Multipole symmetries are of interest both as a window on fracton physics and as a crucial ingredient in realizing new universality classes for quantum dynamics. Here we address the question of whether and when multipole symmetries can be spontaneously broken, both in thermal equilibrium and at zero temperature. We derive generalized Mermin-Wagner arguments for the total or partial breaking of multipolar symmetry groups and generalized Imry-Ma arguments for the robustness of such multipolar symmetry breaking to disorder. We present both general results and explicit examples. Our results should be directly applicable to quantum dynamics with multipolar symmetries and also provide a useful stepping stone to understanding the robustness of fracton phases to thermal fluctuations, quantum fluctuations, and disorder.
Multipole symmetries are of interest both as a window on fracton physics and as a crucial ingredient in realizing new universality classes for quantum dynamics. Here we address the question of whether and when multipole symmetries can be spontaneously broken, both in thermal equilibrium and at zero temperature. We derive generalized Mermin-Wagner arguments for the total or partial breaking of multipolar symmetry groups and generalized Imry-Ma arguments for the robustness of such multipolar symmetry breaking to disorder. We present both general results and explicit examples. Our results should be directly applicable to quantum dynamics with multipolar symmetries and also provide a useful stepping stone to understanding the robustness of fracton phases to thermal fluctuations, quantum fluctuations, and disorder.
Multipole symmetries are of interest in multiple contexts, from the study of fracton phases, to nonergodic quantum dynamics, to the exploration of new hydrodynamic universality classes. However, prior explorations have focused on continuum systems or hypercubic lattices. In this work, we systematically explore multipole symmetries on arbitrary crystal lattices. We explain how, given a crystal structure (specified by a space group and the occupied Wyckoff positions), one may systematically construct all consistent multipole groups. We focus on two-dimensional crystal structures for simplicity, although our methods are general and extend straightforwardly to three dimensions. We classify the possible multipole groups on all two-dimensional Bravais lattices, and on the Kagome and breathing Kagome crystal structures to illustrate the procedure on general crystal lattices. Using Wyckoff positions, we provide an in-principle classification of all possible multipole groups in any space group. We explain how, given a valid multipole group, one may construct a consistent lattice Hamiltonian and a low-energy field theory. We then explore the physical consequences, beginning by generalizing certain results originally obtained on hypercubic lattices to arbitrary crystal structures. Next, we identify two apparently novel phenomena: an emergent, robust subsystem symmetry on the triangular lattice, and an exact multipolar symmetry on the breathing Kagome lattice that does not include conservation of charge (monopole), but instead conserves a vector charge. This makes clear that there is new physics to be found by exploring the consequences of multipolar symmetries on arbitrary lattices, and this work provides the map for the exploration thereof, as well as guiding the search for emergent multipolar symmetries and the attendant exotic phenomena in real materials based on nonhypercubic lattices.
Fractonic superfluids are exotic phases of matter in which bosons are subject to mobility constraints, resulting in features beyond those of conventional superfluids. These exotic phases arise from the spontaneous breaking of higher-rank symmetry (HRS) in many-body systems with higher-moment conservation, such as dipoles, quadrupoles, and angular moments. The aim of this paper is to introduce exciting developments on the theory of spontaneous symmetry breaking in such systems, which we refer to as ``many-fracton systems''. More specifically, we introduce exciting progress on general aspects of HRS, minimal model construction, realization of symmetry-breaking ground states, order parameter, off-diagonal long-range order (ODLRO), Noether currents with continuity equations, Gross-Pitaevskii equations, quantum fluctuations, Goldstone modes, specific heat, generalized Mermin-Wagner theorem, critical current, Landau criterion, symmetry defects, and Kosterlitz-Thouless (KT)-like physics, hydrodynamics, and dipolar Hubbard model realization. This paper is concluded with several future directions.
We examine the hydrodynamics of systems with spontaneously broken multipolar symmetries using a systematic effective field theory. We focus on the simplest nontrivial setting: a system with charge and dipole symmetry, but without momentum conservation. When no symmetries are broken, our formalism reproduces the quartic subdiffusion ($\ensuremath{\omega}\ensuremath{\sim}\ensuremath{-}i{k}^{4}$) characteristic of ``fracton hydrodynamics'' with conserved dipole moment. Our formalism also captures spontaneous breaking of charge and/or dipole symmetry. When charge symmetry is spontaneously broken, the hydrodynamic modes are quadratically propagating and quartically relaxing ($\ensuremath{\omega}\ensuremath{\sim}\ifmmode\pm\else\textpm\fi{}{k}^{2}\ensuremath{-}i{k}^{4}$). When the dipole symmetry is spontaneously broken but the charge symmetry is preserved, then we find quadratically relaxing (diffusive) transverse modes, plus another mode which, depending on parameters, may be either purely diffusive ($\ensuremath{\omega}\ensuremath{\sim}\ensuremath{-}i{k}^{2}$) or quadratically propagating and quadratically relaxing ($\ensuremath{\omega}\ensuremath{\sim}\ifmmode\pm\else\textpm\fi{}{k}^{2}\ensuremath{-}i{k}^{2}$). Our work provides concrete predictions that may be tested in near-term cold atom experiments, and also lays out a general framework that may be applied to study systems with spontaneously broken multipolar symmetries.
We present an effective field theory approach to the fracton phases. The approach is based on the notion of a multipole algebra. It is an extension of space(time) symmetries of a charge-conserving matter that includes global symmetries responsible for the conservation of various components of the multipole moments of the charge density. We explain how to construct field theories invariant under the action of the algebra. These field theories generally break rotational invariance and exhibit anisotropic scaling. We further explain how to partially gauge the multipole algebra. Such gauging makes the symmetries responsible for the conservation of multipole moments local, while keeping rotation and translations symmetries global. It is shown that upon such gauging one finds the symmetric tensor gauge theories, as well as the generalized gauge theories discussed recently in the literature. We refer to all such theories as multipole gauge theories. The outcome of the gauging procedure depends on the choice of the multipole algebra. In particular, we show how to construct an effective theory for the U(1) version of the Haah code based on the principles of symmetry and provide a two-dimensional example with operators supported on a Sierpinski triangle. We show that upon condensation of charged excitations, fracton phases of both types as well as various Symmetry-protected topological phases emerge. Finally, the relation between the present approach and the formalism based on polynomials over finite fields is discussed.
In many statistical models (but not all) one can introduce a local order parameter associated with a finite, discrete or continuous group of symmetries. The higher the temperature, the more important the fluctuations. One expects therefore in general to find pure phases at low temperature, with a reduced symmetry group, as a consequence of a nonvanishing expectation value of some order parameter. This situation is referred to as spontaneous symmetry breaking. A typical example is the classical Heisenberg model describing short-range interactions of an n -vector field ϕ, with an orthogonal O(n) symmetry group. For n = 1, 2, 3, this can account for a Curie transition from a ferromagnetic to a paramagnetic phase. In particle physics the σ-model of Gell-Mann and Levy involves a spontaneous symmetry breaking of chiral invariance, typical of massless spinor fields, accompanied by soft excitation modes, the so-called Goldstone modes, associated to a π-meson triplet and leading to a nonvanishing dynamical fermion mass. In a first and rather crude approximation, one can analyse the action itself or an effective one incorporating some fluctuation effects, and look for extrema as a function of field configurations, generally translationally invariant. The remaining fluctuations are then treated perturbatively. This mean field method is common to a great variety of domains, ranging from the Clausius–Mossotti formula for a polarizable medium, the Weiss molecular field in the theory of magnetism, Landau's effective action in various statistical contexts, the effective medium approximation in disordered systems, the Hartree–Fock method in atomic or many-body physics, to the semiclassical approximation in the study of quantum systems.