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arXiv:2405.14949 · Physical review. B./Physical review. B

Lieb-Schultz-Mattis theorem with long-range interactions

Ruochen Ma

Published Sep 5, 2024• 6 citations
arXivHTMLPDF

Abstract

The Lieb-Schultz-Mattis theorem states that a finite-range Hamiltonian in translationally and spin-rotationally invariant spin-1/2 systems cannot have a unique gapped ground state. Here, the author extends the theorem to include Hamiltonians with terms that act on a finite number of sites, but which may be far apart, provided that the interaction strength decays with distance faster than a certain power law. As the spatial dimension increases, the power law required to guarantee the result demands faster decay.

DOI
10.1103/physrevb.110.104412

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The Lieb-Schultz-Mattis theorem states that a finite-range Hamiltonian in translationally and spin-rotationally invariant spin-1/2 systems cannot have a unique gapped ground state. Here, the author extends the theorem to include Hamiltonians with terms that act on a finite number of sites, but which may be far apart, provided that the interaction strength decays with distance faster than a certain power law. As the spatial dimension increases, the power law required to guarantee the result demands faster decay.
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 establish multiple interrelated, fundamental results in quantum many-body systems that can have long-range interactions. For a sufficiently long quantum spin chain, we first show that if the multispin interactions in the Hamiltonian decay fast enough as their ranges increase and the Hamiltonian is gapped, then the ground states satisfy the entanglement area law, even if there is a ground-state degeneracy due to a spontaneously broken discrete symmetry. This area law also holds for certain excited states. Second, if such a long-range interacting Hamiltonian has an anomalous symmetry, then the Lieb-Schultz-Mattis theorem applies, i.e., the Hamiltonian cannot have a unique gapped symmetric ground state. If the Hamiltonian contains only 2-spin interactions, then these results hold when the interactions decay faster than 1/r21/{r}^{2}1/r2, with rrr the distance between the two interacting spins. Third, we show that pure states with an anomalous symmetry, which may not be a ground state of any natural Hamiltonian, must be long-range entangled. The symmetries we consider include on-site internal symmetries combined with lattice translation symmetries, and they can also extend to purely internal but non-on-site symmetries. Moreover, these internal symmetries can be discrete or continuous. We explore the applications of these results through various examples.
We review the Lieb–Schultz–Mattis theorem and its variants, which are no-go theorems that state that a quantum many-body system with certain conditions cannot have a locally-unique gapped ground state. We restrict ourselves to one-dimensional quantum spin systems and discuss both the generalized Lieb–Schultz–Mattis theorem for models with U(1) symmetry and the extended Lieb–Schultz–Mattis theorem for models with discrete symmetry. We also discuss the implication of the same arguments to systems on the infinite cylinder, both with the periodic boundary conditions and with the spiral boundary conditions. For models with U(1) symmetry, we here present a rearranged version of the original proof of Lieb, Schultz, and Mattis based on the twist operator. As the title suggests, we take a modern topological point of view and prove the generalized Lieb–Schultz–Mattis theorem by making use of a topological index (which coincides with the filling factor). By a topological index, we mean an index that characterizes a locally-unique gapped ground state and is invariant under continuous (or smooth) modification of the ground state. For models with discrete symmetry, we describe the basic idea of the most general proof based on the topological index introduced in the context of symmetry-protected topological phases. We start from background materials such as the classification of projective representations of the symmetry group. We also review the notion that we call a locally-unique gapped ground state of a quantum spin system on an infinite lattice and present basic theorems. This notion turns out to be natural and useful from the physicists’ point of view. We have tried to make the present chapter readable and almost self-contained. We only assume basic knowledge about quantum spin systems.
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.
We propose a geometric approach to the Lieb-Schultz-Mattis theorem for quantum many-body systems with discrete spin-rotation symmetries and lattice inversion or rotation symmetry, but without translation symmetry assumed. Under symmetry twisting on a $(d\ensuremath{-}1)$-dimensional plane, we find that any ddd-dimensional inversion-symmetric spin system possesses a doubly degenerate spectrum when it hosts a half-integer spin at the inversion-symmetric point. We also show that any rotation-symmetric generalized spin model with a projective representation at the rotation center has a similar degeneracy under symmetry twisting. We argue that these degeneracies imply that a unique symmetric gapped ground state that is smoothly connected to product states is forbidden in the original untwisted systems---generalized inversional or rotational Lieb-Schultz-Mattis theorems without lattice translation symmetry imposed. The traditional Lieb-Schultz-Mattis theorems with translations also fit in the proposed framework.