Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

We define and compute many-body topological invariants of interacting fermionic symmetry-protected topological phases, protected by an orientation-reversing symmetry, such as time-reversal or reflection symmetry. The topological invariants are given by partition functions obtained by a path integral on unoriented spacetime which, as we show, can be computed for a given ground state wave function by considering a nonlocal operation, “partial” reflection or transpose. As an application of our scheme, we study the ${\mathbb{Z}}_8$ and ${\mathbb{Z}}_{16}$ classification of topological superconductors in one and three dimensions.

Figure 1

(a) Introducing a cross cap in the spacetime. (b) Partial reflection and (c) partial time inversion. First and second columns show the original connectivity after cutting and the twisted bonding after gluing, respectively.

Figure 2

Phase and amplitude of the partition function in the presence of spatial [Fig. 1] and temporal [Fig. 1] cross caps. (Details in Appendix B of the Supplemental Material [35]. In short, we write the partition function in terms of a Pfaffian and extract the SPT phase by evaluating the ratio of the two Paffafians in the presence and absence of a cross cap.) Here, we set $\beta =10$, $N_t=200$, and $N=40$. We put $N_{t,\text{part}}=100$ and $N_{\text{part}}=20$ for time inversion (BDI) and spatial reflection ($D+R_{-}$), respectively.

Figure 3

Schematic representation of the ground state overlap (a) partial reflection $⟨\mathrm{\Psi }|{\mathcal{R}}_{\text{part}}|\mathrm{\Psi }⟩$, and (b) partial transpose $\mathrm{tr}({\rho }_I{U}_{I_1}{\rho }_I^{T_1}{U}_{I_1}^{†})$. Solid squares represent physical sites and vertical bonds represent how physical degrees of freedom are contracted between $|\mathrm{\Psi }⟩$ and $⟨\mathrm{\Psi }|$.

Figure 4

Partial reflection (5) for class $\mathrm{D}+{\mathrm{R}}_{-}$ and partial transpose (6) for class BDI in the Kitaev Majorana chain. Here, $N=120$ and $N_{\text{part}}=60$.

Figure 5

SPT invariant for the three-dimensional inversion symmetric superconductor (class D), Eq. (8). Top: I (II) corresponds to the phase with odd (even) number of gapless Majorana surface states. Here, $N=12^3$ and $N_{\text{part}}=6^3$.