The q-Dependent Susceptibility of Quasiperiodic Ising Models

By

Helen Au-Yang andJacques H.H. Perk

Supported by NSF Grant PHY 01-00041

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

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**

The q-Dependent Susceptibility is defined as***

the Fourier transform of the connected pair correlation function.

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.**

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

Regular Pentagrid

The pentagrid is a superposition of 5 grids, each of which consistsof parallel equidistanced lines.1 These grid lines are the five differentkinds of rapidity lines in a Z-invariant Ising model.2

Penrose TilesEach line in the jth grid is given by (for some integer kj)

Mapping that turnsthe pentagrid intoPenrose Tiles:

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.**

Shift: 0+ 1+2 +3 +4= c

j Kj(z)=1, 2, 3, 4, 5 : No simple matching rules

Penrose showed these tiles fill the whole plane aperiodically.

Shifts

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

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.

Silver mean Sequence 1= 1+ √2: 1/ (q): (T<Tc) ( =1,2)

Silver mean Sequence 1/ (q): (T<Tc) ( =4,8)

Fibonacci Sequence 1= (1+ √5)/2: 1/ (q): (T>Tc) ( =1,2)

Fibonacci Sequence 1/ (q): (T>Tc) ( =4,8)

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.

Fibonacci Ising Model: T<Tc: =4,20

Fibonacci Ising Model: T>Tc: =4,20

Fibonacci Ising Model: T<Tc: =4,20

Fibonacci Ising Model: T>Tc: =4,20

Fibonacci and Silver Mean =16 (T>Tc)

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

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.

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

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

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

Detail near central intensity peak:Average correlation length 2,

less far below critical temperature.

Detail near central intensity peak:Average correlation length 4,

lest far below critical temperature.

Central intensity peak: T>Tc (=4)