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arXiv:2107.09073 · Physical review. B./Physical review. B

Subdimensional criticality: Condensation of lineons and planons in the X-cube model

Ethan Lake, Michael Hermele

Published Oct 11, 2021• 26 citations
arXivHTMLPDF

Abstract

We study quantum phase transitions out of the fracton ordered phase of the ${\mathbb{Z}}_{N}$ 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.

DOI
10.1103/physrevb.104.165121

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We study quantum phase transitions out of the fracton ordered phase of the ZN{\mathbb{Z}}_{N}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=2D=2D=2 or 3 deconfined gauge theories, where DDD 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≥4N\ensuremath{\ge}4N\ensuremath≥4 we find stable critical points described by decoupled stacks of D=2D=2D=2 conformal field theories. Upon condensing lineon excitations, when N>4N>4N>4 we find a gapless phase intermediate between the X-cube and condensed phases, described as an array of D=1D=1D=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.
Subsystem symmetry has emerged as a powerful organizing principle for unconventional quantum phases of matter, most prominently fracton topological orders. Here, we focus on a special subclass of such symmetries, known as higher-form subsystem symmetries, which allow us to adapt tools from the study of conventional topological phases to the fracton setting. We demonstrate that certain transitions out of familiar fracton phases, including the X-cube model, can be understood in terms of the spontaneous breaking of higher-form subsystem symmetries. We find simple pictures for these seemingly complicated fracton topological phase transitions by relating them in an exact manner, via gauging, to spontaneous higher-form subsystem symmetry breaking phase transitions of decoupled stacks of lower-dimensional models. We harness this perspective to construct a sequence of unconventional subdimensional critical points in two and three spatial dimensions based on the stacking and gauging of canonical models with higher-form symmetry. Through numerous examples, we illustrate the ubiquity of coupled layer constructions in theories with higher-form subsystem symmetries.
As an unconventional realization of topological orders with an exotic interplay of topology and geometry, fracton (topological) orders feature subextensive topological ground-state degeneracy and subdimensional excitations that are movable only within a certain subspace. It has been known, in the exactly solvable three-dimensional X-cube model that universally represents the type-I fracton orders, that mobility constraints on subdimensional excitations originate from the absence of spatially deformable stringlike operators. To unveil the interplay of topology and geometry, in this paper, we study the dynamical signature in the X-cube model in the presence of external Zeeman fields via large-scale quantum Monte Carlo simulation and stochastic analytic continuation. We compute both real-space correlation functions and dynamic structure factors of subdimensional excitations (i.e., fractons, lineons, and planons) in the fracton phase and their evolution into the trivial paramagnetic phase by increasing external fields. We find, in the fracton phase, that the correlation functions and the spectral functions show clear anisotropy exactly caused by the underlying mobility constraints. On the other hand, the external fields successfully induce quantum fluctuations and offer mobility to excitations along the subspace allowed by mobility constraints. These numerical results provide the evolution of a dynamical signature of subdimensional particles in fracton orders, indicating that the mobility constraints on local dynamical properties of subdimensional excitations are deeply related to the existence of fracton topological order. The results will also be helpful in potential experimental identifications in spectroscopy measurements such as neutron scattering and nuclear magnetic resonance.
There are now many examples of gapped fracton models, which are defined by the presence of restricted-mobility excitations above the quantum ground state. However, the theory of fracton orders remains in its early stages, and the complex landscape of examples is far from being mapped out. Here we introduce the class of planon-modular (p-modular) fracton orders, a relatively simple yet still rich class of quantum orders that encompasses several well-known examples of type I fracton order. The defining property is that any non-trivial point-like excitation can be detected by braiding with planons. From this definition, we uncover a significant amount of general structure, including the assignment of a natural number (dubbed the weight) to each excitation of a p-modular fracton order. We identify simple new phase invariants, some of which are based on weight, which can easily be used to compare and distinguish different fracton orders. We also study entanglement renormalization group (RG) flows of p-modular fracton orders, establishing a close connection with foliated RG. We illustrate our general results with an analysis of several exactly solvable fracton models that we show to realize p-modular fracton orders, including Z_n versions of the X-cube, anisotropic, checkerboard, 4-planar X-cube and four color cube (FCC) models. We show that each of these models is p-modular and compute its phase invariants. We also show that each example admits a foliated RG at the level of its non-trivial excitations, which is a new result for the 4-planar X-cube and FCC models. We show that the Z_2 FCC model is not a stack of other better-studied models, but predict that the Z_n FCC model with n odd is a stack of 10 4-planar X-cubes, possibly plus decoupled layers of 2d toric code. We also show that the Z_n checkerboard model for n odd is a stack of three anisotropic models.
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 matter can undergo unconventional phase transitions driven by the condensation of particles that move along subdimensional manifolds. We propose that this type of quantum critical point can be realized in a bilayer of crossed Rydberg chains. This system exhibits a transition between a disordered phase and a charge-density-wave phase with subextensive ground-state degeneracy. We show that this transition is described by a stack of critical Ising conformal field theories which become decoupled in the low-energy limit. We also analyze the transition using a Majorana mean-field approach for an effective lattice model, which confirms the picture of a fixed point of decoupled critical chains. We discuss the unusual scaling properties and derive anisotropic correlators that provide signatures of subdimensional criticality in this realistic setup.
The authors present a generic lattice construction for a generalized X-cube model of fracton topological order and propose a coarser definition of phase of matter for fracton orders. Similar to other fracton models, this model hosts emergent subdimensional excitations, which are topologically robust excitations that are confined to only move along certain lines or surfaces. The model is constructed from stacks of intersecting surfaces, and the subdimensional excitations are confined to move only along these (possibly curved) surfaces or their intersections. Surprisingly, for certain curved intersecting surfaces, a robust ground-state degeneracy on a manifold with trivial topology can result.