Renormalization Method Research Articles

Projector-based renormalization approach to electron-hole-photon systems in their nonequilibrium steady state

We present an extended version of the projector-based renormalization method that can be used to address not only equilibrium but also non-equilibrium situations in coupled fermion-boson systems. The theory is applied to interacting electrons, holes and photons in a semiconductor microcavity, where the loss of cavity photons into vacuum is of particular importance. The method incorporates correlation and fluctuation processes beyond mean-field theory in a wide parameter range of detuning, Coulomb interaction, light-matter coupling and damping, even in the case when the number of quasiparticle excitations is large. This enables the description of exciton and polariton formation, and their possible condensation through spontaneous phase symmetry breaking by analyzing the ground-state, steady-state and spectral properties of a rather generic electron-hole-photon Hamiltonian, which also includes the coupling to two fermionic baths and a free-space photon reservoir. Thereby, the steady-state behavior of the system is obtained by evaluating expectation values in the long-time limit by means of the Mori-Zwanzig projection technique. Tracking and tracing different order parameters, the fully renormalized single-particle spectra and the steady-state luminescence, we demonstrate the Bose-Einstein condensation of excitons and polaritons and its smooth transition when the excitation density is increased.

Bloch bound state of spin-orbit-coupled fermions in an optical lattice

Understanding fundamentals of few-body physics provides an interesting bottom-up approach for the clarification of many-body properties. The remarkable experimental progress in realizing spin-orbit coupling (SOC) in optical Raman lattices offers a renewed thrust towards discovering novel few-body features induced by the interplay between SOC and optical lattices. Using the Wilson renormalization method to account for high-band effects, we study the low-energy two-body scattering processes of spin-$1/2$ fermions in spin-orbit coupled optical lattices. We demonstrate that, under weak SOC, adding a small lattice potential would destabilize shallow two-body bound states, contrary to conventional wisdom. On the other hand, when lattice is sufficiently deep, two-body bound states are always stabilized by increasing the lattice depth. This intriguing non-monotonic behavior of the bound-state stability derives from the competition between SOC and optical lattices, and can be explained by analyzing the low-energy density of states. We also discuss the impact of high-band effects on such a behavior, as well as potential experimental detections.

Study on the two-dimensional kinetic Ising model with the dynamic Monte Carlo renormalization group method

Using a modified dynamic Monte Carlo renormalization group method, the two- dimensional kinetic Ising model is studied, and the dynamic critical exponent is obtained. The critical temperature of phase transition can be obtained by the renormalization method for the correlation function. In the method we used, the correlation function is replaced with the absolute value of the magnetization, and it is found that the evolution of the absolute value of the magnetization over time satisfies the power-law form. It is found that the value of the dynamic critical exponent tends to be a stable value in the form of a power-law function as the scale of the system increases. The dynamic critical exponent obtained is z≃2.15. When the modified dynamic Monte Carlo renormalization group method is applied to the two-dimensional Glauber model, the obtained dynamic critical exponent is z≃2.25.

Enriched Conformational Sampling of DNA and Proteins with a Hybrid Hamiltonian Derived from the Protein Data Bank.

In this article, we present a method for the enhanced molecular dynamics simulation of protein and DNA systems called potential of mean force (PMF)-enriched sampling. The method uses partitions derived from the potentials of mean force, which we determined from DNA and protein structures in the Protein Data Bank (PDB). We define a partition function from a set of PDB-derived PMFs, which efficiently compensates for the error introduced by the assumption of a homogeneous partition function from the PDB datasets. The bias based on the PDB-derived partitions is added in the form of a hybrid Hamiltonian using a renormalization method, which adds the PMF-enriched gradient to the system depending on a linear weighting factor and the underlying force field. We validated the method using simulations of dialanine, the folding of TrpCage, and the conformational sampling of the Dickerson–Drew DNA dodecamer. Our results show the potential for the PMF-enriched simulation technique to enrich the conformational space of biomolecules along their order parameters, while we also observe a considerable speed increase in the sampling by factors ranging from 13.1 to 82. The novel method can effectively be combined with enhanced sampling or coarse-graining methods to enrich conformational sampling with a partition derived from the PDB.

Mapping Topological to Conformal Field Theories through strange Correlators.

We extend the concept of strange correlators, defined for symmetry-protected phases in You etal. [Phys. Rev. Lett. 112, 247202 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.247202], to topological phases of matter by taking the inner product between string-net ground states and product states. The resulting two-dimensional partition functions are shown to be either critical or symmetry broken, since the corresponding transfer matrices inherit all matrix product operator symmetries of the string-net states. For the case of critical systems, these nonlocal matrix product operator symmetries are the lattice remnants of topological conformal defects in the field theory description. Following Aasen etal. [J. Phys. A 49, 354001 (2016)JPAMB51751-811310.1088/1751-8113/49/35/354001], we argue that the different conformal boundary conditions can be obtained by applying the strange correlator concept to the different topological sectors of the string net obtained from Ocneanu's tube algebra. This is demonstrated on the lattice by calculating the conformal field theory spectra in the different topological sectors for the Fibonacci (hard-hexagon) and Ising string net. Additionally, we provide a complementary perspective on symmetry-preserving real-space renormalization by showing how known tensor network renormalization methods can be understood as the approximate truncation of an exactly coarse-grained strange correlator.

The impact of viscosity contrast on the error of heterogeneity loss in upscaling of geological models

Multiphase flow simulation is widely used to evaluate different production plans in petroleum engineering. In order to solve the flow equation efficiently, upscaling is applied to transform a fine grid geological model into a coarser one. The coarse model result, however, may be inaccurate due to heterogeneity loss, connectivity distortion and numerical dispersion. Each source of upscaling error has its own influential factors. Knowledge about these factors and error behavior is useful in design of appropriate sizes for coarse grid blocks. Although numerous methods have been developed for upscaling, few works may be found on the related error itself.This paper studies how a major source of upscaling error, heterogeneity loss, changes with viscosity contrast in water-oil displacement. The fBm process is used to create very heterogeneous fine models that are upscaled by Renormalization method. For separating the effect of heterogeneity loss from numerical dispersion, refined upscaled models are derived. Water flooding is then simulated on fine and refined upscaled models at 3 different ratios of oil to water viscosity. For each ratio, a heterogeneity loss plot is prepared. This plot is a powerful tool for evaluating the performance of an upscaling method. The result shows the heterogeneity loss error heavily depends on viscosity contrast. Moreover, the limit of coarsening is determined for each case and it is discussed why that limit is lower when water has higher viscosity than oil.

Study on the phase transition of the fractal scale-free networks**Project supported by the Natural Science Foundation of Shandong Province, China (Grant No. ZR2014EL002).

Based on the Ising spin, the phase transition on fractal scale-free networks with tree-like skeletons is studied, where the loops are generated by local links. The degree distribution of the tree-like skeleton satisfies the power-law form . It is found that when , the renormalized scale-free network will have the same degree distribution as the original network. For a special case of δ = 4.5, a ferromagnetic to paramagnetic transition is found and the critical temperature is determined by the box-covering renormalization method. By keeping the structure of the fractal scale-free network constant, the numerical relationship between the critical temperature and the network size is found, which is the form of power law.

The effect of two distinct fast time scales in the rotating, stratified Boussinesq equations: variations from quasi-geostrophy

Inspired by the use of fast singular limits in time-parallel numerical methods for a single fast frequency, we consider the limiting, nonlinear dynamics for a system of partial differential equations when two fast, distinct time scales are present. First-order slow equations are derived via the method of multiple time scales when the two small parameters are related by a rational power. We find that the resultant system depends only on the relationship of the two fast time scales, i.e. which fast time is fastest? Using the theory of cancellation of fast oscillations, we show that with the appropriate assumptions on the nonlinear operator of the full system, this reduced slow system is exactly that which the solution will converge to if each asymptotic limit is considered sequentially. The same result is also obtained via the method of renormalization. The specific example of the rotating, stratified Boussinesq equations is explored in detail, indicating that the most common distinguished limit of this system—quasi-geostrophy, is not the only limiting asymptotic system.

A novel method for renormalization in quantum-field theory in curved spacetime

In quantum-field theory in curved spacetime, two important physical quantities are the expectation value of the stress-energy tensor [Formula: see text] and of the square of the field operator [Formula: see text]. These expectation values must be renormalized, which is usually performed via the so-called point-splitting prescription. However, the renormalization method that is usually implemented in the literature, in principle, only applies to static, spherically-symmetric spacetimes, and does not readily generalize to other types of spacetime. We present a novel implementation of the renormalization procedure which may be used in the future for more general spacetimes, such as Kerr black hole spacetime. As an example, we apply our method to the renormalization of [Formula: see text] for a massless scalar field in Bertotti–Robinson spacetime.

Multigrid renormalization

We combine the multigrid (MG) method with state-of-the-art concepts from the variational formulation of the numerical renormalization group. The resulting MG renormalization (MGR) method is a natural generalization of the MG method for solving partial differential equations. When the solution on a grid of N points is sought, our MGR method has a computational cost scaling as O(log⁡(N)), as opposed to O(N) for the best standard MG method. Therefore MGR can exponentially speed up standard MG computations. To illustrate our method, we develop a novel algorithm for the ground state computation of the nonlinear Schrödinger equation. Our algorithm acts variationally on tensor products and updates the tensors one after another by solving a local nonlinear optimization problem. We compare several different methods for the nonlinear tensor update and find that the Newton method is the most efficient as well as precise. The combination of MGR with our nonlinear ground state algorithm produces accurate results for the nonlinear Schrödinger equation on N=1018 grid points in three spatial dimensions.

A local and scalable lattice renormalization method for ballistic quantum computation

A recent proposal has shown that it is possible to perform linear-optics quantum computation using a ballistic generation of the lattice. Yet, due to the probabilistic generation of its cluster state, it is not possible to use the fault-tolerant Raussendorf lattice, which requires a lower failure rate during the entanglement-generation process. Previous work in this area showed proof-of-principle linear-optics quantum computation, while this paper presents an approach to it which is more practical, satisfying several key constraints. We develop a classical measurement scheme that purifies a large faulty lattice to a smaller lattice with entanglement faults below threshold. A single application of this method can reduce the entanglement error rate to 7% for an input failure rate of 25%. Thus, we can show that it is possible to achieve fault tolerance for ballistic methods.

Holographic calculation of two-point correlation functions of CFT stress tensor in the BMS gauge and flat-space holography

In this paper, we use the holographic renormalization method to calculate the two-point correlation functions of the CFT stress tensor. In the gravity side, we write the asymptotically AdS spacetimes in the BMS gauge and apply the standard holographic renormalization method for these space-times. The significance of using multi-point functions in this gauge is that its flat-space limit is well-defined. We discuss this point in the last section of the paper.

The renormalization method from continuous to discrete dynamical systems: asymptotic solutions, reductions and invariant manifolds

The renormalization method based on the Taylor expansion for asymptotic analysis of differential equations is generalized to difference equations. The proposed renormalization method is based on the Newton–Maclaurin expansion. Several basic theorems on the renormalization method are proven. Some interesting applications are given, including asymptotic solutions of quantum anharmonic oscillator and discrete boundary layer, the reductions and invariant manifolds of some discrete dynamics systems. Furthermore, the homotopy renormalization method based on the Newton–Maclaurin expansion is proposed and applied to those difference equations including no a small parameter. In addition, some subtle problems on the renormalization method are discussed.

Ballistic transport in aperiodic Labyrinth tiling proven through a new convolution theorem

In this article, we report a distinct convolution theorem developed for the Kubo-Greenwood formula in Labyrinth tiling by transforming the two-dimensional lattice into a set of independent chains with rescaled Hamiltonians. Such transformation leads to an analytical solution of the direct-current conductance spectra, where quantized steps with height of 2g0 are found in Labyrinth tiling with periodic order along the applied electric field direction, in contrast to the step height of g0 observed in the corresponding square lattices, being g0 the conductance quantum. When this convolution theorem is combined with the real-space renormalization method, we are able to address in non-perturbative way the electronic transport in macroscopic aperiodic Labyrinth tiling based on generalized Fibonacci chains. Furthermore, we analytically demonstrate the existence of ballistic transport states in aperiodic Labyrinth tiling. This finding suggests that the periodicity should not be a necessary condition for the single-electron ballistic transport even in multidimensional fully non-periodic lattices.

The analytic solutions of Schrödinger equation with Cubic-Quintic nonlinearities

Abstract This paper deals with the nonlinear Schrodinger equation with Cubic-Quintic nonlinearities. After transforming the non-autonomous nonlinear Schrodinger equation into a stationary equation by adopting the BPVK idea, we give the classification of the solutions to the Cubic-Quintic nonlinear Schrodinger equation by utilizing the complete discrimination system for polynomial method. As an auxiliary result, we use the homotopy renormalization method (HTR) to obtain the global solutions to the Ermakov-Pinney equation satisfied by the width function.

Energy Renormalization Method for the Coarse-Graining of Polymer Viscoelasticity.

Developing temperature transferable coarse-grained (CG) models is essential for the computational prediction of polymeric glass-forming (GF) material behavior, but their dynamics are often greatly altered from those of all-atom (AA) models mainly because of the reduced fluid configurational entropy under coarse-graining. To address this issue, we have recently introduced an energy renormalization (ER) strategy that corrects the activation free energy of the CG polymer model by renormalizing the cohesive interaction strength ε as a function of temperature T, i.e., ε(T), thus semiempirically preserving the T-dependent dynamics of the AA model. Here we apply our ER method to consider-in addition to T-dependency-the frequency f-dependent polymer viscoelasticity. Through smallamplitude oscillatory shear molecular dynamics simulations, we show that changing the imposed oscillation f on the CG systems requires changes in ε values (i.e., ε(T, f)) to reproduce the AA viscoelasticity. By accounting for the dynamic fragility of polymers as a material parameter, we are able to predict ε(T, f) under coarse-graining in order to capture the AA viscoelasticity, and consequently the activation energy, across a wide range of T and f in the GF regime. Specifically, we showcase our achievements on two representative polymers of distinct fragilities, polybutadiene (PB) and polystyrene (PS), and show that our CG models are able to sample viscoelasticity up to the megahertz regime, which approaches state-of-the-art experimental resolutions, and capture results sampled via AA simulations and prior experiments.

Resonant thermoelectric transport in atomic chains with Fano defects

Abstract

Computing eigenfunctions and eigenvalues of boundary-value problems with the orthogonal spectral renormalization method

The spectral renormalization method was introduced in 2005 as an effective way to compute ground states of nonlinear Schr\"odinger and Gross-Pitaevskii type equations. In this paper, we introduce an orthogonal spectral renormalization (OSR) method to compute ground and excited states (and their respective eigenvalues) of linear and nonlinear eigenvalue problems. The implementation of the algorithm follows four simple steps: (i) reformulate the underlying eigenvalue problem as a fixed point equation, (ii) introduce a renormalization factor that controls the convergence properties of the iteration, (iii) perform a Gram-Schmidt orthogonalization process in order to prevent the iteration from converging to an unwanted mode; and (iv) compute the solution sought using a fixed-point iteration. The advantages of the OSR scheme over other known methods (such as Newton's and self-consistency) are: (i) it allows the flexibility to choose large varieties of initial guesses without diverging, (ii) easy to implement especially at higher dimensions and (iii) it can easily handle problems with complex and random potentials. The OSR method is implemented on benchmark Hermitian linear and nonlinear eigenvalue problems as well as linear and nonlinear non-Hermitian $\mathcal{PT}$-symmetric models.

Exact solutions and asymptotic solutions of one-dimensional domain walls in nonlinearly coupled system

We use homotopy renormalization method to investigate one-dimensional radial domain wall modes in the system of two nonlinear Schrodinger or Gross–Pitaevskii equations, which are coupled by linear mixing and by nonlinear XPM (cross-phase-modulation). The asymptotic solutions satisfying the boundary conditions are given by taking the inhomogeneous variable coefficient homotopy equations. To our knowledge, this is the first time that the inhomogeneous variable coefficient homotopy equations are taken. Moreover, the comparisons with numerical results show that our explicit function solutions have good performances in both local and large scale. In particular, in some conditions with special parameters, the asymptotic solutions are actually the exact solutions.

Correlation matrix renormalization theory for correlated-electron materials with application to the crystalline phases of atomic hydrogen

Developing accurate and computationally efficient methods to calculate the electronic structure and total energy of correlated-electron materials has been a very challenging task in condensed matter physics and materials science. Recently, we have developed a correlation matrix renormalization (CMR) method which does not assume any empirical Coulomb interaction $U$ parameters and does not have double counting problems in the ground-state total energy calculation. The CMR method has been demonstrated to be accurate in describing both the bonding and bond breaking behaviors of molecules. In this study, we extend the CMR method to the treatment of electron correlations in periodic solid systems. Using a linear hydrogen chain as a benchmark system, we show that the results from the CMR method compare very well with those obtained recently by accurate quantum Monte Carlo (QMC) calculations. We also study the equation of states of three-dimensional crystalline phases of atomic hydrogen. We show that the results from the CMR method agree much better with the available QMC data in comparison with those from density functional theory and Hartree-Fock calculations.