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Zhongguo kexue. Wulixue Lixue Tianwenxue

Phase diagram of the Hubbard model: A cluster slave-spin study

Wenchang Wu, Minghuan Zeng, Liang Ying, WANG YongJun, Tianxing Ma

Published Apr 20, 2022• 0 citations
Publisher

Abstract

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 $(S=1/2)$, 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 ($U$)-doping plane ($\delta$) is presented.

DOI
10.1360/sspma-2022-0055

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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.
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.
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 study the two-dimensional one-band Hubbard model on the square lattice through the spin rotation invariant Kotliar-Ruckenstein slave-boson method, with an emphasis on the electron-doped regime. The phase diagram determined within mean field approximation reveals three different magnetically ordered phases separated by first-order transitions, and a wide instability region as indicated by a negative compressibility. We show that this conundrum of the unstable mean field domain is resolved by spatially mixed phases of neighboring stable states. We also find charge density wave type instabilities in the antiferromagnetically ordered (AFM) state at wave vectors commensurate with the doping. The corresponding charge and spin density response functions are calculated from a Gaussian fluctuations formalism about a magnetic saddle point, for which we provide a detailed derivation. Further, we present density and spin excitation spectra in the AFM phase, allowing for a quantitative calculation of the spin wave parameters, the gap in the longitudinal spin excitation spectrum, and the density excitations in the upper Hubbard band.
We use the Kotliar-Ruckenstein slave boson formalism to study the phase diagram of the doped antiferromagnetic Mott insulator at zero and finite temperatures. We obtain the commensurate magnetic phase diagram as a function of doping and temperature. At higher doping and temperatures we find a paramagnetic metallic phase. Reducing doping and temperature, we find antiferromagnetic phases with both metallic and insulating solutions. We calculate the electronic density of states in various regions of the phase diagram. We use our results for the electronic compressibility and magnetic susceptibility to analyze the stability of the various phases obtained within this mean field formalism. We find various parameter regimes where the method fails to provide physical solutions. We argue that those regions correspond to more exotic phases that cannot be described within the current mean field theory.
We present a comprehensive study of the two-dimensional one-band Hubbard model applying the spin-rotation-invariant slave-boson method. We utilize a spiral magnetic mean field and fluctuations around a paramagnetic mean field to determine the magnetic phase diagram and find the two approaches to be in good agreement. Apart from the commensurate phases characterized by ordering wave vectors $\mathbf{Q}=(\ensuremath{\pi},\ensuremath{\pi})$, $(0,\ensuremath{\pi})$, and (0,0) we find incommensurate phases where the ordering wave vectors Q=(Q,Q)\mathbf{Q}=(Q,Q)Q=(Q,Q) and $(Q,\ensuremath{\pi})$ vary continuously with filling, interaction strength, or temperature. The mean-field quantities magnetization and effective mass are found to change discontinuously at the phase boundaries separating the (Q,Q)(Q,Q)(Q,Q) and $(Q,\ensuremath{\pi})$ phases, indicating a first-order transition. The band structure and Fermi surface is shown in selected cases. The dynamic spin and charge susceptibilities as well as the structure factors are calculated and discussed, including the emergence of collective modes of the zero sound and Mott insulator type. The dynamical conductivity is calculated in dependence of doping, interaction strength, and temperature. Finally, a temperature-interaction strength phase diagram is established.
The phase diagram of the strongly correlated Hubbard model with intrinsic spin-orbit coupling on the honeycomb lattice is explored here. We obtain the low-energy effective model describing the spin degree of freedom. The resulting model is then studied by the Schwinger boson and Schwinger fermion approaches. The Schwinger boson method elucidates the boundary between the spin liquid phase and the magnetically ordered phases, Neel order, and incommensurate Neel order. Increasing the strength of the spin-orbit coupling is shown to narrow the width of the spin liquid region. The Schwinger fermion approach sheds further light on the nature of the spin liquid phase. We obtained three different candidates for the spin liquid phase within the mean-field approximation, namely, the gapless spin liquid, topological Mott insulator (fractionalized topological insulator), and chiral spin liquid phases. However, we argue that the gauge fluctuations and the instanton effect may suppress the first two spin liquids, while the chiral spin liquid is stable against gauge fluctuations due to its nontrivial topology.