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International Journal of Modern Physics B

Raman Scattering in Mott-Hubbard Systems

B. Sriram Shastry, Boris I. Shraiman

Published Jan 1, 1991• 111 citations
Publisher

Abstract

We present a theory of Raman Scattering in the Hubbard model. The scattering of light has resonant and non-resonant contributions. The resonant term gives rise to scattering by spin degrees of freedom in the insulating case, for which we derive a general form of the effective scattering Hamiltonian. The later involves four terms with distinct symmetry properties, one of them being the “chiral” spin operator ∑ Si · (Sj×Sk). The corresponding four distinct correlation function can be measured directly in the experiments with different scattering geometries. The nonresonant term contributes to the scattering in the doped case and is shown to probe the fluctuations of the “stress tensor”. This quantity is not conserved and hence its fluctuations at small q inherent in optical experiments, need not be small, in striking contrast with the density fluctuation in usual metals.

DOI
10.1142/s0217979291000237

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We present a theory of Raman Scattering in the Hubbard model. The scattering of light has resonant and non-resonant contributions. The resonant term gives rise to scattering by spin degrees of freedom in the insulating case, for which we derive a general form of the effective scattering Hamiltonian. The later involves four terms with distinct symmetry properties, one of them being the “chiral” spin operator ∑ Si · (Sj×Sk). The corresponding four distinct correlation function can be measured directly in the experiments with different scattering geometries. The nonresonant term contributes to the scattering in the doped case and is shown to probe the fluctuations of the “stress tensor”. This quantity is not conserved and hence its fluctuations at small q inherent in optical experiments, need not be small, in striking contrast with the density fluctuation in usual metals.
We present a theory of Raman scattering in the Hubbard model. The scattering of light has two contributions. One gives rise to scattering by spin degrees of freedom in the insulating case where the general form of the scattering Hamiltonian is derived. The fluctuations of the ``chiral'' spin operator ${\mathrm{\ensuremath{\Sigma}}}_{{\mathit{S}}_{\mathit{i}}}$\ensuremath{\cdot}(Si{\mathrm{S}}_{\mathit{i}}Si​\ifmmode\times\else\texttimes\fi{}Sk{\mathrm{S}}_{\mathit{k}}Sk​) are shown to contribute in the B2g{\mathit{B}}_{2\mathit{g}}B2g​ scattering geometry. The other contributes in the doped case and is shown to probe the fluctuations of the ``stress tensor.'' This quantity is not conserved, and hence its fluctuations at small q inherent in optical experiments need not be small, in striking contrast to density fluctuations in usual metals.
We solve for the electronic Raman scattering response functions on an infinite-dimensional hypercubic lattice employing dynamical mean-field theory. This contribution extends previous work on the nonresonant response to include the mixed and resonant contributions. We focus our attention on the infinite-dimensional spinless Falicov-Kimball model, where the problem can be solved exactly, and the system can be tuned to go through a Mott-Hubbard-like metal-insulator transition. Resonant effects vary in different scattering geometries, corresponding to the symmetries of the charge excitations scattered by the light. We do find that the Raman response is large near the double resonance, where the transfered frequency is close to the incident photon frequency. We also find a joint resonance of both the charge-transfer peak and the low-energy peak when the incident photon frequency is on the order of the interaction strength. In general, the resonance effects can create order of magnitude (or more) enhancements of features in the nonresonant response, especially when the incident photon frequency is somewhat larger than the frequency of the nonresonant feature. Finally, we find that the resonant effects also exhibit isosbestic behavior, even in the A1g{A}_{1\mathrm{g}}A1g​ and B2g{B}_{2\mathrm{g}}B2g​ sectors, and it is most prominent when the incident photon frequency is on the order of the interaction energy.
Motivated by the lack of an obvious spectroscopic probe to investigate nonconventional order such as quadrupolar orders in spin S>12S>\frac{1}{2}S>21​ systems, we present a theoretical approach to inelastic light scattering for spin-1 quantum magnets in the context of a two-band Hubbard model. In contrast to the S=12S=\frac{1}{2}S=21​ case, where the only type of local excited state is a doubly occupied state of energy UUU, several local excited states with occupation up to four electrons are present. As a consequence, we show that two distinct resonating scattering regimes can be accessed depending on the incident photon energy. For $\ensuremath{\hbar}{\ensuremath{\omega}}_{\text{in}}\ensuremath{\lesssim}U$, the standard Loudon-Fleury operator remains the leading term of the expansion as in the spin-12\frac{1}{2}21​ case. For $\ensuremath{\hbar}{\ensuremath{\omega}}_{\text{in}}\ensuremath{\lesssim}4U$, a second resonant regime is found with a leading term that takes the form of a biquadratic coupling $\ensuremath{\sim}{({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{j})}^{2}$. Consequences for the Raman spectra of S=1S=1S=1 magnets with magnetic or quadrupolar order are discussed. Raman scattering appears to be a powerful probe of quadrupolar order.
Inelastic light scattering from electrons is a symmetry-selective probe of the charge dynamics within correlated materials. Many measurements have been made on correlated insulators, and recent exact solutions in large dimensions explain a number of anomalous features found in experiments. Here we focus on the correlated metal, as described by the Hubbard model away from half filling. We can determine the B1g{B}_{1\mathrm{g}}B1g​ Raman response and the inelastic x-ray scattering along the Brillouin-zone diagonal exactly in the large dimensional limit. We find a number of interesting features in the light-scattering response, which should be able to be seen in correlated metals such as the heavy fermions.
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.