Triplet superconductivity in coupled odd-gon rings

Shedding light on the nature of spin-triplet superconductivity has been a long-standing quest in condensed matter physics since the discovery of superfluidity in liquid 3He. Nevertheless, the mechanism of spin-triplet pairing is much less understood than that of spin-singlet pairing explained by the Bardeen-Cooper-Schrieffer theory or even observed in high-temperature superconductors. Here we propose a versatile mechanism for spin-triplet superconductivity which emerges through a melting of macroscopic spin polarization stabilized in weakly coupled odd-gon (e.g., triangle, pentagon, etc) systems. We demonstrate the feasibility of sustaining spin-triplet superconductivity with this mechanism by considering a new class of quasi-one-dimensional superconductors A2Cr3As3 (A = K, Rb, and Cs). Furthermore, we suggest a simple effective model to easily illustrate the adaptability of the mechanism to general systems consisting of odd-gon units. This mechanism provides a rare example of superconductivity from on-site Coulomb repulsion.

www.nature.com/scientificreports www.nature.com/scientificreports/ spin-triplet SC state 25,26 or an inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov state at high fields 27,28 . This scenario raises even more fascinating challenges in this field of research.
At present, the most promising material for spin-triplet superconductivity is the ruthenate Sr 2 RuO 4 . No sooner was discovered the SC transition at T 1 5 c . K 29 , the similarity between the superconductivity of Sr 2 RuO 4 and the spin-triplet superfluidity of 3 He was theoretically pointed out 30 . After that, the spin-triplet pairing state having spontaneous time-reversal symmetry breaking with d-vector perpendicular to the conducting plane was experimentally confirmed 31,32 . Thus, analogous to the 3 He-A phase, superconductivity in Sr 2 RuO 4 has been generally concluded to be of the spin-triplet p-wave type 33 .
Another candidate is water-intercalated sodium cobalt dioxides Na x CoO 2 ⋅1.3H 2 O (x 0 35 . ), which exhibits a SC transition at T c ≈ 5 K 34 . By controlling x using the trigonal CoO 2 distortion, two SC phases appear and they are separated by a narrow magnetic phase 35 . A theoretical multi-band tight-binding calculation speculated that the SC state for larger x is of s-wave spin-singlet and that for smaller x is of p-or f-wave spin-triplet. Experimentally, the relation between the two SC states as well as the origins of them are still highly controversial 36,37 .
Very recently, a new family of Q1D superconductors A 2 Cr 3 As 3 (A = K, Rb, and Cs) has been recognized. At ambient pressure, a SC transition was observed at T c = 6.1 K for A = K 38 , 4.8 K for A = Rb 39 , and 2.2 K for A = Cs 40 . It has been confirmed that the extrapolated upper critical field H c2 (T = 0) largely exceeds the Pauli limit [38][39][40] , which strongly supports spin-triplet pairings in the family compounds. Furthermore, the existence of nodes in the SC gap has been indicated by several experiments; linear behavior of the London penetration depth λ(T) 41 , absence of the Hebel-Slichter coherence peak below T c in the nuclear spin-lattice relaxation rate 1/T 1 42,43 , and Volovik-like field dependence ( H ∝ ) of the zero-temperature Sommerfeld coefficients γ(H) in the SC mixed state 39 . From the theoretical aspect, spin-triplet Cooper pairs are considered to be formed simply by a ferromagnetic (FM) interaction 44 . In fact, both the spin-lattice relaxation rate divided by temperature, 1/T 1 T, and the Knight shift K increase when the temperature is decreased from ~100 K down to T c , suggesting significant FM spin fluctuations 45 .
Remarkably, the conduction pathway consists of coupled triangles in all the above four materials (see also Supplementary Information). It is known that a three-site Hubbard ring with two electrons generates a local FM interaction forming a triplet ground state rather than a singlet pairing 46 . When these rings with pre-formed triplet pairs are coupled, a spin-triplet superconductivity might be expected just as the pre-formed singlets on rungs give rise to singlet-SC in ladder systems 47,48 . Previously, one of the present authors proposed a similar SC mechanism for a limited case of the TMTSF salts with anisotropic triangular structure including long-range Coulomb repulsions 49 . The aim of this paper is to construct a more general theory of spin-triplet SC mechanism for coupled odd-gons systems: a Hubbard ring with odd number of sites can provide a local FM interaction, and then spin-triplet superconductivity emerges through a melting of macroscopic ferromagnetism stabilized in the weakly coupled odd-gons systems. In this paper, we demonstrate the applicability of our SC mechanism by studying a twisted triangular Hubbard system as a model for Q1D superconductors A 2 Cr 3 As 3 . The strong Q1D nature of this system enables us to perform very precise and unbiased numerical analyses using the density-matrix renormalization group (DMRG) method 50 . Nevertheless, since the macroscopic ferromagnetism is expected to occur for any geometry of connections between the odd-gons (see below), this SC mechanism would be widely applicable to other real materials having crystal structure like triangular, kagome, pyrochlore lattices, and fullerenes, etc (see also Supplementary Information). It is also worth noting that our spin-triplet SC is mediated only by on-site Coulomb repulsion.

Ferromagnetic Mechanism
Before performing the numerical analysis, we explain the FM mechanism which is generalized via a spin-triplet formation of two fermions in an isolated odd-gon Hubbard ring. We consider a Hubbard ring with odd number of sites l o containing two fermions [see Fig. 1(a)]. In momentum space the Hamiltonian is written as where c k,σ is an annihilation operator of fermion with spin σ at momentum k and ε(k) = 2t cosk for uniform hopping integrals t. We assume t > 0. The allowed momenta are The energy dispersion is shown Fig. 1(b). When the spins of two fermions are parallel, namely in a spin-triplet state, the correlation term ′ vanishes. Thus, this spin-triplet state is exactly described as a two-fermion occupancy at k = ±(l o − 1)π/l o levels [see Fig. 1 On the other hand, the correlation is involved in a spin-singlet state with anti-parallel spins. The total energy of the spin-singlet E s is plotted as a function of U for, l o = 3, 5 and 7 in Fig. 1(c). We find that E t is always lower than E s for U > 0. This immediately confirms the emergence of attractive interaction between two fermions in a spin-triplet state. Note that the total energy of a spin-triplet state with anti-parallel spins is also E t since the Hamiltonian (1) has the SU(2) symmetry.
We further consider the case of general l o by perturbation theory. In the weak-coupling limit (  U 1) we take ′ as a perturbation to the unperturbed Hamiltonian  0 . The unperturbed ground state (U = 0) is exactly the same as the spin-triplet state; namely, the total energy is E t . When the correlation is involved in a spin-singlet state the total energy is modified as www.nature.com/scientificreports www.nature.com/scientificreports/ up to the second order of U. Similarly, in the strong-coupling limit (U 1  ) the total energy is expressed as is the total energy at U = ∞. Since the weak-and strong-coupling regimes are expected to be smoothly connected, a spin-triplet state would be always the ground state for any odd l o at U > 0. The perturbation results (2) and (3) are compared to the numerical results for l o = 3, 5 and 7 in Fig. 1(c). This type of a spin-triplet formation has been frequently discussed in the open-shell problem of a finite-size cluster. Here, the existence of (nearly-) degeneracy in the lowest-lying shells is essential to obtain a spin-triplet ground state; and, such the situation is easily realized for odd l o and U, t > 0. So, the question we address here is whether a triplet SC is stabilized when these spin-triplet odd-gon units are hybridized.

Model
Let us then couple the odd-gon units. A good playground with coupled odd-gon units is provided by the chromium arsenide A 2 Cr 3 As 3 . The crystal structure consists of double-walled subnanotubes [Cr 3 As 3 ] 2 [see Fig. 2(a)]. They form Q1D chains by stacking of triangular units along the -axis and these Q1D chains are separated by columns of A + ions. First-principles calculations 45,51 found that the conductivity is dominated by the d z 2, − d x y 2 2 , and d xy orbitals of Cr atom, and three energy bands are involved at the Fermi level; two Q1D and a 3D. Besides, experimentally, the strong 1D nature of conduction along the c-axis has been confirmed by detecting the Tomonaga-Luttinger liquid behaviors 42,52 . Therefore, it would be reasonable to consider the so-called "twisted triangular Hubbard tube" as a model to study the conduction properties of A 2 Cr 3 As 3 by taking the highest occupied molecular orbital (HOMO) in Cr to be a Hubbard site 53 [see Fig. 2(b)]. The Hamiltonian in real space is where c i,σ is an annihilation operator of electron with spin σ at site i, t ij is hopping integral between sites i and j, U(>0) is on-site Coulomb interaction, and n c c Using the Slater-Koster tight-binding method, the sign of intra-triangle hopping integral t 1 is estimated to be positive in hole notation (see Supplementary Information). www.nature.com/scientificreports www.nature.com/scientificreports/ This meets the FM condition introduced in the previous section. We take t 1 = 1 as the unit of energy hereafter. Note that the system is gauge-invariant under a transformation of inter-triangle hopping integral t 2 → −t 2 . We use U = 10 as a typical value for strongly correlated electron systems.

Numerical Calculations
We now present the DMRG results for the twisted triangular Hubbard tube. We studied clusters up to L × 3 = 60 × 3 sites under the open boundary conditions with keeping m = 1000 to 12000 density-matrix eigenstates. All the physical quantities have been extrapolated to the m → ∞ and L → ∞ limits. First, we consider the system at n = 2/3 filling where the spin-triplet (S = 1) triangle units are coupled by t 2 along the tube direction. It is found the the system remains insulating with large charge gap and can be treated as S = 1 spin chain system 54 . The situation drastically changes away from n = 2/3 filling. Let us consider the case of a smaller filling n = 1/2. First, we calculate the total spin S tot defined as S S S S S ( 1 ) Fig. 3 the L → ∞ extrapolated result of total spin per site is plotted as a function of t 2 . Quite interestingly, an infinitesimal t 2 induces a fully-polarized FM (FF) order even though there is no explicit FM interactions between the triangles. This implies that the geometry of inter-triangle network is not essential for the appearance of the FF state; in other words, the spins are aligned only if hole(s) can hop from odd-gon to odd-gon. This may be similar to the Nagaoka mechanism 55 in the sense that hole propagation stabilizes a ferromagnetism. The dependence of the FM stabilization on the on-site Coulomb interaction U is demonstrated in Supplementary Information. With increasing t 2 , the FF order starts to melt at t 2 = 0.092 and the system goes into a narrow partially-polarized FM (PF) phase at 0.092 < t 2 < 0.121. The FM polarization completely vanishes at t 2 = 0.121 and a global singlet state characterized by S tot = 0 follows. Nonetheless, the short-range (at least within the odd-gon units) spin-spin correlation is naturally expected to be FM even after the melting of FM order.
Thus, the next question is whether the short-range FM correlation stabilizes spin-triplet bound pairs of fermions. We determine this by examining binding energy of fermions, defined as  . We have also confirmed that the charge gap, defined as , and the spin gap, defined as Δ = , are always zero for all t 2 at n = 1/2. Therefore, the region with Δ B > 0 is characterized as a spin-triplet SC phase. The same quantities are also calculated for 0 < n < 1 and the results are summarized as a phase diagram in Fig. 4. We found a wide region of the spin-triplet SC phase and the binding energy is most enhanced around = n 2 3 . In fact, the Q1D HOMO band of A 2 Cr 3 As 3 is nearly 2 3 -filled with holes (or 4 3 -filled with electrons) 45 . For further confirmation of the opening of SC gap, it would be interesting to calculate the anomalous Green's functions for Bogoliubov quasiparticle excitations 56 in future.

Effective Model
In this section, an effective model is constructed for the twisted triangular Hubbard tube to explain the phase diagram [ Fig. 4] more intuitively. Here we illustrate its construction for a coupled triangular unit system but a similar consideration is possible for any coupled odd-gon units system. Each of the triangles can be mapped onto a site with two orbitals at 0 < n < 1 which correspond to the two lowest-lying shells in the momentum space [see Fig. 1(b)]. The intra-triangle FM interaction is captured as Hund's coupling J eff (<0) between the intra-site orbitals.  www.nature.com/scientificreports www.nature.com/scientificreports/ The amount is given by the energy difference between the spin-triplet ground state and first spin-singlet excited state of the isolated triangle including two fermions, i.e., J eff = E s − E t [see Fig. 1(c)]. Further, the on-site Coulomb repulsion U eff , which is the origin of Mott state at = n 2 3 , is approximately estimated by the single-particle gap of the isolated triangle. The estimation at n 2 3 = is given in Supplementary Information. The inter-site hopping parameter t eff is simply proportional to t 2 . A schematic picture of our effective model is sketched in Fig. 2(c).
We now give an interpretation of the phase diagram using the effective model. Let us start in the small t eff limit. At n 2 3 = the system is insulating because two 'parallel' spins are confined in each of the sites. The degrees of freedom of each the site is spin-1. An AFM interaction between the inter-site orbitals is induced by nonzero t eff and the magnetism is described by the Affleck-Kennedy-Lieb-Tasaki state in the spin-1 Heisenberg chain 57 . Surprisingly, the ground state is drastically changed when the system is doped by fermions or holes. In either case the spins polarize to avoid feeling U eff and also to gain J eff in the inter-triangle hopping processes. It is similar to the Nagaoka ferromagnetism 55 in the sense that the spins are ferromagnetically aligned due to the movement of particles in the system. Connecting with infinitesimally small t eff , Therefore, the total spin is + ( ) > . In the latter case the spins are partially screened because more than two electrons on average are contained in a site. Now consider what happens when we increase t eff (∝t 2 ). With increasing t eff , a collapse of the FM polarization is expected since the AFM interaction between the inter-site orbitals In fact, the both FF and PF phases disappear at larger t eff for all n. Especially, the FM polarization is immediately destroyed by small t eff near = n 2 3 where the AFM interaction is maximized like in the half-filled Hubbard model.
As t eff increases, the FF and PF phases depending on electronic fillings are followed by a global singlet (S tot = 0) phases; spin-triplet SC (TS), PM, and spin-singlet SC (SS) phases. The appearance of the TS phase may be rather naturally expected because the increase of t eff enhances only the inter-site AFM interaction and does not affect the on-site triplet pairings. However, the SS phase can not be explained within our effective model because it does not include an AFM interaction with the triangular unit to generate singlet pairs. But the coupled triangular units system at n = 1 is magnetically frustrated and in some cases the ground state is spontaneously dimerized, namely, the spin-singlet pairs are locally formed to relax the magnetic frustration 58,59 . Thus, our SS state may be interpreted as a doped valence-bond-solid state. So, the PM would be just a crossover between TS and SS phases. They are mostly distributed near n = 1. The minimized binding energy and the appearance of PM near t 1 = t 2 , where the frustration is maximal, at < n 2 3 can be also explained in the same sense. The binding energy goes up again with increasing t 2 for t 2 ≥ t 1 .

pairing symmetry
Finally, we discuss the pairing symmetry of our twisted triangular Hubbard model. So far, the possibilities of f-wave and p z -wave symmetries have been suggested both theoretically and experimentally for the superconductivity in K 2 Cr 3 As 3 . To determine it numerically, we calculated the pair-field correlation function † for singlet pairs and Δ i = c i1↑ c i2↓ + c i1↓ c i2↑ for triplet pairs. Here, c iλσ annihilates an electron of spin σ on a site λ(=1 or 2) of the bond indexed by i. For reference, the correlation function |D(i, j)| is plotted on a log-log scale as a function of distance |i − j| − 1 in Fig. 5(a,b). Although the system length is too short to estimate the decay ratio accurately, the spin-triplet pair-field correlation is obviously much more dominant than the spin-singlet one. Let us then see the sign distribution of the spin-triplet pair-field correlation function. The results are shown in Fig. 5(c,d), where the twisted triangular tube is expanded along the side of t 1 triangles parallel to the xy-plane (crystallographic ab-plane) and the conducting c-direction is taken to be the z-axis. Positive correlations are represented by blue color and negative correlations are represented by red color. We found that the correlation function keeps the sign unchanged for a rotation about the z-axis, namely, all D(i, j)'s parallel to the xy plane have the same sign. This means that the pairing symmetry is isotropic with a rotation about the z-axis. While, D(i, j) changes its sign alternately along the z line, e.g., on a dotted line in Fig. 5(c,d). This is consistent with the p z -wave pairing symmetry. More details are given in Supplementary Information.

Conclusion and Discussion
We proposed a universal mechanism for spin-triplet superconductivity (SC) in a coupled odd-gons (e.g., triangular unit) Hubbard system. First, we show that two fermions on a Hubbard ring with odd number of sites form a spin-triplet pair in the ground state. When the odd-gons are weakly coupled by hopping integral, a global FM order is induced by particle moving between the odd-gons; and with increasing the hopping integral, a spin-triplet SC state appears through a melting of the FM order. We demonstrated the validity of this mechanism by considering the twisted triangular Hubbard tube as a model of Q1D superconductors A 2 Cr 3 As 3 using the DMRG technique. From the analogy of the high-temperature SC which is located next to the AFM insulating phase, it is interesting that the SC pairing energy is most enhanced in the vicinity of the Mott-Hubbard metal-to-insulator transition at filling n = 2/l o . We then derived a simple effective model, namely, two-orbital Hubbard model with inter-orbital FM interaction, to provide a general application to coupled odd-gons system. We also confirmed that the spin-triplet pairing of the twisted triangular Hubbard tube occurs predominantly in the p z -wave channel. The Nagaoka 55 and flat-band 60 mechanisms are well known as the origin of ferromagnetism. However, both of them are unsuitable for explaining the spin-triplet SC since they lead not to Cooper pairs but only to a saturated magnetization. On the other hand, in our model, a global ferromagnetism is generated which melts away to triplet SC when the local spin-triplet pairs in odd-gons are coupled. It is similar to the situation where preformed singlet pairs on rungs give rise to singlet SC when these are coupled to form a two-leg ladder system. Since the geometry of hopping network between the odd-gons is not essential for the appearance of global ferromagnetism, there Figure 5. Pair-field correlation functions D(i, j) between (a) two intra-triangle bonds and (b) intra-triangle and inter-triangle bonds, calculated with L = 60 × 3 cluster for t 2 = 0.25 at n = 1/2. In the inset, the circle and square denote the sites λ = 1 and λ = 2 within a pair bond, respectively. Horizontal axis |i − j| + 1 counts the distance between the circles along the conduction (z) direction. (c,d) Spin-triplet pair-field correlation function of the twisted triangular Hubbard tube, where the twisted triangular tube is expanded along the side of t 1 triangles parallel to the xy-plane and the conducting c direction is taken to be the z-axis. The reference bonds A and B are indicated by thick-black lines. The width of each ellipse for a bond at position i is proportional to 1/log|D(i, A)| and 1/log|D(i, B)| in (c) and (d), respectively. Blue (red) color represents positive (negative) value of the correlation function.