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arXiv:cond-mat/0503033 · Journal of the Physical Society of Japan

Exact Analysis of δ-Function Attractive Fermions and Repulsive Bosons in One-Dimension

Toshiaki Iida, Miki Wadati

Published Jun 1, 2005• 45 citations
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Abstract

The Gaudin integral equation for the ground state of a one-dimensional delta-function attractive spin-1/2 fermions is solved in the form of power series. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities such as the ground state energy are expressed in terms of a single dimensionless parameter gamma =c/D, where c is the coupling constant and D is the number density. The results agree with those obtained from the perturbation calculations, which include the one in the classical electrostatics originally by Kirchhoff. In the strong coupling limit, the connection to the solutions of the Lieb-Liniger integral equation for the ground state of a one-dimensional delta-function repulsive bose gas is shown explicitly.

DOI
10.1143/jpsj.74.1724

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The Gaudin integral equation for the ground state of a one-dimensional delta-function attractive spin-1/2 fermions is solved in the form of power series. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities such as the ground state energy are expressed in terms of a single dimensionless parameter gamma =c/D, where c is the coupling constant and D is the number density. The results agree with those obtained from the perturbation calculations, which include the one in the classical electrostatics originally by Kirchhoff. In the strong coupling limit, the connection to the solutions of the Lieb-Liniger integral equation for the ground state of a one-dimensional delta-function repulsive bose gas is shown explicitly.
The ground state of a one-dimensional δ-function attractive spin-1/2 fermions is studied by the nested Bethe ansatz method. Explicitly, the Gaudin integral equation for the system is solved in the form of power series with arbitrary spin polarization. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities, such as the ground state energy and the chemical potentials, are expressed in terms of the dimensionless coupling constant and the polarization , where c is the coupling constant and are the number densities of the spin-up (down) particles. While in the limiting case P = 1, the system consists of spinless free fermions, the other limit P = 0 describes the BCS–BEC crossover in one dimension. As a function of P , the evolution is continuous and smooth.
The Fredholm equations for one-dimensional two-component fermions with repulsive and with attractive $\ensuremath{\delta}$-function interactions are solved by an asymptotic expansion for strong repulsion, weak repulsion, weak attraction, and strong attraction. Consequently, we obtain the first few terms of the expansion of the ground-state energy for the Fermi gas with polarization for these regimes. We also prove that the two sets of Fredhom equations for weakly repulsive and attractive interactions are identical as long as the integration boundaries match each other between the two types. Thus the asymptotic expansions of the energies of repulsive and attractive fermions are identical to all orders in this region. The identity of the asymptotic expansions may not mean that the energy connects analytically.
Proposed is a method of solving the Lieb–Liniger integral equation for a one-dimensional δ-function bose gas at zero temperature. The integral equation reduces to a set of algebraic equations for the coefficients of the series expansion. For strong coupling case, a few coefficients are enough to have a good approximation. On the contrary, all the coefficients are in the same order for weak coupling case. The expansion in λ= c / K , where c is the coupling constant and K is the cutoff momentum, is a formal one in the sense that the coefficients may include the divergent sums. However, the physical quantities such as the ground state energy are shown to be finite in γ= c / ρ, where ρ is the number density, which agree with the results of the Bogoliubov theory.
1D multicomponent Fermions with delta function interaction in strong and weak coupling limits: κκκ-component Fermi gas
We derive the first few terms of the asymptotic expansion of the Fredholm equations for one-dimensional κκκ-component fermions with repulsive and with attractive delta-function interaction in strong and weak coupling regimes. We thus obtain a highly accurate result for the ground state energy of a multicomponent Fermi gas with polarization for these regimes. This result provides a unified description of the ground state properties of the Fermi gas with higher spin symmetries. However, in contrast to the two-component Fermi gas, there does not exist a mapping that can unify the two sets of Fredholm equations as the interacting strength vanishes. Moreover, we find that the local pair correlation functions shed light on the quantum statistic effects of the κκκ-component interacting fermions. For the balanced spin case with repulsive interaction, the ground state energy obtained confirms Yang and You's result [Chin. Phys. Lett. {\bf 28}, 020503 (2011)] that the energy per particle as κ→∞κ\to \inftyκ→∞ is the same as for spinless Bosons.
We derive the first few terms of the asymptotic expansion of the Fredholm equations for one-dimensional $\ensuremath{\kappa}$-component fermions with repulsive and attractive $\ensuremath{\delta}$-function interaction in strong- and weak-coupling regimes. We thus obtain a highly accurate result for the ground-state energy of a multicomponent Fermi gas with polarization for these regimes. This result provides a unified description of the ground-state properties of the Fermi gas with higher spin symmetries. However, in contrast to the two-component Fermi gas, there does not exist a mapping that can unify the two sets of Fredholm equations as the interacting strength vanishes. Moreover, we find that the local pair-correlation functions shed light on the quantum statistic effects of the $\ensuremath{\kappa}$-component interacting fermions. For the balanced spin case with repulsive interaction, the ground-state energy obtained confirms Yang and You's result [Chin. Phys. Lett. 28, 020503 (2011)] that the energy per particle as $\ensuremath{\kappa}\ensuremath{\rightarrow}\ensuremath{\infty}$ is the same as for spinless Bosons.