396 Quantum Monte Carlo methods
Now perform two independent DMC calculations, one withφ−and the other
withφ+as a starting distribution, whereφis a trial fermion wave function. What
will happen? Applying the (exact) imaginary-time evolution operatorT(X→Y;τ)
toφwe obtain
φ(Y;τ)=∫
dXT(X→Y;τ)φ(X;0)=
∫
dXT(X→Y;τ)φ+(X,0)−∫
dXT(X→Y;τ)φ−(X,0)=φ+(Y,t)−φ−(Y,t). (12.65)This suggests that we can follow the time evolution ofφby subtractingφ+(t)and
φ−(t)as produced in the two simulations. Asφ−( 0 )andφ+( 0 )are both positive,
and as the imaginary time-evolution operator is always positive, the application of
the DMC approach causes no problems. In fact, one could also say that if the initial
wave function is positive everywhere, it contains no fermion character and hence
we have an unambiguous bosonic time evolution for such an initial state. A guide
function approach can be used in the two boson simulations.
As the time-evolution operator contains no fermion-like features (see above),
both simulations will tend to the bosonic ground state solution for long times. The
fermion ground state wave function is an excited state solution of the many-particle
Hamiltonian, so the boson ground state contribution to the solution at imaginary
timeτwill dominate the fermion contribution by a factor exp[τ(EF−EB)], where
EBandEFare the fermion and boson ground state energies respectively. Note that
this factor grows exponentially with time. The fermion ground state wave function
is thedifferencebetween the two distributions resulting fromφ−andφ+, which
because of the foregoing analysis are both essentially boson-like. If we are to find
a fermion wave function as a small difference of two large, essentially boson wave
function distributions we must be prepared for large statistical errors. The analysis
given here is represented pictorially in Figure 12.2.
The analysis so far leads to the conclusion that, at the beginning, the difference
between the distributions is equal to the trial functionφ, and for large times it
converges to the exact fermion wave function, but it will be buried in the noise of
the boson solutions forming the bulk of the two distributions. We might be lucky: if
the trial function relaxes to the exact Fermi wave function quickly enough, before
the latter is buried in the ‘boson noise’, then we have an intermediate (‘transient’)
regime in imaginary time during which we might extract useful data from the
simulation. The trial energy which is adjusted to keep the respective population
sizes stable is no longer a suitable energy estimator as this will converge to the