Lieb-Schultz-Mattis Theorem and Its Relation to Topological States

The theorem of Lieb, Schultz and Mattis (LSM), which states that the S=1/2 XXZ spin chain has gapless or degenerate ground states, can be applied to broader models. However, in the original LSM theorem the non-frustrating system was assumed. Independently, Kolb considered the relation between the wave number $q$ and the twisting boundary condition. In our previous paper, reconsidering the LSM theorem and combining Kolb's discussion, we have extended the LSMK theorem for frustrating or non-symmetric cases. However, there remains a complexity in the proof of continuity. In this paper, we will simplify the proof than the previous work. Keywords: Lieb-Schultz-Mattis, rigorous theorem, frustration, one-dimension, topological aspect, Dzyalosinskii-MoriyaThe theorem of Lieb, Schultz and Mattis (LSM), which states that the S=1/2 XXZ spin chain has gapless or degenerate ground states, can be applied to broader models. However, in the original LSM theorem the non-frustrating system was assumed. Independently, Kolb considered the relation between the wave number $q$ and the twisting boundary condition. In our previous paper, reconsidering the LSM theorem and combining Kolb's discussion, we have extended the LSMK theorem for frustrating or non-symmetric cases. However, there remains a complexity in the proof of continuity. In this paper, we will simplify the proof than the previous work. Keywords: Lieb-Schultz-Mattis, rigorous theorem, frustration, one-dimension, topological aspect, Dzyalosinskii-Moriya

Aug 22, 20160 citations

Anomaly manifestation of Lieb-Schultz-Mattis theorem and topological phases

Gil Young Cho, Chang-Tse Hsieh, Shinsei Ryu

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.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.

arXiv:1705.03892Physical review. B./Physical review. BNov 2, 2017129 citations

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and the twisting boundary condition. In our previous paper, reconsidering the LSM theorem and combining Kolb's discussion, we have extended the LSMK theorem for frustrating or non-symmetric cases. However, there remains a complexity in the proof of continuity. In this paper, we will simplify the proof than the previous work. Keywords: Lieb-Schultz-Mattis, rigorous theorem, frustration, one-dimension, topological aspect, Dzyalosinskii-Moriya