We investigate the behavior of higher-form symmetries at various quantum phase transitions. We consider discrete 1-form symmetries, which can be either part of the generalized concept “categorical symmetry” (labelled as \tilde{Z}_N^{(1)} Z ̃ N ( 1 ) ) introduced recently, or an explicit Z_N^{(1)} Z N ( 1 ) 1-form symmetry. We demonstrate that for many quantum phase transitions involving a Z_N^{(1)} Z N ( 1 ) or \tilde{Z}_N^{(1)} Z ̃ N ( 1 ) symmetry, the following expectation value \langle \left( O_\mathcal{C}\right)^2 \rangle ⟨ ( O 𝒞 ) 2 ⟩ takes the form \langle \left( \log O_\mathcal{C} \right)^2 \rangle \sim - \frac{A}{\epsilon} P + b \log P ⟨ ( log O 𝒞 ) 2 ⟩ ∼ − A ϵ P + b log P , where O_\mathcal{C} O 𝒞 is an operator defined associated with loop \mathcal{C} 𝒞 (or its interior \mathcal{A} 𝒜 ), which reduces to the Wilson loop operator for cases with an explicit Z_N^{(1)} Z N ( 1 )
In recent years, new phases of matter that are beyond the Landau paradigm of symmetry breaking have been accumulating, and to catch up with this fast development, new notions of global symmetry are introduced. Among them, the higher-form symmetry, whose symmetry charges are spatially extended, can be used to describe topologically ordered phases as the spontaneous breaking of the symmetry, and consequently unify the unconventional and conventional phases under the same conceptual framework. However, such conceptual tools have not been put into quantitative tests except for certain solvable models, therefore limiting their usage in the more generic quantum many-body systems. In this work, we study
Z2 higher-form symmetry in a quantum Ising model, which is dual to the global (zero-form) Ising symmetry. We compute the expectation value of the Ising disorder operator, which is a nonlocal order parameter for the higher-form symmetry, analytically in free scalar theories and through unbiased quantum Monte Carlo simulations for the interacting fixed point in
(2+1)d. From the scaling form of this extended object, we confirm that the higher-form symmetry is indeed spontaneously broken inside the paramagnetic, or quantum disordered phase (in the Landau sense), but remains symmetric in the ferromagnetic or ordered phase. At the Ising critical point, we find that the disorder operator also obeys a ``perimeter'' law scaling with possibly multiplicative power-law corrections. We discuss examples where both the global zero-form symmetry and the dual higher-form symmetry are preserved, in systems with a codimension-1 manifold of gapless points in momentum space. These results provide nontrivial working examples of higher-form symmetry operators, including the direct computation of one-form order parameter in an interacting conformal field theory, and open the avenue for their generic implementation in quantum many-body systems.
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 $\ensuremath{\ell}$ 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.
Abstract We study the concept of ‘categorical symmetry’ introduced recently, which in the most basic sense refers to a pair of dual symmetries, such as the Ising symmetries of the 1 d quantum Ising model and its self-dual counterpart. In this manuscript we study discrete categorical symmetry at higher-dimensional critical points and gapless phases. At these selected gapless states of matter, we can evaluate the behavior of categorical symmetries analytically. We analyze the categorical symmetry at the following examples of criticality: (i) (2 + 1) d Lifshitz critical point of a quantum Ising system; (ii) (3 + 1) d photon phase as an intermediate gapless phase between the topological order and the confined phase of 3 d Z 2 quantum gauge theory; (iii) 2 d and 3 d examples of systems with both categorical symmetries (either zero-form or one-form categorical symmetries) and subsystem symmetries. We demonstrate that at some of these gapless states of matter the categorical symmetries have very different behavior from the nearby gapped phases.
Higher-form symmetries are a valuable tool for classifying topological phases of matter. However, emergent higher-form symmetries in interacting many-body systems are typically not exact due to the presence of topological defects. In this paper, we develop a systematic framework for building effective theories with approximate higher-form symmetries. We focus on a continuous U(1) q -form symmetry and study phases with various patterns of spontaneous and explicit symmetry breaking. We uncover a web of dualities between such phases and highlight their role in describing the presence of dynamical higher-form topological defects. In order to study the out-of-equilibrium dynamics of these phases of matter, we formulate respective hydrodynamic theories and study the spectra of excitations exhibiting higher-form charge relaxation and Goldstone relaxation effects. We show that our framework is able to describe various phase transitions due to proliferation of vortices or defects. This includes the melting transition in smectic crystals, the plasma phase transition from polarized gases to magnetohydrodynamics, the spin-ice transition, the superfluid to neutral fluid transition, and the Meissner effect in superconductors, among many others. Published by the American Physical Society 2024
Higher-form symmetries act on subdimensional spatial manifolds of a quantum system. They can emerge as an exact symmetry at low energies even when they are explicitly broken at the microscopic level, making them difficult to characterize. In this work, we propose that the emergence of 1-form symmetries is information-theoretic in nature, characterized by the preservation of information about a specific bare (microscopic) 1-form symmetry. As a consequence, the loss of the emergent 1-form symmetry is an information-theoretic transition which we argue to be revealed from the long-range entanglement in the ensemble of post-measurement states. We analytically determine the regimes in which a 1-form symmetry emerges in product states on one- and two-dimensional lattices. In analytically intractable regimes, we demonstrate how to efficiently detect 1-form symmetries with a global quantum error correction (QEC) decoder and numerically examine the information-theoretic transition of the 1-form symmetry, including systems with Z 2 topological order. As a practical application of our framework, we show that once the 1-form symmetry is detected to exist, a topological quantum phase transition characterized by the spontaneous breaking of the 1-form symmetry can be accurately determined by a disorder parameter. We further argue that our proposed theory for emergent 1-form symmetries offers new perspectives on particle condensation and suggests sharp information-theoretic phase boundaries between Higgs and confining regimes in the Z 2 lattice gauge theory.