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arXiv:1311.3430 · Physical Review B

Distinct-symmetry spin-liquid states and phase diagram of the Kitaev-Hubbard model

Long Liang, Ziqiang Wang, Yue Yu

Published Aug 11, 2014• 6 citations
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Abstract

We report the finding of a series of distinct-symmetry spin-liquid (SL) states and a rich phase diagram in a half-filled honeycomb lattice Hubbard model with spin-dependent hopping amplitude ${t}^{\ensuremath{'}}$. We first study the magnetic instability of the system and find two antiferromagnetic orders beyond a critical Hubbard $U$ which increases with the ratio ${t}^{\ensuremath{'}}/t$. For ${t}^{\ensuremath{'}}$ approaching $t$, the semimetal transforms to a $U(1)$ SL and then to the Kitaev ${Z}_{2}$ SL as $U$ increases. In a wide middle range of ${t}^{\ensuremath{'}}/t$, the latter is replaced by a $U(1)$ SL to $SU(2)$ SL transition. The physical properties of the stable SL phases are discussed.

DOI
10.1103/physrevb.90.075119

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We report the finding of a series of distinct-symmetry spin-liquid (SL) states and a rich phase diagram in a half-filled honeycomb lattice Hubbard model with spin-dependent hopping amplitude t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′. We first study the magnetic instability of the system and find two antiferromagnetic orders beyond a critical Hubbard UUU which increases with the ratio t\ensuremath′/t{t}^{\ensuremath{'}}/tt\ensuremath′/t. For t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ approaching ttt, the semimetal transforms to a U(1)U(1)U(1) SL and then to the Kitaev Z2{Z}_{2}Z2​ SL as UUU increases. In a wide middle range of t\ensuremath′/t{t}^{\ensuremath{'}}/tt\ensuremath′/t, the latter is replaced by a U(1)U(1)U(1) SL to SU(2)SU(2)SU(2) SL transition. The physical properties of the stable SL phases are discussed.
Recent numerical work [Z. Y. Meng et al., Nature (London) 464, 847 (2010)] indicates the existence of a spin liquid (SL) phase that intervenes between the antiferromagnetic and semimetallic phases of the half filled Hubbard model on a honeycomb lattice. To better understand the nature of this exotic phase, we study the quantum J(1)-J(2) spin model on the honeycomb lattice, which provides an effective description of the Mott insulating region of the Hubbard model. Employing the variational Monte Carlo approach, we analyze the phase diagram of the model. We find three phases-antiferromagnetic, an unusual Z(2) SL state, and a dimerized state with spontaneously broken rotational symmetry. We identify the Z(2) SL state as the likely candidate for the SL phase of the Hubbard model.
The Kitaev-Hubbard model of interacting fermions is defined on the honeycomb lattice and, at strong coupling, interpolates between the Heisenberg model and the Kitaev model. It is basically a Hubbard model with ordinary hopping ttt and spin-dependent hopping ${t}^{\ensuremath{'}}$. We study this model in the weak to intermediate coupling regime, at half filling, using the cellular dynamical impurity approximation (CDIA), an approach related to dynamical mean field theory but based on Potthoff's variational principle. We identify four phases in the $(U,{t}^{\ensuremath{'}})$ plane: two semimetallic phases with different numbers of Dirac points, an antiferromagnetic insulator, and an algebraic spin liquid. The last two are separated by a first-order transition. These four phases all meet at a single point and could be realized in cold atom systems.
In this paper we generalized the slave-particle technique to study the phase diagram of the Hubbard model on honeycomb lattice which may contain charge fluctuations. For large UUU, we have antiferromagnetic order phase. As we decrease UUU below Uc2≃3tU_{c2}\simeq 3tUc2​≃3t, the system undergoes a first order phase transition into a gapped spin rotation invariant phase. Under a semiclassical approximation of the slave-particle approach, we find that such phase breaks the translation symmetry, the parity and the lattice rotation symmetry. However, beyond the semiclassical approximation, a Z2Z_2Z2​ spin liquid that does not break any lattice symmetry is also possible.
The Hubbard model encapsulates the physics of strongly correlated quantum systems in its most basic form. It has been studied intensively in the context of the high-temperature superconductivity. A number of novel phases were recently proposed for Hubbard-like models on the honeycomb lattice, the structure of graphene. We analyzed the Hubbard model of spin- 12\frac{1}{2}21​ fermions on the honeycomb lattice at half-filling using large-scale quantum Monte Carlo simulations. We find that the weak coupling semimetal and the antiferromagnetic Mott insulator at strong interaction are separated by an extended gapped phase in an intermediate coupling regime. Exploring excitation gaps, various correlation functions as well as probing for flux quantization, we conclude that a quantum spin liquid, lacking any conventional order, emerges with local charge and spin correlations, best described by a resonating valence bonds state.
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.
A spin liquid is a novel quantum state of matter with no conventional order parameter where a finite charge gap exists even though the band theory would predict metallic behavior. Finding a stable spin liquid in two or higher spatial dimensions is one of the most challenging and debated issues in condensed matter physics. Very recently, it has been reported that a model of graphene, i.e., the Hubbard model on the honeycomb lattice, can show a spin liquid ground state in a wide region of the phase diagram, between a semi-metal (SM) and an antiferromagnetic insulator (AFMI). Here, by performing numerically exact quantum Monte Carlo simulations, we extend the previous study to much larger clusters (containing up to 2592 sites), and find, if any, a very weak evidence of this spin liquid region. Instead, our calculations strongly indicate a direct and continuous quantum phase transition between SM and AFMI.