402 Quantum Monte Carlo methods
for largeM. The error is then given by [11, 12]
α^2
M∑
m>m′|[Hm,Hm′]|e−α∑
m|Hm|, (12.74)where|...|denotes the norm of an operator.
It is very easy to get confused with many physical quantities having different
meaning according to whether we address the time-evolution operator, the quantum
partition function, or the classical partition function. Therefore we summarise the
different interpretations in Table 12.2. The classical time in the last row of Table 12.2
is the time that elapses in the classical system and is analogous to the time in a Monte
Carlo simulation. This quantity has no counterpart in quantum mechanics or in the
statistical partition function.
The quantum partition function is now simulated simply by performing a standard
Monte Carlo simulation on the classical system. The PIMC algorithm is
Put theNMparticles at random positions;
REPEAT
FORm=1TOMDO
Select a time slicem ̃at random;
Select one of theNparticles at time slicem ̃at random;
Generate a random displacement of that particle;
Calculater=exp[−
τ (Hnew−Hold)];
Accept the displacement with probability min(1,r);
END FOR;
UNTIL finished.In this algorithm we have usedHto denote the Hamiltonian of the classical system,
which is equal to the Lagrangian occurring in the exponent of the path integral –
seeEq. (12.71).
Let us compare the path integral method with the diffusion Monte Carlo approach.
In the latter we start with a given distribution and let time elapse. At the end of the
simulation the distribution of walkers reflects the wave function at imaginary time
τ. Information about the history is lost: physical time increases with simulation
time. The longer our simulation runs, the more strongly will the distribution be
projected onto the ground state. In the path integral method, we change the positions
of the particles along the imaginary-time (inverse-temperature) axis. Letting the
simulation run for a longer time does not project the system more strongly onto the
ground state – the extent to which the ground state dominates in the distribution is
determined by the temperatureβ=M
τ, i.e. for fixed
τ, it is determined by the
length of the chain. The PIMC method is not necessarily carried out in imaginary