Dynamical Renormalization Group Research Articles

Dynamical mean-field theory and numerical renormalization group study of superconductivity in the attractive Hubbard model

We present a study of the attractive Hubbard model based on the dynamical mean-field theory (DMFT) combined with the numerical renormalization group (NRG). For this study, the NRG method is extended to deal with self-consistent solutions of effective impurity models with superconducting symmetry breaking. We give details of this extension and validate our calculations with DMFT results with antiferromagnetic ordering. We also present results for static and integrated quantities for different filling factors in the crossover from weak to strong-coupling superfluidity. We study the evolution of the single-particle spectra throughout the crossover regime. Although the DMFT does not include the interaction of the fermions with the Goldstone mode, we find strong deviations from the mean-field theory in the intermediate and strong-coupling regimes. In particular, we show that low-energy charge fluctuations induce a transfer of spectral weight from the Bogoliubov quasiparticles to a higher-energy incoherent hump.

Analytic response theory for the density matrix renormalization group

We propose an analytic response theory for the density matrix renormalization group, whereby response properties correspond to analytic derivatives of density matrix renormalization group observables with respect to the applied perturbations. Both static and frequency-dependent response theories are formulated and implemented. We evaluate our pilot implementation by calculating static and frequency-dependent polarizabilities of short oligodiacetylenes. The analytic response theory is competitive with dynamical density matrix renormalization group methods and yields significantly improved accuracies when using a small number of density matrix renormalization group states. Strengths and weaknesses of the analytic approach are discussed.

Propagation and localization of acoustic and elastic waves in heterogeneous materials: renormalization group analysis and numerical simulations

We describe and discuss the recent progress in the study of propagation and localization of acoustic and elastic waves in heterogeneous media. The heterogeneity is represented by a spatial distribution of the local elastic moduli. Both randomly distributed elastic moduli as well as those with long-range correlations with a nondecaying power-law correlation function, are considered. The motivation for the study is twofold. One is that recent analysis of experimental data for the spatial distribution of the elastic moduli of rock indicated that the distribution is characterized by the type of long-range correlations that we consider in this study. The second motivation for the problem is to understand whether localization of electrons (which, in quantum mechanics, are described by wave functions) has any analogy in the propagation of classical waves in disordered media. The problem is studied by two approaches. One of them is based on developing a dynamic renormalization group (RG) approach to analytical analysis of the governing equations for wave propagation. The RG analysis indicates that, depending on the type of the disorder (correlated vs. uncorrelated), one may have a transition between localized and extended regimes in any spatial dimension. The second approach utilizes numerical simulations of the governing equations in two- and three-dimensional media. The results obtained by the two approaches are in agreement with each other. Using numerical simulations, we also describe how the characteristics of a propagating wave may be used for probing the differences between heterogeneous media with short- and long-range correlations. To do so, we study the evolution of several distinct characteristics of the waves, such as the amplitude of the coherent wave front, its width, the spectral densities, the scalogram (wavelet transformation of the waves’ amplitudes at different scales and times), and the dispersion relation. It is demonstrated that such properties have completely different characteristics in uncorrelated and correlated media. Finally, it is shown how wave propagation may be used for establishing a link between the static and dynamical properties of heterogeneous media.

Thermodynamic properties of triethylamine + water liquid mixture from shear viscosity measurements

Densities ρ and viscosities η, for the binary mixture triethylamine–water have been measured over the entire range of mole fractions at different temperatures, both near and far away from the critical temperature T c (0.076 K ≤ ( T − T c) ≤ 8.276 K). By calculating the corresponding activation parameters Δ H* and Δ S*, it has been shown that a possible connection may be presented between the evolution of these parameters and the critical region. Furthermore, in the one-phase region just below T c of the critical (TEA–W) binary fluid, the viscosity measurements yield an enhancement as expected for Ising criticality with a crossover to regular behaviour. The regular background is obtained by interpolation from the measurements at non-critical concentrations. Shear viscosity data are consistent with a power law divergence η = η 0( Qξ 0) z t − y F predicted by the mode-coupling and the dynamic renormalization group theories. Different equations are used in order to determine the critical exponent y, the obtained values are in good agreement with the renormalization group-predicted value. The best fit gives y = νz = 0.0391 ± 0.0002 (i.e., z = 0.0679). Moreover, the resulting value for the wave number Q = 1.0152 ± 0.0016 nm −1 is in the region found for many other systems.

Dynamical renormalization group methods in theory of eternal inflation

Dynamics of eternal inflation on the landscape admits a description in terms of the Martin-Siggia-Rose (MSR) effective field theory, that is, in one-to-one correspondence with vacuum dynamics equations. On those sectors of the landscape, where transport properties of the probability measure for eternal inflation are important, renormalization group fixed points of the MSR effective action determine late-time behavior of the probability measure. I argue that these RG fixed points may be relevant for the solution of the gauge invariance problem for eternal inflation.

Scaling approach to the conservation-law growth equations in anomalous surface roughening

We employ an analytical approach introduced by López to determine the anomalous scaling exponent of the growth equation with a generalized conservation law in both the weak-and strong-coupling regimes, which included the Kardar-Parisi-Zhang（KPZ）, Sun-Guo-Grant（SGG）, and molecular beam epitaxy （MBE）equations as special cases and allows for a unified treatment of growth equations. Analysis shows that KPZ equation and SGG equation exhibit normal Family-Vicsek scaling behavior, whether in the weak-coupling or strong-coupling regime. Differently, MBE equation exists intrinsic anomalous scaling in the strong-coupling regime and normal Family-Vicsek scaling behavior in weak coupling regime. All the conclusions obtain here are well consistent with the corresponding results derived by the dynamic renormalization group theory, numerical simulation and experiment.

Geometric Principles of Surface Growth

I introduce a new equation describing epitaxial growth processes. This equation is derived from a simple variational geometric principle and it has a straightforward interpretation in terms of continuum and microscopic physics. It is also able to reproduce the critical behavior already observed, mound formation and mass conservation, but however does not fit a divergence form as the most commonly spread models of conserved surface growth. This formulation allows me to connect the results of the dynamic renormalization group analysis with intuitive geometric principles, whose generic character may well allow them to describe surface growth and other phenomena in different areas of physics.

Effect of electron–phonon interaction on optical response in one-dimensional cuprates

We examine the single-particle excitation and linear optical absorption spectra in the one-dimensional (1D) extended Hubbard–Holstein model. We perform dynamical density matrix renormalization group calculations with use of pseudo-site representation of phonons. We focus on the interplay among phonons and elementary excitations in 1D Mott insulators. The excitations in the Mott insulators are easily modified by the phonons. We discuss implications of the present results in light of spectroscopic measurements in 1D cuprates.

Dynamic renormalization-group analysis of the d+1 dimensional Kuramoto-Sivashinsky equation with both conservative and nonconservative noises

The long-wavelength properties of the (d + 1)-dimensional Kuramoto-Sivashinsky (KS) equation with both conservative and nonconservative noises are investigated by use of the dynamic renormalization-group (DRG) theory. The dynamic exponent z and roughness exponent α are calculated for substrate dimensions d = 1 and d = 2, respectively. In the case of d = 1, we arrive at the critical exponents z = 1.5 and α = 0.5 , which are consistent with the results obtained by Ueno et al. in the discussion of the same noisy KS equation in 1+1 dimensions [Phys. Rev. E 71, 046138 (2005)] and are believed to be identical with the dynamic scaling of the Kardar-Parisi-Zhang (KPZ) in 1+1 dimensions. In the case of d = 2, we find a fixed point with the dynamic exponents z = 2.866 and α = -0.866 , which show that, as in the 1 + 1 dimensions situation, the existence of the conservative noise in 2 + 1 or higher dimensional KS equation can also lead to new fixed points with different dynamic scaling exponents. In addition, since a higher order approximation is adopted, our calculations in this paper have improved the results obtained previously by Cuerno and Lauritsen [Phys. Rev. E 52, 4853 (1995)] in the DRG analysis of the noisy KS equation, where the conservative noise is not taken into account.

Disorder on the landscape

Disorder on the string theory landscape may significantly affect dynamics of eternalinflation leading to the possibility for some vacua on the landscape to become dynamicallypreferable over others. We systematically study effects of a generic disorder on thelandscape, starting by identifying a sector with built-in disorder—a set of de Sitter vacuacorresponding to compactifications of the type IIB string theory on Calabi–Yau manifoldswith a number of warped Klebanov–Strassler throats attached randomly to the bulk part ofthe Calabi–Yau. Further, we derive a continuum limit of the vacuum dynamics equationson the landscape. Using methods of the dynamical renormalization group we determine thelate-time behavior of the probability distribution for an observer to measure agiven value of the cosmological constant. We find the diffusion of the probabilitydistribution to significantly slow down in sectors of the landscape where the number ofnearest-neighboring vacua for any given vacuum is small. We discuss the relationof this slowdown with the phenomenon of Anderson localization in disorderedmedia.

Excitations in two-component Bose gases

In this paper, we study a strongly correlated quantum system that has become amenable to experiment by the advent of ultracold bosonic atoms in optical lattices, a chain of two different bosonic constituents. Excitations in this system are first considered within the framework of bosonization and the Luttinger liquid theory which are applicable if the Luttinger liquid parameters are determined numerically. The occurrence of a bosonic counterpart of fermionic spin–charge separation is signalled by a characteristic two-peak structure in the spectral functions found by dynamical density-matrix renormalization group (DMRG) in good agreement with analytical predictions. Experimentally, single-particle excitations as probed by spectral functions are currently not accessible in cold atoms. Therefore we consider the modifications needed for current experiments, namely the investigation of the real-time evolution of density perturbations instead of single-particle excitations, a slight inequivalence between the two intraspecies interactions in actual experiments, and the presence of a confining trap potential. Using time-dependent DMRG, we show that only quantitative modifications occur. With an eye to the simulation of strongly correlated quantum systems far from equilibrium, we detect a strong dependence of the time-evolution of entanglement entropy on the initial perturbation.

Charge and spin excitation spectra in the one-dimensional Hubbard model with next-nearest-neighbor hopping

The dynamical density-matrix renormalization group technique is used to calculate spin and charge excitation spectra in the one-dimensional (1D) Hubbard model at quarter filling with nearest-neighbor $t$ and next-nearest-neighbor $t'$ hopping integrals. We consider a case where $t$ ($>0$) is much smaller than $t'$ ($>0$). We find that the spin and charge excitation spectra come from the two nearly independent $t'$-chains and are basically the same as those of the 1D Hubbard (and t-J) chain at quarter filling. However, we find that the hopping integral $t$ plays a crucial role in the short-range spin and charge correlations; i.e., the ferromagnetic spin correlations between electrons on the neighboring sites is enhanced and simultaneously the spin-triplet pairing correlations is induced, of which the consequences are clearly seen in the calculated spin and charge excitation spectra at low energies.

Dynamic renormalization group analysis of propagation of elastic waves in two-dimensional heterogeneous media

We study localization of elastic waves in two-dimensional heterogeneous solids with randomly distributed Lam\'e coefficients, as well as those with long-range correlations with a power-law correlation function. The Matin-Siggia-Rose method is used, and the one-loop renormalization group (RG) equations for the the coupling constants are derived in the limit of long wavelengths. The various phases of the coupling constants space, which depend on the value $\rho$, the exponent that characterizes the power-law correlation function, are determined and described. Qualitatively different behaviors emerge for $\rho<1$ and $\rho>1$. The Gaussian fixed point (FP) is stable (unstable) for $\rho<1$ ($\rho>1$). For $\rho<1$ there is a region of the coupling constants space in which the RG flows are toward the Gaussian FP, implying that the disorder is irrelevant and the waves are delocalized. In the rest of the disorder space the elastic waves are localized. We compare the results with those obtained previously for acoustic wave propagation in the same type of heterogeneous media, and describe the similarities and differences between the two phenomena.

Effect of Electron-Phonon Interaction on Optical Properties in One-Dimensional Mott Insulators

Interplay of strong electron correlation and electron-phonon interaction is one of important issues in the physics of low dimensional correlated electron systems. In order to understand the interplay, we examine the optical properties of the one-dimensional (1D) Hubbard-Holstein model at half-filling, by employing the dynamical density matrix renormalization group (DMRG) technique. We discuss the effect of the electron-phonon interaction on the single-particle spectral function and the optical absorption spectrum. The dynamical DMRG data are compared with experimental data of 1D copper oxide Mott insulators.

Thermal Conductivity and Critical Boundary Resistance of Helium Near the Lambda Point

We report measurements of the temperature dependence of the thermal conductivity of helium in the region just above the lambda point and the critical boundary resistance on both sides of the transition. The data were obtained using a cell with a gap that was varied from 0.12 to 0.73 mm. The range of the data was from about 2 mK below to 8 mK above the transition, with a maximum resolution of about 5×10−9 K, well within the gravity rounded region. We find good agreement with the expected behavior of the thermal conductivity based on the dynamic renormalization group theory. The boundary resistance results above the transition are in fair agreement with the theoretical prediction of Frank and Dohm. Below the transition the wide range results are in approximate agreement with additional predictions and other measurements at higher powers. Closer to the transition on the low temperature side phenomena are observed which appear to be apparatus-dependent.

DYNAMICS IN ONE-DIMENSIONAL SPIN SYSTEMS – DENSITY MATRIX RENORMALIZATION GROUP STUDY

We study the one-dimensional S = 1/2 Heisenberg model with an uniform and a staggered magnetic fields, using the dynamical density-matrix renormalization group (DDMRG) technique. The DDMRG enables us to investigate the dynamical properties of chain with lengths up to a few hundreds, and the results are numerically exact in the same sense as 'exact diagonalization' results are. Thus, we can analyze the low-energy spectrum almost in the thermodynamic limit. In this work, we calculate the dynamical spin structure factor and demonstrate the performance of the DDMRG method applying the open-end boundary conditions as well as the periodic boundary conditions.

Edge singularities in high-energy spectra of gapped one-dimensional magnets in strong magnetic fields

We use the dynamical density matrix renormalization group technique to show that the high-energy part of the spectrum of an S=1 Heisenberg chain, placed in a strong external magnetic field H exceeding the Haldane gap Delta, contains edge singularities, similar to those known to exist in the low-energy spectral response. It is demonstrated that in the frequency range omega greater than or similar to Delta the longitudinal (with respect to the applied field) dynamical structure factor is dominated by the power-law singularity S-parallel to(q=pi,omega)proportional to(omega-omega(0))(-alpha'). We study the behavior of the high-energy edge exponent alpha(') and the edge omega(0) as functions of the magnetic field. The existence of edge singularities at high energies is directly related to the Tomonaga-Luttinger liquid character of the ground state at H>Delta and is expected to be a general feature of one-dimensional gapped spin systems in high magnetic fields.

Spectral Properties of One-Dimensional Diffusive Systems Subject to Stochastic Forcing

Abstract The vertical wavenumber and frequency spectra of horizontal wind and temperature in stochastically driven systems with diffusion, either due to uniform background eddy and molecular transport, or due to adjustment processes associated with shear or convective instability, are studied. Because of the dominating role of vertical transport in a stratified fluid, one-dimensional Langevin-type equations could be ascribed to such systems in the vertical direction. The linear equation with uniform diffusion is solved explicitly, and the spectra follow power-law distributions if the stochastic force is Gaussian. The nonlinear equations with gradient (either shear or lapse rate) dependent diffusion coefficients are shown to support scale invariance, and the power-law indices of the spectra are determined from dynamic renormalization group (DRG) analysis under rather general conditions. The exact power-law indices vary with the spectrum of the stochastic force and the nonlinearity of the systems. If the wavenumber spectrum of the force is moderately red (between k0 and k−2), the spectral indices of horizontal wind and temperature and the range of their variability are in general agreement with those inferred from wind and temperature measurements. The indices in both linear and nonlinear cases are confirmed by numerical simulations. This theory may suggest an alternative explanation to the universal vertical wavenumber and frequency spectra and their variability. By relating the universal spectra to systems characterized by stochastic forcing and background diffusion or diffusive adjustment due to shear or convective instability, which are ubiquitous in a stratified fluid, the difficulty to associate the time- and location-independent spectral features directly with the highly time- and location-dependent gravity waves or wave-breaking events is avoided. If such systems are suggestive of the real atmosphere, there is a need to be cautious in making assumptions regarding gravity waves solely based on the universal spectra when analyzing and interpreting wind and temperature observations.

Electron-phonon coupling and spin-charge separation in one-dimensional Mott insulators

We examine the single-particle excitation spectrum in the one-dimensional Hubbard-Holstein model at half-filling by performing the dynamical density matrix renormalization group (DDMRG) calculation. The DDMRG results are interpreted as superposition of spectra for a spinless carrier dressed with phonons. The superposition is a consequence of robustness of the spin-charge separation against electron-phonon coupling. The separation is in contrast to the coupling between phonon and spin degrees of freedom in two-dimensional systems. We discuss implication of the results of the recent angle-resolved photoemission spectroscopy measurements on SrCuO${}_{2}$.

Direct Photoexcitation of Appreciable Size of Domains without Lattice Motion in Neutral-Ionic Transition Systems

In a recent ultrafast pump-probe experiment on a typical neutral-(N-)ionic (I) transition material, tetrathiafulvalene-p-chloranil [S. Iwai, Phys. Rev. Lett. 96, 057403 (2006)10.1103/PhysRevLett.96.057403], it was claimed that an I-phase domain in the N-phase background was very quickly formed even before the lattice started to move. This study proves this idea in terms of an extended Hubbard model that has site-potential alternation and has been frequently used so far. Our focus is on the photoexcited states of the N phase and the optical conductivity spectrum. As a result of accurate spectrum calculation based on a dynamical density-matrix renormalization group technique, we clearly observe photoexcited ionic domains accompanied by no lattice dimerization. Their contribution almost dominates the spectrum near the N-I phase boundary, and we relate this property to the long-tailed spectrum observed in experiments.