[Paper Review] Non-Bloch PT symmetry breaking: Universal threshold and dimensional surprise

Fei Song, Hongyi Wang+1 more|arXiv (Cornell University)|Feb 3, 2021

Quantum Mechanics and Non-Hermitian Physics48 references7 citations

TL;DR

This paper reveals that non-Bloch PT symmetry breaking in two and higher dimensions exhibits a universal threshold that vanishes as system size increases, unlike in one dimension where the threshold remains finite. The phenomenon arises from the non-perturbative growth of effective non-Hermiticity, quantified by the product of bare non-Hermiticity and system size, which drives a transition to complex eigenenergies even for infinitesimal non-Hermiticity in large systems.

ABSTRACT

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.

Motivation & Objective

• To investigate the behavior of non-Bloch PT symmetry breaking in two and higher-dimensional non-Hermitian lattices.
• To understand the origin of the universal threshold behavior in higher dimensions, contrasting with the size-independent threshold in one-dimensional systems.
• To identify the role of the non-Hermitian skin effect and spatial dimensionality in enabling a vanishing threshold for PT symmetry breaking.
• To establish a universal scaling parameter—the product of bare non-Hermiticity and system size—as a measure of effective non-Hermiticity.
• To confirm the generality of the phenomenon across different models and geometries, including Chern bands and disordered systems.

Proposed method

• Analyzing a 2D single-band non-Hermitian lattice model with real hoppings under open boundary conditions (OBC), where non-Bloch bands emerge due to the non-Hermitian skin effect.
• Using the generalized Brillouin zone (GBZ) formalism to characterize eigenstates and eigenenergies under OBC, replacing the standard Bloch band theory.
• Introducing a product measure, $ \gamma L $, where $ \gamma $ is the bare non-Hermiticity and $ L $ is the system size, to quantify the effective non-Hermiticity.
• Applying perturbation theory to analyze the stability of real eigenenergies and identifying its breakdown when $ \gamma L \gg 1 $, which signals the onset of PT symmetry breaking.
• Studying size-dependent energy spectra in square and disk geometries for both tight-binding models and non-Hermitian Chern bands to confirm universal scaling.
• Confirming robustness by introducing weak disorder on boundary sites and observing similar threshold collapse, indicating insensitivity to symmetry-breaking perturbations.

Experimental results

Research questions

• RQ1 Does non-Bloch PT symmetry breaking in two and higher dimensions exhibit a size-dependent threshold, and if so, how does it scale with system size?
• RQ2 What is the origin of the universal vanishing threshold in higher dimensions, and how does it differ from the finite threshold observed in one-dimensional systems?
• RQ3 How does the product of bare non-Hermiticity and system size ($ \gamma L $) govern the effective non-Hermiticity and the breakdown of perturbation theory?
• RQ4 Is the universal threshold behavior robust against disorder and geometric variations, such as square vs. disk-shaped lattices?
• RQ5 Can the phenomenon be generalized beyond simple tight-binding models, such as in non-Hermitian Chern bands with intrinsic topology?

Key findings

• In two and higher dimensions, the threshold for non-Bloch PT symmetry breaking universally approaches zero as system size increases, in stark contrast to one-dimensional systems where the threshold remains finite.
• The product $ \gamma L $, representing the effective non-Hermiticity, grows with system size, causing the breakdown of perturbation theory and enabling real-to-complex eigenenergy transitions even for infinitesimal $ \gamma $.
• For a 2D square lattice with $ t=1 $, $ s=0.3 $, and $ \gamma=0.1 $, the proportion of complex eigenenergies increases from $ L=10 $ to $ L=70 $, indicating PT symmetry breaking at large sizes.
• In a 3D model with $ t=1 $, $ s=0.5 $, the threshold for PT breaking also decreases toward zero with increasing system size, confirming the universality of the phenomenon beyond 2D.
• The non-Bloch PT symmetry breaking in non-Hermitian Chern bands shows increasing proportions of complex eigenenergies with system size in both square and disk geometries, with disorder enhancing the effect in square lattices.
• The phenomenon persists even under weak boundary disorder and is not reliant on non-reciprocal hoppings, as demonstrated in a model with only onsite gain/loss.