The q -Dependent Susceptibility of Quasiperiodic Ising Models

1. The q-Dependent Susceptibility of Quasiperiodic Ising Models By Helen Au-Yang andJacques H.H. Perk Supported by NSF Grant PHY 01-00041

2. Outline • Introduction: • Quasicrystals: • q-Dependent Susceptibility: • Regular lattice with Quasi-periodic interactions : • Quasi-Periodic Sequences – Aperiodic Ising lattices. • Quasi-periodicity in the lattice structure: • Pentagrid–Penrose tiles • Results

3. In 1984, Shechtman et al. found five-fold symmetry in the diffraction pattern of some alloys. As such symmetry is incompatible with periodicity, their crystalline structure must be aperiodic.* The q-Dependent Susceptibility is defined as*** Quasicrystals Diffraction Pattern: Structure Function = Fourier Transform of the density- density correlation functions** the Fourier transform of the connected pair correlation function.

4. The Lattice of Z-invariant Ising model • The rapidity lines on the medial graph are represented by oriented dashed lines. • The positions of the spins are indicated by small black circles, the positions of the dual spins by white circles. Each spin takes two values, =1.* • The interactions are only between the black spins, and are function of the two rapidities line sandwiched between them. • Boltzmann weight P=e K’is the probability for the pair.**

5. Quasiperiodic sequences: Quasi-periodic Ising model: un = uA if pj(n)=0, and un = uB if the pj(n)=1. Knm=K if pj(n)=0, and Knm=-K if pj(n)=1

6. Regular Pentagrid The pentagrid is a superposition of 5 grids, each of which consists of parallel equidistanced lines.1 These grid lines are the five different kinds of rapidity lines in a Z-invariant Ising model.2

7. Penrose Tiles Each line in the jth grid is given by (for some integer kj) Mapping that turns the pentagrid into Penrose Tiles:

8. Shifts Shift: 0+1+2 +3 +4=0* The index of a Mesh: j Kj(z)=1, 2, 3, 4. Odd sites = index 1,3 Even sites = index 2,4.** Penrose showed these tiles fill the whole plane aperiodically. Shift: 0+1+2 +3 +4= c j Kj(z)=1, 2, 3, 4, 5 : No simple matching rules

9. Half of the sites of a Penrose tiling interact as indicated by the lines. The other sites play no role.

10. Results: Regular latticesFerromagnetic Interactions • The q-dependent susceptibilities (q) of the models, on regular lattices, are always periodic. • When the interactions between spins are quasi-periodic, but ferromagnetic, (q) has only commensurate peaks, similar to the behavior of regular Ising models. • The intensity of the peaks depend on temperature, and increases as T approaches Tc.

11. Silver mean Sequence1= 1+ √2: 1/ (q): (T<Tc) ( =1,2)

12. Silver mean Sequence1/ (q): (T<Tc) ( =4,8)

13. Fibonacci Sequence1= (1+ √5)/2:1/ (q): (T>Tc) ( =1,2)

14. Fibonacci Sequence1/ (q): (T>Tc) ( =4,8)

15. Mixed Interactions:Ferro & Anti-ferromagnetic • The susceptibilities (q) is periodic and has everywhere dense incommensurate peaks in every unit cell, when both ferro and anti-ferromagnetic interactions are present. • These peaks are not all visible when the temperature is far away from the critical temperature Tc. The number of visible peaks increases as TTc. • For T above Tc, (the disorder state), the number of peaks are more dense*. • Structure function are different for different aperiodic sequences.

16. Fibonacci Ising Model: T<Tc:  =4,20

17. Fibonacci Ising Model: T>Tc:  =4,20

18. Fibonacci Ising Model: T<Tc:  =4,20

19. Fibonacci Ising Model: T>Tc:  =4,20

20. Fibonacci and Silver Mean  =16 (T>Tc)

21.  =16 (T>Tc) j=2: …0010001001… j=3: …0001000100001…

22. Quasiperiodic LatticePentagrid-Penrose Tiles • When the lattice is quasiperiodic --- such as Z-invariant Ising model on the Penrose tiles ---  (q) is no longer periodic but quasiperiodic. • Even if interactions between spins are regular and ferromagnetic,  (q) exhibits everywhere dense and incommensurate peaks. • These peaks are not all visible when the temperature is far away from the critical temperature. The number of visible peaks increases as T approaches the critical temperature Tc. • For T above Tc,, when the system is in the disordered state, there are more peaks.

23. Ising Model on Penrose Tiles:T<Tc (=4)

24. Ising Model on Penrose Tiles:T>Tc (=4)

25. Detail near central intensity peak:Average correlation length 1,far below critical temperature.

26. Detail near central intensity peak:Average correlation length 2,less far below critical temperature.

27. Detail near central intensity peak:Average correlation length 4,lest far below critical temperature.

28. Central intensity peak:T>Tc (=4)