Lecture 4 Phase transitions (part 2)
I The Ising model : mean field treatment
It is a lattice model of a ferromagnet
(E. Ising, Zeits. fur Phys 31. p.253 (1925)). The simplest lattice is
some finite subset Lambda in Z^D (D = number of dimensions).
At each lattice site there is a "spin")
capable of two positions, "up" and "down". Usually the spin at site
x is denoted s_x , capable of the values +/- 1. There is no kinetic
energy, so the phase space is just the set of 2^|Lambda|
configurations. A commonly used Hamiltonian
(this is the usual word, even though
there is no Hamiltonian dynamics on a lattice) is
H = 0.5 sum_{x,y} U_{x,y} s_x s_y
the simplest non-triv interaction is
U_{x,y} = -J if x,y are nearest neighbours, 0 otherwise
Note added after lecture: the following method of arriving at
formula (1) for the free energy is not a good one.
There is no need to calculate the entropy and indeed
it is rather confusing because the definition of entropy used
in earlier lectures for systems of particles does not work well
for lattice systems (because the energy of a lattice system
is bounded above). It is much better to get the formula (1)
for the free energy directly from the partition function Z
using the formula -F/LkT = lim {L -> infty} L^{-1} log Z ,
not bothering to consider the entropy.
For J = 0 the entropy per site is easily found; for E = 0 it is
S(0,M,L)/kL = lim (1/L) log [L!/N_+! N_-!] where N_{+/-} = (L +/- M)/2
= - 0.5[(1-m) log(1-m) + (1+m) log (1+m)] + const if E = 0,
but S(E,M,L)/kL = - infty if E <> 0, since there are no states with E <>0
where L = |Lambda| is the number of lattice sites, M = sum_x s_x
is the total magnetization and m = M/L is the magnetization per site.
(show convex graph of S/L vs m)
The free energy for J = 0 is
F(T,M,L) = min_E (E - TS(E,M,L))
= 0.5 kT [(1-m) log(1-m) + (1+m) log (1+m)] (1)
(show concave graph of F/L versus m)
The mean field approxn for the interaction energy is
- J times (M/L)^2 times number of pairs of sites
= - 0.5 zJL m^2
where z := coordination number (number of nearest neighbours of a given site)
so the approximate total free energy is
"F"/L = 0.5 kT [(1-m) log(1-m) + (1+m) log (1+m)] - 0.5 Jz m^2
to check convexity, differentiate:
(d/dm) "F"/L = kT arc tanh m - Jm (interpretation: a magnetic field)
d^2/dm^2 = kT/(1-m^2) - Jz
which is no longer convex at m = 0 if kT < Jz. The true mean-field free
energy curve is the convex envelope of the graph of "F".
Critical temp. is Jz/k.
II. The 1-D Ising model
To test the mean-field prediction, work out the exact free energy for
a 1-D Ising model with nearest neighbour interaction
(the problem that Ising did as a PhD student of Lenz). To make it
a bit more interesting I'll assume that the end spins are held in the
up position, i.e. s1 = s_L = + 1 (the "plus" boundary condition)
Z = Sum_{s2 ...s_{L-1}} exp (beta sum{i = 1}^{L-1} J s_i s_{i+1})
= [1 0] A^(L-1) [1 0]^T
where A is the **transfer matrix**
exp beta J exp -beta J
exp -beta J exp beta J
Hence - F /kT = log lambda
where lambda is the largest eigenvalue of A, which is 2 cosh J/kT and so
F = - kT log (2 cosh J/kT)
This is analytic, so there is NO phase phase transition,contrary to what
mean-field theory says.
The transfer matrix method is very important in higher dimensions,
where this matrix becomes infinite-dimensional in the thermodynamic limit,
so that its largest eigenvalue need not be analytic.
III. Exact solutions, critical exponents, etc.
Onsager (1944) Phys Rev 65 117-149, using the transfer matrix method
for the 2-D Ising model, found an exact formula for F.
The critical temp is not 2J/k as given by mean field theory but
2J/k arc sinh 1 = 0.88 x 2J/k. [Other predictions of MFT are also
wrong, e.g. the MFT says m* proportional to (Tc-T)^{1/2}
but the correct exponent is 1/8. There was a big critical exponents
industry, at first phenomenological and after 1971 based on
the "renormalization group" of K G Wilson (like Onsager, a Nobel laureate)]
IV The Peierls argument
R E Peierls (1936) Proc Camb Phil Soc 32, 477-481
R B Griffiths (1964) Phys Rev 136A, 437-439
R L Dobrushin (1965) Teor Verojatn i ee Prim 10, 209-230
I will use this argument to show that the 2-D Ising model with
nearest neighbour-interactions has a spontaneous magnetization
at sufficiently low temps.
The argument can be generalized to other cases,
e.g. if second-nearest neighbour interaction is added in,
or if we go to 3 or more dimensions.
Theorem: at sufficiently low temp, the 2-D Ising ferromagnet
with nearest neighbour interactions has a spontaneous magnetization
(i.e. magnetization in the absence of an external magnetic field)
Its sign depends on the boundary conditions : this dependence
is a sign of **long-range order**.
proof:
Consider Ising model in a square box Lambda with minus
boundary conditions (i.e. spins on the boundary of Lambda
are required to be minus). The configurations are sets
of plus spins on Lambda, subject to the BC that
no plus spins are next to the boundary of Lambda.
Draw the dual lattice Lambda' such that each lattice
site of Lambda is a cell of Lambda'.
By a contour we mean a closed polygon made up of edges on the dual lattice.
Any configuration omega is characterized by a set of
minus spins on Lambda, or equivalently by a set omega' of
contours on Lambda' made up of edges on the dual lattice
that have opposite spins on opposite sides.
Because of the minus BCs the contours are all closed polygons.
The energy of the configuration omega is
E_0 + sum_{gamma \in omega'} 2J |gamma|
where E_0 is the energy of the config with no contours at all
(i.e. all spins are minus) and |gamma| is the length of contour gamma.
Let gamma be any contour on the dual lattice, and define
A_gamma is the set of polygon configurations that include gamma
B_gamma is the set of configs each obtained from a config in
A_gamma by removing the one polygon gamma.
Then the configs in A_gamma and B_gamma are in 1-1 correspondence
and each of the A configs has energy 2J|gamma| more than the corresponding
B config. It follows that
Prob(A_gamma))/Prob(B_gamma) = exp( - beta 2J |gamma|)
so that the probability that a given contour gamma
is a part of omega' satisfies
Prob(A_gamma) <= exp( - beta 2J |gamma|)
Consider any site x on Lambda.
Prob (s_x = +1) = sum_{gamma: x in gamma} prob(gamma) times
prob(s_x = +1 conditional on gamma being present)
+ prob( x is inside 3 countours) + etc
<= sum_{gamma: x in gamma} prob(gamma)
<= sum_{gamma: x in gamma} exp(-beta 2J |gamma|)
= sum_n C_n(x) exp(- beta 2Jn)
where C_n = number of possible contours of length n surrounding x.
But C_n is less than the max area of a contour of length n times the number
of polygons of length n starting from a given point, so that
C_n < (n/4)^2 3^n
Hence the series converges for 2 beta J > log 3 = 1.1 and can be made
less than 0.5 by making beta large enough.
With such a choice of beta, the magnetization at x is then
Expectation of s_x = prob(s_x = +1) - prob(s_x = -1)
= 2 prob(s_ = +1) - 1
< 0
conclusion: at low enough temperatures there is a non-zero
spontaneous magnetization, having the same sign as the imposed
magnetization at the boundary. (see Ruelle, Statistical Mechanics
(Benjamin 1969), theorem 5.3.1)
V. Other theorems about this and similar systems, not mentioned in lecture:
If there is a spontaneous magnetization m*, then the curve of
free energy versus m is flat for -m* <= m <= m* (Ruelle, theorem 5.3.3)
At sufficiently *high* temperatures there is *no* phase transition
(Ruelle, theorem 5.2.1)
Various inequalities, the simplest being that of Griffiths
J Math Phys 8 (1967) 478-483 showing that for an Ising
ferromagnet (in which the interactions are not required to
be nearest-neighbour) the correlation
of the spins at different sites, E(s_x s_y) - E(s_x)E(s_y)
where E(...) denotes an expectation, is non-negative.
See Ruelle theorem 5.4.1
The Lee-Yang circle theorem (Phys Rev 87 (1952) 410-419), which
implies that in any Ising ferromagnet with a non-zero magnetic field
(i.e. an extra term in the Hamiltonian proportional to the magnetization),
there is no phase transition. See Ruelle theorem 5.1.2