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arXiv:1205.5702 · Journal of Physics A Mathematical and Theoretical

Finite temperature correlation functions from discrete functional equations

Britta Aufgebauer, Andreas Klümper

Published Aug 8, 2012• 19 citations
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Abstract

We present a new approach to the static finite temperature correlation functions of the Heisenberg chain based on functional equations. An inhomogeneous generalization of the n -site density operator is considered. The lattice path integral formulation with a finite but arbitrary Trotter number allows us to derive a set of discrete functional equations with respect to the spectral parameters. We show that these equations yield a unique characterization of the density operator. Our functional equations are a discrete version of the reduced q-Knizhnik–Zamolodchikov equations which played a central role in the study of the zero temperature case. As a natural result, and independent of the arguments given by Jimbo et al (2009 J. Phys. A: Math. Theor. 42 304018), we prove that the inhomogeneous finite temperature correlation functions have the same remarkable structure as for zero temperature: they are a sum of products of nearest-neighbour correlators.

DOI
10.1088/1751-8113/45/34/345203

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We present a new approach to the static finite temperature correlation functions of the Heisenberg chain based on functional equations. An inhomogeneous generalization of the n -site density operator is considered. The lattice path integral formulation with a finite but arbitrary Trotter number allows us to derive a set of discrete functional equations with respect to the spectral parameters. We show that these equations yield a unique characterization of the density operator. Our functional equations are a discrete version of the reduced q-Knizhnik–Zamolodchikov equations which played a central role in the study of the zero temperature case. As a natural result, and independent of the arguments given by Jimbo et al (2009 J. Phys. A: Math. Theor. 42 304018), we prove that the inhomogeneous finite temperature correlation functions have the same remarkable structure as for zero temperature: they are a sum of products of nearest-neighbour correlators.
We present a new approach to the static finite temperature correlation functions of the Heisenberg chain based on functional equations. An inhomogeneous generalization of the n-site density operator is considered. The lattice path integral formulation with a finite but arbitrary Trotter number allows to derive a set of discrete functional equations with respect to the spectral parameters. We show that these equations yield a unique characterisation of the density operator. Our functional equations are a discrete version of the reduced q-Knizhnik-Zamolodchikov equations which played a central role in the study of the zero temperature case. As a natural result, and independent of the arguments given by Jimbo, Miwa, and Smirnov (2009) we prove that the inhomogeneous finite temperature correlation functions have the same remarkable structure as for zero temperature: they are a sum of products of nearest-neighbor correlators. 1
Using the properties of the local Boltzmann weights of integrable interaction-round-a-face (IRF or face) models we express local operators in terms of generalized transfer matrices. This allows for the derivation of discrete functional equations for the reduced density matrices in inhomogeneous generalizations of these models. We apply these equations to study the density matrices for IRF models of various solid-on-solid type and quantum chains of non-Abelian \mathbold{su(2)_3} 𝐬 𝐮 ( 2 ) 3 or Fibonacci anyons. Similar as in the six vertex model we find that reduced density matrices for a sequence of consecutive sites can be ‘factorized’, i.e. expressed in terms of nearest-neighbour correlators with coefficients which are independent of the model parameters. Explicit expressions are provided for correlation functions on up to three neighbouring sites.
Using the properties of the local Boltzmann weights of integrable interaction-round-a-face (IRF or face) models we express local operators in terms of generalized transfer matrices. This allows for the derivation of discrete functional equations for the reduced density matrices in inhomogeneous generalizations of these models. We apply these equations to study the density matrices for IRF models of various solid-on-solid type and quantum chains of non-Abelian su(²)³ or Fibonacci anyons. Similar as in the six vertex model we find that reduced density matrices for a sequence of consecutive sites can be’factorized’, i.e. expressed in terms of nearest-neighbour correlators with coefficients which are independent of the model parameters. Explicit expressions are provided for correlation functions on up to three neighbouring sites.
We communicate results on correlation functions for the spin-1/2 Heisenberg chain in two particularly important cases: (a) for the infinite chain at arbitrary finite temperature T, and (b) for finite chains of arbitrary length L in the ground state. In both cases we present explicit formulas expressing the short-range correlators in a range of up to seven lattice sites in terms of a single function ω encoding the dependence of the correlators on T (L). These formulas allow us to obtain accurate numerical values for the correlators and derived quantities like the entanglement entropy. By calculating the low T (large L) asymptotics of ω we show that the asymptotics of the static correlation functions at any finite distance are T(2) (1/L(2)) terms. We obtain exact and explicit formulas for the coefficients for up to eight sites.