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arXiv:1910.06336

Systematic Constructions of Fracton Theories

Djordje Radicevic

Published Oct 14, 2019• 29 citations
arXivHTMLPDF

Abstract

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 $\mathfrak{F}_p$ gauge theories, in which gauge transformations of rank-$k$ fields are specified by rank-$(k - p)$ gauge parameters. The $\mathbb{Z}_2$ rank-two theory of type $\mathfrak{F}_2$, 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 $\mathrm{U}(1)$ rank-two theory of type $\mathfrak{F}_2$ 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 $\mathfrak{F}_2$ theory also show gapless fracton behavior; on a cubic lattice they have a conserved $\mathrm{U}(1)$ charge and dipole moment, but these particular charges are not necessarily conserved on more general lattices. The construction straightforwardly generalizes to $\mathfrak{F}_2$ theories of nonabelian rank-two gauge fields, giving first examples of pure nonabelian higher-rank theories.

DOI
10.48550/arxiv.1910.06336

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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.
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 order is an intriguing new type of order which shares many common features with topological order, such as topology-dependent ground-state degeneracies, and excitations with mutual statistics.However, it also has several distinctive geometrical aspects, such as excitations with restricted mobility, which naturally lead to effective descriptions in terms of higher-rank gauge fields.In this paper, we investigate possible effective field theories for three-dimensional fracton order, by presenting a general philosophy whereby topological-like actions for such higher-rank gauge fields can be constructed.Our approach draws inspiration from Chern-Simons and BF theories in 2+1 dimensions, and imposes constraints binding higher-rank gauge charge to higher-rank gauge flux.We show that the resulting fractonic Chern-Simons and BF theories reproduce many of the interesting features of their familiar two-dimensional cousins.We analyze one example of the resulting fractonic Chern-Simons theory in detail, and show that upon quantization it realizes a gapped fracton order with quasiparticle excitations that are mobile only along a subset of one-dimensional lines, and display a form of fractional self-statistics.The ground-state degeneracy of this theory is both topology and geometry dependent, scaling exponentially with the linear system size when the model is placed on a three-dimensional torus.By studying the resulting quantum theory on the lattice, we show that it describes a Z s generalization of the Chamon code.
We study novel three-dimensional gapped quantum phases of matter which support quasiparticles with restricted mobility, including immobile ``fracton'' excitations. So far, most existing fracton models may be instructively viewed as generalized Abelian lattice gauge theories. Here, by analogy with Dijkgraaf-Witten topological gauge theories, we discover a natural generalization of fracton models, obtained by twisting the gauge symmetries. Introducing generalized gauge transformation operators carrying an extra phase factor depending on local configurations, we construct a plethora of exactly solvable three-dimensional models, which we dub ``twisted fracton models.'' A key result of our approach is to demonstrate the existence of rich non-Abelian fracton phases of distinct varieties in a three-dimensional system with finite-range interactions. For an accurate characterization of these novel phases, the notion of being inextricably non-Abelian is introduced for fractons and quasiparticles with one-dimensional mobility, referring to their new behavior of displaying braiding statistics that is, and remains, non-Abelian regardless of which quasiparticles with higher mobility are added to or removed from them. We also analyze these models by embedding them on a 3-torus and computing their ground-state degeneracies, which exhibit a surprising and novel dependence on the system size in the non-Abelian fracton phases. Moreover, as an important advance in the study of fracton order, we develop a general mathematical framework which systematically captures the fusion and braiding properties of fractons and other quasiparticles with restricted mobility.
We demonstrate the existence of a fundamentally new type of excitation, fractonic lines, which are linelike excitations with the restricted mobility properties of fractons. These excitations, described using an amalgamation of higher-form gauge theories with symmetric tensor gauge theories, see direct physical realization as the topological lattice defects of ordinary three-dimensional quantum crystals. Starting with the more familiar elasticity theory, we show how theory maps onto a rank-4 tensor gauge theory, with phonons corresponding to gapless gauge modes and disclination defects corresponding to linelike charges. We derive flux conservation laws which lock these linelike excitations in place, analogous to the higher moment charge conservation laws of fracton theories. This way of encoding mobility restrictions of lattice defects could shed light on melting transitions in three dimensions. This new type of extended object may also be a useful tool in the search for improved quantum error-correcting codes in three dimensions.
Fractonic phases are new phases of matter that host excitations with restricted mobility. We show that a certain class of gapless fractonic phases are realized as a result of spontaneous breaking of continuous higher-form symmetries whose conserved charges do not commute with spatial translations. We refer to such symmetries as nonuniform higher-form symmetries. These symmetries fall within the standard definition of higher-form symmetries in quantum field theory, and the corresponding symmetry generators are topological. Worldlines of particles are regarded as the charged objects of 1-form symmetries, and mobility restrictions can be implemented by introducing additional 1-form symmetries whose generators do not commute with spatial translations. These features are realized by effective field theories associated with spontaneously broken nonuniform 1-form symmetries. At low energies, the theories reduce to known higher-rank gauge theories such as scalar/vector charge gauge theories, and the gapless excitations in these theories are interpreted as Nambu-Goldstone modes for higher-form symmetries. Due to the nonuniformity of the symmetry, some of the modes acquire a gap, which is the higher-form analogue of the inverse Higgs mechanism of spacetime symmetries. The gauge theories have emergent nonuniform magnetic symmetries, and some of the magnetic monopoles become fractonic. We identify the ’t Hooft anomalies of the nonuniform higher-form symmetries and the corresponding bulk symmetry-protected topological phases. By this method, the mobility restrictions are fully determined by the choice of the commutation relations of charges with translations. This approach allows us to view existing (gapless) fracton models such as the scalar/vector charge gauge theories and their variants from a unified perspective and enables us to engineer theories with desired mobility restrictions.