A large binding energy implies that the dimer cannot be broken in scattering processes and can be treated as a single degree of freedom. For systems with a finite density $n$, this binding energy should be compared with typical kinetic energies that are given by ${\hslash }^2{n}^2/m$. Our effective theory is applicable when $n/|\tilde{g}|\ll 1$. As will become clear in the following, the difference between exact energies and effective theory results is less than one percent when $n/|\tilde{g}|<0.04$.

We define $r_{\mathrm{D}}$ as the length scale that contains most of the probability to find two fermions, forming a bound dimer, close to each other, i.e., ${\int }_{-r_{\mathrm{D}}}^{r_{\mathrm{D}}}{\mathrm{\Phi }}_{\mathrm{D}}(z{)}^2\mathrm{d}z=1-e^{-1}\approx 0.63$, where ${\mathrm{\Phi }}_{\mathrm{D}}\propto \exp \{-|\tilde{g}z|/2\}$ is the wave function of the dimer.

More precisely, a strongly attractive regime in our Letter implies that the size of the dimer, and, hence, the effective range is unresolved in scattering, i.e., $k{r}_{\mathrm{D}}\ll 1$ for all relevant values of the momentum $k$.

Other possibilities appear impossible because identical fermions cannot form bound states with more than two particles. This follows from few-body calculations [15, 16] or distribution of momenta in the GY model [17, 18]. Although these considerations are based on a homogeneous geometry, the conclusion holds also for the external potentials $v_{\mathrm{ext}}$ that change slowly on the length scales given by the size of the dimer state.

In the situation where the dimer consists of fermions 1 and 2, the relative distance reads $x_{\mathrm{rel}}=(x_1+x_2)/2-x_3$. In the case of two dimers made of fermions 1 and 3 and 2 and 4, respectively, the relative distance reads $x_{\mathrm{rel}}=(x_1+x_3)/2-(x_2+x_4)/2$.

The vanishing scattering length $a_{\mathrm{o}}^{\mathrm{FD}}$ implies the existence of a zero-energy virtual state [37], which in the fermion-dimer system corresponds to a trimer state of negative parity at the threshold for binding [15, 16].

The use of finite-range potentials is, in general, necessary to satisfy the Wigner lower limit [41] for one-dimensional scattering phase shifts. In detail, the Wigner bound implies that there can be no causality-preserving zero-range interaction for a nonvanishing effective range $r_{\mathrm{o}}\ne 0$ [34].

As this boundary condition corresponds to a zero-range interaction, it must violate causality for $1/\tilde{g}\ne 0$ as the Wigner lower limit indicates for $r_{\mathrm{o}}\ne 0$. This manifests itself as non-Hermiticity of the resulting problem, which can be demonstrated by nonorthogonality of the odd solutions of a two-body problem. As we illustrate in this Letter, this does not preclude us from estimating the energies in the first order of $1/\tilde{g}$ [22].

We remark here that the boundary conditions introduced in Eq. (8) do not correspond to the odd-channel interaction ${\delta }^{\prime }$ introduced in Ref. [38]. Indeed, the potential ${\delta }^{\prime }$ connects the derivative of the wave function to the discontinuity of the wave function itself, whereas Eq. (8) connects the derivative of the wave function to the discontinuity of the second derivative of the wave function. It is worth noting, however, that in the limit $1/g=0$ both Eq. (8) and ${\delta }^{\prime }$ demand that the derivative of the wave function vanishes, which leads to the important conclusion that strongly interacting odd channel can be mapped onto a weakly interacting even channel [44].

This numerical estimate is in agreement with the construction of our model, exact in the order $1/\gamma$. A detailed investigation of the beyond-$1/\gamma$ physics is outside the scope of the present Letter. We remark, however, that in some cases the effective model is accurate even at the level $1/{\gamma }^2$. One example is a spin-balanced limit of the GY model. Indeed, the approximate energy of the Lieb-Liniger gas of dimers [46] $(E_0/N{E}_{\mathrm{F}})\simeq (1/12)\{1-({\hslash }^{2}N/mgL)+[3({\hslash }^{2}N{)}^2/4(mgL{)}^2]\}$ is in agreement with the direct solution of the GY model [6]. Another example is actually the present fermion-dimer system; see S.4 of [22].

Contrast also the uniqueness of the ground state for $1/{\gamma }^{\mathrm{HO}}\to 0^{-}$ with triple degeneracy of the ground state in the limit $1/{\gamma }^{\mathrm{HO}}\to 0^{+}$ [49, 50, 51]. This illustrates the fact that one cannot consider orderings of particles (e.g., $x_1<x_2<x_3$) as independent for strongly attractive systems.

This result can be benchmarked against a Bethe ansatz solution for a system in a box trap of length $a$. Indeed, the energy of the noninteracting mixture in a box trap is given by $({\hslash }^2/24m)({\pi }^2/a^2)M(M+1)(2M+1)+({\hslash }^2/12m)({\pi }^2/a^2)N_{\downarrow }(N_{\downarrow }+1)(2N_{\downarrow }+1)$ in agreement with Ref. [59].

The conjecture follows from contradiction: Let us assume that it is not possible to map such a system onto a weakly interacting mass-imbalanced model. This implies the existence of a Bethe ansatz solvable mass-imbalanced system, since the underlying GY model is Bethe ansatz solvable with the wave function describing an effective system of constituents (unpaired fermions, dimers, trimers, etc.) in the limit of strong attractive interactions. However, mass-imbalanced systems are not Bethe ansatz solvable; see Refs. [71, 72] illustrating this point using the smallest three-body problem.