Protected qubits and Chern-Simons theories in Josephson junction arrays

Abstract

We present general symmetry arguments that show the appearance of doubly denerate states protected from external perturbations in a wide class of Hamiltonians. We construct the simplest spin Hamiltonian belonging to this class and study its properties both analytically and numerically. We find that this model generally has a number of low energy modes which might destroy the protection in the thermodynamic limit. These modes are qualitatively different from the usual gapless excitations as their number scales as the linear size (instead of volume) of the system. We show that the Hamiltonians with this symmetry can be physically implemented in Josephson junction arrays and that in these arrays one can eliminate the low energy modes with a proper boundary condition. We argue that these arrays provide fault tolerant quantum bits. Further we show that the simplest spin model with this symmetry can be mapped to a very special Z Z2 Chern-Simons model on the square lattice. We argue that appearance of the low energy modes and the protected degeneracy is a natural property of lattice Chern-Simons theories. Finally, we discuss a general formalism for the construction of discrete Chern-Simons theories on a lattice.

The emergence of resonating valence bond physics in spin–orbital models

“…So the Hamiltonian is only invariant if one simultaneously performs the same rotation in real space and in pseudo-spin space. For purely orbital models, this is known to have remarkable consequences [34,35,36,37]. For spin-orbital models, this implies that dimers with different orientations involve different orbital wave-functions, as can be clearly seen in phases C and C'.…”

Quantum phase transition in the quantum compass model

“…6,7 The twofold degenerate state has been shown to be gapped from low-energy excitations based on the results calculated in a small size system. 3 However, the results cannot be extended to large size systems. The results from semiclassical spin-wave study, exact diagonalization, and Green's function Monte Carlo calculation on finite-size clusters have suggested that the system develops a spontaneously symmetry-broken state in the thermodynamic limit.…”

“…More recently, it was also proposed as a realistic model to generate protected qubits. 3 It has been argued that the eigenstates of the quantum compass model are two-and only twofold degenerate. 3 The twofold degenerate ground state can be an implementation of a protected qubit if it is separated from the lowenergy excitations by a finite gap.…”

“…[2][3][4][5] It was originally proposed as a simplified model to describe some Mott insulators with orbit degeneracy described by a pseudospin. In particular, the compass model describes a system where the anisotropy of the spin coupling is related to the orientation of bonds.…”

Anyonic order parameters for discrete gauge theories on the lattice

“…Experimental realizations of discrete gauge theories are still modest, though there are interesting proposals for implementing them in Josephson junction networks [3,4,5] and in systems of polar molecules in optical lattices [6]. Globally symmetric implementations were explored in [7,8].…”