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

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 and completely asymmetric chain that corresponds to a reaction-diffusion model with input and outflow of particles the smallest energy gap correponding directly to the inverse of the temporal correlation length shows the same properties as the spatial correlation length of the stationary state. For the antiferromagnetic chain with both boundary terms, we find a conformal invariant spectrum where the partition function corresponds to the one of a Coulomb gas with only magnetic charges shifted by a purely imaginary and a lattice-length dependent constant. Similar results are obtained by studying a toy model that can be diagonalized analytically in terms of free fermions.

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“…On the transition line α = 0.5, we recover z = 3/2 in the large system size limit. Besides, from our numerical results, the infinite size dynamical exponent seems to be equal to 3/2 in the whole maximal current phase, confirming the extrapolation in [18].…”

“…Here, we calculate explicitly the relaxation times of the process, in order to compare with the predictions of the DW method. The same has been done already by using the non-symmetric Arnoldi method [18,19], but the system sizes that one can treat by the non-symmetric Arnoldi method are too small (L ≤ 16 in [19]) in order to obtain convergence with the DW predictions. Our aim is to improve this convergence thanks to the ability of DMRG to treat larger systems.…”

“…This has been done by U. Bilstein and B. Wehefritz [18], and later by M. Dudzińsky and G.M. Schütz [19] who calculated the relaxation times by using exact diagonalization.…”

“…This method is, however, restricted to very short chains (less than twenty), that are not in the asymptotic limit. In this work we used the density matrix renormalization group (DMRG) technique [20], that enables us to calculate the relaxation times for much larger system sizes, compared to [18,19]. By treating large chains we have obtained more conclusive results for the dynamical exponent z in the maximal current phase and could also achieve convergence between the exact numerical values of the relaxation times and the estimates of the phenomenological approach.…”

Quantum chimera states

“…It is well known that spin chains can exhibit kinds of quantum disorder and of quantum chaos [8,9,10,11], and that quantum synchronization is related to the entanglement [12,13,14,15]. To involve a kind of chimera states, our model consists of a non-hermitian spin chain [16,17,19,22] which can be assimilated to a spin chain in contact with an environment. This model is presented next section.…”