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 one-dimensional disordered systems with average non-invertible symmetries, where quenched disorder may locally break part of the symmetry while preserving it upon disorder averaging. A canonical example is the random transverse-field Ising model, which at criticality exhibits an average Kramers-Wannier duality. We consider the general setting in which the full symmetry is described by a
G-graded fusion category
B, whose identity component
A remains exact, while the components with nontrivial
G-grading are realized either exactly or only on average. We develop a topological holographic framework that encodes the symmetry data of the 1D system in a 2D topological order
Z[A] (the Drinfeld center of
A), enriched by an exact or, respectively, average
G symmetry. Within this framework, we obtain a complete classification of anomalies and average symmetry-protected topological (SPT) phases: when the components with nontrivial
G-grading are realized only on average, the symmetry is anomaly-free if and only if
Z[A] admits a magnetic Lagrangian algebra that is invariant under the permutation action of
G on anyons. When an anomaly is present, we show that the ground state of a single disorder realization is long-range entangled with probability one in the thermodynamic limit, and is expected to exhibit power-law Griffiths singularities in the low-energy spectrum. Finally, we present an explicit, exactly solvable lattice model based on a symmetry-enriched string-net construction. It yields trivial ground state ensemble in the anomaly-free case, and exhibits exotic low-energy behavior in the presence of an average anomaly.
We prove a trade-off theorem for order and disorder parameters in one-dimensional quantum spin systems with quenched disorder. For a disordered ensemble with exact Ising symmetry and average translation symmetry, any gapped ensemble must have one and only one of the following: an
O(1) order parameter or an
O(1) disorder parameter with even parity, both of the Edwards-Anderson type. The result extends to nearly gapped ensembles that accommodate Griffiths-type rare-region effects. These results offer a powerful and rigorous framework to understand the disorder effects beyond perturbative approaches. As applications, we (1) establish the existence of string order parameters for SPT phases; (2) derive a Lieb-Schultz-Mattis-type constraint for disordered ensembles, which requires a nearly gapped ensemble to spontaneously break the symmetry; and (3) discuss similar trade-off relations for disordered fermion chains, leading to an improved understanding of certain "intrinsically disordered" topological phases.
In mixed states of quantum systems, symmetries come in two types: strong and weak. Furthermore, it has been argued that in quantum many-body systems, strong symmetries can be "spontaneously broken" down to weak symmetries. An issue is that as previously formulated, such "strong-to-weak symmetry breaking" appears to be a fairly non-local effect. In this paper, we show how to understand and diagnose strong symmetries and strong-to-weak symmetry breaking in an explicitly local way. Our main technical tool is a rigorous definition of strong symmetry in the limit of infinite volume, which generalizes the conventional finite-volume definitions, and for which we give several equivalent formulations, including one involving the concept of "local charge coherence". Finally, we introduce von Neumann systems, which in infinite-volume symmetries are intermediate between strong and weak symmetries. We derive a Lieb-Schultz-Mattis type anomaly constraint for von Neumann symmetries (and therefore, in particular, strong symmetries) in quantum spin chains.
Local moment formation is ubiquitous in disordered semiconductors such as Si:P, where it is observed both in the metallic and the insulating regimes. Here, we focus on local moment behavior in disordered insulators, which arises from short-ranged, repulsive electron-electron interactions. Using density matrix renormalization group and strong-disorder renormalization group methods, we study paradigmatic models of interacting insulators: one-dimensional Hubbard chains with quenched randomness. In chains with either random fermion hoppings or random chemical potentials, both at and away from half-filling, we find exponential decay of disorder-averaged charge and fermion two-point correlations, but power-law decay of disorder-averaged spin correlations that are indicative of the random singlet phase. The numerical results can be understood qualitatively by appealing to the large-interaction limit of the Hubbard chain, in which a remarkably simple picture emerges.
A bstract Symmetries and their anomalies are powerful tools for understanding quantum systems. However, realistic systems are often subject to disorders, dissipation and decoherence. In many circumstances, symmetries are not exact but only on average. This work investigates the constraints on mixed states resulting from non-commuting average symmetries. We will focus on the cases where the commutation relations of the average symmetry generators are violated by nontrivial phases, and call such average symmetry anomalous. We show that anomalous average symmetry implies degeneracy in the density matrix eigenvalues, and present several lattice examples with average symmetries, including XY chain, Heisenberg chain, and deformed toric code models. In certain cases, the results can be further extended to reduced density matrices, leading to a new lower bound on the entanglement entropy. We discuss several applications in the contexts of many body localization, quantum channels, entanglement phase transitions and also derive new constraints on the Lindbladian evolution of open quantum systems.