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 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.

Sep 18, 20200 citations

Symmetry Protection of Critical Phases and a Global Anomaly in 1+1 Dimensions

Shunsuke C. Furuya, Masaki Oshikawa

We derive a selection rule among the (1+1)-dimensional SU(2) Wess-Zumino-Witten theories, based on the global anomaly of the discrete Z_{2} symmetry found by Gepner and Witten. In the presence of both the SU(2) and Z_{2} symmetries, a renormalization-group flow is possible between level-k and level-k^{'} Wess-Zumino-Witten theories only if k≡k^{'} mod 2 . This classifies the Lorentz-invariant, SU(2)-symmetric critical behavior into two "symmetry-protected" categories corresponding to even and odd levels, restricting possible gapless critical behavior of translation-invariant quantum spin chains.We derive a selection rule among the (1+1)-dimensional SU(2) Wess-Zumino-Witten theories, based on the global anomaly of the discrete Z_{2} symmetry found by Gepner and Witten. In the presence of both the SU(2) and Z_{2} symmetries, a renormalization-group flow is possible between level-k and level-k^{'} Wess-Zumino-Witten theories only if k≡k^{'} mod 2 . This classifies the Lorentz-invariant, SU(2)-symmetric critical behavior into two "symmetry-protected" categories corresponding to even and odd levels, restricting possible gapless critical behavior of translation-invariant quantum spin chains.

arXiv:1503.07292Physical Review LettersJan 11, 201786 citations

Non-invertible translation from Lieb-Schultz-Mattis anomaly

Tsubasa Oishi, Takuma Saito, Hiromi Ebisu

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.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.

cond-mat.str-elCross-listedarXiv:2601.21625Physical review. B./Physical review. BJan 30, 20260 citations

Fermionic Lieb-Schultz-Mattis theorems and weak symmetry-protected phases

Meng Cheng

The celebrated Lieb-Schultz-Mattis theorem excludes a nondegenerate gapped ground state of a spin system with an odd number of $s=\frac{1}{2}$ 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.The celebrated Lieb-Schultz-Mattis theorem excludes a nondegenerate gapped ground state of a spin system with an odd number of $s=\frac{1}{2}$ 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.

arXiv:1804.10122Physical review. B./Physical review. BFeb 21, 201968 citations

Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases

Shenghan Jiang, Meng Cheng, Yang Qi +1 more

We propose and prove a family of generalized Lieb-Schultz-Mattis~(LSM) theorems for symmetry protected topological~(SPT) phases on boson/spin models in any dimensions. The ``conventional'' LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin-1/2 particles per unit cell, forbids a symmetric short-range-entangled ground state in such a system. Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a non-trivial SPT phase with anomalous boundary excitations. Depending on models, they can be either strong or ``higher-order'' crystalline SPT phases, characterized by non-trivial surface/hinge/corner states. Furthermore, given the symmetry group and the spatial assignment of fractional spins, we are able to determine all possible SPT phases for a symmetric ground state, using the real space construction for SPT phases based on the spectral sequence of cohomology theory. We provide examples in one, two and three spatial dimensions, and discuss possible physical realization of these SPT phases based on condensation of topological excitations in fractionalized phases.We propose and prove a family of generalized Lieb-Schultz-Mattis~(LSM) theorems for symmetry protected topological~(SPT) phases on boson/spin models in any dimensions. The ``conventional'' LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin-1/2 particles per unit cell, forbids a symmetric short-range-entangled ground state in such a system. Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a non-trivial SPT phase with anomalous boundary excitations. Depending on models, they can be either strong or ``higher-order'' crystalline SPT phases, characterized by non-trivial surface/hinge/corner states. Furthermore, given the symmetry group and the spatial assignment of fractional spins, we are able to determine all possible SPT phases for a symmetric ground state, using the real space construction for SPT phases based on the spectral sequence of cohomology theory. We provide examples in one, two and three spatial dimensions, and discuss possible physical realization of these SPT phases based on condensation of topological excitations in fractionalized phases.

arXiv:1907.08596SciPost PhysicsAug 6, 202122 citations

Exact Quantization of Topological Order Parameter in SU ( N ) Spin Models, N -ality Transformation and Ingappabilities

Hang Su, Yuan Yao, Akira Furusaki

We show that the ground-state expectation value of twisting operator is a topological order parameter for U(1)- and Z_{N}-symmetric symmetry-protected topological (SPT) phases in one-dimensional "spin" systems-it is quantized in the thermodynamic limit and can be used to identify different SPT phases and to diagnose phase transitions among them. We prove that this (nonlocal) order parameter must take values in Nth roots of unity, and its value can be changed by a generalized lattice translation acting as an N-ality transformation connecting distinct phases. This result also implies the Lieb-Schultz-Mattis (LSM) ingappability for SU(N) spins if we further impose a general translation symmetry. Furthermore, our exact result for the order parameter of SPT phases can predict a large number of LSM ingappabilities by the general lattice translation. We also apply the N-ality property to provide an efficient way to construct possible multicritical phase transitions starting from a single Hamiltonian with a unique gapped ground state.We show that the ground-state expectation value of twisting operator is a topological order parameter for U(1)- and Z_{N}-symmetric symmetry-protected topological (SPT) phases in one-dimensional "spin" systems-it is quantized in the thermodynamic limit and can be used to identify different SPT phases and to diagnose phase transitions among them. We prove that this (nonlocal) order parameter must take values in Nth roots of unity, and its value can be changed by a generalized lattice translation acting as an N-ality transformation connecting distinct phases. This result also implies the Lieb-Schultz-Mattis (LSM) ingappability for SU(N) spins if we further impose a general translation symmetry. Furthermore, our exact result for the order parameter of SPT phases can predict a large number of LSM ingappabilities by the general lattice translation. We also apply the N-ality property to provide an efficient way to construct possible multicritical phase transitions starting from a single Hamiltonian with a unique gapped ground state.

arXiv:2406.05407Physical Review LettersDec 31, 20241 citation

The celebrated Lieb-Schultz-Mattis theorem excludes a nondegenerate gapped ground state of a spin system with an odd number of

s=\frac{1}{2}

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.