Theory of Confined Quantum Systems - Part One
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Taking the three-particle system as an example, we have also demonstrated that the ansatz works equally well for mass-imbalanced systems. A future extension of this study might investigate mass-imbalanced systems of four or more particles. It is an open problem to find a general method that yields exact eigenstates in the strongly interacting regime for mass-imbalanced systems.
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A drawback of our ansatz is that it is in general not perturbatively correct in the strongly-interacting regime. The modified ansatz has great simplicity and accuracy at a level that is competitive with state-of-the-art numerical methods for obtaining the energy of the ground state for arbitrary g. Due to its simplicity it should provide a very useful tool. We note that although the results presented here assume that the particles are trapped in an external harmonic trap, the formalism is completely general and can be applied for arbitrary external potentials with at least N bound single-particle states for N -body systems.
As we have shown, the relative deviation of the energy obtained from the ansatz grows only very slowly with N , and there should be no problem in extending the technique to even larger systems than considered here. This overlap may be computed using the same methods that have recently been used to compute spin chain models for strongly interacting fermions 34 and thus scaling to larger N of order 30 or 40 is certainly within reach. In addition, it is relatively straightforward to apply the interpolatory ansatz in the case of strongly interacting bosons 35 , 36 , or mixed systems 22 , The requirements are knowledge of states in the two limits and their overlaps so that the interpolation can be performed.
The formulas for the interpolated energy given here still apply. An example could be an impurity interacting strongly with a Tonks-Girardeau gas of hard-core bosons, which is a topic of great recent interest 37 , 38 , In the following, we provide the details of the methods used in applying the interpolatory ansatz to two-, three- and many-particle systems. We consider a system of two distinguishable fermions and we define. Exact solutions of the two-particle problem are available for arbitrary values of g 31 , The energy of an exact energy eigenstate is given indirectly by Hence, odd harmonic eigenstates do not change as we introduce a non-zero interaction.
The even harmonic eigenstates do, however, change; the correct even eigenstate in the infinite-interaction limit is. Before we employ the interpolatory ansatz, we first separate out the center-of-mass motion using hyperspherical coordinates 35 , This is done merely for convenience and is not in any way essential for the approach.
The Hamiltonian of the relative motion becomes. The eigenstate wave function in both limits has the general form Between the two remaining eigenstates in the triplet, one is odd and the other is even.
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The coefficients in the four remaining regions follow by symmetry considerations. The transformation into hyperspherical coordinates proceeds along the same lines with a modified Jacobian. This transformation allows us to separate the center-of-mass motion from the relative motion, the solutions of the former being the well-known harmonic eigenstates. The interaction potential can then be written as. As for the equal-mass case, the energy is given by Eq.
The angular part of the wave function for the ground state and the first excited state in the infinite-interaction limit are.
The angular part for the non-degenerate second excited state is. In addition, the wave function has the same form as Eq. For the first and second excited states, the wave function has the form of Eq. How to cite this article : Andersen, M. An interpolatory ansatz captures the physics of one-dimensional confined Fermi systems. Part of this work is based on the Bachelor thesis of M.
The authors thank C. Fedorov, and A. Jensen for discussions, as well as C. Cui for feedback on the manuscript. We thank the experimental team of the S. Jochim group for extended discussions and for sharing their results. Author Contributions A.
The numerical calculations were carried out by A. The initial draft of the paper was written by M. All authors contributed to the revisions that led to the final version. National Center for Biotechnology Information , U. Sci Rep.
Epub Theory Of Confined Quantum Systems Part One
Published online Jun Andersen , 1 A. Dehkharghani , 1 A. Volosniev , 1, 2 E. Lindgren , 3, 4 and N. Zinner a, 1.
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Author information Article notes Copyright and License information Disclaimer. Received Apr 15; Accepted Jun 1. This work is licensed under a Creative Commons Attribution 4. Abstract Interacting one-dimensional quantum systems play a pivotal role in physics.
An interpolatory ansatz captures the physics of one-dimensional confined Fermi systems
Open in a separate window. Figure 1. Energy spectrum for the two-particle system. Figure 2. Energy spectrum of the interpolatory ansatz for the three-particle system. Figure 3. Energy spectrum for the three-particle system—modified ansatz. Mass-imbalanced systems In the case of different masses, one typically uses another length scale given by where , and m 1 is the mass of the spin-down fermion while m 2 and m 3 are the masses of the spin-up fermions.
Figure 4. Energy spectrum for the three-particle system with different masses. Figure 5. Energy spectrum of the impurity system. Figure 6. Ansatz accuracy comparison. Figure 7. Anderson overlap according to the interpolatory ansatz. Outlook We have proposed a simple interpolatory ansatz for approximating the energy eigenstate of a confined, one-dimensional system of interacting particles.
Methods In the following, we provide the details of the methods used in applying the interpolatory ansatz to two-, three- and many-particle systems. Details of the two-particle system We consider a system of two distinguishable fermions and we define. Details of the three-particle system Before we employ the interpolatory ansatz, we first separate out the center-of-mass motion using hyperspherical coordinates 35 , Details of the mass-imbalanced system The transformation into hyperspherical coordinates proceeds along the same lines with a modified Jacobian.
Additional Information How to cite this article : Andersen, M. Supplementary Material Supplementary Information: Click here to view. Acknowledgments Part of this work is based on the Bachelor thesis of M. Footnotes Author Contributions A. References Giamarchi T. The resonating valence bond state in La 2 CuO 4 and superconductivity. Science , — Doping a mott insulator: Physics of high-temperature superconductivity. Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond. Advances in Physics 56 , — Many-body physics with ultracold gases.
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