quantum critical properties of a metallic spin-density-wave transition · 2017. 1. 17. · physical...

PHYSICAL REVIEW B 95, 035124 (2017)

Quantum critical properties of a metallic spin-density-wave transition

Max H. Gerlach,1 Yoni Schattner,2 Erez Berg,2 and Simon Trebst11Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany

2Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot, 76100, Israel(Received 4 October 2016; published 17 January 2017)

We report on numerically exact determinantal quantum Monte Carlo simulations of the onset of spin-density-wave (SDW) order in itinerant electron systems captured by a sign-problem-free two-dimensional lattice model.Extensive measurements of the SDW correlations in the vicinity of the phase transition reveal that the criticaldynamics of the bosonic order parameter are well described by a dynamical critical exponent z = 2, consistentwith Hertz-Millis theory, but are found to follow a finite-temperature dependence that does not fit the predictedbehavior of the same theory. The presence of critical SDW fluctuations is found to have a strong impact on thefermionic quasiparticles, giving rise to a dome-shaped superconducting phase near the quantum critical point.In the superconducting state we find a gap function that has an opposite sign between the two bands of themodel and is nearly constant along the Fermi surface of each band. Above the superconducting Tc, our numericalsimulations reveal a nearly temperature and frequency independent self-energy causing a strong suppression ofthe low-energy quasiparticle weight in the vicinity of the hot spots on the Fermi surface. This indicates a clearbreakdown of Fermi liquid theory around these points.

DOI: 10.1103/PhysRevB.95.035124

I. INTRODUCTION

Metallic spin-density-wave (SDW) transitions are ubiq-uitous to strongly correlated materials such as the electron-doped cuprates [1], the Fe-based superconductors [2], heavyfermion systems [3], and organic superconductors [4]. In allthese materials, unconventional superconductivity is found toemerge near the onset of SDW order, with the maximum of thesuperconducting Tc occuring either at or near the underlyingSDW quantum phase transition (QPT). In addition, the vicinityof the SDW transition is often characterized by strongdeviations from Fermi liquid theory—both in thermodynamicand in single electron properties.

More broadly, understanding the properties of quantum crit-ical points (QCPs) in itinerant fermion systems has attractedmuch interest over the past several decades [5–29]. Unlikethermal critical phenomena and QCPs in insulating systems,here the critical order parameter fluctuations are stronglyinteracting with low-energy fermionic quasiparticles near theFermi surface. In the traditional approach to this problemdue to Hertz [5], later refined by Millis [6], the fermions areintegrated out from the outset, leading to an effective bosonicaction for the order parameter fluctuations. The dynamics ofthe order parameter is found to be overdamped due to thecoupling to the fermions, that act as a bath. The effectivebosonic action is then treated using conventional renormaliza-tion group (RG) techniques. While physically appealing, thisapproach has the drawback that integrating out low-energymodes is dangerous, since it generates nonanalytic terms inmomentum and frequency that are difficult to treat within theRG scheme. An alternative popular approach has been to use a1/N expansion [8], where N is the number of fermion flavors.However, in the important case of two spatial dimensions,this approach turns out to be uncontrolled, as well [12,15].Alternative expansion parameters have been introduced inorder to control the problem [13,18], or have been proposed toemerge naturally [29]. A fully controlled analytical treatmentof QCPs in itinerant electron systems has remained one of thegrand challenges in strongly correlated electron physics.

In addition to the bosonic critical fluctuation dynamics,an important open question regards the behavior of thefermionic quasiparticles in the vicinity of the transition.The exchange of SDW critical fluctuations can mediate asuperconducting instability; however, the same fluctuationsalso strongly scatter the quasiparticles, causing them to losetheir coherence and leading to the formation of a non-Fermiliquid metal. It is not clear which of these effects dominates,i.e., is there a well-defined non-Fermi liquid regime thatprecedes superconductivity, or does pairing always preemptthe formation of a non-Fermi liquid [30]?

In this work, we perform extensive numerically exactdeterminantal quantum Monte Carlo (QMC) simulations [31–35] of a metallic system in the vicinity of an SDW transition.We use the approach of Refs. [17,28], that introduced atwo-dimensional multiband lattice model that captures thegeneric structure of the “hot spots”—points on the Fermisurface where quasiparticles can scatter off critical SDWfluctuations resonantly. The universal properties of metallicSDW transitions are believed to depend only on the vicinityof the hot spots. At the same time, the model is amenableto QMC simulations without a sign problem [17]. Ourgoal here is both to understand the generic properties ofthe transition, and to provide a controlled benchmark toanalytic theories. We present detailed information about thebosonic and fermionic correlations and the interplay withunconventional superconductivity in the vicinity of the QCP.

Previously, the finite-temperature phase diagram of themodel has been characterized, and a dome-shaped supercon-ducting phase peaked near the SDW transition was found[28,36]. Here, we measure the SDW correlations in the vicinityof the transition, above the superconducting Tc. We find that,over a broad range of parameters, the SDW susceptibility iswell described by the following form:

χ0(q,ωn,r,T )

= 1aq(q − Q)2 + aω|ωn| + ar (r − rc0) + f (r,T ) . (1)

2469-9950/2017/95(3)/035124(17) 035124-1 ©2017 American Physical Society

https://doi.org/10.1103/PhysRevB.95.035124

GERLACH, SCHATTNER, BERG, AND TREBST PHYSICAL REVIEW B 95, 035124 (2017)

Here, Q is the ordering wave vector [chosen to be (π,π ) inour model], ωn = 2πnT is a Matsubara frequency, and r is aparameter used for tuning through the SDW QCP, while aq ,aω, ar , and rc0 are nonuniversal constants. The function f (r,T )extrapolates to 0 as T → 0. Importantly, Eq. (1) capturesthe behavior of both the bosonic SDW correlations and thesusceptibility of a fermion bilinear operator with the samesymmetry, establishing the consistency of our analysis.

Interestingly, χ0(q,ωn,r,T → 0) has precisely the formpredicted by Hertz and Millis; in particular, the bosonic criticaldynamics are characterized by a dynamic critical exponentz = 2. The function f (r,T ) does not follow the predictedbehavior, however. In a window of temperatures above Tc, wefind a power-law dependence f (r ≈ rc0,T ) ∝ T α with α � 2,in contrast to the linear behavior predicted by Millis [6,37].

The single-fermion properties above Tc are found to dependstrongly on the distance from the hot spots. Away from thehot spots, a behavior consistent with Fermi liquid theoryis observed. At the hot spots, a substantial loss of spectralweight is seen upon approaching the QCP. In a temperaturewindow above Tc, the fermionic self-energy is only weaklyfrequency and temperature dependent, corresponding to anearly-constant lifetime of quasiparticles at the hot spots. Thisbehavior indicates a strong breakdown of Fermi liquid theoryat these points. It is not clear, however, whether it representsthe asymptotic behavior at the putative underlying SDW QCP,since superconductivity intervenes before the QCP is reached.

Finally, in order to probe the interplay between magneticquantum criticality and superconductivity, we measure thesuperconducting gap, �k, and the momentum-resolved su-perconducting susceptibility, Pk,k′ , across the phase diagram.No strong feature is found in �k at the hot spots. Rather,�k and Pk,k′ vary smoothly on the Fermi surface. While thepairing interaction may be strongly peaked at wave vectorQ, the resulting gap function does not reflect such a strongmomentum dependence.

This paper is organized as follows. In Sec. II, we describethe model and review its phase diagram. Section III presentsa detailed analysis of the SDW susceptibility across the phasediagram. In Sec. IV, we study the single-fermion properties,providing evidence for the breakdown of Fermi liquid theoryin the vicinity of the hot spots. Section V analyzes the gapstructure in the superconducting state. The cumulative resultsare put into perspective in Sec. VI. Supplementary data setsand some technical details are presented in the appendices.

II. THE MODEL AND THE PHASE DIAGRAM

Our lattice model is composed of two flavors of spin- 12fermions, ψx and ψy , that are coupled to a real bosonic vectorfield �ϕ, which represents fluctuations of a commensurate SDWorder parameter at wave vector Q = (π,π ). The two typesof fermions exhibit quasi one-dimensional dispersions alongmomenta kx and ky , respectively, which in the absence ofintreractions give rise to the Fermi surfaces illustrated in Fig. 1.It is precisely this two-flavor structure that fundamentallyallows us to set up an action completely devoid of the fermionsign problem in QMC simulations [17]. Yet, the Fermi surfacesof this model capture the generic structure of the hot spots,

FIG. 1. Noninteracting Fermi surfaces of the ψx and ψy fermions.Hot spots (indicated by the black dots) are linked by the vector Q =(π,π ) with points on the other band. The ψx and ψy fermions havestronger dispersion in direction kx and ky , respectively.

which is generally believed to determine the universal physicsnear the QCP.

As in our previous work on this model [28], we assumean O(2) symmetric SDW order parameter, i.e., �ϕ is restrictedto the XY plane. In contrast to the case of an O(3) orderparameter (studied in Refs. [17,36]), the easy-plane orderparameter implies the existence of a finite-temperature SDWphase transition of Berezinskii-Kosterlitz-Thouless (BKT)character, which we can track in our numerics. In addition,the reduced dimensionality of the order parameter brings awelcome computational benefit as it enables a reduction ofthe dimensions of all single-fermion matrices by half, greatlyimproving the efficiency of the numerical linear algebra.

Our lattice model is given by the action S = SF + Sϕ =∫ β0 dτ (LF + Lϕ) with1

LF =∑i,j,s

α = x,y

ψ†iαs[(∂τ − μ)δij − tαij ]ψjαs

+ λ∑i,s,s ′

[�s · �ϕi]ss ′ψ†ixsψiys ′ + H.c.,

Lϕ = 12

∑i

1

c2

(d �ϕidτ

)2+ 1

2

∑〈i,j〉

(eiQ·ri �ϕi − eiQ·rj �ϕj )2

+∑

i

[r

2�ϕ2i +

u

4

( �ϕ2i )2]. (2)

This action is defined on a square lattice with sites labeledby i,j = 1, . . . ,Ns , where 〈i,j 〉 are nearest neighbors. Thetwo fermion flavors are indexed by α = x,y, while s,s ′ = ↑,↓index the spin polarizations and �s are the Pauli matrices.Imaginary time is denoted by τ and β = 1/T is the inversetemperature. The fermionic dispersions are implemented by

1Note a slight change in notation compared to our previouspublication [28].

035124-2

QUANTUM CRITICAL PROPERTIES OF A METALLIC . . . PHYSICAL REVIEW B 95, 035124 (2017)

-1.6 -1.5 -1.4 -1.3 -1.2tuning parameter r

0

0.05

0.1

0.15

0.2

tem

pera

ture

T

SDW

TSDW

(a) λ = 1

0.2 0.3 0.4 0.5 0.6 0.7tuning parameter r

0

0.05

0.1

0.15

0.2

tem

pera

ture

SDW

SC

TSDWTc

(b) λ = 1.5

2.5 3 3.5 4 4.5tuning parameter r

0

0.05

0.1

0.15

0.2

tem

pera

ture

SDW

TSDWTc

SC

(c) λ = 2

FIG. 2. Finite-temperature phase diagram of model (2) for three different values of the Yukawa coupling λ and bare bosonic velocity c = 3.Shown are the transition temperature TSDW to magnetic spin density wave (SDW) order as well as the estimated location of the zero-temperaturephase transition point rc (indicated by the red star). The superconducting (SC) transition temperature Tc is shown where applicable. In the sameunits the Fermi energy is EF = 2.5. Dashed lines are a guide to the eye.

setting different hopping amplitudes along the horizontal andvertical lattice directions. For the ψx fermions they are given bytx,h = 1 and tx,v = 0.5, respectively, and for the ψy fermionsby ty,h = 0.5 and ty,v = 1. Note that our model is fullyC4-symmetric with a π/2 rotation mapping the ψx band tothe ψy band and vice versa. The tuning parameter r allowsto tune the system to the vicinity of an SDW instability.In an experimental system, r could be proportional to aphysical tuning parameter such as pressure or doping. Weset the chemical potential to μ = −0.5, such that the Fermienergy, measured relative to the band bottom, is EF = 2.5. Thequartic coupling is set to u = 1. The second term in Lϕ favorscommensurate antiferromagnetic order in �ϕ with Q = (π,π ).In the following we mostly focus on the case of a bare bosonicvelocity of c = 3 and a Yukawa coupling between fermionsand bosons of λ = 1.5. Occasionally, we also consider othervalues of c and λ.

Our numerical analysis of model (2) is based on exten-sive finite-temperature DQMC [31–35] simulations. For thegeneral setup and technical details on the implementation ofour DQMC simulations, we refer to our earlier paper [28]and its detailed supplemental online material. Here we wantto single out a few conceptual aspects of our setup, whichhave allowed to push our simulations down to temperatures ofT = 1/40 for system sizes up to 16 × 16 sites. First, usinga replica-exchange scheme [38,39] in combination with aglobal update procedure [28] thermal equilibration of oursimulations is decidedly improved. Most of our simulationswere performed in the presence of a weak fictitious perpen-dicular “magnetic” field (designed not to break time-reversalsymmetry), which serves to greatly speed up convergenceto the thermodynamic limit for metallic systems [28,40,41].Since this technique breaks translational invariance of thefermionic Green’s function, we cannot make use of it tostudy k-resolved fermionic observables. For this reason wehave carried out additional simulations without the magneticflux, but with twisted boundary conditions, which allows toincrease the momentum space resolution. We give details onthese procedures in Appendix A.

To set the stage for our discussion in the followingsections, we summarize the finite-temperature phase diagramof model (2) for c = 3 and three different values of theYukawa coupling λ = 1, 1.5, and 2 in Fig. 2. Besides a

paramagnetic regime for sufficiently high temperature, thedominant feature of these phase diagrams is a quasi-long-rangeordered SDW phase whose transition temperature TSDW issuppressed with increasing tuning parameter r . Extrapolatingthe SDW transition down to the zero temperature providesan estimate of the location of the quantum phase transitionat r = rc (indicated by the red star). At low temperatures,the SDW transition may become weakly first order [28].However, in the temperature range considered here, thetransition is either continuous or (possibly) very weakly firstorder. While for λ = 1, this SDW phase is the only orderedphase down to temperatures of T = 1/40, there is an additionalsuperconducting phase emerging in the vicinity of the QPT forthe two larger values of the Yukawa coupling. For λ = 1.5, webarely observe the tip of this quasi-long-range ordered phasewith a maximum critical temperature of T maxc ≈ 1/40, whichis our numerical temperature limit. For λ = 2, we can clearlymap out a superconducting dome with the maximum of thecritical temperature T maxc ≈ 1/20.

At finite temperatures, both the SDW and the SC finite-temperature transitions are expected to be of BKT type.The SDW susceptibility χ = ∫ dτ ∑i eiQ·ri 〈 �ϕi(τ ) �ϕ0(0)〉 isfound to follow a scaling law χ ∝ L2−η with a continuouslychanging exponent η. We identify the transition temperatureTSDW with the point where the exponent takes the universalvalue η = 1/4. The SC transition temperature can be bothdetermined via a similar η fit or by the point where thesuperfluid density ρs obtains the universal value of 2Tc/π .Both estimates are found to agree. The error bars in Fig. 2are mostly due to finite-size effects. For further details on theprocedures employed to identify the different phase transitionssee Ref. [28] and its accompanying supplementary onlinematerial.

A common feature to all three phase diagrams is a changeof slope of the SDW phase boundary curve at low temperatures(T ≈ 0.07). This apparent “bending” is more pronounced atlarger Yukawa coupling λ. Such behavior is generally expectedto occur at the onset of superconductivity [28,42]. Here,however, the bending does not visibly track the SC transitiontemperature. The curvature of the TSDW line is also reflectedin the SDW susceptibility in the paramagnetic region of thephase diagram, see Fig. 3 for λ = 1 and Fig. 18 in Appendix Bfor λ = 1.5,2.

035124-3


FIG. 3. Inverse SDW susceptibility χ−1 across the phase diagramfor λ = 1. We show numerical data obtained at L = 14 at thetemperatures that are indicated by the ticks on the inside of the plot.Intermediate temperatures are interpolated linearly, while the highresolution in the tuning parameter r is achieved by reweighting.Contour lines of χ−1 are marked gray.

Over a broader range of parameters, the maximum SCtransition temperature T maxc grows monotonically with in-creasing of either the Yukawa coupling or the boson velocityc, as illustrated in Fig. 4. Up to an intermediate couplingstrength λ ≈ 3, T maxc rapidly grows as T maxc ∝ λ2, eventuallysaturating at stronger coupling. Qualitatively, these trends arein agreement with results from Eliashberg theory [43].

III. MAGNETIC CORRELATIONS

We start our discussion of the quantum critical behaviorof model (2) with an examination of magnetic fluctuationsacross its entire finite-temperature phase diagram. We probethe formation of magnetic correlations both through thesusceptibility of the bosonic order parameter, �ϕ, and througha fermionic bilinear of the same symmetry, which we evaluate

12 22 32 42 52 62

Yukawa coupling λ 2

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

Tm

axc

c = 1c = 2c = 3

FIG. 4. The maximal superconducting transition temperatureT maxc for different values of the Yukawa coupling λ and (bare)boson velocity c. The hatched gray region indicates temperaturesT < 0.025, which are beyond our numerically accessible temperaturerange.

independently in the same numerical simulations of the action(2). As we show below, both susceptibilities exhibit the samebehavior, supporting the robustness of our conclusions to bepresented.

A. Bosonic SDW susceptibility

We first consider the bosonic susceptibility calculated fromthe SDW order parameter �ϕ in action (2)

χ (q,iωn,r,T ) =∑

i

∫ β0

dτeiωnτ−iq·ri 〈 �ϕi(τ ) · �ϕ0(0)〉 (3)

for a given momentum q and Matsubara frequency ωn =2πnT . The expectation values are estimated in a DQMCsimulation run at finite temperature T and for a specific valueof the tuning parameter r , indicated here as explicit parameters.At low temperature, we use the following form to fit the data,inspired by Hertz theory [5]:

χ−10 (q,iωn,r,T → 0)= aq(q − Q)2 + aω|ωn| + ar (r − rc0), (4)

where aq, aω, and ar are nonuniversal fitting parameters thatdescribe the momentum dependence in the vicinity of theordering wave vector Q = (π,π ), Landau damping, and thedependence on the tuning parameter r , respectively. The fittingparameter rc0 indicates the location of the divergence of χ0.Due to the appearance of a superconducting phase at lowtemperatures, rc0 may differ from the actual location of theQPT at r = rc. However, within our numerical resolution,we find rc0 ≈ rc, where rc is obtained by extrapolating thefinite-temperature transition line TSDW → 0, as shown in thephase diagrams of Fig. 2.

Running extensive DQMC simulations for system sizesL = 8,10, . . . 14, we have evaluated χ across the threeprincipal phase diagrams of Fig. 2 for different values ofthe Yukawa coupling λ = 1, 1.5, 2 and bare bosonic velocityc = 3. Restricting our analysis to the magnetically disorderedside for each coupling and to temperature scales above thesuperconducting phase, we find that our calculated susceptibil-ities are in good agreement with the functional form of Eq. (1).The consistency with Eq. (4) is illustrated in the panels ofFig. 5, which show data collapses for a range of small momentaq − Q, small Matsubara frequencies, low temperatures T �0.1 and tuning parameters r � rc0. Finite-size effects are rathersmall. Considering the variation of the Yukawa coupling λ, wefind that the fit to the functional form (4) is slightly worsefor stronger coupling λ, which is also indicated by the largerspread of the data points. This decreasing fit quality may bea consequence of the smaller temperature window availableabove the superconducting Tc, as well as the associatedregime of superconducting fluctuations at T � Tc [28], whichincreases with Yukawa coupling (see also Fig. 4).

With the data collapse of Fig. 5 asserting the general validityof the functional form (4), we now take a closer look at itsindividual dependence on tuning parameter, frequency, andmomentum. First, the dependence on the tuning parameter r isillustrated for the inverse susceptibility χ−1(q = Q,iωn = 0)in Fig. 6 (for λ = 1.5 and T = 0.1). For tuning parametersr � rc0 = 0.6 we find that the data for different system sizes

035124-4


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.54(q−Q)2 +0.29|ωn|+0.36[r +1.31]

0.00.20.40.60.81.01.21.4

SDW

susc

eptib

ility

χ−1 0.025 ≤ T ≤ 0.1

−1.30 ≤ r ≤ 0.408 ≤ L ≤ 140.00 ≤ |ωn| ≤ 1.100.00 ≤ |q−Q| ≤ 1.05

99,696 data points, χ2dof =3.3

(a) λ = 1

510152025303540

0.0 0.5 1.0 1.5 2.00.62(q−Q)2 +0.57|ωn|+0.40[r−0.60]

0.0

0.5

1.0

1.5

2.0

SDW

susc

eptib

ility

χ−1 0.025 ≤ T ≤ 0.1

0.65 ≤ r ≤ 2.408 ≤ L ≤ 140.00 ≤ |ωn| ≤ 1.10 *0.00 ≤ |q−Q| ≤ 1.05


(b) λ = 1.5

369121518212427

0.0 0.5 1.0 1.5 2.0 2.50.71(q−Q)2 +0.94|ωn|+0.43[r−3.03]

0.0

0.5

1.0

1.5

2.0

2.5

SDW

susc

eptib

ility

χ−1 excluding superconducting dome

0.029 ≤ T ≤ 0.13.15 ≤ r ≤ 4.708 ≤ L ≤ 140.00 ≤ |ωn| ≤ 1.08 *0.00 ≤ |q−Q| ≤ 1.05


(c) λ = 2

24681012141618

FIG. 5. Comparison between the inverse SDW susceptiblity χ−1

and the functional form χ−10 = aq (q − Q)2 + aω|ωn| + ar (r − rc0),which has been fitted for small frequencies ωn and momenta q − Q atlow temperatures T and tuning parameters r > rc0 in the magneticallydisordered phase, for (a) λ = 1, (b) 1.5, and (c) 2. Data insidethe superconducting phase have been excluded from the fit. Fortemperatures T � 2T maxc we restrict the fit to finite frequencies |ωn| >0. The correspondence of χ−1 with the fitted form is shown in the formof 2D histograms over all data points, which are normalized over the

total area. In each fit we have minimized χ 2dof = 1Ndof∑

[χ−1−χ−10

ε]2,

where Ndof is the number of degrees of freedom of the fit and ε is thestatistical error of the data.

follows a linear dependence. The moderate deviation from aperfect kinklike behavior at rc0 is likely a combination of finite-size and finite-temperature effects (see also the finite-size trendshown in the inset of Fig. 6). A very similar picture emergesfor the two other coupling parameters λ = 1 and λ = 2, forwhich we show analogous plots in Fig. 19 of Appendix B.

Turning to the frequency dependence of χ−1(q,iωn) next,we find that for a range of values r � rc0 the frequencydependence is linear for small Matsubara frequencies ωn withan apparent cusp at ωn = 0, signaling overdamped dynamics

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0tuning parameter r

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

SDW

susc

eptib

ility

χ−1

λ = 1.5, T = 0.1L = 8L = 10L = 12L = 14

L = 8L = 10L = 12L = 14

0.6 0.7 0.8

0.00

0.05

FIG. 6. Bosonic SDW susceptibility χ−1(q = Q,iωn = 0) as afunction of the tuning parameter r for λ = 1.5 at T = 0.1. The blackline is a linear fit for r > 0.7 and L = 14. Continuous colored linesthrough data points have been obtained by a reweighting analysis.

of the order parameter field. This holds both for q = Q and forsmall finite momentum differences q − Q. See Fig. 7 for anillustration at λ = 1.5 and Appendix B with Fig. 20 for λ = 1and λ = 2. At finite Matsubara frequencies ωn, finite-sizeeffects are negligibly small, as evident in the data collapseof χ−1 for different system sizes in the left panel in Fig. 7.

To establish the presence of a |ωn| term in χ−1, we fit itat low frequencies to the form b0 + b1|ωn| + b2ω2n. The fitsare shown in Fig. 7. The |ωn| contribution is clearly dominantin this frequency range. Inside the superconducting phase, the|ωn| term is suppressed (see Fig. 24 in Appendix B). This ispresumably due to gapping out of the fermions.

Third, for the same range of r , the momentum dependenceof χ−1(q,iωn) is consistent with a quadratic form in q − Q,which holds both for ωn = 0 and small finite frequencies ωn.

−1 0 1frequency ωn

0.0

0.5

1.0

1.5

SDW

susc

eptib

ility

χ−1

(a) r = 0.7L = 14

|q| = 0.45|q| = 0.63

|q| = 0.90|q| = 1.00

q = 0L = 8L = 10L = 12L = 14

−1 0 1

0.0

0.5

1.0

1.5

(b) r ≥ 0.7q = 0, L = 14

r = 0.7r = 0.9r = 1.1

r = 1.3r = 1.5r = 1.7

FIG. 7. Frequency dependence of the inverse bosonic SDWsusceptibility χ−1 for λ = 1.5 at T = 1/40 (a) shown at r ≈ rc0for various momenta q = Q + q̃ and (b) shown at various valuesr > rc0 for q = Q. The black line is the best fit of a second degreepolynomial b0 + b1|ωn| + b2ω2n to the q = Q, L = 14 low-frequencydata, yielding a basically straight line.

035124-5


0 1 2momentum (q−Q)2

0.0

0.5

1.0

1.5

SDW

susc

eptib

ility

χ−1

(a) r = 0.7

L = 14ωn = 0.16ωn = 0.31ωn = 0.47ωn = 0.63ωn = 0.79ωn = 0.94

ωn = 0L = 8L = 10L = 12L = 14

0 1 2

0.0

0.5

1.0

1.5

(b) r ≥ 0.7

ωn = 0, L = 14r = 0.7r = 0.9r = 1.1

r = 1.3r = 1.5r = 1.7

FIG. 8. Inverse bosonic SDW susceptibility χ−1 as a function ofmomentum q = Q + q̃ for λ = 1.5 at T = 1/40 (a) shown at r ≈ rc0for various frequencies ωn and (b) shown at various values r � rc0for ωn = 0. The black line is the best fit of a0 + a2q̃2 to the ωn = 0,L = 14 small-momentum data.

(See Fig. 8 for λ = 1.5 and Appendix B with Fig. 21 for λ = 1and λ = 2.) Note that due to the discretization of the Brillouinzone, finite-size effects are more pronounced here than for thefrequency dependence.

B. Fermion bilinear SDW susceptibility

An important independent confirmation that the form (4)is generic to the quantum critical regime is to affirm that italso holds for other SDW order parameters that have the samesymmetry. We have examined the correlations of a fermionbilinear order parameter:

Sxx(q,iωn,r,T ) =∑

i

∫ β0

dτeiωnτ−iq·ri〈Sxi (τ )S

x0 (0)

〉. (5)

In the estimation of Sxx , we make use of spin rotationalsymmetry around the z axis, 〈Sxi (τ )Sx0 (0)〉 = 〈Syi (τ )Sy0 (0)〉.Here, Sxi and S

y

i are interflavor fermion spin operators, whichare given by

�Si =(Sxi ,S

y

i ,Szi

) = ∑s,s ′

�sss ′ψ†xisψyis ′ + H.c. (6)

Indeed, we find that at small frequencies and momenta, thefermion bilinear SDW susceptibility Sxx follows the samefunctional form (4) as the bosonic SDW susceptibility χdiscussed above. The momenta and frequency dependenciesof the fermionic bilinear susceptibility at λ = 1.5 are shownin Fig. 9, with the respective dependencies of the bosonicsusceptibility appearing in Figs. 7 and 8. Additional data forthe fermionic SDW susceptibility at λ = 1 and λ = 2 is givenin Fig. 22 of Appendix B. In summary, the dependence of boththe bosonic and fermionic SDW susceptibilities on the tuningparameter, frequency, and momentum stand in good agreementwith the form (4).


0.0

0.5

1.0

ferm

ioni

cSD

Wsu

scep

tibili

tyS−

1xx L = 14

|q| = 0.45|q| = 0.63

|q| = 0.90|q| = 1.00

q = 0L = 8L = 10L = 12L = 14


0.0

0.5

1.0

L = 14ωn = 0.16ωn = 0.31ωn = 0.47

ωn = 0.63ωn = 0.79ωn = 0.94

ωn = 0L = 8L = 10L = 12L = 14

FIG. 9. Inverse fermionic SDW susceptibility S−1xx for λ = 1.5 atT = 1/40 and r = 0.7 ≈ rc0. (Left-hand side) Frequency dependencefor various momenta q = Q + q̃. The black line is a fit of the seconddegree polynomial b0 + b1|ωn| + b2ω2n to the q = Q, L = 14 low-frequency data, yielding a basically straight line. (Right-hand side)Momentum dependence for various frequencies ωn. The black line isa fit of a0 + a2q̃2 to the ωn = 0, L = 14 small-momentum data.

C. Temperature dependence

We now turn to the temperature dependence of the nu-merically computed bosonic and fermionic SDW susceptibil-ities χ−1 and S−1xx . Our numerical data for the temperaturedependence of χ−1 and S−1xx are shown in Fig. 10 for fixedYukawa coupling λ = 1.5 and two different values of thetuning parameter on the paramagnetic side of the QCP, i.e.,for r > rc0. These data are complemented with similar resultsfor λ = 1 and 2 in Fig. 23 of Appendix B.

Evidently, the data show different scaling regimes withincreasing temperature. At sufficiently high temperatures, T �0.35, the susceptibilities χ−1(q = Q,iωn = 0) and S−1xx (q =Q,iωn = 0) are approximately linearly dependent on temper-ature, as shown in the insets of Fig. 10. In an intermediatetemperature regime, however, we observe a crossover to adifferent functional temperature dependence as shown in themain panels of Fig. 10. In this intermediate temperaturewindow 0.05 � T � 0.35, our numerical data are found toreasonably fit functions of the quadratic form a0 + a2T α withα � 2 ± 0.3. Unlike the leading dependencies on the tuningparameter, frequency and momentum discussed in the previoussection, this power-law dependence is not as robust. Notethat the crossover temperature between the high-T linear andintermediate-T quadratic behaviors does not depend stronglyon the tuning parameter r . Notably, even for r ≈ rc0, thisintermediate regime does not disappear.

At still lower temperatures T � 0.05, our data mightindicate a second crossover to yet different behavior. Withthe tuning parameter r tuned close to its critical value rc0both χ and Sxx are found to be nonmonotonic for the smallesttemperatures and largest system sizes accessed in this study.The apparent upturn, whose precise location is hard to deter-mine due to finite-size effects (which are strongest for r ≈ rc0)and the enhanced statistical uncertainty at low temperatures, ismost likely a precursor effect of superconductivity [28], which

035124-6


0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35temperature T

0.00

0.05

0.10

0.15

0.20

inve

rse

SDW

susc

eptib

ility

(a) r = 0.7

S−1xxL = 8L = 10L = 12L = 14

χ−1L = 8L = 10L = 12L = 14

0.0 1.00.0

0.7

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35temperature T

0.00

0.05

0.10

0.15

0.20

0.25

0.30

inve

rse

SDW

susc

eptib

ility

(b) r = 1.0

S−1xxL = 8L = 10L = 12L = 14

χ−1L = 8L = 10L = 12L = 140.0 1.0

0.0

0.8

FIG. 10. Inverse bosonic SDW susceptibility χ−1(q = Q,iωn =0) and inverse fermionic SDW susceptibility S−1xx (q = Q,iωn = 0) asa function of temperature for λ = 1.5 at (a) r = 0.7 ≈ rc0 and at (b)r = 1.0 > rc0. Solid lines indicate fits of a0 + a2T 2 to the L = 14data at intermediate temperatures. Dashed lines are linear fits to thehigh-temperature data. In each figure the inset shows the same dataas the main plot over a more extended temperature range.

for λ = 1.5 sets in just at the lowest temperature we haveaccessed in this work, Tc ≈ 1/40. For larger Yukawa couplingλ = 2, where Tc is higher, this nonmonotonic behavior isindeed found to be more pronounced as shown in Fig. 23of Appendix B.

Note that over the range of temperatures displayed in Fig. 3the leading temperature dependence of χ−1 is quadratic. Thisis reflected in the contour lines of χ−1 in the r-T plane, whichhave a form T ∼

√χ−1 − ar (r − rc0), approaching infinite

slope at low temperatures.Since the data do not allow us to identify a simple functional

form for the temperature dependence of χ−1, we have optedagainst taking into account any temperature dependence in thefits for the data collapses shown in Fig. 5. Instead we haveconstrained the included data to T < 0.1 where the overalltemperature dependence is rather weak.

IV. SINGLE-FERMION CORRELATIONS

We now turn to examine the fermionic spectral propertiesin the metallic state above the superconducting Tc. As ourDQMC simulations are performed in imaginary time, there

is an inherent difficulty in probing real-time dynamics. Topartially circumvent this issue, we use the relation [44]

Gk(τ ) =∫ ∞

−∞dω

e−ω(τ−β/2)

2 cosh βω/2Ak(ω) , (7)

which connects the readily available imaginary-time orderedGreen’s function Gk(τ ) = 〈ψk(τ )ψ†k(0)〉, where 0 � τ � β,with the spectral function Ak(ω) of interest. Here and inthe following, we focus on a single flavor of fermions ψy ,suppressing band and spin indices. Close inspection of (7)reveals that the behavior of the Green’s function Gk(τ ) at longtimes, i.e., for imaginary times close to τ = β/2, providesinformation about the spectral function integrated over afrequency window of width T .

In Fig. 11, we present the evolution of the Fermi surfaceacross the phase diagram. For orientation, the Fermi surfacesof the noninteracting system are shown in panel (a). Inpanels (b)–(d) we show Gk(τ = β/2) across a quadrant of theBrillouin zone for a low temperature T = 0.05 ≈ 2T maxc . Nearthe magnetic QCP (panel c), there is a clear loss of spectralweight in the immediate vicinity of the hot spots, as comparedto the magnetically disordered phase (panel d). Upon enteringthe magnetic phase (panel b), a gap opens around the hot spots.In this section we focus on the parameter set λ = 1.5 and c = 3.The DQMC simulations are carried out with different sets oftwisted boundary conditions (see Appendix A for details),providing a four-fold enhancement in k-space resolution.

A Fermi liquid is usually characterized by the quasiparticleweight ZkF and the Fermi velocity vkF . We note that thesequantities are only strictly defined at zero temperature. Giventhat the zero-temperature ground state of our model is probablyalways superconducting, our strategy is to consider finite-temperature proxies for ZkF and vkF , and study their behaviorover an intermediate temperature range EF > T > Tc. Suchproxies, ZτkF (T ) and v

τkF (T ), can be extracted by considering

the imaginary time dependence of Gk(τ ) near τ = β2 and fittingit to the Fermi liquid form [41]

Gk(τ ∼ β/2) = Zτk(T )e−�k(τ−

β

2 )

2 cosh(

β�k2

) , (8)where �k = vτkF (T ) · (k − kF ).

In a complementary approach we consider the Matsubarafrequency dependence of the Green’s function Gk(ωn) =∫ β

0 dτ eiωnτGk(τ ). In a Fermi liquid at low temperatures, we

have [45]

Gk(ωn) ≈ Zk[iωn − vkF · (k − kF )]−1 (9)up to higher-order terms in temperature, frequency, or thedistance from the Fermi surface. It is then natural to define thefinite-temperature quantities

ZωkF (T ) =ω1

Im G−1kF (ω1)(10)

and

vωkF (T ) = ω1∂

∂k

Re Gk(ω1)

Im Gk(ω1)

∣∣∣∣k=kF

, (11)

where ω1 = πT is the first Matsubara frequency at tem-perature T . In the zero-temperature limit, ZωkF (T → 0) =

035124-7


−−π

π

Q

(a)

0π π π0

π (b) r = 0.6 < rc

0 π0

π (c) r = 0.7 ≈ rc

0 π0

π (d) r = 1 > rc

0.000.050.100.150.200.250.300.350.400.450.50

FIG. 11. (a) Noninteracting Fermi surfaces. A pair of hot spots is connected by the magnetic ordering wave vector Q. The dashed curvecorresponds to the Fermi surface of the ψx band, shifted by Q, with a hot spot now at the intersection with the ψy band. (b)–(d) Color-codedGreen’s function Gk(τ = β/2) evaluated for the ψy fermions on a quadrant of the Brillouin zone, dotted in (a), for three values of the tuningparameter r . The dashed curve in panel (c) corresponds to the shifted noninteracting ψx Fermi surface. The parameters used here are L = 16,T = 0.05, λ = 1.5, and c = 3. Results of simulations with different boundary conditions are combined for enhanced momentum resolution.

ZτkF (T → 0) = ZkF , and similarly for vkF . We therefore usethe finite-temperature observables (10) and (11) as alternativeproxies for the quasiparticle spectral weight and Fermivelocity, respectively.

Figure 12 shows the momentum dependence of Zτk fortemperature T = 1/20. With r tuned close to the location ofthe QCP at rc, Zτk is suppressed in the vicinity of the hot spots,as shown for one quadrant of the Brillouin zone in Fig. 12(a)and along the Fermi surface in Fig. 12(c). This stands in sharp

0.0 0.2 0.4 0.6 0.8 1.0momentum kx/π

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

quas

ipar

ticle

wei

ghtZ

hs

(c) r = 0.7

0.2 0.4 0.6 0.8 1.0momentum kx/π

hs

(d) r = 1

Z τk(T = 0.05)Z ωk (T = 0.05)

0 π0

π

Z τk(T = 0.05)

(a) r = 0.7 ≈ rc

0 π

Z τk(T = 0.05)

(b) r = 1 > rc

0.320.400.480.560.640.720.800.880.96

FIG. 12. (a) and (b) The quasiparticle weight Zτk(T = 0.05) ina quadrant of the Brillouin zone. The dashed line in panel (a)corresponds to the noninteracting Fermi surface of the ψx fermions,shifted by Q. (c) and (d) The quasiparticle weights Zτk(T = 0.05)and Zωk (T = 0.05) along the Fermi surface. The location of the hotspot is indicated by the red marker. Here, we show data obtained forL = 16.

contrast to the featureless behavior of Zτk in the magneticallydisordered phase, as shown in Figs. 12(b) and 12(d). We findqualitative agreement between the two proxies Zτk and Z

ωk

throughout, as illustrated in panels (c) and (d) of Fig. 12. Here,we numerically identify and track the Fermi surface as themaxima of Gk(τ = β/2) at fixed kx .

The temperature dependence of Zτk(T ) is shown in Fig. 13.For momenta away from the hot spots, we find Zτk(T ) tobe nearly flat in temperature and to approach a constant asT → 0. A different picture emerges at the hot spots, i.e., fork = khs . Here, Zτk=khs (T ) remains flat only in the magneticallydisordered phase r > rc, whereas the quasiparticle weightZτk=khs (T ) decreases substantially at the critical couplingrc ≈ 0.7 as the temperature is lowered towards the QCP, seeFig. 13(a). While our numerical data do not allow for a simpleextrapolation towards T = 0, the results are not inconsistentwith a vanishing of Zk=khs , indicating a breakdown of Fermi-liquid behavior at this point.

Figure 14 shows the velocity vkF (T = 0.05) along theFermi surface. The qualitative behavior of the velocity doesnot differ substantially between r = 0.7 ≈ rc [Fig. 14(a)]and r = 1 > rc [Fig. 14(b)]. The insets show the ratio

0.0 0.1 0.2 0.3temperature T

0.0

0.2

0.4

0.6

0.8

1.0

quas

ipar

ticle

wei

ghtZ

τ k(T

)

(a) r = 0.7 ≈ rc

0.1 0.2 0.3temperature T

(b) r = 1 > rc

kx = 0kx = khskx = π

FIG. 13. Temperature dependence of the quasiparticle weightZτk(T ) for different momenta kx along the Fermi surface (a) in thevicinity of the QCP at rc and (b) in the disordered side. Here we showdata obtained for L = 16.

035124-8


1.0

1.2

1.4

1.6

1.8

2.0

Ferm

ivel

ocity

v(a) r = 0.7 ≈ rc

hs

(b) r = 1 > rc

hs

v τk(T = 0.05)v ωk (T = 0.05)

0.0 0.2 0.4 0.6 0.8 1.0momentum kx/π

0.50

0.75

1.00

vω k/

vnon

int

k

(c) r = 0.7

hs0.2 0.4 0.6 0.8 1.0momentum kx/π

(d) r = 1

hs

FIG. 14. (a) and (b) The finite-temperature proxies to the velocityvk along the Fermi surface. (c)-(d) The velocity renormalizationvωk (T = 0.05)/vnonintk . The location of the hot spot is indicated bythe red marker. Here we show data obtained for L = 16 at T = 0.05.

vkF (T = 0.05)/vnonintk , where vnonintk is the Fermi velocity ofthe noninteracting system. A small feature might be visible inthe vicinity of the hot spot, but there is certainly no evidenceof a substantial suppression of vkhs close to rc.

Having found a substantial suppression of the quasiparticleweight tuned close to the QCP, we now directly examine thefrequency dependence of the self-energy, �k(ωn), defined viaGk(ωn) = (iωn − �k − �k(ωn))−1. The imaginary part of theself-energy is shown in Fig. 15. With r = 0.7 ≈ rc tuned closeto the QCP, see Fig. 15(a), the self-energy close to the hot spotsis found to be nearly frequency independent, consistent witha constant, yet small, scattering rate γ = − Im �khs (ωn →0+) ≈ 0.13, which is only weakly dependent on temperature.

0.0 0.5 1.0 1.5 2.0Matsubara frequency ω

0.00

0.05

0.10

0.15

self

ener

gy-I

m(Σ

)

khs

kx = 0

(a) r = 0.7 ≈ rc

T = 0.05T = 0.07T = 0.1T = 0.2

0.5 1.0 1.5 2.0Matsubara frequency ω

khs

kx = 0

(b) r = 1 > rc

FIG. 15. The imaginary part of the Matsubara self-energy Im �for different temperatures and two momenta along the Fermi surface,(a) in the vicinity of the QCP at rc and (b) on the disordered side.Here we show data obtained for L = 14. The data for k = khs areindicated by full circles, the momentum away from the hotspot isindicated by empty squares.

Away from the hot spot the self-energy is linear with frequency,with a substantially smaller intercept. Moving away fromthe critical point [Fig. 15(b)], the self-energy at all momentadecreases rapidly as the frequency is lowered.

V. SUPERCONDUCTING STATE

After concentrating our discussion on the manifestation ofquantum critical behavior in the normal state, we now considerthe effect of the QCP on the superconducting state that emergesin its vicinity. We begin by considering the fermionic Green’sfunction for temperatures T � Tc. In this regime, the single-particle excitation energy Ek can be extracted, as demonstratedin Appendix D, from the decay of the single-particle Green’sfunction at intermediate times τ0 < τ < β/2 (where τ0 is amicroscopic scale). The so-extracted excitation energy Ek isplotted in Fig. 16, which shows that, across the Brillouin zone,Ek has a broad minimum in the vicinity of the noninteractingFermi surface. From these momentum-resolved energy bands,we extract the superconducting gap �kx as the minimum of Ekwith respect to ky . As seen in Fig. 16(b), the superconductinggap �kx varies smoothly across momentum space, without anysignificant features at the hot spots. In this section we chooseparameters λ = 3 and c = 2, as in Ref. [28]. The maximalTc for this value of λ is high enough to allow us to exploreproperties of the superconducting state significantly below Tc.

At higher temperatures, close to Tc, additional informationcan be obtained by considering the momentum-resolvedsuperconducting susceptibility

Pkα;k′α′ =∫ β

0dτ 〈�kα(τ )�†k′α′ (0)〉, (12)

where �kα = 12 (ψkα↑ψ−kα↓ − ψkα↓ψ−kα↑) is the singlet su-perconducting pair amplitude on the band α = x,y. Here wefocus on the intraband, spin-singlet channel since it is theleading instability [17,28]. Figure 17 shows the optimal pairamplitude �optkα , corresponding to the maximal eigenvalue ofthe matrix Pkα;k′α′ at a temperature slightly above Tc. The pairamplitude of the band α = y, shown in Fig. 17(a), is of theopposite sign to the amplitude on the band α = x, shown in

−π π−π

π(a)

0.0 0.2 0.4 0.6 0.8 1.0momentum kx/π

0.000.050.100.150.200.250.30

gap

Δ kx

hs

(b)

0.300.450.600.750.901.051.201.351.50

FIG. 16. (a) Single-particle excitation energy Ek of the ψyfermions, as extracted from the imaginary-time evolution of theGreen’s function Gk(τ ) across the Brillouin zone, cf. Fig. 26 of theappendix. (b) Single-particle gap �kx . For both panels, data are forparameters λ = 3, c = 2, r = 10.2 and T = 0.025 ≈ 0.3Tc [28] anda system size of L = 12. Several twisted boundary conditions werecombined for a fourfold enhancement of the resolution in k space,see Appendix A.

035124-9


−−π

π (a)

Ψy−π π π π

−π

π (b)

Ψx0.0150.0300.0450.0600.0750.0900.1050.120

−0.120−0.105−0.090−0.075−0.060−0.045−0.030−0.015

FIG. 17. Optimal pair amplitude �optkα with the band α = y shownin (a) and the band α = x shown in (b). Data are calculated forparameters λ = 3, c = 2, r = 10.2, and T = 0.1 ≈ 1.2Tc and systemsize L = 14.

Fig. 17(b). In fact, the two amplitudes are related preciselyby a π/2 rotation, highlighting the d-wave symmetry of thesuperconducting order parameter. The optimal pair amplitudeis found to be maximal around the (noninteracting) Fermisurface. The variation of �optkα along the Fermi surface is weak,again showing no strong features at the hot spots.

VI. DISCUSSION

In this work, we have explored the properties of a metalon the verge of an SDW transition. We focused on the criticalregime upon approaching the transition, characterized by arapid growth of the SDW correlations, but still above thesuperconducting transition temperature. Our main conclusionis that, in this regime, the SDW correlations are remarkablywell described by a form similar to that predicted by Hertz-Millis theory, Eq. (1) (although the temperature dependenceof the SDW susceptibility deviates from the expected form).This holds both for the correlations of the bosonic SDWorder parameter field, and for an SDW order parameterdefined in terms of a fermion bilinear. In the same regime,we find evidence for strong scattering of quasiparticles nearthe hotspots, leading to a breakdown of Fermi liquid theoryat these points on the Fermi surface. The scattering rateat the hotspots (extracted from the fermion self-energy) isonly weakly temperature and frequency dependent, down toT ≈ 2Tc, where we suspect that superconducting fluctuationsbegin to play a role; it is out of this unusual metallic state thatthe superconducting phase emerges.

In addition, we have studied the structure of the super-conducting gap near the SDW transition. Unlike the single-fermion Green’s function in the normal state, it does not havea sharp feature at the hot spots; rather, it is found to varysmoothly across the Fermi surface. Experimentally, a broadmaximum of the superconducting gap near the hot spots wasobserved in a certain electron doped cuprate [46]. Eliashbergtheory predicts a peak of the gap function at the hot spots atweak coupling [47] and it remains to be seen whether suchbehavior appears in our model at weaker coupling.

It is interesting to discuss our results in the context of theexisting theories of metallic SDW transitions. First, the factthat Hertz-Millis theory successfully describes many featuresof our data is nontrivial, in view of the fact that it hasno formal justification, even in the large N limit [12,14].However, as we saw, an extension of the Hertz-Millis analysis

to finite temperature predicts that at criticality, χ (T ) ∼ 1/T , inapparent disagreement with our data. This may be due to thelimited temperature window we can access without hittingthe superconducting Tc, or to effects beyond the one-loopapproximation.

An important conclusion of our study is that the SDWcritical point is always masked by a superconducting phase.2

As a result, it seems likely that the critical metallic regimeis never parametrically broad, and one cannot sharply definescaling exponents within the metallic phase.3 As mentionedabove, the SDW correlations follow a Hertz-Millis form—andhence it is tempting to associate with them critical exponents,i.e., a mean-field value ν = 1/2 for the correlation lengthexponent, and a dynamical critical exponent z = 2. Thefermionic quasiparticles at the hot spots, however, do notexhibit this scaling behavior. In particular, the expected scalinglaw �(ωn) ∼ √ωn for the fermion self-energy at the hot spotsis not seen within our accessible temperature range.

One can imagine trying to access the “bare” metallicquantum critical point by suppressing the superconductingtransition. Presumably, this can be done by adding to the modela term that breaks time-reversal and inversion symmetries(such a term would lift the degeneracy of fermionic states withopposite momenta, and hence remove the Cooper instability).Breaking time-reversal symmetry, however, gives rise to a signproblem, so it is not clear whether the critical behavior can beaccessed within the QMC technique.

Alternatively, one could try to understand the metallicregime above Tc in our model as a crossover regime ofan underlying “nearby” metallic critical point, where somecorrelators already exhibit their asymptotic behavior (suchas the SDW order parameter correlations), while others donot (e.g., the single-fermion Green’s function). Interestingly,a simple, non-self-consistent one-loop calculation of thefermionic self-energy in our model does show a broad range oftemperature and frequency where the self-energy at the hot spotis nearly constant, before eventually settling into the expected√

ωn behavior (see Appendix C). This calls for a more detailedcomparison between our numerically exact DQMC resultsand a detailed self-consistent one-loop analysis. Preliminaryresults show that this approximation is surprisingly successfulin capturing at least some of the physics of our model [43].

To what extent such a crossover behavior, characterizedby a nearly-constant fermionic lifetime at the hotspots, isubiquitous across different models, as well as in real materials,remains to be seen. It is interesting to note, however, thata similar behavior has been observed in a study of anematic transition in a metal [48]. It would be interestingto systematically look for such behavior in angle-resolvedphotoemission spectroscopy in the electron-doped cuprates,where anomalously large broadening of the quasiparticle peaksis seen near the hot spots [49].

2Even for the smallest value of λ that we have studied, where themaximal Tc is smaller than the lowest temperature in our study, wesee enhancement of diamagnetic fluctuations, indicating that we arenot too far above Tc.

3In the superconducting phase, we expect criticality of the d =2 + 1 XY universality class, since the Fermi surface is gapped.

035124-10


Another important aspect of the metallic state in thecritical regime, which we have not addressed in this work,is the electrical conductivity. The optical conductivity may bestrongly affected by the presence of an SDW critical point, evenwithout quenched disorder [16,50]. Extracting the conductivityfrom quantum Monte Carlo simulations requires an analyticcontinuation, and is therefore intrinsically more difficult (andinvolves more uncertainties) than calculating thermodynamicand imaginary-time quantities. Nevertheless, we have obtainedpreliminary results showing strong effects of the criticalfluctuations on the low-frequency optical conductivity [51].A full analysis of the conductivity is deferred to future work.

ACKNOWLEDGMENTS

We thank A. Chubukov and C. Varma for useful discussionson this work and S.-S. Lee for his comments on the manuscript.E. B. and Y. S. also thank R. Fernandes, S.A. Kivelson,S. Lederer, and X. Wang for numerous discussions andcollaborations on related topics. E.B. thanks the hospitalityof the Aspen Center for Physics, where part of this workwas done. The numerical simulations were performed on theCHEOPS cluster at RRZK Cologne, the JUROPA/JURECAclusters at the Forschungszentrum Jülich, and the ATLAScluster at the Weizmann Institute. Y.S. and E.B. were supportedby the Israel Science Foundation under Grant No. 1291/12, bythe US-Israel BSF under Grant No. 2014209, and by a MarieCurie reintegration grant. E. B. was supported by an Alonfellowship. M. G. thanks the Bonn-Cologne Graduate Schoolof Physics and Astronomy (BCGS) for support.

M.H.G. and Y.S. have contributed equally to this work.

APPENDIX A: DQMC SIMULATIONS

In this appendix, we elaborate on a number of specifictechnical aspects of our numerical implementation of thedeterminantal quantum Monte Carlo (DQMC) approach. Werefer readers looking for a more comprehensive discussion ofthe general DQMC setup to our previous paper [28] and inparticular its supplementary online material.

We study the lattice model described by the action (2)at finite temperature in the grand canonical ensemble. Afterdiscretizing imaginary time and integrating out the fermionicdegrees of freedom, the partition function reads

Z =∫

D �ϕ e−�τ∑

τ Lϕ (τ ) det G−1ϕ , (A1)

where Gϕ is the equal-time Green’s function matrix for afixed configuration of the bosonic order parameter field �ϕ. Weuse an imaginary time step of �τ = 0.1 in all calculations.The DQMC method samples configurations of �ϕ accordingto their weight exp (−�τ ∑τ Lϕ(τ )) · det G−1ϕ . For efficientMonte Carlo sampling, is it highly advantageous to considermodels in which the determinant in (A1) is guaranteed to bepositive, thereby avoiding the notorious fermion sign problem.For the two-band model that has been proposed in Ref. [17] thisis ensured by an antiunitary symmetry of the action [52–54],written in first quantization as

U = isyτzK with U2 = −1 . (A2)

Here, K is the complex conjugation operator and sy (τz) arePauli matrices acting in spin (flavor) space, respectively.

In this manuscript we have modified the model of Ref. [17]in two regards. First, as in our previous paper [28], we consideran easy-plane SDW order parameter, rather than an O(3)symmetric order parameter used in [17]. Second, we couplethe system to a fictitious, spin and band dependent orbital“magnetic” field, whose flux through the system is givenby ��α,s = �(x)α,s �e x + �(y)α,s �e y + �(z)α,s �e z. Here, we place thetwo-dimensional lattice (x,y directions) on a torus. The x andy components of the flux twist the boundary conditions alongthe y and x directions, respectively, while the z component actsas an orbital magnetic field with a uniform flux per plaquettethrough the torus. As in the definition of the action (2), α = x,yis a fermion flavor index and s = ↑,↓ is a spin index. The flux�z is restricted to be an integer multiple of the flux quantum�0.

In order to preserve the symmetry (A2), fermions ofdifferent spin and flavor are coupled to this fictitious flux as

��α,↑ = − ��α,↓ and��x,s = − ��y,s, (A3)

with ��x,↑ = ��. Note that the interband sign change is notstrictly necessary to avoid the fermion sign problem. Impor-tantly, in the thermodynamic limit L → ∞ the fictitious fieldvanishes. In our previous paper [28] and for some of the resultsin the present one, we have chosen �� = (0,0,�0). Althoughsuch a flux is useful in reducing spurious finite-size effects atlow temperatures [40,41], it also breaks lattice translationalsymmetry, hampering the analysis of nonlocal, fermioniccorrelation functions such as the momentum-resolved Green’sfunction. In order to measure such quantities, for the presentpaper, we have run additional simulations without applying aperpendicular flux �(z), but instead with in-plane fields, suchthat �� = (nx,ny,0)�04 , where ni = 0,1,2,3. This procedure isequivalent to having twisted boundary conditions, such thatthe allowed momenta for the ψx,↑ fermions are

k = 2π4L

(4jx − ny,4jy + nx), (A4)

where jx , jy are integers, thereby enhancing the momentum-space resolution of fermionic observables fourfold.

APPENDIX B: MAGNETIC CORRELATIONS FOR λ = 1AND λ = 2, AND INSIDE THESUPERCONDUCTING PHASE

In this appendix, we present additional data for thebosonic and fermionic SDW susceptibilities χ (q,iωn,r,T )and Sxx(q,iωn,r,T ). We begin with Fig. 18, which showsχ−1(q = Q,iωn = 0) across the phase diagrams for λ = 1.5and λ = 2. It illustrates how the bending of the SDW finite-temperature transition line is also visible in χ−1 for r > rc0,similarly to the behavior for λ = 1 in Fig. 3, where, however,no superconducting phase has been observed within ourtemperature resolution.

Next, to complement the discussion in Sec. III, where wehave focused on the Yukawa coupling λ = 1.5 and a bosonicvelocity of c = 3, we here show detailed data for values λ = 1

035124-11


FIG. 18. Companion figure to Fig. 3 for (a) λ = 1.5 and (b) λ = 2.

and 2 at the same velocity c. In Fig. 19, we show χ−1(q =Q,iωn = 0) as a function of r for constant T = 0.1, where thesame linear dependence as for λ = 1 (Fig. 6) is apparent. Aswe show in Fig. 20 both for r ≈ rc0 and a range of r > rc0the leading frequency dependence of χ−1 is clearly linear,similarly to λ = 1.5 (Fig. 7), whereas the leading momentumdependence shown in Fig. 21 is quadratic in q − Q, whichis again comparable to λ = 1.5 (Fig. 8). Note that for λ = 2

we show data at a higher temperature T = 1/20 rather thanat T = 1/40, because otherwise the system would be in thesuperconducting phase, where the frequency dependence issignificantly altered. For small frequencies and momenta thefermionic susceptibility S−1xx in Fig. 22 behaves similarly to thebosonic χ−1 (see Fig. 9 for λ = 1.5).

The temperature dependence of both χ−1 and S−1xx isdemonstrated in Fig. 23. As in the case of λ = 1.5 (Fig. 10),we observe a linear regime at high temperatures, shown in theinsets, and a crossover region, where we can fit a quadraticlaw. For λ = 1, this second region extends down to lowertemperatures than for λ = 1.5, where Tc is higher. At λ = 2, thedata for T < 0.05 are from within the superconducting phase.Moreover, at r = 3.1 ≈ rc0 the system is partially inside themagnetic quasi-long-range order phase (cf. the phase diagramin Fig. 2).

Finally, to illustrate the influence of superconductivity onthe frequency dependence of the SDW susceptibility, we showdata from deep within the superconducting phase in Fig. 24.Here we have chosen a data set with different values of theYukawa coupling λ = 3 and the bosonic velocity c = 2 (asin Ref. [28]) since T maxc ≈ 0.08 is about twice as high forthese parameters as for λ = 2, c = 3. In contrast to the data atT > Tc shown in Figs. 7 and 20, the low-frequency behavioris clearly no longer purely linear—indicative of a suppressionof Landau damping in the superconducting phase.

APPENDIX C: COMPARISON WITH A ONE LOOPAPPROXIMATION FOR THE FERMION SELF-ENERGY

In this Appendix, we consider the fermionic self-energyin a one-loop approximation. To this order, the self-energy isgiven by

�k,α=y(ωn) = λ2

βL2

∑q,m

χq(�m)G0k+q,α=x(ωn + �m), (C1)

where G0 is the noninteracting Green’s function and �m =2πmT is a bosonic Matsubara frequency. Deferring moresystematic calculations for future work, here we do not attempt

−1.5 −1.2 −0.9 −0.6 −0.3 0.0 0.3tuning parameter r

0.0

0.2

0.4

0.6

SDW

susc

eptib

ility

χ−1

(a) λ = 1, T = 0.1L = 8L = 10L = 12L = 14

L = 8L = 10L = 12L = 14

−1.3 −1.2 −1.10.00

0.07

2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6tuning parameter r

−0.2

0.0

0.2

0.4

0.6

SDW

susc

eptib

ility

χ−1

(b) λ = 2, T = 0.1L = 8L = 10L = 12L = 14

L = 8L = 10L = 12L = 14

3.1 3.2 3.3

0.00

0.07

FIG. 19. Companion figure of Fig. 6 for Yukawa couplings (a) λ = 1 and (b) λ = 2. The black lines are linear fits of the L = 14 data for(a) r > −1.2 and (b) r > 3.2.

035124-12



−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

SDW

susc

eptib

ility

χ−1

(a) r = −1.3L = 14

|q| = 0.45|q| = 0.63

|q| = 0.90

q = 0L = 8L = 10L = 12L = 14

−1 0 1

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2(b) r ≥−1.3q = 0, L = 14

r = −1.3r = −1.1r = −0.9

r = −0.7r = −0.5


0.0

0.5

1.0

1.5

2.0

2.5

SDW

susc

eptib

ility

χ−1

(c) r = 3.2L = 14

|q| = 0.45|q| = 0.63

|q| = 0.90|q| = 1.00

q = 0L = 8L = 10L = 12L = 14

−1 0 1

0.0

0.5

1.0

1.5

2.0

2.5

(d) r ≥ 3.2q = 0, L = 14

r = 3.2r = 3.4r = 3.6

r = 3.8r = 4.0r = 4.2

FIG. 20. Companion figure of Fig. 7 for Yukawa couplings λ = 1 at T = 1/40 (left) and λ = 2 at T = 1/20 (right).

a full, self-consistent solution of the coupled Eliashbergequations [8] for the SDW correlations and the fermionicGreen’s function. Instead, we use the noninteracting Green’sfunction and χ taken from a lattice, discretized imaginary timeversion of Eq. (4):

χ−1q (�m) = ar (r − rc)

+ 4aq[

sin2(

qx − Qx2

)+ sin2

(qy − Qy

2

)]

+ 2aω�τ

∣∣∣∣ sin(

�τ�m

2

)∣∣∣∣, (C2)where the parameters ar,aq,aω are taken from a fit to theDQMC data, see Section III A. Strictly speaking, this proce-dure is not justified. However, since within a self-consistentEliashberg theory, χ has the form (4, C2), we expect oursimplified approximation to capture the general behavior ofthe self-consistent theory.

In Fig. 25, we show the imaginary part of the self-energy.The results bear some similarities to the DQMC data, shown in

Fig. 15. Whereas at moderate r − rc or away from the hotspotsthe self-energy is rapidly diminished as the frequency ωn islowered, the behavior at the hotspots as r approaches rc isdifferent. There, as a function of temperature, a change of slopeoccurs in the frequency dependence of the self-energy, whereat intermediate temperatures T ≈ 0.1 the self-energy is nearlyfrequency independent. Only at lower temperatures, �khs (ωn)starts resembling the expected

√(ωn) form. In comparing with

the DQMC results in Fig. 15, we note the similar magnitudeof the self-energy. However, the DQMC results show a farweaker temperature dependence at the hotspots for r closeto rc.

APPENDIX D: EXTRACTING THE SUPERCONDUCTINGGAP

In this appendix, we provide a detailed description of theprocedure by which we extract the single-particle excitationenergy Ek in the superconducting state, which we discuss inSec. V of the main text. The single-particle Green’s function


−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

SDW

susc

eptib

ility

χ−1

(a) r = −1.3

L = 14ωn = 0.16ωn = 0.31ωn = 0.47ωn = 0.63ωn = 0.79ωn = 0.94

ωn = 0L = 8L = 10L = 12L = 14

0 1 2

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

(b) r ≥−1.3

ωn = 0, L = 14r = −1.3r = −1.1r = −0.9

r = −0.7r = −0.5


0.0

0.5

1.0

1.5

2.0

SDW

susc

eptib

ility

χ−1

(c) r = 3.2

L = 14ωn = 0.31ωn = 0.63ωn = 0.94

ωn = 0L = 8L = 10L = 12L = 14

0 1 2

0.0

0.5

1.0

1.5

2.0

(d) r ≥ 3.2

ωn = 0, L = 14r = 3.2r = 3.4r = 3.6

r = 3.8r = 4.0r = 4.2


035124-13



0.0

0.5

1.0

ferm

ioni

cSD

Wsu

scep

tibili

tyS−

1xx L = 14

|q| = 0.45|q| = 0.63

|q| = 0.90|q| = 1.00

q = 0L = 8L = 10L = 12L = 14


0.0

0.5

1.0

L = 14ωn = 0.16ωn = 0.31ωn = 0.47

ωn = 0.63ωn = 0.79

ωn = 0L = 8L = 10L = 12L = 14


−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

ferm

ioni

cSD

Wsu

scep

tibili

tyS−

1xx L = 14

|q| = 0.45|q| = 0.63

|q| = 0.90|q| = 1.00

q = 0L = 8L = 10L = 12L = 14


−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2L = 14ωn = 0.31ωn = 0.63

ωn = 0.94

ωn = 0L = 8L = 10L = 12L = 14


Gk(τ ) is found to exhibit a characteristic imaginary-timeevolution as shown in Fig. 26. At intermediate times, τ0 <τ < β/2, where τ0 ∼ 1 is some microscopic time scale, thesingle-particle Green’s function decays exponentially as

Gk(τ ) ∝ e−Ep

k τ , (D1)

and similarly, for times τ0 < β − τ < β/2,

Gk(τ ) ∝ e−Ehk (β−τ ) . (D2)At long times, τ ≈ β/2 the Green’s function is substantiallysuppressed and statistical errors dominate the signal. Weextract the decay constants Epk and E

hk from appropriate

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35temperature T

0.00

0.02

0.04

0.06

0.08

0.10

inve

rse

SDW

susc

eptib

ility

(a) λ = 1, r = −1.3

S−1xxL = 8L = 10L = 12L = 14

χ−1L = 8L = 10L = 12L = 14

0.0 1.00.0

0.6

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35temperature T

0.00

0.05

0.10

0.15

0.20

inve

rse

SDW

susc

eptib

ility

(b) λ = 1, r = −1.0

S−1xxL = 8L = 10L = 12L = 14

χ−1L = 8L = 10L = 12L = 14

0.0 1.00.0

0.7

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35temperature T

0.00

0.05

0.10

0.15

0.20

0.25

0.30

inve

rse

SDW

susc

eptib

ility

(c) λ = 2, r = 3.1

S−1xxL = 8L = 10L = 12L = 14χ

−1

L = 8L = 10L = 12L = 14

0.0 1.00.0

1.2

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35temperature T

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

inve

rse

SDW

susc

eptib

ility

(d) λ = 2, r = 3.4

S−1xxL = 8L = 10L = 12L = 14

χ−1L = 8L = 10L = 12L = 14

0.0 1.00.0

1.2

FIG. 23. Companion figure of Fig. 10 for for (top) λ = 1 at (a) r = −1.3 ≈ rc0 and at (b) r = −1.0 > rc0, and for (bottom) λ = 2 at (c)r = 3.1 ≈ rc0 and at (d) r = 3.4 > rc0.

035124-14


−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5Matsubara frequency ωn

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5SD

Wsu

scep

tibili

tyχ−

1L = 14

|q| = 0.449|q| = 0.635

|q| = 0.898|q| = 1.004

q = 0L = 8L = 10L = 12L = 14

FIG. 24. Frequency dependence of the SDW susceptibility χ−1

for λ = 3, c = 2 at T = 1/30 < Tc and r = 10.37 ≈ ropt, close towhere Tc is highest for this set of parameters, shown for variousmomenta q = Q + q̃. The black line is the best fit of a second degreepolynomial b0 + b1|ωn| + b2ω2n to the q = Q, L = 14 low-frequencydata.

exponential fits and define the single-particle excitation energyas their minimum Ek = min {Epk ,Ehk}.

For a qualitative understanding of these results, we considerthe behavior of the Green’s function in a Fermi liquid and ina Bardeen-Cooper-Schrieffer (BCS) superconductor. The factthat Gk(τ ) exhibits exponential behavior can be interpreted asarising from a peak in the spectral function Ak(ω), occurringat a nonzero frequency, as can be seen from (7). In a Fermiliquid, the spectral function at a given momentum has asingle, sharp peak at the energy of the quasiparticle. It thenfollows that Gk(τ ) has the form of a single exponential,with holelike quasiparticles obeying (D2) and particlelike

0.0 0.5 1.0 1.5 2.0Matsubara frequency ω

0.00

0.05

0.10

0.15

0.20

0.25

0.30

self

ener

gy-I

m(Σ

)

khs

kx = 0

(a) r− rc = 0.05

0.5 1.0 1.5 2.0Matsubara frequency ω

khs

kx = 0

(b) r− rc = 0.35T = 0.025T = 0.05T = 0.1T = 0.2

FIG. 25. Imaginary part of the self-energy (C1), calculated withina one-loop approximation for several temperatures for two momentaon the Fermi surface and (a) close to the QCP at r = rc + 0.05, (b)further away from the QCP. An SDW correlator of the form (C2) wasused, with the parameters taken from the fit to the DQMC data forλ = 1.5, c = 3, with �τ = 0.1, and L = 200, as shown in Fig. 5(b).The data for k = khs are indicated by full circles, the momentumaway from the hot spot is indicated by empty squares.

0 5 10 15 20 25 30 35 40imaginary time τ

10−2

10−1

100

Gk(

τ)

(a) k = (0,0.62)πk = (0.67,0.62)π ≈ khsk = (1,0.62)π

0 5 10 15 20 25 30 35 40imaginary time τ

10−2

10−1

100

Gk(

τ)

(b) k = (0,0.62)π . E p −Eh = −0.01k = (0,0.58)π . E p −Eh = 0.04k = (0,0.67)π . E p −Eh = −0.06

FIG. 26. Imaginary time evolution of the single-particle Green’sfunction Gk(τ ) for (a) several momenta along the noninteractingFermi surface and (b) several momenta along a cut perpendicularto the Fermi surface. Here, L = 12, T = 1/40, λ = 3, and c = 2.Shaded regions indicate the statistical uncertainty. The solid lines areexponential fits.

quasiparticles obeying (D1). Indeed, in our simulations in thenormal state we find monotonic behavior of Gk(τ ) (not shown).It is illuminating to contrast this behavior with the BCS state,where a superposition of holelike and particlelike excitationsis allowed. In this case, the spectral function consists oftwo delta-function peaks at ω = ±Ek, such that the Green’sfunction takes the form

Gk(τ ) = 11 + e−βEk

(u2ke

−Ekτ + v2ke−Ek(β−τ )). (D3)

Here, Ek =√

�2k + �2k with the quasiparticle dispersion �kand the gap �k, and uk, vk are particle and hole amplitudes,respectively. The resulting Green’s function is nonmonotonic,showing a minimum at a finite imaginary time.

While our numerical data below Tc share some similaritieswith the BCS form (D3), they differ in two notable ways.First, clear exponential behavior is not seen at short timesτ � τ0 ∼ 1. Second, the particle and hole excitation energiesdiffer, i.e., Epk �= Ehk , with the difference more pronouncedaway from the Fermi surface, as illustrated in Fig. 26(b).

035124-15


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