Electron Pairing Induced by Repulsive Interactions in Tunable One-Dimensional Platforms

The setup we suggest is comprised of three important ingredients: (i) A spinful interacting semiconducting quantum wire, or any other 1D system with two electron species; (ii) A secondary one-dimensional system which is either gapless, or has an experimentally controllable gap; (iii) Interactions between electrons in the primary wire and the (quasi-) particles of the secondary part.

Overall, the setup we propose is described by a Hamiltonian of the form

The primary wire degrees of freedom are described by

$\mathcal{H}^{0}_{\rm pri}$, the non-interacting part, is given by the continuum description

where $\psi_{R/L\sigma}=\psi_{R/L\sigma}\left(x\right)$ annihilates a right/left-moving electron with spin $\sigma$ at position $x$. The parameters, $v_{F},\mu$ are the Fermi velocity and chemical potential, respectively. Interactions can be accounted for in the general form

with greek indices generalizing chirality and spin, and repeated indices are summed over. The elements of the interaction matrix $g^{\alpha\beta}_{\gamma\delta}$ determines the nature of interactions in the system, and we assumed that the system is away from half-filling so we do not include umpklapp processes. Notice that for simplicity we have also implicitly assumed the existence of short-range interactions only, this will not have an important effect on our results.

The Hamiltonian term $\mathcal{H}_{\rm sec}$ describes the physics of the secondary system. In this work, we consider a 1D system hosting strongly interacting spinless fermions, with chiral annihilation operators $c_{R/L}\left(x\right)$ at position $x$,

where $v_{\rm sec}$, $\mu_{\rm sec}$, $m_{\rm sec}$ are the secondary Fermi velocity, chemical potential, and mass, respectively. We have explicitly allowed here for finite-range interactions, (so that the corresponding Luttinger liquid parameter that we introduce later can be smaller than 1/2) , with the charge density in the secondary wire $\rho_{\rm sec}\equiv c_{R}^{\dagger}c_{R}+c_{L}^{\dagger}c_{L}$, and the interaction strength determined by the function $U$. An experimentally accessible platform to implement such a strongly-correlated effectively spinless 1D system is a carbon nanotube (CNT), tuned by gates and magnetic flux. Details for such an implementation are given in Appendix A.

We now include the most crucial part in our model, capacitive coupling between the primary electrons and the secondary fermions,

with $V>0$ the interaction strength, and $\rho_{\sigma}\equiv\psi_{R\sigma}^{\dagger}\psi_{R\sigma}+\psi_{L\sigma}^{\dagger}\psi_{L\sigma}$. Again, we assume short-range interactions, which we find a posteriori to be sufficient in order to capture the most important consequences of this term and the way it determines the phases of the system.

The model Eq. (1) is best treated in the framework of abelian bosonization [38, 39, 40]. This is achieved by expressing the chiral fermionic operators in terms of bosonic variables,

with $r=\pm$ corresponding to $R/L$, $\alpha$ is the short-distance cutoff of our continuum model, $k_{F},k$ are the Fermi momenta of the electrons (with spin up and down) in the primary system and the fermions in the secondary system, respectively, $\eta,\tilde{\eta}$ are Klein factors ensuring fermionic commutation relations, and the bosonic fields obey the algebra

with the indices $i,j$ specifying one of the two primary $\uparrow/\downarrow$ sectors or the secondary sector. In terms of bosons, we may write $\mathcal{H}_{\rm{pri}}=\mathcal{H}_{c}+\mathcal{H}_{s}$, accounting for the charge and spin part ($c,s$) of the Hamiltonian,

where the charge and spin sectors were defined by $\phi_{c/s}=\frac{\phi_{\uparrow}\pm\phi_{\downarrow}}{\sqrt{2}}$ (and similarly for $\theta_{c/s}$). Eqs. (9)–(10) feature the famous spin-charge separation. For repulsive electron-electron interactions one usually finds ¹¹1often it is assumed $K_{s}=1$, $g_{s}=0$, which is the fixed point of the RG flow for a system with repulsive interactions and SU(2) symmetry [40]. $K_{c}<1$, $K_{s}>1$, and $g_{s}>0$. For the secondary sector we have

where $u,K$ are determined by $v_{\rm sec}$ and $U$. In this work we will mostly assume strong (or sufficiently long-range) repulsive $U$, such that $K$ may be much smaller than 1. If $m_{\rm sec}\neq 0$, changing $\mu_{\rm sec}$ (and as a consequence $k$) leads to a commensurate-incommensurate transition, where the gap induced by the mass term is opened or closed.

Finally, the potentially relevant parts of (6) may be written in the general form

with $k_{\pm}=k_{F}\pm k$. The first line of Eq. (12) is the so-called forward scattering interactions, whereas the last two terms involve the momentum transfer of $2k_{F}$ and $2k$. We note that for the kind of interaction shown in Eq. (6), which is of a pure density-density type, one finds $V_{\phi}=\sqrt{2}V$, $V_{\theta}=0$, $g_{1}=g_{2}=\frac{V}{2}$.

We proceed with a short qualitative discussion of our model, Eqs. (1),(9)–(12) and the possible phases that may be obtained in limiting cases.

If $k_{F}$ and $k$ are grossly mismatched, such that $k_{\pm}$ times a typical correlation length are much larger than 1, or if the secondary sector has a gap, the $g_{1},g_{2}$ terms are irrelevant as they oscillate rapidly in space. Then, the spin sector decouples from the other two, and for generic repulsive interactions it will also be gapless ($g_{s}$ is marginally irrelevant). The remaining bilinear interaction terms between the charge of the primary wire and secondary degrees of freedom cannot open any gaps, yet they cause some mixing of these sectors, changing the relevant Luttinger liquid (LL) susceptibilities power-laws (see, e.g., Ref. [42]). We refer to this phase as the decoupled LL phase.

What happens when either $g_{1}$ or $g_{2}$ are relevant? From a semi-classical point of view, one sees that since $g_{s}>0$, the $g_{1,2}$ term competes with the $g_{s}$ one. The minima of $\cos\left(\sqrt{8}\phi_{s}\right)$ at $\sqrt{8}\phi_{s}=\pi(2n+1)$, with $n$ an integer, lead to $\cos\left(\sqrt{2}\phi_{s}\right)=0$. Thus, there is no simple $\phi_{s}$ configuration which minimizes the classical energy of both such terms. As will be shown, this is a signature of the competition between the pairing tendency in the primary system and the intrinsic electron-electron repulsion within it. In the limit $g_{1/2}\to\infty$ the terms in the corresponding cosines are pinned. The spin sector is gapped, as well as an additional sector combining the charge and secondary degrees of freedom. More concretely, if it is $g_{1}$ that is at the strong coupling fixed point, the only remaining gapless sector is $\propto\phi_{\uparrow}+\phi_{\downarrow}-\phi$. We refer to this phase as the “trion” phase, since the gapless sector may be interpreted as a Luttinger liquid of composite particles comprised of a pair of electrons bound to a secondary hole. This is to be contrasted with bound electrons composite particles induced by attraction in other 1D platforms [43].

In our analysis below we will establish an additional intermediate phase, where electrons with opposite spins indeed form bound pairs, yet $g_{1,2}$ are irrelevant at low enough energy scales, leaving the secondary sector essentially unaltered. The spin-gapped “paired” phase that develops in the primary wire is reminiscent of the Luther-Emery liquid [44], which is generic in spinful systems with attractive interactions, e.g., in the negative-$U$ 1D Hubbard model, however, here it is established for repulsive interaction only.

It turns out to be useful to introduce a unitary transformation to our Hamiltonian $H=\mathcal{H}_{\rm pri}+\mathcal{H}_{\rm sec}+\mathcal{H}_{\rm int}$ in their bosonized form. To do so consider

where $Q$ parameterizes the transformation. Applying this transformation to the Hamiltonian $\tilde{U}_{Q}^{\dagger}H\tilde{U}_{Q}$ has the effect (c.f. Ref. [45])

and modifies the Hamiltonian accordingly. We now specialize our theory to a specific form of the repulsive interactions between the primary and the secondary system and assume

Along this line in the $V_{\phi}-V_{\theta}$ plane, we choose $Q=\frac{V_{\phi}K_{c}}{\pi u_{c}}$, allowing us to completely eliminate the forward scattering part of $\mathcal{H}_{\rm int}$. This will simplify our analysis significantly, and enable a non-perturbative treatment of $V_{\phi}$. Generic repulsive interactions that somewhat deviate from Eq. (13) will influence the phase diagram only quantitatively and not qualitatively. Defining the dimensionless interaction parameter $\upsilon\equiv\frac{K_{c}K}{\pi\sqrt{u_{c}u}}V_{\phi}$, we can sum up the changes to the Hamiltonian as

where in the last cosine terms the $k_{\pm}$ dependence is implicit.

We analyze our model using perturbative renormalization group (RG), up to second order in all couplings. Before deriving the full RG equations, we should address the issue of the oscillating cosines which appear throughout the theory. A fair approximation is to treat the momenta $k_{\pm}$ (and $k$ for the mass term, if it exists) as the inverse of the length scale at which the corresponding cosine is cut off, and the overall system “realizes” it. This is supported by the results of the rigorous RG equations describing the commensurate-incommensurate transition [46]. If we parameterize the short-distance cutoff as $\alpha=\alpha_{0}e^{\ell}$, where in each RG step $\ell$ increases incrementally, one may approximate the threshold at which the $g_{1,2}$ cosines are cut-off as

Generically, $\alpha_{0}\sim{\cal O}\left(k_{F}^{-1}\right)$, and one of the cosine terms, depending on the sign of $k$, will have a small $\ell^{*}$, influencing the RG flow in only a negligible way. The important consequences of our theoretical model can be well understood even for $m_{\rm sec}=0$, and we leave the discussion on the role of the mass term in the RG flow to Appendix B.

Finally, we derive the RG equations using operator product expansion (OPE) [47]. We define the dimensionless coupling constants $y_{s}=\frac{g_{s}}{\pi u_{s}}$, $y_{1,2}=\frac{g_{1,2}}{\pi\bar{u}}$, with $\bar{u}=\frac{u_{c}+u}{2}$. In the following we also neglect the flow due to velocity differences of the different sectors, which only impact the flow in higher orders. Our equations are thus

The scale dependent factors $c_{1}$ and $c_{2}$ impose a smooth cutoff of scale $\chi$ for the incommensurate cosines, where we use $c_{1,2}\left(\ell\right)=\frac{1}{2}\left[1-\tanh\left(\frac{\ell-\ell_{1,2}^{*}}{\chi}\right)\right]$. We note that we do not include in the RG flow generated bilinear cross terms such as $\partial_{x}\phi\partial_{x}\phi_{c}$, as feeding these back into the RG equations results in higher (third) order corrections for $y_{1,2}$. We stress once more that such modifications will not lead to a qualitative alteration of the phase diagram.

We proceed to make some observations regarding the derived RG equations. It is apparent from these equations that the forward-scattering primary-secondary interactions, embodied entirely by $Q$, “favor” $y_{1}$ over $y_{2}$, or vice versa. An intuitive understanding of this may be obtained by considering the secondary sector as comprised of either electrons with $k>0$, or holes with effectively $k<0$ ²²2Strictly speaking, the Fermi momenta values of electrons and holes with the same densities are identical, but the Fermi velocities are opposite. This implies that electron states with positive velocities (right moving electrons) cross the Fermi energy at $+\left|k_{F}\right|$, while hole states with positive velocities (right moving holes) cross the Fermi energy at $-\left|k_{F}\right|$. We use this convention throughout the manuscript [see Eq. (7)], see Fig. 2. The fugacity $y_{1}$ for example, describes the interaction between a secondary hole and an electron pair. It is thus clear why repulsive electron-electron interaction ($Q>0$) would favor it and decrease its scaling dimension (and why they would have the opposite effect on $y_{2}$). Without loss of generality, we shall henceforth assume the secondary part was tuned such that $k$ is negative and in the vicinity of $-k_{F}$, so we neglect $y_{2}$ and denote $y=y_{1}$, $\ell^{*}=\ell^{*}_{1}$. The analysis for a setup with positive $k$ (”particle-like”) can thus be extracted from the ”hole-like” case by simply taking $Q\to-Q$.

We are now in the position to examine which of the three possible phases is established as a function of experimentally controllable parameters, namely the incommensurability cutoff $\ell^{*}$ (controlled by, e.g., a gate voltage applied to the secondary wire), and the interwire bare interaction with high momentum transfer $y^{0}$ (typically tuned by the distance between the primary and secondary parts). This is explored in Fig. 4, where the different phase boundaries are found. We observe that moving towards the commensurate point (increasing $\ell^{*}$) tends to favor the “locked” trion phase over the paired one. The role of the Luttinger parameter $K$ is also apparent: as it gets smaller (reflecting stronger interaction in the secondary system), the region in phase space where one can easily tune between the non-trivial phases is greatly enhanced.

The presented RG analysis allows us to deduce a schematic finite temperature phase diagram, see Fig. 5. As a function of a single parameter, e.g., the incommensurability cutoff $\ell^{*}$, we calculate the RG scale at which $y_{s}$ reaches strong coupling $\bar{\ell}$, as well as the scale corresponding to the pinning of the $y$ term, $\bar{\bar{\ell}}$. We estimate the gaps associated to these coupling constants as $\propto\Lambda e^{-\bar{\ell}},\Lambda e^{-\bar{\bar{\ell}}}$, allowing us to determine an appropriate energy scale for the transition. It can be seen that for a certain range of the detuning parameter (depending on interaction strength, etc.), as we lower the temperature the system may experience a crossover from the Luttinger liquid phase, through the paired Luther-Emery liquid, and finally to a trion phase where the primary pairs are bound to secondary fermions.

We now turn to discuss the properties of the system in its three possible phases.

Inducing attraction between electrons mediated by extrinsic repulsive interactions of these electrons and an auxiliary system was originally proposed by Little [8]. Little considered a one-dimensional system coupled by Coulomb interactions with strength $V$ to local polarizable side-chains along the system (that was termed “spine”). Excitations of these so-called polarizers, which cost a finite energy $\tilde{E}$, mediate attractive interactions between system electrons, proportional to $\sim\frac{V^{2}}{\tilde{E}}$. This attraction then competes with the intrinsic electron-electron repulsion in the chain hosting the electrons.

The model Little put forward was studied more carefully by Hirsch and Scalapino (HS) [19, 53]. HS studied this model by employing perturbation theory in the weak and strong electron-polarizer coupling limits, as well as using numerical methods, namely small-scale quantum Monte-Carlo. HS found certain regimes of parameters where superconductivity occurs in such a system, yet concluded that $2k_{F}$ CDW of pairs generally has larger susceptibility, even when pairing occurs. This is reminiscent of the conclusions we draw in Sec. II.3 for the setup proposed here, which similarly has a SC order or CDW of pairs in the spin gapped paired phase.

At first glance, our model might seem to have very little to do with Little’s model. The pairing mechanism we discuss in this work relies on interaction with secondary fermions, which are either in a Luttinger liquid phase or gapped (such that the gap does not overwhelm the interactions with the electronic system). In fact, Little’s model presented in Ref. [8] can be shown to coincide with a certain limit of the model we presented here. It corresponds to extremely localized fermions, i.e., $v_{\rm sec}\rightarrow 0$ in our description, which are gapped, with the energy required to polarize/excite one of the side-chains polarizers being the gap. A solitonic excitation in our version of secondary fermions Hamiltonian, Eq. (11), is the analog of an excited polarizer (in a very non-itinerant limit).

To better illustrate the connection between Little’s proposal and some particular sector of our model, we consider one possible implementation of a secondary system that may be employed: a two-leg Hubbard ladder exactly at half-filling, illustrated in Fig. 6. For our purposes, it embodies a minimal representation of a strongly correlated Mott insulator. Strong interactions pin the system into a state of one electron per site, with a charge gap proportional to the amplitude of the on-site interactions. We assume that the Hubbard ladder is in its arguably most generic phase, the so-called “D-Mott” phase, or the rung singlet, appropriate when the on-site interactions are large compared to the nearest-neighbor repulsion and the exchange energy [54]. In this phase, all charge and spin sectors of the system are gapped.

We now consider the “relative-charge” sector of the ladder, i.e., the one that measures the total difference in charge between the different legs, $\phi_{\rho-}\propto\phi_{\rho,1}-\phi_{\rho,2}$, where $\phi_{\rho,i}$ represents the charge sector of the $i$’th leg, and argue that it is the most important sector in terms of its effects on the nearby electronic system we wish to induce pairing in. The interaction-induced gap in this sector is considerably smaller compared to that of the total charge sector, since it has a smaller effective interaction parameter ($K_{\rho-}$ is much closer to 1, whereas $K_{\rho+}$ can be rather small, as it accounts for the total charge sector).

Assuming the dominant interaction of the primary electronic system is with the leg closest to the system (the leg with $i=1$), the interactions between the primary spinful wire (described by the sectors $\phi_{c}$ and $\phi_{s}$) and the Hubbard ladder secondary system are accounted for by

with $k_{F}$ and $k_{1}$ corresponding to the Fermi momenta of the primary wire and closest Hubbard leg, respectively. We have also absorbed a factor of $\left\langle\cos\left(\sqrt{2}\phi_{\sigma,1}\right)\right\rangle$ into the definition of the interaction coupling constants $\tilde{g}_{1,2}$, with $\phi_{\sigma,1}$ corresponding to the spin sector of the leg interacting with the electron wire. Assuming that the total charge sector is gapped, such that $\phi_{\rho+}$ is strongly pinned to its semi-classical strong-coupling value, we arrive at an expression similar to Eq. (12). One important difference is that the $\phi_{\rho-}$ appears inside the cosine with a smaller prefactor, for example in $~{}\tilde{g}_{1}\cos\left(\sqrt{2}\phi_{c}+\phi_{\rho,1}\right)$ (1 instead of $\sqrt{2}$). This in turn makes the non-linear perturbations in Eq. (22) significantly more relevant as compared to those in Eq. (12). (This is not surprising, as we have assumed that some of the degrees of freedom in the secondary ladder are already frozen by the interactions in the Mott insulator.)

A chemical potential difference between the two legs of the ladder can be generated, for example with back gate potential, leading to a term $\sim\mu^{*}\partial_{x}\phi_{\rho-}$. This will allow one to experimentally control $\ell^{*}$ (as it controls the commensurability condition) as well as the mass gap in such a setup, and tune it towards the desired phase for the overall system.

Let us briefly comment on the nature of the $\phi_{\rho-}$ solitons. As mentioned, in the D-Mott phase the density is one electron per site, with the spins forming a rung-singlet. A relative-charge particle (hole) would correspond to an electron being displaced from leg 2 to leg 1 (or vice-versa), without perturbing the spin-singlet nature of the electrons on the rung. Notice that such an excitation costs a finite amount of energy, approximately proportional to the difference between the on-site repulsion and inter-leg repulsion energies in the strong coupling limit. The analogy to Little’s model now becomes a bit more transparent: the ground state and excited state of the polarizers correspond to a rung with one electron per site and a $\phi_{\rho-}$ hole, respectively. A key difference in our proposed setup is that we can account for interactions between these secondary fermions, and that we allow for their excitations to delocalize. These two effects, as can be inferred from our RG analysis and results, can be quite important in determining and manipulating the phase diagram.

On top of allowing us to make a concrete connection to Little’s model, this specific realization of the secondary wire has additional significance. The discussion above suggests that under certain conditions, proximity of a spinful wire to a Mott insulator may result in pairing of electrons in that 1D system. Realistically, this Mott insulator does not necessarily have to be one-dimensional: If its gaps are sufficiently larger than the interaction with the primary wire, the physics may be well-described by that of the insulator sites nearest to (and strongest interacting with) the electronic wire. Thus, our toy-model may be an appropriate effective description of a simple experimentally accessible setup, namely placing a nanowire on top of a strongly correlated Mott insulator.

The model we have presented here concerns the pairing one may induce in a spinful one-dimensional system, even when it has non-negligible intrinsic repulsive interactions. Due to the Mermin-Wagner theorem, “true” superconductivity cannot occur in a 1D system. Moreover, we have discussed superconducting tendencies brought on by our setup, and shown that they were generically expected to be sub-dominant to CDW order developing in the wire (whether for spin-singlet pairs or the so-called trions). We would therefore like to consider the extension of the setup presented here to a higher dimension.

Let us consider an array of spinful quantum wires, as depicted in Fig. 7, each with its own secondary system, all decoupled from one another. This is a very anisotropic limit of a 2D system, ubiquitous in many coupled wires studies (c.f. Ref. [55]). One can then tune each such individual primary-secondary duo to the paired phase by changing the relevant controllable parameters, and allow weak hopping between neighboring primary wires, such that the tunneling amplitude $t_{\perp}$ is smaller than the induced spin gap $\Delta_{s}$. [We neglect the tunneling between adjacent secondary wires, which is justified if: (i) Strong interactions within them makes such tunneling irrelevant, (ii) each secondary system is individually tuned such that there is a Fermi momentum mismatch between neighboring secondary wires, or (iii) the secondary wire has a single particle gap rendering such hopping non-important (such a case is discussed in Sec. III.1).]

For the sake of completeness, the remainder of this Section mostly recapitulates results discussed in Refs. [36, 56, 45], in the terminology used in Sec. II. Namely, we assume the spin-gapped paired phase in the primary wires (induced solely by repulsion) and discuss the conditions under which it leads to long-range phase coherent superconductivity.

Since $t_{\perp}\ll\Delta_{s}$ we can restrict our discussion to higher order processes of ${\cal O}(t_{\perp}^{2}/\Delta_{s})$ between adjacent (primary) electron wires. The key points in this discussion will be best illustrated by considering just two spinful spin-gapped wires, with labels 1,2. We note that the results we outline below are somewhat different than those discussed in Ref. [57] for a similar setup, as large-momentum backscattering events (of the kind that lead to the formation of the pairing spin gap) were not considered in [57]. Defining the sectors $\phi_{\alpha\pm}=\frac{\phi_{\alpha 1}\pm\phi_{\alpha 2}}{\sqrt{2}}$, with $\alpha=c,s,$ the intra-wire charge and spin sectors, we write the Hamiltonian density of the two-particle hopping processes as

which are the Josephson and particle-hole couplings, respectively. The intra-wire spin-gaps render ${\cal H}_{p-h}$ irrelevant, as it contains a cosine of variables dual to the gapped ones, whereas ${\cal H}_{J}$ is relevant in the RG sense if the Luttinger parameter corresponding to the relative charge sector is not too small, i.e., $K_{c-}>\frac{1}{2}$. (This is because in the spin-gapped regime we replace cosines of the individual $\phi_{1/2s}$ by their expectation values.) Interestingly, this condition does not exclude very strongly interacting systems, as long as electron-electron interactions between wires is not too small compared with the intra-wire one. In other words, inter-wire repulsion actually makes ${\cal H}_{J}$ more relevant [45]. Presumably, ${\cal H}_{J}$ can now flow to strong coupling, with the $\theta_{c-}$ term inducing inter-wire phase coherence throughout the system, causing a Kosterlitz-Thouless transition into the superconducting regime, similarly to the studied phenomena in Refs. [36, 56].

In fact, it is well established that in considering such setups one should include an additional term, proportional to the inter-wire interaction $U_{\perp}$,

Comparing it to ${\cal H}_{J}$, a clear competition between pinning of the dual variables $\phi_{c-}$ and $\theta_{c-}$ is evident. The $\pi$-CDW interaction triumphs over the Josephson pair hopping, i.e., more relevant in an RG sense, if $K_{c-}<1$, facilitating a $\pi$-CDW order in the higher dimensional system [36]. This term is typically neglected when considering, e.g., stripes in high-$T_{c}$ superconductors, since it is assumed that each stripe can effectively have different carrier densities compared to its neighbors. This in turn causes spatial oscillation in the relative charge cosine term,

with $\delta$ being proportional to the density difference between neighboring primary wires. On length scales much larger than $\delta^{-1}$, the $\pi$-CDW is thus rendered irrelevant.

In the absence of this kind of dephasing, superconductivity would prevail only if $U_{\perp}$ is sufficiently small as compared to $\frac{t_{\perp}^{2}}{\Delta_{s}}$, or if $K_{c-}$ is large enough. The latter requires that the long-wavelength part of the inter-wire repulsive interaction be rather large. Conversely, engineering a fully tunable quasi-1D system, each having a separate spin-gap, the CDW tendency may be greatly diminished (dephased) by modulating the electronic densities between adjacent wires, effectively sending ${\cal H}_{\pi{\rm CDW}}\rightarrow 0$ at low enough energy scales.

We note here that the discussion above, which centered around coupling of two spinfull wires in the paired (spin-gap) phase, alludes to the fact that introducing further complexity to the primary system may lead to dominant superconducting tendencies even in 1D setups. As an example, one may consider CNTs, which host two spinfull sectors (one in each “valley”) [58, 59, 59]. Manipulating the properties of such CNTs, one may induce pairing and superconducting tendencies by proximitizing them to a proper secondary system.

To conclude this section, the role of the scheme proposed in this work and explored in Sec. II is essentially providing the intra-wire pairing (spin-gap) mechanism, which in turn, under the right conditions, can be used to construct the anisotropic superconducting phase. Conventionally, this mechanism is provided by effective attractive interactions due to interactions with phonons or other degrees of freedom, and we present here a feasibly controllable way of inducing this gap. Then, manipulating additional experimental parameters, e.g., different gate-voltages, one can achieve long range coherent superconductivity in a two-dimensional system driven solely by repulsive electron-electron interactions.