Three $s$-wave-interacting fermions under anisotropic harmonic confinement: Dimensional crossover of energetics and virial coefficients

We present highly accurate solutions of the Schrödinger equation for three fermions in two different spin states with zero-range $s$-wave interactions under harmonic confinement. Our approach covers spherically symmetric, strictly two-dimensional, strictly one-dimensional, cigar-shaped, and pancake-shaped traps. In particular, we discuss the transition from quasi-one-dimensional to strictly one-dimensional and from quasi-two-dimensional to strictly two-dimensional geometries. We determine and interpret the eigenenergies of the system as a function of the trap geometry and the strength of the zero-range interactions. The eigenenergies are used to investigate the dependence of the second- and third-order virial coefficients, which play an important role in the virial expansion of the thermodynamic potential, on the geometry of the trap. We show that the second- and third-order virial coefficients for anisotropic confinement geometries are, for experimentally relevant temperatures, very well approximated by those for the spherically symmetric confinement for all $s$-wave scattering lengths.

Figure 1

Relative three-body energies $E_{3\mathrm{b}}/E_z$ as a function of the inverse scattering length $a_z/a^{3\mathrm{D}}$ for a cigar-shaped trap with aspect ratio $\eta =2$ and (a) $M=0$ and ${\Pi }_z=+1$, and (b) $M=0$ and ${\Pi }_z=-1$.

Figure 2

“Crossover curve” of the three-body system with $M=0$, shifted by $2E_{\rho }=2\eta E_z$, for cigar-shaped traps with $\eta =2$ (solid line), 4 (dotted line), 6 (dashed line), 8 (dash-dot-dotted line), and 10 (dash-dotted line) as a function of $a_z/a^{3\mathrm{D}}$. The scattering lengths at which the parity of the corresponding eigenstate changes from ${\Pi }_z=-1$ (“left side of the graph”) to ${\Pi }_z=+1$ (“right side of the graph”) are marked by asterisks. At these points, the derivative of the crossover curve is discontinuous; the discontinuities are not visible on the scale shown. The inset shows the (unshifted) crossover curve as a function of $a_{\rho }/a^{3\mathrm{D}}$.

Figure 3

(a) Relative two-body energies $E_{2\mathrm{b}}/E_z$ as a function of the inverse scattering length $a_z/a^{3\mathrm{D}}$ for a cigar-shaped trap with aspect ratio $\eta =10$, $M=0$, and ${\Pi }_z=+1$. (b) The solid curve from panel (a) is replotted and compared with the energy obtained by solving the strictly one-dimensional eigenequation with renormalized one-dimensional scattering length $a_{\mathrm{ren}}^{1\mathrm{D}}$ (dotted line; the energy $E_{\rho }=\eta E_z$ has been added to allow for a comparison with the full three-dimensional energy). The inset shows the difference between the dotted and solid lines as a function of the inverse scattering length $a_z/a^{3\mathrm{D}}$. The scale of the $y$ axis is identical to that of the inset of Fig. 4b.

Figure 4

(a) Relative three-body energies $E_{3\mathrm{b}}/E_z$ as a function of the inverse scattering length $a_z/a^{3\mathrm{D}}$ for a cigar-shaped trap with aspect ratio $\eta =10$, $M=0$, and ${\Pi }_z=+1$. (b) The solid curve from panel (a) is replotted and compared with the energy obtained by solving the strictly one-dimensional eigenequation with renormalized one-dimensional scattering length $a_{\mathrm{ren}}^{1\mathrm{D}}$ (the energy of $2E_{\rho }=2\eta E_z$ has been added to allow for a comparison with the full three-dimensional energy). For comparison, the dashed line shows one of the three-dimensional energy curves for $M=0$ and ${\Pi }_z=-1$ [not shown in panel (a)]; the corresponding strictly one-dimensional energy is shown by a dotted line. The difference between the full three-dimensional and strictly one-dimensional descriptions is hardly visible on the scale shown. Solid and dashed lines in the inset show the differences between the strictly one-dimensional energies [dotted lines in panel (b)] and the full three-dimensional energies [solid and dashed lines in panel (b)] as a function of the inverse scattering length $a_z/a^{3\mathrm{D}}$ for ${\Pi }_z=+1$ and ${\Pi }_z=-1$, respectively.

Figure 5

Relative three-body energies $E_{3\mathrm{b}}/E_{\rho }$ as a function of the inverse scattering length $a_{\rho }/a^{3\mathrm{D}}$ for a pancake-shaped trap with aspect ratio $\eta =1/2$ and (a) $M=0$ and ${\Pi }_z=+1$, and (b) $M=\pm 1$ and ${\Pi }_z=+1$.

Figure 6

“Crossover curve” of the three-body system with ${\Pi }_z=+1$, shifted by $E_z=E_{\rho }/\eta$, for various aspect ratios of the trap, $\eta =1/2$ (solid line), $1/4$ (dotted line), $1/6$ (dashed line), $1/8$ (dash-dot-dotted line), and $1/10$ (dash-dotted line), as a function of $a_{\rho }/a^{3\mathrm{D}}$. The scattering lengths at which the $M$ quantum number of the corresponding eigenstate changes from $M=\pm 1$ (“left side of the graph”) to $M=0$ (“right side of the graph”) are marked by asterisks. The inset shows the (unshifted) crossover curve as a function of $a_z/a^{3\mathrm{D}}$.

Figure 7

(a) Relative two-body energies $E_{2\mathrm{b}}/E_{\rho }$ as a function of the inverse scattering length $a_{\rho }/a^{3\mathrm{D}}$ for a pancake-shaped trap with aspect ratio $\eta =1/10$, $M=0$, and ${\Pi }_z=+1$. (b) The solid curve from panel (a) is replotted and compared with the energy obtained by solving the strictly two-dimensional eigenequation with renormalized two-dimensional scattering length $a_{\mathrm{ren}}^{2\mathrm{D}}$ (dotted line; the energy $E_z/2$ has been added to allow for a comparison with the full three-dimensional energy). The inset shows the difference between the dotted and solid lines as a function of the inverse scattering length $a_{\rho }/a^{3\mathrm{D}}$. The scale of the $y$ axis is identical to that of the inset of Fig. 8b.

Figure 8

(a) Relative three-body energies $E_{3\mathrm{b}}/E_{\rho }$ as a function of the inverse scattering length $a_{\rho }/a^{3\mathrm{D}}$ for a pancake-shaped trap with aspect ratio $\eta =1/10$, $M=0$, and ${\Pi }_z=+1$. (b) The solid curve from panel (a) is replotted and compared with the energy obtained by solving the strictly two-dimensional eigenequation with renormalized two-dimensional scattering length $a_{\mathrm{ren}}^{2\mathrm{D}}$ (dotted line; the energy of $E_z$ has been added to allow for a comparison with the full three-dimensional energy). The inset shows the difference between the dotted and solid lines as a function of the inverse scattering length $a_{\rho }/a^{3\mathrm{D}}$.

Figure 9

(a) Third-order virial coefficient $b_3$ at unitarity as a function of the inverse temperature ${\tilde{\omega }}_z$ for $\eta =1$ (solid line), 2 (dashed line), and 3 (dash-dotted line). (b) Third-order virial coefficient $b_3$ at unitarity as a function of the inverse temperature ${\tilde{\omega }}_{\rho }$ for $\eta =1$ (solid line), $1/2$ (dashed line), and $1/3$ (dash-dotted line). For $\eta \ne 1$, $b_3$ terminates at the inverse temperature of about 0.25 since our calculations include a finite number of three-body energies; obtaining the behavior of $b_3$ in the high-temperature limit requires the inclusion of infinitely many three-body energies. For $\eta \ne 1$, dotted lines show $b_3$ for $\eta =1$, calculated using the average frequency of the respective anisotropic system. This approximate description is quite good. The insets of panels (a) and (b) show the same data as the main figure, but now as a function of ${\tilde{\omega }}_{\mathrm{ave}}$ as opposed to ${\tilde{\omega }}_z$ and ${\tilde{\omega }}_{\rho }$. The insets show that the third-order virial coefficients for different $\eta$ collapse to a universal curve for all $\eta$ (deviations arise in the low-temperature regime, i.e., for ${\tilde{\omega }}_{\mathrm{ave}}\gtrsim 1$).

Figure 10

Second-order virial coefficient $b_2$ for the two-body system under isotropic confinement as a function of the inverse scattering length $a_{\mathrm{ave}}/a^{3\mathrm{D}}$ ($a^{3\mathrm{D}}$ negative) and the inverse temperature ${\tilde{\omega }}_{\mathrm{ave}}$. The smallest ${\tilde{\omega }}_{\mathrm{ave}}$ considered is $0.0003$. In the ${\tilde{\omega }}_{\mathrm{ave}}\to 0$ limit, $b_2$ approaches $1/2$ for all $a_{\mathrm{ave}}/a^{3\mathrm{D}}$ ($a^{3\mathrm{D}}<0$; see text for further discussion).

Figure 11

(a) Relative two-body energies $E_{2\mathrm{b}}/E_{\mathrm{ave}}$, shifted by $2(n-1)$, for isotropic confinement ($\eta =1$ and $s$-wave channel) as a function of the inverse scattering length $a_{\mathrm{ave}}/a^{3\mathrm{D}}$. (b) Relative two-body energies $E_{2\mathrm{b}}/E_{\rho }$, shifted by $2(n-1)$, for pancake-shaped confinement ($\eta =1/10,M=0$, and ${\Pi }_z=+1$) as a function of the inverse scattering length $a_{\rho }/a^{3\mathrm{D}}$. Solid, dotted, and dashed lines show the energies for $n=1,100$, and $10000$, respectively.

Figure 12

Second-order virial coefficient $b_2$ of the trapped two-body system as a function of the inverse temperature ${\tilde{\omega }}_{\mathrm{ave}}$ for three different values of the inverse scattering length ($a_{\mathrm{ave}}/a^{3\mathrm{D}}=-1$, 0, and $+1$; see labels) for $\eta =1$ (solid lines), $\eta =1/5$ (dashed lines), and $\eta =1/100$ (dotted lines).

Figure 13

Convergence study. (a) The dots show the coefficient $C_{n_z,n_z^{\prime }}^k$, Eq. (A7), for $n_z=2$ and $n_z^{\prime }=80$ as a function of $k$. (b) The circles and triangles show the sum $I_{n_z,n_z^{\prime }}^{\mathrm{c}}({\epsilon }_{{\lambda }^{\prime }},j)$, Eq. (A6), for $({\epsilon }_{{\lambda }^{\prime }}-2\eta j)/2=25.745$ and $40.234$, respectively, as a function of the cutoff $k_{\max }$. As in panel (a), we used $n_z=2$ and $n_z^{\prime }=80$.