Renormalized Perturbation Theory Research Articles

The Wess Zumino consistency condition: a paradigm in renormalized perturbation theory

Talk given at Symposium in Honor of Julius Wess on the Occasion of his 70th Birthday, 10-11 January 2005 at Max Planck Institute for Physics (Werner Heisenberg Institut) Fohringer Ring 6 - D-80805 MUENCHEN

Spin and charge dynamics in a renormalized perturbation theory

We calculate the spin and charge dynamical susceptibilities of a strongly correlatedimpurity model in a renormalized perturbation theory. The irreducible vertices for thequasiparticle scattering are deduced from the renormalized parameters, which have beencalculated by fitting the low-lying levels of a numerical renormalization group(NRG) calculation to those of an effective Anderson model. The susceptibilities areasymptotically exact in the low frequency limit and satisfy the Korringa–Shiba relation.By comparing the results with those calculated from a direct NRG calculation,we show that the renormalized perturbation theory (RPT) description gives avery good description of the spin dynamics for all values of the local interactionU, not only at low frequencies but over the whole relevant frequency range.In the presence of a magnetic field the approach can be generalized using field dependentrenormalized parameters to calculate the transverse and parallel spin dynamicsusceptibilities. The RPT gives accurate results for the spin dynamics over the wholefrequency range and for all values of the magnetic field.

Non-equilibrium differential conductance through a quantum dot in a magneticfield

We derive an exact expression for the differential conductance for a quantum dot in an arbitrarymagnetic field for small bias voltage. The derivation is based on the symmetric Anderson modelusing renormalized perturbation theory and is valid for all values of the on-site interactionU including the Kondo regime. We calculate the critical magnetic field for the splitting of theKondo resonance to be seen in the differential conductivity as a function of bias voltage.Our calculations for small field show that the peak positions of the component resonancesin the differential conductance are reduced substantially from estimates using theequilibrium Green function. We conclude that it is important to take the voltagedependence of the local retarded Green function into account in interpreting experimentalresults.

Choice of the uncritical manifold in renormalization group calculations for dilute polymersolutions

We discuss the dependence of the results of renormalized perturbation theory for dilutepolymer solutions on the choice of the uncritical manifold where the perturbation series areevaluated. Special emphasis is given to the influence of polydispersity correctionson the results of one and two loop calculations. For monodisperse solutions weestablish that after a Borel resummation the dependence on the choice of theuncritical manifold decreases when higher orders of the expansion are taken intoaccount.

Spintronic Transport and Kondo Effect in Quantum Dots

We investigate the spin-dependent transport properties of quantum-dot based structures where Kondo correlations dominate the electronic dynamics. The coupling to ferromagnetic leads with parallel magnetizations is known to give rise to nontrivial effects in the local density of states of a single quantum dot. We show that this influence strongly depends on whether charge fluctuations are present or absent in the dot. This result is confirmed with numerical renormalization group calculations and perturbation theory in the on-site interaction. In the Fermi-liquid fixed point, we determine the correlations of the electric current at zero temperature (shot noise) and demonstrate that the Fano factor is suppressed below the Poissonian limit for the symmetric point of the Anderson Hamiltonian even for nonzero lead magnetizations. We discuss possible avenues of future research in this field: coupling to the low energy excitations of the ferromagnets (magnons), extension to double quantum dot systems with interdot antiferromagnetic interaction and effect of spin-polarized currents on higher symmetry Kondo states such as SU(4).

A local formulation of lattice Wess-Zumino model with exact U(1)Rsymmetry

A lattice Wess-Zumino model is formulated on the basis of Ginsparg-Wilson fermions. In perturbation theory, our formulation is equivalent to the formulation by Fujikawa and Ishibashi and by Fujikawa. Our formulation is, however, free from a singular nature of the latter formulation due to an additional auxiliary chiral supermultiplet on a lattice. The model posssesses an exact U(1)R symmetry as a supersymmetric counterpart of the L?scher lattice chiral U(1) symmetry. A restration of the supersymmetric Ward-Takahashi identity in the continuum limit is analyzed in renormalized perturbation theory. In the one-loop level, a supersymmetric continuum limit is ensured by suitably adjusting a coefficient of a single local term *. The non-renormalization theorem holds to this order of perturbation theory. In higher orders, on the other hand, coefficents of local terms with dimension ? 4 that are consistent with the U(1)R symmetry have to be adjusted for a supersymmetric continuum limit. The origin of this complexicity in higher-order loops is clarified on the basis of the Reisz power counting theorem. Therefore, from a view point of supersymmetry, the present formulation is not quite better than a lattice Wess-Zumino model formulated by using Wilson fermions, although a number of coefficients which require adjustment is much less due to the exact U(1)R symmetry. We also comment on an exact non-linear fermionic symmetry which corresponds to the one studied by Bonini and Feo; an existence of this exact symmetry itself does not imply a restoration of supersymmetry in the continuum limit without any adjustment of parameters.

Particle abundance in a thermal plasma: Quantum kinetics versus Boltzmann equation

We study the abundance of a particle species in a thermalized plasma by introducing a quantum kinetic description based on the nonequilibrium effective action. A stochastic interpretation of quantum kinetics in terms of a Langevin equation emerges naturally. We consider a particle species that is stable in the vacuum and interacts with heavier particles that constitute a thermal bath in equilibrium. Asymptotic theory suggests a definition of a fully renormalized single particle distribution function. Its real time dynamics is completely determined by the nonequilibrium effective action which furnishes a Dyson-like resummation of the perturbative expansion. The distribution function reaches thermal equilibrium on a time scale $\ensuremath{\sim}1/2{\ensuremath{\Gamma}}_{k}(T)$ with ${\ensuremath{\Gamma}}_{k}(T)$ being the quasiparticle relaxation rate. The equilibrium distribution function depends on the full spectral density as a consequence the fluctuation-dissipation relation. Such dependence leads to off shell contributions to the particle abundance. A specific model of a bosonic field $\ensuremath{\Phi}$ in interaction with two heavier bosonic fields ${\ensuremath{\chi}}_{1,2}$ is studied. The decay of the heaviest particle and its recombination lead to a width of the spectral function for the particle $\ensuremath{\Phi}$ and to off shell corrections to the abundance. We find substantial departures from the Bose-Einstein result both in the high temperature and the low temperature but high momentum region. In the latter the abundance is exponentially suppressed but larger than the Bose-Einstein result. We obtain the Boltzmann equation in renormalized perturbation theory and highlight the origin of the differences. Cosmological consequences are discussed: we argue that the corrections to the abundance of cold dark matter candidates are observationally negligible and that recombination erases any possible spectral distortions of the cosmic microwave background (CMB). However we expect that the enhancement at high temperature may be important for baryogenesis.

Structure functions of the 2d [formula omitted] non-linear sigma models

We investigate structure functions in the 2-dimensional (asymptotically free) non-linear O ( n ) σ-models using the non-perturbative S-matrix bootstrap program. In particular, the exact small (Bjorken) x behavior is derived. Structure functions in the special case of the n = 3 model are accurately computed over the whole x range for − q 2 / M 2 < 10 5 , and some moments are compared with results from renormalized perturbation theory. Some results concerning the structure functions in the 1 / n approximation are also presented.

The Strong Coupling Fixed-Point Revisited

In recent work we have shown that the Fermi liquid aspects of the strong coupling fixed point of the s-d and Anderson models can brought out more clearly by interpreting the fixed point as a renormalized Anderson model, characterized by a renormalized level $\tilde\epsilon_d$, resonance width, $\tilde\Delta$, and interaction $\tilde U$, and a simple prescription for their calculation was given using the numerical renormalization group (NRG). These three parameters completely specify a renormalized perturbation theory (RPT) which leads to exact expressions for the low temperature behaviour. Using a combination of the two techniques, NRG to determine $\tilde\epsilon_d$, $\tilde\Delta$, and $\tilde U$, and then substituting these in the RPT expressions gives a very efficient and accurate way of calculating the low temperature behaviour of the impurity as it avoids the necessity of subtracting out the conduction electron component. Here we extend this approach to an Anderson model in a magnetic field, so that $\tilde\epsilon_d$, $\tilde\Delta$, and $\tilde U$ become dependent on the magnetic field. The de-renormalization of the renormalized quasiparticles can then be followed as the magnetic field strength is increased. Using these running coupling constants in a RPT calculation we derive an expression for the low temperature conductivity for arbitrary magnetic field strength.

Fermi Liquid Theory for the Nonequilibrium Kondo Effect at Low Bias Voltages

In this paper, we describe a recent development in a Fermi liquid theory for the Kondo effect in quantum dots under a finite bias voltage V . Applying the microscopic theory of Yamada and Yosida to a nonequilibrium steady state, we derive the Ward identities for the Keldysh Green's function, and determine the low-energy behavior of the differential conductance d I /d V exactly up to terms of order ( e V ) 2 for the symmetric Anderson model. These results are deduced from the fact that the Green's function at the impurity site is a functional of a nonequilibrium distribution f eff (ω), which at e V =0 coincides with the Fermi function. Furthermore, we provide an alternative description of the low-energy properties using a renormalized perturbation theory (RPT). In the nonequilibrium state the unperturbed part of the RPT is determined by the renormalized free quasiparticles, the distribution function of which is given by f eff (ω). The residual interaction between the quasiparticles \(\widetilde{U}\), which is defined by the full vertex part at zero frequencies, is taken into account by an expansion in the power series of \(\widetilde{U}\). We also discuss the application of the RPT to a high-bias region beyond the Fermi-liquid regime.

Structure functions of 2d integrable asymptotically free models

We investigate structure functions in the 2-dimensional (asymptotically free) non-linear O(n) σ-models using the non-perturbative S-matrix bootstrap program. In particular the exact small (Bjorken) x behavior is exhibited; the structure is rather universal and is probably the same in a wide class of (integrable) asymptotically free models. Structurally similar universal formulae may also hold for the small x behavior of QCD in 4-dimensions. Structure functions in the special case of the n=3 model are accurately computed over the whole x range for −q2/M2<105, and some moments are compared with results from renormalized perturbation theory. Some remarks concerning the structure functions in the 1/n approximation are also made.

Renormalization Methods---A Guide For Beginners

The stated goal of this book is to fill a perceived gap between undergraduate texts on critical phenomena and advanced texts on quantum field theory, in the general area of renormalization methods. It is debatable whether this gap really exists nowadays, as a number of books have appeared in which it is made clear that field-theoretic renormalization group methods are not the preserve of particle theory, and indeed are far more easily appreciated in the contexts of statistical and condensed matter physics. Nevertheless, this volume does have a fresh aspect to it, perhaps because of the author's background in fluid dynamics and turbulence theory, rather than through the more traditional migration from particle physics. The book begins at a very elementary level, in an effort to motivate the use of renormalization methods. This is a worthy effort, but it is likely that most of this section will be thought too elementary by readers wanting to get their teeth into the subject, while those for whom this section is apparently written are likely to find the later chapters rather challenging. The author's particular approach then leads him to emphasise the role of renormalized perturbation theory (rather than the renormalization group) in a number of problems, including non-linear systems and turbulence. Some of these ideas will be novel and perhaps even surprising to traditionally trained field theorists. Most of the rest of the book is on far more familiar territory: the momentum-space renormalization group, epsilon-expansion, and so on. This is standard stuff, and, like many other textbooks, it takes a considerable chunk of the book to explain all the formalism. As a result, there is only space to discuss the standard 4 field theory as applied to the Ising model (even the N-vector model is not covered) so that no impression is conveyed of the power and extent of all the applications and generalizations of the techniques. It is regrettable that so much space is spent on rather oversimplified and unrelated models in the first part of the book that, in the end, the reader is left breathless on the threshold of the really interesting material. Despite the earlier emphasis on the application of renormalization ideas in dynamics, in the end the full power of the field-theoretical approach is not applied to the obvious arena of dynamic critical phenomena (where there certainly is currently a gap in the literature) but to the Navier--Stokes equations. The development of the book is somewhat illogical in places. Mean field theory interrupts discussion of block spin methods and scaling arguments---the two are distinct approaches. The Callan--Symanzik equation is introduced before Feynman diagrams are explained, so that there is a hiatus before the actual results for the critical exponents can be found. I think this book is too broad in some respects, and too limited in others, to be a really useful textbook for a course on renormalization methods. Those who have learned these ideas either from field theory, or from nonlinear systems, will find it more rewarding for the sections covering the topics with which they are less familiar. For this reason alone, the book should at least find a place on most library reference shelves.

On the Structure of the Superfluid State and Quasiparticle Spectrum in a Bose Liquid with a Suppressed Bose–Einstein Condensate

A self-consistent model of the superfluid (SF) state of a Bose liquid with strong interaction between bosons and a weak single-particle Bose–Einstein condensate (BEC) is considered. The ratio of the BEC density n 0 to the total particle density n of the Bose liquid is used as a small parameter of the model, n 0/n≪1, unlike in the Bogolyubov theory of a quasi-ideal Bose gas, in which the small parameter is the ratio of the number of supracondensate excitations to the number of particles in an intensive BEC, (n−n 0)/n 0≪1. A closed system of nonlinear integral equations for the normal ~Σ11(p, ω) and anomalous ~Σ12(p, ω) self-energy parts is obtained with account for terms of first order in the BEC density. A renormalized perturbation theory is used, which is built on combined hydrodynamic (at p→0) and field (at p≠0) variables with analytic functions ~Σ ij (p, e) at pe0 and e→0 and a nonzero SF order parameter ~Σ12(0, 0)≠0, proportional to the density ρ s of the SF component. Various pair interaction potentials U(r) with inflection points in the radial dependence and with an oscillating sign-changing momentum dependence of the Fourier component V(p) are considered. Collective many-body effects of renormalization (“screening”) of the initial interaction, which are described by the bosonic polarization operator Π(p, ω), lead to a suppression of the repulsion [V(p)<0] and an enhancement of the effective attraction [V(p)<0] in the respective domains of nonzero momentum transfer, due to the negative sign of the real part of Π(p, ω) on the “mass shell” ω=E(p). In the framework of the “soft spheres” model with the single fitting parameter—the value of the repulsion potential at r=0—the quasiparticle spectrum E(p) is calculated, which is in good accordance with the experimental spectrum E exp(p) of elementary excitations in superfluid 4He. It is shown that the roton minimum in the quasiparticle spectrum is directly associated with the first negative minimum of the Fourier component of the renormalized (“screened”) potential of pair interaction between bosons.

Interaction-induced Fermi surface deformations in quasi-one-dimensional electronic systems

We consider serious conceptual problems with the application of standard perturbation theory, in its zero-temperature version, to the computation of the dressed Fermi surface for an interacting electronic system. In order to overcome these difficulties, we set up a variational approach which is shown to be equivalent to the renormalized perturbation theory where the dressed Fermi surface is fixed by recursively computed counterterms. The physical picture that emerges is that couplings that are irrelevant tend to deform the Fermi surface in order to become more relevant (irrelevant couplings being those that do not exist at vanishing excitation energy because of kinematical constraints attached to the Fermi surface). These insights are incorporated in a renormalization group approach, which allows for a simple approximate computation of Fermi surface deformation in quasi-one-dimensional electronic conductors. We also analyze flow equations for the effective couplings and quasiparticle weights. For systems away from half-filling, the flows show three regimes corresponding to a Luttinger liquid at high energies, a Fermi liquid, and a low-energy incommensurate spin-density wave. At half-filling Umklapp processes allow for a Mott insulator regime where the dressed Fermi surface is flat, implying a confined phase with vanishing effective transverse single-particle coherence. The boundary between the confined and Fermi liquid phases is found to occur for a bare transverse hopping amplitude of the order of the Mott charge gap of a single chain.

Conformational properties of bottle-brush polymers.

General and renormalized perturbation theories are used to study the conformational properties of a bottle-brush molecule, composed of multiarmed polymer stars grafted regularly onto a flexible backbone. The end-to-end distances of the backbone and of an arm of the middle star are calculated within the first order of perturbation theory. For the high grafting densities of stars, the calculated expressions are generalized with the help of the scaling arguments to give the equivalent power laws. According to these laws, the molecule may adopt a sequence of three different conformations (star-rod-coil) as the length of the backbone grows.

Renormalized perturbation theory for Fermi systems: Fermi surface deformation and superconductivity in the two-dimensional Hubbard model

Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the noninteracting) Hamiltonian. We describe such a renormalized perturbation expansion for interacting Fermi systems, which treats Fermi surface shifts and superconductivity with an arbitrary gap function via additive counterterms. The expansion is formulated explicitly for the Hubbard model to second order in the interaction. Numerical solutions of the self-consistency condition determining the Fermi surface and the gap function are calculated for the two-dimensional case. For the repulsive Hubbard model close to half-filling we find a superconducting state with d-wave symmetry, as expected. For Fermi levels close to the van Hove singularity a Pomeranchuk instability leads to Fermi surfaces with a broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second-order calculation yields an s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.

The dielectric constant of polarizable fluids from the renormalized perturbation theory

A perturbation theoretical equation for the dielectric constant of polarizable dipolar fluids is proposed. For the fluctuation of the dipole moment, namely for the Kirkwood g-factor, a formula is given on the basis of Wertheim's renormalized perturbation theory. Using this formula, a series expansion for ε(p) is suggested on the basis of the Kirkwood equation, which gives an implicit function for ε as a function of ¶. The same series expansion can be derived from the Clausius-Mosotti equation—thus it proves to be independent of the boundary conditions. The resulting equation gives excellent results for the dielectric constant of the polarizable Stockmayer fluid producing good agreement with computer simulation data. The series expansion gives better results than the Kirkwood equation itself.

Kondo effect in quantum dots at high voltage: universality and scaling.

We examine the properties of a dc-biased quantum dot in the Coulomb blockade regime. For voltages V that are large compared to the Kondo temperature T(K), the physics is governed by the scales V and gamma, where gamma approximately V/ln(2)(V/T(K)) is the nonequilibrium decoherence rate induced by the voltage-driven current. Based on scaling arguments, self-consistent perturbation theory, and perturbative renormalization group, we argue that due to the large gamma the system can be described by renormalized perturbation theory in 1/ln(V/T(K))<<1. However, in certain variants of the Kondo problem, two-channel Kondo physics is induced by a large voltage V.

RENORMALIZATION GROUP APPROACHES TO STRONGLY CORRELATED ELECTRON SYSTEMS

In recent years the numerical renormalization group (NRG) method has been extended to the calculation of dynamic response functions and transport properties of magnetic impurity models. The approach can now be applied more widely to lattice models of strongly correlated electron systems by the use of dynamical mean field theory (DMFT), in which the lattice problem is transformed into one for an effective impurity with an additional self-consistency constraint. We review these developments and assess the potential for further applications of this approach. We also discuss an alternative approach to renormalization, renormalized perturbation theory, in which the leading asymptotically exact results for the low temperature regime for a number of magnetic impurity models can be obtained within finite order perturbation theory.

Asymptotic analysis and renormalized perturbation theory of the non-Hermitian dynamics of an inviscid vortex.

An analysis of the non-Hermitian fluid systems described by the Rayleigh equation in an unbounded domain has been carried out in the regime of large wave numbers. The evolution of a special class of localized vorticities is also discussed. Asymptotic and perturbative approaches lead to the same final result. In the limit considered, the system is stable. The perturbation analysis reveals interesting pathologies of the non-Hermitian systems. Under specific conditions, the expansion is found to show secular growth. A discussion about the mechanism of insurgence of such singular behavior is presented. It is also shown that the divergent expansion is renormalizable by means of the renormalization group method-the renormalized results are in complete conformity with the asymptotic solutions.