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Physical Review Letters

Exact solution of the Perk-Schultz model

Hector J. de Vega, Esther Lopes

Published Jul 22, 1991• 55 citations
Publisher

Abstract

In the Perk-Schultz vertex model links can have n+1 different colors and the weights are ferroelectric or antiferroelectric depending on a discrete parameter ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{a}}$=\ifmmode\pm\else\textpm\fi{}1 (1\ensuremath{\le}a\ensuremath{\le}n). We compute the exact ground-state spins and find ferrielectric behavior for general signs ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{a}}$. The exact free energy and excitation spectrum are found as well as the finite-size corrections yielding the central charge and conformal dimensions. The scaling limit yields a quantum field theory whose mass spectrum and S matrix are explicitly obtained through the light-cone approach. The mass spectrum surprisingly depends on the anisotropy parameter and the signs ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{a}}$.

DOI
10.1103/physrevlett.67.489

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In the Perk-Schultz vertex model links can have n+1 different colors and the weights are ferroelectric or antiferroelectric depending on a discrete parameter \ensuremathϵa{\mathrm{\ensuremath{\epsilon}}}_{\mathit{a}}\ensuremathϵa​=\ifmmode\pm\else\textpm\fi{}1 (1\ensuremath{\le}a\ensuremath{\le}n). We compute the exact ground-state spins and find ferrielectric behavior for general signs \ensuremathϵa{\mathrm{\ensuremath{\epsilon}}}_{\mathit{a}}\ensuremathϵa​. The exact free energy and excitation spectrum are found as well as the finite-size corrections yielding the central charge and conformal dimensions. The scaling limit yields a quantum field theory whose mass spectrum and S matrix are explicitly obtained through the light-cone approach. The mass spectrum surprisingly depends on the anisotropy parameter and the signs \ensuremathϵa{\mathrm{\ensuremath{\epsilon}}}_{\mathit{a}}\ensuremathϵa​.
The exact solution of the A (2) n integrable lattice models is presented. The free energy and excitation energies of the vertex models are found as functions of the spectral and anisotropy parameters as well as the finite-size corrections yielding the central charges and conformal dimensions. These lattice models yield a new massive QFT where the mass spectrum and the S-matrix are obtained through the light-cone approach. The ground state and excitation energies of a new anisotropic spin-one integrable chain associated to the A (2) 2 model are computed.
In [1] (to which we refer to hereafter by I), based on numerical evidence, we present a series of conjectures about the eigenspectra of the SUq(N) invariant Perk-Schultz Hamiltonians, given by I.1 with p = 0. Subsequent extensive numerical work indicate that the conjecture (3) of I (I.85) is just a particular case of a more general one, relating the ground-state energy of diferent SUq(N) quantum chains. This new conjecture can be stated as follows: Conjecture: The difference of the ground-state energy of the SUq(N) and SUq ′(K) model in an open chain of lenght L is given by E0[SUq(N), q = e iπK/(N+K) ] − E0[SUq(N), q = e iπN/(N+K) ] = π(N − K) 2(1 − L) sin (), (1) 2(N + K) where N = K = 1, 2,... and E0[SUq(1), q)] = 0. The particular case N> 1 and K = 1 gives E0[SUq(N), q = e iπ/(N+1) π(N − 1)] = 2(1 − L) sin (), (2) 2(N + 1) recovering the earlier announced conjecture (I.85). The results (1) and (2) give exact finite-size corrections suporting earlier conjectures about the op-1 erator content of the Perk-Schultz models. As is well known [2] as a consequence of conformal invariance the finite-size corrections of the ground-state energy of critical chains with free boundary conditions are given by E0(L)/L = e ∞ + f∞/L − πc 24L 2 + o(L−2), (3) where e ∞ (f∞) is the ground-state energy per site (surface energy) at the bulk limit L → ∞, and c is the conformal anomaly of the effective underlying conformal field theory defined on a semi-infinte plane. The relations (3) and (2) imply that in the case of the SUq(N) model with q = exp(iπN/(N + 1)) the conformal anomaly has the value c = 0 for all N ≥ 2 and π(N − 1) e ∞ = −f ∞ = −2 sin (. (4) 2(N + 1) Moreover all the other finite-size corrections appearing in (3) are identically zero! This result can be understood from the conjectured operator content of the model. The conformal dimensions of the SUq(N) model are expected to be given by a generalized coulomb gas description x(n, m) = xp
The finite-size SU(3) Perk-Schultz model with deformation parameter q = exp (2iπ\piπ/3)
From extensive numeric diagonalizations of the SU (3) Perk-Schultz Hamiltonian with a special value of the anisotropy and different boundary conditions, we have observed simple regularities for a significant part of its eigenspectrum. In particular, the ground-state energy and nearby excitations belong to this part of the spectrum. Our simple formulae describing these regularities recall, apart from some selection rules, the eigenspectrum of the free-fermion model. Based on the numerical observations we formulate several conjectures. Using explicit solutions of the associated nested Bethe-ansatz equations, guessed from an analysis of the functional equations of the model, we provide evidence for some of them.
The eigenspectrum of the critical anisotropic Heisenberg model (or XXZ model), with arbitrary spin S in its disordered-ferroelectric regime (0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}\ensuremath{\pi}/2S) is solved by the Bethe-Ansatz method. The amplitudes of the leading finite-size corrections are calculated analytically and numerically. Using conformal invariance we give exact results for the conformal anomaly and scaling dimensions. Our results indicate that, for all spin S, the critical behavior is governed by a conformal field theory with central charge c=1.
Extensive numerical analysis of the eigenspectra of the SUq(N)SU_q(N)SUq​(N) invariant Perk-Schultz Hamiltonian shows some simple regularities for a significant part of the eigenspectrum. Inspired by those results we have found two set of solutions of the associated nested Bethe-ansatz equations. The first set is obtained at a special value of the anisotropy (q=exp(i2pi(N−1)/N)q = \\exp(i2\\pi (N-1)/N)q=exp(i2pi(N−1)/N)) and describes in particular the ground state and nearby excitations as a sum of free-fermion quasienergies. The second set of solutions provides the energies in the sectors whose number nin_ini​ of particles of distinct species (i=0,>...,N−1i =0, >..., N-1i=0,>...,N−1) are less or equal to the unity except for one of the species. For this last set we obtain the eigenspectra of a free fermion model for arbitrary values of the anisotropy.
Extensive numerical analysis of the eigenspectra of the SU q ( N ) invariant Perk–Schultz Hamiltonian shows some simple regularities for a significant part of the eigenspectrum. Inspired by those results we have found two sets of solutions of the associated nested Bethe ansatz equations. The first set is obtained at a special value of the anisotropy ( q = exp(iπ( N − 1)/ N )) and describes, in particular, the ground state and nearby excitations as a sum of free-fermion quasienergies. The second set of solutions provides the energies in the sectors whose number n i of particles of distinct species ( i = 0, ..., N − 1) are less than or equal to unity except for one of the species. For this last set we obtain the eigenspectra of a free-fermion model for arbitrary values of the anisotropy.