Limit Of Infinite Dimensions Research Articles

Crossover to Fermi-liquid behavior for weakly coupled Luttinger liquids in the anisotropic large-dimension limit

We study the problem of the crossover from one- to higher-dimensional metals by considering an array of Luttinger liquids (one-dimensional chains) coupled by a weak interchain hopping {\tp.} We evaluate the exact asymptotic low-energy behavior of the self-energy in the anisotropic infinite-dimension limit. This limit extends the dinamical mean field concept to the case of a chain embedded in a self-consistent medium. The system flows to a Fermi-liquid fixed point for energies below the dimensional crossover temperature, and the anomalous exponent $\al$ renormalizes to zero, in the case of equal spin and charge velocities. In particular, the single-particle spectral function shows sharp quasiparticle peaks with nonvanishing weight along the whole Fermi surface, in contrast to the lowest-order result. Our result is obtained by carring out a resummation of all diagrams of the expansion in \tp contributing to the anisotropic $D\to\infty$ limit. This is done by solving, in an almost completely analytic way, an asymptotically exact recursive equation for the renormalized vertices, within a skeleton expansion. Our outcome shows that perturbation expansions in \tp restricted to lowest orders are unreliable below the crossover temperature. The extension to finite dimensions is discussed. This work extends our recent Letter [Phys. Rev. Lett. {\bf 83}, 128 (1999)], and includes all mathematical details.

A Self-Consistent Ornstein–Zernike Approximation for the Edwards–Anderson Spin-Glass Model

We propose a self-consistent Ornstein–Zernike approximation for studying the Edwards–Anderson spin glass model. By performing two Legendre transforms in replica space, we introduce a Gibbs free energy depending on both the magnetizations and the overlap order parameters. The correlation functions and the thermodynamics are then obtained from the solution of a set of coupled partial differential equations. The approximation becomes exact in the limit of infinite dimension and it provides a potential route for studying the stability of the high-temperature phase against replica-symmetry breaking fluctuations in finite dimensions. As a first step, we present the predictions for the freezing temperature Tf and for the zero-field thermodynamic properties and correlation length above Tf as a function of dimensionality.

Comment on “Robustness of a Local Fermi Liquid against Ferromagnetism and Phase Separation”

In this paper we comment on the "Local Fermi Liquid Theory", proposed by Engelbrecht and Bedell [1]. We emphasize several important properties of the limit of infinite dimensions, in particular the k-dependence of the irreducible vertex function, the anisotropy of the Fermi surface and the necessary conditions for phase separation. It appears that these issues have not been fully appreciated in Ref. [1].

Landau theory of the Mott transition in the fully frustrated Hubbard model in infinite dimensions

We discuss the solution of the Mott transition problem in a fully frustrated lattice with a semicircular density of states in the limit of infinite dimensions from the point of view of a Landau free energy functional. This approach provides a simple relation between the free energy of the lattice model and that of its local description in terms of an impurity model. The character of the Mott transition in infinite dimensions, (as reviewed by Georges Kotliar Krauth and Rozenberg, RMP 68, 1996, 13) follows simply from the form of the free energy functional and the physics of quantum impurity models. At zero temperature, below a critical value of the interaction U, a Mott insulator with a finite gap in the one particle spectrum, becomes unstable to the formation of a narrow band near the Fermi energy. Using the insights provided by the Landau approach we answer questions raised about the dynamical mean field solution of the Mott transition problem, and comment on its applicability to three dimensional transition metal oxides.

Spinless two-band model in infinite dimensions

A spinless two-band model is studied in infinite-dimension limit. Starting from the atomic limit, the formal exact solution of the model is obtained by means of a perturbative treatment of the hopping and hybridization terms. The model is solved in closed form in high dimensions assuming no local spin fluctuations. The non-Fermi-liquid properties appearing in the metallic phase are analyzed through the behavior of the density of states and the self-energy near the Fermi level.

Thermal and dynamical properties of the two-band Hubbard model compared with FeSi

We study the two-band Hubbard model introduced by Fu and Doniach as a model for FeSi, which is suggested to be a Kondo insulator. Using the self-consistent second-order perturbation theory combined with the local approximation that is justified in the limit of infinite dimensions, we calculate the specific heat, the spin susceptibility, and the dynamical conductivity, and point out that the reduction of the energy gap due to correlation is not significant in contrast to the previous calculation. It is also demonstrated that the gap at low temperatures in the optical conductivity is filled up at a lower temperature than the gap size, which is consistent with the experiment.

Metal-Insulator Crossover in the Boson-Fermion Model in Infinite Dimensions

The Boson-Fermion model, describing a mixture of tightly bound electron pairs and quasi-free electrons hybridized with each other via a charge exchange term, is studied in the limit of infinite dimensions, using the Non-Crossing Approximation within the Dynamical Mean Field Theory. It is shown that a metal-insulator crossover, driven by strong pair fluctuations, takes place as the temperature is lowered. It manifests itself in the opening of a pseudogap in the electron density of states, accompanied by a corresponding effect in the optical and dc conductivity.

Effects of the nearest-neighbor Coulomb interactions on the ground state of the periodic Anderson model

The magnetic and non-magnetic ground states of the periodic Anderson model with Coulomb interaction between $f$-electrons on the nearest-neighbour(NN) sites are investigated using a variational method, which gives exact calculation of the expectation values in the limit of infinite dimensions. It is shown that for a critical value of NN Coulomb interactions the magnetic ground state of the periodic Anderson model in the Kondo regime is unstable. Factors in terms of the physical processes responsible for instability of the magnetic ground state are also discussed. Our study indicates the importance of the NN Coulomb interactions for correlated two band models.

Evolution of Spectral Functions in Doped Transition Metal Oxides

We discuss the experimental photoemission and inverse photoemission of early transition metal oxides, in the light of the dynamical mean field theory of correlated electrons which becomes exact in the limit of infinite dimensions. We argue that a comprehensive description of the experimental data requires spatial inhomogeneities and present a calculation of the evolution of the spectral function in an inhomogeneities and present a calculation of the evolution of the spectral function in an inhomogenous system with various degrees of inhomogeneity. We also point out that comparison of experimental results and large d calculations require that the degree of correlation and disorder is larger in the surface than in the bulk.

Persistent currents in high dimensions

We introduce a model for the persistent current carried by spinless fermions moving in a ring with a high-dimensional cross section. The effects of both disorder and electron - electron interaction are considered. It is found that the non-interacting system behaves like previously considered low-dimensional models. The more complicated interacting/disordered case is analysed by means of a functional integral which is evaluated in the limit of infinite dimensionality by expanding in the number of transverse channels. To leading order in this expansion scheme, the Coulomb interaction does not affect the persistent current. It is found that the insensitivity to interaction effects is due to the absence of local contributions to the Coulomb vertex in our model (which in turn is a consequence of the neglect of the electron spin). It is argued that the physical mechanism suppressing the interaction in the high-dimensional spinless model applies to the analogous low-dimensional case as well.

Dynamical mean-field theory of the small polaron

A dynamical mean-field theory of the small polaron problem is presented, which becomes exact in the limit of infinite dimensions. The ground state properties and the one-electron spectral function are obtained for a single electron interacting with Einstein phonons by a mapping of the lattice problem onto a polaronic impurity model. The one-electron propagator of the impurity model is calculated through a continued fraction expansion (CFE), both at zero and finite temperature, for any electron-phonon coupling and phonon energy. In contrast to the ground state properties such as the effective polaron mass, which have a smooth behaviour, spectral properties exhibit a sharp qualitative change at low enough phonon frequency: beyond a critical coupling, one energy gap and then more and more open in the density of states at low energy, while the high energy part of the spectrum is broad and can be explained by a strong coupling adiabatic approximation. As a consequence narrow and coherent low-energy subbands coexist with an incoherent featureless structure at high energy. The subbands denote the formation of quasiparticle polaron states. Also, divergencies of the self-energy may occur in the gaps. At finite temperature such effect triggers an important damping and broadening of the polaron subbands. On the other hand, in the large phonon frequency regime such a separation of energy scales does not exist and the spectrum has always a multipeaked structure.

Invading interfaces and blocking surfaces in high-dimensional disordered systems

We study the high-dimensional properties of an invading front in a disordered medium with random pinning forces. We concentrate on interfaces described by bounded slope models belonging to the quenched Kardar-Parisi-Zhang [Phys. Rev. Lett. 56, 889 (1986)] universality class. We find a number of qualitative transitions in the behavior of the invasion process as dimensionality increases. In low dimensions $d<6$ the system is characterized by two different roughness exponents: the roughness of individual avalanches and the overall interface roughness. We use the similarity of the dynamics of an avalanche with the dynamics of invasion percolation to show that above $d=6$ avalanches become flat and the invasion is well described as an annealed process with correlated noise. In fact, for $d>~5$ the overall roughness is the same as the annealed roughness. In very large dimensions, strong fluctuations begin to dominate the size distribution of avalanches; this phenomenon is studied on the Cayley tree, which serves as an infinite dimensional limit. We present numerical simulations in which we measured the values of the critical exponents of the depinning transition, both in finite-dimensional lattices with $d<~6$ and on the Cayley tree, which support our qualitative predictions. We find that the critical exponents in $d=6$ are very close to their values on the Cayley tree and we conjecture on this basis the existence of a further dimension, where mean-field behavior is obtained.

Magnetism in the single-band Hubbard model

A self-consistent spectral density approach (SDA) is applied to the Hubbard model to investigate the possibility of spontaneous ferro- and antiferromagnetism. The starting point is a two-pole ansatz for the single-electron spectral density, the free parameter of which can be interpreted as energies and spectral weights of respective quasiparticle excitations. They are determined by fitting exactly calculated spectral moments. The resulting self-energy consists of a local and a non-local part. The higher correlation functions entering the spin-dependent local part can be expressed as functionals of the single-electron spectral density. Under certain conditions for the decisive model parameters (Coulomb interaction U, Bloch bandwidth W, band occupation n, temperature T) the local part of the self-energy gives rise to a spin-dependent band shift, thus allowing for spontaneous band magnetism. As a function of temperature, second-order phase transitions are found away from half-filling, but close to half-filling, the system exhibits a tendency towards first-order transitions. The non-local self-energy part is determined by use of proper two-particle spectral densities. Its main influence concerns a (possibly spin-dependent) narrowing of the quasiparticle bands with the tendency to stabilize magnetic solutions. The non-local self-energy part disappears in the limit of infinite dimensions. We present a full evaluation of the Hubbard model in terms of quasiparticle densities of states, quasiparticle dispersions, magnetic phase diagram, critical temperatures ( T c, T N) as well as spin and particle correlation functions. Special attention is focused on the non-locality of the electronic self-energy, for which some rigorous limiting cases are worked out.

Simplified periodic Anderson model: Exact solution in infinite dimensions

We present a diagrammatic perturbative treatment of the hybridization for the periodic Anderson model that recovers the dynamical mean-field equations in the limit of infinite dimensions. The resulting effective single-site problem is naturally addressed by perturbation theory on the dynamical mean field. We introduce a simplified version of the model in which only electrons with a given spin orientation hybridize. The perturbation series can be summed in this case, yielding an exact solution for the single-particle Green's functions. Electronic and transport properties are analyzed, showing the existence of a metallic regime with non-Fermi-liquid behavior.

Phase Diagram of the Two-Channel Kondo Lattice

The phase diagram of the two-channel Kondo lattice model is examined with a Quantum Monte Carlo simulation in the limit of infinite dimensions. Commensurate (and incommensurate) antiferromagnetic and superconducting states are found. The antiferromagnetic transition is very weak and continuous; whereas the superconducting transition is discontinuous to an odd-frequency channel-singlet and spin-singlet pairing state.

Evidence for a metastable state of the fundamental dianionH2−

A resonance state with three equivalent electrons and one proton has been identified. Large-scale calculations have been performed using the complex-rotation technique, and the results indicate that the investigated $^{4}\mathrm{S}^{\mathrm{o}}$ state of ${\mathrm{H}}^{2\mathrm{\ensuremath{-}}}$ is strongly correlated. The findings are rationalized by comparing with bound doubly charged anions, isoelectronic states of the Li atom, and calculations in the infinite-dimension limit. In addition, the emerging picture is sustained by an analysis of the complex-rotated wave function.

Non-Fermi-liquid dynamics of the two-channel Kondo lattice

The paramagnetic phase of the two-channel Kondo lattice model is examined with a quantum Monte Carlo simulation in the limit of infinite dimensions. We find non-Fermi-liquid behavior at low temperatures including a finite low-temperature single-particle scattering rate, no Fermi distribution discontinuity, and zero Drude weight. Both the optical and quasiparticle mass enhancement and scattering relaxation rate show consistent evidence of non-fermi-liquid behavior. However, the low-energy density of electronic states is finite.

The Anharmonic Electron-Phonon Problem.

The anharmonic electron-phonon problem is solved in the infinite-dimensional limit using quantum Monte Carlo simulation. Charge-density-wave order is seen to remain at half filling even though the anharmonicity removes the particle-hole symmetry of the model. Superconductivity is strongly favored away from half filling but the anharmonicity does not enhance transition temperatures over the maximal values found in the harmonic limit.

Exact Single Spin Flip for the Hubbard Model in d=

It is shown that the dynamics of a single $\ensuremath{\downarrow}$-electron interacting with a band of $\ensuremath{\uparrow}$-electrons can be calculated exactly in the limit of infinite dimension. The corresponding Green function is determined as a continued fraction. It is used to investigate the stability of saturated ferromagnetism and the nature of the ground state for two generic nonbipartite infinite dimensional lattices. Non-Fermi-liquid behavior is found. For certain dopings the $\ensuremath{\downarrow}$-electron is bound to the $\ensuremath{\uparrow}$-holes.

Kosterlitz-Thouless Transition and Short Range Spatial Correlations in an Extended Hubbard Model.

We study the competition between intersite and local correlations in a spinless two-band extended Hubbard model by taking an alternative limit of infinite dimensions. We find that the intersite density fluctuations suppress the charge Kondo energy scale and lead to a Fermi liquid to non-Fermi liquid transition for repulsive on-site density-density interactions. In the absence of intersite interactions, this transition reduces to the known Kosterlitz-Thouless transition. We show that a new line of non-Fermi liquid fixed points replace those of the zero intersite interaction problem.