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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)