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arXiv:1411.4767 · Journal of the Physical Society of Japan

Density-Matrix Renormalization Group Study of Third Harmonic Generation in One-Dimensional Mott Insulator Coupled with Phonon

Shigetoshi Sota, Seiji Yunoki, Takami Tohyama

Published Apr 16, 2015• 3 citations
arXivHTMLPDF

Abstract

We examine the third-order non-linear optical response of a one-dimensional Mott insulator coupled with phonons. The Mott insulator is described by an extended Hubbard-Holstein model. The third harmonic generation (THG) of the model is calculated by dynamical density-matrix renormalization group. We find that the electron-phonon interaction enhances the intensity of THG if the on-site Coulomb interaction is comparable to the band width, but the enhancement is small for a realistic parameter value of a typical Mott insulator Sr$_2$CuO$_3$. The low-energy spin excitation does not contribute to the optical response if the electron-phonon coupling is neglected. We find that the THG spectrum can detect the spin excitations even for no electron-phonon coupling. The introduction of the electron-phonon coupling leads to a slight increase of low-energy weight in THG without peak structure through phonon-assisted spin excitation. To fully understand the phonon-assisted spin excitation, we additionally calculate the linear susceptibility of a doped Hubbard-Holstein model, showing a spectral distribution different from that at half filling.

DOI
10.7566/jpsj.84.054403

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We examine the third-order non-linear optical response of a one-dimensional Mott insulator coupled with phonons. The Mott insulator is described by an extended Hubbard-Holstein model. The third harmonic generation (THG) of the model is calculated by dynamical density-matrix renormalization group. We find that the electron-phonon interaction enhances the intensity of THG if the on-site Coulomb interaction is comparable to the band width, but the enhancement is small for a realistic parameter value of a typical Mott insulator Sr2_22​CuO3_33​. The low-energy spin excitation does not contribute to the optical response if the electron-phonon coupling is neglected. We find that the THG spectrum can detect the spin excitations even for no electron-phonon coupling. The introduction of the electron-phonon coupling leads to a slight increase of low-energy weight in THG without peak structure through phonon-assisted spin excitation. To fully understand the phonon-assisted spin excitation, we additionally calculate the linear susceptibility of a doped Hubbard-Holstein model, showing a spectral distribution different from that at half filling.
Applying newly developed dynamical density matrix renormalization group techniques at zero and finite temperatures to a Hubbard-Holstein model at half filling, we examine the optical conductivity of a typical one-dimensional Mott insulator Sr2CuO3{\text{Sr}}_{2}{\text{CuO}}_{3}Sr2​CuO3​. We find a set of parameters in the Hubbard-Holstein model, which can describe optical conductivity for both Mott-gap excitation in the high-energy region and phonon-assisted spin excitation in the low-energy region. We also find that electron-phonon interaction gives additional broadening in the temperature dependence of the Mott-gap excitation.
Interplay of strong electron correlation and electron-phonon interaction is one of important issues in the physics of low dimensional correlated electron systems. In order to understand the interplay, we examine the optical properties of the one-dimensional (1D) Hubbard-Holstein model at half-filling, by employing the dynamical density matrix renormalization group (DMRG) technique. We discuss the effect of the electron-phonon interaction on the single-particle spectral function and the optical absorption spectrum. The dynamical DMRG data are compared with experimental data of 1D copper oxide Mott insulators.
The real part of optical conductivity, Reσ(ω)\text{Re}\sigma(\omega)Reσ(ω), of the Mott insulators has a large amount of information on how spin and charge degrees of freedom interact with each other. By using the time-dependent density-matrix renormalization group, we study Reσ(ω)\text{Re}\sigma(\omega)Reσ(ω) of the two-dimensional Hubbard model on a square lattice at half filling. We find an excitonic peak at the Mott-gap edge of Reσ(ω)\text{Re}\sigma(\omega)Reσ(ω) not only for the two-dimensional square lattice but also for two- and four-leg ladders. For the square lattice, however, we do not clearly find a gap between an excitonic peak and continuum band, which indicates that a bound state is not well-defined. The emergence of an excitonic peak in Reσ(ω)\text{Re}\sigma(\omega)Reσ(ω) implies the formation of a spin polaron. Examining the dependence of Reσ(ω)\text{Re}\sigma(\omega)Reσ(ω) on the on-site Coulomb interaction and next-nearest neighbor hoppings, we confirm that an excitonic peak is generated from a magnetic effect. Electron scattering due to an electron-phonon interaction is expected to easily suppress an excitonic peak since spectral width of an excitonic peak is very narrow. Introducing a large broadening in Reσ(ω)\text{Re}\sigma(\omega)Reσ(ω) by modeling the electron-phonon coupling present in La2_{2}2​CuO4_{4}4​ and Nd2_{2}2​CuO4_{4}4​, we obtain Reσ(ω)\text{Re}\sigma(\omega)Reσ(ω) comparable with experiments.
The real part of optical conductivity $\text{Re}\ensuremath{\sigma}(\ensuremath{\omega})$ of the Mott insulators has a large amount of information on how spin and charge degrees of freedom interact with each other. By using the time-dependent density-matrix renormalization group, we study $\text{Re}\ensuremath{\sigma}(\ensuremath{\omega})$ of the two-dimensional Hubbard model on a square lattice at half filling. We find an excitonic peak at the Mott-gap edge of $\text{Re}\ensuremath{\sigma}(\ensuremath{\omega})$ not only for the two-dimensional square lattice but also for two- and four-leg ladders. For the square lattice, however, we do not clearly find a gap between an excitonic peak and continuum band, which indicates that a bound state is not well defined. The emergence of an excitonic peak in $\text{Re}\ensuremath{\sigma}(\ensuremath{\omega})$ implies the formation of a spin polaron. Examining the dependence of $\text{Re}\ensuremath{\sigma}(\ensuremath{\omega})$ on the on-site Coulomb interaction and next-nearest-neighbor hoppings, we confirm that an excitonic peak is generated from a magnetic effect. Electron scattering due to an electron-phonon interaction is expected to easily suppress an excitonic peak since spectral width of an excitonic peak is very narrow. Introducing a large broadening in $\text{Re}\ensuremath{\sigma}(\ensuremath{\omega})$ by modeling the electron-phonon coupling present in La2CuO4{\mathrm{La}}_{2}{\mathrm{CuO}}_{4}La2​CuO4​ and Nd2CuO4{\mathrm{Nd}}_{2}{\mathrm{CuO}}_{4}Nd2​CuO4​, we obtain $\text{Re}\ensuremath{\sigma}(\ensuremath{\omega})$ comparable with experiments.
We study the third-order nonlinear optical susceptibility ${\ensuremath{\chi}}^{(3)}$ and photoexcited states of two-dimensional (2D) Mott insulators by using an effective model in the strong-coupling limit of a half-filled Hubbard model. In the numerically exact diagonalization calculations on finite-size clusters, we find that the coupling of charge and spin degrees of freedom plays a crucial role in the distribution of the dipole-allowed states with odd parity and the dipole-forbidden states with even parity in the photoexcited states. This is in contrast with the photoexcited states in one dimension, where the charge and spin degrees of freedom are decoupled. In the third-harmonic generation (THG) spectrum, the main contribution is found to come from the process of three-photon resonance associated with the odd-parity states. As a result, the two-photon resonance process is less pronounced in the THG spectrum. The calculated THG spectrum is compared with recent experimental data. We also find that ${\ensuremath{\chi}}^{(3)}$ with a cross-polarized configuration of pump and probe photons shows spectral distributions similar to ${\ensuremath{\chi}}^{(3)}$ with a copolarized configuration, although the weight is small. These findings will help analyses of the experimental data of ${\ensuremath{\chi}}^{(3)}$ in 2D Mott insulators.
We examine the single-particle excitation spectrum in the one-dimensional Hubbard-Holstein model at half-filling by performing a dynamical density matrix renormalization group calculation. The spin-charge separation is robust against the electron-phonon (EP) coupling, although both spinon and holon branches are affected by phonons. We find that this robustness is in sharp contrast to a rather strong influence of the EP coupling on quasiparticle properties in two dimensions. We discuss the implication of the results of recent angle-resolved photoemission spectroscopy measurements on SrCuO2\mathrm{Sr}\mathrm{Cu}{\mathrm{O}}_{2}SrCuO2​.