Gauge fixing for strongly correlated electrons coupled to quantum light

Abstract

We discuss the problem of gauge fixing for strongly correlated electrons coupled to quantum light, described by projected low-energy models such as those obtained within tight-binding methods. Drawing from recent results in the field of quantum optics, we present a general approach to write down a quantum light-matter Hamiltonian in either dipole or Coulomb gauge which is explicitly connected by a unitary transformation, thus ensuring gauge equivalence even after projection. The projected dipole gauge Hamiltonian features a linear light-matter coupling and an instantaneous self-interaction for the electrons, similar to the structure in the full continuum theory. On the other hand, in the Coulomb gauge the photon field enters in a highly nonlinear way, through phase factors that dress the electronic degrees of freedom. We show that our approach generalizes the well-known Peierls approximation, to which it reduces when only local, on-site orbital contributions to lightmatter coupling are taken into account. As an application we study a two-orbital model of interacting electrons coupled to a uniform cavity mode, recently studied in the context of excitonic superradiance and associated no-go theorems. Using both gauges we recover the absence of a superradiant phase in the ground state and show that excitations on top of it, described by polariton modes, contain instead nontrivial light-matter entanglement. Our results highlight the importance of treating the nonlinear light-matter interaction of the Coulomb gauge nonperturbatively, to obtain a well-defined ultrastrong coupling limit and to not spoil gauge equivalence.

Gauge Principle and Gauge Invariance in Two-Level Systems

“…It is our hope that the results and the connection between the QRM and lattice gauge theory presented here can stimulate the development of lattice gauge models for the study of USC cavity QED in 1D and 2D sys-tems, as well as of interacting electron systems [29][30][31][32]. It would also be interesting to apply lattice gauge theory to investigate cavity QED systems beyond the dipole approximation [33].…”

Non-linear optical processes in cavity light-matter systems

“…(A5) below. In a subsequent step, we will perform yet another unitary transformation which restores translational invariance and establishes the equivalence between the Hamiltonian forms encountered in some recent literature 22,32,50 and this work. Despite the fact that these transformations are based on the infinite volume limit, our starting Hamiltonian does not break translational invariance and can therefore be studied with periodic boundary conditions.…”

A non-perturbative no-go theorem for photon condensation in approximate models

“…where dr(∇ × A 0 (r)) 2 /(2π) = drΠ 2 0 (r)/(2π) = ω ph , where ω ph is the resonance frequency of the cavity mode, and A 0 (r), Π 0 (r) are the mode functions 42 .…”

“…We show in this Article that this is not case. While enforcing the TRK sum-rule alone was a reasonable approach at that time, nowadays more refined techniques to enforce gauge invariance in systems with an arbitrary but finite number of levels have been developed [39][40][41] and applied to a few solid-state systems 42,43 . Such methods can be viewed as an application of lattice gauge theories 44 .…”