Topological Order in the ( 2 + 1 ) D Compact Lattice Super - conductor

– We study topological aspects of a compact lattice superconductor, and show that the characteristic energy splitting, ∆, between almost degenerate ground states, is simply related to a novel order parameter W̃ , which is closely related to large Wilson loops. Using Monte Carlo methods, we study the scaling properties of W̃ close to the deconfining phase transition, and conclude that ∆ ∼ e, where L is the size of the system, thus giving quantitative support to the vortex tunneling scenario proposed by Wen. The concept of topological order was first introduced to describe the ground states of the quantum Hall system [1]. Although the present interest in topological order to large extent derives from the quest for exotic non-Fermi liquid states of relevance for the high-Tc superconductors [2–6], the concept is of much more general interest. Both spin liquids [7, 8], believed to be of relevance for frustrated magnets, and ordinary BCS superconductors with dynamical electromagnetism [9, 10], are examples of topologically ordered systems. Two characteristics of topological order are particularly striking; fractionally charged quasiparticles, and a ground state degeneracy depending on the topology of the underlying space, which is lifted by tunneling processes. In the case of the most celebrated example – the Laughlin FQH states – both these properties are well understood [11], but for superconductors, the situation is less clear. It was only in 1990 that Kivelson and Rokshar pointed out that the quasiparticles are indeed fractional [12], and to our knowledge there has been no quantitative study of ground state degeneracy and splitting. We improve on this situation by a numerical study of a 2D compact lattice superconductor, using a novel order parameter W̃ , which is sensitive to vortex tunneling processes. Our results are in quantitative agreement with predictions based on topological order, and rules out the possibility of a spontaneously broken discrete symmetry. A compact lattice superconductor captures important properties of real type II superconductors [10], and the 2D case is also of intrinsic interest in the context of effective gauge theories for the high-Tc cuprates [2–6]. Examples of recent proposals are the U(1) slave-boson theory [2], the nodal liquid and the spinon-chargeon Z2 theory [3, 4], and compact QED3 [5].