Composite particle theory of three-dimensional gapped fermionic phases: Fractional topological insulators and charge-loop excitation symmetry

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 ${\mathbb{Z}}_2$ gauge theory) is a $\text{U}\left(1\right)\times \text{U}\left(1\right)$ Chern-Simons theory in which gauge charges (i.e., $e$ and $m$ 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 $\text{U}\left(1\right)\times \text{U}\left(1\right)$ gauge fields, we construct gapped phases of interacting fermions by condensing two linearly independent bosonic composite particles consisting of partons and $\text{U}\left(1\right)\times \text{U}\left(1\right)$ magnetic monopoles. This can be regarded as a 3D generalization of the 2D Bais-Slingerland condensation mechanism. Charge fractionalization results from a Debye-Hückel–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 $\mathrm{\Theta }\ne \pi$, the fractional topological insulator. Second, we generalize the notion of anyonic symmetry of 2D Abelian topological phases to the charge-loop excitation symmetry ($\mathsf{Charles}$) of 3D Abelian topological phases. We show that line twist defects, which realize $\mathsf{Charles}$ transformations, exhibit non-Abelian fusion properties.

Figure 1

Parton construction of electron operators in this work. The wavy lines denote interactions mediated by gauge bosons. $A_{\mu }$ is the external nondynamical EM (electromagnetic) field, serving as a probe of the electromagnetic response of the system. $a_{\mu }$ and $b_{\mu }$ are two dynamical, compact $\text{U}\left(1\right)$ gauge fields, belonging to the $\text{U}{\left(1\right)}_a$ and $\text{U}{\left(1\right)}_b$ gauge groups, respectively. The partons $f^1$ and $f^2$ carry 1 and $-1$ gauge charges of the $\text{U}{\left(1\right)}_a$ gauge group, respectively. The partons $f^3$ and $f^2$ carry 1 and $-1$ gauge charges of the $\text{U}{\left(1\right)}_b$ gauge group, respectively. The EM electric charges carried by the partons $f^1, f^2, f^3$ are $e, e, -e$, respectively.

Figure 2

Schematic representation of composite particle condensation. Before condensation, the system is an electromagnetic plasma of composites in the $\text{U}{\left(1\right)}_{\mathrm{EM}}\times \text{U}{\left(1\right)}_a\times \text{U}{\left(1\right)}_b$ gauge group. There are many composite particles (denoted by solid circles) including ${\phi }_1$ and ${\phi }_2$. There are also two gapless photons (denoted by $a$ and $b$ in the figure), indicating that the phase before condensation is a gapless Coulomb phase for both internal dynamical gauge fields. After condensing ${\phi }_1$ and ${\phi }_2$, the system enters a gapped phase in the absence of photons. All composites (denoted by black solid circles on the left) that have nonzero mutual statistics with both ${\phi }_1$ and ${\phi }_2$ are confined. Otherwise, those composites that have trivial (zero) mutual statistics with both condensates survive as excitations (denoted by blue solid circles) of the gapped phase. The red loops on the right represent loop excitations due to the two condensates.

Figure 3

Schematic representation of the charge-screening mechanism. Consider a composite particle carrying $N_A$ units of the EM ($A_{\mu }$) electric charge [i.e., $\text{U}\left(1\right)$ symmetry charge], $N_m^a$ units of magnetic charge of the $a_{\mu }$ field, and $N_m^b$ units of magnetic charge of the $b_{\mu }$ gauge field. Due to the condensates, $N_A$ is partially screened such that the net EM electric charge $Q$ is given by Eq. (28), which is different from $N_A$. In (a) the physics of Aharonov-Bohm effect in Eq. (25) is illustrated. An excitation (denoted by the blue ball) adiabatically moves along a closed trajectory and feels the EM magnetic flux ${\mathrm{\Phi }}_M^A$, the electric flux ${\mathrm{\Phi }}_e^a$ of the $a_{\mu }$ gauge field, and the electric flux of the $b_{\mu }$ gauge field. In (b), condensed particles ${\phi }_1$ and ${\phi }_2$ form a Debye-Hückel–type charge cloud around an excitation, providing the screening charge $Q_{\mathrm{Debye}}$ in Eq. (29).

Figure 4

Throwing three external EM magnetic monopoles (denoted by blue balls) in vacuum into topological materials TI and FTI. Only the EM magnetic monopole with $3k, k\in \mathbb{Z}$, magnetic charge shown in Eq. (45) can penetrate the FTI boundary in our example. An EM magnetic monopole with $M=1,2$ will be completely reflected on the FTI boundary, as shown by the leftward arrows. The shadow of each ball pictorially denotes the polarization charge cloud induced by the Witten effect.

Figure 5

Self-statistics distribution on the 4D charge lattice. (a) Self-statistics distribution as a function of $M$ and $Q$ by turning off $\mathrm{\Theta }$ in Eq. (58). (b) Self-statistics distribution as a function of $M$ and $Q$ as shown in Eq. (58). The allowed values of $M$ and $Q$ are determined by Eqs. (51) and (52). Geometrically, $\mathrm{\Theta }=2\pi tan\alpha$, where $tan\alpha =\frac{1/6}{3}=\frac{1}{18}$. Thus, $\mathrm{\Theta }$ angle can be viewed as a consequence of a shear deformation from (a) to (b). During the shear deformation, the area of “Dirac unit cell” (denoted by the shaded area) is invariant. Since the charge lattice is 4D (Definition t4), each site in (b) on the $(M-Q)$ parameter space corresponds to more than one excitation. An example is shown in (c), where $N_m^a,N_m^b$ are used to label excitations that have the same $Q$ and $M$: $Q=\frac{1}{6},M=3$.

Figure 6

Extrinsic twist defects in 3D. (a) Line defect; (b) point defect. The two cubic boxes denote the 3D bulk of an underlying quantum many-body system. The shaded plane in (a) denotes a 2D branch cut/plane ending at the line defect, while the dashed line in (b) denotes a 1D branch cut ending at the point defect. A line defect can act on both composite particles denoted by a black dot, and loops denoted by a red circle. Once a pointlike excitation and a loop excitation move around a line defect, the defect performs the $\mathsf{Charles}$ symmetry transformation $\mathrm{\Omega }$ and $W$ on the pointlike excitation and the loop excitation, respectively. In (b), the loop moves around the point defect such that the branch line intersects at the loop's spatial trajectory (a torus) only once. A $\mathsf{Charles}$ transformation induced by a point defect can only be $\mathcal{G}=W\oplus \mathrm{\Omega }=\mathbb{I}\oplus \mathrm{\Omega }$, which acts only on the loop excitations. However, due to Eq. (63), the only candidate for $\mathrm{\Omega }$ is $\mathbb{I}$. This means that point defects can only behave like the identity element ${\mathcal{G}}_{\mathbb{I}}$ of $\mathsf{Charles}$. Therefore, in 3D, we only consider line defects.

Figure 7

Diagrammatic description of equivalence classes of fusion rules. (a) Shows the equivalent defect-anyon composites in 2D Abelian topological phases [74]. The bare point defect is labeled by a group element $\mathcal{G}$ of the anyonic symmetry group. The dashed lines are branch cuts that end at the defect. The vertical solid lines are quasiparticle (i.e., anyon, denoted by $qp$) strings (e.g., a string operator in the Wen-plaquette model), ending at the point defect. An anyon is transformed to another anyon when passing through the branch cut ($q{p}_1\to \mathcal{G}q{p}_1$). The fusion of a bare defect denoted as ${\mathcal{D}}_{\mathcal{G}}^0$ and anyon ($qp$) forms a defect-anyon composite denoted as “${\mathcal{D}}_{\mathcal{G}}^0\times qp$.” It is topologically equivalent to the defect-anyon composite denoted as ${\mathcal{D}}_{\mathcal{G}}^0\times [qp+(\mathbb{I}-\mathcal{G})q{p}_1]$ where $q{p}_1$ denotes all anyons of the 2D Abelian topological phase. In 3D systems where $\mathsf{Charles}$ replaces the anyonic symmetry of 2D systems, (b) and (c) show line defects (denoted as solid blue circles) on which the 2D branch branes (denoted as the surface of a cylinder) terminate. By passing through the 2D branch branes, a particle and a loop are transformed to another particle and loop, respectively ($q{p}_1\to Wq{p}_1, {\text{loop}}_1\to \mathrm{\Omega }{\text{loop}}_1$). In (b), the defect-charge (i.e., $qp$) composite denoted ${\mathcal{D}}_{\mathcal{G}}^0\times qp$ is topologically equivalent to ${\mathcal{D}}_{\mathcal{G}}^0\times [qp+(\mathbb{I}-W)q{p}_1]$ where $q{p}_1$ denotes all topologically distinct charge excitations. In (c), the red cylinder denotes the membrane operator that creates loop excitations and end on the line defect (the solid blue circles). The defect-loop composite denoted ${\mathcal{D}}_{\mathcal{G}}^0\times \text{loop}$ is topologically equivalent to ${\mathcal{D}}_{\mathcal{G}}^0\times [\text{loop}+(\mathbb{I}-\mathrm{\Omega }){\text{loop}}_1]$ where $\forall {\text{loop}}_1$ denotes all topologically distinct loop excitations.

Figure 8

Tensor-network-type graphical representations of $\mathsf{Charles}$ transformations. (a) Represents Eq. (63) where $K$ is a fixed-point matrix. (b) Represents Eq. (68) where $\mathrm{\Lambda }$ is a fixed-point tensor with a bond dimension no less than two. (c) Represents Eq. (70) where $\mathrm{\Xi }$ is a fixed-point tensor with a bond dimension no less than four.