Approximate Renormalization Research Articles

Donnan equilibrium in charged slit-pores from a hybrid nonequilibrium molecular dynamics/Monte Carlo method with ions and solvent exchange.

Ion partitioning between different compartments (e.g., a porous material and a bulk solution reservoir), known as Donnan equilibrium, plays a fundamental role in various contexts such as energy, environment, or water treatment. The linearized Poisson-Boltzmann (PB) equation, capturing the thermal motion of the ions with mean-field electrostatic interactions, is practically useful to understand and predict ion partitioning, despite its limited applicability to conditions of low salt concentrations and surface charge densities. Here, we investigate the Donnan equilibrium of coarse-grained dilute electrolytes confined in charged slit-pores in equilibrium with a reservoir of ions and solvent. We introduce and use an extension to confined systems of a recently developed hybrid nonequilibrium molecular dynamics/grand canonical Monte Carlo simulation method ("H4D"), which enhances the efficiency of solvent and ion-pair exchange via a fourth spatial dimension. We show that the validity range of linearized PB theory to predict the Donnan equilibrium of dilute electrolytes can be extended to highly charged pores by simply considering renormalized surface charge densities. We compare with simulations of implicit solvent models of electrolytes and show that in the low salt concentrations and thin electric double layer limit considered here, an explicit solvent has a limited effect on the Donnan equilibrium and that the main limitations of the analytical predictions are not due to the breakdown of the mean-field description but rather to the charge renormalization approximation, because it only focuses on the behavior far from the surfaces.

New-Generation Electron-Propagator Methods for Molecular Electron-Binding Energies.

A new generation of electron-propagator methods for the calculation of electron binding energies has surpassed its antecedents with respect to accuracy, efficiency, and interpretability. No adjustable parameters are introduced in these fully ab initio procedures. Numerical tests versus several databases of valence, vertical electron binding energies of closed-shell molecules and atoms have been performed. Easily interpreted self-energy approximations with cubic arithmetic scaling produce mean absolute errors (MAEs) of 0.2 and 0.3 eV for electron detachments and attachments, respectively. The most accurate explicitly renormalized methods with fifth-power arithmetic scaling yield MAEs below 0.1 eV for detachments and attachments. Approximate renormalization leads to more efficient fifth-power alternatives for electron detachments that achieve similar accuracy with fewer bottleneck operations. Composite protocols generate excellent predictions versus highly accurate basis-extrapolated standards and experiments. The validity of the diagonal self-energy approximation and the accuracy of the approximate renormalizations are confirmed. The success of these perturbative methods based on canonical Hartree-Fock orbitals rests on a Hermitized, intermediately normalized superoperator metric. The results of all of the new-generation calculations may be analyzed in terms of final-state orbital relaxation and differential correlation effects.

Statistics of the Inertial Energy Transfer Range in d-Dimensional Turbulence (2 ≤ d ≤ 3) in a Lagrangian Renormalized Approximation

Statistics in the inertial energy transfer range (IETR) of d-dimensional turbulence ( 2≤d≤3) are studied using a Lagrangian renormalized approximation (LRA). The LRA suggests that the energy spectrum in the IETR is given by Kd|ε¯|2/3k−5/3, where Kd is a constant and ε¯ is the energy flux across wave-number k; the energy transfer is forward for dc<d≤3 but inverse for 2≤d<dc, where dc≈2.065; at d=dc, Kd diverges and the skewness of the longitudinal velocity difference vanishes; and the d-dependence of the two-time Lagrangian velocity correlation spectra under appropriate normalization is weak in the IETR.

A convergence criterion for the unstable manifolds of the MacKay approximate renormalisation

We give an explicit arithmetical condition which guarantees the existence of the unstable manifold of the MacKay approximate renormalisation scheme for the breakup of invariant tori in one and a half degrees of freedom Hamiltonian systems, correcting earlier results. Furthermore, when our condition is violated, we give an example of points on which the unstable manifold does not converge.

Adiabatically deformed ensemble: Engineering nonthermal states of matter

We propose a route towards engineering non-thermal states of matter, which show largely unexplored physics. The main idea relies on the adiabatic passage of a thermal ensemble under slow variations of the system Hamiltonian. If the temperature of the initial thermal ensemble is either zero or infinite the ensemble after the passage is a simple thermal one with the same vanishing or infinite temperature. However, for any finite non-zero temperature intriguing non-thermal ensembles can be achieved. We exemplify this in: (a) a single oscillator (b) a dimerized interacting one dimensional chain of spinless fermions, (c) a BCS-type superconductor and (d) the topological Kitaev chain. We solve these models with a combination of methods; either exactly, numerically using the density matrix renormalization group (DMRG) or within an approximate functional renormalization group (FRG) scheme. The designed states show strongly non-thermal behavior in each of the considered models. For example, for the chain of spinless fermions we exemplify how long ranged non-thermal power-law correlations can be stabilized and for the Kitaev chain we elucidate how the non-thermal ensemble can largely alter the transition temperature separating topological and trivial phases.

Duality, Gauge Symmetries, Renormalization Groups and the BKT Transition

In this chapter, I will briefly review, from my own perspective, the situation within theoretical physics at the beginning of the 1970s, and the advances that played an important role in providing a solid theoretical and experimental foundation for the Berezinskii–Kosterlitz–Thouless theory (BKT). Over this period, it became clear that the Abelian gauge symmetry of the 2D-XY model had to be preserved to get the right phase structure of the model. In previous analyses, this symmetry was broken when using low order calculational approximations. Duality transformations at that time for two-dimensional models with compact gauge symmetries were introduced by José, Kadanoff, Nelson and Kirkpatrick (JKKN). Their goal was to analyze the phase structure and excitations of XY and related models, including symmetry breaking fields which are experimentally important. In a separate context, Migdal had earlier developed an approximate Renormalization Group (RG) algorithm to implement Wilson’s RG for lattice gauge theories. Although Migdal’s RG approach, later extended by Kadanoff, did not produce a true phase transition for the XY model, it almost did asymptotically in terms of a non-perturbative expansion in the coupling constant with an essential singularity. Using these advances, including work done on instantons (vortices), JKKN analyzed the behavior of the spin–spin correlation functions of the 2D XY-model in terms of an expansion in temperature and vortex-pair fugacity. Their analysis led to a perturbative derivation of RG equations for the XY model which are the same as those first derived by Kosterlitz for the two-dimensional Coulomb gas. JKKN’s results gave a theoretical formulation foundation and justification for BKT’s sound physical assumptions and for the validity of their calculational approximations that were, in principle, strictly valid only at very low temperatures, away from the critical [Formula: see text] temperature. The theoretical predictions were soon tested successfully against experimental results on superfluid helium films. The success of the BKT theory also gave one of the first quantitative proofs of the validity of the RG theory.

Towards rigorous analysis of the Levitov–Mirlin–Evers recursion

This paper aims to develop a rigorous asymptotic analysis of an approximate renormalization group recursion for inverse participation ratios Pq of critical powerlaw random band matrices. The recursion goes back to the work by Mirlin and Evers (2000 Phys. Rev. B 62 7920) and earlier works by Levitov (1990 Phys. Rev. Lett. 64 547, 1999 Ann. Phys. 8 697–706) and is aimed to describe the ensuing multifractality of the eigenvectors of such matrices. We point out both similarities and dissimilarities between the LME recursion and those appearing in the theory of multiplicative cascades and branching random walks and show that the methods developed in those fields can be adapted to the present case. In particular the LME recursion is shown to exhibit a phase transition, which we expect is a freezing transition, where the role of temperature is played by the exponent q. However, the LME recursion has features that make its rigorous analysis considerably harder and we point out several open problems for further study.

Berezinskii-Kosterlitz-Thouless phase transitions in two-dimensional non-Abelian spin models.

It is argued that two-dimensional U(N) spin models for any N undergo a Berezinskii-Kosterlitz-Thouless (BKT)-like phase transition, similarly to the famous XY model. This conclusion follows from the Berezinskii-like calculation of the two-point correlation function in U(N) models, approximate renormalization group analysis, and numerical investigations of the U(2) model. It is shown, via Monte Carlo simulations, that the universality class of the U(2) model coincides with that of the XY model. Moreover, preliminary numerical results point out that two-dimensional SU(N) spin models with the fundamental and adjoint terms and N>4 exhibit two phase transitions of BKT type, similarly to Z(N) vector models.

Many-body localization phase transition: A simplified strong-randomness approximate renormalization group

We present a simplified strong-randomness renormalization group (RG) that captures some aspects of the many-body localization (MBL) phase transition in generic disordered one-dimensional systems. This RG can be formulated analytically, and the critical fixed point distribution and critical exponents (that satisfy the Chayes inequality) are obtained to numerical precision by solving integro-differential equations. This reproduces many, but not all, of the qualitative features of the MBL phase transition that are suggested by previous numerical work and approximate RG studies: our RG might serve as a "zeroth-order" approximation for future RG studies. One interesting feature that we highlight is that the rare Griffiths regions are fractal. For thermal Griffiths regions within the MBL phase, this feature might be qualitatively correctly captured by our RG. If this is correct beyond our approximations, then these Griffiths effects are stronger than has been previously assumed.

Correlation matrix renormalization approximation for total-energy calculations of correlated electron systems

We generalized the commonly used Gutzwiller approximation for calculating the electronic structure and total energy of strongly correlated electron systems. In our method, the evaluation of one-body and two-body density matrix elements of the Hamiltonian is simplified using a renormalization approximation to achieve better scaling of the computational effort as a function of system size. To achieve a clear presentation of the concept and methodology, we describe the detailed formalism for a finite hydrogen system with minimal basis set. We applied the correlation matrix renormalization approximation approach to a H${}_{2}$ dimer and H${}_{8}$ cubic fragment with minimal basis sets, as well as a H${}_{2}$ molecule with a large basis set. The results compare favorably with sophisticated quantum chemical calculations. We believe our approach can serve as an alternative way to build up the exchange-correlation energy functional for an improved density functional theory description of systems with strong electron correlations.

Continuum percolation of overlapping disks with a distribution of radii having a power-law tail

We study the continuum percolation problem of overlapping disks with a distribution of radii having a power-law tail; the probability that a given disk has a radius between R and R+dR is proportional to R(-(a+1)), where a>2. We show that in the low-density nonpercolating phase, the two-point function shows a power-law decay with distance, even at arbitrarily low densities of the disks, unlike the exponential decay in the usual percolation problem. As in the problem of fluids with long-range interaction, we argue that in our problem, the critical exponents take their short-range values for a>3-η(sr) whereas they depend on a for a<3-η(sr) where η(sr) is the anomalous dimension for the usual percolation problem. The mean-field regime obtained in the fluid problem corresponds to the fully covered regime, a≤2, in the percolation problem. We propose an approximate renormalization scheme to determine the correlation length exponent ν and the percolation threshold. We carry out Monte Carlo simulations and determine the exponent ν as a function of a. The determined values of ν show that it is independent of the parameter a for a>3-η(sr) and is equal to that for the lattice percolation problem, whereas ν varies with a for 2<a<3-η(sr). We also determine the percolation threshold of the system as a function of the parameter a.

On the Renormalization of the Covariance Operators

Many background error correlation (BEC) models in data assimilation are formulated in terms of a smoothing operator [Formula: see text], which simulates the action of the correlation matrix on a state vector normalized by respective BE variances. Under such formulation, [Formula: see text] has to have a unit diagonal and requires appropriate renormalization by rescaling. The exact computation of the rescaling factors (diagonal elements of [Formula: see text]) is a computationally expensive procedure, which needs an efficient numerical approximation. In this study approximate renormalization techniques based on the Monte Carlo (MC) and Hadamard matrix (HM) methods and on the analytic approximations derived under the assumption of the local homogeneity (LHA) of [Formula: see text] are compared using realistic BEC models designed for oceanographic applications. It is shown that although the accuracy of the MC and HM methods can be improved by additional smoothing, their computational cost remains significantly higher than the LHA method, which is shown to be effective even in the zeroth-order approximation. The next approximation improves the accuracy 1.5–2 times at a moderate increase of CPU time. A heuristic relationship for the smoothing scale in two and three dimensions is proposed for the first-order LHA approximation.

Computer simulation of surface-enhanced Raman scattering in nanostructured metamaterials

The simulation of local field fluctuations and surface-enhanced Raman scattering in percolation systems at the percolation threshold is described. An approximate real-space renormalization group method was used in the simulation. It allows one to radically reduce the computation time compared to an exact calculation and to obtain detailed information about the electromagnetic field. The local fields in real macroscopic systems can be calculated by using this approximation. A computer simulation of the local fields in metal island (percolation) films has been performed by the developed method. The calculation has confirmed the existence of giant local field fluctuations. In turn, the local electric field excites Raman scattering. The local fields of surface-enhanced Raman scattering have been calculated for the first time. The dependence of the Raman scattering enhancement factor on the reference frequency and Stokes shift has been derived. An experimental observation of this dependence could be considered as a confirmation of the electromagnetic nature of the enhancement.

Transport properties of a molecule embedded in an Aharonov-Bohm interferometer

We theoretically investigate the transport properties of a molecule embedded in one arm of a mesoscopic Aharonov-Bohm interferometer. Due to the presence of phonons the molecule level position ($\epsilon_d$) and the electron-electron interaction ($U$) undergo a \emph{polaronic shift} which affects dramatically the electronic transport through the molecular junction. When the electron-phonon interaction is weak the linear conductance presents Fano-line shapes as long as the direct channel between the electrodes is opened. The observed Fano resonances in the linear conductance are originated from the interference between the spin Kondo state and the direct path. For strong enough electron-phonon interaction, the electron-electron interaction is renormalized towards negative values, {\it i.e.} becomes effectively attractive. This scenario favors fluctuations between the empty and doubly occupied charge states and therefore promotes a charge Kondo effect. However, the direct path between the contacts breaks the electron-hole symmetry which can efficiently suppress this charge Kondo effect. Nevertheless, we show that a proper tuning of the gate voltage is able to revive the Kondo resonance. Our results are obtained by using the Numerical Renormalization approximation to compute the electronic spectral function and the linear conductance.

OFF-SHELL EFFECTS IN FEW-BODY SYSTEMS WITH COULOMB FORCE

The exact treatment of the two-body problem in momentum space in the presence of Coulomb forces is discussed. Convergence of the traditionally used renormalization approximation is investigated with respect to increasing ranges of the screened Coulomb potentials. The permissible energy range for this approximation is obtained for the first time in this work.

DIRTY BOSONS: TWENTY YEARS LATER

A concise and somewhat personal review of the problem of superfluidity and quantum criticality in regular and disordered interacting Bose systems is given, concentrating on general features and important symmetries that are exhibited in different parts of the phase diagram, and that govern the different possible types of critical behavior. A number of exact results for various insulating phase boundaries, which may be used to constrain the results of numerical simulations, can be derived using large rare region type arguments. The nature of the insulator-superfluid transition is explored through general scaling arguments, exact model calculations in one dimension, numerical results in two dimensions, and approximate renormalization group results in higher dimensions. Experiments on 4 He adsorbed in porous Vycor glass, on thin film superconductors, and magnetically trapped atomic vapors in a periodic optical potential are used to illustrate many of the concepts.

Running Neutrino Mass Parameters in See-Saw Scenarios

We systematically analyze quantum corrections in see-saw scenarios, including effects from above the see-saw scales. We derive approximate renormalization group equations for neutrino masses, lepton mixings and CP phases, yielding an analytic understanding and a simple estimate of the size of the effects. Even for hierarchical masses, they often exceed the precision of future experiments. Furthermore, we provide a software package allowing for a convenient numerical renormalization group analysis, with heavy singlets being integrated out successively at their mass thresholds. We also discuss applications to model building and related topics.

Hybrid RANS/LES modelling with an approximate renormalization group. II: Applications

The hybrid RANS/LES model developed previously is extended with near-wall modifications. The model is validated, in its low-Reynolds form, for channel flow and for flow over a periodic hill. The results for the channel flow are (indirectly) compared with DES results for the same flow, and are qualitatively similar. The separated flow over the periodic hill shows good agreement with the reference LES data, although performed with about 20 times less grid points. Finally the model is applied in high-Reynolds form with wall functions to a sudden pipe expansion; good agreement with experimental data is also obtained.

Hybrid RANS/LES modelling with an approximate renormalization group. I: Model development

A hybrid RANS/LES model based on renormalization group (RG) calculations is presented. The result of the RG approach is a one-equation LES subgrid model in well-resolved regions of the flow, and a two-equation RANS model otherwise. The two modes of the model are linked by comparing the filter width with a length scale constructed from subgrid quantities. The one-equation subgrid model uses a transport equation for the mean rate of dissipation, in which the inverse time scale is explicitly filter width dependent. The model is free of adjustable parameters (after the truncations of the approximate RG are made) in its high-Reynolds formulation, has a clear physical interpretation and is easy to implement.

Stability ofU(1)spin liquids in two dimensions

We establish that spin liquids described in terms of gapless fermionic (Dirac) spinons and gapless U(1) gauge fluctuations can be stable in two dimensions, at least when the physical SU(2) spin symmetry is generalized to SU(N). Equivalently, we show that compact QED3 has a deconfined phase for a large number of fermion fields, in the sense that monopole fluctuations can be irrelevant at low energies. A precise characterization is provided by an emergent global topological U(1) symmetry corresponding to the conservation of gauge flux. Beginning with an SU(N) generalization of the S=1/2 square lattice Heisenberg antiferromagnet, we consider the pi-flux spin liquid and, via a systematic analysis of all operators, show that there are NO relevant perturbations (in the renormalization group sense) about the large-N spin liquid fixed point, which is thus a stable phase. We provide a further illustration of this conclusion with an approximate renormalization group calculation that treats the gapless fermions and the monopoles on an equal footing. This approach directly points out some of the flaws in the erroneous "screening" argument for the relevance of monopoles in compact QED3.