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arXiv:2109.04259 · Physical review. B./Physical review. B

Arboreal topological and fracton phases

Nandagopal Manoj, Vijay B. Shenoy

Published Apr 20, 2023• 9 citations
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Abstract

We describe topologically ordered and fracton-ordered quantum systems on geometries which do not have an underlying manifold structure. Using tree graphs such as the $k$-coordinated Bethe lattice $\text{B}(k)$ and a hypertree called the $(k,n)$-hyper-Bethe lattice $\text{HB}(k,n)$ consisting of $k$-coordinated hyperlinks (links defined by $n$ sites), we construct multidimensional arboreal arenas such as $\text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2})$ using a generalized notion of a graph Cartesian product $\ensuremath{\square}$. We study various quantum systems such as the ${\mathbb{Z}}_{2}$ gauge theory, generalized quantum Ising models (GQIMs), the fractonic X-cube model, and related X-cube gauge theory defined on these arboreal arenas, finding several fascinating results. Even the simplest ${\mathbb{Z}}_{2}$ gauge theory on a two-dimensional arboreal arena is found to be fractonic---an isolated monopole excitation is rendered fully immobile on an arboreal arena. The X-cube model on a generic three-dimensional arboreal arena is found to be fully fractonic in the magnetic sector, i.e., all multipoles of magnetic excitations are rendered immobile on the arboreal arena. We obtain variational ground state phase diagrams of the gauge theories (both ${\mathbb{Z}}_{2}$ and X-cube gauge theories) which are shown to have deconfined and confined phases. These phases are usually separated by a first-order transition, while continuous transitions are obtained in some cases. Further, we find an intriguing class of dualities in arboreal arenas, as illustrated by the ${\mathbb{Z}}_{2}$ gauge theory defined on $\text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2})$ being dual to a GQIM defined on $HB(2,{k}_{1})\ensuremath{\square}HB(2,{k}_{2})$. Finally, we discuss different classes of topological and fracton orders that appear on arboreal arenas. We find three distinct classes of arboreal toric code orders on two-dimensional arboreal arenas, those that occur on $\text{B}(2)\ensuremath{\square}\text{B}(2), \text{B}(k)\ensuremath{\square}\text{B}(2),k>2$, and $\text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2}), {k}_{1},{k}_{2}>2$. Likewise, four classes of X-cube fracton orders are found on three-dimensional arboreal arenas which correspond to those on $\text{B}(2)\ensuremath{\square}\text{B}(2)\ensuremath{\square}\text{B}(2), \text{B}(k)\ensuremath{\square}\text{B}(2)\ensuremath{\square}\text{B}(2),k>2, \text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2})\ensuremath{\square}\text{B}(2),{k}_{1},{k}_{2}>2$, and $\text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2})\ensuremath{\square}\text{B}({k}_{3}),{k}_{1},{k}_{2},{k}_{3}>2$.

DOI
10.1103/physrevb.107.165136

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We describe topologically ordered and fracton-ordered quantum systems on geometries which do not have an underlying manifold structure. Using tree graphs such as the kkk-coordinated Bethe lattice B(k)\text{B}(k)B(k) and a hypertree called the (k,n)(k,n)(k,n)-hyper-Bethe lattice HB(k,n)\text{HB}(k,n)HB(k,n) consisting of kkk-coordinated hyperlinks (links defined by nnn sites), we construct multidimensional arboreal arenas such as B(k1)\ensuremath□B(k2)\text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2})B(k1​)\ensuremath□B(k2​) using a generalized notion of a graph Cartesian product \ensuremath□\ensuremath{\square}\ensuremath□. We study various quantum systems such as the Z2{\mathbb{Z}}_{2}Z2​ gauge theory, generalized quantum Ising models (GQIMs), the fractonic X-cube model, and related X-cube gauge theory defined on these arboreal arenas, finding several fascinating results. Even the simplest Z2{\mathbb{Z}}_{2}Z2​ gauge theory on a two-dimensional arboreal arena is found to be fractonic---an isolated monopole excitation is rendered fully immobile on an arboreal arena. The X-cube model on a generic three-dimensional arboreal arena is found to be fully fractonic in the magnetic sector, i.e., all multipoles of magnetic excitations are rendered immobile on the arboreal arena. We obtain variational ground state phase diagrams of the gauge theories (both Z2{\mathbb{Z}}_{2}Z2​ and X-cube gauge theories) which are shown to have deconfined and confined phases. These phases are usually separated by a first-order transition, while continuous transitions are obtained in some cases. Further, we find an intriguing class of dualities in arboreal arenas, as illustrated by the Z2{\mathbb{Z}}_{2}Z2​ gauge theory defined on B(k1)\ensuremath□B(k2)\text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2})B(k1​)\ensuremath□B(k2​) being dual to a GQIM defined on HB(2,k1)\ensuremath□HB(2,k2)HB(2,{k}_{1})\ensuremath{\square}HB(2,{k}_{2})HB(2,k1​)\ensuremath□HB(2,k2​). Finally, we discuss different classes of topological and fracton orders that appear on arboreal arenas. We find three distinct classes of arboreal toric code orders on two-dimensional arboreal arenas, those that occur on B(2)\ensuremath□B(2),B(k)\ensuremath□B(2),k>2\text{B}(2)\ensuremath{\square}\text{B}(2), \text{B}(k)\ensuremath{\square}\text{B}(2),k>2B(2)\ensuremath□B(2),B(k)\ensuremath□B(2),k>2, and B(k1)\ensuremath□B(k2),k1,k2>2\text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2}), {k}_{1},{k}_{2}>2B(k1​)\ensuremath□B(k2​),k1​,k2​>2. Likewise, four classes of X-cube fracton orders are found on three-dimensional arboreal arenas which correspond to those on B(2)\ensuremath□B(2)\ensuremath□B(2),B(k)\ensuremath□B(2)\ensuremath□B(2),k>2,B(k1)\ensuremath□B(k2)\ensuremath□B(2),k1,k2>2\text{B}(2)\ensuremath{\square}\text{B}(2)\ensuremath{\square}\text{B}(2), \text{B}(k)\ensuremath{\square}\text{B}(2)\ensuremath{\square}\text{B}(2),k>2, \text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2})\ensuremath{\square}\text{B}(2),{k}_{1},{k}_{2}>2B(2)\ensuremath□B(2)\ensuremath□B(2),B(k)\ensuremath□B(2)\ensuremath□B(2),k>2,B(k1​)\ensuremath□B(k2​)\ensuremath□B(2),k1​,k2​>2, and B(k1)\ensuremath□B(k2)\ensuremath□B(k3),k1,k2,k3>2\text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2})\ensuremath{\square}\text{B}({k}_{3}),{k}_{1},{k}_{2},{k}_{3}>2B(k1​)\ensuremath□B(k2​)\ensuremath□B(k3​),k1​,k2​,k3​>2.
Strongly correlated many-body systems provide a platform for novel phases of matter where constituent particles organize themselves in a variety of ways. At low temperature, these particles interact quantum mechanically and generate entanglement building up exotic quantum phases, such as topologcial order, where there can be emergent excitations which cannot be created locally. Such excitations, if gapped, are also called topological excitations.Fracton is one of such gapped point-like topological excitation in three dimensional system. Different from conventional topological excitation, it is immobile and was firstly discovered in exact solvable models exhibiting fracton topological order. This new order has sub-extensive topological ground-state degeneracy and generically also possesses other mobile excitations restricted to move in sub-dimensional spaces. Also, it has been noticed that the charges of symmetric-tensor U(1) gauge theories can be fractons. Its immobility is due to the existence of multiple conservation laws.In this thesis, I will present the relation among the gapped fracton topological order, gapless fracton phase described by U(1) symmetric tensor gauge theories, and ordinary topological ordered phases. Particularly, the fracton topological order exhibited in an exact solvable model called X- cube model can be constructed by coupling toric code layers. The mechanism leads to fracton topological orders is dubbed “p-string condensation” or “p-membrane condensation,” in which strings or membranes built up from particle excitations from layers of topological orders condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the resulting fracton order to be easily understood in terms of more familiar degrees of freedom. And the fracton topological order in the X-cube model can also be obtained from a particular rank-2 symmetric tensor gauge theory called scalar charge theory by a partial confinement transition followed by Higgs mechanism removing the gapless photon modes.
Fracton topological order describes a remarkable phase of matter, which can be characterized by fracton excitations with constrained dynamics and a ground-state degeneracy that increases exponentially with the length of the system on a three-dimensional torus. However, previous models exhibiting this order require many-spin interactions, which may be very difficult to realize in a real material or cold atom system. In this work, we present a more physically realistic model which has the so-called X-cube fracton topological order [Vijay, Haah, and Fu, Phys. Rev. B 94, 235157 (2016)] but only requires nearest-neighbor two-spin interactions. The model lives on a three-dimensional honeycomb-based lattice with one to two spin-1/2 degrees of freedom on each site and a unit cell of six sites. The model is constructed from two orthogonal stacks of Z2{Z}_{2}Z2​ topologically ordered Kitaev honeycomb layers [Kitaev, Ann. Phys.321, 2 (2006)], which are coupled together by a two-spin interaction. It is also shown that a four-spin interaction can be included to instead stabilize 3+1D Z2{Z}_{2}Z2​ topological order. We also find dual descriptions of four quantum phase transitions in our model, all of which appear to be discontinuous first-order transitions.
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, ZNZ_{N}ZN​ generalization of this fracton phase, which we introduce. We describe the rich phase structure of our layered ZNZ_{N}ZN​ system. By invoking a lattice duality, we demonstrate that when N≥5N \ge 5N≥5, there is an intermediate phase that appears between the decoupled, layered system and the fracton topologically-ordered state, which opens the possibility of a continuous transition into the fracton topological phase. We conclude by presenting a solvable model, that interpolates between the fracton phase and a confined phase in which the phase transition is first-order.
We introduce hybrid fracton orders: three-dimensional gapped quantum phases that exhibit the phenomenology of both conventional three-dimensional topological orders and fracton orders. Hybrid fracton orders host both (i) mobile topological quasiparticles and loop excitations, as well as (ii) pointlike topological excitations with restricted mobility, with nontrivial fusion rules and mutual braiding statistics between the two sets of excitations. Furthermore, hybrid fracton phases can realize either conventional three-dimensional topological orders or fracton orders after undergoing a phase transition driven by the condensation of certain gapped excitations. Therefore they serve as parent orders for both long-range-entangled quantum liquid and nonliquid phases. We study the detailed properties of hybrid fracton phases through exactly solvable models in which the resulting orders hybridize a three-dimensional Z2{\mathbb{Z}}_{2}Z2​ topological order with (i) the X-cube fracton order or (ii) Haah's code. The hybrid orders presented here can also be understood as the deconfined phase of a gauge theory whose gauge group is given by an Abelian global symmetry GGG and subsystem symmetries of a normal subgroup NNN along lower-dimensional subregions.
We introduce lattice gauge theories which describe three-dimensional, gapped quantum phases exhibiting the phenomenology of both conventional three-dimensional topological orders and fracton orders, starting from a finite group GGG, a choice of an Abelian normal subgroup NNN, and a choice of foliation structure. These hybrid fracton orders, examples of which were introduced in a previous paper [Tantivasadakarn et al., Phys. Rev. B 103, 245136 (2021)], can also host immobile, pointlike excitations that are non-Abelian, and therefore give rise to a protected degeneracy. We construct solvable lattice models for these orders which interpolate between a conventional, three-dimensional GGG gauge theory and a pure fracton order, by varying the choice of normal subgroup NNN. We demonstrate that certain universal data of the topological excitations and their mobilities are directly related to the choice of GGG and NNN, and also present complementary perspectives on these orders: certain orders may be obtained by gauging a global symmetry which enriches a particular fracton order, by either fractionalizing on or permuting the excitations with restricted mobility, while certain hybrid orders can be obtained by condensing excitations in a stack of initially decoupled, two-dimensional topological orders.
We introduce a generalization of conventional lattice gauge theory to describe fracton topological phases, which are characterized by immobile, pointlike topological excitations, and subextensive topological degeneracy. We demonstrate a duality between fracton topological order and interacting spin systems with symmetries along extensive, lower-dimensional subsystems, which may be used to systematically search for and characterize fracton topological phases. Commutative algebra and elementary algebraic geometry provide an effective mathematical tool set for our results. Our work paves the way for identifying possible material realizations of fracton topological phases.