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The Heisenberg Hamiltonian H=J∑SiSj with antiferromagnetic interactions between spins (J>0) can lead to different kinds of ground state spin configurations, depending on the type of lattice.

 

Classical picture of an antiferromagnetic state. Néel order : spins take on an alternating up-down arrangement . This state breaks the rotational symmetry. This is possible for structures that have even-sided loops such as the honeycomb or square lattices which are bipartite lattices . This is not the case for the triangular lattice, which is said to be frustrated.

 

For large values of the spin quantum number S, and bipartite lattices, the ground state is close to the classical state , while in low dimensions and for small S, quantum fluctuations become more important and lead to new effects. Much is known, for example, about the properties of the square lattice or the honeycomb lattice spin S=1/2 antiferromagnets which are two-dimensional systems. Quantum fluctuations lead to a reduction of the local spin average value. At T=0 in the square lattice the order parameter is reduced by about 30 %. In the honeycomb lattice (fewer neighbors and therefore bigger fluctuations) by about 50 %.

 

The low energy excitations correspond to spin waves – magnons . In ferro- and antiferromagnets these give rise to characteristic power laws in thermodynamic properties at low temperatures.

 

a microscopic view of spin orientations in a ferromagnetic spin wave state

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Quantum fluctuations in a quasiperiodic antiferromagnet

 

We considered the ground states of the Heisenberg model in the Penrose tiling.

 

Some questions one can ask :

 

  • What is the nature of the ground state of a Heisenberg antiferromagnet on this tiling ?
  • Are quantum fluctuations stronger or weaker than in a comparable periodic system ?
  • What are the Goldstone modes - the low energy spin excitations , if they exist - like in such a system ?

 

Our calculations measured the local spin expectation values as a function of environment, using spin wave theory and Quantum Monte Carlo. The calculations are performed in real space, and do not utilize the property of self similarity (excepting in the case of a different slightly simpler tiling, for which a renormalization group transformation has been proposed).

 

Results for perfect Penrose tilings

 

The density of states of magnons showing the peak at E=3 due to localized states

 

Some Results :

 

The ground state has an inhomogeneous Néel (2 sublattice) order, with a complex pattern of magnetizations in real space which simplifies when mapped into internal space coordinates.

 

Quantum fluctuations are globally weaker than in the square lattice (which has the same coordination number).

 

The quantum fluctuations are smallest for spins with 3 neighbors -these sites have the largest staggered moments. There is a different mean value of the moment for each coordination number, following the predictions for Heisenberg star clusters.

 

Spin waves propagate with a bigger velocity in the random tiling, which has a more smeared out and overall smaller magnetization distribution.

 

The wavefunctions of spin excitations can be extended or multifractal or localized depending on the energy of the excitation. E=3 is an exact eigenenergy, whose states can be written using the known geometrical properties of the tiling.

 

References :

A. Jagannathan, A. Szallas, S.Wessel, M. Duneau, PRB 75 212407 (2007) A. Szallas, A. Jagannathan, PRB 77 104427 (2008) A. Szallas, A. Jagannathan, Stefan Wessel, PRB 79 172406 (2009)

 

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Some geometrical facts about the perfect Penrose tiling

 

The Penrose rhombus tiling shown has many interesting geometrical properties, of which a few are listed here.

 

A sample of the undisordered Penrose rhombus tiling

 

Fourier transform : The structure factor consists of Dirac peaks . The main peaks nearest to the origin are often used to define a quasi-Brillouin zone, but there are an infinity of lesser peaks distributed densely in the reciprocal plane, with five-fold symmetry.

 

Repetitivity : Any given local environment of linear size R is guaranteed to repeat a little further on in the tiling within a distance of about 2R. The repetition is not, of course, periodic as in a crystal. Concerning nearest neighbor environments, a spin can have three, four, five, six or seven neighbors, with a frequency of occurrence of each type of local environment that can be exactly determined in terms of the golden mean, τ = (√5 – 1)/2

 

Inflation : One can transform the tiling into a scaled up or scaled down version of itself by eliminating sites or introducing sites according to well-defined rules. The scale factor is the golden mean τ.

 

Mapping into internal space : Each site has a counterpart in the internal – or perpendicular – space representation of the tiling, in which sites map into four plane pentagonal regions, depending on their spatial characteristics. The two sublattices map into different planes. Planes 1 and 3 for sublattice A and planes 2 and 4 correspond to sublattice B, which is equivalent to A upto a rotation. Sites of different coordination number z map into different domains, as shown in the figure below which uses a different color for each z. Each of the colored domains can in turn can be divided into smaller subdomains depending on the next neighbors, and so on . The invariance of the structure under inflation-deflation leads to a self-similarity of the tiling in the internal space representation.

 

Sites of the tiling as seen in the physical plane…

 

Tiling with spins colored according to number of neighbors (yellow=3,blue=4,red=5, green=6,violet=7)

 

…and as seen in « perpendicular » space (the figure is schematic. It represents parallel planes at four different heights along the vertical axis of this 3D space)

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Random systems

 

The best samples of quasicrystals have been measured to have extremely long structural coherence lengths, but they also have the possibility of allowing new types of vibrational excitations : phason modes, in addition to the phonons always present. In an alternative scenario for quasicrystals, it has been proposed that many quasicrystals are intrinsically disordered, and that they belong to an ensemble of random tilings. This should be reflected in a difference in physical properties of the disordered tilings compared to the perfect ly ordered one. We have therefore considered the magnetism of randomized tilings . They are obtained by disordering a perfect Penrose quasicrystal by introducing « phason flips » where a single site is shifted to a new position.

 

A phason flip from the old (blue) to the new (red) position. Bonds are accordingly redefined so that the result is a local shuffling of rhombuses

 

Two local configurations that are not present in the perfect tiling :– the ten-fold star and the vertex of four fat rhombi- that are produced after randomization

 

Staggered magnetization in a randomized tiling showing the greater spatial homogeneity, of the ground state in this case, as well as a lower global average value

 

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Frustrated systems

 

In an isolated frustrated triangular unit, for example, three spins can minimize energy by aligning at 120° angles between neighboring spins. When many triangles assemble together in a crystal, this can give rise to many different degenerate configurations.This occurs, for example, when the triangular lattice is diluted, giving rise to the Kagome lattice. In three dimensions, a similar situation obtains in the pyrochlores, tetrahedron-based structures. Thermal and/or quantum fluctuations can lead to phenomena, such as order-by-disorder, or new types of nonmagnetic states.

 

Kagome lattice showing spin degrees of freedom An example of such a spin system studied in our laboratory is the mineral Herbertsmithite ZnCu3(OH)6Cl2.

 

Model (green bonds are antiferromagnetic. The addition of the red bonds leads to frustration) SrCu2(BO3)2 is a compound whose spin-spin interactions are well approximated in terms of the Shastry-Sutherland model.

 

Materials such as Ho2Ti2O7 (« spin ice « ) have corner-sharing tetrahedra. The spin degrees of freedom in this compound are Ising-like, and their ferromagnetic couplings lead to frustration.

 

Ising spin representation of a two dimensional spin ice.

 

Many recent experimental and theoretical studies on some of the systems illustrated above are aimed at understanding the low temperature properties of such materials.