Efficient Methods for Solving the Time-Dependent Schrödinger Equation de Raedt, Hans

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1 University of Groningen Efficient Methods for Solving the Time-Dependent Schrödinger Equation de Raedt, Hans Published in: Default journal DOI: / /3/2/002 IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 1987 ink to publication in University of Groningen/UMCG research database Citation for published version (APA): Raedt, H. D. (1987). Efficient Methods for Solving the Time-Dependent Schrödinger Equation: Application to Anderson ocalization. Default journal. DOI: / /3/2/002 Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date:

2 EUROPHYSICS ETTERS 15 January 1987 Europhys. ett., 3 (2), pp (1987) Efficient Methods for Solving the Time-Dependent Schrödinger Equation: Application to Anderson ocalization. H. DE RAEDT Physics Department, University of Antwerp Universiteitsplein 1, B-261O Wilrijk, Belgium (*) Natuurkundig aboratorium, University of Amsterdam Valckenierstraat 65, 1018XE Amsterdam, The Netherlands (received 31 July 1986; accepted in final form 22 September 1986) PACS Quantum theory; quantum mechanics. PACS General theories and computational techniques. PACS J - ocalization in disordered structures. Abstract. - Algorithms for solving the time-dependent Schrödinger equation, based on symmetrized versions of generalized Trotter-Suzuki formulae, are proposed. It is demonstrated that these stable, explicit algorithms are more accurate and efficient than the popular Cranck- Nicholson method. They are used to study the dynamic of the Anderson model for localization. Numerical integration of the time-dependent Schrödinger equation (TDSE) ψ(t)/ t = = - ihψ(t), where H is the Hamiltonian and ψ(t) is a normalized complex-valued wave function (units are such that ħ = 1), is usually accomplished by means of the Crank-Nicholson (CN) method [1]. Appealing features of the CN method are that the norm of the wave function is preserved during the integration process, the energy is constant and, most importantly, it is unconditionally stable [1,2]. For one time step τ, the error of the CN methods is proportional to τ3. A serious drawback of the CN method is that each time step requires the inversion of a matrix [1,2]. Consequently, explicit and stable methods of at least the same order of accuracy should be preferred over the CN method. It is the aim of this paper to introduce a new class of explicit, stable and efficient algorithms for solving the TDSE. For simplicity of notation the algorithm for the one-dimensional system will be developed first. It is then used as a building block to construct algorithms for solving twoand three-dimensional problems. The motion of a particle moving on a chain of lattice sites is described by the Hamiltonian H = V Σ (cl Cl+l + c l+l Cl) + W Σ εlnl, (1) l=l 1=1 where c l(cl) creates (annihilates) a particle at the site l, nl cl Cl counts the number of particles at site l (i.e. nl = 0, 1), V (V = 1 in the following) sets the kinetic-energy scale and (*) Permanent address.

3 140 EUROPHYSICS ETTERS WεI is the potential at site l felt by the particle. Since there is only one particle, c land Cl may be chosen to obey fermion operator algebra. To specify lattice model (1) completely, a particular choice for the εl has to be made. Taking εl random with uniform distribution over the interval [-½,½] yields the d-dimensional Anderson model for localization [3], an electron moving in a random potential. In this letter, the algorithms to be proposed will be tested on this model. It will be demonstrated that the technique can be used to study the time-dependent properties of the Anderson model for much longer time periods than previously possible. Obviously, the usual discretized form of the TDSE of a particle moving in continuum space is identical to the difference equation that follows from (1), provided proper identification of V and W in terms of particle mass, mesh length, etc. is made. The advantage of working with lattice models such as (1) is that all efforts can be directed toward the solution of the time integration problem. Problems resulting from approximating derivatives by finite differences are not present. The first step in the development of a method for solving the TDSE is to make an approximation for the time evolution operator exp [- iτh]. The standard approach is to select some rational approximation R(x) to the function exp [x] and replace x by - iτh [2]. The Crank-Nicholson method corresponds to R(x) = (1+ x/2)(1- X/2)-1[2]. The generalized Trotter formulae introduced by SUZUKI[4, 5] provide a convenient framework for constructing systematic approximations to exponential operators that do not suffer from the same deficiencies [1, 2] as rational approximations. The simplest of these product formulae have proven to be useful in performing Monte Carlo simulations of quantum statistical systems [6]. A symmetrized version [7] of one of Suzuki's formulae [4, 5] will be the starting point for development of algorithms for solving the TDSE. The exponential operator exp [X + Y] is approximated by the product of exponential operators exp [X + Y] E(4)(X, Y) exp [X/2] exp [Y/2] exp [C(X, Y) exp [Y/2] exp[x/2], (2) where C(X, Y) [X + 2Y, [X, Y]]/24. For H = A + B, the r.m.s.-error on the wave function at time t = mτ is bounded by ψexact(t) - ψ(4)(t) C(4)tτ4, where ψexact(t) = = exp [- imτh] ψ(t = 0), ψ(4)(t)= (E(4)(- iτa, - iτb))mψ(t = 0) and c(4) is a finite constant. From this result and unitarity of E(4)(- iτa, - iτb) it follows directly that algorithms based upon (2) are unconditionally stable [2]. The superscript «4» indicates that employing (2) will result in algorithms that are correct to fourth order in the time step τ. A scheme, correct to second-order in τ, is obtained from (2) by putting C(X, Y) = O. The usefulness of the secondorder formula for solving parabolic partial differential equations such as the TSDE has long been recognized in electron-diffraction calculations [8]. Since there is no restriction on the choice of A or B except that H = A + B, this flexibility can be exploited to considerable extent. The most obvious choice is to put H = K + U, where K = V Σ (c l Cl+l + h.c.) is the kinetic and U = W Σ εl nl, the potential energy. For arbitrary 1=1 1=1 lattice dimensionality, the expression for C(- iτk, - iτu) is most readily obtained by first transforming to Fourier space, working out the commutators and transforming back to real space. The results for the one-dimensional case read [K, [K, UJ] = WV2 Σ (εl+2+ εl- 2εl+l)(c! Cl+2 + h.c.) + 2WV 2 Σ (2εl- εl+l - εl-l)nl (3a) and from which C(- iτk, 1=1 1=1 [U, [K, UJ] = - VW2 Σ (εl - εl+1)2(c l Cl+l + h.c.), (3b) 1=1 - iτu) follows immediately.

4 H. DE RAEDT: NEW METHOD FOR SOVING THE SCHRÖDINGER EQUATION 141 At first sight the complexity of these expressions is not encouraging. However, noting that each term appearing in exp [C( - iτk, - iτu)] is proportional to τ3 and that for λ real[5] exp[ia(al+a Ap)]-exp[iλA1]exp[iλA2]... exp[iλap]ii A 2 p Σ [Ar,As], it r,s=l is allowed to replace exp [C( - iτk, - iτu)] by any product of exponential operators, p provided C(- iτk, - iτu) = Σ Ar. Thereby, it is entirely irrelevant how this is done, since r=l the errors introduced are of order τ [6] and do not affect the fourth-order correctness of the approximant E(4)(-iτK, - iτu). The simplest choice for the Als is to take the local two-site contributions that appear in (3) (terms in nl can be absorbed in exp [- iτu/2]). As usual, the wave function is written down in the occupation number representation. The procedure outlined above reduces propagation with exp [C( - iτk, - iτu)] to multiplication by two-by-two matrices, each propagation requiring 2 (d = 1), 6 (d = 2) or 12 (d = 3) passes over the lattice. Propagation with exp [- iτu/2] is almost trivial because one merely has to multiply the value of the wave function at site l by exp [- iτwεl/2]. Since the kinetic energy can be diagonalized by Fourier transformation, propagation with exp [- iτk/2] is as easy, provided the wave number representation of the wave function is known. Transformation from one representation to the other is most effectively done by invoking the fast Fourier transform (FFT). Although use of the FFT dramatically decreases the number of operations, this approach still has some serious deficiencies [8]. Methods based on more or less heuristic arguments have been proposed to circumvent these problems [8]. Here I propose to solve this problem by invoking (2) once again, not on H but on K. First note that the kinetic-energy operator of a d-dimensional system is the sum of d commuting one-dimensional kinetic-energy operators, each one representing the motion in a different direction. Consequently it is sufficient to concentrate on the one-dimensional case. Just as for exp [C( - iτk, - iτu)], propagation with exp [- iτk/2] will be replaced by multiplications with two-by-two matrices. To this end K is written as a sum of odd (Ko) and even (K e ) numbered blocks of two sites [9], a decomposition frequently used in Monte Carlo simulations of quantum systems [6]. Analytical evaluation of C(- iτk o, - iτke) and application of the same strategy as the one used for exp [C( - iτk, - iτu)] completes the construction of the algorithm. Propagation with exp [- iτk/2] requires 4 lattice passes per direction. This method does not require FTTs. It operates directly on wave functions in realspace (RS) representation. The modularity of the present approach should be clear. Extending the algorithm in two- or three-dimensional systems requires evaluation of expressions such as (3) and repeating the steps outlined above. Memory requirements and computation time remain proportional to the number of lattice sites. An important advantage of these RS algorithms is that they exhibit a very high degree of parallelism: in each operation, each pair of lattice points can be treated simultaneously and independently. Table I shows how approximate results compare with exact results, obtained by numerical diagonalization of Hamiltonian (1). From the point of view of accuracy, secondorder Trotter-formula algorithms (C(X, Y) = 0 in (2» are superior to the CN method. As one time step for both the CN and τ2-rs methods require one pass over the lattice, the latter is approximately five times as efficient as the former. Although the τ4-rs algorithm requires approximately four times more computation time than τ2-rs, this loss is more than just compensated by the increase in accuracy. The τ4-rs method is the most efficient one and outperforms the CN method by at least a factor of twenty. Figure 1 shows some results for the case of the 1D Anderson model, obtained by solving the TDSE by means of the τ4-rs method, averaged over 10 different realizations of the random potential. The function p00(t) is the time-dependent return probability for a particle,

5 142 EUROPHYSICS ETTERS TABE I. - Typical r.m.s.-errors on the wave function, obtained by comparing results of the Cranck- Nicholson method and algorithms based on Trotter formula (2) with the exact answers for the twodimensional Anderson model. The energy of the wave packet moving on a square lattice of 64 sites is zero, the time step, = 0.1 and the number of time steps is 100. The potential strength W is expressed in units of V (V = 1 in this work). CN: Cranck-Nicholson method, FFT:free-particle propagation using FFT, RS: full real-space algorithms. Method Accuracy W=O W=2 W=4 W=6 W=8 W= 10 CH τ FFT τ RS τ FFT τ RS τ Fig Return probability P00(t) (0) and the function ξ(t) ( ) (see text) for the 1D Anderson model obtained by solving the time-dependent Schrödinger equation for a particle with energy E = O. Data points represent averages over 10 independent runs. as introduced by ANDERSON[3]. Within the statistical noise, poo(t) is constant and distinctly larger than what one would expect if diffusive motion takes place (then Poo(t) ~ -d for large t). Assuming exponential decay, ψ(r, t) 2 ~ exp [- r/ξ(t)], the function ξ(t) can be determined. For a localized state one expects lim ξ(t) = ξ, where ξ is the localization length. t oo From fig. 1 it sollows that for, W = 6 (V = 1) and E = 0, ξ = 3 ± 1. As opposed to numerical methods that rely on the Kubo formula, it seems easy to find numerical evidence for localization in a ID system by following the time evolution of a wave packet. Estimates of ξ for W 2 obtained in this manner are in reasonable agreement with recent results [10]. Some data for the 2D Anderson model is depicted in fig. 2. For W = 10 and t 500, ξ(t) is constant, showing that the particle is localized, as is also confirmed by the behaviour of Poo(t). For W = 10, ξ 4 lattice spacings, in good agreement with established results [11]. For W = 8 and W = 6 one cannot conclude from the data that the particle is in a localized state. Detailed examination of the time dependence suggests that also in these cases ξ(t) will settle to a constant value, but, to make more definite statements, longer runs are necessary. If ψ(r) 2 ~ exp [- r/ξ], ξ must be at least 10 times smaller than the linear size, before size effects become unimportant. For W = 2 and W = 4, lim ξ(t) 10, whereas = 128, 'uoo leading to the conclusion that for W 4 a system of N = 2 = lattice sites is too small to exhibit localization [11]. Work to perform calculations for longer times and larger systems is in progress.

6 H. DE RAEDT: NEW METHOD FOR SOVING THE SCHRÖDINGER EQUATION 143 Fig Function ξ(t) for the 2D Anderson model for several values of disorder Wand energy E = 0. Dashed curves are results for systems that are too small to exhibit localization. Fluctuations are too small to be seen on this scale. Previous work on localization, based on the time evolution of wave packets [12, 13] was limited to shorter time intervals (t::o::; 250) and/or smaller systems (N::o::; 10000). For 0::0::; t ::0::; 250 the present results are in satisfactory agreement with those of ref. [12]. As fig. 2 shows, for t::o::; 300 and W::o::; 10 the system is still in a transient state. Thus results obtained in these earlier simulation studies [12] should not be used to argue for or against the existence of localization in the 2D Anderson model. Following the method of ref. [12], the (time dependent) diffusion coefficient D(t) has been extracted from the raw data. imiting the analysis to t 200 [12] yields results for D(200) that are in good agreement with those of ref. [12]. Repeating this analysis using all the data currently available shows, that for W::o::; 6, D(200) is much too large compared to D(1000). Extrapolation of D(t) for 2 W 10 suggests that the diffusion coefficient tends to zero as t 00. This may explain why the results of ref. [12] do not support current consensus that in 2D all states are localized [14, 15]. * * * Iwould like to thank W. GÖTZEand A. AGENDIJKfor stimulating discussions and useful suggestions. This work is partially supported by the «Supercomputer Project» of the National Science Foundation, Belgium and NATO Grant No. RG.86/0026. The author thanks the National Science Foundation, Belgium, for financial support. REFERENCES [1] KOONIN S. E., Computational Physics (Benjamin/Cummings, Melo Park California, Cal.) [2] SMITH G. D., Numerical Solution of Partial Differential Equations: Finite Difference Methods (Clarendon Press, Oxford) 1985.

7 144 EUROPHYSICSETTERS [3] ANDERSONP. W., Phys. Rev., 109 (1958) [4] SUZUKI M., Prog. Theor. Phys., 56 (1976) [5] SUZUKI M., J. Math. Phys., 26 (1985) 601. [6] DE RAEDT H. and AGENDIJKA., Phys. Rep., 127 (1985) 233 and references therein. [7] DE RAEDT H. and DE RAEDT B., Phys. Rev. A, 28 (1983) [8] VAN DYCKD., in Advances in Electronics and Electron Physics, Vol. 65 (Academic Press, New York, N. Y.) 1985 and references therein. [9] BARMAM. and SHASTRYB. S., Phys. ett. A, 61 (1977) 15. [10] ROMANE. and WIECKOC., Z. Phys. B, 62 (1986) 163. [11] MACKINNONA. and KRAMERB., Z. Phys. B, 53 (1983) 1 and references therein. [12] PREOVŠEKP., Phys. Rev. B, 18 (1978) [13] KRAMERB., MACKINNONA. and WEAIRE D., Phys. Rev. B, 23 (1981) [14] ABRAHAMSE., ANDERSONP. W., ICCIARDEOD. C. and RAMAKRISHNANT. V., Phys. Rev. ett., 42 (1979) 673. [15] GÖTZEW., in Modern Problems in Solid State Physics, Vol. 1, edited by Yu. E. OZOVIKand A. A. MARADUDIN(North-Holland, Amsterdam) to be published.