arXiv subjects

Uwe Grimm

Publications and source records attributed to Uwe Grimm.

At least 91 records · Page 5Linked to original sources

arXiv1999

Planar quasiperiodic Ising models

We investigate zero-field Ising models on periodic approximants of planar quasiperiodic tilings by means of partition function zeros and high-temperature expansions. These are obtained by employing a determinant expression for the partition function. The partition function zeros in the complex temperature plane yield precise estimates of the critical temperature of the quasiperiodic model. Concerning the critical behaviour, our results are compatible with Onsager universality, in agreement with the Harris-Luck criterion based on scaling arguments.

cond-mat.stat-mech↗

arXiv1999

Universal level-spacing statistics in quasiperiodic tight-binding models

We study statistical properties of the energy spectra of two-dimensional quasiperiodic tight-binding models. The multifractal nature of the eigenstates of these models is corroborated by the scaling of the participation numbers with the systems size. Hence one might have expected `critical' or `intermediate' statistics for the level-spacing distributions as observed at the metal-insulator transition in the three-dimensional Anderson model of disorder. However, our numerical results are in perfect agreement with the universal level-spacing distributions of the Gaussian orthogonal random matrix ensemble, including the distribution of spacings between second, third, and forth neighbour energy levels.

cond-mat.dis-nn↗

arXiv1999

Trace maps, invariants, and some of their applications

Trace maps of two-letter substitution rules are investigated with special emphasis on the underlying algebraic structure and on the existence of invariants. We illustrate the results with the generalized Fibonacci chains and show that the well-known Fricke character I(x,y,z) = x^2 + y^2 + z^2 - 2 x y z - 1 is not the only type of invariant that can occur. We discuss several physical applications to electronic spectra including the gap-labeling theorem, to kicked two-level systems, and to the classical 1D Ising model with non-commuting transfer matrices.

math-ph↗

arXiv1999

Some remarks on the visible points of a lattice

We comment on the set of visible points of a lattice and its Fourier transform, thus continuing and generalizing previous work by Schroeder and Mosseri. A closed formula in terms of Dirichlet series is obtained for the Bragg part of the Fourier transform. We compare this calculation with the outcome of an optical Fourier transform of the visible points of the 2D square lattice.

math-ph↗

arXiv1999

Modelling Quasicrystal Growth

Understanding the growth of quasicrystals poses a challenging problem, not the least because the quasiperiodic order present in idealized mathematical models of quasicrystals prohibit simple local growth algorithms. This can only be circumvented by allowing for some degree of disorder, which of course is always present in real quasicrystalline samples. In this review, we give an overview of the present state of theoretical research, addressing the problems, the different approaches and the results obtained so far.

cond-mat.dis-nn↗

arXiv1999

Aperiodic Tilings on the Computer

We briefly review the standard methods used to construct quasiperiodic tilings, such as the projection, the inflation, and the grid method. A number of sample Mathematica programs, implementing the different approaches for one- and two-dimensional examples, are discussed. Apart from small examples, the corresponding programs are not contained in the text, but will be made available in electronic form.

cond-mat.mtrl-sci↗

arXiv1999

Non-periodic Ising quantum chains and conformal invariance

A while ago, Luck (J. Stat. Phys. 72 (1993) 417) investigated the critical behaviour of one-dimensional Ising quantum chains with couplings constants modulated according to general non-periodic sequences. In this short note, we take a closer look at the case where the sequences are obtained from (two-letter) substitution rules and at the consequences of Luck's results at criticality. They imply that only for a certain class of substitution rules the long-distance behaviour is still described by the c=1/2 conformal field theory of a free Majorana fermion as for the periodic Ising quantum chain, whereas the general case does not lead to a conformally invariant scaling limit.

cond-mat.stat-mech↗

arXiv1999

Ising spins on the labyrinth

We consider a zero-field Ising model defined on a quasiperiodic graph, the so-called Labyrinth tiling. Exact information about the critical behaviour is obtained from duality arguments and the subclass of models which yield commuting transfer matrices. For the latter, the magnetization is independent of the position and the phase transition between ordered and disordered phase belongs to the Onsager universality class. In order to obtain information about the generic case, we calculate the magnetization for a series of couplings by standard Monte-Carlo methods.

cond-mat.stat-mech↗

arXiv1999

Lee-Yang zeros for substitutional systems

Qualitative and quantitative information about critical phenomena is provided by the distribution of zeros of the partition function in the complex plane. We apply this idea to Ising models on non-periodic systems based on substitution. In 1D we consider the Thue-Morse chain and show that the magnetic field zeros are filling a Cantor subset of the unit circle, the gaps being related to a general gap labeling theorem. In 2D we study the temperature zeros of the Ising model on the Ammann-Beenker tiling. The use of corner transfer matrices allows an efficient calculation of the partition function for rather large patches which results in a reasonable estimate of the critical temperature.

cond-mat.stat-mech↗

arXiv1999

High-temperature expansion for Ising models on quasiperiodic tilings

We consider high-temperature expansions for the free energy of zero-field Ising models on planar quasiperiodic graphs. For the Penrose and the octagonal Ammann-Beenker tiling, we compute the expansion coefficients up to 18th order. As a by-product, we obtain exact vertex-averaged numbers of self-avoiding polygons on these quasiperiodic graphs. In addition, we analyze periodic approximants by computing the partition function via the Kac-Ward determinant. For the critical properties, we find complete agreement with the commonly accepted conjecture that the models under consideration belong to the same universality class as those on periodic two-dimensional lattices.

cond-mat.stat-mech↗

arXiv1998

Electronic states in topologically disordered systems

Networks generated from Voronoi tessellations of space are prototypes for topologically disordered systems. In order to assess the effects of random coordination on the statistical properties of energy spectra, we analyze tight-binding models for planar topologically disordered systems. To this end, the networks are generated by a simple topological model covering a wide range of naturally observed random structures. We find that the energy-level-spacing distributions exhibit level repulsion, similar to the spacings in the energetically disordered Anderson model of localization in the metallic regime.

cond-mat.dis-nn↗

arXiv1998

Energy Levels of Quasiperiodic Hamiltonians, Spectral Unfolding, and Random Matrix Theory

We consider a tight-binding Hamiltonian defined on the quasiperiodic Ammann-Beenker tiling. Although the density of states (DOS) is rather spiky, the integrated DOS is quite smooth and can be used to perform spectral unfolding. The effect of unfolding on the integrated level-spacing distribution is investigated for various parts of the spectrum which show different behaviour of the DOS. For energy intervals with approximately constant DOS, we find good agreement with the distribution of the Gaussian orthogonal random matrix ensemble (GOE) even without unfolding. For energy ranges with fluctuating DOS, we observe deviations from the GOE result. After unfolding, we always recover the GOE distribution.

cond-mat.dis-nn↗

arXiv1998

Lax pair formulation for a small-polaron chain with integrable boundaries

Using a fermionic version of the Lax pair formulation, we construct an integrable small-polaron model with general open boundary conditions. The Lax pair and the boundary supermatrices $K_{\pm}$ for the model are obtained. This provides a direct proof of the integrability of the model.

cond-mat.str-el↗

arXiv1998

Application of random matrix theory to quasiperiodic systems

We study statistical properties of energy spectra of a tight-binding model on the two-dimensional quasiperiodic Ammann-Beenker tiling. Taking into account the symmetries of finite approximants, we find that the underlying universal level-spacing distribution is given by the Gaussian orthogonal random matrix ensemble, and thus differs from the critical level-spacing distribution observed at the metal-insulator transition in the three-dimensional Anderson model of disorder. Our data allow us to see the difference to the Wigner surmise.

cond-mat.dis-nn↗

arXiv1998

Coordination Sequences and Critical Points

Coordination sequences of periodic and quasiperiodic graphs are analysed. These count the number of points that can be reached from a given point of the graph by a number of steps along its bonds, thus generalising the familiar coordination number which is just the first member of this series. A possible application to the theory of critical phenomena in lattice models is outlined.

cond-mat.stat-mech↗

arXiv1998

The Entropy of Square-Free Words

Finite alphabets of at least three letters permit the construction of square-free words of infinite length. We show that the entropy density is strictly positive and derive reasonable lower and upper bounds. Finally, we present an approximate formula which is asymptotically exact with rapid convergence in the number of letters.

math-ph↗

arXiv1998

Magnetotransport in Quasilattices

The dc conductance and the Hall voltage of planar arrays of interconnected quantum wires are calculated numerically. Our systems are derived from finite patches of aperiodic graphs, with completely symmetric scatterers placed on their vertices which are interconnected by ideal quantum wires. Already in a periodic square lattice arrangement, quantum interference effects lead to complicated magnetotransport properties related to the Hofstadter butterfly. For rectangular Fibonacci grids and other quasiperiodic lattices, we obtain still more complex fractal patterns. In particular, irrational ratios of edge lengths and of tile areas in our samples destroy the periodicities with respect to the Fermi wave vector and the magnetic flux, respectively.

cond-mat.mes-hall↗

arXiv1998

Localization of Electronic Wave Functions on Quasiperiodic Lattices

We study electronic eigenstates on quasiperiodic lattices using a tight-binding Hamiltonian in the vertex model. In particular, the two-dimensional Penrose tiling and the three-dimensional icosahedral Ammann-Kramer tiling are considered. Our main interest concerns the decay form and the self-similarity of the electronic wave functions, which we compute numerically for periodic approximants of the perfect quasiperiodic structure. In order to investigate the suggested power-law localization of states, we calculate their participation numbers and structural entropy. We also perform a multifractal analysis of the eigenstates by standard box-counting methods. Our results indicate a rather different behaviour of the two- and the three-dimensional systems. Whereas the eigenstates on the Penrose tiling typically show power-law localization, this was not observed for the icosahedral tiling.

cond-mat.dis-nn↗