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Physical review. B, Condensed matter

Conductivity of a Luttinger liquid

Thierry Giamarchi, Andrew J. Millis

Published Oct 15, 1992• 96 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.46.9325

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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 \ensuremathω3{\mathrm{\ensuremath{\omega}}}^{3}\ensuremathω3 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 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} 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 ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$, 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.
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 analyze the uniform conductivity of a one-dimensional degenerate fermion system placed in a random disorder potential so smooth that backward scattering can be neglected. We use the nonlinear Luttinger liquid model to consider effects of both interaction and the curvature of fermionic dispersion. The finite frequency conductivity, calculated in the lowest order of disorder potential, consists of two parts. First one is the elastic contribution, largely independent of temperature and interaction. Second one is the inelastic contribution, strongly dependent on temperature and frequency and appearing upon simultaneous presence of curvature, disorder and interaction. We argue that apart from such finite frequency conductivity, there should always remain the δδδ-function peak of conductivity at zero frequency, whose weight is weakly dependent on the disorder.
An accurate numerical consideration of one-dimensional (1D) spinless fermion model with next-nearest-neighbor interactions is carried out for the electron concentration \ensuremath{\rho}=4/7. It is shown that depending on the parameters of the model, it can be either Luttinger liquid or bipolaron liquid. In the former case, competing interactions can result in a smooth behavior of one-electron distribution function at the Fermi surface with a divergence in the second derivative with respect to the quasimomentum (and not in the first one, as usual). In this connection, uncommon photoemission spectra for some 1D conductors can be explained qualitatively.
The low--energy excited states of a system of interacting one--dimensional fermions in a conducting state are collective charge and spin density oscillations. The unusual physical properties of such a system (called ``Luttinger liquid'') are characterized by the velocities uρu_ρuρ​ and uσu_σuσ​ of the charge and spin excitations, as well as by a parameter KρK_ρKρ​ that determines the power law behavior of correlation functions. Umklapp scattering occuring at half--filling or other commensurate bandfilling can lead to a transition into an insulating state, characterized in particular by a gap in the charge excitations (the Mott--Hubbard gap). The properties in the vicinity of the transition are shown to depend on both the way the transition is approached (constant bandfilling and varying interaction, or constant interaction and varying bandfilling) and on the ``order'' of the commensurability. In particular, even and odd fractional fillings show quite different behavior. This behavior is illustrated in detail using lattice models like the Hubbard model and its extensions.
One-particle interchain hopping in a system of coupled Luttinger liquids is investigated using exact diagonalizations techniques. First, the two-chain problem of spinless fermions is studied in order to see the behavior of the band splitting as a function of the exponent $\ensuremath{\alpha}$ which characterizes the one-dimensional (1D) Luttinger liquid. Moderate intrachain interactions can lead to a strong reduction of this splitting. The onset of the confinement within the individual chains (defined by a vanishing splitting) seems to be governed by $\ensuremath{\alpha}$. We give numerical evidence that interchain coherent hopping can be totally suppressed for $\ensuremath{\alpha}\ensuremath{\sim}0.4$ or even smaller $\ensuremath{\alpha}$ values. The transverse conductivity is shown to exhibit a strong incoherent part. Even when coherent interchain hopping is believed to occur (at small $\ensuremath{\alpha}$ values), it is shown that the coherent Drude weight is always significantly smaller than the incoherent weight. Implications for the optical experiments in quasi-1D organic or high-Tc{T}_{c}Tc​ superconductors are outlined.