Lecture 2: Meanfield approximation, variational
principle and Landau expansion
Naoki KAWASHIMA1
1 ISSP, U. Tokyo

April 15, 2019

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

April 15, 20191 / 16

[2-1] Mean-field approximation
In this lecture we will see:Molecular-field theory does not gives us the free energy.
Gibbs-Bogoliubov-Feynman inequality give us a very systematic andfrexible framework for constructiong the mean-field-type
approximations.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

April 15, 20192 / 16

Molecular field theory revisitedIn the molecular-field theory, the effect of environment is replaced byan additional term, in the case of Ising ferromagnet, we focus on asingle spin, say S0 , and replace the Hamiltonian asXX
H = −JSi Sj − H
Si → HMF = −HMF S0 − HS0ij

i

It is also argued that the right choice of the effective field isX
HMF = JhSj ij

The uniformity condition m = hSi i (independent of i) yields,m = tanh(β(H + zJm))

Naoki KAWASHIMA (ISSP)(z = (number of nearest-neighbors))

Statistical Machanics IApril 15, 20193 / 16

Why should we complain?
In principle, we have multiple solutions of the self-consistent equation.
The molecular-field theory does not tell us which solution we shouldchoose.
If the theory allowed us to compute the free energy for each solution,
we would be able to tell which one to take.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

April 15, 20194 / 16

Gibbs-Bogoliubov-Feynman (GBF) inequality
Theorem 1 (GBF inequality)
For two Hamiltonians H(S) and H0 (S) defined on the same space S ∈ Ω,Fv ≡ F0 + hH − H0 i0 ≥ F ,(1)where F and F0 are the free-energies of H and H0 respectively and h· · · i0
is the thermal average with respect to H0 .

Variational calculation
When H(S) is the Hamiltonian of the system that we want to study but is
not solvable, by taking H0 (Λ, S) for H0 in (1) that also depends on a listof parameters Λ and is solvable for any Λ, Fv (Λ) gives us the upperbound of the correct free energy. (And it is computable!)

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

April 15, 20195 / 16

GBF inequality from information-scientific view-point
Theorem 2 (Relation to Kullback-Leibler divergence)
The “error” in the variational free-energy is proportional to theKullback-Leibler divergence of the thermodynamic distribution of H0
relative to that of H.
More precisely,
Fv − F = kB T IKL [ρ0 |ρ]

(2)

where
ρ0 ≡ e −βH0 /Z0 ,and ρ ≡ e −βH /Z

(3)

P(S)
Q(S)

(4)

andIKL [P|Q] ≡XS

P(S) log(Z and Z0 are partition functions of H and H0 respectively.)
Naoki KAWASHIMA (ISSP)

Statistical Machanics I

April 15, 20196 / 16

Proof of Theorem 2hi
IKL Z0−1 e −βH0 Z −1 e −βH=XS

e −βH0 (S)logZ0e −βH0 (S)Z0!

e −βH0 (S)−logZ0e −βH(S)Z!!

= − log Z0 + h−βH0 i0 + log Z − h−βHi0= βF0 − βhH0 i0 − βF + βhHi0= β(Fv − F )Naoki KAWASHIMA (ISSP)

Statistical Machanics I

April 15, 20197 / 16

Kullback-Leibler information measure is positive
IKL [P|Q]
X
X
P(S)
Q(S)
=
P(S) log=−P(S) log
Q(S)
P(S)
SS

X
Q(S)
≥−P(S)−1(because log(x) ≤ x − 1)
P(S)SX=(Q(S) − P(S)) = 1 − 1 = 0S

RemarkThis inequality together with Theorem 2 proves Theorem 1.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

April 15, 20198 / 16

Quantum extention
Though the theorems have been proved for classical systems, thecorresponding quantum version of them can be also proved.
For the extension, the KL divergence must be generaized toIKL [P|Q] ≡ Tr (P(log P − log Q))(5)where P and Q are now density operators satisfying Tr (P) =Tr (Q) = 1.
Only non-trivial part in the proof of the quantum extension is thepositivity of theP KL information. The rest is straight-forward simply
by replacing S by Tr.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

April 15, 20199 / 16

Proof of quantum extention (1)Theorem 3 (Positivity of quantum KL divergence)
For any density operators P and Q,
IKL [P|Q] ≡ Tr (P log P − P log Q) ≥ 0(6)Proof:Let us take the basis set in which P is diagonal, i.e., Pij = pi δij .
∗For some unitary operator U, Qij = uik qk ujk

With this uij ,X
X
IKL [P|Q] =pi log pi − piaij log qjiNaoki KAWASHIMA (ISSP)(aij ≡ |uij |2 )j

Statistical Machanics I

April 15, 201910 / 16

Proof of quantum extention (2)P
PNow, notice that aij ≥ 0,i aij =j aij = 1.
Let us define pij0 = pi aij , qij0 = qj aij .
Then, these can be regarded as the classical distribution function in
the squared Hilbert space H × H ≡ { (ij) | i, j ∈ H}:
X
X
pij0 ≥ 0, qij0 ≥ 0,pij0 =qij0 = 1
ij

ij

Now, we can seeXXX
pi log pi +pi aij log aij
pij0 log pij0 =ijXi

pij0 log qij0 =ij

X
ij

ij

pi aij log qj +X

pi aij log aij ,ijPThus we have obtained IKL [P|Q] = ij pij0 log(pij0 /qij0 ) ≥ 0(because the RHS is the classical KL information).
(QED)
Naoki KAWASHIMA (ISSP)

Statistical Machanics I

April 15, 201911 / 16Variational approximation to the Ising model (1)PP
For the target Hamiltonian H ≡ −JSS−Hijiji Si , let us takePthe “trial” Hamiltonian H0 ≡ −Λ i Si . where Λ is a variationalparameter.
Then, our variational free energy isFv = F0 + hH − H0 i0= hHi0 − S0 T
where S0 ≡ T −1 (hH0 i0 − F0 ) is the entropy of the H0 system
By introducing m ≡ hSi i0 = tanh βΛ,zhH0 i0 = − NJm2 − HNm
S0 = Nσ(m)21+m 1−m1−m1+mlog+logσ(m) ≡ −kB
2222

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

April 15, 2019(7)(8)

12 / 16

Variational approximation to the Ising model (2)Variational free-energy densityfv ≡FvzJm2=−− Hm − T σ(m)N2The GBF inequality tells us that we should minimize fv with respect
to λ. Since fv depends on λ only through m, the stational condition∂fv /∂λ = 0 leads to ∂fv /∂m = 0.
From this, we obtain the same as the molecular-field approx.:∂fv=0∂m

Naoki KAWASHIMA (ISSP)⇒m = tanh βHMF

Statistical Machanics I(HMF ≡ zJm + H).

April 15, 2019(9)13 / 16

Mean-field free energy — Landau expansion
We have also obtained the explicit expression for the free energy.
Since its behavior near m ≈ 0 is most important for the criticalphenomena, let us expand fv with respect to m.
m2 m4zJ 2−
fv = − m − Hm − kB T log 2 −2212(10)

From the condition (coefficient of m2 ) = 0, we obtain kB Tc = zJ.
Near T ≈ Tc , we can get the Landau expansion:
fv ≈ fv0 + tm2 + um4 − Hm(11)where fv0 ≡ −zJ log 2, t ≡ (kB T − zJ)/2 and u ≡ zJ/12

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

April 15, 201914 / 16[2-2] Summary of Lecture 2
Molecular-field theory does not give us the free energy.
Gibbs-Bogoliubov-Feynman inequality give us a very systematic andfrexible framework for constructiong the mean-field-type
approximations that also provides the free energy.
Landau expansion is useful in having a clear view of the phasetransitions.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

April 15, 201915 / 16

Homework
Veryfy (7) and (8).
At the point where we have arrived at (9), m is just a variationalparameter and its physical meaning is not yet clear. Give the reason
why we can interprete it as the magnetization.
Following the example of the Ising model in the lecture, obtain theLandau expansion of the 3-state Potts model by taking the trial
HamiltonianX
H0 ≡ −ΛδSi ,1 .
(12)
iThis time, the order parameter should be m ≡ hδSi ,1 i − 1/3. What isthe essential difference from the Ising case?
Submit your report on one of these problems at the beginning of thenext lecture.
Naoki KAWASHIMA (ISSP)

Statistical Machanics I

April 15, 201916 / 16