Computational Physics

534 Computational methods for lattice field theories

(c) To diagonalise the Hamiltonian we introduce the operators

aˆk=

1

√

4 πωk

[ωkφ(ˆk)+iπ(ˆk)];

aˆ†k=

1

√

4 πωk

[ωkφˆ†(k)−iπˆ†(k)].

Show that

[ak,ak′]=[ak,a†−k′]=δ(k−k′).

(d) Show thatHcan be written in the form

H=

1

2

∫ 2 π

0

dkωk(a†kak+aka†k)=

∫ 2 π

0

dkωk

(

a†kak+

1

2

)

.

15.3 Consider the path integral for the harmonic chain of the previous problem. We have
seen that the Lagrangian could be written as ak-integral over uncoupled
harmonic-oscillator Lagrangians:

L=

∫ 2 π

0

dkL(k)=

1

2

∫ 2 π

0

dk

[

|φ( ̇k)|^2 −ω^2 k|φ(k)|^2

]

.

We discretise the time with time step 1 so that

φ( ̇k,t)→φ(k,t+ 1 )−φ(k,t).

(a) Show that the Lagrangian can now be written as a two-dimensional Fourier

integral of the form:

L=−

1

2

∫

d^2 q

( 2 π)^2

ω ̃q^2 |φ(q)|^2

with

ω ̃^2 q=m^2 + 2 ( 1 −cosq 0 )+ 2 ( 1 −cosq 1 );

q 0 corresponds to the time component andq 1 to the space component.

(b) Show that in the continuum limit (smallq), the two-point Green’s function in

q-space reads

〈φqφq′〉=

1

m^2 +q^2

δq,−q′.

15.4 [C] The multigrid Monte Carlo program for theφ^4 field theory can be extended
straightforwardly to theXYmodel. It is necessary to work out the coarsening of the
Hamiltonian. The Hamiltonian of theXYmodel reads
H=−

∑

〈n,n′〉

Jcos(φn−φn′).

In the coarsening procedure, the new coupling constant will vary from bond to bond,

and apart from the cosines, sine interactions will be generated. The general form

which must be considered is therefore

H=−

∑

〈nn′〉

[Jnn′cos(φn−φn′)+Knn′sin(φn−φn′)].