- Email: [email protected]
Finite Size Scaling Effects in Dynamics
Nuclear Physics B280 [FS18] (1987) 340-354 North-Holland, Amsterdam
FINITE SIZE SCALING EFFECTS IN DYNAMICS Yadin Y. GOLDSCHMIDT Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA Received 23 June 1986 (Revised 28 July 1986)
We calculate the linear relaxation time for a finite size system with a cubic geometry and " model A" dynamics both above Tc in a 4 - ε expansion and below Tc in a 2 + ε expansion, and express the results in a scaling form. The universal scaling functions are obtained to one-loop order. We use the method of the effective hamiltonian for the homogeneous modes. The quantum mechanical hamiltonian is supersymmetric. The large n limit (infinitely many spin components) is also considered both above and below Tc.
1. Introduction In recent years there has been much interest in the theory of finite size scaling (FSS) [1]. The theory has been formulated for the first time by Fisher [2] and studied extensively in subsequent papers. It is very useful as an efficient extrapola tion for numerical calculations which are typically performed on small systems. It is also the basis for a real space renormalization procedure [3]. Recently Brezin and Zinn-Justin [4] (BZ) developed a method to analytically perform a calculation of the size-dependent universal scaling functions in an ε expansion, which is singular about four dimensions [5]. This means that the series generated is in powers of ε1/2 or ε1/3 depending upon the geometry of the sample. The first case corresponds to a cubic geometry of volume Ld and periodic boundary conditions, the second case to a "cylindrical" geometry for which d — 1 dimensions are of size L and one dimension is infinite. A calculation of the correlation length in powers of 2 + ε has also been performed [4]. In this paper we extend their method for dynamics [6], and calculate the linear relaxation time for a sample with the cubic geometry. In sect. 2 the scaling form of the relaxation time is calculated in powers of ε1/2 in 4 - ε dimensions above and at Tc. The quantum mechanical hamiltonian for the homogeneous modes is supersym metric. In sect. 3 we calculate the linear relaxation time below Tc in 2 + ε dimensions using the nonUnear σ-model. We also discuss the behavior of the relaxation time below Tc for an Ising spin system. In sect. 4 we discuss the large n limit of infinitely many spin components and show that it is consistent with the ε 0619-6823/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
CPSC 2 - paper 1.8
113
Υ. Υ. Goldschmidt / Finite size scaling effects
341
expansion. We restrict ourselves in this paper to model A dynamics [6] - a nonconserved order parameter.
2. Expansion about the upper critical dimension (four) Our aim is to calculate the linear relaxation time for an «-component spin system. The relaxation time is defined, for example, by Fisher and Racz [7]: Let ψ(τ) be a thermodynamic property which relaxes to zero with time τ. Let Ψ0 = Ψ(0) be its initial value. Then
τ<"(Γ)= lim /
dr^-Λ
(2.1)
We will not consider here the nonlinear relaxation time which is defined for nonzero ψ0 as Γ-» Tc [7,8]. We will omit the superscript (/). We denote by rR(oo, T) the bulk relaxation time and by T R (L, T) the relaxation time of the finite system. The relaxation time can also be defined [6] as [o)^(k = 0)] - 1 where Ψψ(*) = —ΓΓΤ χ (Λ) > ' ψ
=/ ΤΓΤ7Τ Γψ(Λ)
(2.2)
8ω
Here χψ(Λ, ω) is the linear response function and χψ(Λ) = χψ(Α, ω = 0). Near Tc the bulk relaxation time diverges like t~vz where t = (T— Tc)/Tc, v is the correla tion length exponent and z is the dynamical exponent. Model A dynamics is defined by the following markovian equations of motion
m dTi{x,r)= -λ0—
T
+
ξι(χ,τ),
2 2 2 2 = H = j ddx Κ^φ) +^ 0 φ +-Μ 0 (φ )
(Ux, T)> = 0,
(S,(x, r)Sj(x', τ')> = 2X^(x-
χ')δ(τ-
τ')β„· (2.3)
The function ξ(χ, τ) is the gaussian white noise. All correlations and response functions can be calculated from the generating functional [9]
Z[J,J] = /[αφ][άφ]βχρ|- {άάχάτ[λ0φ2-φ[3τφ -lτλ0u0φφφ2-^
114
CPSC 2 - paper 1.8
+ λ0(-ν2φ
+ Γ0φ)}
+ ^S(dKO)λ0u0φ2-Jφ-Jφ}).
(2.4)
342
Υ. Υ. Goldschmidt / Finite size scaling effects
The term involving 8{d\x = 0) originates from the jacobian [9] D: D = det
d dT
δ2Η δφδφ
(2.5)
It serves as a counterterm and subtracts the contribution of the response function (φ(χ, τ)φ(χ, τ)> at equal points in space and time whenever it occurs. In (2.3) the average over the noise ξ has already been taken and vector indices have been suppressed. It is now useful to integrate out the auxiliary field <> / and consider the generating functional
Φ + 4 λ ο ( " ν2φ + Γ φ + ±ιι φφ2) Z[J] = }[άφ]&φ\- f ddxdr -7Γ 0 0 4λ - ^ ( « + 2)δ<^(0)λ οΜο φ 2 -7φ
(2.6)
This functional was derived in a different manner by Munoz Sudupe and AlvarezEstrada [10] who constructed the renormalized perturbation theory in 4 — ε. More generally it was used as a basis for renormalization of dynamical models by Zinn-Justin [11]. In passing from (2.4) to (2.6) we have dropped the term linear in φ = 3τφ in the action which is a total derivative and contributes only to boundary terms. We now expand the fields in Fourier modes in the definite dimensions of the box:
Φ(
(2.7)
iq x
%(r),
where the components of q are quantized in units of cannot be treated perturbatively. Defining
2TT/L.
The modes with q =
φ(0ΞΦ,=ο(Ό, these modes are an arbitrary function of τ and are not damped by the ^-dependent terms in the action. The q Φ 0 modes can be treated perturbatively, and one can define an effective action ^ [ φ ί τ ) ] by tracing out the q Φ 0 modes up to a given order in the loop expansion. We can consider first the simpler situation above the upper critical dimension which is four for the model described by (2.6). In the case d > 4 one can neglect loop contributions to the effective action for the q = 0 modes. One has to be careful about the role of the jacobian in this case. This can be done by returning to eq. (2.3) and considering the equation of motion for the
CPSC 2 - paper 1.8
115
Y.Y. Goldschmidt / Finite size scaling effects
343
q = 0 mode only
m
( ^ τ Κ , ( τ ' ) ) = 2λ 0 ΖΓ*δ(τ-τ')δ,,.;
(2.8)
repeating the steps that led to eq. (2.6) one obtains: Z[J] = f[d]exp[-Seti[]+J], 56„[φ]=^/<1τ|^-(φ)2+μ0(Γ0φ+^φ2φ)2 - μ - " λ 0 ( ™ · 0 + Κ » + 2)« 0 φ 2 )).
(2.9)
We have kept the constant term for reasons which will be explained below. It is obvious now that this effective action corresponds to a quantum mechanical problem in n dimensions as formulated by the Feynman path integral in imaginary time. It is useful to realize that this action can alternatively be written in the form SM
= \fdr {1Φ2+ \W2{*) ~ \*W'(*)} ,
(2.10)
with h = 2X0L-d,
(2.11)
^ ( φ ) = λ 0 φ(>ο+ΚΦ 2 )>
(2-12)
Τν'Μ^ΣΜΤ,Μ/άφ,.
(2.13)
i
This action is related to the action of a supersymmetric quantum mechanical model
116
CPSC 2 - paper 1.8
344
Υ. Υ. Goldschmidt / Finite size scaling effects
[17,18] of the form
SSUSYU)
= ^ / < * τ { μ 2 - ψ * 3 τ ψ + \W\x)
+ ΐΛ[ψ·, ψ ] » " ( φ ) ,
(2.14)
where ψ is an anticommuting Grassmann variable. When the ψ field is integrated out the theory is equivalent to the bosonic action (2.10). This is provided supersym metry is not broken as it is for the potential under consideration [17,18]. We can associate with this action a quantum-mechanical hamiltonian in n dimensions by identifying φ ( τ ) with the coordinates q(r)
H< P
^^
=
2(J/2\ )
+
4 ^ 0 < 7 2 ( r 0 + ^q2]j
- ±(n + 2)X0u0q2-
\n\,r0. (2.15)
In particular the relaxation time, governing the exponential decay of correlations with time, is related to the gap of the hamiltonian (2.15), i.e. TR(L) = ( £ 1 - £ 0 ) - 1 ,
(2.16)
where E0, Ελ are the ground state and first excited state of the hamiltonian respectively. Since the potential is O(n) symmetric, and so are the lowest lying states, the gap can be found by solving a one-dimensional radial problem. Because of the underlying supersymmetry the ground state energy E0 vanishes. It is useful to make the following changes of variables
p^L"*uY*p,
(2.17)
and express the hamiltonian in the new variables H(p, q) = 2\A/2L-"/2
\\P2 + k 2 ( X
1 / 2
^
/ 2
' + ^
- £ ( « + 2)q2 - \n{W^L^t)\,
2
)
2
(2.18)
where we have replaced rQ by t( 4). Thus the relaxation time T R ( L ) must
CPSC 2 - paper 1.8
111
Y.Y. Goldschmidt / Finite size scaling effects
345
be of the scaling form TR(L) =
^"o1/2^/2/(K1/2^/20,
(2.19)
where /(JC) is the inverse gap of the hamiltonian h(p,q)
= i2p2+\q2{x
+ ±q1)1-Un
+ l)q2-hnx.
(2.20)
Relation (2.19) can be also cast in the form T R ( L , 0 = r R (oo, 0 g ( ^ y i ( r f - 4 ) / 4 ) ,
(2.21)
Τκ(«Μ) = τ -
(2.22)
where
A0t
is the mean-field relaxation time for the infinite system, and ^(t) oc t~l/1 is the mean-field correlation length. Eq. (2.21) is consistent with the breaking of the usual form of finite size scaling above four dimensions, which is well known [5]. Loop corrections which result from integrating out the q Φ 0 modes are irrelevant above four dimensions. All they cause is a shift of Tc, a finite renormalization of u0 and λ 0 , and they also generate terms in the action that lead to contributions which are down by powers of L compared to the leading contribution to the scaHng function calculated above. We now turn to the case d < 4. In this case loop corrections are important since they are responsible for long-distance singularities. Their effect can be taken into account using' the renormalization group and ε expansion. To one-loop order an infinite renormaUzation as well as a finite renormalization of λ 0 is not required (this occurs only at the two-loop order) but a renormalization of u0 and a shift of t is required. We realize that the renormaUzation of these parameters can be obtained from the statics properties of the system. (These statements can be verified by performing the loop expansion associated with the generating functional (2.6).) The shifts in t and g have been calculated previously [4] for the system with the cubic geometry. Introducing the dimensionless coupling constant g = ^uR
(2.23)
y = tL}/\
(2.24)
and the scaling variable
118
CPSC 2 - paper 1.8
346
Y.Y. Goldschmidt / Finite size scaling effects
the quantity x = \tKg~l/1Ld/1 g * = 877 2
*
=
6ε H+ 8
2*R^
becomes at the fixed point
+ ο(ε2),
8
(2.25)
I fixed point
ε
r°°
+
2 ( « + 2)
+ ε-
«+8
,Λ
,00
2
A4(u)-l
due-uy/4n A \ u ) - \ - ^
■/
+ ο(ε 2 ) , (2.26)
where
A(u)=
e"
Σ
(2.27)
Hence, below four dimensions we find that TR
^
(L) "
1 =
^
( g
, „ *
}
/(Χ)(ΐ +
θ(ε))
(2.28)
'
where f(x) is the inverse gap of the hamihonian (2.20), x is a function of the variable y = tLl/v as expressed in eq. (2.26) and g* is given in (2.25). At T= Tc (y = 0) one obtains 'R(£)
■^•»"'('"'ίΑί) 0 **»' i
(2 29)
"
with ,00
/
1
77"
7Γ « Ό
- = - /
/
772\
\
U ]
dwL44(w)-l
= -1.7650....
(2.30)
In higher loop order the denominator L2 on the l.h.s. of (2.28) and (2.29) will be replaced by U where z is the dynamical critical exponent, since it is well known [12,8] that at Tc, TR(L) ~ U. z has been calculated in 4 - ε and found to be [13]
ζ=2+0 726ΐ
·
·2τ!^) ε 2 + ο ( ε 3 ) ·
(2.31)
CPSC 2 - paper 1.8
119
Y. Y. Goldschmidt / Finite size scaling effects
347
Dividing (2.28) by (2.29) we find TR(L,0
/(*)
rR(L,r=rc)
/(^.i(n
+
2)/A/3ÖTf8)") '
(2.32)
which is a universal function of tL1/v. Eq. (2.28) can also be written in the form I
^ r \ = L2(g*y1/1y"f(x)(l
TD
+ o(e)),
(2.33)
(00)
where to order one-loop (Λ + 2 )
vz = l + — — e, 2(/i + 8)
(2.34)
V
}
and y was defined in (2.24). 3. Expansion about the lower critical dimension In order to investigate the behavior of the relaxation time below the critical temperature it is useful to investigate the nonlinear σ-model [14,15] in a 2 + ε expansion. Considering the case n > 2 near two dimensions the critical temperature is very small (ο(ε)) and hence the expansion parameter for T < Tc is the temperature divided by the spin stiffness constant, which we will denote in this section by t. (This is not the temperature difference from Tc as it was in the previous section.) The dynamics of the non-linear σ-model has been first investigated by de Dominicis, Ma and Peliti [16]. It was shown later by Bausch, Janssen and Yamuzaki [19], that the same dynamics can be derived from the generating func tional
{[άφ][άφ]Π[8(φ2-ΐ)δ(φ'φ)]^χρ{-Ξ(φ,φ)-7φ-/φ},
Ζ(79Ι) = 1
Ξ(φ,φ) = - jddxdr
λ0φ2 + /φ
δΗ
Η=±}άάχ(νφ)2.
(3.1)
It is now possible to use the identity Π δ ( φ · φ ) = /[dw]expl/Jd^dTw<£-
120
CPSC 2 - paper 1.8
(3.2)
348
Υ. Υ. Goldschmidt / Finite size scaling effects
and integrate out the field φ for the case / = 0 to obtain
Ζ[/] = /[αφ][α<ο]Πδ(φ2-ΐ) Xexpj -—— jddxdT
[ φ 2 - X20(v2f - 2λ 0 ίωφ· ν2φ + tW\ +J\, (3.3)
where we again dropped a total time derivative. Note that because φ2 = 1 one has φ · ν2φ = — (νφ)2. The field ω can now be integrated over to yield Z[J] = / [ α φ ] Π « ( φ 2 - l)exp{ - 5 ε „ ( φ ) + J) , 5 ε „(φ) = ^ 7 / ^ χ ( 1 τ { ( 3 τ φ ) 2 + λ 2 0 [ ( ^ φ ) 2 - ( ^ 7 φ ) 2 · ( ^ φ ) 2 ] } .
(3.4)
This form of the action coincides with the form derived by Lebedev [20]. We will now restrict the spatial integration to a box of volume Ld, and consider the action
S(*) = ^JLdddxfoTdT{(*)2
+
\ll(v2*)2-(V*)2(V*)2},
(3.5)
with the constraint φ2 = 1. Here we denoted by T some finite time cutoff, and it should not be confused with the temperature that in our present formalism appears in the form of the variable t (in units of the spin stiffness constant). We can now follow similar steps used by BZ [4], the difference being the appearance of the laplacian operator in (3.5) instead of the gradient operator in their treatment of the cylindrical geometry. Defining φ ( Χ ) Τ = = 0 ) = ιι1,
φ(χ,τ=Γ)=ιι2,
(3.6)
with ir1-ii2 = cose,
| if! | = |ir2| = 1 ,
(3.7)
the generating functional Z(J = 0) defined in (3.5) has to be of the form Ee / i > /(cos0)e- 7 ' e '. (3.8) / Here ε, are the eigenvalues of the quantum hamihonian H, P;(cos Θ) are Gegenbauer Z{Tj)
= (u2\e-TH\Ul)=
CPSC 2 - paper 1.8
121
349
Υ. Υ. Goldschmidt / Finite size scaling effects
polynomials, orthogonal with respect to the measure sin"~20 and 8l are degeneracy factors. Parametrizing the fields φ in an appropriate way [4] that takes into account the condition φ2 = 1 and the boundary conditions (3.6), we can re-write the action (3.5) keeping only quadratic terms in the fields, which is legitimate for a calculation carried out to one-loop order. We can thus integrate out over the fields φ in the expression for Z(J = 0) in (3.5) and obtain, after some labor: In
θ2
Ζ(Γ,Ο)
Λ
4λ0ί
Η\η\-γ2+\\(ν2)2-—2
}
"
,
\\\(V2Y-
-trln
dM
dr 2
• (3-9)
The eigenvalues of the operator X20(V2)2 - d 2 /dr 2 with the appropriate boundary conditions for the eigenfunctions (periodic boundary conditions of x and vanishing eigenfunctions for τ = 0, T) are: 2 2
^
where pv..., In
+
mV
(3.10)
pd are integers and m are positive integers. Thus (3.9) takes the form
Z(TJ) A = θΗ Z(T,0)
L"
« —2 2
4λ 0 ίΓ
IT
S. ^
1 2
2
2
+ ir2m2/T2 ,
w t x λ\{4ττ ρ /L f
+ O(0 4 ). (3.11)
Here an expansion in Θ about 0 = 0 has been performed which is the relevant region of Θ for small t. For large T one has
yy
pZX20(4v2p2/L2)2
= ir2+y
+ v2m2/T2
6
,
1 TL2 I~0 U2p2 '
(3.12)
where the prime denotes all values of p Φ 0. The sum over p is dimensionally regularized. Σ~2= p P
ΓάηΣε-ρ2=
J
0
p
J
Γάη[Α"(η)-ΐ]9
(3.13)
0
where A(u) has been defined in eq. (2.27). For small w, A(u) - (m/u)l/1, hence the
122
CPSC 2 - paper 1.8
350
Ύ. Υ. Goldschmidt / Finite size scaling effects
integral converges for d < 2. Expanding in e = u f - 2 w e obtain
Yl—r=*d/1\ , P2
7H+f
J
o {c"-l)d/2
d« U " ( « ) - l
J
o
\
V
'
= - —-77lnw + o(e)+ /"°°dwM2(w)-l ε -Ό \ 2w
=
Α = -πΙην
^77
(e"-l)d/2/ — ) + 0(ε) e" - 1 /
+ Λ+0(ε),
+)
2
Λ
(3.14)
( Η ) - 1 - - — -
d « = - 2 . 8247.
(3.15)
Collecting together the previous results we obtain Ζ(Τ,Θ)
Ld
A
2 ln^f=0 U(«-2)Z(T,0)
+
1 n - 2 1 L1
n-2 L2A
ε 8ττ . λ 0 Τ + Ιβπ2 λ0Τ
4λ0ίΓ
θ\θ4;ί,-,\.
+ 0(ε)
(3.16)
To project the two lowest eigenvalues we have to multiply (3.16) by 1 and cos Θ (the first Gegenbauer polynomials) and integrate with the measure sin n - 2 0. The integrals can be evaluated by the steepest descent technique. We finally obtain: r^(L)
= (n-l)
X0t .
1+
n— 2
( 2π
^ - Μ - τ + ^ Ι + 0(ε)
(3.17)
The ε pole is cancelled by introducing the renormalized temperature parameter [15] n-2 ί = ίρ +
2πε
■ti + o(ti)
(3.18)
to one-loop order. λ 0 is not renormaUzed and one can set λ = λ 0 . Introducing the running coupling constant tR(L) [15] which is, at this order, given by the relation 1
1
'R
/*= R
/ '£ 2πε n-2
1 'R(^)
+0(ε2), V '
(3.19)
CPSC 2 - paper 1.8
123
Y.Y. Goldschmidt / Finite size scaling effects
351
we can write (3.17) in the form TR(£)
(n-l)XtR(L)
l-A^tR(L) 477"
+ 0{ti)
(3.20)
For large L, tK(L) vanishes. It is related to the correlation length by
tR(L) =
K'R)
d-2
1 η-ΙΙζγ-2 1-— +o 2w —— d-2\L
(3.21)
lKd-1)
Hence for large L, T < Tc we can rewrite (3.20) in the form TR(L)
1
1
λη-lU
d-2
1+
n-21 ξ 2π XL
d-2
1
d-2
-Λ 2π
\
+0
/
1 L2(^-2)
(3.22)
For an Ising system below Tc the situation is different. Returning back to the hamiltonian (2.20) with « = 1 we show in fig. 1 the qualitative shapes of the potential for various values of the parameter x. We know, because of the underlying supersymmetry, that the ground state is at E = 0. For x < 0, i.e. T < Tc we see that there is a barrier of the potential rising above E = 0, and thus the ground state is a linear combination of states lying in the two outermost wells. This indicates the existence of an exponentially small gap between the ground state and first excited state which can be calculated using instanton calculus. (Notice that had we failed to include the terms originating from the jacobian in the hamiltonian (2.20), the potential for x < 0 would have had three minima of equal depth. But the frequency of oscillation in the two outer wells would have been twice as large as the frequency in the middle well. Thus the ground state would originate from the state in the middle well which is contrary to the correct situation below Tc.) Thus we observe, that as opposed to the O(N) case for which the relaxation time grows as a power of L (see eq. (3.22)), for the Ising case the relaxation time grows exponentially with increasing L. A detailed instanton calculation to determine the relaxation time is beyond the scope of this article. It ought to include also the contribution of the q Φ 0 modes, since the reversal of the magnetization of the sample occurs via the development of a bubble of the opposite phase, thus one is forced to consider inhomogeneous configurations*. * See the note added at the end of the paper.
124
CPSC 2 - paper 1.8
Υ. Υ. Goldschmidt / Finite size scaling effects
352
Fig. 1. Qualitative shape of the potential of the quantum hamiltonian for the case (a) JC> \, (b) 0 < x < \, (c) — \ < x < 0, (d) x < - \ . The horizontal line denotes the zero energy ground state.
4. The relaxation time in large n limit In the large n limit there is a simple relation between the relaxation time for the cubic geometry and the correlation length both above and below Tc. We will follow the notation of Brezin [5]. In the large n limit we can show that the time dependent correlation function for the model defined by eq. (2.3) is given by the expression 1
r dco
(
2λ
(4.1)
where mL depends on r0 and u0 but not on momentum and frequency. It is the same self-energy part as that appearing in the static calculation. As τ -> oo the leading contribution to C(q, ω) comes from# = 0. Hence C(T)
1
e'WT
,d
~ τ*hz
2 i2,
λ
2
e XomL|T| > 2,2~ -
(4·2)
ml\t).
(4.3)
from which we can infer that T
R
(L)=— Λ
ο
CPSC 2 - paper 1.8
125
Y. Y. Goldschmidt / Finite size scaling effects
353
The quantity mLl(Tc) has been evaluated in ref. [5] and identified with the correlation length. Hence we obtain ε= 4-Λ^0Η 2 (lnL) 1/2 rM)cc{ -U Λ
rf=4
J_ L 2 L «/-4)/2 5
d>4
ο
λ
0
(4.4)
These results are consistent with our previous results above and below four dimensions. Below Tc9 relation (4.3) still holds, but mL has not been evaluated for the cubic geometry. Brezin evaluated mL for the cylindrical geometry with one infinite dimension [5]. It is possible to modify his equation (60) to our case. The second term on the r.h.s. has to be replaced by
Σνττ^^'^^^^'^^^ττι^Γ 6 "'^ 0 ^ 72 ^' (4·5) where
m = Σν-2/4'
(4.6)
and y = Lmh. For yL«1 one can replace g(t) by its asymptotic behavior ~ (4mtY/2 and hence one finds
ß-ßc
= L-"ml2,
(4.7)
from which we obtain at T < Tn TK(L)=-LdΛ(ß-ßc). ο
(4.8)
From eq. (3.19) and the fact that ηβ ~ \/t we see that for large L
ß-ßc~-L-^\L) and thus eq. (4.8) agrees with our result (3.20).
126
CPSC 2 - paper 1.8
(4.9)
354
Υ.Ύ. Goldschmidt / Finite size scaling effects
5. Conclusions We have carried here a computation of the scaling behavior of the relaxation time for a system with a cubic geometry and periodic boundary conditions. We have used model A dynamics. It will be useful in the future to consider other geometries and also other types of dynamics. Our results should be useful for an estimation of the equilibration time in Monte Carlo calculations as a function of the system size. This work was supported by the National Science Foundation under grant #DMR-8302323. Note added After this work had been submitted for publication we learnt from Dr. J. Zinn-Justin (private communication), that a similar work has just been completed by himself together with J.C. Niel. They obtained many of the results of this paper, and also carried out a detailed instanton calculation to evaluate the relaxation time of the Ising model below Tc. They found TR ~ exp(Z/ -1 · 2σ(Γ)). References [1] M.N. Barber, in Phase transitions and critical phenomena, vol. VIII, eds. C. Domb and J. Lebowitz (Academic Press, NY, 1984) and references therein [2] M.E. Fisher, in Critical phenomena, Proc. 51st Enrico Fermi Summer School, Varena, ed. M.S. Green (Academic Press, NY, 1972); M.E. Fisher and M.N. Barber, Phys. Rev. Lett. 28 (1972) 1516 [3] M.P. Nightingale, Physica 83 (1976) A561 [4] E. Brezin and J. Zinn-Justin, Nucl. Phys. B257 [FS14] (1985) 868 [5] E. Brezin, J. de Phys. 43 (1982) 15 [6] P.C. Hoenberg and B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435 [7] M.E. Fisher and Z. Racz, Phys. Rev. B13 (1976) 5039 [8] J.M. Sancho et al., J. Phys. A13 (1980) L443 [9] R. Bausch et al., Z. Phys. B24 (1976) 113 [10] A. Munoz Sudupe and R.F. Alvarez-Estrada, J. Phys. A16 (1983) 3049 [11] J. Zinn-Justin, Nucl. Phys. B275 [FS17] (1986) 135 [12] M. Suzuki, Prog. Theor. Phys. 58 (1977) 1142 [13] B. Halperin et al, Phys. Rev. Lett. 29 (1972) 1548; C. de Dominicis et al, Phys. Rev. B12 (1975) 4945 [14] A.M. Polyakov, Phys. Lett. 59B (1975) 79 [15] E. Brezin and J. Zinn-Justin, Phys. Rev. Lett. 36 (1976) 691; Rev. B14 (1976) 3110 [16] C. de Dominicis et al, Phys. Rev. B15 (1977) 4313 [17] E. Witten, Nucl. Phys. B188 (1981) 513 [18] F. Cooper and B. Freedman, Ann. of Phys. 146 (1983) 262 [19] R. Bausch, H.K. Janssen and Y. Yamazaki, Z. Phys. B37 (1980) 163 [20] V. Lebedev, Phys. Lett. 105A (1984) 173
CPSC 2 - paper 1.8
127
FINITE SIZE SCALING EFFECTS IN DYNAMICS Yadin Y. GOLDSCHMIDT Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA Received 23 June 1986 (Revised 28 July 1986)
We calculate the linear relaxation time for a finite size system with a cubic geometry and " model A" dynamics both above Tc in a 4 - ε expansion and below Tc in a 2 + ε expansion, and express the results in a scaling form. The universal scaling functions are obtained to one-loop order. We use the method of the effective hamiltonian for the homogeneous modes. The quantum mechanical hamiltonian is supersymmetric. The large n limit (infinitely many spin components) is also considered both above and below Tc.
1. Introduction In recent years there has been much interest in the theory of finite size scaling (FSS) [1]. The theory has been formulated for the first time by Fisher [2] and studied extensively in subsequent papers. It is very useful as an efficient extrapola tion for numerical calculations which are typically performed on small systems. It is also the basis for a real space renormalization procedure [3]. Recently Brezin and Zinn-Justin [4] (BZ) developed a method to analytically perform a calculation of the size-dependent universal scaling functions in an ε expansion, which is singular about four dimensions [5]. This means that the series generated is in powers of ε1/2 or ε1/3 depending upon the geometry of the sample. The first case corresponds to a cubic geometry of volume Ld and periodic boundary conditions, the second case to a "cylindrical" geometry for which d — 1 dimensions are of size L and one dimension is infinite. A calculation of the correlation length in powers of 2 + ε has also been performed [4]. In this paper we extend their method for dynamics [6], and calculate the linear relaxation time for a sample with the cubic geometry. In sect. 2 the scaling form of the relaxation time is calculated in powers of ε1/2 in 4 - ε dimensions above and at Tc. The quantum mechanical hamiltonian for the homogeneous modes is supersym metric. In sect. 3 we calculate the linear relaxation time below Tc in 2 + ε dimensions using the nonUnear σ-model. We also discuss the behavior of the relaxation time below Tc for an Ising spin system. In sect. 4 we discuss the large n limit of infinitely many spin components and show that it is consistent with the ε 0619-6823/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
CPSC 2 - paper 1.8
113
Υ. Υ. Goldschmidt / Finite size scaling effects
341
expansion. We restrict ourselves in this paper to model A dynamics [6] - a nonconserved order parameter.
2. Expansion about the upper critical dimension (four) Our aim is to calculate the linear relaxation time for an «-component spin system. The relaxation time is defined, for example, by Fisher and Racz [7]: Let ψ(τ) be a thermodynamic property which relaxes to zero with time τ. Let Ψ0 = Ψ(0) be its initial value. Then
τ<"(Γ)= lim /
dr^-Λ
(2.1)
We will not consider here the nonlinear relaxation time which is defined for nonzero ψ0 as Γ-» Tc [7,8]. We will omit the superscript (/). We denote by rR(oo, T) the bulk relaxation time and by T R (L, T) the relaxation time of the finite system. The relaxation time can also be defined [6] as [o)^(k = 0)] - 1 where Ψψ(*) = —ΓΓΤ χ (Λ) > ' ψ
=/ ΤΓΤ7Τ Γψ(Λ)
(2.2)
8ω
Here χψ(Λ, ω) is the linear response function and χψ(Λ) = χψ(Α, ω = 0). Near Tc the bulk relaxation time diverges like t~vz where t = (T— Tc)/Tc, v is the correla tion length exponent and z is the dynamical exponent. Model A dynamics is defined by the following markovian equations of motion
m dT
T
+
ξι(χ,τ),
2 2 2 2 = H = j ddx Κ^φ) +^ 0 φ +-Μ 0 (φ )
(Ux, T)> = 0,
(S,(x, r)Sj(x', τ')> = 2X^(x-
χ')δ(τ-
τ')β„· (2.3)
The function ξ(χ, τ) is the gaussian white noise. All correlations and response functions can be calculated from the generating functional [9]
Z[J,J] = /[αφ][άφ]βχρ|- {άάχάτ[λ0φ2-φ[3τφ -lτλ0u0φφφ2-^
114
CPSC 2 - paper 1.8
+ λ0(-ν2φ
+ Γ0φ)}
+ ^S(dKO)λ0u0φ2-Jφ-Jφ}).
(2.4)
342
Υ. Υ. Goldschmidt / Finite size scaling effects
The term involving 8{d\x = 0) originates from the jacobian [9] D: D = det
d dT
δ2Η δφδφ
(2.5)
It serves as a counterterm and subtracts the contribution of the response function (φ(χ, τ)φ(χ, τ)> at equal points in space and time whenever it occurs. In (2.3) the average over the noise ξ has already been taken and vector indices have been suppressed. It is now useful to integrate out the auxiliary field <> / and consider the generating functional
Φ + 4 λ ο ( " ν2φ + Γ φ + ±ιι φφ2) Z[J] = }[άφ]&φ\- f ddxdr -7Γ 0 0 4λ - ^ ( « + 2)δ<^(0)λ οΜο φ 2 -7φ
(2.6)
This functional was derived in a different manner by Munoz Sudupe and AlvarezEstrada [10] who constructed the renormalized perturbation theory in 4 — ε. More generally it was used as a basis for renormalization of dynamical models by Zinn-Justin [11]. In passing from (2.4) to (2.6) we have dropped the term linear in φ = 3τφ in the action which is a total derivative and contributes only to boundary terms. We now expand the fields in Fourier modes in the definite dimensions of the box:
Φ(
(2.7)
iq x
%(r),
where the components of q are quantized in units of cannot be treated perturbatively. Defining
2TT/L.
The modes with q =
φ(0ΞΦ,=ο(Ό, these modes are an arbitrary function of τ and are not damped by the ^-dependent terms in the action. The q Φ 0 modes can be treated perturbatively, and one can define an effective action ^ [ φ ί τ ) ] by tracing out the q Φ 0 modes up to a given order in the loop expansion. We can consider first the simpler situation above the upper critical dimension which is four for the model described by (2.6). In the case d > 4 one can neglect loop contributions to the effective action for the q = 0 modes. One has to be careful about the role of the jacobian in this case. This can be done by returning to eq. (2.3) and considering the equation of motion for the
CPSC 2 - paper 1.8
115
Y.Y. Goldschmidt / Finite size scaling effects
343
q = 0 mode only
m
( ^ τ Κ , ( τ ' ) ) = 2λ 0 ΖΓ*δ(τ-τ')δ,,.;
(2.8)
repeating the steps that led to eq. (2.6) one obtains: Z[J] = f[d
(2.9)
We have kept the constant term for reasons which will be explained below. It is obvious now that this effective action corresponds to a quantum mechanical problem in n dimensions as formulated by the Feynman path integral in imaginary time. It is useful to realize that this action can alternatively be written in the form SM
= \fdr {1Φ2+ \W2{*) ~ \*W'(*)} ,
(2.10)
with h = 2X0L-d,
(2.11)
^ ( φ ) = λ 0 φ(>ο+ΚΦ 2 )>
(2-12)
Τν'Μ^ΣΜΤ,Μ/άφ,.
(2.13)
i
This action is related to the action of a supersymmetric quantum mechanical model
116
CPSC 2 - paper 1.8
344
Υ. Υ. Goldschmidt / Finite size scaling effects
[17,18] of the form
SSUSYU)
= ^ / < * τ { μ 2 - ψ * 3 τ ψ + \W\x)
+ ΐΛ[ψ·, ψ ] » " ( φ ) ,
(2.14)
where ψ is an anticommuting Grassmann variable. When the ψ field is integrated out the theory is equivalent to the bosonic action (2.10). This is provided supersym metry is not broken as it is for the potential under consideration [17,18]. We can associate with this action a quantum-mechanical hamiltonian in n dimensions by identifying φ ( τ ) with the coordinates q(r)
H< P
^^
=
2(J/2\ )
+
4 ^ 0 < 7 2 ( r 0 + ^q2]j
- ±(n + 2)X0u0q2-
\n\,r0. (2.15)
In particular the relaxation time, governing the exponential decay of correlations with time, is related to the gap of the hamiltonian (2.15), i.e. TR(L) = ( £ 1 - £ 0 ) - 1 ,
(2.16)
where E0, Ελ are the ground state and first excited state of the hamiltonian respectively. Since the potential is O(n) symmetric, and so are the lowest lying states, the gap can be found by solving a one-dimensional radial problem. Because of the underlying supersymmetry the ground state energy E0 vanishes. It is useful to make the following changes of variables
p^L"*uY*p,
(2.17)
and express the hamiltonian in the new variables H(p, q) = 2\A/2L-"/2
\\P2 + k 2 ( X
1 / 2
^
/ 2
' + ^
- £ ( « + 2)q2 - \n{W^L^t)\,
2
)
2
(2.18)
where we have replaced rQ by t(
CPSC 2 - paper 1.8
111
Y.Y. Goldschmidt / Finite size scaling effects
345
be of the scaling form TR(L) =
^"o1/2^/2/(K1/2^/20,
(2.19)
where /(JC) is the inverse gap of the hamiltonian h(p,q)
= i2p2+\q2{x
+ ±q1)1-Un
+ l)q2-hnx.
(2.20)
Relation (2.19) can be also cast in the form T R ( L , 0 = r R (oo, 0 g ( ^ y i ( r f - 4 ) / 4 ) ,
(2.21)
Τκ(«Μ) = τ -
(2.22)
where
A0t
is the mean-field relaxation time for the infinite system, and ^(t) oc t~l/1 is the mean-field correlation length. Eq. (2.21) is consistent with the breaking of the usual form of finite size scaling above four dimensions, which is well known [5]. Loop corrections which result from integrating out the q Φ 0 modes are irrelevant above four dimensions. All they cause is a shift of Tc, a finite renormalization of u0 and λ 0 , and they also generate terms in the action that lead to contributions which are down by powers of L compared to the leading contribution to the scaHng function calculated above. We now turn to the case d < 4. In this case loop corrections are important since they are responsible for long-distance singularities. Their effect can be taken into account using' the renormalization group and ε expansion. To one-loop order an infinite renormaUzation as well as a finite renormalization of λ 0 is not required (this occurs only at the two-loop order) but a renormalization of u0 and a shift of t is required. We realize that the renormaUzation of these parameters can be obtained from the statics properties of the system. (These statements can be verified by performing the loop expansion associated with the generating functional (2.6).) The shifts in t and g have been calculated previously [4] for the system with the cubic geometry. Introducing the dimensionless coupling constant g = ^uR
(2.23)
y = tL}/\
(2.24)
and the scaling variable
118
CPSC 2 - paper 1.8
346
Y.Y. Goldschmidt / Finite size scaling effects
the quantity x = \tKg~l/1Ld/1 g * = 877 2
*
=
6ε H+ 8
2*R^
becomes at the fixed point
+ ο(ε2),
8
(2.25)
I fixed point
ε
r°°
+
2 ( « + 2)
+ ε-
«+8
,Λ
,00
2
A4(u)-l
due-uy/4n A \ u ) - \ - ^
■/
+ ο(ε 2 ) , (2.26)
where
A(u)=
e"
Σ
(2.27)
Hence, below four dimensions we find that TR
^
(L) "
1 =
^
( g
, „ *
}
/(Χ)(ΐ +
θ(ε))
(2.28)
'
where f(x) is the inverse gap of the hamihonian (2.20), x is a function of the variable y = tLl/v as expressed in eq. (2.26) and g* is given in (2.25). At T= Tc (y = 0) one obtains 'R(£)
■^•»"'('"'ίΑί) 0 **»' i
(2 29)
"
with ,00
/
1
77"
7Γ « Ό
- = - /
/
772\
\
U ]
dwL44(w)-l
= -1.7650....
(2.30)
In higher loop order the denominator L2 on the l.h.s. of (2.28) and (2.29) will be replaced by U where z is the dynamical critical exponent, since it is well known [12,8] that at Tc, TR(L) ~ U. z has been calculated in 4 - ε and found to be [13]
ζ=2+0 726ΐ
·
·2τ!^) ε 2 + ο ( ε 3 ) ·
(2.31)
CPSC 2 - paper 1.8
119
Y. Y. Goldschmidt / Finite size scaling effects
347
Dividing (2.28) by (2.29) we find TR(L,0
/(*)
rR(L,r=rc)
/(^.i(n
+
2)/A/3ÖTf8)") '
(2.32)
which is a universal function of tL1/v. Eq. (2.28) can also be written in the form I
^ r \ = L2(g*y1/1y"f(x)(l
TD
+ o(e)),
(2.33)
(00)
where to order one-loop (Λ + 2 )
vz = l + — — e, 2(/i + 8)
(2.34)
V
}
and y was defined in (2.24). 3. Expansion about the lower critical dimension In order to investigate the behavior of the relaxation time below the critical temperature it is useful to investigate the nonlinear σ-model [14,15] in a 2 + ε expansion. Considering the case n > 2 near two dimensions the critical temperature is very small (ο(ε)) and hence the expansion parameter for T < Tc is the temperature divided by the spin stiffness constant, which we will denote in this section by t. (This is not the temperature difference from Tc as it was in the previous section.) The dynamics of the non-linear σ-model has been first investigated by de Dominicis, Ma and Peliti [16]. It was shown later by Bausch, Janssen and Yamuzaki [19], that the same dynamics can be derived from the generating func tional
{[άφ][άφ]Π[8(φ2-ΐ)δ(φ'φ)]^χρ{-Ξ(φ,φ)-7φ-/φ},
Ζ(79Ι) = 1
Ξ(φ,φ) = - jddxdr
λ0φ2 + /φ
δΗ
Η=±}άάχ(νφ)2.
(3.1)
It is now possible to use the identity Π δ ( φ · φ ) = /[dw]expl/Jd^dTw<£-
120
CPSC 2 - paper 1.8
(3.2)
348
Υ. Υ. Goldschmidt / Finite size scaling effects
and integrate out the field φ for the case / = 0 to obtain
Ζ[/] = /[αφ][α<ο]Πδ(φ2-ΐ) Xexpj -—— jddxdT
[ φ 2 - X20(v2
where we again dropped a total time derivative. Note that because φ2 = 1 one has φ · ν2φ = — (νφ)2. The field ω can now be integrated over to yield Z[J] = / [ α φ ] Π « ( φ 2 - l)exp{ - 5 ε „ ( φ ) + J
(3.4)
This form of the action coincides with the form derived by Lebedev [20]. We will now restrict the spatial integration to a box of volume Ld, and consider the action
S(*) = ^JLdddxfoTdT{(*)2
+
\ll(v2*)2-(V*)2(V*)2},
(3.5)
with the constraint φ2 = 1. Here we denoted by T some finite time cutoff, and it should not be confused with the temperature that in our present formalism appears in the form of the variable t (in units of the spin stiffness constant). We can now follow similar steps used by BZ [4], the difference being the appearance of the laplacian operator in (3.5) instead of the gradient operator in their treatment of the cylindrical geometry. Defining φ ( Χ ) Τ = = 0 ) = ιι1,
φ(χ,τ=Γ)=ιι2,
(3.6)
with ir1-ii2 = cose,
| if! | = |ir2| = 1 ,
(3.7)
the generating functional Z(J = 0) defined in (3.5) has to be of the form Ee / i > /(cos0)e- 7 ' e '. (3.8) / Here ε, are the eigenvalues of the quantum hamihonian H, P;(cos Θ) are Gegenbauer Z{Tj)
= (u2\e-TH\Ul)=
CPSC 2 - paper 1.8
121
349
Υ. Υ. Goldschmidt / Finite size scaling effects
polynomials, orthogonal with respect to the measure sin"~20 and 8l are degeneracy factors. Parametrizing the fields φ in an appropriate way [4] that takes into account the condition φ2 = 1 and the boundary conditions (3.6), we can re-write the action (3.5) keeping only quadratic terms in the fields, which is legitimate for a calculation carried out to one-loop order. We can thus integrate out over the fields φ in the expression for Z(J = 0) in (3.5) and obtain, after some labor: In
θ2
Ζ(Γ,Ο)
Λ
4λ0ί
Η\η\-γ2+\\(ν2)2-—2
}
"
,
\\\(V2Y-
-trln
dM
dr 2
• (3-9)
The eigenvalues of the operator X20(V2)2 - d 2 /dr 2 with the appropriate boundary conditions for the eigenfunctions (periodic boundary conditions of x and vanishing eigenfunctions for τ = 0, T) are: 2 2
^
where pv..., In
+
mV
(3.10)
pd are integers and m are positive integers. Thus (3.9) takes the form
Z(TJ) A = θΗ Z(T,0)
L"
« —2 2
4λ 0 ίΓ
IT
S. ^
1 2
2
2
+ ir2m2/T2 ,
w t x λ\{4ττ ρ /L f
+ O(0 4 ). (3.11)
Here an expansion in Θ about 0 = 0 has been performed which is the relevant region of Θ for small t. For large T one has
yy
pZX20(4v2p2/L2)2
= ir2+y
+ v2m2/T2
6
,
1 TL2 I~0 U2p2 '
(3.12)
where the prime denotes all values of p Φ 0. The sum over p is dimensionally regularized. Σ~2= p P
ΓάηΣε-ρ2=
J
0
p
J
Γάη[Α"(η)-ΐ]9
(3.13)
0
where A(u) has been defined in eq. (2.27). For small w, A(u) - (m/u)l/1, hence the
122
CPSC 2 - paper 1.8
350
Ύ. Υ. Goldschmidt / Finite size scaling effects
integral converges for d < 2. Expanding in e = u f - 2 w e obtain
Yl—r=*d/1\ , P2
7H+f
J
o {c"-l)d/2
d« U " ( « ) - l
J
o
\
V
'
= - —-77lnw + o(e)+ /"°°dwM2(w)-l ε -Ό \ 2w
=
Α = -πΙην
^77
(e"-l)d/2/ — ) + 0(ε) e" - 1 /
+ Λ+0(ε),
+)
2
Λ
(3.14)
( Η ) - 1 - - — -
d « = - 2 . 8247.
(3.15)
Collecting together the previous results we obtain Ζ(Τ,Θ)
Ld
A
2 ln^f=0 U(«-2)Z(T,0)
+
1 n - 2 1 L1
n-2 L2A
ε 8ττ . λ 0 Τ + Ιβπ2 λ0Τ
4λ0ίΓ
θ\θ4;ί,-,\.
+ 0(ε)
(3.16)
To project the two lowest eigenvalues we have to multiply (3.16) by 1 and cos Θ (the first Gegenbauer polynomials) and integrate with the measure sin n - 2 0. The integrals can be evaluated by the steepest descent technique. We finally obtain: r^(L)
= (n-l)
X0t .
1+
n— 2
( 2π
^ - Μ - τ + ^ Ι + 0(ε)
(3.17)
The ε pole is cancelled by introducing the renormalized temperature parameter [15] n-2 ί = ίρ +
2πε
■ti + o(ti)
(3.18)
to one-loop order. λ 0 is not renormaUzed and one can set λ = λ 0 . Introducing the running coupling constant tR(L) [15] which is, at this order, given by the relation 1
1
'R
/*= R
/ '£ 2πε n-2
1 'R(^)
+0(ε2), V '
(3.19)
CPSC 2 - paper 1.8
123
Y.Y. Goldschmidt / Finite size scaling effects
351
we can write (3.17) in the form TR(£)
(n-l)XtR(L)
l-A^tR(L) 477"
+ 0{ti)
(3.20)
For large L, tK(L) vanishes. It is related to the correlation length by
tR(L) =
K'R)
d-2
1 η-ΙΙζγ-2 1-— +o 2w —— d-2\L
(3.21)
lKd-1)
Hence for large L, T < Tc we can rewrite (3.20) in the form TR(L)
1
1
λη-lU
d-2
1+
n-21 ξ 2π XL
d-2
1
d-2
-Λ 2π
\
+0
/
1 L2(^-2)
(3.22)
For an Ising system below Tc the situation is different. Returning back to the hamiltonian (2.20) with « = 1 we show in fig. 1 the qualitative shapes of the potential for various values of the parameter x. We know, because of the underlying supersymmetry, that the ground state is at E = 0. For x < 0, i.e. T < Tc we see that there is a barrier of the potential rising above E = 0, and thus the ground state is a linear combination of states lying in the two outermost wells. This indicates the existence of an exponentially small gap between the ground state and first excited state which can be calculated using instanton calculus. (Notice that had we failed to include the terms originating from the jacobian in the hamiltonian (2.20), the potential for x < 0 would have had three minima of equal depth. But the frequency of oscillation in the two outer wells would have been twice as large as the frequency in the middle well. Thus the ground state would originate from the state in the middle well which is contrary to the correct situation below Tc.) Thus we observe, that as opposed to the O(N) case for which the relaxation time grows as a power of L (see eq. (3.22)), for the Ising case the relaxation time grows exponentially with increasing L. A detailed instanton calculation to determine the relaxation time is beyond the scope of this article. It ought to include also the contribution of the q Φ 0 modes, since the reversal of the magnetization of the sample occurs via the development of a bubble of the opposite phase, thus one is forced to consider inhomogeneous configurations*. * See the note added at the end of the paper.
124
CPSC 2 - paper 1.8
Υ. Υ. Goldschmidt / Finite size scaling effects
352
Fig. 1. Qualitative shape of the potential of the quantum hamiltonian for the case (a) JC> \, (b) 0 < x < \, (c) — \ < x < 0, (d) x < - \ . The horizontal line denotes the zero energy ground state.
4. The relaxation time in large n limit In the large n limit there is a simple relation between the relaxation time for the cubic geometry and the correlation length both above and below Tc. We will follow the notation of Brezin [5]. In the large n limit we can show that the time dependent correlation function for the model defined by eq. (2.3) is given by the expression 1
r dco
(
2λ
(4.1)
where mL depends on r0 and u0 but not on momentum and frequency. It is the same self-energy part as that appearing in the static calculation. As τ -> oo the leading contribution to C(q, ω) comes from# = 0. Hence C(T)
1
e'WT
,d
~ τ*hz
2 i2,
λ
2
e XomL|T| > 2,2~ -
(4·2)
ml\t).
(4.3)
from which we can infer that T
R
(L)=— Λ
ο
CPSC 2 - paper 1.8
125
Y. Y. Goldschmidt / Finite size scaling effects
353
The quantity mLl(Tc) has been evaluated in ref. [5] and identified with the correlation length. Hence we obtain ε= 4-Λ^0Η 2 (lnL) 1/2 rM)cc{ -U Λ
rf=4
J_ L 2 L «/-4)/2 5
d>4
ο
λ
0
(4.4)
These results are consistent with our previous results above and below four dimensions. Below Tc9 relation (4.3) still holds, but mL has not been evaluated for the cubic geometry. Brezin evaluated mL for the cylindrical geometry with one infinite dimension [5]. It is possible to modify his equation (60) to our case. The second term on the r.h.s. has to be replaced by
Σνττ^^'^^^^'^^^ττι^Γ 6 "'^ 0 ^ 72 ^' (4·5) where
m = Σν-2/4'
(4.6)
and y = Lmh. For yL«1 one can replace g(t) by its asymptotic behavior ~ (4mtY/2 and hence one finds
ß-ßc
= L-"ml2,
(4.7)
from which we obtain at T < Tn TK(L)=-LdΛ(ß-ßc). ο
(4.8)
From eq. (3.19) and the fact that ηβ ~ \/t we see that for large L
ß-ßc~-L-^\L) and thus eq. (4.8) agrees with our result (3.20).
126
CPSC 2 - paper 1.8
(4.9)
354
Υ.Ύ. Goldschmidt / Finite size scaling effects
5. Conclusions We have carried here a computation of the scaling behavior of the relaxation time for a system with a cubic geometry and periodic boundary conditions. We have used model A dynamics. It will be useful in the future to consider other geometries and also other types of dynamics. Our results should be useful for an estimation of the equilibration time in Monte Carlo calculations as a function of the system size. This work was supported by the National Science Foundation under grant #DMR-8302323. Note added After this work had been submitted for publication we learnt from Dr. J. Zinn-Justin (private communication), that a similar work has just been completed by himself together with J.C. Niel. They obtained many of the results of this paper, and also carried out a detailed instanton calculation to evaluate the relaxation time of the Ising model below Tc. They found TR ~ exp(Z/ -1 · 2σ(Γ)). References [1] M.N. Barber, in Phase transitions and critical phenomena, vol. VIII, eds. C. Domb and J. Lebowitz (Academic Press, NY, 1984) and references therein [2] M.E. Fisher, in Critical phenomena, Proc. 51st Enrico Fermi Summer School, Varena, ed. M.S. Green (Academic Press, NY, 1972); M.E. Fisher and M.N. Barber, Phys. Rev. Lett. 28 (1972) 1516 [3] M.P. Nightingale, Physica 83 (1976) A561 [4] E. Brezin and J. Zinn-Justin, Nucl. Phys. B257 [FS14] (1985) 868 [5] E. Brezin, J. de Phys. 43 (1982) 15 [6] P.C. Hoenberg and B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435 [7] M.E. Fisher and Z. Racz, Phys. Rev. B13 (1976) 5039 [8] J.M. Sancho et al., J. Phys. A13 (1980) L443 [9] R. Bausch et al., Z. Phys. B24 (1976) 113 [10] A. Munoz Sudupe and R.F. Alvarez-Estrada, J. Phys. A16 (1983) 3049 [11] J. Zinn-Justin, Nucl. Phys. B275 [FS17] (1986) 135 [12] M. Suzuki, Prog. Theor. Phys. 58 (1977) 1142 [13] B. Halperin et al, Phys. Rev. Lett. 29 (1972) 1548; C. de Dominicis et al, Phys. Rev. B12 (1975) 4945 [14] A.M. Polyakov, Phys. Lett. 59B (1975) 79 [15] E. Brezin and J. Zinn-Justin, Phys. Rev. Lett. 36 (1976) 691; Rev. B14 (1976) 3110 [16] C. de Dominicis et al, Phys. Rev. B15 (1977) 4313 [17] E. Witten, Nucl. Phys. B188 (1981) 513 [18] F. Cooper and B. Freedman, Ann. of Phys. 146 (1983) 262 [19] R. Bausch, H.K. Janssen and Y. Yamazaki, Z. Phys. B37 (1980) 163 [20] V. Lebedev, Phys. Lett. 105A (1984) 173
CPSC 2 - paper 1.8
127