Absence of a four-body Efimov effect in the $2+2$ fermionic problem

In the free three-dimensional space, we consider a pair of identical $\uparrow$ fermions of some species or in some internal state and a pair of identical $\downarrow$ fermions of another species or in another state. There is a resonant $s$-wave interaction (that is, of zero range and infinite scattering length) between fermions in different pairs and no interaction within the same pair. We study whether this $2+2$ fermionic system can exhibit (as the $3+1$ fermionic system) a four-body Efimov effect in the absence of three-body Efimov effect, that is, the mass ratio $\alpha$ between $\uparrow$ and $\downarrow$ fermions and its inverse are both smaller than 13.6069…. For this purpose, we investigate scale invariant zero-energy solutions of the four-body Schrödinger equation, that is, positively homogeneous functions of the coordinates of degree $s-7/2$, where $s$ is a generalized Efimov exponent that becomes purely imaginary in the presence of a four-body Efimov effect. Using rotational invariance in momentum space, it is found that the allowed values of $s$ are such that $M\left(s\right)$ has a zero eigenvalue; here the operator $M\left(s\right)$, that depends on the total angular momentum $\ell$, acts on functions of two real variables (the cosine of the angle between two wave vectors and the logarithm of the ratio of their moduli), and we write it explicitly in terms of an integral matrix kernel. We have performed a spectral analysis of $M\left(s\right)$, analytical and for an arbitrary imaginary $s$ for the continuous spectrum and numerical and limited to $s=0$ and $\ell \le 12$ for the discrete spectrum. We conclude that no eigenvalue of $M\left(0\right)$ crosses zero over the mass ratio interval $\alpha \in [1;13.6069\cdots ]$, even if, in the parity sector ${(-1)}^{\ell }$, the continuous spectrum of $M\left(s\right)$ has everywhere a zero lower border. As a consequence, there is no possibility of a four-body Efimov effect for the $2+2$ fermions. We also enunciated a conjecture for the fourth virial coefficient of the unitary spin-$1/2$ Fermi gas, inspired from the known analytical form of the third cluster coefficient and involving the integral over the imaginary $s$ axis of $s$ times the logarithmic derivative of the determinant of $M\left(s\right)$ summed over all angular momenta. The conjectured value is in contradiction with the experimental results.

Figure 1

Positions and parametrizations of the wave vectors appearing in the angular integration in the functionals ${\mathcal{E}}_{24,ij}$. The vectors ${\mathbf{k}}_2$ and ${\mathbf{k}}_4$ are given by Eqs. (30) and (31). (a) For $(i,j)=(2,3), {\mathbf{k}}_1=-({\mathbf{k}}_2+{\mathbf{k}}_3+{\mathbf{k}}_4)$ and one integrates over ${\mathbf{k}}_3$ using spherical coordinates of polar axis $x$ and azimuthal axis $y$. (b) For $(i,j)=(1,4), {\mathbf{k}}_3=-({\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_4)$ and one integrates over ${\mathbf{k}}_1$ using the polar axis $Z$ (direction of ${\mathbf{k}}_4$) and the azimuthal axis $X$ (direction of ${\mathbf{e}}_z$) as defined by Eq. (38), leading to the polar angle ${\theta }_{14}$ and the azimuthal angle $\varphi$ ($<0$ in the figure). The dashed line gives the direction of the component ${\mathbf{e}}_1^{\mathrm{XY}}$ of ${\mathbf{e}}_1={\mathbf{k}}_1/k_1$ in the $XY$ plane. (c) For $(i,j)=(1,3)$, one integrates over the rotation $\mathcal{R}$, such that ${\mathbf{k}}_1$ and ${\mathbf{k}}_3$ are given by the action of $\mathcal{R}$ on vectors ${\mathbf{k}}_1^{\mathrm{fix}}$ and ${\mathbf{k}}_3^{\mathrm{fix}}$ in the $xy$ plane as in Eqs. (43) and (44), using the parametrization in terms of Euler angles associated to the convenient axes $X, Y$, and $Z$ of Eq. (48).

Figure 2

Analytically obtained borders of the continuous spectrum of $M\left(s\right)$ ($s\in i{\mathbb{R}}^{+}$) corresponding to the limits $x\to \pm \infty$. It is a collection of components characterized by an angular-momentum quantum number $L$ of a three-body asymptotic problem (not to be confused with the total angular momentum $\ell$ of the four-body states), leading to a continuous set of eigenvalues ranging from ${\mathrm{\Lambda }}_L(0,\alpha )$ to $cos\left[\nu \right(\alpha \left)\right]$ for $x\to -\infty$ and ranging from ${\mathrm{\Lambda }}_L(0,1/\alpha )$ to $cos\left[\nu \right(1/\alpha \left)\right]$ for $x\to +\infty$, where $\alpha$ is the mass ratio given by Eq. (56), the ${\mathrm{\Lambda }}_L$ function is given by Eq. (78), and the mass angle $\nu$ is given by Eq. (79). We plot $cos\left[\nu \right(\alpha \left)\right]$ (black solid line) and ${\mathrm{\Lambda }}_L(0,\alpha )$ (colored dashed lines, with $L=0,2,4$ from top to bottom above the solid line and $L=1,3$ from bottom to top below the solid line) as functions of $\alpha \in [0,{\alpha }_c(2;1)]$, where ${\alpha }_c(2;1)$ is the critical mass ratio (83) for the Efimov effect in the $\uparrow \uparrow \downarrow$ three-body problem. Due to the $\alpha \leftrightarrow 1/\alpha$ symmetry of the $2+2$ fermionic problem, one can restrict to $\alpha \ge 1$ (that is, to the right of the vertical dotted line); the borders of the $x\to -\infty$ continuum can then be directly read on the figure, and the ones of the $x\to +\infty$ continuum can be obtained by mentally folding back the $\alpha \le 1$ part of the figure into the $\alpha \ge 1$ part.

Figure 3

Eigenvalues of $M(s=0)$ as functions of the mass ratio $\alpha \in [1,13.6]$ for various values of the angular momentum $\ell$ and the parity [left half for the parity ${(-1)}^{\ell }$, right half for the parity ${(-1)}^{\ell +1}]$, obtained numerically after discretization and truncation of the variables $x$ and $\theta$ in the zone $\rho \equiv {[x^2+{(\pi -\theta )}^2]}^{1/2}>{\rho }_0$ and of their log-polar versions $t\equiv ln(\rho /{\rho }_0)$ and $\psi$ in the zone $\rho <{\rho }_0$ (see text): $x_{\max }=-x_{\min }=12, dx=1/5, d\theta \simeq \pi /15, {\rho }_0=2/5, t_{\min }=-12, dt=1/5, d\psi =\pi /15$, with the Gauss-Legendre integration scheme for the integrals over $\theta$ and $\psi$, and the midpoint integration formula for the integrals over $x$ and $t$. For the ${(-1)}^{\ell +1}$ parity channels the $\rho <{\rho }_0$ zone is not useful and is not included in the numerics. The boundaries of the continuous spectrum of $M\left(s\right)$ are given by black thick dashed curves: For the ${(-1)}^{\ell +1}$ parity channels, this corresponds to the $x\to +\pm \infty$ continua of Eqs. (77) and (81) with all $L\ge 1$; for the ${(-1)}^{\ell }$ parity channels, this corresponds to the $x\to +\pm \infty$ continua of Eqs. (77) and (81) with all $L\ge 0$ and to the $(x,\theta )\to (0,\pi )$ (that is, $t\to -\infty$) continuum of Eq. (84). In contrast to the $3+1$ case, no eigenvalue of $M(s=0)$ is found to cross zero for $\alpha <{\alpha }_c(2;1)=13.6069...$: No four-body Efimov effect is found for the $2+2$ fermionic problem.

Figure 4

Histogram of the eigenvalues $\mathrm{\Omega }$ of $M(s=0)$ for a mass ratio $\alpha =10$ and an angular momentum $\ell =1$ in the ${(-1)}^{\ell +1}$ parity channels, obtained numerically after discretization and truncation of the $x$ and $\theta$ variables. The numerical grid is the same as in Fig. 3, except that much larger values of $x_{\max }=-x_{\min }$ are used to reveal the emergence of the continuous part of the spectrum in the $x_{\max }\to +\infty$ limit: $x_{\max }=48$ (red bars in the foreground) and $x_{\max }=96$ (blue bars in the background). The black vertical dashed lines indicate the analytically predicted borders of the continuous spectrum (as in Fig. 3); between the first two ones and between the last two ones, it is indeed observed that the number of eigenvalues per bin is approximately multiplied by 2 when $x_{\max }$ is doubled. On the contrary, the histogram is unaffected by the change of $x_{\max }$ in the bins strictly in between the second and third dashed lines, indicating that the corresponding eigenvalues belong to the discrete spectrum of $M(s=0)$, with localized eigenfunctions in $x$ space.

Figure 5

Numerically determined minimal eigenvalue ${\mathrm{\Omega }}_{\min }$ of $M^{\left(\ell \right)}(s=0)$ almost at the three-body critical mass ratio, $\alpha =13.6069\simeq {\alpha }_c(2;1)$, as a function of the numerical cutoff $t_{\min }=-x_{\max }$. For each given cut-off value, each angular momentum $\ell$ from 0 to 12 and parity sector ${(-1)}^{\ell }$ and ${(-1)}^{\ell +1}$ contributes as a point in the figure: The fact that the points (in red) are superimposed and cannot be distinguished shows that ${\mathrm{\Omega }}_{\min }$ does not depend on $\ell$ nor on the parity. Furthermore, ${\mathrm{\Omega }}_{\min }$ is always positive; it is linear in $1/x_{\max }^2$ and extrapolates to zero for infinite cutoff (see the blue line): This is perfectly consistent with the fact that ${\mathrm{\Omega }}_{\min }$ corresponds to the lower border of the $2+1$ continuum ${\mathrm{\Lambda }}_{L=1}(ik,\alpha )$, where $k$ has a minimal, discrete value scaling as $1/x_{\max }$ in the presence of the numerical cutoff, and ${\mathrm{\Lambda }}_{L=1}(ik,{\alpha }_c(2;1))$ vanishes quadratically at $k=0$. In other words, there is no negative ${\mathrm{\Omega }}_{\min }$ and no four-body Efimov effect for $2+2$ fermions.

Figure 6

Diagram giving the structure of the eigenstates of the third continuum (84), in terms of the variable $t$ of Eq. (91): $k>0$ is the wave vector of the incoming wave and $-k$ that of the reflected wave with a phase shift $\theta (k,S)$. The reflection due to the physics at $t=O\left(1\right)$ is, in a toy model, represented by a Dirac scattering potential at $t=0$ and a hard wall acting as a mirror at $t=L$; the toy model exemplifies the expected low-$k$ behavior (B32) of $\theta (k,S)$.

Figure 7

Convergence in the integration over $S$ and summation over $\ell$ in Eq. (B11) (we recall that the mass ratio is $\alpha =1$). (a) The logarithmic derivative of the determinant of $M^{\left(\ell \right)}\left(iS\right)$ is a rapidly decreasing function of $S$; the figure takes as an example (a1) the $\ell =0$ channel of the $2+2$ fermionic problem (for a numerical cutoff $t_{\min }=-9$, thus without extrapolation; note the minus sign in the vertical axis) and (a2) the $\ell =0$ channel of the $3+1$ fermionic problem. (b) The sum over $\ell$ also seems to converge well, see the contribution of each angular-momentum channel to the result (B11) for (b1) the $2+2$ problem and (b2) the $3+1$ problem: black disks for the parity sector ${(-1)}^{\ell }$ and red disks for the parity sector ${(-1)}^{\ell +1}$. The plus signs indicate the corresponding cumulative sums. Convincing evidence is even shown in (b3) for the $2+2$ problem in the ${(-1)}^{\ell +1}$ parity sector and in (b4) for the $3+1$ problem in both parity sectors, where the numerical results [black disks for parity ${(-1)}^{\ell }$ and red disks for parity ${(-1)}^{\ell +1}]$ are compared to the perturbative results (B54) and (B60) [red asterisks for parity ${(-1)}^{\ell }$ and black asterisks for parity ${(-1)}^{\ell +1}]$ that extend to four bodies a technique developed for three bodies in Ref. [28] and are expected to be exact asymptotic equivalents for $\ell \to +\infty$ [what is actually plotted is the absolute value of the results to allow for a log scale, but their sign is indicated with the label “$<0$” of the same color as the corresponding disks when they are negative: The negative black (red) disks are indicated with a black (red) “$<0$” label. Note that the black (red) asterisks always have the same sign as the corresponding red (black) disks].

Figure 8

Fourth-order cluster coefficient (B67) for a harmonically trapped unpolarized spin-$1/2$ unitary Fermi gas at temperature $T$, as a function of $\beta \hslash \omega$, with $\beta =1/\left(k_{B}T\right), \omega$ the angular oscillation frequency of a fermion in the trap, and $\alpha =m_{\uparrow }/m_{\downarrow }=1$. Blue line with symbols: results of Ref. [30] obtained by brute-force numerical calculation of the up-to-four-body spectra in the trap (disks: actually calculated values; circles: values resulting from an extrapolation). Red lines: our conjecture (B66) (the values slightly differ depending on the linear or cubic extrapolation to the numerical cut-off limit $1/t_{\min }\to 0^{-}$).