Abstract
The Lefschetz-thimble approach to path integrals is applied to a one-site model of electrons, i.e., the one-site Hubbard model. Since the one-site Hubbard model shows a non-analytic behavior at the zero temperature and its path integral expression has the sign problem, this toy model is a good testing ground for an idea or a technique to attack the sign problem. Semiclassical analysis using complex saddle points unveils the significance of interference among multiple Lefschetz thimbles to reproduce the non-analytic behavior by using the path integral. If the number of Lefschetz thimbles is insufficient, we found not only large discrepancies from the exact result, but also thermodynamic instabilities. Analyzing such singular behaviors semiclassically, we propose a criterion to identify the necessary number of Lefschetz thimbles. We argue that this interference of multiple saddle points is a key issue to understand the sign problem of the finite-density quantum chromodynamics.
Highlights
Understanding strongly-correlated quantum many-body systems has been an ultimate goal in contemporary physics
Many thermodynamic quantities can be computed for various systems using this method, such as finite-temperature quantum chromodynamics (QCD) in hadron physics [1], and liquid helium [2], ultracold atomic gases [3], Bose–Fermi mixtures [4] in condensed matter physics
One-site repulsive Hubbard model shows a non-analytic behavior by changing the chemical potential at the zero temperature β = ∞
Summary
Understanding strongly-correlated quantum many-body systems has been an ultimate goal in contemporary physics. QCD at finite quark densities attracts much attention for understanding the interior structure of neutron stars [8,9,10], but we have no ab initio simulation due to the sign problem [11]. We apply the Lefschetz-thimble approach to the one-site model of electrons. The Hamiltonian of the one-site model can be diagonalized and we can calculate any expectation value exactly Since this model has the severe sign problem in its path-integral expression, it is hard to calculate expectation values by the conventional Monte Carlo method. This toy model provides us a good playground to study theoretical structures of the Lefschetz-thimble approach.
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