An investigation of ${\cal PT}$-symmetry breaking in tight-binding chains

TL;DR

This paper investigates how the threshold for ${ m PT}$-symmetry breaking in non-Hermitian tight-binding chains depends on chain length and potential configuration, revealing power-law decay and optimal configurations.

Contribution

It provides a detailed analysis of the ${ m PT}$-symmetry breaking threshold in various configurations, including random and periodic potentials, and identifies optimal chains with maximal robustness.

Findings

Threshold decays as a power of chain length, with exponents depending on potential configuration.

Random potential sequences have a mean threshold decaying as chain length to the power -3/2.

Optimal chains exhibit irregular threshold dependence, possibly decaying as chain length to the power -1.

Abstract

We consider non-Hermitian ${\cal PT}$-symmetric tight-binding chains where gain/loss optical potentials of equal magnitudes $\pm{\rm i}\gamma$ are arbitrarily distributed over all sites. The main focus is on the threshold $\gamma_c$ beyond which ${\cal PT}$-symmetry is broken. This threshold generically falls off as a power of the chain length, whose exponent depends on the configuration of optical potentials, ranging between 1 (for balanced periodic chains) and 2 (for unbalanced periodic chains, where each half of the chain experiences a non-zero mean potential). For random sequences of optical potentials with zero average and finite variance, the threshold is itself a random variable, whose mean value decays with exponent 3/2 and whose fluctuations have a universal distribution. The chains yielding the most robust ${\cal PT}$-symmetric phase, i.e., the highest threshold at fixed chain length, are obtained by exact enumeration up to 48 sites. This optimal threshold exhibits an irregular dependence on the chain length, presumably decaying asymptotically with exponent 1, up to logarithmic corrections.