Subsystem Symmetries Research Articles

Topological properties of subsystem-symmetry-protected edge states in an extended quasi-one-dimensional dimerized lattice

We investigate theoretically the topological properties of dimerized quasi-one-dimensional (1D) lattice comprising of multi legs $(L)$ as well as multi sublattices $(R)$. The system has main and subsidiary exchange symmetries. In the basis of latter one, the system can be divided into $L$ 1D subsystems each of which corresponds to a generalized $SSH_R$ model having $R$ sublattices and on-site potentials. Chiral symmetry is absent in all subsystems except when the axis of main exchange symmetry coincides on the central chain. We find that the system may host zero- and finite-energy topological edge states. The existence of zero-energy edge state requires a certain relation between the number of legs and sublattices. As such, different topological phases, protected by subsystem symmetry, including zero-energy edge states in the main gap, no zero-energy edge states, and zero-energy edge states in the bulk states are characterized. Despite the classification symmetry of the system belongs to $BDI$ but each subsystem falls in either $AI$ or $BDI$ symmetry class.

Hilbert space fragmentation in a 2D quantum spin system with subsystem symmetries

We consider a 2D quantum spin model with ring-exchange interaction that has subsystem symmetries associated to conserved magnetization along rows and columns of a square lattice, which implies the conservation of the global dipole moment. The model is not integrable, but violates the eigenstate thermalization hypothesis through an extensive Hilbert space fragmentation, including an exponential number of inert subsectors with trivial dynamics, arising from kinetic constraints. While subsystem symmetries are quite restrictive for the dynamics, we show that they alone cannot account for such a number of inert states, even with infinite-range interactions. We present a procedure for constructing shielding structures that can separate and disentangle dynamically active regions from each other. Notably, subsystem symmetries allow the thickness of the shields to be dependent only on the interaction range rather than on the size of the active regions, unlike in the case of generic dipole-conserving systems.

Dynamics in Systems with Modulated Symmetries.

We extend the notions of multipole and subsystem symmetries to more general spatially modulated symmetries. We uncover two instances with exponential and (quasi)periodic modulations and provide simple microscopic models in one, two, and three dimensions. Seeking to understand their effect on the long-time dynamics, we numerically study a stochastic cellular automaton evolution that obeys such symmetries. We prove that, in one dimension, the periodically modulated symmetries lead to a diffusive scaling of correlations modulated by a finite microscopic momentum. In higher dimensions, these symmetries take the form of lines and surfaces of conserved momenta. These give rise to exotic forms of subdiffusive behavior with a rich spatial structure influenced by lattice-scale features. Exponential modulation, on the other hand, can lead to correlations that are infinitely long-lived at the boundary while decaying exponentially in the bulk.

Multicritical point with infinite fractal symmetries

Recently, ``Pascal's triangle model'' constructed with $\mathrm{U}(1)$ rotor degrees of freedom was introduced and it was shown that ($\mathit{i}$) this model possesses an infinite series of fractal symmetries and ($\mathit{ii}$) it is the parent model of a series of ${Z}_{p}$ fractal models each with its own distinct fractal symmetry. In this work we discuss a multicritical point of Pascal's triangle model that is analogous to the Rokhsar-Kivelson point of the better known quantum dimer model. We demonstrate that the expectation value of the characteristic operator of each fractal symmetry at this multicritical point decays as a power law of space and this multicritical point is shared by the family of descendent ${Z}_{p}$ fractal models. Afterwards, we generalize our discussion to a $(3+1)d$ model termed ``Pascal's tetrahedron model'' that has both planar and fractal subsystem symmetries. We also establish a connection between Pascal's tetrahedron model and the $\mathrm{U}(1)$ Haah's code.

Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps.

It has recently been established that clusterlike states-states that are in the same symmetry-protected topological phase as the cluster state-provide a family of resource states that can be utilized for measurement-based quantum computation. In this Letter, we ask whether it is possible to prepare clusterlike states in finite time without breaking the symmetry protecting the resource state. Such a symmetry-preserving protocol would benefit from topological protection to errors in the preparation. We answer this question in the positive by providing a Hamiltonian in one higher dimension whose finite-time evolution is a unitary that acts trivially in the bulk, but pumps the desired cluster state to the boundary. Examples are given for both the 1D cluster state protected by a global symmetry, and various 2D cluster states protected by subsystem symmetries. We show that even if unwanted symmetric perturbations are present in the driving Hamiltonian, projective measurements in the bulk along with feed-forward correction is sufficient to recover a clusterlike state.

Anomaly inflow for subsystem symmetries

We study 't Hooft anomalies and the related anomaly inflow for subsystem global symmetries. These symmetries and anomalies arise in a number of exotic systems, including models with fracton order such as the $X$-cube model. As is the case for ordinary global symmetries, anomalies for subsystem symmetries can be canceled by anomaly inflow from a bulk theory in one higher dimension; the corresponding bulk is therefore a nontrivial subsystem symmetry protected topological (SSPT) phase. We demonstrate these phenomena in several examples with continuous and discrete subsystem global symmetries, as well as time-reversal symmetry. For each example we describe the boundary anomaly, and present classical continuum actions for the corresponding bulk SSPT phases, which describe the response of background gauge fields associated with the subsystem symmetries. Interestingly, we show that the anomaly does not uniquely specify the bulk SSPT phase. In general, the latter may also depend on how the symmetry and the associated foliation structure on the boundary are extended into the bulk.

Non-Fermi liquid induced by Bose metal with protected subsystem symmetries

Understanding non-Fermi liquids in dimensions higher than one has been a subject of great interest. Such phases may serve as parent states for other unconventional phases of quantum matter, in a similar manner that conventional broken symmetry states can be understood as instabilities of the Fermi liquid. In this work, we investigate the emergence of an exotic non-Fermi liquid in two dimensions, where the fermions with quadratic band-touching dispersion interact with Bose metal. The bosonic excitations in Bose metal possess an extended nodal-line spectrum in momentum space, which arises due to the subsystem symmetry or the restricted motion of bosons. Using renormalization group analysis and direct computations, we show that the extended infrared singularity of Bose metal leads to a line of interacting fixed points of an exotic non-Fermi liquids, where the anomalous dimension of the fermions varies continuously, akin to the Luttinger liquid in one dimension. Further, the generalization of the model with multiple low-energy excitations is used to explore other unusual features of the resulting ground state.

Spontaneously broken supersymmetric fracton phases with fermionic subsystem symmetries

We construct a purely fermionic system with spontaneously broken supersymmetry that shares the common feature with a fracton phase of matter. Our model is gapless due to the Nambu-Goldstone mechanism. It shows a ground-state degeneracy with the “Area-law” entropy due to fermionic subsystem symmetries. In the strongly coupled limit, it becomes a variant of the Nicolai model, and we conjecture that the ground-state degeneracy shows the “Volume-law” entropy. Gauging the fermionic subsystem symmetry has an t’Hooft anomaly by itself, but the would-be gauged theory may possess a fermionic defect that is immobile in certain spatial directions.

Fractionalization of subsystem symmetries in two dimensions

The fractionalization of global symmetry charges is a striking hallmark of topological quantum order. Here, we discuss the fractionalization of subsystem symmetries in two-dimensional topological phases. In line with previous no-go arguments, we show that subsystem symmetry fractionalization is not possible in many cases due to the additional rigid geometric structure of the symmetries. However, we identify a new mechanism that allows fractionalization, involving global relations between macroscopically many symmetry generators. We find that anyons can fractionalize such relations, meaning that the total charge carried under all generators involved in the global relation is non-trivial, despite the fact that these generators multiply to the identity. We first discuss the general algebraic framework needed to characterize this new type of fractionalization, and then explore this framework using a number of exactly solvable models with $\mathbb{Z}_2$ topological order, including models having line and fractal symmetries. These models all showcase another necessary property of subsystem symmetry fractionalization: fractionalized anyons must have restricted mobility when the symmetry is enforced, such that they are confined to a single line or point in the case of line and fractal symmetries, respectively. Looking forward, we expect that our identification of the importance of global relations in fractionalization will hold significance for the classification of phases with subsystem symmetries in all dimensions.

Fracton topological order at finite temperature

As new kinds of stabilizer code models, fracton models have been promising in realizing quantum memory or quantum hard drives. However, it has been shown that the fracton topological order of 3D fracton models occurs only at zero temperature. In this Letter, we show that higher dimensional fracton models can support a fracton topological order below a nonzero critical temperature $T_c$. Focusing on a typical 4D X-cube model, we show that there is a finite critical temperature $T_c$ by analyzing its free energy from duality. We also obtained the expectation value of the 't Hooft loops in the 4D X-cube model, which directly shows a confinement-deconfinement phase transition at finite temperature. This finite-temperature phase transition can be understood as spontaneously breaking the $\mathbb{Z}_2$ one-form subsystem symmetry. Moreover, we propose a new no-go theorem for finite-temperature quantum fracton topological order.

Interaction-enabled fractonic higher-order topological phases

In this work, we present a collection of three-dimensional higher-order symmetry protected topological phases (HOSPTs) with gapless hinge modes that exist only in strongly interacting systems subject to subsystem symmetry constraints. We use a coupled wire construction to generate three families of microscopic lattice models: insulators with helical hinge modes, superconductors with chiral Majorana hinge modes, and fractionalized insulators with helical hinge modes that carry fractional charge. In particular, these HOSPTs do not require spatial symmetry protection, but are instead protected by subsystem symmetries, and support "fractonic" quasiparticle excitations that move within only a low-dimensional sub-manifold of the system. We analyze the anomaly structure for the boundary theory and the entanglement Hamiltonian, and show that the side surfaces of these HOSPTs, despite being partially gapped, exhibit symmetry anomalies, and can only be realized as the boundary of three-dimensional HOSPT phases.

Spontaneously broken subsystem symmetries

We investigate the spontaneous breaking of subsystem symmetries directly in the context of continuum field theories by calculating the correlation function of charged operators. Our methods confirm the lack of spontaneous symmetry breaking in some of the existing continuum field theories with subsystem symmetries, as had previously been established based on a careful analysis of the spectrum. We present some novel continuum field theory constructions that do exhibit spontaneous symmetry breaking whenever allowed by general principles. These interesting patterns of symmetry breaking occur despite the fact that all the theories we study are non-interacting.

Gapless edge modes in (4+1)-dimensional topologically massive tensor gauge theory and anomaly inflow for subsystem symmetry

Abstract We consider the (4 + 1)D topologically massive tensor gauge theory. This theory is an analog of the (2 + 1)D topologically massive Maxwell–Chern–Simons theory. If the space has a boundary, we find that a (3 + 1)D gapless theory appears at the boundary. This gapless theory is a chiral version of the (3 + 1)D φ theory. This gapless theory is protected by the anomaly inflow mechanism for subsystem symmetry. We also consider the corner of our topologically massive tensor gauge theory, and find that an infinite number of (1 + 1)D chiral bosons appear at the corner.

Infinite families of fracton fluids with momentum conservation

We construct infinite families of new universality classes of fracton hydrodynamics with momentum conservation, both with multipole conservation laws and/or subsystem symmetry. We explore the effects of broken inversion and/or time-reversal symmetry at the ideal fluid level, along with momentum relaxation. In the case of one-dimensional multipole-conserving models, we write down explicit microscopic Hamiltonian systems realizing these new universality classes. All of these hydrodynamic universality classes exhibit instabilities and will flow to new nonequilibrium fixed points. Such fixed points are predicted to exist in arbitrarily large spatial dimensions.

Worldline description of fractons

We formulate the word-line approach of the field theory of fractons and their symmetries. The distinction between the different models is based on their dispersion relations for the energy. In order to study the sub-system symmetries, we construct the Routhians associated with the particle Lagrangians considered. We also construct the pseudoclassical description of spinning fractons.

Non-Abelian fracton order from gauging a mixture of subsystem and global symmetries

We demonstrate a general gauging procedure of a pure matter theory on a lattice with a mixture of subsystem and global symmetries. This mixed symmetry can be either a semidirect product of a subsystem symmetry and a global symmetry, or a non-trivial extension of them. We demonstrate this gauging procedure on a cubic lattice in three dimensions with four examples: $G=\mathbb{Z}_3^{\text{sub}} \rtimes \mathbb{Z}_2^{\text{glo}}$, $G=(\mathbb{Z}_2^{\text{sub}} \times \mathbb{Z}_2^{\text{sub}}) \rtimes \mathbb{Z}_2^{\text{glo}}$, $1\to \mathbb {Z}_2^\text {sub}\to G\to \mathbb {Z}_2^\text {glo}\to 1$, and $1\to \mathbb {Z}_2^\text {sub}\to G\to K_4^\text {glo}\to 1$. The former two cases and the last one produce the non-Abelian fracton orders. Our construction of the gauging procedure provides an identification of the electric charges of these fracton orders with irreducible representations of the symmetry. Furthermore, by constraining the local Hilbert space, the magnetic fluxes with different geometry (tube-like and plaquette-like) satisfy a subalgebra of the quantum double models (QDMs). This algebraic structure leads to an identification of the magnetic fluxes to the conjugacy classes of the symmetry.

Fractons and Exotic Symmetries from Branes

AbstractThe emerging study of fractons, a new type of quasi‐particle with restricted mobility, has motivated the construction of several classes of interesting continuum quantum field theories with novel properties. One such class consists of foliated field theories which, roughly, are built by coupling together fields supported on the leaves of foliations of spacetime. Another approach, which we refer to as exotic field theory, focuses on constructing Lagrangians consistent with special symmetries (like subsystem symmetries) that are adjacent to fracton physics. A third framework is that of infinite‐component Chern‐Simons theories, which attempts to generalize the role of conventional Chern‐Simons theory in describing (2+1)D Abelian topological order to fractonic order in (3+1)D. The study of these theories is ongoing, and many of their properties remain to be understood. Historically, it has been fruitful to study QFTs by embedding them into string theory. One way this can be done is via D‐branes, extended objects whose dynamics can, at low energies, be described in terms of conventional quantum field theory. QFTs that can be realized in this way can then be analyzed using the rich mathematical and physical structure of string theory. In this paper, we show that foliated field theories, exotic field theories, and infinite‐component Chern‐Simons theories can all be realized on the world‐volumes of branes. We hope that these constructions will ultimately yield valuable insights into the physics of these interesting field theories.

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

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 sub-dimensional excitations and their composites are condensed. The condensed phases are either trivial paramagnets, or are built from stacks of $d=2$ or $d=3$ 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 $N \geq 4$ 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.

Quivers and Fractons.

We discuss certain structural analogies between supersymmetric quiver gauge theories and lattice models leading to fracton phases of matter. In particular, classes of quiver models can be viewed as lattice models having subsystem symmetries, dimensions of moduli spaces growing linearly with the size of the lattice, and having excitations with limited mobility (with "excitations" and "mobility" properly defined).

Comments on foliated gauge theories and dualities in 3+1d

We investigate the properties of foliated gauge fields and construct several foliated field theories in 3+1d that describe foliated fracton orders both with and without matter, including the recent hybrid fracton models. These field theories describe Abelian or non-Abelian gauge theories coupled to foliated gauge fields, and they fall into two classes of models that we call the electric models and the magnetic models. We show that these two classes of foliated field theories enjoy a duality. We also construct a model (using foliated gauge fields and an exactly solvable lattice Hamiltonian model) for a subsystem-symmetry protected topological (SSPT) phase, which is analogous to a one-form symmetry protected topological phase, with the subsystem symmetry acting on codimension-two subregions. We construct the corresponding gauged SSPT phase as a foliated two-form gauge theory. Some instances of the gauged SSPT phase are a variant of the X-cube model with the same ground state degeneracy and the same fusion, but different particle statistics.