$Z_N$ Berry Phases in Symmetry Protected Topological Phases

We show that the $Z_N$ Berry phase (Berry phase quantized into $2\pi /N$) provides a useful tool to characterize symmetry protected topological phases with correlation that can be directly computed through numerics of a relatively small system size. The $Z_N$ Berry phase is defined in a $N-1$-dimensional parameter space of local gauge twists, which we call the “synthetic Brillouin zone,” and an appropriate choice of an integration path consistent with the symmetry of the system ensures exact quantization of the Berry phase. We demonstrate the usefulness of the $Z_N$ Berry phase by studying two 1D models of bosons, SU(3) and SU(4) Affleck-Kennedy-Lieb-Tasaki models, where topological phase transitions are captured by $Z_3$ and $Z_4$ Berry phases, respectively. We find that the exact quantization of the $Z_N$ Berry phase at the topological transitions arises from a gapless band structure (e.g., Dirac cones or nodal lines) in the synthetic Brillouin zone.

Figure 1

(a) Schematic picture of the biquadratic model and the local gauge twist. The bonds $J_i$ represent the biquadratic interactions. (b) Synthetic Brillouin zone and the integration path $C$ leading to the Berry phase quantization. (c)–(e) The energy spectra for the ground state and the first excited state on the Brillouin zone. The gap at $\mathrm{\Gamma }$ point is always nonzero due to the finite size effect. At the phase transition, the gap closes at $K$ and $K^{\prime }$ points forming Dirac cones.

Figure 2

(a) The Berry phase with $e^{i{\widehat{A}}_3\varphi }$ for several system sizes ($L$ denotes the number of the spins). (Right) The energy spectra as functions of $\varphi$ at $\mathrm{\Delta }=0$. (b) The Berry phase with $e^{i\sum _i{\widehat{A}}_i{\varphi }_i}$ using $C$ in Fig. 1 as an integration path. (Right) The energy spectrum along the high symmetry lines in the synthetic Brillouin zone. $L$ is set to 12. The energies are in the unit of $J$.

Figure 3

(a) Schematic picture of the SU(4) model. $q$ and $\overline{q}$ denote the fundamental representation and the conjugate representation, respectively. (b),(c) The Berry phases obtained with (b) the path $W_1\text{−}\mathrm{\Gamma }\text{−}{W}_2$ and (c) the straight integration path along the ${\varphi }_1$ axis. (d),(e) The synthetic Brillouin zone and the integration path. (f) The energy spectrum on the symmetric lines in the synthetic BZ for $\delta J=0$. (Inset) The location of the nodal lines. (g) The band structure of the single orbital tight-binding model on the diamond lattice.