Braiding statistics approach to symmetry-protected topological phases

TL;DR

The paper constructs two-dimensional bosonic Ising-paramagnet phases, distinguishing a conventional paramagnet from a nontrivial symmetry-protected paramagnet by coupling to a Z2 gauge field and examining π-flux braiding statistics. It demonstrates that the nontrivial phase hosts symmetry-protected gapless edge modes, tightly linked to the semionic statistics of π-flux excitations in the gauged theory, and provides a detailed microscopic edge analysis yielding a non-chiral Luttinger liquid description. Dualities connect the spin models to toric-code and doubled-semion string-net models, suggesting a classification of 2D bosonic SPT phases via H^3(G,U(1)) and a general string-net construction for finite groups G. The work extends to higher-dimensional and non-abelian scenarios and outlines criteria for edge stability, offering a general framework in which bulk braiding data determines edge physics. Overall, it presents braiding statistics as a robust bulk diagnostic and a route to systematically characterizing SPT phases beyond free-fermion paradigms.

Abstract

We construct a 2D quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a "symmetry-protected topological phase." We describe a simple physical construction that distinguishes this system from a conventional paramagnet: we couple the system to a Z_2 gauge field and then show that the π-flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular microscopic model for the edge and derive a field theoretic description of the low energy excitations. We believe that the braiding statistics approach outlined in this paper can be generalized to a large class of symmetry-protected topological phases.

Figures (12)

• Figure 1: The Hamiltonians $H_0, H_1$ (\ref{['Hconv']}-\ref{['Hnew']}) for the two spin models. (a) The Hamiltonian $H_0$ is a sum of single spin terms, $\sigma^x_p$. (b) The Hamiltonian $H_1$ is a sum of seven spin terms $B_p = -\sigma^x_p \prod_{\langle pqq'\rangle} i^{\frac{1-\sigma^z_{q}\sigma^z_{q'}}{2}}$ where the product runs over the six triangles $\langle pqq'\rangle$ containing $p$.
• Figure 2: A schematic plot of the ground states $\Psi_0$ and $\Psi_1$ for the two paramagnets $H_0, H_1$. (a) In terms of domain wall configurations, the ground state $\Psi_0$ is a equal weight superposition of all configurations. (b) The ground state $\Psi_1$ is also a superposition of all domain wall configurations, but each configuration enters with a sign $(-1)^{N_{dw}}$ where $N_{dw}$ is the total number of domain walls.
• Figure 3: The Hamiltonians $\widetilde{H}_0,\widetilde{H}_1$ (\ref{['gaugeH']}) for the two gauged spin models. (a) The Hamiltonian $\widetilde{H}_0$ is a sum of two terms. The first term is the gauge flux term $\mu^z_{pq}\mu^z_{qr}\mu^z_{rp}$ (thick triangle) where $\mu^z_{pq}$ denotes the $\mathbb{Z}_2$ gauge field on the link $\langle pq\rangle$. The second term is the spin interaction $\sigma^x_p O_p$ where $O_p = \prod_{\langle pqr\rangle} (1 + \mu^z_{pq}\mu^z_{qr}\mu^z_{rp})/2$ and the product runs over the six triangles adjacent to $p$. (b) The Hamiltonian $\widetilde{H}_1$ includes the same gauge flux term $\mu^z_{pq}\mu^z_{qr}\mu^z_{rp}$ but has a more complicated seven spin interaction $\widetilde{B}_p O_p$ (\ref{['Bgauge']}).
• Figure 4: The string operator $V^0_{\beta}$ (\ref{['string0']}) is defined for any path $\beta$ on the dual honeycomb lattice and is given by a product of $\mu^x_{pq}$ over all links $\langle pq\rangle$ crossing $\beta$ (thickened lines). Applying this operator to the ground state $|\Psi_0\rangle$ creates two $\pi$-fluxes at the endpoints of $\beta$ (shaded triangles).
• Figure 5: A schematic picture of the two states $V^0_\beta V^0_\gamma |\Psi_0\rangle$, $V^0_\gamma V^0_\beta |\Psi_0\rangle$. The first state (left) is obtained from a process in which two $\pi$-fluxes are created at the endpoints of $\gamma$, and then two more fluxes are created, braided around the path $\beta$ and then annihilated. The second state (right) corresponds to executing these two steps in the opposite order. We expect these two states to differ by the Berry phase $e^{2i\theta}$ associated with braiding one $\pi$-flux excitation around another. The same is true for $V^1_\beta, V^1_\gamma$.