Abstract
We formulate the physics of two species of non-relativistic hard-core bosons with attractive or repulsive delta function interactions on a space-time lattice in the worldline approach. We show that worm algorithms can efficiently sample the worldline configurations in any fixed particle-number sector if the chemical potential is tuned carefully. Since fermions can be treated as hard-core bosons up to a permutation sign, we apply this approach to study non-relativistic fermions. The fermion permutation sign is an observable in this approach and can be used to extract energies in each particle-number sector. In one dimension, non-relativistic fermions can only permute across boundaries, and so our approach does not suffer from sign problems in many cases, unlike the auxiliary field method. Using our approach, we discover limitations of the recently proposed complex Langevin calculations in one spatial dimension for some parameter regimes. In higher dimensions, our method suffers from the usual fermion sign problem. Here we provide evidence that it may be possible to alleviate this problem for few-body physics
Highlights
Computing the properties of quantum systems containing fermions remains challenging especially when perturbative techniques begin to fail
We find that the complex Langevin (CL) method yields incorrect results as the repulsive coupling strength grows, implying that the observed flattening is unphysical and an artifact of the method
We proposed a worldline based approach to few-body physics where fermions are formulated as hardcore bosons, since they incorporate one of the ingredients of the Pauli exclusion principle, which is that two identical fermions cannot exist at the same spacetime point
Summary
Computing the properties of quantum systems containing fermions remains challenging especially when perturbative techniques begin to fail. Even in the case of few-body physics, where each particle is described by a large dimensional vector space, the free and the interacting parts of the Hamiltonian may be diagonalized by two very different basis vectors and the ground state in a given particle-number sector may be severely entangled in both these bases with no apparent small parameters This problem is even more severe in quantum field theories, where these particles are created out of a vacuum that can itself be nontrivial, like in quantum chromodynamics (QCD). We can adapt our approach to study fermionic particles even in higher dimensions by treating the fermionic permutation sign as an observable, but it becomes difficult to compute it accurately As expected, this observable suffers from a severe signal-to-noise ratio problem, especially at low temperatures and when the number of particles becomes large.
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