BASICS OF STATISTICAL MECHANICS

Statistical Mechanics is the theory of

the physical behaviour of macroscopic systems

starting from a knowledge of the microscopic

forces between the constituent particles.

A model of a physical system is a car-

icature of the system obtained by extracting

only the essential features of the phenomenon

to be studied so that it becomes manageable

for mathematical investigation.

1

GAS LAW AND LIQUID-GAS TRANSITION

Equation of State (Ideal Gas Law):

p V = nR0 T

R0 = 8.3 J K−1 is the gas constant. T is the

absolute temperature. Absolute zero is at 0 K

= 273 0C.

Energy equation:

U = CV T

CV is called the heat capacity at constant vol-

ume. In general, it is defined by

CV =(

∂U

∂T

)

V

,

2

and can be a function of T and V .

Real gases become liquid as the tempera-

ture is lowered and the p-V diagram is more

complicated.

First Law of Thermodynamics:

∆U = Q + W

For a fluid,

W = −∫

Γ

p dV

where the integral depends on the path in the

p-V diagram.

Second Law of Thermodynamics:

δQ = T dS

3

MAGNETISM

Magnetic Induction: ~B = µ0( ~H+ ~m). Here~H is the external magnetic field and ~m is the

magnetisation.

The susceptibility is defined by

χT =(

∂m

∂H

)

T

.

For a paramagnet, χT > 0,

for a diamagnet, χT < 0.

Paramagnets satisfy Curie’s Law:

χT = C T−1.

4

PHASE DIAGRAMS

Liquid-Gas transition:

5

Experimental vapour-liquid curve

for a variety of substances

Ferromagnetic transition:

T

m 0

Tc

(T)

6

THERMODYNAMICS

Fundamental equation: s = s(u, v),

where v = V/N and u = U/N .

Properties:

1. s(u, v) is a concave function.

2. s(u, v) is continuously differentiable.

3.∂s

∂u(u, v) > 0 for all v.

Basic formulae:

1T

=(

∂s

∂u

)

v

p

T=

(∂s

∂v

)

u

Example. For the ideal gas,

s(u, v) = cV ln u + kB ln v + s0.

7

FREE ENERGY

The (Helmholtz) free energy is defined by

f(v, T ) = infu

[u− T s(u, v)]

Differentiation gives: df = −p dv − s dT.

Similarly for magnets: df = −mdH − s dT .

Partition Function:

ZN (T ) =∑

s

exp(−EN (s)

kBT

),

where the sum runs over all possible microstatesof the system with corresponding energy EN (s).

Basic identity:

f(v, T ) = −kBT limN→∞

V/N=v

1N

ln ZN (T )

8

Notation: β =1

kBT.

The average of an observable X(s) is

〈X〉 =1

ZN (β)

∑s

X(s)e−βEN (s).

In particular,

U = 〈E〉 = − ∂

∂βln ZN (β),

or, in terms of the free energy,

u = −T 2 ∂

∂T

(f(T )

T

)=

∂

∂β(βf(β)).

9

Example: Independent Spins.

E1(s1, . . . , sN ) = −HN∑

i=1

si.

With si = ±1 we get

ZN (β) = (2 coshβH)N

and hence

f(β) = − 1β

ln 2 cosh βH.

Therefore

u(β) = −H tanh βH and m(β) = tanh βH.

Consequently,

χT =∂m

∂H=

β

cosh2 βH∼ β,

for βH ¿ 1, which is Curie’s Law.

10

In general, there are interactions between

the spins and E(s) = E0(s) + E1(s), where we

expect that E0(−s) = E0(s). Then

ZN (β) =∑

s

e−β(E0(s)−H∑N

i=1si).

Hence,

m(H, T ) = − ∂f

∂H=

1N

⟨ N∑

i=1

si

⟩

and

χT =1

NkBT

[〈M2〉 − 〈M〉2] .

11

THE ISING MODEL

A special case is the nearest-neighbour

Ising model given by

E0(s) = −J∑

(i,j)n.n.si sj ,

where the sum runs over pairs nearest neighbour

point of some regular (Bravais) lattice. Exam-

ples are the square lattice and the triangular

and hexagonal lattices in 2 and 3 dimensions.

This model can also be interpreted as a

model of a lattice gas, where molecules are

restricted to hop between sites of a lattice. In-

troducing the interaction potential φi,j between

sites i and j we have

E =∑

(i,j)

φi,jni nj ,

12

where ni denotes the number of molecules at

site i.

For a square well, φi,j = +∞ if i = j, −ε

if i and j are n.n. and 0 otherwise. This im-

poses the condition ni = 0, 1 and we can change

variables to si = 2ni − 1.

13

For fluids, one introduces the grand

canonical partition function

ZV (β, µ) =∑

{ni}exp [β(µn− E)],

and the pressure is given by

p(β, µ) = limV→∞

1βV

ln ZV (β, µ).

N.B. To get the equation of state one has

to solve for µ in the equation

ρ =1

βV

∂

∂µln ZV (β, µ) =

∂p(β, µ)∂µ

.

A comparison yields: V = N , ε = 4J ,

µ = 2H − 2zJ ,

p = −fI + H − 12zJ ,

ρ = 12 (1 + m) and

ρ2κT = 14χT .

14

HIGH-TEMPERATURE EXPANSION

Write t = tanh βH and expand in powers

of K = βJ :

ZN (β) = (2 cosh βH)N

{1 +

12qNt2K+

12N

[ (14q2N − q(q − 1

2))

t4

+ q(q − 1)t2 +12q

]K2 +O(K3)

}.

It follows that

m = t{1 + q(1− t2)K

× (1 + (q − 1− (2q − 1)t2)K +O(K2)

) }.

This means that m behaves like tanh βH for

high temperatures and m = 0 at H = 0.

15

LOW-TEMPERATURE EXPANSION

Write u = e−4βJ . Then

ZN (β) = eNψ(β,J,H) + eNψ(β,J,−H),

where

ψ(β, J,H) = 2βJ + βH + ude−2βH

+ du2d−1e−4βH − 12(2d + 1)u2de−4βH

+ d(2d− 1)u3d−2e−6βH

+12d(d− 1)u4d−4e−8βH +O(u5).

It follows that for H > 0,

m ∼ 1− 2ude−2βH + . . . .

16

THE 1-DIM. ISING MODEL

For periodic boundary conditions:

ZN (β) =∑

s

exp

{βJ

N∑

i=1

sisi+1 + βHN∑

i=1

si

}.

Transfer matrix:

V =(

eβ(J+H) e−βJ

e−βJ eβ(J−H)

).

ZN (β) = Trace V N = λN+ + λN

− .

Here

λ± = eβJ cosh βH ±√

e2βJ sinh2 βH + e−2βJ .

In particular,

m(β, J,H) =sinhβH√

sinh2 βH + e−4βJ.

17

THE MEAN FIELD MODEL

For this model the energy is given by

E(s) = − qJ

N − 1

∑

(i,j)

sisj −HN∑

i=1

si

= − qJ

2(N − 1)(M2 −N)−HM,

where M =∑N

i=1 si.

Result:

f(β) = − supm∈[−1,1]

{12qJm2 + Hm +

1β

s(m)}

,

where

s(m) = −1 + m

2ln

1 + m

2− 1−m

2ln

1−m

2.

18

u(β) = −12qJm(β)2 −Hm(β)

where m(β) satisfies:

m(β) = tanh [β(qJm(β) + H)]

The solution depends on whether β > βc or

β < βc, where

βc =1qJ

tanh(2βJs)

-2 -1 0 1 2

0

-1

1

s

m (β)0

19

QUANTUM MECHANICS OF

FINITE SYSTEMS

The state of a quantum system is given by

a vector in a Hilbert space H. For finite spin

systems this Hilbert space is finite-dimensional.

Observables of the system are self-adjoint oper-

ators on H. Bounded observables are elements

of the algebra B(H) of bounded linear operators

on H. In particular, the operator corresponding

to the total energy of the system is called the

Hamiltonian. It plays a special role in that

it determines the time-evolution of the system.

In the Heisenberg representation the oper-

ators evolve in time according to

αt(A) = eiHt Ae−iHt

20

MIXED STATES AND ENTROPY

Given a vector state Φ ∈ H with ||Φ|| =

1, the expectation value of an operator A ∈B(H) is given by

ρΦ(A) = 〈Φ |AΦ〉.

More general expectation values are given by a

map ρ : B(H) → C such that

1. ρ(λ1A1 + λ2A2) = λ1ρ(A1) + λ2ρ(A2)

2. ρ(A) ≥ 0 if A ≥ 0

3. ρ(1) = 1. If H is finite-dimensional such

mixed states are given by a density matrix

ρ such that

ρ(A) = Trace(ρA)

Density matrices satisfy

ρ ≥ 0 and Trace(ρ) = 1.

21

Given a mixed state ρ with density matrix

ρ the (von Neumann) entropy of ρ is defined

by

S(ρ) = −kB Trace(ρ ln ρ).

Lemma 1 If A and B are positive definite

matrices,

Trace(A ln A)− Trace(A ln B) ≥ Trace(A−B).

It follows that

0 ≤ S(ρ) ≤ kB ln n

if n = dim (H).

22

If n = dim(H) < ∞, let H be a Hamilto-

nian. Then we define the free energy F (β, H)

by

F (β,H) = − 1β

lnTrace e−βH .

We also define the Gibbs state at temperature

T = 1/kBβ by

ρ =1

Z(β, H)exp [−βH].

Theorem 1 (Variational Principle).

For any mixed state ρ,

F (β, H) ≤ ρ(H)− T S(ρ).

Moreover, for a given H and β > 0, the Gibbs

state is the unique state for which the equality

holds.

23

QUANTUM LATTICE SYSTEMS

Let Zd be a d-dimensional square lattice.

At each x ∈ Zd we assume given an n-dimen-

sional Hilbert space Hx. For any finite subset

Λ ⊂ Zd we define

H(Λ) =⊗

x∈Λ

Hx,

and we write A(Λ) = B(H(Λ)). If Λ1 ⊂ Λ2 then

H(Λ2) = H(Λ1) ⊗ H(Λ2 \ Λ1) and A(Λ1) ⊂A(Λ2) if we identifyA(Λ1) withA(Λ1)⊗1Λ2\Λ1 .

Thus, we also have that if Λ1 ∩ Λ2 = ∅ then

A(Λ1) and A(Λ2) commute inside A(Λ) if Λ ⊃Λ1 ∪ Λ2. The union

AL =⋃

Λ finite

A(Λ)

is the algebra of local observables. The

norm completion A = AL is a C∗-algebra.

24

For the infinite lattice there is no Hamilto-

nian. However, we can define a potential Φ as

a map X 7→ Φ(X) from the finite subsets of Zd

to the self-adjoint elements of A such that

HΛ(Φ) =∑

X⊂Λ

Φ(X).

In order that this decomposition is unique, we

require that

TraceY (Φ(X)) = 0 if Y ⊂ X.

In the following we consider in particular trans-

lation invariant potentials, such that

τx(Φ(X)) = Φ(X + x).

25

A potential is said to have finite range if

RΦ = sup{diam(X) : Φ(X) 6= 0} < +∞.

The linear space of potentials with finite range

will be denoted B0. More generally, we consider

potentials which have infinite range but decay

as X increases. We define Banach spaces B1

and Bf of translation-invariant potentials with

norms given by

||Φ||1 =∑

0∈X

||Φ(X)|||X|

and

||Φ||f =∑

0∈X

||Φ(X)|| f(X),

where f is a positive function, increasing in |X|.

26

If the limit

limΛ→Zd

eiHΛ(Φ)tAe−iHΛ(Φ)t =: αΦt (A)

exists for all A ∈ A and gives rise to a strongly

continuous 1-parameter automorphism group of

A, then Φ is called a dynamical potential

and αΦt is called the corresponding dynamical

automorphism group.

Theorem 2 If f(X) = e|X| then the po-

tentials in Bf are dynamical.

This is proved using the fact that B0 is

dense in Bf and the following lemma:

27

Lemma 2 If Φ ∈ Bf and A ∈ A(Λ0) with

Λ0 ⊂ Λ then

||[HΛ(Φ), A](n)|| ≤ ||A|| e|Λ0|n! (2||Φ||f )n,

where [B, A](n) stands for the repeated commu-

tator

[B,A](n) = [B, [B, . . . , [B, A] . . .]].

28

THE MEAN FREE ENERGY

We prove the existence of the thermody-namic limit of the free energy density

fΛ(Φ) = − 1β|Λ| lnTrace e−βHΛ(Φ).

First note that ||HΛ(Φ)|| ≤ |Λ| ||Φ||1. Nextwe use the following lemmas:

Lemma 3. (Peierls’ inequality)

Let A be a hermitian n×n matrix and {ψn}ni=1

an arbitrary orthonormal basis. Then

n∑

i=1

e〈ψi |Aψi〉 ≤ Trace eA.

Lemma 4.

For hermitian matrices A and B,

| lnTrace eA − lnTrace eB | ≤ ||A−B||.

29

We conclude that

|FΛ(Φ)− FΛ(Ψ)| ≤ |Λ| ||Φ−Ψ||1. (∗)

Given finite Λ1 and Λ2, and Φ ∈ B0, we

denote

N(Λ1, Λ2, Φ)

=∣∣∣∪{X ⊂ Zd : X ∩ Λi 6= ∅, Φ(X) 6= 0}

∣∣∣ .

Then

Lemma 5. If Λ1 ∩ Λ2 = ∅ then

|FΛ1∪Λ2(Φ)− FΛ1(Φ)− FΛ2(Φ)|≤ N(Λ1, Λ2,Φ) ||Φ||1.

We now first consider (hyper-)cubes K(a)

of side a.

30

Corollary. If Φ ∈ B0, a ∈ N and Ki(a)

(i = 1, . . . , n) are n non-overlapping cubes, then

lima→∞

1ad

[1n

F∪Ki(a)(Φ)− FK(a)(Φ)]

= 0

uniformly in n and the position of the cubes.

It easily follows that

Theorem 2. For Φ ∈ B0, the limit

lima→∞

1ad

FK(a)(Φ) = f(Φ)

exists.

This result can be generalised to more gen-

eral shapes of domains Λ. The most general se-

quence of domains Λ(n) (n = 1, 2, . . .) for which

this holds was proposed by Van Hove:

31

Divide the lattice Zd into cubes of a fixed

length a ∈ N. For any finite Λ ⊂ Zd, denote

Λ−a the union of all such cubes contained in Λ,

and Λ+a the union of all such cubes with non-

empty intersection with Λ. Then a sequence

{Λ(n)}n ∈ N is said to tend to Zd in the

sense of Van Hove if |Λ(n)| → ∞, and for all

a ∈ N, |Λ(n)+a |/|Λ(n)−

a | → 1.

We can generalise the above theorem to

Theorem 3. For Φ ∈ B1 the thermody-

namic limit

limn→∞

1|Λ(n)|FΛ(n)(Φ) = f(Φ)

exists if Λ(n) → Zd in the sense of Van Hove.

The limit f(Φ) is a convex function with the

property

|f(Φ)− f(Ψ)| ≤ ||Φ−Ψ||1.

32

The continuity of f(Φ) follows from

inequality (*) on page 30.

The convexity of f(Φ) follows from Peierls’ in-

equality together with Holder’s inequality. In-

deed, these imply that

Lemma 5. The function a 7→ lnTrace eA

from the hermitian matrices to R is an increas-

ing convex function.

33

THE MEAN ENTROPY

Given a translation-invariant state ρ on A,

its restriction to A(Λ) defines a density matrix

ρΛ. Its entropy is the local entropy

SΛ(ρ) = −Trace(ρΛ ln ρΛ).

It has the following properties:

Proposition 1.

1. 0 ≤ SΛ(ρ) ≤ |Λ| ln m, where m is the dim-

nension of the single-site Hilbert space;

2. (Subadditivity) If Λ1 ∩ Λ2 = ∅ then

SΛ1∪Λ2(ρ) ≤ SΛ1(ρ) + SΛ2(ρ).

3. If Λ ⊂ Λ′ then SΛ′(ρ)− SΛ(ρ) ≤|Λ′ \ Λ| ln m.

34

The existence of the thermodynamic limit

follows from

Lemma 6. If Λ 7→ G(Λ) is a positive func-

tion on the finite subsets of Zd such that

a. There exists a constant c > 0 such that 0 ≤G(Λ) ≤ c|Λ|,

b. If Λ1 ∩Λ2 = ∅ then G(Λ1 ∪Λ2) ≤ G(Λ1) +

G(Λ2),

c. G(Λ + x) = G(Λ) for all x ∈ Zd,

then

limΛ→Zd

G(Λ)|Λ| = g

exists for any Van Hove sequence with the addi-

tional property that |Λ|/|K(Λ)| ≥ δ > 0, where

K(Λ) is the smallest cube containing Λ.

35

Given this lemma, it is easy to prove that

the thermodynamic limit of the entropy den-

sity exists for Van Hove sequences with the ad-

ditional property mentioned. This property is

in fact not necessary, but that requires a much

more involved proof.

Theorem 4. For any translation-

invariant state ρ on A, the mean entropy per

lattice site

s(ρ) = limΛ→Zd

1|Λ|SΛ(ρ)

exists if Λ tends to Zd in the sense of Van Hove.

Moreover, it has the following properties:

i. 0 ≤ s(ρ) ≤ ln m,

ii. s(ρ) is an affine, upper semicontinuous

function of the state ρ.

36

THE VARIATIONAL PRINCIPLE

We denote the set of translation-invariant

states on A by Sτ (A) (omitting the A in most

instances).

Proposition 2.

For any translation-invariant state ρ ∈ Sτ (A)

and Φ ∈ B1, the limit

limΛ→Zd

1|Λ|ρ(HΛ(Φ)) = ρ(eΦ),

where

eΦ =∑

X: 0∈X

Φ(X)|X| .

The main theorem of the thermodynamics

of quantum spin systems is now:

37

Theorem 5. For any potential Φ ∈ B1

and β > 0,

f(Φ, β) = infρ∈Sτ

[ρ(eΦ)− β−1s(ρ)].

Moreover, the infimum is attained at a non-

empty set of points SΦτ (β), which is a closed and

convex subset of Sτ .

If SΦτ (β) contains more than one point for

a given potential Φ and inverse temperature β,

then Φ has a first-order phase transition at

β. The extremal points of SΦτ (β) represent the

pure phases. These are in fact also extremal

translation invariant states.

38

MEAN-FIELD THEORY

Let M be the algebra of all m×m complex

matrices,

AN =N⊗

n=1

M

and

A = ∪∞N=1AN .

Let ρ be a state on M with density matrix

ρ =e−βh

Trace e−βh,

where h ∈ M is hermitian. Define the product

state ωρ = ⊗∞n=1ρ on A. We define the relative

entropy of a state φ on AN by

S(φ, (ωρ)N ) = Trace(φ ln ρ⊗N )− Trace(φ ln φ).

39

As above, the mean relative entropy then exists

for permutation-invariant states φ on A:

Theorem 6. For any permutation-

invariant state φ on A, the mean relative en-

tropy

s(φ, ωρ) = limN→∞

1N

S(φN , (ωρ)N )

exists. Moreover, it has the following proper-

ties:

i. ln λmin ≤ s(φ, ωρ) ≤ 0, where l; ambdamin

is the smallest eigenvalue of ρ;

ii. s(ρ) is an affine upper semicontinuous

function of the state ρ.

40

Given a hermitian x ∈ M, we denote xn a

copy of x in the n-th factor of A. We denote

also

xN =1N

(x1 + . . . + xN ).

Lemma 7. Suppose that φ is a

permutation-invariant state on A and g is a

real-valued polynomial function on R. Then the

limit

eg(φ) = limN→∞

φ(g(xN ))

exists.

This leads to an analogue of the variational

principle for permutation-invariant states:

41

Theorem 7. Suppose that ρ is a state on

M with density matrix

ρ =e−βh

Trace e−βh.

Put HN = h1 + . . . + hN . Let g be a real-valued

polynomial and x ∈ M a fixed hermitian ma-

trix. Then

limN→∞

− 1βN

lnTrace e−β[HN+Ng(xN )]

= infφ∈Sπ(A)

[eg(φ)− β−1s(φ, ωρ)

]+ f0(β),

where

f0(β) = − 1β

ln Trace e−βh.

This theorem can be further refined to the

following:

42

Theorem 8. Suppose that ρ is a state on

M with density matrix

ρ =e−βh

Trace e−βh.

Put HN = h1 + . . . + hN . Let g be a real-valued

polynomial and x ∈ M a fixed hermitian ma-

trix. Then

limN→∞

− 1βN

lnTrace e−β[HN+Ng(xN )]

= infφ∈S(M)

[g(φ(x))− β−1S(φ, ρ)] + f0(β)

= infu∈[−||x||,||x||]

[g(u) + β−1I(u)],

where

I(u) = supt∈R

[tu−G(t)]

43

and

G(t) = ln Trace e−βh+tx.

This theorem is proved using the following

two results:

Theorem 9. (Størmer) The extremal

permutation-invariant states of A are the prod-

uct states. Hence every permutation-invariant

state is a convex combination of product states.

This reduces the expression for the free en-

ergy density to the first expression in Theorem

8. To obtain the second expression, we use the

following lemma:

44

Lemma 8. For any state φ ∈ S(M),

S(φ, ρ) ≤ −I(φ(x)) + βf0(β).

Moreover, if S(φ, ρ) = −I(φ(x)) + βf0(β)

> −∞ and I(z±) = +∞, where

z± = limt→±∞G′(t), then there exists t0 ∈ Rsuch that

φ(A) =Trace Ae−βh+t0x

Trace e−βh+t0x

for all A ∈M. It is given by G′(t0) = φ(x).

45