Download Problem 1. Domain walls of ϕ theory. [10 pts]

Homework # 4, Quantum Field Theory I: 7640
Due Monday, November 17
Problem 1. Domain walls of ϕ4 theory. [10 pts]
Spontaneous breaking of discrete symmetry, such as in a ϕ4 theory with real-valued field
ϕ, does not lead to gapless modes below Tc . In this case states with different expectation
values hϕi = ±ϕ1 are separated by sharp domain walls.
R
To describe this physics, we start with SLG = dxdy
ρ
(∂x ϕ)2 + ρ2 (∂y ϕ)2 + 2t ϕ2 + u4 ϕ4
2
. The
field ϕ(x, y) is subject to boundary condition ϕ(x = −∞, y) = −ϕ1 and ϕ(x = +∞, y) = ϕ1 ,
where ϕ1 =
q
−t/u is the equilibrium expectation value for t < 0.
The boundary condition is chosen so as to force the domain wall solution ϕ(x) interpolating between ±ϕ1 values at the boundaries. From now on we assume that everything is
uniform along y-axis and consider the Lagrangian corresponding to SLG as y-independent.
2
ϕ(x)
(a) Show that ϕ(x) obeys equation ρ d dx
= −|t|ϕ(x) + uϕ(x)3 . Use trial function
2
ϕ(x) = ϕ1 tanh[(x − x0 )/w] and determine the width w in terms of the parameters of the
problem.
(b) Estimate the free energy cost of creating the domain wall by calculating the difference
∆S between the system with one domain wall and the one with a uniform magnetization ϕ1 ,
∆S = SLG (ϕ(x)) − SLG (ϕ1 ). Express ∆S in terms of t, ϕ1 and w, as well as cross-sectional
area A.
Problem 2. XY model; spin waves. [20 pts]
In the XY-model of planar magnetism, a unit planar vector s(i) = (sx (i), sy (i)) =
(cos θi , sin θi ) is placed on each site i of a d-dimensional lattice.
−βH = K
P
hi,ji
The Hamiltonian is
s(i)·s(j), and the notation hi, ji is conventionally used to indicate summing
over all nearest-neighbour pairs (i, j).
R
(a) Rewrite the partition function Z = Πi ds(i) exp(−βH) as an integral over the set of
angles {θi } of the spins {s(i)}.
(b) At low temperature (K 1), the angles {θi } vary slowly from site to site. In this
case expand −βH to get a quadratic form in {θi }.
(c) For d = 1, consider L sites with periodic boundary conditions (i.e. forming a closed
chain). Find the normal modes θq that diagonalize the quadratic form, and the corresponding
eigenvalues K(q). Pay careful attention to whether the modes are real or complex, and to
the allowed values of q.
(d) Generalize the results from the previous part to a d-dimensional simple cubic lattice
with periodic boundary conditions.
(e) Calculate the contribution of these modes to the free energy and heat capacity. (Evaluate the classical partition function, i.e. do not quantize the modes.)
(f) Find an expression for hs(0) · s(x)i = Rehexp[iθx − iθ0 ]i by adding contributions from
different Fourier modes. Convince yourself that for |x| → ∞, only q → 0 modes contribute
appreciably to this expression, and hence calculate the asymptotic limit.