Interacting electron gas in a strong magnetic field

The density response function of an electron gas in a strong magnetic field shows logarithmic singularities due to scattering across the Fermi surface. We analyze the parquet equations for the vertex function in leading logarithmic order for a general interaction potential. The parquet equations are solved for a special interaction potential (Schulz and Keiter model). The divergence of the density response function at extremely high fields is discussed in connection with a possible transition to a Wigner lattice.     

Leading temperature corrections to Fermi-liquid theory in two dimensions

We calculate the basic parameters of the Fermi liquid: the scattering vertex, the Landau interaction function, the effective mass, and physical susceptibilities for a model of two-dimensional (2D) fermions with a short-ranged interaction at nonzero temperature. The leading temperature dependences of the spin components of the scattering vertex, the Landau function, and the spin susceptibility are found to be linear. T-linear terms in the effective mass and in the "charge-sector" quantities are found to cancel to second order in the interaction, but the cancellation is argued not to be generic. The connection with previous studies of the 2D Fermi-liquid parameters is discussed.     

Fermi liquid theory: a renormalization group approach

We show how Fermi liquid theory results can be systematically recovered using a renormalization group (RG) approach. Considering a two-dimensional system with a circular Fermi surface, we derive RG equations at one-loop order for the two-particle vertex function in the limit of small momentum () and energy () transfer and obtain the equation which determines the collective modes of a Fermi liquid. The density-density response function is also calculated. The Landau function (or, equivalently, the Landau parameters F _l ^s and F _l ^a ) is determined by the fixed point value of the -limit of the two-particle vertex function (). We show how the results obtained at one-loop order can be extended to all orders in a loop expansion. Calculating the quasi-particle life-time and renormalization factor at two-loop order, we reproduce the results obtained from two-dimensional bosonization or Ward Identities. We discuss the zero-temperature limit of the RG equations and the difference between the Field Theory and the Kadanoff-Wilson formulations of the RG. We point out the importance of n-body () interactions in the latter. Received: 27 June 1997 / Received in final form: 17 December 1997 / Accepted: 26 January 1998     

Antiferromagnetism in metals: from the cuprate superconductors to the heavy fermion materials

The critical theory of the onset of antiferromagnetism in metals, with concomitant Fermi surface reconstruction, has recently been shown to be strongly coupled in two spatial dimensions. The onset of unconventional superconductivity near this critical point is reviewed: it involves a subtle interplay between the breakdown of fermionic quasiparticle excitations on the Fermi surface and the strong pairing glue provided by the antiferromagnetic fluctuations. The net result is a logarithm-squared enhancement of the pairing vertex for generic Fermi surfaces, with a universal dimensionless coefficient independent of the strength of interactions, which is expected to lead to superconductivity at the scale of the Fermi energy. We also discuss the possibility that the antiferromagnetic critical point can be replaced by an intermediate 'fractionalized Fermi liquid' phase, in which there is Fermi surface reconstruction but no long-range antiferromagnetic order. We discuss the relevance of this phase to the underdoped cuprates and the heavy fermion materials.     

Optical conductivity in a 2D model of the pseudogap state

A 2D model of the pseudogap state is considered on the basis of the scenario of strong electron scattering by short-range-order fluctuations of the “dielectric” (antiferromagnetic or charge density wave) type. A system of recurrence relations is constructed for a one-particle Green’s function and the vertex part, describing the interaction of electrons with an external field. This system takes into account all Feynman diagrams for electron scattering at short-range-order fluctuations. The results of detailed calculations of optical conductivity are given for various geometries (topologies) of the Fermi surface, demonstrating both the effects of pseudogap formation in the electron spectrum and the localization effects. The obtained results are in qualitative agreement with experimental data for underdoped HTSC cuprates.     

On the possibility of superconductivity in electron-hole liquids

It is shown that an electron-hole liquid under suitable conditions can become superconducting. This conclusion is reached by using an effective electron-electron interaction which includes vertex corrections and multiple electron-hole scattering. The mechanism for superconductivity is novel. The intermediate bosons responsible for the superconducting pairing are not acoustic plasmons but correlated pair excitations from the Fermi sea of the holes. Some materials are proposed for an experimental verification of the ideas presented here.     

Exact Renormalization Group for the Brazovskii Model of Striped Patterns

We consider the coarse graining of the generalized Brazovskii free energy functional for striped patterns. The technique developed by Shankar for the Fermi liquids is combined with the irreducible version of the exact renormalization group to calculate the recursion relations for interaction vertices. We perform the one-loop calculations from this method taking the eight-point vertex into account.     

Nonadiabatic superconductivity in K3C60 and Rb3C60

We discuss the validity of Migdal–Eliashberg theory applied to the superconductor fullerides K₃C₆₀ and Rb₃C₆₀. Recently, the relevant superconductor properties have been measured, like the isotope coefficient, the energy gap and critical temperatures for these compounds and compared with their optical properties. They all present a very disperse band of phonon frequencies, running from very small to very large energies, the latter being close to the Fermi edge. Therefore, these materials exceed the limit of validity of the adiabatic Migdal theorem, measured with a nonadiabatic parameter m=w₀/E_F, where w₀ is a characteristic phonon frequency and E_F=250 meV, the Fermi level. We examine previous theories incorporating vertex corrections into the Eliashberg equations to deal with such a situation. We compare these approaches by calculating the critical temperatures using a multimodal Eliashberg spectral function ²F(w) to study the contribution of the various phononic modes. We arrive at the conclusion that the optical modes, not the intramolecular ones, are among those which maximize T_c independently of including vertex corrections or not. This result goes in the direction to understand why doped fullerides A₃C₆₀ are superconductors.     

Magnetism and electrical transport in Kondo lattices

The Anderson lattice model is studied via time ordered perturbation theory in order to derive approximate results for dynamical susceptibility and electrical conductivity in the Kondo regime. A classification of processes on the lattice contributing to the susceptibility leads to expressions containing renormalized band Green's functions and local vertex parts. These quantities are determined by integral equations. Explicit results are obtained via a decoupling procedure for the local parts, which can be motivated in physical terms. It is shown that the formation of the Abrikosov-Suhl resonance near the Fermi level works against, and may actually suppress the tendency towards formation of a magnetic phase. Using a simple, but well founded form for the temperature dependent self energy of band electrons near the Fermi level the influence of coherence on the electrical conductivity at low temperatures can be demonstrated.Dedicated to B. Mühlschlegel on the occasion of the 60th birthday     

Full fermion-boson vertex function derived in terms of symmetry relations in Abelian gauge theory

Nonperturbative studies such as confinement and dynamical chiral symmetry breaking need the nonperturbative interacting vertex functions. In this paper, an approach to determining the full fermion-boson vertex function in four-dimensional Abelian gauge theory is presented: this full vertex function is derived in terms of a set of normal (longitudinal) and transverse Ward-Takahashi relations for the fermion-boson (vector) and axial-vector vertices in the momentum space in the case of massless fermion. Such a derived fermion-boson vertex function should be satisfied both perturbatively and nonperturbatively. The fact that such a derived full fermion-boson vertex function to one-loop order holds indeed is proven and the nonperturbative form of this vertex is also under discussion.     

A comment on coherent states of the isotonic oscillator

Landau's theory of density and pair fluctuations in a Fermi liquid is extended to include also fluctuations in the vertex functions, i.e. in the Landau parameters themselves. This is a natural consequence of the higher effective action Г[G, α] which depends explicitly on the full propagator G and the vertex α and whose extreme in G and α determine physical configurations of these quantities.     

A renormalization-group approach to the coulomb gap

The Coulomb Gap is a phenomenon in which the long-ranged Coulomb repulsions between localized electrons cause a soft gap to appear in the density of states at the Fermi surface, dominating transport processes. A standard diagrammatic technique and a renormalization group philosophy are applied to this problem. The density of states (which corresponds to the two-point vertex function in our field theory) is renormalized by interactions, with a correction that is logarithmic in 1D. Thus the RG is valid and an epsilon-expansion gives the density of states in other dimensionalities. It is found that the standard Efros-Shklovskii theory of the Coulomb Gap is identical to the TAP theory of spin glasses. We speculate that by means of a diagrammatic expansion for the beta-function, one can study the weak-disorder case, for which the standard theory breaks down.     

Intrinsic spin Hall effect in platinum: first-principles calculations

Spin Hall effect (SHE) is studied with first-principles relativistic band calculations for platinum, which is one of the most important materials for metallic SHE and spintronics. We find that intrinsic spin Hall conductivity (SHC) is as large as approximately 2000(variant Planck's over 2 pi/e)(Omega cm)(-1) at low temperature and decreases down to approximately 200(variant Planck's over 2 pi/e)(Omega cm)(-1) at room temperature. It is due to the resonant contribution from the spin-orbit splitting of the doubly degenerated d bands at high-symmetry L and X points near the Fermi level. By modeling these near degeneracies by an effective Hamiltonian, we show that SHC has a peak near the Fermi energy and that the vertex correction due to impurity scattering vanishes. We therefore argue that the large SHE observed experimentally in platinum is of intrinsic nature.     

Coupled electron-phonon system in two dimensions and its implications for inversion layers

The first order electron-phonon vertex in two dimensions is calculated in the regime (q_o/qvf)?1 and (q_o/qvf)?1 where, q_o and q are the phonon frequency and wave vector, and v_f is the electron Fermi velocity. In the former case, the vertex correction is found to be λ. (Ωph/E_f)^{$\text{1}\text{2}$} and in the latter case it is λ, where λ is the dimensionless electron-phonon coupling, Ωph a characteristic phonon energy and E_f the Fermi energy. It is shown that for the relevant range of carrier density and phonon branches in the inversion layer, the term (Ωph/E_f)^{$\text{1}\text{2}$} is of order unity, possibly implying significant higher order effects.     

Small distance behaviour in field theory and power counting

For infinitesimal changes of vertex functions under infinitesimal variation of all renormalized parameters, linear combinations are found such that the net infinitesimal changes of all vertex functions are negligible relative to those functions themselves at large momenta in all orders of renormalized perturbation theory. The resulting linear first order partial differential equations for the asymptotic forms of the vertex functions are, in quantum electrodynamics, solved in terms of one universal function of one variable and one function of one variable for each vertex function whereby, in contrast to the renormalization group treatment of this problem, the universal function is obtained from nonasymptotic considerations. A relation to the breaking of scale invariance in renormalizable theories is described.     

Transverse Vector Vertex Function and Transverse Ward-Takahashi Relations in QED

The transverse vector vertex function in momentum space in four-dimensional QED is derived in terms of a set of transverse Ward-Takahashi relations for the vector and the axial-vector vertices in the case of massless fermion. It is demonstrated explicitly that the transverse vector vertex function derived this way to one-loop order leads to the same result as one obtained in perturbation theory. This provides a basic approach to determine the transverse part of basic vertex function from the symmetry relations of the system.     

Transverse Ward—Takahshi Relation for the Vector Vertex in Quantum Field Theory

The transverse Ward-Takahashi(W-T) realtion for the Vector vertex in quantum filed theory is derived by calculation the coul of the time-ordered product of the three-point function inclduing the vector current operator.This provides the constraint on the transverse part of the vertex.By combining the transverse and normal (longitudinal)W-T identities,we obtain the expression for the full vector vertex function.     

Transverse Vector Vertex Function and Transverse Ward-Takahashi Relations in QED

The transverse vector vertex function in momentum space in four-dimensional QED is derived in terms of a set of transverse Ward-Takahashi relations for the vector and the axial-vector vertices in the case of massless fermion.It is demonstrated explicitly that the transverse vector vertex function derived this way to one-loop order leads to the same result as one obtained in perturbation theory. This provides a basic approach to determine the transverse part of basic vertex function from the symmetry relations of the system.     

Traffic of indistinguishable particles in complex networks

In this paper, we apply a simple walk mechanism to the study of the traffic of many indistinguishable particles in complex networks. The network with particles stands for a particle system, and every vertex in the network stands for a quantum state with the corresponding energy determined by the vertex degree. Although the particles are indistinguishable, the quantum states can be distinguished. When the many indistinguishable particles walk randomly in the system for a long enough time and the system reaches dynamic equilibrium, we find that under different restrictive conditions the particle distributions satisfy different forms, including the Bose--Einstein distribution, the Fermi--Dirac distribution and the non-Fermi distribution (as we temporarily call it). As for the Bose--Einstein distribution, we find that only if the particle density is larger than zero, with increasing particle density, do more and more particles condense in the lowest energy level. While the particle density is very low, the particle distribution transforms from the quantum statistical form to the classically statistical form, i.e., transforms from the Bose distribution or the Fermi distribution to the Boltzmann distribution. The numerical results fit well with the analytical predictions.