Computational Physics

410 Quantum Monte Carlo methods

It will be clear that for a full boson simulation, moving particles is not sufficient:
we must also include permutation moves, in which we swap two springs between
particles at subsequent beads, for example. However, the configurations are usually
equilibrated for a particular permutation, and changing this permutation can be so
drastic a move that permutations are never accepted. In that case it is possible to
combine a permutation with particle displacements which adjust the positions to
the new permutation [4].

12.5 Quantum Monte Carlo on a lattice

There are several interesting quantum systems which are or can be formulated on
a lattice. First of all, we may consider quantum spin systems as generalisations of
the classical spin systems mentioned in Chapter 7. An example is the Heisenberg
model, with Hamiltonian

HHeisenberg=−J

∑

〈ij〉

si·sj (12.89)

where the sum is over nearest neighbour sites〈ij〉of a lattice (in any dimensions),
and the spins satisfy the standard angular momentum commutation relations on the
same site (
≡1):

[sx,sy]=

isz

2

. (12.90)

Another example is the second quantised form of the Schrödinger equation. This
uses the ‘occupation number representation’ in which we have creation and anni-
hilation operators for particles in a particular state. If the Schrödinger equation is
discretised on a grid, the basis states are identified with grid points, and the cre-
ation and annihilation operators create and annihilate particles on these grid points.
These operators are calledc†iandcirespectively, and they satisfy the commutation
relations
[ci,cj]=[c†i,c†j]=0; [ci,c†j]=δij. (12.91)

In terms of these operators, the Schrödinger equation for a one-dimensional,
noninteracting system reads[26]
∑

i

−t(ci†ci+ 1 +c†i+ 1 ci)+

∑

i

Vini (12.92)

whereniis the number operatorc†ici, and where appropriate boundary conditions
are to be chosen.
A major advantage of this formulation over the original version of the Schrödinger
equation is that the boson character is automatically taken into account: there is no
need to permute particles in the Monte Carlo algorithm. A disadvantage is that the
lattice will introduce discretisation errors.