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arXiv:1008.4102 · Physical Review B

Spontaneous PT symmetry breaking and quantum phase transitions in dimerized spin chains

Gian Luca Giorgi

Published Aug 24, 2010• 50 citations
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Abstract

The occurrence of parity-time reversal $(\mathcal{P}\mathcal{T})$ symmetry breaking is discussed in a non-Hermitian spin chain. The Hermiticity of the model is broken by the presence of an alternating, imaginary, transverse magnetic field. A full real spectrum, which occurs if and only if all the eigenvectors are $\mathcal{P}\mathcal{T}$ symmetric, can appear only in presence of dimerization, i.e., only if the hopping amplitudes between nearest-neighbor spins assume alternate values along the chain. In order to make a connection between such system and the Hermitian world, we study the critical magnetic properties of the model and look for the conditions that would allow to observe the same phase diagram in the absence of the imaginary field. Such procedure amounts to renormalizing the spin-spin coupling amplitudes.

DOI
10.1103/physrevb.82.052404

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The occurrence of parity-time reversal (PT)(\mathcal{P}\mathcal{T})(PT) symmetry breaking is discussed in a non-Hermitian spin chain. The Hermiticity of the model is broken by the presence of an alternating, imaginary, transverse magnetic field. A full real spectrum, which occurs if and only if all the eigenvectors are PT\mathcal{P}\mathcal{T}PT symmetric, can appear only in presence of dimerization, i.e., only if the hopping amplitudes between nearest-neighbor spins assume alternate values along the chain. In order to make a connection between such system and the Hermitian world, we study the critical magnetic properties of the model and look for the conditions that would allow to observe the same phase diagram in the absence of the imaginary field. Such procedure amounts to renormalizing the spin-spin coupling amplitudes.
A quantum system governed by a non-Hermitian Hamiltonian may exhibit zero-temperature phase transitions that are driven by interactions, just as its Hermitian counterpart, raising the fundamental question of how non-Hermiticity affects quantum criticality. In this context, we consider a non-Hermitian system consisting of an X Y model with a complex-valued four-state clock interaction that may or may not have parity-time-reversal ( PT ) symmetry. When the PT symmetry is broken, and time-evolution becomes nonunitary, a scaling behavior similar to the Berezinskii-Kosterlitz-Thouless phase transition ensues, but in a highly unconventional way, as the line of fixed points is absent. From the analysis of the d -dimensional renormalization group equations, we obtain that the unconventional behavior in the PT broken regime follows from the collision of two fixed points in the d → 2 limit, leading to walking behavior or pseudocriticality. For d = 2 + 1 the near critical behavior is characterized by a correlation length exponent ν = 3 / 8 , a value smaller than the mean-field one. These results are in sharp contrast with the PT -symmetric case where only one fixed point arises for 2 < d < 4 and in d = 1 + 1 three lines of fixed points occur with a continuously varying critical exponent ν . Published by the American Physical Society 2024
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.
We study the properties of a parity- and time-reversal- (PT\mathcal{P}\mathcal{T}PT) symmetric tight-binding chain of size NNN with position-dependent hopping amplitude. In contrast to the fragile PT\mathcal{P}\mathcal{T}PT-symmetric phase of a chain with constant hopping and imaginary impurity potentials, we show that, under very general conditions, our model is always in the PT\mathcal{P}\mathcal{T}PT-symmetric phase. We numerically obtain the energy spectrum and the density of states of such a chain, and show that they are widely tunable. By studying the size dependence of inverse participation ratios, we show that although the chain is not translationally invariant, most of its eigenstates are extended. Our results indicate that tight-binding models with non-Hermitian, PT\mathcal{P}\mathcal{T}PT-symmetric hopping have a robust PT\mathcal{P}\mathcal{T}PT-symmetric phase and rich dynamics which may be explored in coupled waveguides.
The study of electromagnetic waves in engineering optical materials presents opportunities to manipulate light transmission through various optical media such as fiber optic cables. Recent experiments in this field have explored materials that have balanced gain and absorption that occur at different locations. We can model such a system as a chain with nearest neighbor hopping, and balanced, parity and time reversal symmetric (PT-symmetric) impurities, and it is represented by a non-Hermitian Hamiltonian matrix. The emergence of complex energy eigenvalues for such a matrix corresponds to the PT-symmetry breaking. We numerically and analytically investigate the eigenvalues of such a non-Hermitian Hamiltonian for an open chain with N sites. We find that the critical impurity strength, when the PT-symmetry breaks, is determined by the hopping amplitude between the impurities t_b, and hopping amplitude outside the impurities t_o in a non-trivial way. As a consequence, we show that the PT-breaking in such an open chain can be dramatically tuned by changing the hopping amplitude. Our results suggest that small changes in such systems, created in optical waveguides, can lead to significant changes for wave propagation through them (Joglekar, Y. N.,& Barnett, J. L., (2011). Origin of maximal symmetry breaking in even PT-symmetric lattices, Physical Review A, 84, 024103.)
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.
We study the non-Hermitian quantum mechanics for the discrete system. This paper gives an exact analytic single-particle solution for an NNN-site tight-binding chain with two conjugated imaginary potentials $\ifmmode\pm\else\textpm\fi{}i\ensuremath{\gamma}$ at two end sites, which Hamiltonian has parity-time symmetry (PT\mathcal{P}\mathcal{T}PT symmetry). Based on the Bethe ansatz results, it is found that, in single-particle subspace, this model is comprised of two phases: an unbroken symmetry phase with a purely real energy spectrum in the region $\ensuremath{\gamma}\ensuremath{\prec}{\ensuremath{\gamma}}_{c}$ and a spontaneously broken symmetry phase with $N\ensuremath{-}2$ real and two imaginary eigenvalues in the region $\ensuremath{\gamma}\ensuremath{\succ}{\ensuremath{\gamma}}_{c}$. The behaviors of eigenfunctions and eigenvalues in the vicinity of ${\ensuremath{\gamma}}_{c}$ are investigated. It is shown that the boundary of two phases possesses the characteristics of exceptional point. We also construct the equivalent Hermitian Hamiltonian of the present model in the framework of metric-operator theory. We find out that the equivalent Hermitian Hamiltonian can be written as another bipartite lattice model with real long-range hoppings.