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arXiv:0906.4070 · Journal of Physics A Mathematical and Theoretical

A spin chain model with non-Hermitian interaction: the Ising quantum spin chain in an imaginary field

Olalla A. Castro-Alvaredo, Andreas Fring

Published Oct 26, 2009• 71 citations
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Abstract

We investigate a lattice version of the Yang–Lee model which is characterized by a non-Hermitian quantum spin chain Hamiltonian. We propose a new way to implement -symmetry on the lattice, which serves to guarantee the reality of the spectrum in certain regions of values of the coupling constants. In that region of unbroken -symmetry, we construct a Dyson map, a metric operator and find the Hermitian counterpart of the Hamiltonian for small values of the number of sites, both exactly and perturbatively. Besides the standard perturbation theory about the Hermitian part of the Hamiltonian, we also carry out an expansion in the second coupling constant of the model. Our constructions turn out to be unique with the sole assumption that the Dyson map is Hermitian. Finally, we analyse the magnetization of the chain in the z - and x -direction.

DOI
10.1088/1751-8113/42/46/465211

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Using finite-size-scaling methods, we study the quantum chain version of the spin-111-Blume-Capel model coupled to an imaginary field. The aim is to realize higher order non-unitary conformal field theories in a simple Ising-type spin model. We find that the first ground-state level crossing in the high-temperature phase leads to a second-order phase transition of the Yang-Lee universality class (central charge c=−22/5c=-22/5c=−22/5). The Yang-Lee transition region ends at a line of a new type of tricriticality, where the {\em three} lowest energy levels become degenerate. The analysis of the spectrum at two points on this line gives good evidence that this line belongs to the universality class of the M2,7{\cal M}_{2,7}M2,7​-conformal theory with c=−68/7c=-68/7c=−68/7.
An open Uq(sl2)U_q(sl_2)Uq​(sl2​)-invariant spin chain of spin SSS and length NNN with inhomogeneous coupling is investigated as an example of a non-Hermitian (quasi-Hermitian) model. For several particular cases of such a chain, the ranges of the deformation parameter γ\gammaγ are determined for which the spectrum of the model is real. For a certain range of γ\gammaγ, a universal metric operator is constructed, and thus, the quasi-Hermitian nature of the model is established. This universal metric operator is nondynamical, its structure is determined only by the symmetry of the model. The results apply, in particular, to all known homogeneous Uq(sl2)U_q(sl_2)Uq​(sl2​)-invariant integrable spin chains with nearest-neighbor interaction. In addition, the most general form of a metric operator for a quasi-Hermitian operator in finite-dimensional spaces is discussed.
Using finite-size-scaling methods, we study the quantum chain version of the spin-1-Blume-Capel model coupled to an imaginary field. The aim is to realize simple non-unitary minimal conformal field theories in an Ising-type spin model. We find that the first ground-state level crossing in the high-temperature phase leads to a second-order phase transition of the Yang-Lee universality class (central charge c=−22/5). The Yang-Lee transition region ends at a line of a new type of tricriticality, where the three lowest energy levels become degenerate. The analysis of the spectrum at two points on this line gives good evidence that this line realizes the universality class of the ℳ2,7-conformal theory with c=−68/7.
We systematically study the non-Hermitian version of the one-dimensional anisotropic XYXYXY model, which in its original form is a unique, exactly solvable, quantum spin model for understanding the quantum phase transition. The distinguishing features of this model are that it has a full real spectrum if all of the eigenvectors are intrinsic rotation-time-reversal symmetric rather than parity-time-reversal symmetric, and that its Hermitian counterpart is shown approximately to be an experimentally accessible system, i.e., an isotropic XYXYXY spin chain with nearest-neighbor coupling. Based on the exact solution, exceptional points which separate the unbroken- and broken-symmetry regions are obtained and lie on a hyperbola in the thermodynamic limit. This provides a nice paradigm to elucidate the complex quantum-mechanics theory for a quantum spin system.
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.
Since the realization of quantum systems described by non-Hermitian Hamiltonians with parity-time (PT\mathcal{PT}PT) symmetry, interest in non-Hermitian, quantum many-body models has steadily grown. Most studies to date map to traditional quantum spin models with a non-Hermiticity that arises from making the model parameters complex or purely imaginary. Here we present a set of models with non-Hermiticity generated by splitting a Hermitian term into two Jordan-normal form parts, and the perturbations are confined to one or two sites. We present exact diagonalization results for a finite PT\mathcal{PT}PT threshold in such models and provide an analytical approach for understanding the numerical results. Surprisingly, with non-Hermitian potentials confined to two or even a single site, the PT\mathcal{PT}PT threshold seems insensitive to the size of the quantum spin chain. Our results provide a pathway to experimentally feasible non-Hermitian quantum spin chains where the confluence of many-body effects and non-Hermiticity effects can be observed.