On the integrable structure of the Ising model

Physica A: Statistical Mechanics and its Applications

2013

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We consider the 2D critical Ising model with spatially periodic 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 periodic Temperley-Lieb Algebra by introducing a lattice version of the Local Integrals of Motion.

“…for example µ = (1, −1, −1, −1, 1, 1, −1, 1) is associated to the partition P = (1,5,6,8). Now, by calling Λ r 0 the eigenvalue corresponding to µ = (1, 1, 1, 1, 1, .…”

Section: Inversion Identity and Eigenvaluesmentioning

confidence: 99%

Section: Local Integrals Of Motionmentioning

confidence: 99%

Lattice integrals of motion of the Ising model on the cylinder

Physica A: Statistical Mechanics and its Applications

2013

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“…In 2 previous papers on the Ising model by the author [3] [4], the integrals of motion (IOM) of Bhazanov Lukyanov and Zamoldchikov (BLZ) [1] were analized by means of Themodynamic Bethe ansatz, and in the most recent work the model on a chain with spatially periodic boundary conditions was analyzed by introducing a finitized version of the BLZ integrals of motion, taking values in the enveloping algebra of the periodic Temperley Lieb algebra with periodic boundaries. In this work it was recognized that the eigenvalues for this finitized IOM admit expansions in 1/N (where N is the size of the system) whose coefficients are related to the continuum eigenvalues of the BLZ IOM.…”

Section: Introductionmentioning

confidence: 99%

A q-Virasoro algebra at roots of unity, free fermions, and Temperley-Lieb hamiltonians

Journal of Mathematical Physics

2016

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In this work we introduce a novel q-deformation of the Virasoro algebra expressed in terms of free fermions, we then realize that this algebra, when the deformation parameter is a root of unity can be realized exactly on the lattice. We then study the relations existing between this lattice deformed Virasoro algebra at roots of unity and the tower of commuting Temperley-Lieb hamiltonians introduced in a previous work.

“…In contrast with unitary minimal models, which are realized on the lattice for example by the RSOS models [19] [20] [21], the transfer matrix may exhibit a Jordan indecomposable structure for some choice of Cardy-type boundary conditions. It is well known [5] [6] that the CFT corresponding to critical dense polymers has central charge c = −2.…”

Section: Introductionmentioning

confidence: 99%

The BaxterQoperator of critical dense polymers

2009

J. Stat. Mech.

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We consider critical dense polymers L 1,2 , corresponding to a logarithmic conformal field theory with central charge c = −2. An elegant decomposition of the Baxter Q operator is obtained in terms of a finite number of lattice integrals of motion. All local, non local and dual non local involutive charges are introduced directly on the lattice and their continuum limit is found to agree with the expressions predicted by conformal field theory. A highly non trivial operator Ψ(ν) is introduced on the lattice taking values in the Temperley Lieb Algebra. This Ψ function provides a lattice discretization of the analogous function introduced by Bazhanov, Lukyanov and Zamolodchikov. It is also observed how the eigenvalues of the Q operator reproduce the well known spectral determinant for the harmonic oscillator in the continuum scaling limit.