The analysis of phase transitions of gauge theories has relied heavily on simplifications that arise at the boundaries of phase diagrams, where certain excitations are forbidden. Taking $2+1$-dimensional ${\mathbb{Z}}_{2}$ gauge theory as an example, the simplification can be visualized geometrically: on the phase diagram boundaries the partition function is an ensemble of closed membranes. More generally, however, the membranes have ``holes'' in them, representing worldlines of virtual anyon excitations. If the holes are of a finite size, then typically they do not affect the universality class, but they destroy microscopic (higher-form) symmetries and microscopic (string) observables. We demonstrate how these symmetries and observables can be restored using a ``membrane patching'' procedure, which maps the ensemble of membranes back to an ensemble of closed membranes. (This is closely related to the idea of gauge fixing in the ``minimal gauge,'' though not equivalent.) Membrane patching makes the emergence of higher symmetry concrete. Performing patching in a Monte Carlo simulation with an appropriate algorithm, we show that it gives access to numerically useful observables. For example, the confinement transition can be analyzed using a correlation function that is a power law at the critical point. We analyze the quasi-locality of the patching procedure and discuss what happens at a self-dual multicritical point in the gauge-Higgs model, where the lengthscale $\ensuremath{\ell}$ characterizing the holes diverges. The patching approach described here generalizes to many other statistical ensembles with a representation in terms of fluctuating membranes or loops, in various dimensions, and related constructions could be implemented in experiments on quantum devices.
The analysis of phase transitions of gauge theories has relied heavily on simplifications that arise at the boundaries of phase diagrams, where certain excitations are forbidden. Taking 2+1-dimensional Z2 gauge theory as an example, the simplification can be visualized geometrically: on the phase diagram boundaries the partition function is an ensemble of closed membranes. More generally, however, the membranes have holes'' in them, representing worldlines of virtual anyon excitations. If the holes are of a finite size, then typically they do not affect the universality class, but they destroy microscopic (higher-form) symmetries and microscopic (string) observables. We demonstrate how these symmetries and observables can be restored using a membrane patching'' procedure, which maps the ensemble of membranes back to an ensemble of closed membranes. (This is closely related to the idea of gauge fixing in the ``minimal gauge,'' though not equivalent.) Membrane patching makes the emergence of higher symmetry concrete. Performing patching in a Monte Carlo simulation with an appropriate algorithm, we show that it gives access to numerically useful observables. For example, the confinement transition can be analyzed using a correlation function that is a power law at the critical point. We analyze the quasi-locality of the patching procedure and discuss what happens at a self-dual multicritical point in the gauge-Higgs model, where the lengthscale \ensuremathℓ characterizing the holes diverges. The patching approach described here generalizes to many other statistical ensembles with a representation in terms of fluctuating membranes or loops, in various dimensions, and related constructions could be implemented in experiments on quantum devices.
Self-dual criticality in three-dimensional Z2 gauge theory with matter
The simplest topologically ordered phase in 2+1D is the deconfined phase of Z2 gauge theory, realized for example in the toric code. This phase permits a duality that exchanges electric and magnetic excitations (``e'' and ``m'' particles). Condensing either particle while the other remains gapped yields a phase transition with 3D Ising exponents. More mysterious, however, is the transition out of the deconfined phase when self-duality symmetry is preserved. If this transition is continuous, which has so far been unclear, then it may be the simplest critical point for which we still lack any useful continuum Lagrangian description. This transition also has a soft matter interpretation, as a multicritical point for classical membranes in 3D. We study the self-dual transition with Monte Carlo simulations of the Z2 gauge-Higgs model on cubic lattices of linear size L≤96. Our results indicate a continuous transition: for example, cumulants show a striking parameter-free scaling collapse. We estimate scaling dimensions by using duality symmetry to distinguish the leading duality-odd/duality-even scaling operators A and S. All local operators have large scaling dimensions, making standard techniques for locating the critical point ineffective. We develop an alternative using ``renormalization group trajectories'' of cumulants. We check that two- and three-point functions, and temporal correlators in the Monte-Carlo dynamics, are scale-invariant, with scaling dimensions xA and xS and dynamical exponent z. We also give a picture for emergence of 1-form symmetries, in some parts of the phase diagram, in terms of ``patching'' of membranes/worldsurfaces. We relate this to the percolation of anyon worldlines in spacetime. Analyzing percolation yields a fourth exponent for the self-dual transition. We propose variations of the model for further investigation.
Gauging a finite Abelian normal subgroup Γ of a nonanomalous 0-form symmetry G of a theory in (d+1)D spacetime can yield an unconventional critical point if the original theory has a continuous transition where Γ is completely spontaneously broken and if G is a nontrivial extension of G/Γ by Γ. The gauged theory has symmetry G/Γ×Γ^(d−1), where Γ^(d−1) is the (d−1)-form dual symmetry of Γ, and a 't Hooft anomaly between them. Thus it can be viewed as a boundary of a topological phase protected by G/Γ×Γ^(d−1). The ordinary critical point, upon gauging, is mapped to a deconfined quantum critical point between two ordinary symmetry-breaking phases (d=1) or an unconventional quantum critical point between an ordinary symmetry-breaking phase and a topologically ordered phase (d≥2) associated with G/Γ and Γ^(d−1), respectively. Order parameters and disorder parameters, before and after gauging, can be directly related. As a concrete example, we gauge the Z2 subgroup of Z4 symmetry of a 4-state clock model on a 1D lattice and a 2D square lattice. Since the symmetry of the clock model contains D8, the dihedral group of order 8, we also analyze the anomaly structure which is similar to that in the compactified SU(2) gauge theory with θ=π in (3+1)D and its mixed gauge theory. The general case is also discussed.
For a zero-temperature Landau symmetry breaking transition in n-dimensional space that completely breaks a finite symmetry G, the critical point at the transition has the symmetry G. In this paper, we show that the critical point also has a dual symmetry - a (n−1)-symmetry described by a higher group when G is Abelian or an algebraic (n−1)-symmetry beyond higher group when G is non-Abelian. In fact, any G-symmetric system can be viewed as a boundary of G-gauge theory in one higher dimension. The conservation of gauge charge and gauge flux in the bulk G-gauge theory gives rise to the symmetry and the dual symmetry respectively. So any G-symmetric system actually has a larger symmetry called categorical symmetry, which is a combination of the symmetry and the dual symmetry. However, part (and only part) of the categorical symmetry must be spontaneously broken in any gapped phase of the system, but there exists a gapless state where the categorical symmetry is not spontaneously broken. Such a gapless state corresponds to the usual critical point of Landau symmetry breaking transition. The above results remain valid even if we expand the notion of symmetry to include higher symmetries and algebraic higher symmetries. Thus our result also applies to critical points for transitions between topological phases of matter. In particular, we show that there can be several critical points for the transition from the 3+1D Z2 gauge theory to a trivial phase. The critical point from Higgs condensation has a categorical symmetry formed by a Z2 0-symmetry and its dual - a Z2 2-symmetry, while the critical point of the confinement transition has a categorical symmetry formed by a Z2 1-symmetry and its dual - another Z2 1-symmetry.
Self-dual S3 gauge theory in 2+1d: lattice model and topological phase transitions
Electric-magnetic self-duality of the Z2 gauge theory, realized microscopically as a half-lattice-translation exchanging electric charge and magnetic flux, has been an influential example of a duality symmetry with an exact lattice realization. We construct the first non-Abelian generalization of this construction: a lattice model of the S3 quantum double D(S3) on a tensor product Hilbert space in which the Z2em anyon-permutation symmetry, exchanging the non-Abelian chargeon C and fluxon F, is realized via lattice translation. Consequently we find that the zigzag boundary termination of the model realizes, without fine-tuning, a gapless critical edge state described by the tetracritical Ising CFT. The bulk admits three independent Z2em-preserving bosonic perturbations, driving D(S3) into distinct gapped phases. We analyze these transitions by three independent methods: category-theoretic anyon condensation, microscopic lattice Hamiltonians, and Chern-Simons-Higgs theory, which all agree, yielding a unified picture. These examples motivate a minimal-condensation principle: proliferating a bosonic anyon generically drives condensation of a minimal condensable algebra containing it, with symmetry-related condensates appearing as degenerate vacua that spontaneously break the anyon-permutation symmetry. Our model construction extends to an infinite family of self-dual dihedral quantum doubles D(D2n). Notably, each model is sign-problem-free, opening the door to large-scale numerical exploration of the phases of non-Abelian Chern-Simons-Higgs theories.
The deconfined phase of (2+1)-dimensional Zk gauge theory exhibits topological order, with e and m anyons that have a $2\ensuremath{\pi}/k$ braiding phase. Proliferating either e or m drives Higgs or confinement transitions, respectively. At the multicritical point where these transitions meet, the theory enjoys an additional duality symmetry that exchanges e and m anyons. This symmetry forces anyons with nontrivial braiding to close their gaps simultaneously, giving rise to a critical theory that mixes strong interactions with mutual statistics. We propose an effective $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ gauge theory with a mutual Chern-Simons term at level k to describe the vicinity of the multicritical point for $k\ensuremath{\ge}4$. The emergence of a global $\mathrm{U}{(1)}^{\mathrm{top}}\ifmmode\times\else\texttimes\fi{}\mathrm{U}{(1)}^{\mathrm{top}}$ symmetry at the critical point imposes powerful constraints on universal properties of the phase transition. In particular, we show that (1) the lattice magnetic flux operator embeds as a conserved U(1) current with protected scaling dimension; (2) the first-order line emanating from the critical point for k=2 disappears generically for sufficiently large k; (3) the correlation length exponent approaches that of the three-dimensional XY model with corrections of order 1/k2 in the large-k limit. These predictions can be tested in near-term numerical simulations and pave the way for a more general exploration of topological quantum criticality enriched with anyon-permuting symmetries.
We outline a holographic framework that attempts to unify Landau and beyond-Landau paradigms of quantum phases and phase transitions. Leveraging a modern understanding of symmetries as topological defects/operators, the framework uses a topological order to organize the space of quantum systems with a global symmetry in one lower dimension. The global symmetry naturally serves as an input for the topological order. In particular, we holographically construct a String Operator Algebra (SOA) which is the building block of symmetric quantum systems with a given symmetry G in one lower dimension. This exposes a vast web of dualities which act on the space of G-symmetric quantum systems. The SOA facilitates the classification of gapped phases as well as their corresponding order parameters and fundamental excitations, while dualities help to navigate and predict various corners of phase diagrams and analytically compute universality classes of phase transitions. A novelty of the approach is that it treats conventional Landau and unconventional topological phase transitions on an equal footing, thereby providing a holographic unification of these seemingly-disparate domains of understanding. We uncover a new feature of gapped phases and their multi-critical points, which we dub fusion structure, that encodes information about which phases and transitions can be dual to each other. Furthermore, we discover that self-dual systems typically posses emergent non-invertible, i.e., beyond group-like symmetries. We apply these ideas to 1+1d 1 + 1 d quantum spin chains with finite Abelian group symmetry, using topologically-ordered systems in 2+1d 2 + 1 d . We predict the phase diagrams of various concrete spin models, and analytically compute the full conformal spectra of non-trivial quantum phase transitions, which we then verify numerically.