Lecture 3: ϕ4 Theory
Naoki KAWASHIMAISSP, U. Tokyo

April 22, 2024

Statistical Mechanics I: Lecture 3

April 22, 2024

1 / 11In this lecture we see ...
The mean-field theory discussed in the previous section does not tell
us about the spatial correlation because the variational Hamiltonian
has trivial spatial structure, i.e., zero correlation among fluctuationsin spins at different locations.
In this lecture, starting from the Ising model, we derive ϕ4 model.
While it inherits the same essential properties from the Ising model, it
is defined with continuous degrees of freedom in contrast to the Ising
spins.
The advantage of the continuous degrees of freedom is that they
allow as to define a variational Hamiltonian that has non-trivial
spatial structure, which will be exploited in the next lecture.

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ϕ4 field theory
We first see a very “hand-waving” derivation of ϕ4 field theory
starting from the Ising model and using the coarse-graining.
We next see an alternative derivation which looks less hand-waving,based on the Hubbard-Stratonovich transformation.
Since the ϕ4 theory is obtained by the coarse-graining of the Isingmodel, they are supposed to share the same long-range behavior,
while they may differ quantitatively for short-range physics.
In particular, we expect, ϕ4 model belongs to the same universality
class as the Ising model, as has been verified by a number ofarguments and numerical calculations.

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Coarse-graining
Let us consider the Ising model on d-dimensional hyper-cubic lattice.
(Hereafter, we use symbols like x and y, instead of i and j, to specifylattice points.)
Divide the whole lattice into cells of size ab, where a is the lattice
constant and b ≫ 1, and denote the one located at X as Ωb (X).
d PConsider the “cell average” of spins ϕX = 1bx∈Ωb (X) Sx
Consider the coarse-grained Hamiltonian H̃ defined as
Xe−H̃(ϕ) ≡∆(S|ϕ)e−H(S)Swhere ϕ ≡ {ϕX }, S ≡ {Sx }, and ∆(S|ϕ) (= 0, 1) takes 1 if and only
if ϕX equals the cell average of original spins in the cell at every X.
The function H̃ is a very complicated one. So, we try to construct a
simple approximation by intuition.
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An intuitive approximation of H̃
H̃ must have two parts: a single-cell part reflecting the physics insideeach cell and a multiple-cell part for inter-cell interactions.
The single-cell part itself consists of two parts: the energy and theentropy. The internal energy tends to align spins parallel to eachother, giving rise to −ϕ2 term, while the internal entropy favors ϕ ∼ 0state, producing the terms like +ϕ2 and +ϕ4 .
For the multiple-cell part, since the total energy should be larger forlarge spatial inhomogeneity. It would be represented by terms like(∇ϕ)2 , etc, while the odd order terms like ∇ϕ should not appear
because of the symmetry of the system.
Putting these together and including the Zeeman term,XH̃(ϕ) =
ρ|∇ϕ|2 + tϕ2 + uϕ4 − hϕX(ρ, u > 0. The sign of t depends on the temperature.)
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Derivation by the Hubbard-Stratonovich transformation
X KP
X K T′ ) Sx Sx ′(x,xZIsing =e∝e 2 S CS Cx,x′ =S∝

ZS==

⇒T

1T−1 ψ+Dψ e− 2K ψ CDϕ eZ

1Dψ e− 2K ψ CZ
∝



SXZ

−1 ψ+ψ T S−( KϕT C −1 ϕ−2PP c(|x−x′ |=0)

1(|x−x′ |=a)0(otherwise)· · · HS transformation

x log cosh ψx· · · trace over S

x log cosh(Kϕx ))· · · ϕ ≡ K −1 ψDϕ e−HHS (ϕ)

XK T −1
HHS (ϕ) = ϕ C ϕ −log cosh(Kϕx )2xStatistical Mechanics I: Lecture 3

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Relevant part of HHS
HHS (ϕ) =XK T −1
ϕ C ϕ−log cosh(Kϕx )2xFor the first term, we can express C −1 as C −1 = (cI + ∆)−1 = 1c I − ∆c + ···where ∆ is the lattice Laplacian. For c > z, the series converges exponentially.
So, we take up to the first order to obtain
X c − z
1T −1
2
2
ϕ C ϕ≈
ϕ + (∇ϕx )(see supplement)
c2 x c2x

For the second term, by expending it w.r.t. K (log cosh(x) ≈ x2 /2 − x4 /12), we obtain
1 ( K − K 2 ), u ≡ K 4 ):Hϕ4 as the relevant part of HHS (with ρ ≡ K2 c−z, t≡ 212c2c2
HHS ≈ Hϕ4 ≡Xρ(∇ϕ)2 + tϕ2 + uϕ4

(1)xThough the expansions used here may not be justified by the smallness ofvariables, they are justified by the renormalization group arguments.
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Supplement: Hubbard-Stratonovich transformation
For an arbitrary positive definite symmetric matrix A and a vector B, wecan show the following,ZPP− 21 x,x′ Ax,x′ ψx ψx′ + r Br ψxDψ eZ1 TT= Dψ e− 2 ψ Aψ+B ψZ
1 TT= Dξ|A|−1/2 e− 2 ξ ξ+η ξ (ξ ≡ A1/2 ψ, η ≡ A−1/2 B)Z
12 1
2= Dξ|A|−1/2 e− 2 (ξ−η) + 2 (η)
N

12N1T

= (2π) 2 |A|−1/2 e 2 (η) = (2π) 2 |A|−1/2 e 2 B A−1 B

By taking KC for A−1 and S for B,ZK T
1T −1TSCSe2∼ Dψ e− 2K ψ C ψ+ψ S

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Supplement: The matrix ∆ and derivatives
In the derivation of theϕ4 action, we considered the inverse of C ≡ cI + ∆, i.e.,
C −1 = 1c I − ∆c + · · · where ∆ is the lattice connectivity matrix
1 (if x′ and x are nearest neighbors)∆ x′ x ≡.
0 (othewise)
We used the follwoing formula:XXT′ϕ ∆ϕ = 2
ϕx ϕx =
ϕ2x′ + ϕ2x − (ϕx′ − ϕx )2(x′ ,x)(x′ ,x) X1X 222=ϕx+δ + ϕx − (ϕx+δ − ϕx ) =zϕ2x − (∇ϕx )22xx,δ

where δ is a vector pointing to nearest neighbors, and ∇ϕx is the lattice gradientvector. (It would correspond to the regular nabla operator in the limit of finespace discretization.)

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The meaning of ϕ in the HS derivation
Remembering the HS transformation, for any real symmetric matrix A,∂ X K S T (C+ 2η A)S−1TK⟨S AS⟩Ising = ZIsinge2∂ηSη→0ZP∂ −( K ϕT (C+ 2η A)−1 ϕ− x log cosh(Kϕx ))−1Ke 2= ZHS
Dϕ∂η
η→0Z
∂ −( K ϕT (C −1 + 2η C −1 AC −1 )ϕ−Px log cosh(Kϕx ))−1Ke 2= ZHS Dϕ∂ηη→0Z
PK T −1−1= ZHS
Dϕ e−( 2 ϕ C ϕ− x log cosh(Kϕx )) ϕT C −1 AC −1 ϕ= ⟨ϕT C −1 AC −1 ϕ⟩HSThis means
S corresponds to C −1 ϕ. In other words, ϕx representsPS̃x ≡ x′ Cxx′ Sx′ , i.e., a weighted sum of spins in a local cluster. (Thus,
we recover the original simple and intuitive derivation of ϕ4 theory.)
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Exercise 3.1: Consider an Ising model with only 4 spins.
H = −K(S1 S2 + S3 S4 ) − K ′ (S1 S3 + S2 S4 + S1 S4 + S2 S3 )By coarse-graining ϕ1 ≡ 21 (S1 + S2 ) and ϕ2 ≡ 21 (S3 + S4 ), obtain theexact effective Hamiltonian in terms of ϕ1 and ϕ2 , and verify theexistence of terms proportional to ϕ2 , ϕ4 and |∇ϕ|2 (= (ϕ1 − ϕ2 )2 ),respectively. (If necessary, solve numerically by setting some numericalvalues of your choice to K and K ′ .)Statistical Mechanics I: Lecture 3

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