Homework 4: Effective Action Formulation, due on april 30

Note: Additional information on Quantum Montecarlo and optional excercises can be found at http://www.physics.rutgers.edu/ udo/qmc.html

In this homework we will apply the effective action reasoning, that leads to the density functional theory, to a simpler model, namely the classical Ising system.

We want to study

Z[J] = ∑_{{S_{i}}} e^{−(∑_{ij} J_{ij}S_{i}S_{j}+∑_{i} J_{i}S_{i})} = e^{−W [J]}.  (1)

Note that average magnetization is defined as

\frac{δW [J]}{δJ_{i}} = ⟨ S_{i} ⟩ = m_{i},  (2)

and the Legendre transform

Γ[m] = W [J] − ∑_{i} J_{i}m_{i}.  (3)

In the last equation J_{i} is understood as a function of m_{i}, i.e, J = J(m) via equation (2).

Our goal is to provide simple approximations for Γ[m], following the ideas discussed in class (inversion method) :

1. Multiply J_{ij} by α, where α is a parameter which will be set equal to one, like the charge of the electron in DFT. Evaluate Γ_{0}[m], using equation (3), when α = 0.

2. Now expand Γ[m] in a series of α,

Γ[m] = Γ_{0}[m] + αΓ_{1}[m] + \frac{α^{2}}{2} Γ_{2}[m] + ⋯

Evaluate exactly the first two non trivial orders and discuss the physical meaning.