A form factor series for dynamical correlation functions of integrable lattice models at finite temperature Frank G¨ ohmann Bergische Universit¨ at Wuppertal Fakult¨ at f¨ ur Mathematik und Naturwissenschaften Annecy 11.9.2018
Introduction Quantum StatMech Quantum (spin) chain: � C d � ⊗ L H L ∈ End Hamiltonian L length of chain d dimension of local Hilbert space � C d � x j = id ⊗ ( j − 1 ) ⊗ x ⊗ id ⊗ ( L − j ) , x ∈ End local operator j ∈ { 1 ,..., L } lattice sites Frank G¨ ohmann TFA to dynamical correlation functions 11.9.2018 2 / 44
Introduction Quantum StatMech Quantum (spin) chain: � C d � ⊗ L H L ∈ End Hamiltonian L length of chain d dimension of local Hilbert space � C d � x j = id ⊗ ( j − 1 ) ⊗ x ⊗ id ⊗ ( L − j ) , x ∈ End 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 � e − H L / T x 1 e i tH L y m + 1 e − i H L t � tr 1 ,..., L � x 1 y m + 1 ( t ) � T = lim � e − H L / T � L → ∞ tr 1 ,..., L Frank G¨ ohmann TFA to dynamical correlation functions 11.9.2018 2 / 44
Introduction Motivation Example: Large distance asymptotics of � σ z 1 σ z m + 1 ( t ) � T = 0 for the XXZ chain at 1 < ∆ and 0 ≤ h < h c 1 [from Dugave, G, Kozlowski, Suzuki 16] 0 . 004 0 . 01 0 . 002 Longitudinal spin-spin correlation func- 0 tion of XXZ at T = 0 (static part sub- − 0 . 002 0 . 005 I 2 ( m, t ) stracted) as a function of t for fixed m = 4 5 6 45 and ∆ = 2 . 375 (real part blue, imag- 0 inary part red). Vertical lines separate different asymptotic regimes t = m / v c 2 − 0 . 005 and t = m / v c 1 0 5 10 15 20 t Frank G¨ ohmann TFA to dynamical correlation functions 11.9.2018 3 / 44
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¨ ohmann TFA to dynamical correlation functions 11.9.2018 4 / 44
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¨ ohmann TFA to dynamical correlation functions 11.9.2018 4 / 44
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¨ ohmann TFA to dynamical correlation functions 11.9.2018 4 / 44
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¨ ohmann TFA to dynamical correlation functions 11.9.2018 4 / 44
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 ‘ n p -particle n h -hole’ contributions to form-factor series into multiple integrals (technique develloped for static case in [DGKS 15, 16]) Frank G¨ ohmann TFA to dynamical correlation functions 11.9.2018 4 / 44
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 ‘ n p -particle n h -hole’ contributions to form-factor series into multiple integrals (technique develloped for static case in [DGKS 15, 16]) Based on joint work with M. K ARBACH , A. K L ¨ UMPER , K. K. K OZLOWSKI AND J. S UZUKI , J. Stat. Mech.: Theor. Exp. (2017) 113106 Frank G¨ ohmann TFA to dynamical correlation functions 11.9.2018 4 / 44
Foundations Integrable lattice models Integrable lattice models � C d ⊗ C d � Defined in terms of their R -matrices R : C 2 → End which are solutions of the Yang-Baxter equation R 12 ( λ , µ ) R 13 ( λ , ν ) R 23 ( µ , ν ) = R 23 ( µ , ν ) R 13 ( λ , ν ) R 12 ( λ , µ ) We shall assume that the R -matrix has the following additional properties: R ( λ , λ ) = P regularity R t ( λ , µ ) = R ( λ , µ ) symmetry R 12 ( λ , µ ) R 21 ( µ , λ ) = id unitarity Here the superscript t denotes transposition and P is the permutation matrix Frank G¨ ohmann TFA to dynamical correlation functions 11.9.2018 5 / 44
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 ) = R aL ( λ , 0 ) ... R a 1 ( λ , 0 ) on L lattice sites and the corresponding transfer matrix t ⊥ ( λ ) = tr a { T ⊥ , a ( λ ) } Then typically h R ∈ C exists, such that H 0 = h R t ′ ⊥ ( 0 ) t − 1 ⊥ ( 0 ) can be interpreted as a local lattice Hamiltonian. Locality is clear from L � � ∑ H 0 = h R ∂ λ ( PR ) j − 1 , j ( λ , 0 ) λ = 0 j = 1 where periodic boundary conditions, ( PR ) 0 , 1 = ( PR ) L , 1 , are understood Frank G¨ ohmann TFA to dynamical correlation functions 11.9.2018 6 / 44
Foundations Integrable lattice models Transfer matrix realization of statistical operator and time evolution operator Need to realize the exponential of H 0 in terms of transfer matrices. For this purpose t ⊥ ( λ ) = tr a { T − 1 ⊥ , a ( λ ) } = tr a { R 1 a ( 0 , λ ) ... R La ( 0 , λ ) } Then H 0 = − h R t ⊥ ( 0 ) t ′ t ⊥ ( 0 ) = t − 1 ⊥ ( 0 ) ⊥ ( 0 ) and � � � h R � � N − 2 � − h R = id − 2 H 0 NT + O t ⊥ t ⊥ NT NT For every even N let � �� N � � � h R − h R 2 ρ N , L ( 1 / T ) = t ⊥ t ⊥ NT NT Then N → ∞ ρ N , L ( 1 / T ) = e − H 0 / T lim For finite N the product of transfer matrices ρ N , L ( 1 / T ) is an approximation to the statistical operator e − H 0 / T , where T is the temperature. We shall call N the Trotter number and the limit N → ∞ the Trotter limit Frank G¨ ohmann TFA to dynamical correlation functions 11.9.2018 7 / 44
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 ( C d ) 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] x 1 = t ⊥ ( 0 ) tr a { x a T − 1 ε → 0 t ⊥ ( − ε ) tr a { x a T − 1 ⊥ , a ( 0 ) } = lim ⊥ , 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¨ ohmann TFA to dynamical correlation functions 11.9.2018 8 / 44
Foundations Integrable lattice models External fields We include a simple class of external fields, fields which do not break integrability. ϕ such that ϕ ∈ End ( C d ) a local operator, let Θ( α ) = e α ˆ Let ˆ [ R 12 ( λ , µ ) , Θ 1 ( α )Θ 2 ( α )] = 0 Then [ t ( λ ) ⊥ , Θ 1 ( α ) ... Θ L ( α )] = [ t ⊥ ( λ ) , Θ 1 ( α ) ... Θ L ( α )] = 0 Define L ˆ ∑ Φ = ˆ ϕ j j = 1 Then [ H 0 , ˆ Φ] = 0 Frank G¨ ohmann TFA to dynamical correlation functions 11.9.2018 9 / 44
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