Solvable real space renormalization group for the kinetic Ising chain

A real-space renormalization group for the one-dimensional kinetic Ising model is established. The parameter space of the model must be enlarged to include non-Markovian kernels in the equation of motion. The recursion relations for these kernels can be iterated analytically so that the global flow under the renormalization group can be traced exactly. The resulting fixed-point equation is non-Markovian.     

Differential real space renormalization: The linear Ising chain

The differential real space renormalization method, recently introduced by Hillhorst et al., is applied to the linear Ising chain. It is shown that chains with spatially homogeneous as well as inhomogeneous or quenched random interactions can be treated. For the first two cases the free energy is computed by renormalization. The discussion includes also the case with a magnetic field, higher order interactions and the behavior of correlation functions under renormalization.     

Real space renormalization group investigation of three-dimensional Ising spin glasses

Using real space renormalization group techniques we determine the phase diagram of bond dilute frustrated nearest-neighbor Ising three-dimensional simple cubic (sc) and body-centered cubic (bcc) systems.     

The XY model and Kadanoff's variational real space renormalization group

The XY model in d dimensions is studied by means of a variational real space renormalization group transformation. Contrary to an earlier computation in the same framework for the d = 2 case, we find that a low order operator basis truncation is highly unstable. For certain values of the variational parameter p the renormalization group flow can display period doubling sequences towards a chaotic regime. The behavior in the d = 3 case is very similar.     

Self-consistent calculation of real space renormalization group flows and effective potentials

We show how to compute real space renormalization group flows in lattice field theory by a self-consistent method which is designed to preserve the basic stability properties of a Boltzmann factor. Particular attention is paid to controlling the errors which come from truncating the action to a manageable form. In each step, the integration over the fluctuation field (high frequency components of the field) is performed by a saddle point method. The saddle point depends on the block spin. Higher powers of derivatives of the field are neglected in the actions, but no polynomial approximation in the field is made. The flow preserves a simple parameterization of the action. In the first part the method is described and numerical results are presented. In the second part we discuss an improvement of the method where the saddle point approximation is preceded by self-consistent normal ordering, i.e. solution of a gap equation. In the third part we describe a general procedure to obtain higher order corrections with the help of Schwinger-Dyson equations.In this paper we treat scalar field theories as an example. The basic limitations of the method are also discussed. They come from a possible breakdown of stability which may occur when a composite block spin or block variables for domain walls would be needed.     

Higher-order calculations in the real space dynamic renormalization group: One dimension

The application of real space dynamic renormalization group methods to the one-dimensional kinetic Ising model, discussed in an earlier paper, is extended to one order higher in perturbation theory than was done previously. It is shown that the treatment of short-range, local quantities is improved in going to higher order in the perturbation expansion, while that of the long-range properties remains largely unaffected. Arbitrariness in the real space mapping function and how it may be exploited to our advantage is duscussed. It is shown that the renormalized Hamiltonian continues to be characterized by one coupling through second order. We find that the single spin-flip kinetic Ising model generates at second-order new spin-flip mechanisms in the renormalized dynamical operator but that their effects are small (at most 2%) over the entire temperature range.     

Quantum spins and quasiperiodicity: a real space renormalization group approach

We study the antiferromagnetic spin-1/2 Heisenberg model on a two-dimensional bipartite quasiperiodic structure, the octagonal tiling, the aperiodic equivalent of the square lattice for periodic systems. An approximate block spin renormalization scheme is described for this problem. The ground state energy and local staggered magnetizations for this system are calculated and compared with the results of a recent quantum Monte Carlo calculation for the tiling. It is conjectured that the ground state energy is exactly equal to that of the quantum antiferromagnet on the square lattice.     

Variational real space renormalization group and its application to frustrated systems

We propose an improved variational version of the Migdal-Kadonoff approximation. We test this method on the two-dimensional triangular lattice Ising model in the ferromagnetic, antiferromagnetic (fully frustrated) and spin glass (randomly frustrated) cases.     

Interface free energy and transition temperature of the square-lattice Ising antiferromagnet at finite magnetic field

We use a simple method to calculate interface properties of the square-lattice Ising antiferromagnet with nearest neighbour interaction. The method bypasses the more complicated bulk problem by taking into account only interface configurations of spins and allows the inclusion of a finite magnetic field. From this we derive two new results: 1) the interface free energy associated with the coexistence of the two antiferromagnetic phases at finite magnetic field, and 2) the transition temperature as a function of the magnetic field which determines the phase boundary.     

Density matrix renormalization group numerical study of the kagome antiferromagnet

We numerically study the spin-1/2 antiferromagnetic Heisenberg model on the kagome lattice using the density-matrix renormalization group method. We find that the ground state is a magnetically disordered spin liquid, characterized by an exponential decay of spin-spin correlation function in real space and a magnetic structure factor showing system-size independent peaks at commensurate magnetic wave vectors. We obtain a spin triplet excitation gap DeltaE(S=1)=0.055+/-0.005 by extrapolation based on the large size results, and confirm the presence of gapless singlet excitations. The physical nature of such an exotic spin liquid is also discussed.     

Application of the real space dynamic renormalization group method to the one-dimensional spin-exchange model

We apply the real space dynamic renormalization group method to the one-dimensional spin-exchange kinetic Ising model. We show that the conservation of magnetization property of this model is preserved directly under renormalization. We also demonstrate that one can derive recursion relations for the space-and time-dependent correlation functions and that the iterated solutions of these recursion relations lead to the appropriate hydrodynamic forms in the small-wavenumber and -frequency regime.     

Heisenberg ferromagnet with Dzyaloshinsky-Moriya interactions: A real space renormalization group approach

We study the anisotropic Heisenberg ferromagnet with anti-symmetric Dzyaloshinsky-Moriya (DM) interactions for arbitrary dimensions. We use a real space renormalization group approach of the Migdal-Kadanoff type in order to obtain the phase diagrams and critical exponents. The effect of the Dzyaloshinsky-Moriya term in the global phase diagram is worked out in detail. Our results suggest that in all dimensions the effect of the DM interaction is to renormalize the parameters of the anisotropic exchange Hamiltonian. Finally we discuss the modification of hyperscaling associated with the zero temperature Heisenberg-like fixed point.     

Phase diagram of the square-lattice three-state Potts antiferromagnet with a staggered polarization field

We study a square-lattice three-state Potts antiferromagnet with a staggered polarization field at finite temperature. Numerically treating the transfer matrices, we determine two phase boundaries separating the model-parameter space into three parts. We confirm that one of them belongs to the ferromagnetic three-state Potts criticality, which is in accord with a recent prediction, and another to the Ising-type; these are both corresponding to the massless renormalization-group flows stemming from the Gaussian fixed points. We also discuss a field theory to describe the latter Ising transition.     

Application of the real space dynamic renormalization group method to the one-dimensional kinetic ising model

We apply the recently developed real space dynamic renormalization group method to the one-dimensional kinetic Ising model. We show how one can develop block spin methods that lead to recursion relations for the space and time dependent correlation functions that correspond to the observables for this system. We point out the importance of carefully choosing the appropriate parameters governing the behavior of individual blocks of spins and the necessity of worrying about the high temperature properties of the temperature recursion relations if one is to obtain the proper exponential decay of correlation functions at large distances away from the critical point at zero temperature. We systematically investigate the accuracy of our approximate recursion relations for various correlation functions by checking them against the known exact results. Our simple methods work surprisingly well over a wide range of temperatures, wavenumbers and frequencies.     

Phase transitions and renormalization group hamiltonian densities in the 80 diperiodic space groups

Phase transitions for systems with diperiodic symmetry are discussed. Direct group-theoretical methods are employed to obtain a list of possible commensurate lower-symmetry phases (subgroups) which are induced by a single order parameter. The lower-symmetry phases for all 80 diperiodic space groups are given, along with specific details of the group-subgroup relationships. Results for the 17 two-dimensional space groups are also contained in our list. The renormalization-group Hamiltonian densities for the diperiodics are calculated. The 12 densities listed constitute the complete set of densities which may arise in the diperiodic space groups. Critical properties for the diperiodics can thus be obtained from analysis of these densities.     

Effective medium approach to real space renormalization

We use the ideas of effective medium theory to present a self-consistent decimation which yields the exact fixed point for the q-component Potts model in a square lattice and also generates the exact critical surface for anisotropic interactions.     

Diluted transverse Ising model within the mean field renormalization group

Clusters of different size and symmetry are exploited in the study of the diluted transverse Ising model on several lattices within the mean field renormalization group approach. It is noticed that the critical exponents depend both on the size of clusters as well as on the cluster symmetry. Harris' conjecture is verified for all lattices studied.