Universal Entanglement Transitions of Free Fermions with Long-range Non-unitary Dynamics
Pengfei Zhang, Chunxiao Liu, Shao-Kai Jian, Xiao Dong Chen
Abstract
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<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>r</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mi>&#x03B1;</mml:mi></mml:mrow></mml:msup></mml:math>in non-Hermitian free-fermion systems. We first study two solvable Brownian models with long-range non-unitary dynamics: a large-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>SYK<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi/><mml:mn>2</mml:mn></mml:msub></mml:math>chain and a single-flavor fermion chain and we show that they share the same phase diagram. When<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B1;</mml:mi><mml:mo>&#x003E;</mml:mo><mml:mn>0.5</mml:mn></mml:math>, we observe two critical phases with subvolume entanglement scaling: (i)<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B1;</mml:mi><mml:mo>&#x003E;</mml:mo><mml:mn>1.5</mml:mn></mml:math>, a logarithmic phase with dynamical exponent<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math>and logarithmic subsystem entanglement, and (ii)<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0.5</mml:mn><mml:mo>&#x003C;</mml:mo><mml:mi>&#x03B1;</mml:mi><mml:mo>&#x003C;</mml:mo><mml:mn>1.5</mml:mn></mml:math>, a fractal phase with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>&#x03B1;</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:math>and subsystem entanglement<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>S</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo>&#x221D;</mml:mo><mml:msubsup><mml:mi>L</mml:mi><mml:mi>A</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>z</mml:mi></mml:mrow></mml:msubsup></mml:math>, where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:math>is the length of the subsystem<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>. These two phases cannot be distinguished by the purification dynamics, in which the entropy always decays as<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>T</mml:mi></mml:math>. We then confirm that the results are also valid for the static SYK<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi/><mml:mn>2</mml:mn></mml:msub></mml:math>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.