Discussion about the presence of a phase transition in a ferromagnetic spin lattice with the aid of statistical models Leipzig, June/July 2007
Alexandre Barrat Professor Manfred Salmhofer by
Tutor :
1
Contents 1
Phase transition and critical point
3
2
The Nearest Neighbour Ising model - Peirls argument
5
3
A more realistic model for the spins
13
4
Conclusion
17
2.1 2.2 2.3 2.4
Nearest Neighbour Ising model . . . . . . . . . Finite lattice case . . . . . . . . . . . . . . . . . Peirls argument - Physical point of view . . . . Peirls argument - The proof with contour theory
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5 6 8 9
3.1 One and two-dimensional systems - Mermin Wagner proof . . . . 13 3.2 Three and more dimensional systems - Fröhlich Simon Spencer proof 15
References
17
My internship took place in the 'Institut für Theoretische Physik' in Leipzig (Germany) during 7 weeks in June and July 2007. It consisted in reading and understanding articles or parts of books about phase transition in ferromagnets. First, I learned more about some physical quantities and some experimental facts. Then, I studied many mathematical models on this subject. After a short presentation of the physical problem of phase transition in a ferromagnet, we shall here present some important results of these models.
2
1
Phase transition and critical point
Consider a ferromagnet (for example an iron bar) in a magnetic eld B. What we shall study is its magnetization M in the same direction as B, as a function of B and the temperature T. M is expected to be an odd function of B in every cases. But the interesting phenomenon is that at su ciently low temperatures, if B decreases to zero, then M decreases, but not to zero (see g. 1). That is to say there is a spontaneous magnetization M0 at zero eld, or spontaneous ”symmetry breaking”.
Figure 1 : M as a function of B for T su ciently small T < TC
If we increase slightly the temperature of the system, we obtain the same graph but M0 decreases. And as we reach a limit temperature TC (called the Curie temperature), there is no spontaneous magnetization anymore : M0 = 0 (see g. 2).
Figure 2 : M as a function of B for T = TC
For higher temperatures also, M(B) is a continuous function, and becomes analytic at B = 0 (see g. 3). 3
Figure 3 : M as a function of B for T > TC
At low temperatures (T < TC ), it is said that the ferromagnet undergoes a as B changes from negative to positive values. The function M(B) has a hysteresis behaviour. At high temperatures, we cannot observe such a phase transition. We can summarize these observations by representing the set of points where the function M(B, T ) is discontinuous. They are all contained in the plane M = 0, i.e. the (B,T) plane (see g. 4). phase transition
Figure 4 : Set of points where M(B, T ) is discontinuous in the plane M = 0 . There exists two phases to the left of the vertical line, only one to the right
The particular point (TC , 0) is called the critical point. The function M(B, T ) has a singularity at this point, whose study is very interesting. But we shall now concentrate on the phenomenon of phase transition.
4
2
The Nearest Neighbour Ising model - Peirls argument
2.1
Nearest Neighbour Ising model
The general Ising model is a model of a magnet. Suppose the N molecules of the magnet are placed on a regular lattice, labelled by i = 1,2...,N. Notice that N is to tend to in nity, according to the statistical limit [In the subsection 2.2 we shall see why we must take this limit in the case of magnetization]. Each site is caracterized by a spin value up to two possibilities : +1 or -1 (respectively parallel or anti-parallel to the axis of B). We note these con gurations brie y + and -. So, a magnet con guration is speci ed by a set of N values of spins σ = {σ1 , ..., σN }, where each σi veri es σi ∈ {−, +}. The Hamiltonian of the system is a function E(σ1 , ..., σN ) = E(σ). It is made of two parts : E(σ) = E0 (σ) + E1 (σ),
where E0 (σ) is the contribution from the intermolecular forces inside the magnet and E1 (σ) = −B
N X
σi is the contribution from the interactions between the spins
i=1
and the external magnetic eld. The Nearest Neighbour approximation consists in simplifying E0 by considering that only the nearest neighbours in the lattice interact with each other, i.e. : N X
E0 (σ) = −J
σi σj .
i=1,j=1
|i−j|=1
If the coupling constant J is positive, then the lowest energy state occurs when all spins point the same way, that is to say the model is ferromagnetic. J < 0 would be an anti-ferromagnetic model. 2.2
Finite lattice case
We shall consider the average magnetization < M >Λ of the system Λ, while B = 0. Let us remember that if X(σ) is a variable depending on the con guration σ , its statistical average value on Λ is, in the most general case : 5
R < X >Λ =
σ∈S
X(σ)e−βE(σ) ZΛ
Q
x∈Λ
dµ(σx )
,
• S is the set of possible spin con gurations • β = 1/kB T Y • dµ(σx ) is an appropriate measure on S x∈Λ
• ZΛ is the partition function (i.e. the normalizing factor) : Z Y dµ(σx ) . e−βE(σ) ZΛ = σ∈S
x∈Λ
If Λ ⊂ Zd is a lattice system, then : P
X(σ)e−βE(σ) . −βE(σ) σ∈S e
σ∈S
< X >Λ =
P
Suppose now that Λ ⊂ Zd is a nite set. Then we can prove that < M >Λ = 0 as follows. We use a symmetry argument : if −σ is the image of σ through the re exion S = {−1, +1} → {−1, +1} ∀x ∈ Λ, σx 7→ −σx ,
then E0 (σ) = E0 (−σ)
i.e. E(σ) = E(−σ), since E1 = 0 Indeed, changing the sense of every spins does not change the energy if B = 0. So, because Λ is nite, P < M >Λ =
σ∈{−1,+1}|Λ|
M(σ)e−βE(σ)
ZΛ
The sum is nite and can be decomposed into : X
ZΛ < M >Λ =
M(σ)e−βE(σ) +
σ∈{−1,+1}|Λ|
X
M(σ)e−βE(σ)
σ∈{−1,+1}|Λ|
σ(0)=+1
=
X σ(0)=−1
(M(σ)e−βE(σ) + M(−σ)e−βE(−σ) )
σ∈{−1,+1}|Λ|
σ(0)=+1
= 0
because M is an odd function of σ . As a conclusion, the spontaneous magnetization can never happen if Λ is nite. 6
2.3
Peirls argument - Physical point of view
Consider an Ising system in a zero- eld. Then the Hamiltonian is still E(σ) = E0 (σ) = −J
N X
σi σj , with J > 0 .
i=1,j=1
|i−j|=1
The ordered state is reached when every spins are aligned, either in the positive or in the negative direction of the axis. We want to test whether or not the ordered state is an equilibrium state. To do this we assume that the system is initially ordered and then perturb the state slightly while observing the change of the free energy F , where F = U − T S • U : internal energy • S : entropy of the state • T : temperature
If the ordered state is an equilibrium state, then F must be minimal, so the little perturbation must increase F (∂F > 0). On the other hand, if there exists a con guration that can drop the free energy, the ordered system will spontaneously tend towards this new con guration, preferring it energetically. That would mean that the ordered state is no equilibrium state. In one-dimension
In the 1D Nearest-Neighbour Ising model, the Hamiltonian is E(σ) = −J
N X
σi σi+1 .
i=1
The considered perturbation is a random ip of one spin, which realises the smallest possible perturbation. The change of internal energy is ∂U = Hf lipped − Hordered = −J(N − 2) − (−JN ) = 2J
because every spin is connected to two nearest neighbours. There are N dierent way to ip a single spin so the variation of entropy is ∂S = Sf lipped − Sordered = kB ln N − kB ln 1 = kB ln N
7
Thus, ∂F = ∂U − T ∂S = 2J − kB ln N . For all T > 0, we obtain ∂F < 0 as N goes to in nity. So the ordered state is no equilibrium state in the one-dimension case and there is no phase transition phenomenon. In two-dimension
For the 2D-model (and higher dimension) the result is dierent. If we start from the ordered state, the smallest perturbation must cut the lattice into two domains. Let us calculate the best scenario for such a cut : the change of energy is at least 2N J . The entropy change is approximatively kB ln(3N ) because after each already ipped spin we have at most 3 choices to proceed with the cut. So the change of energy is ∂F = ∂U − T ∂S = 2N J − kB T N ln 3 = N (2J − kB T ln 3) .
So ∂F ≥ 0 ⇐⇒ T ≤ TC ,
where TC =
J 2J ≈ 2.27 . kB ln 3 kB
One has the stability of the ordered phase if T ≤ TC , and the instability for T ≥ TC . Moreover, this value of TC is surprisingly close to the real measured temperature, especially given the simplicity of the above argument. 2.4
Peirls argument - The proof with contour theory
Consider the same model, in a 2D lattice Λ ⊂ Z2 , Λ nite set. We will be interested in the limit Λ → Z2 . Denote σ a spin con guration σ = {σ(x), x ∈ Λ}, with σ(x) or −1, and σ 0 a boundary condition σ 0 = {σ 0 (y), y ∈ ∂Λ}, where = +1 ∂Λ = z ∈ Z2 \Λ/d(z, Λ) = 1 . Only neighbour spins interact, so the Hamiltonian is EΛ (σ/σ 0 ) =
X
σ(x1 )σ(x2 ) +
(x1 ,x2 )∈Λ2
X x∈Λ,y∈∂Λ
σ(x)σ 0 (y) + B
X
σ(x) .
x∈Λ
|x−y|=1
|x1 −x2 |=1
We suppose < 0 to have the ferromagnetic model, = −1 for simplicity. The probability of having the state σ knowing σ 0 (denoted by Pλ,β (σ/σ 0 )) is given by the statistical formula : 8
0
σ (σ) = Pλ,β (σ/σ 0 ) ≡ Pλ,β
1 0 e−βEΛ (σ/σ ) 0 Z(Λ, β, σ )
where Z(Λ, β, σ 0 ) is the partition function of the system and β = 1/kB T . For all su ciently large β (i.e. all su ciently small T ), and B = 0, there are at least two equilibrium states for the two-dimensional ferromagnetic Ising model. Theorem 1
Proof. Let us consider a sequence of squares Λ1 ⊂ Λ2 ⊂ ...Λn ⊂ ... such that [ Λn = Z2 , and two sequences of boundary conditions on ∂Λ : n 0 • σn,+ ≡ +1 everywhere in ∂Λ (see g. 5) 0 • σn,− ≡ −1 everywhere in ∂Λ
Figure 5 : Example of distribution in dimension d = 2 with the boundary 0 condition σ 0 = σn,+ 0
0
Each of the two sequences of distribution {Pλσnn,β,+ , n = 1, 2, ...} and {Pλσnn,β,− , n = 1, 2, ...} contains a convergent subsequence (we shall not prove this fact here, see + − [4], Lecture 9, Lemma 9.2) we denote them P∞,β and P∞,β . What we need to prove is that these two con gurations are dierent for su ciently large β and B = 0. Let Λ be a square centered at the origin, σ any con guration and σ 0 ˆ 2 be the ”dual” lattice, that is the a constant boundary con guration. Let Z ˆ 2 is called a boundary edge if translated lattice Z2 + (1/2, 1/2). An edge b ∈ Z b separates two spins x1 ∈ Λ and x2 ∈ Λ such that σ(x1 ) 6= σ(x2 ), or x1 ∈ Λ and x2 ∈ ∂Λ such that σ(x1 ) 6= σ(x2 ). The set of boundary edges of the con guration σ is called the boundary Γ(σ/σ 0 ) of this con guration. Each contour γ of the ˜ 2 . What boundary Γ(σ/σ 0 ) is a connected polygonal path composed of edges of Λ 9
is important to notice is that any nonintersecting collection of such contours uniquely determines the con guration σ . Thus, we have EΛ (σ/σ 0 ) = −
X
σ ˜ (x)˜ σ (y)
˜2 (x,y)∈Λ
= −
X
X
1+
˜ 2 \Γ(σ/σ 0 ) (x,y)∈Λ 2 0 ˜
1
(x,y)∈Γ(σ/σ 0 )
= |Λ | + 2|Γ(σ/σ )| ,
where σ ˜ (x) = σ(x) if x ∈ Λ = σ 0 (x) if x ∈ ∂+ Λ .
Then for B = 0 we get Q −2β|γi | 0 e e−2β|Γ(σ/σ )| = = , 0 Z(Λ, β, σ ) Ξ(Λ, β, σ 0 )
σ0 Pλ,β (σ)
with Ξ(Λ, β, σ 0 ) =
XY
e−2β|γi | , where the summation is over all collections of
γ0
˜ 2 . As in the 2.2, let us consider two events (sets nonintersecting contours on Λ of con gurations inside Λ) : V± = {σ/σ(0) = ±1} . By the same symmetry principle as in 2.2, we have 0
0
σ ≡+1 σ ≡−1 Pλ,β (V− ) = Pλ,β (V+ ) .
In both cases, the origin is contained inside an odd number of contours. Thus, the theorem will be established if we prove that σ ≡+1 Pλ,β (V− ) ≤ k xed < 1/2 . 0
Indeed, the inequality is still ful lled after passage to the limit Λ → Z2 : + − (V− ) = P∞,β (V+ ) ≤ k , P∞,β
and then + (V+ ) ≥ 1 − k , P∞,β
whence
+ + P∞,β (V+ ) 6= P∞,β (V− ) .
Denote Intγ the interior of a contour γ . We have 10
+ P∞,β (V− ) ≤ P (there is a contour γ ⊂ Γ(σ/σ 0 ) such that 0 ∈ Intγ ) X ≤ P (γ ⊂ Γ(σ/σ 0 )) γ/ 0∈Intγ
Moreover, P Q −2β|γi | γi e−2β|γ0 | e γi ∩γ0 =∅ 0 P Q P (γ ⊂ Γ(σ/σ )) = ≤ e−2β|γ0 | . −2β|γ | i e γi
Two contours are said equivalent if they dier by a shift along a vector in Z2 . Let us choose one representative of each equivalence class and remark that the number of contours equivalent to a given contour γ and containing the origin is precisely |Intγ |. Thus, X
X
P (γ ⊂ Γ(σ/σ 0 )) ≤
|Intγ |e−2β|γ| ,
γ∈Γ(σ/σ 0 )/∼
γ/ 0∈Intγ
where the sum is on the quotient space. Further, the number of dierent (nonequivn alent) contours of length n (|γ| = n) is less than 4 ∗ 3n−1 , and |Intγ | < ( )2 . 4 Hence, X X |Intγ |e−2β|γ| ≤ 4
γ∈Γ(σ/σ 0 )/∼
n ( )2 3n e−2βn . 4 n≥4
1
The last series converges for β > ln 3 an its sum is O(e−8β ) for β → ∞ . So 2 it is less than k for a su ciently large β . The theorem is proved. Remark 1 : + Typical con gurations for P∞,β,B=0 has the following form : there is an ”ocean” of values σ(x) = +1, in which nite ”continents” of values σ(x) = −1 are ”swimming”. Also, ”lakes” of values σ(x) = +1 can appear inside these ”continents”, and then there can be smaller ”islands” of values σ(x) = −1, and so forth... − Of course the same phenomenon is observed for P∞,β,B=0 with the opposite dominance. There is coexistence of two dierent phases.
Remark 2 :
An explicit calculation of the partition function of the two-dimensional Ising model for B = 0 and all values of B has been performed by Onsager. 11
3
A more realistic model for the spins
The problem of the Ising model is that it is physically incorrect : a single spin can rotate in every directions in our 3D space, so its projection along an axis can take every value between -1 and +1, and not only -1 and +1. Considering this fact, we shall see that the result of the existence of a phase transition is modi ed in 2D (and only in 2D). 3.1
One and two-dimensional systems - Mermin Wagner proof
Consider a model involving a two-dimensional spins on lattice of dimensionality d ≤ 2. Mermin and Wagner proved in 1967 that there is no spontaneous symmetry breaking. We could present their proof also for a three-dimensional spins but the idea is the same. Consider a classical system, described by canonical variables q1 , ..., qn , p1 , ..., pn (respectively positions and impulsions) and their Hamiltonian H(q1 , ..., pn ). We de ne the average of any function A(q1 , ..., pn ) as before : R Z dΓAe−βH = dΓAe−β(H−F ) < A >= R dΓe−βH
with • dΓ the space volume element dΓ = dq1 ...dqn dp1 ...dpn • β = 1/kB T • F ≡
−1 ln Z per de nition β
n X ∂A ∂B ∂B ∂A − ) . The Mermin We de ne also the Poisson bracket [A, B] := ( ∂qi ∂pi ∂qi ∂pi i=1
Wagner proof uses the Bogoliubov inequality which implies : < |A|2 > ≥
kB T | < [C, A∗ ] > |2 < [C, [C ∗ , H]] >
for all variables A and C depending on the canonical variables. Let us consider the statistical mechanical system de ned by the free energy 12
−βF
e
Z
2π
=
Y
0
and H=−
X
dθ(~r)e−βH
~ r
J(~r − ~r 0 ) cos(θ(~r) − θ(~r 0 )) − B
~ r,~ r0
X
cos θ(~r)
~ r
where ~r runs over the sites of a lattice of N sites (with periodic boundary conditions), θ(~r) is the angle between the local spin and the direction of the magnetic ~ , and J(~r) is the interaction energy. The canonical variables are now θ(~r) eld B and P (~r), the angular momentum along the axis. We take X
A(~k) =
~ r X
C(~k) =
~
sin θ(~r)e−ik.~r ~
P (~r)e−ik.~r
~ r
(Fourier transformations : ~k runs over [−π, π]d as ~r runs over Rd ) and hence the Poisson bracket is written as follows : [C, H] = −
X
~
e−ik.~r
~ r
∂H . ∂θ(~r)
Then < [C, [C ∗ , H]] > =
X
~
0
eik.(~r−~r )
+B
~ r,~ r0 X
≤ N(
∂ 2H > ∂θ(~r)∂θ(~r 0 ) X ~ r
r2 |J(~r)|k 2 + |B| |m|)
~ r
where m is the magnetization per particle : m =
1 X < cos θ(~r) > . Moreover, N ~ r
< [A∗ , C] >= N m . Therefore, from the Bogoliubov inequality we get X
~
0
< sin θ(~r) sin θ(~r 0 ) > eik.(~r−~r ) ≥
~ r,~ r0
k
N kB T m2 . 2 r)| + |B| |m| ~ r r |J(~
P 2
But 1 XX 1 X 0 i~k.(~ r−~ r0 ) ( < sin θ(~ r ) sin θ(~ r ) > e ) = < sin2 θ(~r) > < 1 , 2 N N 0 ~k
~ r,~ r
~ r
13
< cos θ(~r) >
so kB T m2 X 1 P 2 obeys c ≥ 1 − N I(d)T , where 2
Z d~k 1 I(d) = dd k . Pd d (2π) |ki |≤π i=1 (1 − cos ki )
In particular, c 6= 0 if T < TC , where TC =
2 . N I(d)
Proof. We shall use a lemma. Lemma 1
For all h with values in RN , < σ(h)σ(−∆h) >Λ ≤ T
X
|h(α)|2 ,
α,i
where σ(h) :=
X
h(α)σα and −∆ is the operator de ned by
α
(−∆h)(α) = 2dh(α) −
d X i=1
14
[h(α + i ) + h(α − i )] .
We shall not prove the lemma (see [6], pp. 81-84). Let Fj (α) =< σα(j) σ0(j) > . De ne a measure dωj by Z
eiαk dωj (k) .
Fj (α) = |ki |≤π
dωj is a positive measure because F is positive de nite. By the lemma, we get Z Z X d 2 d ˆ ˆ (1 − cos ki )|h(k)|dωj (k) ≤ T |h(k)| d k 2(2π) d
i=1
Z X X 1 2 d −ikα ˆ |h(α)|2 . | h(k)| d k = e h(α) is normalized so that (2π)d/2 α α d ˆ ˆ The inegality is true for all h, so by peaking h near any k0 ∈ [−π, π] , k0 6= 0, we ˆ where h(k) =
obtain that
dωj (k) = [Dj δ(k) + gj (k)]dd k
with d X d~k T (1 − cos ki )gj (k) ≤ . d 2 (2π) i=1
By summation over j , 1=
N X
= c+
j=1
N X
gj (k)dd k
j=1
1 ≤ c + N I(d)T . 2
which establishes the theorem. As a conclusion, there is a phase transition if and only if the dimensionality d ≥ 3.
15
4
Conclusion
The concepts and constructions used above are fundamental in modern mathematical approach to problems of statistical physics. These models are also useful for studying other physical systems. For example, by replacing on a lattice the spins -1 and +1 by 1 and 0 symbolizing respectively the presence or the absence of a particle, one can modelize a uid, for example a gas, thanks to the analogy of the corresponding Hamiltonian with the one we have studied here. So, all the discussions that we have made about dimensionality in a lattice are important, because they often apply in statistical physics. I wish to thank rst Manfred Salmhofer, professor and researcher at the Leipzig University, for everything he taught me. I also would like to thank Christoph Dehne, Christoph Husemann, Mrs. Menge and Mrs. Voigt for their warm welcome.
References
[1] Baxter, Exactly Solved Models in Statistical Mechanics [2] Manfred Salmhofer, Renormalization, ed. Springer [3] Janet Anders, Damian Markham, Vlatko Vedral and Michal Hajdusek, How much of one-way computation is just thermodynamics ?, Feb. 2007 [4] Minlos, Lecture 10 : Nonuniqueness of Gibbs Distribution [5] N.D.Mermin, Absence of Ordering in Certain Classical Systems, Journal of Mathematical Physics, Dec. 1966 [6] J.Fröhlich, B.Simon and T.Spencer, Infrared Bounds, Phase Transitions and Continuous Symmetry Breaking, Communications in Mathematical Physics, ed. Springer, 1976
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