Emergent fermionic gauge theory and foliated fracton order in the Chamon model

The Chamon model is an exactly solvable spin Hamiltonian exhibiting nontrivial fracton order. In this work, we dissect two distinct aspects of the model. First, we show that it exhibits an emergent fractonic gauge theory coupled to a fermionic subsystem symmetry-protected topological state under four stacks of $\mathbb{Z}_2$ planar symmetries. Second, we show that the Chamon model hosts 4-foliated fracton order by describing an entanglement renormalization group transformation that exfoliates four separate stacks of 2D toric codes from the bulk system.Comment: 18 pages, 9 figure

Fracton phases of matter

Fractons are a new type of quasiparticle which are immobile in isolation, but can often move by forming bound states. Fractons are found in a variety of physical settings, such as spin liquids and elasticity theory, and exhibit unusual phenomenology, such as gravitational physics and localization. The past several years have seen a surge of interest in these exotic particles, which have come to the forefront of modern condensed matter theory. In this review, we provide a broad treatment of fractons, ranging from pedagogical introductory material to discussions of recent advances in the field. We begin by demonstrating how the fracton phenomenon naturally arises as a consequence of higher moment conservation laws, often accompanied by the emergence of tensor gauge theories. We then provide a survey of fracton phases in spin models, along with the various tools used to characterize them, such as the foliation framework. We discuss in detail the manifestation of fracton physics in elasticity theory, as well as the connections of fractons with localization and gravitation. Finally, we provide an overview of some recently proposed platforms for fracton physics, such as Majorana islands and hole-doped antiferromagnets. We conclude with some open questions and an outlook on the field

Many-body physics of spontaneously broken higher-rank symmetry: from fractonic superfluids to dipolar Hubbard model

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.Comment: Title changed, references updated. A short review on recent progress on higher rank symmetry breaking, fractonic superfluids, dipole (and other higher moments) conservation, and related topic

Boundary theory of the X-cube model in the continuum

We study the boundary theory of the $\mathbb{Z}_N$ X-cube model using a continuum perspective, from which the exchange statistics of a subset of bulk excitations can be recovered. We discuss various gapped boundary conditions that either preserve or break the translation/rotation symmetries on the boundary, and further present the corresponding ground state degeneracies on $T^2\times I$. The low-energy physics is highly sensitive to the boundary conditions: even the extensive part of the ground state degeneracy can vary when different sets of boundary conditions are chosen on the two boundaries. We also examine the anomaly inflow of the boundary theory and find that the X-cube model is not the unique (3+1)d theory that cancels the 't Hooft anomaly of the boundary.Comment: v

Layer Codes

The surface code is a two dimensional topological code with code parameters that scale optimally with the number of physical qubits, under the constraint of two dimensional locality. In three spatial dimensions an analogous simple yet optimal code was not previously known. Here, we introduce a construction that takes as input a stabilizer code and produces as output a three dimensional topological code with related code parameters. The output codes have the special structure of being topological defect networks formed by layers of surface code joined along one dimensional junctions, with a maximum stabilizer check weight of six. When the input is a family of good low density parity check codes, the output is a three dimensional topological code with optimal scaling code parameters and a polynomial energy barrier.Comment: 63 pages, 16 figure

Engineering Floquet codes by rewinding

Floquet codes are a novel class of quantum error-correcting codes with dynamically generated logical qubits, which arise from a periodic schedule of non-commuting measurements. We engineer new examples of Floquet codes with measurement schedules that $\textit{rewind}$ during each period. The rewinding schedules are advantageous in our constructions for both obtaining a desired set of instantaneous stabilizer groups and for constructing boundaries. Our first example is a Floquet code that has instantaneous stabilizer groups that are equivalent -- via finite-depth circuits -- to the 2D color code and exhibits a $\mathbb{Z}_3$ automorphism of the logical operators. Our second example is a Floquet code with instantaneous stabilizer codes that have the same topological order as the 3D toric code. This Floquet code exhibits a splitting of the topological order of the 3D toric code under the associated sequence of measurements i.e., an instantaneous stabilizer group of a single copy of 3D toric code in one round transforms into an instantaneous stabilizer group of two copies of 3D toric codes up to nonlocal stabilizers, in the following round. We further construct boundaries for this 3D code and argue that stacking it with two copies of 3D subsystem toric code allows for a transversal implementation of the logical non-Clifford $CCZ$ gate. We also show that the coupled-layer construction of the X-cube Floquet code can be modified by a rewinding schedule such that each of the instantaneous stabilizer codes is finite-depth-equivalent to the X-cube model up to toric codes; the X-cube Floquet code exhibits a splitting of the X-cube model into a copy of the X-cube model and toric codes under the measurement sequence. Our final example is a generalization of the honeycomb code to 3D, which has instantaneous stabilizer codes with the same topological order as the 3D fermionic toric code.Comment: 20+3 pages, 27 figures, $\texttt{Mathematica}$ files are available at https://github.com/dua-arpit/floquetcodes, v2 changes: added more details on the rewinding X-cube Floquet code and made minor updates in color code figures in the appendi