Device description<\/p>\n
The device is fabricated in a Si-doped GaAs\/AlGaAs heterostructure grown by molecular-beam epitaxy. The 2DEG resides 110\u2009nm below the surface, with electron density 2.8\u2009\u00d7\u20091011\u2009cm\u22122 and mobility 9\u2009\u00d7\u2009105\u2009cm2\u2009V\u22121\u2009s\u22121. Metallic gates (Ti, 3\u2009nm; Au, 14\u2009nm) are deposited on the surface of the semiconductor using electron-beam lithography. All measurements are performed at a temperature of about 20\u2009mK in a 3He\/4He dilution refrigerator. The sample and measurement scheme are the same as in ref.\u200918<\/a>. A set of negative gate voltages is applied to the surface gates to deplete the 2DEG underneath and create the nanostructures, including four QDs, four QPCs and two guiding rails, which are fully depleted. These rails connect the source QDs to the detector QDs and merge in the centre to form a single 40-\u03bcm-long channel, equipped with a narrow barrier gate in the middle to tune the shape of the confining potential from a single well to a double well.<\/p>\n SAW generation<\/p>\n The SAW is generated using a double-finger IDT deposited on the surface and placed at a distance of 1.5\u2009mm from the device. The metallic fingers are fabricated using electron-beam lithography and thin-film evaporation (Ti, 3\u2009nm; Al, 27\u2009nm) on the heterostructure. The IDT consists of 111 cells with a periodicity of 1\u2009\u03bcm and a resonance frequency of 2.86\u2009GHz at low temperature. The aperture of the transducer is 50\u2009\u03bcm. To perform electron transport by SAW, a radiofrequency signal is applied on the IDT at its resonance frequency for a duration of 60\u2009ns. To have a strong SAW confinement potential, the signal is amplified to 28\u2009dB using a high-power amplifier before being injected into a coaxial line of the cryostat through a series of attenuators. The velocity of the SAW is 2,860\u2009m\u2009s\u22121.<\/p>\n Electron transfer<\/p>\n Each single-shot experiment corresponds to the transfer of one or a few electrons from the source QDs to the detector QDs using the SAW as the transport carrier. To prepare a given number N of electrons in a source QD, we use a sequence of fast voltage pulses to the channel gate and reservoir gate controlling the tunnel barriers of the QD. This sequence consists of three steps: initializing the QD, loading the electrons into the QD and preparing the QD for electron transfer. To initialize the source QD, electrons previously present in the QD are removed. Then, a given number of electrons are loaded into the QD by accessing a particular loading position in the charge-stability diagram of the QD. Finally, these electrons are trapped within the QD by switching to a holding configuration, from which they will be taken away by the SAW. At the same time, the two detector QDs are set in a configuration for which the electrons transported by the SAW will be captured with high fidelity. For more details, see Supplementary Note\u00a01<\/a>. By sensing the QPC currents of both the source and detector QDs with single-electron resolution and comparing their values before and after the electrons are transferred by the SAW, the precise number of electrons transferred to each detector can be determined. When calculating the partitioning probability, the very few events that do not conserve the total number of electrons are excluded by a post-selection routine.<\/p>\n In our experiment, the electrons are deterministically loaded into specific locations within the SAW train. The plunger gate of the QD is used to trigger the sending of the electrons into a precise minimum of the periodic SAW potential with a 30\u2009ps resolution. This precise control allows for the formation of an electron droplet containing up to five electrons, using the two source QDs of the device. To synchronize the two trigger pulses with the radiofrequency signal generating the SAW, we use two arbitrary waveform generators combined with a synchronization module. The outputs of the arbitrary waveform generators are connected to the plunger gates by means of high-bandwidth bias tees for voltage pulsing and dc biasing.<\/p>\n Electron partitioning<\/p>\n The electron droplet is partitioned at the Y-junction located at the end of the central channel, after a flight time of 14\u2009ns. By applying a voltage detuning \u0394\u2009=\u2009VU\u2009\u2212\u2009VL, in which VU and VL are the voltages applied to the side gates of the central channel, we can control the partitioning ratio between the two detectors D1 and D2. For all partitioning experiments reported here, the barrier gate voltage is set to VB\u2009=\u2009\u22121.25\u2009V to have a single central channel with a weak double-well potential profile. Careful analysis of the double-well potential and the electron number equilibration in the central channel is described in Supplementary Notes\u00a02<\/a> and 3<\/a>.<\/p>\n Statistical uncertainty<\/p>\n The error in estimating the probability pn from the counting statistics is dictated by the distribution of independent Bernoulli trials. The corresponding likelihood function of measuring exactly Nn outcomes of n particles in detector D1 out of Nrep repetitions is \\(\\left(\\begin{array}{c}{N}_{{\\rm{rep}}}\\\\ {N}_{n}\\end{array}\\right){p}_{n}^{{N}_{n}}\\)\\({(1-{p}_{n})}^{{N}_{{\\rm{rep}}}-{N}_{n}}\\). We use the mean Nn\/Nrep as the statistical estimate for pn. The boundaries of the corresponding confidence interval (CI) at confidence level 1\u2009\u2212\u2009c\u2009=\u200995% are determined from the likelihood function by inverting a one-tailed binomial test at significance level c\/2, for the lower bound and the upper bound separately. At the extreme count Nn\u2009=\u20090 (Nn\u2009=\u2009Nrep), the lower (upper) boundary is set to 0 (1) and the other boundary is computed at significance level c. These CIs of probability measurements are used in the Ising model parameter estimation. For 0.06\u2009pn\u2009Nrep\u2009=\u20093,000, the CIs are almost symmetric and approximately given by \\(\\pm 1.96\\sqrt{{p}_{n}(1-{p}_{n})\/{N}_{{\\rm{rep}}}}\\). Because the maximum width of the CI is about 0.036 (at pn\u2009=\u20090.5), a value smaller than the size of the data points, we did not represent the error bars on the graphs of probabilities.<\/p>\n Symmetrized multivariate cumulants<\/p>\n In our experiment, the observable is the number n of electrons measured at detector D1, which can be expressed as a sum of binary variables Tj. From \\(n={\\sum }_{j=1}^{N}{T}_{j}\\) and \\({T}_{j}^{2}={T}_{j}\\), we derive the general relation<\/p>\n $${m}_{k}={\\left(\\begin{array}{c}N\\\\ k\\end{array}\\right)}^{-1}\\mathop{\\sum }\\limits_{n=k}^{N}\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right){p}_{n}$$<\/p>\n \n (2)\n <\/p>\n between the probabilities pn\u2009=\u2009P(N\u2212n,n) of the full counting statistics (FCS)21<\/a> and the kth-order symmetrized multivariate moments mk defined as averages of all permutations of k distinct variables,<\/p>\n $${m}_{k}={\\left(\\begin{array}{c}N\\\\ k\\end{array}\\right)}^{-1}\\sum _{1\\le {j}_{1} <\/p>\n \n (3)\n <\/p>\n in which \\(\\left(\\begin{array}{c}N\\\\ k\\end{array}\\right)=N!\/[k!(N-k)!]\\) is the binomial coefficient. The corresponding symmetrized multivariate cumulants<\/p>\n $${\\kappa }_{k}={\\left(\\begin{array}{c}N\\\\ k\\end{array}\\right)}^{-1}\\sum _{1\\le {j}_{1} <\/p>\n \n (4)\n <\/p>\n are, in general, not uniquely determined by FCS probabilities and their calculation requires further information (such as symmetry constraints or a microscopic model).<\/p>\n For statistically equivalent particles (that is, full permutational symmetry of the multivariate probability distribution), all terms in equations (3<\/a>) and (4<\/a>) are equal, and the moments mk can be related to cumulants \u03bak through standard univariate relations51<\/a>, \\(\\text{ln}(1+{\\sum }_{k=1}^{\\infty }{m}_{k}{z}^{k}\/k!)={\\sum }_{k=1}^{\\infty }{\\kappa }_{k}{z}^{k}\/k!\\). Using an explicit formula in terms of Bell polynomials52<\/a>, we can write<\/p>\n $${\\kappa }_{k}=\\mathop{\\sum }\\limits_{j=1}^{k}(j-1)!{(-1)}^{j-1}{B}_{kj}({m}_{1},{m}_{2},\\ldots ,{m}_{k-j+1}).$$<\/p>\n \n (5)\n <\/p>\n See Supplementary Note\u00a05<\/a> for the derivation of equation (2<\/a>) and explicit formulas for \u03bak for k\u2009=\u20091\u20135. An example of correlated partitioning, in which equation (5<\/a>) is not valid and the general combinatorial expressions for multivariate cumulants39<\/a>,53<\/a> need to be used, is shown in Fig. 2e<\/a> and described in detail in Supplementary Note\u00a06<\/a>.<\/p>\n There is an important distinction between our method for extracting interaction signatures and the approach of so-called factorial cumulants considered in the context of electron transport32<\/a> and particle physics54<\/a>. The multivariate moments defined by equation (3<\/a>) can be written mk\u2009=\u2009\u27e8(n)k\u27e9\/(N)k, in which (x)k\u2009=\u2009x(x\u2009\u2212\u20091)\u2009\u00d7\u2026\u00d7\u2009(x\u2009\u2212\u2009k\u2009+\u20091) is the falling factorial and \u27e8(n)k\u27e9 is known as the factorial moment of FCS32<\/a>. The k-dependent denominator (N)k in this expression for mk makes the \u03bak distinct from the factorial cumulants; see further discussion in Supplementary Note\u00a05<\/a>.<\/p>\n Ising model on a complete graph<\/p>\n The Ising model on a complete graph is exactly solvable55<\/a> and, hence, equilibrium fluctuations at any freeze-out quench temperature T can be computed for any N. The Ising Hamiltonian of equation (1<\/a>) can be expressed as a quadratic form of the observable \\(n={\\sum }_{j=1}^{N}{T}_{j}\\),<\/p>\n $${\\mathcal{H}}=U{n}^{2}+(\\mu -NU)n+UN(N-1)\/4.$$<\/p>\n \n (6)\n <\/p>\n The corresponding exact counting statistics in a canonical ensemble is pn\u2009=\u2009cn\/Z with the partition function \\(Z={\\sum }_{n=0}^{N}{c}_{n}\\) and the statistical weights<\/p>\n $${c}_{n}=\\left(\\begin{array}{c}N\\\\ n\\end{array}\\right){e}^{-\\beta Un(n-N)-\\beta \\mu n},$$<\/p>\n \n (7)\n <\/p>\n in which \u03b2\u2009=\u20091\/kBT. Together with equations (2<\/a>) and (5<\/a>), this gives a way to calculate the exact multivariate cumulants \u03bak of all orders k\u2009\u2264\u2009N at any N.<\/p>\n To make the connection with the thermodynamic phase diagram in terms of \u03bc and T in the large-N limit, explicit analytic expressions are obtained following ref.\u200955<\/a>. We apply the lowest-order Stirling\u2019s formula \\(m!\\approx {m}^{m}{e}^{-m}\\sqrt{2\\pi m}\\) to the factorials in the binomial coefficient \\(\\left(\\begin{array}{c}N\\\\ n\\end{array}\\right)\\) in equation (7<\/a>) and perform expansion of ln(cn) near its maximum n\u2009\u2248\u2009\u27e8n\u27e9 up to quadratic order. This results in a Gaussian approximation to pn of the form<\/p>\n $${p}_{n}\\propto {e}^{-(\\beta +{\\beta }^{{\\prime} })U{(n-{\\kappa }_{1}N)}^{2}},$$<\/p>\n \n (8)\n <\/p>\n in which \u03b2\u2032\u2009=\u2009[4\u03ba1(1\u2009\u2212\u2009\u03ba1)kBTN]\u22121 and kBTN\u2009=\u2009UN\/2 is the zero-field N\u00e9el temperature for the antiferromagnetic crossover. The relation between the effective magnetic field \u03bc and the effective magnetization \u03ba1\u2009=\u2009\u27e8n\u27e9\/N in the large-N limit is given by the transcendental equation56<\/a><\/p>\n $$2{\\kappa }_{1}-1=\\tanh \\left[-\\frac{{T}_{{\\rm{N}}}}{T}\\left(2{\\kappa }_{1}-1-\\frac{1}{2}\\frac{\\mu }{{k}_{{\\rm{B}}}{T}_{{\\rm{N}}}}\\right)\\right],$$<\/p>\n \n (9)\n <\/p>\n which has only one solution for the antiferromagnetic sign of the coupling (U\u2009>\u20090).<\/p>\n To quantify the antiferromagnetic correlations in the thermodynamic limit, we choose the pair correlation function \u27eaT1T2\u27eb\u2009=\u2009\u03ba2 as the order parameter. It is obtained from the identity \u27e8n2\u27e9\u2009\u2212\u2009\u27e8n\u27e92\u2009=\u2009N\u03ba1(1\u2009\u2212\u2009\u03ba1)\u2009+\u2009N(N\u2009\u2212\u20091)\u03ba2 in the large-N limit. Treating n as a continuous variable and using the Gaussian approximation of equation (8<\/a>), this gives the leading-order behaviour of \u03ba2 at a fixed T\/TN and N\u2009\u2192\u2009\u221e,<\/p>\n $${\\kappa }_{2}N=-\\frac{4{\\kappa }_{1}^{2}{(1-{\\kappa }_{1})}^{2}}{4{\\kappa }_{1}(1-{\\kappa }_{1})+T\/{T}_{{\\rm{N}}}}.$$<\/p>\n \n (10)\n <\/p>\n A numerical solution to equation (9<\/a>) together with equation (10<\/a>) is used for the phase diagram in Fig. 4b<\/a>.<\/p>\n As there is no lattice on a full graph favouring a particular pattern of staggered magnetization, the antiferromagnetic transition here is not a second-order phase transition but a crossover. The corresponding change in free energy has a weaker divergence (logN) in the thermodynamic limit than at the ferromagnetic transition. The corresponding singular part46<\/a> of the free energy change between T\u2009=\u2009\u221e and T\u2009\u2192\u20090+ is \u03b2\u0394FU>0\u2009=\u2009(1\/2)ln(1\u2009+\u2009TN\/T).<\/p>\n On the ferromagnetic side (U\u2009\u03ba2N diverges when the temperature T approaches the Curie temperature TC\u2009=\u2009\u2212TN\u2009>\u20090 as \u03ba2N\u2009\u221d\u2009(T\u2009\u2212\u2009TC)\u22121 for \\(T\\to {T}_{{\\rm{C}}}^{+}\\). At T\u2009\u2264\u2009TC, the Gaussian approximation of equation (8<\/a>) breaks down and strong ferromagnetic order sets in. This corresponds to the droplet scattering at the Y-junction as a whole (without partitioning), with probability \u03ba1 to go to detector D1 and with \u03ba2\u2009=\u2009\u03ba1(1\u2009\u2212\u2009\u03ba1)\u2009>\u20090 in the large-N limit. For a large but finite droplet, there is no symmetry breaking, hence \u03bak>1\u2009=\u2009O(1), unlike O(N\u2212k+1) in the antiferromagnetic case.<\/p>\n Universal scaling of partitioning cumulants<\/p>\n The interaction-dominated partitioning of a large droplet at U\u2009>\u20090 is described by the antiferromagnetic phase of the effective Ising model with T\/TN\u2009\u2192\u20090 and N\u2009\u2192\u2009\u221e. The Boltzmann factor in equation (7<\/a>) suppresses the fluctuations of n around \u27e8n\u27e9\u2009=\u2009\u03ba1N and caps the large-N asymptotics of univariate cumulants from \u27eank\u27eb\u2009=\u2009O(N) (Gaussian limit of binomial distribution) to \u27eank\u27eb\u2009=\u2009O(1). From the latter condition (which is independent of the specifics of the Ising model), we derive the asymptotics \u03bak\u2009=\u2009Gk(\u03ba1)N\u2212k+1\u2009+\u2009O(N\u2212k) for k\u2009\u2265\u20092, in which the prefactor<\/p>\n $${G}_{k}({\\kappa }_{1})=-\\frac{(k-1)!}{2}{C}_{k}^{(-1\/2)}\\,(2{\\kappa }_{1}-1)$$<\/p>\n \n (11)\n <\/p>\n is universal and given by the ultraspherical (Gegenbauer) polynomials \\({C}_{k}^{(a)}\\) of degree k and parameter a\u2009=\u2009\u22121\/2. The first polynomials up to k\u2009=\u20095 are plotted in Fig. 3a<\/a>, to show the universal strong-correlation asymptotics of the scaled cumulants \u03bakNk\u22121. Note that G2\u2009=\u2009\u2212\u03ba1(1\u2009\u2212\u2009\u03ba1) is also the zero-temperature limit of equation (10<\/a>).<\/p>\n The polynomials Gk(\u03ba1) have exactly k\u2009\u2212\u20092 zeros for 0\u2009\u03ba1\u200931<\/a>. We note that a similar generic N\u2212k+1 scaling has been discussed for cumulants of initial density perturbations in heavy-ion collisions57<\/a>, in which it arises for different reasons (dominance of autocorrelations in the independent point-sources model).<\/p>\n In contrast to antiferromagnetic correlations decaying with N as \u03bak\u2009\u221d\u2009N\u2212k+1, the fluctuations in the ferromagnetic case are between n\u2009=\u20090 and n\u2009=\u2009N only, hence \u03bak\u2009=\u2009O(1), and the limiting form for the unpartitioned scattering (T\/TC\u2009\u2192\u20090 in the Ising model) is the polynomial \\({\\kappa }_{k}=-{{\\rm{Li}}}_{1-k}\\left(\\frac{{\\kappa }_{1}}{{\\kappa }_{1}-1}\\right)\\), in which Li is the polylogarithm.<\/p>\n Coulomb liquid simulations<\/p>\n We model a finite droplet of Coulomb plasma in 2D using a confining single-electron potential V1e and an unscreened Coulomb potential47<\/a>, which results in the total potential<\/p>\n $$U({{\\bf{r}}}_{1},\\ldots ,{{\\bf{r}}}_{N})=\\mathop{\\sum }\\limits_{i=1}^{N}{V}_{1e}({{\\bf{r}}}_{i})+\\sum _{i <\/p>\n \n (12)\n <\/p>\n in which r<\/b>i\u2009=\u2009(xi,\u2009yi) is the in-plane coordinate of the ith electron and \u03f5r\u2009=\u200912.1 is the relative dielectric permittivity in GaAs. The equilibrium distribution of electron coordinates is determined by a classical canonical ensemble at an effective temperature T. We sample electron positions {r<\/b>i} using a random walk Metropolis Monte Carlo algorithm designed to sample the canonical distribution. The convergence of the corresponding Markov chain is controlled by the Kolmogorov\u2013Smirnov test58<\/a>. For each set of parameters, a statistics of positions is collected with the estimated effective sample size ranging from 103 to 105 depending on parameters. The statistics of positions is translated to partitioning statistics of a sudden quench using binary variables Ti\u2009=\u2009\u0398(yi), in which \u0398 is the Heaviside step function. This corresponds to an observable n\u2009=\u2009T1\u2009+\u2026+\u2009TN counting the number of particles in the y\u2009>\u20090 half-plane.<\/p>\n The confining electrostatic potential of our experiment can be approximated as a double-well quartic-parabolic 2D potential<\/p>\n $${V}_{1e}({\\bf{r}})={V}_{{\\rm{b}}}+{\\mu }_{{\\rm{q}}}\\frac{y}{{y}_{0}}-8{V}_{{\\rm{b}}}\\frac{{y}^{2}}{{y}_{0}^{2}}+16{V}_{{\\rm{b}}}\\frac{{y}^{4}}{{y}_{0}^{4}}+\\frac{m{\\omega }_{x}^{2}{x}^{2}}{2},$$<\/p>\n \n (13)\n <\/p>\n in which Vb is the height of the central barrier, y0 is the distance between the two minima, \u03bcq is the transverse energy detuning proportional to the side-gates voltage difference \u0394\u2009\u2212\u2009\u03940 (which controls the partitioning of the droplet) and \u03c9x is the oscillation frequency in the longitudinal direction, resulting from the confinement potential of the SAW.<\/p>\n The barrier height Vb\u2009=\u200927.5\u2009meV and the distance between minima y0\u2009=\u2009220\u2009nm are estimated from an electrostatic simulation of the gate-controlled potential as explained in Supplementary Note\u00a02<\/a>. The transverse oscillation frequency in the two potential minima is then calculated as \\({\\omega }_{y}=\\sqrt{32{V}_{{\\rm{b}}}\/(m{y}_{0}^{2})}=7.0\\,{\\rm{THz}}\\) using the effective mass m\u2009=\u20090.067me for electrons in GaAs. The longitudinal oscillation frequency in the SAW potential is estimated from the peak-to-peak amplitude ASAW\u2009=\u200942\u2009meV and wavelength \u03bbSAW\u2009=\u20091\u2009\u03bcm, using the relation \u03c9x\u2009=\u2009(\u03c0\/\u03bbSAW)(2ASAW\/m)1\/2\u2009=\u20091.5\u2009THz (see Supplementary Note 4 in ref.\u200918<\/a>). The aspect ratio of the 2D confinement is thus \u03c9x\/\u03c9y\u2009=\u20090.21.<\/p>\n The potential being entirely determined, the effective electron temperature T is the only free parameter to be chosen for good agreement with the experimental data, as shown by the solid lines in Fig. 3b\u2013e<\/a> using T\u2009=\u200925\u2009K. This value is also consistent with the one extracted from the barrier-height dependence of the thermally activated hopping rate between the two wells of the quartic potential, as estimated in Supplementary Note\u00a02<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"Device description The device is fabricated in a Si-doped GaAs\/AlGaAs heterostructure grown by molecular-beam epitaxy. 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