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Free Fermion Branches in some Quantum Spin Models

Francisco Castilho Alcaraz, Yu. G. Stroganov

Published Jan 1, 2002• 4 citations
OpenAlex

Abstract

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 set of solutions of the associated nested Bethe-ansatz equations. The first set is obtained at a special value of the anisotropy ($q = \\exp(i2\\pi (N-1)/N)$) 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 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.

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Free-fermion branches in some quantum spin models

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We conjecture that the free-fermion part of the eigenspectrum observed recently for the SU q ( N ) Perk–Schultz spin chain Hamiltonian in a finite lattice with q = exp(iπ( N − 1)/ N ) is a consequence of the existence of a special simple eigenvalue for the transfer matrix of the auxiliary inhomogeneous SU q ( N − 1) vertex model which appears in the nested Bethe ansatz approach. We prove that this conjecture is valid for the case of the SU q (3) spin chain with periodic boundary condition. In this case we obtain a formula for the components of the eigenvector of the auxiliary inhomogeneous 6-vertex model ( q = exp(2iπ/3)), which permits us to find one by one all components of this eigenvector and consequently to find the eigenvectors of the free-fermion part of the eigenspectrum of the SU q (3) spin chain. Similarly, as in the known case of the SU q (2) case at q = exp(i2π/3) our numerical and analytical studies induce some conjectures for special rates of correlation functions.

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New conjecture for the SUq(N) Perk-Schultz models

Francisco Castilho Alcaraz, Yu. G. Stroganov

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

•Jan 1, 2004•0 citations
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.
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.
We conjecture that the free-fermion part of the eigenspectrum observed recently for the SU q ( N ) Perk–Schultz spin chain Hamiltonian in a finite lattice with q = exp(iπ( N − 1)/ N ) is a consequence of the existence of a special simple eigenvalue for the transfer matrix of the auxiliary inhomogeneous SU q ( N − 1) vertex model which appears in the nested Bethe ansatz approach. We prove that this conjecture is valid for the case of the SU q (3) spin chain with periodic boundary condition. In this case we obtain a formula for the components of the eigenvector of the auxiliary inhomogeneous 6-vertex model ( q = exp(2iπ/3)), which permits us to find one by one all components of this eigenvector and consequently to find the eigenvectors of the free-fermion part of the eigenspectrum of the SU q (3) spin chain. Similarly, as in the known case of the SU q (2) case at q = exp(i2π/3) our numerical and analytical studies induce some conjectures for special rates of correlation functions.
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