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Register a new userUniversal Features of Higher-Form Symmetries at Phase Transitions
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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 $tilde{Z}_N^{(1)}$) introduced recently, or an explicit $Z_N^{(1)}$ 1-form symmetry. We demonstrate that for many quantum phase transitions involving a $Z_N^{(1)}$ or $tilde{Z}_N^{(1)}$ 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 $Z_N^{(1)}$ 1-form symmetry. $P$ is the perimeter of $mathcal{C}$, and the $b log P$ 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 $O_mathcal{C}$: $S_{mathcal{C}} = langle 0 | O_mathcal{C} | 1 rangle / langle 0 | 1 rangle$ where $|0rangle$ and $|1rangle$ are many-body states with different topological nature.
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In recent years, new phases of matter that are beyond the Landau paradigm of symmetry breaking are mountaining, 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 test except for certain solvable models, therefore limiting its usage in the more generic quantum manybody systems. In this work, we study Z2 higher-form symmetry in a quantum Ising model, which is dual to the global (0-form) Ising symmetry. We compute the expectation value of the Ising disorder operator, which is a non-local 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/ordered phase. At the Ising critical point, we find that the higher-form symmetry is also spontaneously broken, even though the 0-form symmetry is preserved. We discuss examples where both the global 0-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 non-trivial working examples of higher-form symmetry operators, including the first 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.
We study higher-form symmetries in 5d quantum field theories, whose charged operators include extended operators such as Wilson line and t Hooft operators. We outline criteria for the existence of higher-form symmetries both from a field theory point of view as well as from the geometric realization in M-theory on non-compact Calabi-Yau threefolds. A geometric criterion for determining the higher-form symmetry from the intersection data of the Calabi-Yau is provided, and we test it in a multitude of examples, including toric geometries. We further check that the higher-form symmetry is consistent with dualities and is invariant under flop transitions, which relate theories with the same UV-fixed point. We explore extensions to higher-form symmetries in other compactifications of M-theory, such as $G_2$-holonomy manifolds, which give rise to 4d $mathcal{N}=1$ theories.
We investigate the effects of quenched randomness on topological quantum phase transitions in strongly interacting two-dimensional systems. We focus first on transitions driven by the condensation of a subset of fractionalized quasiparticles (`anyons) identified with `electric charge excitations of a phase with intrinsic topological order. All other anyons have nontrivial mutual statistics with the condensed subset and hence become confined at the anyon condensation transition. Using a combination of microscopically exact duality transformations and asymptotically exact real-space renormalization group techniques applied to these two-dimensional disordered gauge theories, we argue that the resulting critical scaling behavior is `superuniversal across a wide range of such condensation transitions, and is controlled by the same infinite-randomness fixed point as that of the 2D random transverse-field Ising model. We validate this claim using large-scale quantum Monte Carlo simulations that allow us to extract zero-temperature critical exponents and correlation functions in (2+1)D disordered interacting systems. We discuss generalizations of these results to a large class of ground-state and excited-state topological transitions in systems with intrinsic topological order as well as those where topological order is either protected or enriched by global symmetries. When the underlying topological order and the symmetry group are Abelian, our results provide prototypes for topological phase transitions between distinct many-body localized phases.
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 $1d$ 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: (1) Lifshit critical point of a $(2+1)d$ quantum Ising system; (2) $(3+1)d$ photon phase as an intermediate gapless phase between the topological order and the confined phase of 3d $Z_2$ quantum gauge theory; (3) $2d$ and $3d$ examples of systems with both categorical symmetries (either 0-form or 1-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.
Theories of photoinduced phase transitions have developed along with the progress in experimental studies, especially concerning their nonlinear characters and transition dynamics. At an early stage, paths from photoinduced local structural distortions to global ones are explained in classical statistical models. Their dynamics are governed by transition probabilities and inevitably stochastic, but they were sufficient to describe coarse-grained time evolutions. Recently, however, a variety of dynamics including ultrafast ones are observed in different electronic states. They are explained in relevant electronic models. In particular, a coherent lattice oscillation and coherent motion of a macroscopic domain boundary need appropriate interactions among electrons and lattice displacements. Furthermore, some transitions proceed almost in one direction, which can be explained by considering relevant electronic processes. We describe the history of theories of photoinduced phase transitions and discuss a future perspective.
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