Computational Physics

12.5 Quantum Monte Carlo on a lattice 411
Finally, this model may be formulated for interacting fermions. A famous model
of this type is the so-calledHubbard model, which models the electrons which
are tightly bound to the atoms in a crystalline material. The Coulomb repulsion is
restricted to an on-site effect; electrons on different sites do not feel it. The creation
and annihilation operators are now calledc†i,σ,ci,σ, whereσ=±labels the spin.

They anticommute, except for[ci†,σ,cj,σ′]+ =δijδσσ′. The standard form of the
Hubbard model in one dimension reads

H=

∑

i,σ

−t[c†i,σci+1,σ+c†i+1,σci,σ]+U

∑

i

ni,σni,−σ (12.93)

whereni,σis the number operator which counts the particles with spinσat sitei:
ni=c†i,σci,σ. The first term describes hopping from atom to atom, and the second
one represents the Coulomb interaction between fermions at the same site.
We shall outline the quantum path-integral Monte Carlo analysis for one-
dimensional lattice quantum systems, taking the Heisenberg method as the principal
example. Extensions to other systems will be considered only very briefly. For a
review,seeRef.[5];seealsoRef.[27].
The quantum Heisenberg model is formulated on a chain consisting ofNsites,
which we shall number by the indexi. We have already discussed this model in
Section 11.5. The Hilbert space has basis states|S〉=|s 1 ,s 2 ,...,sN〉, where thesi
assume values±1; they are the eigenstates of thez-component of the spin operator.
The Heisenberg Hamiltonian can be written as the sum of operators containing
interactions betweentwo neighbouringsites. Let us callHithe operator−Jsi·si+ 1 ,
coupling spins at sitesiandi+1. Suppose we haveNsites and thatNis even. We
now partition the Hamiltonian as follows:

H=Hodd+Heven=(H 1 +H 3 +H 5 +···+HN− 1 )

+(H 2 +H 4 +H 6 +···+HN). (12.94)

HiandHi+ 2 commute as theHicouple only nearest neighbour sites. This makes the
two separate HamiltoniansHoddandHeventrivial to deal with in the path integral.
However,HoddandHevendo not commute. It will therefore be necessary to use the
short-time approximation.
The time-evolution operator is split up as follows:

e−τH≈e−

τHodde−

τHevene−

τHodde−

τHeven...e−

τHodde−

τHeven (12.95)

with a total number of 2Mexponents in the right hand side;
τ=τ/M. In calculat-
ing the partition function, we insert a unit operator of the form

∑

S|S〉〈S|between