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 kth smallest eigenvalue λ (k) of a large dimensional Gaussian random matrix. We find that this probability behaves for large N as, where β is the Dyson index of the ensemble. The rate function ψ(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.Introduction -Recent spectacular progresses in the fabrication of devices for the manipulation of cold atoms [1,2] have given a formidable impulse towards the theoretical understanding of the behavior of quantum many-body systems using tools and techniques borrowed from classical statistical physics. Perhaps the simplest conceivable setting is a system of fermions confined by optical laser traps into a limited region of space [2][3][4][5][6][7][8]. In this experimentally rather common setup, there is a well-known connection between the ground state (T = 0) many-body wavefunction and the statistics of eigenvalues of a certain class of random matrices. More precisely, in presence of a harmonic potential U (x) =

Systematic time expansion for the Kardar–Parisi–Zhang equation, linear statistics of the GUE at the edge and trapped fermions

“…With no boundary term using that g(0) = 0 as long as √ xg(σe −x ) → 0 for x → +∞. We precisely obtain (50).…”

Large Deviation Function for the Number of Eigenvalues of Sparse Random Graphs Inside an Interval

“…(9) and (10) (12) and (13) in Eqs. (15) and (16), we obtain self-consistent equations for ω p (u) and ω a (u, v), whose solutions depend on y. As the y-dependent factors play the role of reweighting terms in Eqs.…”

“…As the y-dependent factors play the role of reweighting terms in Eqs. (15) and (16), these are solved numerically by a weighted population dynamics algorithm, discussed in [48]. The subsequent numerical solutions are used to evaluate the CGF's of Eqs.…”

Periodic Airy process and equilibrium dynamics of edge fermions in a trap

“…where G(x, τ +1 |y, τ ) is defined in (32). The transition probability of the Markov process defined in [44] (in their notations, and using superscripts for different times) Prob(X (1) , .…”

“…Another interesting application of this process, which we will develop extensively in this paper, concerns the quantum mechanical problem of N non-interacting fermions in a harmonic trap at zero temperature. In a recent series of papers we have studied the connection between the position of the trapped fermions x j and the eigenvalues of the GUE random matrix [25][26][27][28][29][30] (see also [31,32]). Indeed the quantum JPDF for the positions of the fermions in the many-body ground state is identical to the JPDF of the GUE eigenvalues, given in (5).…”