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arXiv:1103.3534 · Physical Review A

Validity of the scattering-length approximation in strongly interacting Fermi systems

Sizhe Zhou, David M. Ceperley, Shiwei Zhang

Published Jul 29, 2011• 8 citations
arXivHTMLPDF

Abstract

We investigate the energy spectrum of systems of two, three, and four spin-$\frac{1}{2}$ fermions with short range attractive interactions both exactly and within the scattering length approximation. The formation of molecular bound states and the ferromagnetic transition of the excited scattering state are examined systematically as a function of the two-body scattering length. Identification of the upper branch (scattering states) is discussed and a general approach valid for systems with many particles is given. We show that an adiabatic ferromagnetic transition occurs, but at a critical transition point ${k}_{F}a$ much higher than predicted from previous calculations, almost all of which use the scattering length approximation. In the four-particle system the discrepancy is a factor of 2. The exact critical interaction strength calculated in the four-particle system is consistent with that reported by experiment. To make comparisons with the adiabatic transition, we study the quench dynamics of the pairing instability using the eigenstate wave functions.

DOI
10.1103/physreva.84.013625

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We investigate the energy spectrum of systems of two, three, and four spin-12\frac{1}{2}21​ fermions with short range attractive interactions both exactly and within the scattering length approximation. The formation of molecular bound states and the ferromagnetic transition of the excited scattering state are examined systematically as a function of the two-body scattering length. Identification of the upper branch (scattering states) is discussed and a general approach valid for systems with many particles is given. We show that an adiabatic ferromagnetic transition occurs, but at a critical transition point kFa{k}_{F}akF​a much higher than predicted from previous calculations, almost all of which use the scattering length approximation. In the four-particle system the discrepancy is a factor of 2. The exact critical interaction strength calculated in the four-particle system is consistent with that reported by experiment. To make comparisons with the adiabatic transition, we study the quench dynamics of the pairing instability using the eigenstate wave functions.
The control afforded by Feshbach resonance phenomena has enabled the exploration of strongly interacting degenerate regimes in dilute ultracold atomic alkali-gases. In these dilute systems, interactions are characterized by a single parameter, the s-wave scattering length. In this dissertation, we review the physics of quantum degenerate atomic gases from a theoretical perspective and present the applications of non-perturbative numerical methods ranging from exact diagonalization to quantum Monte Carlo techniques. Emphasis is given to the effect of interactions. A major goal of this work is to compare theoretical predictions with available experimental results. We begin by introducing the effective interactions in the many-body alkali-gas system. As simplifications of the real interaction between two alkali atoms, the resonance properties of various short range models, the zero-range model and the two-channel model are investigated. The fundamental result is that under appropriate conditions, the true interaction potential V(r) of two atoms may be replaced by a regularized delta-function. In collaboration with experimentalists in our department, we developed a unique method to construct localized single-particle wave functions using imaginary time projection and thereby determine lattice Hamiltonian parameters. Our method enables an efficient coarse-grained mapping from a continuum system to a lattice model. We apply the method to a specific disordered potential generated by an optical lattice experiment and calculate for each instance of disorder the equivalent lattice model parameters. Detailed statistical analysis is performed on the resulting probability distributions of the Hubbard parameters. In the final part, we study the pairing and ferromagnetic instabilities in systems of spin-1/2 fermions. To interpret a recent experimental study of the possibility of itinerant ferromagnetism in cold atom systems, we investigate the energy spectrum of a system of four spin-1/2 fermions with short range attractive interactions both exactly and within the scattering length approximation. The formation of molecular bound states and the ferromagnetic transition of the excited scattering state are examined systematically as a function of the two-body scattering length. We show that an adiabatic ferromagnetic transition occurs, but at a critical transition point k_{F}a much higher than predictions from the scattering length approximations. The exact critical interaction strength calculated in the four-particle system is consistent with that reported by experiment. Finally, by constructing a many-body wavefunction satisfying the Bethe-Peierls boundary conditions, we attempted the first variational Monte Carlo calculations of the zero-range model of unitary Fermion gases, eliminating the need for short-range approximations employed by existing QMC calculations.
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Quantum degenerate Fermi gases can be created in the laboratories using alkali atoms. These gases can be in different regimes of density and interaction strength and provide an ideal test bed for the basic properties of the quantum and statistical mechanics. Also some astrophysical objects such as neutron stars can be idealized as interacting fermion systems. In this thesis work, the ground state properties of dilute Fermi gases with attractive short range two-body interactions are reported. Main results of this work are produced by application of quantum Monte Carlo methods. We introduce variational nodal structure to the trial wave function in order to deal with the fermion sign problem. We consider the possibility of Cooper pairing between fermions of different species. The strength of the interaction is varied to study different pairing regimes which are characterized by the product of the s-wave scattering length a and the Fermi wave vector kF. We also review some of the conventional variational methods such as the Lowest Order Constrained Variational which is suitable for Fermi fluids without pairing correlation. The ground state energy, pairing gap Δ, quasi-particle spectrum, two-body distribution function, and momentum distribution of the superfluid were studied with greater accuracy using ab initio quantum Monte Carlo techniques. In the case of the two component Fermi fluid, we find that in the weak coupling regime, 1/akF < -1, the system is a Bardeen-Cooper-Schrieffer type superfluid and the energy gap Δ is much smaller than the Fermi gas energy EFG. For 1/akF ≥ 0:5 we find that weakly interacting composite bosons or molecules are formed in the superfluid gas with Δ and gas energy per particle approaching half of the molecular energy jEmolj=2. In this region we seem to have Bose-Einstein condensation of molecules. The behavior of the energy, chemical potential and energy gap in the transition region, -0:5 < 1/akF < 0:5, is discussed. The study is extended to the three component Fermi fluid in relation to the spin polarized alkali gases and color superconductivity of the quark matter. Unlike the two component fluids, zero potential range limit cannot be reached without causing collapse of the system. We also study the scaling behavior of the three-body system. We calculate the threshold of the three-body bound state as a function of the potential range and strength. For the many-body systems, we find that ground state can be achieved by broken symmetry pairing. However, the restored symmetry pairing scenario in the strongly interacting regime is not conclusive. We also discuss the pairing gap as well as the structural properties such as two-body distribution functions.
We report results of quantum Monte Carlo calculations of the ground state of dilute Fermi gases with attractive short-range two-body interactions. The strength of the interaction is varied to study different pairing regimes which are characterized by the product of the sss-wave scattering length and the Fermi wave vector, akFa{k}_{F}akF​. We report results for the ground-state energy, the pairing gap $\ensuremath{\Delta}$, and the quasiparticle spectrum. In the weak-coupling regime, $1∕a{k}_{F}<\ensuremath{-}1$, we obtain Bardeen-Cooper-Schrieffer (BCS) superfluid and the energy gap $\ensuremath{\Delta}$ is much smaller than the Fermi gas energy EFG{E}_{\mathrm{FG}}EFG​. When a>0a>0a>0, the interaction is strong enough to form bound molecules with energy Emol{E}_{\mathrm{mol}}Emol​. For $1∕a{k}_{F}\ensuremath{\gtrsim}0.5$, we find that weakly interacting composite bosons are formed in the superfluid gas with $\ensuremath{\Delta}$ and gas energy per particle approaching $\ensuremath{\mid}{E}_{\mathrm{mol}}\ensuremath{\mid}∕2$. In this region, we seem to have Bose-Einstein condensation (BEC) of molecules. The behavior of the energy and the gap in the BCS-to-BEC transition region, $\ensuremath{-}0.5<1∕a{k}_{F}<0.5$, is discussed.
The properties of cold interacting Fermi gases in terms of scattering length are investigated. The scattering length is calculated by exact solution of Schrödinger equation. A Cosine hyperbolic potential with variable strength is employed in this study. The different interaction regimes are provided by choosing different strength of potential, which are characterized by an inverse of S-wave scattering length and Fermi wave number. The energy per particle for 3He system is calculated within the lowest order constrained variational (LOCV) method. It is found that the bound bosonic pairs as well as bound molecules are formed at positive scattering length, only by considering two-body interactions.
Two-component fermions are known to behave like a gas of molecules in the limit of Bose-Einstein condensation of diatomic pairs tightly bound with zero-range interactions. We discover that the formation of cluster states occurs when the effective range of two-body interaction exceeds roughly 0.46 times the scattering length, regardless of the details of the short-range interaction. Using an explicitly correlated Gaussian basis set expansion approach, we calculate the binding energy of cluster states in trapped few-body systems and show the difference of structural properties between cluster states and gaslike states. We identify the condition for cluster formation and discuss the potential observation of cluster states in experiments.