Exactly Solvable Phase Transition in a Cavity-Coupled 1D Ising Chain

TL;DR

This work shows that a one-dimensional Ising chain, which alone does not exhibit a finite-temperature phase transition, attains a finite-T superradiant phase transition when coupled to a single cavity mode. By applying a polaron transformation, the photon mode decouples as a free boson and induces a photon-mediated all-to-all interaction among spins, reducing the problem to an exactly solvable 1D Ising chain in an external field via a Hubbard-Stratonovich transformation and a saddle-point analysis. The authors derive a closed self-consistent equation for the magnetization and an explicit expression for the critical temperature $T_c = \frac{2J}{W\left( \frac{\omega J}{g^{2}} \right)}$, providing an exact, minimal model of SRPT coexisting with material interactions and illustrating the role of long-range coupling in cavity-QED systems. The results connect to the Dicke model and contrast with zero-temperature studies of related Hamiltonians, highlighting how symmetry and dimensionality influence the order of the transition and suggesting experimental platforms in anisotropic magnets or cold-atom setups.

Abstract

Although one-dimensional classical spin chains do not exhibit phase transitions, we found that a phase transition does occur when they are coupled to a cavity photon mode. This provides the simplest exactly solvable examples demonstrating that finite-temperature superradiant phase transitions can emerge from long-range fully connected interactions mediated by photons and interactions within the material.

Figures (3)

• Figure 1: The spin system is coupled to cavity photons with a coupling constant $g$. The figure shows seven spins depicted in blue, coupled to photons illustrated in red.
• Figure 2: When the spin system couples to a cavity photon, a long-range, all-to-all ferromagnetic interaction emerges between the spins. In the figure, a specific spin highlighted in red interacts with the other spins not only through the original internal spin-spin interactions $J_{ij}$, but also via a fully-connected intractions $-\frac{g^2}{N\omega}$ mediated by photons.
• Figure 3: (a) Magnetization $m$ and (b) normalized photon number $n = g^2 m^2 / \omega^2$ for a nearest-neighbor interaction $J=1$. (c) Normalized photon number $n$ for a stronger interaction, $J=10$. All quantities are plotted as a function of temperature $T$ and light--matter coupling strength $g$. The colored regions represent the ordered ferromagnetic, superradiant phase ($m, n \neq 0$), while the white region corresponds to the disordered paramagnetic phase ($m=n=0$). The red line is the exact critical temperature $T_c$ for the interacting chain, and the black line shows $T_c$ for the non-interacting ($J=0$) case for comparison. All calculations are performed with cavity frequency $\omega=1$.