Extended tail states in an imaginary random potential

Abstract

Non-Hermitean operators may appear during the calculation of a partition function in various models of statistical mechanics. The tail eigen-states, having anomalously small real part of energy Re(ε), became naturally important in this case. We consider the distribution of such states and the form of eigenfunctions for the particle propagating in an imaginary random potential (the model motivated by the statistics of polymer chains). Unlike it is in the Hermitean quantum mechanics, our tail states are sufficiently extended. Such state appear if the values of random potential turns out to be anomalously close inside the large area. Results of numerical simulations in the case of strong coupling confirm the analytic estimates.PACS numbers: 72.15. Rn, 73.20.Jc The phenomena of Anderson localization [1] of a quantum particle in the random potential attracted a permanent interest during last 40 years. On the other hand, a considerable attention have been paid last years to the investigation of features of eigenfunctions of non-Hermitean Hamiltonians with disorder [2][3][4][5][6][7][8][9][10][11]. A popular example of this kind is the Hatano-Nelson model [6], where in the presence of the imaginary vector-potential the transition between real and complex spectrum takes place [6,7]. This transition is usually referred as the mobility edge in 1d problem (which is already surprising [12]). In this paper we consider another example, where the localized(delocalized) eigenfunctions of non-Hermitean Hamiltonian behave in the "inverted", compared to the Hermitean case, way.Two recent papers [10,11] deals with the non-Hermitean quantum mechanical Hamiltonian with an imaginary random potentialThis study was motivated by the observation that the Euclidean evolution operator r| exp[−tH]|0 with H (1) after averaging over δ-correlated disordered potential V coincides with the probability distribution Z(r, t) for the Edwards self-repulsing polymer [13]. Many applications of non-Hermitean operators (like both our examples) come from the statistical physics and naturally deal with the imaginary time(Euclidean) evolution. In its turn, for the Euclidean evolution operator the ground state contribution (contribution from the states with lowest Reε) is naturally enhanced [18]. Therefore in this paper we are going to consider the tail of the eigenfunctions of the Hamiltonian (1), having smallest real part of energy Reε. In the Hermitean quantum mechanics eigen-states with anomalously small ε (so called "Lifshitz tails" [15]) originates from the rare localized fluctuations of the disorder V and are themselves well localized. For the Hamiltonian eq. (1) first of all the energy is bounded from below Reε > 0 [16]. The most surprising is the fact that for smaller and smaller Reε in eq. (1) (closer to Reε = 0) one finds (exponentially rare) the more and more extended states. Investigation of this "delocalization at the tail" in imaginary disordered potential will be the main goal of this paper [17].In all cases the tail states appear only due t...

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