Phase Transitions

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Phase Transitions and Critical Phenomena Steven
T Bramwell London Centre for Nanotechnology, UCL
Introduction to the concepts

• Q What should you see in experiment?

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Plan of Lecture 1 Phase Transitions
Introduction 1. Ising model 2. Spin and
Lattice Dimensionality 3. Effect of Dipolar
Interactions
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Microscopic interactions

• Exchange Hex
• Dipole-dipole Hdip
• Single ion HSI
• External Field HZeeman

? Try and capture with simple spin models
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Frustration versus (lattice) disorder
Frustration a combined effect of lattice and
interaction
spin ice spin liquid
Periodic order
spin glass
percolation
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Order parameter, correlation function and
susceptibility
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Intensive versus non-intensive order parameter
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(1) The Ising Model
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Ising paramagnet in a field
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Ising Ferromagnet with long range exchange
interactions
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1D Ising ferromagnet with short range exchange
interactions
N spins N-1 bonds
Partition Function
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Correlation function
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2D Ising ferromagnet with short range exchange
interactions
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Two Real Ising Magnets
K2CoF4 - nn AFM interactions
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K2CoF4 HoRh4B4
M C
M
T
T
Hirakawa Ikeda Ott et al.
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(2) Spin and lattice dimensionality
Nine (D,n) classes
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Size dependence of Order Parameter at finite T
(Classical ferromagnetic models Mermin-Wagner
Theorem)
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Phase Transition at finite T (classical
ferromagnetic models)
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Phase diagram 2D Ising vs 2D-XY
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(3) Effect of Dipolar Interactions
Dipole interaction D lt 3 convergent (short
ranged, weak) D 3 conditionally convergent
(long ranged)
Note of caution observation of the following
effects depends on the details of the system!
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Effect of Dipole-dipole interactions in 3D
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Order parameter M Classical ferromagnetic models
with dipolar interaction at equilibrium
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Phase diagram 3D Heisenberg vs
3D-Heisenberg-dipolar (soft FM)
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Conclusions of Lecture 1

• Intensive and non intensive order parameters
• Nine D-n classes various partitions

- But what should we see in experiment??!
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Plan of Lecture 2 Critical Phenomena

• Introduction
• Intensive Order Parameter
• - critical exponents, RG theory, crossover
• 2. Non-Intensive order parameter
• - 1D models, 2D-XY model
• 3. Conclusions

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Critical exponents and exponent equalities

• Power law behaviour arises from scale invarience
• Exponents depend on certain symmetries (not on
  fine details)
• (D,n) Universality classes define sets of
  exponents

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Renormalisation Procedure
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Renormalisation group flow
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Crossover

• Exponents associated with linear flow around
  fixed points
• Thermal properties determined by fluctuations on
  scale of x
• ? crossover phenomena (small relevant variable)

Interpreting experiment is about assessing the
relevance of variables
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Ising-like K2CoF4 Phase diagram
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Fluctuations 2D or 3D ?

• Fluctuations become
• 3D for x2 gt J/J

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Finite size behaves as a small relevant variable
Apart from very close to TC inter-layer coupling
is equivalent to an effective finite size
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(2) Non-intensive order parameters
x1/T 1D Heisenberg 1D XY
xexp(1/T) 1D Ising 2D Heisenberg
xexp(1/(T-Tc)1/2) 2D XY
M e-LT
M exp(-elnL-1/T)
M (cN)-T/8?J
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Magnetization for N108
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The 2D-XY Model - KTB Transition
VL Berezinzkii, Sov. Phys. JETP 12 480 (1971) JM
Kosterlitz, DJ Thouless, J Phys C 6 1181
(1973) JM Kosterlitz, J Phys C 7 1046 (1974) AP
Young J Phys C 11 L453 (1978)
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Free Vortex
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Free vortex energy
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Free vortex free energy
N places to put the vortex

• Free vortices generated above Tc - destroy local
  magnetic order
• ? What happens below Tc, ?

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Vortex Pair Below Tc
Eqiqjln(rij)
? Finite energy local magnetization preserved
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KTB vortex unbinding transition
Local magnetization Spin waves
Increasing T
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Low temperature spin wave regime
Keff ? 2/p at Tc/J lt p/2 universal jump
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Above Tc paramagnetic regime
xexp(b/(T-Tc)1/2 ) cx2-h
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2D-XY Phase diagram
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Non Intensive magnetization in critical region

• To account for vortex pairs replace K by
  Keff(T,L) in spin wave theory

? Then analyse the behaviour of Keff(T,L)
ST Bramwell and PCW Holdsworth JPCM 5 L53 (1993)
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Linearised RG equation for Keff
LC Keff? 0
LKeff? 2/p
LCL2

• Stop flow at b L (good approximation for 2D-XY
  model, because it is critical even at small
  length scales).

Bramwell Holdsworth PRB 49 8811 (1994)
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Two temperature scales emerge
2/p
TC-TKT4(T-TKT)
? L
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Calculation of Magnetization
A universal effective exponent!
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Crossover in quasi-2D XY magnets
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Crossover to 2DXY behaviour in layered Ising
Antiferromagnet
Rb2MnCl4 Ising D 3?10-3J
Spin flop
H
Van de Kamp et al, Physica B 241-243 570 (1998)
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Ultrathin films 2.2 ML Fe(100) on W(100)
Elmers et al. J Appl. Phys., 79 4984 (1998)
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Ni films on Cu(100), Cu(111), W(110)
Huang et al, PRB 49 3962 (1994)
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Conclusions for Lecture 2

• Universality classes
• Relevent, irrelevant, marginal parameters
• Crossover phenomena
• 2D-XY model - critical phase and effective b
  0.23

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Overall Conclusion

• Q What should you see in experiment?
• A It all depends on the length scale!