Abstract
In this paper, we develop an exotic fractonic superfluid phase in $d$-dimensional space where subdimensional particles -- their mobility is \emph{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 unified description of gapless phonons and gapped rotons. With the coherent-path-integral representation, we also derive the long-wavelength 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 point vortex excitations whose structure is 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. Several future directions are discussed.
Highlights
As exotic states of matter, fracton topological order can be characterized by noise-immune ground state degeneracy that unconventionally depends on the system size on a nontrivial compact manifold [1–4]
As a series of works, here we focus on dcomponent fields = ( ˆ 1, · · ·, ˆ d ) in d spatial dimensions in the Hamiltonian H = dd xH where Hamiltonian density H reads
Since quantum fluctuations are weaker in higher dimensions, a fractonic superfluid phase dSF1 stays stable in two spatial dimensions d = 2 and higher d > 2
Summary
As exotic states of matter, fracton topological order can be characterized by noise-immune ground state degeneracy that unconventionally depends on the system size on a nontrivial compact manifold [1–4]. Instead of the interpretation as “topological excitations,” one can regard all these strange particles, i.e., fractons, lineons, and planeons as original bosons, which leads to unconventional many-body physics In this context, the restriction on mobility is ascribed to the implementation of so-called “higher-rank symmetry.”. When the chemical potential is turned from a negative to positive value, the system undergoes a quantum phase transition from the normal state to the superfluid phase The latter is manifested by occupation of a macroscopic number of fractons on the same quantum state, which leads to the formation of an off-diagonal long range order (ODLRO) [66]. Gaussian terms such that mobility restriction of lineons is correctly encoded Both angular charge moments and particle numbers of each component are conserved due to the presence of quartic terms that respect a higher-rank symmetry.
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