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arXiv:1211.2714

Lattice Integrals of Motion of the Ising Model on the Strip

Alessandro Nigro

Published Nov 12, 2012• 0 citations
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Abstract

We consider the 2D critical Ising model on a strip with fixed boundary conditions. It is shown that for a suitable reparametrization of the known Boltzmann weights the transfer matrix becomes a polynomial in the variable $\csc(4u)$, being $u$ the spectral parameter. The coefficients of this polynomial are decomposed on the fixed boundaries Temperley-Lieb Algebra by introducing a lattice version of the Local Integrals of Motion.

DOI
10.48550/arxiv.1211.2714

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We consider the 2D critical Ising model on a strip with fixed boundary conditions. It is shown that for a suitable reparametrization of the known Boltzmann weights the transfer matrix becomes a polynomial in the variable csc⁡(4u)\csc(4u)csc(4u), being uuu the spectral parameter. The coefficients of this polynomial are decomposed on the fixed boundaries Temperley-Lieb Algebra by introducing a lattice version of the Local Integrals of Motion.
We consider critical dense polymers L(1,2){\cal L}(1,2)L(1,2). We obtain for this model the eigenvalues of the local integrals of motion of the underlying Conformal Field Theory by means of Thermodynamic Bethe Ansatz. We give a detailed description of the relation between this model and Symplectic Fermions including the indecomposable structure of the transfer matrix. Integrals of motion are defined directly on the lattice in terms of the Temperley Lieb Algebra and their eigenvalues are obtained and expressed as an infinite sum of the eigenvalues of the continuum integrals of motion. An elegant decomposition of the transfer matrix in terms of a finite number of lattice integrals of motion is obtained thus providing a reason for their introduction.
Starting from the lattice A 3 realization of the Ising model defined on a strip with integrable boundary conditions, the exact spectrum (including excited states) of all the local integrals of motion is derived in the continuum limit by means of thermodynamic Bethe ansatz techniques. It is also possible to follow the massive flow of this spectrum between the UV c = 1/2 conformal fixed point and the massive IR theory. The UV expression of the eigenstates of such integrals of motion in terms of Virasoro modes is found to have only rational coefficients and their fermionic representation turns out to be simply related to the quantum numbers describing the spectrum.
We consider critical dense polymers . We obtain for this model the eigenvalues of the local integrals of motion of the underlying conformal field theory by means of a thermodynamic Bethe ansatz. We give a detailed description of the relation between this model and symplectic fermions including some examples of the indecomposable structure of the transfer matrix in the continuum limit. Integrals of motion are defined directly on the lattice in terms of the Temperley–Lieb algebra and their eigenvalues are obtained and expressed as an infinite sum of the eigenvalues of the continuum integrals of motion. An elegant decomposition of the transfer matrix in terms of a finite number of lattice integrals of motion is obtained, thus providing a reason for their introduction.
The square lattice Ising model under the Brascamp-Kunz boundary conditions is a well-known exactly solvable lattice model. The exact solution of this system has been derived within the framework of Pfaffian-type method. In this paper, we provide a derivation for the solution by the Schultz-Mattis-Lieb method in the transfer matrix formalism. We set special interactions on the boundaries and take certain limit of these interactions, so that the system under the Brascamp-Kunz boundary conditions is transformed into another system under the toroidal boundary conditions. The Schultz-Mattis-Lieb method is applied to the mapping system and the partition function is exactly solved in the fermionic representation. The Fisher zeros are analytically calculated and the physical critical point is identified. We also discuss the difference between the transfer matrix approaches to the Brascamp-Kunz and to the toroidal boundary conditions. Our work introduces a member to the family of transfer-matrix-based studies for Ising model under various boundary conditions.
Abstract The dimer model on a strip is considered as a Yang–Baxter integrable six vertex model at the free-fermion point with crossing parameter and quantum group invariant boundary conditions. A one-to-many mapping of vertex onto dimer configurations allows for the solution of the free-fermion model to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by compared to their usual orientation. In a suitable gauge, the dimer model is described by the Temperley–Lieb algebra with loop fugacity . It follows that the model is exactly solvable in geometries of arbitrary finite size. We establish and solve transfer matrix inversion identities on the strip with arbitrary finite width N . In the continuum scaling limit, in sectors with magnetization S z , we obtain the conformal weights where . We further show that the corresponding finitized characters decompose into sums of q -Narayana numbers or, equivalently, skew q -binomials. In the particle representation, the local face tile operators give a representation of the fermion algebra and the fermion particle trajectories play the role of nonlocal degrees of freedom. We argue that, in the continuum scaling limit, there exist nontrivial Jordan blocks of rank 2 in the Virasoro dilatation operator L 0 . This confirms that, with quantum group invariant boundary conditions, the dimer model gives rise to a logarithmic conformal field theory with central charge c = −2, minimal conformal weight and effective central charge . Our analysis of the structure of the ensuing rank 2 modules indicates that the familiar staggered c = −2 modules appear as submodules.
According to finite-size scaling theory, the correlation length in an Ising strip with infinite length and a width of L lattice constants has the form ${\ensuremath{\xi}}_{L}$(T)=Lf(c(T-Tc{T}_{c}Tc​)L) in the scaling regime L\ensuremath{\gg}1,\ensuremath{\Vert}T-Tc{T}_{c}Tc​\ensuremath{\Vert}/Tc{T}_{c}Tc​\ensuremath{\ll}1. Here f(x) is a universal scaling function, which does, however, depend on the boundary conditions at the edges of the strip. Utilizing the correspondence between the two-dimensional Ising model and the one-dimensional quantum Ising model in a transverse field, we calculate the explicit form of the function f(x) for spin-spin and energy-energy correlations in strips with periodic, free, fixed, and mixed free-fixed boundary conditions.