Gábor Fáth

Propagation failure of traveling waves in a discrete bistable medium

Abstract Propagation failure (pinning) of traveling waves is studied in a discrete scalar reaction-diffusion equation with a piecewise linear, bistable reaction function. The critical points of the pinning transition, and the wavefront profile at the onset of propagation are calculated exactly. The scaling of the wave speed near the transition, and the leading corrections to the front shape are also determined. We find that the speed vanishes logarithmically close to the critical point, thus the model belongs to a different universality class than the standard Nagumo model, defined with a smooth, polynomial reaction function.

Accuracy of the density-matrix renormalization-group method

Whites density-matrix renormalization-group (DMRG) method has been applied to the one-dimensional Ising model in a transverse field (ITF), in order to study the accuracy of the numerical algorithm. Due to the exact solubility of the ITF for any finite chain length, the errors introduced by the basis truncation procedure could have been directly analyzed. By computing different properties, like the energies of the low-lying levels or the ground-state one- and two-point correlation functions, we obtained a detailed picture of how these errors behave as functions of the various model and algorithm parameters. Our experience with the ITF contributes to a better understanding of the DMRG method, and may facilitate its optimization in other applications.

Probable absence of a quadrupolar spin-nematic phase in the bilinear-biquadratic spin-1 chain

We study numerically the ground-state phase diagram of the bilinear-biquadratic spin-1 chain near the ferromagnetic instability point, where the existence of a gapped or gapless nondimerized quantum nematic phase has been suggested. Our results, obtained by a highly accurate density-matrix renormalization-group calculation, are consistent with the view that the order parameter characterizing the dimer phase vanishes only at the point where the system becomes ferromagnetic, although the existence of a gapped or gapless nondimerized phase in a very narrow parameter range between the ferromagnetic and the dimerized regimes cannot be ruled out.

String order in spin liquid phases of spin ladders

Two-leg spin ladders have a rich phase diagram if rung, diagonal and plaquette couplings are allowed for. Among the possible phases there are two Haldane-type spin liquid phases without local order parameter, which differ, however, in the topology of the short range valence bonds. We show that these phases can be distinguished numerically by two different string order parameters. We also point out that long range string- and dimer orders can coexist.

Solitonic excitations in the Haldane phase of an S=1 chain

We study low-lying excitations in the 1D S=1 antiferromagnetic valence-bond-solid (VBS) model. In a numerical calculation on finite systems the lowest excitations are found to form a discrete triplet branch, separated from the higher-lying continuum. The dispersion of these triplet excitations can be satisfactorily reproduced by assuming approximate wavefunctions. These wavefunctions are shown to correspond to moving hidden domain walls, i.e. to one-soliton excitations.

Luttinger liquid behavior in spin chains with a magnetic field

Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary(Dated: 8 April 2003)Antiferromagnetic Heisenberg spin chains in a sufficiently strong magnetic field are Luttinger liq-uids, whose parameters depend on the actual magnetization of the chain. Here we present precisenumerical estimates of the Luttinger liquid dressed charge Z, which determines the critical expo-nents, by calculating the magnetization and quadrupole operator profiles for S = 1/2 and S = 1chains using the density matrix renormalization group method. Critical amplitudes and the scat-tering length at the chain ends are also determined. Although both systems are Luttinger liquidsthe characteristic parameters differ considerably.

PHASE DIAGRAM OF MAGNETIC LADDERS CONSTRUCTED FROM A COMPOSITE-SPIN MODEL

Whites density matrix renormalization group ({DMRG}) method has been applied to an

A renormalization group theory of cultural evolution

S= 1/2 + 1/2

Commensurate and incommensurate correlations in Haldane-gap antiferromagnets

composite-spin model, which can also be considered as a two-leg ladder model. By appropriate choices of the coupling constants this model allows not only to study how the gap is opened around the gapless integrable models, but also to interpolate continuously between models with different spin lengths. We have found indications for the existence of several different massive phases.

Logarithmic delocalization of end spins in the S=3/2 antiferromagnetic heisenberg chain

We present a theory of cultural evolution based upon a renormalization group scheme. We consider rational but cognitively limited agents who optimize their decision-making process by iteratively updating and refining the mental representation of their natural and social environment. These representations are built around the most important degrees of freedom of their world. Cultural coherence among agents is defined as the overlap of mental representations and is characterized using an adequate order parameter. As the importance of social interactions increases or agents become more intelligent, we observe and quantify a series of dynamic phase transitions by which cultural coherence advances in the society. A similar phase transition may explain the so-called “cultural explosion’’ in human evolution some 50,000 years ago.