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 )
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
Z~N(1)) introduced recently, or an explicit
ZN(1) 1-form symmetry. We demonstrate that for many quantum phase transitions involving a
ZN(1) or
Z~N(1) symmetry, the following expectation value $ \langle \left( \log O_\mathcal{C} \right)^2 \rangle$ takes the form $\langle \left( \log O_\mathcal{C} \right)^2 \rangle \sim - \frac{A}{\epsilon} P+ b \log P $, where $O_\mathcal{C} $ is an operator defined associated with loop $\mathcal{C} $ (or its interior $\mathcal{A} $), which reduces to the Wilson loop operator for cases with an explicit
ZN(1) 1-form symmetry.
P is the perimeter of $\mathcal{C} $, and the
blogP term arises from the sharp corners of the loop $\mathcal{C} $, which is consistent with recent numerics on a particular example.
b is a universal microscopic-independent number, which in
(2+1)d is related to the universal conductivity at the quantum phase transition.
b can be computed exactly for certain transitions using the dualities between
(2+1)d conformal field theories developed in recent years. We also compute the ``strange correlator" of $O_\mathcal{C} $:
SC=⟨0∣OC∣1⟩/⟨0∣1⟩ where
∣0⟩ and
∣1⟩ are many-body states with different topological nature.
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.
Although condensed matter systems usually do not have higher-form symmetries, we show that, unlike 0-form symmetry, higher-form symmetries can emerge as exact symmetries at low energies and long distances. In particular, emergent higher-form symmetries at zero temperature are robust to arbitrary local UV perturbations in the thermodynamic limit. This result is true for both invertible and noninvertible higher-form symmetries. Therefore emergent higher-form symmetries are exact emergent symmetries: they are not UV symmetries but constrain low-energy dynamics as if they were. Since phases of matter are defined in the thermodynamic limit, this implies that a UV theory without higher-form symmetries can have phases characterized by exact emergent higher-form symmetries. We demonstrate this in three lattice models, the quantum clock model and emergent
ZN and
U(1)p-gauge theory, finding regions of parameter space with exact emergent (anomalous) higher-form symmetries. Furthermore, we perform a generalized Landau analysis of a
2+1D lattice model that gives rise to
Z2 gauge theory. Using exact emergent 1-form symmetries accompanied by their own energy/length scales, we show that the transition between the deconfined and Higgs/confined phases is continuous and equivalent to the spontaneous symmetry-breaking transition of a
Z2 symmetry, even though the lattice model has no symmetry. Also, we show that this transition line must always contain two parts separated by multi-critical points or other phase transitions. We discuss the physical consequences of exact emergent higher-form symmetries and contrast them to emergent 0-form symmetries. Lastly, we show that emergent 1-form symmetries are no longer exact at finite temperatures, but emergent
p-form symmetries with $p\ensuremath{\ge}2$ are.
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.
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
We discuss physical constructions and the boundary properties of various symmetry-protected topological phases that involve 1-form symmetries from one spatial dimension to four spatial dimensions (
4d). For example, the prototype three-dimensional (
3d) boundary state of
4d SPT states involving 1-form symmetries can be either a gapless photon phase (quantum electrodynamics) or gapped topological order enriched by 1-form symmetries; that is, the loop excitations of these topological orders carry nontrivial 1-form symmetry charges. This study also serves the purpose of diagnosing anomalies of
3d states of matter. Connection between SPT states with 1-form symmetries and condensed-matter systems such as quantum dimer models at one lower dimension will also be discussed. Whether a quantum dimer model can have a trivial gapped phase depends on the nature of its corresponding bulk state in one higher dimension.