Transition between 2D Symmetry Protected Topological Phases on a Klein Bottle

TL;DR

This work investigates transitions between the 2+1D $Z_2$ SPT phases on a Klein bottle, showing that inserting a parity defect along the orientation-reversing cycle induces an extra symmetry charge in the ground state. At the critical point between the trivial and nontrivial phases, threading a $ ext{π}$ flux yields an exact two-fold ground-state degeneracy on the Klein bottle, providing a robust topological constraint on the spectrum. The analysis combines exactly solvable lattice models, gauging to the doubled semion theory, and a modular-transformation perspective from the 3+1D $Z_2$ gauge theory via the symmetry TFT framework, and generalizes the phenomenon to abelian order-2 SPT phases with parity symmetry. The work also develops a broader notion of emergent parity symmetry in gapped phases, using LEEs and the symmetry-TFT language, and discusses potential extensions to gapless critical theories and other symmetry groups.

Abstract

Manifolds with nontrivial topology play an essential role in the study of topological phases of matter. In this paper, we study the nontrivial symmetry response of the $2+1$D $Z_2$ symmetry-protected topological (SPT) phase when the system is put on a non-orientable manifold -- the Klein bottle. In particular, we find that when a symmetry defect is inserted along the orientation-reserving cycle of the Klein bottle, the ground state of the system gets an extra charge. This response remains well defined at transition points into the trivial SPT phase, resulting in an exact two-fold degeneracy in the ground state independent of the system size. We demonstrate the symmetry response using exactly solvable lattice models of the SPT phase, as well as numerical work across the transition. We explore the connection of this result to the modular transformation of the $3+1$D $Z_2$ gauge theory and the emergent nature of the parity symmetry in the $Z_2$ SPT phase.

Figures (17)

• Figure 1: Changing from a torus to a Klein bottle. A semion $s$ starting at the dot and tunneling across the vertical loop becomes an anti-semion $\bar{s}$ and fuses with $s$ to leave behind a charge $e$.
• Figure 2: A tripartite (red, green, black) triangular lattice with Torus and Klein bottle boundary conditions. The arrows and the numbering of lattice sites indicate how the boundary degrees of freedom are identified. A symmetry defect can be inserted along the nontrivial purple cycle. On the Klein bottle, the defect line runs along the orientation reserving nontrivial cycle.
• Figure 3: Action of the $\pi$ symmetry defect line (purple) on the Hamiltonian terms in $H_{\text{SPT}}$ (with the $CZ$ gates shown in red). The action is given by conjugation by the symmetry operator $\prod_v X_v$ on the vertices to one side of the symmetry defect line. The solid and dotted lines are two different symmetry defect line configurations that yield the same result. Pairs of $Z$'s are attached to the red edges crossed by the symmetry defect line. Only the first configuration, which has an odd number of triangles intersected by the symmetry defect line, picks up a negative sign.
• Figure 4: Action of the single plaquette term in the double semion model on a minimal lattice on the torus (top row) and the Klein bottle (bottom row). The thick (dotted) red line represent a semion string operator above (below) the lattice that is fused with it. The second equality uses the second rule in Fig. \ref{['fig:rules']}. On the Klein bottle part of the semion string lies below the lattice. It is moved up by crossing the qubits on the lattice using the third rule. This results in a product of $Z$s in the fourth equality.
• Figure 5: Rules for deforming the loop configurations in the double semion model.