Higher-group symmetry in finite gauge theory and stabilizer codes

• URL: http://arxiv.org/abs/2211.11764v3
• Date: Wed, 6 Mar 2024 02:10:11 GMT
• Title: Higher-group symmetry in finite gauge theory and stabilizer codes
• Authors: Maissam Barkeshli, Yu-An Chen, Po-Shen Hsin, Ryohei Kobayashi
• Abstract summary: A large class of gapped phases of matter can be described by topological finite group gauge theories. We derive the $d$-group global symmetry and its 't Hooft anomaly for topological finite group gauge theories in $(d+1) space-time dimensions. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups.
• Score: 3.8769921482808116
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper we show how such gauge theories possess a higher-group global symmetry, which we study in detail. We derive the $d$-group global symmetry and its 't Hooft anomaly for topological finite group gauge theories in $(d+1)$ space-time dimensions, including non-Abelian gauge groups and Dijkgraaf-Witten twists. We focus on the 1-form symmetry generated by invertible (Abelian) magnetic defects and the higher-form symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetry-protected topological (SPT) phases. We show that due to a generalization of the Witten effect and charge-flux attachment, the 1-form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such higher-group symmetry in various lattice model examples. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups, where we also derive a simpler formula for the $[O_5] \in H^5(BG, U(1))$ obstruction that has appeared in prior work. We also show how the $d$-group symmetry is related to fault-tolerant non-Pauli logical gates and a refined Clifford hierarchy in stabilizer codes. We discover new logical gates in stabilizer codes using the $d$-group symmetry, such as a Controlled-Z gate in (3+1)D $\mathbb{Z}_2$ toric code.

Related papers

• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly [33.49184078479579]
  The interplay between symmetry and topological properties plays a very important role in modern physics. How to realize all these fermionic SET (fSET) phases in lattice models remains to be a difficult open problem.
  arXiv  Detail & Related papers  (2024-10-24T19:52:27Z)
• Strong-to-weak spontaneous symmetry breaking meets average symmetry-protected topological order [17.38734393793605]
  We propose a new class of phases, termed the double ASPT phase, which emerges from a nontrivial extension of these two orders. This new phase is absent from prior studies and cannot exist in conventional closed systems.
  arXiv  Detail & Related papers  (2024-10-17T16:36:53Z)
• Gapless symmetry-protected topological phases and generalized deconfined critical points from gauging a finite subgroup [0.6675805308519986]
  Gauging a finite subgroup of a global symmetry can map conventional phases and phase transitions to unconventional ones. In this work, we study an emergent $mathbbZ$-gauged system with global $U(1)$. We also discuss the stability of these phases and the critical points to small perturbations and their potential experimental realizations.
  arXiv  Detail & Related papers  (2024-01-22T05:46:49Z)
• Higher-group symmetry of (3+1)D fermionic $\mathbb{Z}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry [0.0]
  We study the higher-group structure in (3+1)D $mathbbZ$ gauge theory with an emergent fermion. Our considerations also imply the possibility of a logical $T$ gate by placing the code on $mathbbRP3$ and pumping a $p+ip$ topological state.
  arXiv  Detail & Related papers  (2023-11-09T19:00:00Z)
• Classification of Dipolar Symmetry-Protected Topological Phases: Matrix Product States, Stabilizer Hamiltonians and Finite Tensor Gauge Theories [0.0]
  We classify one-dimensional symmetry-protected topological phases protected by dipole symmetries. For each phase in the classification, we explicitly construct a stabilizer Hamiltonian to realize the SPT phase. These field theories generalize the Dijkgraaf-Witten theories to twisted finite tensor gauge theories.
  arXiv  Detail & Related papers  (2023-11-08T19:00:00Z)
• Emergence of non-Abelian SU(2) invariance in Abelian frustrated fermionic ladders [37.69303106863453]
  We consider a system of interacting spinless fermions on a two-leg triangular ladder with $pi/2$ magnetic flux per triangular plaquette. Microscopically, the system exhibits a U(1) symmetry corresponding to the conservation of total fermionic charge, and a discrete $mathbbZ$ symmetry. At the intersection of the three phases, the system features a critical point with an emergent SU(2) symmetry.
  arXiv  Detail & Related papers  (2023-05-11T15:57:27Z)
• Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles [55.41644538483948]
  We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the transformations symmetry and the corresponding generators. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.
  arXiv  Detail & Related papers  (2023-01-13T16:25:25Z)
• Codimension-2 defects and higher symmetries in (3+1)D topological phases [2.410842777583321]
  (3+1)D topological phases of matter can host a broad class of non-trivial topological defects of codimension-1, 2, and 3. We study several examples of such higher codimension defects from distinct perspectives.
  arXiv  Detail & Related papers  (2022-08-15T17:59:59Z)
• Non-Hermitian $C_{NH} = 2$ Chern insulator protected by generalized rotational symmetry [85.36456486475119]
  A non-Hermitian system is protected by the generalized rotational symmetry $H+=UHU+$ of the system. Our finding paves the way towards novel non-Hermitian topological systems characterized by large values of topological invariants.
  arXiv  Detail & Related papers  (2021-11-24T15:50:22Z)
• Classification of (2+1)D invertible fermionic topological phases with symmetry [2.74065703122014]
  We classify invertible fermionic topological phases of interacting fermions with symmetry in two spatial dimensions for general fermionic symmetry groups $G_f$. Our results also generalize and provide a different approach to the recent classification of fermionic symmetry-protected topological phases by Wang and Gu.
  arXiv  Detail & Related papers  (2021-09-22T21:02:07Z)
• The Geometry of Time in Topological Quantum Gravity of the Ricci Flow [62.997667081978825]
  We continue the study of nonrelativistic quantum gravity associated with a family of Ricci flow equations. This topological gravity is of the cohomological type, and it exhibits an $cal N=2$ extended BRST symmetry. We demonstrate a standard one-step BRST gauge-fixing of a theory whose fields are $g_ij$, $ni$ and $n$, and whose gauge symmetries consist of (i) the topological deformations of $g_ij$, and (ii) the ultralocal nonrelativistic limit of space
  arXiv  Detail & Related papers  (2020-11-12T06:57:10Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.