Renormalization group and Fermi liquid theory

We give a Hamiltonian-based interpretation of microscopic Fermi liquid theory within a renormalization group framework. The Fermi liquid fixed-point Hamiltonian with its leading-order corrections is identified and we show that the mean field calculations for this model correspond to the Landau phenomenological approach. This is illustrated first of all for the Kondo and Anderson models of magnetic impurities which display Fermi liquid behaviour at low temperatures. We then show how these results can be deduced by a reorganization of perturbation theory, in close parallel to that for the renormalized φ⁴ field theory. The Fermi liquid results follow from the two lowest order diagrams of the renormalized perturbation expansion. The calculations for the impurity models are simpler than for the general case because the self-energy depends on frequency only. We show, however, that a similar renormalized expansion can be derived also for the case of a translationally invariant system. The parameters specifying the Fermi liquid fixed-point Hamiltonian are related to the renormalized vertices appearing in the perturbation theory. The collective zero sound modes appear in the quasiparticle-quasihole ladder sum of the renormalized perturbation expansion. The renormalized perturbation expansion can in principle be used beyond the Fermi liquid regime to higher temperatures. This approach should be particularly useful for heavy fermions and other strongly correlated electron systems, where the renormalization of the single-particle excitations are particularly large.

We briefly look at the breakdown of Fermi liquid theory from a renormalized perturbation theory point of view. We show how a modified version of the approach can be used in some situations, such as the spinless Luttinger model, where Fermi liquid theory is not applicable. Other examples of systems where the Fermi liquid theory breaks down are also briefly discussed.     

Self-consistent perturbation theory of the anderson model: Ground state properties

The zero temperature limit of the self-consistent perturbation theory of a degenerate Anderson impurityin a metal is partially solved with analytical methods. In particular, the energy scales relevant to the system are made explicit and closed expressions are derived for the local level propagator around the Fermi energy and for the dynamical susceptibility. Results are compared with what is known from Fermi liquid theory. The argument that the theory is valid for large degeneracy is disturbed by certain pathologies.Supported by DFG through the research program of Sonderforschungsbereich 125 (Aachen-Jülich-Köln)     

Phase theory for binary one-dimensional Fermi gases in harmonic traps

The phase theory of two interacting one-dimensional degenerate Fermi gases confined to different harmonic traps is developed based on the concept of bosonization. The excitation spectra of mass and composition waves are calculated and stability boundaries as function of interaction strength and frequency detuning of the traps are found. The experimentally accessible non-local mobilities and their relation to the excitation frequencies are also studied. The problem of the Kohn mode within phase theory is discussed and a solution is proposed. As an example of the simplicity and effectiveness of the new approach the propagation of a composition perturbation across the Fermi seas is calculated analytically.     

Perturbation theory of the Fermi surface in a quantum liquid. A general quasiparticle formalism and one-dimensional systems

We develop a perturbation theory formalism for the theory of the Fermi surface in a Fermi liquid of particles interacting via a bounded short-range repulsive pair potential. The formalism is based on the renormalization group and provides a formal expansion of the large-distance Schwinger functions in terms of a family of running couplings consisting of one- and two-body quasiparticle potentials. The flow of the running couplings is described in terms of a beta function, which is studied to all orders of perturbation theory and shown to obey, in thenth order,n! bounds. The flow equations are written in general dimensiond1 for the spinless case (for simplicity). The picture that emerges is that on a large scale the system looks like a system of fermions interacting via a-like interaction potential (i.e., a potential approaching 0 everywhere except at the origin, where it diverges, although keeping the integral bounded); the theory is not asymptotically free in the usual sense and the freedom mechanism is thus more delicate than usual: the technical problem of dealing with unbounded effective potentials is solved by introducing a mathematically precise notion ofquasiparticles, which turn out to be natural objects with finite interaction even when the physical potential diverges as a deltalike function. A remarkable kind of gauge symmetry is associated with the quasiparticles. To substantiate the analogy with the quasiparticle theory we discuss the mean field theory using our notion of quasiparticles: the resulting self-consistency relations are closely reminiscent of those of the BCS model. The formalism seems suited for a joint theory of normal states of Fermi liquids and of BCS states: the first are associated with the trivial fixed point of our flow or with nearby nontrivial fixed points (or invariant sets) and the second may naturally correspond to really nontrivial fixed points (which may nevertheless turn out to be accessible to analysis because the BCS state is a quasi free state, hence quite simple, unlike the nontrivial fixed points of field theory). Thed=1 case is deeply different from thed> 1 case, for our spinless fermions: we can treat it essentially completely for small coupling. The system is not asymptotically free and presents anomalous renormalization group flow with a vanishing beta function, and the discontinuity of the occupation number at the Fermi surface is smoothed by the interaction (remaining singular with a coupling-dependent singularity of power type with exponent identified with the anomalous dimension). Finally, we present a heuristic discussion of the theory for the flow of the running coupling constants in spinlessd> 1 systems: their structure is simplified further and the relevant part of the running interaction is precisely the interaction between pairs of quasiparticles which we identify with the Cooper pairs of superconductivity. The formal perturbation theory seems to have a chance to work only if the interaction between the Cooper pairs is repulsive: and to second order we show that in the spin-0 case this happens if the physical potential is repulsive. Our results indicate the possibility of the existence of a normal Fermi surface only if the interaction is repulsive.     

Calculation of vibrational energy levels of water molecule by summing divergent perturbation theory series

The Rayleigh-Schrödinger perturbation theory is applied to a calculation of vibrational energy levels of the H₂O molecule for isolated states and the states involved in the anharmonic Fermi and Darling-Dennison resonances. It is shown that in spite of the rapid divergence of the perturbation theory series caused by the resonances, the use of the summation methods of Padé, Padé-Borel, and Padé-Hermite and the moments method allows one to obtain quite satisfactory results.     

Energy diffusion in strongly driven quantum chaotic systems

The energy evolution of a quantum chaotic system under a perturbation that harmonically depends on time is studied in the case of a large perturbation in which the transition rate calculated from the Fermi golden rule exceeds the frequency of the perturbation. It is shown that the energy evolution retains its diffusive character, with a diffusion coefficient that is asymptotically proportional to the magnitude of the perturbation and to the square root of the density of states. The results are supported by numerical calculation. Energy absorption by the system and quantum-classical correlations are discussed. The text was submitted by author in English.     

开放量子台球体系散射的赝路径半经典近似

本文讨论正方形量子台球的输运性质，考虑电子以费米能量穿过台球区域，在台球出口和入口处对入射和出射波函数采用基尔霍夫散射．采用微扰论的Dyson方程得到半经典格林函数，并把赝路径半经典近似作微扰展开得到体系的传输矩阵元．比较了传输矩阵元的傅立叶变换谱的峰位置与腔内自由电子经典轨道长度，发现在精度允许范围内它们符合的很好．     

Effect of hole doping on the electronic structure and the Fermi surface in the Hubbard model within norm-conserving cluster pertubation theory

The concentration dependences of the band structure, spectral weight, density of states, and Fermi surface in the paramagnetic state are studied in the Hubbard model within cluster pertubation theory with 2 × 2 clusters. Representation of the Hubbard X operators makes it possible to control conservation of the spectral weight in constructing cluster perturbation theory. The calculated value of the ground-state energy is in good agreement with the results obtained using nonperturbative methods such as the quantum Monte Carlo method, exact diagonalization of a 4 × 4 cluster, and the variational Monte Carlo method. It is shown that in the case of hole doping, the states in the band gap (in-gap states) lie near the top of the lower Hubbard band for large values of U and near the bottom of the upper band for small U. The concentration dependence of the Fermi surface strongly depends on hopping to second (t′) and third (t″) neighbors. For parameter values typical of HTSC cuprates, the existence of three concentration regions with different Fermi surfaces is demonstrated. It is shown that broadening of the spectral electron density with an energy resolution typical of contemporary ARPES leads to a pattern of arcs with a length depending on the concentration. Only an order-of-magnitude decrease in the linewidth makes it possible to obtain the true Fermi surface from the spectral density. The kinks associated with strong electron correlations are detected in the dispersion relation below the Fermi level.     

开放量子台球体系散射的赝路径半经典近似

本文讨论正方形量子台球的输运性质,考虑电子以费米能量穿过台球区域,在台球出口和入口处对入射和出射波函数采用基尔霍夫散射.采用微扰论的Dyson方程得到半经典格林函数,并把赝路径半经典近似作微扰展开得到体系的传输矩阵元.比较了传输矩阵元的傅立叶变换谱的峰位置与腔内自由电子经典轨道长度,发现在精度允许范围内它们符合的很好.     

New correlation potential for the local-spin-density functional formalism II. Hyperfine structure contact term in free atoms

Using the new parametrization for the correlation potential, which seems to be the best that is at present available within the local-spin-density (LSD) functional formalism, the Fermi contact term in light atoms (up to Ni) is calculated. Although an overall improvement of the previous LSD results is obtained, the discrepancy between theory and experiment remains rather large. It seems that the local approximation for exchange and correlation fails to predict such quantities as magnetic-moment density near the nucleus. It is also shown that the self-interaction correction does not remedy this failure. Further, the effect of the nonzero nuclear radius is investigated and found to be most important in the lightest atoms (e.g. a factor of 0.664 appears in the case of Li). This fact has been omitted in all previous calculations and throws doubt on the reported excellent agreement of the results of many-body perturbation theory with experiment. We also verify that the contact approximation of the Fermi contact term is really good enough.Dedicated to Professor Miroslav Trlifaj on the occasion of his sixtieth birthday.We are indebted to Dr. P. Novák for bringing this problem to our attention and for useful discussions.     

Relaxation in a fermi liquid

The initial value problem for fermions is solved by an explicit construction of the time evolution operator in terms of a spectral representation of the linearized Uehling-Uhlenbeck collision integral. Unless the scattering between the fermions is very anisotropic the decay of a perturbation follows a universal law with relaxation frequencies depending on the total cross section and the ratio of the free energy per particle to the thermal broadening of the Fermi surface. The theory is applied to the decay of quasi-particles and quadrupole deformations of the Fermi sphere. Also the thermalization of the energy of relative motion in a heavy ion collision is studied. For an investigation of the general properties of the linearized fermion Uehling-Uhlenbeck collision operator a Levinson theorem for one-dimensional problems is derived.     

Composite elementary particles and self-damping of Fermi interactions

It is shown that the bootstrap and Z=0 mechanisms of composite elementary particle theories emerge from a Fermi theory. The self-damping of Fermi interactions is demonstrated: Such theories are finite at the outset and need no indefinite metric for their definition. A mathematically rigorous peratization scheme is set up rendering unrenormalizable theories finite in terms of perturbation theory.     

Fermi surface of the 2D Hubbard model at weak coupling

We calculate the interaction-induced deformation of the Fermi surface in the two-dimensional Hubbard model within second order perturbation theory. Close to half-filling, interactions enhance anisotropies of the Fermi surface, but they never modify the topology of the Fermi surface in the weak coupling regime.     

Quasiparticle renormalization of the memory function approach for response functions

The problem of renormalization of the double-time Green function method for response functions is treated in the framework of the Mori-theory for the special case of Fermi liquids and finite Fermi systems. It is shown that the quasiparticle-quasihole renormalization of the response function can be carried out under conditions which are physically equivalent to the conditions under which the quasiparticle-quasihole renormalization is performed in the many-time Green function theory of Fermi liquids.     

Superfluid transition temperature in a Fermi gas with repulsion allowing for higher orders of perturbation theory

Higher-order perturbation-theory corrections to the superfluid transition temperature in a weakly nonideal Fermi gas with repulsion are determined. This involves calculating the contribution of third-and fourth-order diagrams in terms of the gas parameter ap _F to the effective interaction which determines the super-fluid transition temperature and also allowing for effects associated with retardation and renormalization of the polar part of the Green’s function. The expressions obtained provide evidence in support of attraction in the effective interaction in the second, third, and fourth orders of perturbation theory. It is shown that the critical temperature is mainly determined by second-and third-order terms of perturbation theory. Calculations are made of the superfluid transition temperature for a gas comprising neutral Fermi particles in a magnetic field. The limits of validity of the theory are analyzed and the possibility of applying the results to dilute solutions of ³He in ⁴He and neutral-particle Fermi gases in magnetic traps is discussed.     

Chiral invariant Gross-Neveu model: Classical versus quantum

We present a unification of different and independently investigated aspects of the chiral invariant Gross-Neveu model. Special emphasis is placed on the relevance of classical (c-number, non Grassmann) spinor solutions of the G-N field equations for the construction, and thus understanding of the respective quantized Fermi model. To get an insight into the “quantum meaning of classical field theory” if specialized to the G-N case, we perform the path integral quantization procedure which first leads to the Fermi oscillator problem, and then, after appropriate generalizations, to the quantum Fermi G-N model. Path integrals are carried out with respect to c-number spinor paths only, and in fact no reference is necessary to the Grassmann algebra methods, which are conventionally used to integrate out fermions.     

Effect of canted antiferromagnetic order on the electronic structure in the <Emphasis Type="Italic">t–J</Emphasis>* model within the cluster perturbation theory

The electronic structure in the two-dimensional t–J* model with canted antiferromagnetic order in an external magnetic field has been calculated within the cluster perturbation theory. In zero external field, the evolution of the Fermi surface with n-type doping has been obtained in good agreement with experimental data on cuprate superconductors. It has been shown that the inclusion of short-range correlations can result in a nonmonotonic dependence of the spectral weight distribution at the Fermi level on the external magnetic field. In contrast to the case of electron doping, such changes in the case of hole doping can be expected at experimentally achievable fields.     

基于自洽输运理论的介观体系动态电导的研究

基于介观体系电子动态输运的自洽理论,讨论了介观结构的动态电导.作为该理论的应用,采用一介观相干平行板电容器模型来进行研究. 结果表明:体系的动态电导与外场频率和体系费米能有关,为一复数且有有限虚部. 当外场频率较小时,动态电导随费米能的变化所呈现的特性和直流情形非常相似,但是随着外场频率的增加,两者差异就变得非常明显,体系动态电导随外场频率的变化呈现一些峰值结构. 在给定体系费米能时,动态电导随着外场频率的变化而产生振荡,并且出现了负的电导虚部,电导虚部的正负表明了体系的电容特性和电感特性. 关键词： 自洽输运理论 相干平行板电容器 电导 介观体系     

Photoemission quasiparticle spectra of Sr2RuO4

Multiband quasiparticle calculations based on perturbation theory and dynamical mean-field methods show that the creation of a photoemission hole state in Sr2RuO4 is associated with a highly anisotropic self-energy. Since the narrow Ru-derived d(xz,yz) bands are more strongly distorted by Coulomb correlations than the wide d(xy) band, charge is partially transferred from d(xz,yz) to d(xy), thereby shifting the d(xy) Van Hove singularity close to the Fermi level.     

Single-particle spectral function of the Hubbard chain: frustration induced

The one-electron spectral function of a frustrated Hubbard chain is computed by making use of the cluster perturbation theory. The spectral weight we found turns out to be strongly dependent on the frustrating next-nearest-neighbor hopping t'. A frustration induced pseudogap arises when the system evolves from a gapful Mott insulator to a gapless conductor for an intermediate value of the frustration parameter |t'|. Furthermore, the opening of a pseudogap in the density of states already in the metallic side leads to a continuous opening of the true gap in the insulator. For the hole-doped case, the pseudogap is pinned at the Fermi energy, while the Mott gap is shifted in energy with increasing Hubbard interaction U. The separation of the pseudogap and Mott gap in the hole-doped system demonstrates the validity of the existence of a pseudogap.