Computer Physics Communications 142 (2001) 418–423www.elsevier.com/locate/cpc

The forced oscillator method incorporating the fasttime-evolution algorithm

H. Shima∗, H. Obuse, K. Yakubo, T. NakayamaDepartment of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan

Abstract

The forced oscillator method (FOM) is a powerful algorithm to calculate linear response functions of quantum systemsrepresented by very large Hamiltonian matrices. We demonstrate the efficiency of the FOM by applying for the calculation ofthe ac conductivityσ(ω) for two-dimensional electron systems with spin-orbit interactions, for which conventional numericalmethods are inoperative. It is shown that the efficiency of the FOM is greatly enhanced by combining the fast time-evolutionmethod. 2001 Elsevier Science B.V. All rights reserved.

PACS: 02.70.-c; 02.60.Dc; 72.15.-v

1. Introduction

The eigenvalue analysis of very large matrices is crucial in computational physics. As sizes of matrices increase,calculations in terms of conventional methods become difficult since computing times as well as required memoryspaces grow rapidly. So far, many algorithms suitable to treat very large matrices have been developed [1]. Amongthese, the forced oscillator method (FOM) [2,3] has offered a quite efficient scheme for computing spectraldensities, eigenvalues and eigenvectors of large matrices. The FOM utilizes a principle of mechanics: a linear latticedynamical system driven by a periodic external force of frequencyΩ will respond with large amplitudes in thoseeigenmodes close to this frequency. Namely, the eigenvalue analysis is reduced to solving the time-developmentof the equation of motion. It is now possible to treat not only the eigenvalue problems, but also linear responsefunctions [4]. The efficiency of the FOM is greatly enhanced when adopting the fast time-evolution method(FEM) [5,6] for the time development calculation. The FEM is based on the Chebyshev polynomial expansionof the formal operator solution of the equation of motion.

In this article we demonstrate the efficiency of the FOM by calculating the linear response function of two-dimensional (2D) electron systems. We also stress how the FEM accelerates the efficiency of the FOM. Althoughwe present a single example of applications, it is accepted that the FOM is broadly applicable and of importance ina various physical system. (See a review in Ref. [7].)

* Corresponding author.E-mail address: shima@eng.hokudai.ac.jp (H. Shima).

0010-4655/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0010-4655(01)00379-4

H. Shima et al. / Computer Physics Communications 142 (2001) 418–423 419

2. Mapping onto a lattice dynamical problem

Assume that the Hermitian matrixDmn has an eigenvectore(λ) belonging to an eigenvalueελ, i.e.

ελem(λ) =∑n

Dmnen(λ). (1)

If the matrixDmn has negative eigenvalues, we add an appropriate amount ofε0 |εmin| to the diagonal elementsof the matrixDmn so that the minimum eigenvalueεmin + ε0 can be always positive. The mapping of Eq. (1) ontothe equations of lattice dynamical motion is done by

d2

dt2xm(t) = −

∑n

D′mnxn(t), (2)

where

D′mn = Dmn + δmnε0, (3)

andxm(t) denotes the displacement of the particle on themth site. Each displacementxm(t) can be decomposedinto a sum of normal modes as

xm(t) =∑λ

Qλ(t)em(λ), (4)

where Qλ(t) is the time-dependent amplitude with which the modeλ contributes toxm(t), and varies as∝ exp(−iµλt) (µ

2λ ≡ ελ+ε0). Sinceµ2

λ should be positive, the matrixDmn should be modified intoD′mn expressed

by Eq. (3). Hereafter we callµλ andελ as eigenfrequency and eigenvalue for clarity.The spectral density is calculated by the following procedure. The displacementxm(t) and the velocityxm(t)

are set to be zero att = 0 in Eq. (2), and the periodic forceFm cos(Ωt) is imposed as

d2

dt2xm(t) = −

∑n

D′mnxn(t) + Fm cos(Ωt). (5)

Here we should setFm = F0 cos(φm), whereF0 is a constant, andφm is a random quantity distributed uniformlyin the range 0 φm 2π .

As a next step, we introduce the energy functionE(t) of the system given by

E(t) = 1

2

∑m

x2m(t) +

∑mn

xm(t)D′mnxn(t)

. (6)

With a straightforward calculation, one yields the averaged value ofE(t) overφm as [8]

⟨E(t)

⟩ = F 20

4

∑λ

sin2(µλ − Ω)t/2(µλ − Ω)2

. (7)

After a sufficiently large timet = T , only modesλ’s belonging to eigenfrequenciesµλ in the vicinity of Ωcontribute to the sum in Eq. (7). For a proper time-intervalT , Eq. (7) yields

⟨E(T )

⟩ πT F 20

8

∑λ

δ(µλ − Ω) = πTNF 20

8ρ(Ω), (8)

whereρ(Ω) is the density of states for the mapped system characterized byD′mn. The spectral densityρ(ε) for the

original matrixDmn is obtained by multiplyingρ by the Jacobian|dµ(ε)/dε|.

420 H. Shima et al. / Computer Physics Communications 142 (2001) 418–423

3. Time development

FOM does not require the time development of dynamical variables during the time interval 0< t < T . Weneed only displacements and velocities att = T . For this purpose, the FEM [5,6] is remarkably suitable becauseit enables us to obtain directly displacements and velocities of a dynamical system at an arbitrary timet withoutpursuing displacements (or velocities) during the time interval 0< t < T . The method is based on the Chebyshevpolynomial expansion of the formal solution of the ordinary differential equations in the eigenfrequency domain.By incorporating this method, the efficiency of the FOM is extraordinarily enhanced. Indeed, the computing timefor the time development ofxm(t) in Eq. (5) is an order of magnitude less than the case using the conventionalVerlet method.

Eq. (5) can be symbolically written as(d2

dt2+ D

)x = f cos(Ωt), (9)

whereD, x, andf are the matrix and the vectors whose elements areD′mn, xm, andFm, respectively. The initial

conditions are set asx(0) = x0, and x(0) = z0. The formal solution of Eq. (9) under these initial conditions isgiven by

x(t) = cos(√

D t)x0 + sin(

√D t)√D

z0 +[

cos(Ωt)I − cos(√

D t)√D − Ω2I

]f , (10)

whereI is the unit matrix. Functions of the matrixD should be understood as power series such as a Taylorexpansion.

In the FEM the timet is regarded as a parameter and we expand Eq. (10) in terms of the Chebyshev polynomialsTp(D). Since the domain of the Chebyshev polynomials is[−1,1], the matrixD should be converted to the matrixG having a spectrum bounded in the range[−1,1] as

G = 2

εmax− εminD − εmax+ εmin

εmax− εminI , (11)

where bothεmax andεmin, the upper and lower bounds of eigenvalues, are estimated by the Gerschgorin’s theorem.Functions of the matrixD in Eq. (10) are expanded as

cos(√

D t) =

∞∑p=0

ap(t)Tp(G). (12)

(In the followings, only the first term of Eq. (10) is discussed for simplicity.) In actual calculations, the infiniteupper limit of the summation in Eq. (12) should be truncated up toP − 1, whereP is an appropriate cutoff so that|ap(t)| with p P are much smaller than unity. Using the orthonormal conditions forTp(x), one can obtain theexplicit form of the coefficientap(t) defined by Eq. (12) as follows,

ap(t) = 2

(1+ δp0)P

P−1∑q=0

cos

[πp(q + 1

2)

P

]cos(ξq t), (13)

where

ξq =√

εmax+ εmin

2+ εmax− εmin

2cos

[πp(q + 1

2)

P

]. (14)

Notice that the expression given by Eq. (13) takes the form of the Fourier cosine transform. Using the fast Fouriertransform techniqueap(t) can be obtained within a quite short time.

H. Shima et al. / Computer Physics Communications 142 (2001) 418–423 421

Fig. 1. Spectral densities of states for 2D electron systems with spin-orbit interactions. The system sizeL is fixed to be 600 and disorderstrengths are taken asW = 0 (solid) andW = 5.86 (dotted). The spectral density of states forE > 0 is symmetric aroundE = 0 due to theparticle-hole symmetry.

Let us next consider the calculation of the Chebyshev polynomialsTp(G) of the matrixG. The recurrenceformulae of the Chebyshev polynomials leads to the relations of the vectorsxp ≡ Tp(G)x0 given by

xp+1 = 2Gxp − xp−1, x1 = Gx0. (15)

We should note that there are no matrix–matrix multiplications in Eq. (15), which implies the computational timeis saved considerably.

In Fig. 1, spectral densities of states of 2D disordered electron systems with spin-orbit interactions are shown.The Hamiltonian of the system is given by [9,10]

H =∑m,σ

Wm,σ |mσ 〉〈mσ | +∑

m,σ ;n,σ ′Vm,σ ;n,σ ′ |mσ 〉〈nσ ′|, (16)

wherem andσ denote the lattice site and the spin, respectively. We set the lattice constant to be unity and onlythe nearest neighbor coupling is taken into account. The on-site potentialWi is randomly distributed in the range[−W/2,W/2]. The hopping energy is described by

Vm,σ ;m−k,σ ′ = V[exp(−iθσk)

]σ,σ ′ , k ≡ x, y, z (17)

whereσk are Pauli matrices. We choose the hopping amplitudeV as a unit of energy. This system is quite interestingin the sense that it is the only 2D non-interacting electron system exhibiting the metal-insulator transition [11]. TheHamiltonian matrix of this system becomes very large due to the spin-degree of freedom. So the FOM is suitableto calculate spectral densities of this system with high speed and accuracy.

4. Linear response functions

Linear response functions are quite important to gain insight into dynamical properties of large-scaleHamiltonian matrices. In this section, we present the algorithm based on the FOM to calculate linear responsefunctions of quantum systems such as the ac conductivityσ(ω). Let us consider a system described by a tight-binding HamiltonianH = ∑

mn Hmn|m〉〈n|. For calculating the ac conductivity, we impose a perturbation

H ′ = −J

2

(A0e−iωt + c.c.

), (18)

422 H. Shima et al. / Computer Physics Communications 142 (2001) 418–423

whereJ is the current operator andA0 is the amplitude of the external vector potential. c.c. indicates a complexconjugate. Expanding the time dependent electron states as|Ψ (t)〉 = ∑

n an(t)|n〉 and applying the time-dependent

first-order perturbation theory by puttingan(t) = a(0)n (t) + a

(1)n (t), one has the linear differential equation with a

periodic external force [12],

ihda(1)

m (t)

dt−

∑n

Hmna(1)n (t) = − h

2Fme−i(ωλ+ω)t , (19)

where

Fm ≡∑n

A0

hJmnϕn(ωλ), (20)

andϕn(ωλ) ≡ 〈n|ωλ〉 is the initial eigenmode belonging to the eigenvaluehωλ of the matrixHmn. We have usedin Eq. (19) the definition given bya(0)

n (t) = ϕn(ωλ)e−iωλt .We introduce the resonance function defined by

E(ω,ωλ, t) ≡∑n

∣∣a(1)n (t)

∣∣2. (21)

Substituting the solution of Eq. (19) under the initial conditiona(1)n (0) = 0 into Eq. (21), one has

E(ω,ωλ, t) =∑λ′

∣∣∣∣∑m

Fmϕ∗m(ωλ′)

∣∣∣∣2 sin2(ωλ′ −ωλ − ω)t/2

(ωλ′ − ωλ − ω)2 . (22)

Fig. 2. Calculated results ofσ(ω) for 2D electron systems in the presence of spin-orbit interactions with various disorder strengthsW . Thesolid line indicates a Drude-like behavior, and the dotted line shows a constant∼1.8.

H. Shima et al. / Computer Physics Communications 142 (2001) 418–423 423

After a sufficient large time intervalt , we have

E(ω,ωλ, t) = πt|A0|22h2

∑λ′

∣∣〈ωλ′ |J |ωλ〉∣∣2δ(ωλ′ − ωλ − ω). (23)

The resonance function expressed by Eq. (23) can be related to the ac conductivityσ(ω) as

σ(ω) = 4h

ωtLd

ωF∑ωλ=ωF −ω

E(ωλ,ω, t), (24)

whereωF = EF/h and|A0| is chosen to be unity without loss of generality. Fig. 2 presents the calculated results ofσ(ω) for 2D electron systems with spin-orbit interactions for various disorder strengthW . The system is describedby the Hamiltonian (16). We setθ in Eq. (17) to beπ/6, and the Fermi energyEF at the band center. For a weakdisorder (W = 2.0),σ(ω) exhibits a Drude-like behavior. At the critical valueW = 5.86,σ(ω) does not depend onω, which is consistent with the scaling theory [13], The constant value ofσ(ω) also agrees with previous numericalresults [14,15].

5. Summary

We have demonstrated that the FOM is quite efficient, in particular when combined with the FEM, fortreating physical systems described by very large matrices. The scheme enables us to compute spectral densities,eigenvalues, eigenvectors, and linear response functions of large matrices with high speed and accuracy. Althoughwe have demonstrated only a conductivity problem as its application, the present method is rather general andapplicable for various types of physical problems.

Acknowledgements

This work was supported in part by a Grant-in-Aid for Scientific Research from the Japan Ministry of Education,Science, Sports and Culture. Numerical calculations were performed on the SR8000 of the Supercomputer Center,Institute of Solid State Physics, University of Tokyo.

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