The fusion model of the L-state cyclic solid-on-solid (CSOS) model is considered at criticality. Each Z L -charge sector of the fusion CSOS model is related to a sum of total spin sectors of the fusion vertex model with a seam on which the vertex weights are modified by a phase factor. The latter in turn becomes the higher-spin XXZ quantum chain with a twisted boundary condition in the extremely anisotropic limit. Using the known operator content of the higher-spin XXZ chain, the authors deduce that of the fusion CSOS model. The modular invariant partition function of the fusion CSOS at fusion level k is a sum of products of the Z(k) parafermionic and free boson sectors with the effective coupling g'=L 2 (k-lambda/pi) for L odd and g"=L 2 (1/k-lambda/pi)/4 for L even, where lambda is the crossing parameter. Where L is odd and a multiple of k, or when L/2 is a multiple of k for L even, the modular invariant partition function becomes a simple product of the Z(k) parafermionic and the Gaussian partition function.
The fusion model of the L-state cyclic solid-on-solid (CSOS) model is considered at criticality. Each Z L -charge sector of the fusion CSOS model is related to a sum of total spin sectors of the fusion vertex model with a seam on which the vertex weights are modified by a phase factor. The latter in turn becomes the higher-spin XXZ quantum chain with a twisted boundary condition in the extremely anisotropic limit. Using the known operator content of the higher-spin XXZ chain, the authors deduce that of the fusion CSOS model. The modular invariant partition function of the fusion CSOS at fusion level k is a sum of products of the Z(k) parafermionic and free boson sectors with the effective coupling g'=L 2 (k-lambda/pi) for L odd and g"=L 2 (1/k-lambda/pi)/4 for L even, where lambda is the crossing parameter. Where L is odd and a multiple of k, or when L/2 is a multiple of k for L even, the modular invariant partition function becomes a simple product of the Z(k) parafermionic and the Gaussian partition function.
The authors are concerned with the critical properties of the anisotropic Heisenberg chain, or XXZ model, with arbitrary integer or half-integer spin. The eigenspectra of these Hamiltonians, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified, for lattices with an even and odd number of sites. The results for spin 1 and spin 3/2 indicate that the conformal anomaly c for these models, in the gapless regime, has the value c=3S/(1+S), independent of the anisotropy, and the exponents vary continuously with the anisotropy as in the eight-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-S models is described by composite fields formed by the product of Gaussian and Z(N) fields, with N=2S. Finally some of the irrelevant operators which produce the leading finite-size corrections of the eigenenergies are identified.
Given a critical quantum spin chain described by a conformal field theory (CFT) at long distances, it is crucial to understand the universal conformal data. One most important ingredient is the operator product expansion (OPE) coefficients, which describe how operators fuse into each other. It has been proposed in [Zou and Vidal, Phys. Rev. B 105, 125125 (2022)] that the OPE coefficients can be computed from overlaps of low-energy wave functions of the spin chain. In this paper, we establish that all conformal data including central charge, conformal dimensions, and OPE coefficients are encoded in the wave-function overlaps, with universal finite-size corrections that depend on the operator content of the cyclic orbifold CFT. Thus this method allows us to numerically compute all the conformal data based solely on the low-energy eigenstates. The predictions are verified in the Ising and XXZ model. As an application, we study the recently proposed Haagerup model built from the Haagerup fusion category. We find that the CFT has central charge $c\ensuremath{\approx}2.1$ and the lowest spin-1 operator in the twisted sector has scaling dimension $1<{\mathrm{\ensuremath{\Delta}}}_{J}\ensuremath{\le}1.4$.
Using a Lanczos method, we study the spectra of Heisenberg chains with spin S=1, 3/2, 2, 5/2, and 3, as a function of the anisotropy parameter \ensuremath{\lambda} and the number of sites of the chain. We use periodic, twisted, and open boundary conditions. We found that for all values of the spin when -1\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}0, these models are massless with critical behavior described by a conformal theory with central charge c=1. We discuss the whole operator content. In particular, we found that the critical exponent ${\mathrm{\ensuremath{\eta}}}_{\mathit{x}}$ follows the behavior ${\mathrm{\ensuremath{\eta}}}_{\mathit{x}}$=(\ensuremath{\pi}-\ensuremath{\gamma})/2\ensuremath{\pi}S and ${\mathrm{\ensuremath{\eta}}}_{\mathit{z}}$=2 where \ensuremath{\gamma}=${\mathrm{cos}}^{\mathrm{\ensuremath{-}}1}$(\ensuremath{\lambda}). For \ensuremath{\lambda}>0, our results indicate that the S=1 model is massive, while for S>1, the massless behavior characterized by the critical exponents described above continues up to \ensuremath{\lambda}=${\ensuremath{\lambda}}^{\mathrm{*}}$(S). For ${\ensuremath{\lambda}}^{\mathrm{*}}$\ensuremath{\le}1, our results are consistent with a massive phase for the integer-spin models and a massless phase with a rapid variation of the critical exponents for the half-integer-spin models.
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
We consider the L-state cyclic solid-on-solid lattice models under a class of open boundary conditions. The integrable boundary face weights are obtained by solving the reflection equations. Functional relations for the fused transfer matrices are presented for both periodic and open boundary conditions. The eigenspectra of the unfused transfer matrix is obtained from the functional relations using the analytic Bethe ansatz. For a special case of crossing parameter , the finite-size corrections to the eigenspectra of the critical models are obtained, from which the corresponding conformal dimensions follow. The calculation of the surface free energy away from criticality yields two surface specific heat exponents, and , where coprime to L. These results are in agreement with the scaling relations and .