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arXiv:1403.6762 · Physical Review B

Unconventional pairing and electronic dimerization instabilities in the doped Kitaev-Heisenberg model

Daniel D. Scherer, Michael M. Scherer, Giniyat Khaliullin, Carsten Honerkamp A, Bernd Rosenow

Published Jul 28, 2014• 29 citations
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Abstract

We study the quantum many-body instabilities of the $t -J_{\mathrm{K}} - J_{\mathrm{H}}$ Kitaev-Heisenberg Hamiltonian on the honeycomb lattice as a minimal model for a doped spin-orbit Mott insulator. This spin-$1/2$ model is believed to describe the magnetic properties of the layered transition-metal oxide Na$_2$IrO$_3$. 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 ($\mathbb{Z}_2$ quantum spin liquids and magnetically ordered phases). We corroborate superconducting triplet $p$-wave instabilities driven by ferromagnetic exchange and various singlet pairing phases. For filling $δ> 1/4$, the $p$-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.

DOI
10.1103/physrevb.90.045135

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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.
Interacting fermions on the half-filled honeycomb lattice with short-range repulsions have been suggested to host a variety of interesting many-body ground states, e.g., a topological Mott insulator. A number of recent studies of the spinless case in terms of exact diagonalization, the infinite density matrix renormalization group, and the functional renormalization group, however, indicate a suppression of the topological Mott insulating phase in the whole range of interaction parameters. Here, we complement the previous studies by investigating the quantum many-body instabilities of the physically relevant case of spin-1/2 fermions with onsite, nearest-neighbor, and second-nearest-neighbor repulsion. To this end, we employ the multipatch functional renormalization group for correlated fermions with refined momentum resolution observing the emergence of an antiferromagnetic spin-density wave and a charge-density wave for dominating onsite and nearest-neighbor repulsions, respectively. For dominating second-nearest neighbor interaction our results favor an ordering tendency towards a charge-modulated ground state over the topological Mott insulating state. The latter evades a stabilization as the leading instability by the additional onsite interaction.
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.
Spinless fermions on the honeycomb lattice with repulsive nearest-neighbor interactions are known to harbour a quantum critical point at half-filling, with critical behavior in the Gross-Neveu (chiral Ising) universality class. The critical interaction strength separates a weak-coupling semimetallic regime from a commensurate charge-density-wave phase. The phase diagram of this basic model of correlated fermions on the honeycomb lattice beyond half-filling is, however, less well established. Here, we perform an analysis of its many-body instabilities using the functional renormalization group method with a basic Fermi surface patching scheme, which allows us to treat instabilities in competing channels on equal footing also away from half-filling. Between half-filling and the Van Hove filling, the free Fermi surface is holelike and we again find a charge-density wave instability to be dominant at large interactions. Moreover, its characteristics are those of the half-filled case. Directly at the Van Hove filling, the nesting property of the free Fermi surface stabilizes a dimerized bond-order phase. At lower filling, the free Fermi surface becomes electronlike and a superconducting instability with fff-wave symmetry is found to emerge from the interplay of intra-unit-cell repulsion and collective fluctuations in the proximity to the charge-density wave instability. We estimate the extent of the various phases and extract the corresponding order parameters from the effective low-energy Hamiltonians.
We provide analytical and numerical evidence of spin-triplet Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductivity in the itinerant Kitaev-Heisenberg model (antiferromagnetic Kitaev coupling and ferromagnetic Heisenberg coupling) on the honeycomb lattice around quarter filling. The strong spin-orbit coupling in our model leads to the emergence of six inversion symmetry centers for the Fermi surface at nonzero momenta in the first Brillouin zone. We show how the Cooper pairs condense into these nontrivial momenta, causing spatial modulation of the superconducting order parameter. Applying a Ginzburg-Landau expansion analysis, we find that the superconductivity has three separated degenerate ground states with three different spin-triplet pairings. Exact diagonalizations on finite clusters support this picture while ruling out a spin (charge) density wave.
We investigate the quantum many-body instabilities of the extended Hubbard model for spinless fermions on the honeycomb lattice with repulsive nearest-neighbor and second-nearest-neighbor density-density interactions. Recent exact diagonalization and infinite density matrix renormalization group results suggest that a putative topological Mott insulator phase driven by the second-nearest-neighbor repulsion is suppressed, while other numerically exact approaches support the topological Mott insulator scenario. In the present work, we employ the functional renormalization group (fRG) for correlated fermionic systems. Our fRG results hint at a strong suppression of the scattering processes stabilizing the topological Mott insulator. From analyzing the effects of fermionic fluctuations, we obtain a phase diagram, which is the result of the competition of various charge ordering instabilities.