Computational Physics

394 Quantum Monte Carlo methods

12.3.4 Problems with fermion calculations

We have described how the simulation of a diffusion process can generate an average
distribution of random walkers which is proportional to the ground state wave
function or (in the case of guide function DMC) to the product of this function and a
trial function. But a distribution of walkers can only represent wave functions which
are positive everywhere. For bosons, this property is satisfied by the ground state,
but the same does not hold in the case of fermions. The difficulties associated with
treating fermions in quantum Monte Carlo are generally denoted as ‘the fermion
problem’. It should be noted that there is no fermion problem in VMC.

The fixed-node method

There are several approaches to the fermion problem. The simplest approximation is
thefixed-nodemethod, in which the diffusion process is simulated as before, except
for steps crossing a node of the trial function being forbidden. The nodes of the
trial function divide the configuration space up into simply connected volumes in
which the trial wave function has a unique sign. These volumes are separated from
each other by nodal surfaces: hypersurfaces on which the wave function vanishes.
To understand why the fixed-node method is useful, suppose that we know the
nodes of the exact ground state wave function. If we could solve the ground state
of the Schrödinger equation in each simply connected region bounded by the nodal
surfaces of the ground state wave function with vanishing boundary conditions
on these surfaces, this solution would be proportional to the exact ground state
of the full Hamiltonian in each region. In the fixed-node solution, we solve the
Schrödinger equation in connected regions bounded by the nodal surfaces of the
trial function instead of the exact function, and therefore the quality of the solution
depends on how close these surfaces are to those of the exact ground state. It can be
shown that the resulting energy is a variational upper bound to the exact ground state
energy[2]. It should be noted that the fixed-node method often gives a substantial
improvement over the variational Monte Carlo method (which does not suffer from
the fermion problem).
An additional problem with the fixed-node method is the fact that moves in which
two (or any even number of) nodal surfaces are crossed are accepted. This introduces
an error as the number of walkers in two regions separated by an even number of
node crossings does not necessarily represent the norm of the wave functions on
those regions. The degree to which we suffer from this increases with the time step,
as a larger time step will result in larger steps to be taken. It introduces an extra
time-step bias error which goes by the namecross–recross error.
Let us study the nodes more carefully. The requirement thatψ(x 1 ,...,xN)= 0
(xidenotes the spin-orbit coordinate of electroni) defines the nodal surfaces. If