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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)