Advanced Statistical Mechanics Phys. 705 Take-Home Final Exam

PH 705 Final Exam
Winter 2005
April 5, 2007
Directions
Please make sure you put your name on every page that you hand in, and make sure that you
show all of your work in order to receive full credit. Remember also that all questions are not
worth the same. Your solutions are due in my office (or my email box) by midnight Thursday
April 26th.
1 Ferroelectricity. [10 points]
Consider a dielectric with N atoms in a volume V . An external electric field E leads to an induced
dipole moment of the i th atom: p i = αE. Now the polarization of all other atoms enhances the
effect of the external field: p i = αE + ∑ i≠j λ i,j p j . Use a mean field approximation to calculate
the dielectric constant ǫ = 1 + 4πP/E, where P =| P |= 1 V | ∑ i p i | is the polarization, and
E =| E |. Give the condition for the occurrence of a spontaneous polarization in terms of the
λ i,j .
2 A model of a 1st order phase transition. [10 points]
Consider a mean-field theory with a cubic term in the Landau free energy:
F ∼ r 0
2 m2 + s 0
3 m3 + 1 4 m4 .
Sketch the shape of F for various values of r 0 to show how the lowest minimum of F depends
on r 0 . Show that there is a first order phase transition at a certain value of r 0 . Determine this
value and determine the jump of the order parameter at the transition.
3 Functional Integrals [10 points]
Calculate I 1 /I 0 for the following examples:
3.1
I 1 =
∫ ∞
−∞
dφ 1
∫ ∞
−∞
dφ 2
∫ ∞
−∞
dφ 3 φ 2 1 φ 2φ 3 e −(φ 1φ 2 +φ 2 φ 3 +φ 1 φ 3 +φ 2 1 +φ2 2 +φ2 3 ) ,
1

PH 705 Final Exam, Winter 2005 2
I 0 =
∫ ∞
−∞
dφ 1
∫ ∞
−∞
dφ 2
∫ ∞
−∞
dφ 3 e −(φ 1φ 2 +φ 2 φ 3 +φ 1 φ 3 +φ 2 1 +φ2 2 +φ2 3 ) .
3.2
I 1 =
∫ ∞
I 0 =
−∞
∫ ∞
dφ 1
∫ ∞
−∞
−∞
dφ 1
∫ ∞
dφ 2
∫ ∞
−∞
−∞
dφ 2
∫ ∞
dφ 3
∫ ∞
−∞
−∞
dφ 3
∫ ∞
dφ 4 φ 1 φ 4 e −[φ 1φ 2 +φ 2 φ 3 +φ 3 φ 4 +φ 1 φ 4 + 3 2 (φ2 1 +φ2 2 +φ2 3 +φ2 4 )] ,
−∞
dφ 4 e −[φ 1φ 2 +φ 2 φ 3 +φ 3 φ 4 +φ 1 φ 4 + 3 2 (φ2 1 +φ2 2 +φ2 3 +φ2 4 )] .
4 Gaussian approximation with an external field. [15
points]
Calculate the longitudinal and transverse correlation functions for T

PH 705 Final Exam, Winter 2005 3
7 One dimensional Ising model [15 points]
Consider a 1-D Ising model with N particles in an external field h:
N∑
N∑
H(σ 1 , ..., σ n ) = −TK σ i σ i+1 − TH σ i ,
i=1
i=1
with σ i = ±1, and H = h/T, and cyclic boundary conditions: σ N+1 = σ 1 .
7.1
Show that the partition function Z N can be written as
Z N = ∑ σ 1
... ∑ σ N
V σ1 ,σ 2
V σ2 ,σ 3
...V σN−1 ,σ N
V σN ,σ 1
where
V σi ,σ j
= V σj ,σ i
= e Kσ iσ j +(H/2)[σ i +σ j ] .
7.2
Show that Z N = λ N 1 + λ N 2 , where λ 1,2 are the eigenvalues of the 2 × 2 matrix whose matrix
elements are the V σi ,σ j
.
7.3
Show that for N → ∞, lim N→∞ (1/N) lnZ N = lnλ 1 , where λ 1 is the larger of the two eigenvalues.
7.4
Calculate the magnetization M = lim N→∞
1
N ∂ ln Z N/∂H and the susceptibility χ. Is there a
phase transition?
8 Beta function. [10 points]
Consider a β function of the form
β ∼ g2 − bg 4 + g 6
1 + g 8
for some field theory. Discuss the physical scaling behavior of the system in detail for various
ranges of the parameter b. Make sure you identify all the important information you can from
the β function, and include sketches in your discussion.