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arXiv:0710.4213 · Journal of Statistical Mechanics Theory and Experiment

On the integrable structure of the Ising model

Alessandro Nigro

Published Jan 18, 2008• 9 citations
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Abstract

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.

DOI
10.1088/1742-5468/2008/01/p01017

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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 show how a lattice approach can be used to derive Thermodynamic Bethe Ansatz (TBA) equations describing all excitations for boundary flows. The method is illustrated for a prototypical flow of the tricritical Ising model by considering the continuum scaling limit of the A4 lattice model with integrable boundaries. Fixing the bulk weights to their critical values, the integrable boundary weights admit two boundary fields ξ and η which play the role of the perturbing boundary fields ϕ1,3 and ϕ1,2 inducing the renormalization group flow between boundary fixed points. The excitations are completely classified in terms of (m,n) systems and quantum numbers but the string content changes by certain mechanisms along the flow. For our prototypical example, we identify these mechanisms and the induced map between the relevant finitized Virasoro characters. We also solve the boundary TBA equations numerically to determine the flows for the leading excitations.
We show how a lattice approach can be used to derive Thermodynamic Bethe Ansatz (TBA) equations describing all excitations for boundary flows. The method is illustrated for a prototypical flow of the tricritical Ising model by considering the continuum scaling limit of the A4 lattice model with integrable boundaries. Fixing the bulk weights to their critical values, the integrable boundary weights admit two boundary fields xi\\xixi and eta\\etaeta which play the role of the perturbing boundary fields phi1,3\\phi_{1,3}phi1,3​ and phi1,2\\phi_{1,2}phi1,2​ inducing the renormalization group flow between boundary fixed points. The excitations are completely classified in terms of (m,n) systems and quantum numbers but the string content changes by certain mechanisms along the flow. For our prototypical example, we identify these mechanisms and the induced map between the relevant finitized Virasoro characters. We also solve the boundary TBA equations numerically to determine the flows for the leading excitations.
The integrable perturbation of the degenerate boundary condition ( d ) by the φ 1,3 boundary field generates a renormalization group flow down to the superposition of Cardy boundary states (+)&(−). Exact thermodynamic Bethe ansatz (TBA) equations for all the excited states are derived here extending the results of Feverati et al (2003 Nucl. Phys. B 675 469) to this case. As an intermediate step, the non-Cardy boundary conformal sector (+)&(−) is also described as the scaling limit of an A 4 lattice model with appropriate integrable boundary conditions and produces the first example of superposition of finitized Virasoro characters.
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 . 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 exact perturbation approach is used to derive the (seven) elementary correlation lengths and related mass gaps of the two-dimensional dilute A4 lattice model in regime 2− from the Bethe Ansatz solution. This model provides a realization of the integrable φ(1,2) perturbation of the c=710 conformal field theory, which is known to describe the off-critical thermal behavior of the tricritical Ising model. The E7 masses predicted from purely elastic scattering theory follow in the approach to criticality. Universal amplitudes for the tricritical Ising model are calculated.