NON-MARKOVIAN DIFFUSION OF A QUANTUM PARTICLE IN A FLUCTUATING MEDIUM

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1 JOURNAL DE PHYSIQUE Colloque C7, suppl6ment au nolo, Tome 46, octobre 1985 page C7-35 NON-MARKOVIAN DIFFUSION OF A QUANTUM PARTICLE IN A FLUCTUATING MEDIUM P. Reineker and K. Kassner Abtei lung Theoretische Physik, Universittit illm, Ulm, F. R. G. Abstract - A model for diffusion of a quantum particle on a fluctuating lattice is considered. The Hamiltonian contains a coherent transfer matrix element between nearest neighbors and local energy fluctuations described by a dichotomic Markov process with coloured noise. The diffusion constant is calculated up to the fourth order in J. The result is compared to recent findings of Kitahara-Haus and Inaba. Anderson localization emerges in the l i m i t of static fluctuations. The dynamics of electronic excitations interacting with vibrations is of importance in various fields of condensed matter physics Examples are the investigation of optical and spin resonance line shapes, of relaxation phenomena or of charge and energy transport phenomena. Because the full quantum mechanical problem is difficult to treat, its Hamiltonian is often replaced by a stochastic process /4,5/ modeling the vibrations. In the Haken-Strobl model for the coupled coherent and incoherent exciton motion local and non-local fluctuations are allowed for in the stochastic part and described by a Gaussian 6 correlated (white noise) Markov process. With this Hamiltonian the diffusion constant /7,9,10/ and the time dependence of the mean square'displacement of the particle have been derived The same results have been obtained subsequently in The influence of exponentially decaying correlation functions (coloured noise) has been investigated in 1171 by expanding the diffusion constant in powers of the correlation time. In a recent approach the stochastic part was represented by a dichotomic Markov process 1181 with exponentially decaying correlation functions. Within this model an approximate analytical expression for the diffusion tensor was obtained by considering only diagonal elements and their nearest neighbors in the density matrix 1191, which may be justified in the case of strong fluctuations with rapidly decaying correlation functions. In a following paper 101 the diffusion constant was calculated numerically using the dynamical coherent potential method. In this letter we use the same model, but the method of solution is quite different from those used in /19,0/ because we use an expansion in powers of the coherent part of the Hamiltonian. The mean square displacement is finally described by a continued fraction the convergence behaviour of which is analyzed with respect to the amplitude of the fluctuations. Anderson localization is obtained in the l i m i t of static fluctuations. Article published online by EDP Sciences and available at

2 C7-36 JOURNAL DE PHYSIQUE The Hamiltonian of the model H = H. + Hl(t) is given by In> describes a state localized at site n. In the coherent part H. of the Hamilto- nran J describes the particle transfer between neighboring sites; n runs over all sites, over nearest neighbors only. hn(t) in the stochastic part Hl(t) is a clas- sical fluctuating quantity with <h (t)>z 0 (angular brackets denote stochastic averaging) and correlation functions " with tl t t3,t4,..., t is characteristic 1181 for a dichotomic P A describes the strength of the fluctuations and y the decay rate. Markov process. Using the disentanglement theorem from the equation of motion for the density operator of the particle 6 = -i[~,p] equations determining the Laplace transform of the - mean square displacement <R (S)> are derived (for details see 11,1). Explicitly we obtain ^ <R(s)>=J/s C where the matrix K ^ ^ a$'% (S) aa' afar -- A (S) is determined from {S + ;(S)} w(s) = 1. (5) This equation has the structure of equatins determining Green's functions with playing the role of a self-energy. After z has been obtained up to second order in J, (5) has been solved analytically 11 for a linear chain. The diffusion constant, correct to fourth order in J, is calculated from 1 - D = lim s <R (S)> s-to and represented in Fig.. For more complicated lattices, however, it is more convenient to derive recurrence relations for the Fean square displacement start!ng from (5) (with the second-order approximation for :(S) ) and the definition of <R'(s)>. To that end it is useful to define (all the sums run over nearest neighbors) 0 (6) - It is obvious that <R (S)> = Ro(s). We furthermore define lattice sums 5, = C 6(0,a +a a ) which describe the number of ways of retur n al,a,...,an ning to the origin after n nearest neighbor steps and obtain (details of the derivation will be published in // )

3 Subtracting two consecutive equations we have unxrn - (x-on4j) - 4' = 0 with an = (n+l) ~,+~/(n+3) The functions rp(j,s), x(j,s) and $(J,s) have been calculated in 11. For the following calculation of the diffusion constant, however, we need and give them below for S = 0 only. At the moment we have tg know that x(j,s) contains terms independent of J whereas $(J,s) is proportional to J and thus a small quantity for small J. The system of equations (9) is solved by a continued fraction 131. Using (8) for n=o, the fact that gl=z (number of nearest neighbors), the con- nection between Ro(s) and the mean square displacement, and (6) the diffusion constant is given by with y-3 qo = l i m q(j,s) = J l' s+o xo = lim x(j,s) = l' + zq 0 C - ($ A +73y+4a 7 Y } s+o 1 where r-' = (y/a+y-l). To evaluate (10) in N-th order approximation we have to know ol...an for the lattice under consideration which are determined by the c~...<~+~. It is easy to give combinatorial formulae for these lattice sums. For a linear chain Zn we have ( = ( ) and for a square lattice C = (:). In the case of a simple cubic or a bodyncenteped cubic lattice the an may Be calculated up to N = 30 within some seconds of CPU time on a minicomputer. For a face centered cubic lattice, however, one has to evaluate 8 nested sums which makes the calculation rather time consuming. Truncating the continued fractions by neglecting the 4J0 term in the second denominator of (10) gives This result may be compared with expressions (1,13) by Kitahara and Haus 1171 and by Inaba 1191, respectively, 8. after expanding all res Its up to fourth order in J. For a linear chain (z=) the terms proportional to J In (11,13) are identical, whereas in (1) a term - y-l is missing. In the expression proportional to J~ both in (1) and in (13) terms are missing. For z > only (11) and (13) can be compared. The expressions do not agree because Inaba's result is proportional to z whereas (9) contains a contribution proportional to z. The reason for this difference is that in Inaba's treatment all the memory of jumps of the diffusing particle beyond nearest neighbors ist lost, whereas our result contains memory effects of closed paths consisting of four jumps (to a nearest neighbor each).

4 JOURNAL DE PHYSIQUE Fig. 1 - D/(Ja ) as a function of y/j for several values of A/J for a linear chain (z = ).. Kitahara- Haus /15/; -----: Inaba /17/; -: eq. (9) of this paper. Single points: solution of the stochastic Schrodinger equation /17/. Fig. - Several approximants of the continued fraction (8) for the diffusion constant as compared to the solution /0/ of (,4) for a linear chain. For A/J = 1 this solution does not exist over the whole range of y/~.... approximation (9). Only the third and fifth approximants are plotted for A/> > 1. Fig. 1 illustrates the difference between the three results in the case of a linear chain. It shows the reduced diffusion constant D/(a J) corresponding to (11-13) as a function of the reduced switching rate y/j for several values of A/J. The figure shows that approximation (1) becomes rather poor for small values of y/j and that the other approximations agree pretty well for large values of A/J. The single points together with characteristic error bars represent solutions of the stochastic Schrodinger equation obtained by Inaba 101. Fig. shows (11) and additional approximants of the continued fraction (10) together with the solution obtained from (5). The figure shows that for A/J = 1 the latter solution does not exist over the whole range of y/j. In the same interval the continued fraction diverges as is seen from the behaviour of the plotted approximants. The figure shows also that for A/J 1.5 there is convergence over the whole range of y/j and that already the fifth approximant gives a rather good description. A rather crude estimate 11 in order to investigate the convergence behaviour of the continued fraction (10) shows that it converges for A ~ / > J z(z+l) ~ independent of the switching rate y/j. For a linear chain (z=) we expect convergence for A/J > V@.4. From Fig. we see that there is perfect convergence even for A/J 1.5. l I In the case of convergence we can consider the limit y/j+o, i.e. we approach the case of a lattice with static random potential fluctuations (Anderson problem/4/). The figures show that the diffusion constant vanishes for y/j -t 0. This means that in the case of static fluctuations the particle is immobile, i.e. we obtain Anderson localization /4/. To the knowledge of the authors this is the first proof of Anderson localization starting from a time dependent stochastic model with coloured noise. For large enough values of A/J the continued fraction result (10) for the diffusion constant converges also for higher dimensional lattices and shows Anderson localization for y/j + 0. These investigations are represented in detail in //.

5 REFERENCES Silbey, R., Ann. Rev. Phys. Chem. 7 (1976) 03 - Kenkre, V.M., Springer Tracts in Modern Physics 9 (198) 1 Reineker, P,, Springer Tracts in Modern Physics 3 (198) 111 Anderson, P.W., J. Phys. Soc. 9 (1954) 316 Kubo, R., and Tornita, K., J. Phys. Soc. Jap. - 9 (1954) 888 Haken, H., and Strobl, G., in: The Triplet State, ed. by Zahlan, A., (Cambridge, England, 1967) Haken, H., and Reineker, P., Z. Physik 49 (197) 53 Haken, H., and Strobl, G., Z. Physik 6 (1973) 135 Reineker, P,, and Kuhne, R., Z. Physik B (1975) Kuhne, R., and Reineker, P.,. Physik B (1975) 01 Schwarzer, E., and Haken, H., Phys. Lett. 4A (197) Reineker, P., Phys. Lett. %A (1973) 385 Reineker, P.,. Physik 61 (1973) 187 Ovchinnikov, A.A., and Erikrnan, N.S., Sov. Phys. JETP 40 (1975) 733 Madhukar, A., and Post, W., Phys. Rev. Lett (1977) 144; - 40 (1978) 70 Rips, I.B., Theor. Math, Phys. 40 (1979) 74 - Kitahara, K., and Haus, J.W.,. Physik B3 (1979) 419 Bourett, R.C., Frisch, U., and Pouquet, A., Physica 65 (1973) Inaba, Y., J. Phys. Soc. Jap. (1981) 473 Inaba, Y., J. Phys. Soc. Jap. 5 (1983) 3144 Kassner, K., and Reineker, P., Z. Physik B59 (1985) 357 Kassner, K., and Reineker, P., Z. Physik BE,in print Wall,H.S., Continued Fractions (Chelsea, New York 1973) p. 17 Anderson, P.W., Phys. Rev. 109 (1958) 149

Coherent and incoherent motion in a one-dimensional lattice

PHYSICAL REVIEW B VOLUME 62, UMBER 6 Coherent and incoherent motion in a one-dimensional lattice AUGUST 2-II M. A. Palenberg and R. J. Silbey Department of Chemistry and Center for Materials Science and