Emergent Replica Conformal Symmetry in Non-Hermitian SYK$_2$ Chains

Pengfei Zhang1, Shao-Kai Jian2, Chunxiao Liu3, and Xiao Chen4

1Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA
2Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, MD 20742, USA
3Department of Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA
4Department of Physics, Boston College, Chestnut Hill, MA 02467, USA

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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) ${O(2)\times O(2)}$ 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 the 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 ${S\sim \rho_s \log L_A}$, where ${\rho_s}$ is the total stiffness of the Goldstone modes. We also discuss special limits with more than one branch of Goldstone modes and comment on interaction effects.

Quantum system, subject to repeated measurements, is undergoing non-unitary dynamics. For many-body free fermions, this gives rise to critical steady states with a logarithmic subsystem entanglement and power law decay of correlation functions. We use solvable large N models to reveal the underlying mechanism: the emergence of criticality due to Goldstone modes in the replicated Hilbert space. We can then map the calculation of the entanglement entropy to the energy of a half-vortex pair, which shows the logarithmic scaling.

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