga particular vacuum polarization (order parameter), say Ω,thereby breaking the symmetry of the isometry group Gdown to the stability subgroup H ⊂ G of Ω, determined bythe condition

hΩh−1 ¼ Ω; h ∈ H: ð34Þ

We are interested specifically in homogeneous (i.e., trans-lation-invariant) ferromagnetic chains (of length L), wherethe global (pseudo)vacuum is simply a product state

ΩL ¼ ⊗L

l¼1Ω. It is therefore sufficient to carry out the

analysis at the level of local Hilbert spaces, and (with noloss of generality) we can assume Ω≡ j0ih0j.At the algebraic level, the symmetry-breaking pattern

G → H implies the following splitting of the generators,g ¼ h ⊕ m, with subalgebra h constituting the generatorsof H (which contains the Cartan subalgebra t) and mdenoting the linear span of the “broken generators” (thesedo not enclose an algebra). The number of broken gen-erators is

nb ¼ dimm ¼ dimðGÞ − dimðHÞ: ð35Þ

Let Xσ ∈ m, σ ∈ f1; 2;…; dimðG=HÞg be a Hermitianbasis of m. By the Goldstone counting theorem, thenumber of independent Goldstone modes nGB is

nG ¼ nb −1

2rankV; ð36Þ

with matrix

Vab ¼ −ih0j½Xa;Xb�j0i ð37Þ

containing the vacuum expectation values of the commu-tators amongst the broken generators Xσ.Goldstone modes come in two varieties, depending on

the type of dispersion law at long wavelengths (k → 0),ωðkÞ ∼ kν: type I (or type A) with ν odd and type II (or typeB) with ν even. In general, we thus have

nG ¼ nI þ nII; ð38Þ

with

nI ¼ nb − 2nII; nII ¼1

2rankV: ð39Þ

In relativistic systems, Lorentz invariance forces V to beidentically zero. In contrast, target spaces of nonrelati-vistic ferromagnets are symplectic manifolds. They haveeven (real) dimension, and consequently, V has full rank.Therefore, nI ¼ 0 and

nG ≡ nII ¼1

2nb: ð40Þ

In simple terms, every canonically conjugate pair of brokengenerators contributes an independent magnon branch, aquadratically dispersing, long-wavelength, spin-wave exci-tation of the ferromagnetic order parameter.

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B. Symmetry-breaking patterns

We now describe the spectra of Goldstone modes for theclass of G-invariant ferromagnetic chains. The structure ofcoset spaces depends on the choice of a finite-dimensionalirreducible on-site representation VΛ. Let us, mostly forsimplicity, suppose for the moment that the local degrees offreedom transform in the defining representation of g. Thehighest weight is the first fundamental weight Λ ¼ ω1 ¼½1; 0;…; 0� of g, and the stabilizer H ⊂ G of the vacuumstate Ω has the structure Uð1Þ ×H0 (i.e., H is not semi-simple). For the classical series, the coset structure issummarized in Table I.Next, we specialize to the family of fundamental

representations with Λ ¼ ωa. In this case, the symmetry-breaking pattern can be inferred directly from the enumer-ated Dynkin diagrams with the use of a graphical recipe bysimply “breaking the bonds” attached to the node with thenonzero label. This case is illustrated in Fig. 3 on twosimple examples. The list of possibilities for all the classicalseries is summarized in Table II, with concrete examplesprovided in Table III. Employing the same trick allows us toalso determine the structure of coset spaces for G-invariantclassical ferromagnets corresponding to the exceptionalgroups G2; F4; E6−8, specializing to the fundamental

representations of g (for completeness, we include themin the Supplemental Material).The outlined recipe likewise applies for generic (i.e.,

nonfundamental), finite-dimensional, irreducible represen-tations VΛ of g with Dynkin labels Λ ¼ ½ω1;…;ωr�; therule is to break all the bonds connecting to nodes withnonzero Dynkin labels. The resulting coset spaces aregeneralized flag manifolds F i1;i2;…;ik ≡G=Hi1;i2;…ik

, where

the indices ii mark the nodes of the Dynkin diagram of gwith nonvanishing Dynkin labels (that is, ii ¼ 1 if and onlyif mi ≠ 0). Compact complex manifolds F i1;i2;…;ik indeedexhaust all Kähler manifolds (i.e., possess compatibleRiemannian metric and symplectic structures), which arehomogeneous spaces with a transitive action of G.Furthermore, points on such flag manifolds are in one-to-one correspondence with generalized coherent states in aspecific representation of group G, namely,

jψi ¼ exp

�Xα∈Δþ

w−αE−α

�jΛi; ð41Þ

for the highest weight state jΛi of VΛ. However, if somema ≠ 0, not all coordinates w−

α are independent, and somecan be eliminated. For instance, in the unitary case (Aseries), the family of generalized flag manifolds has thestructure

SUðnÞ=S½Uðn1Þ × Uðn2Þ × � � � × UðnjÞ × Uð1Þ×k�; ð42Þ

for integers j, k ≥ 1 and n1 ≥ n2 ≥ … ≥ nj > 1, subjected

to kþPji¼1 ni ¼ n. For example, for the case of G ¼

SUð3Þ, there are two possible isotropy groups: H1;0 ¼H0;1 ¼ Uð2Þ [as, e.g., for ð3Þ, ð3Þ, or ð6Þ] orH1;1 ¼ Uð1Þ×2[as, e.g., for ð8Þ, ð15Þ, or ð27Þ]. The upshot of that is thatfor quantum spin chains with different local degrees of

TABLE I. Target spaces associated with classical continuousferromagnets, describing the low-energy sector of quantumferromagnetic chains with local degrees of freedom realized inthe defining representation Vω1

of a Lie algebras g. Here, nG is thenumber of Goldstone modes in the spectrum.

g Vω1Target space G=H nG

An ðnÞ SUðnÞ=Uð1Þ × SUðn − 1Þ n − 1Bn ð2nþ 1Þ SOð2nþ 1Þ=Uð1Þ × SOð2n − 1Þ 2n − 1Cn ð2nÞ USpð2nÞ=Uð1Þ × USpð2n − 2Þ 2n − 1Dn ð2nÞ SOð2nÞ=Uð1Þ × SOð2n − 2Þ 2n − 2

FIG. 3. Various symmetry-breaking patterns in fundamental ferromagnets with on-site irreducible representations Vωa, depicted for

Lie algebras C4 ¼ spð4Þ (top) and D5 ¼ soð10Þ (bottom). The vacuum stability subgroup H always includes a U(1) factor. The nthDynkin node with label 1 corresponds to the ath fundamental irreducible representation with highest weight Λ ¼ ωa. Dashed linesdesignate the broken bonds.

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freedom (i.e., transforming in different irreducible repre-sentations), the classical degrees of freedom of theiremergent, effective, low-energy theories can live on thesame target coset manifold. Below, we explain this factfrom another perspective by directly examining the struc-ture of the semiclassical spectrum.Hermitian symmetric spaces. An important subclass of

flag manifolds includes irreducible compact Hermitiansymmetric spaces. Amongst cosets (42), one finds (inthe special cases of Λ ¼ ωa) complex Grassmannian mani-folds SUðnÞ=S½UðkÞ × Uðn − kÞ�. The remaining infinitefamilies of classical (i.e., nonexceptional) Hermitian sym-metric spaces comprise (i) Lagrangian GrassmanniansUSpð2nÞ=UðnÞ, associated with representations Λ ¼ ωn ¼½0; 0;…; 1� from the Cn series, SOð2nÞ=UðnÞ of complexdimension 1

2nðn − 1Þ, obtained from the fundamental spinor

representations ofDn (with Dynkin labels Λ ∈ fωn−1;ωng),and (ii) orthogonal Grassmannians SOðnÞ=½SOðn − 2Þ ×SOð2Þ� of real dimension 2n, obtained from the definingSOðnÞ representations Λ ¼ ω1 of Bn or Dn [notice, how-ever, that compact Riemannian symmetric spaces, such as,

e.g., SUðnÞ=SOðnÞ and SUð2nÞ=USpð2nÞ, do not appear].Concerning charge transport, we note that a recent numericalstudy [42] demonstrated that a class of integrable, non-relativistic, classical sigma models (of the Landau-Lifshitztype) on complex and Lagrangian Grassmannians supportsanomalous charge transport with the dynamical exponentz ¼ 3=2 (and KPZ scaling profiles).

C. Goldstone modes in the Bethe-ansatz spectrum

So far in this section, we have shown how to countGoldstone modes for a wide class of ferromagnets. Thesearguments do not rely on integrability. However, when themodel is integrable, one can express all eigenstates usingthe Bethe ansatz, so the Goldstone modes must correspondto some type of object in the Bethe-ansatz spectrum. Forthe SU(2) case, the correspondence is obvious: the Betheansatz magnons are precisely the expected quadraticGoldstone modes. One might be tempted to conjecturethat the same correspondence holds for other symmetrygroups, namely, that the total number of internal degreesof freedom in generic classical field configurations willexactly match the number of distinct flavors of theBethe roots, that is, r ¼ rankðgÞ. However, one canquickly realize that there has to be a flaw in such a naiveidentification. There are two apparent inconsistenciesthat arise:(1) First, consider the fundamental SUðnÞ ferromagnetic

chains. For these chains, the Goldstone theorempredicts n − 1 magnonic branches, and indeed, thisnumber precisely agrees with the rank of SUðnÞ,which also coincides with the total number of flavorsin the SUðnÞ magnets. Nevertheless, one cannotdirectly identify the magnon branches with Gold-stone branches since all but one of the magnonbranches are auxiliary quasiparticles. In generic,excited, quantum eigenstates, auxiliary quasipar-ticles cannot be excited on their own without firsthaving physical (i.e., momentum-carrying) excita-tions in a state. (By contrast, all the Goldstonebranches are manifestly physical.)

(2) Consider, next, as another example, the SO(5) chain(which has rank r ¼ 2) made out of the fundamentaldegrees of freedom. The low-energy theory is associ-ated with the coset space SOð5Þ=½SOð2Þ × SOð3Þ�,which is a real Grassmannian manifold of realdimension 6. Here, we expect to find three Gold-stone modes, which now obviously exceeds the totalnumber of flavors. More generally, one would expectthe number of distinct types of macroscopic semi-classical strings to always equal r ¼ rankðgÞ, whichis typically vastly smaller than the number ofindependent Goldstone modes (cf. Sec. IV B).

To evade an apparent paradox, the only viable option isto look for additional degrees of freedom. We illustrate theresolution of this paradox first in the simplest example of

TABLE II. Symmetry-breaking patterns for all the fundamentalrepresentations Λ ¼ ωk for the family of classical Lie algebras g.

g irrep Λ Stabilizer H dimðG=HÞAn ωk Ak−1 × Uð1Þ × An−k 2kðnþ 1 − kÞBn ωk≤n−2 Ak−1 × Uð1Þ × Bn−k kð4nþ 1 − 3kÞBn ωn−1 An−2 × Uð1Þ × A1 ðnþ 4Þðn − 1ÞBn ωn An−1 × Uð1Þ nðnþ 1ÞCn ωk≤n−2 Ak−1 × Uð1Þ × Cn−k kð4nþ 1 − 3kÞCn ωn−1 An−2 × Uð1Þ × A1 ðnþ 4Þðn − 1ÞCn ωn An−1 × Uð1Þ nðnþ 1ÞDn ωk≤n−3 Ak−1 × Uð1Þ × Cn−k kð4n − 1 − 3kÞDn ωn−2 An−3 × Uð1Þ × A1 × A1 ðnþ 5Þðn − 2ÞDn ωn−1;ωn An−1 × Uð1Þ nðn − 1Þ

TABLE III. Coset spaces for all the fundamental representa-tions Λ ¼ ωa for classical Lie algebras of rank 4.

g On-site irrep Target space G=H nG

A4 ð5Þ; ð5Þ SUð5Þ=Uð1Þ × SUð4Þ 4ð10Þ; ð10Þ SUð5Þ=Uð1Þ × SUð2Þ × SUð3Þ 6

B4 ð9Þ SOð9Þ=SOð2Þ × SOð7Þ 7ð36Þ SOð9Þ=Uð1Þ × SUð2Þ × SOð5Þ 11ð84Þ SOð9Þ=Uð1Þ × SUð2Þ × SUð3Þ 12ð16Þ SOð9Þ=Uð1Þ × SUð4Þ 10

C4 ð8Þ USpð8Þ=SOð2Þ × USpð6Þ 7ð27Þ USpð8Þ=Uð1Þ × SUð2Þ × SOð5Þ 11ð48Þ USpð8Þ=Uð1Þ × SUð2Þ × SUð3Þ 12ð42Þ USpð8Þ=Uð1Þ × SUð4Þ 10

D4 ð8vÞ; ð8cÞ; ð8sÞ SOð8Þ=Uð1Þ × SUð3Þ 6ð28Þ SOð8Þ=Uð1Þ × SUð2Þ×3 9

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SU(3) magnets, in an elementary way, and then extend ouranalysis to the general case in the following section.

1. Example: SU(3) ferromagnet

In the SU(3) ferromagnet, there are two Goldstonemodes. The single-site Hilbert space has three states,which we label j0i, j1i, j2i. We choose our ferromagneticground state so that all spins are polarized in state 0, i.e.,jGSi ¼ ΩL ¼ j00…00i. Recall that the Hamiltonian sim-ply permutes the labels on neighboring sites, so one canidentify the low-lying excitations above the ferromagnet byinspection. There are two excitation branches; they corre-spond, respectively, to Bloch states of the form jM1;ni ¼ð1= ffiffiffiffi

Lp Þðj100…0i þ e2niπ=Lj010…0i þ…Þ and jM2;ni ¼

ð1= ffiffiffiffiL

p Þðj20…00i þ e2niπ=Lj020…0i þ…Þ (with n ¼ 1;…; L − 1, excluding the zero-momentum states n ¼ 0).These branches are symmetry related but do not mixunder the Hamiltonian, so they are both quadraticallydispersing (in the large-system limit) with identical dis-persion relations.As we noted above, there are manifestly two orthogonal

magnon branches, but only one physical magnon in the Betheansatz spectrum. The resolution to this apparent paradox is asfollows: When a non-Abelian symmetry is present, only thehighest-weight eigenstates are directly constructed by theBethe ansatz. All the remaining eigenstates are descendantsand can be reached from these by acting with appropriatelowering operators. In the present case, one of the magnons isa descendant of the other and therefore is not explicitlypresent in the Bethe ansatz enumeration.We now turn to the next simplest states, which are those

with two flipped spins. There are two families of thesestates, one in which both flipped spins have the same flavor,i.e., involve configurations of the form j01010…0i, andanother in which they have different flavors, i.e., involveconfigurations of the form j01020…0i. The latter type ofstate is distinguished from the former (in the Bethe ansatzenumeration) by containing two physical magnons and oneauxiliary magnon. It is important to note that neither thetype-1 nor the type-2 magnon is a well-defined propagatingobject in this state, as the magnons can “exchange” theirflavor index. In other words, the flavor is not “bound” toeach quasiparticle but has its own autonomous dynamics.(This phenomenon is analogous to spin-charge separationin the Hubbard model.)Now, supposewe have a very long spin chain that contains

two excitations of this form. A typical configuration close tothe ground state would be j000…00100…00200…000i,i.e., the type-1 and type-2 magnons would be so widelyseparated that they could be regarded as effectively inde-pendent objects. In this scenario, it is natural to think of thetype-1 magnon as “just” a physical magnon, whereas thetype-2 magnon is a composite object containing one physicaland one auxiliary magnon, which propagate together. This is

the basic intuition underlying the concept of stacks, whichwe will discuss in more generality in what follows.

D. General Goldstone-stack correspondence

Let us remind the reader that magnon modes arise asplane-wave solutions to linear partial differential equations.On the other hand, classical field theories are genuinelyinteracting systems that are governed by nonlinear equationsof motion. The latter exhibit a vastly richer spectrum ofsolutions. As mentioned earlier, a hallmark feature of non-linear, completely integrable PDEs are soliton solutions,referring to nonlinear, ballistically propagating, particlelikefield configurations whose stability is ensured by integra-bility. In the isotropic ferromagnets considered in this work,magnetic solitons assume a nontrivial internal structure.Indeed, from the perspective of quantum spin chains,solitons emerge as certain macroscopic coherent super-positions of magnons. Such states represent highly excitedsemiclassical eigenstates in the low-energy spectrum of themodel, and their time evolution is generated by an effectiveclassical Hamiltonian. There are various ways to establishthis fact; the most standard and direct route is to make use ofthe standard path-integral techniques to project the quantummany-body Hamiltonian onto the manifold of (generalized)coherent states jψi, formally achieved by rotating the(pseudo)vacuum, jψi ¼ gjΛi, as prescribed by Eq. (41).This way one can deduce a classical evolution for theferromagnetic order parameter that takes values on a quotientmanifold G=H, where H is the vacuum stability subgroupdefined above in Eq. (34) (see also, e.g., Refs. [42,132,133]).Classical fields can thus be identified with coordinates ofgeneralized (Perelomov) coherent states [134].Every admissible classical solution is a particular long-

wavelength (i.e., low-momentum) solution of the Betheansatz equations with a macroscopically large number ofquanta. This particular regime, often referred to as the“asymptotic Bethe ansatz,” was first studied by Sutherland[48] and, subsequently, more thoroughly examined inRef. [135]. Nevertheless, an exact identification with classi-cal field solutions was only made precise afterwards inRef. [136], which provided a prescription to construct theassociated classical spectral curve (the characteristic equa-tion associated with the classical monodromy matrix; see,e.g., Refs. [137,138]) by transcribing the asymptotic Betheequations as a Riemann-Hilbert problem. This methodpermits us to describe and classify the nonlinear modesfor the class of so-called finite-gap solutions [139].In the remainder of this section, we explain how the

spectrum of both linear and nonlinear modes of classicalintegrable ferromagnets arises out of the nested Betheansatz magnon spectra in integrable G-invariant ferromag-netic quantum chains. A complete characterization ofclassical algebraic curves that emerges in this picture isa technical matter that falls outside of the scope of thisstudy, and we hence do not attempt to undertake it here.

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Rather, we focus here on a deceptively innocent problem ofidentifying the relevant emergent classical degrees offreedom. The task turns out to be quite subtle, and inorder to resolve it, we need a basic description of theasymptotic Bethe ansatz.

1. Asymptotic Bethe ansatz

We begin by noting that every finite-volume (highest-weight) Bethe eigenstate in a quantum chain can beuniquely resolved in terms of magnon excitations.Individual unbound magnons correspond to real-valuedBethe roots with rapidities θ ∼OðL0Þ. As describedearlier, complex-valued rapidities of constituent magnons,which are part of a bound state, take the form of Bethestrings. Semiclassical eigenstates in the spectrum belong tohighly excited states with OðLÞ excitations and largerapidities θ ∼OðLÞ, which can be thought of as condensinginto a finite number of coherent macroscopic modes. Inthis low-energy regime, magnon momenta are inverselyproportional to the system size, pðθÞ ∼Oð1=LÞ. By

accordingly introducing rescaled rapidities λðaÞj via

θðaÞj ¼ LλðaÞj , the Bethe equations in the leading orderOðLÞ assume the form

2πnðaÞj þ ma

λðaÞj

¼Xrb¼1

XNb

k¼1;k≠j

t−1a Kab

λðaÞj − λðbÞk

: ð43Þ

As usual, nðaÞj represent (integer) mode numbers due to themultivaluedness of the complex logarithm.Classical nonlinear modes pertain to bound states made

of Na ∼OðLÞ constituent low-momentum magons.Crucially, rapidities of two nearby magnons are separatedby an amount oðLÞ; however, since θ ∼OðLÞ, magnons arenot, in general, spaced equidistantly as in the Bethe stringswith θ ∼Oð1Þ. What generically happens is that, upon

taking the large-L limit, rapidities λðaÞj distribute denselyalong certain one-dimensional disjoint segments (i.e.,

contours) CðaÞ ¼ ∪iCðaÞi in the spectral λ plane, described

by some smooth (in general, nonuniform) densities ϱaðλÞ ofBethe roots λðaÞj with support on CðaÞ. Such macroscopiclow-energy configurations assume a purely classicaldescription, appearing as interacting “nonlinear Fouriermodes” of nonlinear classical field configurations. Thelong-wavelength spectrum of a classical field may, inaddition, involve a continuous spectrum of small fluctua-tions, namely, modes that carry infinitesimal charges—these are the aforementioned Goldstone modes.

2. Classical degrees of freedom: Stacks

As we discussed in the previous subsection, the countingof Goldstone modes does not match that of magnons butrather that of stacks. We now discuss these emergent

objects in detail. To the best our knowledge, stack exci-tations were first encountered in Refs. [140–142] in thestudies of gauge-string duality in certain unitary super-groups that underlie the superconformal Yang-Mills theory[143,144]. Other than that, there has not been any generalor systematic discussion devoted to these emergent andinherently classical objects so far in the literature. Weproceed by explaining how the notion of stacks naturallyarises in the context of integrable quantum ferromagnetswith continuously degenerate ground states.In the most basic terms, stacks pertain to excitations

with an internal multiflavor structure. They are producedby merging together a subset of Bethe roots of distinctflavors that share the same rapidity (not merely just the realpart). As we already emphasized previously, in a quantumspin chain, all rapidities must be pairwise distinct, andhence such configurations are prohibited. However, thisrestriction gets lifted in the low-momentum scaling limitsince the Bethe roots of different flavors are allowed toapproach arbitrarily close to each other and eventuallyrecombine into an independent mode. It is clear that, bysuch a composition of magnons, one will be able to pro-duce a large number of stacks. But only those stacks thatinvolve momentum-carrying roots show up as physicalfluctuations, whereas all the other stacks made solelyfrom auxiliary excitations are likewise called auxiliary.Moreover, similar to the elementary excitations, stacks alsoexert attractive forces on themselves and thus can grow intononlinear modes that carry finite filling fractions andcharge densities.The upshot is that for the complete classification of

classical modes, one requires additional mode numbers toproperly account for the internal structure of all theadmissible stacks. This extra piece of information, whichcan be thought of various polarization directions in which aclassical configuration can vibrate, depends on both thesymmetry algebra and on-site representation under con-sideration. For illustration, let us look at our main exampleof the fundamental SUðnÞ chains. The asymptotic Betheansatz equations [see Eq. (43)], in the continuum descrip-tion, turn into an n-sheeted Riemann surface whose sheetsare “stitched together” by finitely many (square-root-type)branch cuts (see, e.g., Refs. [136,137] for additionaldetails). The standard choice is to place the cuts along

the contours CðaÞi on which the Bethe roots condense.Creating a single excitation, say, of flavor a, amounts toconnecting the ath and (aþ 1)th sheets with an infinitesi-mal cut. On the other hand, exciting a stack, say, of typeða; bÞ, corresponds to joining two nonadjacent sheets of thesurface indexed by a and bþ 1. Equations (43) need to beappropriately amended to incorporate these extra stackexcitations. In Fig. 4, we illustrate the process of stackformation on a four-sheeted Riemann surface associatedwith the asymptotic Bethe equations of the SU(4) quantumchain made out of fundamental spins ð4Þ.

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In summary, the number of independent classical degreesof freedom equals the number of distinct physical stacks.We reemphasize that stack excitations are not permitted inthe spectrum of a quantum chain due to kinematic con-straints. In other words, upon including quantum fluctua-tions [even perturbatively at the level of asymptoticequations (43)] stacks immediately disintegrate into indi-vidual magnons.In the next section, we describe how to determine the

admissible symmetry-breaking patterns and classify thecorresponding physical stacks.

3. Giant quasiparticles as classical solitons

In the previous section, we have not mentioned howsoliton modes enter the picture. While classical solitons are,strictly speaking, not part of the nonlinear (finite-gap)spectrum, they nevertheless emerge as from elliptic wavesin the limit of unit elliptic modulus. Thus, they are onlymade possible after decompactifying the spatial dimension[136,145]. By tracing this process in the spectral plane,one finds that solitons emerge out of a “singular event” inwhich two density contours close by merge with oneanother, resulting in a single, vertically oriented contourwith constant unit density of Bethe roots. Such objects havebeen described before in certain integrable classical sigmamodels—see, e.g., Refs. [136,137,141,145]—where theyare normally called condensates. It is important to stress,however, that these are distinct from ordinary condensatesthat constitute square-root branch cuts of finite-genusRiemann surfaces. From the viewpoint of an algebraiccurve, such uniform condensates arise from coalescing twosquare-root branch cuts into a logarithmic branch cut. The

giant quasiparticles from the preceding discussion, namely,bound-state excitations with a macroscopic number ofquanta carrying low momenta, are therefore none otherthan quantized solitons.

4. Counting stacks

Now, we finally address the “counting problem.” Theremaining task is to extract the correct number of Goldstone

TABLE IV. Positive roots systems for Lie algebras A3 ¼ suð4Þand B3 ¼ soð7Þ of rank 3. In the α basis, any positive root α ∈Δþ can be decomposed as an integral linear combination ofsimple roots αi (level equals the sum of coefficients), whereas theω basis provides an expansion in terms of the fundamentalweights (with coefficients being the Dynkin labels).

α basis ω basis Level

(1,1,1) ½1; 0; 1� 3(0,1,1) ½−1; 1; 1� 2(1,1,0) ½1; 1;−1� 2(0,0,1) ½0;−1; 2� 1(0,1,0) ½−1; 2;−1� 1(1,0,0) ½2;−1; 0� 1(1,2,2) ½1; 0; 1� 5(1,1,2) ½1;−1; 2� 4(0,1,2) ½−1; 0; 2� 3(1,1,1) ½1; 0; 0� 3(0,1,1) ½−1; 1; 0� 2(1,1,0) ½1; 1;−2� 2(0,0,1) ½0;−1; 2� 1(0,1,0) ½−1; 2;−2� 1(1,0,0) ½2;−1; 0� 1

FIG. 4. Elementary and auxiliary excitations in the fundamental SU(4) quantum spin chain with on-site representation ð4Þ, representedon the associated four-sheeted Riemann surface. Left: physical momentum-carrying magnons (blue) emanating from the upper sheet,and two flavors of auxiliary magnons (red and green). Middle: semiclassical limit, depicting quantized stack excitations with nearlycoinciding rapidities. Right: four-sheeted Riemann surface encoding the classical spectral curve of an effective low-energy field theory,with two types of excitations—isolated stacks, representing various branches of linear Goldstone modes, and condensates of stacks(square-root branch cuts) pertaining to nonlinear interacting modes carrying a finite amount of energy. Physical stacks (dark green) andcondensates (blue and purple) involve momentum-carrying excitations. Condensates of stacks of a uniform density are identified withclassical solitons.

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modes from the semiclassical spectrum of quantum ferro-magnetic chains whose excitations, both physical andauxiliary, carry r distinct flavor numbers (one per eachsimple root αi of g). Different types of stacks can begenerated simply by forming all linear combinations ofsimple roots αi with non-negative integral coefficients, seeTable IV for examples. In other words, excitation operatorsassociated with stacks correspond to positive rootsα ∈ ΔþðgÞ. The subset of physical roots corresponds tothose roots that involve at least one physical simpleroot, associated with physical (i.e., momentum-carrying)excitations. In the fundamental models, for example, it isonly α1.By making use of partially ordered sets known as

Hasse diagrams, we, in turn, describe a helpful graphicalrepresentation for all the physical stacks, as exemplified inFig. 5. Let us first outline the construction by specializingto fundamental irreducible on-site representations Λ ¼ ωa.We start by picking one of the bottom nodes (associatedwith simple roots) and proceed by “climbing up’ in thedirection of the top node (the highest root) by successively

adding one simple root at a time. With this rule, one canfind several inequivalent paths through the Hasse diagram.Since each vertex is an independent Goldstone mode,their total number nG equals the total number of verticesvisited along the way. More generally, namely, for thenonfundamental irreducible on-site representations VΛ withΛ ¼ P

ra¼1 maωa involving multiple nonvanishing Dynkin

labels ma, one finds multiple momentum-carrying speciesin the spectrum. In this case, one has to simply superimposeall the sublattices that emanate from base nodes αi for alla ∈ Ir for which ma ≠ 0. The maximal breaking corre-sponds to all ma being nonzero (in which case, the iso-tropy group H coincides with the maximal torus subgroupT of G), resulting in the maximal number of Goldstonemodes nG ¼ 1

2dimðG=TÞ ¼ jΔþj, rendering all stacks

physical.Finally, in cases when a Lie algebra g involves a node of

connectivity 3 (that is, types Dr and E families), the Hasselattices extend into the third dimension [this is illustrated inFig. 6 for the example of soð10Þ].

FIG. 5. Left: Hasse diagrams associated with the sublattices Δþ of positive roots of g, shown for Lie algebras suð4Þ (top) and soð7Þ(bottom) of rank 3. The nodes represent positive roots α ∈ Δþ, with the simple ones (at the bottom) marked with different colors.Moving upwards along a colored edge amounts to adding a simple root corresponding to that color. Positive roots are separated into llevels, given by the sum of all positive integral coefficients in the expansion of α over the basis of simple roots αi. The total number ofpositive roots yields the maximal number of Goldstone modes nG ¼ jΔþj ¼ 1

2dimðG=TÞ for the full breaking of symmetry G → T.

Middle: Hasse sublattices pertaining to all the distinct fundamental irreducible representations Λ ¼ ωa (i.e., Young diagrams with asingle column), connecting the base node αa to the upper node (maximal root). The total number of nodes in a given sublattice equals thenumber of distinct branches of magnon excitations (Goldstone bosons). Right: Hasse sublattices for nonfundamental on-siterepresentations (shown for Λ ¼ ½1; 0; 1�) obtained by superimposition of the fundamental sublattices, each belonging to a singlenonvanishing Dynkin label.

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V. CONCLUSION

In this work, we initiated a systematic study of anoma-lous transport at finite temperature, which was recentlyuncovered in a number of quantum integrable systems. Weaimed mainly at settling whether all integrable models withlocal and isotropic interactions—that, in addition, possessa global continuous symmetry of the non-Abelian Liegroup—display superdiffusion of the Noether charges. Byestablishing the link between the thermodynamic quasi-particle content and representations of the correspondingquantum groups, we first obtained universal algebraicdressing equations and subsequently performed a scalinganalysis of these dressing equations within the frameworkof generalized hydrodynamics. Our conclusion is thatanomalous charge transport with a superdiffusive dynami-cal exponent z ¼ 3=2 is a common feature of all integrableferromagnetic lattice models with short-range interactions,provided the Hamiltonian exhibits invariance under a non-Abelian continuous symmetry. This conclusion holdsirrespective of the symmetry group or unitary irreduciblerepresentations associated with (local) physical degrees offreedom. Each such subclass indeed constitutes an infi-nitely large family of commuting integrable Hamiltonians,all of which exhibit z ¼ 3=2 superdiffusion. Moreover,we have argued that any other anomalous exponent is

incompatible with the underlying quasiparticle structure.In addition, the same type of anomalous charge trans-port persists for group-valued Noether currents, even inLorentz-invariant and nonrelativistic integrable quantumfield theories. Because of its remarkable level of robust-ness, we dubbed this phenomenon as superuniversal.Let us briefly restate the key steps that lead to this

general conclusion. We begin by recalling that anomaloustransport of some charge Q occurs only when Q is oneof the generators of a non-Abelian subgroup H ⊆ G suchthat the equilibrium state is invariant under H. If oneconsiders generic polarized states (realized by applyingchemical potentials to the Cartan charges), the anomalousconductivity gets regularized, and one recovers ballistictransport with subleading diffusive corrections, as onegenerically expects within generalized hydrodynamics.Upon approaching the G-symmetric state, however, onefinds that the Drude weights governing the ballistic trans-port vanish, whereas the diffusion constants simultaneouslydiverge. In this regime, the quasiparticles that dominatethe magnetic susceptibility and transport are macroscopi-cally large, coherent bound states of magnons, which wecall giant quasiparticles. All other nongiant quasiparticlesbecome effectively depolarized and do not contribute eitherto transport or to susceptibility. Transport of the Noethercharges is, in effect, described by a dense gas of interactinggiant quasiparticles. Each of these giant quasiparticlesmoves with a characteristic velocity inversely proportionalto its width: We established this result by explicit analysisof the GHD equations and interpreted it as indicatingthat these giant quasiparticles are solitons made up ofnonlinearly interacting, quadratically dispersing Goldstonemodes. When these modes are present at finite density, theydress each other’s charge in a nonperturbative way, andthis nontrivial dressing leads to the fractional dynamicalexponent z ¼ 3=2.To further elucidate the nature of the giant quasiparticles,

we, in addition, examined the structure of semiclassicaleigenstates from the viewpoint of an effective low-energytheory with respect to a (continuously degenerate) ferro-magnetic pseudovacuum. At the level of an integrableclassical field theory, giant quasiparticles can be iden-tified with soliton modes. The latter may be thought of asthe nonlinear counterparts of quadratically dispersingGoldstone bosons. As we explain, the number of internaldegrees of freedom of classical soliton modes does notequal the number of distinct flavors of the elementaryexcitations that constitute the quasiparticle content of anintegrable quantum chain, contrary to naive expectations.Instead, the total number of polarization directions of aclassical field typically exceeds the rank of the group G.This mismatch is due to the formation of emergent classicalmultiflavored degrees of freedom called stacks, which areproduced by gluing together magnons of different flavors(which dissolve upon introducing quantum corrections).

FIG. 6. Hasse diagram for the positive root lattice Δþ(soð10Þ).There are 20 roots in total (black dots), arranged into 7 levels. Thelattice extends into three dimensions as the simple root α3 hasconnectivity 3. To find the α pattern corresponding to a simpleroot αi, one begins at αi and also collects the nodes by alwaysproceeding forward in the direction of the highest root. The Z2

automorphism of the Dynkin graph reveals itself as the reflectionsymmetry with respect to the main diagonal of the cube passingthough nodes 4 and 5.

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Stacks that contain momentum-carrying magnons shouldbe regarded as independent physical excitations, and weoutlined how they can be naturally arranged on vertices ofthe Hasse diagrams. The total number of physical stacks isfound to be in perfect agreement with the prediction of theGoldstone theorem. As a by-product of our work, weprovided a full classification of ferromagnets with theglobal symmetry of a simple Lie algebra.At the formal level, the anomalous nature of charge

transport can be attributed to a particular structure of fusionidentities amongst the characters of Yangian symmetryalgebras, which enforces rigid algebraic constraints on thethermodynamic quasiparticle spectra. The list of modelsdiscussed in this study is therefore unlikely to be exhaus-tive, and other classes of integrable models not included inthis study are likewise expected to feature a superdiffusivecharge transport with characteristic exponent z ¼ 3=2.Specifically, we anticipate that the phenomenon extendsbeyond Yangians associated with simple Lie algebra toother classes of models that arise from rational solutions tothe Yang–Baxter equations as, for example, the Fermi–Hubbard model [38], and to other integrable spin chainsthat realize Yangian symmetries associated with unitary ororthosymplectic Lie superalgebras, that is, suðnjmÞ andospðnj2mÞ (see, e.g., Refs. [79,146,147]), whose bosonicNoether charges are known to display singular diffusionconstants [46].To benchmark our predictions, we have performed

numerical simulations on a handful of representativeinstances of integrable quantum chains, restricting theanalysis to classical simple Lie algebras and only tofundamental local degrees of freedom. We confirmed thepredicted anomalous algebraic dynamical exponent z ¼3=2 with great numerical precision. Unfortunately, wecould not reliably discern the anticipated KPZ scalingprofiles. Quantum spin chains with the symmetry ofexceptional Lie algebras have not been included in thepresent numerical study, largely for technical reasons.Since constructing integrable quantum chains based onthe “quantized symmetry” of an exceptional Lie algebra isalgebraically rather laborious (see, e.g., Ref. [148]), weprefer to leave them for a separate technical study. Theother practical difficulty is that dimensions of the definingirreducible representations are, possibly with the exceptionof ð7Þ and ð14Þ of G2, conceivably too large to beefficiently simulated with current DMRG techniques.Finally, a central piece of the full puzzle is still missing:

Despite integrability, a first-principles derivation of theKPZ asymptotic scaling profiles of dynamical structurefactors remains elusive. At this time, there only exists aphenomenological picture, which, under certain plausibleassumptions, justifies the emergence of a noisy Burgersequation for the basic SU(2) case [45]. One obviousdrawback of such an approach is that it cannot yieldquantitative predictions such as the value of the KPZ

coupling. Understanding quantitatively, from ab initioprinciples, how KPZ universality emerges (and not onlyz ¼ 3=2) remains a major challenge for future works, evenin the case of the SU(2) symmetric Heisenberg XXX spinchain. Another open problem is to appropriately extend theproposal of Ref. [44] to systems that are invariant undersymmetries of higher rank in order to extract the KPZnonlinearity constant.

ACKNOWLEDGMENTS

We thank Vir Bulchandani, Michele Fava, Žiga Krajnik,Fedor Levkovich-Maslyuk, Sid Parameswaran, TomažProsen, and Marko Žnidarič for comments, stimulatingdiscussions, and collaboration on related topics. This workwas supported by the National Science Foundation underNSF Grant No. DMR-1653271 (S. G.), the US Departmentof Energy, Office of Science, Basic Energy Sciences, underEarly Career Award No. DE-SC0019168 (R. V.), the AlfredP. Sloan Foundation through a Sloan Research Fellowship(R. V.), the Research Foundation Flanders (F.W. O. andJ. D. N.), and the Slovenian Research Agency (ARRS)program No. P1-0402 (E. I.).

APPENDIX A: DRESSED CHARGEFLUCTUATIONS

For a general equilibrium macrostate characterized byquasiparticle densities ρAðθÞ, dressed charge susceptibil-ities (in the Cartan sector) can be linked to densityfluctuations via

hqðiÞdrA qðjÞdrA i ¼XB;B0

Zdθdθ0

δqðiÞdrA

δρBðθÞδqðjÞdrA

δρB0 ðθÞ× hδρBðθÞδρB0 ðθ0Þi: ðA1Þ

Variations of qðiÞdrA with respect to quasiparticle densitiesρAðθÞ can be expressed as

δqðiÞdrA

δρBðθÞ¼

XB

KdrABðθÞqðiÞdrB ; ðA2Þ

where KdrABðθÞ are the dressed scattering differential phases.

Using that [149]

hδρAðθÞδρA0 ðθ0Þi ¼ 1

lCAA0 ðθ; θ0Þ; ðA3Þ

and employing, for convenience, the following compactmatrix notations,

C ¼ Ω½n�−1ρð1 − nÞΩ½n�; ðA4Þ

Kdr ¼ Ω½n�K; ðA5Þ

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where Ω½n�≡ ð1þKnÞ−1, we arrive at

hqðiÞdrA qðjÞdrA i ¼ 1

lðKdrqdrCqdrKdrÞAA; ðA6Þ

where qdr stands for the diagonal matrix of dressedcharges qdrA .For the Cartan charges QðiÞ, in an unpolarized back-

ground (i.e., hi ¼ 0 for all i ¼ 1; 2;…; r), we have

qðiÞdra;s ¼ 0, with an exception in the limit s → ∞ wherethe above expression simplifies to give Eq. (25).

APPENDIX B: ANALYTIC SOLUTIONS

In order to infer the nature of charge transport in anunpolarized thermal ensemble, it is essential to deduce theasymptotic properties of the TBA state functions for thegiant quasiparticles, namely, for the large Bethe strings inthe s → ∞ limit. To this end, we subsequently examinecertain formal properties of the dressing equations for thespecific algebraic case of the grand-canonical Gibbsequilibrium ensembles in the limit of infinite temperature.We study states with unbroken symmetry G, requiring us toswitch all the U(1) chemical potentials off.To keep our presentation succinct, we only provide the

essential information. For more details and definitions, werefer the reader to the Supplemental Material [58].

1. Asymptotic analysis

We first carry an asymptotic analysis at large s at aformal level.Fermi functions. Integrable quantum chains that are

associated with the class of evaluation representations ofthe Yangian algebra exhibit Fermi-Dirac statistics. Theoccupation function scales at large s as na;s ∼ 1=s2 (and,accordingly, Ya;s ∼ s2). This is a direct corollary of theHirota relation (in this particular context, known also as theT system) for the “quantum characters” T a;s, correspond-ing to the large-θ limits of the thermodynamic T functions;see Ref. [58] for details. The key property in this respect isthat the T functions T a;s are, by construction, polynomialsin s (for all a ∈ Ir), which readily implies that the Yfunctions Ya;s are rational functions that only depend onquantum numbers ða; sÞ.State densities. The above scaling of na;s immediately

implies the asymptotic relation ρtota;s ∼ s2ρa;s. Notice thatwhile both state densities ρa;sðθÞ and ρtota;sðθÞ likewisedepend algebraically on s, in distinction to Fermi functions,they also depend nontrivially on θ (even in the limit ofinfinite temperature).We are now in a position to analyze the large-s behavior

of the dressing equations. We start with the simple case ofthe spin-1=2 Heisenberg chain SU(2), involving only onespecies of magnons. The analysis can be performed on thecanonical form of the Bethe-Yang equations,

ρtots ¼ Ks −Xs0≥1

Ks;s0ns0ρtots0 : ðB1Þ

Two observations can readily be made: (i) For a fixed valueof θ, the kernels fall off as KsðθÞ ¼ 2πp0

sðθÞ ∼ 1=s, and(ii) because of the telescopic cancellation of poles, thedifferential scattering phase shiftsKs;s0 involve 2‐minðs; s0Þterms. Thus, we can make the following estimate at large s:

ρtots ≲ 1

sþXs

s0¼1

2s0ns0ρtots0 þ 2sX∞

s0¼sþ1

ns0ρtots0 ; ðB2Þ

where each term under the sum has been upper bounded bya constant using the sum rule 1 ⋆ Ks ¼ 1. The second sumcan be understood as the residual contribution coming fromlarge s0 strings with s0 > s, which gets suppressed in the∼1=s fashion. The first sum converges in the s → ∞ limit,provided lims→0 ρ

tots ¼ 0, when the total density of states

ρtots decays to zero at large s (in an algebraic fashion).Lastly, using that the bulk recurrence relations pertaining tothe algebraic dressing equations involve only rationalfunctions of s, this implies ρtots ∼ 1=s scaling at large s.In more general integrable models (with nested spectra),

one has to additionally account for quasiparticles of differ-ent flavors, whose mutual interactions are prescribed by theCartan matrixK (cf. Ref. [58] for the definition). The aboveargument can be easily adapted by including an extra sumover the flavor index.

2. Analytic solutions in the high-temperature limit

To solidify our statements and conclusions, here weexplicitly work out the exact closed-form solution to thealgebraic TBA equations. The full solution for the basiccases of the SU(2) spin-S (Babujian-Takhtajan) quantumchains has already been obtained in Ref. [109]. Here, weare primarily interested in models that possess Lie sym-metries of higher rank.For definiteness, we specialize to SUðnÞ quantum chains

with fundamental on-site degrees of freedom. While othercases can be treated in a similar fashion, for compactness ofpresentation, we postpone a complete and comprehensiveanalysis for a separate technical study.Let us consider, as our starting point, the algebraic limit

of the group-theoretic dressing equations [58], which, inFourier k space, turn into a system of coupled algebraicequations of the form (with implicit summation overrepeated indices)

s−1 ⋆ Fa;s − Is;s0 na;s0Fa;s0 − Ia;a0na0;sFa0;s ¼ δa;s; ðB3Þ

with s−1ðkÞ ¼ 2 cosh ðk=2Þ and adjacency matrix Ia;b ¼δa;b−1 þ δa;bþ1. The first step is to solve the homogeneouspart, which can be achieved with the following ansatz:

Fa;sðkÞ ¼ Aa;sðkÞKaþs−1ðkÞ þ Ba;sðkÞKaþsþ1ðkÞ; ðB4Þ

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where KsðkÞ ¼ e−sjkj=2 are the elementary scattering ker-nels in Fourier space. Plugging this ansatz into Eq. (B3),performing the Laplace transform from the k plane to the zplane, and demanding the null condition for all the residueslocated at zi ∈ 1

2N, we end up with the following system of

recurrence relations:

Aa;s ¼ na;s−1Aa;s−1 þ na−1;sAa−1;s; ðB5Þ

Ba;s ¼ na;sþ1Ba;sþ1 þ naþ1;sBaþ1;s; ðB6Þ

Aa;s − Ba;s ¼ na;sþ1Aa;s−1 þ naþ1;sAaþ1;s

− na;s−1Ba;s−1 þ na−1;sBa−1;s: ðB7Þ

The Fermi functions can be conveniently expressed interms of ratios of suðnÞ characters χa;s,

na;s ¼χa−1;sχaþ1;s

χ2a;s; na;s ¼

χa;s−1χa;sþ1

χ2a;sðB8Þ

(see Ref. [58] for additional information). The solution hasthe form

Aa;s ¼ A0

χa;sχa−1;s−1χa−1;sχa;s−1

; ðB9Þ

Ba;s ¼ B0

χa;sχaþ1;sþ1

χaþ1;sχa;sþ1

; ðB10Þ

with coefficients A0 and B0 yet to be determined. Byplugging these back into Eqs. (B7), we retrieve the Hirotarelation for classical characters. To fix the undeterminedconstants, we find the particular solution of Eq. (B3), withthe source term attached at the first node at ða; sÞ ¼ ð1; 1Þ,yielding A0 ¼ −B0 ¼ 1=χ1;1. Transforming back to the θplane, we finally obtain

Fa;sðθÞ ¼χa;sχ1;1

�χa−1;s−1

χa−1;sχa;s−1Kaþs−1ðθÞ

−χaþ1;sþ1

χaþ1;sχa;sþ1

Kaþsþ1ðθÞ�: ðB11Þ

In the context of anomalous transport, we are solelyinterested in the xi → 1 limit. In this limit, the suðnÞcharacters become the dimensions da;s of rectangularirreducible representations Vsωa

; more details are providedin Ref. [58]. Specifically, the total state densities read

ρtota;sðθÞ ¼sþ n − a

n

�Ksþa−1ðθÞsþ a − 1

−Ksþaþ1ðθÞsþ aþ 1

�: ðB12Þ

There is an analogous expression (up to a multiplicativeprefactor) for the dressed differential quasiparticle energiesε0a;sðθÞ, which can be obtained by simply replacing KsðθÞ

with K0sðθÞ. In rapidity space, the latter is given by a

discrete family of Cauchy distributions

KsðθÞ ¼1

2πi∂θ log SsðθÞ ¼

1

2π

sθ2 þ ðs=2Þ2 : ðB13Þ

Notice that for any finite fixed rapidity θ, the densityof states ρtota;s decays as about 1=s2 in the large-s limit.However, when integrated against any dummy integrablefunction fðθÞ as in GHD, we have the following large-sproperties: Z

RdθfðθÞρtota;sðθÞ ∼

1

s; ðB14Þ

ZRdθfðθÞveffa;sðθÞ ∼

1

s; ðB15Þ

for every quasiparticle flavor a.As explained in the main text, these properties allow us

to infer the superdiffusive scaling of charge dynamics withan algebraic dynamical exponent z ¼ 3=2.Nonfundamental on-site representations. The large-s

scaling properties (B14) and (B15) likewise hold in modelswith nonfundamental local degrees of freedom, namely, forany on-site finite-dimensional unitary irreducible represen-tations VΛ. At the level of algebraic dressing equations, thisamounts to placing the source terms in a different (generic)position.As a quick illustration, we derive the explicit solution

to the SU(3) chain in the adjoint representation ð8Þ, withDynkin labels Λp ¼ ω1 þ ω2 ≡ ½1; 1�. By virtue of Z2

invariance of the dressing equations (under interchangingthe flavors), we have F1;s ¼ F2;s, and therefore

s−1 ⋆ Fa;s − IA∞s;s0 ns0Fa;s0 − nsFa;s ¼ δs;1; ðB16Þ

where IA∞s;s0 ¼ δs;s0−1 þ δs;s0þ1. The solution reads

Fa;s ¼1

3

�sþ 2

sKs −

sþ 1

sþ 3Ksþ3

�

þ 1

3½Ksþ1 − Ksþ2�: ðB17Þ

APPENDIX C: INTEGRABLE QUANTUMFIELD THEORIES

In the main text, we confined ourselves exclusively tointegrable ferromagnetic quantum chains. Nonetheless,these do not exhaust the list of integrable quantum modelsinvariant under global symmetries of a non-Abelian Liegroup. As briefly mentioned in the conclusions, there areother types of quantum chains, such as most prominentlyintegrable fermionic models that exhibit Lie supersymme-tries (Z2-graded Lie algebras), which are on the same

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grounds expected to reveal charge superdiffusion with theexponent z ¼ 3=2; see, e.g., Refs. [38,46] for concreteexamples.More remarkably, however, anomalous charge transport

likewise takes place in integrable quantum field theoriesin two space-time dimensions, provided that elementaryparticles carry internal isotropic degrees of freedom, which,in the classical action, take values either in Lie groups G orcoset manifolds G=H. The two most prominent classes ofintegrable quantum field theories (IQFT) of this kind arethe OðnÞ nonlinear sigma models (NLSM) and the principalchiral fields (PCF) on G ×G; cf. Refs. [150–155] for moredetails.Even though Goldstone modes of Lorentz-invariant

systems always comprise linearly dispersing (i.e., type-I)modes, one should keep in mind that we are interested hereexclusively in the time evolution of the Noether currentsand not correlation functions amongst individual compo-nents of quantum fields. A crucial observation in thisrespect is that the temporal components of the Noether two-currents may be understood as (quantized) spin waves.Moreover, in finite-temperature ensembles, excitations ofthese internal interacting degrees of freedom undergonontrivial dressing. The situation in fact mirrors that ofthe Heisenberg spin chains or their higher-rank analogueswe investigated earlier. We do not attempt to give acomprehensive exposition but rather demonstrate the basicprinciples on the O(3) NLSM, representing a paradigmatic,integrable, interacting QFT with nondiagonal scattering.We briefly outline the differences in a few other IQFTsthat display SU(2) symmetry while postponing a detailedquantitative study of other integrable nonlinear sigmamodels on Riemannian symmetric spaces G=H [156]and their classical limits for another study.Another widely studied class of integrable system that

exhibit global non-Abelian symmetries are nonrelativisticQFTs commonly known as the Yang-Gaudin models[157–161]. They describe one-dimensional interactingBose or Fermi gases with internal degrees of freedom(see, e.g., Refs. [162,163] for recent applications of GHD).

1. O(3) nonlinear sigma model

To begin, we spell out some of the main propertiesof the O(3) NLSM quantum field theory, employing theHamiltonian formalism. The quantum field n ¼ ðnx; ny; nzÞ,subjected to the nonlinear constraint n · n ¼ 1, trans-forms in the vector representation of O(3). Since thevacuum stability subgroup with respect to the polari-zation axis (say, nz ¼ 1) is O(2), the target manifoldfor O(3) NLSM is Oð3Þ=Oð2Þ ≅ S2. Introducing themomentum π conjugate to n and the angular-momentumfield m ¼ n × π perpendicular to n, m · n ¼ 0, theHamiltonian density (here, without the topological Θ term)has the form

HOð3Þ ¼v2

Zdx½gm2 þ g−1n2

xÞ�: ðC1Þ

The conserved Noether two-current associated with globalO(3) rotations is given by

∂μjμ ¼ 0; jμ ¼ g−1n × ∂μn; ðC2Þ

for μ ∈ fx; tg. The integrated magnetization density pro-vides a local conservation law, ðd=dtÞ R dxmðxÞ ¼ 0.The O(3) NLSM arises as the effective low-energy theory

of (Haldane) spin-S antiferromagnets, where one identifiesv ¼ 2JS, while the coupling strength is given by g ¼ 2=S,assuming large S; the S2 field n governs fluctuations of thestaggered order parameter above the antiferromagnetic(Neel) ground state, whereas m pertains to ferromagnetic(spin-wave) fluctuations. Elementary excitations of Eq. (C1)are anSU(2) triplet ofmassive spinful bosonswith relativistic(bare) dispersion eðpÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 þ Δ2

p. As is customary,

relativistic dispersion of massive particles can be parame-trized by a rapidity variable θ,

eðθÞ ¼ Δ coshðθÞ; pðθÞ ¼ Δ sinhðθÞ: ðC3Þ

The mass Δ gets generated dynamically via dimensionaltransmutation and vanishes in the large-S limit. At weakcoupling (g → 0), the theory becomes effectively free withdimðG=HÞ ¼ 2 massless (Goldstone) bosons.Because of integrability, the model possesses infinitely

many “hidden” conserved currents and a completelyfactorizable many-body scattering matrix. Notice, however,that exchange of isotropic degrees of freedom upon elasticinterparticle interactions renders the scattering matrix non-diagonal, signifying that particles, upon collision, do notonly pick up a phase factor but can also change their state(internal quantum numbers). This feature has a directanalogy with the nested spin chains.The O(3) NLSM can be diagonalized by means of the

algebraic Bethe ansatz, initially carried out in Ref. [59] (seealso Refs. [156,164] for a more comprehensive discussion).The standard trick to resolve the exchange of isotropicdegrees of freedom is to introduce additional (calledauxiliary) spin-wave excitations (magnons)—the extendedbasis of excitations that renders the scattering diagonal.These auxiliary magnons are, despite being massless,regarded as independent quasiparticles. Imposing thefinite-volume boundary condition, one arrives at the fol-lowing nested Bethe Ansatz equations [59]:

eipðθjÞLYNθ

k¼1

Sðθj; θkÞYNλ

l¼1

S−1ðθj; λlÞ ¼ −1; ðC4Þ

YNθ

k¼1

S−1ðλj; θkÞYNλ

l¼1

Sðλj; λlÞ ¼ −1; ðC5Þ

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belonging to the sector with Nθ momentum-carryingphysical excitations and Nλ magnons (with associatedrapidity λ). There is a single rational scattering amplitudereading SðθÞ¼ðθ− iπ=2Þ=ðθþ iπ=2Þ. Looking at Eq. (C5),it is evident that interactions among the magnons areidentical to those in the isotropic Heisenberg SU(2) chain,resulting in the formation of bound states (Bethe strings).Taking the thermodynamic limit, obtained by sending

L → ∞ with ratios Nθ=L and Nλ=L finite, and introducingthe physical density ρ0ðθÞ and auxiliary densities of theBethe s-strings ρs≥1ðθÞ, one arrives at the following(decoupled) Bethe-Yang equations

ρtot0 ¼ p0

2πþ s ⋆ ρ2; ðC6Þ

ρtots ¼ δs;2s ⋆ ρ0 þ s ⋆ IA∞s;s0 ρs0 : ðC7Þ

By performing an extra particle transformation on thephysical node, Y0 ↦ 1=Y0 (i.e., identifying Y0 ≡ ρ0=ρ0),the TBA equations can, once again, be brought into thecanonical Y-system format

logYs ¼ −δs;0βeþ s ⋆ ID∞s;s0 logð1þ Ys0 Þ; ðC8Þ

with the incidence matrix of the D∞ Dynkin diagram (seeFig. 7). It is worth noticing that the U(1) chemical potentialh enters only implicitly via the asymptotics of the Yfunctions, lims→∞ logð1=sÞ logYs ¼ h. The O(3) modelis, in fact, a member of an infinite family of integrablesigma models with SUðnÞ=SOðnÞ target spaces, with n − 1massive nodes whose thermodynamic particle content getsarranged according to An−1 ×D∞ Dynkin diagrams [156].Since QFT are not UV complete, there is no regime in

which the TBA equations would be rendered algebraic.Despite the fact that there is no issue with computing thethermodynamic Y functions in the β → 0 limit, say, at halffilling h ¼ 0, the equations for the densities, Eqs. (C7),yield UV-divergent rapidity integrals. This type of diver-gence is often cured by hand by imposing a momentumcutoff, such as, e.g., in Ref. [165]. We purposely avoid ithere since this would invariably break integrability andthereby spoil the late-time decay of the charge correlations.To ensure convergence of rapidity integrals, temperatureβ > 0 has to be taken into account in a nonperturbativeway. The only safe way to introduce a UV cutoff whilepreserving integrability is via a full light-cone discretiza-tion. This discretization has been achieved for the case ofSU(2) PCF in Ref. [153], which demonstrates how theintegrable QFT arises from a particular continuum scalinglimit of a staggered quantum spin chain.In the scope of physics applications, one is typically

interested in the opposite regime of low temperatures. Thelow-temperature expansion of Eqs. (C7) and (C8) can befound in Ref. [109]. We do not reproduce the results herebut merely reserve a few remarks. In order to properly

account for the effects of thermal fluctuations in a finite-density state, even at arbitrarily low temperatures T > 0, itis crucial to properly account for all the contributions of thespin-wave excitations. Magnonic excitations turn out to beinconsequential only in the regime h=T ≫ 1, where thetheory can be well approximated by the semiclassicaldescription developed in Refs. [166–168]. In stark contrast,when approaching the unmagnetized (half-filled) state (i.e.,for h=T ≪ 1), the contributions to the spin diffusion due togiant magnons become amplified, and, in analogy to theisotropic Heisenberg chain, one accordingly finds a diverg-ing spin diffusion constant Ds ∼ 1=h accompanied by avanishing-spin Drude weight.Topological term. It is well known that the O(3) NLSM

admits an integrable deformation with the inclusion of thetopological Θ term with angle Θ ¼ π [59,169] [describingthe low-energy limit of SU(2)-invariant spin chains withodd spin S; in general, Θ ¼ 2πS [170] ]. This term has aprofound effect on the low-energy spectrum; instead of amassive triplet, one instead finds an SU(2) doublet ofmassless elementary excitations, with bare dispersion

e�ðθÞ ¼ �pðθÞ ¼ M2e�θ; ðC9Þ

FIG. 7. TBA incidence diagrams depicting the completethermodynamic quasiparticle spectra for a number of paradig-matic examples of massive and massless IQFTs with isometrygroups of rank one: SU(2) chiral Gross-Neveu model, O(3)nonlinear sigma model, with and without the topological Θ termand SU(2) principal chiral field, appearing from top to bottom inthe respective order. Massive physical excitations, with baredispersion [Eq. (C3)], are designated by gray nodes. Masslessphysical excitations, with bare dispersion [Eq. (C9)], are markedin blue. The structure of TBA equations is a direct product of twotypes of Dynkin diagrams: a finite one associated with theisometry group of the elementary physical excitations (markedwith color nodes), and an infinite one of type A associated with atower of auxiliary magnons (white nodes), whose bijectives cor-respond to finite-dimensional irreducible suð2Þ representations.

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where � designate the right (p > 0) and left (p < 0)moving fields. The theory is massless but not scaleinvariant; M provides a mass scale that only affects theleft-right scattering processes. At the level of the TBAdescription, the physical species are comprised of the leftand right movers with densities ρ�ðθÞ [59]. The internalmagnon structure, however, remains intact [59]. For in-stance, the TBA equations in the quasilocal form are now ofthe form

logY� ¼ βM2e�θ þ s ⋆ logð1þ Y1Þ; ðC10Þ

logYs ¼ δs;1s ⋆ ð1þ YþÞð1þ Y−Þþ s ⋆ IA∞

s;s0 logð1þ Ys0 Þ: ðC11Þ

Further details can be found in Ref. [109] and referencestherein.Classical picture. We stress that magnons in QFTs are

not due to quantization but are rather a consequence of therotational symmetry. Thus, magnons are already featuredat the classical level. The O(3) nonlinear sigma modelrepresents an integrable classical field theory [171]. TheEuler-Lagrange equation for the classical field n ∈ S2 reads(in dimensionless units) [172]

ntt − nxx þ ðn2t − n2

xÞn ¼ 0: ðC12Þ

Owing to Lorentz invariance, linear fluctuations above adegenerate ground state are comprised of two, transversal,linearly dispersing (type-I) Goldstone modes. However,we are interested in the time evolution of the temporalcomponent of the conserved Noether two-current, whichpresently corresponds to the ferromagnetic order para-meter m ¼ n × π. Its equation of motion simply readsmt ¼ n × πt. In terms of the Hamiltonian equations,

nt ¼ π; πt ¼ nxx þ ðn2x − π2Þn; ðC13Þ

one can readily deduce the equation of motion for theangular-momentum field,

mt ¼ n × nxx; m2 ¼ π2: ðC14Þ

This result reveals the mechanism for how spatial fluctua-tions of field nðx; tÞ induce the dynamics of magnetizationmðx; tÞ. In the quantum version of the O(3) NLSM,fluctuations of these magnetization waves carrying integerquanta scatter completely elastically off one another, asdescribed by Eqs. (C5).

2. Other integrable QFTs

There are many other integrable QFTs with non-Abelianisometry groups that can be treated along the lines of theO(3) NLSM; see, e.g., Refs. [155,173]. The best-studied

examples are OðnÞ NLSMs on a hypersphere Sn−1 ≅OðnÞ=Oðn − 1Þ, with the Lagrangians

LOðnÞ ¼1

2g

Zdxð∂μnÞ2; n2 ¼ 1: ðC15Þ

In the simply laced cases, that is, Oð2rÞ with r ≥ 4 [174],the thermodynamic spectrum of excitations comprises rflavors of quasiparticles (one per node in the Dr Dynkindiagram), each of which further binds into bound states (theordinary s strings). The O(4) NSLM in the fundamental(vector) representation is special as it can be represented asthe SUð2ÞL × SUð2ÞR principal chiral field with particlestransforming in the bifundamental representation of suð2Þ[175]. Its spectrum involves massive spinful particles withtwo types of SU(2) spins—these are elementary excitationsabove the Fermi sea (antiferromagnetic ground state) in anSU(2) spin-S chain. Its thermodynamic particle content isgraphically represented in Fig. 7. In this respect, one canview (using a loose analogy) the SU(2) PCF as the bosonicQFT counterpart of the Fermi-Hubbard model [72,73]. Inthe strong-coupling limit, the SU(2) PCF model splits upinto two copies of the isotropic Heisenberg chain. Theclassical limit and the associated Riemann-Hilbert equa-tions can, once again, be derived from the asymptotic Betheequations [173], governing the regime with N → ∞ par-ticles with large rapidities θ ∼OðNÞ [with the quantityΔL ¼ exp ð−2πNÞ playing the role of a small parameter].Another prominent class of integrable QFTs are the

SUðnÞ chiral Gross-Neveu models (cGN) [176]. We take abrief look at the simplest SU(2) case, which describes twointeracting Dirac fermions expressed in terms of two-component spinors ψa (a ¼ 1, 2) with Lagrangian density

LcGN ¼ i ˆψa=∂ψa þ 1

2g2sðð ˆψaψ

aÞ2 − ð ˆψaγ5ψaÞ2Þ

−1

2g2vð ˆψaγμψ

aÞ2; ðC16Þ

with =∂ ≡ γμ∂μ and γ matrices γ0 ¼ σ1, γ1 ¼ iσ2, γ5 ¼ γ0γ1obeying the Clifford algebra γμ; γν ¼ 2ημν with metrictensor η ¼ diagð1;−1Þ. Lagrangian (C16) is symmetricunder Uð2Þ × Uð1Þc; spinors transform in the fundamentalrepresentation of U(2), whereas Uð1Þc is associated withthe chiral symmetry ψ → eiθγ5 ψ. The spectrum of the modelinvolves a single SU(2) multiplet of massive fermions, withrelativistic dispersion eðθÞ ¼ Δ cosh ðπθ=2Þ and pðθÞ ¼Δ sinh ðπθ=2Þ [aside from the massless excitation chargedunder Uð1Þc that completely decouples]. The amplitudeof a two-fermion scattering is given by SðθÞ ¼ −Γ½1−ðθ=4iÞ�Γ½1

2þ ðθ=4iÞ�=Γ½1þ ðθ=4iÞ�Γ½1

2− ðθ=4iÞ�; see, e.g.,

Refs. [76,177]. It is of primary importance that the scatteringof fermions carrying opposite spin projections is, onceagain, identical to the spin exchange of the SU(2)Heisenberg chain. To exhibit this similarity, we performthe particle-hole transformation on the zeroth node (assigned

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to physical, i.e., fermionic, density), Y0 ↦ 1=Y0, yieldingthe Dynkin-type TBA equations with the incidencematrix ofthe A1 × A∞ diagram [177],

logYs ¼ −δs;0βeþ s ⋆ logð1þ Ys−1Þð1þ Ysþ1Þ: ðC17Þ

In this labeling, the s strings sit at nodes s ≥ 1. Apart froman extra massive particle assigned to the initial node s ¼ 0,the obtained equations are indeed those of the isotropicHeisenberg spin-1=2 chain (see Fig. 7). A family of inte-grable Gross-Neveu models with SUðnÞ symmetry withn − 1 coupled copies of Eqs. (C17), resulting in the TBAincidence matrices of the type An−1 × A∞. There are also theGross-Neveu models with Oð2nÞ symmetry, which havebeen described in Ref. [156].Numerical analysis. In order to extract the large-s

asymptotic properties of thermodynamic state functions,we have numerically solved the TBA equations for theabove SU(2)-invariant IQFTs with a common magnonstructure. The deduced scaling properties match those oftheir spin-chain counterparts, as specified by Eqs. (B14)and (B15), indicating, once again, superdiffusive transportwith the dynamical exponent z ¼ 3=2.

3. Multicomponent Fermi gases

For completeness, we finally give another representativeexample of an integrable QFT. We consider the Yang-Gaudin field theory of spinful fermions—the fermioniccousin of the integrable two-component Bose gas. Our pur-pose here is mainly to achieve unification with otherintegrable systems characterized by the anomalous dynami-cal exponents z ¼ 3=2. There are two conserved Noethercharges involved, representing the conserved electron chargeand total magnetization. As we clarify below, anomaloustransport occurs only in the spin sector, provided the non-Abelian symmetry is not explicitly broken by the finitechemical potential. On the contrary, transport of electroncharge [Noether charge associated to a uð1Þ subalgebra]behaves ordinarily; i.e., it is characterized by a finite-chargeDrude weight with a regular diffusive correction.To elucidate the underlying algebraic structure, we note

that integrable multicomponent Fermi (and Bose) gasesarise as continuum limits of integrable lattice models withUq(slðnÞ) symmetry (commonly known in the integra-bility literature as the Perk-Schultz models) in the unde-formed q → 1 limit [upon which the global SUð2Þinvariance gets restored]. Which field theory one endsup with depends, besides the rank r ¼ n − 1 of g, also onthe choice of Z2 grading (without the grading, one cansimilarly produce multicomponent integrable Bose gases).The specific instance of a two-component Fermi gas thatwe are revisiting here arises as a continuum limit of therank-2 Perk-Schultz model with Uq(slð2j1Þ) symmetry inð−;þ;þÞ grading [161,178,179]. As we clarify below, theresulting TBA equations algebraically match those of an

integrable fermionic chain with the global symmetry ofSUð2j1Þ Lie superalgebra, representing the t − J model ofspin-1=2 fermions at the supersymmetric point J ¼ 2t,

HSUð2j1Þ ¼ P

�−t

XLj¼1

Xσ∈↑;↓

ðc†j;σcjþ1;σ þ c†jþ1;σcj;σÞ�P

þ JXLj¼1

�Sj · Sjþ1 −

1

4njnjþ1

�; ðC18Þ

where P ¼ QLj¼1ð1 − nj;↑nj;↓Þ projects out the doubly

occupied states.Two-component Yang-Gaudin Fermi gas. The

Hamiltonian density of the Yang-Gaudin model withcoupling constant c > 0 reads (in units m ¼ ℏ ¼ 1)

HYG ¼ 1

2Ψ†

xΨxþc2∶ðΨ†ΨÞ2∶− μΨ†Ψ− hΨ†σzΨ; ðC19Þ

expressed in terms of a two-component spinor fieldΨðxÞ ¼ (ψ↑ðxÞ; ψ↓ðxÞ)T, with external fields μ and hcoupling to charge and spin degrees of freedom, respec-tively. At the level of grand-canonical Gibbs ensembles, weintroduced the corresponding chemical potentials μ≡ βμand h≡ βh. Notice that, sometimes, the model includes anadditional (integrability-breaking) potential term VðxÞΨ†Ψ,which we have to exclude.The model is exactly solvable by the nested Bethe

ansatz. Consider an eigenstate of N spin-1=2 electronsin a periodic system of length L, of which N↑ and N↓

correspond to the number of electrons with up and down zcomponents. The electron bare dispersion is not non-relativistic, and it assumes a quadratic dependence onthe momentum θ, written e0ðθÞ ¼ θ2. The variable θ canthus be regarded as the rapidity, and subsequently, werelabel it as θ → θð1Þ, with the superscript label referring tothe physical (i.e., momentum-carrying) excitations. Thetotal energy of an eigenstate in the presence of fields reads

E ¼ PNi¼1 e0ðθð1Þi Þ − μN − hðN↑ − N↓Þ. In analogy to the

O(3) NLSM and other systems with nested spectra, adynamical lattice of propagating electrons supports inter-acting spin waves. The latter are associated with auxiliary

rapidities θð2Þj . For a periodic system of size L with N ≡N↑ þ N↓ electrons, the two sets of rapidities are deter-mined as solutions to the following nested Bethe equations:

eiθð1Þi L

YN↓

j¼1

S1ðθð1Þi ; θð2Þj Þ ¼ 1; ðC20Þ

YN↓

j≠iS1;1ðθð2Þi ; θð2Þj Þ

YNj¼1

S−11;1ðθð2Þi ; θð1Þj Þ ¼ 1; ðC21Þ

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where in the first (second) equation, the index i runs from 1toN (N↓), and SjðθÞ ¼ ðθ − jic=2Þ=ðθ þ jic=2Þ, S1;1ðθÞ≡S2ðθÞ are the standard elementary scattering amplitudes,with the associated scattering kernels

KjðθÞ ¼1

2πi∂θ log SjðθÞ ¼

1

2π

cjθ2 þ ðjc=2Þ2 ; ðC22Þ

or equivalently, in Fourier space, KjðkÞ ¼ e−jjkjc=2.Passing to the thermodynamic limit, and introducing

rapidity densities of unbound electrons ρ0ðθÞ and thestandard Bethe s strings ρsðθÞ, the Bethe-Yang equationstake the following quasilocal form:

ρtot0 ¼ 1

2πþ ðK1 ⋆ sÞ ⋆ ρ0 − s ⋆ ρ1; ðC23Þ

ρtot1 ¼ s ⋆ ðρ0 þ ρ2Þ; ðC24Þ

ρtots≥2 ¼ s ⋆ ðρs−1 þ ρsþ1Þ; ðC25Þ

with the s kernel sðθÞ ¼ (2c cosh ðπθ=cÞ)−1, satisfying1 ⋆ s ¼ 1=2 and sðkÞ ¼ ðecjkj=2Þ=ð1þ ecjkjÞ. Noticethat the source term comes from ð1=2πÞ∂θk0ðθÞ ¼ 1=2π.The energy density of an equilibrium macrostate is thengiven by

e ¼Z

dθðe0ðθÞ − μÞρ0ðθÞ þXs≥1

2shZ

dθρsðθÞ; ðC26Þ

whereas electron charge and magnetization density readne ¼ 1 ⋆ ρ0 and m ¼ 1

2ð1 ⋆ ρ0Þ −

Ps≥1 sð1 ⋆ ρsÞ. Unlike

the density of magnetization, electron density is a strictlypositive quantity in any equilibrium macrostate, whichreadily implies that the Drude weight of the electron chargealways stays strictly positive.The TBA equations in terms of the thermodynamic Y

functions YAðθÞ ¼ ρAðθÞ=ρAðθÞ, on the other hand, read

logY0 ¼ βθ2 − μ − ðK1 ⋆ sÞ ⋆ logð1þ 1=Y0Þ− s ⋆ logð1þ Y1Þ; ðC27Þ

logY1 ¼ −s ⋆ logð1þ 1=Y0Þ þ s ⋆ logð1þ Y2Þ; ðC28Þ

logYs≥2 ¼ s ⋆ IA∞s;s0 logð1þ Ys0 Þ; ðC29Þ

with IA∞s;s0 ¼ δs;s0þ1 þ δs;s0−1. Importantly, the chemical

potential h coupling to total magnetization enters, as usual,through the large-s asymptotics

lims→∞

logYs

s¼ 2h: ðC30Þ

Finally, the equilibrium free energy density in the grand-canonical Gibbs states reads

−βf ¼ 1

2π

Zdθ log (1þ 1=Y0ðθÞ): ðC31Þ

Let us mention, in passing, that the above TBA equationscoincide, apart from the dispersion-dependent sourceterms, with those of the fundamental SUð2j1Þ quantumchain [the supersymmetric t − J model given by Eq. (C18)]in the distinguished grading . For further details,we refer the reader to Ref. [46], where the completesolution to the algebraic TBA equations for the SUð2j1Þchain is derived.

a. Dressing equations

In the scope of our study, we are primarily interested inthe dressed dispersion relation and the dressed Noethercharges. There are three different limits of the above Bethe-Yang and TBA equations that are analytically tractable, alltreated originally in Ref. [160]: (i) the zero-temperaturelimit T → 0, (ii) the weak-coupling limit c → 0, and (iii) thestrong-coupling limit c → ∞. Since none of these limitscan be regarded as generic local equilibrium states, they arenot really relevant to us. On the other hand, there is not UVcompleteness in the high-temperature limit, as in relativisticintegrable QFTs. Even though the Y functions become flat(i.e., independent of rapidity θ) in the β → 0 limit, there isno regulator at large rapidities to ensure convergence of thefree energy density.It is nevertheless instructive to exhibit a closed-form

solution to the TBA equation in the limit β → 0 whilekeeping chemical potentials μ and h general. Denoting theconstant thermodynamic Y functions by Ys≥0, and using1 ⋆ s ¼ 1=2 and likewise ðK1 ⋆ sÞ ⋆ 1 ¼ 1=2, the TBAequations are readily rendered algebraic:

e2μY20 ¼ 1=½ð1þ 1=Y0Þð1þ Y1Þ�; ðC32Þ

Y21 ¼ ð1þ Y2Þ=ð1þ 1=Y0Þ; ðC33Þ

Y2s≥2 ¼ ð1þ Ys−1Þð1þ Ysþ1Þ: ðC34Þ

Notice, once more, that the charge chemical potential μremains explicitly present while h implicitly hides in theasymptotic condition (C30). The solution to the three-termrecursion for Ys≥1 in the bulk that satisfies the prescribedasymptotics at large s is given by

Ys≥1 ¼ y2s − 1; Y0 ¼y20

1 − y20: ðC35Þ

We have introduced the functions

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ysða; bÞ ¼bas − b−1a−s

a − a−1; ðC36Þ

which obey the Hirota bilinear relation, y2s − ys−1ysþ1 −1 ¼ 0. The functions ys are none other than classicalcharacters associated to a non-semi-simple Lie algebrasuð2Þ ⊕ uð1Þ, which is the bosonic subalgebra of the Liesupergroup SUð2j1Þ, specifically, ys specialized to suð2Þcharacters if b ¼ a. The two remaining free constants a andb, which are functions of h and μ, can be determined fromthe boundary condition Y0ð1þ Y0Þð1þ Y1Þ ¼ e−2μ. Thelarge-s asymptotics is satisfied by a ¼ e−h. The coefficientb is then obtained from

a2ðb2 − 1Þ2ða2 − b2Þ2 ¼ c2; c≡ e−μ; ðC37Þ

yielding

bðμ; hÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ca1þ c=a

s; Y0 ¼

ac2

aþ cþ a2c: ðC38Þ

Reparametrizing the y characters in terms of chemicalpotentials μ and h, we have the form ysðμ; hÞ ¼sinh2 ðshþ μÞ= sinh2ðhÞ.In the β → 0 limit, the Fermi occupation functions,

ns≥0ðμ; hÞ ¼ 1=(1þ Ysðμ; hÞ), can be expressed in theform

n0ðμ; hÞ ¼ 1 − y20ðμ; hÞ; nsðμ; hÞ ¼ y−2s ðμ; hÞ: ðC39Þ

In particular, taking h → 0with μ general, we again find theanticipated universal behavior at large s,

limh→0

nsðμ; hÞ ¼ð1þ eμÞ2

1þ sð1þ eμÞ2 ∼ 1=s2: ðC40Þ

The dressed electronic charge in the high-temperaturelimit and general h and μ is thus

ndr0 ðμ; hÞ ¼ −∂μ logY0ðh; μÞ ¼1þ e2h þ 2ehþμ

1þ e2h þ ehþμ ; ðC41Þ

and, as already emphasized previously, is manifestlystrictly positive. On the other hand, the dressed magneti-zation is

mdr0 ðμ; hÞ ¼ −∂h logY0ðh; μÞ ¼

e2h − 1

1þ e2h þ ehþμ ; ðC42Þ

which vanishes in the h → 0 limit, namely, mdrðμ; hÞ ¼ð2=2þ eμÞhþOðh3Þ. The dressed magnetization forthe s strings displays an analogous qualitative behavior;in the μ → 0 limit, we find limμ→0mdr

s ðμ; hÞ ¼16ð2sþ 1Þ2hþOðh3Þ.

Next, we consider the dressed dispersion relations forthe quasiparticles. The dressing equations can be put in thefollowing compact algebraic form:

F0 ¼ d0 þ ðK1 ⋆ sÞ ⋆ n0F0 − s ⋆ n1F1; ðC43Þ

F1 ¼ s ⋆ ðn0F0 þ n2F2Þ; ðC44Þ

Fs≥2 ¼ s ⋆ ðns−1Fs−1 þ nsþ1Fsþ1Þ: ðC45Þ

Specifically, for FA → ∂θεA with d0 → ∂θe0ðθÞ ¼ 2θ, oneobtains the dressed energy derivatives, and likewise forFA → ρtotA with d0 ¼ 1=2π, the total state densities. Recallthat these are related to the dressed momentum derivativesvia ½p0

AðθÞ�dr ¼ 2πρtotA ðθÞ. Despite the UV catastrophe, therapidity densities are well defined in the β → 0 limit wherethey (in distinction to the NLSM) become flat.The solution to the homogeneous “bulk” recurrence

relation, ρtots ¼ 12ðns−1ρtots−1þ nsþ1ρ

totsþ1Þ, with the prescribed

large-s asymptotics, reads

ρtots≥1 ¼α

2π

�ysys−1

−ysysþ1

�: ðC46Þ

This formula is in fact the same bulk solution governing theHeisenberg SU(2) chain upon replacing suð2Þ charactersχsðhÞ with ysðμ; hÞ. The difference shows up in the initialconditions, i.e., in the equations for densities ρtot0 and ρtot1 ,which provide two extra equations to uniquely determinethe unknown function ρtot0 and coefficient α. We findα ¼ −ðy−1y0Þ=2π ¼ ac=½ðaþ cÞð1þ acÞ�, and

ρtot0 ¼ 1

2π

ða2 − 1Þðb2 þ 1Þ1 − a2b2

: ðC47Þ

In the h → 0 limit, the latter simplifies to

ρtots →2cðcþ sþ csÞ

ð1þ cÞ½sðcþ 1Þ − 1�½sðcþ 1Þ þ 2cþ 1� ; ðC48Þ

whence we can extract the anticipated scaling at large s,

ρtots ðμ; hÞ ¼ 2eμ

ð1þ eμÞ21

sþOð1=s2Þ: ðC49Þ

Furthermore, for μ → 0, the result is even simpler:limh→0 ρ

tots ð0; hÞ ¼ ð2sþ 1Þ=ð4s2 þ 4s − 3Þ.

Since the bare energies are not UV-finite quantities, thecorresponding dressed energies are well defined only atfinite inverse temperature β. Similarly as in other QFTs,the solutions to the dressing equations cannot be foundin a closed algebraic form; instead, one has to resort tonumerical solutions. We find the anticipated scaling rela-tion for the effective velocities of giant magnons at large s,

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veffs ∼ 1=s; ðC50Þ

irrespectively of the coupling c > 0. This result can, in fact,be consistently deduced from the strong-coupling limitc → ∞, where the electron excitations effectively decouplefrom magnons; the latter interact precisely as spins in theisotropic Heisenberg spin-1=2 chain [up to Oð1=cÞ cor-rections], which, themselves, already give rise to super-diffusive magnetization transport at any finite density athalf filling.

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