[Paper Review] Abelian combinatorial gauge symmetry

Hongji Yu, Dmitry Green+2 more|arXiv (Cornell University)|Dec 7, 2022

Physics of Superconductivity and Magnetism4 citations

TL;DR

This paper generalizes combinatorial gauge symmetry (CGS) to all finite Abelian groups, enabling exact, non-perturbative lattice gauge theories with only two-body interactions. By constructing Hamiltonians using monomial matrices $ L $ and diagonal matrices $ R $, the framework ensures exact gauge symmetry and topological order, with a physical realization proposed in superconducting wire arrays, offering a path to experimentally stable topological quantum phases.

ABSTRACT

Combinatorial gauge symmetry is a principle that allows us to construct lattice gauge theories with two key and distinguishing properties: a) only one- and two-body interactions are needed; and b) the symmetry is exact rather than emergent in an effective or perturbative limit. The ground state exhibits topological order for a range of parameters. This paper is a generalization of the construction to any finite Abelian group. In addition to the general mathematical construction, we present a physical implementation in superconducting wire arrays, which offers a route to the experimental realization of lattice gauge theories with static Hamiltonians.

Motivation & Objective

• To extend the framework of combinatorial gauge symmetry (CGS) from small Abelian groups like $\mathbb{Z}_2$ and $\mathbb{Z}_3$ to all finite Abelian groups.
• To provide a systematic mathematical construction of CGS Hamiltonians using matrix representations of group actions and permutations of invariant sets.
• To demonstrate that topological order and exact gauge symmetry can be realized with only two-body interactions, avoiding perturbative or emergent symmetry approaches.
• To propose a physical implementation in superconducting wire arrays, enabling experimental realization of static, gapped topological quantum phases.

Proposed method

• Constructing CGS Hamiltonians via two-body interaction matrices $ W_{ij} $ that satisfy $ W = L^T W R $, where $ L $ and $ R $ are monomial matrices preserving quantum commutation relations.
• Enforcing $ R $ to be diagonal due to lattice geometry, while $ L $ acts as a permutation matrix on the group action, linking group structure to invariant set permutations.
• Classifying allowed $ W $ matrices through the interplay of group actions and permutation symmetries on quantum variables indexed by lattice sites.
• Using the $ \mathbb{Z}_2 $ triangular lattice as a motivating example to illustrate how gauge symmetry arises from $ \pm 1 $-valued matrices with even parity.
• Deriving the ground state spectrum by minimizing energy under $ \sigma = 1 $ constraint, with degeneracy splitting via uniform field on matter spins.
• Proposing a physical implementation using superconducting wire arrays, where Josephson phases realize the required two-body interactions and gauge symmetry.

Experimental results

Research questions

• RQ1 How can combinatorial gauge symmetry be systematically generalized from $\mathbb{Z}_2$ and $\mathbb{Z}_3$ to all finite Abelian groups?
• RQ2 What are the necessary and sufficient conditions on two-body interaction matrices $ W $ to realize exact, non-perturbative gauge symmetry in lattice gauge theories?
• RQ3 Can topological order be stabilized in a model with only two-body interactions and exact gauge symmetry, without relying on emergent or perturbative constructions?
• RQ4 How can spurious degeneracies in $\mathbb{Z}_2$ CGS theories on odd-coordination lattices be lifted to stabilize the topological ground state?
• RQ5 What is a feasible physical implementation of such a CGS model in a controllable quantum platform like superconducting circuits?

Key findings

• The ground state degeneracy in $\mathbb{Z}_2$ CGS theories on odd-coordination lattices is lifted by a uniform field on matter spins, as the total $\mu$-magnetization is non-zero and equal to $(-1)^{(q-1)/2}\binom{q-1}{(q-1)/2}$, ensuring a gapped phase.
• The framework generalizes to all finite Abelian groups by classifying $ W $ matrices through group actions and permutation symmetries of invariant sets, ensuring exact gauge symmetry.
• For $\mathbb{Z}_2$ theories on odd-coordination lattices, the number of flux-0 states with minimal energy is reduced due to non-degenerate ground state under uniform field, resolving spurious degeneracies.
• The construction ensures that only two-body interactions are required, and the gauge symmetry is exact across all parameters, avoiding perturbative or effective symmetry breaking.
• A physical implementation is proposed using superconducting wire arrays, where Josephson phases realize the required two-body interactions and gauge symmetry via engineered coupling matrices.
• The local gauge symmetry group of $ A $-type operators in Haah’s code is shown to be isomorphic to $ \mathbb{Z}_2^7 $, generated by 7 independent $ P_{i,i+1} $ operations, each expressible as products of $ B $-type operators.