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arXiv:1010.5744

Phase diagram of the Hubbard model on honeycomb lattice

Abolhassan Vaezi, Xiao-Gang Wen

Published Oct 27, 2010• 1 citation
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Abstract

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 $U$, we have antiferromagnetic order phase. As we decrease $U$ below $U_{c2}\simeq 3t$, 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 $Z_2$ spin liquid that does not break any lattice symmetry is also possible.

DOI
10.48550/arxiv.1010.5744

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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.
In this paper we investigate possible spin disordered phase in the Hubbard model on the honeycomb lattice. Using a slave-particle theory that include the charge fluctuations, we find a meanfield spin disordered phase in a range of on-site repulsion UUU. The spin disordered state is described by gapped fermions coupled to compact U(1) gauge field. We study the confinement/deconfinement problem of the U(1) gauge theory due to the instantons proliferation. We calculate all allowed instanton terms and compute their quantum numbers. It is shown that the meanfield spin disordered phase is unstable. The instantons proliferation induce a translation symmetry breaking.
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 cluster slave-spin method is implemented to research the ground state properties of the honeycomb lattice Hubbard model with doping $\ensuremath{\delta}$ and coupling UUU being its parameters. At half filling, a single direct and continuous phase transition between the semimetal and antiferromagnetic (AFM) insulator is found at UAFM=2.43t{U}_{\text{AFM}}=2.43tUAFM​=2.43t that is in the Gross-Neveu-Yukawa universality class, where a relation between the staggered magnetization MMM and the AFM energy gap ${\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}$ is established as $M\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}$, compared to $M\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}{(ln{\mathrm{\ensuremath{\Delta}}}_{\text{AFM}})}^{2}$ in the square lattice case. A first-order semimetal to the underlying paramagnetic (PM) insulator Mott transition is corroborated at UMott=8.36t{U}_{\text{Mott}}=8.36tUMott​=8.36t, which is responsible for a broad crossover around Uc=5.4t{U}_{c}=5.4tUc​=5.4t between the weak- and strong-coupling regimes in the AFM state that increases with $\ensuremath{\delta}$, in contrast to the square lattice case. In the doped system, the compressibility $\ensuremath{\kappa}$ near the van Hove singularity at $\ensuremath{\delta}=1/4$ is suppressed substantially by the interaction before the semimetal to AFM transition occurs, whereas $\ensuremath{\kappa}$ near the Dirac points is very close to the noninteracting one, indicating that the Dirac cone structure of the energy dispersion is rather robust. An overall phase diagram in the $U\text{\ensuremath{-}}\ensuremath{\delta}$ plane is presented, consisting of four regimes: the AFM insulator at $\ensuremath{\delta}=0$ for U>UAFMU>{U}_{\text{AFM}}U>UAFM​, the AFM metal with compressibility $\ensuremath{\kappa}>0$ or $\ensuremath{\kappa}<0$, and the PM semimetal; the AFM metal with $\ensuremath{\kappa}<0$ only exists in an extremely small area near the phase boundary between the AFM and PM state.
We review the study of a cluster spin method in the Hubbard model. Based on the spin chargeseparation theory, the electron operator is decomposed into the product of a fermionic spinon and aslave spin (S=1/2)(S=1/2)(S=1/2), which represents the physical charge and spin degrees of freedom, respectively.Zeng et al. confirmed the first-order metal-insulator Mott transition in the paramagnetic statein a square lattice based on the saddle point approximation. Moreover, the relationship betweenthe staggered magnetization and the antiferromagnetic gap is established through asymptoticexpansion. They systematically investigated the staggered magnetization with doping and couplingstrength as parameters. Zeng et al. confirmed the absence of the intermediate spin liquid statein the honeycomb lattice and the first-order metal-insulator Mott transition in the paramagneticstate. The compressibility near van Hove singularity is suppressed most drastically by interaction,while that at low energy (i.e., near the Dirac points) remains close to the noninteracting one,indicating the robustness of the Dirac cone structure. Similarly, the overall phase diagram ofcoupling strength (UUU)-doping plane (δ\deltaδ) is presented.
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 UUU which increases with the ratio ${t}^{\ensuremath{'}}/t$. For ${t}^{\ensuremath{'}}$ 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 ${t}^{\ensuremath{'}}/t$, 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.
Motivated by recent experiments on the triangular lattice Mott-Hubbard system $\ensuremath{\kappa}\text{\ensuremath{-}}{(\mathrm{BEDT}\text{\ensuremath{-}}\mathrm{TTF})}_{2}{\mathrm{Cu}}_{2}{(\mathrm{C}\mathrm{N})}_{3}$, we develop a general formalism to investigate quantum spin liquid insulators adjacent to the Mott transition in Hubbard models. This formalism, dubbed the SU(2) slave-rotor formulation, is an extension of the SU(2) gauge theory of the Heisenberg model to the case of the Hubbard model. Furthermore, we propose the honeycomb lattice Hubbard model (at half filling) as a candidate for a spin liquid ground state near the Mott transition; this is an appealing possibility, as this model can be studied via quantum Monte Carlo simulation without a sign problem. The pseudospin symmetry of Hubbard models on bipartite lattices turns out to play a crucial role in our analysis, and we develop our formalism primarily for the case of a bipartite lattice. We also sketch its development for a general Hubbard model. We develop a mean-field theory to describe spin liquids and some competing states, and apply it to the honeycomb lattice. On the insulating side of the Mott transition, we find an SU(2) algebraic spin liquid (ASL), described by gapless S=1∕2S=1∕2S=1∕2 Dirac fermions (spinons) coupled to a fluctuating SU(2) gauge field. This result contrasts with that obtained via a U(1) slave-rotor approach, which instead found a U(1) ASL. That formulation does not respect the pseudospin symmetry, and is therefore not correct on the honeycomb lattice. We construct a low-energy effective theory describing the ASL phase, the conducting semimetal phase and the Mott transition between them. This physics can be detected in numerical simulations via the simultaneous presence of substantial antiferromagnetic and valence-bond solid correlations. In the SU(2) ASL, these observables have slowly decaying fluctuations in space and time, described by power laws with the same critical exponent. We recover the results of the U(1) slave-rotor formulation in the presence of a strong breaking of pseudospin symmetry. Our analysis suggests that both a third-neighbor electron hopping, and/or pseudospin-breaking terms such as a nearest-neighbor density interaction, may help to stabilize a spin liquid phase.