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arXiv:1212.5218 · Physical Review B

Global phase diagram of a doped Kitaev-Heisenberg model

Satoshi Okamoto

Published Feb 22, 2013• 70 citations
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Abstract

The global phase diagram of a doped Kitaev-Heisenberg model is studied using an $SU(2)$ slave-boson mean-field method. Near the Kitaev limit, $p$-wave superconducting states which break the time-reversal symmetry are stabilized as reported by You et al. [Phys. Rev. B 86, 085145 (2012)] irrespective of the sign of the Kitaev interaction. By further doping, a $d$-wave superconducting state appears when the Kitaev interaction is antiferromagnetic, while another $p$-wave superconducting state appears when the Kitaev interaction is ferromagnetic. This $p$-wave superconducting state does not break the time-reversal symmetry as reported by Hyart et al. [Phys. Rev. B 85, 140510 (2012)], and such a superconducting state also appears when the antiferromagnetic Kitaev interaction and the ferromagnetic Heisenberg interaction compete. This work, thus, demonstrates the clear difference between the antiferromagnetic Kitaev model and the ferromagnetic Kitaev model when carriers are doped while these models are equivalent in the undoped limit, and how novel superconducting states emerge when the Kitaev interaction and the Heisenberg interaction compete.

DOI
10.1103/physrevb.87.064508

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The global phase diagram of a doped Kitaev-Heisenberg model is studied using an SU(2)SU(2)SU(2) slave-boson mean-field method. Near the Kitaev limit, ppp-wave superconducting states which break the time-reversal symmetry are stabilized as reported by You et al. [Phys. Rev. B 86, 085145 (2012)] irrespective of the sign of the Kitaev interaction. By further doping, a ddd-wave superconducting state appears when the Kitaev interaction is antiferromagnetic, while another ppp-wave superconducting state appears when the Kitaev interaction is ferromagnetic. This ppp-wave superconducting state does not break the time-reversal symmetry as reported by Hyart et al. [Phys. Rev. B 85, 140510 (2012)], and such a superconducting state also appears when the antiferromagnetic Kitaev interaction and the ferromagnetic Heisenberg interaction compete. This work, thus, demonstrates the clear difference between the antiferromagnetic Kitaev model and the ferromagnetic Kitaev model when carriers are doped while these models are equivalent in the undoped limit, and how novel superconducting states emerge when the Kitaev interaction and the Heisenberg interaction compete.
The possible ground states of the undoped and doped Kitaev–Heisenberg model on a triangular lattice are studied. For the undoped system, a combination of the numerical exact diagonalization calculation and the four-sublattice transformation analysis suggests one possible exotic phase and four magnetically ordered phases, including a collinear stripe pattern and a noncollinear spiral pattern in the global phase diagram. The exotic phase near the antiferromagnetic (AF) Kitaev point is further investigated using the Schwinger-fermion mean-field method, and we obtain an energetically favorable Z 2 chiral spin liquid with a Chern number ±2 as a promising candidate. At finite doping, we find that the AF Heisenberg coupling supports an s -wave or a -wave superconductivity (SC), while the AF and the ferromagnetic Kitaev interactions favor a -wave SC and a time-reversal invariant topological p -wave SC, respectively. Possible experimental realizations and related candidate materials are also discussed.
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.
We study the Kitaev-Heisenberg-$\mathrm{\ensuremath{\Gamma}}$ model with antiferromagnetic Kitaev exchanges in the strong anisotropic (toric code) limit to understand the phases and the intervening phase transitions between the gapped Z2{Z}_{2}Z2​ quantum spin liquid and the spin-ordered (in the Heisenberg limit) as well as paramagnetic phases (in the pseudodipolar, $\mathrm{\ensuremath{\Gamma}}$, limit). We find that the paramagnetic phase obtained in the large $\mathrm{\ensuremath{\Gamma}}$ limit has no topological entanglement entropy and is proximate to a gapless critical point of a system described by equal superposition of differently oriented stacked one-dimensional ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry protected topological phases. Using a combination of exact diagonalization calculations and field theoretic analysis we map out the phases and phase transitions to reveal the complete phase diagram as a function of the Heisenberg, the Kitaev, and the pseudodipolar interactions. Our work shows a rich plethora of unconventional phases and phase transitions and provides a comprehensive understanding of the physics of anisotropic Kitaev-Heisenberg-$\mathrm{\ensuremath{\Gamma}}$ systems along with our recent paper [Phys. Rev. B 102, 235124 (2020)] where the ferromagnetic Kitaev exchange was studied.
We explore the phase diagram of the Kitaev-Heisenberg model with nearest neighbor interactions on the honeycomb lattice using the exact diagonalization of finite systems combined with the cluster mean field approximation, and supplemented by the insights from analytic approaches: the linear spin-wave and second-order perturbation theories. This study confirms that by varying the balance between the Heisenberg and Kitaev term, frustrated exchange interactions stabilize in this model either one of four phases with magnetic long range order: N\'eel phase, ferromagnetic phase, and two other phases with coexisting antiferromagnetic and ferromagnetic bonds, zigzag and stripy phase, or one of two distinct spin-liquid phases. Out of these latter disordered phases, the one with ferromagnetic Kitaev interactions has a substantially broader range of stability as the neighboring competing ordered phases, ferromagnetic and stripy, have very weak quantum fluctuations. Focusing on the quantum spin-liquid phases, we study spatial spin correlations and dynamic spin structure factor of the model by the exact diagonalization technique, and discuss the evolution of gapped low-energy spin response across the quantum phase transitions between the disordered spin liquid and phases with long range magnetic order.
The ground-state phase diagram of the two-dimensional t-J model is investigated in the context of the tensor network algorithm in terms of the graded Projected Entangled-Pair State representation of the ground-state wave functions. There is a line of phase separation between the Heisenberg anti-ferromagnetic state without hole and a hole-rich state. For both J=0.4t and J=0.8t, a systematic computation is performed to identify all the competing ground states for various dopings. It is found that, besides a possible Nagaoka's ferromagnetic state, the homogeneous regime consists of four different phases: one phase with charge and spin density wave order coexisting with a p_x (p_y)-wave superconducting state, one phase with the symmetry mixing of d+s-wave superconductivity in the spin-singlet channel and p_x (p_y)-wave superconductivity in the spin-triplet channel in the presence of an anti-ferromagnetic background, one superconducting phase with extended s-wave symmetry, and one superconducting phase with p_x (p_y)-wave symmetry in a ferromagnetic background.
We investigate the hole-doped Kitaev-Heisenberg (ttt-JJJ-KKK) model on a two-leg ladder geometry using the density-matrix renormalization group (DMRG). We first consider the behavior of the antiferromagnetic Kitaev (AFK) spin-liquid phase as a function of hopping strength ttt and doping level. This reveals intriguing pairing tendencies only for tK≲0.65\frac{t}{K} \lesssim 0.65Kt​≲0.65, consistent with prior results on three-leg ladders, and firmly supports the emerging picture that the physics of doped Kitaev spin liquids strongly depends on the kinetic energy of the doped holes. Analysis of one- and two-hole doping uncovers close links between the spatial profiles of the plaquette operator and the charge density. We construct a doping-dependent phase diagram for antiferromagnetic Heisenberg interactions and intermediate hopping t=1t=1t=1. Upon doping, the rung-singlet region develops dominant superconducting correlations. Charge-density-wave correlations dominate at weak doping near the transition to the stripy phase. Spin-density wave-like behavior is found in the AFK and ferromagnetic Kitaev limits, and in the stripy phase.