Limit Of Anisotropy Research Articles

Extended spectral scaling laws for shear-Alfvén wave turbulence

In this paper, resonant interactions among three shear-Alfvén waves is investigated in weak incompressible magnetohydrodynamics (MHD) turbulence, neglecting interactions involving the pseudo-Alfvén waves. It is found that these interactions lead to a wave kinetic equation that has the same compact form as the full nonlinear MHD system in the highly anisotropic limit (k⊥⪢k‖). These results have important implications for interpreting the effects of resonant three-wave interactions among shear-Alfvén waves throughout k space. In particular, it shows that resonant three-wave interactions involving only shear-Alfvén waves attempt to set up a k⊥nsf(k‖) 3D shear-Alfvén-wave spectrum (with −4<ns<−2, and f an arbitrary function depending on the initial conditions). The results also provide a framework for estimating the time scale on which the three shear-Alfvén-wave interactions transfer energy to larger k⊥ everywhere in the k⊥-k‖ plane. The compatibility of the results with previous predictions and the possible astrophysical implications of these extended spectral solutions for anisotropic plasmas is discussed.

Anisotropic pyrochlores and the global phase diagram of the checkerboard antiferromagnet

We study the phase diagram of two models of spin-1/2 antiferromagnets composed of corner-sharing tetrahedra, the basis of the pyrochlore structure. Primarily, we focus on the Heisenberg antiferromaget on the checkerboard lattice (also called the planar pyrochlore and crossed-chains model). This model has an anisotropic limit, when the dimensionless ratio of two exchange constants, J_\times/J << 1, in which it consists of one-dimensional spin chains coupled weakly together in a frustrated fashion. Using recently developed techniques combining renormalization group ideas and one-dimensional bosonization and current algebra methods, we show that in this limit the model enters a crossed dimer state with two-fold spontaneous symmetry breaking but no magnetic order. We complement this result by an approximate ``quadrumer triplet boson'' calculation, which qualitatively captures the physics of the ``plaquette valence bond solid'' state believed to obtain for J_\times/J = 1. Using these known points in parameter space, the instabilities pointed to by the quadrumer boson calculation, and the simple limit J_\times/J >> 1, we construct a few candidate global phase diagrams for the model, and discuss the nature of the quantum phase transitions contained therein. Finally, we apply our quasi-one-dimensional techniques to an anisotropic limit of the three-dimensional pyrochlore antiferromagnet, an approximate model for magnetism in GeCu2O4. A crossed dimer state is predicted here as well.

A note on the extensional viscosity of elastic liquids under strong flows

In this article a parametric study based on a balance between viscous drag and restoring Brownian forces is used in order to construct a nonlinear dumbbell model with a finite spring and a drag correction for a dilute polymer solution. The constitutive equations used are reasonable approximation for describing flows of very dilute polymer solutions such as those used in turbulent drag reduction. We investigate the response of an elastic liquid under extensional flows in order to explore the roles of a stress anisotropy and of elasticity in strong flows. It is found that for low Reynolds numbers, the extensional viscosity of a dilute polymer solution is governed by two parameters: a Deborah number representing the importance of the elasticity on the flow and the macromolecule extensibility that accounts for the viscous anisotropic effects caused by the macromolecule orientation. Two different asymptotic regimes are described.The first corresponds to an elastic limit in which the extensional viscosity is a function of the Deborah number and the particle volume fraction. The second is an anisotropic regime with the extensional viscosity independent of Deborah number but strongly dependent on macromolecule aspect ratio. The analysis may explain from a phenomenological point of view why few ppms of macromolecules of high molecule weight or a small volume fraction of long fibres produce important attenuation of the pressure drop in turbulent flows. On the basis of our analysis it is seen that the anisotropic limit of the extensional viscosity caused by extended polymers under strong flows should play a key role in the attenuation of flow instability and in the mechanism of drag reduction by polymer additives.

The Hamiltonian limit of (3+1)D SU(3) lattice gauge theory

The extreme anisotropic limit of Euclidean SU(3) lattice gauge theory is examined to extract the Hamiltonian limit, using standard path integral Monte Carlo (PIMC) methods. We examine the mean plaquette and string tension and compare them to results obtained within the Hamiltonian framework of Kogut and Susskind. The results are a significant improvement upon previous Hamiltonian estimates, despite the extrapolation procedure necessary to extract observables. We conclude that the PIMC method is a reliable method of obtaining results for the Hamiltonian version of the theory. Our results also clearly demonstrate the universality between the Hamiltonian and Euclidean formulations of lattice gauge theory.

Anisotropic limit of the bond-percolation model and conformal invariance in curved geometries.

We investigate the anisotropic limit of the bond-percolation model in d dimensions, which is equivalent to a (d-1) -dimensional quantum q-->1 Potts model. We formulate an efficient Monte Carlo method for this model. Its application shows that the anisotropic model fits well with the percolation universality class in d dimensions. For three-dimensional rectangular geometry, we determine the critical point as t(c) =8.6429(4), and determine the length ratio as alpha(0) =1.5844(3), which relates the anisotropic limit of the percolation model and its isotropic version. On this basis, we simulate critical systems in several curved geometries including a spheroid and a spherocylinder. Using finite-size scaling and the assumption of conformal invariance, we determine the bulk and surface magnetic exponents in two and three dimensions. They are in good agreement with the existing results.

Hamiltonian limit of (3+1)-dimensional SU(3) lattice gauge theory on anisotropic lattices

The extreme anisotropic limit of Euclidean SU(3) lattice gauge theory is examined to extract the Hamiltonian limit, using standard path integral Monte Carlo (PIMC) methods. We examine the mean plaquette and string tension and compare them to results obtained within the Hamiltonian framework of Kogut and Susskind. The results are a significant improvement upon previous Hamiltonian estimates, despite the extrapolation procedure necessary to extract observables. We conclude that the PIMC method is a reliable method of obtaining results for the Hamiltonian version of the theory. Our results also clearly demonstrate the universality between the Hamiltonian and Euclidean formulations of lattice gauge theory. It is particularly important to take into account the renormalization of both the anisotropy, and the Euclidean coupling $ \beta_E $, in obtaining these results.

Driven depinning of strongly disordered media and anisotropic mean-field limits.

Extended systems driven through strong disorder are modeled generically using coarse-grained degrees of freedom that interact elastically in the directions parallel to the drive and slip along at least one of the directions transverse to the motion. In the limit of infinite-range elastic and viscous coupling this model has a tricritical point separating a region where the depinning is continuous, in the universality class of elastic depinning, from a region where depinning is hysteretic. Many of the collective transport models discussed in the literature are special cases of the generic model.

Path integral Monte Carlo approach to the U(1) lattice gauge theory in 2+1 dimensions

Path Integral Monte Carlo simulations have been performed for U(1) lattice gauge theory in (2+1) dimensions on anisotropic lattices. We extractthe static quark potential, the string tension and the low-lying "glueball" spectrum.The Euclidean string tension and mass gap decrease exponentially at weakcoupling in excellent agreement with the predictions of Polyakov and G{\" o}pfert and Mack, but their magnitudes are five times bigger than predicted. Extrapolations are made to the extreme anisotropic or Hamiltonian limit, and comparisons are made with previous estimates obtained in the Hamiltonian formulation.

Undrained Stability of Braced Excavations in Clay

Short-term undrained stability often controls the design of braced excavations in soft clays. This paper summarizes the formulation of numerical limit analyses that compute rigorous upper and lower bounds on the exact stability number and include anisotropic yielding, typical of K0-consolidated clays and bending failure of the wall. Calculations for braced cuts bound the actual failure conditions within ±5%, and highlight limitations of existing basal stability equations. The analyses clarify how wall embedment and bending capacity improve the stability of well braced excavations. Careful selection of mobilized strengths at shear strains in the range 0.6–1.0% are necessary to match the predictions of anisotropic limit analyses with nonlinear finite-element predictions of failure for the embedded walls. Two example applications from recent projects in Boston highlight the practicality of the numerical limit analyses for modeling realistic soil profiles and lateral earth support systems, but also focus atte...

Bulk and surface critical behavior of the three-dimensional Ising model and conformal invariance

Using a continuous cluster Monte Carlo algorithm, we investigate the critical three-dimensional Ising model in its anisotropic limit. From the ratio of the magnetic correlations in the strong- and the weak-coupling directions, we determine the length ratio relating the isotropic Ising model and the anisotropic limit. On this basis, we simulate the critical Ising model on a spherocylinder S2 x R1, i.e., a curved geometry obtained from a conformal mapping of the infinite space R3. From correlation lengths along the spherocylinder, combined with the prediction of conformal invariance, we estimate the magnetic and thermal scaling dimensions as X(h)=0.5182(6) and X(t)=1.419(7), respectively. The behavior of the Binder cumulant is also determined in the limit of an infinitely long spherocylinder. Next, free boundary conditions are imposed on the equators of the spherocylinder, and thus the geometry S1 x S+ x R1 is obtained. The surface magnetic scaling dimension is estimated as X(s)(h)=1.263(5). The consistency of the aforementioned estimations and existing results confirms that the three-dimensional Ising model is conformally invariant. Further, the precision of these results reveals that, as in two dimensions, conformal mappings provide a powerful tool to investigate critical phenomena. With the continuous cluster algorithm, we also perform simulations of systems inside a conventional solid cylinder. The surface magnetic correlation length differs, within the estimated error margin, by a factor pi/2 from that along a half spherocylinder S1 x S+ x R1 with the same radius.

Cluster Monte Carlo simulation of the transverse Ising model.

We formulate a cluster Monte Carlo method for the anisotropic limit of Ising models on (d+1)-dimensional lattices, which in effect, are equivalent with d-dimensional quantum transverse Ising models. Using this technique, we investigate the transverse Ising models on the square, triangular, Kagome, honeycomb, and simple-cubic lattices. The Monte Carlo data are analyzed by finite-size scaling. In each case we find, as expected, that the critical behavior fits well in the (d+1)-dimensional Ising universality class. For the transverse Ising model on the square lattice, we determine the Binder cumulant of the classical counterpart for a range of aspect ratios between the system sizes in the third or "classical" direction and that in the other two directions. Matching this universal function with the case of the isotropic Ising model yields the length ratio relating the isotropic Ising model with the anisotropic limit. The efficiency of the present algorithm is reflected by the precision of its results, which improves significantly on earlier analyses.

Conformal invariance and simulations in curved geometries

Conformal mappings serve as useful tools for the determination of universal properties of critical models. Typical applications are subject to a major restriction, namely that the pertinent conformal mapping should lead to a geometry that can be investigated by means of numerical methods such as Monte Carlo simulations. Since conformal mappings of 3-D systems usually lead to curved geometries which are difficult to investigate numerically, most applications have thus far been restricted to 2-D systems. We present a solution of this problem for discrete spin models, by taking the anisotropic or Hamiltonian limit which renders one of the lattice directions continuous, such that the dimensionality in effect reduces by one. Applications to the 3-D Ising and percolation models confirm the predictions obtained from the assumption of conformal invariance, and lead to accurate numerical results for the scaling dimensions.

Phase transition of KCl under shock compression

It had been reported that for potassium chloride (KCl) the B1–B2 phasetransition (PT) occurs under shock and static compressions, but the measuredtransition points showed large scatter. In this study, Hugoniot measurementexperiments were performed on KCl single crystals by the inclined-mirror methodcombined with use of a powder gun. The anisotropic Hugoniot elastic limits andPT points were observed. The PT points along the ⟨100⟩, ⟨110⟩ and ⟨111⟩axis directions were determined as 2.5, 2.2 and 2.1 GPa, respectively.The anisotropic transition was reasonably explained in terms ofthe displacement mechanism along the ⟨111⟩ axisdirection.

Conformal Invariance of the Ising Model in Three Dimensions

We investigate a critical Ising-like model in the curved geometry S2 x R1 obtained by a conformal mapping of the infinite 3D space R3. The incompatibility of regular lattices with this geometry is avoided by use of the anisotropic limit of the lattice Ising model, which renders one of the space coordinates continuous. We determine magnetic and energylike correlation lengths of this model by means of a cluster Monte Carlo algorithm. From these data, and the assumption of conformal invariance, we obtain the magnetic and temperature scaling dimensions as X(h) = 0.5178(12) and X(t) = 1.423(19), respectively. These numbers are in a good agreement with the existing results for the 3D Ising universality class.

Capacitance of a quantum dot from the channel-anisotropic two-channel Kondo model

We investigate the charge fluctuations of a large quantum dot coupled to a two-dimensional electron gas via a quantum point contact following the work of Matveev. We limit our discussion to the case where exactly two channels enter the dot and we discuss the role of an anisotropy between the transmission coefficients (for these two channels) at the constriction. Experimentally, a channel-anisotropy can be introduced applying a relatively weak in-plane magnetic field to the system when only one ``orbital'' channel is open. The magnetic field leads to different transmission amplitudes for spin-up and spin-down electrons. In a strong magnetic field the anisotropic two-channel limit corresponds to two (spin-polarized) orbital channels entering the dot. The physics of the charge fluctuations can be captured using a mapping on the channel-anisotropic two-channel Kondo model. For the case of weak reflection at the point contact this has already briefly been stressed by one of us in PRB {\bf 64}, 161302R (2001). This mapping is also appropriate to discuss the conductance behavior of a two-contact set-up in strong magnetic field. Here, we elaborate on this approach and also discuss an alternative solution using a mapping on a channel-isotropic Kondo model. In addition we consider the limit of weak transmission. We show that the Coulomb-staircase behavior of the charge in the dot as a function of the gate voltage, is already smeared out by a small channel-anisotropy both in the weak- and strong transmission limits.

Geometric frustration and magnetization plateaus in quantum spin and Bose-Hubbard models on tubes

We study XXZ Heisenberg models on frustrated triangular lattices wrapped around a cylinder. In addition to having interesting magnetic phases, these models are equivalent to Bose-Hubbard models that describe the physical problem of adsorption of noble gases on the surface of carbon nanotubes. We find analytical results for the possible magnetization plateau values as a function of the wrapping vectors of the cylinder, which in general introduce extra geometric frustration besides the one due to the underlying triangular lattice. We show that for particular wrapping vectors $(N,0)$, which correspond to the zig-zag nanotubes, there is a macroscopically degenerate ground state in the classical Ising limit. The Hilbert space for the degenerate states can be enumerated by a mapping first into a path in a square lattice wrapped around a cylinder (a Bratteli diagram), and then to free fermions interacting with a single ${\bf Z}_N$ degree of freedom. From this model we obtain the spectrum in the anisotropic Heisenberg limit, showing that it is gapless. The continuum limit is a $c=1$ conformal field theory with compactification radius $R=N$ set by the physical tube radius. We show that the compactification radius quantization is exact in the projective $J_\perp/J_z \ll 1$ limit, and that higher order corrections reduce the value of $R$. The particular case of a $(N=2,0)$ tube, which corresponds to a 2-leg ladder with cross links, is studied separately and shown to be gapped because the fermion mapped problem contains superconducting pairing terms.

General expression for the angular dependence of the critical field in layered superconductors

Starting from the Ginzburg-Landau equation we derive an analytical expression for the angular and temperature dependence of the critical field in layered superconducting materials. This formula is shown to reproduce as the limiting cases the well known results of Tinkham for the isolated film of finite thickness and of Lawrence-Doniach for the stacked array of films of zero thickness with its particular anisotropic Ginzburg-Landau limit. Since this general formula is very simple, we suggest that it should be used in all cases where the dimensional crossover of the critical field is studied.

Random-bond Ising model in two dimensions: The Nishimori line and supersymmetry

We consider a classical random-bond Ising model (RBIM) with binary distribution of $\pm K$ bonds on the square lattice at finite temperature. In the phase diagram of this model there is the so-called Nishimori line which intersects the phase boundary at a multicritical point. It is known that the correlation functions obey many exact identities on this line. We use a supersymmetry method to treat the disorder. In this approach the transfer matrices of the model on the Nishimori line have an enhanced supersymmetry osp($2n+1|2n$), in contrast to the rest of the phase diagram, where the symmetry is osp($2n|2n$) (where $n$ is an arbitrary positive integer). An anisotropic limit of the model leads to a one-dimensional quantum Hamiltonian describing a chain of interacting superspins, which are irreducible representations of the osp($2n+1|2n$) superalgebra. By generalizing this superspin chain, we embed it into a wider class of models. These include other models that have been studied previously in one and two dimensions. We suggest that the multicritical behavior in two dimensions of a class of these generalized models (possibly not including the multicritical point in the RBIM itself) may be governed by a single fixed point, at which the supersymmetry is enhanced still further to osp($2n+2|2n$). This suggestion is supported by a calculation of the renormalization-group flows for the corresponding nonlinear sigma models at weak coupling.

Decoupling of layers in the three-dimensional Abelian Higgs model

The Abelian Higgs model with anisotropic couplings in 2+1 dimensions is studied in both the compact and non-compact formulations. Decoupling of the space-like planes takes place in the extreme anisotropic limit, so charged particles and gauge fields are presumably localized within these planes. The behaviour of the model under the influence of an external magnetic field is examined in the compact case and yields further characterization of the phases.

Superconductivity, Upper Critical Field and Anomalous Normal State in CePd2Si2 Near the Quantum Critical Point

We report resistivity measurements performed on a high quality single crystal of CePd2Si2 under hydrostatic pressure. At ambient pressure the de Haas–van Alphen effect has been also studied. Two different frequencies with weak angular dependence were detected with magnetic field lying in the basal plane, while another frequency was found with magnetic field parallel to the tetragonal c axis. Near the critical pressure, Pc∼27 kbar, where the antiferromagnetic transition vanishes, the normal state resistivity does not follow the usual Fermi-liquid (FL) behavior and is described by a T1.3 law, while just below Pc, the resistivity shows clearly separated spin wave and electron–electron contributions. At Pc, the FL form of ρ(T) is not restored even at magnetic field up to 6T. The first appearance of superconductivity is observed at P=19 kbar, and the critical temperature increases with pressure up to 27 kbar. The analysis of the upper critical field at Pc shows that the superconducting state is well described by a weak coupling, clean limit model with a slightly anisotropic orbital limit and a strongly anisotropic paramagnetic one.