Small matrix path integral in imaginary time.

Area of Science:

• Quantum mechanics
• Computational physics
• Statistical mechanics

Background:

• Thermal equilibrium properties are typically calculated using imaginary-time path integrals and Monte Carlo methods.
• The sign problem, arising from alternating signs in the Boltzmann matrix, hinders convergence for certain quantum systems like identical fermions or frustrated Hamiltonians.

Purpose of the Study:

• To develop a robust and efficient quadrature-based method for computing the Boltzmann matrix.
• To address the convergence issues caused by the sign problem in quantum system simulations.

Main Methods:

• Developed a Small Matrix Path Integral (SMatPI) decomposition by expressing the discretized path integral with the influence functional as a sum of matrix products.
• Implemented iterative propagation in imaginary time, avoiding the need to store large tensors.
• Applied the method to discrete systems coupled to harmonic baths, including two- and three-level systems.

Main Results:

• The SMatPI algorithm provides a stable and efficient solution for calculating Boltzmann matrices.
• Demonstrated the method's effectiveness on systems exhibiting the sign problem, such as cyclic tight-binding Hamiltonians.
• Successfully circumvented the storage limitations of previous tensor-based approaches.

Conclusions:

• The SMatPI method offers a significant advancement for simulating thermal equilibrium properties in challenging quantum systems.
• This approach overcomes the limitations of Monte Carlo methods when faced with the sign problem.
• The SMatPI algorithm is a powerful and efficient tool for computational quantum physics research.