Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAnomaly indicator of rotation symmetry in (3+1)D topological order
122
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We examine (3+1)D topological ordered phases with $C_k$ rotation symmetry. We show that some rotation symmetric (3+1)D topological orders are anomalous, in the sense that they cannot exist in standalone (3+1)D systems, but only exist on the surface of (4+1)D SPT phases. For (3+1)D discrete gauge theories, we propose anomaly indicator that can diagnose the $mathbb{Z}_k$ valued rotation anomaly. Since (3+1)D topological phases support both point-like and loop-like excitations, the indicator is expressed in terms of the symmetry properties of point and loop-like excitations, and topological data of (3+1)D discrete gauge theories.
rate research
Read More
In this short paper, we argue that the chiral central charge $c_-$ of a (2+1)d topological ordered state is sometimes strongly constrained by t Hooft anomaly of anti-unitary global symmetry. For example, if a (2+1)d fermionic TQFT has a time reversal anomaly with $T^2=(-1)^F$ labeled as $ uinmathbb{Z}_{16}$, the TQFT must have $c_-=1/4$ mod $1/2$ for odd $ u$, while $c_-=0$ mod $1/2$ for even $ u$. This generalizes the fact that the bosonic TQFT with $T$ anomaly in a particular class must carry $c_-=4$ mod $8$ to fermionic cases. We also study such a constraint for fermionic TQFT with $U(1)times CT$ symmetry, which is regarded as a gapped surface of the topological superconductor in class AIII.
Certain patterns of symmetry fractionalization in (2+1)D topologically ordered phases of matter can be anomalous, which means that they possess an obstruction to being realized in purely (2+1)D. In this paper we demonstrate how to compute the anomaly for symmetry-enriched topological (SET) states of bosons in complete generality. We demonstrate how, given any unitary modular tensor category (UMTC) and symmetry fractionalization class for a global symmetry group $G$, one can define a (3+1)D topologically invariant path integral in terms of a state sum for a $G$ symmetry-protected topological (SPT) state. We present an exactly solvable Hamiltonian for the system and demonstrate explicitly a (2+1)D $G$ symmetric surface termination that hosts deconfined anyon excitations described by the given UMTC and symmetry fractionalization class. We present concrete algorithms that can be used to compute anomaly indicators in general. Our approach applies to general symmetry groups, including anyon-permuting and anti-unitary symmetries. In addition to providing a general way to compute the anomaly, our result also shows, by explicit construction, that every symmetry fractionalization class for any UMTC can be realized at the surface of a (3+1)D SPT state. As a byproduct, this construction also provides a way of explicitly seeing how the algebraic data that defines symmetry fractionalization in general arises in the context of exactly solvable models. In the case of unitary orientation-preserving symmetries, our results can also be viewed as providing a method to compute the $mathcal{H}^4(G, U(1))$ obstruction that arises in the theory of $G$-crossed braided tensor categories, for which no general method has been presented to date.
We discuss a way to construct a commuting projector Hamiltonian model for a (3+1)d topological superconductor in class DIII. The wave function is given by a sort of string net of the Kitaev wire, decorated on the time reversal (T) domain wall. Our Hamiltonian is provided on a generic 3d manifold equipped with a discrete form of the spin structure. We will see how the 3d spin structure induces a 2d spin structure (called a Kasteleyn direction on a 2d lattice) on T domain walls, which makes possible to define fluctuating Kitaev wires on them. Upon breaking the T symmetry in our model, we find the unbroken remnant of the symmetry which is defined on the time reversal domain wall. The domain wall supports the 2d non-trivial SPT protected by the unbroken symmetry, which allows us to determine the SPT classification of our model, based on the recent QFT argument by Hason, Komargodski, and Thorngren.
We study anomalies in time-reversal ($mathbb{Z}_2^T$) and $U(1)$ symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly $(2+1)$-D system but can be realized on the surface of a $(3+1)$-D symmetry-protected topological (SPT) phase. To detect these anomalies we propose several anomaly indicators --- functions that take as input the algebraic data of a symmetric topological order and that output a number indicating the presence or absence of an anomaly. We construct such indicators for both structures of the full symmetry group, i.e. $U(1)rtimesmathbb{Z}_2^T$ and $U(1)timesmathbb{Z}_2^T$, and for both bosonic and fermionic topological orders. In all cases we conjecture that our indicators are complete in the sense that the anomalies they detect are in one-to-one correspondence with the known classification of $(3+1)$-D SPT phases with the same symmetry. We also show that one of our indicators for bosonic topological orders has a mathematical interpretation as a partition function for the bulk $(3+1)$-D SPT phase on a particular manifold and in the presence of a particular background gauge field for the $U(1)$ symmetry.
We study t Hooft anomalies of symmetry-enriched rational conformal field theories (RCFT) in (1+1)d. Such anomalies determine whether a theory can be realized in a truly (1+1)d system with on-site symmetry, or on the edge of a (2+1)d symmetry-protected topological phase. RCFTs with the identical symmetry actions on their chiral algebras may have different t Hooft anomalies due to additional symmetry charges on local primary operators. To compute the relative anomaly, we establish a precise correspondence between (1+1)d non-chiral RCFTs and (2+1)d doubled symmetry-enriched topological (SET) phases with a choice of symmetric gapped boundary. Based on these results we derive a general formula for the relative t Hooft anomaly in terms of algebraic data that characterizes the SET phase and its boundary.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
