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arXiv:2508.16720 · Physical review. B./Physical review. B

Searching for superconductivity in doped triangular lattice Kitaev magnets

Andrew Hardy, Ryan Levy, Arun Paramekanti

Published Apr 22, 2026• 2 citations
arXivHTMLPDF

Abstract

Motivated by exploring correlated metals with frustrating bond-dependent exchange interactions, we study hole and electron doped Kitaev Mott insulators on the triangular lattice. Using homogeneous parton mean field theory, we find that the stripe antiferromagnetic (AFM) order for Kitaev coupling $K>0$ and the ferromagnetic (FM) order for $K<0$, both vanish at sufficiently large doping, beyond which we find regimes with chiral $d\pm i d$ singlet pairing and $p\pm ip$ triplet pairing respectively. Our tensor network computations however reveal that the superconducting correlations are strongly suppressed; while FM order stubbornly persists for the doped $K<0$ model, the doped $K>0$ model features emergent spin-charge modulated stripe orders. At higher hole doping for $K > 0$, where AFM order is more strongly suppressed than for the electron doped case, incorporating a sufficiently strong nearest-neighbor attraction yields evidence for singlet $d$-wave superconductivity with Luttinger parameter $K_{\rm sc} < 1$. Our work sets the stage for a broader exploration of doping effects in triangular lattice magnets such as NaRuO$_2$ which feature bond-dependent exchange interactions.

DOI
10.1103/wftz-1syr

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Motivated by exploring correlated metals with frustrating bond-dependent exchange interactions, we study hole and electron doped Kitaev Mott insulators on the triangular lattice. Using homogeneous parton mean field theory, we find that the stripe antiferromagnetic (AFM) order for Kitaev coupling K>0K>0K>0 and the ferromagnetic (FM) order for K<0K<0K<0, both vanish at sufficiently large doping, beyond which we find regimes with chiral d±idd\pm i dd±id singlet pairing and p±ipp\pm ipp±ip triplet pairing respectively. Our tensor network computations however reveal that the superconducting correlations are strongly suppressed; while FM order stubbornly persists for the doped K<0K<0K<0 model, the doped K>0K>0K>0 model features emergent spin-charge modulated stripe orders. At higher hole doping for K>0K > 0K>0, where AFM order is more strongly suppressed than for the electron doped case, incorporating a sufficiently strong nearest-neighbor attraction yields evidence for singlet ddd-wave superconductivity with Luttinger parameter Ksc<1K_{\rm sc} < 1Ksc​<1. Our work sets the stage for a broader exploration of doping effects in triangular lattice magnets such as NaRuO2_22​ which feature bond-dependent exchange interactions.
We investigate the evolution of the Mott insulators in the triangular lattice Hubbard Model, as a function of hole doping $\ensuremath{\delta}$ in both the strong and intermediate coupling limits. Using the advanced density matrix renormalization group (DMRG) method, at light hole doping $\ensuremath{\delta}\ensuremath{\lesssim}10%$, we find a significant difference between strong and intermediate couplings. Notably, at intermediate coupling an unusual metallic state emerges, with short ranged spin correlations but long ranged spin-chirality order. Moreover, no clear Fermi surface or wave vector is observed, this chiral metal also exhibits staggered loop current, which breaks the translational symmetry. These features disappear on increasing interaction strength or on further doping. At strong coupling, the 120 degree magnetic order of the insulating magnet persists for light doping, and produces hole pockets with a well-defined Fermi surface. On further doping, $\ensuremath{\delta}\ensuremath{\approx}10%\ensuremath{\sim}20%$ SDW order and coherent hole Fermi pockets are found at both strong and intermediate couplings. At even higher doping $\ensuremath{\delta}\ensuremath{\gtrsim}20%$, the SDW order is suppressed and the spin-singlet Cooper pair correlations are simultaneously enhanced. We also briefly comment on the strong particle-hole asymmetry of the model.
We study the quantum many-body instabilities of the t−JK−JHt -J_{\mathrm{K}} - J_{\mathrm{H}}t−JK​−JH​ Kitaev-Heisenberg Hamiltonian on the honeycomb lattice as a minimal model for a doped spin-orbit Mott insulator. This spin-1/21/21/2 model is believed to describe the magnetic properties of the layered transition-metal oxide Na2_22​IrO3_33​. We determine the ground-state of the system with finite charge-carrier density from the functional renormalization group (fRG) for correlated fermionic systems. To this end, we derive fRG flow-equations adapted to the lack of full spin-rotational invariance in the fermionic interactions, here represented by the highly frustrated and anisotropic Kitaev exchange term. Additionally employing a set of Ward identities for the Kitaev-Heisenberg model, the numerical solution of the flow equations suggests a rich phase diagram emerging upon doping charge carriers into the ground-state manifold (Z2\mathbb{Z}_2Z2​ quantum spin liquids and magnetically ordered phases). We corroborate superconducting triplet ppp-wave instabilities driven by ferromagnetic exchange and various singlet pairing phases. For filling δ>1/4δ> 1/4δ>1/4, the ppp-wave pairing gives rise to a topological state with protected Majorana edge-modes. For antiferromagnetic Kitaev and ferromagnetic Heisenberg exchange we obtain bond-order instabilities at van Hove filling supported by nesting and density-of-states enhancement, yielding dimerization patterns of the electronic degrees of freedom on the honeycomb lattice. Further, our flow equations are applicable to a wider class of model Hamiltonians.
We study the hole-doped Kitaev-Heisenberg chain using the density-matrix renormalization group. In the Kitaev-only limit, the bond-directional exchange itself promotes pairing, favoring spin-singlet and spin-triplet superconducting tendencies for antiferromagnetic and ferromagnetic Kitaev couplings, respectively, together with finite-size Majorana edge correlations suggestive of topological superconductivity. In the full Kitaev-Heisenberg chain, cooperative J - and K exchanges broadly stabilize superconductivity, while competition between J and K induces a strong filling dependence and enables superconductivity even when both J and K are weak. At quarter filling, this competition produces a Mott insulator with spontaneous hopping dimerization. These results identify magnetic frustration as a common mechanism underlying superconducting and interaction-driven insulating phases in doped Kitaev systems.
Unravelling competing orders emergent in doped Mott insulators and their interplay with unconventional superconductivity is one of the major challenges in condensed matter physics. To explore the possible superconducting state in a doped Mott insulator, we study the square-lattice t-J model with both the nearest-neighbor and next-nearest-neighbor electron hoppings and spin interactions. By using the state-of-the-art density matrix renormalization group calculation with imposing charge U(1) and spin SU(2) symmetries on the six-leg cylinders, we establish a quantum phase diagram including three phases: a stripe charge density wave phase, a superconducting phase without static charge order, and a superconducting phase coexistent with a weak charge stripe order. Crucially, we demonstrate that the superconducting phase has a power-law pairing correlation that decays much slower than the charge density and spin correlations, which is a quasi-1D descendant of the uniform d-wave superconductor in two dimensions. These findings reveal that enhanced charge and spin fluctuations with optimal doping is able to produce robust d-wave superconductivity in doped Mott insulators, providing a foundation for connecting theories of superconductivity to models of strongly correlated systems.
We study the prerequisites for realizing superconductivity in doped triangular-lattice Mott insulators by considering three distinct parent spin backgrounds, i.e., ${120}^{\ensuremath{\circ}}$ antiferromagnets, quantum spin liquid, and stripy antiferromagnets, and all possible sign combinations $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$ of nearest-neighbor hopping and next-nearest-neighbor hopping (t1,t2)({t}_{1},{t}_{2})(t1​,t2​). Based on density matrix renormalization group calculations, we find that, with finite t2{t}_{2}t2​ and specific sign combinations $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$, the quasi-long-range superconductivity order can always be achieved, regardless of the nature of the parent spin backgrounds. Besides specific hopping signs $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$, these superconductivity phases in triangular lattices are commonly characterized by short-ranged spin correlations and two charges per stripe. In the robust superconductivity phase realized at larger t2/t1{t}_{2}/{t}_{1}t2​/t1​, flipping the signs ${\ensuremath{\tau}}_{2}$ and ${\ensuremath{\tau}}_{1}$ gives rise to the stripe phase without strong pairing and a pseudogaplike phase without Cooper-pair phase coherence, respectively. Interestingly, the roles of the two hopping signs are switched at smaller t2/t1{t}_{2}/{t}_{1}t2​/t1​. Moreover, different sign combinations $({\ensuremath{\tau}}_{1},{\ensuremath{\tau}}_{2})$ would stabilize distinct phases including superconductivity, charge density waves, spin density waves, and pseudogaplike phases accordingly. Our findings suggest the important role of charge kinetic energy in realizing superconductivity in doped triangular-lattice Mott insulators.
We investigate superconductivity in a doped Kitaev quantum spin liquid by applying the many-variable variational Monte Carlo method to the hole-doped ttt-JJJ-type Kitaev model. Using a projected pair-product wave function that can exactly represent the Kitaev quantum spin liquid, we examine the stability of superconducting phases on isotropic two-dimensional clusters. For the ferromagnetic Kitaev interaction, robust triplet ppp-wave superconductivity coexists with ferromagnetism in the low-to-intermediate doping regime but is suppressed as the system approaches the fully polarized ferromagnetic phase. For the antiferromagnetic Kitaev interaction, superconductivity exhibits a change in the dominant pairing symmetry from spin-dependent triplet ppp-wave at low doping to singlet d+idd+idd+id at intermediate doping. By varying the strength of the ferromagnetic Kitaev interaction at fixed doping, we show that the triplet superconductivity increases together with the ferromagnetic moment and becomes strongest slightly below full polarization. Our results provide a theoretical basis for experimental searches for unconventional superconductivity, such as triplet superconductivity coexisting with ferromagnetism, in carrier-doped Kitaev candidate materials.