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arXiv:1310.4332 · Physical Review B

Gapless metallic charge-density-wave phase driven by strong electron correlations

R. Lemański, K. Ziegler

Published Feb 4, 2014• 20 citations
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Abstract

We analyze the transformation from insulator to metal induced by thermal fluctuations within the Falicov-Kimball model. Using the dynamic mean field theory (DMFT) formalism on the Bethe lattice we find rigorously the temperature dependent density of states (DOS) at half filling in the limit of high dimensions. At zero temperature ($T=0$) the system is ordered to form the checkerboard pattern and the DOS has the gap $\ensuremath{\Delta}$ at the Fermi level ${\ensuremath{\varepsilon}}_{F}=0$, which is proportional to the interaction constant $U$. With an increase of $T$ the DOS evolves in various ways that depend on $U$. For $U>{U}_{cr}$ the gap persists for any $T$ (then $\ensuremath{\Delta}>0$), so the system is always an insulator. However, if $U<{U}_{cr}$, two additional subbands develop inside the gap. They become wider with increasing $T$ and at a certain $U$-dependent temperature ${T}_{\mathrm{MI}}$ they join with each other at ${\ensuremath{\varepsilon}}_{F}$. Since above ${T}_{\mathrm{MI}}$ the DOS is positive at ${\ensuremath{\varepsilon}}_{F}$, we interpret ${T}_{\mathrm{MI}}$ as the transformation temperature from insulator to metal. It appears that ${T}_{\mathrm{MI}}$ approaches the order-disorder phase transition temperature ${T}_{\mathrm{O}\text{\ensuremath{-}}\mathrm{DO}}$ when $U$ is close to 0 or ${U}_{cr}$, but ${T}_{\mathrm{MI}}$ is substantially lower than ${T}_{\mathrm{O}\text{\ensuremath{-}}\mathrm{DO}}$ for intermediate values of $U$. Moreover, using an analytical formula we show that ${T}_{\mathrm{MI}}=0$ at $U=\sqrt{2}<{U}_{cr}$, so we prove that the quantum critical point exists for the ordered metal at $(T=0,\phantom{\rule{0.28em}{0ex}}U=\sqrt{2})$. Having calculated the temperature dependent DOS we study thermodynamic properties of the system starting from its free energy $F$. Then we find how the order parameter $d$ and the gap $\ensuremath{\Delta}$ change with $T$ and we construct the phase diagram in the variables $T$ and $U$, where we display regions of stability of four different phases: ordered insulator, ordered metal, disordered insulator, and disordered metal. Finally, we use a low temperature expansion to demonstrate the existence of a nonzero DOS at a characteristic value of $U$ on a general bipartite lattice.

DOI
10.1103/physrevb.89.075104

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We analyze the transformation from insulator to metal induced by thermal fluctuations within the Falicov-Kimball model. Using the dynamic mean field theory (DMFT) formalism on the Bethe lattice we find rigorously the temperature dependent density of states (DOS) at half filling in the limit of high dimensions. At zero temperature (T=0T=0T=0) the system is ordered to form the checkerboard pattern and the DOS has the gap \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ at the Fermi level \ensuremathεF=0{\ensuremath{\varepsilon}}_{F}=0\ensuremathεF​=0, which is proportional to the interaction constant UUU. With an increase of TTT the DOS evolves in various ways that depend on UUU. For U>UcrU>{U}_{cr}U>Ucr​ the gap persists for any TTT (then \ensuremathΔ>0\ensuremath{\Delta}>0\ensuremathΔ>0), so the system is always an insulator. However, if U<UcrU<{U}_{cr}U<Ucr​, two additional subbands develop inside the gap. They become wider with increasing TTT and at a certain UUU-dependent temperature TMI{T}_{\mathrm{MI}}TMI​ they join with each other at \ensuremathεF{\ensuremath{\varepsilon}}_{F}\ensuremathεF​. Since above TMI{T}_{\mathrm{MI}}TMI​ the DOS is positive at \ensuremathεF{\ensuremath{\varepsilon}}_{F}\ensuremathεF​, we interpret TMI{T}_{\mathrm{MI}}TMI​ as the transformation temperature from insulator to metal. It appears that TMI{T}_{\mathrm{MI}}TMI​ approaches the order-disorder phase transition temperature TO\ensuremath-DO{T}_{\mathrm{O}\text{\ensuremath{-}}\mathrm{DO}}TO\ensuremath-DO​ when UUU is close to 0 or Ucr{U}_{cr}Ucr​, but TMI{T}_{\mathrm{MI}}TMI​ is substantially lower than TO\ensuremath-DO{T}_{\mathrm{O}\text{\ensuremath{-}}\mathrm{DO}}TO\ensuremath-DO​ for intermediate values of UUU. Moreover, using an analytical formula we show that TMI=0{T}_{\mathrm{MI}}=0TMI​=0 at U=2<UcrU=\sqrt{2}<{U}_{cr}U=2​<Ucr​, so we prove that the quantum critical point exists for the ordered metal at (T=0,U=2)(T=0,\phantom{\rule{0.28em}{0ex}}U=\sqrt{2})(T=0,U=2​). Having calculated the temperature dependent DOS we study thermodynamic properties of the system starting from its free energy FFF. Then we find how the order parameter ddd and the gap \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ change with TTT and we construct the phase diagram in the variables TTT and UUU, where we display regions of stability of four different phases: ordered insulator, ordered metal, disordered insulator, and disordered metal. Finally, we use a low temperature expansion to demonstrate the existence of a nonzero DOS at a characteristic value of UUU on a general bipartite lattice.
In this work, we study the Anderson-Falicov-Kimball model within the dynamical mean field theory for the Bethe lattice, restricting our analysis to the nonmagnetic case. The one-particle density of states is obtained by both arithmetic and geometric averages over disorder, since only the latter can detect localization in the absence of an energy gap. Varying the strengths of Coulomb interaction and disorder at zero temperature, we construct phase diagrams for this model, where we distinguish spectral regions with localized states, with extended states, or with a correlation-induced gap. With this, we identify metal-insulator transitions driven by correlation and disorder, as well as the competition between these effects. This is done for various band fillings, since our main interest here is to study how the variation of the electron density affects the phase diagrams previously obtained for half-filling. The picture revealed by the density of states is further checked by evaluating the static and dynamic conductivities, including temperature effects.
Correlated electrons in a binary alloy ${A}_{x}{B}_{1\ensuremath{-}x}$ are investigated within the Hubbard model and dynamical mean-field theory (DMFT). The random energies ${\ensuremath{\epsilon}}_{i}$ have a bimodal probability distribution and an energy separation $\ensuremath{\Delta}.$ We solve the DMFT equations by the numerical renormalization-group method at zero temperature and calculate the spectral density as a function of disorder strength $\ensuremath{\Delta}$ and interaction U at different fillings. For filling factors $\ensuremath{\nu}=x$ or 1+x1+x1+x the lower or upper alloy subband is half filled and the system becomes a Mott insulator at strong interactions, with a correlation gap at the Fermi level. At the metal-insulator transition hysteresis is observed. We also analyze the effective theory in the $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Delta}}\ensuremath{\infty}$ limit and find good agreement between analytical and numerical results for the critical interaction Uc{U}_{c}Uc​ at which the metal-insulator transition occurs.
The ground-state density of states of the half-filled Falicov-Kimball model contains a charge-density-wave gap. At finite temperature, this gap is not immediately closed, but is rather filled in by subgap states. For a specific combination of parameters, this leads to a stable phase where the system is in an ordered charge-density-wave phase, but there is high density of states at the Fermi level. We show that this property can be, in finite dimensions, traced to a crossing of sharp states resulting from the single particle excitations of the localized subsystem. The analysis of the inverse participation ratio points to a strong localization in the discussed regime. However, the pronounced subgap density of states can still lead to a notable increase of charge transport through a finite size system. We show this by focusing on the transmission in heterostructures where a Falicov-Kimball system is sandwiched between two metallic leads.
We study the magnetic field-driven metal-to-insulator transition in half-filled Hubbard model on the Bethe lattice, using the dynamical mean-field theory by solving the quantum impurity problem with density-matrix renormalization group algorithm. The method enables us to obtain a high-resolution spectral densities in the presence of a magnetic field. It is found that the Kondo resonance at the Fermi level splits at relatively high magnetic field: the spin-up and -down components move away from the Fermi level and finally form a spin-polarized band insulator. By calculating the magnetization and spin susceptibility, we clarify that an applied magnetic field drives a transition from a paramagnetic metallic phase to a band insulating phase. In the weak interaction regime, the nature of the transition is continuous and captured by the Stoner's description, while in the strong interaction regime the transition is very likely to be metamagnetic, evidenced by the hysteresis curve. Furthermore, we determine the phase boundary by tracking the kink in the magnetic susceptibility, and the steplike change of the entanglement entropy and the entanglement gap closing. Interestingly, the phase boundaries determined from these two different ways are largely consistent with each other.
The Falicov-Kimball model was introduced in 1969 as a statistical model for metal-insulator transitions; it includes itinerant and localized electrons that mutually interact with a local Coulomb interaction and is the simplest model of electron correlations. It can be solved exactly with dynamical mean-field theory in the limit of large spatial dimensions, which provides an interesting benchmark for the physics of locally correlated systems. In this review, the authors develop the formalism for solving the Falicov-Kimball model from a path-integral perspective and provide a number of expressions for single- and two-particle properties. Many important theoretical results are examined that show the absence of Fermi-liquid features and provide a detailed description of the static and dynamic correlation functions and of transport properties. The parameter space is rich and one finds a variety of many-body features like metal-insulator transitions, classical valence fluctuating transitions, metamagnetic transitions, charge-density-wave order-disorder transitions, and phase separation. At the same time, a number of experimental systems have been discovered that show anomalies related to Falicov-Kimball physics [including YbInCu4,{\mathrm{YbInCu}}_{4},YbInCu4​, ${\mathrm{EuNi}}_{2}({\mathrm{Si}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}{)}_{2},$ NiI2,{\mathrm{NiI}}_{2},NiI2​, and TaxN].{\mathrm{Ta}}_{x}\mathrm{N}].Tax​N].
The extended Falicov-Kimball model is analyzed exactly in the ground state at half filling in the limit of large dimensions. In the model the on-site and the intersite density-density interactions between all particles are included. We determined the model's phase diagram and found a discontinuous transition between two different charge-ordered phases. Our analytical calculations show that the ground state of the system is insulating for any nonzero values of the interaction couplings. We also show that the dynamical mean-field theory and the static broken-symmetry Hartree-Fock mean-field approximation give the same results for the model at zero temperature. In addition, we prove using analytical expressions that at infinitesimally small, but finite, temperatures the system can be metallic.