Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B./Physical review. B

Extraction of important degrees of freedom in quantum dynamics using singular value decomposition: Application to linear optical spectrum in two-dimensional Mott insulators

Jun Tokimoto, Shu Ohmura, Akira Takahashi, Kaoru Iwano, Hiroshi Okamoto

Published May 15, 2024• 1 citation
Publisher

Abstract

We propose a new approach to extract the important degrees of freedom in quantum dynamics induced by an external stimulus. We calculate the coefficient matrix numerically, where the $i\ensuremath{-}l$ element of the matrix is the coefficient of the $l\mathrm{th}$ basis state at the $i\mathrm{th}$ discretized time in the solution of the time-dependent Schr\"odinger equation induced by the external stimulus. By performing a randomized singular value decomposition of the coefficient matrix, a practically exact solution is obtained using a linear combination of the important modes, where the number of modes is much smaller than the dimensions of the Hilbert space in many cases. We apply this method to the analysis of the light absorption spectrum in two-dimensional (2D) Mott insulators using an effective model of the 2D Hubbard model in the strong interaction case. From the dynamics induced by an ultrashort weak light pulse, we find that the practically exact light absorption spectrum can be reproduced by as few as 1000 energy eigenstates in the $1.7\ifmmode\times\else\texttimes\fi{}{10}^{7}$-dimension Hilbert space of a 26-site cluster. These one-photon active energy eigenstates are classified into free holon and doublon (H-D) and localized H-D states. In the free H-D states, the main effect of the spin degrees of freedom on the transfer of a holon (H) and a doublon (D) is the phase shift, and the H and the D move freely. In the localized H-D states, an H and a D are localized with relative distances of $\sqrt{5}$ or $\sqrt{13}$. The antiferromagnetic (AF) spin orders in the localized H-D states are much stronger than those in the free H-D states, and the charge localization is of magnetic origin. There are sharp peaks caused by excitations to the localized H-D states below the broad band caused by excitations to the free H-D states in the light absorption spectrum.

DOI
10.1103/physrevb.109.195150

Related papers

New approach to extract important degrees of freedom in quantum dynamics using singular value decomposition: Application to linear optical spectrum in two-dimensional Mott insulators

Jun Tokimoto, Shu Ohmura, Akira Takahashi +2 more

We propose a new approach to extract the important degrees of freedom in quantum dynamics induced by an external stimulus. We calculate the coefficient matrix numerically, where the $i-l$ element of the matrix is the coefficient of the lth basis state at the ith discretized time in the solution of the time-dependent Schrödinger equation induced by the external stimulus. By performing a randomized singular value decomposition of the coefficient matrix, a practically exact solution is obtained using a linear combination of the important modes, where the number of modes is much smaller than the dimensions of the Hilbert space in many cases. We apply this method to analysis of the light absorption spectrum in two-dimensional (2D) Mott insulators using an effective model of the 2D Hubbard model in the strong interaction case. From the dynamics induced by an ultrashort weak light pulse, we find that the practically exact light absorption spectrum can be reproduced by as few as 1000 energy eigenstates in the $1.7 \times 10^{7}$-dimension Hilbert space of a 26-site cluster. These one-photon active energy eigenstates are classified into free holon and doublon (H-D) and localized H-D states. In the free H-D states, the main effect of the spin degrees of freedom on the transfer of a holon (H) and a doublon (D) is the phase shift, and the H and the D move freely. In the localized H-D states, an H and a D are localized with relative distances of $\sqrt{5}$ or $\sqrt{13}$. The antiferromagnetic (AF) spin orders in the localized H-D states are much stronger than those in the free H-D states, and the charge localization is of magnetic origin. There are sharp peaks caused by excitations to the localized H-D states below the broad band caused by excitations to the free H-D states in the light absorption spectrum.

arXiv:2312.06300•Dec 11, 2023•0 citations

Light absorption spectrum of two-dimensional Mott insulators

Hisashi Itoh, Akira Takahashi, Masaki Aihara

We theoretically investigate the light absorption spectrum of two-dimensional (2D) Mott insulators. We employ the numerical diagonalization method to calculate the light absorption spectrum with using the effective Hamiltonian for the extended Hubbard model, which is valid when the on-site Coulomb interaction energy $U$ is much larger than the nearest-neighbor transfer integral $t$ and the nearest-neighbor Coulomb interaction energy $V$. For $V=0$, the absorption spectrum consists of a broad band with a width of about $16t$ and a sharp central peak, and the energy eigenstates contributing to the absorption spectrum do not have the antiferromagnetic (AF) spin order, when $U$ is sufficiently larger than $t$. These features result from the spin-charge interplay inherent in the very strong correlation region where charge transfer term is dominant. With decreasing $U∕t$, the peak structure in absorption spectrum becomes unclear and some low energy eigenstates have the AF spin order, as a result of the spin-spin interaction term. For large $V∕t$, the dominant peak arises in the lower energy region of the spectrum. A large number of holon-doublon bound states, which have nearly the same charge but different spin structures, are responsible for this peak, in contrast to the conventional exciton state. This is also in contrast to the charge bound states in one-dimensional (1D) Mott insulators, where a single energy eigenstate dominates the optical transition moment as a result of the spin-charge separation. The essentially different absorption spectra between the 1D and 2D Mott insulators originate from the difference in the coupling between spin and charge degrees of freedom.

Physical Review B•Feb 16, 2006•18 citations

Dynamics of photoinduced insulator-metal transition in one-dimensional Mott insulators

Akira Takahashi

We theoretically investigate the dynamics of insulator-metal transition induced by light-pulse excitation in one-dimensional (1D) and two-dimensional (2D) Mott insulators. We adopt the Pariser-Parr-Pople model, and pump-probe signal is obtained by numerically calculating the time development of the state excited by a light pulse. In the 1D case, when photoexcitation density is below about 10%, an extremely small portion of the energy eigenstates dominates the optical process, and the spin-charge separation holds in these dominant energy eigenstates. As a result, the Mott gap and the short-range antiferromagnetic (AF) spin order are preserved, and the spin relaxation does not occur. When the density is above the value, a metallic state without the Mott gap is photogenerated, and the magnitude of the short-range AF spin order is significantly reduced by photoexcitation. This is consistent with the experimentally observed photoinduced Mott transition. Furthermore, the spin relaxation occurs in the metallic state. This photoinduced Mott transition is a manifestation of the spin-charge coupling in the intensely photoexcited states. In the 2D case, the spin and the charge degrees of freedom are coupled even in the weakly photoexcited states, and the characteristic crossover associated with the excitation density is not observed.

Journal of Physics Conference Series•Feb 1, 2009•0 citations

Nonlinear optical response in two-dimensional Mott insulators

Makoto Takahashi, Takami Tohyama, Sadamichi Maekawa

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.

arXiv:cond-mat/0201190•Physical review. B, Condensed matter•Sep 11, 2002•19 citations

Charge dynamics of the antiferromagnetically ordered Mott insulator

Xing-Jie Han, Yu Liu, Zhi-Yuan Liu +6 more

We introduce a slave-fermion formulation in which to study the charge dynamics of the half-filled Hubbard model on the square lattice. In this description, the charge degrees of freedom are represented by fermionic holons and doublons and the Mott-insulating characteristics of the ground state are the consequence of holon–doublon bound-state formation. The bosonic spin degrees of freedom are described by the antiferromagnetic Heisenberg model, yielding long-ranged (Néel) magnetic order at zero temperature. Within this framework and in the self-consistent Born approximation, we perform systematic calculations of the average double occupancy, the electronic density of states, the spectral function and the optical conductivity. Qualitatively, our method reproduces the lower and upper Hubbard bands, the spectral-weight transfer into a coherent quasiparticle band at their lower edges and the renormalisation of the Mott gap, which is associated with holon–doublon binding, due to the interactions of both quasiparticle species with the magnons. The zeros of the Green function at the chemical potential give the Luttinger volume, the poles of the self-energy reflect the underlying quasiparticle dispersion with a spin-renormalised hopping parameter and the optical gap is directly related to the Mott gap. Quantitatively, the square-lattice Hubbard model is one of the best-characterised problems in correlated condensed matter and many numerical calculations, all with different strengths and weaknesses, exist with which to benchmark our approach. From the semi-quantitative accuracy of our results for all but the weakest interaction strengths, we conclude that a self-consistent treatment of the spin-fluctuation effects on the charge degrees of freedom captures all the essential physics of the antiferromagnetic Mott–Hubbard insulator. We remark in addition that an analytical approximation with these properties serves a vital function in developing a full understanding of the fundamental physics of the Mott state, both in the antiferromagnetic insulator and at finite temperatures and dopings.

arXiv:1602.01276•New Journal of Physics•Oct 6, 2016•19 citations

Temperature and dimensionality dependencies of optical absorption spectra in Mott insulators

H. Onodera, Takami Tohyama, Sadamichi Maekawa

We investigate the temperature dependence of optical absorption spectra of one-dimensional (1D) and 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 in 1D the energy position of the absorption edge is almost independent of temperature, while in 2D the edge position shifts to lower energy with increasing temperature. The different temperature dependence between 1D and 2D is attributed to the difference of the coupling of the charge and spin degrees of freedom. The implications of the results on experiments are discussed in terms of the dimensionality dependence.

arXiv:cond-mat/0312625•Physical Review B•Jun 28, 2004•9 citations
We propose a new approach to extract the important degrees of freedom in quantum dynamics induced by an external stimulus. We calculate the coefficient matrix numerically, where the i\ensuremath−li\ensuremath{-}li\ensuremath−l element of the matrix is the coefficient of the lthl\mathrm{th}lth basis state at the ithi\mathrm{th}ith discretized time in the solution of the time-dependent Schr"odinger equation induced by the external stimulus. By performing a randomized singular value decomposition of the coefficient matrix, a practically exact solution is obtained using a linear combination of the important modes, where the number of modes is much smaller than the dimensions of the Hilbert space in many cases. We apply this method to the analysis of the light absorption spectrum in two-dimensional (2D) Mott insulators using an effective model of the 2D Hubbard model in the strong interaction case. From the dynamics induced by an ultrashort weak light pulse, we find that the practically exact light absorption spectrum can be reproduced by as few as 1000 energy eigenstates in the 1.7\ifmmode×\else\texttimes\fi1071.7\ifmmode\times\else\texttimes\fi{}{10}^{7}1.7\ifmmode×\else\texttimes\fi107-dimension Hilbert space of a 26-site cluster. These one-photon active energy eigenstates are classified into free holon and doublon (H-D) and localized H-D states. In the free H-D states, the main effect of the spin degrees of freedom on the transfer of a holon (H) and a doublon (D) is the phase shift, and the H and the D move freely. In the localized H-D states, an H and a D are localized with relative distances of 5\sqrt{5}5​ or 13\sqrt{13}13​. The antiferromagnetic (AF) spin orders in the localized H-D states are much stronger than those in the free H-D states, and the charge localization is of magnetic origin. There are sharp peaks caused by excitations to the localized H-D states below the broad band caused by excitations to the free H-D states in the light absorption spectrum.
We propose a new approach to extract the important degrees of freedom in quantum dynamics induced by an external stimulus. We calculate the coefficient matrix numerically, where the i−li-li−l element of the matrix is the coefficient of the lth basis state at the ith discretized time in the solution of the time-dependent Schrödinger equation induced by the external stimulus. By performing a randomized singular value decomposition of the coefficient matrix, a practically exact solution is obtained using a linear combination of the important modes, where the number of modes is much smaller than the dimensions of the Hilbert space in many cases. We apply this method to analysis of the light absorption spectrum in two-dimensional (2D) Mott insulators using an effective model of the 2D Hubbard model in the strong interaction case. From the dynamics induced by an ultrashort weak light pulse, we find that the practically exact light absorption spectrum can be reproduced by as few as 1000 energy eigenstates in the 1.7×1071.7 \times 10^{7}1.7×107-dimension Hilbert space of a 26-site cluster. These one-photon active energy eigenstates are classified into free holon and doublon (H-D) and localized H-D states. In the free H-D states, the main effect of the spin degrees of freedom on the transfer of a holon (H) and a doublon (D) is the phase shift, and the H and the D move freely. In the localized H-D states, an H and a D are localized with relative distances of 5\sqrt{5}5​ or 13\sqrt{13}13​. The antiferromagnetic (AF) spin orders in the localized H-D states are much stronger than those in the free H-D states, and the charge localization is of magnetic origin. There are sharp peaks caused by excitations to the localized H-D states below the broad band caused by excitations to the free H-D states in the light absorption spectrum.
We theoretically investigate the light absorption spectrum of two-dimensional (2D) Mott insulators. We employ the numerical diagonalization method to calculate the light absorption spectrum with using the effective Hamiltonian for the extended Hubbard model, which is valid when the on-site Coulomb interaction energy UUU is much larger than the nearest-neighbor transfer integral ttt and the nearest-neighbor Coulomb interaction energy VVV. For V=0V=0V=0, the absorption spectrum consists of a broad band with a width of about 16t16t16t and a sharp central peak, and the energy eigenstates contributing to the absorption spectrum do not have the antiferromagnetic (AF) spin order, when UUU is sufficiently larger than ttt. These features result from the spin-charge interplay inherent in the very strong correlation region where charge transfer term is dominant. With decreasing U∕tU∕tU∕t, the peak structure in absorption spectrum becomes unclear and some low energy eigenstates have the AF spin order, as a result of the spin-spin interaction term. For large V∕tV∕tV∕t, the dominant peak arises in the lower energy region of the spectrum. A large number of holon-doublon bound states, which have nearly the same charge but different spin structures, are responsible for this peak, in contrast to the conventional exciton state. This is also in contrast to the charge bound states in one-dimensional (1D) Mott insulators, where a single energy eigenstate dominates the optical transition moment as a result of the spin-charge separation. The essentially different absorption spectra between the 1D and 2D Mott insulators originate from the difference in the coupling between spin and charge degrees of freedom.
We theoretically investigate the dynamics of insulator-metal transition induced by light-pulse excitation in one-dimensional (1D) and two-dimensional (2D) Mott insulators. We adopt the Pariser-Parr-Pople model, and pump-probe signal is obtained by numerically calculating the time development of the state excited by a light pulse. In the 1D case, when photoexcitation density is below about 10%, an extremely small portion of the energy eigenstates dominates the optical process, and the spin-charge separation holds in these dominant energy eigenstates. As a result, the Mott gap and the short-range antiferromagnetic (AF) spin order are preserved, and the spin relaxation does not occur. When the density is above the value, a metallic state without the Mott gap is photogenerated, and the magnitude of the short-range AF spin order is significantly reduced by photoexcitation. This is consistent with the experimentally observed photoinduced Mott transition. Furthermore, the spin relaxation occurs in the metallic state. This photoinduced Mott transition is a manifestation of the spin-charge coupling in the intensely photoexcited states. In the 2D case, the spin and the charge degrees of freedom are coupled even in the weakly photoexcited states, and the characteristic crossover associated with the excitation density is not observed.
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 introduce a slave-fermion formulation in which to study the charge dynamics of the half-filled Hubbard model on the square lattice. In this description, the charge degrees of freedom are represented by fermionic holons and doublons and the Mott-insulating characteristics of the ground state are the consequence of holon–doublon bound-state formation. The bosonic spin degrees of freedom are described by the antiferromagnetic Heisenberg model, yielding long-ranged (Néel) magnetic order at zero temperature. Within this framework and in the self-consistent Born approximation, we perform systematic calculations of the average double occupancy, the electronic density of states, the spectral function and the optical conductivity. Qualitatively, our method reproduces the lower and upper Hubbard bands, the spectral-weight transfer into a coherent quasiparticle band at their lower edges and the renormalisation of the Mott gap, which is associated with holon–doublon binding, due to the interactions of both quasiparticle species with the magnons. The zeros of the Green function at the chemical potential give the Luttinger volume, the poles of the self-energy reflect the underlying quasiparticle dispersion with a spin-renormalised hopping parameter and the optical gap is directly related to the Mott gap. Quantitatively, the square-lattice Hubbard model is one of the best-characterised problems in correlated condensed matter and many numerical calculations, all with different strengths and weaknesses, exist with which to benchmark our approach. From the semi-quantitative accuracy of our results for all but the weakest interaction strengths, we conclude that a self-consistent treatment of the spin-fluctuation effects on the charge degrees of freedom captures all the essential physics of the antiferromagnetic Mott–Hubbard insulator. We remark in addition that an analytical approximation with these properties serves a vital function in developing a full understanding of the fundamental physics of the Mott state, both in the antiferromagnetic insulator and at finite temperatures and dopings.
We investigate the temperature dependence of optical absorption spectra of one-dimensional (1D) and 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 in 1D the energy position of the absorption edge is almost independent of temperature, while in 2D the edge position shifts to lower energy with increasing temperature. The different temperature dependence between 1D and 2D is attributed to the difference of the coupling of the charge and spin degrees of freedom. The implications of the results on experiments are discussed in terms of the dimensionality dependence.