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arXiv:0708.0590 · Physical Review B

Nonperturbative interaction effects in the thermodynamics of disordered wires

D. A. Pesin, Anton V. Andreev

Published Dec 6, 2007• 0 citations
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Abstract

We study nonperturbative interaction corrections to the thermodynamic quantities of multichannel disordered wires in the presence of the Coulomb interactions. Within the replica nonlinear $\ensuremath{\sigma}$-model $(\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ formalism, they arise from nonperturbative soliton saddle points of the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ action. The problem is reduced to evaluating the partition function of a replicated classical one-dimensional Coulomb gas. The state of the latter depends on two parameters: the number of transverse channels in the wire ${N}_{\mathit{ch}}$ and the dimensionless conductance $G({L}_{T})$ of a wire segment of length equal to the thermal diffusion length ${L}_{T}$. At relatively high temperatures, $G({L}_{T})\ensuremath{\gtrsim}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{N}_{\mathit{ch}}$, the gas is dimerized, i.e., consists of bound neutral pairs. At lower temperatures, $\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{N}_{\mathit{ch}}\ensuremath{\gtrsim}G({L}_{T})\ensuremath{\gtrsim}1$, the pairs overlap and form a Coulomb plasma. The crossover between the two regimes occurs at a parametrically large conductance $G({L}_{T})\ensuremath{\sim}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{N}_{\mathit{ch}}$ and may be studied independently from the perturbative effects. Specializing on the high-temperature regime, we obtain the leading nonperturbative correction to the wire heat capacity. Its ratio to the heat capacity for noninteracting electrons, ${C}_{0}$, is $\ensuremath{\delta}C∕{C}_{0}\ensuremath{\sim}{N}_{\mathit{ch}}{G}^{2}({L}_{T}){e}^{\ensuremath{-}2G({L}_{T})}$.

DOI
10.1103/physrevb.76.235108

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We study nonperturbative interaction corrections to the thermodynamic quantities of multichannel disordered wires in the presence of the Coulomb interactions. Within the replica nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ-model (NL\ensuremathσM)(\mathrm{NL}\ensuremath{\sigma}\mathrm{M})(NL\ensuremathσM) formalism, they arise from nonperturbative soliton saddle points of the NL\ensuremathσM\mathrm{NL}\ensuremath{\sigma}\mathrm{M}NL\ensuremathσM action. The problem is reduced to evaluating the partition function of a replicated classical one-dimensional Coulomb gas. The state of the latter depends on two parameters: the number of transverse channels in the wire Nch{N}_{\mathit{ch}}Nch​ and the dimensionless conductance G(LT)G({L}_{T})G(LT​) of a wire segment of length equal to the thermal diffusion length LT{L}_{T}LT​. At relatively high temperatures, G(LT)\ensuremath≳lnNchG({L}_{T})\ensuremath{\gtrsim}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{N}_{\mathit{ch}}G(LT​)\ensuremath≳lnNch​, the gas is dimerized, i.e., consists of bound neutral pairs. At lower temperatures, lnNch\ensuremath≳G(LT)\ensuremath≳1\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{N}_{\mathit{ch}}\ensuremath{\gtrsim}G({L}_{T})\ensuremath{\gtrsim}1lnNch​\ensuremath≳G(LT​)\ensuremath≳1, the pairs overlap and form a Coulomb plasma. The crossover between the two regimes occurs at a parametrically large conductance G(LT)\ensuremath∼lnNchG({L}_{T})\ensuremath{\sim}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{N}_{\mathit{ch}}G(LT​)\ensuremath∼lnNch​ and may be studied independently from the perturbative effects. Specializing on the high-temperature regime, we obtain the leading nonperturbative correction to the wire heat capacity. Its ratio to the heat capacity for noninteracting electrons, C0{C}_{0}C0​, is \ensuremathδC∕C0\ensuremath∼NchG2(LT)e\ensuremath−2G(LT)\ensuremath{\delta}C∕{C}_{0}\ensuremath{\sim}{N}_{\mathit{ch}}{G}^{2}({L}_{T}){e}^{\ensuremath{-}2G({L}_{T})}\ensuremathδC∕C0​\ensuremath∼Nch​G2(LT​)e\ensuremath−2G(LT​).
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Recent experiments have revealed that the temperature dependence of the conductance of quasiballistic quantum wires bears characteristic features of the Luttinger-liquid state. In this paper, the conductance of an N-channel quantum wire is calculated within the model of N coupled Luttinger liquids and under the assumption of weak disorder. It is shown that as the number of channels increases, a crossover from the Luttinger-liquid to the Fermi-liquid behavior occurs. This crossover manifests itself in the 1/N decrease of the scaling exponent of the temperature dependence. An exact expression for the scaling exponent for the case of N coupled Luttinger chains is obtained, and the large N limit is studied for the case of a quantum wire. The case of N=2 for electrons with spin is analyzed in detail, and a qualitative agreement with the experiment is achieved.
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We study Coulomb drag in a pair of parallel one-dimensional electron systems within the framework of the Tomonaga-Luttinger model. We find that Coulomb coupling has a much stronger effect on one-dimensional wires than on two-dimensional layers: At zero temperature the transresistivity diverges, due to the formation of locked charge density waves. At temperature well above a crossover temperature T∗{T}^{*}T∗ the transresistivity follows a power law $\ensuremath{\rho}\ensuremath{\propto}{T}^{x},$ where the interaction-strength dependent exponent x is determined by the Luttinger liquid parameter ${K}_{c\ensuremath{-}}$ of the relative charge mode. At temperature below T∗{T}^{*}T∗ relative charge displacements are enabled by solitonic excitations, reflected by an exponential temperature dependence. The crossover temperature T∗{T}^{*}T∗ depends sensitively on the wire width, interwire distance, Fermi wavelength and the effective Bohr radius. For wire distances $\overline{d}\ensuremath{\gg}{k}_{F}^{\ensuremath{-}1}$ it is exponentially suppressed with ${T}^{*}{/E}_{F}\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}\overline{d}{k}_{F}/(1\ensuremath{-}{K}_{c\ensuremath{-}})].$ The behavior changes drastically if each of the two wires develop spin gaps. In this case we find that the transresistivity vanishes at zero temperature. We discuss our results in view of possible experimental realizations in $\mathrm{GaAs}\ensuremath{-}{\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ semiconductor structures.
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