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arXiv:0905.0285 · Journal of Statistical Mechanics Theory and Experiment

The BaxterQoperator of critical dense polymers

Alessandro Nigro

Published Oct 9, 2009• 6 citations
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Abstract

We consider critical dense polymers , 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, nonlocal and dual nonlocal 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 nontrivial 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.

DOI
10.1088/1742-5468/2009/10/p10008

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We consider critical dense polymers , 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, nonlocal and dual nonlocal 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 nontrivial 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.
A lattice model of critical dense polymers is solved exactly on a cylinder with finite circumference. The model is the first member of the Yang–Baxter integrable series of logarithmic minimal models. The cylinder topology allows for non-contractible loops with fugacity α that wind around the cylinder or for an arbitrary number of defects that propagate along the full length of the cylinder. Using an enlarged periodic Temperley–Lieb algebra, we set up commuting transfer matrices acting on states whose links are considered distinct with respect to connectivity around the front or back of the cylinder. These transfer matrices satisfy a functional equation in the form of an inversion identity. For even N , this involves a non-diagonalizable braid operator J and an involution R = − ( J 3 − 12 J )/16 = (−1) F with eigenvalues . This is reminiscent of supersymmetry with a pair of defects interpreted as a fermion. The number of defects thus separates the theory into Ramond ( even), Neveu–Schwarz ( odd) and ( odd) sectors. For the case of loop fugacity α = 2, the inversion identity is solved exactly sector by sector for the eigenvalues in finite geometry. The eigenvalues are classified according to the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity. The finite-size corrections are obtained from Euler–Maclaurin formula. In the scaling limit, we obtain the conformal partition functions as sesquilinear forms and confirm the central charge c = − 2 and conformal weights . Here and in the even sectors with Kac labels r = 1, 2, 3,...; s = 1, 2 while in the odd sectors. Strikingly, the odd sectors exhibit a -extended symmetry but the even sectors do not. Moreover, the naive trace summing over all even sectors does not yield a modular invariant.
A lattice model of critical dense polymers is solved exactly on a cylinder with finite circumference. The model is the first member of the Yang–Baxter integrable series of logarithmic minimal models. The cylinder topology allows for non-contractible loops with fugacity α that wind around the cylinder or for an arbitrary number of defects that propagate along the full length of the cylinder. Using an enlarged periodic Temperley–Lieb algebra, we set up commuting transfer matrices acting on states whose links are considered distinct with respect to connectivity around the front or back of the cylinder. These transfer matrices satisfy a functional equation in the form of an inversion identity. For even N, this involves a non-diagonalizable braid operator J and an involution R = − (J3 − 12J)/16 = (−1)F with eigenvalues . This is reminiscent of supersymmetry with a pair of defects interpreted as a fermion. The number of defects thus separates the theory into Ramond ( even), Neveu–Schwarz ( odd) and ( odd) sectors. For the case of loop fugacity α = 2, the inversion identity is solved exactly sector by sector for the eigenvalues in finite geometry. The eigenvalues are classified according to the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity. The finite-size corrections are obtained from Euler–Maclaurin formula. In the scaling limit, we obtain the conformal partition functions as sesquilinear forms and confirm the central charge c = − 2 and conformal weights . Here and in the even sectors with Kac labels r = 1, 2, 3,...;s = 1, 2 while in the odd sectors. Strikingly, the odd sectors exhibit a -extended symmetry but the even sectors do not. Moreover, the naive trace summing over all even sectors does not yield a modular invariant.
Working in the dense loop representation, we use the planar Temperley–Lieb algebra to build integrable lattice models called logarithmic minimal models . Specifically, we construct Yang–Baxter integrable Temperley–Lieb models on the strip acting on link states and consider their associated Hamiltonian limits. These models and their associated representations of the Temperley–Lieb algebra are inherently non-local and not (time-reversal) symmetric. We argue that, in the continuum scaling limit, they yield logarithmic conformal field theories with central charges c = 1−(6( p − p ') 2 / pp '), where p , p ' = 1, 2, ... are coprime. The first few members of the principal series are critical dense polymers ( m = 1, c = −2), critical percolation ( m = 2, c = 0) and the logarithmic Ising model ( m = 3, c = 1/2). For the principal series, we find an infinite family of integrable and conformal boundary conditions organized in an extended Kac table with conformal weights Δ r , s = ((( m +1) r − ms ) 2 −1)/4 m ( m +1), r , s = 1, 2, .... The associated conformal partition functions are given in terms of Virasoro characters of highest-weight representations. Individually, these characters decompose into a finite number of characters of irreducible representations. We show with examples how indecomposable representations arise from fusion.
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.
In this paper we derive the finite size corrections to the energy eigenvalues of the energy for 2D dimers on a square lattice. These finite size corrections, as in the case of Critical Dense Polymers, are proportional to the eigenvalues of the Local Integrals of Motion of Bazhanov Lukyanov and Zamolodchikov for central charge c=−2c=-2c=−2. This sheds more light on the status of the Dimer model as a conformal field theory with this value of the certral charge.
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.