[Paper Review] Non-Hermitian entanglement dip from scaling-induced exceptional criticality

Sirui Liu, Hui Jiang+5 more|arXiv (Cornell University)|Aug 5, 2024

Advanced NMR Techniques and Applications4 citations

TL;DR

This paper introduces scaling-induced exceptional criticality (SIEC), a novel non-Hermitian critical phenomenon where entanglement entropy exhibits dramatic, scale-specific dips due to exceptional crossings that emerge only at certain system sizes. The mechanism arises from a strongly size-dependent generalized Brillouin zone (GBZ) that sweeps through exceptional points, breaking conventional logarithmic entanglement scaling and enabling exotic quantum information signatures in non-Hermitian systems.

ABSTRACT

It is well established that the entanglement entropy of a critical system generally scales logarithmically with system size. Yet, in this work, we report a new class of non-Hermitian critical transitions that exhibit dramatic divergent dips in their entanglement entropy scaling, strongly violating conventional logarithmic behavior. Dubbed scaling-induced exceptional criticality (SIEC), it transcends existing non-Hermitian mechanisms such as exceptional bound states and non-Hermitian skin effect (NHSE)-induced gap closures, which are nevertheless still governed by logarithmic entanglement scaling. Key to SIEC is its strongly scale-dependent spectrum, where eigenbands exhibit an exceptional crossing only at a particular system size. As such, the critical behavior is dominated by how the generalized Brillouin zone (GBZ) sweeps through the exceptional crossing with increasing system size, and not just by the gap closure per se. We provide a general approach for constructing SIEC systems based on the non-local competition between heterogeneous NHSE pumping directions, and show how a scale-dependent GBZ can be analytically derived to excellent accuracy. Beyond 1D free fermions, SIEC is expected to occur more prevalently in higher-dimensional or even interacting systems, where antagonistic NHSE channels generically proliferate. SIEC-induced entanglement dips generalize straightforwardly to kinks in other entanglement measures such as Renyi entropy, and serve as spectacular demonstrations of how algebraic and geometric singularities in complex band structures manifest in quantum information.

Motivation & Objective

• To identify and characterize a new class of non-Hermitian critical transitions that deviate from conventional logarithmic entanglement scaling.
• To explain how exceptional criticality can emerge only at specific system sizes due to scale-dependent spectral evolution.
• To develop a general framework for constructing SIEC systems based on competing non-Hermitian skin effect (NHSE) pumping directions.
• To demonstrate that entanglement dips in SIEC are rooted in the dynamic sweeping of the generalized Brillouin zone (GBZ) through exceptional points.
• To generalize the entanglement dip phenomenon to other measures like Renyi entropy and link it to algebraic and geometric singularities in complex band structures.

Proposed method

• Constructing a generalized Brillouin zone (GBZ) that evolves with system size to capture the scale-dependent spectrum in non-Hermitian systems.
• Identifying exceptional criticality through the emergence of exceptional points (EPs) only at specific system sizes $L_c$, where eigenbands cross non-trivially.
• Using non-local competition between heterogeneous NHSE pumping directions to engineer systems with size-dependent GBZs and exceptional crossings.
• Deriving the GBZ analytically with high accuracy by modeling the interplay between complex momentum bands and system size.
• Computing entanglement entropy and Renyi entropy via two-point correlation functions obtained from biorthogonal expectation values using the $oldsymbol{ ilde{oldsymbol{ ho}}}$-operator formalism.
• Validating the framework in 1D non-Hermitian free fermion models and extending its applicability to higher dimensions and interacting systems.

Experimental results

Research questions

• RQ1 Can non-Hermitian critical systems exhibit entanglement scaling that deviates fundamentally from the logarithmic $S \sim \log L$ behavior?
• RQ2 What mechanism enables exceptional criticality to emerge only at specific system sizes, leading to entanglement dips?
• RQ3 How does the generalized Brillouin zone (GBZ) evolve with system size in such systems, and how does it govern the critical behavior?
• RQ4 To what extent can the entanglement dip phenomenon be generalized beyond von Neumann entropy to other measures like Renyi entropy?
• RQ5 What role do competing non-Hermitian skin effect (NHSE) channels play in enabling scale-dependent exceptional crossings?

Key findings

• SIEC introduces a new class of non-Hermitian critical transitions where entanglement entropy exhibits strong, scale-specific dips, violating the conventional logarithmic scaling $S \sim \log L$.
• The entanglement dip arises from the GBZ sweeping through an exceptional point only at a critical system size $L_c$, with the dip depth and width controlled by the rate of GBZ evolution.
• The generalized Brillouin zone (GBZ) is analytically derived with high accuracy, showing that its size-dependent evolution is the key driver of the non-logarithmic scaling.
• The mechanism transcends existing non-Hermitian phenomena like NHSE-induced gap closures and exceptional bound states, which still obey logarithmic entanglement scaling.
• The entanglement dip generalizes to kinks in Renyi entropy, demonstrating robustness across quantum information measures.
• SIEC is expected to be prevalent in higher-dimensional and interacting non-Hermitian systems due to the proliferation of antagonistic NHSE channels.