Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance

We define for quantum many-body systems a quasiadiabatic 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 the distinction of 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.

Figure 1

(Color online) 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

(Color online) A dual closed string.

Figure 4

The energy levels of $N$ spin-$1∕2$ spins described by $H_0+H_1+H_2$, assuming $\mid J_1\mid ,\mid J_2\mid ⪡\mid g\mid ⪡U⪡N\mid g\mid$.

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 a 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 nondegenerate ground state on the torus. In general, the confined phase and the deconfined phase are distinguished by the zero law and the perimeter law of certain loop operators.

Figure 6

(Color online) The empty dots represent rotors with $L^z=0$—the no-string states. A closed string is formed by a loop connecting neighboring vertices. A closed-string state is obtained by alternately increasing or decreasing $L^z$ by 1 along the closed string. The filled dots represent rotors with $L^z=\pm 1$. The arrows on the links all point from even vertices to odd vertices.