Long-range interactions from $U\left(1\right)$ gauge fields via dimensional mismatch

TL;DR

This paper establishes a duality between certain long-range fermionic models in lower dimensions and $U(1)$ gauge theories in higher dimensions, enabling the engineering of long-range interactions like $1/r$ potentials.

Contribution

It introduces an exact mapping for $d=1$ models, linking fermionic interactions to gauge theories in higher dimensions, and discusses applications including Hamiltonian construction and long-range potential engineering.

Findings

Exact mapping for $d=1$ models relating fermionic interactions to Laplacian powers.

Diagrammatic representation of theories with dimensional mismatch.

Possibility to engineer $1/r$ long-range potentials via gauge field and dimensional mismatch.

Abstract

We show how certain long-range models of interacting fermions in $d+1$ dimensions are equivalent to $U\left(1\right)$ gauge theories in $D+1$ dimensions, with the dimension $D$ in which gauge fields are defined larger than the dimension $d$ of the fermionic theory to be simulated. For $d=1$ it is possible to obtain an exact mapping, providing an expression of the fermionic interaction potential in terms of half-integer powers of the Laplacian. An analogous mapping can be applied to the kinetic term of the bosonized theory. A diagrammatic representation of the theories obtained by dimensional mismatch is presented, and consequences and applications of the established duality are discussed. Finally, by using a perturbative approach, we address the canonical quantization of fermionic theories presenting non-locality in the interaction term to construct the Hamiltonians for the effective theories found by dimensional reduction. We conclude by showing that one can engineer the gauge fields and the dimensional mismatch in order to obtain long-range effective Hamiltonians with $1/r$ potentials.