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arXiv:0805.3988 · Physical Review B

Phase diagram of the asymmetric Hubbard model

Pavol Farkašovský

Published Feb 12, 2008• 21 citations
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Abstract

The ground-state phase diagram of the asymmetric Hubbard model is studied in one and two dimensions by a well-controlled numerical method. The method allows us to calculate directly the probabilities of particular phases in the approximate ground state and thus to specify the stability domains corresponding to phases with the highest probabilities. Depending on the electron filling $n$ and the magnitude of the asymmetry ${t}_{f}∕{t}_{d}$ between the hopping integrals of $f$ and $d$ electrons two different scenarios in formation of ground states are observed. At low electron fillings $(n\ensuremath{\leqslant}1∕3)$ the ground states are always phase segregated in the limit of strong asymmetry $({t}_{d}⪢{t}_{f})$. With decreasing asymmetry the system undergoes a transition to the phase-separated state and then to the homogeneous state. For electron fillings $n>1∕3$ and weak Coulomb interactions the ground state is homogeneous for all values of asymmetry, while for intermediate and strong interactions the system exhibits the same sequence of phase transitions as for $n$ small. Moreover, it is shown that the segregated phase is significantly stabilized with increasing electron filling, while the separated phases disappear gradually from the ground-state phase diagrams.

DOI
10.1103/physrevb.77.085110

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The ground-state phase diagram of the asymmetric Hubbard model is studied in one and two dimensions by a well-controlled numerical method. The method allows us to calculate directly the probabilities of particular phases in the approximate ground state and thus to specify the stability domains corresponding to phases with the highest probabilities. Depending on the electron filling nnn and the magnitude of the asymmetry tf∕td{t}_{f}∕{t}_{d}tf​∕td​ between the hopping integrals of fff and ddd electrons two different scenarios in formation of ground states are observed. At low electron fillings (n\ensuremath⩽1∕3)(n\ensuremath{\leqslant}1∕3)(n\ensuremath⩽1∕3) the ground states are always phase segregated in the limit of strong asymmetry (td⪢tf)({t}_{d}⪢{t}_{f})(td​⪢tf​). With decreasing asymmetry the system undergoes a transition to the phase-separated state and then to the homogeneous state. For electron fillings n>1∕3n>1∕3n>1∕3 and weak Coulomb interactions the ground state is homogeneous for all values of asymmetry, while for intermediate and strong interactions the system exhibits the same sequence of phase transitions as for nnn small. Moreover, it is shown that the segregated phase is significantly stabilized with increasing electron filling, while the separated phases disappear gradually from the ground-state phase diagrams.
We study the ground state of the two-dimensional Anderson-Hubbard model using a quantum real space renormalization group method. We obtain the phase diagram near half filling. The system is always insulating with disorder. At half filling, the system undergoes a transition from a gapless (Anderson) insulator to an incompressible (Mott-Hubbard) insulator as the interaction UUU reaches a critical value UcU_cUc​. Away from half filling, the insulating phase is always gapless and is found to be controlled by the Anderson fixed point at half filling. This result is similar to that obtained in the corresponding one dimensional system and suggests strongly the importance of the electron-electron correlation in this gapless insulating phase.
We study the ground state of the two-dimensional Anderson-Hubbard model using a quantum real space renormalization group method. We obtain the phase diagram near half filling. The system is always insulating with disorder. At half filling, the system undergoes a transition from a gapless (Anderson) insulator to an incompressible (Mott-Hubbard) insulator as the interaction U reaches a critical value Uc{\mathit{U}}_{\mathit{c}}Uc​. Away from half filling, the insulating phase is always gapless and is found to be controlled by the Anderson fixed point at half filling. This result is similar to that obtained in the corresponding one-dimensional system and suggests strongly the importance of the electron-electron correlation in this gapless insulating phase.
We address some open questions regarding the phase diagram of the one-dimensional Hubbard model with asymmetric hopping coefficients and balanced species. In the attractive regime we present a numerical study of the passage from on-site pairing dominant correlations at small asymmetries to charge-density waves in the region with markedly different hopping coefficients. In the repulsive regime we exploit two analytical treatments in the strong- and weak-coupling regimes in order to locate the onset of phase separation at small and large asymmetries, respectively.
The extrapolation of small-cluster exact-diagonalization calculations is used to examine the zero-temperature phase diagram of the simplified Hubbard model in an external magnetic field. Using a general expression for hopping matrix elements ${(t}_{\mathrm{ij}}\ensuremath{\sim}{q}^{|i\ensuremath{-}j|})$ the influence of the long-range hopping (the asymmetry of the density of states) on ground-state properties, and particularly on ferromagnetism is studied. It is found that the long-range hopping (the asymmetric density of states) stabilizes the ferromagnetic state for a wide range of electron interactions UUU and electron concentrations nnn. The critical value of magnetic field hhh below which the ferromagnetic state becomes unstable is calculated numerically. It is shown that below this critical value the ferromagnetic state breaks down into a metallic state for weak coupling and an insulating state for strong coupling. The phase diagram of the model in the $h\ensuremath{-}U$ plane is presented for physically the most interesting cases.
The ground-state phase diagram of the one-dimensional asymmetric Hubbard model at half-filling is studied by means of a perturbation expansion in the large-U limit and White's density matrix renormalization-group method for various values of the couplings. We argue that the introduction of asymmetry in the hopping parameters is a relevant perturbation for any repulsive U and drives the system into an ordered N\'eel antiferromagnet. The transition belongs to the Kosterlitz-Thouless universality class.