Symmetry and ergodicity breaking, mean-field study of the
Ising model
Rafael Mottl, ETH Zurich Advisor: Dr. Ingo Kirsch
Topics
Introduction Formalism The model system: Ising model
Solutions in one and more dimensions Symmetry and ergodicity breaking Conclusion
Introduction
- phase A
- phase B
Tc
temperature T phase transition
order parameter
ꢀ Critical exponents
T
Tc
Formalism
Statistical mechanical basics
Ω
sample region
d
i) certain dimension ii) volume
V (Ω) N(Ω)
iii) number of particles/sites iv) boundary conditions
Hamiltonian defined on the sample region
ꢀ
HΩ
−
=
KnΘn
kBT
n
Depending on the degrees of freedom
Coupling constants
Partition function
1
β =
ZΩ[{Kn}] = Tr exp−βH ({K },{Θ })
Ω
- n
- n
kBT
FΩ[{Kn}] = FΩ[K] = −kBTlogZΩ[{Kn}]
Free energy Free energy per site
FΩ[K] fb[K] = lim
N(Ω)→∞ N(Ω)
i) Non-trivial existence of limit
Ω
ii) Independent of iii)
lim
N(Ω)
= const
N(Ω)→∞ V (Ω)
Phases and phase boundaries
fb[K]
- Supp.: -
- exists
{K1, . . . , KD}
- there exist D coulping constants:
- -
- is analytic almost everywhere
fb[K]
f [{Kn}] are points, lines, planes, hyperplanes in the
- non-analyticities of b phase diagram
Ds = 0, 1, 2, . . .
ꢀ Dimension of these singular loci:
Codimension C for each type of singular loci:
C = D − Ds
fb[K]
Phase: region of analyticity of
Phase boundaries: loci of codimension C = 1
Types of phase transitions
fb[K]
is everywhere continuous.
Two types of phase transitions:
∂fb[K]
a) Discontinuity across the phase boundary of
∂Ki
first-order phase transition
b) All derivatives of the free energy per site are continuous across the phase boundaries
Continuous phase transition
Overview
ZΩ[K] = Tr exp−βH (K,{Θ })
Ω
n
FΩ[K] = −kBTlogZΩ[K]
FΩ[K] fb[K] = lim
N(Ω)→∞ N(Ω)
∂
∂fb[K]
ǫin[K] = (βfb[K])
∂β
M[K] = −
∂H
- internal energy
- magnetization
∂M[K]
∂ǫin[K]
χT [K] =
C[K] =
∂H
∂T
magnetic susceptibility heat capacity
Critical exponents
How do the measurable quantities change in the neighbourhood of a critical point?
T − TC t =
TC
Critical temperature
TC
C ∼|t|−α
heat capacity
β
for H ≡ 0
M ∼|t|
Order parameter (t < 0) susceptibility
χ ∼|t|−γ M ∼ H1/δ
Equation of state
Correlation length
- Length scale over which the fluctuations of the microscopic degrees of freedom are significantly correlated with each other
(Cardy: Scaling and Renormalization in Statistical Physics)
- Strong dependence on temperature near a phase transition diverging to infinity at the transition itself
Critical exponents for the correlation function
t → 0
Γ(r) −→ r−pe−r/ξ
ξ ∼|t|−ν
Correlation length
p = d − 2 + η
Power-law decay (t=0)
The model system: Ising model
- attempt to simulate a domain in a ferromagnetic material - Spontaneous magnetization in absence of an external magnetic field - Critical temperature: Curie temperature
Importance of Ising model
- Equivalent models (lattice gas, binary alloy) - The only non-trivial example of a phase transition that can be worked out be mathematical rigor in statistical mechanics
- Compare computer simulations of the model with exact solution
Characterization of the Ising model
i) Periodic lattice ii) Lattice contains in d dimensions
Ω
fixed points called lattice sited
N(Ω)
iii) For each site: classical spin variable Si = +/- 1 (i = 1, …, N) in a definite direction (degrees of freedom)
iv) Most general Hamiltonian
- ꢀ
- ꢀ
- ꢀ
−HΩ =
- HiSi +
- JijSiSj +
KijkSiSjSk + . . .
i,j
i∈Ω
i,j,k
2N(Ω)
v) Number of possible configurations:
- ꢀ ꢀ
- ꢀ
- ꢀ
Tr ≡
- · · ·
- ≡
S1=±1 S2=±1
S
N(Ω)=±1
{Si=±1}
Assumptions
Kijk = 0, . . .
i) two-spin coupling only
Hi ≡ H
ii) External magnetic field spatially constant iii) Nearest-neighbour interactions only
- ꢀ
- ꢀ
→
- i,j
- <ij>
iv) Isotropic interactions v) Hypercubic lattice
Jij ≡ J z = 2d J > 0
vi) Ferromagnetic material
- ꢀ
- ꢀ
−HΩ = H Si + J
SiSj
<i,j>
i∈Ω
ꢀ
- ꢀ
- ꢀ
i∈Ω Si+J
<i,j> SiSj)
- ZΩ[H, T, J] =
- expβ(H
{Si=±1}
Arguments for phase transition in d = 1,2 dimensions with H = 0
ꢀ
FΩ[K] = Ein[K] − TSΩ[K] = −J
SiSj − TkBlog(♯(states))
<i,j>
A
B
d = 1
N → ∞
FN,B − FN,A = 2J − kBTlog(N − 1) → −∞
d = 2
A
B
n bonds
FN,B − FN,A = [2J − log(z − 1)kBT]n
2J
TC =
kBlog(z − 1)
Long range order ꢀ short range order
- nearest neighbour interaction: short range interaction - What about:
J
Jij =
σ
|ri − rj|
Answer:
σ < 1
thermodynamic limit does not exist
1 ꢀ σ ꢀ 2
long range order persist for 0 < T < TC short-range interaction: no ferromagnetic state for T > 0
2 ꢀ σ
Solutions in one and more dimensions
- ad hoc methods - recursion method
H = 0 d = 1
- transfer matrix method
H = 0
(Kramers, Wannier 1941)
- low temperature expansion
- H = 0
- d = 2
- Onsager solution (1944) - mean-field method (Weiss)
d = 1, 2, 3
H = 0
- d = 1
- H = 0
Transfer matrix method
Reduce the problem of calculating the partition function to the problem of finding the eigenvalues of a matrix
SN+1 = S1
SN+1 = S1
Assumption:
SN
S2
ZΩ[H, J] = Tr exp−βH (H,J,{S })
S3
Ω
i
- ꢀ
- ꢀ
- HΩ(H, J, {Si}) = −H
- Si − J
SiSj
<i,j>
i∈Ω
- h := βH
- K := βJ
ꢀ ꢀ
- ꢀ
- ꢀ
- i Si+K
- i SiSi+1
ZΩ[h, K] = . . . eh
S1
SN
- ꢁ
- ꢂ
eh+K − λ
e−K
e−h+K − λ det
≡ 0
Calculation:
e−K
ꢃ
- ⇒ λ1,2 = eK(cosh(h) ± sinh(h)2 + e−4K
- )
ꢃ
⇒ fb(h, K, T) = −JkBTlog(cosh(h) + sinh(h)2 + e−4K)
Spatial correlation in one dimension (T > 0)
- H = 0
- d = 1
Definition: two point correlation function
G(i, j) := ꢁ(Si − ꢁSiꢂ)(Sj − ꢁSjꢂ)ꢂ = ꢁSiSjꢂ − ꢁSiꢂꢁSjꢂ
G(i, i + j) = ꢁSiSi+jꢂ = (tanh(K))j
G(i, i + j) = e−jlog(coth(K)) ≡ e−j/ξ
1
J/(kBT)
∼
⇒ ξ =
1/2 e
=
log(coth)K))
- H = 0
- d = 2
Onsager solution
Columns →
3
Notation and boundary conditions:
...
n
2
1
n + 1 ≡ 1
1
2
µα = {s1, s2, . . . , sn}αth row
sn+1 = s1 , µn+1 = µ1
3
N = n2
n
Elements of the Hamiltonian:
n + 1 ≡ 1
n
ꢀ
E(µ, µ′) = −ǫ
sks′k
k=1
- n
- n
- ꢀ
- ꢀ
E(µ) = −ǫ
sksk+1 − H
sk
$$2$$
- k=1
- k=1
Partition function:
ꢀ ꢀ
ꢀ
n
α=1[E(µα,µα+1)+E(µα)]
ZΩ[H, T] = . . . e−β
µ1
µn
−β[E(µ,µ′)+E(µ)]
ꢁµ|P|µ′ꢂ := e
- Properties of the matrix P:
- -
--
P : 2n × 2nmatrix ZΩ[H, T] = TrPn
- 1
- 1
⇒ lim log(ZΩ[H = 0, T]) = lim log(λmax)
- N→∞ N
- n→∞ n
largest eigenvalue of P
- FΩ[0, T]
- −kBTlogZΩ[0, T ]
N(Ω)
1
= −kBT lim log(λmax
n→∞ n
- fb[H = 0, T] = lim
- = lim
- )
N(Ω)→∞ N(Ω)
N(Ω)→∞
ꢄ
π
ꢅ
- 1
- 1
- ⇒ βfb[0, T] = − log (2 cosh 2βǫ) −
- dφ log 1 + 1 − κ2sin2φ
2π
2
0
κ = 2[cosh 2φ coth2φ]−1