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arXiv:2212.14605 · Physical review. B./Physical review. B

Obstructions to gapped phases from noninvertible symmetries

Anuj Apte, Clay Córdova, Ho Tat Lam

Published Jul 20, 2023• 77 citations
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Abstract

Our understanding of quantum symmetry of systems has considerably broadened over the last decade. The idea of symmetry has become intrinsically linked with topology described and algebraically characterized by higher categories. Studying three-dimensional systems with noninvertible symmetries, the authors show here that these symmetries are in general incompatible with a unique gapped ground state. Their results extend the ideas behind the Lieb-Shultz-Mattis theorem to the arena of higher-dimensional field theories invariant under a novel class of symmetries.

DOI
10.1103/physrevb.108.045134

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Our understanding of quantum symmetry of systems has considerably broadened over the last decade. The idea of symmetry has become intrinsically linked with topology described and algebraically characterized by higher categories. Studying three-dimensional systems with noninvertible symmetries, the authors show here that these symmetries are in general incompatible with a unique gapped ground state. Their results extend the ideas behind the Lieb-Shultz-Mattis theorem to the arena of higher-dimensional field theories invariant under a novel class of symmetries.
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.
Symmetry provides powerful non-perturbative constraints in quantum many-body systems. A prominent example is the Lieb-Schultz-Mattis (LSM) anomaly -- a mixed 't Hooft anomaly between internal and translational symmetries that forbids a trivial symmetric gapped phase. In this work, we investigate lattice translation operators in systems with an LSM anomaly. We construct explicit lattice models in two and three spatial dimensions and show that, after gauging the full internal symmetry, translation becomes non-invertible and fuses into defects of the internal symmetry. The result is supported by the anomaly-inflow in view of topological field theory. Our work extends earlier one-dimensional observations to a unified higher-dimensional framework and clarifies their origin in mixed anomalies and higher-group structures, highlighting a coherent interplay between internal and crystalline symmetries.
The celebrated Lieb-Schultz-Mattis theorem excludes a nondegenerate gapped ground state of a spin system with an odd number of s=12s=\frac{1}{2}s=21​ spins per unit cell. Here, the author presents a new family of Lieb-Schultz-Mattis type theorems for fermionic systems that have no analog in spin systems. An example is a particle-hole symmetric system of spinless fermions on a lattice with an odd number of fermionic modes per unit cell. The author shows that a symmetric gapped ground state must be topologically ordered, and the lattice translation symmetries must permute anyon types, resulting in non-Abelian zero modes bound to lattice dislocations. These results can be interpreted as quantum anomalies on the surface of three-dimensional fermionic symmetry-protected phases.
Anomalies are renormalization group invariants that constrain the dynamics of quantum field theories. We show that certain anomalies for discrete global symmetries imply that the underlying theory either spontaneously breaks its generalized global symmetry or is gapless. We identify an obstruction, formulated in terms of the anomaly inflow action, that must vanish if a symmetry preserving gapped phase, i.e. a unitary topological quantum field theory, exits with the given anomaly. Our result is similar to the 2d2d2d Lieb-Schultz-Mattis theorem but applies more broadly to continuum theories in general spacetime dimension with various types of discrete symmetries including higher-form global symmetries. As a particular application, we use our result to prove that certain 4d4d4d non-abelian gauge theories at θ=πθ=πθ=π cannot flow at long distances to a phase which simultaneously, preserves time-reversal symmetry, is confining, and is gapped. We also apply our obstruction to 4d4d4d adjoint QCD and constrain its dynamics.
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.