An official website of the United States government
A spontaneous symmetry-breaking order is conventionally described by a tensor-product wavefunction of some few-body clusters; some standard examples include the simplest ferromagnets and valence bond solids. We discuss a type of symmetry-breaking orders, dubbed entanglement-enabled symmetry-breaking orders, which cannot be realized by any such tensor-product state. Given a symmetry breaking pattern, we propose a criterion to diagnose if the symmetry-breaking order is entanglement-enabled, by examining the compatibility between the symmetries and the tensor-product description. For concreteness, we present an infinite family of exactly solvable gapped models on one-dimensional lattices with nearest-neighbor interactions, whose ground states exhibit entanglement-enabled symmetry-breaking orders from a discrete symmetry breaking. In addition, these ground states have gapless edge modes protected by the unbroken symmetries. We also propose a construction to realize entanglement-enabled symmetry-breaking orders with spontaneously broken continuous symmetries. Under the unbroken symmetries, some of our examples can be viewed as symmetry-protected topological states that are beyond the conventional classifications.
more »
« less
Topological Symmetry Groups of the Petersen Graphs
The topological symmetry group of an embedding Γ of an abstract graph γ in S3 is the group of automorphisms of γ that can be realized by homeomorphisms of the pair (S3,Γ). These groups are motivated by questions about the symmetries of molecules in space. The Petersen family of graphs is an important family of graphs for many problems in low-dimensional topology, so it is desirable to understand the possible groups of symmetries of their embeddings in space. In this paper, we find all the groups that can be realized as topological symmetry groups for each of the graphs in the Petersen family. Along the way, we also complete the classification of the realizable topological symmetry groups for K3,3.
more »
« less
- Award ID(s):
- 1916494
- PAR ID:
- 10463575
- Date Published:
- Journal Name:
- Symmetry
- Volume:
- 15
- Issue:
- 6
- ISSN:
- 2073-8994
- Page Range / eLocation ID:
- 1267
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
The Ginzburg-Landau (GL) theory is very successful in describing the pairing symmetry, a fundamental characterization of the broken symmetries in a paired superfluid or superconductor. However, GL theory does not describe fermionic excitations such as Bogoliubov quasiparticles or Andreev bound states that are directly related to topological properties of the superconductor. In this work, we show that the symmetries of the fermionic excitations are captured by a Projective Symmetry Group (PSG), which is a group extension of the bosonic symmetry group in the superconducting state. We further establish a correspondence between the pairing symmetry and the fermion PSG. When the normal and superconducting states share the same spin rotational symmetry, there is a simpler correspondence between the pairing symmetry and the fermion PSG, which we enumerate for all 32 crystalline point groups. We also discuss the general framework for computing PSGs when the spin rotational symmetry is spontaneously broken in the superconducting state. This PSG formalism leads to experimental consequences, and as an example, we show how a given pairing symmetry dictates the classification of topological superconductivity.more » « less
-
We investigate fractionalization of non-invertible symmetry in (2+1)D topological orders. We focus on coset non-invertible symmetries obtained by gauging non-normal subgroups of invertible0 -form symmetries. These symmetries can arise as global symmetries in quantum spin liquids, given by the quotient of the projective symmetry group by a non-normal subgroup as invariant gauge group. We point out that such coset non-invertible symmetries in topological orders can exhibit symmetry fractionalization: each anyon can carry a “fractional charge” under the coset non-invertible symmetry given by a gauge invariant superposition of fractional quantum numbers. We present various examples using field theories and quantum double lattice models, such as fractional quantum Hall systems with charge conjugation symmetry gauged and finite group gauge theory from gauging a non-normal subgroup. They include symmetry enrichedS_3 andO(2) gauge theories. We show that such systems have a fractionalized continuous non-invertible coset symmetry and a well-defined electric Hall conductance. The coset symmetry enforces a gapless edge state if the boundary preserves the continuous non-invertible symmetry. We propose a general approach for constructing coset symmetry defects using a “sandwich” construction: non-invertible symmetry defects can generally be constructed from an invertible defect sandwiched by condensation defects. The anomaly free condition for finite coset symmetry is also identified.more » « less
-
Abstract For over 100 years, the group-theoretic characterization of crystalline solids has provided the foundational language for diverse problems in physics and chemistry. However, the group theory of crystals with commensurate magnetic order has remained incomplete for the past 70 years, due to the complicated symmetries of magnetic crystals. In this work, we complete the 100-year-old problem of crystalline group theory by deriving the small corepresentations, momentum stars, compatibility relations, and magnetic elementary band corepresentations of the 1,421 magnetic space groups (MSGs), which we have made freely accessible through tools on the Bilbao Crystallographic Server. We extend Topological Quantum Chemistry to the MSGs to form a complete, real-space theory of band topology in magnetic and nonmagnetic crystalline solids – Magnetic Topological Quantum Chemistry (MTQC). Using MTQC, we derive the complete set of symmetry-based indicators of electronic band topology, for which we identify symmetry-respecting bulk and anomalous surface and hinge states.more » « less
-
Abstract A recent development in our understanding of the theory of quantum fields is the fact that familiar gauge theories in spacetime dimensions greater than two can have non-invertible symmetries generated by topological defects. The hallmark of these non-invertible symmetries is that the fusion rule deviates from the usual group-like structure, and in particular the fusion coefficients take values in topological field theories (TFTs) rather than in mere numbers. In this paper we begin an exploration of the associativity structure of non-invertible symmetries in higher dimensions. The first layer of associativity is captured by F-symbols, which we find to assume values in TFTs that have one dimension lower than that of the defect. We undertake an explicit analysis of the F-symbols for the non-invertible chiral symmetry that is preserved by the massless QED and explore their physical implications. In particular, we show the F-symbol TFTs can be detected by probing the correlators of topological defects with ’t Hooft lines. Furthermore, we derive the Ward–Takahashi identity that arises from the chiral symmetry on a large class of four-dimensional manifolds with non-trivial topologies directly from the topological data of the symmetry defects, without referring to a Lagrangian formulation of the theory.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/sym15061267
