Bosonization in 2+1 dimensions via Chern-Simons bosonic particle-vortex duality

Abstract

Dualities provide deep insight into physics by relating two seemingly distinct theories. Here we consider a duality between lattice fermions and bosons in (2 þ 1) spacetime dimensions, relating free massive Dirac fermions to Abelian Chern-Simons Higgs (ACSH) bosons. To establish the duality, we represent the exact partition function of the lattice fermions in terms of the writhe of fermionic worldlines. On the bosonic side, the partition function is expressed in the writhe of the vortex loops of the particle-vortex dual of the ACSH Lagrangian. In the continuum and scaling limit, we show these to be identical. This result can be understood from the closed fermionic worldlines being direct mappings of the ACSH vortex loops, with the writhe keeping track of particle statistics.

Deconfined criticality and bosonization duality in easy-plane Chern-Simons two-dimensional antiferromagnets

“…However, the analysis of Ref. [35] and previous ones [18,32,[41][42][43][44] requires massive fermions.…”

“…where we sum over all loops L a and L b , not excluding a = b contributions, which will turn out to be a crucial point [32,35]. For a = b the double integral above yields a contribution e i2π 2 N ab κ , N ab ∈ Z, in virtue of the Gauss linking number formula [36,37].…”

“…The sign factor (−1) N ab can be readily associated to the one we have obtained on the lattice, but the writhe did not arise in the lattice formulation. In the literature the writhe contribution is sometimes referred to as the Polyakov spin factor [32,35,[41][42][43][44]. Equation (20) relates to the representation of the partition function of a Dirac fermion in 2+1 Euclidean dimensions in terms of loops [35,[42][43][44], with the difference that in our case the parity anomaly factor implies that the fermions are massless [30,[45][46][47].…”

“…The presence of ǫ µνλ ∆ ν b jλ inside the cosines above requires b jµ to be compact, something that is not implied by the original duality transformation of the Villain action, but is required when one tries to formulate the critical continuum dual action (17) on the lattice in a more precise way. As far as the writhe is concerned, it is worth to recall that it arises quite naturally in the partition function of Wilson fermions on an euclidean cubic spacetime lattice [35]. However, the analysis of Ref.…”

The fermion-boson map for large $d$ and its connection to lattice transformations

“…Three-dimensional bosonization physics [1] via statistical transmutation [2,3] is a very interesting and recurrent subject in field theory and condensed matter physics, and it has been connected (see e.g. [4,5,6,7,8]) to particlevortex duality e.g. [9].…”