発表ファイル数理物理・物性基礎論セミナー Goehmann

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Thermal form factors and form factor series

for correlation functions of the Heisenberg-Ising chain

Frank Göhmann

Bergische Universität Wuppertal

Tokyo 20.5.2017

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XXZ chain

Hamiltonian and ground state phase diagram

More than a decade of steady progress in our understanding of correlation functions of integrable lattice models, chief example XXZ chain

H=J

L

∑

j₌₋L+1

σ^x_j₋₁σ^x_j + σ^y_j₋₁σ^y_j + ∆ σ^z_j₋₁σ^z_j ₋1₋^h 2

L

∑

j₌₋L+1

σ^z_j

∆ = (q+q⁻¹)/2

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XXZ chain

Hamiltonian and ground state phase diagram

H=J

L

∑

j₌₋L+1

L

∑

j₌₋L+1

σ^z_j

∆ = (q+q⁻¹)/2

0 2 4 6 8 10 12

-2 -1 0 1 2 3 4 5

h / J

∆ ferromagnetic massive

antiferromagnetic massive antiferromagnetic critical

Phase boundaries (Yang & Yang 66) hu/J=4(∆ +1)

h_ℓ/J=2(q⁻¹₋q)ϑ²₄(0,q)

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XXZ chain

Hamiltonian and ground state phase diagram

H=J

L

∑

j₌₋L+1

L

∑

j₌₋L+1

σ^z_j

∆ = (q+q⁻¹)/2

0 2 4 6 8 10 12

-2 -1 0 1 2 3 4 5

h / J

Phase boundaries (Yang & Yang 66) hu/J=4(∆ +1)

h_ℓ/J=2(q⁻¹₋q)ϑ²₄(0,q) Physical content to calculate

h^AiT,h= lim

L_→∞

Tr_{e⁻^H^/^TA_} Tr_{e⁻^H^/^T_} In particular e.g. A= σ^z_j or A= σ^z₁σ^z_m(t)

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Quantum transfer matrix approach to correlation functions

XXZ-chain is integrable with R-matrix

R(x,y) =







1 0 0 0

0 b(x,y) c(x,y) 0 0 c(x,y) b(x,y) 0

0 0 0 1





 ,

b(x,y) =_sin^sin₍_y⁽₋^y_x⁻_+iγ)^x⁾ c(x,y) =_sin₍^sin_y^(iγ)

−^x^+iγ)

In this parameterization q= e^−γ

R-matrix can be used to definee⁻^H^/^T. For this purpose let Tj(x_{|κ) =}q^κσ^z^jR_jN x,^iβ_NR_N^t¹

−^1j ⁻ iβ N^,^x

. . .R_j2 x,^iβ_NR_¯_1j^t¹ ₋^iβ_N,x

N_∈2^Nis the ‘Trotter number’, indices^¯j= ¯1, . . . ,N refer to N auxiliary sites. β = −^{2J sh}^(γ)/^{T ,}κ = −^h^/²^γT are rescaled inverse temperature and magnetic field

Defining

ρN,L=Tr¯₁_..._N{^T−^L⁺1⁽⁰|κ)...^TL(0_|κ)} it is easy to see thate⁻^H^/^T=limN_→∞ρN,L

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Quantum transfer matrix approach to correlation functions

We callρN,La finite Trotter number approximant to the statistical operator. Using ρ_N_,_Lwe can calculate approximations to thermal expectation values which become exact in the limit N_{→ ∞}. In particular, the expectation value of any product of local operators^O⁽^j⁾_∈End^C², j=1, . . . ,m+1, m_{∈ N}, acting on m+1 consecutive sites of the infinite chain, is approximated by [G, KLÜMPER, SEEL04]

O⁽₁¹⁾_{. . . O}⁽^m⁺¹⁾

m+1

N⁼_L^lim_→∞

Tr₋L+1...L

ρN,L^O⁽₁¹⁾. . . O⁽_m^m₊⁺₁¹⁾ Tr₋L+1...L{ρN,L}

=^hκ|^Tr^{O

(1)_T₍₀_|κ)}..._Tr_{O(m+1)_T₍₀_|κ)}|κi

hκ|κiΛ^m₀⁺¹⁽⁰|κ)

whereΛ₀(0_|κ)is the unique eigenvalue of largest modulus of the quantum transfer matrix t_{(λ|κ) =}Tr T_(λ|κ)atλ =0, and where_{|κi = |}0_{; κi}is the corresponding eigenvector. We callΛ0(0_|κ)the dominant eigenvalue and_|κithe dominant eigenstate. All other states will be called ‘excited states’. Below we shall be dealing with sequences of excited states and their eigenvalues which will be denoted_|n_{; κi} andΛn(λ|κ), respectively.

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Two-point correlation functions

An important class of correlation functions areα-twisted two-point functions with finite Trotter number approximant

X1q^{α ∑}^m^j⁼²^σ^z^jYm+1_N=^hκ|^Tr^{^XT⁽⁰^|κ)}^Tr^{^q

ασ^z_T₍₀_|κ)}m₋1_Tr_{_YT₍₀_|κ)}|κi

hκ|κiΛ^m0⁺¹⁽⁰|κ) where X and Y are local operators.

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Two-point correlation functions

ασ^z_T₍₀_|κ)}m₋1_Tr_{_YT₍₀_|κ)}|κi

Expanding in a basis of eigenstates of theα-twisted quantum transfer matrix t_{(ξ|κ + α)} we obtain the ‘form factor expansion’

X₁q^{α ∑}^m^j⁼²^σ^z^jYm+1_N⁼

_∑

n

hκ|^Tr{^XT(ξ|κ)}|ⁿ^{; κ}^′i Λ_n_(ξ|κ^′_)hκ|κi

hⁿ^{, κ}^′|^Tr{^YT(ξ|κ)}|κi Λ₀_(ξ|κ)hn, κ^′_|n; κ^′_i

Λn(ξ|κ^′⁾ Λ₀_(ξ|κ)

m

whereκ^′= κ + α. Sending_{α →}0, N_{→ ∞}and_{ξ →}0 we obtain the two-point functions h^X1Ym+1iT,h.

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Two-point correlation functions

ασ^z_T₍₀_|κ)}m₋1_Tr_{_YT₍₀_|κ)}|κi

Expanding in a basis of eigenstates of theα-twisted quantum transfer matrix t_{(ξ|κ + α)} we obtain the ‘form factor expansion’

X₁q^{α ∑}^m^j⁼²^σ^z^jYm+1_N⁼

_∑

n

hκ|^Tr{^XT(ξ|κ)}|ⁿ^{; κ}^′i Λ_n_(ξ|κ^′_)hκ|κi

hⁿ^{, κ}^′|^Tr{^YT(ξ|κ)}|κi Λ₀_(ξ|κ)hn, κ^′_|n; κ^′_i

Λn(ξ|κ^′⁾ Λ₀_(ξ|κ)

m

whereκ^′= κ + α. Sending_{α →}0, N_{→ ∞}and_{ξ →}0 we obtain the two-point functions h^X1Ym+1iT,h.

In [DUGAVE, G, KOZLOWSKI13] we have called this a thermal form factor expansion. It has the following appealing features

no multiple integrals (at first sight)

has already the form of a large-distance asymptotic series as_|Λ_n/Λ0_{| <}1 for T>0 Difficulty: All excited states have to be summed over!

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Eigenvalue ratios

For finite N the eigenvaluesΛn(x_|κ)and eigenstates_|n_{; κi}of the quantum transfer matrix are parameterized by sets_{x_j⁽ⁿ⁾_}_j^M₌₁, M=N/2₋s, of Bethe roots. These are defined with the aid of an auxiliary function

a _x_{_x_k_}_k^M₌₁₌_q⁻²^κ

_{sin x}₊iγ 2⁻^iβN

sin x+³₂îγ+îβ_N sin x+îγ₂+îβ_Nsin x₋îγ₂₋îβ_N

^N₂ _M

∏

k=1

sin(x₋xk− iγ) sin(x₋xk+ iγ) as the solutions of the Bethe Ansatz equations

a _x⁽ⁿ⁾

j

{^xk⁽ⁿ⁾}^Mk=1= −¹^, ^j⁼¹^{, . . . ,}^M (∗)

We write^a_n(x|κ) = a x_{ x_k⁽ⁿ⁾}^M_k₌₁^if{x_k⁽ⁿ⁾}^M_k₌₁satisfies (_∗).

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Eigenvalue ratios

For finite N the eigenvaluesΛn(x_|κ)and eigenstates_|n_{; κi}of the quantum transfer matrix are parameterized by sets_{x_j⁽ⁿ⁾_}_j^M₌₁, M=N/2₋s, of Bethe roots. These are defined with the aid of an auxiliary function

a _x_{_x_k_}_k^M₌₁₌_q⁻²^κ

_{sin x}₊iγ 2⁻^iβN

sin x+³₂îγ+îβ_N sin x+îγ₂+îβ_Nsin x₋îγ₂₋îβ_N

^N₂ _M

∏

k=1

sin(x₋xk− iγ) sin(x₋xk+ iγ) as the solutions of the Bethe Ansatz equations

a _x⁽ⁿ⁾

j

{^xk⁽ⁿ⁾}^Mk=1= −¹^, ^j⁼¹^{, . . . ,}^M (∗)

We write^a_n(x|κ) = a x_{ x_k⁽ⁿ⁾}^M_k₌₁^if{x_k⁽ⁿ⁾}^M_k₌₁satisfies (_∗).

Any auxiliary function^an(·|κ)associated with a set of Bethe roots satisfies a nonlinear integral equation [KLÜMPER92, 93] allowing one to to obtain the eigenvalue ratios

ρ_n(x_{|α) =}^Λⁿ⁽^x^{+ iγ/}²^|κ

′₎

Λ0(x+ iγ/2_|κ) in the Trotter limit

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Task list

Calculation of the thermal form factor series consists of three major steps Step 1: Analyse the spectral problem of the quantum transfer matrix Step 2: Calculate the amplitudes in the Trotter limit

Step 3: Sum the form factor series

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Task list

We have been working on these problems jointly with MAXIMEDUGAVE, KAROLK KOZLOWSKI(since 2012) and J SUZUKI(since 2014)

DGK 2013: J. Stat. Mech. P07010^✔ DGK 2014a: SIGMA 10 043

DGK 2014b: J. Stat. Mech. P04012 DGKS 2015a: J. Stat. Mech. P05037 DGKS 2015b: J. Phys. A 48 334001^✔ DGKS 2016a: J. Phys. A 49 07LT01 DGKS 2016b: J. Phys. A 49 394001^✔

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Task list

We have been working on these problems jointly with MAXIMEDUGAVE, KAROLK KOZLOWSKI(since 2012) and J SUZUKI(since 2014)

DGK 2013: J. Stat. Mech. P07010^✔ DGK 2014a: SIGMA 10 043

DGK 2014b: J. Stat. Mech. P04012 DGKS 2015a: J. Stat. Mech. P05037 DGKS 2015b: J. Phys. A 48 334001^✔ DGKS 2016a: J. Phys. A 49 07LT01 DGKS 2016b: J. Phys. A 49 394001^✔

Summation of critical form factors in low-T limit [DGK 13, 14] relied on the same formula as obtained by the Lyon group for the usual transfer matrix [KITANINE, KOZLOWSKI, MAILLET, SLAVNOV, TERRAS2011] (but analysis of the amplitudes different from [KKMST 09, 11])

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Structure of amplitudes in form factor expansion

Amplitudes in the Trotter limit

Here we consider the amplitudes

A^xyn (ξ|α) =^hκ|^Tr_Λ^{^XT^(ξ|κ)}|ⁿ^{; κ}^′ⁱ

n(ξ|κ^′)hκ|κi

hⁿ^{, κ}^′|^Tr{^YT(ξ|κ)}|κi Λ₀_(ξ|κ)hn, κ^′_|n; κ^′_i where

X= σ^zif x=z, X= σ^±if x_{= ±,} X=q^ασ^z if x= α , X=id if x=1 and similarly for Y and y.

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Amplitudes in the Trotter limit

Here we consider the amplitudes

A^xyn (ξ|α) =^hκ|^Tr_Λ^{^XT^(ξ|κ)}|ⁿ^{; κ}^′ⁱ

n(ξ|κ^′)hκ|κi

hⁿ^{, κ}^′|^Tr{^YT(ξ|κ)}|κi Λ₀_(ξ|κ)hn, κ^′_|n; κ^′_i where

X= σ^zif x=z, X= σ^±if x_{= ±,} X=q^ασ^z if x= α , X=id if x=1 and similarly for Y and y.

We have obtained explicit expressions for A^zz_n = lim

α→0N^lim_→∞^A zz

n ⁽⁰|α), ^A⁻⁺n ⁼ ^lim α→0N^lim_→∞^A

−+n ⁽⁰|α) In the longitudinal case we can use the generating function

A^α_n¹(0_{|α) =}^hκ|ⁿ^{; κ}^′^ihⁿ^{, κ}^′^|κi hκ|κihⁿ^{, κ}^′|ⁿ^{; κ}^′i since

A^zz_n = lim

N_→∞

1 2 ^ρ

1/2 n − ρ⁻n¹^/²

2

∂²_γαA^α_n¹(0_|α) α=0

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General structure appropriate for taking Trotter limit

In [DGK 13] we considered A^α_n¹_(ξ|α)and A⁻⁺n (ξ|α)for finite Trotter number and in the Trotter limit. In both cases the amplitudes consist of three factors

A^xyn (ξ|α) =^Un,s(α)Fn^xy(ξ|α)^Dn^xy(α)

the universal part Un,s(α), the determinant part Dn^xy(α)and the factorizing part Fn^xy(ξ|α)^.

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General structure appropriate for taking Trotter limit

Conjecture: This structure holds in general, and the factorization of F_n^xy_(ξ|α)is related to the ‘hidden Fermionic structure’ of [BOOS, JIMBO, MIWA, SMIRNOV AND

TAKEYAMA2006-10]

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General structure appropriate for taking Trotter limit

Conjecture: This structure holds in general, and the factorization of F_n^xy_(ξ|α)is related to the ‘hidden Fermionic structure’ of [BOOS, JIMBO, MIWA, SMIRNOV AND

TAKEYAMA2006-10]

Derivation based on

Scalar product formula [Slavnov 89]

‘Cauchy extraction’ [Izergin, Kitanine, Maillet, Terras 99] Factorization of multiple integrals [Boos, G, Klümper, Suzuki 06]

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General structure appropriate for taking the Trotter limit

The universal part Un,s(α)does not depend on the details of the operators X , Y , but only on the spin.

Un,s(α) = ^∏

N/2 j=1^ρⁿ⁽^x

(n) j |α)

∏^N_j₌^/₁²⁻^sρ_n(y_j⁽ⁿ⁾_|α)

in terms of Bethe roots_{x_j⁽ⁿ⁾_}^N_j₌^/₁²of the dominant state and_{y_j⁽ⁿ⁾_}^N_j₌^/₁²⁻^sof an excited state of spin s

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General structure appropriate for taking the Trotter limit

The universal part Un,s(α)does not depend on the details of the operators X , Y , but only on the spin.

Un,s(α) = ^∏

N/2 j=1^ρⁿ⁽^x

(n) j |α)

∏^N_j₌^/₁²⁻^sρ_n(y_j⁽ⁿ⁾_|α)

in terms of Bethe roots_{x_j⁽ⁿ⁾_}^N_j₌^/₁²of the dominant state and_{y_j⁽ⁿ⁾_}^N_j₌^/₁²⁻^sof an excited state of spin s

In the longitudinal case the factorizing part is simply F_n^α¹(ξ|α) =¹ In the transversal case the factorizing part is of the form

F_n⁻⁺_{(ξ|α) =} ^G +−^(ξ)^G

+

−^(ξ)

(q^α−¹₋q¹^−α)(q^α₋q^−α)

where the functions in the numerator are determined by linear integral equations [DGK 13]

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General structure appropriate for taking the Trotter limit

The determinant part consists of four determinants

D_n^xy(α) =

det_N_/₂ⁿδ^j_k+^ρ_a⁻ⁿ_′¹⁽^x^j^|α)

0⁽^x^j|κ)^U x₍_x

j,xk)^o det_N_/₂ⁿδ^j_k+_a_′ ¹

0⁽^x^j|κ)^K⁽^x^j⁻^x^k⁾

o ^det^N^/²⁻^s n

δ^j_k+_a^ρ_′ⁿ⁽^y^j^|α)

n(yj_|κ^′)^U y₍_y

j,yk)^o det_N_/₂₋_sⁿδ^j_k+_a_′ ¹

n(yj_|κ^′)^K⁽^y^j⁻^y^k⁾

o

Here the primes inâ^′₀andâ^′_ndenote the derivative with respect to the first argument,â₀is the auxiliary function of the dominant state. The kernel functions in the denominator are defined by K(x) =2πiK0(x), where

Kα(x) = ¹

2πi ^q^−α^ctg⁽^x^{− iγ) −}^q

α_ctg₍_x_{+ iγ)} The kernels in the numerator depend on the operators X , Y

U^±₍_x_,_y_{) =}₂_πi_K_α±₁₍_x₋_y₎ while

U^α₍_x_,_y_{) =}₂_πi_K_α₍_x₋_y_{) + i}_q^−α_{− i}_q^α U¹₍_x_,_y_{) =}₂_πi_K_α₍_x₋_y_{) − i}_q^−α_{+ i}_q^α

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Trotter limit in contour formulation

Contour and eigenvalue ratios ρ

n

The eigenvalue ratiosρn(λ|α)as integrals over the auxiliary functions: ρn(λ|α) =^q^α+^s^exp

_Z

C_n

dµ

2πi^e^{(µ − λ)}^ln

1+ an(µ|κ^′⁾ 1+ a0(µ|κ)

whereλis located inside the contour^Cnandeis the ‘bare energy’ e(λ) = ctg (λ) − ctg(λ − iγ)

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Contour and eigenvalue ratios ρ

n

The eigenvalue ratiosρn(λ|α)as integrals over the auxiliary functions: ρn(λ|α) =^q^α+^s^exp

_Z

C_n

dµ

2πi^e^{(µ − λ)}^ln

1+ an(µ|κ^′⁾ 1+ a0(µ|κ)

whereλis located inside the contour^Cnandeis the ‘bare energy’ e(λ) = ctg (λ) − ctg(λ − iγ)

The Contour^Cn

−iγ/2 0

λ^p

λ^h

C_n

−^π₂ ^π₂

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Generating function

Trotter limit of the generating function

Nlim_→∞^A α1

n ⁽⁰|α) =^exp

− Z

C_n

dλ

2πi^ln ^ρⁿ^(λ|α)

∂_λln

1+ a_n_(λ|κ^′) 1+ a0(λ|κ)

×^det^d^m

α +^,Cⁿ

1_{− b}^U^αdetdm^α₋,Cn

1_{− b}^U¹ det_d_m^α

0^,Cⁿ

1_{− b}Kdet_d_m_,C_n1_{− b}K^. for the amplitudes in the Trotter limit [DGK 13]

Here we have introduced the measures

dm₋^α(λ) = ^{dλ ρ}⁻

n1(λ|α) 2πi(1+ a₀_(λ|κ))^, ^d^m

α +^{(λ) =}

dλ ρn(λ|α) 2πi(1+ an(λ|κ^′⁾⁾ dm(λ) = ^dλ

2πi(1+ a0(λ|κ))^, ^d^m

α0^{(λ) =}

dλ 2πi(1+ an(λ|κ^′⁾⁾

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Transversal case

We obtain the following formula in the Trotter limit [DGK 13]

Nlim_→∞^A

−+n (ξ|α) = ^G

−+^(ξ)^G +

−^(ξ)

(q¹^+α₋q⁻¹^−α)(q^α₋q^−α)

×^exp

− Z

C_n

dλ

2πi ^ln ^ρⁿ^(λ|α)

∂_λln

1+ a_n_(λ|κ^′) 1+ a0(λ|κ)

×^det^d^m

α +^,Cn

1_{− b}^U⁺det_d_m^α

−^,Cⁿ

1_{− b}^U⁻ det_d_m^α

0^,Cⁿ

1_{− b}Kdet_d_m_,C_n1_{− b}K Here, for s_{= ±},

G^±_s(ξ) = lim

Re_λ→±∞^G^s^{(λ, ξ)} and Gs(λ, ξ)is the solution of the linear integral equation

Gs(λ, ξ) = −^cth(λ − ξ) +^q^α−^s^ρ^sn(ξ|α)^cth(λ − ξ − η) +

Z C_n^d^m

α

s^(µ)^G^s^{(µ, ξ)}^Kα−^s(µ − λ)

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Preliminary summary

Summary

Amplitudes in form factor series have simple expressions in the Trotter limit depending on

auxiliary functionsa_n kernel function K_α eigenvalue ratiosρn generalized G function integration contours^C_n

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Preliminary summary

Summary

Amplitudes in form factor series have simple expressions in the Trotter limit depending on

auxiliary functionsa_n kernel function K_α eigenvalue ratiosρn generalized G function integration contours^C_n

A few degenerate amplitudes determine the asymptotics of the static correlation functions at finite T

hX1Ym+1iT⁼

_∑

n

A^XY_n exp

−_ξ^m_XY

n

, ¹

ξ^XY_n ⁼^ln

_Λ

0⁽⁰⁾

Λn(0)

Typically

1

|ξ1|^< 1

|ξ2|^{< . . .}

Amplitudes and correlation length must be calculated numerically by solving NLIEs.

Not much is known about full spectrum for finite T . In general cross-overs occur in the lower part [FABRICIUS, KLÜMPER, MCCOY99]

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Low temperatures

Summation

Under certain circumstances, however, we have to sum over infinitely many contributions

in the low temperature limit of the static correlation functions if we want the full correlation functions at all distances in the dynamical case

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Low temperatures

Summation

Then the closed contour formulation with contours^Cnbecomes

inappropriate. The contours have to be straightened and the resulting (bulky) expressions depend explicitly on particle and hole parameters_{x_j⁽ⁿ⁾_},_{y_k⁽ⁿ⁾_}

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Low temperatures

Summation

Then the closed contour formulation with contours^Cnbecomes

inappropriate. The contours have to be straightened and the resulting (bulky) expressions depend explicitly on particle and hole parameters_{x_j⁽ⁿ⁾_},_{y_k⁽ⁿ⁾_}

0 2 4 6 8 10 12

-2 -1 0 1 2 3 4 5

h / J

Static correlations for T_→0 massless case: infinitely manyξn→ ∞

massive antiferromagnetic case: infinitely manyξn→ ξmax(h)^✔

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Form factor series in the antiferromagnetic massive regime, low T

The antiferromagnetic massive regime _{∆ >} 1

This regime is determined by∆ >1,_|h_{| <}h_ℓ

Corner transfer matrix and q-vertex operator formalisms apply, and expressions for the form factors of the usual transfer matrix have been obtained by representation theoretic means [JIMBO, MIKI, MIWA, NAKAYASHIKI92]

Form factors of the usual transfer matrix were also obtained within the ‘Algebraic Bethe Ansatz method’ [DGKS 15a] (_→ABA method and non-real Bethe roots)

Both approaches support a ‘spinon picture’ for the elementary excitations

Both approaches yield form factor series for the longitudinal correlation functions that are of the form

hσ^z1^σ^zm+1i = (−¹⁾^m ⁽^q

2_;_q2₎4

(−^q²^;^q²⁾⁴⁺_n

^∑

_∈N

k=0,1

(−¹⁾^km (2n)!

Z ^π

2

−^π2

d²ⁿu (2π)²ⁿ^e

2πim∑²ⁿj=1^p⁽^u^j⁾_F^zz_({_u_i_}²ⁿ

i=1|^k⁾

with expressions for the amplitude densities^F^zz_({ui}²ⁿi=1|^k⁾which are of manifestly different form, but are written as multiple residues in both cases

We have re-analysed the problem within the thermal form factor approach sending T_→0 [DGKS 15b, 16b]

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Low-temperature spectrum of correlation length

Functions that determine the physical properties of the XXZ chain for T=0+: dressed momentum p, dressed energyεand dressed phaseϕ In the antiferromagnetic massive regime we can express these functions explicitly in terms of known special functions.

p(x) =¹ 4⁺

x 2π⁺

1 2πi^ln

ϑ4(x+ iγ/2,q²) ϑ₄(x_{− iγ/}2,q²)

ε(x) =^h 2⁻

4JK sh(γ)

π ^dn

2Kx π

^k

whereϑ₄is a Jacobian theta function and dn denotes the Jacobian elliptic dn-function. Note that the dressed energy depends explicitly on the magnetic field h

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Low-temperature spectrum of correlation length

Functions that determine the physical properties of the XXZ chain for T=0+: dressed momentum p, dressed energyεand dressed phaseϕ In the antiferromagnetic massive regime we can express these functions explicitly in terms of known special functions.

p(x) =¹ 4⁺

x 2π⁺

1 2πi^ln

ϑ4(x+ iγ/2,q²) ϑ₄(x_{− iγ/}2,q²)

ε(x) =^h 2⁻

4JK sh(γ)

π ^dn

2Kx π

^k

whereϑ₄is a Jacobian theta function and dn denotes the Jacobian elliptic dn-function. Note that the dressed energy depends explicitly on the magnetic field h

These are special cases of functions that can be expressed in terms of (infinite) q-multi factorials which, for_|qj| <^{1 and a}∈ C, are defined as

(a;q₁, . . . ,qp) =

∞

∏

n1,...,np=0

(1₋aqⁿ₁¹. . .qⁿ_p^p)

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Low-temperature spectrum of correlation length

It turns out that we only need certain combinations which have a limit for q_→1. These are the q-Gamma and the q-Barnes functions

Γq(x) = (1₋q)¹⁻^x ⁽^q^;^q⁾ (q^x;q)

Gq(x) = (1₋q)⁻¹²⁽¹⁻^x⁾⁽²⁻^x⁾(q;q)^x⁻¹⁽^q

x_;_q_,_q₎

(q;q,q)

These functions together with their functional equations are used to obtain an expression for the universal part of the amplitudes in the low-T limit

(36)

Low-temperature spectrum of correlation length

It turns out that we only need certain combinations which have a limit for q_→1. These are the q-Gamma and the q-Barnes functions

Γq(x) = (1₋q)¹⁻^x ⁽^q^;^q⁾ (q^x;q)

Gq(x) = (1₋q)⁻¹²⁽¹⁻^x⁾⁽²⁻^x⁾(q;q)^x⁻¹⁽^q

x_;_q_,_q₎

(q;q,q)

These functions together with their functional equations are used to obtain an expression for the universal part of the amplitudes in the low-T limit The q-Gamma function is needed to define the dressed phase

ϕ(x1,x2) = i^π 2⁺^x¹²

+ln

(Γ_q4 1+ⁱ^x₂¹²_γ Γ_q4 ¹₂₋ⁱ^x₂¹²_γ

Γ_q4 1₋ⁱ^x₂¹²_γΓ_q4 ¹₂+ⁱ^x₂¹²_γ )

where x₁₂=x₁₋x₂,_|Imx₂_{| < γ}

(37)

Low-temperature spectrum of correlation length

In [DGKS 15b] we have analyzed the solutions of the Bethe equations for the quantum transfer matrix in the massive antiferromagnetic regime∆ >1, 0_≤h<hℓfor h>0 and T_→0

Conjecture [DGKS 15b]: at low enough temperatures all excitations of the quantum transfer matrix can be parameterized by an even number of complex parameters located inside the strip_|Imx_{| < γ/}2.

Parameters in the upper half plane ‘particles’ yi, i=1, . . . ,np; parameters in the lower half plane ‘holes’ xj, j=1, . . . ,nh.

nh−ⁿp=2s

s_{∈ Z}is the conserved pseudo spin of the quantum transfer matrix, s=0 longitudinal case, s=1 transversal case

(38)

Low-temperature spectrum of correlation length

Up to corrections of the order T^∞, the particles and holes are determined by the higher-level Bethe Ansatz equations

ε(yj^{) =}²^πiT ℓ_j+F(yj⁾

, j=1, . . . ,np

ε(xj) =2πiT mj+F(xj), j=1, . . . ,nh

whereℓj,mj∈ Zand where we assume theℓjand mjto be mutually distinct. F is the ‘shift function’ defined by

2πiF(x) = iπk+ αγ +

np

ℓ=

∑

1

ϕ(x,yℓ) −

nh

ℓ=

∑

1

ϕ(x,xℓ) .

Hereαis an auxiliary twist related to the magnetic field, which serves as a regularization parameter and will be set equal to zero at the end of the calculation. The parameter k_{∈ {}0,1_}distinguishes between two sectors of excitations of the quantum transfer matrix corresponding to staggered and non-staggered contributions to the form factor series below

(39)

Low-temperature spectrum of correlation length

In the limit T_→0+at finite npand nhthe higher-level Bethe Ansatz equations decouple,iπℓjT andiπmjT turn into independent continuous variables, and the particles and holes become free parameters on the curves

B_±₌_x_{∈ C}_Reε(_x_{) =}₀_{, −π/}₂_{≤ Re}_x_{≤ π/}₂_,₀_{< ±Im}_x_{< γ}

-1.5 -1.0 -0.5 ^0.5 ^1.0 ^1.5

Re_λ

-0.6 -0.4 -0.2 0.2 0.4 0.6 Im_λ

Reε(x) =0 for various values of the magnetic field.

∆ =1.7, h_ℓ/J=0.76. Val- ues of magnetic field de- crease proceeding from in- ner to outer curve: h/h_ℓ= 1.34,1,2/3,1/3,0.

In the critical regime h_ℓ<h<hu=4J(1+ ∆)the curves are closed. In this regimeεis defined as a solution of a linear integral equation

(40)

Low-temperature spectrum of correlation length

At low enough temperatures all excitations are parameterized by solutions of the higher-level Bethe Ansatz equations. Hence, we write

ρ = ρ {^xⁱ}ⁿ_i₌^h₁, {^yj}ⁿ_j₌^p₁|^k. instead ofρn. Then the eigenvalue ratios at finite magnetic field are

ρ {^xⁱ}ⁿi=^h1, {^yj}jⁿ=^p1|^k= (−¹⁾^k^exp

2πi^h

np

∑

j=1

p(yj) −

nh

∑

j=1

p(xj)ⁱ

= (−1)^k

np

j

∏

=1

ϑ₁(yj− iγ/²^,^q²⁾ ϑ₄(yj− iγ/²^,^q²⁾

nh

j

∏

=1

ϑ₄(xj− iγ/²^,^q²⁾ ϑ₁(xj− iγ/²^,^q²⁾

this being valid up to multiplicative corrections of the order 1+ O(T^∞). Note that the value of the magnetic field h enters here through the particle and hole parameters

(41)

Low-temperature spectrum of correlation length

In the special case when there are neither particles nor holes, np=nh=0, the eigenvalue ratios reduce to_{ρ( /0, /0|}k_{) = (−}1)^k. The value k=0 corresponds to the case when the dominant state eigenvalue is divided by itself, while k=1 corresponds to an eigenvalue of an excited state which is almost degenerate in absolute value with the dominant state, meaning that up to sign the two eigenvalues differ only by a factor of 1+ O(T^∞)

-1.5 -1.0 -0.5 ^0.5 ^1.0 ^1.5 ^{Re x}

-1.0 -0.8 -0.6 -0.4 -0.2 Im x

Bethe roots of dominant state depicted as the intersections of the curves Reε(x) =0 andImε(x) =nπT for T/J=0.01, n_{= ±}1_{, ±}3, . . . , ±^11, h/hℓ⁼2/3,∆ =1.7, hℓ^/J=0.76.

(42)

Low-temperature spectrum of correlation length

Using the properties of the dressed momentum function p in the complex plane we conclude that for all other states, for which npor nhis non-zero,_{|ρ| <}1 For small finite temperature the eigenvalue ratios form a sequence of discrete values corresponding to a discrete spectrum of correlation lengths. The largest correlation length for s=0 corresponds to a single particle-hole excitation with np=nh=1 andℓ₁_{= −}m₁=1

(43)

Low-temperature spectrum of correlation length

Using the properties of the dressed momentum function p in the complex plane we conclude that for all other states, for which npor nhis non-zero,_{|ρ| <}1 For small finite temperature the eigenvalue ratios form a sequence of discrete values corresponding to a discrete spectrum of correlation lengths. The largest correlation length for s=0 corresponds to a single particle-hole excitation with np=nh=1 andℓ₁_{= −}m₁=1

As T_→0+the eigenvalue ratio corresponding to the largest correlation length converges to

ρmax(h) = lim

T_→0+^max

ρ({^x},{^y}|⁰⁾

=

"s 1 k²⁻

1 k²⁻¹

h hℓ

2

− s₁

k²⁻¹

1₋

h hℓ

2#²

where k=k(q)is the elliptic modulus. For h=0 this simplifies to

ρmax(0) =¹⁻

√1₋k² 1+^√1₋k² ⁼^k⁽^q

2₎

compare [JOHNSON, KRINSKY, MCCOY73]

発表ファイル 数理物理・物性基礎論セミナー Goehmann

Thermal form factors and form factor series

for correlation functions of the Heisenberg-Ising chain

Hamiltonian and ground state phase diagram

∑

∑

Hamiltonian and ground state phase diagram

∑

∑

Hamiltonian and ground state phase diagram

∑

∑

Quantum transfer matrix approach to correlation functions

Quantum transfer matrix approach to correlation functions

Two-point correlation functions

Two-point correlation functions

∑

Two-point correlation functions

∑

Eigenvalue ratios

∏

Eigenvalue ratios

∏

Task list

Task list

Task list

Amplitudes in the Trotter limit

Amplitudes in the Trotter limit

General structure appropriate for taking Trotter limit

General structure appropriate for taking Trotter limit

General structure appropriate for taking Trotter limit

General structure appropriate for taking the Trotter limit

General structure appropriate for taking the Trotter limit

General structure appropriate for taking the Trotter limit

Contour and eigenvalue ratios ρ

Contour and eigenvalue ratios ρ

Generating function

Transversal case

Summary

Summary

∑

Summation

Summation

Summation

The antiferromagnetic massive regime ∆ > 1

∑

Low-temperature spectrum of correlation length

Low-temperature spectrum of correlation length

∏

Low-temperature spectrum of correlation length

Low-temperature spectrum of correlation length

Low-temperature spectrum of correlation length

Low-temperature spectrum of correlation length

∑

∑

Low-temperature spectrum of correlation length

Low-temperature spectrum of correlation length

∑

∑

∏

∏

Low-temperature spectrum of correlation length

Low-temperature spectrum of correlation length

Low-temperature spectrum of correlation length

発表ファイル数理物理・物性基礎論セミナー Goehmann

_∑

_∑

_∑

The antiferromagnetic massive regime _{∆ >} 1

^∑