Renormalized Perturbation Theory Research Articles

Uncovering Hidden Patterns: Approximate Resurgent Resummation from Truncated Series

We analyze truncated series generated as divergent formal solutions of non-linear ordinary differential equations. Motivating the study is a specific non-linear, first-order differential equation, which is the basis of the resurgent formulation of renormalized perturbation theory in quantum field theory. We use the Borel–Padé approximant and classical analysis to determine the analytic structure of the solution using the first few terms of its asymptotic series. Afterward, we build an approximant, consistent with the resurgent properties of the equation. The procedure gives an approximate expression for the Borel–Ecalle resummation of the solution useful for practical applications. Connections with other physical applications are also discussed.

RealTimeTransport: An open-source C++ library for quantum transport simulations in the strong coupling regime.

The description of quantum transport in the strong system-reservoir coupling regime poses a significant theoretical and computational challenge that demands specialized tools for accurate analysis. RealTimeTransport is a new open-source C++ library that enables the computation of both stationary and transient transport observables for generic quantum systems connected to metallic reservoirs. It computes the Nakajima-Zwanzig memory kernels for both dynamics and transport in real-time, going beyond traditional expansions in the bare system-reservoir couplings. Currently, several methods are available as follows: (i) A renormalized perturbation theory in leading and next-to-leading order, which avoids the low-temperature breakdown that limits the traditional theory. (ii) Starting from this well-behaved reference solution, a two- and three-loop, self-consistent renormalization-group transformation of the memory kernels is implemented. This allows refined quantitative predictions even in the presence of many body resonances, such as the Kondo enhancement of cotunneling. This paper provides an overview of the theory, the architecture of RealTimeTransport, and practical demonstrations of the currently implemented methods. In particular, we analyze the stationary transport through a serial double quantum dot and showcase for the T = 0 interacting Anderson model the complete time-development of single-electron tunneling (SET), cotunneling-assisted SET, and inelastic cotunneling resonances throughout the entire gate-bias stability diagram. We discuss the range of applicability of the implemented methods and benchmark them against other advanced approaches.

Statistical Dynamics and Subgrid Modelling of Turbulence: From Isotropic to Inhomogeneous

Turbulence is the most important, ubiquitous, and difficult problem of classical physics. Feynman viewed it as essentially unsolved, without a rigorous mathematical basis to describe the statistical dynamics of this most complex of fluid motion. However, the paradigm shift came in 1959, with the formulation of the Eulerian direct interaction approximation (DIA) closure by Kraichnan. It was based on renormalized perturbation theory, like quantum electrodynamics, and is a bare vertex theory that is manifestly realizable. Here, we review some of the subsequent exciting achievements in closure theory and subgrid modelling. We also document in some detail the progress that has been made in extending statistical dynamical turbulence theory to the real world of interactions with mean flows, waves and inhomogeneities such as topography. This includes numerically efficient inhomogeneous closures, like the realizable quasi-diagonal direct interaction approximation (QDIA), and even more efficient Markovian Inhomogeneous Closures (MICs). Recent developments include the formulation and testing of an eddy-damped Markovian anisotropic closure (EDMAC) that is realizable in interactions with transient waves but is as efficient as the eddy-damped quasi-normal Markovian (EDQNM). As a similarly efficient closure, the realizable eddy-damped Markovian inhomogeneous closure (EDMIC) has been developed. Moreover, we present subgrid models that cater for the complex interactions that occur in geophysical flows. Recent progress includes the determination of complete sets of subgrid terms for skilful large-eddy simulations of baroclinic inhomogeneous turbulent atmospheric and oceanic flows interacting with Rossby waves and topography. The success of these inhomogeneous closures has also led to further applications in data assimilation and ensemble prediction and generalization to quantum fields.

Turbulence and Rossby Wave Dynamics with Realizable Eddy Damped Markovian Anisotropic Closure

The theoretical basis for the Eddy Damped Markovian Anisotropic Closure (EDMAC) is formulated for two-dimensional anisotropic turbulence interacting with Rossby waves in the presence of advection by a large-scale mean flow. The EDMAC is as computationally efficient as the Eddy Damped Quasi Normal Markovian (EDQNM) closure, but, in contrast, is realizable in the presence of transient waves. The EDMAC is arrived at through systematic simplification of a generalization of the non-Markovian Direct Interaction Approximation (DIA) closure that has its origin in renormalized perturbation theory. Markovian Anisotropic Closures (MACs) are obtained from the DIA by using three variants of the Fluctuation Dissipation Theorem (FDT) with the information in the time history integrals instead carried by Markovian differential equations for two relaxation functions. One of the MACs is simplified to the EDMAC with analytical relaxation functions and high numerical efficiency, like the EDQNM. Sufficient conditions for the EDMAC to be realizable in the presence of Rossby waves are determined. Examples of the numerical integration of the EDMAC compared with the EDQNM are presented for two-dimensional isotropic and anisotropic turbulence, at moderate Reynolds numbers, possibly interacting with Rossby waves and large-scale mean flow. The generalization of the EDMAC for the statistical dynamics of other physical systems to higher dimension and higher order nonlinearity is considered.

Jackson R. Herring and the Statistical Closure Problem of Turbulence: A Review of Renormalized Perturbation Theories

The pioneering applications of the methods of theoretical physics to the turbulence statistical closure problem are summarised. These are: the direct-interaction approximation (DIA) of Kraichnan, the self-consistent-field theory of Edwards, and the self-consistent-field theory of Herring. Particular attention is given to the latter, in terms of its elegance and its pedagogical value. We then concentrate on the assessment of these theories and take the historical route of Kraichnan’s diagnosis of the failure of DIA, followed by Edwards’s analysis of the failure of his self-consistent theory, when compared to the Kolmogorov spectrum. As all three theories are closely related, these analyses also shed light on Herring’s theory. The second-generation theories that grew out of this assessment are then discussed. First, there were the Lagrangian theories, initially stemming from the work of Kraichnan and Herring, and later the purely Eulerian local energy-transfer (LET) theory. The latter is significant because its development exposes the underlying problems with the pioneering theories in terms of the basic physics of the inertial energy transfer. In particular, later work allows us to assign a unified explanation of the incompatibility of all three pioneering theories with the Kolmororov spectrum, in that they are all Markovian approximations (in wavenumber) to the non-Markovian phenomenon of fluid turbulence. In the interests of completeness, we briefly review the formalisms of Wyld and Martin, Siggia, and Rose. More recent developments are also discussed, in order to bring the subject up to the present day.

Renormalized perturbation theory for fast evaluation of Feynman diagrams on the real frequency axis

We present a method to accelerate the numerical evaluation of spatial integrals of Feynman diagrams when expressed on the real frequency axis. This can be realized through use of a renormalized perturbation expansion with a constant but complex renormalization shift. The complex shift acts as a regularization parameter for the numerical integration of otherwise sharp functions. This results in an exponential speed up of stochastic numerical integration at the expense of evaluating additional counter-term diagrams. We provide proof of concept calculations within a difficult limit of the half-filled 2D Hubbard model on a square lattice.

Correlated second-order Dirac semimetals with Coulomb interactions

We investigate the effects of long range Coulomb interactions on the low-temperature properties of a second-order Dirac semimetal in terms of the renormalization group. In contrast to the first-order Dirac semimetal, the full rotation symmetry is broken even in the continuum limit, and thus the low-energy physics is controlled by two dimensionless parameters: the dimensionless coupling constant and the ratio of the anisotropy parameters. We show that the former flows to zero and the latter flows to a fixed value at low energies. Thus, one may calculate physical quantities in terms of the renormalized perturbation theory. As an application, we determine the temperature dependence of the specific heat by solving the renormalization-group equations. Following from the breaking of the full rotation symmetry, there exists a crossover temperature scale $T_c$ (and a length scale $L_c$). Physical quantities approach the values for the first-order Dirac semimetal only when the temperature is much smaller than $T_c$. Similarly, the screened Coulomb potential will become anisotropic when the distance is smaller than $L_c$, while the unscreened form is recovered at its tail.

Renormalized Schwinger\u2013Dyson functional

We consider the perturbative renormalization of the Schwinger–Dyson functional, which is the generating functional of the expectation values of the products of the composite operator given by the field derivative of the action. It is argued that this functional plays an important role in the topological Chern–Simons and BF quantum field theories. It is shown that, by means of the renormalized perturbation theory, a canonical renormalization procedure for the Schwinger–Dyson functional is obtained. The combinatoric structure of the Feynman diagrams is illustrated in the case of scalar models. For the Chern–Simons and the BF gauge theories, the relationship between the renormalized Schwinger–Dyson functional and the generating functional of the correlation functions of the gauge fields is produced.

Coordinate space representation for renormalization of quantum electrodynamics

$ $In this paper we present a systematic treatment for fundamental renormalization of quantum electrodynamics in real space. Although the standard renormalization is an old school problem in this case, it has not yet been completely done in position space. The most important difference with well-known differential renormalization is that we do the whole procedure in coordinate space without need to transformation to momentum space. Specially, we directly derive the conterterms in real space. This problem becomes important when the translational symmetry of the system breaks somehow explicitly (for example by nontrivial boundary condition (BC) on the fields). In this case, one is not able to move to momentum space by a simple Fourier transformation. Therefore, in the context of renormalized perturbation theory, by imposing the renormalization conditions, counterterms in coordinate space will depend directly on the fields BCs (or background topology). Trivial BC or trivial background lead to the usual standard conterterms. If the counterterms modify then the quantum corrections of any physical quantity are different from those in free space where we have the translational invariance. We also show that, up to order $\alpha$, our counterterms are reduced to usual standard terms derived in free space.

Renormalized perturbation theory at large expansion orders

We present a general formalism that allows for the computation of large-order renormalized expansions, effectively doubling the numerically attainable perturbation order of renormalized Feynman diagrams. We show that this formulation compares advantageously to the currently standard techniques due to its high efficiency, simplicity, and broad range of applicability. Our formalism permits to easily complement perturbation theory with non-perturbative information, which we illustrate by implementing expansions renormalized by the addition of a gap or the inclusion of Dynamical Mean-Field Theory. As a result, we present numerically exact results for the square-lattice hole-doped Fermi-Hubbard model in the low-temperature non-Fermi-liquid regime, relevant to study the pseudogap of cuprate superconductors, and show the momentum-dependent suppression of fermionic excitations in the antinodal region.

Exact two-body expansion of the many-particle wave function

Progress toward the solution of the strongly correlated electron problem has been stymied by the exponential complexity of the wave function. Previous work established an exact two-body exponential product expansion for the ground-state wave function. By developing a reduced density matrix analogue of Dalgarno-Lewis perturbation theory, we prove here that (i) the two-body exponential product expansion is rapidly and globally convergent with each operator representing an order of a renormalized perturbation theory, (ii) the energy of the expansion converges quadratically near the solution, and (iii) the expansion is exact for both ground and excited states. The two-body expansion offers a reduced parametrization of the many-particle wave function as well as the two-particle reduced density matrix with potential applications on both conventional and quantum computers for the study of strongly correlated quantum systems. We demonstrate the result with the exact solution of the contracted Schr\"odinger equation for the molecular chains H$_{4}$ and H$_{5}$.

Cooperative spontaneous emission via a renormalization approach: Classical versus semiclassical effects

We address the many-atom emission of a dilute cloud of two-level atoms through a renormalized perturbation theory. An analytical solution for the truncated coupled-dipole equations is derived, which contains an effective spectrum associated to the initial conditions. Our solution is able to distinguish precisely classical from semi-classical predictions for large atomic ensembles. This manifests as a reduction of the cooperativity in the radiated power for higher atomic excitation, in disagreement with the fully classical prediction from linear optics. Moreover, the second-order cooperative emission appears accurate over several single atom lifetimes and for interacting regimes stronger than those permitted in conventional perturbation theory. We can compute the semiclassical dynamics of hundred thousand of interacting atoms with ordinary computational resources, which makes our formalism particularly promising to probe the nonlinear dynamics of quantum many-body systems that emerge from cumulant expansions.

Quantum BRST charge in gauge theories in curved space-time

Renormalized gauge-invariant observables in gauge theories form an algebra which is obtained as the cohomology of the derivation QL,−, with QL as the renormalized interacting quantum BRST charge. For a large class of gauge theories in Lorentzian globally hyperbolic space-times, we derive an identity in renormalized perturbation theory which expresses the commutator [QL, −] in terms of a new nilpotent quantum BRST (Becchi, Rouet, Stora, Tyutin) differential and a new quantum anti-bracket which differ from their classical counterparts by certain quantum corrections. This identity enables us to prove different manifestations of gauge symmetry preservation at the quantum level in a model-independent fashion.

Dissipative quantum mechanics beyond the Bloch-Redfield approximation: A consistent weak-coupling expansion of the Ohmic spin boson model at arbitrary bias

We study the time dynamics of the ohmic spin boson model at arbitrary bias $\epsilon$ and small coupling $\alpha$ to the bosonic bath. Using perturbation theory and the real-time renormalization group (RG) method we present a consistent zero-temperature weak-coupling expansion for the time evolution of the reduced density matrix one order beyond the Bloch-Redfield solution. We develop a renormalized perturbation theory and present an analytical solution covering the whole range from small to large times, including further results for exponentially small or large times. Resumming all secular terms in all orders of perturbation theory we find exponential decay for all terms of the time evolution. We determine the preexponential functions and find slowly varying logarithmic terms with the renormalized Rabi frequency $\Omega$ as energy scale together with strongly varying parts falling off asymptocially as $1/t$ in leading order, in contrast to the unbiased case. Resumming all logarithmic terms in all orders of perturbation theory via real-time RG we find the correct renormalized tunneling and a power-law behaviour for the oscillating modes with exponent crossing over from $2\alpha$ for exponentially small times to a bias-dependent value $2\alpha \epsilon^2/\Omega^2$ for exponentially large times. Furthermore, we present a degenerate perturbation theory to calculate consistently the purely decaying mode one order beyond Bloch-Redfield.

Nematic phase in a two-dimensional Hubbard model at weak coupling and finite temperature

We apply the self-consistent renormalized perturbation theory to the Hubbard model on the square lattice, at finite temperatures in order to study the evolution of the Fermi-surface (FS) as a function of temperature and doping. Previously, a nematic phase for the same model has been reported to appear at weak coupling near a Lifshitz transition from closed to open FS at zero temperature where the self-consistent renormalized perturbation theory was shown to be sensitive to small deformations of the FS. We find that the competition with the superconducting order leads to a maximal nematic order appearing at non-zero temperature. We explicitly observe the two competing phases near the onset of nematic instability and, by comparing the grand canonical potentials, we find that the transitions are first-order. We explain the origin of the interaction-driven spontaneous symmetry breaking to a nematic phase in a system with several symmetry-related Van Hove points and discuss the required conditions.

From semilocal density functionals to random phase approximation renormalized perturbation theory: A methodological assessment of structural phase transitions

The structural phase transitions of different materials, including metal to metal, metal to semiconductor, and semiconductor to semiconductor transitions, were explored using methods based on the random phase approximation. Transition pressures for Si, Ge, SiC, GaAs, ${\mathrm{SiO}}_{2}$, Pb, C, and BN from their stable low-pressure phases to certain high-pressure phases were computed with several semilocal density functionals and from the adiabatic connection fluctuation-dissipation formulation of density functional theory at zero temperature. In addition to the random phase approximation (RPA), three approximate beyond-RPA methods were also investigated to determine the impact of exchange-correlation kernel corrections. Results at finite temperature were obtained with the inclusion of zero-point energy contributions from the phonon spectra. We find that including temperature effects is most important for systems with nearly degenerate phases such as for boron nitride and carbon. In combination with thermal corrections, the kernel-corrected correlation methods deliver high accuracy compared to experimental data and can serve as a useful benchmark method in place of more expensive correlated calculations.

Efficient calculation of beyond RPA correlation energies in the dielectric matrix formalism.

We present efficient methods to calculate beyond random phase approximation (RPA) correlation energies for molecular systems with up to 500 atoms. To reduce the computational cost, we employ the resolution-of-the-identity and a double-Laplace transform of the non-interacting polarization propagator in conjunction with an atomic orbital formalism. Further improvements are achieved using integral screening and the introduction of Cholesky decomposed densities. Our methods are applicable to the dielectric matrix formalism of RPA including second-order screened exchange (RPA-SOSEX), the RPA electron-hole time-dependent Hartree-Fock (RPA-eh-TDHF) approximation, and RPA renormalized perturbation theory using an approximate exchange kernel (RPA-AXK). We give an application of our methodology by presenting RPA-SOSEX benchmark results for the L7 test set of large, dispersion dominated molecules, yielding a mean absolute error below 1 kcal/mol. The present work enables calculating beyond RPA correlation energies for significantly larger molecules than possible to date, thereby extending the applicability of these methods to a wider range of chemical systems.

Effect of Interaction on Reservoir-Parameter-Driven Adiabatic Charge Pumping via a Single-Level Quantum Dot System

We formulate adiabatic charge pumping via a single-level quantum dot (QD) induced by reser- voir parameter driving, i.e., temperature and electrochemical potential driving. Our formulation describes arbitrary strength of dot-reservoir coupling and Coulomb interaction in the QD, and is applicable to the low-temperature regime, where the Kondo effect becomes important. The adia- batic charge pumping is expressed by the Berry connection and is related to delayed response of the QD. We calculate the pumped charge by the renormalized perturbation theory, and discuss how the Coulomb interaction affects the charge pumping.

The two and three-loop matter bispectrum in perturbation theories

We evaluate for the first time the dark matter bispectrum of large-scale structure at two loops in the Standard Perturbation Theory and at three loops in the Renormalised Perturbation Theory (MPTBREEZE formalism), removing in each case the leading divergences in the integrals in order to make them infrared-safe. We show that the Standard Perturbation Theory at two loops can be employed to model the matter bispectrum further into the quasi-nonlinear regime compared to the one loop, up to kmax ∼ 0.1 h/Mpc at z = 0, but without reaching a high level of accuracy. In the case of the MPTBREEZE method, we show that its bispectra decay at smaller and smaller scales with increasing loop order, but with smaller improvements decreases with loop order. At three loops, this model predicts the bispectrum accurately up to scales kmax ∼ 0.17 h/Mpc at z = 0 and kmax ∼ 0.24 h/Mpc at z = 1.

Leading temperature dependence of the conductance in Kondo-correlated quantum dots

Using renormalized perturbation theory in the Coulomb repulsion, we derive an analytical expression for the leading term in the temperature dependence of the conductance through a quantum dot described by the impurity Anderson model, in terms of the renormalized parameters of the model. Taking these parameters from the literature, we compare the results with published ones calculated using the numerical renormalization group obtaining a very good agreement. The approach is superior to alternative perturbative treatments. We compare in particular to the results of a simple interpolative perturbation approach.