[Paper Review] Lieb-Schultz-Mattis Theorem in Open Quantum Systems

Kohei Kawabata, Ramanjit Sohal+1 more|arXiv (Cornell University)|May 25, 2023

Spectroscopy and Quantum Chemical StudiesPhysics and Astronomy9 citations

TL;DR

This paper extends the Lieb-Schultz-Mattis (LSM) constraint to open quantum systems described by Lindbladians, proving that a unique gapped steady state is forbidden under translation invariance and U(1) symmetry at noninteger filling. It demonstrates an open-system analog of the Haldane gap, with S=1/2 vs S=1 dissipative Heisenberg models showing gapless vs gapped behavior respectively.

ABSTRACT

The Lieb-Schultz-Mattis (LSM) theorem provides a general constraint on quantum many-body systems and plays a significant role in the Haldane gap phenomena and topological phases of matter. Here, we extend the LSM theorem to open quantum systems and establish a general theorem that restricts the steady state and spectral gap of Liouvillians based solely on symmetry. Specifically, we demonstrate that the unique gapped steady state is prohibited when translation invariance and U (1) symmetry are simultaneously present for noninteger filling numbers. As an illustrative example, we find that no dissipative gap is open in the spin-1/2 dissipative Heisenberg model while a dissipative gap can be open in the spin-1 counterpart -- an analog of the Haldane gap phenomena in open quantum systems. Furthermore, we show that the LSM constraint manifests itself in a quantum anomaly of the dissipative form factor of Liouvillians. We also find the LSM constraints due to symmetry intrinsic to open quantum systems, such as Kubo-Martin-Schwinger symmetry. Our work leads to a unified understanding of phases and phenomena in open quantum systems.

Motivation & Objective

• Generalize the LSM constraint to open quantum systems described by Lindbladians.
• Show that a unique gapped steady state is prohibited when translation invariance and U(1) symmetry coexist at noninteger filling.
• Illustrate the theorem with dissipative spin chains and connect to Haldane-like gap phenomena in open systems.
• Explore how quantum anomalies and symmetry (including KMS symmetry) manifest in open-system LSM constraints.

Proposed method

• Map the Lindblad master equation to a non-Hermitian operator on the doubled (ket and bra) Hilbert space to obtain the Lindbladian spectrum.
• Assume lattice translation invariance and strong U(1) symmetry to define filling number nu and analyze flux insertion via U(1) twists.
• Apply adiabatic insertion of U(1) flux and twist operators to show that, for noninteger nu, a second eigenstate is generated with shifted translation eigenvalue, implying gaplessness or degeneracy.
• Demonstrate the LSM constraint as a quantum anomaly in open systems via twisted boundary conditions and dissipative form factors.
• Use illustrative dissipative spin models (S=1/2 and S=1 XXZ with dephasing) to show gapless vs gapped behavior under flux insertion.
• Discuss the role of weak vs strong symmetry and KMS symmetry in yielding LSM constraints.

Experimental results

Research questions

• RQ1 Does translation invariance plus strong U(1) symmetry prohibit a unique gapped steady state in open quantum systems at noninteger filling?
• RQ2 How does the dissipative spectrum respond to U(1) flux insertion, and does this reveal gaplessness or degeneracy?
• RQ3 What are the differences between half-integer and integer spin cases in open-system LSM constraints, and how do they relate to the Haldane gap?
• RQ4 How do additional symmetries, like KMS or discrete symmetries, constrain the Lindbladian spectrum and steady states?

Key findings

• A general theorem: in open quantum systems with translation invariance and strong U(1) symmetry, a unique gapped steady state is prohibited at noninteger filling nu.
• In dissipative spin chains, a S=1/2 (half-integer) system shows dissipative gap closing under U(1) flux insertion, while a S=1 (integer) system can maintain a dissipative gap.
• The LSM constraint manifests as a quantum anomaly in the dissipative form factor under twisted boundary conditions.
• KMS symmetry and other intrinsic open-system symmetries yield LSM constraints, indicating widespread applicability beyond continuous U(1) symmetry.
• The results provide an open-system counterpart to the Haldane gap, with half-integer spins forced to be gapless or degenerate while integer spins can support a gapped steady state under the theorem's conditions.