Statistical physics of phase transitions

72 Pages·2001·1.055 MB·English

by  Dodgson M.J.W.

Statisti
al Physi
s of Phase Transitions
TCM graduate le
tures, Mi
haelmas term 2001
Matthew J. W. Dodgson
First le
ture: 5/11/01
1. Classi
al examples: Mean-(cid:12)eld theory of the ferromagneti
phase transition.
Van der Waals model/theory of liquid-gas transition. Landau theory of
ontinuous
phase transitions; interfa
es and domain walls; di(cid:11)erent order-parameter symmetries.
2. Stat. me
h. basis of phase transitions. Why do phase transitions o

ur?
(symmetry, separation of phase spa
e). Simple model of a (cid:12)rst-order phase transi-
tion. From stat. me
h. to MFT: Coarse graining and saddle-point approximation.
Example: the Ising model; exa
t solution for 1D nearest neighbours and for in(cid:12)nite-
range; simulations in 2D.
3. E(cid:11)e
t of thermal (cid:13)u
tuations on ordered phase. Importan
e of the thermo-
dynami
limit. Impossibility of symmetry breaking in 1D. Gaussian approximation,

ontinuous symmetry breaking and Goldstone modes. Lower
riti
al dimensions and
the Hohenberg-Mermin-Wagner theorem.
4. Field theory of phase transitions. Destru
tion of mean-(cid:12)eld theory by (cid:13)u
-
tuations near the
riti
al point; diverging spe
i(cid:12)
heat from Gaussian (cid:13)u
tuations.
Upper
riti
al dimension and Ginzburg
riterion. Perturbation theory and Feynman
diagrams. Breakdown of pert. theory below the upper
riti
al dimension.
5. S
aling and Renormalization Group approa
h. S
ale-invarian
e at the
rit-
i
al point. Review of
riti
al exponents. Renormalization Group transformations;
from (cid:12)xed points to
riti
al exponents and s
aling laws. Momentum-shell RG and
the Gaussian(cid:12)xed point. A new (cid:12)xed point below four dimensions;
riti
alexponents
to order (cid:15) = 4(cid:0)d.
6. The 2D-XY model. Expansions at low and high temperatures. Spin-waves and
vorti
es as relevant ex
itations. Simple Kosterlitz-Thouless argument for vortex-pair
unbinding transition. Mapping from XY-model to Sine-Gordon model, and RG anal-
ysis.
1
Books
P. M. Chaikin and T. C. Lubensky, Prin
iples of Condensed Matter Physi
s, (Cam-
bridge University Press, 1995).
Covers mu
h more than just phase transitions, but still manages to introdu
e mu
h
in these le
tures in an understandable way.
N.Goldenfeld,Le
tures onPhaseTransitionsandtheRenormalizationGroup,(Addison-
Wesley, 1992).
Ex
ellent text book on the statisti
al physi
s of phase transitions.
S.-K. Ma, Modern Theory of Criti
al Phenomena, (Addison-Wesley, 1976).
Possibly the (cid:12)rst textbook explaining (cid:12)eld theory and RG of
ontinuous phase tran-
sitions. Goes through the basi
examples in good detail, but perhaps not the ideal
(cid:12)rst introdu
tion to the subje
t.
C. Domb, The Criti
al Point, (Taylor and Fran
is, London, 1996).
Histori
al overview of the development of the theory of
riti
al phenomena.
J. Zinn-Justin, Quantum Field Theory and Criti
al Phenomena, (Oxford University
Press, 1989).
Heavy goingbookwhi
h
overs alotofthemathemati
alaspe
tsofthe
ourse, aswell
asshowingthelinksbetween thetheoryofphasetransitionsandquantum(cid:12)eldtheory.
2
Le
ture One
1 Early understanding of phase transitions
In this le
ture we review the
lassi
al understanding of phase transitions, dating from
over a
entury ago. We start withthe Weiss mean-(cid:12)eldtheory offerromagnetism. We
then look at the Van der Waals model of a liquid-gas phase transition. Both theories
have similar features, in parti
ular a
riti
al end-point to a
oexisten
e line. We
therefore look at the general des
ription of this
riti
al point, the Landau theory, and
(cid:12)nd that a diverging length s
ale is a
hara
teristi
of
ontinuous phase transitions.
1.1 Classi
al theory of the ferromagneti
phase transition
1.1.1 Statisti
al physi
s of a magneti
moment: paramagnetism
Consider a
lassi
al magneti
dipole moment of (cid:12)xed strength m0, whi
h is free to
rotate. Inamagneti
(cid:12)eldBthedipolehasanintera
tionenergy, E((cid:18);(cid:30)) = (cid:0)m(cid:1)B =
(cid:0)m0B
os(cid:18). In thermalequilibriumthe dipolewillsampleallpossibledire
tionswith
a Boltzmann probability distribution, P((cid:18);(cid:30)) = exp[(cid:0)(cid:12)E((cid:18);(cid:30))℄ (with (cid:12) = 1=T).
We
an
al
ulate the average magneti
moment in the (cid:12)eld dire
tion m~ = hmzi =
R
(cid:0)1
Z d(cid:18)d(cid:30)sin(cid:18) P((cid:18);(cid:30))m0
os(cid:18), where
Z Z
(cid:25) 2(cid:25) 4(cid:25)
Z = d(cid:18) d(cid:30)sin(cid:18)P((cid:18);(cid:30)) = sinh((cid:12)m0B): (1)
0 0 (cid:12)m0B
We qui
kly (cid:12)nd m~ by di(cid:11)erentiating this \partition fun
tion",
" #
1 1 Z 1
m~ = = m0
oth((cid:12)m0B)(cid:0) (cid:17) m0L((cid:12)m0B); (2)
Z (cid:12) B (cid:12)m0B
where we have de(cid:12)ned the Langevin fun
tion as L(x) =
oth(x)(cid:0)1=x. Usually we
are interested in the linear response, i.e as B ! 0 we should have M = (cid:26)m~ = (cid:31)H
(where H = B=(cid:22)0 and (cid:26) is the density of the magneti
moments). Expanding the
Langevin fun
tion we (cid:12)nd limx!0L(x) = x=3 so that we have,
(cid:22)0 2 1
(cid:31) = (cid:26)m0(cid:12) / ; (3)
3 T
(Note, a positive sus
eptibility is de(cid:12)ned as paramagneti
).
An inverse temperature dependen
e of sus
eptibility was systemati
ally measured
by P. Curie (1895) for a range of paramagneti
substan
es (oxygen, palladium, and
1
various salts su
h as FeCl2). The above derivation, whi
h ignores the intera
tions
between a set of (cid:12)xed magneti
moments, is due to Langevin (1905). Of
ourse, we
have also ignored the quantization of spin and angular momentum. This is important
in restri
ting the allowed values of mz of the individual moments. However, the
1 2
general \Curie law" of (cid:31) / still holds.
T
3
MM
1
0 1
0
-1 HH
0.0 0.5TT 1.0 -1
Figure 1: The self-
onsistent mean-(cid:12)eld solution for the magnetization of a ferromag-
net as a fun
tion of temperature and external (cid:12)eld.
1.1.2 Weiss mean-(cid:12)eld des
ription of ferromagnets
Typi
al materials
ontain many su
h magneti
moments (eg. atomi
spins), whi
h
have some intera
tion. A general form whi
h
an often be justi(cid:12)ed (eg.
lassi
al
dipole intera
tion, Heisenberg ex
hange) is the intera
tion energy,
X
Eint = (cid:0) V(Ri (cid:0)Rj)mi (cid:1)mj: (4)
i<j
The thermodynami
s of this problem is now non-trivial. However, a reasonable ap-
proximation is to look at the energeti
s of a single moment intera
ting with the
average moment on the other spins. This is equivalent to the i-th moment seeing an
e(cid:11)e
tive (cid:12)eld,
X
B = (cid:22)0H+ V(Ri (cid:0)Rj)hmji: (5)
i=6 j
This \mean-(cid:12)eld theory" was (cid:12)rst used by Weiss (1907) to explain the behaviour of
ferromagnets observed by Curie. In a ferromagnet the intera
tions tend to align the
P
(cid:0)1
spins, so that (cid:26) i6=jV(Ri (cid:0)Rj) (cid:17) (cid:2) > 0, and the magnetization M = (cid:26)hmji lies
in the same dire
tion as the external (cid:12)eld, B = (cid:22)0H +(cid:2)M. The e(cid:11)e
tive (cid:12)eld now
enters into the Langevin model for the i-th moment,
M = m0(cid:26)L[(cid:12)m0((cid:22)0H +(cid:2)M)℄: (6)
This is a relation between H and M, known as the equation of state. From this, all
thermodynami
properties
an be obtained! The self-
onsistent solution of M(T;H)
for this equation is shown in Fig. 1. At low temperatures there is a spontaneous
magnetization at zero (cid:12)eld. The size of M(T;H ! 0) falls
ontinuously to zero at a
3

riti
altemperature T
. We
an see this by expanding the inverse Langevin fun
tion,
1
(cid:0)1
(cid:22)0H = (cid:0)(cid:2)M + L [M=m0(cid:26)℄ (7)
(cid:12)m0
1
E.C. Stoner, Magnetism and Matter, (Methuen, London, 1934).
2
In this
ourse on phase transitions, we will not be too interested in the e(cid:11)e
ts of quantum
me
hani
s. The reason for this (
oarse-graining)will be justi(cid:12)ed later.
3 (cid:0)1 9 3
we use the expansion L (x)=3x+ x
5
4
Figure 2: Experimental values of sus
eptibility of an Iron alloy, from J.E. Noakes et
al., J. Appl. Phys. 37, 1265 (1966). This log-log plot shows a linear dependen
e over
several orders of magnitude. However, the slope is not the
lassi
al value of one, but
(cid:13) = 1:33(cid:6)0:015.
3 9
3
= ((cid:0)(cid:2)+ )M + M ; (8)
2 4 3
(cid:12)(cid:26)m0 5(cid:12)m0(cid:26)
and we have a sus
eptibility near the
riti
al point (or Curie point) of,
a
(cid:31) = ; (9)
T (cid:0)T
1 2
with T
= 3(cid:2)(cid:26)m0. This is the famous Curie-Weiss law (observed by Curie and
explained by Weiss).
Thisverysimpletheoryalreadyhasari
hnessofbehaviour. Thezero(cid:12)eldsolution
below T
displays symmetry breaking: There are two equivalent solutions, ea
h of
whi
h has a lower symmetry than the intera
tions in the system. As one passes from
positive to negative (cid:12)elds, there is a jump in the magnetization, i.e. the H = 0 line
below T
marks a (cid:12)rst-order phase transition (jump in G=H). The
riti
al point
itself is very spe
ial, where the magnetization has a divergent slope. This also leads
to a dis
ontinuity in the spe
i(cid:12)
heat as we in
rease T through T
at zero (cid:12)eld.
Finally we note that Fig. 1 does not show all of the solutions to (6). Considering
by hand a graphi
al solution of the self-
onsistent equation shows two other solu-
tion bran
hes, one of whi
h is unstable, the other whi
h is metastable (a metastable
solution has higher free energy than the true solution, and so is probabilisti
ally
unfavourable in thermodynami
equilibrium).
1.1.3 An experimental dis
repan
y
The above Curie-Weiss law has proved a very useful way for experimentalists to
har-
a
terize the high-temperature behaviour of magneti
materials, and it is an experi-
mental fa
t that in ferromagnets the sus
eptibility diverges at the Curie temperature.
However, when one looks very
losely at this divergen
e one (cid:12)nds that the divergen
e
5
Figure 3: The phase diagram of Argon in the pressure-temperature plane.
has a di(cid:11)erent form from Curie-Weiss. In Fig. 2, data is shown that obeys a form,
(cid:18) (cid:19)(cid:0)(cid:13)
T (cid:0)T
(cid:31) = a ; (10)
T
with (cid:13) (cid:25) 1:33, rather than the mean-(cid:12)eld predi
tion of (cid:13)MF = 1. We therefore have
a hint that we may need to go beyond mean-(cid:12)eld theory. In Le
ture 4 we will (cid:12)nd
out why the mean-(cid:12)eld approximation has broken down.
1.2 Van der Waals theory of a liquid-gas transition
Fig. 3 shows the phase diagram of Argon. Of interest to us is the boundary between
liquid and gas phases. What is the di(cid:11)eren
e between a liquid and a gas? In a liq-
uid the density is approximately (cid:12)xed in order to minimize the attra
tive intera
tion
between atoms/mole
ules. In a gas the density is mu
h lower, but the system gains
more entropy. The (cid:12)rst understanding of a phase transition between these two states
is due to Van der Waals (1873). Consider the energy of pair-wise intera
tions be-
P
tween identi
al parti
les, E = i<jVint(Rj (cid:0) Ri), and let the intera
tion potential
be written as a sum of a long-range attra
tion (e.g. from the Van der Waals attra
-
tion of (cid:13)u
tuating dipoles) and a hard-
ore repulsion from the parti
le of radius a,
Vint(R) = Vattr(R)+Vh
(R). For example, an attra
tive intera
tion of range (cid:21) may be
of the form Vattr(R) / (cid:0)exp((cid:0)R=(cid:21)). A typi
al form for Vint(R) is shown in Fig. 4.
The thermodynami
s of this intera
ting system is governed by the partition fun
-
tion,
Z P
Y
Z = Tre(cid:0)(cid:12)E = ddRi0e(cid:0)(cid:12) i<jVint(Rj(cid:0)Ri): (11)
i0
1.2.1 Mean-(cid:12)eld theory
If the range of the intera
tion, (cid:21) is mu
h larger than the typi
al parti
le separation
(cid:0)d
(cid:26) (where (cid:26) = N=V is the density), then to a good approximation an individual
d
parti
le feels the average attra
tive potential from many ((cid:24) (cid:21) (cid:26)) parti
les. Therefore
we
an write,
6
V(R)
a λ
R
Figure 4: Typi
al inter-parti
le intera
tion potential with a hard-
ore repulsion and
a long-range attra
tion.
Z
X 1 X 1 X
d 2
Vatt(Rj (cid:0)Ri) = Vattr(Rj (cid:0)Ri) (cid:25) (cid:26) d rVattr(r) = (cid:0)JN =V; (12)
2 2
i<j i6=j i
R
1 d
where we de(cid:12)ne J = 2 d rVattr(r). Now the partition fun
tion be
omes,
Z P
Z = e(cid:12)JN2=V YddRi0e(cid:0)(cid:12) i<jVh
(Rj(cid:0)Ri) = e(cid:12)JN2=VZh
: (13)
i0
Van der Waals assumed the e(cid:11)e
t of the hard
ore is just to ex
lude some of phase
spa
e, so that,
N
Zh
= (V (cid:0)Nb) (14)
d
(b (cid:25) a is the ex
luded volumeofone parti
le). This resultforZh
isonlyexa
t inone
dimension, but should be a goodapproximationatlow densities inhigher dimensions.
The Van der Waals free energy per parti
le now follows,
T J
f(v;T) = (cid:0) lnZ = (cid:0) (cid:0)T ln(v (cid:0)b); (15)
N v
where v = V=N is the volume per parti
le. Using the formula for pressure, P =
(cid:0) f=vj , we get the Van der Waals equation of state,
T
J T
P(v;T) = (cid:0) + : (16)
2
v (v (cid:0)b)
Thefreeenergyandpressureasfun
tionsofvolumeat(cid:12)xedtemperatureareshown
in Fig. 5. Note the presen
e of a
riti
al temperature, T
= (8=27)J=b below whi
h
2 2
there is always a thermodynami
ally unstable range of volumes (be
ause  f=v =
(cid:0)p=v
annot be negative). In this range the system will need a
oexisten
e of a
high-density liquid and a low-density gas. If we instead
onsider systems of (cid:12)xed
pressure, we need to look at the Gibbs free energy G(P;T) = F[V(P;T);T℄+ PV.
The result at T = 0:75T
is shown in Fig. 6 where we see that as a fun
tion of
7
T>T
c
T>T
c
f(v) P(v)
T=T
c
T=T
c
T<T
c T<T
c
v v
Figure 5: (a) The Van der Waals free energy as a fun
tion of volume per parti
le for
temperatures above, below and at T
. (b) The pressure P = (cid:0)f=v at the same
temperatures. Note the unstable region (dashed line) where P=v is positive.
T=0.75 T
c
g(P)
Liquid
Gas
P
Figure 6: The Gibbs free energy g = f +Pv of the Van der Waals system at T =
0:75T
. The
urve has
usps where P=v = 0. The equilibrium system takes the
solution of lowest g, so there is a (cid:12)rst-order transition where the liquid
urve
rosses
the gas
urve.
temperature the Gibbs free energies of the two stable solutions
ross ea
h other.
The
orre
t thermodynami
phase is the one of lowest G. It is this whi
h leads to
a phase diagram similar to that in Fig. 3 (but without a solid phase). (Note, the

oexisten
e region
an also be determined from the P-v
urve using the Maxwell
equal-area
onstru
tion, or from the f-v
urve, using the
onvexity property of the
free energy.)
Again, this simple theory
ontains a wealth of phenomena: On
rossing the
oex-
isten
e line there is a density jump and a latent heat obeying the Clausius-Clapeyron
relation (cid:1)s = (cid:1)v(dP=dT), whi
h in turn must give a peak in the spe
i(cid:12)
heat.
These (cid:12)rst-order jumps redu
e to zero as the
riti
al end-point of the
oexisten
e line
is approa
hed.
8
Question 1
Find the lo
ation of the
riti
al point P
= P(v
;T
), de(cid:12)ned by setting the
(cid:12)rst and se
ond derivatives of (16) to zero. Show that near the
riti
al
point,
" #1=3
v (cid:0)v
2(P (cid:0)P
)
= (cid:0) : (17)
v
3 P
on the
riti
al isotherm T = T
(note, the same exponent of 1=3 is found
for the mean-(cid:12)eld ferromagnet on the
riti
al isotherm.) Show that on the

oexisten
e line,
(cid:18) (cid:19)1=2
T
(cid:0)T
v = v
(cid:6)2v
; (18)
T
with the plus sign for approa
hing from the gas phase and the minus sign

orresponding to the liquid. We see that the density di(cid:11)eren
e (cid:1)v between
gas and liquid falls
ontinuously to zero at the
riti
al point (again, the
same exponent of 1=2 is found for the spontaneous magneti
moment in a
mean-(cid:12)eld ferromagnet). Finally, show that the
ompressibility
(cid:12)
(cid:12)
1 v (cid:12)
(cid:20) = (cid:0) (cid:12) : (19)
v P(cid:12)
T
diverges at the
riti
al point. Therefore we
an expe
t violent density (cid:13)u
-
tuations at
riti
ality (observed in
arbon dioxide as \
riti
al opales
en
e"
by Andrews in 1863).
9
a> 0 h=0 a< 0 h=0
F F
L L
φ φ
a> 0 h>0 a< 0 h>0
F F
L L
φ φ
2 4
Figure 7: The Landau \free energy" fun
tion FL((cid:30)) = a(cid:30) + b(cid:30) (cid:0) h(cid:30) for di(cid:11)erent
values of the parameter a and a (cid:12)eld h. For a > 0 and h = 0 there is only one
minimum at the origin (cid:30) = 0. This
orresponds to the disordered phase. For a < 0
and h = 0 there are two minima at (cid:12)nite (cid:30), and the system orders into one of the
states with a spontaneous breaking of symmetry. With a symmetry breaking (cid:12)eld
h > 0 the a > 0
urve has a minimum at non-zero (cid:30), while for a < 0 the degenera
y
of the minima is lifted.
1.3 Landau theory of
riti
al point
Both examples above share the feature of a
oexisten
e line ending in a
riti
al point.
In ea
h
ase the
oexisting phases have a quantity (i.e. the magnetization or the
density) that is di(cid:11)erent, but where the di(cid:11)eren
e goes to zero at the
riti
al point.
Apart from the qualitative similarities, their are
ertain quantities that are exa
tly
the same in both systems, e.g. the exponent of the vanishing magnetization/density
di(cid:11)eren
e. These features
an be explained with a simple phenomenologi
al model
4
due to Landau (1937).
The
riti
al point is to be regarded as a
ontinuous phase transition (the thermo-
dynami
fun
tions of state su
h as entropy, energy, volume are
ontinuous), whi
h
separates two phases of di(cid:11)erent symmetry. Usually the low-temperature phase has
a lower symmetry than the high-temperature phase. Landau suggested that su
h
transitions are
hara
terized by an order parameter (cid:30) whi
h is nonzero in the less-
symmetri
al phase, and zero in the symmetri
al phase. (For the ferromagnet (cid:30) = M,
a ve
tor, while for the liquid-gas transition (cid:30) = vgas (cid:0) vliq, a s
alar.) The Landau
theory then assumes that the free energy FL of the system is a fun
tion of this pa-
rameter, and that the thermodynami
ally stable state will have the value of (cid:30) that
minimizes this fun
tion, i.e. F = min[FL((cid:30))℄.
As the order parameter goes to zero at the transition T
, we
an expand FL((cid:30))
4
The standard exposition is in L.D. Landau and G.M. Lifshitz, Statisti
al Physi
s, (Pergamon
Press, Oxford 1958).
10