Quantitative approach for the Dicke-Ising chain with an effective self-consistent matter Hamiltonian

TL;DR

This work establishes a thermodynamic-limit framework for the Dicke-Ising chain by mapping it to a self-consistent effective matter Hamiltonian in which the cavity mode acts as a self-consistent field. The authors solve this Hamiltonian using a combination of numerical linked-cluster expansions and DMRG (NLCE+DMRG), enabling high-precision determination of phase boundaries in one dimension. They reveal a ferromagnetic multicritical point, and for antiferromagnetic Ising couplings confirm the existence of a narrow antiferromagnetic superradiant phase, whose microscopic origin they attribute to Dicke-type polariton condensation within the effective spin model. The results provide a thermodynamic-limit, nonperturbative benchmark that complements prior finite-size QMC/DMRG studies and suggests directions for extensions to higher dimensions and finite temperatures, with implications for cavity-QED and quantum simulators.

Abstract

In the thermodynamic limit, the Dicke-Ising chain maps exactly onto an effective self-consistent matter Hamiltonian with the photon field acting solely as a self-consistent effective field. As a consequence, no quantum correlations between photons and spins are needed to understand the quantum phase diagram. This enables us to determine the quantum phase diagram in the thermodynamic limit using numerical linked-cluster expansions combined with density matrix renormalization group calculations (NLCE+DMRG) to solve the resulting self-consistent matter Hamiltonian. This includes magnetically ordered phases with significantly improved accuracy compared to previous estimates. For ferromagnetic Ising couplings, we refine the location of the multicritical point governing the change in the order of the superradiant phase transition, reaching a relative accuracy of $10^{-4}$. For antiferromagnetic Ising couplings, we confirm the existence of the narrow antiferromagnetic superradiant phase in the thermodynamic limit. The effective matter Hamiltonian framework identifies the antiferromagnetic superradiant phase as the many-body ground state of an antiferromagnetic transverse-field Ising model with longitudinal field. This phase emerges through continuous Dicke-type polariton condensation from the antiferromagnetic normal phase, followed by a first-order transition to the paramagnetic superradiant phase. Thus, NLCE+DMRG provides a precise determination of the Dicke-Ising phase diagram in one dimension by solving the self-consistent effective matter Hamiltonian.

Figures (8)

• Figure 1: Mean-field phase diagram of the Dicke-Ising model at fixed longitudinal field $\varepsilon/\omega_c = 0.3$, plotted as a function of the spin-photon coupling $g/\omega_c$ and the Ising interaction strength $J/\omega_c$Zhang2014. Four phases are identified: paramagnetic normal (PN), paramagnetic superradiant (PS), antiferromagnetic normal (AN), and antiferromagnetic superradiant (AS). The AS phase breaks two discrete symmetries and emerges only for antiferromagnetic couplings. Additonally, the sweep in $g$ from \ref{['subsec:af_numerical']} is visualized as a dashed blue line.
• Figure 2: Visualization of the iterative procedure used to solve the self-consistency condition for ferromagnetic Ising interactions. The parameters $J$ and $\varepsilon$ are fixed, and the range of $g$ is specified. The procedure starts from an initial guess for $m_x$ and iterates until convergence. For the antiferromagnetic case, one has the same iterative procedure but an additional self-consistency loop for $m_s$.
• Figure 3: Absolute value of the energy difference $\Delta e = \left| e_{\mathrm{NLCE+DMRG}} - e_{\mathrm{weak}}\right|$ obtained from NLCE+DMRG at the mean-field critical coupling $g_{\mathrm{crit}}$. Calculations are performed on clusters of size $N=100$ and $N=101$. Finite values $\Delta e >0$ signal a first-order transition, while $\Delta e \approx 0$ corresponds to a continuous transition. The curves show results starting from $x$-polarized and $-z$-polarized initial states, plotted in the same panel to highlight their near-perfect agreement over a wide parameter range. Since the data are shown on a logarithmic scale, values below the numerical accuracy ($\sim 10^{-12}$) are effectively rendered as zero. The vertical line marks the first data point at which the two curves become numerically distinguishable, indicating that the numerical accuracy is reached and providing a practical criterion for identifying the multicritical point. DMRG convergence thresholds are $10^{-14}$ in energy and $10^{-13}$ for the self-consistency loop.
• Figure 4: Phase diagram of the Dicke-Ising chain with ferromagnetic interactions ($J=-0.2$). Red dots correspond to first-order transitions; blue dots indicate continuous transitions. The black curve is the mean-field PN-PS boundary. The brown dashed line marks the multicritical point at $\varepsilon/\omega_c = 0.19992 \pm 0.00005$.
• Figure 5: Ground-state energy per site $E_0/N$ as a function of $g/\omega_c$ for the antiferromagnetic Dicke-Ising chain with $J = 0.2$, $\varepsilon = 0.3$. The AN-AS transition is continuous, whereas the AS-PS transition shows a clear first-order signature through a discontinuity between increasing and decreasing sweeps. The inset magnifies the transition region.