At finite temperature, equilibrium properties of quantum systems can be
calculated using the thermal density matrix, .

(47) |

where is the partition function.

The PIMC method is usually implemented in a position-space representation
, where the density
matrix is given by:

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, ,

(51) |

where the units operators are inserted, 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:

(53) |

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 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 [22].

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:

(55) |

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, unexpected state. 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 stable.