Universal Entanglement Transitions of Free Fermions with Long-range Non-unitary Dynamics

Pengfei Zhang1, Chunxiao Liu2, Shao-Kai Jian3, and Xiao Chen4

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

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Non-unitary evolution can give rise to novel steady states classified by their entanglement properties. In this work, we aim to understand the effect of long-range hopping that decays with $r^{-\alpha}$ in non-Hermitian free-fermion systems. We first study two solvable Brownian models with long-range non-unitary dynamics: a large-$N$ SYK$_2$ chain and a single-flavor fermion chain and we show that they share the same phase diagram. When $\alpha\gt0.5$, we observe two critical phases with subvolume entanglement scaling: (i) $\alpha\gt1.5$, a logarithmic phase with dynamical exponent $z=1$ and logarithmic subsystem entanglement, and (ii) $0.5 \lt \alpha \lt 1.5$, a fractal phase with $z=\frac{2\alpha-1}{2}$ and subsystem entanglement $S_A\propto L_A^{1-z}$, where $L_A$ is the length of the subsystem $A$. These two phases cannot be distinguished by the purification dynamics, in which the entropy always decays as $L/T$. We then confirm that the results are also valid for the static SYK$_2$ chain, indicating the phase diagram is universal for general free-fermion systems. We also discuss phase diagrams in higher dimensions and the implication in measurement-induced phase transitions.

We study the effect of long-range hopping in non-unitary non-Hermitian free-fermion systems. We first show that two solvable Brownian models: a large-$N$ SYK$_2$ chain and a single-flavor fermion chain share the same phase diagram, which contains novel critical phases including a logarithmic phase and a fractal phase. We then confirm that the results are also valid for the static SYK$_2$ chain, indicating the phase diagram is universal for general free-fermion systems.

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