Abstract
Recent work on a family of boson-fermion mappings has emphasized the interplay of symmetry and duality: Phases related by a particle-vortex duality of bosons (fermions) are related by time-reversal symmetry in their fermionic (bosonic) formulation. We present exact mappings for a number of concrete models that make this property explicit on the operator level. We illustrate the approach with one- and two-dimensional quantum Ising models, and then similarly explore the duality web of complex bosons and Dirac fermions in (2+1) dimensions.
Highlights
Mapping models of spins or bosons to fermions has a long history in condensed-matter physics
Two important recent works by Seiberg, Senthil, Wang, and Witten [19] and by Karch and Tong [20] have extended this symmetryduality correspondence to 2D systems. These groups established that phases for a free 2D Dirac fermion that are related by time-reversal symmetry are related by particle-vortex duality when expressed in terms of bosons coupled to a Chern-Simons field
This paper aims to elevate the symmetry-duality interplay from the level of quantum phases to explicit properties of operators describing physical d.o.f. in various representations
Summary
Mapping models of spins or bosons to fermions has a long history in condensed-matter physics. Two important recent works by Seiberg, Senthil, Wang, and Witten [19] and by Karch and Tong [20] have extended this symmetryduality correspondence to 2D systems These groups established that phases for a free 2D Dirac fermion that are related by time-reversal symmetry are related by particle-vortex duality when expressed in terms of bosons coupled to a Chern-Simons field. We explore a third set of models that exhibit timereversal symmetry both in the bosonic and fermionic representations— prohibiting Chern-Simons terms for any of the dynamical gauge fields We propose that these wire models yield the relations ijboson · a þ L1⁄2a ↔. The changes of variables that link these representations loosely parallel those that we exploit in later sections for 2D systems
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