404 Quantum Monte Carlo methods
time – there exist versions with real time, which are used to study the dynamics of
quantum systems [ 13 – 15 ].
The analysis so far is correct for distinguishable particles. In fact, we have simply
denoted a coordinate representation state by|R〉. For indistinguishable bosons, we
should read for this state:
|R〉=1
N!
∑
P|r 1 ,r 2 ,...,rN〉, (12.75)where the sum is over all permutations of the positions. The boson character is
noticeable when we impose the periodic boundary conditions along theτ-axis,
where we should not merely identifyrkin the last coordinate|RM〉with the corres-
ponding position in|R 0 〉, but also allow for permutations of the individual particle
positions in both coordinates to be connected.
This feature introduces a boson entropy contribution, which is particularly notice-
able at low temperatures. To see this, let us consider the particles as diffusing from
left (R 0 ) to right (RM). On the right hand side we must connect the particles to
their counterparts on the left hand sides, taking all permutations into account. If the
Boltzmann factor forbids large steps when going from left to right, it is unlikely
that we can connect the particles on the right hand side to the permuted leftmost
positions without introducing a high energy penalty. This is the case whenτ=β
is small, or equivalently when the temperature is high. This can be seen by noticing
that, keeping
τ=β/Nfixed, a decreaseβmust be accompanied by a decrease in
the number of segmentsN. Fewer segments mean less opportunity for the path to
wander away from its initial position. On the other hand, we might keep the num-
ber of segments constant, but decrease
τ. As the spring constants are inversely
proportional to
τ(seeEq. (12.71)), they do not allow, in that case, for large dif-
ferences in position on adjacent time slices; hence permutations are quite unlikely.
When the temperature is high (τ=βsmall), large diffusion steps are allowed and
there is a lot of entropy to be gained from connecting the particles to their starting
positions in a permuted fashion. This entropy effect is responsible for the superfluid
transition in^4 He [ 14 – 16 ]. Path-integral methods also exist for fermion systems. A
reviewcanbefoundinRef. [19].
What type of information can we obtain from the path integral? First of all,
we can calculate ground state properties by takingβvery large (temperature very
small). The system will then be in its quantum ground state. The particles will be
distributed according to the quantum ground state wave function. This can be seen by
considering the expectation value for particle 0 to be at positionR 0. This is given by
P(R 0 )=1
Z
∫
dR 1 dR 2 ...dRM− 1〈R 0 |e−
τH|R 1 〉〈R 1 |e−
τH|R 2 〉...〈RM− 1 |e−
τH|R 0 〉. (12.76)