Reviving the Lieb–Schultz–Mattis theorem in open quantum systems

The Lieb-Schultz-Mattis (LSM) theorem is an important result in condensed matter physics. The LSM theorem states that a spin chain with half-integer spin and translation and rotation symmetry can not have a non-degenerate gapped ground state. It provides powerful constraints on the low-energy physics based on symmetry and basic microscopic data. The theorem also has huge impacts on experimental condensed matter physics, particularly in guiding the search for quantum spin liquids. Over the past decades, the LSM theorem has been generalized to various scenarios including higher dimensions, discrete symmetries, and fermionic systems. Extensive research on the LSM theorem has illuminated its critical role in understanding quantum magnetism and strongly correlated physics.

In recent years, open quantum many-body system is a subject attracting considerable interests. When the system inevitably interacts with the environment, energy is no longer conserved, and the applicability of the original LSM theorem diminishes. This raises a crucial question on the possibility of establishing an LSM-type theorem for open quantum systems. The inquiry is based on the point of view that the LSM theorem, to a large extent, does not depend on microscopic details, but instead concerns more general properties including symmetry and the Hilbert space. Therefore, it is conceivable that a counterpart of the LSM theorem can be established in open systems.

Recently, Yi-Neng Zhou, Xingyu Li, Hui Zhai, Chengshu Li, and Yingfei Gu from the Institute for Advanced Study at Tsinghua University generalized the LSM theorem to open quantum systems. Their work shows that, in open systems, the most natural approach is to consider the entanglement Hamiltonian – when system-environment coupling renders the system short-range correlated, the non-degenerate minimum of the entanglement spectrum can not have a spectral gap from other states. This work extends the topological constraints imposed by the LSM theorem to open quantum systems and to entanglement Hamiltonian, shedding light on the behavior of entanglement in the presence of interactions with the environment. It also contributes to the new frontiers about the topological physics in the mixed states. This result has been published in National Science Review Issue 1 (2025), titled “Reviving the Lieb–Schultz–Mattis Theorem in Open Quantum Systems”. Prof. Chengshu Li and Prof. Yingfei Gu are corresponding authors of the paper.

The research team clarifies the conditions under which the open system LSM theorem holds. Similar to the original LSM theorem, the system should have half-integer spin and translation and rotation symmetries. In open quantum systems, there are two types of symmetries, strong (similar to the canonical ensemble) and weak (similar to the grand canonical ensemble). This work shows that the open system LSM theorem requires weak symmetry. Apart from these two conditions that are similar to the original LSM theorem, the open system LSM theorem also requires that the the system-bath coupling renders the system short-range correlated. The research team uses techniques from quantum information theory (quantum conditional mutual information) to show that this condition suffices to guarantee the short-rangeness of the entanglement Hamiltonian, which in turn validates the open system LSM theorem. Note that in the original LSM theorem, (quasi) long-range correlation is a consequence of the absence of a gap. Hence, short-range correlation plays a dual role in open systems. On the one hand, it announces the failure of the original LSM theorem; on the other hand, it guarantees the open system LSM theorem holds. Apart from the analytic proof, the research team also carries out numerical simulations to verify the open system LSM theorem. They consider quantum spin ladder models which simulate the system-bath coupling, and focus on the scenarios of gaplessness and degenerate ground states. The numerical results agree very well with the open system LSM theorem. Currently, quantum simulation platforms including cold atoms, trapped ions, superconducting qubits, and NV centers are developing quickly, where system-bath coupling plays an important role. Hence, this work is also relevant to experimental platforms.

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