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arXiv:cond-mat/9709080 · Journal of Physics Condensed Matter

The Mott metal - insulator transition in the two-dimensional Hubbard model at half-filling with lifetime effects within the moment approach

José Jesús Rodríguez-Núñez, Sandra Schafroth

Published Jun 15, 1998• 3 citations
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Abstract

We explore the effect of the self-energy, , having a single pole, , with spectral weight and quasi-particle lifetime , on the density of states. We obtain the set of parameters , , and by means of the moment approach (exact sum rules) of Nolting. Due to our choice of self-energy, the system is not a Fermi liquid for any value of the interaction, a result which also holds in the moment approach of Nolting without lifetime effects. Our self-energy satisfies the Kramers - Kronig relationships since it is analytic in one of the complex half-planes. By increasing the value of the local interaction, , at half-filling , there is a transition from a paramagnetic metal to a paramagnetic insulator (a Mott metal - insulator transition) for values of of the order of ( W is the bandwidth) which is in agreement with numerical results for finite lattices and for an infinite number of dimensions . These results expose the main weakness of the spherical approximation of Nolting: a finite gap for any finite value of the interaction, i.e., an insulator for any finite value of . Lifetime effects are absolutely indispensable to making our scheme work better than that based on improving the narrowing band factor, , beyond that obtained from the spherical approximation of Nolting.

DOI
10.1088/0953-8984/10/23/002

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We explore the effect of the self-energy, , having a single pole, , with spectral weight and quasi-particle lifetime , on the density of states. We obtain the set of parameters , , and by means of the moment approach (exact sum rules) of Nolting. Due to our choice of self-energy, the system is not a Fermi liquid for any value of the interaction, a result which also holds in the moment approach of Nolting without lifetime effects. Our self-energy satisfies the Kramers - Kronig relationships since it is analytic in one of the complex half-planes. By increasing the value of the local interaction, , at half-filling , there is a transition from a paramagnetic metal to a paramagnetic insulator (a Mott metal - insulator transition) for values of of the order of ( W is the bandwidth) which is in agreement with numerical results for finite lattices and for an infinite number of dimensions . These results expose the main weakness of the spherical approximation of Nolting: a finite gap for any finite value of the interaction, i.e., an insulator for any finite value of . Lifetime effects are absolutely indispensable to making our scheme work better than that based on improving the narrowing band factor, , beyond that obtained from the spherical approximation of Nolting.
We study the Hubbard model in dimension d= infinity , within an approximation that simultaneously reproduces the atomic limit and the weak-coupling limit up to second order in the interaction strength U. This method, which is a slightly modified version of an approximation suggested by Edwards and Hertz, is applied to the calculation of the self-energy and of the spectral function in the paramagnetic regime for different U, band fillings n, and temperatures T. We find Fermi-liquid behaviour for small U and (for half-filling n=0.5) a metal-insulator (Mott) transition at a critical value U cr , but also a metallic non-Fermi-liquid phase for intermediate values U<U cr . We compare our results with results obtained within the second-order perturbation theory in U and (at finite temperature) with the (essentially exact) quantum Monte Carlo (QMC) results. Qualitatively, the approximation reproduces the correct tendencies obtained by QMC, but in detail there are still discrepancies, in particular for intermediate values of U.
We evaluate the one-particle and double-occupied Green functions for the Hubbard model at half-filling using the moment approach of Nolting [Z. Phys. 255, 25 (1972); Grund Kurs: Theoretische Physik. 7 Viel-Teilchen-Theorie (Verlag Zimmermann-Neufang, Ulmen, 1992)]. Our starting point is a self-energy, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ which has a single pole, $\ensuremath{\Omega}(\stackrel{\ensuremath{\rightarrow}}{k}),$ with spectral weight, $\ensuremath{\alpha}(\stackrel{\ensuremath{\rightarrow}}{k}),$ and quasiparticle lifetime, $\ensuremath{\gamma}(\stackrel{\ensuremath{\rightarrow}}{k})$ [J. J. Rodr\'{\i}guez-N\'u\~nez and S. Schafroth, J. Phys. Condens. Matter 10, L391 (1998); J. J. Rodr\'{\i}guez-N\'u\~nez, S. Schafroth, and H. Beck, Physica B (to be published); (unpublished)]. In our approach, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ becomes the central feature of the many-body problem and due to three unknown $\stackrel{\ensuremath{\rightarrow}}{k}$ parameters we have to satisfy only the first three sum rules instead of four as in the canonical formulation of Nolting [Z. Phys. 255, 25 (1972); Grund Kurs: Theoretische Physik. 7 Viel-Teilchen-Theorie (Verlag Zimmermann-Neufang, Ulmen, 1992)]. This self-energy choice forces our system to be a non-Fermi liquid for any value of the interaction, since it does not vanish at zero frequency. The one-particle Green function, $G(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ shows the fingerprint of a strongly correlated system, i.e., a double peak structure in the one-particle spectral density, $A(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ vs $\ensuremath{\omega}$ for intermediate values of the interaction. Close to the Mott insulator-transition, $A(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ becomes a wide single peak, signaling the absence of quasiparticles. Similar behavior is observed for the real and imaginary parts of the self-energy, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}).$ The double-occupied Green function, ${G}_{2}(\stackrel{\ensuremath{\rightarrow}}{q},\ensuremath{\omega}),$ has been obtained from $G(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ by means of the equation of motion. The relation between ${G}_{2}(\stackrel{\ensuremath{\rightarrow}}{q},\ensuremath{\omega})$ and the self-energy, $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ is formally established and numerical results for the spectral function of ${G}_{2}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ ${\ensuremath{\chi}}^{(2)}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})\ensuremath{\equiv}\ensuremath{-}(1/\ensuremath{\pi}){limop}_{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\delta}}{0}^{+}}\mathrm{Im}[{G}_{2}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})],$ are given. Our approach represents the simplest way to include (1) lifetime effects in the moment approach of Nolting, as shown in the paper, and (2) Fermi or/and marginal Fermi liquid features as we discuss in the conclusions.
We have postulated a single pole for the self-energy looking for the consequences on the one-particle Green function, G(k,w), in the Hubbard model. We find that G(k,w) satisfies the first two sum rules or moments of Nolting (Z. Physik 225, 25 (1972)) for any values of the two unknown parameters of the self-energy. In order to find these two parameters we have used the third and four sum rules of Nolting. G(k,w) turns out to be identical to the one of Nolting which is beyond a Hubbard-I solution. Furthermore, we have also postulated a two-pole Ansatz for the self-energy which is equivalent to a three-pole ansatz for G(k,w). In the later case, we have fixed two of the frequencies. This Green function is also beyond the Hubbard-I solution. We present numerical results for the band structure and the spectral weights.
We investigate the metal-insulator transition in the half-filled Hubbard model on the two-dimensional triangular lattice using both the Kotliar-Ruckenstein slave-boson technique and an exact numerical diagonalization of finite clusters. Contrary to the case of a square lattice, where a perfect nesting of the Fermi surface leads to a metal-insulator transition at arbitrarily small values of U, always accompanied by antiferromagnetic ordering, on a triangular lattice, due to the lack of perfect nesting, the transition takes place at a finite value of U, and frustration induces a nontrivial competition among different magnetic phases. Indeed, within the mean-field approximation in the slave-boson approach, as the interaction grows the paramagnetic metal turns into a metallic phase with incommensurate spiral ordering. Increasing the interaction further, a linear spin-density wave is stabilized, and finally for strong coupling the latter phase undergoes a first-order transition toward an antiferromagnetic insulator. No trace of the intermediate phases is seen in the exact diagonalization results, indicating a transition between a paramagnetic metal and an antiferromagnetic insulator.
Antiferromagnetic and metal to insulator transitions in the ground state of two-dimensional half-filled and nearly half-filled Hubbard models with the nearest neighbor and the second nearest neighbor transfers (- t and t ′ , respectively) are studied by using the Green's function approach with the self-energy up to the second order in the coupling constant U . For a choice of the parameter value t ′ / t =0.2 the half-filled model undergoes the first order phase transition between paramagnetic metallic (PM) phase and antiferromagnetic insulator (AFI) phase at U / t =4.0 while the first order transition between PM and antiferromagnetic metallic (AFM) phases takes place for nearly half-filled band. The one-electron spectral density is calculated for the PM, AFM, and AFI phases near the phase boundaries. In the PM phase the quasi-particle dispersion indicates the band narrowing from the Hartree-Fock results and the flat band behavior as was indicated by previous stadies and was observed in high- T c cuprates. The quasi-particle dispersion in the AFM phase is seen to be close to the Hartree-Fock results.
A new two-pole approximation, which allows to describe the transition from an insulating state to a metallic one at increase of bandwidth, and also the observable in some compounds transition from a metalic state to an insulating one with increasing temperature, is presented. The case of a paramagnetic narrow-band material at half-filling is considered. Single-particle Green function and energy spectrum of electron system are calculated. The expressions for energy gap width and the concentration of polar states (holes or doublons) are obtained. The conditions for metallic and insulating states are found.