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arXiv:2303.15252

Factorization of density matrices in the critical RSOS models

Daniel Westerfeld, Maxime Großpietsch, Hannes Kakuschke, Holger Frahm

Published Jan 1, 2023• 0 citations
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Abstract

We study reduced density matrices of the integrable critical RSOS model in a particular topological sector containing the ground state. Similar as in the spin-$1/2$ Heisenberg model it has been observed that correlation functions of this model on short segments can be `factorized': they are completely determined by a single nearest-neighbour two-point function $ω$ and a set of structure functions. While $ω$ captures the dependence on the system size and the state of the system the structure functions can be expressed in terms of the possible operators on the segment, in the present case representations of the Temperley-Lieb algebra $\text{TL}_n$, and are independent of the model parameters. We present explicit results for the function $ω$ in the infinite system ground state of the model and compute multi-point local height probabilities for up to four adjacent sites for the RSOS model and the related three-point correlation functions of non-Abelian $su(2)_k$ anyons.

DOI
10.15488/16211

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We study reduced density matrices of the integrable critical RSOS model in a particular topological sector containing the ground state. Similar as in the spin-1/21/21/2 Heisenberg model it has been observed that correlation functions of this model on short segments can be `factorized': they are completely determined by a single nearest-neighbour two-point function ωωω and a set of structure functions. While ωωω captures the dependence on the system size and the state of the system the structure functions can be expressed in terms of the possible operators on the segment, in the present case representations of the Temperley-Lieb algebra TLn\text{TL}_nTLn​, and are independent of the model parameters. We present explicit results for the function ωωω in the infinite system ground state of the model and compute multi-point local height probabilities for up to four adjacent sites for the RSOS model and the related three-point correlation functions of non-Abelian su(2)ksu(2)_ksu(2)k​ anyons.
Abstract We study reduced density matrices of the integrable critical restricted solid-on-solid (RSOS) model in a particular topological sector containing the ground state. Similar as in the spin- 1 / 2 Heisenberg model it has been observed that correlation functions of this model on short segments can be ‘factorized’: they are completely determined by a single nearest-neighbour two-point function ω and a set of structure functions. While ω captures the dependence on the system size and the state of the system the structure functions can be expressed in terms of the possible operators on the segment, in the present case representations of the Temperley–Lieb algebra TL n , and are independent of the model parameters. We present explicit results for the function ω in the infinite system ground state of the model and compute multi-point local height probabilities for up to four adjacent sites for the RSOS model and the related three-point correlation functions of non-Abelian s u ( 2 ) k anyons.
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.
Abstract We give a descriptive review of the fermionic basis approach to the theory of correlation functions of the XXZ quantum spin chain. The emphasis is on explicit formulae for short-range correlation functions which will be presented in a way that allows for their direct implementation on a computer. Within the fermionic basis approach a huge class of stationary reduced density matrices, compatible with the integrable structure of the model, assumes a factorized form. This means that all expectation values of local operators and all two-point functions, in particular, can be represented as multivariate polynomials in only two functions ρ and ω and their derivatives with coefficients that are rational in the deformation parameter q of the model. These coefficients are of ‘algebraic origin’. They do not depend on the choice of the density matrix, which only impacts the form of ρ and ω . As an example we work out in detail the case of the grand canonical ensemble at temperature T and magnetic field h for q in the critical regime. We compare our exact results for the fourth-neighbour two-point functions with asymptotic formulae for h , T = 0 and for finite h and T .
The study of the scaling limit of two-dimensional models of statistical mechanics within the framework of integrable field theory is illustrated through the example of the RSOS models. Starting from the exact description of regime III in terms of colliding particles, we compute the correlation functions of the thermal, phi1,2\\phi_{1,2}phi1,2​ and (for some cases) spin operators in the two-particle approximation. The accuracy obtained for the moments of these correlators is analysed by computing the central charge and the scaling dimensions and comparing with the exact results. We further consider the (generally non-integrable) perturbation of the critical points with both the operators phi1,3\\phi_{1,3}phi1,3​ and phi1,2\\phi_{1,2}phi1,2​ and locate the branches solved on the lattice within the associated two-dimensional phase diagram. Finally we discuss the fact that the RSOS models, the dilute qqq-state Potts model at and the O(n) vector model are all described by the same perturbed conformal field theory.
We consider the U q SU(2) invariant spin-1/2 XXZ quantum spin chain at the roots of unity q=exp(i pi /(m+1)), corresponding to different minimal models of conformal field theory. We conduct a numerical investigation of the correlation functions of U q SU(2) scalar two-point operators in order to find which operators in the minimal models they correspond to. Using graphical representations of the Temperley-Lieb algebra we are able to deal with chains of up to 28 sites. Depending on q, the correlation functions show different characteristics and finite-size behaviour. For m=2/3, which corresponds to the Lee-Yang edge singularity, we find the surface and bulk critical exponent -1/5. Together with the known result in the case m=3 (Ising model) this indicates that in the continuum limit the two-point operators involve conformal fields of spin-m-1/m+1. For other roots of unity q the chains are too short to determine the surface and bulk critical exponents.