Thermal renormalization of trapped-ion quantum spin models: a self-consistent approach to a Yukawa-type quantum field theory with $\lambda\phi^4$ interactions

Trapped-ion chains can be used in analog quantum simulations of spin models. They rely on crystal phonons to mediate interactions between pairs of spins that are encoded in their electronic states. The theory describing these interactions is based on a long-wavelength relativistic Klein-Gordon field for the phonons, with a local coupling to the spins. This leads to an analogue of the pion-mediated Yukawa interactions in high-energy physics. However, when the trapped-ion chain undergoes a structural phase transition and becomes a ladder, additional $\lambda\phi^4$ terms that account for phonon scattering must be included,  effectively modifying the range of the spin-spin couplings.

This work proposes a method for revealing the flow of the critical point of the theory with the quartic coupling by using thermal effects that can be controlled by laser cooling. Self-consistent calculations and a non-perturbative treatment are used to predict how measurements on the trapped-ion model can probe properties of the quantum field theory. These calculations, which resum certain Feynman diagrams to evade mean-field theory’s infra-red divergence, are consequently used to numerically predict and explore the behavior of trapped-ion chains.