Towards a Complete Classification of Symmetry-Protected Topological Phases for Interacting Fermions in Three Dimensions and a General Group Supercohomology Theory

The classification and construction of symmetry-protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that (generalized) group cohomology theory or cobordism theory gives rise to a complete classification of SPT phases in interacting boson or spin systems. The construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in three dimensions. In this work, we revisit this problem based on an equivalence class of fermionic symmetric local unitary transformations. We construct very general fixed-point SPT wave functions for interacting fermion systems. We naturally reproduce the partial classifications given by special group supercohomology theory, and we show that with an additional $\tilde{B}{H}^2(G_b,{\mathbb{Z}}_2)$ structure [the so-called obstruction-free subgroup of $H^2(G_b,{\mathbb{Z}}_2)$], a complete classification of SPT phases for three-dimensional interacting fermion systems with a total symmetry group $G_f=G_b\times {\mathbb{Z}}_2^f$ can be obtained for unitary symmetry group $G_b$. We also discuss the procedure for deriving a general group supercohomology theory in arbitrary dimensions.

Popular Summary

Topological insulators, materials that are electrically conductive on the surface but insulating in the interior, are the simplest examples of a classification of matter known as a symmetry-protected topological (SPT) phase. In this phase, the electrons interact in particular ways to give rise to a host of unique electrical properties. The study of SPT phases has expanded to include systems composed of both fermions (particles such as electrons that avoid one another) and bosons (particles such as supercooled helium nuclei that crowd together). Researchers recently came up with a complete classification of SPT phases for interacting bosonic systems; however, no one has yet done the same for systems of interacting fermions. In this work, the authors propose a generic scheme to systematically construct and classify 3D SPT phases in interacting fermion systems.

In contrast to most previous research, which focuses mainly on the reduction of free-fermion SPT phases by interacting effects, our work suggests the existence of new 3D SPT phases that can be realized in neither free-fermion systems nor interacting boson systems. As such, our work greatly extends the knowledge of topological phases of quantum matter for 3D interacting fermion systems. Additionally, the discovered algebraic structure of these phases even leads to new mathematics for a general theory of group supercohomology.

A more challenging and important next step is to figure out how to experimentally realize these SPT phases in interacting fermions.

Figure 1

Bosonic and fermionic degrees of freedom for 1D fixed-point FSPT states on a link. The black dots are bosonic degrees of freedom labeled by $g_i\in G$ on sites. The blue ball represents the complex fermion $c_{(ij)}$ at the center of the link $⟨ij⟩$. The arrow represents the local order of two sites.

Figure 2

Fermionic degrees of freedom in a triangle. The red dots represent Majorana fermions at the two sides of each link. The blue ball represents the complex fermion of the special group supercohomology model at the center of the triangle. The green strip is the decorated Kitaev’s Majorana chain onto the dual lattice $\mathcal{P}$.

Figure 3

Fermionic degrees of freedom in a tetrahedron. The red dots represent Majorana fermions on the two sides of each triangle. The blue ball represents the complex fermion of the (special) group supercohomology model at the center of the tetrahedron. The green line is the decorated Kitaev’s Majorana chain on the dual lattice $\mathcal{P}$.

Figure 4

Example of triangulation $\mathcal{T}$ of torus and Kitaev chain decoration. All vertices $⟨1⟩$, $⟨2⟩$, $⟨3⟩$, and $⟨4⟩$ (blue dots) are singular vertices; i.e., $w_0=⟨1⟩+⟨2⟩+⟨3⟩+⟨4⟩$. We choose link $⟨13⟩$ and $⟨24⟩$ (blue lines) to be singular lines; i.e., $w_0=\partial (⟨13⟩+⟨24⟩)$. The direction of the red links dual to $⟨13⟩$ and $⟨24⟩$ are changed. Vertices $i=1$, 2, 3, and 4 of $\mathcal{T}$ are labeled by group elements $g_i\in G$. Majorana fermions (red dots) reside on the vertices of the resolved dual lattice $\tilde{\mathcal{P}}$ (solid and dashed red links). The solid red links and gray ellipses indicate that the two Majorana fermions at their two ends are paired with respect to the link direction. The green strip is the ${\mathbb{Z}}_2$ domain wall of the “spin” configuration $\{g_i\}$ and is decorated by a Kitaev’s Majorana chain (Majorana fermions along the domain wall are paired differently from the “vacuum”).

Figure 5

Regular pair $v\subseteq {\sigma }^i$ ($i=0$, 1, 2) for vertex $v$.

Figure 6

Triangulation $\mathcal{T}$ (black line), resolved triangulation $\tilde{\mathcal{T}}$ (black and gray line), and resolved dual lattice $\tilde{\mathcal{P}}$ (dashed red line). The resolved triangulation $\tilde{\mathcal{T}}$ was obtained from the original $\mathcal{T}$ by adding a new vertex to the center of each triangle. The links of $\mathcal{P}$ have orientations induced from the link orientations of $\mathcal{T}$ according to the conventions shown in Fig. 7. Red dots on the vertices of $\tilde{\mathcal{P}}$ represent Majorana fermions that split from the complex fermions on each link of $\mathcal{T}$ (see the discussion in Sec. 3b).

Figure 7

Conventions for the orientation of links in $\tilde{\mathcal{P}}$ (dotted red line) from branching structure of triangulation $\mathcal{T}$ (solid black line). Nonsingular (singular) black (blue) links $l\notin S$ ($l\in S$) induce orientation conventions for the dual link in $\tilde{\mathcal{P}}$. We introduce a spinless fermion on each (black or blue) link in $\mathcal{T}$ and split it into two Majorana fermions on the two sides of this link or vertices of $\tilde{\mathcal{P}}$ (red dots).

Figure 8

Shape changing of singular lines $S$ of $\mathcal{T}$ and “gauge transformation” of Kasteleyn orientations of $\tilde{\mathcal{P}}$. We perform gauge transformations on the three red vertices inside a black triangle, which effectively change the shape of $S$ and the directions of the three outreaching red links. The Majorana fermions are mapped from $n$ to $n^{\prime }$ $(n=1,2,\dots ,6)$ with respect to the directions of red links dual to black links under this FSLU.

Figure 9

Standard (2-2) move.

Figure 10

One of the (1-3) moves.

Figure 11

Fermionic pentagon equation.

Figure 12

Regular pairs $l\subseteq {\sigma }^i$ ($i=1$, 2, 3) for link $l$.

Figure 13

Triangulation $\mathcal{T}$ (solid black line), resolved triangulation $\tilde{\mathcal{T}}$ (solid and dashed black lines), and resolved dual lattice $\tilde{\mathcal{P}}$ (red line). The resolved triangulation $\tilde{\mathcal{T}}$ is obtained from the original $\mathcal{T}$ by adding a new vertex to the center of each tetrahedron. The links of $\mathcal{P}$ have orientations induced from the link orientations of $\mathcal{T}$ according to the conventions shown in Fig. 14. Red dots on the vertices of $\tilde{\mathcal{P}}$ represent Majorana fermions, which are split from complex fermions on each face of $\mathcal{T}$.

Figure 14

Conventions for orientations of links in $\tilde{\mathcal{P}}$ (red line) from branching structure of triangulation $\mathcal{T}$ (black line). Nonsingular (singular) black (blue) triangle $f\notin S(f\in S)$ induces orientation for the dual link in $\tilde{\mathcal{P}}$. We introduce a spinless fermion on each (black or blue) triangle in $\mathcal{T}$ and split it into two Majorana fermions on two sides of this triangle or vertices of $\tilde{\mathcal{P}}$ (red dots).

Figure 15

Shape changing of singular surfaces $S$ of $\mathcal{T}$ and gauge transformation of Kasteleyn orientations of $\tilde{\mathcal{P}}$. We perform gauge transformations on the four red vertices inside a black tetrahedron, which effectively changes the shape of $S$ and the directions of the four outreaching red links. The Majorana fermions are mapped with respect to the directions of red links dual to black triangles under this FSLU transformation.

Figure 16

Standard (2-3) move.

Figure 17

A (2-3) move that involves singular surfaces.

Figure 18

A (1-4) move.

Figure 19

Resolvation of four strings of Kitaev’s Majorana chains meeting at one tetrahedron. If four strings meet at one tetrahedron, we should pair the Majorana fermions $\overline{0}$ to $\overline{2}$ and $\overline{1}$ to $\overline{3}$, respectively (gray ellipses). The green lines on $\tilde{\mathcal{P}}$ indicate that these lines are decorated by Kitaev chains.

Figure 20

Fermionic fusion hexagon equation. All of the $F$ moves are of the standard (2-3) move [see Eqs. (40) and (41)]. Colored numbers $i$ and $j$ in the subscript of $F$ indicate that the link $⟨ij⟩$ with the same color is added after this $F$ move. All six $F$ moves do not introduce new singular lines and surfaces. There is a global direction from left to right such that the vertex with the smaller number has the earlier order.

Figure 21

Fermionic fusion hexagon equation on dual lattice $\mathcal{P}$.

Figure 22

One example of Kitaev’s Majorana chain decoration configuration of fermionic fusion hexagon equation. Green lines indicate that these dual links are decorated by Kitaev’s Majorana chains. Among the six $F$ moves for this configuration, only $F_{01345}$ and $F_{12345}$ (red color in the figure) change the Majorana fermion parity. Therefore, this choice of ${\tilde{n}}_2(g_i,g_j,g_k)$ belongs to the (0,1,1,0,0,0) row in Table 3 and has obstruction ${\mathcal{O}}_{\gamma }=i$.