The XYZ spin chain/8-vertex model with quasi-periodic ...From theperiodicSOS model to...

The XYZ spin chain/8-vertex model withquasi-periodic boundary conditions

Exact solution by Separation of Variables

Veronique TERRAS

CNRS & Universite Paris Sud, France

Workshop: Beyond integrability. The mathematics and physics of integrability

and its breaking in low-dimensional strongly correlated quantum phenomena

July 13-17, 2015 – CRM Montreal

In collaboration with G. Niccoli .

8-vertex model

2-d square lattice modellink → εj = ±vertex → Boltzmann weight

R8V (z1 − z2)ε1,ε2

ε′1,ε′2

=

ε1

ε′1

z1

ε2ε′2z2

+

+ +

+

�

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+

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+

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a8V b8V c8V d8V

Veronique TERRAS The quasi-periodic XYZ chain - 2015

8-vertex model


R8V (z1 − z2)ε1,ε2

ε′1,ε′2

=

ε1

ε′1

z1

ε2ε′2z2

R8V (z) =

0BB@a8V (z) 0 0 d8V (z)

0 b8V (z) c8V (z) 00 c8V (z) b8V (z) 0

d8V (z) 0 0 a8V (z)

1CCA z : spectral parameterp = e iπω: ellipticparameter

a8V (z) = ρθ4(z|2ω) θ4(η|2ω)

θ4(z + η|2ω) θ4(0|2ω), b8V (z) = ρ

θ1(z|2ω) θ4(η|2ω)

θ1(z + η|2ω) θ4(0|2ω),

c8V (z) = ρθ4(z|2ω) θ1(η|2ω)

θ1(z + η|2ω) θ4(0|2ω), d8V (z) = ρ

θ1(z|2ω) θ1(η|2ω)

θ4(z + η|2ω) θ4(0|2ω).


8-vertex model


R8V12 (z1 − z2)ε1,ε2

ε′1,ε′2

=

ε1

ε′1

z1

ε2ε′2z2

R8V12 (z) =

0BB@a8V (z) 0 0 d8V (z)

0 b8V (z) c8V (z) 00 c8V (z) b8V (z) 0

d8V (z) 0 0 a8V (z)

1CCA ∈ End(V1 ⊗ V2)Vi ' C2

satisfying the Quantum Yang-Baxter Equation (QYBE) on V1 ⊗ V2 ⊗ V3,Vi ' C2:

R8V12 (z1 − z2) R8V

13 (z1) R8V23 (z2) = R8V

23 (z2) R8V13 (z1) R8V

12 (z1 − z2)


Commuting transfer matrices for the 8-vertex model

Monodromy Matrix:

(on V0 ⊗ VN , VN = V1 ⊗ V2 ⊗ . . .⊗ VN , Vi ' C2)

M(8V)0 (λ) = R

(8V)0N (λ− ξN) · · ·R(8V)

02 (λ− ξ2) R(8V)01 (λ− ξ1)

=

„A(8V)(λ) B(8V)(λ)

C (8V)(λ) D(8V)(λ)

«[0]

satisfying

R(8V)00′ (λ1 − λ2) M

(8V)0 (λ1) M

(8V)0′ (λ2) = M

(8V)0′ (λ2) M

(8V)0 (λ1) R

(8V)00′ (λ1 − λ2)

commutation relations for A(8V), B(8V), C (8V), D(8V)

Transfer Matrix: T (8V)(λ) = tr0

˘M

(8V)0 (λ)

¯ [T (8V)(u),T (8V)(v)] = 0



Monodromy Matrix:

(on V0 ⊗ VN , VN = V1 ⊗ V2 ⊗ . . .⊗ VN , Vi ' C2)

M(8V)0 (λ) = R

(8V)0N (λ− ξN) · · ·R(8V)

02 (λ− ξ2) R(8V)01 (λ− ξ1)

=

„A(8V)(λ) B(8V)(λ)

C (8V)(λ) D(8V)(λ)

«[0]

satisfying

R(8V)00′ (λ1 − λ2) M

(8V)0 (λ1) M

(8V)0′ (λ2) = M

(8V)0′ (λ2) M

(8V)0 (λ1) R

(8V)00′ (λ1 − λ2)

commutation relations for A(8V), B(8V), C (8V), D(8V)

Transfer Matrix: (periodic boundary conditions)

T (8V)(λ) = tr0

˘M

(8V)0 (λ)

¯ [T (8V)(u),T (8V)(v)] = 0



Monodromy Matrix:

(on V0 ⊗ VN , VN = V1 ⊗ V2 ⊗ . . .⊗ VN , Vi ' C2)

M(8V)0 (λ) = R

(8V)0N (λ− ξN) · · ·R(8V)

02 (λ− ξ2) R(8V)01 (λ− ξ1)

=

„A(8V)(λ) B(8V)(λ)

C (8V)(λ) D(8V)(λ)

«[0]

satisfying

R(8V)00′ (λ1 − λ2) M

(8V)0 (λ1) M

(8V)0′ (λ2) = M

(8V)0′ (λ2) M

(8V)0 (λ1) R

(8V)00′ (λ1 − λ2)


˘M

(8V)0 (λ)

¯ [T (8V)(u),T (8V)(v)] = 0

Remark. [R(8V)(λ), σα ⊗ σα] = 0 for α = x , y , z

for any fixed α, T(8V)α (λ) = tr0

˘σα0 M

(8V)0 (λ)

¯defines a

one-parameter family of commuting quasi-periodic transfer matrices

∂ log T(8V)α (λ)

∂λ

˛λ=0ξn=0

= HXYZ =1

2

NXn=1

nJxσ

xnσ

xn+1+Jyσ

ynσ

yn+1+Jzσ

znσ

zn+1

o+

1

2J0,

with σβN+1 = σα1 σβ1 σ

α1

Goal: find the (complete set of) eigenvalues and eigenstates of T(8V)α (λ)



Monodromy Matrix:

(on V0 ⊗ VN , VN = V1 ⊗ V2 ⊗ . . .⊗ VN , Vi ' C2)

M(8V)0 (λ) = R

(8V)0N (λ− ξN) · · ·R(8V)

02 (λ− ξ2) R(8V)01 (λ− ξ1)

=

„A(8V)(λ) B(8V)(λ)

C (8V)(λ) D(8V)(λ)

«[0]

satisfying

R(8V)00′ (λ1 − λ2) M

(8V)0 (λ1) M

(8V)0′ (λ2) = M

(8V)0′ (λ2) M

(8V)0 (λ1) R

(8V)00′ (λ1 − λ2)


˘M

(8V)0 (λ)

¯ [T (8V)(u),T (8V)(v)] = 0

T(8V)α (λ) = tr0

˘σα0 M

(8V)0 (λ)

¯ [T

(8V)α (u),T

(8V)α (v)] = 0

Goal: find the (complete set of) eigenvalues and eigenstates of T(8V)α (λ)

However:

no simple reference state

[X (8V)(λ),X (8V)(µ)] 6= 0 for X = A,B,C ,D

→ not directly solvable by Bethe ansatz nor by separation of variablesBaxter’s solution (Ann.Phys.73) → map onto an IRF model (SOS model)


(8V)SOS model (dynamical 6-vertex model)

2-d square lattice modelvertex → local height tj

tj − tk = ±η (adjacent)face → Boltzmann weight

R(λi − ξj |t)εi ,εjε′i ,ε′j

=

t t + ηε′i

λi

t + ηεj t + η(εi + εj)= t + η(ε′i + ε′j)

ξj

t t + ⌘

t + 2⌘t + ⌘

+

+

+ +

1

t t� ⌘

t� 2⌘t� ⌘

�

�

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1

t t + ⌘

tt� ⌘

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b(u|t)

t t� ⌘

tt + ⌘

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b(u|� t)

t t� ⌘

tt� ⌘

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� +

c(u|t)

t t + ⌘

tt + ⌘

+

�

+ �

c(u|� t)


(8V)SOS model (dynamical 6-vertex model)

2-d square lattice modelvertex → local height tj

tj − tk = ±η (adjacent)face → Boltzmann weight

R(λi − ξj |t)εi ,εjε′i ,ε′j

=

t t + ηε′i

λi

t + ηεj t + η(εi + εj)= t + η(ε′i + ε′j)

ξj

R(λ|t) =

0BB@1 0 0 00 b(λ|t) c(λ|t) 00 c(λ| − t) b(λ| − t) 00 0 0 1

1CCA λ : spectral parametert ∈ t0 + ηZ : dynamical parameter

b(λ|t) =θ(t + η) θ(λ)

θ(t) θ(λ+ η)c(λ|t) =

θ(λ+ t) θ(η)

θ(t) θ(λ+ η)θ(u) = θ1(u|ω)

satisfying the Dynamical Quantum Yang-Baxter Equation:

R12(λ1 − λ2|t + ησz3) R13(λ1 − λ3|t) R23(λ2 − λ3|t + ησz

1)

= R23(λ2 − λ3|t) R13(λ1 − λ3|t + ησz2) R12(λ1 − λ2|t)

Spin conservation, solvable by Bethe Ansatz


Baxter’s Vertex-IRF Transformation

It is equivalent to the following Dynamical Gauge Equivalence:

R(8V)12 (λ12) S1(λ1|t) S2(λ2|t + ησz

1) = S2(λ2|t) S1(λ1|t + ησz2) R(SOS)

12 (λ12|t)

with S(λ|t) =

„θ2(−λ+ t|2ω) θ2(λ+ t|2ω)θ3(−λ+ t|2ω) θ3(λ+ t|2ω)

«λ12 ≡ λ1 − λ2

relation between monodromy matrices:

M(8V)0 (λ) S0(λ|t) Sq(t + ησz

0) = Sq(t) S0(λ|t + ηSz) M(SOS)0 (λ|t)

where M(8V)0 (λ) = R

(8V)0N (λ− ξN) · · ·R(8V)

02 (λ− ξ2) R(8V)01 (λ− ξ1)

M(SOS)0 (λ|t) = R0N(λ− ξN|t + η

N−1Xa=1

σza) · · ·R02(λ− ξ2|t + ησz

1) R01(λ− ξ1|t)

Sq(t) = S1(ξ1|t) S2(ξ2|t + ησz1) . . .SN(ξN|t + η

N−1Xa=1

σza) and Sz =

NXn=1

σzn

−→ Eigenvalues and eigenvectors of the (periodic) 8-vertex transfer matrix(Baxter, 1973)

Restrictions: Lattice size N = 2n evenLη = m1π + m2πω L ∈ N, m1,m2 ∈ Z (cyclic SOS model)


The SOS model: Dynamical Yang-Baxter algebra

Felder, Varchenko (1996) : representations of Eτ,η(sl2)

M0(λ|t) = R0N(λ− ξN|t + η

N−1Xa=1

σza) · · ·R02(λ− ξ2|t + ησz

1) R01(λ− ξ1|t)

=

„A(u|t) B(u|t)C(u|t) D(u|t)

«[0]

satisfyR00′(λ00′ |t + ηSz) M0(λ0|t) M0′(λ0′ |t + ησz

0)= M0′(λ0′ |t) M0(λ0|t + ησz

0′) R00′(λ00′ |t) with Sz =PN

n=1 σzn

introduce dynamical operators bτ and bTτ such that bTτ bτ = (bτ + η) bTτ so asto simplify the commutation relations:

M0(u) =

„A(u) B(u)C(u) D(u)

«[0]

= M0(u|bτ)

bTτ 0

0 bT−1τ

![0]

satisfyR00′(λ00′ |bτ + ηSz)M0(λ0)M0′(λ0′) =M0′(λ0′)M0(λ0) R00′(λ00′ |bτ)

operator algebra: A(u), B(u), C(u), D(u) act on H = VN ⊗ DVN = ⊗N

n=1Vn (with Vn ' C2) : quantum spin space

D : representation space of dynamical operators algebra


The SOS model: Representation spaces of the operatoralgebra

A(u), B(u), C(u), D(u) act on H = VN ⊗ D

VN = ⊗Nn=1Vn (with Vn ' C2) : (2N -dimensional) quantum spin space


basis: | t(j) 〉, j ∈ Z (or j ∈ Z/LZ if Lη = m1π + m2πω: cyclic case)

with τ | t(j) 〉 = t(j) | t(j) 〉, t(j) = t0 − ηjTτ | t(j) 〉 = | t(j + 1) 〉

Periodic transfer matrix: T (u) = tr0{M0(u)} = A(u) +D(u)

commuting family on the subspace H(0) ≡ VN [0]⊗ D of H associatedto the zero eigenvalue of Sz =

PNn=1 σ

zn

Antiperiodic transfer matrix: T (u) = tr0{σx0M0(u)} = B(u) + C(u)

commuting family on the subspace H(0) of H associated to the zeroeigenvalue of Sτ ≡ ηSz + 2τ


The SOS model: Representation spaces of the operatoralgebra

A(u), B(u), C(u), D(u) act on H = VN ⊗ D

VN = ⊗Nn=1Vn (with Vn ' C2) : (2N -dimensional) quantum spin space


basis: | t(j) 〉, j ∈ Z (or j ∈ Z/LZ if Lη = m1π + m2πω: cyclic case)

with τ | t(j) 〉 = t(j) | t(j) 〉, t(j) = t0 − ηjTτ | t(j) 〉 = | t(j + 1) 〉

Periodic transfer matrix: T (u) = tr0{M0(u)} = A(u) +D(u)

commuting family on the subspace H(0) ≡ VN [0]⊗ D of H associatedto the zero eigenvalue of Sz =

PNn=1 σ

zn

Antiperiodic transfer matrix: T (u) = tr0{σx0M0(u)} = B(u) + C(u)

commuting family on the subspace H(0) of H associated to the zeroeigenvalue of Sτ ≡ ηSz + 2τ


From the periodic SOS model to the periodic 8-vertexmodel with N even: solution by Bethe ansatz

Periodic SOS transfer matrix: T (u) = tr0{M0(u)} = A(u) +D(u)

commuting family on the subspace H(0) ≡ VN [0]⊗ D ' Fun(VN [0]) of Hassociated to the zero eigenvalue of Sz

For generic values of η, the physical space of states H(0) of the periodic SOS

model is:

infinite-dimensional (unrestricted SOS model) for N = 2n even

zero-dimensional for N odd

except if Lη = m1π + m2πω (cyclic case considered by Baxter):

dimH(0) = L dimVN [0]

The eigenstates of T (u) can be constructed by (algebraic) Bethe ansatz(Baxter 1973, Felder and Varchenko 1996)

by means of the vertex-IRF transformation, one can construct theeigenstates of the periodic 8-vertex transfer matrix in the case N = 2n even( Baxter 1973 for Lη = m1π + m2πω)

Completeness ?


From the antiperiodic SOS model to the (quasi-periodic)8-vertex model: solution by SOV

Antiperiodic SOS transfer matrix: T (u) = B(u) + C(u)

commuting family on the subspace H(0) of H = VN ⊗ D associated to thezero eigenvalue (or more generally to the eigenvalue xπ + yπω, x, y ∈ {0, 1}) ofSτ ≡ ηSz + 2τ

basis of the physical space of states H(0):`⊗N

n=1 |n, hn〉´⊗ |th〉, h ≡ (h1, . . . , hN) ∈ {0, 1}N hn =

(0 if spin +

1 if spin −

where th = t0 + ηNX

k=1

hk with t0 = −η2N + x

π

2+ y

π

2ω, x, y ∈ {0, 1}

Remark: η generic, (x, y) 6= (0, 0) if N even

H(0) has dimension 2N and is isomorphic to the space of states VN of the

8-vertex model

construction of the eigenstates of the antiperiodic SOS model by separationof variables (Felder and Schorr 99, Niccoli 13, Levy-Bencheton Niccoli V.T. 15)

complete set of eigenstates of the periodic (N odd, x , y = 0) andquasi-periodic ((x , y) 6= (0, 0)) 8-vertex transfer matrices






n=1 |n, hn〉´⊗ |th〉, h ≡ (h1, . . . , hN) ∈ {0, 1}N hn =

(0 if spin +

1 if spin −


k=1


π

2+ y

π

2ω, x, y ∈ {0, 1}



8-vertex model:

P :`⊗N

n=1 |n, hn〉´⊗ |th〉 7→

`⊗N

n=1 |n, hn〉´








n=1 |n, hn〉´⊗ |th〉, h ≡ (h1, . . . , hN) ∈ {0, 1}N hn =

(0 if spin +

1 if spin −


k=1


π

2+ y

π

2ω, x, y ∈ {0, 1}



8-vertex model




Sklyanin’s quantum Separation of Variables (SOV): idea ofthe method

Suppose that the monodromy matrix of the model eM(λ) ≡

eA(λ) eB(λ)eC(λ) eD(λ)

!is

such that eB(λ) is a (usual, trigonometric, elliptic. . . ) polynomial of degree Nand is diagonalizable with simple spectrum, then the operator zeroes Yn,1 ≤ n ≤ N, of eB(λ) can be used to define a basis

|y1, . . . , yN〉, (y1, . . . , yN) ∈ Λ1 × · · · × ΛN , (Λi ∩ Λj = ∅ if i 6= j)

Yn |y1, . . . , yN〉 = yn |y1, . . . , yN〉

of the space of states in which the action of eA(λ) and eD(λ) is quasi-local, inparticular

eA(Yn) |y1, . . . , yn, . . . , yN〉 = a(yn) |y1, . . . , yn + η, . . . , yN〉eD(Yn) |y1, . . . , yn, . . . , yN〉 = d(yn) |y1, . . . , yn − η, . . . , yN〉


Sklyanin’s quantum Separation of Variables (SOV): idea ofthe method

Suppose that the monodromy matrix of the model eM(λ) ≡

eA(λ) eB(λ)eC(λ) eD(λ)

!is

such that eB(λ) is a (usual, trigonometric, elliptic. . . ) polynomial of degree Nand is diagonalizable with simple spectrum, then the operator zeroes Yn,1 ≤ n ≤ N, of eB(λ) can be used to define a basis

|y1, . . . , yN〉, (y1, . . . , yN) ∈ Λ1 × · · · × ΛN , (Λi ∩ Λj = ∅ if i 6= j)

Yn |y1, . . . , yN〉 = yn |y1, . . . , yN〉

of the space of states in which the action of eA(λ) and eD(λ) is quasi-local

The multi-dimensional spectral problem for the transfer matrixeT (λ) = eA(λ) + eD(λ) can be reduced to a set of N one-dimensional spectral

problems by separation of variables: eT (λ)|Ψt 〉 = t(λ)|Ψt 〉 iff

|Ψt 〉 =X

| y1,...,yN 〉

ψt(y1, . . . , yN) | y1, . . . , yN 〉 with ψt(y1, . . . , yN) =NY

n=1

Q(n)t (yn)

where each Q(n)t is solution of a discrete finite-difference equation


Solution of the antiperiodic SOS model by SOV: Spectrumand eigenstates of the antiperiodic transfer matrix

SOV basis of H(0): | h 〉 , h = (h1, . . . , hN) ∈ {0, 1}N

For any fixed N-tuple of inhomogeneities (ξ1, . . . , ξN) ∈ CN satisfying

∀ε ∈ {−1, 0, 1}, ξj − ξk + εη /∈ πZ+ πωZ if j 6= k,

the spectrum Σ(SOS) of the antiperiodic SOS transfer matrix T (λ) in H(0) issimple and coincides with the set of functions of the form

t(λ) =NX

k=j

e iy(ξj−λ) θ(t0 − λ+ ξj)

θ(t0)

Yk 6=j

θ(λ− ξk)

θ(ξj − ξk)t(ξj),

which satisfy the discrete system of equations, ∀j ∈ {1, . . . ,N},

t(ξj) t(ξj − η) = (−1)x+y+xya(ξj) d(ξj − η), with

(a(λ) =

QNn=1θ(λ−ξn +η)

d(λ) = a(λ− η).

The T (λ)-eigenstate associated with the eigenvalue t(λ) ∈ Σ(SOS) is

|Ψ(SOS)t 〉 =

Xh∈{0,1}N

NYj=1

Qt(ξj − ηhj) | h 〉

where the coefficients Qt(ξj − ηhj) are (up to an overall normalization)characterized by Qt(ξj − η)

Qt(ξj)=

t(ξj)

d(ξj − η)= (−1)x+y+xy a(ξj)

t(ξj − η).


From antiperiodic SOS transfer matrix to quasi-periodic8-vertex transfer matrix: the case (x , y) 6= (0, 0)

When (x, y) 6= (0, 0), the vertex-IRF transformation Sq ≡ P ◦ Sq(τ) is anisomorphism from H

(0) (antip. SOS space of states) to VN (8V space of states).

Define the (x, y)-twisted 8-vertex transfer matrix:

T(8V)(x,y)(λ) = tr0

h(σx

0 )y (σz0)x M

(8V)0 (λ)

iIt has the following action on the states | v 〉 of H(0):

T(8V)(x,y)(λ) Sq | v 〉 = (−1)x i xy Sq T

(SOS)(λ) | v 〉.

Complete characterization of the (x, y)-twisted 8-vertex transfer matrixspectrum and eigenstates:

Let (x, y) 6= (0, 0). If |Ψ(SOS)t 〉 ∈ H(0) is an eigenvector of the antiperiodic

SOS transfer matrix T (SOS)(λ) with eigenvalue t(λ) , then

Sq |Ψ(SOS)t 〉 ∈ VN

is an eigenvector of the quasi-periodic 8-vertex transfer matrix T(8V)(x,y)(λ) with

eigenvalue

t(8V)(x,y)(λ) ≡ (−1)x i xy t(λ),

and conversely.Veronique TERRAS The quasi-periodic XYZ chain - 2015





T(8V)(x,y)(λ) = tr0

h(σx

0 )y (σz0)x M

(8V)0 (λ)



(SOS)(λ) | v 〉.






eigenvalue

t(8V)(x,y)(λ) ≡ (−1)x i xy t(λ),






T(8V)(x,y)(λ) = tr0

h(σx

0 )y (σz0)x M

(8V)0 (λ)



(SOS)(λ) | v 〉.






eigenvalue

t(8V)(x,y)(λ) ≡ (−1)x i xy t(λ),


From the antiperiodic SOS transfer matrix to the periodic8-vertex transfer matrix in the case N odd

When (x, y) = (0, 0), Sq is not bijective from H(0) to VN

but, from the symmetry of the periodic (resp. antiper.) transfer matrices:

Γz T(8V)(0,0)(λ) Γz = T

(8V)(0,0)(λ), Γz T

(SOS)(λ) Γz = −T (SOS)

(λ), with Γz = ⊗Nn=1σ

zn

it is possible to define a second vertex-IRF transformation Sq = Γz Sq Γz , suchthat

T(8V)(0,0)(λ) Sq | v 〉 = Sq T

(SOS)(λ) | v 〉

T(8V)(0,0)(λ) Sq | v 〉 = −Sq T

(SOS)(λ) | v 〉

? the spectrum and eigenstates of T (SOS)(λ) can be decomposed into

a ‘+’ part, with eigenvalues t+(λ) and eigenstates |Ψ(SOS)t+

〉,

a ‘−’ part, with eigenv. t−(λ) = −t+(λ) and eigenst. |Ψ(SOS)t− 〉 = Γz |Ψ(SOS)

t+〉

? ker Sq ∩ ker S(0)q = {0} and

ker Sq is generated by the type ‘−’ eigenstates |Ψ(SOS)t− 〉

ker Sq is generated by the type ‘+’ eigenstates |Ψ(SOS)t+

〉





Γz T(8V)(0,0)(λ) Γz = T

(8V)(0,0)(λ), Γz T



zn


T(8V)(0,0)(λ) Sq | v 〉 = Sq T

(SOS)(λ) | v 〉

T(8V)(0,0)(λ) Sq | v 〉 = −Sq T

(SOS)(λ) | v 〉



〉,


t+〉




〉





Γz T(8V)(0,0)(λ) Γz = T

(8V)(0,0)(λ), Γz T



zn


T(8V)(0,0)(λ) Sq | v 〉 = Sq T

(SOS)(λ) | v 〉

T(8V)(0,0)(λ) Sq | v 〉 = −Sq T

(SOS)(λ) | v 〉



〉,


t+〉




〉



Complete characterization of the periodic 8-vertex transfer matrix spectrumand eigenstates for N odd:

The spectrum of the periodic 8-vertex transfer matrix T(8V)(0,0)(λ) for N odd is

Σ(8V)(0,0) = Σ

(SOS)+ , (1)

where Σ(SOS)+ is the ‘+’ part of the antiperiodic SOS transfer matrix spectrum

Σ(SOS).

Each of the 2N−1 T(8V)(0,0)(λ)-eigenvalues t(λ) ∈ Σ

(8V)(0,0) = Σ

(SOS)+ is doubly

degenerated, with two linearly independent T(8V)(0,0)(λ)-eigenvectors given by

Sq |Ψ(SOS)t 〉 and Sq Γz |Ψ(SOS)

t 〉, (2)

where |Ψ(SOS)t 〉 denotes the T (SOS)

(λ)-eigenvector with eigenvalue t(λ).


Going further: from the discrete characterization of thespectrum to a functional T − Q equation

SOV characterization of the spectrum/eigenstates of the transfer matrix:

? eigenvalue t(λ) characterized by

its functional form (“elliptic polynomial” of a certain type)

the fact that it satisfies a discrete system of equations at the(shifted) inhomogeneity parameters ξn − ηhn, hn ∈ {0, 1}, i.e. that,for each n ∈ {1, . . . ,N}, there exists a non-zero vector

Q(n) ≡

q

(0)n

q(1)n

!s.t.

t(ξn−ηhn) q(hn)n = (−1)x+y+xya(ξn−ηhn) q(hn+1)

n + d(ξn−ηhn) q(hn−1)n , hn∈{0, 1}

? The corresponding eigenvector |Ψ(SOS)t 〉 is constructed in terms of Q(n)

Question: Does it exist, for each t(λ) ∈ Σ(SOS), an entire function Q(λ) s.t.

? for each n ∈ {1, . . . ,N}, Q(ξn − ηhn) = q(hn)n

? t(λ) Q(λ) = (−1)x+y+xya(λ) Q(λ− η) + d(λ) Q(λ+ η) ?

If yes, what is the functional form of Q(λ) ( Bethe equations ) ?Veronique TERRAS The quasi-periodic XYZ chain - 2015

From the discrete characterization of the spectrum to afunctional T − Q equation

Question: Does it exist, for each t(λ) ∈ Σ(SOS), an entire function Q(λ) s.t.

t(λ) Q(λ) = (−1)x+y+xya(λ) Q(λ− η) + d(λ) Q(λ+ η) (3)

and (Q(ξn),Q(ξn − η)) 6= (0, 0) ?If yes, what is the functional form of Q(λ) ( Bethe equations ) ?

Remark 1. The algebraic construction of the Q-operator (and the knowledge ofthe functional form of its eigenvalues) provides in principle a solution to thisproblem

Remark 2. From analyticity/periodicity arguments, one can guess thefunctional form of Q(λ).

Ansatz: Q(λ) = eαλNY

j=1

θX (λ− λj)

with θX (λ) =

8><>:θ1

`λ2

˛ω2

íf x = 0,

θ1(λ|2ω) if y = 0,

e i λ2 θ1

`λ2

˛ω´θ1

`λ+π+πω

2

˛ω´

if x = y.

+ restrictions on α andP

i λi .Is it possible to prove the completeness of this solution ?



Let t(λ) be an eigenvalue of the antiperiodic SOS transfer matrix T (λ).Then, if N is even, there exists a unique function Q(λ) of the form

Q(λ) =NY

j=1

θX(λ− λj), (4)

for some set of roots λ1, . . . , λN ∈ C, such that t(λ) and Q(λ) satisfy the T -Qfunctional equation

t(λ) Q(λ) = (−1)x+y+xya(λ) Q(λ− η) + d(λ) Q(λ+ η).

In (4), the notation θX(λ) stands for the function

θX (λ) =

8><>:θ1

`λ2

˛ω2

íf (x, y) = (0, 1),

θ1(λ|2ω) if (x, y) = (1, 0),

e i λ2 θ1

`λ2

˛ω´θ1

`λ+π+πω

2

˛ω´

if (x, y) = (1, 1).

(5)

Remark. In the XXZ limit (y = 1, ω → +i∞) we have shown the completenessof the solution

Q(λ) =NY

j=1

sin“λ− λj

2

”for N even or odd



Let t(λ) be an eigenvalue of the antiperiodic SOS transfer matrix T (λ).Then, if N is even, there exists a unique function Q(λ) of the form

Q(λ) =NY

j=1

θX(λ− λj), (4)

for some set of roots λ1, . . . , λN ∈ C, such that t(λ) and Q(λ) satisfy the T -Qfunctional equation

t(λ) Q(λ) = (−1)x+y+xya(λ) Q(λ− η) + d(λ) Q(λ+ η).

In (4), the notation θX(λ) stands for the function

θX (λ) =

8><>:θ1

`λ2

˛ω2

íf (x, y) = (0, 1),

θ1(λ|2ω) if (x, y) = (1, 0),

e i λ2 θ1

`λ2

˛ω´θ1

`λ+π+πω

2

˛ω´

if (x, y) = (1, 1).

(5)

Complete characterization of the (SOS and hence 8V) spectrum andeigenstates in terms of the solutions of the Bethe equations:

(−1)x+y+xya(λj)NY

k=1k 6=j

θX (λj − λk − η)

θX (λj − λk)+ d(λj)

NYk=1k 6=j

θX (λj − λk + η)

θX (λj − λk)= 0, 1 ≤ j ≤ N


Conclusion

the periodic 8-vertex/XYZ model with an even number of sites canbe solved by relating the periodic 8-vertex transfer matrix with theperiodic SOS transfer matrix

solution by Bethe ansatz (cf. Baxter’s work. . . )

the periodic 8-vertex/XYZ model with an odd number of sites, aswell as the twisted cases, can be solved by relating the (periodic ortwisted) 8-vertex transfer matrix with the antiperiodic SOS transfermatrix

solution by Separation of Variables

complete description of the transfer matrix spectrum and eigenstatesin terms of solutions of discrete equations evaluated at the inhomogeneityparameters of the model

it is possible to reformulate this description in terms of someparticular classes of solutions of a functional T -Q equation

description in terms of Bethe-type equations enabling one to studythe homogeneous/thermodynamic limit (completeness proven for N even)


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