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arXiv:0803.3869 · Journal of Experimental and Theoretical Physics

The optical sum rule in strongly correlated systems

E. Z. Kuchinskii, N. A. Kuleeva, I. A. Nekrasov, M. V. Sadovskii

Published Aug 1, 2008• 1 citation
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Abstract

We discuss the problem of a possible “violation” of the optical sum rule in the normal (nonsuperconducting) state of strongly correlated electronic systems, using our recently proposed DMFT + Σ approach applied to two typical models: the “hot spot” model of the pseudogap state and the disordered Anderson-Hubbard model. We explicitly demonstrate that the general Kubo single-band sum rule is satisfied for both models, but the optical integral itself is in general dependent on temperature and characteristic parameters, such as the pseudogap width, correlation strength, and disorder scattering, leading to an effective “violation” of the optical sum rule, which may be observed in experiments.

DOI
10.1134/s1063776108080128

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We discuss the problem of a possible “violation” of the optical sum rule in the normal (nonsuperconducting) state of strongly correlated electronic systems, using our recently proposed DMFT + Σ approach applied to two typical models: the “hot spot” model of the pseudogap state and the disordered Anderson-Hubbard model. We explicitly demonstrate that the general Kubo single-band sum rule is satisfied for both models, but the optical integral itself is in general dependent on temperature and characteristic parameters, such as the pseudogap width, correlation strength, and disorder scattering, leading to an effective “violation” of the optical sum rule, which may be observed in experiments.
The influence of disorder and pseudogap fluctuations on the Mott insulator-metal transition in strongly correlated systems has been studied in the framework of the generalized dynamic mean field theory (DMFT + Σ approach). Using the results of investigations of the density of states (DOS) and optical conductivity, a phase diagram (disorder-Hubbard interaction-temperature) is constructed for the paramagnetic Anderson-Hubbard model, which allows both the effects of strong electron correlations and the influence of strong disorder to be considered. Strong correlations are described using the DMFT, while a strong disorder is described using a generalized self-consistent theory of localization. The DOS and optical conductivity of the paramagnetic Hubbard model have been studied in a pseudogap state caused by antiferromagnetic spin (or charge) short-range order fluctuations with a finite correlation length, which have been modeled by a static Gaussian random field. The effect of a pseudogap on the Mott insulator-metal transition has been studied. It is established that, in both cases, the static Gaussian random field (related to the disorder or pseudogap fluctuations) leads to suppression of the Mott transition, broadening of the coexistence region of the insulator and metal phases, and an increase in the critical temperature at which the coexistence region disappears.
We theoretically analyze some of the anomalies of the optical sum rules in the high-temperature superconductors. In particular, we address the particularly strong dependence on temperature of the sum rule in the normal state. Both electron-electron correlations and the presence of a Van Hove singularity have been shown to enhance such a dependence. Here, we consider both effects simultaneously by means of dynamical mean-field theory for a two-dimensional Hubbard model with realistic parameters for different cuprates, and we find that the two effects are not cooperative, as they appear to compete with one another in the region of parameters relevant for the experiments.
The optical conductivity of the weakly doped two-dimensional repulsive Hubbard model on the square lattice with the nearest and next-nearest hoppings is calculated within the generalized dynamical mean-field $(\mathrm{DMFT}+{\ensuremath{\sum}}_{\mathbf{p}})$ approach, which includes correlation length scale $\ensuremath{\xi}$ into the standard DMFT equations via the momentum dependent self-energy ${\ensuremath{\sum}}_{\mathbf{p}}$, with a full account of appropriate vertex corrections. This approach takes into consideration the nonlocal dynamical correlations induced, e.g., by short-ranged collective spin-density-wavelike antiferromagnetic spin fluctuations, which (at high enough temperatures) can be viewed as a quenched Gaussian random field with finite correlation length $\ensuremath{\xi}$. The DMFT effective single-impurity problem is solved by numerical renormalization group. We consider both the case of correlated metal with the bandwidth $W\ensuremath{\lesssim}U$ and that of doped Mott insulator with U⪢WU⪢WU⪢W (UUU---the value of local Hubbard interaction). The optical conductivity calculated within $\mathrm{DMFT}+{\ensuremath{\sum}}_{\mathbf{p}}$ demonstrates typical pseudogap behavior within the quasiparticle band, in qualitative agreement with experiments in copper oxide superconductors. For large values of UUU, pseudogap anomalies are effectively suppressed.
This review is devoted to generalization of dynamical mean-field theory (DMFT) for strongly correlated electronic systems towards the account of different types of additional interactions, necessary for correct physical description of many experimentally observed phenomena in such systems. As additional interactions we consider: (1) interaction of electrons with antiferromagnetic (or charge) fluctuations of order parameter in high-Tc superconductors leading to the formation of pseudogap state, (2) scattering of electrons on static disorder and its role in general picture of Anderson-Hubbard metal-insulator transition, (3) electron-phonon interaction and corresponding anomalies of electronic spectra in strongly correlated systems. Proposed DMFT+Sigma approach is based on taking into account above mentioned interactions by introducing additional self-energy Sigma (in general momentum dependent) into conventional DMFT scheme and calculated in a self-consistent way within the standard set of DMFT equations. Here we formulate general scheme of calculation of both one-particle (spectral functions and densities of states) and two-particle (optical conductivity) properties. We examine the problem of pseudogap formation, including the Fermi arc formation and partial destruction of the Fermi surface, metal-insulator transition in disordered Anderson-Hubbard model, and general picture of kink formation within electronic spectra in strongly correlated systems. DMFT+Sigma approach is generalized to describe realistic materials with strong electron-electron correlations based on LDA+DMFT method. General scheme of LDA+DMFT method is presented together with some of its applications to real systems. The LDA+DMFT+Sigma approach is employed to modelling of pseudogap state of electron and hole doped high-T_c cuprates. Comparison with variety of ARPES experiments is given.
We investigate sum-rules applying to the Raman intensity in a strongly correlated system close to the Mott transition. Quite generally, it can be shown that provided the frequency integration is performed up to a cutoff smaller than the upper Hubbard band a sum-rule applies to the nonresonant Raman signal of a doped Mott insulator, resulting in an integrated intensity, which is proportional to the doping level. We provide a detailed derivation of this sum-rule for the $t\text{\ensuremath{-}}J$ model for which the frequency cutoff can be taken to infinity and an unrestricted sum-rule applies. A quantitative analysis of the sum-rule is also presented for the ddd-wave superconducting phase of the $t\text{\ensuremath{-}}J$ model, using slave-boson methods. The case of the Hubbard model is studied in the framework of dynamical mean-field theory, with special attention to the cut-off dependence of the restricted sum-rule and also to the intermediate coupling regime. The sum-rule investigated here is shown to be consistent with recent experimental data on cuprate superconductors, reporting measurements of Raman scattering intensities on an absolute scale.
The DOS, the dynamic (optical) conductivity, and the phase diagram of a strongly correlated and strongly disordered paramagnetic Anderson-Hubbard model are analyzed within the generalized dynamical mean field theory (DMFT + Σ approximation). Strong correlations are taken into account by the DMFT, and disorder is taken into account via an appropriate generalization of the self-consistent theory of localization. The DMFT effective single-impurity problem is solved by a numerical renormalization group (NRG); we consider the three-dimensional system with a semielliptic DOS. The correlated metal, Mott insulator, and correlated Anderson insulator phases are identified via the evolution of the DOS and dynamic conductivity, demonstrating both the Mott-Hubbard and Anderson metal-insulator transition and allowing the construction of the complete zero-temperature phase diagram of the Anderson-Hubbard model. Rather unusual is the possibility of a disorder-induced Mott insulator-to-metal transition.