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arXiv:cond-mat/9611163 · Journal of Physics A Mathematical and General

Spectra of non-Hermitian quantum spin chains describing boundary induced phase transitions

Ulrich Bilstein, Birgit Wehefritz

Published Jul 21, 1997• 33 citations
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Abstract

The spectrum of the non-Hermitian asymmetric XXZ-chain with additional non-diagonal boundary terms is studied. The lowest-lying eigenvalues are determined numerically. For the ferromagnetic chain the temporal correlation length shows the same properties as the spatial correlation length. For the antiferromagnetic chain, we find a conformal invariant spectrum where the partition function corresponds to the one of a Coulomb gas with only magnetic charges. This is in contrast to the case of diagonal boundary terms where one finds only electric charges. Similar results are obtained in a toy model that can be diagonalized analytically in terms of free fermions.

DOI
10.1088/0305-4470/30/14/008

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The spectrum of the non-Hermitian asymmetric XXZ-chain with additional non-diagonal boundary terms is studied. The lowest-lying eigenvalues are determined numerically. For the ferromagnetic chain the temporal correlation length shows the same properties as the spatial correlation length. For the antiferromagnetic chain, we find a conformal invariant spectrum where the partition function corresponds to the one of a Coulomb gas with only magnetic charges. This is in contrast to the case of diagonal boundary terms where one finds only electric charges. Similar results are obtained in a toy model that can be diagonalized analytically in terms of free fermions.
The Bethe ansatz equation is solved to obtain analytically the leading finite-size correction of the spectra of the asymmetric XXZ chain and the accompanying isotropic 6-vertex model near the antiferromagnetic phase boundary at zero vertical field. The energy gaps scale with size N as and their amplitudes are obtained in terms of level-dependent scaling functions. Exactly on the phase boundary, the amplitudes are proportional to a sum of square-root of integers and an anomaly term. By summing over all low-lying levels, the partition functions are obtained explicitly. A similar analysis is performed also at the phase boundary of zero horizontal field in which case the energy gaps scale as . The partition functions for this case are found to be that of a non-relativistic free fermion system. From the symmetry of the lattice model under rotation, several identities between the partition functions are found. The scaling at zero vertical field is interpreted as a feature arising from viewing the Pokrovsky - Talapov transition with the space and time coordinates interchanged.
The low-lying excitations of the asymmetric XXZ spin chain are derived explicitly in the antiferromagnetic regime through the Bethe ansatz. It is found that a massless and conformal invariant phase with central charge c = 1 is separated from a massive phase by a line on which the low-lying excitations surprisingly scale with the lattice length as . The mass gap vanishes with an exponent as one approaches the massless phase. The connection with the asymmetric six-vertex model and some physical consequences are discussed.
The XX spin-chain with non-Hermitian diagonal boundary conditions is shown to be quasi-Hermitian for special values of the boundary parameters. This is proved by an explicit construction of a new inner product employing a ‘quasi-fermion’ algebra in momentum space, where creation and annihilation operators are not related via Hermitian conjugation. For a special example, when the boundary fields lie on the imaginary axis, we show the spectral equivalence of the quasi-Hermitian XX spin-chain with a non-local fermion model, where long-range hopping of the particles occurs as the non-Hermitian boundary fields increase in strength. The corresponding Hamiltonian interpolates between the open XX and the quantum group invariant XXZ model at the free fermion point. For an even number of sites the former is known to be related to a CFT with central charge c = 1, while the latter has been connected to a logarithmic CFT with central charge c = −2. We discuss the underlying algebraic structures and show that for an odd number of sites the superalgebra symmetry can be extended from the unit circle along the imaginary axis. We relate the vanishing of one of its central elements to the appearance of Jordan blocks in the Hamiltonian.
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
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, 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 generalized twisted partition function.
We calculated the spectral properties of two related families of non-Hermitian free-particle quantum chains with NNN-multispin interactions (N=2,3,…N=2,3,\ldotsN=2,3,…). The first family have a Z(N)Z(N)Z(N) symmetry and are described by free parafermions. The second one has a U(1)U(1)U(1) symmetry and are generalizations of XXXXXX quantum chains described by free fermions. The eigenspectra of both free-particle families are formed by the combination of the same pseudo-energies. The models have a multicritical point with dynamical critical exponent z=1z=1z=1. The finite-size behavior of their eigenspectra, as well as the entanglement properties of their ground state wave function, indicate the models are conformally invariant. The models with open and periodic boundary conditions show quite distinct physics due to their non-Hermiticity. The models defined with open boundaries have a single conformal invariant phase while the XXXXXX multispin models show multiple phases with distinct conformal central charges in the periodic case. The critical exponents of the models are calculated for N=3,4,5N=3,4,5N=3,4,5 and 666.