The qDependent Susceptibility of Quasiperiodic Ising Models

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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

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Outline

• Introduction
• Quasicrystals
• q-Dependent Susceptibility
• Regular lattice with Quasi-periodic interactions
  
• Quasi-Periodic Sequences Aperiodic Ising
  lattices.
• Quasi-periodicity in the lattice structure
• PentagridPenrose tiles
• Results

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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.
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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 Pe K?? is the probability for
  the pair.

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Quasiperiodic sequences
Quasi-periodic Ising model un uA if pj(n)0,
and un uB if the pj(n)1. KnmK if pj(n)0,
and Knm-K if pj(n)1
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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
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Penrose Tiles
Each line in the jth grid is given by (for some
integer kj)
Mapping that turns the pentagrid into Penrose
Tiles
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Shifts
Shift ?0 ?1?2 ?3 ?40
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
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Half of the sites of a Penrose tiling interact as
indicated by the lines. The other sites play no
role.
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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.

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Silver mean Sequence ?1 1 v2 1/ ?(q)
(TltTc) (? 1,2)
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Silver mean Sequence 1/ ?(q) (TltTc) (? 4,8)
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Fibonacci Sequence ?1 (1 v5)/2 1/ ?(q)
(TgtTc) (? 1,2)
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Fibonacci Sequence 1/ ?(q) (TgtTc) (? 4,8)
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Mixed InteractionsFerro 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.

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Fibonacci Ising Model TltTc ? 4,20
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Fibonacci Ising Model TgtTc ? 4,20
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Fibonacci Ising Model TltTc ? 4,20
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Fibonacci Ising Model TgtTc ? 4,20
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Fibonacci and Silver Mean ? 16 (TgtTc)
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? 16 (TgtTc) j2 0010001001 j3
0001000100001
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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.

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Ising Model on Penrose TilesTltTc (?4)
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Ising Model on Penrose TilesTgtTc (?4)
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Detail near central intensity peakAverage
correlation length 1,far below critical
temperature.
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Detail near central intensity peakAverage
correlation length 2,less far below critical
temperature.
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Detail near central intensity peakAverage
correlation length 4,lest far below critical
temperature.
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Central intensity peak TgtTc (?4)