Quasi-adiabatic Continuation of Quantum States: The Stability of Topological Ground State Degeneracy and Emergent Gauge Invariance

TL;DR

This work introduces a quasi-adiabatic continuation framework for quantum many-body systems that preserves locality and ground-state expectation values under smooth Hamiltonian deformations with a gap or low-energy density of states. By constructing dressed operators via a finite-cutoff, quasi-adiabatic unitary, the authors show that long-distance structures, including topological order and emergent gauge invariance, persist under perturbations, with explicit zero-law and perimeter-law diagnostics for confinement. The approach yields robust, exponentially small splittings of near-degenerate ground states in gapped systems and demonstrates the resilience of both discrete ($Z_2$) and continuous ($U(1)$) emergent gauge theories against local perturbations, including the protection of gapless gauge bosons in compact, but not non-compact, cases. These results provide a rigorous, general mechanism for the stability of topological phases and confinement diagnostics beyond exactly solvable models, with broad applicability to emergent gauge theories in bosonic lattice systems.

Abstract

We define for quantum many-body systems a quasi-adiabatic continuation of quantum states. The continuation is valid when the Hamiltonian has a gap, or else has a sufficiently small low-energy density of states, and thus is away from a quantum phase transition. This continuation takes local operators into local operators, while approximately preserving the ground state expectation values. We apply this continuation to the problem of gauge theories coupled to matter, and propose a new distinction, perimeter law versus "zero law" to identify confinement. We also apply the continuation to local bosonic models with emergent gauge theories. We show that local gauge invariance is topological and cannot be broken by any local perturbations in the bosonic models in either continuous or discrete gauge groups. We show that the ground state degeneracy in emergent discrete gauge theories is a robust property of the bosonic model, and we argue that the robustness of local gauge invariance in the continuous case protects the gapless gauge boson.

Figures (6)

• Figure 1: A closed-string state. The up-spins are represented by open dots and the down-spin by filled dots.
• Figure 2: The energy levels of $N$ spin-1/2 spins described by $H_0$.
• Figure 3: A dual closed-string.
• Figure 4: The energy levels of $N$ spin-1/2 spins described by $H_0+H_1+H_2$, assuming $|J_1|,|J_2| \ll |g| \ll U \ll N|g|$.
• Figure 5: A likely quantum phase diagram for the spin-1/2 system $H_0+H_1+H_2$. The deconfined phase is characterized by four nearly degenerate ground states on torus with energy splitting of order $e^{-L/\xi}$ where $L$ is the linear size of the torus and $\xi$ a length scale. The confined phase is characterized by a non degenerate ground state on torus. In general, the confined phase and the deconfined phase are distinguished by the zero law and the perimeter law of certain loop operators.