Lieb-Schultz-Mattis (LSM) theorems provide powerful constraints on the problem, i.e. whether a quantum phase or phase transition can emerge in a many-body system. We derive the topological partition functions that characterize the LSM constraints in spin systems with G_s\times G_{int} G s × G i n t symmetry, where G_s G s is an arbitrary space group in one or two spatial dimensions, and G_{int} G i n t is any internal symmetry whose projective representations are classified by \mathbb{Z}_2^k ℤ 2 k with k k an integer. We then apply these results to study the emergibility of a class of exotic quantum critical states, including the well-known deconfined quantum critical point (DQCP), U(1) U ( 1 ) Dirac spin liquid (DSL), and the recently proposed non-Lagrangian Stiefel liquid. These states can emerge as a consequence of the competition between a magnetic state and a non-magnetic state. We identify all possible realizations of these states on systems with SO(3)\times \mathbb{Z}_2^T S O ( 3 ) × ℤ 2 T internal symmetry and either p6m p 6 m or p4m p 4 m lattice symmetry. Many interesting examples are discovered, including a DQCP adjacent to a ferromagnet, stable DSLs on square and honeycomb lattices, and a class of quantum critical spin-quadrupolar liquids of which the most relevant spinful fluctuations carry spin- 2 2 . In particular, there is a realization of spin-quadrupolar DSL that is beyond the usual parton construction. We further use our formalism to analyze the stability of these states under symmetry-breaking perturbations, such as spin-orbit coupling. As a concrete example, we find that a DSL can be stable in a recently proposed candidate material, NaYbO _2 2 .
Lieb-Schultz-Mattis (LSM) theorems provide powerful constraints on the problem, i.e. whether a quantum phase or phase transition can emerge in a many-body system. We derive the topological partition functions that characterize the LSM constraints in spin systems with G_s\times G_{int} G s × G i n t symmetry, where G_s G s is an arbitrary space group in one or two spatial dimensions, and G_{int} G i n t is any internal symmetry whose projective representations are classified by \mathbb{Z}_2^k ℤ 2 k with k k an integer. We then apply these results to study the emergibility of a class of exotic quantum critical states, including the well-known deconfined quantum critical point (DQCP), U(1) U ( 1 ) Dirac spin liquid (DSL), and the recently proposed non-Lagrangian Stiefel liquid. These states can emerge as a consequence of the competition between a magnetic state and a non-magnetic state. We identify all possible realizations of these states on systems with SO(3)\times \mathbb{Z}_2^T S O ( 3 ) × ℤ 2 T internal symmetry and either p6m p 6 m or p4m p 4 m lattice symmetry. Many interesting examples are discovered, including a DQCP adjacent to a ferromagnet, stable DSLs on square and honeycomb lattices, and a class of quantum critical spin-quadrupolar liquids of which the most relevant spinful fluctuations carry spin- 2 2 . In particular, there is a realization of spin-quadrupolar DSL that is beyond the usual parton construction. We further use our formalism to analyze the stability of these states under symmetry-breaking perturbations, such as spin-orbit coupling. As a concrete example, we find that a DSL can be stable in a recently proposed candidate material, NaYbO _2 2 .
Lieb-Schultz-Mattis (LSM) theorems impose non-perturbative constraints on the zero-temperature phase diagrams of quantum lattice Hamiltonians (always assumed to be local in this paper). LSM theorems have recently been interpreted as the lattice counterparts to mixed ’t Hooft anomalies in quantum field theories that arise from a combination of crystalline and global internal symmetry groups. Accordingly, LSM theorems have been reinterpreted as LSM anomalies. In this work, we provide a systematic diagnostic for LSM anomalies in one spatial dimension. We show that gauging subgroups of the global internal symmetry group of a quantum lattice model obeying an LSM anomaly delivers a dual quantum lattice Hamiltonian such that its internal and crystalline symmetries mix non-trivially through a group extension. This mixing of crystalline and internal symmetries after gauging is a direct consequence of the LSM anomaly, i.e., it can be used as a diagnostic of an LSM anomaly. We exemplify this procedure for a quantum spin-1/2 chain obeying an LSM anomaly resulting from combining a global internal \mathbb{Z}^{\,}_{2}×\mathbb{Z}^{\,}_{2} ℤ 2 × ℤ 2 symmetry with translation or reflection symmetry. We establish a triality of models by gauging a \mathbb{Z}^{\,}_{2}\subset\mathbb{Z}^{\,}_{2}×\mathbb{Z}^{\,}_{2} ℤ 2 ⊂ ℤ 2 × ℤ 2 symmetry in two ways, one of which amounts to performing a Kramers-Wannier duality, while the other implements a Jordan-Wigner duality. We discuss the mapping of the phase diagram of the quantum spin-1/2 XYZ X Y Z chains under such a triality. We show that the deconfined quantum critical transitions between Neel and dimer orders are mapped to either topological or conventional Landau-Ginzburg transitions. Finally, we extend our results to \mathbb{Z}^{\,}_{n} ℤ n clock models with \mathbb{Z}^{\,}_{n}×\mathbb{Z}^{\,}_{n} ℤ n × ℤ n global internal symmetry, and provide a reinterpretation of the dual internal symmetries in terms of \mathbb{Z}^{\,}_{n} ℤ n charge and dipole symmetries.
Topological phases and quantum criticality from SU(2) Chern-Simons-matter theories
Motivated by recent numerical studies where various SU(2) Chern-Simons-matter theories emerge, we analytically study topological phases and quantum criticality in two-dimensional systems described by such theories. First, we classify SU(2)k topological orders in all lattice spin systems with a p4×SO(3) symmetry, where k is an arbitrary nonzero integer. We find that for each odd k, the topological order can emerge in systems with an arbitrary Lieb-Schultz-Mattis (LSM) anomaly, and the symmetry cannot permute anyons. If the system has a nontrivial (respectively, trivial) LSM anomaly, then there is exactly one (respectively, nine) symmetry-enriched topological (SET) phases. On the other hand, SU(2)k topological order with any even k can only emerge in systems with a trivial LSM anomaly. If k∈/{6,10,14,⋯}, the symmetry cannot permute anyons, and there are 16 SET phases. If k∈{6,10,14,⋯}, there are 4 different ways how the symmetry can permute anyons, and there are 64 SET phases. Next, we analyze the SU(2)k Chern-Simons theories coupled to Nf flavors of gapless matter fields that can be either bosonic or fermionic. For both types of theories, we consider a joint large-Nf and large-k limit with Nf/k fixed, and compute the scaling dimensions of the bilinear operators of the bosons or fermions to the order of 1/Nf. These results sharpen our understanding of these emergent exotic topological phases and quantum criticality, and provide useful guidance to explore them further.
We study quantum phases and phase transitions in a one-dimensional interacting fermion system with a Lieb-Schultz-Mattis (LSM) type anomaly. Specifically, the inversion symmetry enforces any symmetry-preserving gapped ground state of the system to be a Kitaev chain, following an LSM type theorem that we prove. Alternatively, via the Jordan-Wigner transformation, this system describes a spin system whose gapped ground states must break either the inversion or the Ising symmetry associated with fermion parity. We obtain a phase diagram using analytical methods and variational matrix product state simulations and study the critical behaviors of the quantum phase transitions therein using entanglement entropy, energy variance, and finite-size scaling of order parameters. In particular, we observe a continuous phase transition between two ordered phases that are beyond the Ginzburg-Landau-Wilson paradigm, in analogy to the deconfined quantum critical points in two spatial dimensions. We show this type of a 1D deconfined quantum critical point is described by the Tomonaga-Luttinger liquid theory and extract its Luttinger parameter and critical exponents. We also identify a gapless phase that emerges from a deconfined quantum critical point between two ordered phases, which cannot be described by a U(1) Luttinger liquid.
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.
We study deconfined quantum critical points (DQCP) associated with Lieb-Schultz-Mattis (LSM) anomalies in one-dimensional spin chains. Our starting point is a structural characterization of the LSM anomaly in the Lyndon-Hochschild-Serre spectral sequence: ωLSM∈E∞1,2=H1(Ztrans,H2(Gint,U(1)))⊆H3(Gint⋊ρZtrans,U(1)). Physically, this class decorates a translation defect with a projective representation of the internal symmetry Gint. We show that gauging the internal symmetry in the presence of an E∞1,2-type anomaly necessarily produces a non-invertible dual symmetry. This gives a general mechanism for type-II DQCP: in contrast to type-I examples with E∞2,1-type anomalies which are dual to ordinary group-like symmetry breaking, type-II transitions are dual to spontaneous breaking of a non-invertible symmetry. We illustrate the mechanism using a spin-1/2 chain with an anomalous D8 LSM symmetry. We construct a dimer-to-ferromagnet DQCP candidate, provide numerical evidence for a critical theory with central charge c≈1, and show, using both category theory and explicit lattice constructions, that gauging the internal symmetry yields the non-invertible Rep(H8) dual symmetry.
We present a systematic framework to classify symmetry-enriched topological quantum spin liquids in two spatial dimensions. This framework can deal with all topological quantum spin liquids, which may be either Abelian or non-Abelian and chiral or nonchiral. It can systematically treat a general symmetry, which may include both lattice symmetry and internal symmetry, may contain antiunitary symmetry, and may permute anyons. The framework applies to all types of lattices and can systematically distinguish different lattice systems with the same symmetry group using their quantum anomalies, which are sometimes known as Lieb-Schultz-Mattis anomalies. We apply this framework to classify U ( 1 ) 2 N chiral states and non-Abelian Ising ( ν ) states enriched by a p 6 × SO ( 3 ) or p 4 × SO ( 3 ) symmetry and Z N topological orders and U ( 1 ) 2 N × U ( 1 ) − 2 N topological orders enriched by a p 6 m × SO ( 3 ) × Z 2 T , p 4 m × SO ( 3 ) × Z 2 T , p 6 m × Z 2 T , or p 4 m × Z 2 T symmetry, where p 6 , p 4 , p 6 m , and p 4 m are lattice symmetries while SO(3) and Z 2 T are spin rotation and time-reversal symmetries, respectively. In particular, we identify symmetry-enriched topological quantum spin liquids that are not easily captured by the usual parton-mean-field approach, including examples with the familiar Z 2 topological order. Published by the American Physical Society 2024