Superconductors are topologically ordered

Abstract

We revisit a venerable question: what is the nature of the ordering in a superconductor? We find that the answer is properly that the superconducting state exhibits topological order in the sense of Wen, i.e. that while it lacks a local order parameter, it is sensitive to the global topology of the underlying manifold and exhibits an associated fractionalization of quantum numbers. We show that this perspective unifies a number of previous observations on superconductors and their low lying excitations and that this complex can be elegantly summarized in a purely topological action of the ``$BF$'' type and its elementary quantization. On manifolds with boundaries, the $BF$ action correctly predicts non-chiral edge states, gapped in general, but crucial for fractionalization and establishing the ground state degeneracy. In all of this the role of the physical electromagnetic fields is central. We also observe that the $BF$ action describes the topological order in several other physically distinct systems thus providing an example of topological universality.

References in corpus (13)

• How to detect fluctuating order in the high-temperature superconductors
• Topological orders and Edge excitations in FQH states
• Quantum Orders and Symmetric Spin Liquids
• Z_2 Gauge Theory of Electron Fractionalization in Strongly Correlated Systems
• Quantum orders in an exact soluble model
• Short-ranged RVB physics, quantum dimer models and Ising gauge theories
• Nodal Liquid Theory of the Pseudo-Gap Phase of High-Tc Superconductors
• Three dimensional resonating valence bond liquids and their excitations
• Fractionalization, topological order, and cuprate superconductivity
• Criticality in self-dual sine-Gordon models
• Canonical BF-type Topological Field Theory and Fractional Statistics of Strings
• The Edge States of the BF System and the London Equations
• An Order Parameter Reconciling Abelian and Center Dominance in SU(2) Yang-Mills Theory

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