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アノマリーに基づくLieb-Schultz-Mattis ギャップ不可能性と臨界相の分類

Yuan Yao

Published Sep 18, 2020• 0 citations
Publisher

Abstract

We study (1 + 1)-dimensional SU(N ) spin systems in the presence of global SU(N ) rotation and lattice translation symmetries.Knowing the mixed anomaly of the two symmetries at low energy, we identify, by anomaly matching argument, a topological index for spin model-the total number of Young-tableaux boxes of spins per unit cell modulo N -characterizing the "ingappability" of the system.A nontrivial index implies either a ground-state degeneracy in a gapped phase, which can be thought of as a field-theory version of the Lieb-Schultz-Mattis (LSM) theorem, or a restriction of the possible universality classes in a critical phase, regarded as the symmetry-protected critical phases.As an example of the latter case, we show only a class of SU(N ) Wess-Zumino-Witten theories can be realized in the low-energy limit of the given lattice model in the presence of the symmetries.Similar constraints also apply when a higher global symmetry emerges in the model with a lower symmetry.We also predict a general constraint on the structure factor which is measurable in experiments.To generalize to higher dimensions, we introduce a new boundary condition which renders the flux-insertion argument for the LSM-type theorems in two or higher dimensional lattices free from the specific choice of system sizes.It also enables a formulation of the LSM-type theorems in arbitrary dimensions in terms of the anomaly in field theories of bulk correspondences.Moreover, by our new boundary condition, we also show that the higher-form U(1) LSM theorem and U(1) and SU(N ) LSM theorems in arbitrary higher dimensions have unified form in the bulk viewpoint.At last, we also discuss the discrete-symmetry LSM-type ingappabilities in the lattice language and we find that projective symmetry representations per unit cell implies that the system cannot be gapped with a unique translation-symmetric ground state.I would like to express my sincere gratitude to all those having helped me during my Master and Doctoral courses at the University of Tokyo.First of all, I would like to thank my parents who have supported me financially and encouraged me to do whatever I am keen on in the past 28 years.Also, I am grateful to every one that has taught me great knowledges on life and study at various steps.More specifically, I have to appreciate the guidance and kind supports by my supervisor, Professor Masaki Oshikawa who has given me patient guidance on study and kind financial supports.Moreover, I would thank Prof. Masahiro Sato for his cooperation in our paper, "Theory of electron spin resonance in one-dimensional topological insulators with spinorbit couplings: Detection of edge states".I am also grateful for Dr. Chang-Tse Hsieh for his collaboration on our work "Anomaly Matching and Symmetry-Protected Critical Phases in SU (N ) Spin Systems in 1+1 Dimensions", on which this Thesis is largely based, and Dr. Yoshiki Fukusumi on the work "Bosonization in a background U (1) gauge field".I would also thank Prof. Yuji Tachikawa and Prof. David Tong for patient and kind help on resolving my confusion during study.I would also like to acknowledge with much appreciation the help from our group members.

DOI
10.15083/0002006287

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We study (1 + 1)-dimensional SU(N ) spin systems in the presence of global SU(N ) rotation and lattice translation symmetries.Knowing the mixed anomaly of the two symmetries at low energy, we identify, by anomaly matching argument, a topological index for spin model-the total number of Young-tableaux boxes of spins per unit cell modulo N -characterizing the "ingappability" of the system.A nontrivial index implies either a ground-state degeneracy in a gapped phase, which can be thought of as a field-theory version of the Lieb-Schultz-Mattis (LSM) theorem, or a restriction of the possible universality classes in a critical phase, regarded as the symmetry-protected critical phases.As an example of the latter case, we show only a class of SU(N ) Wess-Zumino-Witten theories can be realized in the low-energy limit of the given lattice model in the presence of the symmetries.Similar constraints also apply when a higher global symmetry emerges in the model with a lower symmetry.We also predict a general constraint on the structure factor which is measurable in experiments.To generalize to higher dimensions, we introduce a new boundary condition which renders the flux-insertion argument for the LSM-type theorems in two or higher dimensional lattices free from the specific choice of system sizes.It also enables a formulation of the LSM-type theorems in arbitrary dimensions in terms of the anomaly in field theories of bulk correspondences.Moreover, by our new boundary condition, we also show that the higher-form U(1) LSM theorem and U(1) and SU(N ) LSM theorems in arbitrary higher dimensions have unified form in the bulk viewpoint.At last, we also discuss the discrete-symmetry LSM-type ingappabilities in the lattice language and we find that projective symmetry representations per unit cell implies that the system cannot be gapped with a unique translation-symmetric ground state.I would like to express my sincere gratitude to all those having helped me during my Master and Doctoral courses at the University of Tokyo.First of all, I would like to thank my parents who have supported me financially and encouraged me to do whatever I am keen on in the past 28 years.Also, I am grateful to every one that has taught me great knowledges on life and study at various steps.More specifically, I have to appreciate the guidance and kind supports by my supervisor, Professor Masaki Oshikawa who has given me patient guidance on study and kind financial supports.Moreover, I would thank Prof. Masahiro Sato for his cooperation in our paper, "Theory of electron spin resonance in one-dimensional topological insulators with spinorbit couplings: Detection of edge states".I am also grateful for Dr. Chang-Tse Hsieh for his collaboration on our work "Anomaly Matching and Symmetry-Protected Critical Phases in SU (N ) Spin Systems in 1+1 Dimensions", on which this Thesis is largely based, and Dr. Yoshiki Fukusumi on the work "Bosonization in a background U (1) gauge field".I would also thank Prof. Yuji Tachikawa and Prof. David Tong for patient and kind help on resolving my confusion during study.I would also like to acknowledge with much appreciation the help from our group members.
We study (1+1)-dimensional SU(N) spin systems in the presence of global SU(N) rotation and lattice translation symmetries. Knowing the mixed anomaly of the two symmetries at low energy, we identify, by the anomaly matching argument, a topological index for the spin model-the total number of Young-tableau boxes of spins per unit cell modulo N-characterizing the "ingappability" of the system. A nontrivial index implies either a ground-state degeneracy in a gapped phase, which can be thought of as a field-theory version of the Lieb-Schultz-Mattis theorem, or a restriction of the possible universality classes in a critical phase, regarded as the symmetry-protected critical phases. As an example of the latter case, we show that only a class of SU(N) Wess-Zumino-Witten theories can be realized in the low-energy limit of the given lattice model in the presence of the symmetries. Similar constraints also apply when a higher global symmetry emerges in the model with a lower symmetry. Our results agree with several examples known in previous studies of SU(N) models, and predict a general constraint on the structure factor which is measurable in experiments.
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 Lieb-Schultz-Mattis (LSM) theorem dictates that emergent low-energy states from a lattice model cannot be a trivial symmetric insulator if the filling per unit cell is not integral and if the lattice translation symmetry and particle number conservation are strictly imposed. In this paper, we compare the one-dimensional gapless states enforced by the LSM theorem and the boundaries of one-higher dimensional strong symmetry-protected topological (SPT) phases from the perspective of quantum anomalies. We first note that they can both be described by the same low-energy effective field theory with the same effective symmetry realizations on low-energy modes, wherein non-on-site lattice translation symmetry is encoded as if it were an internal symmetry. In spite of the identical form of the low-energy effective field theories, we show that the quantum anomalies of the theories play different roles in the two systems. In particular, we find that the chiral anomaly is equivalent to the LSM theorem, whereas there is another anomaly that is not related to the LSM theorem but is intrinsic to the SPT states. As an application, we extend the conventional LSM theorem to multiple-charge multiple-species problems and construct several exotic symmetric insulators. We also find that the (3+1)d chiral anomaly provides only the perturbative stability of the gaplessness local in the parameter space.
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.
The Lieb-Schultz-Mattis (LSM) theorem and its descendants impose strong constraints on the low-energy behavior of interacting quantum systems. In this paper, we formulate LSM-type constraints for lattice translation invariant systems with generalized U(1) symmetries which have recently appeared in the context of fracton phases: U(1) polynomial shift and subsystem symmetries. Starting with a generic interacting system with conserved dipole moment, we examine the conditions under which it supports a symmetric, gapped, and nondegenerate ground state, which we find requires that both the filling fraction and the bulk charge polarization take integer-values. Similar constraints are derived for systems with higher moment conservation laws or subsystem symmetries, in addition to lower bounds on the ground-state degeneracy when certain conditions are violated. Finally, we discuss the mapping between LSM-type constraints for subsystem symmetries and the anomalous symmetry action at boundaries of subsystem symmetric topological (SSPT) states.
The authors construct a Lieb-Schultz-Mattis (LSM)-type theorem applicable for abelian gauge theories without matter. This theorem is based on generalized global symmetries acting on Wilson lines. It provides strong constraints on possible phases, realized in lattice gauge theories. The generalized LSM theorem is demonstrated on the example of a U(1) gauge theory that simulates the quantum dimer model on a bipartite lattice. The authors find a mixed `t Hooft anomaly involving the generalized global symmetry at the Rokhsar-Kivelson critical point that corresponds to the LSM constraint.