[Paper Review] SYK Meets Non-Hermiticity I: Emergent Replica Conformal Symmetry

Pengfei Zhang, Shao-Kai Jian+2 more|arXiv (Cornell University)|Apr 8, 2021

Quantum, superfluid, helium dynamics4 citations

TL;DR

This paper introduces solvable non-Hermitian Sachdev-Ye-Kitaev chains to explain emergent critical phases in non-unitary free fermion dynamics. By analyzing replicated systems with O(2)×O(2) symmetry spontaneously broken to O(2), it derives an effective conformal field theory for Goldstone modes, yielding universal power-law squared correlators and a logarithmic subsystem entanglement entropy scaling as $ S \sim \rho_s \log L_A $, where $ \rho_s $ is the Goldstone mode stiffness.

ABSTRACT

Recently, the steady states of non-unitary free fermion dynamics are found to exhibit novel critical phases with power-law squared correlations and a logarithmic subsystem entanglement. In this work, we theoretically understand the underlying physics by constructing solvable static/Brownian quadratic Sachdev-Ye-Kitaev chains with non-Hermitian dynamics. We find the action of the replicated system generally shows (one or infinite copies of) $O(2) imes O(2)$ symmetries, which is broken to $O(2)$ by the saddle-point solution. This leads to an emergent conformal field theory of the Goldstone modes. We derive their effective action and obtain the universal critical behaviors of squared correlators. Furthermore, the entanglement entropy of a subsystem $A$ with length $L_A$ corresponds to the energy of the half-vortex pair $S\sim ho_s \log L_A$, where $ ho_s$ is the stiffness of the Goldstone mode. We also discuss special limits where more than one Goldstone mode exists and comment on interaction effects.

Motivation & Objective

• To understand the origin of novel critical phases with power-law squared correlations in non-unitary free fermion dynamics.
• To construct solvable static and Brownian quadratic Sachdev-Ye-Kitaev chains with non-Hermitian dynamics.
• To identify the role of replica symmetry, particularly O(2)×O(2), in the emergence of conformal invariance.
• To derive the effective action for Goldstone modes and connect it to universal critical behaviors.
• To relate subsystem entanglement entropy to the stiffness of Goldstone modes in the emergent conformal field theory.

Proposed method

• Constructing replicated systems of non-Hermitian quadratic SYK chains to analyze steady-state dynamics.
• Identifying the action of the replicated system as possessing one or infinite copies of O(2)×O(2) symmetry.
• Analyzing symmetry breaking where O(2)×O(2) is spontaneously broken to O(2), leading to Goldstone modes.
• Deriving the effective action for the Goldstone modes to describe universal critical behavior.
• Computing the entanglement entropy of a subsystem A as proportional to the stiffness $ \rho_s $, yielding $ S \sim \rho_s \log L_A $.
• Considering special limits with multiple Goldstone modes and discussing the impact of interactions.

Experimental results

Research questions

• RQ1 How does non-Hermitian dynamics in quadratic SYK chains lead to emergent conformal symmetry in their steady states?
• RQ2 What is the role of O(2)×O(2) replica symmetry and its spontaneous breaking in generating critical behavior?
• RQ3 How is the subsystem entanglement entropy related to the stiffness of the Goldstone modes in the emergent conformal field theory?
• RQ4 What are the universal scaling behaviors of squared correlation functions in this non-Hermitian critical phase?
• RQ5 How do multiple Goldstone modes and interaction effects modify the critical properties?

Key findings

• The replicated system's action exhibits O(2)×O(2) symmetry, which is spontaneously broken to O(2), leading to the emergence of Goldstone modes.
• The effective action for the Goldstone modes is derived, describing the universal critical behavior of squared correlators.
• The subsystem entanglement entropy scales logarithmically with subsystem size $ L_A $, with $ S \sim \rho_s \log L_A $, where $ \rho_s $ is the stiffness of the Goldstone mode.
• In special limits, multiple Goldstone modes can emerge, indicating richer critical structures.
• The framework provides a field-theoretic explanation for power-law squared correlations and logarithmic entanglement in non-unitary free fermion dynamics.