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arXiv:1901.05918 · Physical review. B./Physical review. B

From coupled wires to coupled layers: Model with three-dimensional fractional excitations

Yohei Fuji, Akira Furusaki

Published Jun 12, 2019• 14 citations
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Abstract

We propose a systematic approach to constructing microscopic models with fractional excitations in three-dimensional (3D) space. Building blocks are quantum wires described by the ($1+1$)-dimensional conformal field theory (CFT) associated with a current algebra $\mathfrak{g}$. The wires are coupled with each other to form a 3D network through the current-current interactions of ${\mathfrak{g}}_{1}$ and ${\mathfrak{g}}_{2}$ CFTs that are related to the $\mathfrak{g}$ CFT by a nontrivial conformal embedding $\mathfrak{g}\ensuremath{\supset}{\mathfrak{g}}_{1}\ifmmode\times\else\texttimes\fi{}{\mathfrak{g}}_{2}$. The resulting model can be viewed as a layer construction of a 3D topologically ordered state, in which the conformal embedding in each wire implements the anyon condensation between adjacent layers. Local operators acting on the ground state create pointlike or looplike deconfined excitations depending on the branching rule. We demonstrate our construction for a simple solvable model based on the conformal embedding $SU{(2)}_{1}\ifmmode\times\else\texttimes\fi{}SU{(2)}_{1}\ensuremath{\supset}U{(1)}_{4}\ifmmode\times\else\texttimes\fi{}U{(1)}_{4}$. We show that the model possesses extensively degenerate ground states on a torus with deconfined quasiparticles, and that appropriate local perturbations lift the degeneracy and yield a 3D ${Z}_{2}$ gauge theory with a fermionic ${Z}_{2}$ charge.

DOI
10.1103/physrevb.99.241107

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The construction of fractional quantum Hall (FQH) states from the two-dimensional array of quantum wires provides a useful way to control strong interactions in microscopic models and has been successfully applied to the Laughlin, Moore-Read, and Read-Rezayi states. We extend this construction to the Abelian and non-Abelian $SU(N\ensuremath{-}1)$-singlet FQH states at filling fraction $\ensuremath{\nu}=k(N\ensuremath{-}1)/[N+k(N\ensuremath{-}1)m]$ labeled by integers kkk and mmm, which are potentially realized in multicomponent quantum Hall systems or SU(N)SU(N)SU(N) spin systems. Utilizing the bosonization approach and conformal field theory (CFT), we show that their bulk quasiparticles and gapless edge excitations are both described by an $(N\ensuremath{-}1)$-component free-boson CFT and the $SU{(N)}_{k}/{[U(1)]}^{N\ensuremath{-}1}$ CFT known as the Gepner parafermion. Their generalization to different filling fractions is also proposed. In addition, we argue possible applications of these results to two kinds of lattice systems: bosons interacting via occupation-dependent correlated hoppings and an SU(N)SU(N)SU(N) Heisenberg model.
Topological phases of matter are usually realized in deconfined phases of gauge theories. In this context, confined phases with strongly fluctuating gauge fields seem to be irrelevant to the physics of topological phases. For example, the low-energy theory of the two-dimensional (2D) toric code model (i.e., the deconfined phase of Z2{\mathbb{Z}}_{2}Z2​ gauge theory) is a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ Chern-Simons theory in which gauge charges (i.e., eee and mmm particles) are deconfined and the gauge fields are gapped, while the confined phase is topologically trivial. In this paper, we point out a route to constructing exotic three-dimensional (3D) gapped fermionic phases in a confining phase of a gauge theory. Starting from a parton construction with strongly fluctuating compact $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ gauge fields, we construct gapped phases of interacting fermions by condensing two linearly independent bosonic composite particles consisting of partons and $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ magnetic monopoles. This can be regarded as a 3D generalization of the 2D Bais-Slingerland condensation mechanism. Charge fractionalization results from a Debye-H\"uckel--type screening cloud formed by the condensed composite particles. Within our general framework, we explore two aspects of symmetry-enriched 3D Abelian topological phases. First, we construct a new fermionic state of matter with time-reversal symmetry and $\mathrm{\ensuremath{\Theta}}\ensuremath{\ne}\ensuremath{\pi}$, the fractional topological insulator. Second, we generalize the notion of anyonic symmetry of 2D Abelian topological phases to the charge-loop excitation symmetry (Charles\mathsf{Charles}Charles) of 3D Abelian topological phases. We show that line twist defects, which realize Charles\mathsf{Charles}Charles transformations, exhibit non-Abelian fusion properties.
Three-dimensional gauge theories with a discrete gauge group can emerge from spin models as a gapped topological phase with fractional point excitations (gauge charge) and loop excitations (gauge flux). It is known that 3D gauge theories can be ``twisted,'' in the sense that the gauge flux loops can have nontrivial braiding statistics among themselves and such twisted gauge theories are realized in models discovered by Dijkgraaf and Witten. A different framework to systematically construct three-dimensional topological phases was proposed by Walker and Wang and a series of examples have been studied. Can the Walker-Wang construction be used to realize the topological order in twisted gauge theories? This is not immediately clear because the Walker-Wang construction is based on a loop condensation picture while the Dijkgraaf-Witten theory is based on a membrane condensation picture. In this paper, we show that the answer to this question is Yes, by presenting an explicit construction of the Walker-Wang models which realize both the twisted and untwisted gauge theories with gauge group ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$. We identify the topological order of the models by performing modular transformations on the ground-state wave functions and show that the modular matrices exactly match those for the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ gauge theories. By relating the Walker-Wang construction to the Dijkgraaf-Witten construction, our result opens up a way to study twisted gauge theories with fermonic charges, and correspondingly strongly interacting fermionic symmetry protected topological phases and their surface states, through exactly solvable models.