One-Dimensional Semirelativistic Bosons with Contact Interactions

Nirav P. Mehta, James A. Shepard

Fatih Erman, O. Teoman Turgut

We study one-dimensional spinless bosons with semirelativistic spinless-Salpeter dispersion and attractive pairwise contact interactions. Because the dispersion becomes linear at large momentum, the contact interaction is marginal by power counting and produces a logarithmic ultraviolet divergence. We construct the renormalized many-body theory using an enlarged Fock space and a Schur-complement representation of the resolvent, eliminating the bare coupling in favor of the physical zero-total-momentum two-body bound-state energy. The resulting cutoff-independent resolvent defines a self-adjoint Hamiltonian in each fixed particle-number sector. We treat the two-body problem explicitly and show, in the norm-resolvent sense, that the nonrelativistic limit reproduces the attractive Lieb--Liniger Hamiltonian. We also formulate a mean-field approximation directly within the renormalized theory. In the massless and deeply bound large-particle-number regimes, it predicts an exponentially increasing binding scale whose exponent is determined by a one-dimensional variational problem.

Sebastian Will

Matteo Rizzi, Christian Miniatura, Anna Minguzzi +1 more

The contact parameter of harmonically trapped, interacting bosons at zero temperature in one dimension is obtained analytically by a simple rescaling of the exact two-boson solution, which further confirms that the ground-state properties of an interacting, one-dimensional Bose gas can be accurately described by an effective two-body contact interaction dressed by the other particles in the many-body system.

Richard L Hall

It is shown that exact solutions may be found for the energy eigenvalue problem generated by the class of semirelativistic Hamiltonians of the form , where is a non-local potential with a separable kernel of the form . Explicit examples in one and three dimensions are discussed, including the Yamaguchi and Gauss potentials. The results are used to obtain lower bounds for the energy of the corresponding N -boson problem, with upper bounds provided by the use of a Gaussian trial function.

Peter A. Ring, Y. K. Gambhir, S. Iwasaki +1 more