Physica B 316–317 (2002) 521–523

Acceleration of the forced oscillator method and its applicationto a model for glasses

H. Shima*, K. Yakubo, T. Nakayama

Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan

Abstract

The forced oscillator method is a powerful algorithm to calculate physical quantities of the system described by very

large Hamiltonian matrices. The efficiency of this algorithm is greatly enhanced by combining the fast time-evolution

method. The article gives a description of these numerical methods, including its applications to a model for three-

dimensional network glasses. r 2002 Elsevier Science B.V. All rights reserved.

Keywords: Forced oscillator method; Fast time-evolution method; Glasses

The forced oscillator method (FOM) [1–3] hasoffered a quite efficient scheme for computingspectral densities, eigenvalues and eigenvectors oflarge matrices. The algorithm utilizes a principle ofmechanics: a linear lattice dynamical system drivenby a periodic external force of frequency O willrespond with large amplitudes in those eigenmodesclose to this frequency. Namely, the eigenvalueanalysis is reduced to solving the time developmentof the equation of motion. The efficiency of theFOM is greatly enhanced when adopting the fasttime-evolution method (FEM) [4,5]. Assume thatthe Hermitian matrix fDmng has an eigenvectoreðlÞ belonging to an eigenvalue el; i.e.,

elemðlÞ ¼X

n

DmnenðlÞ: ð1Þ

The mapping of Eq. (1) onto the equation of thelattice dynamical motion is done by

d2

dt2xmðtÞ ¼ �

Xn

D0mnxnðtÞ; ð2Þ

where D0mn ¼ Dmn þ dmne0; and xmðtÞ denotes the

displacement of the particle on the mth site. Anappropriate amount of e0 is added to the diagonalelements of the matrix fDmng so that the minimumeigenvalue of fD0

mng can be always positive. Eachdisplacement xmðtÞ can be decomposed into a sumof normal modes as xmðtÞ ¼

Pl QlðtÞemðlÞ; where

QlðtÞ is the time-dependent amplitude and variesas pexpð�imltÞ ðm

2l el þ e0Þ:

The spectral density is calculated by the follow-ing procedure [1]. The displacement xmðtÞ and thevelocity ’xmðtÞ are set to be zero at t ¼ 0 in Eq. (2),and the periodic force Fm cosðOtÞ is imposed as

d2

dt2xmðtÞ ¼ �

Xn

D0mnxnðtÞ þ Fm cosðOtÞ: ð3Þ

Here we should set Fm ¼ F0 cosðfmÞ; where F0 is aconstant and fm is a random quantity distributeduniformly in the range ½0; 2p�: We then introduce

*Corresponding author. Tel.: +81-11-706-6624; fax: +81-

11-716-6175.

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

0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 5 6 0 - 4

the energy function EðtÞ given by

EðtÞ ¼1

2

Xm

’x2mðtÞ þ

Xmn

xmðtÞD0mnxnðtÞ

( ): ð4Þ

After a sufficiently large time t ¼ T ; one yields theaveraged value of EðtÞ over fm as

/EðTÞSCpTF 2

0

8

Xl

dðml � OÞ

¼pTNF 2

0

8*rðOÞ; ð5Þ

where *rðOÞ is the density of states for the mappedsystem characterized by D0

mn: The spectral densityrðeÞ for the original matrix Dmn is obtained bymultiplying *r by the Jacobian jdmðeÞ=dej: By settingthe initial amplitude of periodic force Fm in anappropriate form, we can also calculate eigenvec-tors eðlÞ; their eigenvalues el; as well as dynamicalresponse functions in general systems [3].

Note that the FOM does not require the timedevelopment of dynamical variables during thetime interval 0otoT : We need only xmðtÞ and

’xmðtÞ at a proper time T : For this purpose, theFEM [4,5] is remarkably suitable because itenables us to obtain directly xmðtÞ and ’xmðtÞ of adynamical system at an arbitrary time t withoutpursuing them during the time interval 0otoT :The method is based on the Chebyshev polynomialexpansion of the formal solution of the ordinarydifferential equations in the eigenfrequency do-main. By incorporating this method, the efficiencyof the FOM is extraordinarily enhanced. Indeed,the computing time for the time development ofxmðtÞ in Eq. (3) is an order of magnitude less thanthe case using the conventional Verlet method [3].

We apply the method to a numerical investiga-tion of vibrational properties of the three-dimen-sional (3d) model for network glasses proposed byNakayama [6,7]. Our system consists of 3d cubiclattice with constant mass m as a molecular unit,which are connected to their nearest neighbors bylinear springs with constant strength k: In addi-tion, a certain number of extra-potentials are in-troduced to each site [6]. The extra-potential is ge-nerated from internal distortion (buckling) of themolecular structure, and plays an important rolefor the peculiar low-energy dynamics in glasses [6].

The Hamiltonian of this model is expressed by

H ¼X

i;j

p2j

2mþ

P2i

2mþ

k

2ðqi � qi�1 þ tiÞ

2

(

þKi

2ðQj � qi þ uiÞ

2

�; ð6Þ

where qi and Qi are generalized coordinatesrepresenting displacements or changes of anglevariables. The corresponding momenta are de-noted by pi and Pi; respectively. Lower lettersdenote quantities for backbone network structureand capital letters correspond to additional vibra-tions arising from extra-potentials. The symbols ti

and ui express static displacements due to internaltension satisfying the balance of forces [6,7]. Thecurvatures of the extra-potentials are expressed bya randomly distributed force constant Ki at the sitei: The characteristic frequency oi ¼

ffiffiffiffiffiffiffiffiffiffiffiKi=m

pof the

additional potentials are distributed according tothe Gaussian with the average frequency /oiS ¼0:6:

Calculated results of the vibrational density ofstates (DOS) are shown in Fig. 1. The system sizeis set to be 403; and periodic boundary conditionsare applied for all directions. The number of extra-potentials is taken as 20% of the total number ofmain atoms, and the extra-potentials are randomlyattached to the 3d backbone network. One seesfrom Fig. 1 the appearance of the hump in the

Fig. 1. The density of states for the lattice model for 3d

network glasses (solid). The result of a simple cubic lattice is

also shown (dashed).

H. Shima et al. / Physica B 316–317 (2002) 521–523522

DOS at 0:2ooo1:0; whose energy range equalsthat of extra-potentials. This excess DOS shouldcorrespond to the hump in the quantity DðoÞ=o2

observed in inelastic neutron scattering experi-ments.

Further calculations indicate that eigenmodesfor 0:2ooo1:0 are strongly localized. The spatialprofile of the eigenmode with ol ¼ 0:32071 isshown in Fig. 2. The displacements of the atombounded to extra-potential are given on an x–y

sheet in a 3d system. It should be noted that only afew number of attached atoms vibrate signifi-cantly, whereas the amplitudes of main atoms areextremely small (not shown in Fig. 2). Suchstrongly localized modes are thought to be thephysical origin of the Boson peak observed ingeneral glassy systems [6]. Calculated results of

dynamical structure factor Sðq;oÞ of this model isgiven in Ref. [7].

In conclusion, we have demonstrated that theFOM is quite efficient, in particular when com-bined with the FEM, for treating physical systemsdescribed by very large matrices. The schemeenables us to compute spectral densities, eigenva-lues, and their eigenvectors of large matrices withhigh speed and accuracy. Although we have shownonly a single problem as its application, thepresent method is rather general and applicablefor various types of physical problems.

Acknowledgements

This work was supported in part by a Grant-in-Aid for Scientific Research from the JapanMinistry of Education, Science, Sports and Cul-ture. Numerical calculations were performed onthe SR8000 of the Supercomputer Center, Instituteof Solid State Physics, University of Tokyo.

References

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[7] T. Nakayama, in these Proceedings, Physica B 316/317

(2002).

Fig. 2. The profile of the eigenmode belonging to the eigen-

frequency ol ¼ 0:32071: A 2d x–y sheet is taken from our 3d

system, in which one sees that an attached atom has a large

amplitude.

H. Shima et al. / Physica B 316–317 (2002) 521–523 523