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Physical review. B, Condensed matter

Umklapp process and resistivity in one-dimensional fermion systems

Thierry Giamarchi

Published Aug 15, 1991• 222 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.44.2905

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arXiv:cond-mat/9907072•Jul 5, 1999•0 citations
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 e\ensuremath−\ensuremathΔ/T{\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}/\mathit{T}}e\ensuremath−\ensuremathΔ/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.
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 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.
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.
We investigate the Hall effect in two different theoretical models of strongly correlated systems: a system made of weakly coupled Luttinger liquids, in the presence of umklapp scattering, and the 2D triangular lattice, with nearest-neighbor hopping and a local Hubbard interaction. In the first model we use a memory function approach to compute the Hall coefficient as a function of temperature and frequency in the presence of umklapp scattering. We find a power-law correction to the free-fermion value (band value), with an exponent depending on the Luttinger parameter KρK_ρKρ​. At sufficiently high temperature or frequency the Hall coefficient approaches the band value. We discuss the applications to Hall experiments made in quasi 1D organic compounds. In the second model, the complete temperature and doping dependencies of the high-frequency Hall coefficient $\RH$ are evaluated analytically and numerically for small, intermediate, and strong interactions using various approximation schemes. We find that $\RH$ follows the semiclassical 1/qn∗1/qn^*1/qn∗ law near T=0, but exhibits a striking TTT-linear behavior with an interaction- and doping-dependent slope at high temperature. We compare our results with Hall measurements performed in cobaltates.
We review some of our recent results on the potential scattering in a weakly interacting one-dimensional(1D) electron gas. The technique we developed is a poor man's renormalization group procedure in the scattered wave basis. This technique can treat the renormalizations of the scattering on the barrier and the scattering between the electrons in a coherent way, and it allows us to find the scattering amplitudes on a localized potential of arbitrary strength for electrons at any energy. The obtained phase shifts are used to study the Fermi-edge singularity in an interacting 1D electron system, where anomalous exponent of the power-law singularity in the vicinity of the edge is found. The transmission coefficient is directly related to the conductance of a 1D channel by the Landauer formula. Simple formulas that describe the conductance at any temperature are derived. In spin-[Formula: see text] systems, the electron–electron backscattering induces renormalizations of the interaction constants, which causes the low-temperature conductance to deviate from the results of the Luttinger liquid theory. In particular, the temperature dependence of the conductance may become nonmonotonic. In the presence of a magnetic field, backscattering gives rise to a peak in the differential conductance at bias equal to the Zeeman splitting.
Suppression of electron current $ ΔI$ through a 1D channel of length LLL connecting two Fermi liquid reservoirs is studied taking into account the Umklapp interaction induced by a periodic potential. This interaction opens band gaps at the integer fillings and Hubbard gaps 2m2m2m at some rational fillings in the infinite wire: L→∞L \to \inftyL→∞. In the perturbative regime where m≪vc/Lm \ll v_c/Lm≪vc​/L (vc:v_c:vc​: charge velocity), and for small deviations δnδnδn of the electron density from its commensurate values −ΔI/V- ΔI/V−ΔI/V can diverge with some exponent as voltage or temperature V,TV,TV,T decreases above Ec=max(vc/L,vcδn)E_c=max(v_c/L,v_c δn)Ec​=max(vc​/L,vc​δn), while it goes to zero below EcE_cEc​. This results in a non-monotonous behavior of the conductance. In the case when the Umklapp interaction creates a large Mott-Hubbard gap $2m \gg T_L $ inside the wire, the transport is suppressed near half-filling everywhere inside the gap except for an exponentially small region of V,T<TLexp(−2m/TL)V,T < T_L exp(-2m/T_L)V,T<TL​exp(−2m/TL​).