Physical Review Letters

Theory of Raman scattering in Mott-Hubbard systems

B. Sriram Shastry, Boris I. Shraiman

Published Aug 20, 1990• 286 citations

Abstract

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}(${\mathrm{S}}_{\mathit{i}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{S}}_{\mathit{k}}$) are shown to contribute in the ${\mathit{B}}_{2\mathit{g}}$ 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.

DOI: 10.1103/physrevlett.65.1068

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Physical Review Letters

Theory of Raman scattering in Mott-Hubbard systems

B. Sriram Shastry, Boris I. Shraiman

Published Aug 20, 1990• 286 citations

Publisher

Abstract

chiral'' spin operator ${\mathrm{\ensuremath{\Sigma}}}_{{\mathit{S}}_{\mathit{i}}}$\ensuremath{\cdot}(${\mathrm{S}}_{\mathit{i}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{S}}_{\mathit{k}}$) are shown to contribute in the ${\mathit{B}}_{2\mathit{g}}$ 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.

DOI: 10.1103/physrevlett.65.1068

Related papers

Raman Scattering in Mott-Hubbard Systems

B. Sriram Shastry, Boris I. Shraiman

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 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.

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Andrij M. Shvaika, O. Vorobyov, James K. Freericks +1 more

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arXiv:cond-mat/0408400Physical Review BJan 25, 200532 citations

Theory of inelastic light scattering in spin-1 systems: Resonant regimes and detection of quadrupolar order

Frédéric Michaud, F. Vernay, Frédéric Mila

Motivated by the lack of an obvious spectroscopic probe to investigate nonconventional order such as quadrupolar orders in spin $S>\frac{1}{2}$ 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 $S=\frac{1}{2}$ case, where the only type of local excited state is a doubly occupied state of energy $U$ , 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- $\frac{1}{2}$ 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 $S=1$ magnets with magnetic or quadrupolar order are discussed. Raman scattering appears to be a powerful probe of quadrupolar order.Motivated by the lack of an obvious spectroscopic probe to investigate nonconventional order such as quadrupolar orders in spin $S>\frac{1}{2}$ 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 $S=\frac{1}{2}$ case, where the only type of local excited state is a doubly occupied state of energy $U$, 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-$\frac{1}{2}$ 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 $S=1$ magnets with magnetic or quadrupolar order are discussed. Raman scattering appears to be a powerful probe of quadrupolar order.

arXiv:1108.3686Physical Review BNov 16, 201132 citations

Sum-rules for Raman scattering off strongly correlated electron systems

Luca de’ Medici, Antoine Georges, Gabriel Kotliar

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 $d$ -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.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 $d$-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.

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Umklapp scattering in the one-dimensional Hubbard model

Tong Liu, Kang Wang, Runze Chi +3 more

The one-dimensional Mott metal-insulator transition is a typical strong correlation effect triggered by the Umklapp scattering. However, in a physical system, the Umklapp scattering coexists with the normal scattering, including both forward and backward scattering, which conserves the total momentum of scattered electrons. Therefore, it is not easy to quantify the contribution of the Umklapp scattering in a Mott metal-insulator transition. To resolve this difficulty, we propose to explore these scattering processes separately. We study the contribution of each scattering process in the one-dimensional Hubbard model using the momentum-space density-matrix renormalization group (kDMRG) and bosonization methods. Our kDMRG calculation confirms that the Mott charge gap results from the Umklapp scattering, but the normal scattering processes strongly renormalize its value. Furthermore, we present a scaling analysis of the Mott charge gap in the bosonization theory and show that the interplay between the Umklapp and forward scattering dictates the charge dynamics in the half-filled Hubbard model.The one-dimensional Mott metal-insulator transition is a typical strong correlation effect triggered by the Umklapp scattering. However, in a physical system, the Umklapp scattering coexists with the normal scattering, including both forward and backward scattering, which conserves the total momentum of scattered electrons. Therefore, it is not easy to quantify the contribution of the Umklapp scattering in a Mott metal-insulator transition. To resolve this difficulty, we propose to explore these scattering processes separately. We study the contribution of each scattering process in the one-dimensional Hubbard model using the momentum-space density-matrix renormalization group (kDMRG) and bosonization methods. Our kDMRG calculation confirms that the Mott charge gap results from the Umklapp scattering, but the normal scattering processes strongly renormalize its value. Furthermore, we present a scaling analysis of the Mott charge gap in the bosonization theory and show that the interplay between the Umklapp and forward scattering dictates the charge dynamics in the half-filled Hubbard model.

arXiv:2212.10972Physical review. B./Physical review. BSep 21, 20231 citation

Resonant Raman scattering from a spin density wave insulator

A. P. Kampf, W. Brenig

Annalen der PhysikJan 1, 19923 citations

{A}_{1\mathrm{g}}

and

{B}_{2\mathrm{g}}

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>\frac{1}{2}

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=\frac{1}{2}

case, where the only type of local excited state is a doubly occupied state of energy

U

, 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-

\frac{1}{2}

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=1

magnets with magnetic or quadrupolar order are discussed. Raman scattering appears to be a powerful probe of quadrupolar order.

d

-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.