“Linearized” dynamical mean-field theory for the Mott-Hubbard transition

The Mott-Hubbard metal-insulator transition is studied within a simplified version of the Dynamical Mean-Field Theory (DMFT) in which the coupling between the impurity level and the conduction band is approximated by a single pole at the Fermi energy. In this approach, the DMFT equations are linearized, and the value for the critical Coulomb repulsion can be calculated analytically. For the symmetric single-band Hubbard model at zero temperature, the critical value is found to be given by 6 times the square root of the second moment of the free (U=0) density of states. This result is in good agreement with the numerical value obtained from the Projective Selfconsistent Method and recent Numerical Renormalization Group calculations for the Bethe and the hypercubic lattice in infinite dimensions. The generalization to more complicated lattices is discussed. The “linearized DMFT” yields plausible results for the complete geometry dependence of the critical interaction. Received 6 May 1999 and Received in final form 2 July 1999     

Phase diagram of the Mott transition in a two-band Hubbard model in infinite dimensions

The Mott metal-insulator transition in the two-band Hubbard model in infinite dimensions is studied by using the linearized dynamical mean-field theory recently developed by Bulla and Potthoff. The phase boundary of the metal-insulator transition is obtained analytically as a function of the on-site Coulomb interaction at the d-orbital, the charge-transfer energy between the d- and p-orbitals and the hopping integrals between p-d, d-d and p-p orbitals. The result is in good agreement with the numerical results obtained from the exact diagonalization method. Received 5 October 2000 and Received in final form 8 December 2000     

Some comments on Kondo lattices and the Mott transition

The so called exhaustion problem occurs when few electrons have to screen many spins in a metal with magnetic impurities. A singlet Fermi liquid ground state is possible only if all impurities are “isotropized” in such a way as to suppress their entropy. That takes a time and the corresponding energy limits the Fermi liquid range. The present note explores that issue of time and energy scales, and it concludes that is much smaller than the single impurity Kondo temperature. Similarly the relevant energy scale is proportional to the number of electrons. Recent results on the Mott metal insulator transition in infinite dimension are reconsidered in the light of these results: controversies in that respect are shown to reduce to a simple physical question, with no firm answer as to now. Received: 5 May 1998 / Received in final form and Accepted: 29 July 1998     

Quantum charge density fluctuations and the phase transition in Ce

We have calculated the quantum quadrupolar interaction due to charge density fluctuations of localized 4f-electrons in Ce by taking into account the angular dependence, the degeneracy of the localized 4f -orbitals and the spin-orbit coupling. The calculated crystal field of 4 f electronic states is in good agreement with neutron diffraction measurements. We show that orientational ordering of quantum quadrupoles drives a phase transition at K which we assign with the transformation. In the phase the centers of mass of the Ce atoms still form a face centered cubic lattice. The theory accounts for the first order character of the transition and for the cubic lattice contraction which accompanies the transition. The transition temperature increases linearly with pressure. Our approach does not involve Kondo spin fluctuations as the significant process for the phase transition. Received 19 October 1998     

Exact solution for a degenerate Anderson impurity in the limit embedded into a correlated host

We consider the one-dimensional t - J model, which consists of electrons with spin S on a lattice with nearest neighbor hopping t constrained by the excluded multiple occupancy of the lattice sites and spin-exchange J between neighboring sites. The model is integrable at the supersymmetric point, J = t. Without spoiling the integrability we introduce an Anderson-like impurity of spin S (degenerate Anderson model in the limit), which interacts with the correlated conduction states of the host. The lattice model is defined by the scattering matrices via the Quantum Inverse Scattering Method. We discuss the general form of the interaction Hamiltonian between the impurity and the itinerant electrons on the lattice and explicitly construct it in the continuum limit. The discrete Bethe ansatz equations diagonalizing the host with impurity are derived, and the thermodynamic Bethe ansatz equations are obtained using the string hypothesis for arbitrary band filling as a function of temperature and external magnetic field. The properties of the impurity depend on one coupling parameter related to the Kondo exchange coupling. The impurity can localize up to one itinerant electron and has in general mixed valent properties. Groundstate properties of the impurity, such as the energy, valence, magnetic susceptibility and the specific heat coefficient, are discussed. In the integer valent limit the model reduces to a Coqblin-Schrieffer impurity. Received: 31 December 1997 / Accepted: 17 March 1998     

Anisotropy of the energy gap in the insulating phase of the U-t-t' Hubbard model

We apply a diagrammatic expansion method around the atomic limit () for the U-t-t ' Hubbard model at half filling and finite temperature by means of a continued fraction representation of the one-particle Green's function. From the analysis of the spectral function we find an energy dispersion relation with a modulation of the energy gap in the insulating phase. This anisotropy is compared with experimental ARPES results on insulating cuprates. Received 18 May 2000 and Received in final form 9 August 2000     

On the properties of small-world network models

We study the small-world networks recently introduced by Watts and Strogatz [Nature 393, 440 (1998)], using analytical as well as numerical tools. We characterize the geometrical properties resulting from the coexistence of a local structure and random long-range connections, and we examine their evolution with size and disorder strength. We show that any finite value of the disorder is able to trigger a “small-world” behaviour as soon as the initial lattice is big enough, and study the crossover between a regular lattice and a “small-world” one. These results are corroborated by the investigation of an Ising model defined on the network, showing for every finite disorder fraction a crossover from a high-temperature region dominated by the underlying one-dimensional structure to a mean-field like low-temperature region. In particular there exists a finite-temperature ferromagnetic phase transition as soon as the disorder strength is finite. [0.5cm] Received 29 March 1999 and Received in final form 21 May 1999     

Low temperature Kondo reduction of quadrupolar and magnetic moments in the YbCuAl series

We present measurements in the YbCu_5-xAl_x series, down to the 50 mK range, using ¹⁷⁰Yb M?ssbauer absorption spectroscopy and magnetisation measurements. In this series, the hybridisation between the Yb 4 f electrons and the conduction electrons is known to decrease as the Al content x increases. We apply the variational solution of the impurity Kondo problem to the interpretation of our data. We show that the Kondo temperature can be derived from the measured 4 f quadrupole moment and, for the magnetically ordered compounds (), we obtain the exchange energy as a function of the Al content. Our findings are in general agreement with Doniach's model describing the onset of magnetic ordering according to the relative values of the Kondo and exchange energy scales. Received 16 April 1998     

Force distribution in an inhomogeneous sandpile

The force perturbation field in a two-dimensional pile of frictionless gravity-loaded discs or spheres arising from lattice distortions is derived to first order. The starting point is the model proposed by Liffman et al. (Powder Technology (1992) pp. 255-267) and Hong (Phys. Rev. E 47, 760-762 (1993)) in which discs of uniform size are arranged on a regular lattice: this predicts a uniform normal stress distribution at the base of the pile. The analysis is applied to two problems: (i) deformable (rather than rigid) grains that undergo Hertzian deformation at the points of contact; (ii) a pile containing a gradient in particle size from the centre to the free surfaces. The former results in the classical pressure dip at the centre; the latter also produces a dip if the larger particles are at the centre. Received 29 January 1998 and Received in final form 7 September 1998     

Electric quadrupole interactions and the γ–α phase transition in Ce: the role of conduction electrons

We present a theoretical model of the “isostructural" - phase transition in Ce which is based on quadrupolar interactions due to coupled charge density fluctuations of 4f electrons and of conduction electrons. The latter are treated in tight-binding approximation. The - transition is described as an orientational ordering of quadrupolar electronic densities in a structure. The quadrupolar order of the conduction electron densities is complementary to the quadrupolar order of 4f electron densities. The inclusion of conduction electrons leads to an increase of the lattice contraction at the - transition in comparison to the sole effect of 4f electrons. We calculate the Bragg scattering law and suggest synchrotron radiation experiments in order to check the structure. Received 21 September 1999 and Received in final form 2 May 2000     

Critical behaviour near the metal-insulator transition of a doped Mott insulator

We have studied the critical behaviour of a doped Mott insulator near the metal-insulator transition for the infinite-dimensional Hubbard model using a linearized form of dynamical mean-field theory. The discontinuity in the chemical potential in the change from hole to electron doping, for U larger than a critical value U _c, has been calculated analytically and is found to be in good agreement with the results of numerical methods. We have also derived analytic expressions for the compressibility, the quasiparticle weight, the double occupancy and the local spin susceptibility near half-filling as functions of the on-site Coulomb interaction and the doping. Received 15 March 2001 and Received in final form 22 May 2001     

Van der Waals interaction between flux lines in high- superconductors: A variational approach

In pure anisotropic or layered superconductors thermal fluctuations induce a van der Waals attraction between flux lines. This attraction together with the entropic repulsion has interesting consequences for the low field phase diagram; in particular, a first order transition from the Meissner phase to the mixed state is induced. We introduce a new variational approach that allows for the calculation of the effective free energy of the flux line lattice on the scale of the mean flux line distance a, which is based on an expansion of the free energy around the regular triangular Abrikosov lattice. Using this technique, the low field phase diagram of these materials may be explored. The results of this technique are compared with a recent functional RG treatment of the same system. Received: 25 June 1996 / Revised: 18 August 1998 / Accepted: 21 August 1998     

Finite-size scaling in the theory above the upper critical dimension

We derive exact results for several thermodynamic quantities of the O ( n ) symmetric field theory in the limit in a finite d-dimensional hypercubic geometry with periodic boundary conditions. Corresponding results are derived for an O ( n ) symmetric model on a finite d-dimensional lattice with a finite-range interaction. The leading finite-size effects near T_c of the field-theoretic model are compared with those of the lattice model. For 2 < d < 4, the finite-size scaling functions are verified to be universal. For d > 4, significant lattice effects are found. Finite-size scaling in its usual simple form does not hold for d > 4 but remains valid in a generalized form with two reference lengths. The finite-size scaling functions of the field theory turn out to be nonuniversal whereas those of the lattice model are independent of the nonuniversal model parameters. In particular, the field-theoretic model exhibits finite-size effects whose leading exponents differ from those of the lattice model. The widely accepted lowest-mode approach is shown to fail for both the field-theoretic and the lattice model above four dimensions. Received: 20 October 1997 / Accepted: 5 March 1998     

Two interacting particles in a disordered chain II: Critical statistics and maximum mixing of the one body states

For two particles in a disordered chain of length L with on-site interaction U, a duality transformation maps the behavior at weak interaction onto the behavior at strong interaction. Around the fixed point of this transformation, the interaction yields a maximum mixing of the one body states. When (the one particle localization length), this mixing results in weak chaos accompanied by multifractal wave functions and critical spectral statistics, as in the one particle problem at the mobility edge or in certain pseudo-integrable billiards. In one dimension, a local interaction can only yield this weak chaos but can never drive the two particle system to full chaos with Wigner-Dyson statistics. Received: 22 May 1998 / Received in final form: 24 August 1998 / Accepted: 4 September 1998     

Disorder driven roughening transitions of elastic manifolds and periodic elastic media

The simultaneous effect of both disorder and crystal-lattice pinning on the equilibrium behavior of oriented elastic objects is studied using scaling arguments and a functional renormalization group technique. Our analysis applies to elastic manifolds, e.g., interfaces, as well as to periodic elastic media, e.g., charge-density waves or flux-line lattices. The competition between both pinning mechanisms leads to a continuous, disorder driven roughening transition between a flat state where the mean relative displacement saturates on large scales and a rough state with diverging relative displacement. The transition can be approached by changing the impurity concentration or, indirectly, by tuning the temperature since the pinning strengths of the random and crystal potential have in general a different temperature dependence. For D dimensional elastic manifolds interacting with either random-field or random-bond disorder a transition exists for 2<D<4, and the critical exponents are obtained to lowest order in . At the transition, the manifolds show a superuniversal logarithmic roughness. Dipolar interactions render lattice effects relevant also in the physical case of D=2. For periodic elastic media, a roughening transition exists only if the ratio p of the periodicities of the medium and the crystal lattice exceeds the critical value . For p<p _c the medium is always flat. Critical exponents are calculated in a double expansion in and and fulfill the scaling relations of random field models. Received 28 August 1998     

Electronic properties of CuO 2 -planes: A DMFT study

We study one-particle spectra and the electronic band-structure of a CuO ₂ -plane within the three-band Hubbard model. The Dynamical Mean-Field Theory (DMFT) is used to solve the many particle problem. The calculations show that the optical gap is given by excitations from the lower Hubbard band into the so called Zhang-Rice singlet band. The optical gap turns out to be considerably smaller than the bare charge transfer energy () for a typical set of parameters, which is in agreement with experiment. We also investigate the dependence of the shape of the Fermi surface on the different hopping parameters t CuO and t OO. A value t OO / t CuO >0 leads to a Fermi surface surrounding the M point. Received 21 September 1998 and Received in final form 8 June 1999     

Weak-coupling treatment of electronic (anti-)ferroelectricity in the extended Falicov-Kimball model

We study the (spinless) Falicov-Kimball model extended by a finite band width (hopping t _f ) of the localized (f-) electrons in infinite dimensions in the weak-coupling limit of a small local interband Coulomb correlation U for half filling. In the case of overlapping conduction- and f-bands different kinds of ordered solutions are possible, namely charge-density wave (CDW) order, electronic ferroelectricity (EFE) and electronic antiferroelectricity (EAFE). The order parameters are calculated as a function of the model parameters and of the temperature. There is a first-order phase transition from the CDW-phase to the EFE- or EAFE-phase. The total energy is calculated to determine the thermodynamically stable solution. The quantum phase diagrams are calculated.     

The Fermi-sea-like limit of the composite fermion wave function

The experimentally observed filling factors of the fractional quantum Hall effect can be described in terms of the composite fermion wave function of the Jastrow-Slater form [0pt] fully projected into the lowest Landau level. The Slater determinant of the above composite fermion wave function represents the filled Landau levels of composite fermions evaluated at the corresponding reduced magnetic field. For a system of fermions studied in the thermodynamic limit, we prove that in the even-denominator-filled state limit (when the number of filled Landau levels of composite fermions becomes infinite), the above composite fermion wave function exactly transforms into the Rezayi-Read Fermi-sea-like wave function [0pt] constructed by attaching 2m flux quanta to the Slater determinant of two-dimensional free fermions at the density corresponding to that filling. We study the composite fermion wave function and its evolution into the Fermi-sea-like wave function for a range of filling factors very close to the even-denominator-filled state. Received 19 March 1999