In this paper we investigate the spectrum of OSp(n|2m) quantum spin chains with free boundary conditions. We compute the surface free energy of these models which, similar to other properties in the thermodynamic limit including the effective central charge of the underlying conformal field theory, depends on n−2m only. For several models in the regime n−2m<2 we have studied the finite-size properties including the subleading logarithmic corrections to scaling. As in the case of periodic boundary conditions we find the existence of a tower of states with the same conformal dimension as the identity operator. As expected the amplitudes of the corresponding logarithmic corrections differ from those found previously for the models with periodic boundary conditions. We point out however the existence of simple relations connecting such amplitudes for free and periodic boundaries. Based on our findings we formulate a conjecture on the long distance behaviour of the bulk and surface watermelon correlators.
In this paper we investigate the spectrum of OSp(n|2m) quantum spin chains with free boundary conditions. We compute the surface free energy of these models which, similar to other properties in the thermodynamic limit including the effective central charge of the underlying conformal field theory, depends on n−2m only. For several models in the regime n−2m<2 we have studied the finite-size properties including the subleading logarithmic corrections to scaling. As in the case of periodic boundary conditions we find the existence of a tower of states with the same conformal dimension as the identity operator. As expected the amplitudes of the corresponding logarithmic corrections differ from those found previously for the models with periodic boundary conditions. We point out however the existence of simple relations connecting such amplitudes for free and periodic boundaries. Based on our findings we formulate a conjecture on the long distance behaviour of the bulk and surface watermelon correlators.
The low energy spectrum of a spin chain with OSp(3|2) supergroup symmetry is studied based on the Bethe ansatz solution of the related vertex model. This model is a lattice realization of intersecting loops in two dimensions with loop fugacity z=1 which provides a framework to study the critical properties of the unusual low temperature Goldstone phase of the O(N) sigma model for N=1 in the context of an integrable model. Our finite-size analysis provides strong evidence for the existence of continua of scaling dimensions, the lowest of them starting at the ground state. Based on our data we conjecture that the so-called watermelon correlation functions decay logarithmically with exponents related to the quadratic Casimir operator of OSp(3|2). The presence of a continuous spectrum is not affected by a change to the boundary conditions although the density of states in the continua appears to be modified.
The thermodynamic limit of superspin chains can show several intriguing properties, including the emergence of continua of scaling dimensions and the appearance of discrete states when imposing toroidal boundary conditions. Nevertheless, the former’s exhaustive characterization in terms of Conformal Field Theories is still lacking. In order to study the thermodynamic limit of the U_q[sl(2|1)] 3 ⊗ 3̄ superspin chain in the antiferromagnetic regime, we analyze the low lying excitations by means of the model’s exact solution using the Algebraic Bethe Ansatz. In the isotropic limit, this model may be used as a toy model for the description of plateau transitions in Quantum Hall systems. The definition of a quasimomentum operator allows for a characterization of the continua of scaling dimensions, thereby giving rise to a quantum number for the corresponding non-compact component of the spectrum in the thermodynamic limit. The associated degeneracies are lifted on the lattice by logarithmic fine structures. Based on the extrapolation of our finite size data we find that under a variation of the boundary conditions from periodic to antiperiodic for the fermionic degrees of freedom, levels from the continuous part of the spectrum flow into discrete levels and vice versa. Investigating the thermodynamic limit of the q-deformed osp(3|2) superspin chain, corresponding to an intersecting loop model in the rational limit, we seek to uncover its low-lying critical exponents. We present evidences that the latter are built in terms of composites of anomalous dimensions of two Coulomb gases with distinct radii and exponents associated to Z(2) degrees of freedom. This view is supported by the fact that the S = 1 XXZ integrable chain spectrum is present in some of the sectors of the superspin chain at a particular value of the deformation parameter. We find that the fine structure of finite-size effects is very rich for a typical anisotropic spin chain. In fact, we argue on the existence of a family of states with the same conformal dimension whose lattice degeneracies are lifted by logarithmic corrections. On the other hand, we also report on states of the spectrum whose finite-size corrections seem to be governed by a power law behaviour. We finally observe that under toroidal boundary conditions the ground state dependence on the twist angle has two distinct analytical structures.
We obtain long series expansions for the bulk, surface and corner free energies for several two-dimensional statistical models, by combining Enting's finite lattice method (FLM) with exact transfer matrix enumerations. The models encompass all integrable curves of the Q -state Potts model on the square and triangular lattices, including the antiferromagnetic transition curves and the Ising model ( Q = 2) at temperature T , as well as a fully packed O( n ) type loop model on the square lattice. The expansions are around the trivial fixed points at infinite Q , n or 1/ T . By using a carefully chosen expansion parameter, q ≪ 1, all expansions turn out to be of the form , where the coefficients α k and β k are periodic functions of k . Thanks to this periodicity property, we can conjecture the form of the expansions to all orders (except in a few cases where the periodicity is too large). These expressions are then valid for all 0 ⩽ q < 1. We analyse in detail the q → 1 − limit in which the models become critical. In this limit the divergence of the corner free energy defines a universal term which can be compared with the conformal field theory (CFT) predictions of Cardy and Peschel. This allows us to deduce the asymptotic expressions for the correlation length in several cases. Finally we work out the FLM formulae for the case where some of the system's boundaries are endowed with particular (non-free) boundary conditions. We apply this in particular to the square-lattice Potts model with Jacobsen–Saleur boundary conditions, conjecturing the expansions of the surface and corner free energies to arbitrary order for any integer value of the boundary interaction parameter r . These results are in turn compared with CFT predictions.
Abstract. The Ising quantum chain with a peculiar twisted boundary condition is considered. This boundary condition, first introduced in the framework of the spin-1/2 XXZ Heisenberg quantum chain, is related to the duality transformation, which becomes a symmetry of the model at the critical point. Thus, at the critical point, the Ising quantum chain with the duality-twisted boundary is translationally invariant, similar as in the case of the usual periodic or antiperiodic boundary conditions. The complete energy spectrum of the Ising quantum chain is calculated analytically for finite systems, and the conformal properties of the scaling limit are investigated. This provides an explicit example of a conformal twisted boundary condition and a corresponding generalised twisted partition function. PACS numbers: 05.50.+q, 75.10.Jm, 71.10.Fd, 11.25.Hf Recently, there has been a lot of interest in generalised twisted partition functions of conformal field theory [1, 2, 3, 4] and the corresponding conformal twisted boundary conditions in solvable lattice models of statistical mechanics [5]. Examples of such boundary conditions leading to the appearance of “exotic ” spinor operators in the
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