Anomaly in Open Quantum Systems and its Implications on Mixed-State Quantum Phases

In this paper, we develop a systematic approach to characterize the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:msup><a:mi/><a:mo>′</a:mo></a:msup></a:math>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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><d:mi mathvariant="normal">Γ</d:mi></d:math> arises as a central extension of the strong-symmetry group <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><h:mi>K</h:mi></h:math> with the weak-symmetry group <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><k:mi>G</k:mi></k:math>, we find that anomalies of bosonic systems are classified by <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><n:msup><n:mi>H</n:mi><n:mrow><n:mi>d</n:mi><n:mo>+</n:mo><n:mn>2</n:mn></n:mrow></n:msup><n:mo stretchy="false">(</n:mo><n:mi mathvariant="normal">Γ</n:mi><n:mo>,</n:mo><n:mi>U</n:mi><n:mo stretchy="false">(</n:mo><n:mn>1</n:mn><n:mo stretchy="false">)</n:mo><n:mo stretchy="false">)</n:mo><n:mo>/</n:mo><n:msup><n:mi>H</n:mi><n:mrow><n:mi>d</n:mi><n:mo>+</n:mo><n:mn>2</n:mn></n:mrow></n:msup><n:mo stretchy="false">(</n:mo><n:mi>G</n:mi><n:mo>,</n:mo><n:mi>U</n:mi><n:mo stretchy="false">(</n:mo><n:mn>1</n:mn><n:mo stretchy="false">)</n:mo><n:mo stretchy="false">)</n:mo></n:math> in <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><z:mi>d</z:mi></z:math> 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 <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><cb:mo stretchy="false">(</cb:mo><cb:mn>1</cb:mn><cb:mo>+</cb:mo><cb:mn>1</cb:mn><cb:mo stretchy="false">)</cb:mo><cb:mrow><cb:mrow><cb:mi mathvariant="normal">D</cb:mi></cb:mrow></cb:mrow></cb:math> 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 long-range correlation on the boundary, which is enforced by anomalies. We further establish the general relations between mixed-state anomalies and such unconventional boundary correlation. Moreover, we explore the generalization of the “anomaly inflow” mechanism in open quantum systems. We construct <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><ib:mo stretchy="false">(</ib:mo><ib:mn>1</ib:mn><ib:mo>+</ib:mo><ib:mn>1</ib:mn><ib:mo stretchy="false">)</ib:mo><ib:mrow><ib:mrow><ib:mi mathvariant="normal">D</ib:mi></ib:mrow></ib:mrow></ib:math> and <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><ob:mo stretchy="false">(</ob:mo><ob:mn>2</ob:mn><ob:mo>+</ob:mo><ob:mn>1</ob:mn><ob:mo stretchy="false">)</ob:mo><ob:mrow><ob:mrow><ob:mi mathvariant="normal">D</ob:mi></ob:mrow></ob:mrow></ob:math> Lindbladians whose steady states have mixed-state symmetry-protected-topological order in the bulk, with corresponding edge theories characterized by nontrivial anomalies. Finally, we generalize our results to the cases including lattice translation symmetries, which serves as the Lieb-Schulz-Mattis (LSM) theorem for mixed states.