Quantum Anomaly and Thermodynamics of One-Dimensional Fermions with Three-Body Interactions

Abstract

We show that a system of three species of one-dimensional fermions, with an attractive three-body contact interaction, features a scale anomaly directly related to the anomaly of two-dimensional fermions with two-body contact forces. We show, furthermore, that those two cases (and their multispecies generalizations) are the only nonrelativistic systems with contact interactions that display a scale anomaly. While the two-dimensional case is well known and has been under study both experimentally and theoretically for years, the one-dimensional case presented here has remained unexplored. For the latter, we calculate the impact of the anomaly on the equation of state, which appears through the generalization of Tan's contact for three-body forces, and determine the pressure at finite temperature. In addition, we show that the third-order virial coefficient is proportional to the second-order coefficient of the two-dimensional two-body case.

Fermi-Fermi crossover in the ground state of one-dimensional few-body systems with anomalous three-body interactions

“…(3), the ultraviolet (i.e. short-range) behavior of the momentum sum is the same as that of the 2D two-body problem, in the sense that they both diverge logarithmically [29]. We thus see that, even though our problem is in 1D globally, it is in 2D locally.…”

“…To summarize, we have extended the analysis of the three-body problem with a three-body contact interaction of Ref. [29] beyond the 1 + 1 + 1 system. Here, we have studied the higher-body problems up to six particles in all possible non-trivial combinations, namely: 4 particles (2 + 1 + 1); 5 particles (2 + 2 + 1 and 3 + 1 + 1); and 6 particles (2 + 2 + 2, 3 + 2 + 1, and 4 + 1 + 1).…”

“…While the resulting universal relations have not yet been derived for the present case, it is easy to see using the Feynman-Hellman theorem that the contact can be expressed as the derivative of the energy with respect to the logarithm of the scattering length (as shown for the system considered here in Ref. [29]). Here, we work in terms of the binding energy rather than the scattering length, such that the (dimensionless) contact takes the form…”

Three-body interaction near a narrow two-body zero crossing

“…Then, the dimer states given by Eqs. (23)(24)(25) are spurious, consistent with the fact that the zero-range theory can no longer be used at such high momenta.…”

“…In particular, three-body forces have been considered in the context of droplet formation in three dimensions [16][17][18][19] and as a means for stabilizing supersolid phases of quasi-two-dimensional dipolar atoms or molecules [20]. Quite a few recent theory papers have discussed one-dimensional three-bodyinteracting systems, exploring the kinematic equivalence of the three-body scattering in one dimension and the two-body scattering in two dimensions (see, for example [21][22][23][24][25][26][27][28][29][30][31][32]).…”