Nonlinear Renormalization Group Research Articles

Renormalization group improved variational treatment of λφ 4

We use a nonlinear renormalization group equation to sum the 2PPI leading log corrections to GEP (gaussian effective potential). We show that for λφ 4 in 3+1 dimensions, the precarious phase ( λ R < 0) as found in GEP and finite order corrections to GEP, disappears.

Crossover exponent for a weak magnetic field at the localization orthogonal fixed point

We re-examine the non-linear σ-model lagrangian for the soft modes of non-interacting electrons in a disordered system in a weak magnetic field, close to two dimensions. The magnetic field is shown to induce two scaling terms in the low-energy lagrangian. The crossover exponents are computed by using a Wilson-Polyakov non-linear renormalization group. The largest one agrees to lowest order with the exponent expected from naive scaling arguments.

Multicritical unbinding phenomena and nonlinear functional renormalization group.

Using a nonlinear functional renormalization-group approach, we study the problem of unbinding phenomena both for interfaces and membranes. Our numerical and analytical solutions of the differential recursion relation confirm the existence of the ``unusual bifurcation'' of the fixed-point potentials describing unbinding at the upper critical dimension d=${\mathit{d}}^{\mathrm{*}}$. However, in addition to one critical and one purely repulsive potential, we find an infinite family of multicritical potentials bifurcating from the drifting fixed point. The values for the scaling indices, which are singular near ${\mathit{d}}^{\mathrm{*}}$, are calculated here exactly to leading orders in \ensuremath{\epsilon}=${\mathit{d}}^{\mathrm{*}}$-d and are shown to behave as ${\mathrm{\ensuremath{\epsilon}}}^{2/3}$.

Diverse manifolds in random media.

We suggest a unifying perspective of various manifolds subject to quenched correlated disorder. The new picture incorporates seemingly disparate problems as domain walls in impurity-stricken Ising magnets and directed walks upon random lattices. Using Flory-style arguments and a functional nonlinear renormalization group, we obtain new results, including the random-bond interfacial roughening exponent ${\ensuremath{\zeta}}_{\mathrm{RB}}=\frac{2}{9}\ensuremath{\epsilon}$, where $\ensuremath{\epsilon}=5\ensuremath{-}d$, which we believe to be exact, as well as a conjecture for the (2+1)-dimensional directed polymer index ${\ensuremath{\zeta}}_{2+1}=\frac{30}{49}$.

Experimental data analysis on xenon above the critical temperature from nonlinear renormalization group

We analyse the scattered light and specific heat data on xenon using the recent nonlinear renormalization group calculations obtained in the ϕ4 theory in three dimensions. The minimal set of free parameters that the theory imposes is sufficient to describe these data even for the case where, in the usual analysis, many correction terms to scaling are needed. We show that it will be very difficult to determine the influence of transients neglected in the ϕ4 model. Furthermore, we obtain the reduction to dimensionless experimental quantities in agreement with two scale factor universality. We show that the asymptotic amplitudes of the singular quantities are also related to the amplitudes of the first confluent corrections according to the critical path considered.

Critical dynamics near theλtransition inHe3-He4mixtures

The critical dynamics of the $\ensuremath{\lambda}$ transition in $^{3}\mathrm{He}$-$^{4}\mathrm{He}$ mixtures are studied by means of renormalized field theory applied to the model of Siggia and Nelson. A diagonal representation for the equations of motion is introduced, which greatly simplifies the computations in two-loop order. A universal connection is found with the asymptotic critical dynamics of pure $^{4}\mathrm{He}$ in all orders of perturbation theory. The observable critical dynamics of helium mixtures are dominated by nonuniversal crossover effects which can be properly described only within a nonlinear renormalization-group approach. The theory is applied to explain the observable critical and precritical temperature dependence of the mass diffusion $D$, of the thermal conductivity $\ensuremath{\kappa}$, of the thermal diffusion ratio ${k}_{T}$, and of the dynamic structure factor for $T\ensuremath{\ge}{T}_{\ensuremath{\lambda}}(X)$. Recent experimental data for the transport coefficients at the molar $^{3}\mathrm{He}$ concentration $X=0.05$ by Gestrich and Meyer are used to identify the nonuniversal parameters of the theory in the range $X\ensuremath{\ll}1$. Consistency with the dynamics of pure $^{4}\mathrm{He}$ ($X=0$) is verified. Predictions without adjustable parameters are made for the dynamic structure factor and the transport coefficients in very dilute mixtures. The Siggia-Kawasaki problem concerning the leading $X$ dependence of $\ensuremath{\kappa}({T}_{\ensuremath{\lambda}})$ in the $X\ensuremath{\rightarrow}0$ limit is resolved. It is demonstrated that Siggia's prediction $\ensuremath{\kappa}({T}_{\ensuremath{\lambda}})\ensuremath{\sim}{X}^{\ensuremath{-}1}$ is correct but not observable. Theoretical extrapolations to $X&gt;0.05$ without adjustable parameters are presented and compared with measured transport coefficients at $X=0.11 \mathrm{and} 0.15$. The overall agreement is satisfactory. Deviations of order 15% exist with the thermal conductivity $\ensuremath{\kappa}$ at $X=0.15$. This may be attributed to dynamic effects arising from the singular specific heat and mass susceptibility, which are not included in the present analysis.

Nonlinear dynamic renormalization-group analysis above and below the lambda transition in 4He

Recent developments in the theory of the dynamics of liquid 4 He near the lambda transition are reviewed. A nonlinear renormalization-group treatment is presented which provides a quantitative description of the dynamics in the entire region between noncritical background and criticality. The previous discrepancies between asymptotic scaling theories and various experiments above and below T λ are explained in terms of crossover effects. This crossover is governed (i) close to T λ by the effect of the weak-scaling fixed point, and (ii) farther away from T λ by a strong decrease of the effective dynamic coupling. New results are presented which are based on the complete flow equations for model F in two-loop order. They tentatively confirm the conclusions of an earlier analysis based on model E and invalidate part of a recent analysis treating model F in an unsystematic way. Excellent agreement is found with new accurate measurements of second-sound damping over four decades in relative temperature.

Nonlinear dynamic renormalization-group analysis above and below the lambda transition in 4He

Recent developments in the theory of the dynamics of liquid 4He near the lambda transition are reviewed. A nonlinear renormalization-group treatment is presented which provides a quantitative description of the dynamics in the entire region between noncritical background and criticality. The previous discrepancies between asymptotic scaling theories and various experiments above and below Tλ are explained in terms of crossover effects. This crossover is governed (i) close to Tλ by the effect of the weak-scaling fixed point, and (ii) farther away from Tλ by a strong decrease of the effective dynamic coupling. New results are presented which are based on the complete flow equations for model F in two-loop order. They tentatively confirm the conclusions of an earlier analysis based on model E and invalidate part of a recent analysis treating model F in an unsystematic way. Excellent agreement is found with new accurate measurements of second-sound damping over four decades in relative temperature.

Study of Quantum Spin 1/2 X-Y Model by Means of Extended Migdal Approximation

The quantum spin 1/2 X- Y model in dimensions d= 1, 2 and 3 is studied by means of the extended Migdal approximation for the scale factor of the transformation b = 2. It is found, with the imposed conditions to get the recursion equations, that an unstable fixed point between a T=O fixed point and that of T== exists for d>l. However, alternate imposed conditions lead to an unusual behavior that the T = 0 fixed point does not appear in any dimension and that a stable fixed point appears at a lower temperature than that of an unstable one for d > 2. The same model on the simple cubic and face-centered cubic lattices is also treated by the two-step decimation transformation. Recently much attention has been paid to the X- Y model to clarify the nature of a phase transition in two dimensions. As for the quantum X- Y model, it is believed with little doubt that the one-dimensional model remains paramagnetic at all temperatures and that there occurs a second order phase transition in three dimensions to an ordered phase with the long range order (transverse magnetization) at a finite temperature.])<l) The spin 1/2 X- Y model in two dimensions has no long range order at non-zero temperature,4) but shows evidence of a divergent susceptibility and a non-singular specific heat at a finite temperature.3).5),6) For the classical case in two dimensions, it is believed that the topological phase transition with a line of critical points occurs at a finite temperature.7)~l l) Several authorsl2)~16) have studied the X- Y model with spin 1/2 in two dimensions by means of the real-space nonlinear renormalization-group transformations. Nevertheless, their results were contradictory and in­ conclusive. Dekeyser et a1.17) suggested with a real-space linear renormalization transformation and with the imposition of marginality for temperature deviations from criticality that the same model shows a critical behavior of the same kind as that predicted for the classical case. Recently an extension of the Migdal approximation 18 ),19) to quantum spin systems has been made by Suzuki and Takano,20) (hereafter referred to as ST), and Barma et a1. 21l The approximation employed is to perform the quantum­ mechanical calculation only for the one-dimensional decimation transformation (DT) in the Migdal approximation: The bond-moving approximation could be

Differential real-space renormalization of thed-Dimensional Gaussian model

With the aid of the differential real-space method we derive exact renormalization group (RG) equations for the Gaussian model ind dimensions. The equations involved + 1 spatially dependent nearest-neighbor interactions. We locate a critical fixed point and obtain the exact thermal critical indexy T = 2. A special trajectory of the full nonlinear RG transformation is found and the free energy of the corresponding initial state calculated.

Ising model with magnetic field for 3D cubic structures

Results for the Ising model in the presence of a magnetic field and the lattice gas are reported for the SC, BCC and FCC three-dimensional lattices. A non-linear renormalization group transformation is applied to a periodic cluster of cells containing sublattice sites of the original lattice.

Spin-SIsing susceptibility on the triangular lattice

The high-temperature susceptibility of the spin-$S$ Ising model on the triangular net is analyzed as a function of $S$ using tenth-order high-temperature series expansions. In particular the spin dependence of the critical temperature and the amplitude of the dominant singularity are obtained by assuming a universal value $\ensuremath{\gamma}=\frac{7}{4}$ for the critical exponent of the susceptibility. The results, for the critical temperature particularly, provide a valuable guidepost in judging the convergence of recent nonlinear renormalization-group calculations for these models.

Global features of nonlinear renormalization-group equations

The analysis of selected nonlinear problems in the renormalization group is found to show striking contrasts between the usual local linearized fixed-point analysis and the properties of global solutions of nonlinear equations derived from an approximation of the Wegner-Houghton differential formulation. The competition between various fixed points that is incorporated in general global solutions can upset the asymptotically valid critical behavior deduced from the local analysis. In general, the critical-point exponents of such a solution will not satisfy equalities, but rather the corresponding inequalities. However, these nonscaling solutions have extraneous singularities that are not related to the thermodynamic singularities of the system. If singularities of this type are excluded, then the global solution has the same critical-point exponents as the local solution derived by linearizing around the stablest fixed point. It is shown that in this case the critical surface in the Hamiltonian space is closely related to the surface of order-2 critical points in a thermodynamic field space. The boundaries of this surface are correspondingly related to the critical points of higher order in this thermodynamic space. The nonlinear global solution predicts multiple power scaling behavior from a single scaling equation deduced from the renormalization group. Previously such behavior was obtained by postulating the simultaneous validity of two of more "linear" scaling hypotheses.

Renormalization group for Ising spins on a finite lattice

Linear and nonlinear renormalization-group transformations for Ising spins on a two-dimensional triangular lattice are considered; numerical calculations bearing on the convergence of the finite-lattice calculations to the infinite-lattice limit are presented.

Nonlinear renormalization groups

Most renormalization groups studied heretofore are linear, in that the block-spin variables are linearly related to the old spin variables. These renormalization groups all require an adjustable parameter that must be properly fixed in order that the renormalization-group transformation have a useful fixed point. Niemeijer and van Leeuwen, however, have investigated the critical behavior of the two-dimensional Ising model with a nonlinear renormalization group, and find a fixed point without introducing such a parameter. We introduce here another nonlinear renormalization group and study it in detail for the Gaussian model. We find that (i) within certain limits, no such parameter need be adjusted in order to reach a fixed point; (ii) an eigenvalue of the linearized transformation with no physical significance depends on the nonlinearity of the transformation; and (iii) a physically significant eigenvalue is unchanged (to the order examined).