Renormalized Perturbation Theory Research Articles

Modelling the clustering of dark matter haloes in resummed perturbation theories

We address the issue of the cosmological bias between matter and galaxy distributions, looking at dark-matter haloes as a first step to characterize galaxy clustering. Starting from the linear density field at high redshift, we follow the centre of mass trajectory of the material that will form each halo at late times (proto-halo). We adopt a fluid-like description for the evolution of perturbations in the proto-halo distribution, which is coupled to the matter density field via gravity. We present analytical solutions for the density and velocity fields, in the context of renormalized perturbation theory. We start from the linear solution, then compute one-loop corrections for the propagator and the power spectrum. Finally we analytically resum the propagator and we use a suitable extension of the time-renormalization-group method (Pietroni 2008) to resum the power spectrum. For halo masses M<10^{14} Msol/h our results at z=0 are in good agreement with N-body simulations. Our model is able to predict the halo-matter cross spectrum with an accuracy of 5 per cent up to k = 0.1 h/Mpc approaching the requirements of future galaxy redshift surveys.

PERTURBATION THEORY OF THE COSMOLOGICAL LOG-DENSITY FIELD

The matter density field exhibits a nearly lognormal probability density distribution (PDF) after entering into the nonlinear regime. Recently, it has been shown that the shape of the power spectrum of a logarithmically transformed density field is very close to the linear density power spectrum, motivating an analytic study of it. In this paper, we develop cosmological perturbation theory for the power spectrum of this field. Our formalism is developed in the context of renormalized perturbation theory, which helps to regulate the convergence behavior of the perturbation series, and of the Taylor- series expansion we use of the logarithmic mapping. This approach allows us to handle the critical issue of density smoothing in a straightforward way. We also compare our perturbative results with simulation measurements.

Full counting statistics for SU(N) impurity Anderson model

We analyze the full counting statistics of the multiorbital Kondo effect in a quantum dot with SU($N$) symmetry in the framework of renormalized perturbation theory. The current probability distribution function is calculated for an arbitrary dot-site Coulomb repulsion $U$ in the particle-hole-symmetric case. The resulting cumulant up to the leading nonlinear term of applied bias voltages indicates two types of electron transfer, respectively carrying charge $e$ and $2e$, with different $N$ dependences. The cross correlation between different orbital currents shows exponential enhancement with respect to $U$, which directly addresses the formation of the orbital-singlet state.

Kondo crossover in shot noise of a single quantum dot with orbital degeneracy

We investigate out of equilibrium transport through an orbital Kondo system realized in a single quantum dot, described by the multiorbital impurity Anderson model. Shot noise and current are calculated up to the third order in bias voltage in the particle-hole symmetric case, using the renormalized perturbation theory. The derived expressions are asymptotically exact at low energies. The resulting Fano factor of the backscattering current $F_b$ is expressed in terms of the Wilson ratio $R$ and the orbital degeneracy $N$ as $F_b =\frac{1 + 9(N-1)(R-1)^2}{1 + 5(N-1)(R-1)^2}$ at zero temperature. Then, for small Coulomb repulsions $U$, we calculate the Fano factor exactly up to terms of order $U^5$, and also carry out the numerical renormalization group calculation for intermediate $U$ in the case of two- and four-fold degeneracy ($N=2,\,4$). As $U$ increases, the charge fluctuation in the dot is suppressed, and the Fano factor varies rapidly from the noninteracting value $F_b=1$ to the value in the Kondo limit $F_b=\frac{N+8}{N+4}$, near the crossover region $U\sim \pi \Gamma$, with the energy scale of the hybridization $\Gamma$.

Triviality of [formula omitted] theory: Small volume expansion and new data

We study a renormalized coupling g and mass m in four-dimensional ϕ 4 theory on tori with finite size z = m L . Precise numerical values close to the continuum limit are reported for z = 1 , 2 , 4 , based on Monte Carlo simulations performed in the equivalent all-order strong coupling reformulation. Ordinary renormalized perturbation theory is found to work marginally at z = 2 and to fail at z = 1 . By exactly integrating over the constant field mode we set up a renormalized expansion in z and compute three nontrivial orders. These results reasonably agree with the numerical data at small z. In the new expansion, the universal continuum limit exists as expected from multiplicative renormalizability. The triviality scenario is corroborated with significant precision.

A new renormalization group approach for systems with strong electron correlation

The anomalous low energy behaviour observed in metals with strong electron correlation, suchas in the heavy fermion materials, is believed to arise from the scattering of the itinerantelectrons with low energy spin fluctuations. In systems with magnetic impurities this scatteringleads to the Kondo effect and a low energy renormalized energy scale, the Kondo temperatureTK. It has been generally assumed that these low energy scales can only be accessed by anon-perturbative approach due to the strength of the local inter-electron interactions. Herewe show that it is possible to circumvent this difficulty by first suppressing the spinfluctuations with a large magnetic field. As a first step field-dependent renormalizedparameters are calculated using standard perturbation theory. A renormalizedperturbation theory is then used to calculate the renormalized parameters for areduced magnetic field strength. The process can be repeated and the flow of therenormalized parameters continued to zero magnetic field. We illustrate the viabilityof this approach for the single impurity Anderson model. The results for therenormalized parameters, which flow as a function of magnetic field, can be checkedwith those from numerical renormalization group and Bethe ansatz calculations.

Renormalized parameters and perturbation theory for ann-channel Anderson model with Hund’s rule coupling: Asymmetric case

We explore the predictions of the renormalized perturbation theory for the $n$-channel Anderson model, both with and without Hund's rule coupling, in the regime away from particle-hole symmetry. For the model with $n=2$ we deduce the renormalized parameters from numerical renormalization-group calculations and plot them as a function of the local occupation of the impurity site ${n}_{d}$. From the exact relations in terms of the renormalized parameters, we calculate the orbital, spin and charge susceptibilities, Wilson ratios, and quasiparticle density of states at $T=0$, in the different parameter regimes, which gives a comprehensive overview of the low-energy behavior of the model. We compare the difference in Kondo behaviors at the points, where ${n}_{d}=1$ and ${n}_{d}=2$. Some unexpected features of the results are that a strong on-site interaction gives significant renormalization and suppression charge susceptibility in the intermediate valence regimes, and the peaks in the spin susceptibility away from the particle-hole symmetric case do not occur at integer values ${n}_{d}=1,3$

Quantum dynamic localization in the Holstein Hamiltonian at finite temperatures

It is shown that dynamic localization of the charge carrier is present in the simple Holstein Hamiltonian (fully quantum and without random variables) if the polaron binding energy is sufficiently small and the small polaron is not formed at high temperature. Previously, this idea was only explored in semiclassical models of organic semiconductors. Band renormalization cannot be used in the presence of dynamic localization, which is best studied with a vibronic basis set introduced here. The lifetime of these vibronic states can be estimated using second-order renormalized perturbation theory. In the regime of dynamic localization the diffusivity displays a transition between metal-like and insulator-like transport as the temperature increases, similar to that predicted and observed in the nonadiabatic regime.

Multipoint propagators for non-Gaussian initial conditions

We show here how Renormalized Perturbation Theory (RPT) calculations applied to the quasi-linear growth of the large-scale structure can be carried on in presence of primordial non-Gaussian (PNG) initial conditions. It is explicitly demonstrated that the series reordering scheme proposed in Bernardeau, Crocce and Scoccimarro (2008) is preserved for non-Gaussian initial conditions. This scheme applies to the power spectrum and higher order spectra and is based on a reorganization of the contributing terms into sum of products of multi-point propagators. In case of PNG new contributing terms appear, the importance of which is discussed in the context of current PNG models. The properties of the building blocks of such resummation schemes, the multi-point propagators, are then investigated. It is first remarked that their expressions are left unchanged at one-loop order irrespectively of statistical properties of the initial field. We furthermore show that the high-momemtum limit of each of these propagators can be explicitly computed even for arbitrary initial conditions. They are found to be damped by an exponential cutoff whose expression is directly related to the moment generating function of the one-dimensional displacement field. This extends what had been established for multi-point propagators for Gaussian initial conditions. Numerical forms of the cut-off are shown for the so-called local model of PNG.

Renormalized parameters and perturbation theory for ann-channel Anderson model with Hund’s rule coupling: Symmetric case

We extend the renormalized perturbation theory for the single-impurity Anderson model to the $n$-channel model with a Hund's rule coupling, and show that the exact results for the spin, orbital, and charge susceptibilities, as well as the leading low-temperature dependence for the resistivity, are obtained by working to second order in the renormalized couplings. A universal relation is obtained between the renormalized parameters, independent of $n$, in the Kondo regime. An expression for the dynamic spin susceptibility is also derived by taking into account repeated quasiparticle scattering, which is asymptotically exact in the low-frequency regime and satisfies the Korringa-Shiba relation. The renormalized parameters, including the renormalized Hund's rule coupling, are deduced from numerical renormalization-group calculations for the model for the case $n=2$. The results confirm explicitly the universal relations between the parameters in the Kondo regime. Using these results, we evaluate the spin, orbital, and charge susceptibilities, temperature dependence of the low-temperature resistivity, and dynamic spin susceptibility for the particle-hole symmetric regime of the $n=2$ model.

A new model for the full shape of the large-scale power spectrum

We present a new model for the full shape of large-scale the power spectrum based on renormalized perturbation theory. To test the validity of this prescription, we compare this model against power spectra measured in a suite of 50 large volume, moderate resolution N-body simulations. Our results indicate that this simple model provides an accurate description of the full shape of the power spectrum taking into account the effects of non-linear evolution, redshift-space distortions and halo bias for scales k < 0.15 h/Mpc, making it a valuable tool for the analysis of forthcoming galaxy surveys. Even though its application is restricted to large scales, this prescription can provide tighter constraints on the dark energy equation of state parameter w_{DE} than those obtained by modelling the baryonic acoustic oscillations signal only, where the information of the broad-band shape of the power spectrum is discarded. Our model is able to provide constraints comparable to those obtained by applying a similar model to the full shape of the correlation function, which is affected by different systematics. Hence, with accurate modelling of the power spectrum, the same cosmological information can be extracted from both statistics.

Renormalized perturbation theory for a toy model of fluctuating nonlinear hydrodynamics of supercooled liquids

We study the field theoretic renormalized perturbation theory for a toy model of fluctuating nonlinear hydrodynamics (FNH) of compressible liquids. The toy model contains a density-like and a momentum-like variable without any spatial dependence. We present a detailed derivation of a set of coupled equations among correlation and response functions for these variables. In particular, we focus on how the static limit of the correlation and response functions can be achieved in the renormalized perturbation theory. Numerical methods of solving these equations at a given order of the loop expansion are explained and the results for the one-loop theory are given in detail. The simple nature of the toy model enables us to compare the static limit obtained from the exact solution with that of the one-loop order. This shows explicitly the range of validity of the one-loop theory in the field theoretic formulation.

SELF-CONSISTENT THEORY OF ANDERSON LOCALIZATION: GENERAL FORMALISM AND APPLICATIONS

The self-consistent theory of Anderson localization of quantum particles or classical waves in disordered media is reviewed. After presenting the basic concepts of the theory of Anderson localization in the case of electrons in disordered solids, the regimes of weak and strong localization are discussed. Then the scaling theory of the Anderson localization transition is reviewed. The renormalization group theory is introduced and results and consequences are presented. It is shown how scale-dependent terms in the renormalized perturbation theory of the inverse diffusion coefficient lead in a natural way to a self-consistent equation for the diffusion coefficient. The latter accounts quantitatively for the static and dynamic transport properties except for a region near the critical point. Several recent applications and extensions of the self-consistent theory, in particular for classical waves, are discussed.

New Study of Shot Noise Based on the Nonequilibrium Kubo Formula in Mesoscopic Systems: Application to the Kondo Effect in a Quantum Dot

Recently, we have developed a theory of Keldysh formalism for mesoscopic systems. The resulting nonequilibrium Kubo formula for differential conductance makes it possible to propose a new formula of the shot noise S h , and thus to study shot noise in correlated systems at any temperature and any bias voltage. Employing this new approach, we analyze shot noise in the Kondo regime through a quantum dot in a symmetric case at zero temperature. Using the renormalized perturbation theory, we prove that, in the leading order of bias voltage, S h is equal to the noise power at zero temperature conventionally used as shot noise. With S h , we calculate the Fano factor for the backscattering current I b : F b = S h /2 e I b . It is shown that the Fano factor takes the universal form of F b =1+4( R -1) 2 /[1+5( R -1) 2 ] determined by the Wilson ratio R for an arbitrary strength of the Coulomb interaction. Using the Wilson ratio R =2, our result coincides with the fractional value of F b =5/3 derived in the Kondo ...

Cosmological perturbation theory for baryons and dark matter: One-loop corrections in the renormalized perturbation theory framework

We generalize the `renormalized' perturbation theory (RPT) formalism of Crocce & Scoccimarro (2006a) to deal with multiple fluids in the Universe and here we present calculations up to the one-loop level. We apply the approach to the non-linear evolution of baryon and cold dark matter (CDM) perturbations, evolving from distinct sets of initial conditions. In current models of structure formation, it is standard practice to treat baryons and CDM as an effective single component fluid. In this approximation, one uses a weighed sum of late-time baryon and CDM transfer functions to set initial conditions. Here, we explore whether this approach can be used for high precision work. We show that, even for a pure linear treatment, there is a large-scale scale-dependent bias between baryons and CDM for WMAP5 cosmology. This bias is >1% until the present day, when it is driven towards unity through gravitational relaxation. Using the RPT formalism we test this approximation in the non-linear regime, and show that the CDM power spectrum in the 2-component fluid differs from that obtained from a 1-component fluid by ~3% on scales of order k~0.05 h/Mpc at z=10, and by ~0.5% at z=0. However, for the case of baryons the situation is worse and we find that the power spectrum is suppressed by ~15% on scales k~0.05 h/Mpc at z=10, and by ~3-5% at z=0, relative to the total matter. Importantly, besides the suppression of the spectrum, baryonic acoustic oscillations (BAO) are amplified for baryon and damped for CDM spectra. Thus, high precision modeling of baryons can not be probed through CDM only simulations; detection significance of BAO will be amplified in probes that study baryonic matter. Total mass can be modeled accurately using a 1-component fluid approach at all times. (Abridged)

Effect of the Coulomb interaction for shot noise in a quantum dot

The correlation effect of shot noise through a quantum dot is investigated for the symmetric case at zero temperature. Employing the renormalized perturbation theory, we calculate the shot noise S h and the Fano factor F b = S h / 2 eI b for a backscattering current I b . The Fano factor is shown to have the universal form of F b = 1 + 4 ( R − 1 ) 2 / ( 1 + 5 ( R − 1 ) 2 ) with the Wilson ratio R in any strength of the Coulomb interaction.

Renormalized field theory of collapsing directed randomly branched polymers

We present a dynamical field theory for directed randomly branched polymers and in particular their collapse transition. We develop a phenomenological model in the form of a stochastic response functional that allows us to address several interesting problems such as the scaling behavior of the swollen phase and the collapse transition. For the swollen phase, we find that by choosing model parameters appropriately, our stochastic functional reduces to the one describing the relaxation dynamics near the Yang-Lee singularity edge. This corroborates that the scaling behavior of swollen branched polymers is governed by the Yang-Lee universality class as has been known for a long time. The main focus of our paper lies on the collapse transition of directed branched polymers. We show to arbitrary order in renormalized perturbation theory with epsilon expansion that this transition belongs to the same universality class as directed percolation.

Modeling Nonlinear Evolution of Baryon Acoustic Oscillations: Convergence Regime of $N$-body Simulations and Analytic Models

Abstract We used a series of cosmological $N$-body simulations and various analytic models to study the evolution of the matter power spectrum in real space in a $\Lambda$ cold dark matter universe. We compared the results of $N$-body simulations against three analytical model predictions; standard perturbation theory, renormalized perturbation theory, and a closure approximation. We included the effects from a finite simulation box size under comparison. We determined the values of the maximum wavenumbers, $k^{\rm lim}_{1\%}$ and $k^{\rm lim}_{3\%}$, below which the analytic models and the simulation results agree with accuracy to within 1 and 3 percent. We then provided a simple empirical function that describes the convergence regime determined by comparisons between our simulations and the analytical models. We found that if we use the Fourier modes within the convergence regime alone, the characteristic scale of baryon acoustic oscillations can be determined with an accuracy of 1% from future surveys with a volume of a few $h^{-3}$Gpc$^3$ at z$\sim$ 1 or z$\sim$ 3 in the absence of any systematic distortion of the power spectrum.

Universal scaling in nonequilibrium transport through an Anderson impurity

Using nonequilibrium renormalized perturbation theory, we calculate the conductance $G$ as a function of temperature $T$ and bias voltage $V$ for an Anderson model suitable for describing transport properties through a quantum dot. For renormalized parameters that correspond to the extreme Kondo limit, we do not find a simple scaling formula beyond a quadratic dependence in $T$ and $V$. However, if valence fluctuations are allowed, we find excellent agreement with recent experiments.

RENORMALIZED PERTURBATION THEORY: A MISSING CHAPTER

Renormalized perturbation theory à la BPHZ can be founded on causality as analyzed by Epstein and Glaser in the seventies. Here, we list and discuss a number of additional constraints of algebraic character some of which have to be considered as parts of the core of the BPHZ framework.