Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes

TL;DR

This work develops a unifying framework that viewsl ocal topological-order automorphisms as arising from adiabatic paths of gapped Hamiltonians in 2D. It constructs explicit, commuting-projector Hamiltonian paths realizing toric-code automorphisms, and translates these paths into measurement-based automorphism codes (including a novel e↔m Floquet-like code) via Kramers-Wannier duality. The authors propose a broader homotopy-theoretic classification of the space of 2D gapped Hamiltonians realizing a fixed topological order, linking fundamental and higher homotopy groups to invertible states, automorphisms, and abelian anyons, and conjecture a deep connection to the Picard group and its 2-group structure. Overall, the paper bridges concrete code constructions with a topological and categorical framework for understanding dynamical transformations of topological order. This perspective advances both fault-tolerant quantum memory designs and the mathematical classification of topological phases under continuous deformations.

Abstract

The recent "honeycomb code" is a fault-tolerant quantum memory defined by a sequence of checks which implements a nontrivial automorphism of the toric code. We argue that a general framework to understand this code is to consider continuous adiabatic paths of gapped Hamiltonians and we give a conjectured description of the fundamental group and second and third homotopy groups of this space in two spatial dimensions. A single cycle of such a path can implement some automorphism of the topological order of that Hamiltonian. We construct such paths for arbitrary automorphisms of two-dimensional doubled topological order. Then, realizing this in the case of the toric code, we turn this path back into a sequence of checks, constructing an automorphism code closely related to the honeycomb code.

Figures (10)

• Figure 1: The honeycomb code. Qubits are on vertices. Plaquettes are labelled $0,1,2$. Edges of types $0,1,2$ are labelled with red, green, and blue, respectively.
• Figure 2: Phase diagram corresponding to the Hamiltonian Eq. \ref{['eq:KekuleHam']}. In the diagram $J_x + J_y + J_z = 1$. The Hamiltonian has two phases, a gapless spin liquid phase when $J_x = J_y = J_z$ at the center of the triangle, and a gapped Abelian $\mathbb{Z}_2$ spin liquid realizing the toric code topological order everywhere else (exlcuding the boundary of the triangle). The vector ${\bf J}(\theta)$ demonstrates a non-trivial path of gapped Hamiltonians, and as described in the main text can be unrolled into a Kekulé vortex binding a non-Abelian defect.
• Figure 3: Turning a path of quadratic Hamiltonians in $2$ dimensions into a nontrivial one-dimensional quadratic Hamiltonian. There is one Majorana mode $\gamma_j$ on each vertex $j$. Dimers (thick bonds) containing a pair of vertices $j,k$ indicate that the expectation value of $i\gamma_j\gamma_k$ is equal to $\pm 1$. At the top of the left figure, and continuing further above, all type $0$ edges are in a dimer. Then, further down all type $1$ edges are in a dimer, then even further down all type $2$ edges are in a dimer. Finally, on the bottom of the figure, we return to having all type $0$ edges in a dimer. Counting the number of edges that cross a vertical line (shown as a zig-zag pink line), it differs by an odd number from the "trivial path", where all type $0$ edges everywhere in the figure are in a dimer as shown on the right.
• Figure 4: (a) A triangular lattice with qubits at the vertices shown on the left. The "upward" triangles are three-colored according to $r \in \{0,1,2\}$. The lattice hosts a 1-parameter family of toric codes $H(t)$. (b) At integer times $t$ the the terms appearing in the Hamiltonian are shown in the middle and are parameterized by $r = t \mod 3$. The unitary matrix in Eq. \ref{['eq:adunit']} continuously relates the three special points of the Hamiltonian. (c) The circuit implementing Kramers-Wannier duality on three sites. We have labeled the corners of a triangle below the circuit indicating how the circuit is applied to a given plaquette.
• Figure 5: The extended string net model lives on a hexagonal lattice with dangling edges terminating on each plaquette. An element of the Hilbert space is specified by a labeling of the graph as described in the main text. The thick blue lines are labeled by bimodule degrees of freedom, while the thinner black lines are labeled $\mathcal{C}$ degrees of freedom. A diagram satisfying the fusion constraints will provide a vector in $V^{(0)}$.

Theorems & Definitions (1)

• Conjecture 1