开放量子系统中的反常及其对混态量子相的启示
Anomaly in open quantum systems and its implications on mixed-state quantum phases
AI总结：
提出用超算符统一刻画开放量子系统强弱对称性及’t Hooft反常的框架，证明反常强制产生非平庸混态相，并构造新型边界破缺相及SPT Lindbladian。
摘要：
本文中，我们发展了一套系统方法来刻画开放量子系统中的’t Hooft反常。由于与环境存在非平庸耦合，此类系统中的对称性表现为强型或弱型。通过用超算符表示其对称变换，我们将二者纳入一个统一框架，从而能够直接计算其反常。在完全对称群为$K\ imes G$（其中$K$为强对称、$G$为弱对称）的情形下，我们发现，在$d$个空间维度中，玻色系统的反常由$H^{d+2}(K\ imes G,U(1))/H^{d+2}(G,U(1))$分类。为展示开放量子系统中反常的作用，我们一般性地证明，只要施加弱对称性，反常就必然导致非平庸的混态量子相。类似于保证封闭系统中非平庸低能物理的“反常匹配”条件，反常在由Lindbladian支配的开放量子系统中也保证非平庸的稳态和长时间动力学。值得注意的是，我们识别出一种新颖的$(1+1)$维混态量子相，它在封闭系统中没有对应物：其稳态在体内不显示非平庸关联函数，但在边界上呈现由反常强制产生的自发对称破缺序。我们进一步建立了混态反常与此类非常规边界关联之间的一般关系。最后，我们探索了“反常流入”机制在开放量子系统中的推广。我们构造了$(1+1)$维和$(2+1)$维Lindbladian，其稳态在体内具有混态对称保护拓扑序，相应的边界理论则由非平庸反常刻画。
英文摘要：
In this paper, we develop a systematic approach to characterize the 't Hooft anomaly in open quantum systems. Owing to nontrivial couplings to the environment, symmetries in such systems manifest as either strong or weak type. By representing their symmetry transformation through superoperators, we incorporate them in a unified framework that enables a direct calculation of their anomalies. In the case where the full symmetry group is $K\times G$, with $K$ the strong symmetry and $G$ the weak symmetry, we find that anomalies of bosonic systems are classified by $H^{d+2}(K\times G,U(1))/H^{d+2}(G,U(1))$ in $d$ spatial dimensions. To illustrate the power of anomalies in open quantum systems, we generally prove that anomaly must lead to nontrivial mixed-state quantum phases as long as the weak symmetry is imposed. Analogous to the ``anomaly matching" condition ensuring nontrivial low-energy physics in closed systems, anomaly also guarantees nontrivial steady states and long-time dynamics for open quantum systems governed by Lindbladians. Notably, we identify a novel $(1+1)$-D mixed-state quantum phase that has no counterpart in closed systems, where the steady state shows no nontrivial correlation function in the bulk, but displays spontaneous symmetry breaking order on the boundary, which is enforced by anomalies. We further establish the general relations between mixed-state anomalies and such unconventional boundary correlation. Finally, we explore the generalization of the ``anomaly inflow" mechanism in open quantum systems. We construct $(1+1)$-D and $(2+1)$-D Lindbladians whose steady states have mixed-state symmetry-protected-topological order in the bulk, with corresponding edge theories characterized by nontrivial anomalies.
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