Anuradha Jagannathan

Quantum antiferromagnetism in quasicrystals

The antiferromagnetic Heisenberg model is studied on a two-dimensional bipartite quasiperiodic lattice. Using the stochastic series expansion quantum Monte Carlo method, the distribution of local staggered magnetic moments is determined on finite square approximants with up to 1393 sites, and a nontrivial inhomogeneous ground state is found. A hierarchical structure in the values of the moments is observed which arises from the self-similarity of the quasiperiodic lattice. The computed spin structure factor shows antiferromagnetic modulations that can be measured in neutron scattering and nuclear magnetic resonance experiments. This generic model is a first step towards understanding magnetic quasicrystals such as the recently discovered Zn-Mg-Ho icosahedral structure.

Ground state of a two dimensional quasiperiodic quantum antiferromagnet

We consider the antiferromagnetic spin-1/2 Heisenberg model on a two-dimensional bipartite quasiperiodic tiling. The broken symmetry ground state in this model is inhomogeneous, but nevertheless bears interesting similarities with that of the square lattice antiferromagnet. An approximate block spin renormalization scheme developed first for the square lattice is generalized here to the quasiperiodic case. The ground state energy and local staggered magnetizations for this system are calculated, and compared with the results of a Quantum Monte Carlo calculation for the tiling.

An eightfold optical quasicrystal with cold atoms

We propose a means to realize two-dimensional quasiperiodic structures by trapping atoms in an optical potential. The structures have eightfold symmetry and are closely related to the well-known quasiperiodic octagonal (Ammann-Beenker) tiling. We describe the geometrical properties of the structures obtained by tuning parameters of the system. We discuss some features of the corresponding tight-binding models, and experiments to probe quantum properties of this optical quasicrystal.In a recent paper we proposed a means to realize a quasicrystal with eight-fold symmetry by trapping particles in an optical potential created by four lasers. The quasicrystals obtained in this way, which are closely related to the well-known octagonal tiling, offer unique possibilities to study the effects of quasiperiodicity on physical properties. This method allows, furthermore, to transform the structures, to inflate or deflate them, include interactions or disorder and thus realize a large variety of theoretical models, both classical and quantum. In this paper we present the model, derive a number of interesting geometrical properties of the optical quasicrystals, as well as some results obtained by numerical calculations.

MAGNETIC STATES INDUCED BY ELECTRON-ELECTRON INTERACTIONS IN A PLANE QUASIPERIODIC TILING

We consider the Hubbard model for electrons in a two-dimensional quasiperiodic tiling using the Hartree–Fock approximation. Numerical solutions are obtained for the first three square approximants of the perfect octagonal tiling. At half-filling, the magnetic state is antiferromagnetic. We calculate the distributions of local magnetizations and their dependence on the local environments as U is varied. The inflation symmetry of quasicrystals results in a corresponding inflation symmetry of the magnetic configurations when passing from one tiling to the next in the progression towards the infinite quasicrystal. Experimental studies of magnetism in quasicrystal-forming alloys have shown that, for a given composition, the magnetic structure of quasicrystalline, amorphous and crystalline phases were different. The earliest studies, which were carried out on the intrinsically disordered AlMnSi quasicrystals [1], found that the quasicrystalline samples had a small proportion of magnetic atoms and appeared to undergo a spin glass transition. The crystalline phase of similar composition was non-magnetic. The early work stimulated interest in magnetic properties in quasicrystalline media, because they indicated that moments were formed in the quasicrystal whereas they were absent in close-lying crystalline phases. The actual proportion of moment-carrying atoms was difficult to estimate in the absence of any

Energy-level statistics of electrons in a two-dimensional quasicrystal.

A numerical study is made of the spectra of a tight-binding hamiltonian on square approximants of the quasiperiodic octagonal tiling. Tilings may be pure or random, with different degrees of phason disorder considered. The level statistics for the randomized tilings follow the predictions of random matrix theory, while for the perfect tilings a new type of level statistics is found. In this case, the first-, second- level spacing distributions are well described by lognormal laws with power law tails for large spacing. In addition, level spacing properties being related to properties of the density of states, the latter quantity is studied and the multifractal character of the spectral measure is exhibited.

Quantum Simulation of a 2D Quasicrystal with Cold Atoms

We describe a way to obtain a two-dimensional quasiperiodic tiling with eight-fold symmetry using cold atoms. One can obtain a series of such optical tilings, related by scale transformations, for a series of specific values of the chemical potential of the atoms. A theoretical model for the optical system is described and compared with that of the well-known cut-and-project method for the Ammann–Beenker tiling. The relation between the two tilings is discussed. This type of cold atom structure should allow the simulation of several important lattice models for interacting quantum particles and spins in quasicrystals.

Antiferromagnetism in two-dimensional quasicrystals

Abstract Quasiperiodic structures possess long range positional order, but are freed of constraints imposed by translational invariance. For spins interacting via Heisenberg couplings, one may expect therefore to find novel magnetic configurations in such structures. We have studied magnetic properties for simple two dimensional models, as a first step towards understanding experimentally studied magnetic quasicrystals such as the Zn–Mg–R(rare earth) compounds. We analyse properties of the antiferromagnetic groundstate and magnon excitation modes for bipartite tilings such as the octagonal and Penrose tilings. In the absence of frustration, one has an inhomogeneous Neel-ordered ground state, with local quantum fluctuations of the local staggered magnetic order parameter. We study the spin wave spectrum and wavefunctions of such antiferromagnets within the linear spin wave approximation. Some results for spin-spin correlations in these systems are discussed.

Self-similarity under inflation and level statistics: A study in two dimensions

Energy level spacing statistics are discussed for a two dimensional quasiperiodic tiling. The property of self-similarity under inflation is used to write a recursion relation for the level spacing distributions defined on square approximants to the perfect quasiperiodic structure. New distribution functions are defined and determined by a combination of numerical and analytical calculations.

Geometric fluctuations in a two-dimensional quantum antiferromagnet

We consider the effects of random fluctuations in the local geometry on the ground state properties of a two-dimensional quantum antiferromagnet. We analyse the behavior of spins described by the Heisenberg model as a function of what we call phason disorder, following a terminology used for aperiodic systems. The calculations were carried out both within linear spin wave theory and using quantum Monte Carlo simulations. An ”order by disorder”-like phenomenon is observed in this model, wherein antiferromagnetism is found to be enhanced by phason disorder. The value of the staggered order parameter increases with the number of defects, accompanied by an increase in the ground state energy of the system. We furthermore find a long-ranged attractive Casimir-like force between two domain walls of defects separated by a finite distance.

Quasiperiodic Heisenberg antiferromagnets in two dimensions

Abstract We describe some of the properties of 2d quantum quasiperiodic antiferromagnets as reported in studies that have been carried out in the last decade. Many results have been obtained for perfectly ordered as well as for disordered two dimensional bipartite quasiperiodic tilings. The theoretical methods used include spin wave theory, and renormalization group along with Quantum Monte Carlo simulations. These methods all show that the ground state of these unfrustrated antiferromagnets have Néel type order but with a highly complex spatial distribution of local staggered magnetization. The ground state properties, excitation energies and spatial dependence, structure factor, and local susceptibilities are presented and discussed. The effects of introducing geometrical disorder on the magnetic properties are discussed.