Worldline Approach to Few-Body Physics on the Lattice

Abstract

We study the physics of two species of non-relativistic hard-core bosons with attractive or repulsive delta function interactions on a spacetime lattice using the worldline formulation. By tuning the chemical potential carefully we show that worm algorithms can efficiently sample the worldline configurations in any fixed particle-number sector. Since fermions can be treated as hard-core bosons up to a permutation sign, we also apply this approach to non-relativistic fermions. The fermion permutation sign is treated as an observable in this approach and can be used to extract energies for each particle-number sector. Since in one dimension non-relativistic fermions can only permute due to boundary effects, unlike the auxiliary field method, in many cases our approach does not suffer from sign problems. Using our method we discover limitations of the recently proposed complex Langevin calculations in one dimension.

Complex Langevin and other approaches to the sign problem in quantum many-body physics

“…The agreement between these different methods provides confidence that the CL method is suitable to study this problem. Recently, a worldline algorithm was adapted to study mass-imbalanced few-body systems without a sign-problem [229,234] in 1D and its results have been compared to the CL values obtained in [109] (shown in the right panel of Fig. 14).…”

Pairing patterns in one-dimensional spin- and mass-imbalanced Fermi gases

“…However, their theoretical description remains relatively scarce in the literature since the reduced symmetry of the model renders BA studies inapplicable, except for a very few specific values for the mass ratio. Nevertheless, progress on analytic insights has been made for special symmetric configurations in the few-body sector [32][33][34] and numerically the regime was explored by worldline MC [35,36], ED [37] and complex Langevin (CL) studies [38,39]. Beyond the few-body regime, several studies based on effective descriptions have been conducted [40][41][42][43][44] and numerical studies investigated the phase-diagram of the asymmetric 1D Hubbard model (which coincides with the continuum model at low fillings) by means of DMRG [45,46] as well as spin-and mass-imbalanced systems via MC [47], time-evolving block decimation (TEBD) method [48] and DMRG [49] approaches.…”