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arXiv:cond-mat/9609286 · Physical review. B, Condensed matter

Conductivity in a symmetry-broken phase: Spinless fermions with 1/dcorrections

Götz Silvester Uhrig

Published Oct 15, 1996• 8 citations
arXivHTMLPDF

Abstract

The dynamic conductivity \ensuremath{\sigma}(\ensuremath{\omega}) of strongly correlated electrons in a symmetry-broken phase is investigated in the present work. The model considered consists of spinless fermions with repulsive interaction on a simple cubic lattice. The investigated symmetry-broken phase is the charge density wave (CDW) with wave vector Q=(\ensuremath{\pi},\ensuremath{\pi},\ensuremath{\pi}${)}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}}$ which occurs at half-filling. The calculations are based on the high dimensional approach, i.e., an expansion in the inverse dimension 1/d is used. The finite dimensionality is accounted for by the inclusion of linear terms in 1/d and the true finite dimensional DOS. Special care is paid to the setup of a conserving approximation in the sense of Baym/Kadanoff without inconsistencies. The resulting Bethe-Salpeter equation is solved for the dynamic conductivity in the nonsymmetry-broken and in the symmetry-broken phase (AB-CDW). The dc-conductivity is reduced drastically in the CDW. Yet it does not vanish in the limit T\ensuremath{\rightarrow}0 due to a subtle cancellation of diverging mobility and vanishing DOS. In the dynamic conductivity \ensuremath{\sigma}(\ensuremath{\omega}) the energy gap induced by the symmetry breaking is clearly discernible. In addition, the vertex corrections of order 1/d lead to an excitonic resonance lying within the gap. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.10436

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The dynamic conductivity \ensuremath{\sigma}(\ensuremath{\omega}) of strongly correlated electrons in a symmetry-broken phase is investigated in the present work. The model considered consists of spinless fermions with repulsive interaction on a simple cubic lattice. The investigated symmetry-broken phase is the charge density wave (CDW) with wave vector Q=(\ensuremath{\pi},\ensuremath{\pi},\ensuremath{\pi})\ifmmode†\else\textdagger\fi{)}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}})\ifmmode†\else\textdagger\fi which occurs at half-filling. The calculations are based on the high dimensional approach, i.e., an expansion in the inverse dimension 1/d is used. The finite dimensionality is accounted for by the inclusion of linear terms in 1/d and the true finite dimensional DOS. Special care is paid to the setup of a conserving approximation in the sense of Baym/Kadanoff without inconsistencies. The resulting Bethe-Salpeter equation is solved for the dynamic conductivity in the nonsymmetry-broken and in the symmetry-broken phase (AB-CDW). The dc-conductivity is reduced drastically in the CDW. Yet it does not vanish in the limit T\ensuremath{\rightarrow}0 due to a subtle cancellation of diverging mobility and vanishing DOS. In the dynamic conductivity \ensuremath{\sigma}(\ensuremath{\omega}) the energy gap induced by the symmetry breaking is clearly discernible. In addition, the vertex corrections of order 1/d lead to an excitonic resonance lying within the gap. \textcopyright{} 1996 The American Physical Society.
Spinless fermions with repulsion are treated non-perturbatively by classifying the diagrams of the generating functional Φ in powers of the inverse lattice dimension 1/d. The equations derived from the first two orders are evaluated on the one- and on the two-particle level. The order parameter of the AB-charge density wave (AB-CDW) occurring at larger interaction is calculated in d = 3. The Bethe-Salpeter equation is evaluated for the conductivity σ(ω) which is found to have two peaks within the energy gap 2∆ in the AB-CDW: a remnant of the Drude peak and an excitonic resonance. Unexpectedly, σ DC does not vanish for T → 0 1 The limit of infinite dimensions d → ∞ for itinerant fermion systems [1, 2] has proven very useful for the understanding of strongly interacting electron systems [3]. The research focusses on one-particle properties in the strict limit d = ∞. There exist also some results on transport properties [4, 5]. The present contribution extends the knowledge on the AC- and DC-conductivity including 1/d-corrections in the model of spinless fermions with repulsive interaction. The results cover and the non-symmetry broken (homogeneous) phase and the AB-CDW occurring on bipartite lattices. The techniques used in the description of the latter are certainly transferable to other symmetry broken phases in related models. The Hamiltonian of the spinless fermion model at half-filling (n = 0.5) is given in second quantisation
An analytic expression for the dynamical conductivity sigma ( omega ) of spinless fermions with local disorder and nearest-neighbour repulsion is derived on lattices with infinite coordination number Z. The model is exactly solvable in the whole parameter range assuming two possible phases: a homogeneous phase and a checkerboard charge-density wave (CDW). Away from half filling the system displays anomalous behaviour: weak particle-density fluctuations favour spontaneous symmetry breaking. First, the authors investigate the effects of this anomaly on the conductivity in the AC and DC regimes. Second, they focus on the Mott transition occurring at zero temperature at half filling. The critical exponents for the conductivity are computed for the dependence on the interaction U, the disorder gamma , the filling n, the temperature T and the frequency omega .
The Holstein model of spinless fermions interacting with dispersionless phonons in one dimension is studied by a Green's function Monte Carlo technique. The ground-state energy, first fermionic excited state, density wave correlations, and mean lattice displacements are calculated for lattices of up to 16 sites, for one fermion per two sites, i.e., a half-filled band. Results are obtained for values of the fermion hopping parameter of t=0.1\ensuremath{\omega}, \ensuremath{\omega}, and 10\ensuremath{\omega}, where \ensuremath{\omega} is the phonon frequency. At a finite fermion-phonon coupling g there is a transition from a metallic phase to an insulating phase in which there is charge-density-wave order. Finite-size scaling is found to hold in the metallic phase and is used to extract the coupling dependence of the Luttinger liquid parameters, ${\mathit{u}}_{\mathrm{\ensuremath{\rho}}}$ and ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, the velocity of charge excitations, and the correlation exponent, respectively. For free fermions (g=0) and for strong coupling (g2{\mathit{g}}^{2}g2\ensuremath{\gg}t\ensuremath{\omega}) our results agree well with known analytic results. \textcopyright{} 1996 The American Physical Society.
We study spinless fermions with nearest-neighbor repulsive interactions ($t\text{\ensuremath{-}}V$ model) on the two-dimensional three-band Lieb lattice. At half-filling, the free electronic band structure consists of a flat band at zero energy and a single cone with linear dispersion. The flat band is expected to be unstable upon inclusion of electronic correlations, and a natural channel is charge order. However, due to the three-orbital unit cell, commensurate charge order implies an imbalance of electron and hole densities and therefore doping away from half-filling. Our numerical results show that below a finite-temperature Ising transition a charge density wave with one electron and two holes per unit cell and its partner under particle-hole transformation are spontaneously generated. Our calculations are based on recent advances in auxiliary-field and continuous-time quantum Monte Carlo simulations that allow sign-free simulations of spinless fermions at half-filling. It is argued that particle-hole symmetry breaking provides a route to access levels of finite doping, without introducing a sign problem.
We investigate the magnetic order and related strongly correlated effects in a one-dimensional Ising-Kondo lattice with transverse field. This model is the anisotropic limit of the conventional isotropic Kondo lattice model, in the sense that the itinerant electrons interact with the localized magnetic moments via only longitudinal Kondo exchange. Adopting the numerical density-matrix renormalization group method, we map out the ground-state phase diagram in various parameter spaces. Depending on the Kondo coupling and filling number, three distinct phases, including a metallic paramagnetic, a metallic ferromagnetic, and a gapped spin-density wave phase, are obtained. The spin-density wave is characterized by an ordering wave vector, which coincides with the nesting wave vector of the Fermi surface. This makes the corresponding magnetic transition a spin analog of the Peierls transition occurring in the one-dimensional metal. Moreover, by analyzing the momentum distribution function and charge correlation function, the conduction electrons are shown to behave like free spinless fermions in the ferromagnetic phase. We finally discuss the effect of the repulsive Hubbard interaction between conduction electrons. Our paper enriches the Kondo physics and deepens the current understanding of the heavy fermion compounds.
The Drude weight D and the dc conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$(T) of strongly correlated electrons are investigated theoretically. Analytic results are derived for the homogeneous phase of the Hubbard model at d=\ensuremath{\infty}, and for spinless fermions in this limit with 1/d corrections systematically included to lowest order. It is found that ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$(T) is finite for all T>0, displaying Fermi-liquid behavior, ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$\ensuremath{\propto}1/T2{\mathit{T}}^{2}T2, at low temperatures. The validity of this result for finite dimensions is examined by investigating the importance of umklapp scattering processes and vertex corrections. A finite dc conductivity for T>0 is argued to be a generic feature of correlated lattice electrons in not too low dimensions.