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arXiv:cond-mat/0201354 · Journal of Physics A Mathematical and General

The finite-size SU(3) Perk-Schultz model with deformation parameter q = exp (2i$\pi$/3)

Francisco Castilho Alcaraz, Yu. G. Stroganov

Published Apr 19, 2002• 8 citations
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Abstract

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.

DOI
10.1088/0305-4470/35/17/301

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

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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

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Phase diagram and spectral properties of a new exactly integrable spin-1 quantum chain

Francisco Castilho Alcaraz, Gilberto M Nakamura

The spectral properties and phase diagram of the exact integrable spin one quantum chain introduced by Alcaraz and Bariev are presented. The model has a U(1) symmetry and its integrability is associated to an unknown R-matrix whose dependence on the spectral parameters is not of difference form. The associated Bethe ansatz equations, that fix the eigenspectra, are distinct from those associated to other known integrable spin models. The model has a free parameter $t_p$. We show that at the special point $t_p=1$ the model gets an extra U(1) symmetry and reduces to the deformed SU(3) Perk-Schultz model at a special value of its anisotropy $q=\exp(i2π/3)$ andin the presence of an external magnetic field. Our analysis is done either by solving the associated Bethe-ansatz equations or by direct diagonalization of thequantum Hamiltonian for small lattice sizes. The phase diagram is calculated by exploring the consequences of conformal invariance on the finite-size corrections of the Hamiltonian eigenspectrum. The model exhibits a critical phase ruled by $c=1$ conformal field theory separated from a massive phase by first-order phase transitions.

arXiv:1002.4857•Journal of Physics A Mathematical and Theoretical•Mar 25, 2010•8 citations
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.
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.
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.
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 spectral properties and phase diagram of the exact integrable spin one quantum chain introduced by Alcaraz and Bariev are presented. The model has a U(1) symmetry and its integrability is associated to an unknown R-matrix whose dependence on the spectral parameters is not of difference form. The associated Bethe ansatz equations, that fix the eigenspectra, are distinct from those associated to other known integrable spin models. The model has a free parameter tpt_ptp​. We show that at the special point tp=1t_p=1tp​=1 the model gets an extra U(1) symmetry and reduces to the deformed SU(3) Perk-Schultz model at a special value of its anisotropy q=exp⁡(i2π/3)q=\exp(i2π/3)q=exp(i2π/3) andin the presence of an external magnetic field. Our analysis is done either by solving the associated Bethe-ansatz equations or by direct diagonalization of thequantum Hamiltonian for small lattice sizes. The phase diagram is calculated by exploring the consequences of conformal invariance on the finite-size corrections of the Hamiltonian eigenspectrum. The model exhibits a critical phase ruled by c=1c=1c=1 conformal field theory separated from a massive phase by first-order phase transitions.