Path integral Monte Carlo on a lattice: extended states.

Area of Science:

• Quantum mechanics
• Statistical mechanics
• Condensed matter physics

Background:

• Investigating quantum particle (qp) interactions with classical gas across temperature regimes.
• Confining both qp and atoms to a one-dimensional lattice.

Purpose of the Study:

• Develop and test a path integral formalism for canonical ensemble.
• Explore classical and quantum behavior of the system.
• Validate simulation methods against analytical solutions.

Main Methods:

• Path integral formalism with closed, variable-step random walks.
• Monte Carlo simulations (Metropolis algorithm).
• Analytical solutions using Bloch's theorem for periodic potentials.

Main Results:

• Derived analytical expressions for energy, fluctuations, and correlation functions for a free particle.
• Compared Monte Carlo simulations with analytical results for free and striped potentials.
• Demonstrated the path integral formalism's effectiveness under challenging conditions.

Conclusions:

• Path integral formalism and Monte Carlo methods accurately capture quantum particle equilibrium properties.
• Analytical and simulation results show good agreement, validating the approach.
• The study provides a robust framework for investigating quantum-classical systems on lattices.