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Physical review. B, Condensed matter

Resistivity of a one-dimensional interacting quantum fluid

Thierry Giamarchi

Published Jul 1, 1992• 72 citations
Publisher

Abstract

The frequency and temperature dependence of the conductivity of a one-dimensional fermion system with attractive interactions is studied by using a renormalization-group technique. At half filling the real part of the conductivity has both a \ensuremath{\delta}(\ensuremath{\omega}) part and a divergent frequency behavior ${\mathrm{\ensuremath{\omega}}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$ at finite frequencies, where \ensuremath{\nu} is a nonuniversal exponent depending on the interactoins. For the particular case of the attractive Hubbard model, logarithmic corrections appear and the conductivity behaves as 1/[\ensuremath{\omega} ${\mathrm{log}}_{10}^{2}$(\ensuremath{\omega})], plus a \ensuremath{\delta}(\ensuremath{\omega}) part. Away from half filling the conductivity has a \ensuremath{\delta}(\ensuremath{\omega}) part and a gap up to a critical frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$, where ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$ is proportional to the doping with a prefactor depending on the interactions. The results obtained for the fermion model can be straightforwardly extended to the conductivity of an interacting one-dimensional boson model.

DOI
10.1103/physrevb.46.342

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The frequency and temperature dependence of the conductivity of a one-dimensional fermion system with attractive interactions is studied by using a renormalization-group technique. At half filling the real part of the conductivity has both a \ensuremath{\delta}(\ensuremath{\omega}) part and a divergent frequency behavior \ensuremathω\ensuremath−\ensuremathν{\mathrm{\ensuremath{\omega}}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}\ensuremathω\ensuremath−\ensuremathν at finite frequencies, where \ensuremath{\nu} is a nonuniversal exponent depending on the interactoins. For the particular case of the attractive Hubbard model, logarithmic corrections appear and the conductivity behaves as 1/[\ensuremath{\omega} log102{\mathrm{log}}_{10}^{2}log102​(\ensuremath{\omega})], plus a \ensuremath{\delta}(\ensuremath{\omega}) part. Away from half filling the conductivity has a \ensuremath{\delta}(\ensuremath{\omega}) part and a gap up to a critical frequency \ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​, where \ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​ is proportional to the doping with a prefactor depending on the interactions. The results obtained for the fermion model can be straightforwardly extended to the conductivity of an interacting one-dimensional boson model.
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.
We study finite-temperature transport properties of the one-dimensional Hubbard model using the density matrix renormalization group. Our aim is twofold: First, we compute both the charge- and the spin-current correlation function of the integrable model at half filling. The former decays rapidly, implying that the corresponding Drude weight is either zero or very small. Second, we calculate the optical charge conductivity ${\ensuremath{\sigma}}_{r}eg(\ensuremath{\omega})$ in the presence of small integrability-breaking next-nearest-neighbor interactions (the extended Hubbard model). The dc conductivity is finite and diverges as the temperature is decreased below the gap. Our results thus suggest that the half-filled, gapped Hubbard model is a normal charge conductor at finite temperatures. As a test bed for our numerics, we compute ${\ensuremath{\sigma}}_{r}eg(\ensuremath{\omega})$ for the integrable XXZ spin chain in its gapped phase.
We show that the real part of the frequency-dependent conductivity of interacting spinless fermions at incommensurate band fillings in one dimension at T=0 is proportional to ${\mathrm{\ensuremath{\omega}}}^{3}$ for small positive frequencies, contrary to statements in the recent literature. We argue that the dc conductivity of spinless fermions at incommensurate band fillings diverges faster than any power of 1/T as T\ensuremath{\rightarrow}0. We give the extension of our results to the case of commensurate band fillings and fermions with spin.
The one-band two-dimensional Hubbard Hamiltonian is analyzed using a real-space renormalization-group block method. The renormalization-group method, previously applied to the one-dimensional case has been extended to two dimensions. It is also shown how to avoid the proliferation of new terms during the process if the shape of the blocks and the symmetries of the kept states are chosen conveniently. We characterize the ground state by its energy per site, the gap of charge excitations, the double occupancy, and the effective hopping. The system shows an insulating behavior in the case of half filling for all nonzero interaction parameter, U, if the hopping is isotropic (ty{\mathit{t}}_{\mathit{y}}ty​=tx{\mathit{t}}_{\mathit{x}}tx​); that is to say, the gap opens for all positive U, as in the one-dimensional case. If the hopping anisotropy, \ensuremath{\alpha}=ty{\mathit{t}}_{\mathit{y}}ty​/tx{\mathit{t}}_{\mathit{x}}tx​, is different from the unity and zero the ground state is a conductor up to a critical value (U/t)c1{)}_{\mathit{c}1})c1​, which depends on \ensuremath{\alpha}. We have analyzed in the same way another filling, n=7/9. The system behaves as a conductor, paramagnetic up to a value (U/t)c2{)}_{\mathit{c}2})c2​\ensuremath{\sim}2 and weakly ferromagnetic for higher values of U.
The electrical resistivity due only to mutual Coulomb interaction of the d -dimensional lattice electron systems ( d ≥2) in the presence of Umklapp scattering processes is studied on the basis of the Fermi liquid theory. It is shown that normal scattering processes generally contribute to the resistivity if Umklapp scattering processes are present. This fact gives a natural explanation of the resistivity of the d =∞ Hubbard model which is determined only by the damping rate of quasi-particles, γ. In the case of d =2, however, it is shown that the resistivity is proportional to T 2 , T being the temperature, even when normal scattering processes, which give T 2 log T -contribution to γ, contribute to the resistivity. In the special case of the square lattice, where the transfer integrals are only between the nearest neighbors, the coefficient of the T 2 -term in the second order of the interaction is shown to diverge as (1/δ) log (1 / δ) when δ→0, δ being the doping rate away from half-filling.
Nonlinear transport in the one dimensional Hubbard model at half-filling under a finite bias voltage is investigated by the adaptive time-dependent density matrix renormalization group method. For repulsive on-site interaction, dielectric breakdown of the Mott insulating ground state to a current-carrying nonequilibrium steady state is clearly observed when the voltage exceeds the charge gap. It is found that by increasing the voltage further the current-voltage characteristics are scaled only by the charge gap and the scaling curve exhibits almost linear dependence on the voltage whose slope is suppressed by the electron correlation. In the case of attractive interaction the linear conductance is the perfect one 2e2/h2e^2/h2e2/h which agrees with the prediction by the Luttinger liquid theory.