Edge fluctuations and third-order phase transition in harmonically confined long-range systems

Abstract

We study the distribution of the position of the rightmost particle $x_{\max }$ in a $N$-particle Riesz gas in one dimension confined in a harmonic trap. The particles interact via long-range repulsive potential, of the form $r^{-k}$ with $-2<k<\infty$ where $r$ is the inter-particle distance. In equilibrium at temperature $O(1)$, the gas settles on a finite length scale $L_N$ that depends on $N$ and $k$. We numerically observe that the typical fluctuation of $y_{\max }=x_{\max }/L_N$ around its mean is of $O(N^{-{\eta }_k})$. Over this length scale, the distribution of the typical fluctuations has a $N$ independent scaling form. We show that the exponent ${\eta }_k$ obtained from the Hessian theory predicts the scale of typical fluctuations remarkably well. The distribution of atypical fluctuations to the left and right of the mean $⟨y_{\max }⟩$ are governed by the left and right large deviation functions, respectively. We compute these large deviation functions explicitly $\forall k>-2$. We also find that these large deviation functions describe a pulled to pushed type phase transition as observed in Dyson's log-gas ($k\to 0$) and $1d$ one component plasma ($k=-1$). Remarkably, we find that the phase transition remains $3^{\mathrm{rd}}$ order for the entire regime. Our results demonstrate the striking universality of the $3^{\mathrm{rd}}$ order transition even in models that fall outside the paradigm of Coulomb systems and the random matrix theory. We numerically verify our analytical expressions of the large deviation functions via Monte Carlo simulation using an importance sampling algorithm.

42 pages, 13 figures

References in corpus (14)

• Theory of ultracold Fermi gases
• A Quantum Gas Microscope for Fermionic Atoms
• Physics and Mathematics of Calogero particles
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Distributions of Conductance and Shot Noise and Associated Phase Transitions
• Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution
• A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix
• Harmonically confined long-ranged interacting gas in the presence of a hard wall
• Joint statistics of quantum transport in chaotic cavities
• Fractional-order phase transition of charged AdS black holes
• Truncated linear statistics in the one dimensional one-component plasma

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• Dynamically emergent correlations in bosons via quantum resetting
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• Finite temperature equilibrium density profiles of integrable systems in confining potentials
• An exact formula for the variance of linear statistics in the one-dimensional jellium mode
• Current fluctuations in the Dyson Gas
• Full counting statistics of 1d short-range Riesz gases in confinement
• Crossover in densities of confined particles with finite range of interaction
• Equilibrium measures for higher dimensional rotationally symmetric Riesz gases