As a series of work about "fractonic superfluids," in this paper, we develop an exotic fractonic superfluid phase in d-dimensional space where subdimensional particles-their mobility is partially restricted-are condensed.The off-diagonal long range order (ODLRO) is investigated.To demonstrate, we consider "lineons"-a subdimensional particle whose mobility is free only in certain one-dimensional directions.We start with a d-component microscopic Hamiltonian model.The model respects a higher-rank symmetry such that both particle numbers of each component and angular charge moments are conserved quantities.By performing the Hartree-Fock-Bogoliubov approximation, we derive a set of Gross-Pitaevskii equations and a Bogoliubov-de Gennes (BdG) Hamiltonian, which leads to a description of both condensed components and unification of gapless phonons and gapped rotons.With the coherent-path-integral representation, we also derive the longwavelength effective field theory of gapless Goldstone modes and analyze quantum fluctuations around classical ground states.The Euler-Lagrange equations and Noether charges/currents are also studied.In two spatial dimensions and higher, such an ODLRO stays stable against quantum fluctuations.Finally, we study vortex configurations.The higher-rank symmetry enforces a hierarchy of thermal vortex excitations whose structures are dominated by two guiding statements.Specially, we construct two types of vortex excitations, the conventional and dipole vortices.The latter carries a charge with dimension as a momentum.The two statements can be more generally applicable.Further perspectives are discussed.
As a series of work about "fractonic superfluids," in this paper, we develop an exotic fractonic superfluid phase in d-dimensional space where subdimensional particles-their mobility is partially restricted-are condensed.The off-diagonal long range order (ODLRO) is investigated.To demonstrate, we consider "lineons"-a subdimensional particle whose mobility is free only in certain one-dimensional directions.We start with a d-component microscopic Hamiltonian model.The model respects a higher-rank symmetry such that both particle numbers of each component and angular charge moments are conserved quantities.By performing the Hartree-Fock-Bogoliubov approximation, we derive a set of Gross-Pitaevskii equations and a Bogoliubov-de Gennes (BdG) Hamiltonian, which leads to a description of both condensed components and unification of gapless phonons and gapped rotons.With the coherent-path-integral representation, we also derive the longwavelength effective field theory of gapless Goldstone modes and analyze quantum fluctuations around classical ground states.The Euler-Lagrange equations and Noether charges/currents are also studied.In two spatial dimensions and higher, such an ODLRO stays stable against quantum fluctuations.Finally, we study vortex configurations.The higher-rank symmetry enforces a hierarchy of thermal vortex excitations whose structures are dominated by two guiding statements.Specially, we construct two types of vortex excitations, the conventional and dipole vortices.The latter carries a charge with dimension as a momentum.The two statements can be more generally applicable.Further perspectives are discussed.
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.
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.
We study quantum phase transitions out of the fracton ordered phase of the ZN X-cube model. These phase transitions occur when various types of subdimensional excitations and their composites are condensed. The condensed phases are either trivial paramagnets, or are built from stacks of D=2 or 3 deconfined gauge theories, where D is the spatial dimension. The nature of the phase transitions depends on the excitations being condensed. Upon condensing dipolar bound states of fractons or lineons, for $N\ensuremath{\ge}4$ we find stable critical points described by decoupled stacks of D=2 conformal field theories. Upon condensing lineon excitations, when N>4 we find a gapless phase intermediate between the X-cube and condensed phases, described as an array of D=1 conformal field theories. In all these cases, effective subsystem symmetries arise from the mobility constraints on the excitations of the X-cube phase and play an important role in the analysis of the phase transitions.
Phases of matter have always attracted great interest within the scientific community, as they represent emergent phenomena arising from the collective behavior of a large number of interacting degrees of freedom. Their macroscopic properties, often impossible to predict directly from the microscopic scale, can be effectively captured in terms of quasiparticles, \textit{i.e.} particle-like low-energy excitations that emerge from interactions among the underlying microscopic constituents and may exhibit properties entirely different from those of any fundamental particle, reflecting the rich structure of the many-body system. In this Thesis, we investigate two classes of quasiparticles that arise in distinct exotic phases of matter. The first class appears in antiferromagnetic quantum spin chains with discrete symmetries and frustrated boundary conditions, namely periodic boundary conditions with an odd number of sites. In this setting, the competition between global and local orders produces exotic gapless, non-relativistic excitations above the ground state, which can dramatically modify the physics even in the thermodynamic limit. In particular, we focus on how this specific choice of boundary conditions profoundly modifies the phase diagram of one of the most famous integrable models, the XY chain. The second class consists of subdimensional quasiparticles, excitations whose mobility is restricted to lower-dimensional subspaces and which characterize the low-energy sector of emerging fracton phases of matter. Quasiparticles with restricted mobility first appeared in exactly solvable quantum spin models, and it was later realized that they can also emerge from higher-rank gauge theories. In this part of the Thesis, we develop Lorentz-covariant gauge field theories for fractons in 2+1 dimensions, starting from the first principles of Quantum Field Theory (fields and symmetries), with the aim of embedding the features observed in the condensed matter literature into a more formal field theoretical framework.
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.
Strongly correlated many-body systems provide a platform for novel phases of matter where constituent particles organize themselves in a variety of ways. At low temperature, these particles interact quantum mechanically and generate entanglement building up exotic quantum phases, such as topologcial order, where there can be emergent excitations which cannot be created locally. Such excitations, if gapped, are also called topological excitations.Fracton is one of such gapped point-like topological excitation in three dimensional system. Different from conventional topological excitation, it is immobile and was firstly discovered in exact solvable models exhibiting fracton topological order. This new order has sub-extensive topological ground-state degeneracy and generically also possesses other mobile excitations restricted to move in sub-dimensional spaces. Also, it has been noticed that the charges of symmetric-tensor U(1) gauge theories can be fractons. Its immobility is due to the existence of multiple conservation laws.In this thesis, I will present the relation among the gapped fracton topological order, gapless fracton phase described by U(1) symmetric tensor gauge theories, and ordinary topological ordered phases. Particularly, the fracton topological order exhibited in an exact solvable model called X- cube model can be constructed by coupling toric code layers. The mechanism leads to fracton topological orders is dubbed “p-string condensation” or “p-membrane condensation,” in which strings or membranes built up from particle excitations from layers of topological orders condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the resulting fracton order to be easily understood in terms of more familiar degrees of freedom. And the fracton topological order in the X-cube model can also be obtained from a particular rank-2 symmetric tensor gauge theory called scalar charge theory by a partial confinement transition followed by Higgs mechanism removing the gapless photon modes.