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1. Higher-group symmetry of (3+1)D fermionic $$\mathbb{Z}_2$$ gauge theory: Logical CCZ, CS, and T gates from higher symmetry

   https://doi.org/10.21468/SciPostPhys.16.5.122  

   Barkeshli, Maissam; Hsin, Po-Shen; Kobayashi, Ryohei (January 2024, SciPost Physics)

   It has recently been understood that the complete global symmetry of finite group topological gauge theories contains the structure of a higher-group. Here we study the higher-group structure in (3+1)D\mathbb{Z}_2 ${\mathbb{Z}}_2$ gauge theory with an emergent fermion, and point out that pumping chiralp+ip $p+ip$ topological states gives rise to a\mathbb{Z}_{8} ${\mathbb{Z}}_8$ 0-form symmetry with mixed gravitational anomaly. This ordinary symmetry mixes with the other higher symmetries to form a 3-group structure, which we examine in detail. We then show that in the context of stabilizer quantum codes, one can obtain logical CCZ and CS gates by placing the code on a discretization ofT^3 $T^3$ (3-torus) andT^2 \rtimes_{C_2} S^1 $T^2{⋊}_{C_2}{S}^1$ (2-torus bundle over the circle) respectively, and pumpingp+ip $p+ip$ states. Our considerations also imply the possibility of a logicalT $T$ gate by placing the code on\mathbb{RP}^3 ${ℝ\mathbb{P}}^3$ and pumping ap+ip $p+ip$ topological state. 

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2. Generalized symmetry enriched criticality in (3+1)d

   https://doi.org/10.21468/SciPostPhys.19.6.145  

   Moy, Benjamin (January 2025, SciPost Physics)

   We construct two classes of continuous phase transitions in 3+1 dimensions between gapped phases that break distinct generalized global symmetries. Our analysis focuses onSU(N)/\mathbb{Z}_N $SU\left(N\right)/{\mathbb{Z}}_N$ gauge theory coupled toN_f $N_f$ flavors of Majorana fermions in the adjoint representation. ForN $N$ even and sufficiently large oddN_f $N_f$ , upon imposing time-reversal symmetry and anSO(N_f) $SO\left(N_f\right)$ flavor symmetry, the massless theory realizes a quantum critical point between a gapped phase in which a\mathbb{Z}_N ${\mathbb{Z}}_N$ one-form symmetry is completely broken and a phase where it is broken to\mathbb{Z}_2 ${\mathbb{Z}}_2$ , leading to\mathbb{Z}_{N/2} ${\mathbb{Z}}_{N/2}$ topological order. We characterize the possible patterns of symmetry fractionalization in these phases and provide an explicit lattice model that exhibits the transition. The critical point has an enhanced symmetry, which includes non-invertible analogues of time-reversal symmetry. Enforcing a non-invertible time-reversal symmetry and theSO(N_f) $SO\left(N_f\right)$ flavor symmetry, forN $N$ andN_f $N_f$ both odd, we demonstrate that this critical point can appear between a topologically ordered phase and a phase that spontaneously breaks the non-invertible time-reversal symmetry, furnishing an analogue of deconfined quantum criticality for generalized symmetries. 

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3. Higher-group symmetry in finite gauge theory and stabilizer codes

   https://doi.org/10.21468/SciPostPhys.16.4.089  

   Barkeshli, Maissam; Chen, Yu-An; Hsin, Po-Shen; Kobayashi, Ryohei (January 2024, SciPost Physics)

   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 thed $d$ -group global symmetry and its ’t Hooft anomaly for topological finite group gauge theories in(d+1) $(d+1)$ 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] ∈ H^5(BG, U(1)) $\left[O_5\right]\in H^5(BG,U\left(1\right))$ obstruction that has appeared in prior work. We also show how thed $d$ -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 thed $d$ -group symmetry, such as a controlled Z gate in the (3+1) D\mathbb{Z}_2 ${\mathbb{Z}}_2$ toric code. 

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4. Intertwined order of generalized global symmetries

   https://doi.org/10.21468/SciPostPhys.18.5.157  

   Moy, Benjamin; Fradkin, Eduardo (January 2025, SciPost Physics)

   We investigate the interplay of generalized global symmetries in 2+1 dimensions in a lattice model that couples a\mathbb{Z}_N ${\mathbb{Z}}_N$ clock model to a\mathbb{Z}_N ${\mathbb{Z}}_N$ gauge theory via a topological interaction. This coupling binds the charges of one symmetry to the disorder operators of the other, and when these composite objects condense, they give rise to emergent generalized symmetries with mixed ’t Hooft anomalies. These anomalies result in phases with ordinary symmetry breaking, topological order, and symmetry-protected topological (SPT) order, where the different types of order are not independent but intimately related. We further explore the gapped boundary states of these exotic phases and develop theories for phase transitions between them. Additionally, we extend this lattice model to incorporate a non-invertible global symmetry, which can be spontaneously broken, leading to domain walls with non-trivial fusion rules. 

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5. Realizing triality and $$p$$-ality by lattice twisted gauging in (1+1)d quantum spin systems

   https://doi.org/10.21468/SciPostPhys.17.5.136  

   Lu, Da-Chuan; Sun, Zhengdi; You, Yi-Zhuang (January 2024, SciPost Physics)

   In this paper, we study the twisted gauging on the (1+1)d lattice and construct various non-local mappings on the lattice operators. To be specific, we define the twisted Gauss law operator and implement the twisted gauging of the finite group on the lattice motivated by the orbifolding procedure in the conformal field theory, which involves the data of non-trivial element in the second cohomology group of the gauge group. We show the twisted gauging is equivalent to the two-step procedure of first applying the SPT entangler and then untwisted gauging. We use the twisted gauging to construct the triality (order 3) andp $p$ -ality (orderp $p$ ) mapping on the\mathbb{Z}_p× \mathbb{Z}_p ${\mathbb{Z}}_p\times {\mathbb{Z}}_p$ symmetric Hamiltonians, wherep $p$ is a prime. Such novel non-local mappings generalize Kramers-Wannier duality and they preserve the locality of symmetric operators but map charged operators to non-local ones. We further construct quantum process to realize these non-local mappings and analyze the induced mappings on the phase diagrams. For theories that are invariant under these non-local mappings, they admit the corresponding non-invertible symmetries. The non-invertible symmetry will constrain the theory at the multicritical point between the gapped phases. We further give the condition when the non-invertible symmetry can have symmetric gapped phase with a unique ground state. 

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