Evidence of Coulomb liquid phase in few-electron droplets

TL;DR

This work demonstrates an on-chip electron collider that forms few-electron droplets and reveals emergent collective behaviour characteristic of a strongly correlated Coulomb liquid. By harnessing SAW-driven transport through a Y-junction and performing high-order multivariate cumulant analyses up to $N=5$, the authors uncover universal, interaction-dominated partitioning signatures that align with a single collective variable. An effective Ising model on a complete graph captures the gas–liquid crossover in the droplet, supporting a thermodynamic interpretation of the observed correlations, while microscopic Coulomb-plasma simulations corroborate the finite-size scaling and temperature estimates. Overall, the study provides a controllable platform to probe confinement–interaction physics and engineered collective states in mesoscopic electronic systems.

Abstract

Emergence of universal collective behaviour from interactions within a sufficiently large group of elementary constituents is a fundamental scientific paradigm. In physics, correlations in fluctuating microscopic observables can provide key information about collective states of matter such as deconfined quark-gluon plasma in heavy-ion collisions or expanding quantum degenerate gases. Mesoscopic colliders, through shot-noise measurements, have provided smoking-gun evidence on the nature of exotic electronic excitations such as fractional charges, levitons and anyon statistics. Yet, bridging the gap between two-particle collisions and the emergence of collectivity as the number of interacting particles increases remains a challenging task at the microscopic level. Here we demonstrate all-body correlations in the partitioning of electron droplets containing up to N = 5 electrons, driven by a moving potential well through a Y-junction in a semiconductor device. Analyzing the partitioning data using high-order multivariate cumulants and finite-size scaling towards the thermodynamic limit reveals distinctive fingerprints of a strongly-correlated Coulomb liquid. These fingerprints agree well with a universal limit where the partitioning of a droplet is predicted by a single collective variable. Our electron-droplet collider provides critical insight into the interplay of confinement and interaction effects in small electron systems and highlights a new way to study engineered states of matter.

Figures (14)

• Figure 1: Partitioning of an electron droplet.a, Schematic of the experiment. An electron source (S1) delivers a few-electron droplet which is split in-flight at a Y-junction. The output of the partitioning is analysed by two single-shot detectors (D1 and D2). b, Schematic of the electron droplet transport inside the selected potential minimum of a surface acoustic wave (SAW). Electrostatic gates (yellow) are used to guide the electron droplet and create a Y-junction. c, Scanning electron microscope image of the device showing the metallic surface gates (light grey). The electron source (S1) consists of a quantum dot (shown in the top left inset) coupled to a quantum point contact (QPC) for charge sensing. The plunger gate (yellow) is employed to inject a precise number of electrons into a single SAW minimum. A second electron source (S2) is connected to the central channel to inject more electrons. The Y-junction at the end of the central channel (see bottom inset) enables partitioning of the electron droplet.
• Figure 2: Partitioning of an electron droplet containing $N=4$ electrons.a-c, Detection probabilities $P_{(N-n,\,n)}$ versus side-gate detuning voltage $\Delta$. Each data point is extracted from 3000 single-shot measurements. The labels $(N-n,\,n)$ correspond to the events where $n$ electrons are measured at detector D1 and $N-n$ electrons at detector D2. In a, the four electrons are distributed across different SAW minima, as illustrated in the top inset. In b, the four electrons are loaded into adjacent minima, with two electrons in each. In c, all four electrons are confined into a single minimum. The solid lines in a are predictions based on independently measured single electron partitioning probabilities (see Supplementary Note \ref{['supp:reconstruction']}). d-f, Multivariate cumulants $\kappa_1\ldots\kappa_N$ calculated from the measured probabilities shown in a-c. The inset in d shows the evolution of $\kappa_1$ across the entire range, and the solid line is the partitioning probability $P_{(0,1)}$ of a single electron. In e, two non-equivalent cumulants contribute to $\kappa_2$ (see Methods and Supplementary Note \ref{['supp:2e2epartitioning']}). Solid lines in e and f are fits using the Ising model of Eq. \ref{['eq:Hamiltonian']}.
• Figure 3: Scaling of correlation functions with the number of particles.a, Leading-term universal asymptotics of the repulsion-dominated cumulants in the large $N$ limit, as function of $\kappa_1$ according to Eq. \ref{['eq:ultraspherical']}. b-e, Measured cumulants $\kappa_k$ of order $k=2\ldots 5$ for droplets with $N=k\ldots 5$. Lines show corresponding simulations of sudden partitioning of an equilibrium Coulomb plasma confined in a quartic-parabolic potential with realistic microscopic parameters (see Methods).
• Figure 4: a, Interpretation of partitioning in terms of magnetic spin interactions. Uncorrelated partitioning ($U=0$, binomial distribution in the middle), bunching ($U<0$) and anti-bunching ($U>0$) correspond, respectively, to paramagnetic, ferromagnetic and antiferromagnetic phases of the Ising model on a complete graph, for which counting statistics gives the distribution of the total magnetization. b, Phase diagram of the antiferromagnetic crossover in the thermodynamic limit of the Ising model, with appropriately scaled negative pair correlations $\kappa_2 N$ as the order parameter. The axes are given by the temperature $T$ and the magnetic field $\mu$, scaled by the Néel temperature $T_{\text{N}}$. The measured correlations for $N=3$, $4$ and $5$ at $\mu=0$ are shown by colors in small squares. The horizontal position $T/T_\text{N}$ of the squares is obtained from the fits of the Ising model to the partitioning curves (Extended Data Table \ref{['extended-table1']}). The slight deviations in color between the phase diagram and the measured values (36%, 20%, and 11%, respectively) are dominated by the finite-$N$ effect, not by discrepancy with the model. c, Four configurations of the 2D confining potential in the central channel (level lines), together with snapshots of the spatial positions of $N=5$ electrons (red dots) from Monte-Carlo simulations of a classical Coulomb plasma (see Methods). The color scale shows the calculated average electron density. The four panels correspond to (from left to right) $\Delta-\Delta_0=101$, 60, 21, and 0 mV.
• Figure 1.1: Experimental data for partitioning of $N=2$ electrons.a, Partitioning probabilities when the two electrons are distributed in two different minima and are uncorrelated. b, Partitioning probabilities when both electrons are in the same SAW minimum and are interacting. Panels c and d display the multivariate cumulants corresponding to a and b, respectively. Lines in a are reconstructions using single-electron partitioning data, and lines in d are fitting curves from the Ising model using the parameters given in Extended Data Table \ref{['extended-table1']}.