Quantum phase diagrams of Dicke-Ising models by a wormhole algorithm

TL;DR

This work develops a sign-problem-free wormhole quantum Monte Carlo approach for the Dicke-Ising model, enabling quantitative exploration of light-matter coupled quantum phases in 1D and 2D. By integrating out the bosonic mode, the method yields a retarded spin-spin action and allows efficient sampling of large mesoscopic systems, revealing rich phase diagrams with both first- and second-order transitions and a light-matter analogue of lattice supersolidity for antiferromagnetic interactions. The authors demonstrate that second-order superradiant transitions belong to the Dicke/long-range mean-field universality class, while antiferromagnetic cases exhibit 3D Ising criticality in certain regimes; they also characterize multicritical points and apply self-consistent mean-field and $A^2$-term analyses to delineate accessible parameter regions. The results provide precise benchmarks for experiments in cavity and circuit QED and pave the way for studying frustrated and more complex light-matter systems with scalable quantum Monte Carlo methods.

Abstract

We gain quantitative insights on effects of light-matter interactions on correlated quantum matter by quantum Monte Carlo simulations. We introduce a wormhole algorithm for the paradigmatic Dicke-Ising model which combines the light-matter interaction of the Dicke model with Ising interactions. The quantum phase diagram for ferro- and antiferromagnetic interactions on the chain and the square lattice is determined. The occurring superradiant phase transitions are in the same universality class as the Dicke model leading to a well-known peculiar finite-size scaling that we elucidate in terms of scaling above the upper critical dimension. For the ferromagnetic case, the transition between the normal and the superradiant phase is of second order with Dicke criticality (first order) for large (small) longitudinal fields separated by a multicritical point. For antiferromagnetic interactions, we establish the light-matter analogue of a lattice supersolid with off-diagonal superradiant and diagonal magnetic order and determine the nature of all transition lines.

Figures (13)

• Figure 1: Exemplary vertices for the DIM. Diagonal vertices connect two neighboring spins at time $\tau_1=\tau_1'$, while the off-diagonal vertices connect any two spins at time $\tau_2$ and $\tau_2'$. Solid (open) circles symbolize spin up (down).
• Figure 2: Possible processes when hitting an exemplary diagonal vertex with aligned spins in the directed-loop algorithm for the DIM including an extensive amount of processes and integral in imaginary time. The full process visualization leading to the directed-loop equations are shown in sm.
• Figure 3: Ferromagnetic DIM for $Jd/\omega=-0.2$. a) Quantum phase diagram obtained by QMC for the chain (red-toned colors) and square lattice (blue-toned colors). The background corresponds to the results from the variational mean-field ansatz valid in infinite dimensions. For the chain, the exact value for $\epsilon = 0$ from self-consistent mean-field approach is included (see sm). For small $\epsilon \ll \abs{J}$, the transition is of first order, for large $\epsilon \gg \abs{J}$, the transition is of second order and shows Dicke criticality. Both regimes are separated by a multicritical point. b) Order parameter $n_{\rm ph}$ for up to 4096 (darkblue) spins for $\epsilon/\omega=0.15$ where the transition is of first order. c) Order parameter $n_{\rm ph}$ for up to 8192 (darkblue) spins for $\epsilon/\omega=0.3$ where the transition is of second order.
• Figure 4: Antiferromagnetic DIM for $Jd/\omega=0.2$. A/P: Antiferro-/Para-magnetic; S/N: Superradiant/Normal. a) Quantum phase diagram obtained by QMC for the chain (red-toned colors) and square lattice (blue-toned colors). The background shows the phase diagram derived from the variational mean-field ansatz Zhang2014. For the chain, the value for $\epsilon = 0$ from self-consistent mean-field is included (see sm). The first-order character of the transition found for $\epsilon=0$Rohn2020 persists to finite $\epsilon$. b) The order parameters depicted for $\epsilon/\omega = 0.30$ show the existence of the antiferromagnetic superradiant phase for the square lattice with up to $30\cross 30$ (darkblue) spins.
• Figure S1: Visualization of the different diagonal and off-diagonal vertices and their respective weights $w_{\mathrm{up,AF,down}}$ and $w_{\mathrm{od}}$ determined by the matrix element of the spin state ($\uparrow = \bullet$, $\downarrow = \circ$) they act upon. In general, there are four non-vanishing matrix elements per off-diagonal operator $H_{ik}^{\mathrm{od}}(\tau, \tau')$ and up to four non-vanishing matrix elements per diagonal operator $H_{ij}^{\mathrm{d}}(\tau, \tau)$.