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arXiv:2109.11152 · Physical review. B./Physical review. B

Phase diagram of the Hubbard model on a honeycomb lattice: A cluster slave-spin study

Minghuan Zeng, Y.-J. Wang, Tianxing Ma

Published Jan 31, 2022• 9 citations
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Abstract

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 $U$ being its parameters. At half filling, a single direct and continuous phase transition between the semimetal and antiferromagnetic (AFM) insulator is found at ${U}_{\text{AFM}}=2.43t$ that is in the Gross-Neveu-Yukawa universality class, where a relation between the staggered magnetization $M$ 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 ${U}_{\text{Mott}}=8.36t$, which is responsible for a broad crossover around ${U}_{c}=5.4t$ 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 $U>{U}_{\text{AFM}}$, 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.

DOI
10.1103/physrevb.105.035155

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The cluster slave-spin method is implemented to research the ground state properties of the honeycomb lattice Hubbard model with doping \ensuremathδ\ensuremath{\delta}\ensuremathδ 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 \ensuremathΔAFM{\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}\ensuremathΔAFM​ is established as M\ensuremath∝\ensuremathΔAFMM\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}M\ensuremath∝\ensuremathΔAFM​, compared to M\ensuremath∝\ensuremathΔAFM(ln\ensuremathΔAFM)2M\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}{(ln{\mathrm{\ensuremath{\Delta}}}_{\text{AFM}})}^{2}M\ensuremath∝\ensuremathΔAFM​(ln\ensuremathΔAFM​)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 \ensuremathδ\ensuremath{\delta}\ensuremathδ, in contrast to the square lattice case. In the doped system, the compressibility \ensuremathκ\ensuremath{\kappa}\ensuremathκ near the van Hove singularity at \ensuremathδ=1/4\ensuremath{\delta}=1/4\ensuremathδ=1/4 is suppressed substantially by the interaction before the semimetal to AFM transition occurs, whereas \ensuremathκ\ensuremath{\kappa}\ensuremathκ 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\ensuremath-\ensuremathδU\text{\ensuremath{-}}\ensuremath{\delta}U\ensuremath-\ensuremathδ plane is presented, consisting of four regimes: the AFM insulator at \ensuremathδ=0\ensuremath{\delta}=0\ensuremathδ=0 for U>UAFMU>{U}_{\text{AFM}}U>UAFM​, the AFM metal with compressibility \ensuremathκ>0\ensuremath{\kappa}>0\ensuremathκ>0 or \ensuremathκ<0\ensuremath{\kappa}<0\ensuremathκ<0, and the PM semimetal; the AFM metal with \ensuremathκ<0\ensuremath{\kappa}<0\ensuremathκ<0 only exists in an extremely small area near the phase boundary between the AFM and PM state.
The cluster slave-spin method is employed to investigate systematically the ground state properties of the Hubbard model on a square lattice with doping $\ensuremath{\delta}$ and coupling strength UUU being its parameters. At half-filling, a relation between the staggered magnetization MMM and the antiferromagnetic (AFM) gap ${\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}$ is established in the small UUU limit to compare with that from the Hartree-Fock theory, and a first-order metal-insulator Mott transition in the paramagnetic state is substantiated, which is characterized by discontinuities and hystereses at UMott=10t{U}_{\text{Mott}}=10tUMott​=10t. The interaction Uc{U}_{c}Uc​ for the crossover in the AFM state, separating the weak- and strong-coupling regimes, is found to remain almost unchanged with large dopings, and smaller than UMott{U}_{\text{Mott}}UMott​ at half-filling because of long range AFM correlations. Finally, an overall phase diagram in the $U\ensuremath{-}\ensuremath{\delta}$ plane is presented, which is composed of four regimes: the AFM insulator at half-filling, the AFM metal with the compressibility $\ensuremath{\kappa}>0$ or $\ensuremath{\kappa}<0$, and the paramagnetic metal, as well as three phase transitions: (i) From the AFM metal to the paramagnetic metal, (ii) between the AFM metal phases with positive and negative $\ensuremath{\kappa}$, and (iii) separating the AFM insulating phase at $\ensuremath{\delta}=0$ from the AFM metal phase for $\ensuremath{\delta}>0$.
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 study the ground-state properties of the Hubbard model on the honeycomb lattice at half filling. Using the variational cluster approximation, We study the semimetal-insulator transition and antiferromagnetism as functions of the on-site Coulomb interaction U . When U is switched on, a small, but finite, single-particle gap appears; thus, we conclude that a tiny gap appears with arbitrary small U > 0. With further increases in U , the system goes into the antiferromagnetic state. We find that the staggered magnetization increases continuously with increasing U and that the antiferromagnetic phase transition is of the second order.
The cluster slave-spin method is introduced to systematically investigate the solutions of the Hubbard model including the symmetry-broken phases. In this method, the electron operator is factorized into a fermionic spinon describing the physical spin and a slave-spin describing the charge fluctuations. Following the U(1)U(1)U(1) formalism derived by Yu and Si [Phys. Rev. B 86, 085104 (2012)], it is shown that the self-consistent equations to explore various symmetry-broken density wave states can be constructed in general with a cluster of multiple slave-spin sites. We employ this method to study the antiferromagnetic (AFM) state in the single band Hubbard model with the two- and four-site clusters of slave spins. While the Hubbard gap, the charge gap due to the doubly occupied states, scales with the Hubbard interaction UUU as expected, the AFM gap $\mathrm{\ensuremath{\Delta}}$, the gap in the spinon dispersion in the AFM state, exhibits a crossover from the weak- to strong-coupling behaviors as UUU increases. Our cluster slave-spin method reproduces not only the traditional mean-field behavior of $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}U$ in the weak-coupling limit, but also the behavior of $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}{t}^{2}/U$ predicted by the superexchange mechanism in the strong-coupling limit. In addition, the holon-doublon correlator as functions of UUU and doping xxx is also computed, which exhibits a strong tendency toward the holon-doublon binding in the strong coupling regime. We further show that the quasiparticle weight obtained by the cluster slave-spin method is in a good agreement with the generalized Gutzwiller approximation in both AFM and paramagnetic states, and the results can be improved beyond the generalized Gutzwiller approximation as the cluster is enlarged from a single site to four sites. Our results demonstrate that the cluster slave-spin method can be a powerful tool to systematically investigate the strongly correlated system.
We study the ground state properties of the doped Hubbard model with strong interactions on the honeycomb lattice by the density matrix renormalization group (DMRG) method. At half-filling, due to the absence of minus sign problem, it is now well established by large-scale quantum Monte Carlo calculations that a Dirac semimetal to antiferromagnetic Mott insulator transition occurs with the increase of the interaction strength UUU for the Hubbard model on the honeycomb lattice. However, an understanding of the fate of the antiferromagnetic Mott insulator when holes are doped into the system is still lacking. In this work, by calculating the local spin and charge density for width-4 cylinders with DMRG, we discover a half-filled stripe order in the doped Hubbard model on the honeycomb lattice. We also perform complementary large-scale mean-field calculations with renormalized interaction strength. We observe half-filled stripe order and find stripe states with filling close to one half are nearly degenerate in energy.
The Hubbard model on the honeycomb lattice undergoes a quantum phase transition from a semimetallic to a Mott insulating phase and from a disordered to an anti-ferromagnetically phase. We show that these transitions occur simultaneously and we calculate the critical coupling Uc=3.835(14)U_c=3.835(14)Uc​=3.835(14) as well as the critical exponents ν=1.181(43)\nu=1.181(43)ν=1.181(43) and β=0.898(37)\beta= 0.898(37)β=0.898(37) which are expected to fall into the SU(2)SU(2)SU(2) Gross-Neveu universality class. For this we employ Hybrid Monte Carlo simulations, extrapolate the single particle gap and the spin structure factors to the thermodynamic and continuous time limits, and perform a data collapse fit. We also determine the zero temperature values of single particle gap and staggered magnetisation on both sides of the phase transition.