Emergent Replica Conformal Symmetry in Non-Hermitian 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>Chains
Pengfei Zhang, Shao-Kai Jian, Chunxiao Liu, Xiao Chen
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)<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>×</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>symmetries, which is broken to<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>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<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>A</mml:mi></mml:mrow></mml:math>with length<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>L</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow></mml:math>corresponds to the energy of the half-vortex pair<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>S</mml:mi><mml:mo>∼</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mi>log</mml:mi><mml:mo></mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow></mml:math>, where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>ρ</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math>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.