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

arXiv:cond-mat/9504017 · Physical review. B, Condensed matter

Drude weight and dc conductivity of correlated electrons

Götz Silvester Uhrig, D. Vollhardt

Published Aug 15, 1995• 14 citations
arXivHTMLPDF

Abstract

The Drude weight D and the dc conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$(T) of strongly correlated electrons are investigated theoretically. Analytic results are derived for the homogeneous phase of the Hubbard model at d=\ensuremath{\infty}, and for spinless fermions in this limit with 1/d corrections systematically included to lowest order. It is found that ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$(T) is finite for all T>0, displaying Fermi-liquid behavior, ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$\ensuremath{\propto}1/${\mathit{T}}^{2}$, at low temperatures. The validity of this result for finite dimensions is examined by investigating the importance of umklapp scattering processes and vertex corrections. A finite dc conductivity for T>0 is argued to be a generic feature of correlated lattice electrons in not too low dimensions.

DOI
10.1103/physrevb.52.5617

Related papers

Finite-size effects in transport data from quantum Monte Carlo simulations

Rubem Mondaini, Karim Bouadim, Thereza Paiva +1 more

We have examined the behavior of the compressibility, the dc conductivity, the single-particle gap, and the Drude weight as probes of the density-driven metal-insulator transition in the Hubbard model on a square lattice. These quantities have been obtained through determinantal quantum Monte Carlo simulations at finite temperatures on lattices up to $16\ifmmode\times\else\texttimes\fi{}16$ sites. While the compressibility, the dc conductivity, and the gap are known to suffer from ``closed-shell'' effects due to the presence of artificial gaps in the spectrum (caused by the finiteness of the lattices), we have established that the former tracks the average sign of the fermionic determinant ($\ensuremath{\langle}\text{sign}\ensuremath{\rangle}$), and that a shortcut often used to calculate the conductivity may neglect important corrections. Our systematic analyses also show that, by contrast, the Drude weight is not too sensitive to finite-size effects, being much more reliable as a probe to the insulating state. We have also investigated the influence of the discrete imaginary-time interval ($\ensuremath{\Delta}\ensuremath{\tau}$) on $\ensuremath{\langle}\text{sign}\ensuremath{\rangle}$, on the average density ($\ensuremath{\rho}$), and on the double occupancy ($d$): we have found that $\ensuremath{\langle}\text{sign}\ensuremath{\rangle}$ and $\ensuremath{\rho}$ are more strongly dependent on $\ensuremath{\Delta}\ensuremath{\tau}$ away from closed-shell configurations, but $d$ follows the $\ensuremath{\Delta}{\ensuremath{\tau}}^{2}$ dependence in both closed- and open-shell cases.

arXiv:1107.0230•Physical Review B•Mar 26, 2012•27 citations

Spin and Charge Conductivity in the Square Lattice Fermi-Hubbard Model

Linh Pham, Ehsan Khatami

Dynamical properties are notoriously difficult to compute in numerical treatments of the Fermi-Hubbard model, especially in two spatial dimensions. However, they are essential in providing us with insight into some of the most important and less well-understood phases of the model, such as the pseudogap and strange metal phases at relatively high temperatures, or unconventional superconductivity at lower temperatures, away from the commensurate filling. Here, we use the numerical linked-cluster expansions to compute spin and charge optical conductivities of the model at different temperatures and strong interaction strengths via the exact real-time-dependent correlation functions of the current operators. We mitigate systematic errors associated with having a limited access to the long-time behavior of the correlators by introducing fits and allowing for non-zero Drude weights when appropriate. We compare our results to available data from optical lattice experiments and find that the Drude contributions can account for the theory-experiment gap in the DC spin conductivity of the model at half filling in the strong-coupling region. Our method helps paint a more complete picture of the conductivity in the two-dimensional Hubbard model and opens the door to studying dynamical properties of quantum lattice models in the thermodynamic limit.

arXiv:2602.03771•Feb 3, 2026•0 citations

Electrical Conductivity of Fermi Liquids. I. Many-Body Effect on the Drude Weight

Takuya Okabe

On the basis of the Fermi liquid theory, we investigate the many-body effect on the Drude weight. In a lattice system, the Drude weight $D$ is modified by electron-electron interaction due to Umklapp processes, while it is not renormalized in a Galilean invariant system. This is explained by showing that the effective mass $m'$ for $D\propto n/m'$ is defined through the current, not velocity, of quasiparticle. It is shown that the inequality $D>0$ is required for the stability against the uniform shift of the Fermi surface. The result of perturbation theory applied for the Hubbard model indicates that $D$ as a function of the density $n$ is qualitatively modified around half filling $n\sim 1$ by Umklapp processes.

arXiv:cond-mat/9807339•Journal of the Physical Society of Japan•Aug 15, 1998•21 citations

Conductivity of interacting spinless fermion systems via the high dimensional approach

Götz Silvester Uhrig

Spinless fermions with repulsion are treated non-perturbatively by classifying the diagrams of the generating functional Φ in powers of the inverse lattice dimension 1/d. The equations derived from the first two orders are evaluated on the one- and on the two-particle level. The order parameter of the AB-charge density wave (AB-CDW) occurring at larger interaction is calculated in d = 3. The Bethe-Salpeter equation is evaluated for the conductivity σ(ω) which is found to have two peaks within the energy gap 2∆ in the AB-CDW: a remnant of the Drude peak and an excitonic resonance. Unexpectedly, σ DC does not vanish for T → 0 1 The limit of infinite dimensions d → ∞ for itinerant fermion systems [1, 2] has proven very useful for the understanding of strongly interacting electron systems [3]. The research focusses on one-particle properties in the strict limit d = ∞. There exist also some results on transport properties [4, 5]. The present contribution extends the knowledge on the AC- and DC-conductivity including 1/d-corrections in the model of spinless fermions with repulsive interaction. The results cover and the non-symmetry broken (homogeneous) phase and the AB-CDW occurring on bipartite lattices. The techniques used in the description of the latter are certainly transferable to other symmetry broken phases in related models. The Hamiltonian of the spinless fermion model at half-filling (n = 0.5) is given in second quantisation

•Jan 1, 2008•1 citation

Finite-size effects in transport data from quantum Monte Carlo simulations

Raimundo Rocha dos Santos, Rubem Mondaini, Karim Bouadim +1 more

We have examined the behavior of the compressibility, the dc conductivity, the single-particle gap, and the Drude weight as probes of the density-driven metal-insulator transition in the Hubbard model on a square lattice. These quantities have been obtained through determinantal quantum Monte Carlo simulations at finite temperatures on lattices up to 16 × 16 sites. While the compressibility, the dc conductivity, and the gap are known to suffer from “closed-shell” effects due to the presence of artificial gaps in the spectrum (caused by the finiteness of the lattices), we have established that the former tracks the average sign of the fermionic determinant (〈sign〉), and that a shortcut often used to calculate the conductivity may neglect important corrections. Our systematic analyses also show that, by contrast, the Drude weight is not too sensitive to finite-size effects, being much more reliable as a probe to the insulating state. We have also investigated the influence of the discrete imaginary-time interval ( τ ) on 〈sign〉, on the average density (ρ), and on the double occupancy (d): we have found that 〈sign〉 and ρ are more strongly dependent on τ away from closed-shell configurations, but d follows the τ 2 dependence in both closedand open-shell cases.

Bulletin of the American Physical Society•Feb 29, 2012•0 citations

Umklapp process and resistivity in one-dimensional fermion systems

Thierry Giamarchi

The influence of umklapp scattering on the resistivity of an interacting-one-dimensional-fermion system (Luttinger liquid) is studied. By using a renormalization-group calculation and a memory-function approximation for the conductivity, it is possible to obtain its frequency and temperature dependence at arbitrary filling. At high temperature the conductivity behaves as a power law of the temperature with an exponent depending on the interactions. Away from half filling there is a crossover between this behavior and an exponential increase of the conductivity. At half filling, the low-temperature conductivity behaves as ${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}/\mathit{T}}$, where \ensuremath{\Delta} is the gap in the charge spectrum. It is argued that to get such behavior other scattering processes or phase-breaking processes are needed since in the presence of only electron-electron scattering the conductivity should, strictly speaking, be infinite at every finite temperature, even at half filling. Finally some results on the exponents of correlation functions and on the weight of the Drude peak obtained previously for the Hubbard model are shown to be generic features of any Luttinger liquid.

Physical review. B, Condensed matter•Aug 15, 1991•222 citations
The Drude weight D and the dc conductivity \ensuremathσdc{\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}\ensuremathσdc​(T) of strongly correlated electrons are investigated theoretically. Analytic results are derived for the homogeneous phase of the Hubbard model at d=\ensuremath{\infty}, and for spinless fermions in this limit with 1/d corrections systematically included to lowest order. It is found that \ensuremathσdc{\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}\ensuremathσdc​(T) is finite for all T>0, displaying Fermi-liquid behavior, \ensuremathσdc{\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}\ensuremathσdc​\ensuremath{\propto}1/T2{\mathit{T}}^{2}T2, at low temperatures. The validity of this result for finite dimensions is examined by investigating the importance of umklapp scattering processes and vertex corrections. A finite dc conductivity for T>0 is argued to be a generic feature of correlated lattice electrons in not too low dimensions.
We have examined the behavior of the compressibility, the dc conductivity, the single-particle gap, and the Drude weight as probes of the density-driven metal-insulator transition in the Hubbard model on a square lattice. These quantities have been obtained through determinantal quantum Monte Carlo simulations at finite temperatures on lattices up to $16\ifmmode\times\else\texttimes\fi{}16$ sites. While the compressibility, the dc conductivity, and the gap are known to suffer from ``closed-shell'' effects due to the presence of artificial gaps in the spectrum (caused by the finiteness of the lattices), we have established that the former tracks the average sign of the fermionic determinant ($\ensuremath{\langle}\text{sign}\ensuremath{\rangle}$), and that a shortcut often used to calculate the conductivity may neglect important corrections. Our systematic analyses also show that, by contrast, the Drude weight is not too sensitive to finite-size effects, being much more reliable as a probe to the insulating state. We have also investigated the influence of the discrete imaginary-time interval ($\ensuremath{\Delta}\ensuremath{\tau}$) on $\ensuremath{\langle}\text{sign}\ensuremath{\rangle}$, on the average density ($\ensuremath{\rho}$), and on the double occupancy (ddd): we have found that $\ensuremath{\langle}\text{sign}\ensuremath{\rangle}$ and $\ensuremath{\rho}$ are more strongly dependent on $\ensuremath{\Delta}\ensuremath{\tau}$ away from closed-shell configurations, but ddd follows the $\ensuremath{\Delta}{\ensuremath{\tau}}^{2}$ dependence in both closed- and open-shell cases.
Dynamical properties are notoriously difficult to compute in numerical treatments of the Fermi-Hubbard model, especially in two spatial dimensions. However, they are essential in providing us with insight into some of the most important and less well-understood phases of the model, such as the pseudogap and strange metal phases at relatively high temperatures, or unconventional superconductivity at lower temperatures, away from the commensurate filling. Here, we use the numerical linked-cluster expansions to compute spin and charge optical conductivities of the model at different temperatures and strong interaction strengths via the exact real-time-dependent correlation functions of the current operators. We mitigate systematic errors associated with having a limited access to the long-time behavior of the correlators by introducing fits and allowing for non-zero Drude weights when appropriate. We compare our results to available data from optical lattice experiments and find that the Drude contributions can account for the theory-experiment gap in the DC spin conductivity of the model at half filling in the strong-coupling region. Our method helps paint a more complete picture of the conductivity in the two-dimensional Hubbard model and opens the door to studying dynamical properties of quantum lattice models in the thermodynamic limit.
On the basis of the Fermi liquid theory, we investigate the many-body effect on the Drude weight. In a lattice system, the Drude weight DDD is modified by electron-electron interaction due to Umklapp processes, while it is not renormalized in a Galilean invariant system. This is explained by showing that the effective mass m′m'm′ for D∝n/m′D\propto n/m'D∝n/m′ is defined through the current, not velocity, of quasiparticle. It is shown that the inequality D>0D>0D>0 is required for the stability against the uniform shift of the Fermi surface. The result of perturbation theory applied for the Hubbard model indicates that DDD as a function of the density nnn is qualitatively modified around half filling n∼1n\sim 1n∼1 by Umklapp processes.
Spinless fermions with repulsion are treated non-perturbatively by classifying the diagrams of the generating functional Φ in powers of the inverse lattice dimension 1/d. The equations derived from the first two orders are evaluated on the one- and on the two-particle level. The order parameter of the AB-charge density wave (AB-CDW) occurring at larger interaction is calculated in d = 3. The Bethe-Salpeter equation is evaluated for the conductivity σ(ω) which is found to have two peaks within the energy gap 2∆ in the AB-CDW: a remnant of the Drude peak and an excitonic resonance. Unexpectedly, σ DC does not vanish for T → 0 1 The limit of infinite dimensions d → ∞ for itinerant fermion systems [1, 2] has proven very useful for the understanding of strongly interacting electron systems [3]. The research focusses on one-particle properties in the strict limit d = ∞. There exist also some results on transport properties [4, 5]. The present contribution extends the knowledge on the AC- and DC-conductivity including 1/d-corrections in the model of spinless fermions with repulsive interaction. The results cover and the non-symmetry broken (homogeneous) phase and the AB-CDW occurring on bipartite lattices. The techniques used in the description of the latter are certainly transferable to other symmetry broken phases in related models. The Hamiltonian of the spinless fermion model at half-filling (n = 0.5) is given in second quantisation
We have examined the behavior of the compressibility, the dc conductivity, the single-particle gap, and the Drude weight as probes of the density-driven metal-insulator transition in the Hubbard model on a square lattice. These quantities have been obtained through determinantal quantum Monte Carlo simulations at finite temperatures on lattices up to 16 × 16 sites. While the compressibility, the dc conductivity, and the gap are known to suffer from “closed-shell” effects due to the presence of artificial gaps in the spectrum (caused by the finiteness of the lattices), we have established that the former tracks the average sign of the fermionic determinant (〈sign〉), and that a shortcut often used to calculate the conductivity may neglect important corrections. Our systematic analyses also show that, by contrast, the Drude weight is not too sensitive to finite-size effects, being much more reliable as a probe to the insulating state. We have also investigated the influence of the discrete imaginary-time interval ( τ ) on 〈sign〉, on the average density (ρ), and on the double occupancy (d): we have found that 〈sign〉 and ρ are more strongly dependent on τ away from closed-shell configurations, but d follows the τ 2 dependence in both closedand open-shell cases.
The influence of umklapp scattering on the resistivity of an interacting-one-dimensional-fermion system (Luttinger liquid) is studied. By using a renormalization-group calculation and a memory-function approximation for the conductivity, it is possible to obtain its frequency and temperature dependence at arbitrary filling. At high temperature the conductivity behaves as a power law of the temperature with an exponent depending on the interactions. Away from half filling there is a crossover between this behavior and an exponential increase of the conductivity. At half filling, the low-temperature conductivity behaves as ${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}/\mathit{T}}$, where \ensuremath{\Delta} is the gap in the charge spectrum. It is argued that to get such behavior other scattering processes or phase-breaking processes are needed since in the presence of only electron-electron scattering the conductivity should, strictly speaking, be infinite at every finite temperature, even at half filling. Finally some results on the exponents of correlation functions and on the weight of the Drude peak obtained previously for the Hubbard model are shown to be generic features of any Luttinger liquid.