Application of the real space dynamic renormalization group method to the one-dimensional kinetic ising model

ANNALS

OF PHYSICS

132,

Application Method

121-162

of the to the

(1981)

Real Space

Dynamic

One-Dimensional

GENE F. MAZENKO*.+

Renormalization Kinetic

king

Group Model

AND J. LUSCOMBE*

The Department of Physics and the James Franck Ins/ifrrre, The University of Chicago, Chicago, Illinois 60637

ReceivedJuly 30, 1980

Weapplytherecentlydeveloped realspacedynamicrenormalization groupmethodto the one-dimensional kinetic Isingmodel.We showhow onecan developblock spinmethods that lead to recursionrelationsfor the spaceand time dependentcorrelationfunctions that correspond to the observables for thissystem.Wepointout the importanceof carefully choosingthe appropriateparameters governingthe behaviorof individualblocksof spins and the necessity of worrying about the high temperaturepropertiesof the temperature recursion relations if one is to obtain the proper exponential decay of correlation functions at large distances away from the critical point at zero temperature. We systematically investigate the accuracy of our approximate recursion relations for various correlation

functionsby checkingthem againstthe known exact results.Our simplemethodswork surprisingly well over a wide range of temperatures, wavenumbers and frequencies.

I. INTRODUCTION The real space dynamic renormalization group (RSDRG) method has recently [l-4] been introduced in order to investigate the dynamics of spins on a lattice. In particular it has been shown [5, 61 how one can obtain recursion relations relating time and space dependent correlation functions defined on lattices with different lattice spacings. In the case of the two-dimensional square lattice it was shown how one could usethese recursion relations to calculate these correlation functions over a wide range of temperatures, wavenumbers and frequencies. The techniques used in this analysis were somewhat novel. In particular the use of recursion relation methods away from the critical point is a bit unusual. The method led to quite good results for a variety of quantities and has encouraged us to proceed with the further development of these ideas. It is well known that the usual block-spin real spacerenormalization group (RSRG) method works very poorly for the one-dimensional lsing model. This led people [7] to develop the “decimation method” for handling the 1D problem. It is also well known that the one-dimensional kinetic Ising model [8] can be solved exactly. It * Supported by the National Science Foundation + Alfred P. Sloan Foundation Fellow.

Grant DMR77-12637.

121 0003-4916/81/030121-42$05.00/0 Copyright Q 1981 by AcademicPress,Inc. All rights of reproduction in any form reserved.

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therefore seems like a good idea to test the methods developed in Refs. [l-6] on the one-dimensional kinetic Ising model where we expect conventional RSRG methods to be suspect but where we can calculate all static and dynamic quantities exactly. In the course of this work we extend the approach carried out in Refs. [5, 61 by developing a perturbation theory expansion that is considerably more systematic than that developed there. This leads to considerable insight into the structure of the theory. In Section II we define the basic problem of interest and point out various exact solutions. In Section III we outline the basic points in the RSDRG approach. We then set up in Section IV a perturbation theory calculation designed to allow an accurate implementation of the RSDRG using decoupled cells as a zeroth order approximation. In Section V we discuss various methods for determining local parameters which occur in the perturbation theory. Finally in Section VI we discuss the results of detailed calculations for various physical quantities and compare our results with the exact results.

II. THE ONE-DIMENSIONAL

KINETIC

ISINC MODEL

Let us consider a set of N ferromagnetic Ising spins (or ,..., eN) defined on a onedimensional lattice with siteslabeled by index n with lattice spacing a, and located at positions R, = na, . The static or equilibrium properties of this system are governed by the usual Boltzmann probability distribution

where H[a] is the nearest neighbor lsing Hamiltonian (times (--k,T)-l), H[o] = ;

1 *ncJn+a>

n.6

(2.2)

K is the positive coupling, 6 = Al, and Z in (2.1) is the partition function Z

=

C .@[“I 0

(2.3)

where the sum is over all spin configurations. Thermodynamic averagesare given by (A,)

= 1 P[u] A, .

It is straightforward to calculate various static properties in this case exactly. We will be interested in the equilibrium correlation function between spins at different lattice sites: c,,

= (a,o,)

= Ul”-ml,

(2.5)

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123

where u ==: Tanh K. Similarly

Q-6)

we will be interested in the Fourier transform

= (1 - z?)(l + 212- 2u cos 4a,)-l. As 4 + 0, C(q) is related to the magnetic susceptibility 14u c(0) = kBTg = I--u

(2.8)

2 by

e x.

When we turn to dynamics we will be interested in time dependent correlation functions of the form (2.10) G&f> = (o,e &t 4 where D, is a single spin flip operator [9] with matrix elements (2.11) where a is a microscopic relaxation rate, &T$ , sets (T, = CT;at every site except n, and

W,(a) = 1 + 4Jn(%+1 + %L-1) is the spin-flip probability

(2.12)

with

A,--iTanhZK=-&G--%. This o,, is just the usual Glauber spin-flip operator. The primary properties of this operator are that it preserves the time translational invariance of the equilibrium probability distribution D,P[u]

= 0,

(2.14)

where D[u 1 u’] = D[u’ ) a],

(2.15)

and it obeys the symmetry relation D[u 1 u’]P[u’]

= D[u’ ) u]P[u].

(2.16)

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AND LUSCOMBE

It is simple to show that one can calculate the Fourier-Laplace

transform

C(q, z) = -i Joffidt e+-i”tc eiaa~cn-nL)Cn,,L(t) n

(2.17)

exactly in this case to obtain

C(q, -4 = z + ia(l C(q) - ycosqa,)’

(2.18)

In the limit as K -+ co, u + 1 this system displays a phase transition withcritical indices v + co, yl = 1 and P = 2, where z is the dynamic critical index. In the following sections we will be interested in whether we can reproduce these exact results using RSRG methods.

III.

THE RSDRG:

BASIC FORMULATION

A. Cell Specification and Block Spins The first step in a RSRG analysis is to divide the system into cells and associate block spin variables with these cells. There is, of course, a degree of flexibility in the choice of cell and in the nature of the block spin. In this paper we will keep things simple and restrict ourselves to the case of cells containing 2 spins and associate an Ising block spin /* (= 5 1) with each cell [lo]. We adopt the convention that the spins in the ith cell are labeled by cell index i and basis index a = & specifying the particular spin in the cell (see Fig. 1). If a spin is at position R, = na, , measured in terms of the original lattice spacing a,, , then it is in cell I, = (1 + (-1)“) 2

n 02+

(1 - (-1)“) 2

n+ 1 b--1

(3.1)

with basis vector a, = (- 1)“.

(3.2)

R, = a,[2i, + a,/2 - $1.

(3.3)

Thus we have

FIG. 1. o-spins (solid circles) are grouped into two spins per cell. r-spins (crosses) are the block spins associated with each cell. Convention for labelling cells and cell constituents is shown.

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125

B. The Mapping Function T[p ) u] A key quantity in the RSRG approach is the mapping function T[p 101, which basically gives the rule for mapping the primative spins {a} onto the block spins (,u}. For a given T[p 101 the renormalized probability distribution P[p] is related to the original probability distribution P[D] by

&I

= 2 P[ul T~CL I 01.

Since we require that the probability

distributions

p4

(3.4)

be normalized,

= Yp%l = 1, 0

(3.5)

we demand that

c n I 01 = 1. II

(3.6)

We are also interested in the way various spin variables transform under the RSRG. This leads us to introduce the notion of a collective variable. If A, is some spin variable for the original lattice then we define the associated variable A, on the coarse grained lattice via

4A~l = 2 PM A,% I ~1. 0

(3.7)

Note then on summing over p and using Eq. (3.5) that


(3.8) average over P[o]

C. The Eigenvalue Condition

In the original formulations [l l] of the RSRG, T[p / u] was selected out of convenience and physical insight. We have shown [l-3], however, in the case of dynamics that an arbitrary choice for T[~L / u] can lead to strongly non-Markoffian behavior in the renormalized spin-flip operator D,, . We have argued that it is very convenient and that one avoids the problems with non-Markoffian behavior if the renormalized spin-flip operator and the mapping function T[p j u] are determined simultaneously as the solutions of the eigenvalue equation: (3.9) The usefulness of this condition can be appreciated by considering the time evolution matrix Gt[u 1 u’] = eDot ~,,,,P[u’], (3.10)

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where a,,,, is the matrix setting on = C: for all lattice sites n. One can construct any time correlation function of the form (o,, ... u,ebo’aL ... u,) from GJo ( 0’1 by multiplying GJa 1 u’] by o, ..’ u,u; ... cri and summing over all u and u’. Remembering the RG transformation for P[u] given by Eq. (3.4) and noticing that GJu j a’] depends on the set of spins {u) and (u') it is then natural to assume that the time evolution matrix for the p-lattice is defined by

Gtb II-L’] = c c UP I ~1 G[u I ~‘1 UP’ I 4.

(3.11)

G,[u / a’] = 80,0,P[u’].

(3.12)

0 0’

At t = 0

It is very convenient

to preserve

this relation

under renormalization

GAP I $1 = h,u,W =C[p I $1 = mcL I aIT[P

and require

(3.13)

I 4,

which constitutes a normalization condition to be satisfied by the T’s. We have discussed in Refs. [l-3] the difficulties in evaluating G& I ~‘1 using Eq. (3.11) for an arbitrary choice for 7$ j u]. If T[p ( u] satisfies the eigenvalue equation (3.9) we find immediately that

= F P[u] T[p’ 1u] e’“Tb

/ u]

= 1 P[u] Tb’ 1u] eDUtT[p / u] =e Dut &&,u,p[p’l,

(3.14)

where we have used the normalization (3.13). This result gives us some confidence that the eigenvalue D, obtained from Eq. (3.9) corresponds to the appropriate spin-flip operator governing the dynamics of the coarse grained system. D. Recursion Relations for Collective

C’ariables

We indicated above that we could obtain relations between averages on the u-lattice and those on the p-lattice if we defined the appropriate coarse grained variable A, associated with A, . We found that


(3.8)

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127

We will see later that such relations are very useful. It will also be useful to relate space and time dependent correlation functions of the form

to their counterparts on the p-lattice. In developing this relationship it is very useful to introduce the projection operator Y, which projects A, onto its coarse grained equivalent: PA, = 1 Tfp 1u] P-$1

1 P[6] T[p 1 61 A6 3

(3.15)

=;Wh‘L

where, in the second line, we have used Eq. (3.7). We show in Appendix A that B is indeed a projection operator, P2 = 8. We note here that the identification of B as a projection operator assumes that T[p ] u] satisfies the normalization condition given by Eq. (3.13). We also want to introduce the complement to 9, 9=1-P

(3.16)

82 = 0.

(3.17)

so

Let us consider then the time correlation function C,,(t)

= c P[u] B,ebutA, ,

(3.18)

which we can write as

(3.19) where C;,(t)

= c P[u] B,,eBof.9’Ao D

C,,(t)

= 1 P[a] B,&9A, 0

(3.20)

and .

(3.21)

Let us concentrate first on f&,(t). If we use (3.15) in (3.20) we have C;,(t)

595/132/I-9

= c P[u] B,P’~’ c T[p 1CJ]A, . 0 !A

(3.22)

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We then use (3.9) to obtain CA&> = x A,eDwt C Pbl B,71jl,

0

=;A

I ~1

weD~tW’[tLl

I)

or (3.23)

CAB(~)= 1 P[pl ~ue”utA, u

and we see that CiB(t) is just the coarse grained version of CAB(t) (compare (3.23) and (3.18)). Turning then to CAB(t) we can write C,,(t)

= 1 P[o][PB, 0

+ SB,,] e”otZ?A,.

(3.24)

Consider then 1 P[a](i?YB,) eDut2A, 0

z c B,eDut IL

c P[o](2A,) 0

7[p 1u] = 0.

(3.251

Since

= T P[ul T[p I u] A, - c PC,1 T[cL I 01PA,

0

= 4J’bl - c f’bl QJ I 01c TLF.I ~1A, 0 P = A&L] We have then the important

- c 8J’[/T] ii

A, = 0.

(3.26)

result

CAB(t) = G(t)

+ (WC) edutWm)).

(3.27)

The physical interpretation of this result is relatively straightforward. In our problem the block spin variables t.~are assumed to carry the long distance and time information. Thus PA, picks out the slowly varying part of A, which we believe can be best treated using RG methods. Alternatively, 9A, should represent the remaining rapidly varying

RSDRG AND 1-D KINETIC ISING MODEL

129

degrees of freedom which can be treated using standard local perturbation theory methods. These physical assumptionsare at the baseof the quantitative theory which we now develop.

IV. PERTURBATION A. General

THEORY

Structure

In order to implement the formalism discussedin the last section we need to develop approximation methods for solving the eigenvalue problem posed by Eq. (3.9) and supplemented by the normalization conditions given by Eqs. (3.6) and (3.13). Our basic approach will be to divide the spin-flip operator into an intra-cell piece and a piece coupling cells. We can then solve the problem of uncoupled cells exactly and treat the interaction between cells as a perturbation. Written in terms of cell variables the spin-flip operator can be written as

where A, is defined in (2.13). In developing our perturbation theory it will be necessary, just as in the usual RSRG development, to distinguish between the intra- and intercell couplings. Let us write I?, in the more general form

D[u/ u’] = - T c (ly %tLJl + Alui,nai,-a + A$u.i,nui+n,--nJ, (4.2) &,a

where A, is the intra-cell coupling and Az is the inter-cell coupling. We assumethe parameters 01,A, , and A, can be expanded in the form

(4.3b)

7l=O

with A (0) = 0. 2

(4.4)

We can, in a limited sense,think of E as the interaction between cells. We, of course, desire that A, = A, at the end of the analysis. A key point in this development will be the flexibility to determine the various parameters iy, , A?’ and Ap’. We will seelater

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that there are a number of conditions we will want to impose that will determine these parameters. We assume for now that we know CL,, A?‘, and Ap’. B. Zeroth Order Analysis

To zeroth order in E we have the operator D(“)[o I 0’1 = - 7 c A~~laQJ:,,,[l

+ A4(P)oiJJi,-,].

z,a

This form for D,,O is consistent with a probability

(4.5)

distribution (4.6)

where the probability

distribution

for the ith cell is of the form eKo~i,+ui,-

P&T]

=

(4.7)

zo

where K, is the effective intra-cell coupling and Z, = 4 cash K, .

The relationship

(4.8)

between Aj”’ and K,, follows from the general condition ; DO[a ( 61 P,[O] = 0

(4.9)

and is given by A(O) 1

=

-Tanh

K,

= --u 0’

It is easy to show that the only non-trivial

(4.10)

cell average is

(u~,+u~,--)~ = Tanh K. =uo ,

(4.11)


(4.12)

where

The next step in our analysis is to construct the solutions to the eigenvalue equation:

; D[u 161@‘[CT] = -Xn$p[u],

(4.13)

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MODEL

TABLE I Eigenfunctions and Eigenvalues of the Cell Operator Boa

0

Note.

The projection

I

0

of u,,& along the nth eigenfunction

0

~,(a) is also given.

where & depends only on the spins in the ith cell. Using the results Qbi,a

= -N()(ui,n - U”Ui,&J,

Dro’,oI= 0,

D$7i,au.i,--a== -2~oI%.a%,--a - uol

(4.14a) (4.14b)

it is easy to determine the eigenfunctions and eigenvalues given in Table I. We normalize the eigenfunctions such that:

(*~“‘*y’)o = s,,,, .

(4.15)

We have also listed in Table I the values of the quantity

which are useful in our subsequent analysis. It is then straightforward to show that the zeroth order mapping function (4.17) 7 ; z‘)[p ( u] = ‘(1 2 + ~.zy(U)) z z

(4.18)

satisfies the eigenvalue equation (4.19)

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AND

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where (4.20)

It is easy to see that TO satisfies the normalization

conditions

1 Tob I 01 = 1

(4.21)

and

where

Note that we would also satisfy Eq. (3.9) if $I:~‘( 0) is replaced by any of the other $r)(u). We choose $J~“(o) since physically it corresponds to the “slowest” decaying odd eigenfunction. Consequently the block spin pi is to be identified with the most slowly varying degrees of freedom in the cell. C. First Order Analysis

In Ref. [3] we discussed how, if we expand the spin-flip operator about the uncoupled cell B,, = a,0 + f

,qjp

(4.24)

n=d

we can construct T[p j 01 and D, in an expansion in E: TIP I 01 = ToUp I 01 + f

@T@Q

j u],

(4.25)

?k=l

D, = D,O + ‘f PDF).

(4.26)

7t=l

The first step in the analysis is to construct the solution to Eq. (3.9) satisfying the normalization conditions (T(‘W

I 01 Tab’ I ~I>,, = 0

n > 1.

(4.27)

We easily obtain at first order that D-iclYP I P’l ~o[cL’l =
I aI>0

(4.28)

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133

and

= 5 D(l)[t, / ji] To& 1 u] - 4 @l)[u 161 T&L 1 a].

(4.29)

In this paper we will limit our analysis to first order in E. To this order we do not need to explicitly construct 7’(l). It is clear that we can construct the D,“(n) in (4.24) by simply inserting (4.3) in (4.2) and collecting coefficients of E”. This is rather an inefficient way of organizing the calculation and has the disadvantage of mixing together the expansions for 01and the A’s. This is undesirable since 01represents an intrinsically dynamic quantity and the A’s are intrinsically static in nature. We want to develop an expansion that maintains a separation in these different quantities. Let us define the partial sums (4.30a)

and introduce the notation d,(a, A, , A,), where the dependence on the parameters is explicit. We can then:define p

= D&y, ) Al”‘, A?‘) _ @-“,

(4.3 1)

where (4.32) We have then that (4.33) or (4.34) We find then that I&-L I ~‘1 Po[p’l =
I 01 k@i

, AI”, At’) Tab I 4)o + WE’).

(4.35)

It is easy to carry out the average in this expression to obtain

(4.36)

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MAZENKO

AND

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where

r&z) = (z)(l) A q!Jp>,+ A;‘( p(A Q) r&u& )

(4.37a)

r2(a) = v,,@(o,

(4.37b)

and A$:” = #I:‘, - ${$o,,

A #t’),

. Using th e results from Table I we obtain r1 = 1 + A:’

(4.38a)

r2 = A?‘.

(4.38b)

and

Since rI and rz are independent of a we can write

We note at this stage that a[, 1 ~‘1 has the same form as D[u 1 a’] given by (2.11) and (2.12). However, we must remember the distinction between D, and D, . The difference is that B, is associated with T[p 1u] satisfying the normalization (4.27) while D, is associated with a r[p I CT]satisfying (3.13). We assume T and 7 are connected by a rotation in p-space,

UP I 01 = 1 Sb I PI TLPI 4

(4.40)

P

where we assume S is a symmetric matrix. variables)

Then, since T satisfies (sum on barred

m I m[P I 61= mtL I FIW I 4

(4.41)

DIP I $1 = SIP I i@% I Fl WE-i’ I $1.

(4.43)

we have that T satisfies

where

Thus in order to connect D, and EM we must, in principle, construct the matrix S. This analysis is rather involved and is relegated to Appendix B. The basic result is that

SIP I $1 = Ld + l P

I $1 + O(E2),

(4.44)

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KINETIC

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135

MODEL

where (4.45)

We have then that

D[P I ~‘1 = % I ~‘1 + Ec {Jflu I FL]Do& I ~‘1 - D”b I A WF I ~‘1)+ O(c2) P (4.46) = aP I $1 i.e., M and Do commute, so the rotation does not influence D, at first order. If we compare Eqs. (2.1 l), (2.12), (4.39) and (4.46) we see that after renormalization we have an operator of precisely the same form (to this order in l ) but with the renormalized parameters a’ = G,(l + 2:)) (4.47) and (4.48)

It is not clear that this prescription for determining K’ is consistent with that given in Appendix B. We can read off from Eq. (B.21) for P[p] that

$ip I‘?= --a- + O(2).

(4.49)

1

If we rewrite (4.48) as - 1 Tanh 2K’ =

i?,ap 2cr,(l + iiy>

=--

&p 2ru’

(4.50)

and expand the left side in powers of K’ N O(E) and remember that oi’ = h, + O(E) then we recover (4.49). Thus (4.48) is a convenient resummation of higher order terms in E. D. Collective

Variables and Correlation Function Relation

The perturbation theory expansion given by (4.24) can also be used to explicitly evaluate the collective variables defined by (3.7). It is obvious that

In this paper we will restrict ourselves to this lowest order case. We easily obtain the results given in Table II. Using the result for ui,n we have, from (3.8) that

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MAZENKO

AND LUSCOMBE

TABLE II Cell Quantities A,, and the Associated Block Spin Variable A,

Since the magnetization

is given by

m(K) = <%,a)

(4.53)

then, in this approximation, m(K) = lyn(K’),

(4.54)

where m(K) = (pi)’ is the magnetization for a system with the coupling K'. Analysis of this equation shows, as we might guess, that there is no spontaneous magnetization, m(K)= for this one-dimensional

which is a non-trivial

0

(4.55)

system. Using the result for (T~,~u~,~Jin (34,

gives

relation. If we define (4.57)

then it follows that E(1) = 3 + q

E’(1)

(4.58)

and

for n > 0. Similarly

c(2n) = V12E'(n)

(4.59)

c(2n + 1) = J$ (E’(n) + E’(TI + 1))

(4.60)

the Fourier transformed correlation function (4.61)

RSDRG

satisfies the recursion

AND

1-D

KINETIC

ISING

137

MODEL

relation

C(q) = 2v12C’(2q) cos2(qaO/2) + (I - u,) sin2(qa,/2). An important limiting case is that of the magnetic susceptibility the simple recursion relation

(4.62)

(see (2.9)) satisfying

x = 2v,2$.

(4.63)

Of course we are primarily interested in dynamical quantities. Therefore we want to implement (3.27) for the case where A, = ‘J~,~and B, == uj,,’ . In this case, to lowest order,

and

where v,(u) is defined by (4.16). We then easily obtain ((2uj.,,)

e”“t(Sui,a>>O = & C (1 - %,,) v,(a) ~,(a’) e+ 12

and the dynamic recursion Ci,,i,,(t)

+ O(E)

(4.66)

relation

= v12C&(t) + 6i,j C (1 - a,,,) e-‘ntvn(Q) ~,(a’). a

(4.67)

A useful check on this result is that at t = 0 it agrees precisely with (4.56). This follows from the recognition that c %(4 %W n

= (U,Ud>o = Ld

+ (1 - &?4,> *o .

(4.68)

Using the explicit results for v, we obtain Ga.dt) Taking the Laplace-Fourier

(4.69)

= transform

C(q, z) = -i Jamdt @ $ C C e-i”“~(“-““Cn,p(t) n ?I’ = (1 + uo) cos2(qao/2) C’(2q, z) + $$$

(4.70) sin2(qao/2).

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MAZENKO

AND LUSCOMBE

At this stage we have established a number of quite interesting interrelationships between the original and coarse grained problems. In particular we see that, at least to first order in E, the problems are closely related in that they both depend only on the two parameters 01and K. No new couplings are generated. However, it is important to realize that until we specify the various parameters 01~) 01~, Ai” and A?’ we do not have a quantitative theory. This specification will be carried out in the next section. V. SPECIFICATION OF PARAMETERS We see at this stage that we must choose the “bare” parameters 01,) A:“’ and Ap’. We show in this section that it is essential to make intelligent choices for these parameters if we want to obtain quantitative results. We also discuss in this section the several bits of physics that are important in fixing these bare parameters. A. Large Distance Correlations In the development of the RSRG thus far, the emphasis has been on building the collective behavior out of very short range correlations. Indeed we see that our first order recursion relations give us only information about nearest neighbor cells. We expect that this analysis will be very good in treating the short distance behavior and hope that the long distance behavior is treated accurately. In order to check our approximations we consider the simple calculation where we compute the space dependent correlation function c(n), with n = 2” and s an integer, from the recursion relation

l)

=

,‘@‘S-1’) (J+j,

where we use the recursion relation resulting from a direct first order cumulant expansion [ll, 121 K’ = ; (1 + u)

(5.2)

with the intra-cell coupling simply equal to the bulk coupling ug = u. If we iterate the above recursion relation s times we obtain 429 = ‘3 (J+)

E,(l),

i=O

where ui = tanh Ki with Ki the value of the coupling after i RG iterations and e,(l) is the nearest neighbor correlation function evaluated for the temperature KS. The results of this calculation are shown in Table III for the case where u = 4. We

RSDRG

AND

1-D

KINETIC

TABLE Correlations

ISING

139

MODEL

III

between Spins Separated by 25 Lattice Spacings

28 1 2 4 8 16 32 64 128

42”) 0.5

0.3902 0.2788 0.1811 0.1077 0.0598 0.0317 0.0164

0.5 0.293 0.145 0.0604 0.0212 0.00652 0.00183 0.000489

0.5 0.250 0.0625 0.00391 0.0000152 2.33 x IO-‘” 5.42 x 1O-2o 2.94 x lo-=’

Note. Shown are uB and ~(2”), hyperbolic tangent of the coupling after s RG iterations and the correlation function, computed from (5.2), the recursion relation obtained at first order in a cumulant expansion. Also shown is the exact correlation function from the Ising model.

also plot the exact value of 429 = 8. The exact result indicates that the correlation functions decay exponentially with distance

where the correlation length is 5 = (I In u 1)-l.

(5.6)

We find, however, for large distances that our approximate calculation for c(n) leads to algebraic decay,

where, in this case LY= 2. To seethis analytically note first that as

(5.8) Consequently, if (5.9)

140

MAZENKO

AND

LUSCOMBE

then ,im ,(2”-h) -=lim-=-=s-z

2*”

4")

s-m

1 2N

2=1(5+1)

1 4

and we find that a: = 2. This algebraic decay of correlation functions is, as pointed out by Fradkin and Raby [13], a general feature of cumulant expansion developments of the RSRG that shows up in higher dimensions and in treatments of quantum problems, and of course, is in serious disagreement with the known qualitative behavior of these systems. The difficulty in this case is associated with the form of the recursion relations for weak coupling. Looking at Eq. (5.4) in the case where s is large we can write, to a good approximation, that (5.11) since, after s iterations, we see that

the coupling

will be very small. From the analysis

above

(5.12) so

Let us then assume we have a recursion

relation of the form

K’ = aKc

(5.14)

that is valid for small K. For this recursion relation to correspond require c > 1. We have then, after s iterations, KS

=

to a Id model we

a~C'-l)/(C-l)Kc' n

=

and exponential

exp[ln(K~a(l-C-*)/(C-l)

) +l

C/lU

2

1

(5.15)

decay results onZy for c = 2. Letting c = 2, KS = exp[ln(K,,a(1-2-S))~]

(5.16)

KS = exp(n ln(K,,a))

(5.17)

and for large s

and we have exponential

decay.

RSDRG

AND

1-D

KINETIC

TABLE High Temperature

ISING

IV

Recursion Relation and Corresponding of Correlation Function

Recursion relation

141

MODEL

Correlation

Form

function c(n)

Kg’n”

(l/n) exdn In(K,a)l

If c = 1 (as given in the usual cumulant expansions) then KS = exp[ln(K,,a+“)]

(5.18)

= &as

and for a = 4 we recover the cumulant expansion result. In Table IV we compare the long distance behavior of c(n) corresponding to the recursion relations for weak coupling K’ = K/2 and K’ = aK2. We seethat the cumulant expansion result gives completely unphysical behavior. The recursion relation K’ = aK2 is qualitatively correct when compared with the exact result (5.5) except for an overall l/n factor. It seemsnecessary, since in a recursion analysis we always scale to the weak coupling fixed point and it controls the long distance behavior, to arrange our calculation so that we obtain a weak coupling recursion relation K’ - A?. Typically in the RSRG one is concerned with the recursion relations only near the fixed point. We seehowever that when we deal with the spacedependent correlation functions it is necessaryto worry about the form of the recursion relations far away from the fixed point. The important question now arises as to how we can derive recursion relations compatable with these weak coupling requirements. Our strategy will be to focus on the recursion relation we derived for the c(n). Notice that if we take ratios of the c(n) the v12factor cancelsand we obtain an implicit relation between K and K’ if we assume that E is a known function of K and E’ a known function of K’. Then, for example, h n > 1)

424 E’(n) T(G)=? 6 (ml ’ 42n + 1) = 1 e’(n) + E’Cn+ 1) e(2m)

z

e’(m)

(5.19a)

’

c(2n + 1) c’(n) + e’(n + 1) 42m + 1) = E’(m) + c’(m + 1) *

(5.19b) (5.19c)

We know, however, that for small K c(n) = Kn

(5.20)

142

MAZENKO

AND

LUSCOMBE

so (5.19a) gives KW--ml

=

(K’)(n-m,,

(5.21)

(5.19b) gives p(npm,+l

-_ i (0” 2

+ (K’)n+l (K’p

’ (5.22)

= $ (K’)(-)[l

+ K’]

and (5.19~) gives KZ(n-nl) = (K’)n(l (K’)“(l

+ K’) = (K’)n-,n + K’) *

(5.23)

Notice that Eqs. (5.19a) and (5.19~) lead to the desired result K’ = K2. We will choose one of these ratios as a condition determining requires a separate calculation to determine the required correlation use the condition E(4) TV7

43 E (1)

(5.24) K’ = K’(K). This functions. We will

(5.25)

since we believe the recursion relations for the c(n) are more accurate for smaller n, so we choose the smallest possible values of n and m. We also want n and m to be small since they are then, in the more general situations where we do not know the e(n) exactly, more susceptible to accurate approximate treatment. In this simple one-dimensional problem we know the c(n) exactly, e(n) = Mini, and we easily obtain the general recursion

(5.26)

relation

u’ = u2.

(5.27)

A very interesting aspect of this approximate recursion relation is that it agrees precisely with the results of an exact decimation calculation [7]. We will use the recursion relation (5.27) in the rest of our work. B. Determination

qf the Cell Temperature

We turn now to the determination of the “cell temperature” or the coupling appropriate to uncoupled cells KO . The first impulse is to set KO equal to the bulk coupling K. We will now make some strong arguments against this prescription.

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143

MODEL

Let us look at the recursion relation for the nearest neighbor correlation function (1 +4 uo) &(I) *

E(l)++

(5.28)

Remember that K’ = K’(K) is now fixed. If we analyze this equation for weak coupling

41) -4

(5.29)

~‘(1) - K2 and K0 = 2K + O(K’)

(5.30)

we see then that for weak coupling we should choose the cell coupling to be twice the bulk coupling. This makes good physical sense. In the original problem each spin feels two spins of coupling K. When we approximate the system as a set of uncoupled cells each spin interacts only with one spin. If the net interaction on the spin is to approximate that in the bulk then the strength of the interaction in the cell should be twice that in the bulk. In most problms of interest we can determine ~(1) either exactly, in high temperature expansions or using Monte Carlo techniques. We therefore assume ~(1) and therefore also ~‘(1) are known and use (4.58) to determine u,, = u,,(K). In this simple one-dimensional case E(1) = u, (5.31) E’(1) = z?,

(5.32)

u = u(4 - u> 0 24u2 *

(5.33)

and

We see immediately that no - 2~ for small u and u. - u as u --f 1 as desired. In Fig. 2 we plot K, K, , and u. versus K. We also plot 2K and note that K, is approximately given by 2K over the entire temperature range. C. Determination

of the Other Static Parameters

At this point we have determined A 1o = Alo and K’ = K’(u). We still need to specify Ail’ and A, “) . Note that we have the condition from (4.48) that 1

A(l)

A;=21+2A:11

zi

z----=-p. 1 + Ii’2

2.2

1 + u4

(5.34)

where we have used (2.13) and (5.27). We, of course, have the constraint that on summing to all orders in E, A, = A2. We can enforce this condition at this order by requiring A(l)

= 2

p 1

.

(5.35)

144

MAZENKO

AND

LUSCOMBE

FIG. 2. (a) Coupling K, and cell coupling KO as functions of the “temperature” u(=Tanh K). Also the dependence of u,, , the “cell temperature,” on u is given. (b) The relationship between 2K and KO over the entire temperature range. In the limit u --f 1, (2K - KO)/K goes to zero. Though both 2K and KO diverge as II ---f 1, their difference tends to the value ln(2/1/3).

We have then that (5.36) or (5.37) The other quantities of interest are A(1)

1

=

al”

_

A(o) 1

-2u2 u(4 - 24) = (1 + U2)2+ 2 + u2

(5.38)

RSDRG

AND

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KINETIC

TABLE Static

Parameters

ISING

145

MODEL

V

of the Spin-Flip

Operator

to First

Order

-

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0 - 0.099 -0.192 -0.275 -0.345 -0.400 -0.441 -0.467 -0.488 -0.497 1L

0 -0.194 -0.373 -0.531 -0.667 -0.778 -0.864 -0.928 -0.970 -0.993 -1

0 0.174 0.299 0.380 0.429 0.458 0.475 0.486 0.494 0.498 0.500

0 -0.079 -0.118 -0.124 -0.107 -0.080 m-O.052 -0.028 m-o.012 -0.003 0

and (5.39)

We list A,, Ai”, Ai” and A, - All) as functions of u in Table V. The first point is that Ai” ranges from 0 to -I as u goes from 0 to 1. Since A, ranges from 0 to - 2 this requires that A!” range between 0 and 4. The important point is that A, - $) is reasonably small over the entire temperature range. At this point the static problem is completely specified and one can proceed to the calculation of the various static quantities. This will be carried out in Section VI. D.

Determination

of Dynamic

Parameters

We must still specify the parameters 01~and 01~. From Eq. (4.47) we have the relation 01’= &(l + /@‘) G dcr (5.40) and we can eliminate 01~in terms of d and treat 01~and d as our basic parameters. As in the static casewe will use the recursion relations to tie down the parameters d and 01~. In this casewe will usethe time dependent recursion relation given by (4.67). We would like one condition reflecting short-distance and short-time behavior (which should, qualitatively, determine go) and another condition reflecting long-distance and long-time behavior (which should determine d). The short-time short-distance behavior is measured by the initial slope “on-site” I- = (-g W’),=, = = --a (-&$),

(5.41)

146

MAZENKO

AND

LUSCOMBE

where, in the last step, we evaluated the full average explicitly. recursion relation satisfied by r from (4.69) is

r = (LpL)

r’ - F (1 - u$).

The approximate

(5.42)

This equation gives one constraint on the dynamic parameters. We next want to introduce a long-time and -distance condition. The convenient quantity to calculate in a high or low temperature perturbation expansion is the small 4 and z limit of the Fourier-Laplace transformed quantity C(q, z). We can write quite generally [14] that

aq, 4 = z + Q4) i#&, 5 ’ where $(q, z) is the associated memory function. We want the q = z = 0 component of the recursion relation for C(q, z) to “essentially” determine rl. We have from (4.70) C(0, 0) = 2V,2C’(O, 0)

(544)

-= $(o”, 0) 2v12 &Y) .

(5.45)

or, in terms of (5.43),

Using Eq. (4.63) we obtain the simple result

4c0,0) = $‘(O, 0).

(5.46)

It is not difficult to show that (5.47) and c&K) = ol’f(K’)

(5.48)

and, using (5.40) in (5.48),

A = f(K) fo’

(5.49)

In this case we know that, from (2.18)

f(K) = (1 - +J

= ‘; ; ;1’

(5.50)

and therefore A=

1 + u4 (1 + u2)(1 + u>’ .

(5.51)

RSDRG

AND

1-D

KINETIC

TABLE

ISING

VI

Dynamic Parameters of the Spin-Flip

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

147

MODEL

Operator to First Order

1.022 1.081 1.177 1.321 I .538 I .879 2.471 3.689 7.407

-0.187 -0.359 -0.532 -0.729 -0.982 -1.348 -1.956 -3.183 -6.905

1.00 0.834 0.722 0.645 0.592 0.556 0.531 0.516 0.506 0.501

1.00 0.818 0.669 0.547 0.451 0.378 0.324 0.288 0.265 0.253

co

-lx

4

i

1.00

0

Note. The time resealing factor A is also given.

Since, as u ---f I, d + $ we guarantee that we obtain the correct value of the dynamic critical index z = -1n d/in 2 = 2. Using the exact result for r given by (5.41) and the above result for d we can solve (5.42) to obtain

01 (2+u2)(1 +2u+2u” i- u4> CL o=- 2 (1 + 2u)(l - u”) -__.

(5.52)

In Table VI we have listed ~+,/a, aI/ a:, G/M and A as functions of u. We note that the bare dynamic parameters are rather sensitive functions of temperature. The “divergence” of CX,, as u ---f 1 simply reflects the fact that 1 - u, goes to zero faster than 1 - u as u + 1 and the relation (5.42) requires (Y,,to compensate for this effect. We have now specifiedall of the parameters neededin our lowest order calculations. Before investigating the predictions of this theory we should comment on the logical structure of our treatment of these dynamical parameters. We have used the exact values of the memory function at q and z equal zero to determine the time resealing factor A. One could counter that this, to a large degree, begsthe question. After all, if we know the memory function then we know the correlation function and this is the fundamental object of our investigation. There is, however, an important distinction to be made here. In our determination of A we require onZythe ratio of memory functions at different temperatures. Thus in other situations where we do not know the memory functions exactly and where their direct calculation may be very difficult due to critical fluctuations we expect that a direct calculation for their ratio should be well behaved and amenable to approximation.

148

MAZENKO AND LUSCOMBE VI. EVALUATION OF OBSERVABLES USING RECURSION RELATIONS

A. Calculation of Static Quantities Having determined, to first order, the parameters introduced into the model, we are in a position to implement the recursion relations we derived in Section IV. We shall consider first the numerical evaluation of static quantities. The static, space dependent correlation functions are rather easy to compute. This is due to the fact that a pair of spins initially separated by a distance na,, are nearest neighbors on the coarse grained lattice after n/2 or (n - 1)/2 iterations. We are then left with the determination of the nearest neighbor correlation function at the appropriate higher temperature. In Fig. 3 we present e(n) for various n. These were computed using (4.59) and (4.60). The correct qualitative and reasonably close quantitative agreement with the exact values gives us confidence that we understand the transformation of static quantities through renormalization. Next we consider the recursion relation satisfied by the static susceptibility (4.63). We can rewrite (4.63) in the form x ~ 1 = 2l5” - 1 + 2v,Q’

-

1).

(6.1)

We then iterate this expression in the form,

(6.2)

0.1 0.0 Iea. 0.0

FIG. 3. Static correlations between spins separated by n lattice spacings. Filled circles are the exact results, while open circles are the RSRG results.

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149

MODEL

TABLE VII Static Susceptibility Calculated in the king Model, the RSRG, and with the Cumulant Expansion Recursion Relation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.99

1

1 1.231 1.547 2.003 2.715 3.942 6.412 12.840 40.824 615.518 43,244.4

1

1.222 I .500 1.857 2.333 3.000 4.000 5.667 9.000 19.000 39.000 199.000

1.218 1.486 1.829 2.286 2.926 3.889 5.495 8.714 18.379 37.717 192.440

where, by x”, we mean x evaluated at the temperature after two RG iterations. As we continue to iterate we flow to high temperatures, v12 -+ 2, 2v12 - 1 + 0, and the sequence converges. We list in Table VII the exact value of x, the value obtained from the recursion relation given above, and, for comparison, that computed using the cumulant expansion result (5.2). Again we see the inadequacy of the cumulant expansion in one dimension. Let us turn now to the recursion relation (4.62) for the wavenumber-dependent correlation function C(q). With the cell temperature of (5.33) the recursion relation for C(q) can be written as

Ck7)= fh s>C’G?) + g(f4Sk

(6.3)

where

and (64b) Equation (6.3) represents a more general recursion relation than the ones we have considered so far, in that it depends on the wavenumber q. As the rest of the recursion relations we will evaluate have this same structure, we examine the implementation of (6.3) in some detail. Iterating, we will develop a series of terms of the form C(q) = g +fg’

+fs’g”

+fslf”g’”

-- . ..)

150

MAZENKO

AND LUSCOMBE

FIG. 4. Wavenumber-dependent correlation function C(q) as a function of temperature. open circles represent the exact results, while the filled circles are the RSRG results.

The

where at each step of the iteration we let u ---fU’ = u2 and q --f q’ = 29. There are two competing effects in a numerical evaluation of (6.5). Products of the temperature dependent parts of the homogeneous terms are always greater (usually much greater) than 1, while the inhomogeneous terms and all trigonometric factors are less than 1. As we scale to high temperatures where both homogeneous and inhomogeneous terms become temperature independent, convergence is achieved through the action of the trigonometric terms. It may happen however, if q is of the form (2m + 1)42”, integer n and m, that the series is truncated after 12iterations. This is an artifact of choosing two spins per cell in the analysis. We may, in fact, exploit this circumstance by approximating an arbitrary q by one of the above form with suitably chosen integers

C(q)

'o-3o

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

q/lr FIG. 5. Wavenumber dependence of C(q) at different temperatures. The RSRG results are given by open circles for u = 0.1 and 0.5 and by the solid curve for u = 0.95. The exact results are given by filled circles for u = 0.1 and 0.5 and by the dashed line for u = 0.95.

RSDRG

AND

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KINETIC

ISING

MODEL

151

n and m. It can be shown that the error introduced by this procedure is negligible if n and m are large enough.

Returning to the particular case of (6.4), we present in Fig. 4 the temperature dependence of C(q) and in Fig. 5 the wavenumber dependence. In cases where the difference between the exact and RG calculations is too small to graph, we present alternatively one and then the other. In Fig. 4, the q =: 0 case is just the static susceptibility of Table VII. Tt may be seen in Fig. 4 that the smaller the value of q, the lower is the value of the temperature for which the two calculations agree. This effect is also seen in Fig. 5. The exact result is given by Eq. (2.8). The trend of the calculations remaining accurate at small wavenumbers as we go to low temperatures also appears in the dynamics. We reserve comments until the dynamics are presented. The oscillations seen at low temperatures are due to the large number of products of cosines generated in iterating the recursion relation. This effect also appears in the calculation of dynamic quantities. The recursion relations for the static quantities we have considered were derived using the eigenfunction of a dynamic operator. Then it is perhaps surprising at first sight that the statics are successfully treated in a dynamic theory. Recall, however, that the SF0 has detailed balance built into it and that the expansion method employed in perturbation theory carefully keeps separate the statics and dynamics. We now turn our attention to the dynamics. E. Calculation of Dynamic Quantities

Starting with the basic time-dependent recursion relation (4.69) we can analyze various dynamic quantities. A simple dynamic quantity to calculate is the spin-relaxation time, defined via 7=

m dt C”(t)

s0

(6.6)

where C,,(t) is the spin-spin autocorrelation value for this quantity is given by [8]

function. It can be shown that the exact

017= [%]I.

(6.7)

We have seen (5.40) that upon renormalization, the relaxation rate 01is resealed by the quantity d(u) given by (5.51). We may rewrite (4.69), making the 01dependence explicit, as CO(mt) = v12C;(&t)

+ v22e--1\2t,

where h, contains the cell parameter 01~. Defining t’ = At and integrating times, we arrive at a recursion relation for 7:

(6.8)

over all

(6.9)

152

MAZENKO

AND

LUSCOMBE

ar

U

FIG. 6. Spin relaxation time as a function of temperature. Solid line is the RSDRG the dashed line is the exact result.

result and

With cell parameters determined, we have, VI2 ~_ (1 i- 2u)(l + zP)(l + U)” A (2 $ u”)( 1 + z/q-- ’

V2’)

Tq=

1 (1 - U)2(1- U”) -G (2 + u2)(1 + 2u + 224”+

(6.10a)

u4)

.

(6. lob)

The results of iterating (6.9) are presented in Fig. 6. At T = cc, T = l/a. This is to be expected from the Glauber model: (a)-’ representsthe interaction with the heat bath and is the time in which spins flip independently of one another. For lower temperatures T increases, reflecting the effects of correlations among the spins. As T--f 0, T + cc indicating a phasetransition at T = 0. It is seenthat the temperature dependenceis correctly provided for but that the RG value is too large at all temperatures. We may also iterate (6.8) directly, taking care to scale times at each iteration, t --t t’ = At, as well as letting u -+ U’ = ~3.We present in Fig. 7 the time dependence of the autocorrelation function Co(t). Our purpose in presenting a numerical calculatio of the autocorrelation function is that it demonstrates the phenomenon of critical slowing down, which may not be evident from the behavior of the transformed functions. Remember that the exact value (5.41) of the slope of C, at t = 0 as a function of temperature was used to deterline o(”.

RSDRG

AND

1-D

KINETIC

ISING

MODEL

153

01

FIG.

7.

Time

dependence of the spin-spin autocorrelation

function from the RSDRC at various

temperatures.

In order to implement the recursion relation for the transformed correlation function C(q, z), we introduce the scaled frequency Z’ = Z/L in (4.70) such that C(q, z) = y

cos”(qa,:2) C’(2q, 2’)

2v ” -+ -A-sirP(qa,i2). z + A2

(6.11)

Whereas in iterating (6.8) we scaled to short times, in implementing (6.11) we scale to a high frequency limit as we flow to high temperature. In Fig. 8 is shown l/O as a function of temperature. We see that for low temperatures z starts out being multiplied roughly by 4 at each step in the iteration. Thus z flows rapidly away from its original value. Given the determined cell parameters, the temperature dependenceof the inhomogeneous term reads 2v,” 1 2( I - zz)Z(I - 114) = L A2 == -012 f Uf[(Z/01)(1- u4) -I- 1 -{ 211+ 221”+ Z/4]

(6.12)

The temperature dependenceof the homogeneousterm is the sameas that given in (6.10a) (multiplied by two). In Fig. 9 we plot (6.12) for various values of ;. As discussed above, the rate of convergence of the serieswhich results from iterating a recursion relation of the form of (6. I I), is controlled by the size of the inhomogeneous term. As may be seenin Fig. 9, the inhomogeneousterm scalesto I in the high temperature limit only for the caseof z = 0. Any other (finite) z will rapidly scaleto large values

FE. 8. Frequency resealing factor as a function of temperature.

154

MAZENKO

IO

0.9

0.8 0.7

AND LUSCOMBE

06

0.5

3.4

0.3 0.2

0.1

0.0

FIG. 9.r Frequency and temperature dependence of Eq. (6.12), the inhomogeneous recursion~relation for C(q,z).

term in the

in a diminishing inhomogeneous term in the high temperature limit. The z = 0 case is by far the case of the slowest convergence. For this special case, as we flow to high temperatures both homogeneous and inhomogeneous terms scale to 1. Thus we particularly are at the mercy of oscillating products of cosinesto provide convergence. The fluctuations in the low temperature behavior of the solutions to dynamic recursion relations will be worse than in the static case. This is because in the static situation the homogeneous term ranged from 2 to 1 in going from low to high temperature (6.4a) while in the dynamics this same term ranges between 8 and 1 over the same temperature interval. These fluctuations are then especially amplified in the z = 0 case. resulting

FIG. 10. Wavenumber dependence of C(q, z) for z = 0 at several temperatures. The dashed lines and the open circles represent the exact results, while solid lines and filled circles denote the RSDRG results.

RSDRG

0.20

FIG.

AND

1-D

0.24

022

KINETIC

0.26

11. Expanded view of Fig. 10 from

q

ISING

155

MODEL

028

0.30

032

034

= 0.2~ to 0.35~ for the low temperature case.

We now examine the dependence of C(q, z) on each of its arguments through the iteration of (6.11). Figure 10 shows the calculated and the exact results over the first Brillouin zone for z = 0. We have the expected disagreement at low temperature and also the rapid oscillations. On an expanded scale these oscillations are seen to be smooth (Fig. 11). Figure 12 is the same as Fig. 10 except that z = 1. The improvement is evident. In both Figs. 10, 12 the rapid rise in C(q, z) at low temperatures as q --t 0

represents the increasing significance of long wavelength fluctuations as we approach

ciq,z)

!-_L___L_I 3.0

0.1

02

0.3

0.4

05

0.6

0.7

0.8

09

1.0

FIG. 12. Wavenumber dependenceof C(q, z) for z = 1.0. The dashed lines and the filled circles represent the exact results, while solid lines and open circles denote the RSDRG results.

156

MAZENKO

AND LUSCOMBE

the critical point. Note also as we approach low temperatures the exact and approximate calculations agree only at progressively lower wavenumbers. This was observed in the statics. Recall that in developing the general form obeyed by recursion relations (3.27), the slowestcell modes were projected out. These “slowest modes” will represent long wavelength phenomena. The remaining degreesof freedom were left to be dealt with perturbatively. It is expected that in a calculation where we compute the collective variables to higher order and also where we work to higher order in solving the eigenvalue equation (3.9), we will see an improvement in the shorter wavelength behavior. It is also hoped that a higher order calculation will smooth out the Auctuation seenat low temperatures. In Figs. 13 and 14 we plot C(q, z) versus frequency. At q = 0 there is a distinct separation in long-wavelength and long time behavior with temperature (Fig. 13). There are two effects to be observed in Fig. 13. One is that C(q, z) is inversely proportional to frequency; second is the increase in magnitude of C(q, z) at lower temperatures. Both of these support the general picture of second-order phase transitions where long-distance, long-time correlations become larger and larger as one approaches the critical point. Recall that the q = z = 0 limit was used in the determination of d. Otherwise the finite z dependenceis treated quite well in the q -=m0 limit. For q # 0 (Fig. 14), the frequency dependence is correctly treated, but the amplitude is not. Note the discrepancy at z = 0 for the u = 0.95 case. As noted above there are problems associatedwith calculating z identically equal to zero (for q # 0 of course). Finally in Figs. 15 and 16, we present the temperature dependence. We again see the agreement in the calculations in the q = z = 0 limit (Fig. 15). This is certainly expected. We also see the dominatin g influence of zero frequency at q = 0 in its

Z/a 13. Frequency dependence of C(q, z) for q = 0 at several temperatures. The open circles represent the exact results and the dark circles are the RSDRG results. FIG.

RSDRG

AND

1-D

KINETIC

ISING

157

MODEL

FIG. 14. Frequency dependence of C(q, z) for finite wavenumber at several temperatures. For the low temperature case, the exact results are given by the dashed line, while the solid line is the RSRG result. At higher temperatures, the RSDRG result is represented by the open circles and the exact result by the filled circles.

/ 0.4

0:s

1 0.6

/ 07

I 08

I 0.9

1.0

”

FIG. 15. Temperature dependence of C(q, z) at long wavelength. The RSDRG results are given by the open circles, while the exact results are given by the filled circles.

158

MAZENKO

AND LUSCOMBE

u FIG. 16. Temperature dependence of C(q, z) for finite wavenumbers. The RSDRG results are given by the open circles while the exact results are given by the filled circles.

contribution to correlations at the phase transition. We see (Fig. 16) for q # 0, C(q, z) vanishes as u -+ 1, but that the zero frequency contribution still dominates over that for finite frequency. These again are consistent with general ideas about second order phase transitions. Also again, we see that the overall temperature dependence is correct as we approach the critical point, while the amplitude is given incorrectly.

VII.

FINAL

COMMENTS

In this paper we have shown how the RSDRG method can be used to carry out practical calculations for a simple model system. Given the elementary nature of the approximations we find the quantitative power of the theory rather satisfactory. It is not surprising that our methods work best for long wavelength and low frequency behavior and very short wavelength and high frequency behavior. The major errors are in the intermediate regime. It is clear that the extension of these methods to higher order in Eis required from a formal point of view to guarantee that the methods can be systematically improved. Such an extension is currently underway. It is also clear that it is more important to demonstrate the usefulness of these methods in treating problems where we have no exact solutions to guide us.

APPENDIX

A: THE PROJECTION OPERATOR

S

We show here that Yz = 9 by direct analysis. We note that

(A.11

159

RSDRG AND 1-D KINETIC ISING MODEL SO

PA,

= 1 T[p’ ] a] P-lltt’] c P[,] T/JL’ j E](iPAJ II’ (I

64.2)

using the result PA, = 1 T[p 1u] A,

(A-3)

we have

(A.4)

APPENDIX

B: THE FIRST ORDER ROTATION

The “rotation” in EL. - space is associated with the fact that G[p 1 ~‘1 = (T[p 1u]T[p’ 1u]) is not diagonal in p and p’. In this appendix we construct the rotation matrix S defined by Eq. (4.40) correct to O(E). The first step in this analysis is the construction of G[p ) ~‘1. We easily obtain that %

1/“I = %.,,pob‘l + Ec Pl[ul To[p I u] T&’ I u] + O(G),

where we have used (4.27) and PJu] is the O(E) contribution PJa] from the fundamental equation D,P[a] = 0.

(B-1)

to P[u]. We construct

W)

Writing D, = 2 PDF’

(B.3)

7X=0

and (B.4) 595/132/I-11

160

MAZENKO

AND LUSCOMBE

TABLE VIII Coefficients C,,(u) Coefficients Occurring in the Expression for the First Order Probability Distribution (B.7) in -

I__

__--

1

0

1

2

3

0

0

0

0

0

1

0

we obtain, after matching coefficients of powers of E, that D,OP,[a] = 0, D,“Pl[al etc. We can formally

+ LpP&]

= 0,

VW

solve the second equation to obtain

03.6)

Inserting our explicit expression for 0~” given by (4.33) and using the completeness of the cell eigenfunctions #(“)(a) we easily obtain that 03.7)

where the matrix C,,(a) is given in Table VIII. preserved at this order since

Note that the normalization

of P[a] is

03.8)

RSDRG

AND

1-D

KINETIC

ISING

MODEL

161

Using these results we can easily compute

03.9)

We can then write (B. 10)

(B. 11) and d[t*. I ~‘1 is explicitly an off-diagonal contribution

to G[p j ~‘1: (B. 12)

Let us then write (B.13) Since T,, satisfies the appropriate

normalization

conditions we can write (B.14) (B.15)

(B.16) and, from (4.25), (B. 17) we have the requirement

c Mb I p’l = 0.

(B.18)

+ EMrP I CL’1 2~ob’l + O(E2),

(B.19),

We then find that

162

MAZENKO

AND

where we have assumed M is symmetric.

LUSCOMBE

We have first that (B.20)

so the first order probability

distribution

is given by (8.21)

We then diagonalize G by choosing

Mb I $1 ~&I = -iOh

I $1.

(B.22)

This choice leads to a T[p 1G] satisfying (3.13).

1. G. F. MAZENKO, M. J. NOLAN, AND 0. T. VALLS, Phys. Rev. Letr. 41 (1978), 500. 2. G. F. MAZENKO, in “Dynamical Critical Phenomena and Related Topics” (C. P. Enz, Ed.), Springer-Verlag, Berlin, 1979. 3. Cr. F. MA~ENKO, M. J. NOLAN, AND 0. T. VALLS, Phys. Rev. B 22 (1980), 1263. 4. G. F. MAZENKO, M. J. NOLAN, AND 0. T. VALLS, Phys. Rev. B 22 (1980), 1275. 5. G. F. MAZNKO, J. HIRSCH, M. J. NOLAN, AND 0. T. VALLS, Phys. Rev. Letl. 44 (1980), 1083. 6. G. F. MAZENKO, J. HIRSCH, M. J. NOLAN, AND 0. T. VALLS, Phys. Rev. B 23 (1981), 1481. 7. L. P. KADANOFF AND A. HOUGHTON, Phys. Rev. Bll (1975), 377; L. P. KADANOFF, Rev. Mod. Phys. 49 (1977), 267; D. R. NELSON AND M. E. FISHER, Antr. Phys. (N.Y.) 91 (1975), 226: M. NAUENBERG,

8.

9. 10. 11. 12. 13. 14.

J. Math.

Phys.

16 (1975),

703.

R. GLAIJBER, J. Math. Phys. 4 (1963), 294. These operators are discussed in some detail in Ref. [3]. One can vary the number of spins in a cell and one may want to change the nature of the block spins from that of the primitive spins. Thus one could introduce spin-l spins if one wanted to take vacancies into account. T. NIEMEIJER AND J. M. J. VAN LEEUWEN, in “Phase Transitions and Critical Phenomena” (C. Domb and M. S. Green, Ed.), Vol. 6, Academic Press, New York, 1976. Equation (5.2) can be obtained from standard cumulant expansion methods (see, for example, Ref. [ll]) using (4.18). E. FRADKIN AND S. RABY, Phys. Rev. D 20 (1979), 2566. G. F. MAZENKO, in “Correlation Functions and Quasiparticle Interactions in Condensed Matter” (J. W. Halley, Ed.), Plenum, New York, 1978.