Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $σ$ models and a special group supercohomology theory

TL;DR

This work develops a special group super-cohomology framework to classify and construct symmetry-protected topological (SPT) phases of interacting fermions. By encoding fermionic degrees of freedom in a Grassmann tensor-network path integral, the authors derive fixed-point amplitudes governed by cocycle data $( u_d,n_{d-1},u_d^g)$ and show how to obtain ideal ground-state wavefunctions and exactly solvable Hamiltonians for fermionic SPTs. They reveal new 2D and 3D fermionic SPT phases, including ones not achievable by free fermions or bosonic bounds, and analyze boundary anomalies that enforce gapless or topologically ordered edges when symmetry is preserved. The results provide a computational scheme (via $H^{d}[G_f,U_T(1)]$) to identify and classify fermionic SPTs for various symmetry groups, with concrete examples such as $Z_2$ and $Z_2^T imes Z_2^f$. Overall, the paper advances the understanding of interacting fermionic SPTs and offers a robust pathway to realize them in lattice models and study their boundary phenomena.

Abstract

Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry $G$, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of super cocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (ie, non-on-site) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric} boundary must be gapless for 1+1D boundary, and must be gapless or topologically ordered beyond 1+1D. As an application of this general result, we construct a new SPT phase in 3D, for interacting fermionic superconductors with coplanar spin order (which have $T^2=1$ time-reversal $Z_2^T$ and fermion-number parity $Z_2^f$ symmetries described by a full symmetry group $Z_2^T\times Z_2^f$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion-pairs), and thus are not included in the K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in 2D with a full symmetry group $Z_2\times Z_2^f$. Those 2D fermionic SPT phases all have central-charge $c=1$ gapless edge excitations, if the symmetry is not broken.

Figures (22)

• Figure 1: (Color online) (a) The possible gapped phases for a class of Hamiltonians $H(g_1,g_2)$ without any symmetry. (b) The possible gapped phases for the class of Hamiltonians $H_\text{symm}(g_1,g_2)$ with a symmetry. The yellow regions in (a) and (b) represent the phases with long range entanglement. Each phase is labeled by its entanglement properties and symmetry breaking properties. SRE stands for short range entanglement, LRE for long range entanglement, SB for symmetry breaking, SY for no symmetry breaking. SB-SRE phases are the Landau symmetry breaking phases, which are understood by introducing group theory. The SY-SRE phases are the SPT phases, which can be understood by introducing group cohomology theory.
• Figure 2: (Color online) If we extend $\v n(t)$ that traces out a loop to $\v n(t,\xi)$ that covers the shaded disk, then the WZW term $\int_{D^2} \mathrm{d} t\mathrm{d} \xi\; \v n(t,\xi)\cdot [\prt_t \v n(t,\xi) \times \prt_\xi \v n(t,\xi)]$ corresponds to the area of the disk.
• Figure 3: (Color online) (a) The topological term $W$ describes the number of times that $\v n(x,t)$ wraps around the sphere (as we change $t$). (b) On an open chain $x \in [0,L]$, the topological term $W$ in the (1+1)D bulk becomes the WZW term for the end spin $\v n_L(t)=\v n(L,t)$ (where the end spin at $x=0$ is hold fixed).
• Figure 4: (Color online) (a) A branched triangularization of space-time. Each edge has an orientation and the orientations on the three edges of any triangle do not form a loop. The orientations on the edges give rise to a natural order of the three vertices of a triangle $(i,j,k)$ where the first vertex $i$ of a triangle has two outgoing edges on the triangle and the last vertex $k$ of a triangle has two incoming edges on the triangle. $s(i,j,k)=\pm 1$ depending on the orientation of $i\to j \to k$ to be clockwise or anti-clockwise. (b) The phase factor $\nu(\v n_i, \v n_j, \v n_k)$ depends on the image of a space-time triangle on the sphere $S^2$.
• Figure 5: (Color online) A tetrahedron -- the simplest discrete sphere. $\prod \nu^{s(i,j,k)}( \v n_i, \v n_j, \v n_k)=1$ on the tetrahedron becomes 2ccy1. Note that $s(1,2,3)=s(0,1,3)=1$ and $s(0,2,3)=s(0,1,2)=-1$. (b) The total action amplitude of the topological nonlinear $$ model on the complex gives rise to the phase factor in Hi1.