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arXiv:cond-mat/0610426 · Physical Review B

Frequency-temperature crossover in the conductivity of disordered Luttinger liquids

Bernd Rosenow, Andreas Glatz, Thomas Nattermann

Published Oct 10, 2007• 4 citations
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Abstract

The temperature $(T)$ and frequency $(\ensuremath{\omega})$ dependent conductivity of weakly disordered Luttinger liquids is calculated in a systematic way both by perturbation theory and from a finite temperature renormalization group (RG) treatment to leading order in the disorder strength. Whereas perturbation theory results in $\ensuremath{\omega}∕T$ scaling of the conductivity, such scaling is violated in the RG treatment. We also determine the nonlinear field dependence of the conductivity, whose power law scaling is different from that of temperature and frequency dependence.

DOI
10.1103/physrevb.76.155108

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The temperature (T)(T)(T) and frequency (\ensuremathω)(\ensuremath{\omega})(\ensuremathω) dependent conductivity of weakly disordered Luttinger liquids is calculated in a systematic way both by perturbation theory and from a finite temperature renormalization group (RG) treatment to leading order in the disorder strength. Whereas perturbation theory results in \ensuremathω∕T\ensuremath{\omega}∕T\ensuremathω∕T scaling of the conductivity, such scaling is violated in the RG treatment. We also determine the nonlinear field dependence of the conductivity, whose power law scaling is different from that of temperature and frequency dependence.
We calculate the linear response conductance of electrons in a Luttinger liquid with arbitrary interaction g2{g}_{2}g2​, and subject to a potential barrier of arbitrary strength, as a function of temperature. We map the Hamiltonian in the basis of scattering states into an effective low energy Hamiltonian in current algebra form. First the renormalization group (RG) equation for weak interaction is derived in the current operator language both using the operator product expansion and the equation of motion method. To access the strong coupling regime, two methods of deducing the RG equation from perturbation theory, based on the scaling hypothesis and on the Callan-Symanzik formulation, are discussed. The important role of scale-independent terms is emphasized. The latter depend on the regularization scheme used (length versus temperature cutoff). Analyzing the perturbation theory in the fermionic representation, the diagrams contributing to the renormalization group $\ensuremath{\beta}$-function are identified. A universal part of the $\ensuremath{\beta}$-function is given by a ladder series and summed to all orders in g2{g}_{2}g2​. First nonuniversal corrections beyond the ladder series are discussed and are shown to differ from the exact solutions obtained within conformal field theory which use a different regularization scheme. The RG equation for the temperature-dependent conductance is solved analytically. Our result agrees with known limiting cases.
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.
The low-temperature phase diagram of one-dimensional weakly disordered quantum systems such as charge or spin density waves and Luttinger liquids is studied by a full finite temperature renormalization-group (RG) calculation. For vanishing quantum fluctuations this approach is amended by an exact solution in the case of strong disorder and by a mapping onto the Burgers equation with noise in the case of weak disorder, respectively. At zero temperature we reproduce the quantum phase transition between a pinned (localized) and an unpinned (delocalized) phase for weak and strong quantum fluctuations, respectively, as found previously by Fukuyama or Giamarchi and Schulz. At finite temperatures the localization transition is suppressed: the random potential is wiped out by thermal fluctuations on length scales larger than the thermal de Broglie wavelength of the phason excitations. The existence of a zero-temperature transition is reflected in a rich crossover phase diagram of the correlation functions. In particular we find four different scaling regions: a classical disordered, a quantum disordered, a quantum critical, and a thermal region. The results can be transferred directly to the discussion of the influence of disorder in superfluids. Finally we extend the RG calculation to the treatment of a commensurate lattice potential. Applications to related systems are discussed as well.
We consider heat transport in a Luttinger liquid (LL) with weak disorder and study the Lorenz number for this system. We start at a high- T regime, and calculate both the electrical and thermal conductivities using a memory function approach. The resulting Lorenz number L is independent of T but depends explicitly on the LL exponents. Lowering T , however, allows for a renormalization of the LL exponents from their bare values by disorder, causing a violation of the Wiedemann-Franz law. Finally, we extend the discussion to quantum wire systems and study the wire size dependence of the Lorenz number.
The electrical current through an arbitrary junction connecting quantum wires of spinless interacting fermions is calculated in fermionic representation. The wires are adiabatically attached to two reservoirs at chemical potentials differing by the applied voltage bias. The relevant scale-dependent contributions in perturbation theory in the interaction up to infinite order are evaluated and summed up. The result allows one to construct renormalization group equations for the conductance as a function of voltage (or temperature, wire length). There are two fixed points at which the conductance follows a power law in terms of a scaling variable $\ensuremath{\Lambda}$, which equals the bias voltage VVV, if VVV is the largest energy scale compared to temperature TTT and inverse wire length ${L}^{\ensuremath{-}1}$, and interpolates between these quantities in the crossover regimes.
We study the interplay of interactions and disorder in a one-dimensional fermion lattice coupled adiabatically to infinite reservoirs. We employ both the functional renormalization group (FRG) as well as matrix product state techniques, which serve as an accurate benchmark for small systems. Using the FRG, we compute the length- and temperature-dependence of the conductance averaged over 104{10}^{4}104 samples for lattices as large as 105{10}^{5}105 sites. We identify regimes in which non-Ohmic power law behavior can be observed and demonstrate that the corresponding exponents can be understood by adapting earlier predictions obtained perturbatively for disordered Luttinger liquids. In the presence of both disorder and isolated impurities, the conductance has a universal single-parameter scaling form. This lays the groundwork for an application of the functional renormalization group to the realm of many-body localization.