A general approach to Lieb-Schultz-Mattis type results in quantum spin systems

Yoshiko Ogata, Yuji Tachikawa, Hal Tasaki

Anton Kapustin, Nikita Sopenko

Hiromi Ebisu, Bo Han, Weiguang Cao

Symmetries rigidly delimit the landscape of quantum matter. Recently uncovered spatially modulated symmetries, whose actions vary with position, enable excitations with restricted mobility, while Lieb-Schultz-Mattis (LSM) type anomalies impose sharp constraints on which lattice phases are realizable. In a one-dimensional spin chain, gauging procedures have linked modulated symmetry to LSM type anomaly, but a general understanding beyond 1D remains incomplete. We show that generalized LSM anomalies force the emergence of the modulated symmetries upon gauging. We construct explicit lattice models in two and three spatial dimensions and develop complementary field-theoretic descriptions in arbitrary spatial dimensions that connect LSM anomaly inflow to higher-group symmetry structures governing the modulated symmetries. Our results provide a unified, nonperturbative framework that ties together LSM constraints and spatially modulated symmetries across dimensions.

Takaichi Isoyama, Kiyohide Nomura

In this study, we consider one-dimensional (1D) quantum spin systems with translation and discrete symmetries (spin reversal, space inversion, and time reversal symmetries). By combining the continuous U(1) symmetry with the discrete symmetries and using the extended Lieb–Schultz–Mattis (LSM) theorem [E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16, 407 (1961); K. Nomura, J. Morishige, and T. Isoyama, J. Phys. A 48, 375001 (2015)], we investigate the relation between the ground states, energy spectra, and symmetries. For half-integer spin cases, we generalize the dimer and Néel concepts using the discrete symmetries, and we can reconcile the LSM theorem with the dimer or Néel states, since there was a subtle dilemma. Furthermore, a part of discrete symmetries is enough to classify possible phases. Thus we can deepen our understanding of the relation between the LSM theorem and discrete symmetries.

Hal Tasaki

Hal Tasaki

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.