The mixed ferro-antiferromagnetic3-state quantum Potts chain,quantum chiral clock model

and free parafermions

Murray BatchelorChongqing University & Australian National University

INTEGRABILITY IN LOW-DIMENSIONAL QUANTUM SYSTEMSCreswick, 10 July 2017

Outline of this talk

1a) 1D quantum chiral clock model

1b) Mixed ferro-antiferromagnetic 3-state quantum Potts chain

Y-W Dai, S Y Cho, MTB and H-Q Zhou,

Phys Rev B 95, 014419 (2017)

2) ZN free parafermionic chain

F C Alcaraz, MTB and Z-Z Liu,

J Phys A 50, 16LT03 (2017)

1a) 1D quantum chiral clock model

I The N-state asymmetric or chiral clock model was originallyintroduced to provide a simple description of monolayersadsorbed on rectangular substrates.S Ostlund, Phys Rev B 24, 398 (1981)

D Huse, Phys Rev B 24, 5180 (1981)

I The clock models can be considered as discrete versions of thecontinuous XY model.

I The asymmetry or chirality in the model hamiltonian inducesincommensurate floating phases with respect to the periodicityof the underlying lattice, with commensurate-incommensuratephase transitions of the Kosterlitz-Thouless typecorresponding to the melting of the incommensurate phase.

1D quantum version (3-state model)

S Howes, L P Kadanoff and M den Nijs, Nucl Phys B 215, 169 (1983)

H = −∞∑j=1

[cos a(pj −∆p) + β cos a(θj+1 − θj −∆θ)]

where a = 2π/3 and the variables pj , θj take the three eigenvalues0, 1, 2. They obey the commutation relations

eiapj eiaθk = ωδjk eiaθk eiapj

with ω = eia.

We set the chiral parameters to ∆p = ∆θ = ∆.

Rich phase diagram in terms of the parameters ∆ and β, originallymapped out using strong coupling series expansion techniques

β plays the role of inverse temperature.

172 S Howes et al. / Commensurate-mcommensurate transztwns

Based upon the general properties of this perturbation expansion, in sect. 5 we make global statements about our phase diagram.

Before going into detailed analysis, we summarize our results by showing our final, partially derived partially conjectured, phase diagram in fig. 2. (Specific warnings, caveats, etc., about the strength of our conclusions are to be found in the body of the paper.)

The main features of the diagram are: (a) Critical lines, AB, B'A' in the three-state Potts model universality class,

separating disordered phases (with integral values of Qp) from ordered phases (which have integral values of Qe).

(b) Commensurate-incommensurate (C-I) phase boundaries (dashed) of the type discussed by Pokrovskii and Talapov [4]. Notice that these boundaries equally well separate disordered and incommensurate phases as well as ordered and incom- mensurate ones.

(c) Two kinds of Lifshitz points (B and B' as well as D and D'). The specific heat, magnetization and susceptibility show three-state Potts behavior at these points; the mass gap goes linearly to zero as in the Ising model.

(d) Several multicritical points C, C', F and F' where fermion expansions may be applied. The behaviour in the neighborhood of these points is, for most purposes, exactly known.

(e) Some points with Kosterlitz-Thouless critical behavior: G, E and E'.

09

-(~

C / ' , COMMENSURATE COMMENSURATE

_ 0 e :0 "" " ,, ,, 00 1

/ , i A ~ , _ ~ B , INCOMMENSURATE ",[3 A'

~, (IC) " "R,

DISORDERED / DISORDERED Qp : 0 '~" ," Qp : 1 / "o

J C -'-,.~.)/. 0 I/4 o / ~', 3/4

Op:O o, lC, / G ,,zc', Qp:1

o0~'" -~ D' d ,' COMMENSURATE ' , ' ,

Y " Q0: - I ' " "IC," ",IC', ,E '

F'

F~g 2 Phase diagram for the hanultoman (I 5) wath Ae - A0 -- z~ Data points are obtaaned from strong coupling series. The verlacal scale is distorted for [ fl J > l

I For chirality parameter ∆ = 0 and β > 0 the model reducesto the 3-state quantum Potts chain with purely ferromagneticinteractions.

I For ∆ = 0 and β < 0 the model reduces to the much lessstudied quantum Potts model with mixedferro-antiferromagnetic interactions.

I In the antiferromagnetic region of the phase diagram

the line ∆ = 0, β < 0 is dual to the line ∆ = 1/2, 1/β < 0

so results obtained for the mixed ferro-antiferromagnetic3-state quantum Potts chain apply directly to the phasediagram of the more general antiferromagnetic chiral clockmodel at ∆ = 1/2.

I Kosterlitz-Thouless transitions were identified at the criticalpoints (0, βc) and (1/2, 1/βc), where βc = −10± 5.

I These are, respectively, points E and G in the phase diagram.The quantum critical point E is between disordered andincommensurate phases and point G is betweenincommensurate and commensurate phases.

I The only other estimates at ∆ = 0 are of a massless phaseidentified for some critical field value βc , with βc < −5 basedon analysis of small chain sizesH J Herrmann and H O Martin, J Phys A 17, 657 (1984)

and a rough approximate value βc ' −5 estimated using amean-field renormalization group methodM C Marques, J Phys A 21, 1061 (1988)

In terms of the operators

Zj = eiapj , Xj = e

iaθj

the commutation relations are

ZjXk = ωδjkXkZj

where we recall ω = ei2π/3. The hamiltonian is then

2H = −∞∑j=1

[e−iφZj + e

iφZ −1j + β(eiφXjX

−1j+1 + e

−iφX −1j Xj+1

)]with φ = a∆.

In terms of the usual Potts spin-operators

Zj =

1 0 00 ω 00 0 ω2

, Xj = 0 1 00 0 1

1 0 0

acting at site j the model reduces to the 3-state quantum Pottshamiltonian

2H = −∞∑j=1

[Rj + β

(XjX

2j+1 + X

2j Xj+1

)]when ∆ = 0. Here the Potts spin-operator Rj is given as

Rj = Zj + Z−1j =

2 0 00 −1 00 0 −1

with the identities Z 3j = X

3j = 1.

1b) Mixed ferro-antiferromagnetic 3-state quantum Pottschain

The hamiltonian we thus consider is defined by

H =∞∑

j=−∞

[J(XjX

2j+1 + X

2j Xj+1

)− h Rj

]where J/h = −β > 0 is the antiferromagnetic interaction strength and hrepresents the transverse field. This is the mixed ferro-antiferromagnetic

3-state quantum Potts model.

We consider the case of antiferromagnetic coupling J > 0 in the Pottshamiltonian, corresponding to the antiferromagnetic region β < 0 in thephase diagram of the chiral clock model.

Given however, that β = −J/h, we do not restrict ourselves to the valuesh > 0. Rather we also consider the hamiltonian in the wider parameterspace with h < 0.

phase diagram

The previous estimates of hc are

I hc/J ' 0.2I hc/J = 0.1±0.100.03

→ The aim of our study is to estimate hc and to determine thenature of the massless phase.

antiferromagnetic 3-state quantum Potts chain

The point h/J = −1 is the exactly solved (Bethe Ansatz integrable)antiferromagnetic 3-state quantum Potts chain.

The groundstate energy per site for the infinite chain is

e∞ =4

3− 3√

3

2−√

3

π= −1.81607177 . . .

G Albertini, S Dasmahapatra and B M McCoy, Phys Lett A 170, 397 (1992)

The central charge c = 1 has been obtained from the finite-temperaturethermodynamics derived from the Bethe Ansatz solution.R Kedem and B M McCoy, J Stat Phys 71, 865 (1993); R Kedem, J Stat Phys 71, 903 (1993)

G Albertini, Int J Mod Phys A 9, 4921 (1994)

The spin-spin correlation length critical exponent η = 1/3.

AFM Potts model also studied by other authors.

numerical estimate via iMPS

The groundstate energy and wavefunction of the mixedferro-antiferromagnetic three-state quantum Potts model is calculatedusing the infinite Matrix Product State (iMPS) representation with theinfinite time-evolving block decimation (iTEBD) algorithm in order todetermine the critical point hc and the central charge in the masslessphase from the von Neumann entanglement entropy.

We also estimate the critical exponent η of the spin-spin correlation inthe massless phase.

Table: iMPS estimates for e∞ at h/J = −1 with increasing truncationdimension χ. Comparison is with the exact result.

χ 30 60 100 150eχ -1.81606688 -1.81607095 -1.81607153 -1.81607168

error 2.7× 10−6 4.5× 10−7 1.3× 10−7 5.0× 10−8

von Neumann entanglement entropy

In terms of the density matrix % = |ψ〉〈ψ| for the iMPS groundstatewavefunction |ψ〉, the von Neumann entanglment entropy is defined byS = −Tr[%L log %L] = −Tr[%R log %R ] where %L and %R are the reduceddensity matrices of the semi-infinite chains L(−∞, . . . , i) andR(i + 1, . . . ,∞).

The elements of the diagonal matrix λ[i ]αi at site i are used to evaluate the

von Neumann entropy

S(χ) = −χ∑α=1

λ2α log2 λ2α

where χ is the truncation dimension.

L Tagliacozzo, T R de Oliveira, S Iblisdir and J I Latorre, Phys Rev B 78, 024410 (2008)

F Pollmann, S Mukerjee, A Turner and J E Moore, Phys Rev Lett 102, 255701 (2009)

Entanglement measures are a useful means for detecting and classifying

quantum phase transitions.

quantum phase transition

0 0.2 0.4 0.6 0.80.4

0.8

1.2

1.6

2.0

h/J

S(χ)

0 50 100 150 2000.16

0.20

0.24

0.28

0.32

χ

h c/J

χ=18χ=36χ=64χ=100χ=150

(b)

(a)

In this way we estimate the critical point hc/J = 0.143(3) in thethermodynamic (χ→∞) limit.

The corresponding critical point of the 3-state quantum clockmodel can be estimated as βc = −J/hc ' −7.0(1) for ∆ = 0.

The duality symmetry for the 3-state quantum clock model, i.e.,the duality transformation β ↔ 1/β and ∆↔ 1/2−∆, gives thecorresponding critical point βc ' −0.143(3) at ∆ = 1/2.

→ refinement of points E and G in the Howes-Kadanoff-den Njisphase diagram.

172 S Howes et al. / Commensurate-mcommensurate transztwns

Based upon the general properties of this perturbation expansion, in sect. 5 we make global statements about our phase diagram.

Before going into detailed analysis, we summarize our results by showing our final, partially derived partially conjectured, phase diagram in fig. 2. (Specific warnings, caveats, etc., about the strength of our conclusions are to be found in the body of the paper.)

The main features of the diagram are: (a) Critical lines, AB, B'A' in the three-state Potts model universality class,

separating disordered phases (with integral values of Qp) from ordered phases (which have integral values of Qe).

(b) Commensurate-incommensurate (C-I) phase boundaries (dashed) of the type discussed by Pokrovskii and Talapov [4]. Notice that these boundaries equally well separate disordered and incommensurate phases as well as ordered and incom- mensurate ones.

(c) Two kinds of Lifshitz points (B and B' as well as D and D'). The specific heat, magnetization and susceptibility show three-state Potts behavior at these points; the mass gap goes linearly to zero as in the Ising model.

(d) Several multicritical points C, C', F and F' where fermion expansions may be applied. The behaviour in the neighborhood of these points is, for most purposes, exactly known.

(e) Some points with Kosterlitz-Thouless critical behavior: G, E and E'.

09

-(~

C / ' , COMMENSURATE COMMENSURATE

_ 0 e :0 "" " ,, ,, 00 1

/ , i A ~ , _ ~ B , INCOMMENSURATE ",[3 A'

~, (IC) " "R,

DISORDERED / DISORDERED Qp : 0 '~" ," Qp : 1 / "o

J C -'-,.~.)/. 0 I/4 o / ~', 3/4

Op:O o, lC, / G ,,zc', Qp:1

o0~'" -~ D' d ,' COMMENSURATE ' , ' ,

Y " Q0: - I ' " "IC," ",IC', ,E '

F'

F~g 2 Phase diagram for the hanultoman (I 5) wath Ae - A0 -- z~ Data points are obtaaned from strong coupling series. The verlacal scale is distorted for [ fl J > l

central charge in the massless phase

In the iMPS representation, for a critical groundstate, the central chargec can be estimated from the scaling relations

S(χ) =cκ

6log2 χ

ξ(χ) = aξ χκ

where κ is a finite entanglement scaling exponent and aξ is a constant.

L Tagliacozzo, T R de Oliveira, S Iblisdir and J I Latorre, Phys Rev B 78, 024410 (2008)

F Pollmann, S Mukerjee, A Turner and J E Moore, Phys Rev Lett 102, 255701 (2009)

0 50 100 1500.0

50

100

150

200

χ

ξ(χ)

3 4 5 6 7

0.8

1.0

1.2

1.4

1.6

1.8

log2χ

S(χ

)

h=−J

h=−J

10 50 100 1500.0

50

100

150

200

χ

ξ(χ)

3 4 5 6 7

1.0

1.5

2.0

log2χ

S(χ

)

h=0.06J

h=0.1J

h=0.1426J

h=0.06J

h=0.1J

h=0.1426J

(a)

2 3 4 5 6 70.4

0.5

0.6

0.7

0.8

log2χ

S(χ

)

h=0.5J

h=0.8J

0 50 100 150

0.5

1

1.5

χ

ξ(χ) h=0.5J

h=0.8J

(b)

Table: Estimates for the central charge c in the massless phase of themixed ferro-antiferromagnetic 3-state quantum Potts model obtainedfrom the von Neumann entanglement entropy at different values of h.

h/J -1 0.06 0.1 0.143

c 1.02(2) 1.02(3) 1.02(2) 1.03(2)

spin-spin correlation length exponent η

In order to understand more about the physical nature of the masslessphase, we investigate properties of the Potts spin-spin correlation definedby

C12(|i − j |) =〈XiX

2j

〉.

This is expected to scale as C12(r) = a0 r−η as r →∞.

10 50 100 150

0.06

0.08

0.1

0.12

χ

C12(χ

)

101

102

103

104

0.4

0.3

0.2

0.1

0.05

r

C12(r)

χ=12

χ=40

χ=80

χ=150

h=−1J

(b)

(a)

h=−1J

r=3*104

10 50 100 150

0.34

0.35

0.36

0.37

χ

η

(c)

h=−1J

101

102

103

104

0.1

0.2

0.3

0.4

0.5

r

C12(r)

10 50 100 150

0.12

0.16

0.20

χ

C12(χ

)

χ=12

χ=40

χ=80

χ=150

h=0.06J

(a)

(b)

h=0.06J

r=3*104

10 50 100 150

0.26

0.28

0.30

0.32

χ

η

h=0.06J

(c)

Table: Extrapolated estimates for the correlation length exponent η inthe massless phase of the mixed ferro-antiferromagnetic three-statequantum Potts model at different values of h.

h/J −1 −0.9 −0.8 −0.7 −0.6η∞ 0.3346(9) 0.3278(8) 0.3200(4) 0.3119(3) 0.3031(7)

−0.5 −0.4 −0.3 −0.2 −0.10.293(1) 0.2815(8) 0.269(1) 0.258(2) 0.238(2)

−0.06 0.01 0.06 0.120.236(1) 0.237(2) 0.248(1) 0.270(1)

−1 −0.8 −0.6 −0.4 −0.2 0 0.20.22

0.26

0.30

0.34

h/J

η

χ=80χ=150χ→∞

Typical continuously varying shape of Kosterlitz-Thouless exponentestimates in models of this kind.H U Everts & H Röder, J Phys A 22, 2475 (1989)

By combining exact results for the quantum sine-Gordon model with theKosterlitz-Thouless theory of melting, such curves have been predicted tohave a minimum value of 2/q2 and thus 2/9 = 0.222 . . . for q = 3.S Ostlund, Phys Rev B 24, 398 (1981)

H J Schulz, Phys Rev B 22, 5274 (1980); 28, 2746 (1983)

F D M Haldane, P Bak & T Bohr, Phys Rev B 28, 2743 (1983)

However, the plot shows that our estimates may well be higher than theexpected minimum value η = 2/9 and also the expected(Kosterlitz-Thouless) exact value η = 1/4 at the critical point hc .

This discrepancy possibly originates from the difficulty of fittingfinite-truncation dimension data to a power scaling law with a sufficientdegree of accuracy.

Then our data appears to show an overestimation of the Potts spin-spincorrelation exponents in the thermodynamic limit due to the finitetruncation effects in the iMPS approach. The value c ' 1 of the centralcharge is consistent with that of the Kosterlitz-Thouless type.

final note

Our estimate c ' 1 indicates that the known exact value c = 1 at theparticular point h/J = −1 (the antiferromagnetic 3-state quantum Pottsmodel) extends throughout the massless phase of the mixedferro-antiferromagnetic model, and thus into the massless phase of the3-state quantum chiral clock model for β < 0.

It is well known that the ferromagnetic 3-state Potts model plays a keyrole in the ferromagnetic region β > 0 of the phase diagram. It was notclear however, to what extent the antiferromagnetic 3-state Potts modelfeatured in the antiferromagnetic region β < 0 of this phase diagram.

Here we have seen that the antiferromagnetic 3-state Potts modelmanifests itself indirectly through the value c = 1 of the central charge inthe massless incommensurate phase.

This is consistent with a recent DMRG study of the 3-state quantumchiral clock model where it has been shown that the value c = 1 extendsdeep into the incommensurate phase.Y Zhuang, H J Changlani, N M Tubman and T L Hughes, Phys Rev B 92, 035154 (2015)

2) ZN free parafermionic chain

Consider the N × N matrices

(X )`m = δ`,m+1 (mod N)

Z = diag(

1, ω, ω2, . . . , ωN−1)

with ω = e2πi/N . E.g., for N = 3,

X =

0 0 11 0 00 1 0

, Z =1 0 00 ω 0

0 0 ω2

.With e the identity, they satisfy

XN = ZN = e, X † = XN−1, Z † = ZN−1,

ZX = ωXZ .

Some well studied Yang-Baxter integrable N-state quantum spinchains are of the form

H = −L∑

j=1

N−1∑n=1

an(λX nj + Z

nj Z

N−nj+1

)

Xj = e ⊗ e ⊗ · · · ⊗ e ⊗ X ⊗ e ⊗ · · · ⊗ eZj = e ⊗ e ⊗ · · · ⊗ e ⊗ Z ⊗ e ⊗ · · · ⊗ e

where e, X and Z are N×N matrices, X and Z occur in position j .

special cases

• N-state quantum Potts model

an = 1 (1)

• Fateev-Zamolodchikov ZN model

an =1

sin(πn/N)(2)

• N-state superintegrable chiral Potts model

an =2

1− ω−n(3)

Each model reduces to the quantum Ising model for N = 2.

Models (1) and (2) are equivalent for N = 3.

Model (3) has a free fermion solution described by an Onsageralgebra for general N.

free fermions

L-site (open) Ising chain at criticality λ = λc

H = −L∑

j=1

σxj −L−1∑j=1

σzj σzj+1

solution in terms of free fermions (also for general λ) using theJordan-Wigner transformation.

2L eigenvalues of the form

E = ±�1 ± �2 ± · · · ± �L

with

�k = 2 cosπk

2L + 1

excitation spectrum

L = 3 quasi-particle picture

1 2 3 3 2 1

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

𝐸

−𝜀 −𝜀 −𝜀

−𝜀 −𝜀 + 𝜀 = 𝐸 + 2𝜀

−𝜀 + 𝜀 −𝜀 = 𝐸 + 2𝜀

𝜀 −𝜀 −𝜀 = 𝐸 + 2𝜀

−𝜀 + 𝜀 + 𝜀 = 𝐸 + 2𝜀 +2𝜀

𝜀 −𝜀 +𝜀 = 𝐸 + 2𝜀 + 2𝜀

𝜀 +𝜀 −𝜀 = 𝐸 + 2𝜀 + 2𝜀

𝜀 +𝜀 +𝜀 = 𝐸 + 2𝜀 + 2𝜀 + 2𝜀

1 − 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒

2 − 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒

3 − 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒

connection to conformal field theory

For such open boundary conditions, expect (Cardy)

E0(L) = Le∞ + f∞ −πζc

24L+ · · ·

En(L) = E0(L) +πζ(xn + r)

L+ · · · r = 0, 1, 2, . . .

c = central charge, xn = scaling dimensions (critical exponents)and ζ is a scale factor. Easy to show that

E0(L) = −L∑

k=1

�k = 1−1

sin[

π2(2L+1)

]e∞ = lim

L→∞

E0(L)

L= − 4

π, f∞ = 1−

2

π.

In particular,

ζ = 2, c =1

2, xn = n −

1

2

2-particle excitations give xn + xp, n 6= p etc

conformal data of the Ising model:

c =1

2, xσ = x1 =

1

2, x� = x1 + x2 = 2

T W Burkhardt & I Guim J Phys A 18, L33 (1985)

parafermionic algebra

K. Yamazaki (1964) introduced a family of algebras generalising aClifford algebra – characterised by a primitive Nth root of unity,

ω = exp(2πi/N)

and generators cj , j = 1, . . . , L.

Each generator of order N satisfies

cNj = I and cjck = ωckcj for j < k .

Consequently c∗j = cN−1j and cjck = ω

−1ckcj for j > k .

N = 2 reduces to a self-adjoint rep of a Clifford algebra – namelyfermionic co-ordinates.

For N ≥ 3 one obtains the generic algebra of parafermionicco-ordinates.

Long before these definitions, parafermion commutation relationsappeared in both the mathematics and physics literature.

• J.J. Sylvester (1882) introduced matrices satisfyingparafermion commutation relations.

• H.S. Green (1953) proposed such commutators for fields.

Parafermions underpin a range of novel phenomena, particularlywith regard to topological phases in condensed matter physics.

A Jaffe & F L Pedrocchi, Reflection positivity for parafermions, Comm Math Phys 337, 455 (2015)

J Alicea & P Fendley, Topological phases with parafermions: theory and blueprints, Ann Rev of Cond Matt Phys 7

119 (2016)

It was boldly stated

“there is no such thing as a free parafermion in 1D”

P Fendley, J Stat Mech (2011) P11020

Baxter’s ZN chain

A model that received very little attention up until recently wasfound by Baxter in 1989.

For an L-site chain this model can be written as

−H =L∑

j=1

αjXj +L−1∑j=1

γjZ†j Zj+1

It also reduces to the quantum Ising model for N = 2.

This model has open boundary conditions.

The eigenvalues have a simple form which can be written as

−E = ωp1�1 + ωp2�2 + · · ·+ ωpL�L

for any choice of pk = 0, . . . ,N − 1.

• This gives all NL eigenvalues in the spectrum.• The energy levels �k follow from the eigenvalues of a 2L× 2L

matrix just as for the free fermion case.

• The integers pk are independent of this matrix.• Initially a numerical observation.• The model originates as the hamiltonian limit of the τ2 model,

a variant of the chiral Potts model, aka theBazhanov-Stroganov model.

R J Baxter, Phys Lett A 140, 155 (1989); J Stat Phys 57, 1 (1989)

V V Bazhanov and Y G Stroganov, J Stat Phys 59, 799 (1990)

R J Baxter, J Stat Phys 117, 1 (2004)

note I

From Baxter (1989):

Volume 140, number 4 PHYSICS LETTERS A 18 September 1989

When N= 2 the chiral Potts model reduces to the(X)m.n =~5~~+(mod N) Ising model, and .~ to the associated Ising Hamil-

(Z),,,~=ö w”~’, (4) tonian (a one-dimensional Ising model in a trans-verse field: the XY model is equivalent to two such

for m, n = l~...~ N. non-interacting Hamiltonians [10]).The r.h.s. of (1) also involves matrices Z0, Zr. In this case it is well known [11,121 that .J~can

These correspond to the fixed boundary spins of the be expressed as a quadratic form in anti-commutingmodel and are given by fermion operators. (These form a Clifford algebra.

Zo = coi, Z. = ~. (5) Thequadratic form is associated with the matrixB.)Hence .~ can readily be diagonalized by making an

The integer a in (1) and (5) runs from 1 to N. appropriate linear transformation of the fermionIt is quite natural to interlace the ct~and ‘~and to operators.

define b1 b2,_1 by It would be very interesting to find a generaliza-

b23_1 = (y1)N/2, b2,= (a~)N/’2. (6) lion of this technique to N-state spin models. The

simplicity of this result, and the factthat it holds forLet B be the 2r-by-2r bi-diagonal matrix: arbitrary coefficients a

3, y~,suggests that there mayindeed be such a technique.~0 b1 0 0 INote that ZX=coXZ and Z”=X’~z=LIt seems

b1 0 b2 I likely that such “co-commutation” properties wouldbut the author is unaware of any treatment of theB= 0 b~ 0 ~. . (7) play a role: such algebras have been studied [13—15],0 b2r_i problem of diagonalizing matrices such as ~

0 b2r_i 0 The hope isof course that is such a technique couldbe found, then it might have a more general appli-The eigenvalues of B occur in pairs A, —A. Thus cation. For instance, perhaps one could still diagon-

there are r distinct squares A) of eigenvalues. Let alize .~ when periodic (rather than fixed) boundarythese be s~,..., s~’,so that det(B—s7’

2~)=0.One conditions are imposed. (This does not seem as triv-can choose the Nth roots s

1, ...~ S~50 that ial a change as one might think.) If so, then one could

Si ...Sr = )~...~. (8) presumably diagonalize T~01for the superintegrablechiral Potts model on a torus. This would be very

The intriguing result that we find from the lattice interesting, particularly if it led to a simplification ofmodel calculations is that theNr_I eigenvalues of ~ the Bethe-ansatz-type results of Albertini et al. [5—are 7].

E{n}=— ~ s~co”, (9)J= I

Referencesfor all sets of integers ii= { n1, ..., fl~}satisfyingflj +...+flrtl (mod N) , (10) [1]H. Au-Yang, B.M. McCoy, J.H.H. Perk and S. Tang, in:

Algebraic analysis, eds. M. Kashiwara and T. Kawaiand 0~n~~N—1for j=l, ..., r. (Academic Press, NewYork, 1988) p. 29.Thus the eigenvalue spectrum of ~ has a very [2] R.J. Baxter, J.H.H.Perk and H. Au-Yang, Phys. Lett.A 128

(1988) 138.simpledirect sum structure: in fact the diagonalized [3] R.J. Baxter, J. Stat. Phys. 52 (1988) 639.form of ~ is simply [4] R.J. Baxter, Phys. Lett. A 133 (1988) 185.

[5]G. Albertini, B.M. McCoy,J.H.H. Perk and S. Tang, Nuci.Phys.B314(1989)741.(.~*“a)diag ~ S~Z~Z~T’1, (11)

i=’ [6] G. Albertini, B.M. McCoy and J.H.H. Perk, Phys. Lett. A135 (1989) 159.

i.e. the diagonal part of (1), with each y3 replaced by [7] G. Albertini, B.M. McCoy and J.H.H. Perk, Phys. Lett. A139 (1989) 204.Si.

156

From Baxter (2004):

24 Baxter

It appears that the other values of γ1, . . . ,γL that do not satisfy thesum rule also correspond to eigenvalues of τ2(tq), provided we generalizethe model to allow the skewed boundary conditions

aL+1 =a1 + r

in every row of the lattice ( so all spins in column 1 are still the same, asare all spins in column L+1, but now those in the two boundary columnsno longer need be equal). Then

γ1 +· · ·+γL = r.

We can take r to be an integer in the range 0, . . . ,N −1.The eigenvalues therefore have the same simple structure as do direct

products of L matrices, each of size N by N . For N =2 this is the struc-ture of the eigenvalues of the Ising model.(9)

For the Ising model this property follows from Kaufman’s solutionin terms of spinor operators,(10) i.e. a Clifford algebra.(11, p. 189) Whetherthere is some generalization of such spinor operators to handle the τ2(tq)model with open boundaries remains a fascinating speculation.(12)

The results of this section were anticipated in ref. 5. There we con-sidered the superintegable chiral Potts model and rotated it though 90◦ toobtain a model that is in fact the present τN(tq) model. Then we invertedits row-to-row transfer matrix, thereby obtaining the present τ2(tq) model.6

We did in fact note in section 7 of ref. 5. that we could allow the modulusk to be different for different rows: this corresponds to our here allowingap, bp, cp, dp to all vary arbitrarily from column to column.

10. SUMMARY

We have shown that the column-inhomogeneous τ2(tq) model is solv-able for all values of the 8L parameters ap1 , bp1 , cp1 , dp1 , . . . , dp2L , whereapJ , bpJ , cpJ , dpJ are associated with the J th vertical dotted line in Fig. 1.They do not need to satisfy the “chiral Potts” conditions (1). The modelthen has the unusual property that its row-to-row transfer matrices (withdifferent values of tq but the same ap1 , . . . , dp2L ) commute, while the col-umn-to-column transfer matrices do not.

Our results (38), (43), (47a)–(47c), (71) generalize the relations (4.20),(4.21), (4.27a)–(4.27c), (3.46) of ref. 2. The last generalization (71) is essen-tially a conjecture, depending as it does on the identification of (68) with

6Note from (75) that τ2(tq )τN(ωtq )=α, so to within a scalar factor the transfer matrix τ2(tq )is the inverse of τN(ωtq ).

note II

Paul Fendley (2014) made the connection with free parafermions!

(I had read his [Baxter’s 1989] papers, but thought Imisunderstood because the results seemed too good tobe true).

P Fendley, J Phys A 47, 075001 (2014)

free parafermions

Each index k has N possible choices, instead of “filled” or “empty” in the

free fermion (N = 2) case. Now have a “Fermi exclusion circle”.

N = 3(ω = e2πi/3

)𝜔

𝜔

N = 4 (ω = i)

𝜔

𝜔2

𝜔3

NB there can be real excitations for N even, since ω2 = −1.

−E = ωp1�1 + ωp2�2 + · · ·+ ωpL�L

• Fendley derived this result using a generalisation of theJordan-Wigner transformation, namely the Fradkin-Kadanofftransformation to parafermionic operators originallyintroduced for the N-state clock models.

• Baxter (2014) and Au-Yang and Perk (2014,2016) appliedFendley’s parafermionic approach to the more general τ2model with open boundaries.

• The chiral Potts model is related, via the τ2 model, to thesix-vertex model.

R J Baxter, J Phys A 47, 315001 (2014)

H Au-Yang and J H H Perk, J Phys A 47, 315002 (2014); arXiv:1606.06319

We consider the N-state hamiltonian

−H =L∑

j=1

Xj + λL−1∑j=1

Z †j Zj+1

which has eigenspectrum

−E = ωp1�1 + ωp2�2 + · · ·+ ωpL�L

for any choice of pk = 0, . . . ,N − 1 with the above �k andω = e2πi/N .

For the isotropic “critical” case λ = 1

�k =

(2 cos

πk

2L + 1

)2/N

The hamiltonian is non-Hermitian, with complex energyeigenvalues for N ≥ 3.

For any eigenvalue E , there are other eigenvalues ωE , ω2E , . . .

This is the generalisation of the E ↔ −E Ising symmetry(recall ω = −1 for N = 2).

In general non-Hermitian hamiltonians describe the dynamics ofphysical systems that are not conservative.

The properties of the model are still well worth exploring.

ground state energy at λ = 1

The ground state energy is real and given by

E0 = −L∑

k=1

�k = −L∑

k=1

(2 cos

πk

2L + 1

)2/N.

Using the Euler-Maclaurin formula

L∑k=1

f (k) =

∫ L0

f (x)dx + B1[f (L)− f (0)] +∞∑k=1

B2k

(2k)!

[f (2k−1)(L)− f (2k−1)(0)

]

E0(L) = Le∞ + f∞ +γNLν

+ O

(1

Lν+1

)where ν = 2/N, with

e∞ = −2ν√π

Γ( 12 +1N )

Γ(1 + 1N ), f∞ =

1

2e∞ + 2

ν−1.

The amplitudes γN are given by

γN = −

[1

N + 2−∞∑k=1

22k−1B2k(2k)!

2k−2∏`=0

(2

N− `)](π

2

)2/Nwhere B2k are the Bernoulli numbers

B2 =16 , B4 = −

130 , B6 =

142 , . . .

The infinite series only terminates for the case N = 2, whereγ2 = − π24 .

E0(L) = Le∞ + f∞ +γNLν

+ O

(1

Lν+1

), ν =

2

N

excitations at λ = 1

𝜔

𝜔

? 1-particle

E (L)− E0(L) = (2n − 1)ν ε(p)( π

2L

)ν+ O

(1

Lν+1

)for n = 1, . . . , L and p = 1, . . . ,N − 1, with

ε(p) = 1− ωp = 1− cos (2πp/N)− i sin (2πp/N) .

? 2-particle

E (L)− E0(L) = [(2m − 1)ν ε(p) + (2n − 1)ν ε(q)]( π

2L

)ν+ O

(1

Lν+1

)for m, n = 1, . . . , L and p, q = 1, . . . ,N − 1 with m 6= n.

? etc

solution for general λ

2L× 2L Baxter/Fendley determinant satisfied by �1, . . . , �L∣∣∣∣∣∣∣∣∣∣∣∣∣

−�N/2 1 0 0 .. 0 01 −�N/2 λN/2 0 .. 0 00 λN/2 −�N/2 1 .. 0 00 .. .. .. .. .. 0

0 0 0 0 λN/2 −�N/2 10 0 0 0 0 1 −�N/2

∣∣∣∣∣∣∣∣∣∣∣∣∣= 0.

This is a multinomial in integer powers of �2 and λ2, of degree L in�N . This is however, the same multinomial for all values of N andλ if one makes the correspondence

�Nk = �2k(Ising) and λ

N = λ2(Ising)

Thus to find �k for general λ one only needs to understand thesolution of the transverse Ising chain for general λ with openboundary conditions.

The transverse Ising chain for general λ with open bc’s was solvedby Pfeuty (1970), closely following the Lieb-Schultz-Mattis (1961)solution of the XY chain with open bc’s.

P Pfeuty, Ann Phys NY 57, 79 (1970)

E Lieb, T Schultz & D Mattis, Ann Phys NY 16, 407 (1961)

−H =L∑

j=1

Xj + λL−1∑j=1

Z †j Zj+1

−E =L∑

j=1

ωsj �kj , sj = 0, 1, . . . ,N − 1, ω = e2πi/N

�k =(

1 + λN + 2λN/2 cos k)1/N

=(

1 + λN/2)2/N (

1− θ2 sin2 k2

)1/N, θ2 =

4λN/2(1 + λN/2

)2kj satisfy

sin(L + 1)k = −λN/2 sin Lk

for λ = 1, kj =2jπ

2L+1 , j = 1, . . . , L and �k =(2 cos k2

)2/Nas before.

I L real roots in (0, π) for λ ≤ (1 + 1/L)2/N

I L− 1 real roots in (0, π) for λ > (1 + 1/L)2/N

For λ < 1 write

Lkj = πj − πκj + O(

1

L

), j = 1, . . . , L

gives

cot(πκj) =λN/2 + cos(πj/L)

sin(πj/L)

with solution

πκj =πj

2L+ tan−1

[1− λN/2

1 + λN/2tan

(πj

2L

)].

To leading order, the roots are thus approximated for large L by

kj ≈πj

L− 1

1 + λN/2πj

L2, j = 1, . . . , L.

For λ > 1 the complex root is of the form

kL = π + iv

where

sinh(L + 1)v = λN/2 sinh Lv .

Solving for large L the excitation carries energy

�kL =(

1 + λN − 2λN/2 cosh v)1/N

= λ1−L(

1− 2λ−N + λ−2N + . . .)1/N

,

I groundstate is N-fold degenerate for λ > 1.The gap to excitations closes as e−L lnλ.

ground state energy

e∞(λ) = −2

π

(1 + λN/2

)2/N ∫ π/20

(1− θ2 sin2 x

)1/Ndx

withe∞(λ) = λ e∞(1/λ)

For N = 2:

e∞(λ) = −2

π(1 + λ)E

(π2, θ)

Also

e∞(1) = −2

π22/N

∫ π/20

(cos x)2/N dx = −22/N

√π

Γ( 12 +1N )

Γ(1 + 1N )

ground state energy – hypergeometric functions

The change of variable x → arcsin√t gives

e∞(λ) = −(

1 + λN/2)ν

F

(− 1N,

1

2; 1; θ2

).

Moreover, using a quadratic relation for the hypergeometric functionsgives the simple form

e∞(λ) = −F(− 1N,− 1

N; 1;λN

)The series representation of F then gives an expansion in λ,

e∞(λ) = −1−[

1

πsin( πN

)Γ

(1 +

1

N

)]2 ∞∑`=1

[Γ(`+ 1N

)]2Γ(1 + `)

λN`

` !

for λ < 1. The expansion for λ > 1 follows from the duality relation

e∞(λ) = λ e∞(1/λ).

mass gaps

Can also calculate mass gaps etc. E.g., for λ < 1,

E − E0 = ε(p)(

1− λN/2)2/N

as L→∞

where p = 1, . . . ,N − 1, with

ε(p) = 1− ωp = 1− cos (2πp/N)− i sin (2πp/N) .

[N = 2 Ising case E − E0 = 2(1− λ)]

This result implies a correlation length exponent

ν⊥ =2

N

→ same as for the superintegrable chiral Potts model.

specific heat

Another quantity of interest is the specific heat, defined by

C (λ, L) = −λ2

L

d2E0(λ, L)

dλ2,

which at the critical point λ = 1 scales like C ∼ Lα/ν‖ as L→∞.We find

C ∼ L1−2/N as L→∞,implying the result α/ν‖ = 1− 2/N. We confirm that ν‖ = 1from the scaling of the finite-size peak in C (λ, L).

The specific heat critical exponent is thus seen to be

α = 1− 2/N

→ same as for the Fateev-Zamolodchikov ZN model→ critical exponents α, ν‖ = 1 and ν⊥ = 2/N all same as for thesuperintegrable chiral Potts model, both models exhibit anisotropicscaling.

Remarks

I This is an example from the class of models which arenon-Hermitian, with a complex eigenspectrum.

I Nevertheless, the model has a real ground state and aremarkably simple excitation spectrum governed by thestructure of free parafermions.

I The eigenspectrum is seen to share some critical exponentswith the superintegrable chiral Potts model.

I Correlations in this model are also interesting...