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arXiv:2112.06575 · Physical Review Letters

Gappability Index for Quantum Many-Body Systems

Yuan Yao, Masaki Oshikawa, Akira Furusaki

Published Jul 1, 2022• 7 citations
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Abstract

We propose an index I_{G} which characterizes the degree of gappability, namely the difficulty to induce a unique ground state with a nonvanishing excitation gap, in the presence of a symmetry G. I_{G} represents the dimension of the subspace of ambient uniquely gapped theories in the entire G-invariant "theory space." The celebrated Lieb-Schultz-Mattis theorem corresponds, in our formulation, to the case I_{G}=0 (completely ingappable) for the symmetry G including the lattice translation symmetry. We illustrate the usefulness of the index by discussing the phase diagram of spin-1/2 antiferromagnets in various dimensions, which do not necessarily have the translation symmetry.

DOI
10.1103/physrevlett.129.017204

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We propose an index I_{G} which characterizes the degree of gappability, namely the difficulty to induce a unique ground state with a nonvanishing excitation gap, in the presence of a symmetry G. I_{G} represents the dimension of the subspace of ambient uniquely gapped theories in the entire G-invariant "theory space." The celebrated Lieb-Schultz-Mattis theorem corresponds, in our formulation, to the case I_{G}=0 (completely ingappable) for the symmetry G including the lattice translation symmetry. We illustrate the usefulness of the index by discussing the phase diagram of spin-1/2 antiferromagnets in various dimensions, which do not necessarily have the translation symmetry.
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.
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.
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 discuss quantum many-body systems with lattice translation and discrete on-site symmetries. We point out that, under a boundary condition twisted by a symmetry operation, there is an exact degeneracy of ground states if the unit cell forms a projective representation of the on-site discrete symmetry. Based on the quantum transfer matrix formalism, we show that, if the system is gapped, the ground-state degeneracy under the twisted boundary condition also implies a ground-state (quasi)degeneracy under the periodic boundary conditions. This gives a compelling evidence for the recently proposed Lieb-Schultz-Mattis-type ingappability due to the on-site discrete symmetry in two and higher dimensions.
We study quantum many-body systems in the presence of an exotic antiunitary translation or inversion symmetry involving time reversal. Based on a symmetry-twisting method and spectrum robustness, we propose that a half-integer spin chain that respects any of these two antiunitary crystalline symmetries in addition to the discrete Z_{2}×Z_{2} global spin-rotation symmetry must either be gapless or possess degenerate ground states. This explains the gaplessness of a class of chiral spin models not indicated by the Lieb-Schultz-Mattis theorem and its known extensions. Moreover, we present symmetry classes with minimal sets of generators that give nontrivial Lieb-Schultz-Mattis-type constraints, argued by the bulk-boundary correspondence in 2D symmetry-protected topological phases as well as lattice homotopy. Our results for detecting the ingappability of 1D quantum magnets from the interplay between spin-rotation symmetries and magnetic space groups are applicable to systems with a broader class of spin interactions, including Dzyaloshinskii-Moriya and triple-product interactions.