Quantum Monte Carlo study of systems interacting via long-range interactions mediated by a cavity

TL;DR

This work addresses ground-state properties of a one-dimensional continuum quantum gas with cavity-mediated infinite-range interactions, breaking Galilean invariance and introducing a periodic long-range potential. Starting from an exact two-body solution, the authors develop a position-dependent Jastrow-like trial wavefunction and apply Variational Monte Carlo and Diffusion Monte Carlo to bosonic and fermionic systems, including combinations with short-range contact interactions. They identify three regimes—delocalized gas, mesoscopic gas with frustration, and delocalized bound state—along with a qualitative phase diagram that shows how short-range repulsion delays collapse and stabilizes the gas in certain limits. The study provides nonperturbative predictions for density profiles, correlation functions, and superfluid response, offering a platform for understanding cavity-QED–driven many-body physics in continuum geometries and guiding future higher-dimensional extensions and experimental realizations.

Abstract

We study one-dimensional quantum gases in continuous space with cavity-mediated infinite-range interactions using variational and diffusion Monte Carlo methods. Starting from the exact two-body solution, we construct a non-translationally invariant Jastrow wavefunction that accurately captures the spatial structure induced by the cavity field and provides an efficient many-body ansatz for both bosonic and fermionic systems. We analize properties of three characteristic quantum systems, subject to long-range interactions: (i) ideal Bose gas (ii) interacting Bose gas (iii) ideal Fermi gas. In the absence of short-range interactions, we identify a crossover from a stable, weakly modulated phase realized for repulsive interactions to a delocalized bound state for attractive interactions, marked by clustering, loss of superfluidity, and the absence of a thermodynamic limit. Introducing short-range repulsion, either through contact interactions or fermionic statistics, leads to the formation of a mesoscopic gas-like regime that disappears in the thermodynamic limit. A qualitative phase diagram is proposed to illustrate the combined effects of short- and long-range interactions, highlighting the emergence of distinct regimes with characteristic structural properties.

Figures (7)

• Figure 1: Long-range interaction potential and two-body wavefunctions. Panels (a) and (d): cavity-mediated long-range interaction given by the second term of Eq. \ref{['eq:1D_hamiltonian_dimensionless']} for the repulsive and attractive cases, respectively. Panels (b) and (c): ground-state two-body wavefunction $\psi(x_1, x_2)$ for repulsive interactions, obtained via imaginary time evolution, at interaction strengths $V_0 = 10\varepsilon$ and $V_0 = 100\varepsilon$. Panels (e) and (f): ground-state two-body wavefunctions for the corresponding attractive case.
• Figure 2: Density profiles $n(x)$ for various number of particles $N = 2, \dots, 10$ obtained by Metropolis sampling, and the corresponding superfluid fraction estimated by the Leggett bound \ref{['eq:legget_bound']}. The top row, panels (a) and (b), shows the density profiles $n(x)$, while bottom row, panels (c) and (d), displays the superfluid fraction $\rho_s/\rho$. The amplitude of $n(x)$ increases with the number of particles $N$. Larger amplitude corresponds to systems with more particles, both for repulsive ($V_0 > 0$) and attractive ($V_0 < 0$) interactions. Panels (a) and (c): repulsive interactions with $V_0 = 10\varepsilon$. Panels (b) and (d): attractive interactions with $V_0 = -10\varepsilon$. All simulations are performed by fixing the length of the system to $L = 2\lambda$ and changing the density as $\rho=N/L$.
• Figure 3: Ground-state energy of bosons interacting via cavity-mediated long-range potential. Symbols show VMC results. Panel (a): energy per particle $E/N$ as a function of interaction strength $V_0$ for various particle numbers $N = \left\{2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 20, 25, 30, 35, 40\right\}$. In the repulsive case ($V_0 > 0$), an extrapolated estimate of the thermodynamic limit is shown as a dashed maroon line. In contrast, the attractive case ($V_0 < 0$) leads to a divergent energy. Panel (b): Absolute value of the energy, $|E|$, for the specific case of $|V_0|=10\varepsilon$ as a function of the number of particles, $N$, for both interaction signs, compared to the asymptotic scalings law: Eq. \ref{['eq:scaling_attractive']} for the attractive case, and $E_{th} N$ for the repulsive one. All simulations are performed at constant density $\rho = 1 \lambda^{-1}$ and changing the length $L=N/\rho$.
• Figure 4: Density profiles in the presence of short-range correlations for particle numbers $N=2,\dots, 10$. Panel (a): fermionic system with repulsive long-range interaction ($V_0 > 0$). Panel (b): fermionic system with attractive long-range interaction ($V_0 < 0$). Panel (c): bosonic system with contact interaction $\gamma > 0$ and repulsive long-range interaction. Panel (d): bosonic system with contact interaction and attractive long-range interaction. All simulations are performed by fixing the length of the system $L = \lambda$ and changing the density $\rho=N/L$.
• Figure 5: Results for a spinless fermionic system with $N = 5$ particles and attractive cavity-mediated interactions. Panel (a): density profile $n(x)$ for varying $V_0$, exhibiting a shift at weak coupling and recovering the bosonic periodic structure at strong coupling Panels (b) and (c): two-body correlation function $g^{(2)}(x_1, x_2)$ for $V_0 = -10.0\,\varepsilon$ and $V_0 = -30.0\,\varepsilon$, respectively. Panel (d): energy per particle $E/N$ as a function of $V_0$, showing a zero crossing. Panel (e): superfluid fraction obtained from the Leggett bound, Eq. \ref{['eq:legget_bound']}, indicating a crossover in the superfluid response. All simulations are performed by fixing the system size $L = 2\lambda$ and changing the density $\rho=N/L$.