Quantum phase transitions in the S=1/2 distorted diamond chain

By means of the second derivative of the ground-state and first-excited energy, the quantum phase transitions (QPTs) for the distorted diamond chain (DDC) with ferromagnetic and antiferromagnetic frustrated interactions and the trimerized case are investigated, respectively. Our results show the plentiful quantum phases owing to the spin interaction competitions in the model. Meanwhile, by using the transfer-matrix renormalization-group technique, we study the two-site thermal entanglement of the DDC model in the thermodynamic limit for a further understanding of the QPTs.

Thermodynamic properties of Cu-3(CO3)(2)(OH)(2) and the spin-1/2 distorted diamond Heisenberg chain

The thermodynamic properties of the spin-1/2 diamond quantum Heisenberg chain model have been investigated by means of the transfer matrix renormalization group (TMRG) method. Considering different crystal structures, by changing the interactions among different spins and the external magnetic fields, we first investigate the magnetic susceptibility, magnetization, and specific heat of the distorted diamond chain as a model of ferrimagnetic spin systems. The susceptibility and the specific heat show different features for different ferromagnetic (F) and antiferromagnetic (AF) interactions and different magnetic fields. A 1/3 magnetization plateau is observed at low temperature in a magnetization curve. Then, we discuss the theoretical mechanism of the double-peak structure of the magnetic susceptibility and the three-peak structure of the specific heat of the compound Cu-3(CO3)(2)(OH)(2), on which an elegant measurement was performed by Kikuchi [Phys. Rev. Lett. 94, 227201 (2005)]. Our computed results are consistent with the main characteristics of the experimental data. Meanwhile, we find that the double-peak structure of susceptibility can be found in several different kinds of spin interactions in the diamond chain. Moreover, a three-peak behavior is observed in the TMRG results of magnetic susceptibility. In addition, we perform calculations relevant for some experiments and explain the characteristics of these materials. (c) 2007 American Institute of Physics.

Magnetic frustration induced quantum phase transitions in the S=1/2 vertically asymmetric diamond chain

This work was supported by the National Basic Research Program of China (973 Program) grant No. G2009CB929300 and the National Natural Science Foundation of China under Grant Nos. 60521001 and 60776061.

Systematic stability analysis of the renormalization group flow for the normal-superconductor-normal junction of Luttinger liquid wires

We study the renormalization group flows of the two terminal conductance of a superconducting junction of two Luttinger liquid wires. We compute the power laws associated with the renormalization group flow around the various fixed points of this system using the generators of the SU(4) group to generate the appropriate parametrization of an matrix representing small deviations from a given fixed point matrix [obtained earlier in S. Das, S. Rao, and A. Saha, Phys. Rev. B 77, 155418 (2008)], and we then perform a comprehensive stability analysis. In particular, for the nontrivial fixed point which has intermediate values of transmission, reflection, Andreev reflection, and crossed Andreev reflection, we show that there are eleven independent directions in which the system can be perturbed, which are relevant or irrelevant, and five directions which are marginal. We obtain power laws associated with these relevant and irrelevant perturbations. Unlike the case of the two-wire charge-conserving junction, here we show that there are power laws which are nonlinear functions of V(0) and V(2kF) [where V(k) represents the Fourier transform of the interelectron interaction potential at momentum k]. We also obtain the power law dependence of linear response conductance on voltage bias or temperature around this fixed point.

The renormalization group: A perturbation method for the graduate curriculum

In this paper the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. Lett., 73 (1994), pp.1311-1315; Phys. Rev. E, 54 (1996), pp.376-394] is presented in a pedagogical way to increase its visibility in applied mathematics and to argue favorably for its incorporation into the corresponding graduate curriculum.The method is illustrated by some linear and nonlinear singular perturbation problems. Key word. © 2012 Society for Industrial and Applied Mathematics.

A Combined Renormalization Group-Multiple Scale Method for Singularly Perturbed Problems

This paper introduces a straightforward method to asymptotically solve a variety of initial and boundary value problems for singularly perturbed ordinary differential equations whose solution structure can be anticipated. The approach is simpler than conventional methods, including those based on asymptotic matching or on eliminating secular terms. © 2010 by the Massachusetts Institute of Technology.

Reduction of amplitude equations by the renormalization group approach

This article elucidates and analyzes the fundamental underlying structure of the renormalization group (RG) approach as it applies to the solution of any differential equation involving multiple scales. The amplitude equation derived through the elimination of secular terms arising from a naive perturbation expansion of the solution to these equations by the RG approach is reduced to an algebraic equation which is expressed in terms of the Thiele semi-invariants or cumulants of the eliminant sequence { Zi } i=1 . Its use is illustrated through the solution of both linear and nonlinear perturbation problems and certain results from the literature are recovered as special cases. The fundamental structure that emerges from the application of the RG approach is not the amplitude equation but the aforementioned algebraic equation. © 2008 The American Physical Society.

The renormalization group and the implicit function theorem for amplitude equations

This article lays down the foundations of the renormalization group (RG) approach for differential equations characterized by multiple scales. The renormalization of constants through an elimination process and the subsequent derivation of the amplitude equation [Chen, Phys. Rev. E 54, 376 (1996)] are given a rigorous but not abstract mathematical form whose justification is based on the implicit function theorem. Developing the theoretical framework that underlies the RG approach leads to a systematization of the renormalization process and to the derivation of explicit closed-form expressions for the amplitude equations that can be carried out with symbolic computation for both linear and nonlinear scalar differential equations and first order systems but independently of their particular forms. Certain nonlinear singular perturbation problems are considered that illustrate the formalism and recover well-known results from the literature as special cases. © 2008 American Institute of Physics.

Secular series and renormalization group for amplitude equations

We have developed a technique that circumvents the process of elimination of secular terms and reproduces the uniformly valid approximations, amplitude equations, and first integrals. The technique is based on a rearrangement of secular terms and their grouping into the secular series that multiplies the constants of the asymptotic expansion. We illustrate the technique by deriving amplitude equations for standard nonlinear oscillator and boundary-layer problems. © 2008 The American Physical Society.

Renormalization group interpretation of the Born and Rytov approximations

In this paper the method of renormalization group (RG) [Phys. Rev. E 54, 376 (1996)] is related to the well-known approximations of Rytov and Born used in wave propagation in deterministic and random media. Certain problems in linear and nonlinear media are examined from the viewpoint of RG and compared with the literature on Born and Rytov approximations. It is found that the Rytov approximation forms a special case of the asymptotic expansion generated by the RG, and as such it gives a superior approximation to the exact solution compared with its Born counterpart. Analogous conclusions are reached for nonlinear equations with an intensity-dependent index of refraction where the RG recovers the exact solution. © 2008 Optical Society of America.

Velocity ratio-cum-transfer matrix method for the evaluation of a muffler with mean flow

The paper deals with a method for the evaluation of exhaust muffers with mean flow. A new set of variables, convective pressure and convective mass velocity, have been defined to replace the acoustic variables. An expression for attenuation (insertion loss) of a muffler has been proposed in terms of convective terminal impedances and a velocity ratio, on the lines of the one existing for acoustic filters. In order to evaluate the velocity ratio in terms of convective variables, transfer matrices for various muffler elements have been derived from the basic relations of energy, mass and momentum. Finally, the velocity ratiocum-transfer matrix method is illustrated for a typical straight-through muffler.

On the transfer matrix for a uniform tube with a moving medium

Abstract is not available.

A dynamic renormalization group study of active nematics

We carry out a systematic construction of the coarse-grained dynamical equation of motion for the orientational order parameter for a two-dimensional active nematic, that is a nonequilibrium steady state with uniaxial, apolar orientational order. Using the dynamical renormalization group, we show that the leading nonlinearities in this equation are marginally irrelevant. We discover a special limit of parameters in which the equation of motion for the angle field bears a close relation to the 2d stochastic Burgers equation. We find nevertheless that, unlike for the Burgers problem, the nonlinearity is marginally irrelevant even in this special limit, as a result of a hidden fluctuation-dissipation relation. 2d active nematics therefore have quasi-long-range order, just like their equilibrium counterparts.

Relationship Between The Impedance Matrix And The Transfer Matrix With Specific Reference To Symmetrical, Reciprocal And Conservative Systems

Impedance matrix and transfer matrix methods are often used in the analysis of linear dynamical systems. In this paper, general relationships between these matrices are derived. The properties of the impedance matrix and the transfer matrix of symmetrical systems, reciprocal systems and conservative systems are investigated. In the process, the following observations are made: (a) symmetrical systems are not a subset of reciprocal systems, as is often misunderstood; (b) the cascading of reciprocal systems again results in a reciprocal system, whereas cascading of symmetrical systems does not necessarily result in a symmetrical system; (c) the determinant of the transfer matrix, being ±1, is a property of both symmetrical systems and reciprocal systems, but this condition, however, is not sufficient to establish either the reciprocity or the symmetry of the system; (d) the impedance matrix of a conservative system is skew-Hermitian.