The Path Integral Monte Carlo (PIMC)
has become very popular recently. This method is based on
Feynamn's formulation of quantum mechanics
in terms of path integrals .
At finite temperature, equilibrium properties of quantum systems can be
calculated using the thermal density matrix, .
is the inverse temperature.
The expectation value of an observable at equilibrium:
is the partition function.
The PIMC method is usually implemented in a position-space representation
, where the density
matrix is given by:
where the elements of
are positive and can be
interpreted as probabilities.
contains the exponential of the Hamiltonian
, where is the potential energy and
kinetic energy. These operators are non-commutative, which makes them difficult to
evaluate density_matrix. For small (high temperature)
one can approximate
If is large (low temperature), then one divide
into many small segments, , i.e.
, which can be treated
as a case of small .
The partition function, ,
can written in the following form:
where the units operators
and the first and the last elements are identified as
required by the trace operations.
In the short-time approximation the expression for
is given by:
which is the same form as Green_full.
Substituting the result into expas one obtains
The integral PIMC looks like a partition function
of a classical ring polymer, where the nearest neighbors (beads) are connected with springs,
while the is an external potential.
Moving a quantum particle is equivalent
to evolving this polymer.
The system of quantum particles
can be represented by an ensemble of these ring polymers,
as shown in Fig. 5.1.
The trace of the closed paths of 54 helium atoms in
the bcc lattice at a temperature of 1.5 K with M = 25 beads.
Straight lines connect successive beads of the polymers. The
PIMC (upicode 9.0) is used for calculations.
The only difference between the classical and quantum
systems is that the potential interaction between these polymers
occurs only at the same time slices (beads), while in a real polymer all monomers
interact with each other.(See Fig. 5.2).
Cartoon of paths for 3 particles (A, B, C) with M=4,
. The kinetic terms are represented by zig-zag
lines and potential by dotted lines, which connect only beads of the
polymers at the same time, from Bernu and Ceperley 
The expectation values are calculated by sampling the paths with the MC
method. The simplest move is the translation of a whole chain. This move
changes the potential energy of the system. If the relative position of the
monomers is altered, then one changes the kinetic energy.
Several algorithms are used to sample the paths efficiently .
The Bose and Fermi symmetry of the quantum particles increases significantly
the complexity of the PIMC method. In the case of bosons,
the density matrix is written as:
where is the permutation operator,
for example a pair permutation
The action of the permutation operator
is equivalent to opening the ring polymers involved in the permutation
and making a single polymer out of them.
In PIMC (for bosons) no a priori
is needed, one can use only the Hamiltonian and let
the code run for a long time. It
is possible that the paths themselves make the transition to a new,
There is no problems with scalability in the method:
increase of the number of polymers does not lead to
instability, the computational effort increases, the
correlation times become larger, but the system remains