Strong-coupling ansatz for the one-dimensional Fermi gas in a harmonic potential

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Science Advances  24 Jul 2015:
Vol. 1, no. 6, e1500197
DOI: 10.1126/sciadv.1500197


  • Fig. 1 Energy levels in the Tonks-Girardeau limit.

    We display the exact energies (ψl | Embedded Imagel ) (red) and the result of our ansatz 〈Embedded Image〉 (blue dashed), given by E0Embedded Image in this limit, for the case of one ↓ particle and N = 3 ↑ particles. For g < 0, there is also a two-body bound state at negative energies (not shown).

  • Fig. 2 Accuracy of the ansatz.

    Overlaps between our ansatz Embedded Image and the exact wave functions ψl for majority particle numbers N ≤ 8. For the ferromagnetic state, this always equals 1 (black line). The red and blue dots depict the overlap for Embedded Image and Embedded Image (ground state), respectively. These are both 1 for N = 2 because these states are uniquely determined by spin and parity, whereas they are both 0.999993 for N = 3. The extrapolations (dashed lines) are least-squares fits of the data points to cubic polynomials. Inset: The wave function overlap of Girardeau’s proposed state (10) with the exact ground state.

  • Fig. 3 Contact coefficients of the ground-state manifold.

    The contact coefficients (blue dots) of the exact eigenstates ψl in the Tonks-Girardeau regime, controlling the splitting of the energy levels at finite but large coupling. The gray lines represent the approximate relationship Embedded Image; see Eq. 11.

  • Fig. 4 Exchange constants and ground state of the Heisenberg model.

    Illustration of the nearest-neighbor exchange constants (Eq. 14) of the spin Hamiltonian (Eq. 12) for N = 100. We also show the ground-state wave function (Eq. 18) within our ansatz (green dots).

  • Fig. 5 Contact of the ground state in the few- and many-body limit.

    Contact coefficient of the ground state at small positive 1/g as a function of N. The dots are the analytical results for N ≤ 3 and numerical results for 4 ≤ N ≤ 8. We do not show a comparison between the ground-state contact and the perturbation evaluated within our approximate states, Embedded Image, because the relative error between these is less than 0.05% for N ≤ 8. The dashed line is McGuire’s free-space solution mapped to the harmonic potential using the local density approximation—see the discussion in the main text. Inset: The ground-state contact coefficient in units of N3/2 and plotted as a function of 1/N to illustrate the possible convergence to McGuire’s prediction (marked by a triangle). The dashed line is a cubic fit to our data. We also compare with the expectation value of Girardeau’s proposed ground state Embedded Image (green squares).

  • Fig. 6 Emergence of the orthogonality catastrophe.

    Residue of the wave function ψN as a function of N. For N = 1 and N = 2, we find the analytic results 2/π and 81/(16π2), respectively. The dashed line is 0.89/ Embedded Image. We do not show a comparison between the residue of the ground state in our approximation scheme and that of the exact ground state because the relative error is less than 0.07% for N ≤ 7.

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