Renormalization Group Study of Ising Transition in Double－Frequency sine－Gordon Model

We utilize the renormallzation group (RG) technique to analyze the Ising critical behavior in the double frequency Sine-Gordon model. The one-loop RG equations obtained show unambiguously that there exist two Ising critical points besides the trivial Gaussian fixed point. The topology of the RG flows is obtained as well.     

Renormalization Group Study of Ising Transition in Double-Frequency sine-Gordon Model

We utilize the renormalization group (RG) technique to analyze the Ising critical behavior in the doublefrequency sine-Gordon model. The one-loop RG equations obtained show unambiguously that there exist two Isingcritical points besides the trivial Gaussian fixed point. The topology of the RG flows is obtained as well.     

Stability of the critical behavior of weakly disordered systems against replica symmetry breaking

A field-theoretical approach is used to describe the critical behavior of weakly disordered systems with a p-component order parameter. Renormalization group (RG) analysis of the effective replica Hamiltonian of a model with replica-unsymmetrical interaction potential is carried out in a two-loop approximation directly for three-dimensional systems. The Padé-Borel summation technique is used to determine fixed points of the RG equations for the case of a single-step replica symmetry breaking (RSB). Analysis of their stability showed that the type of stable critical behavior of the disordered systems against the RSB effects remains as before.     

Continuous Replica Symmetry Breaking in Critical Phenomena in Disordered Ferromagnets

The replica symmetry breaking (RSB) in critical phenomena in disordered ferromagnets is studied. We modified the assumption of Dotsenko et al. about the local minimum solutions of the Hamiltonian with random temperature. It is shown explicitly that the continuous RSB is generated in the renormalization group (RG). The physical regime of the coupling constants are investigated through the RG equations.     

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.Supported by Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 130 Ferroelektrika.     

The field-theoretic description of dynamics of interfaces in disordered media

The time evolution of an interface in a disordered media is described by using the propagator method. The method enables one to represent the perturbation expansions of different quantities characterizing the interface by means of diagrams which are familiar from the field theory. By the analysis of the divergences in the vicinity of the critical dimension d_c = 4 we found that the regularization of the theory demands the renormalization of the mobility and all moments of the disorder correlator excepting the zero one. The renormalization group (RG) calculations of the average velocity of the interface, the roughness, and the functional RG equation of the disorder correlator are presented to order ? = 4 - d. The latter coincides with the result obtained by D. S. Fisher in the equilibrium case. The RG equations have a pole at the value of the driving force, which coincides with the value of the threshold below which the interface becomes pinned as predicted by Bruinsma and Aeppli. The behavior of the mobility in the vicinity of the pole is discussed.     

Epsilon expansion for multicritical fixed points and exact renormalisation group equations

The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion. The results are compared with the epsilon expansion by showing that the nonlinear differential equations may be linearised at each multicritical point and the epsilon expansion treated as a perturbative expansion. The results for critical exponents are compared with corresponding epsilon expansion results from standard perturbation theory. The results provide a test for the validity of the local potential approximation and also the derivative expansion. An alternative truncation of the exact RG equation leads to equations which are similar to those found in the derivative expansion but which gives correct results for critical exponents to order ε and also for the field anomalous dimension to order ε². An exact marginal operator for the full RG equations is also constructed.     

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t₀=t is naturally understood where t₀ is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink–anti-kink and soliton–soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.     

Renormalization Group Method Applied to Kinetic Equations: Roles of Initial Values and Time

The so-called renormalization group (RG) method is applied to derive kinetic and transport equations from the respective microscopic equations. The derived equations include the Boltzmann equation in classical mechanics, the Fokker-Planck equation, and a rate equation in a quantum field theoretical model. Utilizing the formulation of the RG method which elucidates the important role played by the choice of the initial conditions, the general structure and the underlying assumptions in the derivation of kinetic equations in the RG method are clarified. It is shown that the present formulation naturally leads to the choice for the initial value of the microscopic distribution function at arbitrary time t₀ to be on the averaged distribution function to be determined. The averaged distribution function may be thought of as an integral constant of the solution of the microscopic evolution equation; the RG equation gives the slow dynamics of the would-be initial constant, which is actually the kinetic equation governing the averaged distribution function. It is further shown that the averaging as given above gives rise to a coarse-graining of the time-derivative which is expressed with the initial time t₀, and thereby leads to time-irreversible equations even from a time-reversible equation. It is shown that a further reduction of the Boltzmann equation to fluid dynamical equations and the adiabatic elimination of fast variables in the Fokker-Planck equation are also performed in a unified way in the present method.     

A perturbative nonequilibrium renormalization group method for dissipative quantum mechanics

We study a generic problem of dissipative quantum mechanics, a small local quantum system with discrete states coupled in an arbitrary way (i.e. not necessarily linear) to several infinitely large particle or heat reservoirs. For both bosonic or fermionic reservoirs we develop a quantum field-theoretical diagrammatic formulation in Liouville space by expanding systematically in the reservoir-system coupling and integrating out the reservoir degrees of freedom. As a result we obtain a kinetic equation for the reduced density matrix of the quantum system. Based on this formalism, we present a formally exact perturbative renormalization group (RG) method from which the kernel of this kinetic equation can be calculated. It is demonstrated how the nonequilibrium stationary state (induced by several reservoirs kept at different chemical potentials or temperatures), arbitrary observables such as the transport current, and the time evolution into the stationary state can be calculated. Most importantly, we show how RG equations for the relaxation and dephasing rates can be derived and how they cut off generically the RG flow of the vertices. The method is based on a previously derived real-time RG technique [1-4] but formulated here in Laplace space and generalized to arbitrary reservoir-system couplings. Furthermore, for fermionic reservoirs with flat density of states, we make use of a recently introduced cutoff scheme on the imaginary frequency axis [5] which has several technical advantages. Besides the formal set-up of the RG equations for generic problems of dissipative quantum mechanics, we demonstrate the method by applying it to the nonequilibrium isotropic Kondo model. We present a systematic way to solve the RG equations analytically in the weak-coupling limit and provide an outlook of the applicability to the strong-coupling case.     

Critical exponents predicted by grouping of Feynman diagrams in φ4 model

Different perturbation theory treatments of the Ginzburg‐Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ⁴ model with O(n) symmetry. As a result, equations for calculation of the two‐point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.     

Improvement of Renormalization Group for Barenblatt Equation

We construct a Hartree-Fock (self-consistent)-like algorithm with renormalization group (RG) approach to calculate the anomalous dimension in a nonlinear diffusion equation. We find that our result improves the original RG work because we include the effect of Heaviside function.     

Multiscale theory of fluctuating interfaces: renormalization of atomistic models

We describe a framework for the multiscale analysis of atomistic surface processes which we apply to a model of homoepitaxial growth with deposition according to the Wolf-Villain model and concurrent surface diffusion. Coarse graining is accomplished by calculating renormalization-group (RG) trajectories from initial conditions determined by the regularized atomistic theory. All of the crossover and asymptotic scaling regimes known from computer simulations are obtained, but we also find that two-dimensional substrates show an intriguing transition from smooth to mounded morphologies along the RG trajectory.     

Renormalization group approach to interacting fermion systems in the two-particle-irreducible formalism

We describe a new formulation of the functional renormalization group (RG) for interacting fermions within a Wilsonian momentum-shell approach. We show that the Luttinger-Ward functional is invariant under the RG transformation, and derive the infinite hierarchy of flow equations satisfied by the two-particle-irreducible (2PI) vertices. In the one-loop approximation, this hierarchy reduces to two equations that determine the self-energy and the 2PI two-particle vertex Φ⁽²⁾. Susceptibilities are calculated from the Bethe-Salpeter equation that relates them to Φ⁽²⁾. While the one-loop approximation breaks down at low energy in one-dimensional systems (for reasons that we discuss), it reproduces the exact results both in the normal and ordered phases in single-channel (i.e. mean-field) theories, as shown on the example of BCS theory. The possibility to continue the RG flow into broken-symmetry phases is an essential feature of the 2PI RG scheme and is due to the fact that the 2PI two-particle vertex, contrary to its 1PI counterpart, is not singular at a phase transition. Moreover, the normal phase RG equations can be directly used to derive the Ginzburg-Landau expansion of the thermodynamic potential near a phase transition. We discuss the implementation of the 2PI RG scheme to interacting fermion systems beyond the examples (one-dimensional systems and BCS superconductors) considered in this paper.     

Improvement of Renormalization Group for Barenblatt Equation

We construct a Hartree-Fock (seff-consistent)-like algorithm with renormalization group (RG) approach to calculate the anomalous dimension in a nonlinear diffusion equation. We find that our result improves the original RG work because we include the effect of Heaviside function.     

Anomalous Dimension in the Solution of a Nonlinear Diffusion Equation

A new method is used to obtain the anomalous dimension in the solution of the nonlinear diffusion equation.The result is the same as that in the renormalization group (RG) approach.It gives us an insight into the anomalous dimension in the solution of the nonlinear diffusion equation in the RG approach.Based on this discussion,we can see anomalous dimension appears naturally in this system.``     

Anomalous Dimension in the Solution of a Nonlinear Diffusion Equation

A new method is used to obtain the anomalous dimension in the solution of the nonlinear diffusion equation.The result is the same as that in the renormalization group (RG) approach.It gives us an insight into the anomalous dimension in the solution of the nonlinear diffusion equation in the RG approach.Based on this discussion,we can see anomalous dimension appears naturally in this system.     

非均匀无序系统中Anderson局域化的标度理论——实空间重整化群途径

针对大量具有空间调制的无序系统,本文提出一种实空间重整化群变换方案。这个方案保证了在空间映象下相对的空间调制结构不变,因此可用以研究非均匀无序系统Anderson局域化的临界性质。在有限晶格近似下,我们对无序金属超晶格的一个简化模型求解了RG方程,得到不动点和临界指数的近似值,并发现空间调制在一定程度上引起无序系统电子局域化性质的改变。 关键词：     

Feigenbaum universality in string theory

Brane-like vertex operators, defining backgrounds with ghost-matter mixing in NSR superstring theory, play an important role in the world-sheet formulation of D-branes and M theory as creation operators for extended objects in the second quantized formalism. In this paper, we show that the dilaton beta function in ghost-matter mixing backgrounds becomes stochastic. The renormalization group (RG) equations in ghost-matter mixing backgrounds lead to non-Markovian Fokker-Planck equations whose solutions describe superstrings in curved space-times with brane-like metrics. We show that the Feigenbaum universality constant δ=4.669..., describing transitions from order to chaos in a huge variety of dynamical systems, appears analytically in these RG equations. We find that the appearance of this constant is related to the scaling of relative space-time curvatures at fixed points of the RG flow. In this picture, the fixed points correspond to the period doubling of Feigenbaum iteration schemes.