Spectral order statistics of Gaussian random matrices: large deviations for trapped fermions and associated phase transitions

Abstract

We compute the full order statistics of a one-dimensional gas of fermions in a harmonic trap at zero temperature, including its large deviation tails. The problem amounts to computing the probability distribution of the $k$th smallest eigenvalue ${\lambda }_{(k)}$ of a large dimensional Gaussian random matrix. We find that this probability behaves for large $N$ as $\mathcal{P}[{\lambda }_{(k)}=x]\approx \exp \left(-\beta N^2\psi (k/N,x)\right)$, where $\beta$ is the Dyson index of the ensemble. The rate function $\psi (c,x)$, computed explicitly as a function of $x$ in terms of the intensive label $c=k/N$, has a quadratic behavior modulated by a weak logarithmic singularity at its minimum. This is shown to be related to phase transitions in the associated Coulomb gas problem. The connection with statistics of extreme eigenvalues of random matrices is also elucidated.

5 pages, 3 figures. Version 2: author list changed, acknowledges changed

References in corpus (44)

• Many-Body Physics with Ultracold Gases
• Theory of ultracold Fermi gases
• Level-Spacing Distributions and the Airy Kernel
• On Orthogonal and Symplectic Matrix Ensembles
• Matrix Models for Beta Ensembles
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Gaussian Fluctuation in Random Matrices
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Full counting statistics in a propagating quantum front and random matrix spectra
• Top eigenvalue of a random matrix: large deviations and third order phase transition
• The entanglement entropy of one-dimensional gases
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Statistical distribution of quantum entanglement for a random bipartite state
• Gaussian fluctuations of eigenvalues in the GUE
• Universality in the full counting statistics of trapped fermions
• Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State
• A branching random walk seen from the tip
• Large Deviations of the Maximum Eigenvalue in Wishart Random Matrices
• The Index Distribution of Gaussian Random Matrices
• Statistics at the tip of a branching random walk and the delay of traveling waves
• Distributions of Conductance and Shot Noise and Associated Phase Transitions
• Phase transitions of bipartite entanglement
• Phase transitions and edge scaling of number variance in Gaussian random matrices
• Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution
• How many eigenvalues of a Gaussian random matrix are positive?
• Large deviations of the maximal eigenvalue of random matrices
• Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices
• Universal Order Statistics of Random Walks
• The entanglement entropy of 1D systems in continuous and homogenous space
• Entanglement and particle correlations of Fermi gases in harmonic traps
• Large Deviations of the Smallest Eigenvalue of the Wishart-Laguerre Ensemble
• Equilibrium and off-equilibrium trap-size scaling in 1D ultracold bosonic gases
• Universal Order and Gap Statistics of Critical Branching Brownian Motion
• The Probability of an Eigenvalue Number Fluctuation in an Interval of a Random Matrix Spectrum
• Surface and bulk entanglement in free-fermion chains
• Near-extreme eigenvalues and the first gap of Hermitian random matrices
• Index Distribution of Cauchy Random Matrices
• Gap Probabilities for Edge Intervals in Finite Gaussian and Jacobi Unitary Matrix Ensembles
• Universal quantum behaviors of interacting fermions in 1D traps: from few particles to the trap thermodynamic limit
• Large deviations of the top eigenvalue of large Cauchy random matrices
• Local Central Limit Theorem for Determinantal Point Processes
• Order statistics of 1/f^α signals
• Instantons and Extreme Value Statistics of Random Matrices
• Decomposition of spectral density in individual eigenvalue contributions

Cited by in corpus (19)

• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Random matrices and entanglement entropy of trapped Fermi gases
• Noninteracting fermions in a trap and random matrix theory
• Multicritical edge statistics for the momenta of fermions in non-harmonic traps
• Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions
• Universal ground state properties of free fermions in a $d$-dimensional trap
• Large deviation function for the number of eigenvalues of sparse random graphs inside an interval
• Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble
• Linear statistics and pushed Coulomb gas at the edge of beta random matrices: four paths to large deviations
• Periodic Airy process and equilibrium dynamics of edge fermions in a trap
• Index statistical properties of sparse random graphs
• Universal behavior of the full particle statistics of one-dimensional Coulomb gases with an arbitrary external potential
• A smooth transition towards a Tracy-Widom distribution for the largest eigenvalue of interacting $k$-body fermionic Embedded Gaussian Ensembles
• Theory for the conditioned spectral density of non-invariant random matrices
• Replica-symmetric approach to the typical eigenvalue fluctuations of Gaussian random matrices
• Analytic approach for the number statistics of non-Hermitian random matrices
• Exact eigenvalue order statistics for the reduced density matrix of a bipartite system
• The Tracy-Widom distribution at large Dyson index
• Replica approach to the generalized Rosenzweig-Porter model