Fidelity susceptibility in Gaussian random ensembles

Abstract

The fidelity susceptibility measures sensitivity of eigenstates to a change of an external parameter. It has been fruitfully used to pin down quantum phase transitions when applied to ground states (with extensions to thermal states). Here we propose to use the fidelity susceptibility as a useful dimensionless measure for complex quantum systems. We find analytically the fidelity susceptibility distributions for Gaussian orthogonal and unitary universality classes for arbitrary system size. The results are verified by a comparison with numerical data.The discovery of many body localization (MBL) phenomenon resulting in non-ergodicity of the dynamics in many body systems [1] restored also the interest in purely ergodic phenomena modeled by Gaussian random ensembles (GRE) [2] and in possible measures to characterize them. The gap ratio between adjacent level spacings [3] was introduced precisely for that purpose as it does not involve the so called unfolding [4] necessary for meaningful studies of level spacing distributions and yet often leading to spurious results [5]. Still, the level spacing distribution belongs to the most popular statistical measures used for single particle quantum chaos studies [6][7][8][9] and also in the transition to MBL [10-13]. A particular place among different measures was taken by those characterizing level dynamics for a Hamiltonian H(λ) dependent on some parameter λ. In Pechukas-Yukawa formulation [14,15] energy levels are positions of a fictitious gas particles, derivatives with respect to the fictitious time λ are velocities (level slopes), the second derivatives describe curvatures of the levels (accelerations). Simons and Altschuler [16] put forward a proposition that the variance of velocities distribution is an important parameter characterizing universality of level dynamics. This led to predictions for distributions of avoided crossings [17] and, importantly, curvature distributions postulated first on the basis of numerical data for GRE [18] and then derived analytically via supersymmetric method by von Oppen [19,20] (for alternative techniques see [21,22]). Curvature distributions were recently addressed in MBL studies [23,24].Apart from quantum chaos studies in the eighties and nineties of the last millennium, another "level dynamics" tool has been introduced in the quantum information area, i.e. the fidelity [25]. It compares two close (possibly mixed) quantum states. If these states are dependent on a parameter λ it is customary to introduce a fidelity susceptibility χ. For sufficiently small λ, in a finite system, one hasFidelity susceptibility is directly related to the quantum Fisher information (QFI), G, being directly proportional to the Bures distance between density matrices at slightly differing values of λ [26,27], with G(λ) = 4χ. Fidelity susceptibility emerged as a useful tool to study quantum phase transitions as at the transition point the ground state changes rapidly leading to the enhancement of χ [27][28][29][30][31][32][33][34]. All of these studies w...

Energy level dynamics across the many-body localization transition

“…Then the same distribution holds both for perturbations within time reversal invariant system class as well as for an infinitesimal Aharonov-Bohm flux breaking that symmetry. In a similar way, the fidelity susceptibility distribution in the ergodic regime is faithfully reproduced by the recently found analytic prediction for GOE [64] with small deviations observed for the Aharonov-Bohm flux case.…”

“…The corresponding data are shown in Fig. 3 and compared with the GOE prediction [64]. The GOE prediction (11) seems to be a bad choice at the first glance but after rescaling the universal prediction, (12), represents well the numerical data for the largest considered system size, L = 16.…”

“…fore, even more interesting to inspect the numerical data for this measure for different perturbation schemes. Let us recall that there exists an analytic prediction for the fidelity susceptibility, χ, distribution, P (χ) as a function of x = χ/N for GOE and GUE [64] with N being the matrix rank. Fidelity susceptibility distribution for GOE matrices in N → ∞ limit reads…”

“…The very definition of the fidelity susceptibility, via a Taylor expansion F = 1− 1 2 λ 2 χ 2 shows that a transformation to λ 1 rescales fidelity susceptibility by a factor of a. The analytic prediction derived in [64] assumes the same density of states both for the original Hamiltonian H 0 and its perturbation V . Since we do not intend to estimate the energy scale of the perturbation we add a scaling factor (playing the role of an effective width) defining y = xγ with…”

“…We shall consider an entire fidelity susceptibility distribution for all quantum states of the system, also for generic random matrix representations of Hamiltonians. Recently, analytic predictions for fidelity susceptibility distribution for GOE/GUE dynamics have been derived analytically [64]. We shall consider how this distribution is affected when entering then MBL regime.…”