SMP IOP Hanoi Nov. 18-27, 2002
I.
Applications of computer simulations in condensed matter and
statistical physics

Introduction
In this series of lectures, I will discuss some topics in condensed matter and statistical physics
and to show how to write computer programs to make simulations in these topics.
H. T. Diep

LPTM, UCP
I.
Introduction
II.
Background on Magnetism
Background on Statistical Physics
Background on Monte Carlo Methods
Practice on computer on bulk magnetism
III.
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

Surface Magnetism
Theory
Simulations
Practice on computer


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Frustrated Spin Systems
Theory
Simulations
Practice on computer
IV.
V.
Advanced Monte Carlo Techniques
VI.
Conclusions and Perspectives
II.


For computer simulations, I will concentrate on programs using Monte Carlo methods.

The basic background required for this series of lectures includes the standard theory of
magnetism and elementary statistical theory of phase transition and critical phenomena.

Background on magnetism
Magnetic Interactions and Models
Si : Ising, XY, Heisenberg spins
J(Rij) : exchange interaction
J(Rij) > 0 ferro. ,
Mean Field Theory
Theory
- Spin-Wave
For computer programming, I will use Fortran language in examples shown on the above
topics (but translation to other languages is straightforward).
J(Rij)
Background on statistical physics



H= -  J(Rij) Si.Sj
-
I will limit the discussion on two following topics: the surface magnetism and frustrated spin
systems.
Plan:
Canonical Ensembles
Phase Transition and Universality
Finite Size Scaling
Background on Monte Carlo Method
< 0 antiferro.

Principles

Applications: Programs for Bulk Magnetism
before becoming disordered at higher T. This 'magnetization reorientation' has been shown to be
III. Surface magnetism
a consequence of the competition between perpendicular anisotropy and long-range dipolar
I.
Introduction:
interaction.
Physics at surfaces has been intensively developed over the past thirty years. Many
Note that for magnetic thin films, it has been shown a long time ago
spectacular properties due to the presence of surfaces have been discovered and used in
that low-lying-energy surface-localized modes play an important role in thermodynamic properties
industrial devices. In particular,
of thin films at low T. One of the effects of these surface modes is the low surface magnetization
physics of magnetic thin films and multilayers have been a subject of intensive investigations
with respect to the bulk one. One has to bear this in mind while analyzing numerical results as well
during the last decades.
as analytical ones.
This is due on the one hand to numerous recording applications and on the other hand to
Since the geometry of very small systems (dots, thin films, …) and their surface conditions are
fundamental interests in understanding physical mechanisms involved in those small-scale
complicated, analytical calculations are often impossible. One needs therefore NUMERICAL
systems which are quite different from bulk
SIMULATIONS to study thermodynamic properties of those systems.
properties.
References : A. Zangwill, Physics at Surfaces, Cambridge Univ. Press (1994).
One of the important properties in multilayers is the magnetoresistance hysteresis shape of the
Bland and Heinrich (Editors) Ultrathin Magnetic Structures, Springer-Verlag (1994).
system under an applied field. Another interesting feature in multilayers composed of very thin
films is the perpendicular magnetic anisotropy.
In particular, it has been observed that the film magnetization which is
perpendicular to the film surface at low temperature (T) can change into parallel configuration
at intermediate T
II. Surface Geometries and Surface Interactions:
Clean Surface
III.
Elementary Surface Excitations:
1.
Steps , Islands and Vacancies
Surface Spin-Waves (SW):
Surface Impurities
It is very simple to calculate the spin-wave spectrum for a semi-infinite
Chemisorption
crystal or a thin film.
Adsorption
Surface Interaction Parameters

Surface Density of State (Electronic Band
Structure)

Surface Magnetic Moment

Surface Anisotropy : magnitude, sign, orientation

Surface Exchange Interaction

…
The first approximation consists of
-
for a spin at the surface and for the spins on the beneath layers to
obtain a set of coupled equations
Surface Reconstruction :

Crystalline Structure : dilation or contraction of
lattice constant
Magnetic Structure : Spin Configuration may be non collinear near the surface
writing down the equation of motion for the spin operators S+ and S-
-
using the time and in-plane Fourrier transforms
-
solving the resulting coupled equations to get the spin-wave
frequencies and spin-wave amplitudes
Example :

Surface localized modes lie usually outside the continuum bulk SW band and their
amplitudes are damped from surface toward inside

2.
Usually, surface mode frequencies depend on surface exchange interaction, surface
anisotropy, surface moments, …
Surface Magnetization :
One can calculate the layer magnetization at a given T using the spin-wave spectrum
obtained
Self-consistent solution should be searched for all Mi (i=1,2,3, …). In practice, one retains
only a few different magnetizations M1  M2  M3  M4 = M5= … for simplicity
Example :
Of course, for surface spin-waves one can go to higher approximations
IV.
Surface Phase Transition:
using non linear SW theory or Green function technique .
The SW theory allows to calculate magnetizations at low T (T<T_c/3). The
Green function method permits to go to higher T.
Phase Transitions at surfaces can be investigated by
- Numerical Methods :
High-T and low-T Expansions
However for critical behavior at T_c cannot be obtained with precision by these
methods. NUMERICAL CALCULATIONS and Renormalization Group are needed.
Example : Results from Green function technique
Practice: Programs for Thin Films
Monte Carlo Simulations
Renormalization Group
VI. Frustrated Spin Systems
I.
Introduction :
In 1988, Kawamura has suggested from a renormalization group (RG) calculation and Monte Carlo
In recent years, much effort has been devoted to frustrated spin systems.
(MC) simulations that the transition in XY and Heisenberg models belong each to a new
There are several reasons for this increasing interest. On the one hand,
universality class. Soon after, Azaria et al suggested from a non-linear sigma model that if the
many real systems which show competing interactions cannot be explained
transition is not of first order or mean-field tricritical then it should be O(4) universality. Numerical
by simple model Hamiltonians. An example is planar arrays of
simulations (Diep,Loison,Boubcheur,Dobry) did not confirm these conjectures. Antonenko and
coupled Josephson junctions which can be studied by mapping it
Sokoloff go further in RG calculation and conclude that the transition is of first order. At the time
into a frustrated XY two-dimensional (2D) model.
being, the question remains unsettled.
Another spectacular effect of the frustration is the reentrance phenomenon. In some frustrated
On the other hand, from a fundamental point of view in statistical physics, in particular in the
2D Ising exactly solved models, it has been shown that the frustration can give rise to a partial
domain of phase transitions and critical phenomena, frustrated systems are of great interest
disorder which favors the occurence of a reentrance, i.e. a disordered phase between two ordered
because they serve as testing ground for more sophisticated theories and approximations.
phases on the temperature scale (Azara,Diep).
Among the most controversial subjects, we can mention the nature
of the phase
For reviews on various other aspects of frustrated spin systems, see
transition in non collinear magnets such as helimagnets and in XY and Heisenberg stacked
MAGNETIC SYSTEMS WITH COMPETING INTERACTIONS (FRUSTRATED SPIN SYSTEMS),
triangular antiferromagnets (STA).
Ed. Diep, WS (1994).
new updated Edition to appear in 2003.
We note that while the partial disorder has been discovered very recently in XY and
II. Definition
Heisenberg systems (Quaru,Santamaria,Diep) the reentrance has never been found in
any non-Ising models.
Frustration is due to :
-
the competition between different kinds of interaction
-
the geometry of the lattice
Definition (Toulouse, Villain 1977)
 

sign (Jij)
(around a closed plaquette)
If

= -1  frustrated plaquette
If

= +1  unfrustrated plaquette
Example :
V.
Some consequences of frustration
VI.
Exactly Solved Ising Models :
1.
Kagomé lattice :
-
High degeneracy of the ground state (GS)
2.
Union-Jack lattice (centered square lattice) :
-
GS spin configurations are often non collinear
3.
Centered Honeycomb lattice
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Multiple Phase Transitions
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Reentrance
-
Partial Disorder

-
Disorder Lines
functions of elementary cells
-
Order by Disorder

-
New universality classes ( ?) of criticality
-
…

…..
The method used to solve these 2D Ising models :
Write the partition function as the product of partition
Decimate the centered spin of each cell to transform the
system into vertex models


Frustrated spin systems provide TESTING GROUNDS for
theory and simulation
Verify the free-fermion condition
Obtain critical surfaces in phase space
REMARK : The method does not allow to calculate exactly the order parameter :
one needs simulations to look for the NATURE of the ordering
VII.
Vector Spin Models (XY, Heisenberg) :
1.
Villain’s Lattice :
VIII. PARTIAL DISORDER IN FRUSTRATED NON-ISING
CLASSICAL AND QUANTUM SPIN SYSTEMS
I. Introduction
2.
Stacked Triangular Antiferromagnets :
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Partial Disorder is seen in Exacltly Solved Frustrated Ising Spin Systems
Our Motivation:
Does the partial disorder in Ising systems survive in non-Ising classical and quantum spin systems ?
Does the reentrance occur in disorder-free non-Ising spin models ?
II. MODELS:
We consider two models:
1.
The BCC lattice with Heisenberg spins
2.
The stacked checker-board lattice with XY spins
III. METHOD:
An appropriate method should take into account the temperature dependence of the
FOR A GIVEN SET OF INTERACTION PARAMETERS AND AT A GIVEN T
6.
-assuming now input values for Mi and solving the matrix equation to obtain the eigen spin-wave
angle (s) between neighboring spins so that successive phase transitions can be detected.
The method we propose is:
-
take a value of ,
frequencies for all k in the first Brillouin zone, then using the spectral theorem to compute the output local
A Green function method with decoupling of higher order Green functions, BUT with self- magnetizations Mi . Repeat this until the convergence of Mi is obtained. The free energy F is then
consistent solution, i. e. the most important contributions are retained up to infinite orders ofcalculated with the formula
perturbation
F.
The angle determined at each temperature is determined by the minimum of the free energy
7. Repeat the previous step for another angle 
8.
Plot F versus  for T under consideration:
9.
Changing T and repeat steps 6 , 7 , 8.
the correct angle is at minimum F
The following steps are successively performed:
1.
Expressing the Hamiltonian in the reference of local coordinates
2.
Defining the Green functions G for all types of coupling
3.
Writing the equations of motion for all G
IV. RESULTS:
4.
Decoupling the higher-order Green functions to obtain a matrix equation for G where the
-THE BCC LATTICE WITH QUANTUM HEISENBERG MODEL
coefficients depend on the local magnetizations Mi and the angle  between neighboring spins
5.
Performing the Fourier transforms into wave-vector (k) space
-THE STACKED CHECKER-BOARD LATTICE WITH QUANTUM XY SPINS

Practice
VII. Advanced Monte Carlo Techniques
IX. Conclusions and Perspectives

Cluster Algorithm

Theory and Simulation: complementary roles


Surface Magnetism: much to be done
Histogram Techniques

Frustrated Spin Systems: much to be done

Complicated Systems: long-range interactions, magnetoelastic systems, polymers, arrays of
magnetic dots on substrate, magnetic surface melting, …

Applications of Spin Systems in Other Fields: neural networks, alloy structrues, …