Abstract

We show that the three-dimensional map between fermions and bosons at finite temperature generalises for all odd dimensions d>3. We further argue that such a map has a nontrivial large d limit. Evidence comes from studying the gap equations, the free energies and the partition functions of the U(N) Gross–Neveu and CPN−1 models for odd d≥3 in the presence of imaginary chemical potential. We find that the gap equations and the free energies can be written in terms of the Bloch–Wigner–Ramakrishnan Dd(z) functions analysed by Zagier. Since D2(z) gives the volume of ideal tetrahedra in 3d hyperbolic space our three-dimensional results are related to resent studies of complex Chern–Simons theories, while for d>3 they yield corresponding higher dimensional generalizations. As a spinoff, we observe that particular complex saddles of the partition functions correspond to the zeros and the extrema of the Clausen functions Cld(θ) with odd and even index d respectively. These saddles lie on the unit circle at positions remarkably well approximated by a sequence of rational multiples of π.

Highlights

  • Three-dimensional bosonization [1] via statistical transmutation [2,3] is a recurrent theme in field theory and condensed matter physics, and it has been more recently connected to particle-vortex duality e.g. [8]

  • It is known that for scalars and fermions are coupled to a gauge field Aμ at finite temperature, the temporal component A0 plays the role of an imaginary chemical potential

  • A crucial ingredient in unveiling this property was the presence of an imaginary chemical potential for a U (1) global charge, which plays the role of the Chern–Simons level for theories in a monopole background

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Summary

Introduction

Three-dimensional bosonization [1] via statistical transmutation [2,3] is a recurrent theme in field theory and condensed matter physics, and it has been more recently connected (see e.g. [4,5,6,7]) to particle-vortex duality e.g. [8]. If fermion–boson duality is a fundamental property of three-dimensional quantum physics one may think that it is not necessary to invoke non-abelian gauge fields to make it manifest Having this idea in mind we revisited in [11] the finite temperature phase structure of two 3d systems; the fermionic U (N ) Gross–Neveu model and the bosonic CPN−1 model.

The canonical ensemble
Matter coupled to Chern–Simons in a monopole background
Statistical transmutation
Revisiting the 3d fermion–boson map at imaginary chemical potential
The 3d fermion–boson map
TrI2 Q3
The fermions
S5β4 Cl4
S7 β 6
Spinoff: the zeros and the extrema of Clausen’s functions
The bosons
The fermions–bosons map for large-d
Summary and discussion
Full Text
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