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arXiv:2304.10064 · Physical Review A

PT -symmetry breaking in quantum spin chains with exceptional non-Hermiticities

Jacob Muldoon, Yogesh N. Joglekar

Published Dec 6, 2023• 3 citations
arXivHTMLPDF

Abstract

Since the realization of quantum systems described by non-Hermitian Hamiltonians with parity-time ($\mathcal{PT}$) 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 $\mathcal{PT}$ 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 $\mathcal{PT}$ 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.

DOI
10.1103/physreva.108.062205

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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.
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.
We investigate nonstabilizerness, also known as ``magic,'' to understand criticality and exceptional points in non-Hermitian quantum many-body systems. Our focus is on parity-time (PT\mathcal{PT}PT)-symmetric spin chains, specifically the non-Hermitian transverse-field Ising and XX models. We calculate stabilizer Rényi entropies in their ground states using non-Hermitian matrix product state methods. Our findings show that magic exhibits unique and model-specific signs of phase transitions. In the Ising chain, it peaks near the regular Hermitian-like critical line but disappears across exceptional points. In contrast, in the XX chain, it reaches its maximum at the exceptional line where PT\mathcal{PT}PT symmetry is broken. Finite-size scaling reveals that these effects become more pronounced with larger systems, highlighting nonstabilizerness as a sensitive marker for both quantum criticality and non-Hermitian spectral degeneracies. We also investigate magic in momentum space for the XX model analytically and find that it reaches a minimum around exceptional points. Our results indicate that magic takes extremal values at the exceptional points and serves as a valuable tool for examining complexity, criticality, and symmetry breaking in non-Hermitian quantum matter.
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In the presence of non-Hermitian skin effect, non-Hermitian lattices generally have complex-valued eigenenergies under periodic boundary condition, but they can have non-Bloch PT symmetry and therefore completely real eigenenergies under open boundary condition. This novel PT symmetry and its breaking have been experimentally observed in one dimension. Here, we find that non-Bloch PT symmetry in two and higher dimensions exhibits drastically different behaviors compared to its one-dimensional counterpart. Whereas Bloch PT breaking and one-dimensional non-Bloch PT breaking generally have nonzero thresholds in the large-size limit, the threshold of two and higher-dimensional non-Bloch PT breaking universally approaches zero as the system size increases. A product measure, namely the product of bare non-Hermiticity and system size, is introduced to quantify the PT breaking tendency. This product being small is required for the perturbation theory to be valid, thus its growth with system size causes the breakdown of perturbation theory, which underlies the universal threshold. That the universal behaviors emerge only in two and higher dimensions indicates an unexpected interplay among PT symmetry, non-Hermitian skin effect, and spatial dimensionality. Our predictions can be confirmed on experimentally accessible platforms.
Effects of detuning on PT\mathcal{PT}PT-symmetric, tridiagonal, tight-binding models
Non-Hermitian, tight-binding PT\mathcal{PT}PT-symmetric models are extensively studied in the literature. Here, we investigate two forms of non-Hermitian Hamiltonians to study the PT\mathcal{PT}PT-symmetry breaking thresholds and features of corresponding surfaces of exceptional points (EPs). They include one-dimensional chains with uniform or 2-periodic tunnelling amplitudes, one pair of balanced gain and loss potentials Δ±ıγΔ\pmıγΔ±ıγ at parity-symmetric sites, and periodic or open boundary conditions. By introducing a Hermitian detuning potential, we obtain the dependence of the PT\mathcal{PT}PT-threshold, and therefore the exceptional-point curves, in the parameter space of detuning and gain-loss strength. By considering several such examples, we show that EP curves of a given order generically have cusp-points where the order of the EP increases by one. In several cases, we obtain explicit analytical expressions for positive-definite intertwining operators that can be used to construct a complex extension of quantum theory by re-defining the inner product. Taken together, our results provide a detailed understanding of detuned tight-binding models with a pair of gain-loss potentials.