1. The q-Dependent Susceptibility of Quasiperiodic Ising ModelsBy
Helen Au-Yang and Jacques H.H. Perk
Supported by NSF Grant PHY
Dept. of Physics, Oklahoma State University
Stillwater, O.K

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. Quasicrystals
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.*
Diffraction Pattern:
Structure Function = Fourier Transform of the density-
density correlation functions**
Even before this discovery, great deal is known about the aperiodic tiles by the Mathematician, Penrose, de Bruijm.
*Since then, even more research, both theoretical and experimental are done. Much are due to these great mathematicians.
**At zero temperature, the atoms are frozen and the density-density correlation is 1; S(q) can be calculated for some of the quasiperiodic lattice.
The q-Dependent Susceptibility is what are measured in the Neutron scattering experiment. These quantities so far could not be evaluated for the model is inhomogenous and lattice is irregular, except in certain kind of Ising model.
The q-Dependent Susceptibility is defined as***
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.**
*The Ising model can be related to the lattice gas where  on each site =0,1; by letting  =(1)/2. From the
work of Baxter, one finds that the correlation functions of such inhomogenous Ising lattices, can be calculated,
Not only that, the lattice can be made irregular, or quasi-periodic.
**The experiments were not done at zero temperature, and there are a lot of vacancies, and dislocations in these alloys. These Boltzmann weight gives a proper way to estimate the thermo-excitations.

5. Quasiperiodic sequences:
For j=0, 0= (1+ √5)/2, it is the golden ratio, and the sequence p0(n) has two kinds of blocks consists of 1 or 10.
For j=1, 1= 1+ √2, it is the silver mean, and the sequence p1(n) has two kinds of blocks consists of 10 or 100
For arbitary j, the sequence pj(n) has two kinds of blocks consists of 1, 0…0 (1 followed by j zeroes )or block [1 followed by(j+1) zeroes]
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
1.The pentagrid is called regular if no more than two lines intersect at one point. The area between grid lines are called meshs (or face).
2. We use the difference equation derived by Perk in for the inhomogenous Ising model to calculate correlation to high accuracy.
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:
The jth grid is shifted by j so that no more than two line passing through the origin.
The strip between the kj-1 and kj lines, we assign the integer kj. Thus every point in the same mesh has the same set of integers. This mapping maps each mesh to a point in the complex plane. As the pentagrid is regular, each
Intersection iwhich s surrounded by four mesh, is mapped to the four vertices of the Penrose Tiles.
Here we have : f{z), f(z)+1, f(z)+n, f(z)+1+n (n=2,4).

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.
deBruijn showed that if the sum of the shifts is zero, then it is possible to put arrows on the tiles, and the requirement that arrow of the adjecant tiles match -- this so called matching rule forces the tiles to tile the entire
plane aperiodically. If this matching rule are violated, periodic tiling can be formed. Since molecules that form the quasicrystal do have energetically preferred arrangments, it is believed tiles with matching rules are more closely related to the real world. A great deal is known about this case.
Our calculation will also be concentrated to this case. Since the odd spins only interact with the odd spins, the interaction are along the diagonal of the fat and the thin Penrose tiles. There are four different kind of diagonal, and therefore four different kind of interactions, which are orientationally independent.
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.
As the temperature increases, not all of these sites are occupied. This create vacancies and dislocations. However, the underline pentagrid still satisfied the zero total shift condition. This is rather peculiar espect deserve further investigation.

10. Results: Regular lattices Ferromagnetic 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.
Here we will show the results for two kinds of quasi-periodic Ising models. But these results are true for the other cases that we have examined.

11. Silver mean Sequence 1= 1+ √2: 1/ (q): (T<Tc) ( =1,2)
/2 Is the correlation length, and approaches infinity as T approaches Tc.

12. Silver mean Sequence 1/ (q): (T<Tc) ( =4,8)
As the temperature increases, the peaks remain at the commensurate spots.

13. Fibonacci Sequence 1= (1+ √5)/2: 1/ (q): (T>Tc) ( =1,2)
Now above Tc ,  Is the correlation length. Here is the inverse density plot at very low temperater

14. Fibonacci Sequence 1/ (q): (T>Tc) ( =4,8)
These peaks are at the same commensuate positions (2n, 2m). The intensity increases as T approaches Tc.

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.
In the Ising model, it is well-known that the correlation length of spin correlation function above Tc is double of that below Tc.

16. Fibonacci Ising Model: T<Tc:  =4,20
The interactions have the same aboslute value, but their signs are either + or - , which formed the Fibonacci sequence in the horizontal and vertical directions. The correlation length is /2 for =4,20.

17. Fibonacci Ising Model: T>Tc:  =4,20
The interactions have the same aboslute value, but their signs are either + or - , which formed the Fibonacci sequence in the horizontal and vertical directions.

18. Fibonacci Ising Model: T<Tc:  =4,20
The interactions have the same aboslute value, but their signs are either + or - , which formed the Fibonacci sequence in the two diagonal directions . The correlation length is /2 for =4,20.

19. Fibonacci Ising Model: T>Tc:  =4,20
The interactions have the same aboslute value, but their signs are either + or - , which formed the Fibonacci sequence in the two diagonal directions

20. Fibonacci and Silver Mean  =16 (T>Tc)
The interactions have the same aboslute value, but their signs are either + or - , which formed the Fibonacci sequence in the horizontal and vertical directions on the left, and Silver mean sequence on the right.

21.  =16 (T>Tc) j=2: …0010001001… j=3: …0001000100001…
The interactions have the same aboslute value, but their signs are either + or - , which formed the pj(n) sequence in the two diagonal directions. Shown on the left is for j=2, and j=3 on the right.

22. Quasiperiodic Lattice Pentagrid-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)
This shows that (q) is quasi-periodic:
It does not exactly repeat itself. But, note how near the corners are areas which are quite similar to center. Also, note the ten-fold symmetry around the center. The correlation length is /2 for =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.
More peaks appears as T approaches Tc. They are everywhere dense.

28. Central intensity peak: T>Tc (=4)
There are far more peaks for for T above the critical temperature Tc then below Tc