We describe a coupled-chain construction for chiral spin liquids in two-dimensional spin systems. Starting from a one-dimensional zigzag spin chain and imposing SU(2) symmetry in the framework of non-Abelian bosonization, we first show that our approach faithfully describes the low-energy physics of an exactly solvable model with a three-spin interaction. Generalizing the construction to the two-dimensional case, we obtain a theory that incorporates the universal properties of the chiral spin liquid predicted by Kalmeyer and Laughlin: charge-neutral edge states, gapped spin-1/2 bulk excitations, and ground state degeneracy on the torus signalling the topological order of this quantum state. In addition, we show that the chiral spin liquid phase is more easily stabilized in frustrated lattices containing corner-sharing triangles, such as the extended kagome lattice, than in the triangular lattice. Our field theoretical approach invites generalizations to more exotic chiral spin liquids and may be used to assess the existence of the chiral spin liquid as the ground state of specific lattice systems.

BCS superconductivity is explained by a simple Hamiltonian describing an attractive pairing interaction between pairs of electrons. The Hamiltonian may be treated using a mean field method, which is adequate to study equilibrium properties and a variety of non-equilibrium effects. Nevertheless, in certain non-equilibrium situations, even in a macroscopic, rather than a microscopic, superconductor, the application of mean field may not be valid. In such cases, one may resort to the full solution of the Hamiltonian, as given by Richardson in the 60's. The relevance of Richardson's solution to macroscopic non-equilibrium superconductors was pointed out recently based on the existence of quantum instabilities out of equilibrium. It is then of interest to obtain analytical expressions for expectation values between exact eigenvalues of the pairing Hamiltonian within the Richardson approach for macroscopic systems. We undertake this task in the current paper. It should be noted that Richardson's approach yields the full set of eigenvalues of the Hamiltonian, while BCS theory yields only a subset. The results obtained here, then, generalize the familiar BCS expressions for, e.g., the electron occupation or pairing correlations, to cases where the spectrum of excitations diverges from BCS theory, for example in cases where the spectrum exhibits multiple gaps.

We present a derivation of a previously announced result for matrix elements between exact eigenstates of the pairing Hamiltnonian. Our results, which generalize the well known BCS (BardeenCooper-Schrieffer) expressions for what is known as 'coherence factors', are derived based on the Slavnov formula for overlaps between Bethe-ansatz states, thus making use of the known connection between the exact diagonalization of the BCS Hamiltonian, due to Richardson, and the algebraic Bethe ansatz. The resulting formula has a compact form after a suitable parameterization of the Energy plane. Although we apply our method here to the pairing Hamiltonian, it may be adjusted to study what is termed the 'Sutherland limit' for exactly solvable models, namely where a macroscopic number of rapidities form a large string.arXiv:1309.5606v1 [cond-mat.supr-con] 22 Sep 2013 2

The dynamics of BCS (Bardeen-Cooper-Schrieffer) superconductors is fairly well understood due to the availability of a mean field solution for the pairing Hamiltonian, a solution which gives the quantum state of superconductor as a state of almost-free fermions interacting only with a condensate. As a result, transition probabilities may be computed, and expressed in terms of matrix elements of electron creation and annihilation operators between approximate eigenstates. These matrix elements are also called 'coherence factors'. Mean-field theory is however not sufficient to describe all eigenstates of a superconductor, a deficiency which is hardly important in (or very close) to equilibrium, but one that becomes relevant in certain out of equilibrium situations. We report here on a computation of matrix elements (coherence factors) for the pairing Hamiltonian between any 'two-arc' eigenstates in the thermodynamic limit.