Abstract
Quantum phase transitions occur in non-Hermitian systems. In this work we show that density functional theory, for the first time, uncovers universal critical behaviors for quantum phase transitions and quantum entanglement in non-Hermitian many-body systems. To be specific, we first prove that the non-degenerate steady state of a non-Hermitian quantum many body system is a universal function of the first derivative of the steady state energy with respect to the control parameter. This finding has far-reaching consequences for non-Hermitian systems. First, it bridges the non-analytic behavior of physical observable and no-analytic behavior of steady state energy, which explains why the quantum phase transitions in non-Hermitian systems occur for finite systems. Second, it predicts universal scaling behaviors of any physical observable at non-Hermitian phase transition point with scaling exponent being (1 − 1/p) with p being the number of coalesced states at the exceptional point. Third, it reveals that quantum entanglement in non-Hermitian phase transition point presents universal scaling behaviors with critical exponents being (1 − 1/p). These results uncover universal critical behaviors in non-Hermitian phase transitions and provide profound connections between entanglement and phase transition in non-Hermitian quantum many-body physics.
Highlights
We also need to prove that the non-degenerate steady state also uniquely specifies the control parameter
This is done by reductio ad absurdum
We assume that two different parameters γ and γ′ with γ ≠ γ′ have the same steady state
Summary
Proof of Theorem 1 are based on the following two Lemmas: Lemma 1. We need to prove that the non-degenerate steady state uniquely specifies the control parameter γ. This is done by reductio ad absurdum. We assume that two different parameters γ and γ′ with γ ≠ γ′ have the same steady state, ΨS , we have two eigenvalue equations, (H0 + iγH1) ΨS = E(γ) ΨS and (H0 + iγ′H1) ΨS = E(γ′) ΨS. Subtracting these two equations, we get i(γ − γ′)H1 ΨS = (E(γ) − E(γ′)) ΨS. This means that ΨS is an eigenket of H1 or
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