We propose a coupled-layer construction of a class of fracton topological orders in three spatial dimensions, which is characterized by spatially anisotropic mobility of quasiparticle excitations constrained in subdimensional manifolds. The simplest model is obtained by stacking and coupling layers of the two-dimensional toric codes on the square lattice and can be exactly solved in the strong-coupling limit. The resulting fracton excitations are understood as a consequence of anyon pair condensation induced by the coupling between layers. We also present generalizations of the construction for layers of the Kitaev-honeycomb models, the $Z_N$ toric codes, and the toric codes and the doubled semion models on the honeycomb lattice.
We present a three-dimensional cubic lattice spin model, anisotropic in the $\hat{z}$ direction, that exhibits fracton topological order. The latter is a novel type of topological order characterized by the presence of immobile pointlike excitations, named fractons, residing at the corners of an operator with two-dimensional support. As other recent fracton models, ours exhibits a subextensive ground state degeneracy: On an $L_x\times L_y\times L_z$ three-torus, it has a $2^{2L_z}$ topological degeneracy, and an additional non-topological degeneracy equal to $2^{L_xL_y-2}$. The fractons can be…
Starting from an isotropic configuration of intersecting, two-dimensional toric codes, we construct a fracton topological phase introduced in Ref. [26], which is characterized by immobile, point- like topological excitations ("fractons"), and degenerate ground-states on the torus that are locally indistinguishable. Our proposal leads to a simple description of the fracton excitations and of the ground-state as a "loop" condensate, and provides a basis for building new 3D topological orders such as a natural, $Z_{N}$ generalization of this fracton phase, which we introduce. We describe the rich…
We describe a construction of topological orders from coupled lower dimensional symmetry-protected topological orders, which is closely related to gauging a subsystem symmetry. Our construction yields both conventional topological orders and exotic fracton topological orders of type-I and type-II. In particular, we find a coupled spin chain construction of Haah's cubic code, and a coupled layer construction of Yoshida's fractal spin liquids.
In this work, we develop a coupled layer construction of fracton topological orders in $d=3$ spatial dimensions. These topological phases have sub-extensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple $d=2$ topological phases and from stacks of $d=3$ fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which …
We present a coupled-wire construction of a model with chiral fracton topological order. The model combines the known construction of $ν=1/m$ Laughlin fractional quantum Hall states with a planar p-string condensation mechanism. The bulk of the model supports gapped immobile fracton excitations that generate a hierarchy of mobile composite excitations. Open boundaries of the model are chiral and gapless, and can be used to demonstrate a fractional quantized Hall conductance where fracton composites act as charge carriers in the bulk. The planar p-string mechanism used to construct and analyze …
We introduce a new kind of foliated quantum field theory (FQFT) of gapped fracton orders in the continuum. FQFT is defined on a manifold with a layered structure given by one or more foliations, which each decompose spacetime into a stack of layers. FQFT involves a new kind of gauge field, a foliated gauge field, which behaves similar to a collection of independent gauge fields on this stack of layers. Gauge invariant operators (and their analogous particle mobilities) are constrained to the intersection of one or more layers from different foliations. The level coefficients are quantized and …
Fracton phases feature elementary excitations with fractionalized mobility and are exciting interest from multiple areas of theoretical physics. However, the most exotic 'type-II' fracton phases, like the Haah codes, currently have no known experimental diagnostics. Here, we explain how type-II fracton phases with gapless gauge modes, such as the $\mathrm{U}(1)$ Haah code, may be identified experimentally. Our analysis makes use of the 'multipole gauge theory' description of type-II fracton phases, which exhibits ultraviolet-infrared (UV-IR) mixing. We show that neutron scattering experiments …
It is well-known that 3D Type-I fracton models can be obtained from the condensations of stacked layers of 2D anyons. It is less obvious if 3D Type-II fractons can be understood from a similar perspective. In this paper, we affirm that this is the case: we produce the paradigm Type-II fracton model, Haah's cubic code, from a 2D layer construction. However, this is not a condensation of 2D anyons, but rather we start with stacks of 2D subsystem-symmetry protected topological states (SSPT). As this parent model is not topologically ordered in the strict sense, whereas the final state is, we refe…
Fracton models, a collection of exotic gapped lattice Hamiltonians recently discovered in three spatial dimensions, contain some 'topological' features: they support fractional bulk excitations (dubbed fractons), and a ground state degeneracy that is robust to local perturbations. However, because previous fracton models have only been defined and analyzed on a cubic lattice with periodic boundary conditions, it is unclear to what extent a notion of topology is applicable. In this paper, we demonstrate that the X-cube model, a prototypical type-I fracton model, can be defined on general three-…
We introduce and develop a theory of fusion and statistical processes of gapped excitations in Abelian fracton phases. The key idea is to incorporate lattice translation symmetry via its action on superselection sectors, which results in a fusion theory endowed with information about the non-trivial mobility of fractons and sub-dimensional excitations. This results in a description of statistical processes in terms of local moves determined by the fusion theory. Our results can be understood as providing a characterization of translation-invariant fracton phases. We obtain simple descriptions …
Generally, ``fracton'' topological orders are referred to as gapped phases that support \textit{point-like topological excitations} whose mobility is, to some extent, restricted. In our previous work [Phys. Rev. B 101, 245134 (2020)], a large class of exactly solvable models on hypercubic lattices are constructed. In these models, \textit{spatially extended excitations} possess generalized fracton-like properties: not only mobility but also deformability is restricted. As a series work, in this paper, we proceed further to compute ground state degeneracy (GSD) in both isotropic and anisotropic…
We consider XY-spin degrees of freedom on an FCC lattice, such that the system respects some subsystem global symmetry. We then gauge this global symmetry and study the corresponding $U(1)$ gauge theory on the FCC lattice. Surprisingly, this $U(1)$ gauge theory is dual to the original spin system. We also analyze a similar $\mathbb{Z}_N$ gauge theory on that lattice. All these systems are fractonic. The $U(1)$ theories are gapless and the $\mathbb{Z}_N$ theories are gapped. We analyze the continuum limits of all these systems and present free continuum Lagrangians for their low-energy physics.…
We discuss the procedure for gauging on-site $\mathbb{Z}_2$ global symmetries of three-dimensional lattice Hamiltonians that permute quasi-particles and provide general arguments demonstrating the non-Abelian character of the resultant gauged theories. We then apply this general procedure to lattice models of several well known fracton phases: two copies of the X-Cube model, two copies of Haah's cubic code, and the checkerboard model. Where the former two models possess an on-site $\mathbb{Z}_2$ layer exchange symmetry, that of the latter is generated by the Hadamard gate. For each of these mo…
We introduce a $\mathbb{Z}_N$ stabilizer code that can be defined on any spatial lattice of the form $Γ\times C_{L_z}$, where $Γ$ is a general graph. We also present the low-energy limit of this stabilizer code as a Euclidean lattice action, which we refer to as the anisotropic $\mathbb{Z}_N$ Laplacian model. It is gapped, robust (i.e., stable under small deformations), and has lineons. Its ground state degeneracy (GSD) is expressed in terms of a "mod $N$-reduction" of the Jacobian group of the graph $Γ$. In the special case when space is an $L\times L\times L_z$ cubic lattice, the logarithm o…
Fractional excitations in fracton models exhibit novel features not present in conventional topological phases: their mobility is constrained, there are an infinitude of types, and they bear an exotic sense of 'braiding'. Hence, they require a new framework for proper characterization. Based on our definition of foliated fracton phases in which equivalence between models includes the possibility of adding layers of gapped 2D states, we propose to characterize fractional excitations in these phases up to the addition of quasiparticles with 2D mobility. That is, two quasiparticles differing by a…
We introduce a class of gapped three-dimensional models, dubbed "cage-net fracton models," which host immobile fracton excitations in addition to non-Abelian particles with restricted mobility. Starting from layers of two-dimensional string-net models, whose spectrum includes non-Abelian anyons, we condense extended one-dimensional "flux-strings" built out of point-like excitations. Flux-string condensation generalizes the concept of anyon condensation familiar from conventional topological order and allows us to establish properties of the fracton ordered (equivalently, flux-string condensed)…
Excitation-detector principle and the algebraic theory of planon-only abelian fracton orders
University of Colorado, Boulder（科罗拉多大学博尔德分校）；Center for Theory of Quantum Matter, University of Colorado, Boulder（科罗拉多大学博尔德分校量子物质理论中心）；Kadanoff Center for Theoretical Physics, University of Chicago（芝加哥大学卡达夫理论物理中心）；School of Natural Sciences, Institute for Advanced Study, Princeton（普林斯顿高等研究院自然科学学院）
The crystalline structure, anisotropic magnetoresistance (AMR), and magnetization of La0.7Sr0.3MnO3/SrTiO3 (LSMO/STO) superlattices grown by an rf sputtering system are systematically analyzed to study the spin polarization of manganite at interfaces. A perfectly epitaxial growth with sharp interfaces between LSMO and STO layers is confirmed by the transmission electron microscopy (TEM) image and the x-ray diffraction. The presence of positive low-temperature AMR in LSMO/STO superlattices with thinner LSMO layers or thicker STO layers implies that two bands of majority and minority character c…
Inspired by the coupled-layer construction of the X-Cube model, we introduce the X-Cube Floquet code, a dynamical quantum error-correcting code where the number of encoded logical qubits grows with system size. The X-Cube Floquet code is defined on a three-dimensional lattice, built from intersecting two-dimensional layers in the $xy$, $yz$, and $xz$ directions, and consists of a periodic sequence of two-qubit measurements which couple the layers together. Within a single Floquet cycle, the codespace switches between that of the X-Cube fracton order and layers of entangled, two-dimensional tor…
Inelastic neutron scattering measurements on the low energy spin waves in CaFe2As2 show that the magnetic exchange interactions in the Fe layers are exceptionally large and similar to the cuprates. However, the exchange between layers is ~10% of the coupling in the layers and the magnetism is more appropriately categorized as anisotropic three-dimensional, in contrast to the two-dimensional cuprates. Band structure calculations of the spin dynamics and magnetic exchange interactions are in good agreement with the experimental data.
In the present paper, we study spin-${1 \over 2}$ antiferromagnetic (AF) Heisenberg model on layered anisotropic triangular lattice and obtain its phase structure. We use the Schwinger bosons for representing spin operators and also coherent-state path integral for calculating physical quantities. Finite-temperature properties of the system are investigated by means of the numerical Monte-Carlo simulations. Detailed phase diagram of the system is obtained by calculating internal energy, specific heat, spin correlation functions, etc. There are AF Néel, paramagnetic and spiral states. Turning o…
Paramagnon Heat Capacity and Anomalous Thermopower in Anisotropic Magnetic Systems: Understanding Inter-Layer Spin Correlations in a Magnetically Disordered Phase
