Theory of triangular lattice quasi-one-dimensional charge-transfer solids

TL;DR

This paper uses a 1/4-filled band Hubbard model to better explain the magnetic and superconducting properties of quasi-one-dimensional triangular lattice charge-transfer solids, challenging previous models.

Contribution

It introduces a 1/4-filled band Hubbard model focusing on monomer sites, successfully reproducing experimental results and supporting a valence bond theory of superconductivity.

Findings

Spin gap transition linked to two-dimensional paired-electron crystal formation.

Prediction of charge-ordering and period doubling accompanying the spin gap.

Enhanced superconducting pairing correlations with Hubbard repulsion.

Abstract

Recent investigations of the magnetic properties and the discovery of superconductivity in quasi-one-dimensional triangular lattice organic charge-transfer solids have indicated the severe limitations of the effective 1/2-filled band Hubbard model for these and related systems. Our computational studies of these materials within a 1/4-filled band Hubbard model in which the organic monomer molecules, and not their dimers, constitute the sites of the Hamiltonian are able to reproduce the experimental results. We ascribe the spin gap transition in kappa-(BEDT-TTF)_2B(CN)_4 to the formation of a two-dimensional paired-electron crystal and make the testable prediction that the spin gap will be accompanied by charge-ordering and period doubling in two directions. We find enhancement of the long-range component of superconducting pairing correlations by the Hubbard repulsive interaction for band parameters corresponding to kappa-(BEDT-TTF)_2CF_3SO_3. The overall results strongly support a valence bond theory of superconductivity we have proposed recently.

Equations

$$H=\sum _{⟨ij⟩,\sigma }{t}_{ij}(c_{i,\sigma }^{†}{c}_{j,\sigma }+H.c.)+U\sum _i{n}_{i,\uparrow }{n}_{i,\downarrow },$$

$$S_i^z=\frac{1}{2}(n_{i_1,\uparrow }+n_{i_2,\uparrow }-n_{i_1,\downarrow }-n_{i_2,\downarrow }).$$

$$S(\mathbf{Q})=\frac{1}{N_d}\sum _{j,k}{e}^{i\mathbf{Q}\cdot ({\mathbf{r}}_{\mathbf{j}}-{\mathbf{r}}_{\mathbf{k}})}⟨S_j^z{S}_k^z⟩,$$

Full text

I Introduction

Determination of the mechanism of correlated-electron superconductivity (SC) continues to be a formidable challenge, even after three decades of intensive investigations following the discovery of the phenomenon in doped copper oxides. Discoveries of many other exotic superconductors subsequently have only added to its mystique. One opinion, shared by many scientists, is that the mechanisms of correlated-electron SC in apparently different families are related McKenzie (1997); Yanase et al. (2003); Dagotto (2005); Steglich and Wirth (2016); Clay and Mazumdar (2019). Among the many families of materials that have attracted attention in this context are organic charge-transfer solids (CTS), SC in which was found even before the copper oxides Ishiguro et al. (1998). The proximity of SC to antiferromagnetism (AFM) in some, though not all (see below) CTS, have led to theories of SC in the CTS that are related to their counterparts in copper oxides Vojta and Dagotto (1999); Schmalian (1998); Kino and Kontani (1998); Kondo and Moriya (1998); Powell and McKenzie (2005); Gan et al. (2005); Sahebsara and Sénéchal (2006); Watanabe et al. (2006); Hebert et al. (2015). Furthermore, the occurrence of SC in the CTS at fixed carrier density by application of pressure instead of doping has led to the belief that understanding CTS may be easier and a first step towards understanding correlated-electron SC itself. At the same time though, occurrence of SC at fixed carrier concentration demands that there is both broad and deep understanding of the role of this specific carrier concentration, and this issue has therefore been at the center of the discussions on organic SC, and is also the subject matter of the present work.

Two fundamentally different approaches to correlated-electron superconductivity in organic CTS exist in the literature. The first of these is specific to strongly dimerized quasi-two dimensional (quasi-2D) systems such as $\kappa$-(BEDT-TTF)2X (hereafter $\kappa$-(ET)2X) and R[Pd(dmit)2]2 salts (here X and R are monovalent counteranion and countercation, respectively), in which pairs of cationic ET or anionic Pd(dmit)2 molecules are strongly coupled as dimers, and the dimers form anisotropic triangular lattices (see Fig. 1) Kanoda and Kato (2011). The charge on individual molecules is 0.5 and hence the dimer lattice has been considered as an effective $\frac{1}{2}$-filled band with strong intradimer onsite Coulomb repulsion $U_{\rm eff}$. Theoretical discussions are within the anisotropic triangular lattice Hubbard $U_{\rm eff}$-$t$-$t^{\prime}$ Hamiltonian, with hopping integrals $t$ along x and y, and $t^{\prime}$ along the diagonal y-x direction (see Fig. 1(b)) Kino and Kontani (1998); Schmalian (1998); Kondo and Moriya (1998); Vojta and Dagotto (1999); Baskaran (2003); Liu et al. (2005); Kyung and Tremblay (2006); Yokoyama et al. (2006); Watanabe et al. (2006); Sahebsara and Sénéchal (2006); Nevidomskyy et al. (2008); Sentef et al. (2011); Hebert et al. (2015). Metal-insulator transition occurs at a finite $U^{c}_{\rm eff}$, with $U^{c}_{\rm eff}$ increasing with $t^{\prime}$. For relatively small $|t^{\prime}/t|$ the semiconducting state is AFM, while for moderate to large $t^{\prime}$ spin-liquid (SL) behavior is predicted. The theory gives satisfactory understanding of the magnetic behavior of CTS with small to moderate $|t^{\prime}/t|<1$, including AFM in $\kappa$-(ET)2Cu[N(CN)2]Cl (hereafter $\kappa$-Cl) with $|t^{\prime}/t|\sim 0.44$ and apparent SL behavior in $\kappa$-(ET)2Cu2(CN)3 (hereafter $\kappa$-CN) and $\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2 with $|t^{\prime}/t|\sim 0.8-0.9$ Kanoda and Kato (2011); Kandpal et al. (2009). Very recent experimental work Yamamoto et al. (2017) has concluded that the the simple SL picture as well as the $\frac{1}{2}$-filled band Hubbard Hamiltonian are not appropriate for $\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2, a point to which we return later. Various mean field and dynamical mean field theory calculations find $d$-wave SC within the $\frac{1}{2}$-filled $U_{\rm eff}$-$t$-$t^{\prime}$ model at intermediate $|t^{\prime}/t|$, bounded by an antiferromagnetic semiconductor and a correlated metal in the phase diagram Kino and Kontani (1998); Schmalian (1998); Kondo and Moriya (1998); Vojta and Dagotto (1999); Baskaran (2003); Liu et al. (2005); Kyung and Tremblay (2006); Yokoyama et al. (2006); Watanabe et al. (2006); Sahebsara and Sénéchal (2006); Nevidomskyy et al. (2008); Sentef et al. (2011); Hebert et al. (2015). Numerical investigations by us Clay et al. (2008); Dayal et al. (2012) and others Watanabe et al. (2008); Shirakawa et al. (2017) have, however, demonstrated that nonzero $U_{\rm eff}$ suppresses the long-range component of the superconducting pair-pair correlations, indicating absence of SC within the model. Additionally, the effective $\frac{1}{2}$-filled band approach is inapplicable to CTS in which the organic monomer molecules are not coupled as dimers, and charge order (CO) and not AFM is proximate to SC Kimura et al. (2006); Kobayashi et al. (1986); Tajima et al. (2002); Yamamoto et al. (2011); Shikama et al. (2012).

An alternative theory Gomes et al. (2016); De Silva et al. (2016); Clay and Mazumdar (2019) ascribes each of the competing broken symmetries to the geometrically frustrated monomer lattice of organic molecules with carrier density $\rho=0.5$ per molecule and bandfilling $\frac{1}{4}$. Although AFM and Mott insulating behavior are expected in the strongly dimerized and weakly frustrated $\rho=0.5$ band, for stronger lattice frustration or weaker dimerization there is also a proximate static or dynamically fluctuating CO state, driven by quantum effects rather than classical electrostatics, that is a Wigner crystal of spin-singlet pairs (a paired electron crystal, hereafter PEC) Li et al. (2010); Dayal et al. (2011). Uniquely to $\rho=0.5$, it is possible to have a CO of spin-singlet pairs separated by pairs of vacant sites. Dipolar-spin coupling Hotta (2010) and polar charge fluctuations Sekine et al. (2013) are related to dynamic CO fluctuations. Within the correlated $\frac{1}{4}$-filled band theory, pressure-induced increased lattice frustration drives correlated motion of the spin-pairs into the “paired” vacant sites, giving a wavefunction that is still paired and constitutes the superconducting state Gomes et al. (2016); De Silva et al. (2016); Clay and Mazumdar (2019). Our theory is at the interface of the resonating valence bond (RVB) theory Anderson (1987) and the oldest version of the so-called bipolaron theory of superconductivity Chakraverty and Ranninger (1985). The spin-singlet bonds here arise from antiferromagnetic spin-coupling Gomes et al. (2016); De Silva et al. (2016), as opposed to overly strong electron-phonon couplings, as assumed in the earlier literature Chakraverty and Ranninger (1985). We refer to this as the valence bond theory of SC, distinct from the RVB theory by its focus on $\rho=0.5$.

The PEC concept is related to conjectures of “density wave of Cooper pairs” in the high Tc copper oxides Franz (2004); Tesanovic (2004); Chen et al. (2004); Vojta and Rösch (2008); Hamidian et al. (2016); Cai et al. (2016); Mesaros et al. (2016), where the precise nature of the density wave is still being debated (see however, Reference Mazumdar, 2018). AFM is not the driver of SC within this second approach; SC in $\kappa$-(ET)2X is preceded by an “orbital reordering” that renders unequal charge densities on the monomers within a dimer unit, with mobile spin-singlet bonds forming between the charge-rich molecules of neighboring dimers Gomes et al. (2016); De Silva et al. (2016); Clay and Mazumdar (2019); Sekine et al. (2013). Importantly, PEC formation is not limited to the dimerized $\kappa$-(ET)2X, and the CO phases adjacent to SC in undimerized $\alpha$- $\beta$- and $\theta$-(ET)2X are PECs Clay and Mazumdar (2019). This, in turn, indicates that the mechanism of SC in the dimerized and undimerized CTS is the same. Other calculations that have found SC within the $\frac{1}{4}$-filled band Hubbard model using mean-field and random phase approximations Kondo and Moriya (2001); Kuroki (2006); Sekine et al. (2013); Guterding et al. (2016a), or variational Monte Carlo Watanabe et al. (2019) are not filling-specific; a key aspect of our theory is that the PEC, and hence SC, are absent for bandfilling even modestly far from $\frac{1}{4}$.

In the present paper we discuss a new class of $\kappa$-CTS that are only now beginning to be investigated experimentally. These materials belong to the class $|t^{\prime}/t|>1$ and allow sharp comparisons between the two theoretical starting points described above, as we point out in the present work. We show that while within the effective $\frac{1}{2}$-filled band theory neither the observed strange semiconductors nor superconductivity are expected, the ground state of the $\frac{1}{4}$-filled Hubbard model is a strongly correlated metal that is susceptible to all the observed exotic transitions.

II New Quasi-1D CTS and Theoretical Challenges

Until recently, SC in the $\kappa$-(ET)2X family was limited to materials which lie firmly in the $|t^{\prime}/t|<1$ region of the effective $\frac{1}{2}$-filled Hubbard model. Discoveries of exotic behavior including SC in $\kappa$ compounds with effective $|t^{\prime}/t|$ significantly greater than 1 provide new testing grounds for the two classes of theories. The principal observations are given below.

(i) SL behavior has perhaps been seen in $\kappa$-(ET)2TaF6 (hereafter $\kappa$-TaF6) Kawamoto et al. (2018), with $|t^{\prime}/t|>1.5$. Absence of magnetic ordering has been demonstrated down to 1.6 K Kawamoto et al. (2018).

(ii) $\kappa$-(ET)2B(CN)4 (hereafter $\kappa$-B(CN)4) is an insulator at ambient pressure with effective $|t^{\prime}/t|\sim 1.8$ Yoshida et al. (2015). It undergoes phase transition into a nonmagnetic state with spin-gap (SG) Yoshida et al. (2015) at $\sim 5$ K. Raman measurements have failed to find CO so far, but the experiments were done only down to to 10 K.

(iii) $\kappa$-(ET)2CF3SO3 (hereafter $\kappa$-CF3SO3) contains two inequivalent ET layers, labeled layers A and B, separated by the anion layer Fettouhi et al. (1995); Ito et al. (2016). Within the effective model $|t^{\prime}/t|=1.48$ (1.78) for layer A (B) at low temperatures Ito et al. (2016). Following structural changes at 230 K and 190 K, AFM occurs at ambient pressure at a low Néel temperature TN of 2.5 K Ito et al. (2016). Transition to a metallic state and SC occurs under pressure (Tc,max = 4.8 K at 1.3 GPa) Ito et al. (2016). T${}_{\rm N}<$ Tc (albeit pressure-induced) is unknown in any other correlated electron superconductor that has shown transition from AFM to SC.

The above observations defy explanations within the effective $\frac{1}{2}$-filled band approach. First, even as magnetic measurements on $\kappa$-TaF6 are awaited at lower temperatures ($<1.6$ K), SL behavior is not expected within the effective $\frac{1}{2}$-filled band model for $|t^{\prime}/t|\geq 1.5$. Theory limits the SL phase only to $|t^{\prime}/t|\leq 1$ within the triangular lattice $\frac{1}{2}$-filled Hubbard Hamiltonian Yoshioka et al. (2009); Tocchio et al. (2014); Acheche et al. (2016). What is perhaps more relevant is that very recent experimental works Itoh et al. (2013); Yakushi et al. (2015); Yamamoto et al. (2017); Fujiyama and Kato (2018) have cast doubt on whether even $\kappa$-CN and EtMe3Sb[Pd(dmit)2]2, which were thought to be prime candidates for SL in the parameter region $|t^{\prime}/t|\leq 1$ until very recently Kanoda and Kato (2011), should really be described as such. The original proponents of the SL picture for EtMe3Sb[Pd(dmit)2]2, in particular, have found that the driver of the geometrical frustration here are hidden charge and lattice fluctuations, not expected within the simple $\frac{1}{2}$-filled band Hubbard model Yamamoto et al. (2017).

Second, within the effective $\frac{1}{2}$-filled band model, the SG transition observed in $\kappa$-B(CN)4 can only be due to spin-Peierls transition, or due to a frustration-driven transition to a valence bond solid (VBS). The large ratio of inter-chain to intra-chain hopping, $\sim$ 0.6 in $\kappa$-B(CN)4, implies absence of 1D nesting necessary for the spin-Peierls transition characteristic of quasi-1D systems (note here that spin-density wave, and not the spin-Peierls state characterizes the spatial broken symmetry in (TMTSF)2X where the same ratio at 0.2 is significantly smaller). It is also known from previous theoretical work that the Hubbard repulsion severely reduces bond dimerization in the 2D half-filled band even where nesting would have permitted this in the uncorrelated limit Mazumdar (1987); Pillay et al. (2013). The VBS is also precluded theoretically within the triangular lattice $\frac{1}{2}$-filled band Hubbard model Gomes et al. (2013). We therefore conclude that that the explanation of the SG transition in $\kappa$-B(CN)4 requires going beyond the effective $\frac{1}{2}$-filled band model.

Finally, the observed SC in $\kappa$-CF3SO3 also lies outside the domain of applicability of the effective $\frac{1}{2}$-filled band theories. SC appears for significantly smaller $|t^{\prime}/t|<1$ within these theories Kino and Kontani (1998); Schmalian (1998); Kondo and Moriya (1998); Vojta and Dagotto (1999); Baskaran (2003); Liu et al. (2005); Kyung and Tremblay (2006); Yokoyama et al. (2006); Watanabe et al. (2006); Sahebsara and Sénéchal (2006); Nevidomskyy et al. (2008); Sentef et al. (2011); Hebert et al. (2015). In the present paper we show from explicit numerical calculations based on the competing $\frac{1}{4}$-filled band Hubbard model that the ground state of the latter is a strongly-correlated metal in which none of the above exotic phases are precluded, and hence any one of these can presumably dominate when small but nonzero interactions excluded in the purely electronic Hamiltonian are included. Superconducting correlations, in particular, are enhanced by the Hubbard $U$ in the $\frac{1}{4}$-filled band, relative to the noninteracting $U=0$ limit, which is a necessary condition for correlated-electron SC.

III Theoretical model, parameters and computational techniques

We consider the Hubbard Hamiltonian for the $\kappa$ lattice structure of Fig. 1(a),

[TABLE]

where $c^{\dagger}_{i,\sigma}$ ($c_{i,\sigma}$) creates (annihilates) an electron of spin $\sigma$ on the highest molecular orbital of a monomer ET molecule $i$ (hereafter “site”), and all other terms have their usual meanings. In our calculations for $\kappa$-CF3SO3 we take hopping integrals ($t_{\rm b1}$, $t_{\rm b2}$, $t_{\rm p}$, $t_{\rm q}$; $|t^{\prime}/t|$) in eVs to be (0.234, 0.157, 0.029, -0.077; 1.48) and (0.248, 0.169, 0.048, -0.047; 1.78), for layers A and B, respectively Ito et al. (2016). We report separate calculations for $\kappa$-B(CN)4 with corresponding hopping parameters Yoshida et al. (2015) (0.221, 0.117, 0.035, -0.029; 1.83). These parameters make $\kappa$-B(CN)4 closer to 1D than $\kappa$-CF3SO3, as already noted above. These hopping integrals are for electrons and not holes; the corresponding carrier density is $\rho=1.5$. Our zero temperature computational results are for finite periodic clusters with 32, 64, and 128 monomer sites Sup using the Path Integral Renormalization Group Kashima and Imada (2001); Mizusaki and Imada (2004) and Constrained Path Monte Carlo Zhang et al. (1997) methods Sup . For both methods we fully incorporate lattice and spin-parity symmetries (see References Mizusaki and Imada, 2004 and Shi et al., 2014), which significantly increases their accuracy. Our prior benchmark results of these methods are available in References Dayal et al., 2012, Gomes et al., 2016, and De Silva et al., 2016.

IV Computational results

IV.1 Magnetic behavior

IV.1.1 Weak AFM in $\kappa$-CF3SO3

The tendency to AFM is best discussed within the dimer representation De Silva et al. (2016). We define the total $z$-component of the spin on dimer $i$ as

[TABLE]

In Eq. 2, $i_{1}$ and $i_{2}$ refer to the two different molecules within the dimer $i$ and $n_{j,\sigma}=c^{\dagger}_{j,\sigma}c_{j,\sigma}$. We calculate the dimer spin structure factor,

[TABLE]

where $N_{d}$ is the number of dimers and dimer position vectors rj are defined on a conventional square lattice, whose x and y axes are indicated in Fig. 1. Fig. 2 shows the Q dependence at $\rho=1.5$ of S(Q) for 64 and 128 monomer lattice sites, separately for layers A and B in each case. The Q dependence of $S({\bf Q})$ here is similar to that of the $\frac{1}{2}$-filled Hubbard modelTocchio et al. (2014) with $|t^{\prime}/t|>1$. We find a line of maxima in $S({\bf Q})$ perpendicular to the $t^{\prime}$ direction, with the dominant wavevector $\pi$ along each chain. $S({\bf Q})$ does not increase with system size, indicating the absence of long-range order. Fig. 3 shows that $S({\bf Q})$ at ($\pi,0$) is stronger than that at ($\pi,\pi$) in agreement with results for the $\frac{1}{2}$-filled Hubbard model Clay et al. (2008); Tocchio et al. (2014); Acheche et al. (2016). $S(\pi,\pi)$ here is nearly half of that calculated by us previously for $\kappa$-Cl De Silva et al. (2016). The dominance of $S(\pi,0)$ over $S(\pi,\pi)$ is a consequence of quasi-1D character. Note however that in the 1D limit there is no long-range AFM. Additionally, the absence of true long range order in $S({\bf Q})$ in our calculations indicates that $\kappa$-CF3SO3 is barely insulating and is close to a correlated metallic phase. The enhanced 1D character and the proximity to the metallic state, taken together, are the likely reasons for the small TN in $\kappa$-CF3SO3.

IV.1.2 Transition to the spin-gapped state in $\kappa$-B(CN)4

As mentioned above, simple spin-Peierls transition or VBS formation, as have been suggested within the effective $\frac{1}{2}$-filled model Yoshida et al. (2015), cannot be the origin of the observed SG in $\kappa$-B(CN)4 Mazumdar (1987); Pillay et al. (2013); Gomes et al. (2013). The recent demonstration that the driving forces behind the apparent SL-like behavior in $\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2 are charge and lattice fluctuations Yamamoto et al. (2017), which are precluded in the $\frac{1}{2}$-filled band but are expected within the $\frac{1}{4}$-filled band characterization of the system Clay and Mazumdar (2019), gives a hint to the mechanism of the SG in $\kappa$-B(CN)4. Since similar intradimer charge inequalities can occur in all $\frac{1}{4}$-filled band systems, we investigate the likelihood of PEC formation in the present case.

We have performed numerical calculations of charge densities in which the $t_{\rm b2}$ hopping integrals (see Fig. 1) alternate as $t_{\rm b2}^{0}\pm\delta t_{\rm b2}$, as suggested in Reference Yoshida et al., 2015. All other hopping parameters correspond to those for $\kappa$-B(CN)4, given in Section III above. The PEC charge order pattern (Fig. 4(a)) appears spontaneously for nonzero $\delta t_{\rm b2}$, as shown in Fig. 4(b), with CO $\cdots 0110\cdots$ along the bonds in the y and x-y directions, and $\cdots 1010\cdots$ along the bonds in the third direction x (see Fig. 1(a) and Reference Sup, ). Here ‘1’ and ‘0’ represent charge-rich and charge-poor molecules, respectively. Period doubling along both the crystal $a$ and the $c$-axes are therefore expected at the SG transition, as opposed to only along $t_{\rm b2}$ Yoshida et al. (2015). Note that the pattern and phase of the CO is consistent with ($\pi,0$) periodicity of the AFM, if we make the reasonable assumption of strong antiferromagnetic correlations between nearest neighbor charge-rich molecules.

Fig. 4(b) gives the calculated CO order parameter $\delta n$, the difference in the charge densities between the charge-rich and the charge-poor molecules. The kink at $\delta t_{\rm b2}\sim 0.005$ eV is due to a level crossing; for $\delta t_{\rm b2}>0.005$ eV the difference in $\delta$n for 64 versus 128 sites is less than the size of the symbols in Fig. 4(b). We consider the range of $\delta t_{\rm b2}$ in Fig. 4(b) to be realistic Yoshida et al. (2015), and while the calculated $\delta n$ are small, they should be observable if experiments sensitive to the molecular charge are extended to T$<$TSG. Even more importantly, we predict period doubling along both a and c-directions, consistent with PEC formation, that should also be experimentally verifiable.

IV.2 Superconductivity: $\kappa$-CF3SO3 versus $\kappa$-B(CN)4

IV.2.1 Coulomb enhancement of superconducting correlations: $\kappa$-CF3SO3

We now probe the question of SC. As in previous work De Silva et al. (2016), calculations here are for variable carrier densities $\rho$, since the basic contention of our theory Gomes et al. (2016); De Silva et al. (2016); Clay and Mazumdar (2019) is that SC is specific to $\rho$ exactly equal to or close to 1.5, and is absent in $\rho$ even modestly far from 1.5. We calculate superconducting pair-pair correlations $P_{ij}=\langle\Delta_{i}^{\dagger}\Delta_{j}\rangle$, where $\Delta_{i}^{\dagger}$ creates a superposition of singlet pairs between monomer sites belonging to neighboring dimers. Such interdimer singlet pairs necessarily create unequal intradimer charge densities De Silva et al. (2016).

Considering singlet pairs between monomer molecules on a central dimer and its six neighboring dimers, there occur four types of $d$-wave pairings (see Fig. 3 in Reference De Silva et al., 2016). The first of these pairing symmetries, $d_{1}$, is similar to the usual $d_{x^{2}-y2}$ symmetry and the second, $d_{2}$, is a form of $s+d_{x^{2}-y^{2}}$ pairing also considered by other authors Guterding et al. (2016a, b); Watanabe et al. (2017). For all three clusters we calculate the average long-range pair-pair correlations $\bar{P}=N_{P}^{-1}\sum_{|r_{ij}|>2}P_{ij}$, where $N_{P}$ is the number of terms in the sum with the restriction $r_{ij}>2$ in units of dimer-dimer spacing De Silva et al. (2016). Since the minimal requirement for interaction-driven SC is that nonzero Hubbard $U$ enhances the pairing correlations, we calculate the enhancement factor $\Theta_{P}=[\bar{P}(U)/\bar{P}(U=0)]-1$. Then $\Theta_{P}>0$ implies likelihood of pairing and $\Theta_{P}<0$ indicates suppression of pairing by Hubbard $U$. Previous calculations De Silva et al. (2016) for effective $|t^{\prime}/t|<1$ found enhancement only for $\rho\sim 1.5$. A precise value for $U$ for the actual materials is difficult to estimate. In the quasi-1D (TMTSF)2X materials, $U\sim$ 1 eV has been estimated for the extended Hubbard model Mila (1995). Based on the larger size of the ET molecule we expect a smaller $U$ in the range 0.4 – 0.8 eV in $\kappa$-(ET)2X compared to (TMTSF)2X.

Our calculations find very strong enhancements of pair correlations with $d_{2}$ symmetry for $\rho\sim 1.5$ for both layers A and B of $\kappa$-CF3SO3, for all three cluster sizes ($d_{2}$ gave the most strongly enhanced pair correlations De Silva et al. (2016) also for $\kappa$-Cl and $\kappa$-CN). In Fig. 5 we show plots of $\Theta_{P}(d_{2})$ versus $\rho$ for all three cluster sizes, for $U=0.4$ eV, separately for layers A and B in each case (see Supplementary Materials Sup for results on $\Theta_{P}(d_{1})$). The calculated results are for a wide range of $\rho$ in the 32 and 64-site clusters, spanning $\rho=1.3-1.7$. A narrower density range $1.4-1.6$ was considered for the 128-site cluster, based on the similar behavior of the three clusters. In spite of the quasi-1D character of the lattice, here we find nearly identical results as in the earlier work for the 2D lattice, where an even broader density range $\rho=1.0-2.0$ was considered De Silva et al. (2016). In all cases we find that $\Theta_{P}(d_{2})>0$ occurs over a very narrow density range about 1.5. In Figs. 5(c) and (e) we see dips followed by rise in $\Theta_{P}$ for densities close to but slightly away from density exactly 1.5. However, these dips and increases are tiny compared to the large peaks at density exactly at or very close to 1.5. The widths of the vertical shaded bars in Fig. 5 decrease with increasing lattice size, since between any two densities there are fewer total electron numbers in the smaller lattices. Importantly, (i) $\Theta_{P}(d_{2})<0$ outside of the density range 1.5 $\pm$ 0.1 in lattices larger than 32 sites, (ii) for all cases the largest enhancement in the pairing correlations is found either for $\rho$ exactly equal to 1.5 or for an electron number immediately larger or smaller for a given lattice, and (iii) the range of densities over which pair correlations are enhanced by $U$ decreases with increasing cluster size. Quantitatively speaking, the enhancements of the pair correlations for $\rho=1.5$ here are either larger than (for 32 sites) or comparable to (for 64 sites) De Silva et al. (2016) those for effective $|t^{\prime}/t|<1$. Calculations for the very large 128-site clusters had not been performed before.

For the specific case of $\rho=1.5$ we also performed calculations of the enhancement factor over a broad range of $U$ for the 32- and 64-site clusters (see Fig. 6). For the 32 site layer B lattice a quantum phase transition occurs at $U\approx$ 0.4 eV (see Fig. 6(a)). In this lattice at $U\approx 0.4$ eV a sharp increase in $S(\pi,\pi)$ coupled with a decrease in $\bar{P}$ indicates a transition to an insulating antiferromagnetic state. This is a finite-size effect that is not present in larger lattices. On the other hand, this result does indicate that our strongly-correlated metal is close to an insulating phase, which would be likely a weak AFM, based on our calculations of the magnetic structure factor, see Section IV.1.

IV.2.2 Reduced tendency to superconductivity in $\kappa$-B(CN)4

We have calculated $U$-dependent pair correlations also for the hopping parameters corresponding to $\kappa$-B(CN)4, for clusters consisting of 32 monomer sites. We show these results in Figs. 7(a) and (b), where we have included the computational results for the 32-site $\kappa$-CF3SO3 lattice for comparison. The calculated pair-pair correlations are nearly identical for the two lattices for $\rho$ far from 1.5 (Fig. 7(b)), where there is absence of enhancement, or even suppression of these correlations by the Hubbard $U$. The results for $\rho\sim 1.5$ (Fig. 7(a)) are noticeably different, however. On the one hand, the pair-pair correlations are enhanced by small Hubbard $U$ for $\kappa$-B(CN)4. On the other, the enhancement occurs over a significantly smaller range of $U$ compared to that for $\kappa$-CF3SO3, with the peak in the enhancement occurring at very small ($\lesssim$0.2 eV) $U$. For realistic molecular $U$ ($>$ 0.4 eV) the predicted tendency to SC in $\kappa$-B(CN)4 is thus considerably smaller than in $\kappa$-CF3SO3.

V Conclusions

In summary, we have performed large scale accurate numerical calculations within the Hubbard Hamiltonian, for quasi-1D triangular lattice clusters of three different sizes, corresponding to the monomer $\kappa$-(ET)2X lattice of Fig. 1(a), with realistic parameter values for $\kappa$-CF3SO3 and $\kappa$-B(CN)4. Our results can be summarized as follows.

(i) For hopping parameters corresponding to both the inequivalent layers A and B of $\kappa$-CF3SO3, we find that S(Q) peaks at $(\pi,0)$ instead of $(\pi,\pi)$. This is in agreement with results obtained for the effective $\frac{1}{2}$-filled band Hubbard model for $|t^{\prime}/t|>1$.

(ii) For hopping parameters corresponding to $\kappa$-B(CN)4 we find the smallest alternation in $t_{b2}$, which has been suggested as the origin of the SG transition Yoshida et al. (2015), is accompanied by CO as well as bond distortions in the two other directions, leading to period doublings. This is the classic signature of a tendency to unconditional transition to the PEC, noted previously for the effective $t^{\prime}<t$ materials Li et al. (2010); Dayal et al. (2011).

(iii) Returning to the hopping parameters corresponding to $\kappa$-CF3SO3, we find from electron density-dependent calculations on three different lattice sizes that superconducting pair-pair correlation functions of $d_{2}$ symmetry are enhanced by the Hubbard $U$ only for density close to 1.5, precisely the carrier density in superconducting CTS. The superconducting symmetry is thus the same De Silva et al. (2016) as that for effective $|t^{\prime}/t|<1$. Also in agreement with previous calculations De Silva et al. (2016), for electron densities even weakly away from 1.5 the Hubbard $U$ suppresses pair correlations, indicating absence of SC here.

(iv) For hopping parameters corresponding to $\kappa$-B(CN)4, similar electron density-dependent calculations for the 32-site cluster find enhancement of superconducting pair-pair correlation functions for $\rho=1.5$, but only for $U$ values smaller than the known Hubbard $U$ for the ET molecule. This explains the absence of SC to date in this system, but also suggests that application of pressure, which would increase the bandwidth and decrease the effective $U/|t|$, can induce SC here. This predicted result is reminiscent of the experimental observation of high pressure-induced superconducting transition Adachi et al. (2000) from the spin-Peierls state in (TMTTF)2PF6, which is a PEC at the quasi-1D limit Clay and Mazumdar (2019).

The overall conclusion is that except for the weaker tendency to metal-insulator transition and AFM within the purely electronic Hamiltonian, which presumably is the reason behind the small TN in $\kappa$-CF3SO3, all other behaviors at low temperatures are the same as what would be expected for the effective $|t^{\prime}/t|<1$. The fundamental reason for this is that the intradimer charge degrees of freedom become important at low temperatures in both $\kappa$-(ET)2X and R[Pd(dmit)2]2. In recent years investigators have noted from different experimental studies on effective $|t^{\prime}/t|<1$ dimerized CTS that the lowest excitations in these involve intradimer charge fluctuations and the resultant lattice fluctuations Itoh et al. (2013); Yakushi et al. (2015); Fujiyama and Kato (2018). Here we note that the same is true for effective $|t^{\prime}/t|>1$. Should experiments at T $<1.6$ K confirm the absence of magnetic long-range order in $\kappa$-TaF6, this should be ascribed also to intradimer charge fluctuations. The SG transition in $\kappa$-B(CN)4 is likely due to PEC formation, as in the $|t^{\prime}/t|<1$ compound $\beta$-EtMe3P[Pd(dmit)2]2 Yamamoto et al. (2017); Clay and Mazumdar (2019). We suggest experiments sensitive to CO in $\kappa$-B(CN)4.

Finally, the observation of SC in $\kappa$-CF3SO3 is a strong proof of the validity of our valence bond theory of SC. Elsewhere Clay and Mazumdar (2019) we have drawn attention to the very large number of disparate families of inorganic correlated-electron superconductors that share two features with CTS superconductors, viz., (a) $\frac{1}{4}$-filling on frustrated lattices, and (b) SC proximate to unconventional charge-density waves. The very large number of such families suggests the universality of the proposed PEC-to-SC mechanism. Indeed, one of us has suggested the same mechanism of SC also for high Tc copper oxides, both hole and electron-doped, where an effective correlated $\frac{1}{4}$-filled oxygen band is reached following a dopant-induced Cu${}^{2+}\to$ Cu1+ valence transition Mazumdar (2018).

VI Acknowledgments

S.M. acknowledges partial support from NSF-CHE-1764152. Some calculations in this work used the Extreme Science and Engineering Discovery Environment Towns et al. (2014) (XSEDE), which is supported by National Science Foundation grant number ACI-1548562. Specifically, we used the Bridges system Nystrom et al. (2015) which is supported by NSF award number ACI-1445606, at the Pittsburgh Supercomputing Center (PSC) under award TG-DMR190052.