Fractonic many-body systems with higher-rank symmetry admit exotic conservation of higher moments and have been recently focused on intensively. But microscopic Hamiltonians often look quite intricate, highly anisotropic and fine-tuned. In this paper, we attempt to identify higher-rank symmetry as an emergent phenomenon at low energies. We start with a more realistic Hamiltonian of interacting $d$-component bosons in $d$ dimensions. At symmetric points, the $i$-th component is dispersive only along the $\hat{x}_i$-th direction, which renders the conservation of angular moments -- a typical example of higher moments. We find that, all symmetry-breaking kinetic terms, which are usually unavoidable in real systems, become irrelevant in a wide parameter region. In other words, starting from a less fine-tuned Hamiltonian without symmetry, we can potentially achieve emergent higher-rank symmetry and higher-moment conservation at low energies. A detailed one-loop Wilson's perturbative renormalization-group study is performed, leading to a phase diagram of symmetry-breaking kinetic energy v.s. interaction strength. Such a scenario holds promise for both theoretical and experimental realization of higher-rank symmetry, higher-moment conservation, and the resulting exotic non-equilibrium physics.
Fractonic many-body systems with higher-rank symmetry admit exotic conservation of higher moments and have been recently focused on intensively. But microscopic Hamiltonians often look quite intricate, highly anisotropic and fine-tuned. In this paper, we attempt to identify higher-rank symmetry as an emergent phenomenon at low energies. We start with a more realistic Hamiltonian of interacting d-component bosons in d dimensions. At symmetric points, the i-th component is dispersive only along the x^i-th direction, which renders the conservation of angular moments -- a typical example of higher moments. We find that, all symmetry-breaking kinetic terms, which are usually unavoidable in real systems, become irrelevant in a wide parameter region. In other words, starting from a less fine-tuned Hamiltonian without symmetry, we can potentially achieve emergent higher-rank symmetry and higher-moment conservation at low energies. A detailed one-loop Wilson's perturbative renormalization-group study is performed, leading to a phase diagram of symmetry-breaking kinetic energy v.s. interaction strength. Such a scenario holds promise for both theoretical and experimental realization of higher-rank symmetry, higher-moment conservation, and the resulting exotic non-equilibrium physics.
Higher rank symmetry and higher moment conservation have been drawn considerable attention from, e.g., subdiffusive transport to fracton topological order. In this paper, we perform a one-loop renormalization group (RG) analysis and show how these phenomena emerge at low energies. We consider a d-dimensional model of interacting bosons of d components. At higher rank-symmetric points with conserved angular moments, the ath bosons have kinetic energy only along the ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}}^{a}$ direction. Therefore, the symmetric points look highly anisotropic and fine-tuned. By studying RG in a wide vicinity of the symmetric points, we find that symmetry-disallowed kinetic terms tend to be irrelevant within the perturbative regime, which potentially leads to emergent higher rank symmetry and higher moment conservation at the deep infrared limit. While nonperturbative analysis is called for in the future, by regarding higher rank symmetry as an emergent phenomenon, the RG analysis presented in this paper holds alternative promise for realizing higher rank symmetry and higher moment conservation in experimentally achievable systems.
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.
Symmetry defects, e.g., vortices in conventional superfluids, play a critical role in a complete description of symmetry breaking phases. In this paper, we develop the theory of symmetry defects in fractonic superfluids, i.e., spontaneously higher-rank symmetry (HRS) breaking phases initially proposed by Yuan et al. [Phys. Rev. Res. 2, 023267 (2020)] and by Chen et al. [Phys. Rev. Res. 3, 013226 (2021)]. By Noether's theorem, HRS is associated with the conservation law of higher moments, e.g., dipoles, quadrupoles, and angular moments. We establish finite-temperature phase diagrams by identifying a series of topological phase transitions via the renormalization group flow equations and Debye-H\"uckel approximation. Accordingly, a series of Kosterlitz-Thouless topological transitions are found to occur successively at different temperatures, which are triggered by proliferation of defects, defect bound states, and so on. Such a hierarchical proliferation brings rich phase structures. Meanwhile, a screening effect from sufficiently high density of defect bound states leads to instability and collapse of the intermediate temperature phases, which further enriches the phase diagrams. For concreteness, we consider a fractonic superfluid in which ``angular moments'' are conserved. We then present the general theory, in which other types of HRS can be analyzed in a similar manner. Further directions are present at the end of the paper.
Fractonic superfluids are featured by the interplay of spontaneously broken charge symmetry and mobility constraints on single-particle kinematics due to the conservation of higher moments, such as dipoles, angular charge moments, and quadrupoles. Building on prior studies by Yuan [] and Chen [], we study a class of fractonic superfluids, termed (HFS), in which bosons of multiple species interact while moment hybridization is conserved. We explore the consequences of hybridization via two model series: , conserving total moments of the same order across species, and , conserving total moments of different orders. In Model Series A, we analyze dipole moment hybridization and extend the discussion to higher-order moments, examining the ground state, Goldstone modes, correlation functions, and so on. We compute the minimal spatial dimensions, where the total charge symmetry begins to get partially broken via particle-hole condensation, leading to true off-diagonal long-range order. In Model Series B, we focus on HFS with hybrid dipole-quadrupole conservation. For both series, we introduce Bose-Hubbard–type lattice models that reduce to either of both series in the weak Hubbard interaction regime. We perform a mean-field analysis on the global phase diagram and discuss experimental realizations in strongly tilted optical lattices via a third-order perturbation theory. This work, alongside prior studies, completes a trilogy on fractonic superfluids, uncovering symmetry-breaking physics emerging from higher moment conservation, leaving various promising studies for future investigation.
Fracton emerges from strongly interacting many-body systems whose excitations, referred to as sub-dimensional particles, have restricted mobility or kinetic motions. These entities have garnered significant interest due to their interdisciplinary implications spanning topological quantum codes, quantum field theory, emergent gravity, quantum information, and more, revealing unique nonequilibrium behaviors such as nonergodicity and glassy dynamics. This review presents a structured and educational overview of fracton phenomena, specifically focusing on gapless fracton liquids. Noteworthy for their compressibility and gapless excitations, fracton liquids facilitate collective modes reminiscent of those gauge fluctuations found in Maxwell's electromagnetic framework. However, they are distinct due to an additional higher-moment conservation law that restricts the mobility of individual charges and monopoles. Our exploration begins with the theoretical foundation of 3D fracton liquids, presenting a variety of emergent symmetric tensor gauge theories and reviewing their equilibrium and dynamic properties. Following this theoretical groundwork, we discuss the material realization of various fracton liquids in Yb-based breathing pyrochlore lattices, close-packed tiling structures and other synthetic quantum matter platforms. Additionally, we introduce a general protocol to foster and manipulate emergent fracton spin liquids in the realm of frustrated magnetism.
We extend recent work on hydrodynamics with global multipolar symmetries — known as “fracton hydrodynamics” — to systems in which the multipolar symmetries are gauged. We refer to the latter as “fracton magnetohydrodynamics”, in analogy to conventional magnetohydrodynamics (MHD), which governs systems with gauged charge conservation. We show that fracton MHD arises naturally from higher-rank Maxwell’s equations and in systems with one-form symmetries obeying certain constraints; while we focus on “minimal” higher-rank generalizations of MHD that realize diffusion, our methods may also be used to identify other, more exotic hydrodynamic theories (e.g., with magnetic subdiffusion). In contrast to semi-microscopic derivations of MHD, our approach elucidates the origin of the hydrodynamic modes by identifying the corresponding higher-form symmetries. Being rooted in symmetries, the hydrodynamic modes may persist even when the semi-microscopic equations no longer provide an accurate description of the system.