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arXiv:2310.01475 · National Science Review

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

Yi-Neng Zhou, Xingyu Li, Hui Zhai, Chengshu Li, Yingfei Gu

Published Aug 17, 2024• 17 citations
arXivHTMLPDF

Abstract

In closed systems, the celebrated Lieb-Schultz-Mattis (LSM) theorem states that a one-dimensional locally interacting half-integer spin chain with translation and spin rotation symmetries cannot have a non-degenerate gapped ground state. However, the applicability of this theorem is diminished when the system interacts with a bath and loses its energy conservation. In this letter, we propose that the LSM theorem can be revived in the entanglement Hamiltonian when the coupling to the bath renders the system short-range correlated. Specifically, we argue that the entanglement spectrum cannot have a non-degenerate minimum, isolated by a gap from other states. We further support the results with numerical examples where a spin-[Formula: see text] system is coupled to another spin-[Formula: see text] chain serving as the bath. Compared with the original LSM theorem that primarily addresses UV-IR correspondence, our findings reveal that the UV data and topological constraints also play a pivotal role in shaping the entanglement in open quantum many-body systems.

DOI
10.1093/nsr/nwae287

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In closed systems, the celebrated Lieb-Schultz-Mattis (LSM) theorem states that a one-dimensional locally interacting half-integer spin chain with translation and spin rotation symmetries cannot have a non-degenerate gapped ground state. However, the applicability of this theorem is diminished when the system interacts with a bath and loses its energy conservation. In this letter, we propose that the LSM theorem can be revived in the entanglement Hamiltonian when the coupling to the bath renders the system short-range correlated. Specifically, we argue that the entanglement spectrum cannot have a non-degenerate minimum, isolated by a gap from other states. We further support the results with numerical examples where a spin-[Formula: see text] system is coupled to another spin-[Formula: see text] chain serving as the bath. Compared with the original LSM theorem that primarily addresses UV-IR correspondence, our findings reveal that the UV data and topological constraints also play a pivotal role in shaping the entanglement in open quantum many-body systems.
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.
The Lieb-Schultz-Mattis (LSM) theorem asserts that microscopic details of the system can impose non-trivial constraints on the system's low-energy properties. While traditionally applied to short-range interaction systems, where locality ensures a vanishing spectral gap in large system size limit, the impact of long-range interactions on the LSM theorem remains an open question. Long-range interactions are prevalent in experimental platforms such as Rydberg atoms, dipolar quantum gases, polar molecules, optical cavities, and trapped ions, where the interaction decay exponent can be experimentally tuned. We extend the LSM theorem in one dimension to long-range interacting systems and find that the LSM theorem holds for exponentially or power-law two-body interactions with a decay exponent α>2α> 2α>2. However, for power-law interactions with α<2α< 2α<2, the constraints of the LSM theorem on the ground state do not apply. Numerical simulations of long-range versions of the Heisenberg and Majumdar-Ghosh models, both satisfying the LSM symmetry requirements, are also provided. Our results suggest promising directions for experimental validation of the LSM theorem in systems with tunable long-range interactions.
We construct a Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2Z2​×Z2​ gauge theory coupled to matter on a one-dimensional chain, aiming to study the ground-state physics in the Gauss law subspace. We show that the theory in the Gauss law subspace has a U(1)(1)(1) symmetry whose generator commutes with lattice translations, but anticommutes with the lattice reflection operator. This leads to a Lieb-Schultz-Mattis (LSM) theorem that always rules out a trivial gapped ground state in the Gauss law subspace, if the hamiltonian is invariant under translations and reflection. Any point in the parameter space must realize either a spontaneously symmetry broken (SSB) ground state, or a gapless ground state. Imposing the Gauss law is pivotal for the existence of the U(1)(1)(1) symmetry, and hence of the LSM theorem. We thus demonstrate a novel mechanism to obtain an LSM-type theorem, wherein the symmetry responsible for the theorem originates from the kinematic constraints of a gauge theory. We identify a point in the parameter space at which the system is gapless. At the gapless point, the excitations admit a description in terms of free Dirac fermions with a constraint on the total fermion number. The asymptotic behavior of the two-point correlation function of the simplest local gauge-invariant quantity at the gapless point is found to be $ \propto \cos{(πr)}\,r^{-2/9}$, where rrr is the lattice separation between the two points. This model is also a natural platform to study phase diagram topological defects residing in families of SSB phases.
Quantum anomalies strongly constrain the possible behavior of many-body systems. A prime example is the Lieb-Schultz-Mattis (LSM) theorem, which relates UV symmetry and filling constraints to IR features of the energy spectrum and ground-state structure. Here, we ask how LSM constraints shape dynamical signatures and temporal correlations in open quantum systems. Using a spacetime duality, we show that the Liouvillian of a ddd-dimensional repeated quantum channel with a mixed anomaly between strong SSS and weak GGG symmetry can be mapped to a (d+1)(d{+}1)(d+1)-dimensional mixed-state symmetry-protected topological (mSPT) phase. Under this correspondence, the initial and steady states of the channel are identified with boundary states of the higher-dimensional mSPT in the presence of bulk projection. We further introduce the twisted Renyi-NNN correlator (TRNC) as a probe of temporal correlations in the channel and demonstrate that it is dual to the mSPT strange correlator, providing a direct bulk-boundary route to diagnose long-range temporal order implied by the LSM anomaly. Finally, we identify the \textit{Liouvillian singular spectrum}, rather than the Liouvillian spectrum itself, as a more fundamental diagnostic of quantum anomalies, and show that it is dual to the operator entanglement spectrum of the mSPT.
Following recent developments in the classification of bosonic short-range entangled phases, we examine many-body quantum systems whose ground state fractionalization obeys the Lieb-Schultz-Mattis (LSM) theorem. We generalize the topological classification of such phases by LSM anomalies (arXiv:1907.08204) to take magnetic and non-symmorphic lattice effects into account, and provide direct computations of the LSM anomaly in specific examples. We show that the anomaly-free condition coincides with established filling constraints (arXiv:1705.09298, arXiv:1505.04193), and we also derived new ones on novel crystalline quantum systems.
The Lieb-Schultz-Mattis (LSM) theorem and its higher-dimensional extensions forbid the existence of a unique, symmetric, and gapped ground state at fractional fillings in quantum many-body systems with a conserved particle number (or spin angular momentum) and the conventional translation symmetry of Euclidean lattices. In this work, we propose a generalization of the LSM theorem to quantum many-body systems on hyperbolic lattices, i.e., regular tessellations of two-dimensional negatively curved space. By leveraging concepts from hyperbolic band theory in a many-body setting, we adapt Oshikawa's flux-threading argument to periodic hyperbolic lattices with a non-Euclidean (Fuchsian) translation symmetry and compute a lower-bound to the ground-state degeneracy as a function of filling and lattice geometry. We explore the consequences of LSM constraints for gapped phases of hyperbolic quantum matter and suggest frustrated spin models on hyperbolic analogs of the square and triangular lattices as promising platforms for realizing symmetric spin liquids in hyperbolic space.