Classical, Discrete Spin Models: Symmetry, Duality and Renormalization

258 Pages·1984·3.186 MB·English

by  Hendrik Moraal (eds.)

Lecture Notes
ni Physics
Edited by .H Araki, Kyoto, .J Ehlers, ,nehcni~M .K Hepp, ZBrich
.R Kippenhahn, MiJnchen, .H .A ,relliLmnedieW Heidelberg
and .J Zittartz, nISK
412
Hendrik laaroM
,lacissalC Discrete
Spin :sledoM
Symmetry, Duality and Renormalization
galreV-regnirpS
nilreB Heidelberg New York oykoT 1984
Author
Hendrik Moraal
Institut fLir Theoret~sche Physik, Universit~t zu K~ln
Z~lpicher Stra6e 77, D-5000 K~ln 41, FRG
ISBN 3-540-13896-X Springer-Verlag Berlin Heidelberg New Tokyo York
ISBN 0-387-13896-X Springer-Verlag New Heidelberg York Berlin Tokyo
This work is subject to copyright. All rights are reserved, whethtehre whole or part of the material
is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting,
reproduction by photocopying machine or similar means, and storage in data banks. Under
§ 54 of the German Copyright Law where copies are made for other than private use, a fee is
payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Vedag Bedin Heidelberg 1984
Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
012345-041313512
Nadie puede escribir un libro.
(Borges)
,ECAFERP
The present book is an outgrowth of a series of lectures held for
students at the graduate level at the Institute for Theoretical Physics
of the University of Cologne over the past five years. The purpose of
the book is to present three aspects of the statistical mechanical the-
ory of discrete, classical spin models in a coherent and essentially
self-contained fashion. These three aspects correspond to the three
parts of the book and are:
)i( The theory of the possible (finite) symmetry groups of spin models
and their classification. This is an essentially mathematical sub-
ject and is treated as such in the first six chapters. Nearly all
of the theory of finite groups needed is developed in the text.
(ii) The theory of the duality transformation is introduced for several
reasons; it not only gives exact information on the phase diagrams
of spin models, but also allows for a simple introduction of lattice
gauge theories. These are treated exclusively from a statistical
mechanics point of view.
(iii) The theory of the renormalization of fields and interactions on
recursively defined sequences of graphs gives a qualitative overview
of possible symmetry-breaking patterns and other aspects of the
phase diagrams of spin and gauge systems.
In order to interrupt the flow of reasoning as little as possible,
the number of references in the text has been kept to a minimum. For
further reading, general references are given at the ends of some of
the chapters. These are mostly to review articles, which, in turn, allow
for easy access to the original literature.
The numbering of equations, theorems, figures, etc., is by section.
Reference is made to these as follows: suppose the reader finds, in
Section 3 of Chapter :5 eq. (5); this is the fifth equation of the same
section of the same chapter; Lemma 2.1; this is the first lemma in the
second section of the same chapter; Fig. 3.5.1; this is the first figure
of the fifth section of the third chapter, etc.
The author thanks Professor J. Hajdu for pressing him not only to
start, but also to finish the present work. Thanks are also due to Mrs.
A. Schneider for expertly drawing the many figures.
Cologne, September 1984. H. Moraal
TABLE OF CONTENTS
PART :A THEORY OF THE SYMMETRY GROUPS OF SPIN MODELS, I
.I SPIN MODELS AND THEIR SYMMETRY GROUPS. I
1.1. Statistical mechanics of spin models on graphs. I
1.2. Interactions. 2
1.3. Maximal interactions and permissible groups. 7
1.4. CoMpletely permissible groups and P-algebras. 11
1.5. External fields and representation theory. 14
References and General References. 22
.2 SUBGROUPS AND PRODUCTS OF PERMISSIBLE GROUPS. 23
2.1. Permissible subgroups. 23
2.2. Wreath products. 27
2.3. Direct products. 13
2.4. Semidirect products. 33
2.5. Abelian groups, corresponding permissible groups and
permissible subgroups of these. 34
References. 37
.3 PRIMITIVE AND IMPRIMITIVE PERMISSIBLE GROUPS. 38
3.1. Primitivity and imprimitivity. 38
3.2. Permissible groups on p p( a prime) letters. 40
3.3. The p-wreath product. 43
References. 46
.4 REGULAR GROUPS. 48
4.1 Spin models defined on groups. 48
4.2. Permissible groups corresponding to regular groups. 48
4.3. Permissible class function groups. 59
4.4. Permissible characteristic interaction groups. 64
References. 68
.5 GRAPH-THEORETICAL CONSTRUCTIONS OF PERMISSIBLE GROUPS. 69
5.1. Primitive permissible groups with two-graph MI's. 69
5.2. Platonic graphs. 72
5.3. Covering graphs. 78
References. 81
.6 TABLES OF PERMISSIBLE GROUPS. 82
6.1. The permissible groups on M~IO letters. 82
6.2. The regular groups with MS12 elements. 88
6.3. Special permissible groups. 89
lV
TRAP :B ,YTILAUD
19
.7 DUALITY TRANSFORMATIONS AND DUAL MODELS. 19
7.1. Duality transformations and inequalities in two
dimensions. 19
7.2. Dual permissible groups. 97
7.3. The question of duality for nonabelian, regular groups. 102
7.4. Duality transformations in dimensions higher than two;
k-gauge models. 105
7.5. Duality transformations for k-gauge models in a
magnetic field. 110
7.6. Self-duality for k-gauge models on hypercubic lattices. 112
7.7. General gauge models and Higgs fields. 127
References and General References. 130
PART C: PSEUDO-LATTICES AND RENORMALIZATION, 132
.8 PSEUDO-LATTICES. 132
8.1. Recursive site graph sequences. 132
8.2. Recursive bond graph sequences. 137
8.3. Free energies, thermodynamic limits and renormalization. 140
8.3.1. Spin systems on recursive site graph sequences. 140
8.3.2. Spin systems on recursive bond graph sequences. 143
8.3.3. Gauge models on recursive bond graphs of class r I. 147
8.4. Infinite pseudo-lattices (Bethe lattices) and the
Be£he-Peierls approximation. 149
8.5. Other recursively defined graph sequences: Fractals. 154
References. 157
.9 HOMOGENEOUS SPIN MODELS ON CAYLEY BRANCHES WITH SURFACE FIELD.158
9.1. Small-field phase transitions. 158
9.2. The Ising and Potts models. 160
9.2.1. The small-field phase transitions. 160
9.2.2. Absence of high-field phase transitions for
the Ising model. 160
9.2.3. Explicit solution of the recursion relation for
Ising systems. 161
9.2.4. High-fieldphase transitions for the Potts model. 163
9.3. Phase diagrams for models with permissible groups with
two-graph maximal interactions. 165
9.4. Phase diagrams for models with permissible groups with
3-graph MI's. 171
9.5. Phase diagrams for models with permissible groups, which
do not have P-algebras. 179
IIV
9.6. The closed Cayley branch and the GsG~G symmetry-breaking
phase transition. 182
9.7. The exponential branch. 184
9.8. Reduction of cactus branches to Cayley branches for
small-field phase transitions. 185
References and General References. 186
10. RANDOM SPIN SYSTEMS ON CAYLEY BRANCHES WITH SURFACE FIELD. 188
10.1. Bond-random spin models: cumulant expansion. 188
10.2. The diluted bond case. 192
10.3. The spin glass case. 198
10.4. Site disorder. 201
Reference and General References. 205
11. SPIN MODELS ON RECURSIVE BOND GRAPH SEQUENCES AND FRACTALS. 207
11.1. Fractals: Koch curves and the Sierpinski gasket. 207
11.2. A self-dual diamond hierarchical lattice. 209
11.3. Other recursive bond graph sequences: the Potts model. 217
11.3.1. Absence of the high-temperature fixed point. 217
11.3.2. Absence of the low-temperature fixed point. 219
11.3.3. Absence of both high- and low-temperature fixed
points. 221
References. 222
12. GAUGE MODELS ON PLAQUETTE BRANCHES. 223
12.1. General formulation. 223
12.2. The gauge Ising model. 226
12.3. Gauge Potts models. 229
12.4. The gauge C(4) model. 231
12.5. The Bethe approximation for gauge models. 234
Reference and General References. 236
13. CRITICAL EXPONENTS FOR SYSTEMS ON SELF-SIMILAR PSEUDO-
LATTICES. 237
13.1. Introductory remarks. 237
13.2. General expressions for the critical exponents. 237
13.3. Cayley branches: phase transitions of continuous order. 239
13.4. Critical exponents for spin and gauge Potts models on
recursive bond graph sequences. 241
References. 242
APPENDIX: TWO EXTENSIONS OF DISCRETE, CLASSICAL SPIN MODELS. 243
A.I. The Potts model for noninteger M and percolation. 243
A.2. A model with an infinite symmetry group. 245
General References. 246
SUBJECT INDEX. 248
PART A : THEORY OF THE SYMMETRY GROUPS OF SPIN MODELS,
I. SPIN MODELS AND THEIR SYMMETRY GROUPS.
1.1. Statistical mechanics of spin models on graphs.
In the following, the statistical mechanics of an assembly of spins,
which for the purpose of these lectures are classical variables taking
on a finite number of values (this number will always be denoted by M
in what follows) will be considered. If these spins interact with each
other pairwise only, as will always be assumed to be the case, it is
expedient to consider them as attached to the vertices of a graph, two
vertices of which are connected by an edge if and only if (iff) the
corresponding spins have a nonzero interaction. The mathematical defin-
ition of a graph is (i):
Definition I. A graph G is a triple (V,E,I) consisting of the two dis-
joint sets v (of vertices) and E (of edges) and of the incidence
function I which maps every element eEE on an unordered pair of ver-
tices v I and v 2 from V :
I(e) ={v I ,v2}. )I(
Graphs on which spin systems are considered will, in general, be such
that v1~v 2 holds in eq. )I( and that e1~e 2 implies I(el)~I(e2) .
These two conditions are easily seen to imply that )i( the graph has no
loops, i.e., there are no edges beginning and ending at the same vertex,
and (ii) there are no multiple edges connecting the same pair of vertices.
In graph theory, the class of graphs satisfying conditions )i( and (ii)
is called the class of simple graphs. For a pictorial representation,
see Fig. I below.
(a) (b) (c)
Fig. I. )a( A simple graph. )b( A graph with a loop. )c( A graph with
multiple edges.
Given a graph G, the pair interaction energy of the assembly of
spins on the vertices can be calculated once the spin-spin energy func-
tion E(i,j) is known for all i,j~S, where S is the set of integers
{I,2,..,M}. The total pair interaction energy is
Epair = E e (ivl ) , )2(
eeE ' iv2
I(e)={v I ,v 2}
where provislon has been made for a possible edge-dependence of the
energy function. In order to calculate the total energy, the interaction
with external fields F(i) should also be taken into account:
)3(
Efield= v!V Fv(iv)"
Now the partition function of the spin model on the graph G can be
written down as
Z(G)= ~ H ~ "( ,iv2) H Av(iv) , )4(
{i v } eEE e iv I veV
where the abbreviations (B=I/kBT, k B the Boltzmann constant)
~(i,j)= exp-BE ,i( )j )5(
and
A(i)= exp-~F )i( )6(
have been introduced. The sum in eq. )4( is over the values from S for
all spin variables.
Interesting properties of spin systems on graphs can, of course,
only be expected in the thermodynamic limit, i.e., eq. )4( must be evalu-
ated for a sequence {G n} of graphs such that the number of vertices goes
to infinity for n÷~. Special classes of such sequences, which are ob-
tained from recursive prescriptions, are the subject of Chapter 8 of
this work. In the following sections, the very generally defined inter-
action energy E(i,j) will be studied from a group-theoretical point
of view after certain restrictions have been introduced.
i.2. Interactions.
The M×M matrix E of the interaction energies E(i,j) will now be
taken such that it satisfies the following three requirements:
)i( The pair interaction is invariant with respect to the interchange
of the spins:
E(i,j)=E(j,i) for all i,jsS. (I)
(ii) Let g be a permutation of the set S ; the energy function E(i,j)
is invariant with respect to g if
E ,i( )j =E g( )i( ,g(j)) )2(
holds for all i,jsS. The set of all permutations of S which leave the
energy function invariant form a group G, which is a subgroup of the
group S(M) of all permutations of the M elements of S. The group
S(M) is called the symmetric group on M objects and has M! elements,
denoted as
iS(M) = M! . )3(
The second requirement is now that the symmetry group 6 of the pair
interaction must be transitive, which means that
{g(i) g~6}=S )4(
holds for all iES. This means that for any pair i,j£S, there is at
least one permutation gE G which maps i on j : g(i)=j. Referral to
eq. (1.2) shows that the transitivity of G implies that a certain
value of the total pair interaction energy can be obtained in at least
M ways: a particular spin in state i can be transformed into any other
state j by a permutation gs6 ; if all other spins are also subjected
to the same permutation, the total pair energy must stay invariant. In
particular, the ground state energy in the absence of external fields
will be at least M-fold degenerate, so that one may expect symmetry-
breaking phase transitions to occur at sufficiently low temperatures
(in a suitable thermodynamic limit). This problem is considered at length
in Chapter 9.
(iii) The transitivity requirement (ii) implies that all "diagonal"
energies E(i,i) are equal for all ;i this energy is chosen as zero:
E(i,i)=E(1,1)=O for all ieS. )5(
The above three requirements allow for a characterization of all
possible pair interactions in a graph-theoretical way. Let E k be a
value that E(i,j) can take for i#j. Then the number of times that
E k occurs in this matrix for a fixed value of i is
N (Ek) = Z ~ (E (i, )j ,Ek , )6(
j~i
where ~(a,b)=1 for a=b and &(a,b)=O otherwise. The transitivity
of the group G implies that this number does not depend on the index
i. This, together with the symmetry E(i,j)=E(j,i) , shows that a part-
icular value E k shows up N(E )k times in every row and column of E.
The set E k of different values that E(i,j) can take on for i~j then
has at most M-I elements, since one has
Z N(E k) = Z I = M-1 , (7)
E k j~i
This shows that E(i,j) can be written as
s
(8)
E ,i( )j - -k kEi__ Mk(i,J) ,
where s~M-1 holds and the ~k are given as
M llJi for E(i,j)=Ek,
)9(
otherwise.
Now ~k can be read off from a (simple) graph G k with M vertices,
pairs of which are connected by edges iff the corresponding entries of
Mk(i,j) are nonzero. The simplicity of G k follows from the fact that
the diagonal elements of the energy matrix are zero. Further, since ~k
has exactly N(E )k entries equal to I in each row and column, the
graph G k is regula ~, i.e., every vertex has Zk=N(E )k edges emanating
from it; the number of edges of G k is then ZkM/2.
The energy function of eq. )8( is' therefore, completely defined
once the s graphs G k corresponding to the s matrices ~k are
given. These graphs all have the same vertex set, but their edge sets
are disjoint. By definition, the union of these edge sets must contain
all edges which can be drawn between M vertices, so that the G k "add
up" to the complete graph K(M) on M vertices: