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We consider phase transitions out of a general topological phase in 2 + 1 dimensions. We assume that the transition is triggered by a single Abelian anyon, which becomes light near the transition and whose worldlines proliferate after the transition. (This proliferation is often referred to as “condensation.”) We describe the transition using a continuum field theory obtained by coupling the corresponding topological quantum field theory (TQFT) to a single complex scalar field associated with this anyon. With these assumptions, we find the most general relativistic field theory for such a transition. Even though for a given TQFT and a choice of anyon, there are infinitely many such field theories, the transition theory depends on only a single additional integer parameter. We analyze all these theories, their global symmetries, and their phases. In generic cases, the theory after the transition can be related to the original one via an Abelian hierarchy construction. In special cases, the theory after the transition is gapless, and with a particular deformation, it is related to the original TQFT by gauging an anomaly-free one-form global symmetry. We also explore the enrichment of this setup by a global U(1) symmetry. In some cases, enriching the original TQFT is incompatible with the full transition theory. Finally, we demonstrate our construction with many specific examples. 

We discuss the possible topological order/topological quantum field theory of different quantum Hall systems. Given the value of the Hall conductivity, we constrain the global symmetry of the low-energy theory and its anomaly. Specifically, the one-form global symmetry and its anomaly are presented as the organizing principle of these systems. This information is powerful enough to lead to a unique minimal topological order (or a small number of minimal topological orders). Almost all of the known experimentally discovered topological orders are these minimal theories. Since this work is interdisciplinary, we made a special effort to relate to researchers with different backgrounds by providing translations between different perspectives. 

We construct a family of two-dimensional topological stabilizer codes on continuous variable (CV) degrees of freedom, which generalize homological rotor codes and the toric-GKP code. Our topological codes are built using the concept of boson condensation—we start from a parent stabilizer code based on an R gauge theory and condense various bosonic excitations. This produces a large class of topological CV stabilizer codes, including ones that are characterized by the anyon theories of U(1)_{2n} x U(1)_{-2m} Chern-Simons theories, for arbitrary pairs of positive integers (n,m). Most notably, this includes anyon theories that are nonchiral and nevertheless do not admit a gapped boundary. It is widely believed that such anyon theories cannot be realized by any stabilizer model on finite-dimensional systems. We conjecture that these CV codes go beyond codes obtained from concatenating a topological qudit code with a local encoding into CVs, and thus constitute the first example of topological codes that are intrinsic to CV systems. Moreover, we study the Hamiltonians associated with the topological CV stabilizer codes and show that, although they have a gapless spectrum, they can become gapped with the addition of a quadratic perturbation. We show that similar methods can be used to construct a gapped Hamiltonian whose anyon theory agrees with a U(1)_2 Chern-Simons theory. Our work initiates the study of scalable stabilizer codes that are intrinsic to CV systems and highlights how error-correcting codes can be used to design and analyze many-body systems of CVs that model lattice gauge theories. 

We investigate quantum cellular automata (QCA) on one-dimensional spin systems defined over a subalgebra of the full local operator algebra - the symmetric subalgebra under a finite Abelian group symmetry G. For systems where each site carries a regular representation of G, we establish a complete classification of such subalgebra QCAs based on two topological invariants: (1) a surjective homomorphism from the group of subalgebra QCAs to the group of anyon permutation symmetries in a (2+1)d G gauge theory; and (2) a generalization of the Gross-Nesme-Vogts-Werner (GNVW) index that characterizes the flow of the symmetric subalgebra. Specifically, two subalgebra QCAs correspond to the same anyon permutation and share the same index if and only if they differ by a finite-depth unitary circuit composed of G-symmetric local gates. We also identify a set of operations that generate all subalgebra QCAs through finite compositions. As an example, we examine the Kramers-Wannier duality on a ℤ2 symmetric subalgebra, demonstrating that it maps to the e-m permutation in the two-dimensional toric code and has an irrational index of $$\sqrt{2}$$. Therefore, it cannot be extended to a QCA over the full local operator algebra and mixes nontrivially with lattice translations. 

Incompressible insulating phases of electronic systems at partial filling of a lattice are often associated with charge ordering that breaks lattice symmetry. The resulting phases have an enlarged unit cell with an effective integer filling. Here we explore the possibility of insulating states—which we dub “quantum charge liquids” (QCLs)—at partial lattice filling that preserve lattice translation symmetry. Such QCL phases must necessarily either have gapped fractionally charged excitations and associated topological order or have gapless neutral excitations. We establish some general constraints on gapped fermionic QCL phases that restrict the nature of their topological order. We prove a number of results on the minimal topological order that is consistent with the lattice filling. In particular we show that, at rational fillings 𝜈=𝑝/𝑞 with 𝑞 an even integer, the minimal ground-state degeneracy on a torus of the fermionic QCL is 4⁢𝑞2, four times larger than that of the bosonic QCL at the same filling. We comment on models and physical systems which may host fermionic QCL phases and discuss the phenomenology of these phases. 

We identify a three-dimensional system that exhibits long-range entanglement at sufficiently small but nonzero temperature—it therefore constitutes a quantum topological order at finite temperature. The model of interest is known as the fermionic toric code, a variant of the usual 3D toric code, which admits emergent fermionic pointlike excitations. The fermionic toric code, importantly, possesses an anomalous two-form symmetry, associated with the spacelike Wilson loops of the fermionic excitations. We argue that it is this symmetry that imbues low-temperature thermal states with a novel topological order and long-range entanglement. Based on the current classification of three-dimensional topological orders, we expect that the low-temperature thermal states of the fermionic toric code belong to a phase of matter that, in the context of equilibrium phases of matter, only exists at nonzero temperatures. We conjecture that further examples of topological orders at nonzero temperatures are given by discrete gauge theories with anomalous two-form symmetries. Our Letter therefore opens the door to studying quantum topological order at nonzero temperature in physically realistic dimensions. 

Global symmetries greatly enrich the landscape of topological quantum phases, playing an essential role from topological insulators to fractional quantum Hall effect. Topological phases in mixed quantum states, originating from decoherence in open quantum systems or disorders in imperfect crystalline solids, have recently garnered significant interest. Unlike pure states, mixed quantum states can exhibit average symmetries—symmetries that keep the total ensemble invariant but not on each individual state. In this work, we present a systematic classification and characterization of average symmetry-protected topological (ASPT) phases applicable to generic symmetry groups, encompassing both average and exact symmetries, for bosonic and fermionic systems. Moreover, we formulate the theory of average symmetry-enriched topological (ASET) orders in disordered bosonic systems. Our systematic approach helps clarify nuanced issues in previous literature and uncovers compelling new physics. Notably, we discover that (1) the definition and classification of ASPT phases in decohered and disordered systems exhibit subtle differences, (2) despite these differences, ASPT phases in both settings can be classified and characterized under a unified framework of defect decoration and spectral sequence, (3) this systematic classification uncovers a plethora of ASPT phases that are intrinsically mixed, implying they can exclusively manifest in decohered or disordered systems where part of the symmetry is average, and (4) similarly for ASET, we find intrinsically disordered phases exhibiting exotic anyon behaviors—the ground states of such phases necessarily contain localized anyons, with gapless (yet still localized) excitation blue spectra. 

Symmetry in mixed quantum states can manifest in two distinct forms: strong symmetry, where each individual pure state in the quantum ensemble is symmetric with the same charge, and weak symmetry, which applies only to the entire ensemble. This paper explores a novel type of spontaneous symmetry breaking (SSB) where a strong symmetry is broken to a weak one. While the SSB of a weak symmetry is measured by the long-ranged two-point correlation function, the strong-to-weak SSB (SWSSB) is measured by the fidelity correlator. We prove that SWSSB is a universal property of mixed-state quantum phases, in the sense that the phenomenon of SWSSB is robust against symmetric low-depth local quantum channels. We also show that the symmetry breaking is “spontaneous” in the sense that the effect of a local symmetry-breaking measurement cannot be recovered locally. We argue that a thermal state at a nonzero temperature in the canonical ensemble (with fixed symmetry charge) should have spontaneously broken strong symmetry. Additionally, we study nonthermal scenarios where decoherence induces SWSSB, leading to phase transitions described by classical statistical models with bond randomness. In particular, the SWSSB transition of a decohered Ising model can be viewed as the “ungauged” version of the celebrated toric-code decodability transition. We confirm that, in the decohered Ising model, the SWSSB transition defined by the fidelity correlator is the only physical transition in terms of channel recoverability. We also comment on other (inequivalent) definitions of SWSSB, through correlation functions with higher Rényi indices. 

We study a Z_3 Kitaev model on the honeycomb lattice with nearest neighbor interactions. Based on matrix product state simulations and symmetry considerations, we find evidence that, with ferromagnetic isotropic couplings, the model realizes a chiral spin liquid, characterized by a possible U(1)_{12} chiral topological order. This is supported by simulations on both cylinder and strip geometries. On infinitely long cylinders with various widths, scaling analysis of entanglement entropy and maximal correlation length suggests that the model has a gapped two-dimensional bulk. The topological entanglement entropy is extracted and found to be in agreement with the U(1)_{12} topological order. On infinitely long strips with moderate widths, we find the model is critical with a central charge consistent with the chiral edge theory of the U(1)_{12} topological phase. We conclude by discussing several open questions.