Lieb-Schultz-Mattis (LSM) theorem and its various generalizations provide powerful guidance towards the search for novel phases of matter. Here, the authors propose and prove a generalized LSM theorem suitable for 2+1D lattice models of interacting bosons or spins, with both magnetic flux and fractional spin in the unit cell. One outcome of this theorem is that, under certain conditions, the gapped ground states preserving all symmetries must be a nontrivial symmetry-protected topological (SPT) phase. Such symmetry-enforced SPTs display a dyonic character in that they associate charge with symmetry flux, which is demonstrated using quasiexactly solvable models constructed by decorating quantum dimer models.Lieb-Schultz-Mattis (LSM) theorem and its various generalizations provide powerful guidance towards the search for novel phases of matter. Here, the authors propose and prove a generalized LSM theorem suitable for 2+1D lattice models of interacting bosons or spins, with both magnetic flux and fractional spin in the unit cell. One outcome of this theorem is that, under certain conditions, the gapped ground states preserving all symmetries must be a nontrivial symmetry-protected topological (SPT) phase. Such symmetry-enforced SPTs display a dyonic character in that they associate charge with symmetry flux, which is demonstrated using quasiexactly solvable models constructed by decorating quantum dimer models.

arXiv:1705.05421Physical review. B./Physical review. BSep 10, 201836 citations

The Lieb-Schultz-Mattis Theorem

Christopher Mudry, Claudio Chamon

Chapter 6 ties invertible topological phases to extensions of the original Lieb–Schultz–Mattis theorem. A review is made of the original Lieb–Schultz–Mattis theorem and how it has been refined under the assumption that a continuous symmetry holds. Two extensions of the Lieb–Schultz–Mattis theorem are given that apply to the Majorana chains from Chapter 5 when protected by discrete symmetries. To this end, it is necessary to introduce the notion of projective representations of symmetries and their classifications in terms of the second cohomology group. A precise definition is given of fermionic invertible topological phases and how they can be classified by the second cohomology group in one-dimensional space. Stacking rules of fermionic invertible topological phases in one-dimensional space are explained and shown to deliver the degeneracies of the boundary states that are protected by the symmetries.Chapter 6 ties invertible topological phases to extensions of the original Lieb–Schultz–Mattis theorem. A review is made of the original Lieb–Schultz–Mattis theorem and how it has been refined under the assumption that a continuous symmetry holds. Two extensions of the Lieb–Schultz–Mattis theorem are given that apply to the Majorana chains from Chapter 5 when protected by discrete symmetries. To this end, it is necessary to introduce the notion of projective representations of symmetries and their classifications in terms of the second cohomology group. A precise definition is given of fermionic invertible topological phases and how they can be classified by the second cohomology group in one-dimensional space. Stacking rules of fermionic invertible topological phases in one-dimensional space are explained and shown to deliver the degeneracies of the boundary states that are protected by the symmetries.

Cambridge University Press eBooksJan 9, 20250 citations

Matrix Product States for Modulated Topological Phases: Crystalline Equivalence Principle and Lieb-Schultz-Mattis Constraints

Shang-Qiang Ning, Hiromi Ebisu, Ho Tat Lam

Modulated symmetries are internal symmetries that act in a spatially non-uniform manner. Consequently, when a modulated symmetry $G_{\text{int}}$ is combined with a spatial symmetry $G_{\text{sp}}$ , the total symmetry group takes the form of a semidirect product $G=G_{\text{int}}\rtimes G_{\text{sp}}$ . Using matrix product states, we classify topological phases protected by modulated symmetries together with spatial symmetries in one spatial dimension. We show that these modulated symmetry-protected topological (SPT) phases are classified by $H^{2}(G,U(1)_s)$ , in agreement with the crystalline equivalence principle, which states that SPT phases protected by symmetries involving spatial elements are in one-to-one correspondence with internal SPT phases protected by the same symmetries, viewed as acting internally. Furthermore, we provide a matrix product state derivation of the Lyndon-Hochschild-Serre spectral sequence for the corresponding internal SPT phases, which enables us to construct an explicit correspondence between modulated SPT phases and internal SPT phases. As applications of this classification, we prove a Lieb-Schultz-Mattis (LSM) theorem for modulated symmetries that forbids the existence of symmetric short-ranged entangled ground state, as well as an SPT-LSM constraint that enforces nontrivial entanglement in the SPT ground state. Finally, we use the classification to establish a similar LSM-type constraints for non-invertible Kramers-Wannier reflection symmetries.Modulated symmetries are internal symmetries that act in a spatially non-uniform manner. Consequently, when a modulated symmetry $G_{\text{int}}$ is combined with a spatial symmetry $G_{\text{sp}}$, the total symmetry group takes the form of a semidirect product $G=G_{\text{int}}\rtimes G_{\text{sp}}$. Using matrix product states, we classify topological phases protected by modulated symmetries together with spatial symmetries in one spatial dimension. We show that these modulated symmetry-protected topological (SPT) phases are classified by $H^{2}(G,U(1)_s)$, in agreement with the crystalline equivalence principle, which states that SPT phases protected by symmetries involving spatial elements are in one-to-one correspondence with internal SPT phases protected by the same symmetries, viewed as acting internally. Furthermore, we provide a matrix product state derivation of the Lyndon-Hochschild-Serre spectral sequence for the corresponding internal SPT phases, which enables us to construct an explicit correspondence between modulated SPT phases and internal SPT phases. As applications of this classification, we prove a Lieb-Schultz-Mattis (LSM) theorem for modulated symmetries that forbids the existence of symmetric short-ranged entangled ground state, as well as an SPT-LSM constraint that enforces nontrivial entanglement in the SPT ground state. Finally, we use the classification to establish a similar LSM-type constraints for non-invertible Kramers-Wannier reflection symmetries.

cond-mat.str-elCross-listedarXiv:2603.19381Mar 23, 20260 citations

Topological phases protected by point group symmetry

Sheng-Jie Huang, Hao Song, Michael Hermele

Bulletin of the American Physical SocietyMar 16, 20160 citations

Modulated symmetries are internal symmetries that act in a spatially non-uniform manner. Consequently, when a modulated symmetry

G_{\text{int}}

is combined with a spatial symmetry

G_{\text{sp}}

, the total symmetry group takes the form of a semidirect product

G=G_{\text{int}}\rtimes G_{\text{sp}}

. Using matrix product states, we classify topological phases protected by modulated symmetries together with spatial symmetries in one spatial dimension. We show that these modulated symmetry-protected topological (SPT) phases are classified by

H^{2}(G,U(1)_s)

, in agreement with the crystalline equivalence principle, which states that SPT phases protected by symmetries involving spatial elements are in one-to-one correspondence with internal SPT phases protected by the same symmetries, viewed as acting internally. Furthermore, we provide a matrix product state derivation of the Lyndon-Hochschild-Serre spectral sequence for the corresponding internal SPT phases, which enables us to construct an explicit correspondence between modulated SPT phases and internal SPT phases. As applications of this classification, we prove a Lieb-Schultz-Mattis (LSM) theorem for modulated symmetries that forbids the existence of symmetric short-ranged entangled ground state, as well as an SPT-LSM constraint that enforces nontrivial entanglement in the SPT ground state. Finally, we use the classification to establish a similar LSM-type constraints for non-invertible Kramers-Wannier reflection symmetries.