H.2 Euler-Maclaur in Sum Formula
27.1 Ising Model, Mean Field Treatment
In the presence of a magnetic fieldB, we write the Hamiltonian for the Ising model in the form1
H= −1 2
i,j
Jijσiσj−μ∗B
i
σi, (27.4)
whereμ∗>0 is a magnetic moment (not the chemical potential) and Jij=
J>0 ifiandjare nearest neighbors.
0 otherwise. (27.5)
There is still only one coupling constantJas in Eq. (27.3) but this change of notation will facilitate the sums over nearest neighbors.
We denote byσithe ensemble average value ofσi. Then we substitute the identity
σi=(σi− σi)+ σi (27.6)
and a similar one forσjto obtain
−1 2
i,j
Jijσiσj=1 2
i,j
Jijσiσj −
i,j
Jijσiσj
−1 2
i,j
Jij(σi− σi)(σj− σj), (27.7) where cross terms have been combined after interchange of iand jto give a factor of 2 in the second term. We shall use periodic boundary conditions so all lattice sites are equivalent. Thusσi = σj ≡ σand Eq. (27.7) takes the form
−1 2
i,j
Jijσiσj=1
2JNqσ2−Jqσ
i
σi
−1 2
i,j
Jij(σi− σ)(σj− σ), (27.8) whereN is the number of lattice sites andqis the number of nearest neighbors.
The term on the second line of Eq. (27.8) represents correlations between nearest-neighbor spins. This may be seen because its average would vanish if σiσj = σiσj = σ2 for i = j. This can also be seen because the average of the first two terms is −(1/2)Jqσ2 which would not equal the average of the left-hand side unless nearest-neighbor spins were uncorrelated. The second term on the right-hand side resembles the term in Eq. (27.4) that contains the external magnetic fieldB. This becomes more evident if we introduce the notation
BJ:=Jqσ/μ∗ (27.9)
1The factor of 1/2 avoids double counting of interactions. The reader is cautioned that Ising Hamiltonians are often written without this factor of 1/2 and sometimes also with a factor of 2. If one sums only over nearest- neighborpairs, the factor of 1/2 should be omitted. The sign convention here is thatμ∗σiis the magnetic moment for this pseudo-spin.
in which case Eq. (27.4) takes the form H= 1
2JNqσ2−μ∗(B+BJ)
i
σi−1 2
i,j
Jij(σi− σ)(σj− σ). (27.10) The quantityBJis seen to play the role of a mean field experienced by a given spin due to the presence of the other spins.
Themean-field approximation2consists of ignoring the correlation term, resulting in a mean field Hamiltonian
HM=1
2JNqσ2−μ∗(B+BJ)
i
σi. (27.11)
Many books also ignore the first term in Eq. (27.11) because it depends only on average quantities and plays no role in computing the magnetization. Omitting it, however, leads to an average energy forB=0 that is too large by a factor of 2; this requires “patching” by a factor of 1/2 in a somewhatad hocmanner.
By using the mean field Hamiltonian given by Eq. (27.11), we obtain a problem for which the spins are formally independent. The canonical partition function for a single spin is therefore
z=exp[−β(1/2)Jqσ2]2 cosh[βμ∗(B+BJ)] (27.12) and the probabilities are
p+= exp[βμ∗(B+BJ)]
2 cosh[βμ∗(B+BJ)]; p−= exp[−βμ∗(B+BJ)]
2 cosh[βμ∗(B+BJ)] (27.13) forσi = ±1, respectively. Note that these probabilities do not depend on the exponential factor inz, which came from the first term in Eq. (27.11). The magnetizationM=Nμ∗σ, where
σ =p+−p−=tanh[βμ∗(B+BJ)] (27.14) and the average energy is3
U= HM = −1
2JNqσ2−μ∗NBσ. (27.15) SinceBJis given by Eq. (27.9), we see that Eq. (27.14) can be rewritten in the form
σ =tanh[βμ∗(B+Jqσ/μ∗)], (27.16) which is a self-consistency equation forσ. We can solve this equation graphically by defining a dimensionless parameter
x=βμ∗(B+Jqσ/μ∗). (27.17)
2This is also known as the Bragg-Williams or the Weiss molecular field approximation.
3SinceBJ depends onσwhich in turn depends onβandB, the formulaeM = −NkBT∂lnz/∂Band U= −∂lnz/∂βwill only work ifσis held constant during the differentiation. This inconsistency of the mean field approximation arises because average quantities appear in the mean field Hamiltonian.
472 THERMAL PHYSICS
-3 -2 -1 1 2 3
-1 -0.5 0.5 1
x
FIGURE 27–1 Graphical solution of Eq. (27.19) for B =0. The curve is tanhxand the lines have slopeskBT/Jq of 1.4, 1, and 0.6. There are only solutions forx=0 forkBT/Jq<1.
Then
σ = −μ∗B Jq +kBT
Jq x (27.18)
and Eq. (27.16), which is nowσ =tanhx, becomes
−μ∗B Jq +kBT
Jq x=tanhx. (27.19)
Viewed as a function ofx, the left-hand side of Eq. (27.19) is just a straight line of slope kBT/Jqand intercept−μ∗B/Jqand the right-hand side is a curve that can be drawn once and for all. The caseB=0 is of special importance and is illustrated inFigure 27–1. In that case,xis just proportional toσ. Since the slope of tanhxis one atx=0, we see that there are solutions forx=0 provided thatkBT/Jq<1 and otherwise no solutions. This defines acritical temperature
Tc=qJ/kB (27.20)
below which there is a spontaneous magnetization in the absence of an applied magnetic field.
Note that ifxis a solution,−xis also a solution. This degeneracy arises becauseB =0 so there is no preferred direction for the spontaneous magnetic field. If we started with a finite positive field and then let it shrink to zero, we would create a bias for the positive solution.
Graphical solutions for B > 0 are illustrated in Figure 27–2. We see that positive solutions4 forxexist for all values ofkBT/Jq, but those for large T correspond to small values ofxand therefore to small values ofσ = tanhx. Similar considerations lead to negative solutions for allTwhenB<0.
4For sufficiently small positive values ofkBT/Jq, there can also be negative solutions. These can be shown to correspond to metastable or unstable solutions that represent cases in which a magnetic field is applied in a direction opposite to the spontaneous magnetization that occurs in zero field.
-3 -2 -1 1 2 3
-1 -0.5 0.5 1
x
FIGURE 27–2 Graphical solution of Eq. (27.19) forB>0, namelyμ∗B/Jq=0.5 for the sake of illustration. The curve is tanhxand the lines have slopeskBT/Jqof 1.4 and 0.6. There are positive solutions for all values ofkBT/Jqbut those for largeTcorrespond to small values ofxand therefore to small values ofσ =tanhx. ForB<0, the lines would have a positive intercept and the solutions forxwould be negative.
The foregoing results in the mean field approximation are suggestive butincorrect.
Indeed, it is possible to solve the Ising model exactly in one dimension and two dimen- sions and numerically in three and higher dimensions. There are also better approximate solutions for all dimensions. The most serious discrepancy occurs in one dimension where the mean field model leads to a critical temperature atkBTc/J = 2 but the exact solution displays no spontaneous magnetization. In higher dimensions, there are critical temperaturesTc > 0 but the numerical values ofkBTc/J are different. For instance, the exact two-dimensional solution for a square lattice, due to Onsager, gives
kBTc
J = 2
ln(1+√
2)=2.26919, (27.21)
whereas the mean field approximation gives kBTc/J = 4. Some comparative values of Tc are given inTable 27–1. We see that the mean field model shows the general trend with dimensionality but is certainly wrong in detail because correlations are neglected.
The cluster model of Boethe (see Pathria [8, p. 329]) takes into account the correlations of a given spin with its neighbors but all other interactions (e.g., the interactions of its neighbors with other neighbors) are taken into account by a mean field. The critical temperature for the Boethe model satisfies
kBTc
J = 2
ln[q/(q−2)]. (27.22)
Table 27–1 Values ofkBTc/Jfor the Ising Model for a “Simple Cubic”
Lattice of Various Dimensionality According to Several Theories
Dimensionality 1 2 3 4 5 6 7
Exact 0 2.26919
Numerical 2.26919 4.51153 6.68003 8.77739 10.8348 12.8690 Boethe 0 2.88539 4.93261 6.95212 8.96284 10.9696 12.9743
Mean field 2 4 6 8 10 12 14
Note: Numerical results are from Galam and Mauger [70].
474 THERMAL PHYSICS
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.2
0.4 0.6 0.8 1
T/Tc σ
FIGURE 27–3 Dimensionless magnetization per spin,σ =M/Nμ∗, versus dimensionless temperaturet=T/Tcfor B=0 andb=μ∗B/Jq=0.05 according to the parametric equations Eq. (27.23). ForB=0, the magnetization is zero forT>Tcbut forB>0 it extends beyondTc.
In one dimension,q=2 so Eq. (27.22) givesTc=0, correctly showing that there is no phase transition for anyT>0.
For the mean field model,Figure 27–3shows a plot of the dimensionless magnetization per spin, M/Nμ∗= σ, as a function of T/Tc for B=0 and B>0. These plots were constructed by writing Eqs. (27.14) and (27.19) in the parametric form
σ =tanhx; t=tanhx x +b
x, (27.23)
where the dimensionless temperature t:=T/Tc and the dimensionless magnetic field b:=(μ∗B/Jq)=(μ∗B/kBTc). For a given value ofb, one can assign values of the parameter xand construct a plot ofσversust. Forb=0, we see thatσ =0 forT>Tcbut forb>0 it extends beyondTc, although its value is small.
For the case B=0 we can get a series representation for 1/t in terms ofm≡ σas follows: ForB=0 we can eliminatexfrom Eq. (27.23) to obtain
m=tanh(m/t)=e2m/t−1
e2m/t+1. (27.24)
We can then solve form/t, which amounts to finding the inverse of the hyperbolic tangent function to obtain
1 t = 1
2mln 1+m
1−m
= ∞ p=0
m2p
2p+1, (27.25)
where the series converges for|m|<1. Thus
t= 1
1+m2/3+m4/5+ ã ã ã =1−m2/3−4m4/45+ ã ã ã, (27.26) which can be solved to lowest order to give
m=√
3(1−t)1/2. (27.27)
Equation (27.27) shows how the magnetization rises from zero asTdecreases slightly from Tcand the exponent of 1/2 is referred to as acritical exponent. In one dimension, we know from the exact solution thatTc=0, so there really is no critical exponent in that case. For two dimensions, the correct critical exponent is 1/8 for a square lattice and approximately 0.313 for a simple cubic lattice in three dimensions. As was the case withTcitself, the mean field theory shows some qualitative trends but is certainly wrong in detail.
By differentiation ofUfrom Eq. (27.15) we can calculate the heat capacity, but we have to remember thatσ ≡mdepends onT. Thus
CV =∂U
∂T = −NJq(m+b)∂m
∂T. (27.28)
To calculate the derivative ofm, we write Eq. (27.16) in the form m=tanh
1 t(b+m)
. (27.29)
Then
∂m
∂T =sech2 1
t(b+m) 1 t
∂m
∂T − 1
t2Tc(b+m)
, (27.30)
which we can solve to obtain
∂m
∂T = −(b+m)(1−m2) t2−t(1−m2)
1
Tc, (27.31)
where we have used sech2
(b+m)/t =1−m2. Combining Eqs. (27.28) and (27.31) gives CV
NkB =(b+m)2(1−m2)
t2−t(1−m2) . (27.32)
Equation (27.32), however, is not very enlightening, so we introducex= (b+m)/tas in Eq. (27.23) and obtain, after some algebra, the parametric equations
t=tanhx x +b
x; CV
NkB = x2(tanhx+b)sech2x
(tanhx+b)−xsech2x. (27.33) Figure 27–4shows a plot ofCV/NkB versusT/TcforB =0 andB >0. ForB=0 there is a sharp peak atCV/NkB =3/2 atT = Tcand zero heat capacity forT >Tc. The height of this peak is not obvious from Eq. (27.32) or (27.33) because the function is discontinuous atT =Tcforb=0. It can be computed easily, however, by substituting Eq. (27.27), which holds fort≈1, into Eq. (27.15) forB=0 to obtain
U/N = −(1/2)Jqm2= −(3/2)kB(Tc−T); T≈Tc, (27.34) and then differentiating with respect toT.
The same technique of implicit differentiation can be used to compute the magnetic susceptibility
χ=∂M
∂B =Nμ∗∂m
∂B =Nμ∗2 kBTc
1−m2
t−(1−m2). (27.35)
476 THERMAL PHYSICS
0.2 0.4 0.6 0.8 1 1.2 1.4 0.25
0.5 0.75 1 1.25 1.5 1.75
T/Tc CV/NkB
FIGURE 27–4 Dimensionless heat capacity per spin,CV/NkB, versus dimensionless temperaturet=T/TcforB=0 andb=μ∗B/Jq=0.05 according to the parametric equations Eq. (27.33). ForB=0, the heat capacity rises to a sharp peak and then drops to zero forT>Tc, but forB>0 it extends beyondTc.
0.25 0.5 0.75 1 1.25 1.5 1.75 2 1
2 3 4 5 6 7 8
T/Tc
˜ χ
FIGURE 27–5 Dimensionless magnetic susceptibility,χ˜, versus dimensionless temperaturet = T/Tcforb = μ∗ B/Jq = 0.05 (low peak) andb=0.02 (high peak) according to the parametric equations Eq. (27.37). ForB=0, the susceptibility diverges asT→Tc.
Fort1 we havem1 so
χ≈ Nμ∗2
kB(T−Tc), (27.36)
which is known as theCurie-Weiss law. Parametric equations for the general case are t=tanhx
x +b
x; χ˜ :=χkBTc
Nμ∗2 = xsech2x
(tanhx+b)−xsech2x. (27.37) Figure 27–5shows a plot of the dimensionless susceptibilityχ˜as a function oft=T/Tcfor two positive magnetic fields. As the field strength is decreased, the peak in the vicinity of t=1 becomes progressively higher and ultimately diverges asB→0. We can see the nature of this divergence by substituting Eq. (27.27) into Eq. (27.35) to obtain
˜ χ≈1
2(1−t)−1 fort<1 andt≈1. (27.38)
Thusχ diverges like (1−t)−γ with a critical exponentγ=1. Although the mean field model is incorrect, this value ofγis close to values 1.2-1.4 measured for magnetic systems [8, p. 336].