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SLIDE 1

A form factor series for dynamical correlation functions

f integrable lattice models at finite temperature

Frank G¨

hmann

Bergische Universit¨ at Wuppertal Fakult¨ at f¨ ur Mathematik und Naturwissenschaften

Annecy

11.9.2018

SLIDE 2

Introduction

Quantum StatMech

Quantum (spin) chain: HL ∈ End

Cd⊗L

Hamiltonian L length of chain d dimension of local Hilbert space xj = id⊗(j−1) ⊗x ⊗ id⊗(L−j), x ∈ End

Cd

local operator j ∈ {1,...,L} lattice sites

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 2 / 44

SLIDE 3

Introduction

Quantum StatMech

Quantum (spin) chain: HL ∈ End

Cd⊗L

Hamiltonian L length of chain d dimension of local Hilbert space xj = id⊗(j−1) ⊗x ⊗ id⊗(L−j), x ∈ End

Cd

local operator j ∈ {1,...,L} lattice sites Linear response theory (‘Kubo theory’) connects the response of a large quantum system to time-(= t)-dependent perturbations (= experiments) with dynamical correlation functions at finite temperature T

x1ym+1(t)T = lim

L→∞

tr1,...,L

e−HL/T x1 eitHLym+1e−iHLt

tr1,...,L

e−HL/T

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 2 / 44

SLIDE 4

Introduction

Motivation

Example: Large distance asymptotics of σz

1σz m+1(t)T=0 for the XXZ chain at

1 < ∆ and 0 ≤ h < hc1 [from Dugave, G, Kozlowski, Suzuki 16]

5 10 15 20 −0.005 0.005 0.01 t I2(m, t)

4 5 6 −0.002 0.002 0.004

Longitudinal spin-spin correlation func- tion of XXZ at T = 0 (static part sub- stracted) as a function of t for fixed m = 45 and ∆ = 2.375 (real part blue, imag- inary part red). Vertical lines separate different asymptotic regimes t = m/vc2 and t = m/vc1

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 3 / 44

SLIDE 5

Introduction

Outline of the talk

Goal: design (I) a general method for calculating dynamical correlation functions at finite temperature (in integrable lattice models of Yang-Baxter type)

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 4 / 44

SLIDE 6

Introduction

Outline of the talk

Goal: design (I) a general method for calculating dynamical correlation functions at finite temperature (in integrable lattice models of Yang-Baxter type) Based on lattice representation by [Sakai 07]

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 4 / 44

SLIDE 7

Introduction

Outline of the talk

Goal: design (I) a general method for calculating dynamical correlation functions at finite temperature (in integrable lattice models of Yang-Baxter type) Based on lattice representation by [Sakai 07] Input 1: combine this with the thermal form factor expansion introduced in [Dugave, G, Kozlowski 13]

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 4 / 44

SLIDE 8

Introduction

Outline of the talk

Goal: design (I) a general method for calculating dynamical correlation functions at finite temperature (in integrable lattice models of Yang-Baxter type) Based on lattice representation by [Sakai 07] Input 1: combine this with the thermal form factor expansion introduced in [Dugave, G, Kozlowski 13] Leads to form factor series of the same degree of complexity as in the static case. Only a single (not a double) sum over excited states is involved in series for two-point functions

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 4 / 44

SLIDE 9

Introduction

Outline of the talk

Goal: design (I) a general method for calculating dynamical correlation functions at finite temperature (in integrable lattice models of Yang-Baxter type) Based on lattice representation by [Sakai 07] Input 1: combine this with the thermal form factor expansion introduced in [Dugave, G, Kozlowski 13] Leads to form factor series of the same degree of complexity as in the static case. Only a single (not a double) sum over excited states is involved in series for two-point functions Input 2: for (II) XXZ model use off-shell auxiliary functions from NLIEs for excited states for combining ‘np-particle nh-hole’ contributions to form-factor series into multiple integrals (technique develloped for static case in [DGKS 15, 16])

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 4 / 44

SLIDE 10

Introduction

Outline of the talk

Goal: design (I) a general method for calculating dynamical correlation functions at finite temperature (in integrable lattice models of Yang-Baxter type) Based on lattice representation by [Sakai 07] Input 1: combine this with the thermal form factor expansion introduced in [Dugave, G, Kozlowski 13] Leads to form factor series of the same degree of complexity as in the static case. Only a single (not a double) sum over excited states is involved in series for two-point functions Input 2: for (II) XXZ model use off-shell auxiliary functions from NLIEs for excited states for combining ‘np-particle nh-hole’ contributions to form-factor series into multiple integrals (technique develloped for static case in [DGKS 15, 16]) Based on joint work with M. KARBACH, A. KL ¨

UMPER, K. K. KOZLOWSKI AND J. SUZUKI,

J. Stat. Mech.: Theor. Exp. (2017) 113106

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 4 / 44

SLIDE 11

Foundations Integrable lattice models

Integrable lattice models

Defined in terms of their R-matrices R : C2 → End

Cd ⊗Cd

which are solutions

f the Yang-Baxter equation

R12(λ,µ)R13(λ,ν)R23(µ,ν) = R23(µ,ν)R13(λ,ν)R12(λ,µ) We shall assume that the R-matrix has the following additional properties: regularity R(λ,λ) = P symmetry Rt(λ,µ) = R(λ,µ) unitarity R12(λ,µ)R21(µ,λ) = id Here the superscript t denotes transposition and P is the permutation matrix

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 5 / 44

SLIDE 12

Foundations Integrable lattice models

Row-to-row transfer matrix and Hamiltonian

With a given R-matrix, which has the above properties, we associate an integrable lattice model. We define a (‘row-to-row’) monodromy matrix T⊥,a(λ) = RaL(λ,0)...Ra1(λ,0)

n L lattice sites and the corresponding transfer matrix

t⊥(λ) = tra{T⊥,a(λ)} Then typically hR ∈ C exists, such that H0 = hR t′

⊥(0)t−1 ⊥ (0)

can be interpreted as a local lattice Hamiltonian. Locality is clear from H0 = hR

L

∑

j=1

∂λ(PR)j−1,j(λ,0)

λ=0

where periodic boundary conditions, (PR)0,1 = (PR)L,1, are understood

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 6 / 44

SLIDE 13

Foundations Integrable lattice models

Transfer matrix realization of statistical operator and time evolution operator

Need to realize the exponential of H0 in terms of transfer matrices. For this purpose t⊥(λ) = tra{T −1

⊥,a(λ)} = tra{R1a(0,λ)...RLa(0,λ)}

Then t⊥(0) = t−1

⊥ (0)

H0 = −hR t⊥(0)t′

⊥(0)

and t⊥

− hR

NT

t⊥

hR

NT

= id−2H0

NT +O

N−2

For every even N let

ρN,L(1/T) =

t⊥
− hR

NT

t⊥

hR

NT

N

2

Then lim

N→∞ρN,L(1/T) = e−H0/T

For finite N the product of transfer matrices ρN,L(1/T) is an approximation to the statistical operator e−H0/T , where T is the temperature. We shall call N the Trotter number and the limit N → ∞ the Trotter limit

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 7 / 44

SLIDE 14

Foundations Integrable lattice models

‘Solution of the quantum inverse problem’

For the evaluation of correlation functions we will have to express the action of a local operator x ∈ End(Cd) on the first site of our quantum chain in terms of monodromy and transfer matrix. For this purpose we shall employ the ‘solution of the quantum inverse problem’ formula [Kitanine, Maillet, Terras 99] x1 = t⊥(0)tra{xaT −1

⊥,a(0)} = lim ε→0t⊥(−ε)tra{xaT −1 ⊥,a(ε)}

Following [Sakai 07] we have introduced a regularization parameter ε. The reg- ularization is trivial for the row-to-row transfer matrix. It becomes important only later when we apply a variant of the above formula to the quantum transfer matrix introduced in the next section

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 8 / 44

SLIDE 15

Foundations Integrable lattice models

External fields

We include a simple class of external fields, fields which do not break integrability. Let ˆ

ϕ ∈ End(Cd) a local operator, let Θ(α) = eαˆ

ϕ such that

[R12(λ,µ),Θ1(α)Θ2(α)] = 0

Then

[t(λ)⊥,Θ1(α)...ΘL(α)] = [t⊥(λ),Θ1(α)...ΘL(α)] = 0

Define

ˆ Φ =

L

∑

j=1

ˆ ϕj

Then

[H0, ˆ Φ] = 0

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 9 / 44

SLIDE 16

Foundations Quantum transfer matrix

The quantum transfer matrix

Using Θ(α) we define the staggered, twisted and inhomogeneous monodromy matrix acting on ‘vertical spaces’ with site indices 1,...,2N + 2, Ta(λ|α)

= Θa(α)Rt1

2N+2,a(ν2N+2,λ)Ra,2N+1(λ,ν2N+1)...Rt1 2,a(ν2,λ)Ra,1(λ,ν1)

Here the superscript t1 denotes transposition with respect to the first space R is acting on, and ν1,...,ν2N+2 are 2N + 2 arbitrary complex ‘inhomogeneity para- meters’. The corresponding transfer matrix t(λ|α) = tra{Ta(λ|α)} is called the inhomogeneous quantum transfer matrix of the model

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 10 / 44

SLIDE 17

Foundations Quantum transfer matrix

Solution of quantum inverse problem for quantum transfer matrix

LEMMA: Solution of the quantum inverse problem for the inhomogeneous quantum transfer matrix. Let j ∈ {1,...,2N + 2} odd. Then for any x ∈ EndCd xj = t(ν1|α)t−1(ν2|α)...t−1(νj−1|α)

× tr{xT(νj|α)}t−1(νj|α)t(νj−1|α)...t−1(ν1|α)

this being valid whenever the t(νk|α) are invertible

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 11 / 44

SLIDE 18

Foundations Quantum transfer matrix

Solution of quantum inverse problem for quantum transfer matrix

LEMMA: Solution of the quantum inverse problem for the inhomogeneous quantum transfer matrix. Let j ∈ {1,...,2N + 2} odd. Then for any x ∈ EndCd xj = t(ν1|α)t−1(ν2|α)...t−1(νj−1|α)

× tr{xT(νj|α)}t−1(νj|α)t(νj−1|α)...t−1(ν1|α)

this being valid whenever the t(νk|α) are invertible REMARK: In general invertibility will require that the inhomogeneity parameters on

dd and even lattice site are mutually distinct. For XXZ this is also sufficient as long

as inhomogeneity parameters are close enough to the origin

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 11 / 44

SLIDE 19

Correlation functions

Definition two-point functions

We want to calculate correlation functions of integrable lattice models with Hamiltonian H = H0 −αˆ

Φ

We restrict ourselves to the dynamical two-point functions of two (ultra-) local operators x,y ∈ EndCd defined by

x1ym+1(t)T = lim

L→∞

tr1,...,L{e−H/T x1eitHym+1e−itH} tr1,...,L{e−H/T }

= lim

L→∞

tr1,...,L{e−(1/T+it)Hx1eitHym+1} tr1,...,L{e−H/T } Here we have denoted the spatial distance on the lattice by m and the time by t. The indices 1,...,L indicate that the traces are computed in (Cd)⊗L which is the space of states of the above Hamiltonian Goal is to express the right hand side of the above equation in terms of the quantum transfer matrix and the entries of the corresponding monodromy matrix, then to simplify the resulting expression using the spectral decomposition of the quantum transfer matrix

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 12 / 44

SLIDE 20

Correlation functions

Two-point functions as ratios of partition functions

Inserting ρN,L and eαˆ

Φ we obtain

x1ym+1(t)T = lim

L→∞ lim N→∞

tr1,...,L{ρN,L(1/T +it)e(1/T+it)αˆ

Φx1e−itαˆ ΦρN,L(−it)ym+1}

tr1,...,L{eαˆ

Φ/T ρN,L(1/T +it)ρN,L(−it)}

= lim

L→∞ lim N→∞

tr1,...,L{eαˆ

Φ/T ρN,L(1/T +it)eitαˆ ϕ1x1e−itαˆ ϕ1ρN,L(−it)ym+1}

tr1,...,L{eαˆ

Φ/T ρN,L(1/T +it)ρN,L(−it)}

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 13 / 44

SLIDE 21

Correlation functions

Two-point functions as ratios of partition functions

Inserting ρN,L and eαˆ

Φ we obtain

x1ym+1(t)T = lim

L→∞ lim N→∞

tr1,...,L{ρN,L(1/T +it)e(1/T+it)αˆ

Φx1e−itαˆ ΦρN,L(−it)ym+1}

tr1,...,L{eαˆ

Φ/T ρN,L(1/T +it)ρN,L(−it)}

= lim

L→∞ lim N→∞

tr1,...,L{eαˆ

Φ/T ρN,L(1/T +it)eitαˆ ϕ1x1e−itαˆ ϕ1ρN,L(−it)ym+1}

tr1,...,L{eαˆ

Φ/T ρN,L(1/T +it)ρN,L(−it)}

We assume that the adjoint action of eαˆ

ϕ can be diagonalized. This is rather natu-

ral, since the operator ˆ

ϕ is typically a Cartan element of a Lie algebra acting as a

local symmetry. Then, without loss of generality,

eitαˆ

ϕ1x1e−itαˆ ϕ1 = eitαs(x)x1

where s(x) is the eigenvalue corresponding to x

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 13 / 44

SLIDE 22

Correlation functions

Numerator

We first of all concentrate on the numerator. Using the ‘inversion formula’ for x1 we find tr1,...,L

eαˆ

Φ/T ρN,L(1/T +it)eitαˆ ϕ1x1e−itαˆ ϕ1ρN,L(−it)ym+1

= eitαs(x)

× lim

ε→0tr1,...,L

eαˆ

Φ/T ρN,L(1/T +it)t⊥(−ε)tra{xaT −1 ⊥,a(ε)}ρN,L(−it)ym+1

This can now be written it in terms of the quantum transfer matrix and its mono-

dromy matrix. The easiest way to proceed is to represent the right hand side graph- ically and re-express it in terms of column-to-column rather than row-to-row mon-

dromy matrices. The column-to-column monodromy matrix at hand is a special

case of the staggered monodromy matrix with inhomogeneities

ν2k−1 = −ν2k =      − tR

N

k = 1,..., N

2

ε

k = N

2 + 1 tR+hR/T N

k = N

2 + 2,...,N + 1

where tR = ihRt

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 14 / 44

SLIDE 23

Correlation functions

Change of perspective graphically

y 2N + 2 1 N x m + 1 1 L −ǫ ǫ eα ˆ

ϕ/T

−(tR + hR/T)/N (tR + hR/T)/N −(tR + hR/T)/N (tR + hR/T)/N tR/N −tR/N tR/N −tR/N

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 15 / 44

SLIDE 24

Correlation functions

A change of perspective

Then tr1,...,L

eαˆ

Φ/T ρN,L(1/T +it)t⊥(−ε)tra{xaT −1 ⊥,a(ε)}ρN,L(−it)ym+1

=tr1,...,2N+2
xN+1tm(0|κ)tr{yT(0|κ)}tL−m−1(0|κ)
=tr1,...,2N+2
t
−tR

N

κ
t−1tR

N

κ

N

2

tr{xT(ε|κ)}t−1(ε|κ)

×

t−1

−tR

N

κ
t

tR

N

κ

N

2

tm(0|κ)tr{yT(0|κ)}tL−m−1(0|κ)

where κ = α/T for short and the solution of the inverse problem is used in the

second equation. We reinsert this and take the limit L → ∞ first, then

x1ym+1(t)T = lim

N→∞ lim

ε→0eitαs(x)

Λ0

− tR

N

κ
Λ0

tR

N

κ
N

2

× Ψ0|X(ε|κ)t−1(ε|κ)

t−1

− tR

N

κ
t

tR

N

κ

N

2 tm(0|κ)Y(ε|κ)|Ψ0

Λ0(ε|κ)Λm

0 (0|κ)Ψ0|Ψ0

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 16 / 44

SLIDE 25

Correlation functions

Form factor series for thermal correlation functions

In order to be able to deal with the Trotter limit N → ∞ we insert a complete set of eigenstates |Ψn of t(λ|κ) with corresponding eigenvalues Λn(λ|κ). This brings us to

ur first important result.

THEOREM: With the definitions above the dynamical two-point functions of two local

perators x and y have the form-factor series expansion

x1ym+1(t)T = lim

N→∞ lim

ε→0eitαs(x)∑

n

Ψ0|X(ε|κ)|ΨnΨn|Y(ε|κ)|Ψ0 Λn(ε|κ)Ψ0|Ψ0Λ0(ε|κ)Ψn|Ψn × Λn tR

N

κ
Λ0
− tR

N

κ
Λ0

tR

N

κ
Λn
− tR

N

κ
N

2 Λn(0|κ)

Λ0(0|κ) m

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 17 / 44

SLIDE 26

Correlation functions

Form factor series for thermal correlation functions

In order to be able to deal with the Trotter limit N → ∞ we insert a complete set of eigenstates |Ψn of t(λ|κ) with corresponding eigenvalues Λn(λ|κ). This brings us to

ur first important result.

THEOREM: With the definitions above the dynamical two-point functions of two local

perators x and y have the form-factor series expansion

x1ym+1(t)T = lim

N→∞ lim

ε→0eitαs(x)∑

n

Ψ0|X(ε|κ)|ΨnΨn|Y(ε|κ)|Ψ0 Λn(ε|κ)Ψ0|Ψ0Λ0(ε|κ)Ψn|Ψn × Λn tR

N

κ
Λ0
− tR

N

κ
Λ0

tR

N

κ
Λn
− tR

N

κ
N

2 Λn(0|κ)

Λ0(0|κ) m

REMARK: Sending t → 0 we recover the known form-factor series expansion of the static correlation functions of integrable lattice models [Dugave, G, Kozlowski 13]

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 17 / 44

SLIDE 27

The XXZ chain as an example Hamiltonian and R-matrix

XXZ Hamiltonian and R-matrix

The Hamiltonian of the spin- 1

2 XXZ chain in a magnetic field of strength h is defined by

the local action of Pauli matrices σα, α = x,y,z, on a chain of spins on L lattice sites, HXXZ = J

L

∑

j=1

σx

j−1σx j +σy j−1σy j +∆

σz

j−1σz j − 1

− h

2

L

∑

j=1

σz

j

Here ∆ = (q + q−1)/2 and J > 0. In the following we restrict ourselves to q = e−iγ,

γ ∈ (0,π/2] implying that 0 ≤ ∆ < 1

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 18 / 44

SLIDE 28

The XXZ chain as an example Hamiltonian and R-matrix

XXZ Hamiltonian and R-matrix

The Hamiltonian of the spin- 1

2 XXZ chain in a magnetic field of strength h is defined by

the local action of Pauli matrices σα, α = x,y,z, on a chain of spins on L lattice sites, HXXZ = J

L

∑

j=1

σx

j−1σx j +σy j−1σy j +∆

σz

j−1σz j − 1

− h

2

L

∑

j=1

σz

j

Here ∆ = (q + q−1)/2 and J > 0. In the following we restrict ourselves to q = e−iγ,

γ ∈ (0,π/2] implying that 0 ≤ ∆ < 1

HXXZ is obtained from R(λ,µ) =

   

1 b(λ,µ) c(λ,µ) c(λ,µ) b(λ,µ) 1

    ,

b(λ,µ) =

sh(λ−µ) sh(λ−µ−iγ)

c(λ,µ) =

sh(−iγ) sh(λ−µ−iγ)

It has a U(1) symmetry with Θ(α) = qασz or ˆ

ϕ = −iγσz

The Hamiltonian in our general construction turns into HXXZ if hR = −2iJ sin(γ),

α = ih

2γ Then κ = ih/2γT and tR = 2J sin(γ)t

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 18 / 44

SLIDE 29

The XXZ chain as an example Algebraic Bethe Ansatz

Pseudo vacuum and vacuum expectation values

In order to make our form factor series work we need to know the eigenvalues and eigenvectors of the quantum transfer matrix. For the XXZ chain the eigenvectors can be constructed by means of the algebraic Bethe Ansatz. We consider the staggered inhomogeneous monodromy matrix and write it as a 2× 2 matrix in ‘auxiliary space’ a, Ta(λ|κ) =

A(λ)

B(λ) C(λ) D(λ)

a

This defines the operators A(λ),...,D(λ). The operators B(λ) generate the eigenstates

f the quantum transfer matrix t(λ|κ) = A(λ)+D(λ) by acting on a pseudo vacuum |0

defined by C(λ)|0 = 0. In our case the pseudo vacuum is

|0 =

1
⊗
1

⊗(N+1)

The pseudo vacuum is an eigenvector of the the operators A(λ), with eigenvalue a(λ), and D(λ), with eigenvalue d(λ), where a(λ) = qκ

N+1

∏

k=1

b(ν2k,λ), d(λ) = q−κ

N+1

∏

k=1

b(λ,ν2k−1)

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 19 / 44

SLIDE 30

The XXZ chain as an example Algebraic Bethe Ansatz

Bethe Ansatz equations and auxiliary function

In the context of the quantum transfer matrix formalism it has turned out to be useful [Kl¨ umper 93] to describe the Bethe Ansatz solution in terms of certain auxiliary

functions. For M = 0,...,2N + 2 define a family of functions

a

λ|{λk}M

k=1,κ

= d(λ)

a(λ)

M

∏

k=1

sh(λ−λk −iγ) sh(λ−λk +iγ) depending meromorphically on M complex parameters λj. The equations

a

λj|{λk}M

k=1,κ

= −1,

j = 1,...,M are called the Bethe Ansatz equations. Their solutions {λ(n)

j

}M

j=1 are sets of ‘Bethe

roots’. We have supplied a superscript ‘(n)’ to distinguish the different solutions at fixed M. With every set of Bethe roots we associate its corresponding auxiliary function

an(λ|κ) = a

λ|{λ(n)

k }M k=1,κ

Frank G¨
hmann

TFA to dynamical correlation functions 11.9.2018 20 / 44

SLIDE 31

The XXZ chain as an example Algebraic Bethe Ansatz

Eigenvalues and eigenstates of the QTM

Sets of Bethe roots {λ(n)

j

}M

j=1 parameterize the solutions of the eigenvalue problem

f the quantum transfer matrix. The eigenvalues can be written as

Λn(λ|κ) = a(λ)

M

∏

j=1

sh(λ−λ(n)

j

+iγ)

sh(λ−λ(n)

j

) + d(λ)

M

∏

j=1

sh(λ−λ(n)

j

−iγ)

sh(λ−λ(n)

j

)

The eigenvectors and their ‘duals’ take the form

|Ψn = B(λ(n)

1 )...B(λ(n) M )|0,

Ψn| =

C(λ(n)

1 )...C(λ(n) M )

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 21 / 44

SLIDE 32

The XXZ chain as an example The XX chain

XX finite Trotter number amplitudes and eigenvalue ratios

The XX is simple, because γ = π/2: Then all Bethe root patterns a known sufficiently well and all auxiliary functions degenerate into two functions

an(λ|κ) = (−1)sq−2κ

N+1

∏

k=1

th(λ−ν2k−1) th(λ−ν2k) where s = 0,1

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 22 / 44

SLIDE 33

The XXZ chain as an example The XX chain

XX finite Trotter number amplitudes and eigenvalue ratios

The XX is simple, because γ = π/2: Then all Bethe root patterns a known sufficiently well and all auxiliary functions degenerate into two functions

an(λ|κ) = (−1)sq−2κ

N+1

∏

k=1

th(λ−ν2k−1) th(λ−ν2k) where s = 0,1 For N sufficiently large there are precisely N + 1 roots of 1+a0 in the strip

S =

λ ∈ C
|Imλ| < π/4
. These are the ground state Bethe roots

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 22 / 44

SLIDE 34

The XXZ chain as an example The XX chain

XX finite Trotter number amplitudes and eigenvalue ratios

The XX is simple, because γ = π/2: Then all Bethe root patterns a known sufficiently well and all auxiliary functions degenerate into two functions

an(λ|κ) = (−1)sq−2κ

N+1

∏

k=1

th(λ−ν2k−1) th(λ−ν2k) where s = 0,1 For N sufficiently large there are precisely N + 1 roots of 1+a0 in the strip

S =

λ ∈ C
|Imλ| < π/4
. These are the ground state Bethe roots

The amplidutes for the longitudinal correlation functions σz

1σz m+1(t)T vanish

except for the one-particle one-hole excitations (s = 0, λh ∈ S, λp ∈ S+iπ/2) for which (with e(x) = 2/sh(2x))

Ψ0|A(ξ)|Ψn Ψ0|Ψ0Λn(ξ|κ) Ψn|A(ξ)|Ψ0 Ψn|ΨnΛ0(ξ|κ) = e(ξ−λh) a′

0(λh)

e(ξ−λp) a′

0(λp)

The corresponding eigenvalue ratios are

Λn(ξ|κ) Λ0(ξ|κ) = th(λh −ξ)

th(λp −ξ)

Frank G¨

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SLIDE 35

The XXZ chain as an example The XX chain

Summing the form factor series

Our theorem implies the form factor series representation

σz

1σz m+1(t)T − 4m2(T,h) =

lim

N→∞

ε→0 ∑ λh,λp

4e(λh)e(λp)

a′

0(λh|κ)a′ 0(λp|κ)

th(λh)

th(λp)

m

th(λh − tR/N)th(λp + tR/N) th(λh + tR/N)th(λp − tR/N)

N

2

The individual terms under the sum have N/2-fold poles at λh = −tR/N and at λp =

iπ/2 − tR/N. Fortunately, these can be canceled if we choose the auxiliary functions

appropriately. In our case

a0(λ|κ) = q−2κ th(λ−ε)

th(λ+ε)

th(λ+ tR/N)th(λ−(tR + hR/T)/N)

th(λ− tR/N)th(λ+(tR + hR/T)/N)

N

2

from which we can see that a0 has an N/2-fold zero at −tR/N and an N/2-fold pole at

λ = iπ/2− tR/N. Thus, 1+ 1/a0 has an N/2-fold pole at −tR/N, while 1+a0 has an

N/2-fold pole at λ = iπ/2− tR/N

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 23 / 44

SLIDE 36

The XXZ chain as an example The XX chain

Summing up

Setting a0 = 1/a0 it follows for m > 0 that

σz

1σz m+1(t)T − 4m2(T,h) =

lim

N→∞

ε→0

−

∂S

dλ πi e(λ)

−ith(λ)

m

1+a0(λ|κ)

th(λ− tR/N)

th(λ+ tR/N)

N

2

×

∂S+ iπ

2

dλ πi e(λ)

−ith(λ)

−m

1+a0(λ|κ)

th(λ+ tR/N)

th(λ− tR/N)

N

2

Here the Trotter limit and the limit ε → 0 can be taken:

σz

1σz m+1(t)T = 4m2(T,h)

−

∂S

dλ πi e(λ)

−ith(λ)

me−tRe(λ)

1+e

ε(λ) T

∂S+ iπ

2

dλ πi e(λ)

−ith(λ)

−metRe(λ)

1+e− ε(λ)

T

where

e(λ) =

2 sh(2λ) ,

ε(λ) = h − 2iJe(λ)

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 24 / 44

SLIDE 37

The XXZ chain as an example The XX chain

Substituting momenta for rapidities

This can be transformed into a more familiar form by employing the iπ-periodicity of the integrand in the second integral and by turning to momentum variables. The one-particle momentum is defined as p(λ) = −iln

−ith(λ)
where we understand the logarithm as its principal value, meaning that we provide cuts

in the complex plane from −iπ/2 to zero modulo iπ. The one-particle momentum is real

n the lines Imλ = ±π/4

p(λ) =

− π

2 + 2arctg

e−2Reλ

if Imλ = π/4

−πsign(Reλ)+ π

2 − 2arctg

e−2Reλ

if Imλ = −π/4 Hence the assignment λ → p maps

(−∞−iπ/4,+∞−iπ/4) − → [−π,−π/2]∪[π/2,π] (+∞+iπ/4,−∞+iπ/4) − → [−π/2,π/2]

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TFA to dynamical correlation functions 11.9.2018 25 / 44

SLIDE 38

The XXZ chain as an example The XX chain

Representation as an integral over momenta

Then

σz

1σz m+1(t)T = 4m2(T,h)+

π

−π

dp π ei(mp−tε0(p))

1+eε0(p)/T

π

−π

dp π e−i(mp−tε0(p))

1+e−ε0(p)/T

where we have introduced the energy function in momentum variables

ε0(p) = h − 4J cos(p)

For the sake of completeness let us also recall the expression

m(T,h) =

π

−π

dp

4π th

ε0(p)

2T

for the magnetization as a function of temperature and magnetic field here. Compare

classical papers [Niemeijer 67; Katsura, Horiguchi, Suzuki 70]

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 26 / 44

SLIDE 39

The XXZ chain as an example The XX chain

Representation as an integral over momenta

Then

σz

1σz m+1(t)T = 4m2(T,h)+

π

−π

dp π ei(mp−tε0(p))

1+eε0(p)/T

π

−π

dp π e−i(mp−tε0(p))

1+e−ε0(p)/T

where we have introduced the energy function in momentum variables

ε0(p) = h − 4J cos(p)

For the sake of completeness let us also recall the expression

m(T,h) =

π

−π

dp

4π th

ε0(p)

2T

for the magnetization as a function of temperature and magnetic field here. Compare

classical papers [Niemeijer 67; Katsura, Horiguchi, Suzuki 70] Summary: For XX longitudinal case our novel form factor series appears efficient!

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 26 / 44

SLIDE 40

The XXZ chain as an example The XX chain

Representation as an integral over momenta

Then

σz

1σz m+1(t)T = 4m2(T,h)+

π

−π

dp π ei(mp−tε0(p))

1+eε0(p)/T

π

−π

dp π e−i(mp−tε0(p))

1+e−ε0(p)/T

where we have introduced the energy function in momentum variables

ε0(p) = h − 4J cos(p)

For the sake of completeness let us also recall the expression

m(T,h) =

π

−π

dp

4π th

ε0(p)

2T

for the magnetization as a function of temperature and magnetic field here. Compare

classical papers [Niemeijer 67; Katsura, Horiguchi, Suzuki 70] Summary: For XX longitudinal case our novel form factor series appears efficient! Remark: Applying it to the transversal XX case we obtain a new form factor series representation in the Trotter limit

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 26 / 44

SLIDE 41

The XXZ chain as an example Nonlinear-integral-equation

Characterization of auxiliary functions by nonlinear integral equation

The thermal form factors and eigenvalue ratios are parameterized by the auxiliary func- tions an. For the an several alternative descriptions are available. The formally simplest

ne uses equivalence classes of contours Cn in order to classify the excitations. For

any given auxiliary function an there exists a contour Cn which encircles all the Bethe roots, but no other zeros of 1+an and no other poles of this function than those at ν2k, k = 1,...,N + 1. We shall assume that we can shape the contour Cn in such a way that λ − µ ± iγ remains outside for λ,µ ∈ Cn. For simplicity we also assume that Cn contains all Bethe roots of the dominant state and no additional pole or zero of 1 + a0 as compared to C0

γ i 2 γ i 2

Cn

λ

2

λh λ 1

h

λ 1

p

λ

p 2

Cn Then, for an excitation with M Bethe roots,

Cn

dµ

2πi

a′

n(µ|κ)

1+an(µ|κ) = M − N − 1 = −s ‘monodromy condition’

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 27 / 44

SLIDE 42

The XXZ chain as an example Nonlinear-integral-equation

Functions appearing in nonlinear integral equation

Kernel function and bare energy K(λ) = cth(λ+iγ)− cth(λ−iγ)

eN(λ) = T

hR

N+1

∑

k=1

ln

sh(λ−ν2k)

sh(λ−ν2k−1) sh(λ−ν2k−1 −iγ) sh(λ−ν2k −iγ)

Frank G¨
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TFA to dynamical correlation functions 11.9.2018 28 / 44

SLIDE 43

The XXZ chain as an example Nonlinear-integral-equation

Functions appearing in nonlinear integral equation

Kernel function and bare energy K(λ) = cth(λ+iγ)− cth(λ−iγ)

eN(λ) = T

hR

N+1

∑

k=1

ln

sh(λ−ν2k)

sh(λ−ν2k−1) sh(λ−ν2k−1 −iγ) sh(λ−ν2k −iγ)

Definition of ln(1 + an): We fix a point xn ∈ Cn and, for every λ ∈ Cn, define a contour

Cλ

xn starting at xn and running along Cn up to the point λ. This enables us to define

ln(1+an)(λ|κ) =

Cλ

xn

dµ a′

n(µ|κ)

1+an(µ|κ) which is holomorphic along the contour, except possibly at xn

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 28 / 44

SLIDE 44

The XXZ chain as an example Nonlinear-integral-equation

Functions appearing in nonlinear integral equation

Kernel function and bare energy K(λ) = cth(λ+iγ)− cth(λ−iγ)

eN(λ) = T

hR

N+1

∑

k=1

ln

sh(λ−ν2k)

sh(λ−ν2k−1) sh(λ−ν2k−1 −iγ) sh(λ−ν2k −iγ)

Definition of ln(1 + an): We fix a point xn ∈ Cn and, for every λ ∈ Cn, define a contour

Cλ

xn starting at xn and running along Cn up to the point λ. This enables us to define

ln(1+an)(λ|κ) =

Cλ

xn

dµ a′

n(µ|κ)

1+an(µ|κ) which is holomorphic along the contour, except possibly at xn Choosing xn and the contour Cn is such that we can send Rexn → −∞ we obtain the nonlinear integral equation ln

an(λ|κ)
= iπs + 2iγ(κ+ s)− hReN(λ)/T −
Cn

dµ

2πiK(λ−µ)ln(1+an)(µ|κ)

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 28 / 44

SLIDE 45

The XXZ chain as an example Nonlinear-integral-equation

Trotter limit

Numerical studies suggest that contour Cn associated with ‘nth excitation from below’ stabilizes in the Trotter limit N → ∞ which therefore only affects the term eN(λ). For our Trotter decomposition we also have to send ε to zero. Then lim

N→∞

ε→0

eN(λ) = e(λ) = cth(λ)− cth(λ−iγ)

which is the bare energy function. With this the nonlinear integral equation for the auxil- iary function of the dominant state in the Trotter limit alim takes its familiar form ln

alim

0 (λ|κ)

= 2iγκ− hRe(λ)/T −
C0

dµ

2πiK(λ−µ)ln(1+alim

0 )(µ|κ)

and similar for alim

n .

The function alim determines the thermodynamic properties [Kl¨ umper 92, 93] and the static temperature dependent correlation functions of the XXZ chain at all finite temperatures [G, Kl¨ umper, Seel 04] as it parameterizes the integral representations of dominant eigenvalue and reduced density matrix of the model

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 29 / 44

SLIDE 46

The XXZ chain as an example Nonlinear-integral-equation

Trotter limit

Numerical studies suggest that contour Cn associated with ‘nth excitation from below’ stabilizes in the Trotter limit N → ∞ which therefore only affects the term eN(λ). For our Trotter decomposition we also have to send ε to zero. Then lim

N→∞

ε→0

eN(λ) = e(λ) = cth(λ)− cth(λ−iγ)

which is the bare energy function. With this the nonlinear integral equation for the auxil- iary function of the dominant state in the Trotter limit alim takes its familiar form ln

alim

0 (λ|κ)

= 2iγκ− hRe(λ)/T −
C0

dµ

2πiK(λ−µ)ln(1+alim

0 )(µ|κ)

and similar for alim

n .

The function alim determines the thermodynamic properties [Kl¨ umper 92, 93] and the static temperature dependent correlation functions of the XXZ chain at all finite temperatures [G, Kl¨ umper, Seel 04] as it parameterizes the integral representations of dominant eigenvalue and reduced density matrix of the model The amazing fact, which we would like to emphasize and which was already observed by [Sakai 07], is that no dependence on time t has remained in the Trotter limit. Thus, many results obtained in static case [Dugave, G, Kozlowski, Suzuki 13-16] can be used

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 29 / 44

SLIDE 47

The XXZ chain as an example Eigenvalue ratios and amplidudes

Eigenvalue ratios

Let us recall the expressions for the eigenvalue ratios. Following [Jimbo, Miwa, Smirnov 09; Boos, G 09] we shall consider ratios of eigenvalues with different values of the magnetic field. We will use the function zn(λ|κ,κ′) = ln(1+a0)(λ|κ)− ln(1+an)(λ|κ′) 2πi in order to have more compact expressions. Then

ρn(λ|κ,κ′) = Λn(λ|κ′) Λ0(λ|κ) = qs+κ′−κ exp

−
Cn

dµe(µ−λ)zn(λ|κ,κ′)

The eigenvalue ratios appearing in the form-factor series are recovered by setting κ′ = κ

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 30 / 44

SLIDE 48

The XXZ chain as an example Eigenvalue ratios and amplidudes

Eigenvalue ratios

Let us recall the expressions for the eigenvalue ratios. Following [Jimbo, Miwa, Smirnov 09; Boos, G 09] we shall consider ratios of eigenvalues with different values of the magnetic field. We will use the function zn(λ|κ,κ′) = ln(1+a0)(λ|κ)− ln(1+an)(λ|κ′) 2πi in order to have more compact expressions. Then

ρn(λ|κ,κ′) = Λn(λ|κ′) Λ0(λ|κ) = qs+κ′−κ exp

−
Cn

dµe(µ−λ)zn(λ|κ,κ′)

The eigenvalue ratios appearing in the form-factor series are recovered by setting κ′ = κ

For the amplitudes in the form factor series [Dugave, G, Kozlowski 13] we shall stick with the example of the transverse correlation functions σ−

1 σ+ m+1(t)T for which

Ψ0|B(ξ|κ)|ΨnΨn|C(ξ|κ)|Ψ0 Λn(ξ|κ)Ψ0|Ψ0Λ0(ξ|κ)Ψn|Ψn = lim

κ′→κA−+

n

(ξ|κ,κ′)

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 30 / 44

SLIDE 49

The XXZ chain as an example Eigenvalue ratios and amplidudes

Amplitudes in transverse case

A−+

n

(ξ|κ,κ′) =

G−

+(ξ)G+ −(ξ)

(q1+κ′−κ − q−1−κ′+κ)(qκ′−κ − q−κ′+κ) × exp

Cn

dλ ln

ρn(λ|κ,κ′)
∂λzn(λ|κ,κ′)
×

detdm+,Cn

1−

K1−κ′+κ

detdm−,Cn
1−

K1+κ′−κ

detdm0,Cn
1−

K

detdm,Cn
1−

K

Here

G±

±(ξ) =

lim

Reλ→±∞

G±(λ,ξ) and G±(λ,ξ) is the solution of the linear integral equation G±(λ,ξ) = −cth(λ−ξ)+ qκ′−κ∓1ρ±1

n (ξ|κ,κ′)cth(λ−ξ+iγ)

+

Cn

dm±(µ)G±(µ,ξ)Kκ′−κ∓1(µ−λ)

with deformed kernel Kκ(λ) = q−κ cth(λ+iγ)− qκ cth(λ−iγ)

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 31 / 44

SLIDE 50

The XXZ chain as an example Eigenvalue ratios and amplidudes

Amplitudes in transverse case

The ‘measures’ dmε, ε = −,0,+, and dm are defined by

dm−(λ) = dλρ−1

n (λ|κ,κ′)

2πi(1+a0(λ|κ)) ,

dm+(λ) = dλρn(λ|κ,κ′)

2πi(1+an(λ|κ′))

dm(λ) = dλ

2πi(1+a0(λ|κ)) ,

dm0(λ) = dλ

2πi(1+an(λ|κ′)) The determinants above are Fredholm determinants of integral operators defined by the respective kernels and measures and by the integration contours Cn

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 32 / 44

SLIDE 51

The XXZ chain as an example Eigenvalue ratios and amplidudes

Amplitudes in transverse case

The ‘measures’ dmε, ε = −,0,+, and dm are defined by

dm−(λ) = dλρ−1

n (λ|κ,κ′)

2πi(1+a0(λ|κ)) ,

dm+(λ) = dλρn(λ|κ,κ′)

2πi(1+an(λ|κ′))

dm(λ) = dλ

2πi(1+a0(λ|κ)) ,

dm0(λ) = dλ

2πi(1+an(λ|κ′)) The determinants above are Fredholm determinants of integral operators defined by the respective kernels and measures and by the integration contours Cn CONJECTURE: [DGK 13] Structure is general: universal factor, determinant factor, and factorizing factor

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 32 / 44

SLIDE 52

The XXZ chain as an example Eigenvalue ratios and amplidudes

Amplitudes in transverse case

The ‘measures’ dmε, ε = −,0,+, and dm are defined by

dm−(λ) = dλρ−1

n (λ|κ,κ′)

2πi(1+a0(λ|κ)) ,

dm+(λ) = dλρn(λ|κ,κ′)

2πi(1+an(λ|κ′))

dm(λ) = dλ

2πi(1+a0(λ|κ)) ,

dm0(λ) = dλ

2πi(1+an(λ|κ′)) The determinants above are Fredholm determinants of integral operators defined by the respective kernels and measures and by the integration contours Cn CONJECTURE: [DGK 13] Structure is general: universal factor, determinant factor, and factorizing factor REMARK: Derivation of above formulae is based of excessive use of the formula

Cn

dλ f(λ)

2πi(1+an(λ|κ)) =

M

∑

j=1

f

λ(n)

j

a′

n(λ|κ)

which hold for any functions f holomorphic on and inside Cn. Sum on the rhs is a sum over all roots of a given solution to Bethe Ansatz equations

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 32 / 44

SLIDE 53

The XXZ chain as an example Summation of form factor series

Reminder

THEOREM ⇒ The dynamical transverse two-point functions of of the XXZ chain have the form-factor series expansion

σ−

1 σ+ m+1(t)

T = lim

N→∞ lim

ε→0eitαs(σ−)∑

n

Ψ0|B(ε|κ)|ΨnΨn|C(ε|κ)|Ψ0 Λn(ε|κ)Ψ0|Ψ0Λ0(ε|κ)Ψn|Ψn × Λn(0|κ) Λ0(0|κ) mΛn tR

N

κ
Λ0
− tR

N

κ
Λ0

tR

N

κ
Λn
− tR

N

κ
N

2

= lim

N→∞ lim

ε→0e−iht ∑

n

A−+

n

(ε|κ,κ)ρm

n (0|κ,κ)ρ

N 2

n (tR/N|κ,κ)ρ− N

2

n

(−tR/N|κ,κ)

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 33 / 44

SLIDE 54

The XXZ chain as an example Summation of form factor series

Reminder

THEOREM ⇒ The dynamical transverse two-point functions of of the XXZ chain have the form-factor series expansion

σ−

1 σ+ m+1(t)

T = lim

N→∞ lim

ε→0eitαs(σ−)∑

n

Ψ0|B(ε|κ)|ΨnΨn|C(ε|κ)|Ψ0 Λn(ε|κ)Ψ0|Ψ0Λ0(ε|κ)Ψn|Ψn × Λn(0|κ) Λ0(0|κ) mΛn tR

N

κ
Λ0
− tR

N

κ
Λ0

tR

N

κ
Λn
− tR

N

κ
N

2

= lim

N→∞ lim

ε→0e−iht ∑

n

A−+

n

(ε|κ,κ)ρm

n (0|κ,κ)ρ

N 2

n (tR/N|κ,κ)ρ− N

2

n

(−tR/N|κ,κ)

Here the sum over n is a sum over all solutions of the Bethe ansatz equations. How to deal with such sums? Usual transfer matrix and low-T limit in [Dugave, G, Kozlowski, Suzuki 15, 16]

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 33 / 44

SLIDE 55

The XXZ chain as an example Summation of form factor series

The particle-hole picture

Above description of form factors and eigenvalue ratios based on equivalence classes

f contours is inconvenient for summation. We deform contours Cn into reference con-

tours C0,s which brings about an explicit dependence on particle and hole parameters. After deformation some of the Bethe roots will be outside the new reference contour

C0,s, some zeros of 1 + an which are no Bethe roots will be inside. We call the former

particles, the latter holes and denote their numbers by np, nh, respectively There is a arbitrariness in the choice of the reference contour. A natural choice would be the contour that contains all Bethe roots and no other zeros of the auxiliary function pertaining to an eigenvalue of largest modulus in the pseudo-spin-s sector. Due to the monodromy condition this choice implies that the function ln(1 + an) has nontrivial monodromy along such contour unless s = 0. Our experience with the case s = 1 in the low-T limit and with ∆ = 0 suggests that we obtain simpler formulae if we deform the reference contour such as to include s more holes, if s is positive, or to exclude −s more particles, if s is negative. Then

C0,s

dµ

2πi

a′

n(µ|κ)

1+an(µ|κ) = nh − np − s = 0 connects the numbers of particles and holes defined with respect to C0,s with the pseudo-spin. In the following we shall assume for simplicity that this ’zero-monodromy condition’ is satisfied

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 34 / 44

SLIDE 56

The XXZ chain as an example Summation of form factor series

Eigenvalue ratios and form factors in the Trotter limit

All equations and expressions considered in the previous subsection can be rewritten with respect to the reference contour C0,s. The auxiliary functions an, the eigenvalue ratios ρn, and the amplitudes An then become explicit functions of the particle and hole roots. Instead of equivalence classes of contours Cn we then use sets of holes

{x(n)

j

}nh

j=1 and particles {y(n) k

}

np k=1 to classify the solutions, meaning that we have al-

together three equivalent parameterizations: by sets of Bethe roots, by equivalence classes of contours, or by sets of particles and holes associated with a reference con- tour C0,s

γ i 2 γ i 2

Cn

λ

2

λh λ 1

h

λ 1

p

λ

p 2

Cn Contraction of Cn brings about bare phase

θ(λ) = −iln

sh(iγ+λ)

sh(iγ−λ)

in NLIE

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 35 / 44

SLIDE 57

The XXZ chain as an example Summation of form factor series

Particle-hole form of NLIE

Fix a reference contour C0,s. With respect to this contour define multi-parametric func- tion a(λ|{u},{v},κ) as the solution of the nonlinear integral equation ln

a(λ|{u},{v},κ)
= iπs + 2iγκ− hReN(λ)/T +i

nh

∑

j=1

θ(λ− uj)−i

np

∑

k=1

θ(λ− vk) −

C0,s

dµ

2πi K(λ−µ)ln(1+a)(µ|{u},{v},κ) for which ln(1+a(·|{u},{v},κ) has trivial monodromy along C0,s Function a(λ|{u},{v},κ) depends holomorphically on {u} = {uj}nh

j=1 and {v} =

{vk}

np k=1, where uj take values inside C0,s and vk outside. Solutions {x} = {xj}nh j=1,

{y} = {yk}

np k=1 of ‘subsidiary conditions’ (higher-level Bethe Ansatz eqations?)

a(xj|{x},{y},κ) = a(yk|{x},{y},κ) = −1,

j = 1,...,nh, k = 1,...,np define sets of hole and particle roots which are in one-to-one correspondence with solu- tions {λ(n)

j

}M

j=1 of the Bethe Ansatz equations and with the contours Cn. Thus, we may

label them by the same superscript ‘(n)’ implying

an(λ|κ) = a(λ|{x(n)},{y(n)},κ)

Frank G¨

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TFA to dynamical correlation functions 11.9.2018 36 / 44

SLIDE 58

The XXZ chain as an example Summation of form factor series

A hint toward summation

Consider the determinant detdm0,Cn

1 −

K

in the denominator of the expression for

the amplitude (appears universally!) LEMMA: A Jacobian can be factored out from this term, whose structure suggests to use multiple-residue calculus for the summation over the excitations with a fixed number of particles and holes det

dm0,Cn

1−

K

=

det

dm0,C0,s

1−

K

nh

∏

j=1

1 a′

n(x(n) j

|κ) np

∏

j=1

1 a′

n(y(n) j

|κ)

× det
∂uk a(uj|{u},{v},κ)

∂vk a(uj|{u},{v},κ) ∂uk a(vj|{u},{v},κ) ∂vk a(vj|{u},{v},κ)

{u}={x(n)}

{v}={y(n)}

Here the products over reciprocals of a′

n will be canceled by corresponding terms origi-

nating from the ‘universal factor’

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 37 / 44

SLIDE 59

The XXZ chain as an example Summation of form factor series

Local multiple residue

The above determinant is exactly what is needed to transform a sum over solutions

f the subsidiary conditions into a multiple-contour integral over ‘particle and hole vari-

ables’ uj and vj. It may be interpreted as the Jacobian ∂(z,w)/∂(u,v) of a transforma- tion Cnh+np → Cnh+np, (u,v) → (z,w), where zj(u,v) = 1+a(uj|{u},{v},κ), j = 1,...,nh wk(u,v) = 1+a(vk|{u},{v},κ), k = 1,...,np This transformation maps solutions to the subsidiary conditions to the origin in Cnh+np,

(x(n),y(n)) → (z,w) = (0,0)

We shall assume that the map is invertible in the neighbourhood of (x(n),y(n)), viz. that the Jacobian

∂(z,w) ∂(u,v) (n) = ∂(z,w) ∂(u,v)

(u,v)=(x(n),y(n))

is non-vanishing

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 38 / 44

SLIDE 60

The XXZ chain as an example Summation of form factor series

Local multiple residue

Let D(n)

ε,η =

(u,v) ∈ Cnh+np
|zj| < ε,|wj| < ε,
(x(n),y(n))−(u,v)
< η
where η is sufficiently small so that D(n)

ε,η is included in the domain where the map is

invertible. Then, for any function f(u,v) which is holomorphic in all uj and vk, we obtain

the local residue

∂D(n)

ε,η

dunh (2πi)nh dvnp (2πi)np

f(u,v)

∏nh

j=1 zj(u,v)

∏

np k=1 wj(u,v)

= f(x(n),y(n)) ∂(z,w)

∂(u,v)

(n)

Straightening Cn everywhere summands in in form facor series take the form F−+({x(n)}|{y(n)})

∂(z,w) ∂(u,v) (n)

f local multi-dimensional residues, where

F−+({x(n)}|{y(n)}) =

A−+(ξ|{x(n)}|{y(n)})ρm

n (0|κ,κ)ρ

N 2

n (tR/N|κ,κ)ρ− N

2

n

(−tR/N|κ,κ)

and where A−+(ξ|{x(n)}|{y(n)}) is the ‘amplitude density’

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 39 / 44

SLIDE 61

The XXZ chain as an example Summation of form factor series

Local multiple residue

We now replace the particle and hole parameters {y(n)} and {x(n)} by complex vari- ables {u} and {v} which we do not require to satisfy the subsidiary conditions. This means that an(λ|κ) is replaced by a(λ|{u},{v},κ) everywhere, implying that we obtain z(λ|{u},{v},κ) = ln(1+a0)(λ|κ)− ln(1+a)(λ|{u},{v},κ) 2πi instead of zn(λ|κ,κ) and

ρ(λ|{u},{v},κ) = nh

∏

j=1

sh(λ− uj) sh(λ− uj +iγ)

np

∏

k=1

sh(λ− vk +iγ) sh(λ− vk)

× exp
−
C0,s

dµe(µ−λ)z(µ|{u},{v},κ)

instead of ρn(λ|κ,κ). Consequentially, A−+(ξ|{x(n)}|{y(n)}) is replaced by a function

A−+(ξ|{u}|{v}) and F−+({x(n)}|{y(n)}) by a function F−+({u}|{v})

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 40 / 44

SLIDE 62

The XXZ chain as an example Summation of form factor series

Summation at fixed Trotter number

Taking properly into account the singularities of F−+({u}|{v}), assuming, in particular, m > 0 and taking into account that np + 1 = nh we obtain the following representation for the transverse two-point functions

σ−

1 σ+ m+1(t)T = lim N→∞

ε→0 ∞

∑

ℓ=1

(−1)ℓe−iht ℓ!(ℓ− 1)!

C0,1

duℓ (2πi)ℓ

C0,1

dvℓ−1 (2πi)ℓ−1 A−+(ε|{u}|{v}) × ρm(ε|{u},{v},κ)ρN/2(tR/N|{u},{v},κ)ρ−N/2(−tR/N|{u},{v},κ)

∏ℓ

j=1

1+a
uj|{u},{v},κ
∏ℓ−1

k=1

1+a
vk|{u},{v},κ
The contour C0,1 encloses all particle roots. When calculating the integrals over the

extended contours C0,1, C0,1 we will obtain each local residue with multiplicity ℓ!(ℓ−1)! due to the symmetry of the functions in the denominator under the integral, which is why we divided each summand by this factor

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 41 / 44

SLIDE 63

The XXZ chain as an example Summation of form factor series

Trotter limit of individual terms

After rewriting the form-factor series as a sum over multiple integrals we may finally take the Trotter limit. For this purpose we introduce the function E(λ) = ln

sh(λ)

sh(λ−iγ)

and remark that

lim

N→∞

ε→0

ρm(ε|{u},{v},κ)ρN/2(tR/N|{u},{v},κ)ρ−N/2(−tR/N|{u},{v},κ) = exp nh

∑

j=1

mE(uj)− tRe(uj)
−

np

∑

j=1

mE(vj)− tRe(vj)
−
C0,s

dµ zlim(µ|{u},{v},κ)

me(µ)− tRe′(µ)
which follows from the explicit form of ρ and where the superscript ‘lim’ refers to the

Trotter limit

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 42 / 44

SLIDE 64

The XXZ chain as an example Summation of form factor series

Form factor series in the Trotter limit

Then we end up with the thermal form-factor series representation

σ−

1 σ+ m+1(t)T =

∞

∑

n=1

(−1)n

n!(n − 1)!

C0,1

dun (2πi)n

C0,1

dvn−1 (2πi)n−1 × n

∏

j=1

emE(uj)−tRe(uj)

1+alim uj|{u},{v},κ

n−1

∏

j=1

e−mE(vj)+tRe(vj)

1+alim vj|{u},{v},κ

×A−+

lim (0|{u},{v})e−iht−

C0,1 dµ zlim(µ|{u},{v},κ)
me(µ)−tRe′(µ)
for the transverse correlation functions of the XXZ chain. In this formula alim denotes

the Trotter limit of the function a which is obtained by replacing eN with e in the nonlinear integral equation. Similarly zlim and A−+

lim are obtained from z and A−+ by replacing a

with alim A similar form factor series representation can be also derived for the longitudinal cor- relation functions. In our derivation we used the implicit assumption that the reference contour C0,1 and the contour C0,1 can be chosen independently of the excitation. In the low-temperature limit [DGK 13, DGK 14, DGKS 16] and for ∆ = 0 (see below) we know that this is possible. The general case will need further study

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 43 / 44

SLIDE 65

Summary and discussion

Discussion

How to proceed?

1

Consider transversal correlation functions of the XX chain, calculate the large-time, large-distance asymptotics and the high- and low-T asymptotics, compare with known results (✔)

2

General XXZ in the low-T limit (✔)

3

General XXZ for T → ∞ (non-trivial in the dynamical case!) (✔)

4

XXZ for generic T at long times, large distances

5

Start an exploration into the numerical summation of the series at finite Trotter number (✔) A most interesting question will be if there are cases in which the leading particle- hole contribution determines the long-distance large-time asymptotic behaviour of the correlation functions, and what will be possible otherwise

Frank G¨

hmann

TFA to dynamical correlation functions 11.9.2018 44 / 44