To infinity and back -- $1/N$ graph expansions of light-matter systems

TL;DR

This work develops a full graph-expansion framework for light-matter systems by extending linked-cluster expansions to include light–matter couplings, enabling explicit $1/N$ corrections to the thermodynamic limit and access to the mesoscopic regime. The method combines graph-decomposition with both non-perturbative (ED) and perturbative (pcst++) solvers and introduces a novel light-matter graph type treated via disconnected-matter graphs, yielding observables per site $\,\mathcal{O}(G)/N = \sum_o d_o N^{-o}$. Applying the approach to the Dicke-Ising chain in the paramagnetic normal phase, the authors obtain ground-state and photon observables across system sizes, validate against exact results, and extract $1/N$ corrections to locate the critical point and its exponent using Dlog-Padé extrapolation. The results demonstrate smooth interpolation between microscopic and thermodynamic limits within the normal phase and highlight convergence limitations near the phase transition, outlining a path toward exploring mesoscopic physics in more complex light–matter models. Overall, the paper provides a scalable, size-free framework for mesoscopic quantum optics and strongly correlated light–matter systems with potential for broader applicability beyond the specific model studied.

Abstract

We present a method for performing a full graph expansion for light-matter systems, utilizing the linked-cluster theorem. This method enables us to explore $1/N$ corrections to the thermodynamic limit $N\to \infty$ in the number of particles, giving us access to the mesoscopic regime. While this regime is yet largely unexplored due to the challenges of studying it with established approaches, it incorporates intriguing features, such as entanglement between light and matter that vanishes in the thermodynamic limit. As a representative application, we calculate physical quantities of the low-energy regime for the paradigmatic Dicke-Ising chain in the paramagnetic normal phase by accompanying the graph expansion with both exact diagonalization (NLCE) and perturbation theory (\pcst), benchmarking our approach against other techniques. We investigate the ground-state energy density and photon density, showing a smooth transition from the microscopic to the macroscopic regime up to the thermodynamic limit. Around the quantum critical point, we extract the $1/N$ corrections to the ground-state energy density to obtain the critical point and critical exponent using extrapolation techniques.

Figures (10)

• Figure 1: The three realms of light-matter quantum systems, depending on the number of matter entities $N$. While for the limiting cases of few particles and infinitely many a multitude of methods exist to model these systems, this is not the case for the intermediate mesoscopic regime. This regime is characterized by a large number of particles, such that both a few-particle approach and taking the thermodynamic limit are not valid.
• Figure 2: Visualization of an exemplary lattice structure defined by Eq. \ref{['eq:ham']}. The blue circles represent the matter degrees of freedom, which are coupled with their respective neighbors with $J_{i,j}$.
• Figure 3: Visualization of an exemplary generalized lattice structure from Eq. \ref{['eq:generalized-hamiltonian']}. A chain of $N=10$ matter sites with periodic boundary conditions is coupled to a single light mode. The blue circles represent the matter degrees of freedom, the red circle represents the light mode. While the matter degrees of freedom are coupled only to their nearest neighbors with $J_{i,j}$, the light mode is coupled to all matter sites with $g_{k,i}$.
• Figure 4: Visualization of some of the subgraphs of the lattice depicted in Fig. \ref{['fig:lmlattice']}. On the left side, graphs $g_1,g_2,g_3$ represent pure matter graphs. These have embedding factors that scale linearly with system size and are treated with standard graph-decomposition techniques described in the last section. On the right side, graphs $g_4,g_5,g_6$ represent the new type of graph, stemming from the light-matter interactions. In general, these graphs have embedding factors that scale polynomially in system size. This is grounded in the lattice structure of Fig. \ref{['fig:lmlattice']}, e.g., the light mode is connected to all matter sites.
• Figure 5: Sketch of treating the light-matter graphs as disconnected pure matter graphs. (a): The starting graph $g$ couples two independent matter graphs via the light-matter coupling. (b): When ignoring the light modes and their light-matter interactions, we end up with pure matter graphs. For this exemplary graph, we have two disconnected graphs $g_1,g_2$. For calculating the embedding factor of this pure matter graph, we can use Eq. \ref{['eq:embedding-disconnected']}, as explained in the main text. (c): To take the light-matter interaction back into account, we insert them back for every disconnected graph $g_1,g_2$ individually. With these modified disconnected graphs, we can still use Eq. \ref{['eq:embedding-disconnected']} to calculate the embedding factor, enforcing that the two light sites have to overlap. (d): Overlap partitions $\tilde{g}_1, \tilde{g}_2, \tilde{g}_3$ forming out of the graphs $g_1, g_2$ from (c). The graphs $\tilde{g}_2, \tilde{g}_3$ are isomorphic. Therefore, we only have to take into account one of the two.