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arXiv:1802.10108 · Physical review. B./Physical review. B

Fracton topological order from the Higgs and partial-confinement mechanisms of rank-two gauge theory

Han Ma, Michael Hermele, Xie Chen

Published Jul 10, 2018• 146 citations
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Abstract

Fractons are gapped pointlike excitations in $d=3$ topological ordered phases whose motion is constrained. They have been discovered in several gapped models but a unifying physical mechanism for generating them is still missing. It has been noticed that in symmetric-tensor $\mathrm{U}(1)$ gauge theories, charges are fractons and cannot move freely due to, for example, the conservation of not only the charge but also the dipole moment. To connect these theories with fully gapped fracton models, we study Higgs and partial confinement mechanisms in rank-2 symmetric-tensor gauge theories, where charges or magnetic excitations, respectively, are condensed. Specifically, we describe two different routes from the rank-2 $\mathrm{U}(1)$ scalar charge theory to the X-cube fracton topological order, finding that a combination of Higgs and partial confinement mechanisms is necessary to obtain the fully gapped fracton model. On the other hand, the rank-2 ${\mathbb{Z}}_{2}$ scalar charge theory, which is obtained from the former theory upon condensing charge-2 matter, is equivalent to four copies of the $d=3$ toric code and does not support fracton excitations. We also explain how the checkerboard fracton model can be viewed as a rank-2 ${\mathbb{Z}}_{2}$ gauge theory with two different Gauss' law constraints on different lattice sites.

DOI
10.1103/physrevb.98.035111

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Fractons are gapped pointlike excitations in d=3d=3d=3 topological ordered phases whose motion is constrained. They have been discovered in several gapped models but a unifying physical mechanism for generating them is still missing. It has been noticed that in symmetric-tensor U(1)\mathrm{U}(1)U(1) gauge theories, charges are fractons and cannot move freely due to, for example, the conservation of not only the charge but also the dipole moment. To connect these theories with fully gapped fracton models, we study Higgs and partial confinement mechanisms in rank-2 symmetric-tensor gauge theories, where charges or magnetic excitations, respectively, are condensed. Specifically, we describe two different routes from the rank-2 U(1)\mathrm{U}(1)U(1) scalar charge theory to the X-cube fracton topological order, finding that a combination of Higgs and partial confinement mechanisms is necessary to obtain the fully gapped fracton model. On the other hand, the rank-2 Z2{\mathbb{Z}}_{2}Z2​ scalar charge theory, which is obtained from the former theory upon condensing charge-2 matter, is equivalent to four copies of the d=3d=3d=3 toric code and does not support fracton excitations. We also explain how the checkerboard fracton model can be viewed as a rank-2 Z2{\mathbb{Z}}_{2}Z2​ gauge theory with two different Gauss' law constraints on different lattice sites.
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 theories possess exponentially degenerate ground states, excitations with restricted mobility, and nontopological higher-form symmetries. This paper shows that such theories can be defined on arbitrary spatial lattices in three dimensions. The key element of this construction is a generalization of higher-form gauge theories to so-called Fp\mathfrak{F}_pFp​ gauge theories, in which gauge transformations of rank-kkk fields are specified by rank-(k−p)(k - p)(k−p) gauge parameters. The Z2\mathbb{Z}_2Z2​ rank-two theory of type F2\mathfrak{F}_2F2​, placed on a cubic lattice and coupled to scalar matter, is shown to have a topological phase exactly dual to the well-known X-cube model. Generalizations of this example yield novel fracton theories. In the continuum, the U(1)\mathrm{U}(1)U(1) rank-two theory of type F2\mathfrak{F}_2F2​ is shown to have a perturbatively gapless fracton regime that cannot be consistently interpreted as a tensor gauge theory of any kind. The compact scalar fields that naturally couple to this F2\mathfrak{F}_2F2​ theory also show gapless fracton behavior; on a cubic lattice they have a conserved U(1)\mathrm{U}(1)U(1) charge and dipole moment, but these particular charges are not necessarily conserved on more general lattices. The construction straightforwardly generalizes to F2\mathfrak{F}_2F2​ theories of nonabelian rank-two gauge fields, giving first examples of pure nonabelian higher-rank theories.
Subdimensional motion occurs when excitations of a system can only move in a lower-dimensional subspace of the full system. This phenomenon occurs both in a class of generalized U(1) gauge theories and in ``fracton'' phases found in certain exactly solvable gapped spin models. Here, the authors present and analyze a more general class of symmetric higher-rank U(1) gauge theories and show how these two cases of subdimensional motion are connected via the Higgs mechanism, which often yields conventional phases but in special cases can drive a symmetric higher-rank U(1) gauge theory into a fracton phase.
The objective of the present work -- a literature review on both gapped and gapless fractonic theories -- is to pedagogically fill in the gaps between the research on fractons, and an undergraduate physics education (particularly quantum and statistical mechanics). Some familiarity with classical field theory is assumed. We will begin this work by reviewing the gauging of the Ising model to obtain the toric code. Then, following the chronological order of fracton research, we will establish the gauged spin model picture of gapped fracton theories. Next (after introducing lattice gauge theory) we will cover the developments on the tensor gauge theory front in describing gapless fracton theories. We then explain how conservation of dipole moment is key in the development of field-theoretic descriptions of fracton models, and conclude by providing future plans for work on gauging such field theories. Finally, we point out two more future plans: one being an established theoretical advance which in fact inspired this work, and the other being exciting experimental advances in ultracold atomic physics which could lead to the physical realization of fracton physics. Highlights of this work include: a derivation of the gauge field dynamics (field strength/curvature) term in the gauging of a spin system to obtain the toric code, a reduction of the 3 terms of the Kogut-Susskind lattice QED Hamiltonian to the 2 terms of the toric code, a proof of the correspondence between the invariance of a gauge field and a particular Gauss's law constraint in a tensor gauge theory context, a demonstration that conservation of dipole moment accounts for the same immobility/fractalization phenomenology that we also show from a gauged spin model perspective, and proof of the necessity for polynomial shift symmetries and higher-order spatial derivatives in a Lagrangian in order to have dipole moment conservation.
Motivated by the prediction of fractonic topological defects in a quantum crystal, we utilize a reformulated elasticity duality to derive a description of a fracton phase in terms of coupled vector U(1) gauge theories. The fracton order and restricted mobility emerge as a result of an unusual Gauss law where electric field lines of one gauge field act as sources of charge for others. At low energies this vector gauge theory reduces to the previously studied fractonic symmetric tensor gauge theory. We construct the corresponding lattice model and a number of generalizations, which realize fracton phases via a condensation of stringlike excitations built out of charged particles, analogous to the p-string condensation mechanism of the gapped X-cube fracton phase.
We investigate how symmetry and topological order are coupled in the (2+12+12+1)--dimensional ZN{\mathbb{Z}}_{N}ZN​ rank-2 toric code for general NNN, which is an exactly solvable point in the Higgs phase of a symmetric rank-2 U(1)\text{U}(1)U(1) gauge theory. The symmetry-enriched topological order present has a nontrivial realization of square-lattice translation (and rotation and reflection) symmetry, where anyons on different lattice sites have different types and belong to different superselection sectors. We call such particles ``position-dependent excitations.'' As a result, in the rank-2 toric code anyons can hop by one lattice site in some directions while only by NNN lattice sites in others, reminiscent of fracton topological order in 3+13+13+1 dimensions. We find that while there are N2{N}^{2}N2 flavors of eee charges and 2N2N2N flavors of mmm fluxes, there are not NN2+2N{N}^{{N}^{2}+2N}NN2+2N anyon types. Instead, there are N6{N}^{6}N6 anyon types, and we can use Chern-Simons theory with six U(1)\text{U}(1)U(1) gauge fields to describe all of them. While the lattice translations permute anyon types, we find that such permutations cannot be expressed as transformations on the six U(1)\text{U}(1)U(1) gauge fields. Thus, the realization of translation symmetry in the U6(1){\text{U}}^{6}(1)U6(1) Chern-Simons theory is not known. Despite this, we find a way to calculate the translation-dependent properties of the theory. In particular, we find that the ground-state degeneracy on an ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ torus is N3gcd(Lx,N)gcd(Ly,N)gcd(Lx,Ly,N){N}^{3}gcd({L}_{x},N)gcd({L}_{y},N)gcd({L}_{x},{L}_{y},N)N3gcd(Lx​,N)gcd(Ly​,N)gcd(Lx​,Ly​,N), where gcdgcdgcd stands for ``greatest common divisor.'' We argue that this is a manifestation of UV/IR mixing which arises from the interplay between lattice symmetries and topological order.