Memory functions and relaxation functions of some spin systems

250

Nuclear Physics B (Proc. Suppl ) 5& (1988) 250 254 North-Holland, Amsterdam

MEMORY FUNCTIONS AND RELAXATION FUNCTIONS OF SOME SPIN SYSTEMS Joao FLORENClO Department of Physics, Pennsylvania State University, Altoona, Pennsylvania

16601

M. Howard LEE Department of Physics and Astronomy, University of Georgia, Athens, Georgia The method of recurrence r e l a t i o n s has proved to be evolutions of many-body systems such as an electron systems on a l a t t i c e . We review the method and i t s one-dimensional XY and transverse Ising models, and

30602

a powerful tool in the understanding of time gas, classical harmonlc chains, and spln applications to the dynamlcs of spins in the the spin van der Waals model

The I t a l i a n group have used the Lanczos method

1. INTRODUCTZON The study of dynamics of many-body systems

to t r e a t continued f r a c t i o n s by using matrix

has been of great t h e o r e t i c a l i n t e r e s t in non-

representations.

e q u i l i b r i u m s t a t i s t i c a l mechanics.

varlous problems, including slow motlon EPR

The l i n e a r

response theory of KuboI and the memory function

They applied t h e l r ideas to

spectra, F center in KCl, van Hove c r i t i c a l

approach due to Mori 2 are among the outstanding

points, nonlinear stochastic o s c i l l a t o r s , and

accomplishments in the f i e l d .

d i f f u s i o n in l i q u i d s , among others. I0-16

Mori showed that

The

the Heisenberg equation of motion can be rear-

group at the U. of Georgia developed the method

ranged as a generalized Langevin equation in

of recurrence r e l a t i o n s to study the tlme evolu-

which the time evolution is governed by a time-

tions in many-body systems.

dependent f r i c t l o n c o e f f i c l e n t --the memory

center on H i l b e r t spaces and t h e i r r e a l i z a t i o n .

f u n c t i o n - - and the random forces.

The method has been applied to some physical

Mori's theory

The ideas developed

has been used by many authors in a wide v a r i e t y

models, such as an electron gas, a classical

of dynamical problems. 3-9

Someof these authors

harmonic o s c i l l a t o r chain, spin systems, and to

use approximations to the memory function, which

the study of v e l o c i t y aut oc o r r e la t io n func17-25 tions. In the f o l l o w l n g sections, we out-

i t s e l f was shown to be of the form of continued fractions.

In these treatments, one truncates

l i n e the more important features of the method

the continued f r a c t i o n s , therefore imposing a

of recurrence r e l a t i o n s and t h e i r a p p l i c a t i o n to

f i n i t e number of poles, the three-pole approxi-

some spin systems, namely, the XY and transverse

matlon being the most common. These truncation

Ising models in one dlmenslon, and the spln van

schemes i n v a r i a b l y introduce phenomenologlcal

der Waals model.

parameters so that the o r i g l n a l goal of t r e a t i n g the dynamics from f l r s t

p r i n c i p l e s was never

fulfilled. Attempts to t r e a t the memory function more

2. METHODOF RECURRENCERELATIONS The time evolution of a spin operator S is governed by the generalized Langevin equation:

exactly have been carried out by the group of Pastori P a r r a v i c i n l , Grosso, G r l g o l l n i , and t h e i r co-workers in I t a l y , and the group of Lee and collaborators at the University of Georgia. 0920 5632/88/$03 50 © Elsevxer Science Pubhshers B V (North-Holland Physics Pubhstung Division)

dS(t)

+

f o

t dt' ~ ( t - t ' ) S ( t ' )

= F(t),

(i)

J Florencto, M H Lee / Memory functions and relaxation functtons where F and @ are the random force and the memmory f u n c t i o n , r e s p e c t i v e l y .

The random force F is then glven by

In the method o f

recurrence r e l a t i o n s i t s s o l u t i o n is w r i t t e n as

F(t) =

fo I 1ows : S(t) =

251

d-i z by(t) fv' v:1

(6)

where b y ( t ) s a t i s f i e s the convolution equation

d-i z av(t)fv' v=O

(2)

t f dt' bv(t-t')

av(t ) =

are basis vectors o f a H i l b e r t space S v of d dimensions. The s c a l a r product in S is de-

ao(t'),

v L 1.

(7)

0

where f

The memory f u n c t i o n is given by # ( t ) = A1b1(t). I t is o f t e n convenient to apply the Laplace

f i n e d by

transform to the tlme-dependent q u a n t i t i e s .

B (A,B) = ~1 0 f dx - ,

(3)

where A and B are vectors in S, and B is the i n -

RRII reads:

verse temperature C = 1/kT, the dagger meaning Hermitian conjugation.

1 : Zao(z ) + A l a l ( t ) ,

The q u a n t i t y A(X) =

exp(-xH) A exp(xH), where H is the Hamiltonlan

av_l(Z) = zav(z ) + A + l a v + l ( Z ) ,

(assumed to be H e r m l t i a n ) , and is the ensems a t i s f y the three-term

recurrence r e l a t i o n fv+l = iLfv + Avfv-l'

(8)

1 < v < d-i,

ble average = Tr{A exp(-BH)}/Tr{exp(-BH)}. The basis vectors f

For

instance, upon a Laplace t r a n s f o r m a t i o n , the

where av(z) = L { a v ( t ) } , where L denotes the Laplace transform o p e r a t o r .

0 < v < d-l,

(4)

The above r e l a t ] o n s y i e l d a continued f r a c t ] o n r e p r e s e n t a t i o n f o r aO(z):

where L is the L i o u v i l l e o p e r a t o r , LA = HA - AH. The q u a n t i t y A

lS termed the v - t h r e c u r r a n t ,

aO(z ) =

defined as Av = ( f v , f v ) / ( f v _ l , f v _ l ) . In the above recurrence r e l a t i o n (RRI), the f o l l o w l n g c o n d i t i o n s are i m p l i e d : f - i

1 z+A 1 z+A 2

= O, and A0 = i .

Z

+

(9)

o.o

We choose fo = S, so t h a t the boundary c o n d i t i o n on the a ' s

are ao(t=O) = i , and av(O) = O, f o r

a l l v ~ I.

I t f o l l o w s t h a t a o ( t ) is the r e l a x a -

t i o n f u n c t i o n of l i n e a r response theory. a's V

The

s a t i s f y the f o l l o w ] n g recurrence r e l a t i o n

(RRII):

Notlce t h a t the knowledge of a0 allows f o r the o t h er a ' s

to be d e t e r m n e d r e c u r s i v e l y .

By taking the Laplace transform of Eq. (7) we obtain

(~o)

b (z) = a v ( z ) / a O ( z ) ,

Av+lav+l(t) = _ av[t) + av_z(t), 0 < v < d-i,

from which we obtain the continued f r a c t l o n form

(5)

o f the memory f u n c t i o n : A1

where i v ( t )

: d a v ( t ) / d t and a _ l ( t ) = O.

Hence,

the recurrence r e l a t i o n s RRI and RRII t o g e t h e r with the boundary c o n d i t i o n S(t=O) = S uniquely determine the complete time e v o l u t ] o n of S ( t ) .

#(z) = Alal(z)/aO(z)

-

z+A 2 z

+

A3 z + ...

(ii)

J Florenclo, M H Lee / Memory functtons and relaxation functzons

252

The s i m i l a r i t y between the continued fraction

p h y s i c a l l y d i f f e r e n t systems having H l l b e r t

representations for aO(z) and @(z) = Albl(Z ) im-

spaces with the same structure are dynamically

plies that the memory functions b satisfy a re-

equivalent.

currence relation identical to RRII, but for which the f i r s t recurrant is not present, l . e . ,

The H i l b e r t space basis vectors, which in general are expressed as combinations of operators, can be asslgned to a more tangible physi-

Av+ibv+l(t) = _ i v ( t )

+ bv_l(t),

i < v < d-l.

cal meaning by applying them to some properly

(12)

defined vacuum state.

In that case the tlme

evolutlon of a given physical state is also Some general propertles of av(t) can be inferred from the RRII.

Notice that i t s structure

is not congruent to that o f the recurrence r e l a tions of orthogonal polynomials.

Hence orthogo-

nal polynomials are not admissible solutions f o r n e i t h e r av(t) nor f o r b y ( t ) .

Sinple exponential

given by a r e l a t i o n s l m l l a r to Eq. (2) ( m u l t i plied both sides to a vacuum s t a t e ) .

Since the

length of a vector in S is i n v a r i a n t wlth time, that is , i t s norm must s a t l s f y Bessel e q u a l i t y , one can normalize each of the basls vectors to u n i t y , so that the corresponding expanslon

functions are also ruled out since they cannot

c o e f f i c i e n t s may be regarded as amplltude prob-

s a t i s f y the RRII and i t s boundary conditions

a b i l i t i e s that a given state at t = 0 evolves

ao(t=O) = I , and av(t=O) = O, v ~ i .

These con-

clusions do not apply i f the model Hamiltonian is not Hermitian, nor i f S does not s a t i s f y Heisenberg equation of motion.

In such cases,

into any of the basis vector states at a l a t e r time. The recurrants and the dlmenslonallty of the H i l b e r t space are e n t i r e l y model dependent.

the method of recurrence r e l a t i o n s would have to

F i n i t e - d m e n s i o n a l H i l b e r t spaces y i e l d per i-

be modified to accomodate such a class of prob-

odic or almost-periodlc solutlons f o r the dy-

lems, and the recurrence r e l a t i o n s would take up

namical variables.

a d i f f e r e n t form.

dimensional H i l b e r t spaces give solutlons that

Exponential decay could be

allowed in such cases.

For Hermitian systems,

On the other hand, i n f l n l t e

decay at large times, hence the i n i t l a l

state

the RRII does not exclude long tlme t a i l s or

loses i t s memory as t ~ ~.

power-law decay.

i n f i n i t e - d i m e n s i o n a l H l l b e r t space provldes a

Admissible solutions include

the Gaussian, spherical and c y l i n d r i c a l Bessel functions, and hypergeometric functions, among

I t seems that an

s u f f i c i e n t condltlon f o r i r r e v e r s l b i l l t y . In the f o l l o w l n g section we discuss applications of the method of recurrence r e l a t l o n s to

others. To obtain the solution f o r the av(t) i t is

l a t t i c e spln chains such as the one-dimenslonal

necessary to r e a l i z e the RRII, that i s , to know

spin-½ XY and transverse Islng models, and the

the e x p l i c l t form of the recurrants, which car-

spln van der Waals model, a l l of whlch are

ries the information on the geometry of S, i . e . ,

s t a r t i n g models f o r magnetism.

i t s dimensionality and shape.

Since a given

form of A y i e l d s a p a r t i c u l a r time evolution of

3. APPLICATIONS TO SPIN SYSTEMS ON A LATTICE

a given dynamic v a r i a b l e , we can then c l a s s i f y

3.1. XY model at T =

the tlme evolutlon in d i f f e r e n t systems accord-

Consider the i s o t r o p i c one-dimensional XY

ing to the geometric structure of realized H i l b e r t spaces.

model.

The Hamiltonlan is glven by

For instance, H i l b e r t spaces

with d i f f e r e n t dimensionalities do imply dist i n c t time evolutions.

On the other hand,

H = 2J

N 7 ( Sx~x i ~ l + 1 + sYqY i~1+1,I , i=1

(13)

253

J Florenclo, M.H Lee / Memory functtons and relaxatton functlons

where S~ are spin-½ o p e r a t o r s , ~ = x , y , z ,

and J

l

is the nearest neighbor c o u p l i n g c o n s t a n t . 26

As

n a m i c a l l y e q u i v a l e n t w i t h respect to the time x e v o l u t i o n o f S~. 3.3. Spin van der Waals model

usual, we impose p e r i o d i c boundary c o n d l t l o n s

SN+la = $1,a where N is the t o t a l number of l a t t i c e s i t e s . We are i n t e r e s t e d in the tlme evol u t i o n o f S~ in the i n f i n i t e J

temperature regime

Consider the a n i s o t r o p l c spln-½ Heisenberg model on a hypercubic l a t t i c e D:

and N = ~. We f i n d t h a t A

= ~(2j2),

I < v < ~.

The

r e l a x a t i o n f u n c t i o n s are found to be

where now s e are the spin o p e r a t o r s , ~ = x , y , z , and the c o u p l l n g constant is nonzero only f o r nearest nelghbors.

t ~

(14)

a v ( t ) = ~f, e x p ( - j 2 t 2 ) .

(17)

H = - Z Z (D) s~s ~ e (i,j) aij i j'

Hence,

the H i l b e r t space has i t s dlmension d = ~.

of dlmensionality

sional

The i n f i n i t e

lattice

dimen-

(D ÷ =) is the spin van der Waals mod-

el 21

In p a r t l c u l a r , relation

the time-dependent spin a u t o c o r -

As usual, one takes j x (D) = JY. (D) = " 13 13 J/N and JZ( D ) ~ = Jz/N, where N is the number o f .

f u n c t i o n is glven simply by

spins.

22

J

x

By considerlng Sx = z s i , one finds

that the Hilbert space of this variable has a

j(z)~j = ~ ao(t)

rich structure, i . e . , i t depends on whether the

= ~ exp(-j2t2),

(15)

temperature T is greater or less than Tc, the c r i t i c a l temperature of the model29 and on the

a result obtained previously by Brandt and

r e l a t i v e strength of J and Jz"

Jacoby,27 and also by Capel and Perk28 by using d i f f e r e n t methods. The memory functions are then obtained by solvlng the convolution equa24 tlon numerically.

3.3.1. XY regime (J > Jz ) For T > T = J/2k, one finds that A = vA, C 2 2 v v _> 1,~where A = 4w , where w = (J-Jz)/N and _ = ½N(2-BJz). Since the upper l i m i t of v is unbound, i t follows that the Hilbert space

3.2. Transverse Ising model at T = The Hamiltonlan for the transverse Ising mod-

is inflnite-dlmensional, and the relaxation functions are of the same form as Eq. (14).

el is given by N

N

For T < Tc, the only change in the structure of

H = 2J z Sx~x i ~ i + l - B z Sz l' i=1 i=i

(16)

the recurrants is that now = ½B(J-Jz), owing to the ordered state characterized by Therefore the tlme evolution propertles

where the notation is the same as before, except

.

for the introduction of an external magnetic

are the same as before.

i

f l e l d B.

Again, we consider S~ as the dynamical 3

variable of interest. We flnd that A.± = B2, A2 = 2J2, A3 = 2j2+B2, A4 = 12j2B2/(2j2+B2), etc.

When B = J, however,

3.3.2. Ising regime (Jz > J) For T > Tc, the s t a t l c properties of the Islng and XY cases are Identical, hence one also obtains Av

=

vA.

For T < Tc, the ordered

the recurrants assume the simple form A = vJ 2,

s t a t e i s c h a r a c t e r i z e d by .

which ~s similar to that of the XY model. That

t h a t the r e c u r r a n t s are q u i t e d i f f e r e n t

means that the relaxation functions, memory

those o f the low temperature regime.

functlons, e t c . , have the same time dependence

t h e r e Is only one nonvanlshing r e c u r r a n t ,

as t h e i r counterparts of the XY model, except

A1 = 4~ 2 and Av = 0 f o r v ~ 2.

for a time scale.

f u n c t i o n s are given by

Hence, these models are dy-

It follows from Actually, i.e.,

The r e l a x a t i o n

J Florencto, M H Lee / Mernory Juncttons and relaxation functions

254

ii.

ao(t) = cos ~t, a l ( t ) = ~ sin wt,

(18)

and the memory function is constant, i . e . , 4~2.

# =

A remarkable feature of t h i s case is the

fact that the H l l b e r t space collapses from an

G. Grosso and G. Pastorl P a r r a v l c l n l , Adv. Chem. Phys. 62 (1985).

12. P. G r i g o l i n i , G. Grosso, G. Pastori Parrav i c l n i , and M. Sparpaglione, Phys. Rev. B27 (1983) 7342. 13. M. Giordano, P. G r i g o l l n l , D. Leporlnl, and P. Morln, Phys. Rev. A28 (1983) 2474; Adv. Chem. Phys. 62 (1985) 321.

i n f l n l t e - d i m e n s i o n a l into a two-dimensional mani f o l d as T is lowered through the c r i t i c a l

tem-

14. T. Fonseca, P. G r l g o l l n i , and M. P. Lombardo, Phys. Rev. A33 (1986) 3404.

perature. 15. P. G r i g o l l n l , J. Stat. Phys. 27 (1982) 283. 4. SUMMARY We have reviewed the method of recurrence rel a t i o n s and some of i t s appllcatlons to the dynamics of spin systems on l a t t l c e s . ACKNOWLEDGEMENTS J.F. would l i k e to acknowledge p a r t l a l support from the Endowment Fund of the Altoona Campus Advisory Board, the Altoona Campus Faculty Development Fund, and the PSU Research Development Grant.

M.H.L. was supported in part by the

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