"Level curvature" distribution for diffusive Aharonov-Bohm systems: Analytical results.

We calculate analytically the distributions of ”level curvatures” (LC) (the second derivatives of eigenvalues with respect to a magnetic flux) for a particle moving in a white-noise random potential. We find that the Zakrzewski-Delande conjecture [11] is still valid even if the lowest weak localization corrections are taken into account. The ratio of mean level curvature modulus to mean dissipative conductance is proved to be universal and equal to 2π in agreement with available numerical data. PACS numbers: 05.45.+b, 71.55.J Typeset using REVTEX 1 Nowadays it is considered to be a well established fact that spectral statistics of generic chaotic or disordered systems is adequately described by that typical for eigenvalues of large Gaussian random matrices (RM) [1]. The range of applicability of the ”Gaussian universality” to real disordered systems was considered in the pioneering papers by Efetov [2] and Altshuler and Shklovski [3]. It was demonstrated that statistical properties of the energy levels of a quantum particle moving in a static random potential follow RM predictions as long as effects of Anderson localization on the particle diffusion are neglegible. Another important result is due to Berry who demonstrated how the same universality may arise in globally chaotic ballistic systems ( ”quantum billiards”) [4]. Quite recently an interesting new developement in the study of spectra of disordered systems and their chaotic counterparts has been made. The problem is to study the socalled ”level response statistics”, i.e. to provide a statistical description of sensitivity of the energy levels to external perturbations of different types. This issue attracted a great deal of research interest, both analytically and numerically [5][9], [11][20]. The most frequently studied characteristics are first and second derivatives of the energy levels En(α) with respect to a tunable parameter α characterizing the strength of perturbation. Physically the role of such a parameter can be played by, e.g. an external magnetic field, the strength of a scattering potential for disordered metal, a form of confining potential for quantum billiards, or any other appropriate parameter on which the system Hamiltonian is dependent. The first derivatives vn = ∂En/∂α are frequently called the ”level velocities” (LV) (or ”level currents”), the second ones Kn = ∂ En/∂α 2 are known as ”level curvatures”. In a series of papers by the MIT group [6,8], see also [7], it was found that for a generic chaotic system whose unperturbed spectrum is well described by the universal RM statistics the set of ”level velocities” vn(α) is characterized (after appropriate rescaling) by a universal correlation function 〈vn(α)vn(α)〉 whose form is again dependent only on the symmetry of the unperturbed Hamiltonian and that of the perturbation. The range of applicability of these results to real systems is the same as before: they are valid for systems with completely ”ergodic” chaotic eigenfunctions covering randomly, 2 but uniformly all the available phase space and showing no specific internal structure. The universality is believed to be independent of ”whether the chaos originates in mesoscopic disorder or deterministic instability of the classical trajectories.” [9]. The effects of eigenfunction localization– either due to scarring [10] or due to disorder-induced quantum interference (the Anderson localization) – result in substantial modifications of the LV characteristics, see [11–13,15] Much interest was concentrated on the ”level curvature” (LC) distribution P(K). Gaspard and coworkers [14] discovered the universal asymptotic behaviour P(K) ∝ K in the large curvature limit K → ∞, the parameter β = 1, 2 or 4 depending on the symmetry universality class. This behaviour is a direct consequence of the so-called ”level repulsion” for unperturbed systems. Namely, the probability P(s) for two unperturbed energy levels to be separated by the spacing s much smaller than the mean level spacing ∆ vanishes as P(s) ∝ s, and this fact can be shown to result in the abovementioned large-curvature behaviour [16]. The analytical form of the whole distribution function P(K) was guessed by Zakrzewski and Delande [11] on the basis of numerical results and is given by the following expression: [17] P(k) = C(β)(1 + k) (1) where Cβ is a normalization constant, β = 1, 2, 4 depending on presence or absense of timereversal invariance and Kramers degeneracy and k is the dimensionless level curvature. We will refer to this expression as to the ZD conjecture. Very recently, von Oppen [18] succeeded in demonstrating that eq.(1) is indeed exact for the ensembles of large Gaussian Hermitian (β = 2) and real symmetric matrices. The validity of ZD conjecture for all three classes of large Gaussian matrices was proved in a different way by the present authors [27]. It is natural to suppose, that the status of the distribution eq.(1) within the theory of disordered and chaotic systems is the same as before: it is valid for systems with completely ergodic extended eigenstates. This was indeed found to be the case in a series of interesting numerical experiments [19,20] on quasi-1D as well 3 as 3D periodic random tight-binding models subject to the influence of Aharonov-Bohm magnetic flux, which acts as a time-reversal symmetry breaking parameter. In particular, the distribution eq.(1) was found to persist up to the Anderson localization transition [19]. Unfortunately, the methods used in [18,27] for deriving the form of LC distributions for the Gaussian ensembles were havily based upon the explicit form of the joint probability density of eigenvalues known for all these ensembles [1], which is of course an immense simplification. One therefore has to invent a different technique in order to be able to treat the curvature distribution analytically under more realistic assumptions. It turns out to be possible to find such a technique for the case of time-reversal invariant systems subject to a time-reversal symmetry breaking perturbation. This case seems to be one of the most interesting from the physical point of view as well as relevant experimentally. As a physical realisation one can imagine a disordered mesoscopic sample (e.g. cylinder or ring) pierced by magnetic flux φ. For such a system the ”typical” level curvature is expected to be related to the dimensionless conductance gc of the sample due to the famous Thouless formula: 1 ∆ ∂En/∂φ |φ=0 ∼ Ec/∆ ≡ gc. Initially suggested by Thouless [21], this relation attracted renewed interest recently. Its meaning was reconsidered in a broader context by Akkermans and Montambaux [22], see also a quite detailed discussion in [16,20]. All these facts make the consideration of such systems to be of special interest. A specific feature allowing us to treat the particular case of weak time-reversal symmetry breaking perturbation analytically is vanishing of the first derivatives ∂En/∂φ|φ=0 on reasons of symmetry. This fact allows one to represent the curvature distribution in terms of a product of advanced and retarded Green functions, the average over the disorder being performed nonperturbatively with help of Efetov’s supersymmetry approach [2]. The method provides a unique possibility to derive the level curvature distribution starting from a genuine microscopic Hamiltonian of a quantum particle experiencing elastic scattering: H(φ) = 1 2m ( p − e c A(φ) )2 + U(r) (2) with U(r) being a white noise random potential and A(φ) standing for the vector potential 4 corresponding to the magnetic flux φ. This fact allows one to try to take into account the weak localization effects due to the finite ratio of the Thouless energy Ec to the mean level spacing ∆. The general way of doing this was developed recently by Kravtsov and Mirlin [23]. Exploiting this method one can find that the curvature distribution eq.(1) preserves its form to the first order in ∆/Ec ≪ 1, the width 〈|K|〉 being renormalized. To begin with, we introduce the resolvent operator Ĝ(α; ǫ) = [E −H(φ)± iǫ] where α = φ/φ0, with φ0 = 2πc/e being the flux quanta. Let us now consider the following correlation function: K(u) = limǫ→0 ǫTrĜ (α = 2 √ ǫ/u; ǫ)TrĜ(α = 0; ǫ) ≡ limǫ→0 ∑N n,m=1 ǫ [E−En(α=0)−iǫ][E−Em(α=2 √ ǫ/u)+iǫ] (3) Here we used the expression for the trace of the resolvent TrĜ in terms of eigenvalues of the Hamiltonian Ĥ(φ). It is important to note that the limiting procedure ǫ → 0 in eq. (3) is performed prior to the thermodynamic limit V → ∞, where V is the system volume. Therefore, the parameter ǫ (that plays a role of effective ”level broadening” necessary to regularize the resolvent operator) can be considered as small in comparison with the mean level spacing ∆ ∝ 1/V . It was already mentioned that the probability of having two levels at a distance s ≪ ∆ is vanishingly small due to level repulsion. Taking this fact into account one finds that the only terms that survive the limiting procedure ǫ → 0 are those with coinciding indices m = n. Remembering ∂En/∂φ|φ=0, one obtains: K(u) = π ∑ n u + iuKn u2 +K2 n δ(E − En); Kn = ∂En ∂α2 |α=0 (4) Let us perform formally the averaging over the disorder and introduce the function P(K) =