arXiv:1806.08591 · TURKISH JOURNAL OF PHYSICS

Lattice calculations for two-component fermion systems with unequal masses: one dimension

Serdar Elhatisari

Published Dec 13, 2018• 0 citations

Abstract

We consider systems of two-component fermions with unequal masses and interacting via a short-range attractive potential. We discuss the case where the two-component fermions form a shallow dimer with large scattering length. The three-fermion and four-fermion systems with such properties are universal and characterized by the twofermion scattering length aff and the ratio of the mass of spin-↑ fermion to the mass of spin-↓ fermion, m↑/m↓. In this study, using lattice effective field theory, we analyze fermion-dimer and dimer-dimer systems and calculate the universal fermion-dimer and dimer-dimer scattering lengths for various values of the mass ratio m↑/m↓. We find that these universal scattering lengths increase logarithmically with the mass ratio m↑/m↓.

DOI: 10.3906/fiz-1805-17

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arXiv:1806.08591 · TURKISH JOURNAL OF PHYSICS

Lattice calculations for two-component fermion systems with unequal masses: one dimension

Serdar Elhatisari

Published Dec 13, 2018• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.3906/fiz-1805-17

Related papers

Universal dimer–dimer scattering in lattice effective field theory

Serdar Elhatisari, Kris Katterjohn, Dean Lee +2 more

We consider two-component fermions with short-range interactions and large scattering length. This system has universal properties that are realized in several different fields of physics. In the limit of large fermion–fermion scattering length aff and zero-range interaction, all properties of the system scale proportionally with aff. For the case with shallow bound dimers, we calculate the dimer–dimer scattering phase shifts using lattice effective field theory. We extract the universal dimer–dimer scattering length add/aff=0.618(30) and effective range rdd/aff=−0.431(48). This result for the effective range is the first calculation with quantified and controlled systematic errors. We also benchmark our methods by computing the fermion–dimer scattering parameters and testing some predictions of conformal scaling of irrelevant operators near the unitarity limit.We consider two-component fermions with short-range interactions and large scattering length. This system has universal properties that are realized in several different fields of physics. In the limit of large fermion–fermion scattering length aff and zero-range interaction, all properties of the system scale proportionally with aff. For the case with shallow bound dimers, we calculate the dimer–dimer scattering phase shifts using lattice effective field theory. We extract the universal dimer–dimer scattering length add/aff=0.618(30) and effective range rdd/aff=−0.431(48). This result for the effective range is the first calculation with quantified and controlled systematic errors. We also benchmark our methods by computing the fermion–dimer scattering parameters and testing some predictions of conformal scaling of irrelevant operators near the unitarity limit.

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a

for nonidentical fermions is the only relevant quantity. This is the generalization to fermions with different masses of the problem solved long ago by Skorniakov and Ter-Martirosian for particles with equal masses. We solve this problem analytically in the two limiting cases where the mass of the solitary fermion is very large or very small compared to the mass of the two other identical fermions. This is done for both the value of the scattering length and the function entering the Skorniakov--Ter-Martirosian integral equation, for which simple explicit expressions are obtained.

Two-component Fermi and Bose gases with infinitely large interspecies

s

-wave scattering length

{a}_{s}

exhibit a variety of intriguing properties. Among these are the scale invariance of two-component Fermi gases with equal masses, and the favorable scaling of Efimov features for two-component Bose gases and Bose-Fermi mixtures with unequal masses. This paper builds on our earlier work [Phys. Rev. Lett. 105, 170403 (2010)] and presents a detailed discussion of our studies of small unequal-mass two-component systems with infinite

{a}_{s}

in the regime where three-body Efimov physics is absent. We report on nonuniversal few-body resonances. Just like with two-body systems on resonance, few-body systems have a zero-energy bound state in free space and a diverging generalized scattering length. Our calculations are performed within a nonperturbative microscopic framework and investigate the energetics and structural properties of small unequal-mass two-component systems as functions of the mass ratio $\ensuremath{\kappa}$, and the numbers

{N}_{1}

and

{N}_{2}

of heavy and light atoms. For purely attractive Gaussian two-body interactions, we find that the

({N}_{1},{N}_{2})=(2,1)

and

(3,1)

systems exhibit three-body and four-body resonances at mass ratios $\ensuremath{\kappa}=12.314(2)$ and

10.4(2)

, respectively. The three- and four-particle systems on resonance are found to be large. It seems feasible that the features discussed in this paper can be probed experimentally with present-day technology.

We study the dimer-dimer scattering length

{a}_{4}

for a two-component Fermi mixture in which the different fermions have different masses ${m}_{\ensuremath{\uparrow}}$ and ${m}_{\ensuremath{\downarrow}}$. This is made in the framework of the exact field-theoretic method. In the large mass ratio domain the equations are simplified enough to lead to an analytical solution. In particular we link

{a}_{4}

to the fermion-dimer scattering length

{a}_{3}

for the same fermions and obtain the very simple relation

{a}_{4}={a}_{3}/2

. The result ${a}_{4}\ensuremath{\simeq}{a}_{3}/2$ is actually valid whatever the mass ratio with quite good precision. As a result we find an analytical expression providing

{a}_{4}

with fairly good precision for any mass. To dominant orders for large mass ratio it agrees with the literature. We show that in this large mass ratio domain, the dominant processes are the repeated dimer-dimer Born scatterings, considered earlier by Pieri and Strinati [Phys Rev. B 61, 15370 (2000)]. We conclude that their approximation of retaining only these processes is a fairly good one whatever the mass ratio.