Excerpt
The qstate Potts Model: Partition functions
and their zeros in the complex temperature
and qplane
A Thesis Presented
by
Hubert Ludwig Kl¨
upfel
to
The Graduate School
in Partial Fulfillment of the Requirements
for the Degree of
Master of Arts
in
Physics
State University of New York
at
Stony Brook
August 1999
State University of New York
at Stony Brook
The Graduate School
Hubert Ludwig Kl¨
upfel
We, the thesis committee for the above candidate for the Master of Arts degree,
hereby recommend acceptance of the thesis.
Professor Robert Shrock (Advisor)
Institute for Theoretical Physics
Professor Jacobus Verbaarschot
Department of Physics and Astronomy
Professor Barbara Jacak
Department of Physics and Astronomy
This thesis is accepted by the Graduate School.
Graduate School
ii
Abstract of the Thesis
The qstate Potts Model: Partition functions
and their zeros in the complex temperature
and qplane
by
Hubert Ludwig Kl¨
upfel
Master of Arts
in
Physics
State University of New York at Stony Brook
1999
In this thesis results on the Partition function Z
G
(T, q) for the
qstate PottsModel on finite polygonal lattices G are presented.
These are polynomials in a
e
- J
and q. The first step is to
calculate all the coefficients of Z
G
(a, q) using a transfer matrix
method.
The only points of nonanalyticity are the zeros of the partition
function; in the thermodynamic limit the complex temperature
zeros form a continuous curve
B via coalescence. This is the locus
where the free energy is nonanalytic. The zeros of Z(a, q) in the
iii
complex qplane for finite a and in the complex aplane for integer
and noninteger values of q are plotted.
For a = 0 the partition function reduces to the chromatic poly-
nomial P
G
(q) of the graph and the zeros are called chromatic zeros.
The behavior of those zeros as a increases from zero is investigated.
As for complex temperature the zeros in q form a continuous curve
in the thermodynamic limit. This is the locus where the limit-
ing function W
G
(q) = lim
n
P
G
(q)
1/n
is nonanalytic. W
G
(q) is
the ground state degeneracy and is connected to the ground state
entropy via S
0
(G, q) = k
B
ln(W
G
(q)).
Thus the characteristics of the zeros of q and a for finite lattices
help to understand the properties of the model in the thermody-
namic limit.
iv
Contents
Acknowledgements
viii
1 Introduction
1
1.1
Definition of the model and general properties . . . . . . . . .
4
1.2
Partition functionzeros and critical behavior
. . . . . . . . .
6
1.3
Duality, symmetry and boundary conditions . . . . . . . . . .
8
1.4
Chromatic polynomials and ground state entropy . . . . . . .
11
2 Calculation of the Partition function
15
2.1
Complexity of the problem
. . . . . . . . . . . . . . . . . . .
16
2.2
The transfermatrix method
. . . . . . . . . . . . . . . . . .
17
2.3
The algorithm
. . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.4
The zero temperature limit . . . . . . . . . . . . . . . . . . . .
23
3 Partition function zeros
25
3.1
Lattice types and sizes . . . . . . . . . . . . . . . . . . . . . .
26
3.2
Tests for the correctness of the results . . . . . . . . . . . . . .
29
3.3
Zeros in the qplane
. . . . . . . . . . . . . . . . . . . . . . .
30
3.3.1
Square lattice . . . . . . . . . . . . . . . . . . . . . . .
31
v
3.3.2
Triangular lattice . . . . . . . . . . . . . . . . . . . . .
40
3.3.3
Honeycomb lattice . . . . . . . . . . . . . . . . . . . .
48
3.3.4
Kagom´
e lattice . . . . . . . . . . . . . . . . . . . . . .
53
3.3.5
Scaling of the qzeros with the temperature . . . . . .
55
3.4
Complex temperature zeros
. . . . . . . . . . . . . . . . . . .
58
3.4.1
CTzeros for small non integral q . . . . . . . . . . . .
62
3.4.2
CTzeros for large values of q . . . . . . . . . . . . . .
67
3.4.3
CTzeros for negative q
. . . . . . . . . . . . . . . . .
74
4 Conclusions
75
Bibliography
78
A Lattice size and calculation time
82
B Implementation of the algorithm
88
B.1 Tensor products and direct products of matrices . . . . . . . .
88
B.2 Adding sites and bonds . . . . . . . . . . . . . . . . . . . . . .
89
B.3 Using the S
q
symmetry of the Hamiltonian . . . . . . . . . . .
90
B.4 Simplifications for zero temperature . . . . . . . . . . . . . . .
92
C Zeros for further lattices of different sizes
94
C.1 Complex qplane . . . . . . . . . . . . . . . . . . . . . . . . .
94
C.2 Complex xplane . . . . . . . . . . . . . . . . . . . . . . . . .
103
D Leading terms of some partition functions
109
vi
List of Figures
3.1
Illustration of lattice sizes . . . . . . . . . . . . . . . . . . . .
27
3.2
CTzeros for sq
(4x10),fp
with different numerical precision . . .
30
3.3
sq
(3,18),ff
qzeros for a = 0, ..., a = 0.9
. . . . . . . . . . . . .
32
3.4
sq
(3,18),ff
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . .
32
3.5
sq
(3,18),ff
, a = 0.99, a = 1.01 and scaling with a
. . . . . . . .
33
3.6
sq
(3,18),fp
qzeros for a = 0, ..., a = 0.9
. . . . . . . . . . . . .
34
3.7
sq
(3,18),fp
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . . .
34
3.8
sq
(3,18),fp
, a = 0.99, a = 1.01 and scaling with a
. . . . . . . .
35
3.9
sq
(4,10),ff
qzeros for a = 0, ..., a = 0.9
. . . . . . . . . . . . .
36
3.10 sq
(4,10),ff
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . .
37
3.11 sq
(4,10),ff
, a = 0.99, a = 1.01 and scaling with a
. . . . . . . .
37
3.12 sq
(4,10),fp
qzeros for a = 0, ..., a = 0.9
. . . . . . . . . . . . .
38
3.13 sq
(4,10),fp
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . . .
39
3.14 sq
(4,10),fp
, a = 0.99, a = 1.01 and scaling with a
. . . . . . . .
39
3.15 tri
(3,14),ff
qzeros for a = 0, ..., a = 0.9 . . . . . . . . . . . . .
40
3.16 tri
(3,14),ff
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . .
41
3.17 tri
(3,14),ff
, a = 0.99, a = 1.01 and scaling with a . . . . . . . .
41
3.18 Triangular lattice with periodic boundary conditions
. . . . .
42
vii
3.19 tri
(3,14),fp
qzeros for a = 0, ..., a = 0.9 . . . . . . . . . . . . .
42
3.20 tri
(3,14),fp
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . .
43
3.21 tri
(3,14),fp
, a = 0.99, a = 1.01 and scaling with a . . . . . . . .
43
3.22 tri
(4,8),ff
qzeros for a = 0, ..., a = 0.9 . . . . . . . . . . . . . .
44
3.23 tri
(4,8),ff
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . . .
45
3.24 tri
(4,8),ff
, a = 0.99, a = 1.01 and scaling with a . . . . . . . . .
45
3.25 tri
(4,8),fp
qzeros for a = 0, ..., a = 0.9 . . . . . . . . . . . . . .
46
3.26 tri
(4,8),fp
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . . .
46
3.27 tri
(4,8),fp
, a = 0.99, a = 1.01 and scaling with a . . . . . . . . .
47
3.28 hc
(2,9),ff
qzeros for a = 0, ..., a = 0.9 . . . . . . . . . . . . . .
49
3.29 hc
(2,9),ff
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . . .
49
3.30 hc
(2,9),ff
, a = 0.99, a = 1.01 and scaling with a . . . . . . . . .
50
3.31 Honeycomb lattice with periodic boundary conditions . . . . .
51
3.32 hc
(3,5),fp
qzeros for a = 0, ..., a = 0.9 . . . . . . . . . . . . . .
51
3.33 hc
(3,5),fp
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . . .
52
3.34 hc
(3,5),fp
, a = 0.99, a = 1.01 and scaling with a . . . . . . . . .
52
3.35 kag
(1,10),ff
qzeros for a = 0, ..., a = 0.9 . . . . . . . . . . . . .
53
3.36 kag
(1,10),ff
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . .
54
3.37 kag
(1,10),ff
, a = 0.99, a = 1.01 and scaling with a . . . . . . . .
54
3.38 hc
(5,5),ff
, CT-zeros for q = 3 . . . . . . . . . . . . . . . . . . .
61
3.39 hc
(5,5),ff
, CTzeros for q = 5 . . . . . . . . . . . . . . . . . . .
61
3.40 sq
(4,10),fp
CTzeros for q = 0.1, 0.5, 0.9, 1.1, 1.5, 1.8 . . . . . .
62
3.41 sq
(4,10),fp
CTzeros for q = 2, 2.1, 2.5, 3 . . . . . . . . . . . . .
63
3.42 tri
(4,8),fp
CTzeros for q = 2, 2.1, 2.5, 3 . . . . . . . . . . . . .
64
viii
3.43 hc
(3,5),fp
CTzeros for q = 2, 2.1, 2.5, 3 . . . . . . . . . . . . .
66
3.44 kag
(2,3),fp
CTzeros for q = 2, 2.1, 2.5, 3 . . . . . . . . . . . .
68
3.45 sq
(4,10),fp
CTzeros for q = 4, 10, 25, 100 . . . . . . . . . . . .
69
3.46 CTzeros for the leading coefficient of Z
sq
(4,10),ff
(x, q) in q . . .
70
3.47 tri
(4,8),fp
CTzeros for q = 4, 10, 25, 100 . . . . . . . . . . . .
71
3.48 hc
(3,5),fp
CTzeros for q = 4, 10, 25, 100
. . . . . . . . . . . .
73
3.49 sq
(4,10),fp
CTzeros for q =
-1, -2, -3, -4, -25, -100 . . . .
74
B.1 Honeycomb lattice drawn as a brick lattice . . . . . . . . . . .
91
B.2 Kagom´
e lattice drawn in a brick form . . . . . . . . . . . . . .
91
C.1 sq
(5,5),ff
qzeros for a = 0, ..., a = 0.9 . . . . . . . . . . . . . .
95
C.2 sq
(5,5),ff
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . . .
95
C.3 sq
(5,5),ff
, a = 0.99, a = 1.01 and scaling with a . . . . . . . . .
96
C.4 hc
(2,7),ff
qzeros for a = 0, ..., a = 0.9 . . . . . . . . . . . . . .
96
C.5 hc
(2,7),ff
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . . .
97
C.6 hc
(2,7),ff
, a = 0.99, a = 1.01 and scaling with a . . . . . . . . .
97
C.7 hc
(3,5),ff
qzeros for a = 0, ..., a = 0.9 . . . . . . . . . . . . . .
98
C.8 hc
(3,5),ff
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . . .
98
C.9 hc
(3,5),ff
, a = 0.99, a = 1.01 and scaling with a . . . . . . . . .
99
C.10 kag
(2,3),fp
qzeros for a = 0, ..., a = 0.9 . . . . . . . . . . . . .
99
C.11 kag
(2,3),fp
qzeros for a =
-0.25 and a = -0.5 . . . . . . . . . 100
C.12 kag
(2,3),fp
, a = 0.99, a = 1.01 and scaling with a . . . . . . . .
100
C.13 sq
(4,10),fp
, a = 1.1 and a = 1.25 . . . . . . . . . . . . . . . . . .
101
C.14 tri
(4,8),fp
, a = 1.1 and a = 1.25 . . . . . . . . . . . . . . . . . .
101
ix
C.15 hc
(3,5),fp
, a = 1.1 and a = 1.25 . . . . . . . . . . . . . . . . . .
102
C.16 kag
(2,3),fp
, a = 1.1 and a = 1.25 . . . . . . . . . . . . . . . . .
102
C.17 tri
(4,8),fp
CTzeros for q = 0.1, 0.5, 0.9, 1.1, 1.5, 1.8 . . . . . .
103
C.18 hc
(3,5),fp
CTzeros for q = 0.1, 0.5, 0.9, 1.1, 1.5, 1.8 . . . . . .
104
C.19 sq
(4,10),ff
CTzeros for q = 4, 10, 25, 100 . . . . . . . . . . . .
105
C.20 hc
(3,5),fp
CTzeros for q =
-1, -2, -3, -4, -25, -100 . . . . 106
C.21 tri
(4,8),fp
CTzeros for q =
-1, -2, -3, -4, -25, -100 . . . . 107
C.22 kag
(2,3),fp
CTzeros for q = 4, 10, 25, 100 . . . . . . . . . . . .
108
x
List of Tables
1.1
Table of Symbols . . . . . . . . . . . . . . . . . . . . . . . . .
3
3.1
Properties of the different lattices . . . . . . . . . . . . . . . .
28
3.2
Scaling of the qzeros with 1
- a (square and honeycomb) . .
56
3.3
Scaling of the qzeros with 1
- a (triangular and kagom´e) . . .
57
A.1 Square lattices, size and computation time . . . . . . . . . . .
83
A.2 Triangular lattices, size and computation time . . . . . . . . .
84
A.3 Honeycomb lattices, size and computation time
. . . . . . . .
85
A.4 Kagom´
e lattices, size and computation time . . . . . . . . . .
86
A.5 Computation time for Z
hc
(a, q
0
) . . . . . . . . . . . . . . . . .
86
xi
Acknowledgements
I would like to thank Professor Robert Shrock for his guidance and advice.
He introduced me to the fascinating topic of the Pottsmodel, its rich structure
and its connections to graph theory, in particular chromatic polynomials. He
also gave me the freedom to choose the focus of the work myself and encouraged
me to continue during the difficult stages of this project.
1
Chapter 1
Introduction
Complex behavior and organized structures are ubiquitous in the world
surrounding us. From the viewpoint of physics the most interesting question
is how these evolve from simple laws and how they can be explained by models
that can be treated mathematically.
Two prominent physical systems that show long range order and critical
behavior are the liquidgassystem and magnetic substances. At the critical
point, where the transition line between liquid and gas ends, long range order
can be directly observed as critical opalescence. One way of modeling such
phenomena are lattice spinmodels where a spindirection is assigned to each
vertex and the spins are interacting with each other and with external fields.
In the critical regime where even very distant spins are correlated the details of
such a system are no longer important. It is determined only by its dimension
and symmetry. This is the reason for the similarity in the behavior of different
systems at the transition point.
One of the first models for magnetic substances was the IsingModel [1].
2
(The definitions of the models and detailed explanations of the concepts men-
tioned here will follow in section 1.1.) The onedimensional case is easy to
solve but has zero critical temperature. In two dimensions, however, there
occurs a continuous phase transition at finite temperature. This is the first
nontrivial model which has been solved exactly. Onsager's solution for the
square lattice Isingmodel [2] has been of key importance for the progress of
the theory of phase transitions.
The aim of this work is to investigate the properties of the Standard Potts
model by allowing the temperature and the spin dimensionality q to take on
complex values. There has been a long tradition of using complex magnetic
field, first introduced by Yang and Lee [3] and complex temperature, due to
Fisher [4]. The generalization of q to complex values is motivated by an in-
teresting and deep connection between the ground state entropy of the Potts
antiferromagnet and a concept known in graph theory as chromatic polyno-
mial [5].
One of the most powerful tools in lattice statistical mechanics is the fi-
nite lattice method. In this context it allows to get exact results and make
conjectures about the properties of the system in the thermodynamic limit.
Previous research has concentrated either on the critical properties and
complex temperaturezeros or on chromatic polynomials and ground state
entropy. In this work the attempt is made to combine those two approaches
and obtain results for both T and q variable. However, the size of the lattices
one can reach are considerably smaller in this case.
3
Table 1.1: This table shows the more frequently used symbols
Symbol
Meaning
T
temperature
J
coupling constant
k
B
Boltzmann constant
=
1
k
B
T
inverse temperature
a
e
K
e
- J
Boltzmann weight
q
number of spin directions
G
certain graph or lattice
N
S
number of sites in G
N
B
number of bonds in G
Z
G
(a, q)
partition function for G as polynomial in a and q
P
G
(q)
Z
G
(0, q)
chromatic polynomial for G as polynomial in q
4
1.1
Definition of the model and general prop-
erties
The PottsModel is a generalization of the IsingModel. It was introduced
in its standard form in a thesis by Potts in 1954 [6] and has played an important
role in Theoretical Physics ever since. The IsingHamiltonian depends only
on nearest neighbor interactions:
H
Ising
=
<ij>
J
ij
S
i
S
j
,
S
i
, S
j
{-1, 1}
(1.1)
A natural generalization is to allow a greater number of different spin
directions or interactions between spins that are not nearest neighbors. In the
planar qstate PottsModel the directions of the spins are confined to a plane
perpendicular to the lattice.
n
=
2n
q
,
n = 0, 1, 2, . . . , q
- 1
(1.2)
The interaction between two spins depends only on their relative direction
H =
-
<ij>
I(
ij
)
(1.3)
where I is a 2periodic function: I(x) = I(x+ 2) and
ij
=
i
-
j
. Thus the
partition function is Z
q
symmetric. The standard PottsModel is defined by
I(
ij
) = J
·
S
i
S
j
(
S
i
S
j
is one if S
i
and S
j
have the same value, zero otherwise)
and the Hamiltonian reads
H
Standard
= J
<ij>
S
i
S
j
(1.4)
5
For q=2 (Ising) and q=3 the standard and planar models are equivalent
[7]. The Z
q
symmetry is of course also existing in the StandardModel. In
general the symmetry in this case is even stronger. A permutation of the spin
directions does not change the Hamiltonian and thus the partition function
is invariant under permutations of the spindirections. This explains the be-
forementioned equivalence in the q = 2 and q = 3 case: permutations of two
or three elements are always cyclic or anticyclic. This S
q
symmetry of the
standard model is especially important for the efficient implementation of an
algorithm to calculate the partition function.
In addition to the twosite interaction there could also be multisite inter-
actions and external fields. The general Hamiltonian for the standard model
then takes the following form (with K =
-J):
-H = B
i
S
i
0
+ K
<ij>
S
i
S
j
+ K
2
<ijk>
S
i
S
j
S
k
+ . . .
(1.5)
-B/ is an external field and -K
2
/, . . . are the coupling constants for multi
site interactions.
The physical properties of the system are calculated by taking the thermo-
dynamic limit.
Z
{S
i
}
e
-H
= e
-F
(1.6)
f (q, B, K, K
n
)
-
N
S
F (q, B, K, K
n
) = lim
N
S
1
N
S
ln Z
G
(q, B, K, K
n
)
(1.7)
F = U
- T S
(1.8)
U (q, B, K, K
n
) =
-
f (q, B, K, K
n
)
(1.9)
M (q, K) =
-
B
f (q, B, K, K
n
)
(1.10)
6
The order parameter is defined to be [7]
m(q, B, K, K
n
) =
qM
- 1
q
- 1
.
(1.11)
It takes the value one in the completely ordered and the value zero in the
completely disordered phase. The critical exponents are defined in the usual
fashion.
In this work only the standard model defined by eq. (1.4) is used.
Z
G
(q, K) =
{S
i
}
e
K
<ij>
SiSj
,
S
i
, S
j
{0, 1, . . . , q - 1}.
(1.12)
The Potts model has a first order phase transition for q > 4 and a higher
order phase transition for q
4 [8]. General reviews of the Potts model can
be found in refs. [7, 9].
1.2
Partition functionzeros and critical be-
havior
For finite lattices, the Partition function Z is a polynomial in q and the
Boltzmann weight a
e
K
. This can be seen explicitly after carrying out the
spin summations and rewriting the sum appropriately [7, 8]. Each bond with
aligned spins (e.g. the spins on the adjacent sites in the same direction) con-
tributes just a factor a to the according addend (that represents one distinct
spin configuration) of Z. For a = 1 (K = 0
T = ) all spins are indepen-
dent of each other and Z = q
N
S
.
Thus the highest power of a in Z is N
B
and the highest power of q is N
S
. The
7
partition function then takes the form [8]
Z
G
(q, K) =
G
G
v
b
(G )
q
n
(G )
v = e
K
- 1 = a - 1, K = -J
(1.13)
n is the number of connected components of the subgraph G (1
n(G ) N
s
)
and b is the number of bonds of G (0
b(G ) N
b
=
2
N
s
), where is the
coordination number of the lattice. The summation goes over all subgraphs of
G, including isolated vertices.
It is then obvious that critical behavior can occur only in the thermody-
namic limit. Eq. (1.7) shows that the free energy is an analytical function as
long as the lattice size is finite. Z
G
(K, q) is positive for physical values of K
(e.g. K
0, a 0). Notice that a = 0 is equivalent to T = 0 and J > 0 (anti
ferromagnetic coupling). In this case Z
G
(q, K =
-) = P
G
(q) which is also
positive for sufficiently large values of q. P
G
(q) is the chromatic polynomial
which will be explained in detail in sect. 1.4.
The physical temperature phases can be analytically continued into the
complex plane. The complextemperature phases (CTphases) are separated
from each other by the continuous locus
B where the free energy is non analytic.
Since Z is a polynomial for finite lattices its zeros are the only possibilities
from which those nonanalyticities can arise and
B forms via coalescence of
CTzeros in the thermodynamic limit (excluding isolated zeros). For general
theory and examples cf. [4, 6, 10].
The complex temperature zeros lie all off the real axis as all the coefficients
of Z
q
(a) are positive. Each coefficient corresponds to the number of states with
8
a certain energy. The thermodynamic observables vary smoothly with K for
K real. The critical temperature is the point where
B crosses the real axis.
Of course other supporting arguments such as discontinuities in the change
of the order parameter or energy versus entropy considerations are necessary
to determine the behavior of the free energy and its derivatives at the critical
temperature and to support the inferences made from the distribution of the
partition function zeros.
Various results have been obtained for the q = 2 (e.g. Ising) case for many
different types of lattices. However, there are no analytic results for higher q.
1.3
Duality, symmetry and boundary condi-
tions
Duality means that there is a corresponding face for every site of the orig-
inal lattice G in the dual lattice G
d
D(G) and a corresponding site in G
d
for every face of G. This concept can be generalized to higher dimensions but
for our purpose this explanation is sufficient.
The square lattice is selfdual in the thermodynamic limit. The dual of the
honeycomb lattice is the triangular and the dual of the kagom´
e is the diced
lattice. Some of these lattice types are shown in figure 3.1.
Duality relations allow the connection of the high and low temperature
partition function [7] and are useful in obtaining exact results such as the
transition temperature.
9
Z
G
(q, a) = x
N
B
q
N
S
-1-1/2N
B
Z
G
d
(q, a
d
)
(1.14)
x =
e
K
- 1
q
(1.15)
a
d
D(a) =
a + q
- 1
a
- 1
,
i.e.
x
d
=
1
x
(1.16)
In the thermodynamic limit this duality relation (eqs. (1.14) to (1.16)) for
the Pottsmodel yields the connection between the singularities of the free
energy in the high and low temperature phases or on the lattice G and its
dual G
d
, respectively.
f
G
(q, a)
sing
= f
G
d
(q, a
d
)
sing
(1.17)
Examples for different kinds of duality conserving boundary conditions for
finite square lattices are given in [11]. This approach is not pursued here
because it is computationally unfavorable and it is important only for the
calculation of CTzeros, which is not the exclusive aim of this work.
Periodic boundary conditions in both directions are not duality conserving
because their topology is toroidal and duality relies on the fact that the lattice
can be drawn on a simply connected surface [12]. Still, periodic boundary
conditions in one direction come close to duality conservation, they allow the
use of the interfacemethod as described in section 2.2 and suppress finite
sizeeffects.
There exist two important concepts concerning the symmetry of the Hamil-
tonian: the first one is uniform coordination number and the second one is the
notion of a bipartite lattice.
10
If all lattice sites have even coordination number (
2), then the
partition function contains only even powers of a. This can be seen from a low
temperature expansion where one starts with all spins aligned. Flipping one
spin changes the addend in Z always by a factor a
; Z = a
N
B
(1+N
S
·a
-
+. . . ).
Z
G,
even
(a, q) = Z
G,
even
(
-a, q)
(1.18)
Uniform coordination number is only possible with periodic boundary con-
ditions in both directions or special boundary conditions that use additional
sites. Free boundary conditions never conserve a uniform coordination number
for all lattice sites.
The second important concept of a bipartite lattice is nearly independent
of the boundary conditions. If there are two sublattices G
even
and G
odd
that
have no sites in common and together contain all lattice sites of G and the
spins of G
even
interact only with spins of G
odd
and vice versa, then G is said to
be bipartite. The important consideration is the following: denote the spins
of G
odd
with S
odd
and those of G
even
with S
even
. Then the IsingHamiltonian
is invariant under the transformation [36]
{S
even
, S
odd
, J
} {S
even
,
-S
odd
,
-J},
(1.19)
since the spinspin interaction only involves neares neighbor pairs. This trans-
formation is equivalent to the inversion of a and thus Z
Ising
is symmetric under
a
1/a. In the Pottsmodel notation there occurs an additional prefactor
a
N
B
, because a
l
a
l
-N
B
(l is the number of aligned nearest neighbor spins).
Z
G
bipartite
(a, 2) = a
N
B
· Z
G
bipartite
(1/a, 2)
(1.20)
11
The square lattice is always bipartite for free boundary conditions and is
bipartite for periodic boundary conditions as long as the number of sites in the
periodic direction is even. The honeycomb lattice is bipartite for free boundary
conditions. In the case of periodic boundary conditions it depends on their
actual type. Neither the triangular nor the kagom´
e lattice is bipartite since
they contain triangular subgraphs.
1.4
Chromatic polynomials and ground state
entropy
There is a deep connection between the ground state entropy of the Potts
antiferromagnet and a function known in graph theory as chromatic polyno-
mial. The chromatic polynomial P
G
(q) gives the number of different ways in
which one out of q different colors can be assigned to each vertex of a graph
G such that no two neighboring vertices have the same color. An overview of
the properties of chromatic polynomials can be found in [13, 14, 15, 16].
Consider the free energy in the zero temperature limit K
- (anti
ferromagnetic coupling, J > 0). From eq. (1.7) one obtains
e
f
= lim
N
S
Z
1/N
S
(1.21)
In the ground state only configurations with no two neighboring spins in
the same direction contribute to Z (otherwise there occurs a factor e
-J
which
is zero for T = 0). Hence Z is connected to P by
Z
G
(K =
-, q) = P
G
(q)
(1.22)
12
Since U approaches its ground state value U (0) = 0 exponentially fast, it
holds lim
T
0
U
T
= 0 and the free energy (eq. 1.7) becomes the ground state
entropy per site s
0
s
0
(G, q) = lim
T
0
-
F
N T
= k
B
· lim
T
0
f = k
B
ln(W )
(1.23)
W
G
(q)
lim
N
S
P
G
(q)
1/N
S
(1.24)
The last limit exists because q
N
S
is the number of ways one can color the
N
S
vertex graph G without any constraint (hence P
G
(q)
q
N
S
).
For v =
-1 (e.g. K -) eq. (1.13) reduces to the well known Birkhoff
formula for the chromatic polynomial [17].
P
G
(q) =
G
G
(
-1)
b
(G )
q
n
(G )
(1.25)
After carrying out the summation the formula reads [15].
P
G
(q) =
n
-1
j
=0
(
-1)
j
c
n
-j
q
n
-j
.
(1.26)
where the coefficients obey the following relations:
c
n
-j
=
N
B
j
for j < g
- 1
(1.27)
c
n
-(g-1)
=
N
B
j
- k
g
(1.28)
c
n
-j
<
N
B
j
for j
g.
(1.29)
A lower bound for c
r
is
c
r
(N
B
- n + 2)
n
- 2
5
- 1
+
n
- 2
r
- 2
.
(1.30)
13
k
n
is the number of circuits of length n, g is the girth (length of a shortest
circuit).
This representation of the chromatic polynomial can be used as a starting
point for generalizing q to nonintegral and indeed complex values.
Chromatic polynomials have been the interest of research since the begin-
ning of the century [17, 18]. General references on chromatic polynomials and
graph theory in general are [19, 20]. Algorithms for the calculation of the
chromatic polynomial of a general graph are usually based on the deletion
contractionformula (or addition contraction formula depending on the kind
of graphs that are desired as a final result) [15] and try to find a decomposition
either into empty or complete graphs. Computer algebra systems like Maple
or Mathematica have built in packages that implement such algorithms.
For small lattices it is possible to use them as a check for the general
partition functions in the limit K
- (e.g. a = 0). Except of some special
cases that are very limited in number [13, 15] there are no general results for
chromatic polynomials of certain families of graphs. Moreover the problem
of calculating the chromatic polynomial is known as NPcomplete and the
calculation time grows exponentially with the size of the graph (cf. section
2.1).
To give an example for the deletioncontractionformula and to show one
exact result that can be used for comparison with the (finitelattice) calcula-
tions carried out in this work we cite the chromatic polynomial for the two
dimensional strip (rectangle of height 2 and width m), given by
P
R
2,m
(q) = q(q
- 1)(q
2
- 3q = 3)
m
-1
(1.31)
14
The proof is straightforward and uses the deletioncontractionformula and
induction over the number of columns m and can be found in [13].
Coloring problems have been of interest even to non mathematicians and
the question whether every planar graph is four colorable (which is equivalent
to coloring every map with for colors) remained an outstanding problem for a
long time [21]. The positive answer to it also implies that there is no chromatic
zero q = 4 for any planar graph [22].
15
Chapter 2
Calculation of the Partition function
With the transfermatrix method (cf. section 2.2) used here, it is possible
to calculate Z for fixed q (number of possible spin directions) only. The latter
will be denoted Z
G,q
(a).
It is then necessary to compute Z
G,q
(a) for N
S
(number of sites in the
lattice) different fixed integer values q
0
. . . q
N
S
in order to gain enough infor-
mation to determine all coefficients of Z
G
(a, q). This is a challenging task as
can be seen from the fact that for a = 1 (K = 0) the sum of the coefficients
equals q
N
S
. An explanation for this has been given in section 1.2. The degree
of Z
G
(a, q) in a is N
B
as every bond may contribute a factor a. The highest
coefficient (for a
N
B
) is q, the constant coefficient in a is P
G
(q) (cf. section 1.4).
Notice that it is unavoidable to calculate the exact coefficients of the parti-
tion functions. For obtaining the zeros of a polynomial, which is the final aim,
only a slight perturbation of one of the coefficients can disturb the distribution
of the zeros completely. The standard data types that are usually provided by
a programming language are not capable of storing such large numbers. To
16
handle those large integers that occur during the calculation the C++ class
library NTL was used that allows dealing with arbitrary size integers using
modulo arithmetic [23]. A comprehensive reference for the kind of algorithms
that are implemented in NTL is [24].
2.1
Complexity of the problem
It is interesting and also helpful to get some feeling for the complexity of
the problem and more important the time it requires to calculate the partition
functions. A good measure for the difficulty of a problem is the time needed
to solve it. To make this more quantitative one has to specify a characteristic
parameter as the size s of the problem. In our case this is the number of sites
N
S
.
There is a fundamental distinction between polynomial bounded algorithms
and non polynomial bounded ones (NP problems). The latter will always have
a longer running time above a certain size s than the former ones, which are
called fast or efficient algorithms.
A certain class of NP problems can be transformed into each other in
polynomial time [20]. This class is called "nondeterministic polynomialtime
complete" or NPcomplete. If one of them was solvable in polynomial time
then all would be. This is however very unlikely since they have been studied
intensively for a long time. If there exists no fast algorithm to solve a problem
and its solution provides a solution for a NPcomplete problem (and hence for
all of them) then it is a hard problem. The traveling salespersonproblem is not
17
NPcomplete but hard, since answering the question if there is a Hamiltonian
cycle for a given graph (a cycle that contains all vertices) is NPcomplete.
Finding the shortest Hamiltonian cycle certainly provides an answer to that
question.
Calculating the chromatic polynomial of a graph is a NPcomplete problem
[20]. Since P
G
(q)
Z
G
(0, q) (cf. section 1.4) calculating the partition function
for a given lattice is a hard problem [25].
2.2
The transfermatrix method
The basic idea is to think of a lattice divided into equal patterns and to
build up the lattice out of these patterns (layers). The spins on the boundary
are kept fixed. Their configurations are denoted by i and j. The adding layer
transfer matrix consist then of the elements
M
ij
= i
|M|j = i|e
- H
|j
(2.1)
with the spins of the left and right boundary in the states i and j, respectively.
The internal spins are summed over. Thus M
ij
contains only one term. New
layers can now be added by matrix multiplication which automatically includes
the summation over the internalized spins. The partition function for a lattice
consisting of N layers and the boundary spins on the left in state i
| and on
the right in state
|j is
Z
i,N,j
= i
|M
N
|j
(2.2)
The spins on the left and rightmost layer are still fixed. To get the final
partition function one has to sum over them, either with i and j the same
18
(periodic boundary conditions) or with i and j independent of each other (free
boundary conditions).
Z
pbc,N
= tr(M
N
) =
i
i
|M
N
|j
(2.3)
Z
f bc,N
=
ij
i
|M
N
|j
(2.4)
In this case q
2 W
polynomials must be stored, if the height of the lattice is W
sites.
Another way of calculating partition functions requires less memory: the
interface method that keeps just the boundary information on one side. Thus
the number of polynomials that have to be stored reduces to q
W
. The config-
urations of the other boundary are immediately summed over.
The partition functions related to a certain configuration of the boundary
spins are stored in a vector called partition vector Z. For a lattice of height W
that is build up by columns it has dimension q
W
. The layering direction is then
horizontal. Its entries are the partition functions for fixed spin configurations.
For q = 2 and W = 2 and the spin directions symbolized by + and
- this is
for example.
Z = (Z
++
, Z
+-
, Z
-+
, Z
--
)
(2.5)
For the general case with q allowed spin directions S
i
= 0, . . . , q
- 1
Z = (Z
q
-1,...,q-1
, Z
q
-1,...,q-2
, . . . , Z
0,...,1
, Z
0,...,0
)
(2.6)
Z has q
W
elements and Z
0,1,...,q-1,q-1
represents the configuration with the first
spin on the boundary curve in direction 0, the second in direction 1 and so on
19
and the last W
- q spins in direction q - 1. Thus Z contains one entry for
every possible spin configuration of one column.
Adding a new bond is done by multiplying those entries of Z that corre-
spond to states in which the two sites adjacent to the new bond have the same
spin direction by e
K
. The procedure for adding a new site is similar but a bit
more complicated and will be explained below.
Therefore it is not necessary to calculate the whole transfer matrix, instead
local transfer matrices can be used. An additional advantage is the fact, that
there is a direct relation between adding a bond or site and the algebraic
operation on the vector of the partition functions. Hence the lattices are not
restricted to regular patterns. For the high qvalues that our method requires
the interface method is considerably advantageous.
Above this it is essential to reduce the number of polynomials that have to
be stored as far as possible. This can be achieved by exploiting the equivalence
of states that can be transformed into each other by a permutation of the spin
directions. This S
q
symmetry has already been mentioned in section 1.1. Since
this is merely a programming problem, the inclusion of it in the algorithm is
explained in appendix B.3.
Chapter two of [9] gives a comprehensive treatment of the interface method.
We use the same conventions and refer to this book for further details and
explanations.
The next step is to obtain the local transfer matrices for adding a new site
at position i (t
i
.) and adding a new bond between the sites i and i + 1 (t
ii
+1
)
20
[9].
t
i.
= (a
- 1) · 1
q
W
+ 1
q
. . . M . . . 1
q
(2.7)
t
i i
+1
= 1
q
W
+ (a
- 1) · (1
q
. . . D . . . 1
q
)
(2.8)
M
end(V
q
),
M
ab
= 1
(a, b
V )
(2.9)
D
end(V
q
V
q
),
D
a
b,cd
=
a,b
b,c
c,d
,
(a, b, c, d
V )
(2.10)
Appendix B contains further details concerning the convention for the repre-
sentation of tensor products as direct products of matrices.
In analogy to the completelayer method eq. (2.2) the addition of a new
layer is done by the multiplication of the partition vector with the product of
the local transfer matrices from the right.
Z = Z
· (
N
i
=1
t
i.
)(
N
-1
i
=1
t
ii
+1
)
(2.11)
i denotes the position of M and D in eq. (2.7) and (2.8). As the second term
in eq. (2.8) is the product of only diagonal matrices, all offdiagonal elements
of the matrix t
i i
+1
are zero.
2.3
The algorithm
For the following considerations it is useful to write the partition function
in a more straightforward way.
Z
G
(q, K) =
N
S
i
=1
N
B
j
=0
C
ij
q
i
a
j
(2.12)
The coefficients C
ij
of eq. (2.12) can be calculated by matrix inversion (or
more precise Gaussian elimination or another method for solving systems of
21
linear equations) if sufficiently many partition functions (e.g. their coefficients)
for fixed values of q are known.
The elements of t
i.
and t
ii
+1
from eqs. (2.7) and (2.8) are (cf. appendix
B.1)
(t
i.
)
kl
= (a
- 1) ·
kl
+
l
0
,k
0
l
1
,k
1
· · ·
l
W -i-2
,k
W -i-2
l
W -i
,k
W -i
· · ·
l
W -1
,k
W -1
(2.13)
k
i
and l
i
are given by the representation of k and l to the basis q:
k =
W
-1
i
=0
k
i
q
i
and
l =
W
-1
i
=0
l
i
q
i
.
(2.14)
By the same token one obtains for the elements of t
ii
+1
:
(t
i i
+1
)
kl
=
kl
· (t
i i
+1
)
k
(t
i i
+1
)
k
= 1 + (a
- 1) · 1
k
0
k
0
· · · D
k
i
k
i
,k
i+1
k
i+1
· · · 1
k
W -1
,k
W -1
= 1 + (a
- 1) ·
k
W -2-i
,k
W -1-i
(2.15)
The initial partition vector for just one column is given by:
Z
k
= a
k0,k1
+
k1,k2
+···+
kW -2,kW -1
(2.16)
The procedure for adding a new layer is shown in eq. (2.11), k
i
and l
i
are
defined in eq. (2.14).
That means that the calculation of t
ii
+1
is reduced to the calculation of the
representation of k (0
k q
W
- 1) with basis number q: k =
W
-1
l
=0
k
l
q
l
.
Appendix B treats the algorithmic part of the problem in some more detail.
It is important to remember that the interfacemethod does not allow
to implement periodic boundary conditions in both directions. Therefore one
22
gets the partition function by summing over all elements of the partition vector
after all layers have been added.
Finally the connection between the coefficients of the fixed q Partition func-
tions Z
G,q
(a) and the coefficients of the general partition function Z
G
(q, a) is
given by the following formulas, where q
l
takes on the values q
l
= 0, 1, . . . , N
S
.
Z
G,q
l
(a)
Z
G
(a, q
l
) =
N
B
j
=0
N
S
i
=1
C
ij
q
i
l
A
lj
a
j
(2.17)
Q
· C = A
(2.18)
Q =
1
q
0
q
2
0
. . .
q
N
S
0
1
q
1
q
2
1
. . .
q
N
S
1
..
.
..
.
..
.
. ..
..
.
1 q
N
B
q
2
N
B
. . .
q
N
S
N
B
A =
A
00
A
01
. . .
A
0N
S
..
.
..
.
. ..
..
.
A
N
B
0
A
N
B
1
. . .
A
N
B
N
S
(2.19)
A is calculated row by row (each row corresponds to a certain fixed integer
value of q) and C is obtained by inverting eq. (2.18).
Matrices of the form of Q are known as Vandermondematrices. They
are usually illconditioned and inverting them is very errorprone. However,
this is not a problem here, since the calculations are exact. As this matrix
inversion also requires very large integer arithmetic, it was carried out using
Maple. Standard numerical algorithms for the solution of systems of linear
23
equations usually use floating point numbers and are thus not appropriate for
this task. Of course the entries of C are also integers.
The general partition function Z
G
(q, a) is then given by eq. (2.12). As by
now familiar for a = 0 this reduces to the chromatic polynomial
P
G
(q)
Z
G
(q, a = 0) =
N
S
i
=0
C
i
0
q
i
(2.20)
Finally in the case a = 1 (T =
) the entropy outweighs the energy and
there is no restriction on the spins and all the possible configurations are equal.
Z
G
(q, a = 1) = q
N
S
(2.21)
Besides the limit a = 0 which can be compared to chromatic polynomials
obtained from other calculations and eq. (1.20) these formulas provide a first
check for the correctness of the algorithm.
2.4
The zero temperature limit
In the limit lim
T
0
a = 0 the calculations can be simplified considerably.
The addingsite transfer matrix (eq. (2.13)) becomes:
(t
i.
)
kl
=
-
kl
+
l
0
,k
0
l
1
,k
1
· · ·
l
W -i-2
,k
W -i-2
l
W -i
,k
W -i
· · ·
l
W -1
,k
W -1
(2.22)
The local transfer matrix for connecting sites (eq. (2.15)) reads in this case:
(t
i i
+1
)
kl
= 1
-
k
W -2-i
,k
W -1-i
(2.23)
Note that instead of a vector of polynomials in a where eventually every
polynomial has degree N
B
now just a vector of numbers has to be stored. In
24
the general case Z
G
(a, q) can be looked at as a polynomial in q with coefficients
that are polynomials in a. For the chromatic polynomial only the constant
coefficients of the latter play a role.
Some further considerations about the resemblance of the finitelattice
results to the thermodynamic limit are in place: For periodic boundary condi-
tions and N
y
odd P
sq
(2) = 0. In the thermodynamic limit however the square
lattice is colorable with two colors. Thus one should not use periodic boundary
conditions for the lattices with odd N
y
. This problem does not occur for the
triangular and kagom´
e lattices since they contain triangular subgraphs and
are neither way twocolorable. For the honeycomb lattice, however, a similar
situation occurs.
25
Chapter 3
Partition function zeros
While investigating the properties of the partition functions one has to
distinguish two different cases: One focuses on the the variable q, hence the
zeros for fixed a in the complex qplane are plotted. The other deals with the
CTzeros.
Since these are different aspects of the model they will be treated in dif-
ferent sections. However, there are also similarities in the results. Most of all,
in both cases the zeros form a continuous curve
B in the thermodynamic limit
where either the ground state entropy (qzeros) or the free energy (azeros) is
non analytic.
Before the zerodistributions are shown the convention for the denotation
of the lattices and the latticesizes is presented and some remarks are made
on checks for the validity of the results.
26
3.1
Lattice types and sizes
Figure 3.1 shows the different lattice types that are used in this thesis. It
also gives the convention for the denotation of the lattice sizes.
Note that the labeling for square and triangular lattices is different from this
for honeycomb and kagom´
e lattices. In the first case the number of sites is
used, in the second the number of hexagons.
There are only two possible boundary conditions, either free in both direc-
tions or free in one direction and periodic in the other, which is then taken
as the vertical or ydirection. The general scheme for labelling the lattices
is type
(L
y
,L
x
),bc
. Type is one of sq, tri, hc or kag; L
y
and L
x
are the num-
ber of sites (for hc and kag the number of hexagons, C
y
and C
x
are used
instead). bc is either f f (free in both directions) or f p (free in x and periodic
in ydirection, abbreviated as F BC
x
and P BC
y
).
As the number of sites and bonds determines the degree of Z
G
(a, q) in q
and a, respectively, and those are not obvious for the honeycomb and kagom´
e
lattices, they are given in table 3.1 for further reference.
Especially interesting is the question, how good an approximation finite
lattices are for the thermodynamic limit. There are some exact results for
both cases. These are compared with the finitelattice results where they are
presented.
The next section 3.3 presents the results for the zeros in the complex q
plane, section 3.4 turns then to the complex temperature zeros (CT zeros).
27
a)
b)
c)
d)
Figure 3.1: Lattices to illustrate the convention for the denotation of the
lattice size. a) Square lattice; L
x
= 5 and L
y
= 4, sq
(4,5)
. b) Triangular
lattice; tri
(4,5)
. c) Honeycomb lattice; C
x
= 3 (hexagons in the horizontal
direction) and C
y
= 4 (hexagons in the vertical direction), hc
(4,3)
. d) Kagom´
e
lattice; C
x
= 3 and C
y
= 4, kag
(4,3)
28
Table 3.1: Properties of the lattices: L
x
(C
x
) is the number of sites (polygons)
in horizontal, L
y
(C
y
) the number of sites (polygons) in vertical direction.
The second column of the table contains information about the boundary
conditions, f/f means free in both directions, f/b free in horizontal and periodic
in vertical direction. N
S
and N
B
are the number of sites and the number of
bonds, respectively.
The terms in the rows f/b for N
B
have to be added to the expressions for f/f,
the number of sites is the same for f/f and f/b.
type
bc
N
S
N
B
sq
f/f
N
y
· N
x
(N
y
- 1) · N
x
+ (N
x
- 1) · N
y
f/p
+N
x
tri
f/f
N
y
· N
x
(N
y
- 1) · N
x
+ (N
x
- 1)(2 N
y
- 1)
f/p
+2 N
x
- 1
hc
f/f
2
{(C
x
+ 1)(C
y
+ 1)
- 1}
C
x
· (3 C
y
+ 2) + 2 C
y
- 1
p/f
+C
x
+ 3
kag
f/f
C
y
(3 C
x
+ 2) + 2 C
x
+ 1
2 C
x
(C
y
+ 1) + 2 C
y
(2 C
x
+ 1)
f/p
+2
· C
x
+ 1
29
3.2
Tests for the correctness of the results
Since numerical precision is crucial for calculating the zeros of high order
polynomials, several tests have been carried out to validate the results of the
root finder.
1. Plots of zeros have been rendered with both Maple and Mathematica.
There is no difference in the results.
2. The number of digits used in the calculation of the roots has been in-
creased until the distribution of the zeros showed no dependence on a
further change of the numerical precision. It is necessary to use precisions
up to 100 digits.
3. The polynomials (for fixed a) have been recalculated from the roots.
The coefficients differed at most 10
-3
in absolute value. That means
that this is also the maximum relative aberration since the coefficients
are all integers.
4. The validity of both the algorithm for calculating the Z
G
(a, q) and for
solving for the roots have been checked by comparing the distribution of
the zeros in the complex q and aplane with known results. The best
accordance is reached for chromatic polynomials since there the lattice
sizes for which exact results are known are also small and the results
could be exactly reproduced.
To illustrate the first point in the above list fig. 3.2 shows zerodistributions
calculated with different numerical precision. There is no deviation. Further
30
-2
-1
1
2
Re x
-2
-1
1
2
Im x
Figure 3.2:
The zeros for the 4x10 square lattice with
F BC
x
, P BC
y
, calculated with numerical precision of 40 (stars) and
120 digits (triangles) to provide a check for the reliability of the root
solver.
details are explained in the caption of the figure.
3.3
Zeros of the Partition function in the q
plane for fixed temperature
In this section results are presented on the general partition functions for
the Potts-model on a finite lattice. The method of calculation has been de-
scribed in detail in chapter 2. To investigate the properties of the partition
functions one variable (either a
e
K
or q) is kept fixed and the zeros in the
other one are plotted. Of course the latter is then allowed to take on complex
values.
The first aim is to determine the behavior of the chromatic zeros (zeros in
31
q for a = 0) as one turns on a (a > 0). Notice that a = 0 is equivalent to anti-
ferromagnetic coupling and zero temperature (K =
-). In this case only
those states contribute to the partition function in that no two neighboring
spins have the same direction. This is equivalent to the coloring problem that
leads to the definition of the chromatic polynomial (cf. section 1.4).
The following sections show the chromatic zeros and zeros for different
values of a = 0 in the qplane for different lattice sizes. The temperature is
not confined to physical values, but zeros have also been calculated for values
of a outside the physical range 0
a. Remember also that a > 1 corresponds
to ferromagnetic coupling (J < 0) and thus positive K. The results for the
qzeros have been reported in [26].
3.3.1
Square lattice
The chromatic zeros for some strip graphs are known exactly. In these cases
actually the limiting functions W
G
(q) (eq. 1.24) for strips of infinite length in
one direction have been calculated [27, 28]. Those provide of course a very
valuable check for the results shown here.
The chromatic zeros in fig. 3.3 agree exactly with those of fig. 3 a) in
ref. [27]. We will frequently refer to this paper in the following paragraphs,
since the authors have calculated chromatic zeros for various strip graphs and
it hence provides a very valuable reference for comparison with our results.
The behavior as a is increased from zero can be understood by looking at the
extremal cases. For a = 1 there is only one zero at the origin left. It is then
of order N
S
. The zeros shrink to the origin as a increases from zero to one.
32
0.5
1
1.5
2
2.5
Re q
-1.5
-1
-0.5
0.5
1
1.5
Im q
Figure 3.3: square lattice: L
x
= 18 and L
y
= 3, free boundaries
in both directions; zeros in the q-plane for a = 0 (larger dots),
a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
0.5
1
1.5
2
2.5
3
Re q
-2
-1
1
2
Im q
0.5
1
1.5
2
2.5
3
3.5
Re q
-3
-2
-1
1
2
3
Im q
Figure 3.4: sq
(3,18),ff
(cf. fig. 3.3) zeros in the q-plane for a) a = 0
(stars) and a =
-0.25 (triangles); b) a = 0 (stars) and a = -0.5
(triangles)
33
-0.03 -0.02 -0.01
0.01
0.02
0.03
Re q
-0.015
-0.01
-0.005
0.005
0.01
0.015
Im q
0.1
0.15 0.2
0.3
0.5 0.7
1
1.5
1-a
1
1.5
2
3
q_max
Figure 3.5: sq
(3,18),ff
(cf. fig. 3.3); a) zeros in the q-plane for a =
0.99 (stars) and a = 1.01 (triangles), b) plot of the realpart (larger
dots) of the rightmost and the imaginary part of the uppermost zero
against 1
- a.
Since the change from antiferromagnetic to ferromagnetic coupling (a > 1)
and from physical to complex temperature (a < 0) is continuous there is no
abrupt change in the zerodistributions and the regions a > 1 and a < 0 are
a continuation of the antiferromagnetic region (0
a 1). The flipping of
the order parameter J from its antiferromagnetic value 1 to its ferromagnetic
value
-1 occurs at a = 1 where = and the zeros have shrunk to the origin
anyway. For a > 1 the zeros lie in the left halfplane ( (a)
0) and for a < 0
they grow farther away from the origin compared to the chromatic zeros.
In the case of L
y
= 3 and periodic boundary conditions in ydirection, the
chromatic polynomial factorizes [27, 29]. Hence
B = as shown in fig. 3.6.
P
sq
(3,18),fp
(q) = q (q
- 1) (q - 2) (q
3
- 6 q
2
+ 14 q
- 13)
17
(3.1)
The only zeros apart from the standard one q = 0 and q = 1 are a pair of
34
0.5
1
1.5
2
2.5
Re q
-1.5
-1
-0.5
0.5
1
1.5
Im q
Figure 3.6: square lattice: L
x
= 18 and L
y
= 3, free boundaries
in x and periodic in ydirection; zeros in the q-plane for a = 0
(larger dots), a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
0.5
1
1.5
2
2.5
3
Re q
-2
-1
1
2
Im q
1
2
3
4
Re q
-3
-2
-1
1
2
3
Im q
Figure 3.7: sq
(3,18),fp
(cf. fig. 3.6) zeros in the q-plane for a) a = 0
(stars) and a =
-0.25 (triangles); b) a = 0 (stars) and a = -0.5
(triangles)
35
-0.03 -0.02 -0.01
0.01
0.02
0.03
Re q
-0.02
-0.01
0.01
0.02
Im q
0.1
0.15 0.2
0.3
0.5 0.7
1
1.5
1-a
0.5
1
2
q_max
Figure 3.8: sq
(3,18),fp
(cf. fig. 3.6); a) zeros in the q-plane for a = 0.99
(stars) and a = 1.01 (triangles), b) plot of the realpart (larger
dots) of the rightmost and the imaginary part of the uppermost
zero against 1
- a.
complex conjugate points q = 1.773
± 1.468 i and one point on the real axis
q = 2.453; This degeneracy vanishes as soon as a is increased. Eq. (8.10) in
[29] gives P
sq
(3,),fp
which differs from eq. (3.1) just by the exponent N
S
/3
instead of 17.
It is remarkable, that the similarity between different lattice sizes grows
with a and the differences are most significant for the chromatic zeros (a = 0)
(cf. figs. 3.3 and 3.6).
For a very close to one (fig. 3.11 a)) the plots for 1
- l and 1 + l (l small)
seem to be symmetric about the imaginaryaxis. This is not true, however,
and can easily be seen from the fact, that Z
G
(1
- l, q) = Z(1 + l, q) for any
value of l = 0. For two polynomials to be the same all the coefficients have to
be equal separately and (1
- l)
p
= (1 + l)
p
in general.
36
0.5
1
1.5
2
2.5
Re q
-1.5
-1
-0.5
0.5
1
1.5
Im q
Figure 3.9: square lattice: L
x
= 10 and L
y
= 4, free boundaries
in both directions; zeros in the q-plane for a = 0 (larger dots),
a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
From exact results it is known that the zeros in q for a circular graph
(one dimensional) show a power law behavior [30] and fig. 3.11 b) shows a
loglogplot of q versus 1
- a. Since the plots of the zeros (cf. fig. 3.9) give
hint that the scaling of the real and imaginaryparts of the zeros is different,
the investigation of the scaling behavior is done separately. Hence fig. 3.11 b)
shows ( (q
zero
))
max
and ( (q
zero
))
max
versus (1
-a) separately. Furthermore a
least square fit was carried out for the data (log q and log(1
- a)). The results
are shown at the end of this section in tables 3.2 and 3.3.
The chromatic zeros in fig. 3.9 agree with those that have been calculated
by Ro
cek et al. [28] and are shown in their fig. 3 b). The small differences are
due to the fact that they used a lattice of different size that would be sq
(4,18),ff
in the notation used here. It is possible to infer the form of
B for the limit
C
x
that is also shown in fig. 3 a) of [28] from the chromatic zeros in fig.
37
0.5
1
1.5
2
2.5
3
Re q
-2
-1
1
2
Im q
0.5
1
1.5
2
2.5
3
3.5
Re q
-3
-2
-1
1
2
3
Im q
Figure 3.10: sq
(4,10),ff
(cf. fig. 3.9) zeros in the q-plane for a) a = 0
(stars) and a =
-0.25 (triangles); b) a = 0 (stars) and a = -0.5
(triangles)
-0.03 -0.02 -0.01
0.01
0.02
0.03
Re q
-0.02
-0.01
0.01
0.02
Im q
0.1
0.15 0.2
0.3
0.5 0.7
1
1.5
1-a
0.1
0.2
0.5
1
2
5
q_max
Figure 3.11: sq
(4,10),ff
(cf. fig. 3.9); a) zeros in the q-plane for a =
0.99 (stars) and a = 1.01 (triangles), b) plot of the realpart (larger
dots) of the rightmost and the imaginary part of the uppermost zero
against 1
- a.
38
0.5
1
1.5
2
2.5
3
Re q
-2
-1
1
2
Im q
Figure 3.12: square lattice: L
x
= 10 and L
y
= 4, free boundaries
in x and periodic in ydirection; zeros in the q-plane for a = 0
(larger dots), a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
3.9. Therefore the behavior of
B for the infinite strip as a is increased from
zero should basically be the same as the behavior of the qzeros in figs. 3.9
and 3.12.
Fig. 3.12 compares to fig. 3 a) in [27]. The lattice size used here is again
slightly different from this in the reference. The distribution of the zeros is
however nearly the same and the statements made in the preceding paragraph
also hold in this case. Opposite to L
y
= 3 for L
y
= 4 the chromatic polynomial
for the square lattice does not factorize. Note also that the curve
B for strips
of the square lattice with quadratic transverse cross section (e. g. periodic
boundary conditions in vertical or ydirection) does not enclose regions in the
complex qplane [27]. This is different from the case of triangular lattices that
will be presented in the following section.
39
0.5
1
1.5
2
2.5
3
3.5
Re q
-2
-1
1
2
Im q
1
2
3
4
Re q
-3
-2
-1
1
2
3
Im q
Figure 3.13: sq
(4,10),fp
(cf. fig. 3.12) zeros in the q-plane for a) a = 0
(stars) and a =
-0.25 (triangles); b) a = 0 (stars) and a = -0.5
(triangles)
-0.04
-0.02
0.02
0.04
Re q
-0.02
-0.01
0.01
0.02
Im q
0.1
0.15 0.2
0.3
0.5 0.7
1
1.5
1-a
1
1.5
2
3
q_max
Figure 3.14: sq
(4,10),fp
(cf. fig. 3.12); a) zeros in the q-plane for
a = 0.99 (stars) and a = 1.01 (triangles), b) plot of the real
part (larger dots) of the rightmost and the imaginary part of the
uppermost zero against 1
- a.
40
0.5
1
1.5
2
2.5
3
Re q
-1
-0.5
0.5
1
Im q
Figure 3.15: triangular lattice: L
x
= 14 and L
y
= 3, free boundaries
in both directions; zeros in the q-plane for a = 0 (larger dots),
a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
3.3.2
Triangular lattice
The structure of the plots in this section is the same as for the square
lattice, e.g. first the antiferromagnetic range of a is treated and afterwards
a is allowed to take on ferromagnetic (a > 1) and unphysical (a < 0) values.
The latter corresponds to imaginary temperature. The triangular lattice is
drawn as a square lattice with additional diagonal bonds (cf. fig. 3.1 d), [31]).
The reference for fig. 3.15 is fig. 5 a) in [28]. The zeros that are shown
there are the same as here, since the lattice sizes agree exactly.
As has already been observed in 3.3.1 for L
y
= 3 and periodic boundary
conditions the chromatic polynomial factorizes and
B = .
P
tri
(3,14),fp
(q) = q (q
- 1) (q - 2) (q - 3)
39
(3.2)
The chromatic polynomial that is given in the appendix of [29] is different
41
1
2
3
4
5
Re q
-1
-0.5
0.5
1
Im q
1
2
3
4
5
6
Re q
-1
-0.5
0.5
1
Im q
Figure 3.16: tri
(3,14),ff
(cf. fig. 3.15) zeros in the q-plane for a) a = 0
(stars) and a =
-0.25 (triangles); b) a = 0 (stars) and a = -0.5
(triangles)
-0.03-0.02-0.01
0.01 0.02 0.03
Re q
-0.02
-0.01
0.01
0.02
Im q
0.1
0.15 0.2
0.3
0.5 0.7
1
1.5
1-a
1
1.5
2
3
q_max
Figure 3.17: tri
(3,14),ff
(cf. fig. 3.15); a) zeros in the q-plane for
a = 0.99 (stars) and a = 1.01 (triangles), b) plot of the real
part (larger dots) of the rightmost and the imaginary part of the
uppermost zero against 1
- a.
42
1
2
3
4
5
1
2
3
4
5
Figure 3.18: Triangular lattice with with free boundary conditions
in x (F BC
x
) and periodic in ydirection (P BC
y
). The labelling
for this lattice is tri
(4,5),fb
.
0.5
1
1.5
2
2.5
3
Re q
-1.5
-1
-0.5
0.5
1
1.5
Im q
Figure 3.19: triangular lattice: L
x
= 14 and L
y
= 3, free boundaries
in x and periodic in ydirection; zeros in the q-plane for a = 0
(larger dots), a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
43
1
2
3
4
5
6
7
Re q
-0.4
-0.2
0.2
0.4
Im q
2
4
6
8
10
Re q
-0.6
-0.4
-0.2
0.2
0.4
0.6
Im q
Figure 3.20: tri
(3,14),fp
(cf. fig. 3.19) zeros in the q-plane for a) a = 0
(stars) and a =
-0.25 (triangles); b) a = 0 (stars) and a = -0.5
(triangles)
-0.04
-0.02
0.02
0.04
Re q
-0.03
-0.02
-0.01
0.01
0.02
0.03
Im q
0.1
0.15 0.2
0.3
0.5 0.7
1
1.5
1-a
1
1.5
2
3
5
7
10
q_max
Figure 3.21: tri
(3,14),fp
(cf. fig. 3.19); a) zeros in the q-plane for
a = 0.99 (stars) and a = 1.01 (triangles), b) plot of the real
part (larger dots) of the rightmost and the imaginary part of the
uppermost zero against 1
- a.
44
0.5
1
1.5
2
2.5
3
3.5
Re q
-1.5
-1
-0.5
0.5
1
1.5
Im q
Figure 3.22: triangular lattice: L
x
= 8 and L
y
= 4, free boundaries
in both directions; zeros in the q-plane for a = 0 (larger dots),
a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
from eq. (3.2). This is due to the fact that the implementation of periodic
boundary conditions for a triangular lattice makes it necessary to use a dif-
ferent direction for the diagonal bonds in one row (cf. fig. 3.18). In appendix
B.2 it is explained why this is unavoidable if one wants to use the interface
method. One important consequence of these boundary conditions is that
tri
(3,14),fp
is not colorable with three colors. That would be different if all the
vertical bonds had the same direction.
As before in the case of the square lattice, for a away from zero Z
tri
(3,14),ff
does no longer factorize and in the thermodynamic limit the zeros form again a
continuous locus
B that can be inferred from their distribution in the complex
qplane.
Figures 3.22 and 3.25 show the zeros for the triangular strip with L
y
= 4.
In the case of periodic boundary conditions the chromatic polynomial does
45
1
2
3
4
5
Re q
-1.5
-1
-0.5
0.5
1
1.5
Im q
1
2
3
4
5
6
7
Re q
-1.5
-1
-0.5
0.5
1
1.5
Im q
Figure 3.23: tri
(4,8),ff
(cf. fig. 3.22) zeros in the q-plane for a) a = 0
(stars) and a =
-0.25 (triangles); b) a = 0 (stars) and a = -0.5
(triangles)
-0.04
-0.02
0.02
0.04
Re q
-0.03
-0.02
-0.01
0.01
0.02
0.03
Im q
0.1
0.15 0.2
0.3
0.5 0.7
1
1.5
1-a
1
1.5
2
3
5
7
q_max
Figure 3.24: tri
(4,8),ff
(cf. fig. 3.22); a) zeros in the q-plane for
a = 0.99 (stars) and a = 1.01 (triangles), b) plot of the real
part (larger dots) of the rightmost and the imaginary part of the
uppermost zero against 1
- a.
46
0.5
1
1.5
2
2.5
3
3.5
Re q
-2
-1
1
2
Im q
Figure 3.25: triangular lattice: L
x
= 8 and L
y
= 4, free boundaries
in x and periodic in ydirection; zeros in the q-plane for a = 0
(larger dots), a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
1
2
3
4
5
Re q
-2
-1
1
2
Im q
1
2
3
4
5
6
7
Re q
-2
-1
1
2
Im q
Figure 3.26: tri
(4,8),fp
(cf. fig. 3.25) zeros in the q-plane for a) a = 0
(stars) and a =
-0.25 (triangles); b) a = 0 (stars) and a = -0.5
(triangles)
47
-0.04
-0.02
0.02
0.04
Re q
-0.03
-0.02
-0.01
0.01
0.02
0.03
Im q
0.1
0.15 0.2
0.3
0.5 0.7
1
1.5
1-a
1
1.5
2
3
5
7
q_max
Figure 3.27: tri
(4,8),fp
(cf. fig. 3.25); a) zeros in the q-plane for
a = 0.99 (stars) and a = 1.01 (triangles), b) plot of the real
part (larger dots) of the rightmost and the imaginary part of the
uppermost zero against 1
- a.
no longer factorize and
B = . The case with free boundary conditions in
ydirection can be compared with fig. 5 b) in [28], the chromatic zeros for
P BC
y
with fig. 4 a) in [27]. For the triangular strip with periodic boundaries
in ydirection (e. g. quadratic cross section in transverse direction)
B does
enclose regions. This is different from the square lattice [27].
It is however difficult to infer theses enclosed regions from the plots shown
here and the reader is refered to fig. 4 a) of [27] where the continuous locus
for the triangular strip is shown for the thermodynamic limit in one direction.
There is however a difference in the boundary conditions used here and there
(cf. fig. 3.18).
A further general feature of the qzero distributions is that the gaps be-
tween the different parts of
B become smaller and form arcs separating the
48
otherwise very smooth curve or disappear completely for increasing a. For
values of a very close to one the zeros form a nearly continuous curve. In this
regime there also occur zeros with negative real part of q. This is however
only the case for P BC
y
and non factorizing P
G
(q), e. g. sq
(4,10),fp
(cf. fig.
3.12) and tri
(4,8),fp
(cf. fig. 3.25). For free boundary conditions or cases where
P
G
(q) factorizes there are no zeros with
(q)
0 for any value of a. There are
zeros with
(q) = 0 for the 5x5square lattice (fig. C.1). This feature has to
be noted with caution, though, because those zeros zeros with
(q)
0 could
well be transient and disappear again in the thermodynamic limit.
For the honeycomb lattice treated in the next section the the situation
is different and there are cases with zeros in the left halfplane or on the
imaginary axis for both a = 0 and a = 0.
3.3.3
Honeycomb lattice
The honeycomb lattice is dual to the triangular. However, there exists
nothing similar to the duality relation for a and a
d
that connects the qzeros
of a lattice with that of its dual.
It is of course also of interest to investigate the influence of the lattice size.
In general the features of loci of zeros are very similar. There are some subtle
differences however, which reveal the dependence on the lattice size. Fig. 6 in
ref. [28] shows the locus
B for a strip of infinite length hc
(2,),ff
that is closer
to the curve one can infer from the larger lattice in fig. 3.28 than that in fig.
C.4.
Again the differences for the different lattice sizes (figs. 3.28 and C.4)
49
0.5
1
1.5
2
Re q
-1
-0.5
0.5
1
Im q
Figure 3.28: honeycomb lattice: C
x
= 9 and C
y
= 2, free bound-
aries in both directions; zeros in the q-plane for a = 0 (larger dots),
a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
0.5
1
1.5
2
2.5
Re q
-1.5
-1
-0.5
0.5
1
1.5
Im q
0.5
1
1.5
2
2.5
3
3.5
Re q
-2
-1
1
2
Im q
Figure 3.29: hc
(2,9),ff
(cf. fig. 3.28) zeros in the q-plane for a) a = 0
(stars) and a =
-0.25 (triangles); b) a = 0 (stars) and a = -0.5
(triangles)
50
-0.02
-0.01
0.01
0.02
Re q
-0.01
-0.005
0.005
0.01
Im q
0.1
0.15 0.2
0.3
0.5 0.7
1
1.5
1-a
0.1
0.2
0.5
1
2
q_max
Figure 3.30: hc
(2,9),ff
(cf. fig. 3.28); a) zeros in the q-plane for
a = 0.99 (stars) and a = 1.01 (triangles), b) plot of the real
part (larger dots) of the rightmost and the imaginary part of the
uppermost zero against 1
- a.
are most distinct for the chromatic zeros and become less significant as a is
increased to finite values. Remarkable about the zero distributions shown in
fig. 3.28 is the fact that even for F BC
y
there are qzeros of Z
G
(a, q) with
(q) = 0. This is the only lattice type where this occurs.
To illustrate the periodic boundary conditions for the honeycomb lattices,
an example is shown in fig. 3.31.
The qzeros for periodic boundary conditions form a smoother curve than
those for free boundary conditions (cf. figs. 3.32 and C.7). The most re-
markable fact is however that some zeros of P
G
(q) lie in the left halfplane
( (q) < 0). As can be expected from the experience with the other cases de-
scribed in the previous paragraphs this feature prevails as a is increased away
from zero. On the other hand for decreasing a this does not seem to be true
51
1
2
3
4
5
6
7
1
2
3
4
5
6
7
Figure 3.31: Honeycomb lattice with periodic boundary conditions;
the sites with the same number are identical. The label for this
lattice is hc
(2,3),fp
.
0.5
1
1.5
2
2.5
Re q
-1.5
-1
-0.5
0.5
1
1.5
Im q
Figure 3.32: honeycomb lattice: C
x
= 5 and C
y
= 3, free bound-
aries in x and periodic in ydirection; zeros in the q-plane for a = 0
(larger dots), a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
52
0.5
1
1.5
2
2.5
3
Re q
-2
-1
1
2
Im q
1
2
3
4
Re q
-2
-1
1
2
Im q
Figure 3.33: hc
(3,5),fp
(cf. fig. 3.32) zeros in the q-plane for a) a = 0
(stars) and a =
-0.25 (triangles); b) a = 0 (stars) and a = -0.5
(triangles)
-0.03 -0.02 -0.01
0.01
0.02
0.03
Re q
-0.015
-0.01
-0.005
0.005
0.01
0.015
Im q
0.1
0.15 0.2
0.3
0.5 0.7
1
1.5
1-a
0.5
1
2
q_max
Figure 3.34: hc
(3,5),fp
(cf. fig. 3.32); a) zeros in the q-plane for
a = 0.99 (stars) and a = 1.01 (triangles), b) plot of the real
part (larger dots) of the rightmost and the imaginary part of the
uppermost zero against 1
- a.
53
0.5
1
1.5
2
2.5
Re q
-1
-0.5
0.5
1
Im q
Figure 3.35: kagom´
e lattice: C
x
= 1 and C
y
= 10, free boundaries
in both directions; zeros in the q-plane for a = 0 (larger dots),
a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
since there are zeros with negative real part for a =
-0.25 but not any more
for a =
-0.5 as can bee seen in fig. 3.33. For F BC
y
there are some zeros for
values of a close to one that have negative real part (cf. fig. C.9).
3.3.4
Kagom´
e lattice
The kagom´
elattice is the only lattice that consists of different polygons
and is the most difficult to calculate. Therefore only smaller sizes can be
reached.
Here no zeros with
(q) < 0 occur at all. However the width of the lattice
is very small and no periodic boundary conditions were used. Therefore no
thorough statement can be made about the behavior of the qzeros for the
kagom´
e lattice neither for a = 0 nor for finite values of a.
54
0.5
1
1.5
2
2.5
3
3.5
Re q
-1
-0.5
0.5
1
Im q
1
2
3
4
Re q
-1.5
-1
-0.5
0.5
1
1.5
Im q
Figure 3.36: kag
(1,10),ff
(cf. fig. 3.35) zeros in the q-plane for a)
a = 0 (stars) and a =
-0.25 (triangles); b) a = 0 (stars) and
a =
-0.5 (triangles)
-0.02
-0.01
0.01
0.02
Re q
-0.02
-0.01
0.01
0.02
Im q
0.1
0.150.2
0.3
0.5 0.7
1
1.5
1-a
0.05
0.1
0.5
1
5
q_max
Figure 3.37: kag
(1,10),ff
(cf. fig. 3.35); a) zeros in the q-plane for
a = 0.99 (stars) and a = 1.01 (triangles), b) plot of the real
part (larger dots) of the rightmost and the imaginary part of the
uppermost zero against 1
- a.
55
For the same lattice with P BC
y
, kag
(1,10),fp
, the chromatic polynomial
factorizes and is given by
P
kag
(1,10),fp
(q) = q (q
- 1) (q
2
- 5 q + 7)
10
(q
2
- 3 q + 3)
10
(q
- 2)
11
(3.3)
For a = 0 Z
kag
(1,10),fp
(a, q) does not factorize any more.
A kagom´
estrip with P BC
y
is whown in fig. B.2. The additional vertical
bonds (instead of just identifying the top and bottom line of the lattice with
F BC
y
) are due to the way Z is calculated (section 2.2.
Lattices that are closer to the thermodynamic limit in the sense that they
have a considerable size in both directions are very hard to calculate, because
of the exponential growth of computation time (cf. tables A.1, A.2, A.3,A.4).
3.3.5
Scaling of the qzeros with the temperature
There is strong indication from the preceding figures that the zeros shrink
to the origin like q
c
= r
· (1 - a)
p
, where q
c
stands for the center of the zero
distribution. This notion is motivated by the form of the zeros that seem to
lie on a regular curve with its center on the Reaxis (due to the fact that the
zeros are real or come in complexconjugate pairs). However, the scaling in
x and ydirection is different.
Therefore q
y,max
and q
x,max
were used which are the imaginary part of the
uppermost and the real part of the rightmost zero, respectively. Hence the
scaling law reads
q
x/y,max
= r
x/y
· (1 - a)
p
x/y
(3.4)
The values of r
x/y
and p
x/y
were determined by a least square fit with the
56
Table 3.2: Scaling of the qzeros with 1
- a for the square and
honeycomb lattice. Shown are the coefficients of the scaling law that
describes the shrinking of the zeros to the origin with decreasing
1
- a. The meaning of r and p is explained in the text.
type
size
bc
r
x
p
x
r
y
p
y
sq
3x18
fbc/pbc
4.21
2.81
sq
4x8
fbc/fbc
2.44
2.52
2.71
1.86
sq
4x8
fbc/pbc
2.50
2.91
2.68
2.19
sq
4x10
fbc/fbc
2.44
2.56
2.70
1.89
sq
4x10
fbc/pbc
2.50
2.94
2.67
2.22
sq
5x5
fbc/fbc
2.45
2.49
2.71
1.85
hc
2x7
fbc/fbc
2.59
2.27
2.73
1.33
hc
3x3
fbc/fbc
2.60
2.25
2.75
1.32
hc
3x5
fbc/fbc
2.59
2.33
2.73
1.40
hc
3x5
fbc/pbc
2.60
2.56
2.70
1.68
57
Table 3.3: Scaling of the qzeros with 1
- a for the triangular and
kagom´
e lattice (cf. table 3.2).
type
size
bc
r
x
p
x
r
y
p
y
tri
3x14
fbc/fbc
2.87
3.59
1.78
1.16
tri
3x14
fbc/pbc
2.61
3.42
2.03
1.71
tri
4x8
fbc/fbc
2.85
3.80
1.93
1.49
tri
4x8
fbc/pbc
2.64
3.89
2.21
2.45
tri
4x10
fbc/fbc
1.
7.18
1.
1.60
tri
4x10
fbc/pbc
2.62
3.93
2.20
2.46
tri
4x14
fbc/fbc
1.
7.25
1.
1.69
kag
1x10
fbc/fbc
5.82
2.88
kag
1x10
fbc/pbc
3.75
2.61
1.89
1.11
kag
2x3
fbc/pbc
2.62
2.83
2.51
1.77
58
logarithmic form of eq. (3.4) and are shown for the different lattices presented
in this chapter in tables 3.2 and 3.3.
3.4
Complex temperature zeros of the Parti-
tion function for fixed q
The introduction (section 1.2) gave a brief description of the characteristics
of complextemperature (CT) zeros of the partition function for the Potts
Model (PM) on finite lattices. This section will present some more details.
The only properties that matter in the critical region are symmetry and
dimension. The symmetry of the model is determined by the lattice shape
and the number of possible spin directions q. There is no critical behavior for
finite lattices. However, the distribution of the partition function zeros in the
finite case allows to some extent to infer the critical manifold.
By analytic continuation one can define CT extensions of physicaltemp-
erature phases. These CTphases are separated by boundaries where Z is
non-analytic. The boundaries are collectively denoted as
B and are formed
via coalescence of the accumulating complex temperature zeros of Z in the
thermodynamic limit. The region far away from the origin in the aplane is
the CTextension of the ferromagnetic phase (T
0, J < 0, e.g. K ).
The one that encloses the origin (T
0, J > 0) is the antiferromagnetic
phase. If there is any region in between then it is the paramagnetic phase,
enclosing the point a = 1 (T
).
59
The duality relation eq. (1.16) reads
a
d
(a
- 1) = (a - 1) + q
(3.5)
(a
- 1) (a
d
- 1) = q
(3.6)
It is a general property of the CT phasediagram that the locus
B where
the free energy is nonanalytic is compact, if the model is above its lower
critical dimension, e.g. there is a phase transition associated with symmetry
breaking. Eq. (3.6) connects
B for a lattice and its dual: the locus of the duals
of the zeros is identical to the dual of the locus of zeros in the thermodynamic
limit [32].
In addition to the locus
B further information can be extracted from the
distribution of the zeros. Approaching the thermodynamic limit one can define
a density of zeros that behaves near a singular point a
s
as [4, 33].
g(s)
s
1-
, as s
0
(3.7)
where s is the arclength along
B away from a
s
(s =
|a - a
s
| for s 0) and
the singularity in the free energy at a
s
is f
sing
|a - a
s
|
2-
.
For general theory and examples cf. [4, 9, 34, 35, 36, 37, 38].
In the case of finite lattices there is no critical behavior. However, the
partition function zeros allow to infer the shape of the curve
B at least partially.
Since the square lattice is selfdual if the proper boundaryconditions are used
eq. (3.6) reads (a
- 1)
2
= q and there are two critical points at a = 1 +
q
and a = 1
- q. The second one is of course unphysical for q 1. The zeros
for physical temperatur
(a)
0) in the xplane should hence lie on or close
to the circle
|x| = 1, where x =
a
-1
q
.
60
The critical temperature for the honeycomb lattice is determined by the
following equation [39].
a
3
- 3 a
2
- 3 (q - 1) a - q
2
+ 3 q
- 1 = 0
(3.8)
From this one can deduce the physical transition points PMAFM and PM
FM. As q increases the physical AFMphase is squeezed out. For 2
q 4
the phase transition is of second order, for q
5 it is of first order [7].
For q = 3 the equation reads a
3
- 3 a
2
- 6 a - 1 = 0 and the critical points
are:
a
c
1
= a
P M
-F M
= 1 + 2
3 cos(/18) = 4.41147 . . .
(3.9)
a
c
2
= 1
-
3 cos(/18) + 3 sin(/18) =
-0.1847925 . . .
(3.10)
a
c
3
= 1
-
3 cos(/18)
- 3 sin(/18) = -1.22668 . . .
(3.11)
Figures 3.4 and 3.4 have been used as a check for the CTzeros.
For
the general Z
G
(a, q) these lattice sizes can not be reached. They agree with
the distributions of zeros shown in refs. [40, 41] if one takes into account
the different sizes those authors calculated and more important the different
boundaryconditions.
The critical values are known exactly for the square, triangular and hon-
eycomb lattice and are given by the following equations [42].
x
2
- 1 = 0 square lattice
(3.12)
q x
3
+ 3 x
2
- 1 = 0 triangular lattice
(3.13)
x
3
- 3 x -
q = 0
honeycomb lattice
(3.14)
61
-2
-1
1
2
3
4
Re a
-3
-2
-1
1
2
3
Im a
Figure 3.38: Honeycomb lattice: C
x
= 5 and C
y
= 5,
free boundaries in both directions; zeros in the a-plane
for q = 3.
-2
2
4
6
Re a
-4
-2
2
4
Im a
Figure 3.39: Honeycomb lattice: C
x
= 5 and C
y
= 5,
free boundaries in both directions; zeros in the a-plane
for q = 5.
62
-3
-2.5
-2
-1.5
-1
-0.5
Re x
-2
-1
1
2
Im x
-2.5
-2
-1.5
-1
-0.5
0.5
Re x
-1.5
-1
-0.5
0.5
1
1.5
Im x
-2
-1.5
-1
-0.5
0.5
Re x
-1
-0.5
0.5
1
Im x
-10
-8
-6
-4
-2
Re x
-7.5
-5
-2.5
2.5
5
7.5
Im x
-5
-4
-3
-2
-1
Re x
-3
-2
-1
1
2
3
Im x
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
Re x
-2
-1
1
2
Im x
Figure 3.40: square lattice: L
x
= 10 and L
y
= 4, free boundaries
in x and periodic in ydirection; zeros in the x-plane for q =
0.1, 0.5, 0.9, 1.1, 1.5, 1.8 (from top left).
Eq. (3.13) can be transformed into eq. (3.14) by inverting x, since honeycomb
and triangular lattice are dual to each other. Eq. 3.8 is of course equivalent to
eq. 3.14 with a =
q x + 1.
The results presented in this section have been reported in [43].
3.4.1
CTzeros for small non integral q
For small values of q there are two interesting cases: one is the exactly
solvable Isingmodel (q = 2) and the other one is the connection to a dilute
spinglass for q =
1
2
[7]. There are relations to other problems, too, but
they are only valid for more complex models with multisite interactions and
external fields [7] (cf. eq. (1.5)).
63
-2
-1.5
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
-2
-1.5
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
-2
-1.5
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
-2
-1.5
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
Figure 3.41: square lattice: L
x
= 10 and L
y
= 4, free boundaries
in x and periodic in ydirection; zeros in the x-plane for q =
2, 2.1, 2.5, 3 (from top left).
For q = 0 and q = 1 the partition function reduces simply to 0 and a
N
B
.
q = 2 is the only case where there is an exact solution (Onsager, [2]). This
fact is reflected in the shape of the zerodistribution. In the thermodynamic
limit the zeros for the square lattice Isingpartition function form two circles
of radius one in the xplane centered at x = 0 and x =
-
2 [44, 45].
The questions are therefore:
1. What happens if one goes slightly away from zero or one?
2. What do the zeros look like for q = 1/2 and slightly away from this
value?
3. Is there any region around q = 2 where the zerodistribution shows a
regular shape similar to the case q = 2 where an exact solution exists?
64
-1.5
-1
-0.5
0.5
Re x
-1
-0.5
0.5
1
Im x
-1.5
-1
-0.5
0.5
Re x
-1
-0.5
0.5
1
Im x
-1.5
-1
-0.5
0.5
Re x
-1
-0.5
0.5
1
Im x
-1.5
-1
-0.5
0.5
Re x
-1
-0.5
0.5
1
Im x
Figure 3.42: triangular lattice: L
x
= 8 and L
y
= 4, free boundaries
in x and periodic in ydirection; zeros in the x-plane for q =
2, 2.1, 2.5, 3 (from top left).
65
The exact curves
B in the case q = 2 for the honeycomb (and hence also its
dual the triangular) [46] and the kagom´
e lattice [36] are given by the following
equations.
1
- 2 a + 6 a
2
- 2 a
3
+ a
4
- 2 a(1 - a)
2
w = 0
honeycomb
(3.15)
1 + 3 u
2
- 2 u (1 - u) w = 0 triangular
(3.16)
21 + 24 a
2
+ 18 a
4
+ a
8
- 4 (1 + a
2
+ a
4
- a
6
) w = 0
kagom´
e
(3.17)
where the parameter w takes on the values
-
3
2
w 3 and u = (1/a)
2
. The
equation for the curves in the xplane can immediately be obtained setting
a =
2 x + 1.
They are shown in figs. 3.40 and 3.41 (square) and 3.42 (triangular), 3.43
(honeycomb) and 3.44 (kagom´
e) together with the zerodistributions in the
xplane for q = 2, q = 2.1, q = 2.5 and q = 3.
Fig. 3.40 shows the zerodistributions of the squarelattice Pottsmodel
for small non integral values of q. As mentioned before the case q = 1/2 is
connected to a dilute spinglass that is supposed to have a phase transition
from a paramagnetic to a spinglass phase. The inferred curve
B crosses the
real axis at x =
-0.5 and x = -3.4. The change of B as one goes away from
q = 0.5 is very smooth. Since q = 1.8 is very close to the Isingmodel fig.
3.40 f) shows also the exact locus
B (two circles).
The zeros shown in fig. 3.41 a) resemble the known unit circles for the
thermodynamic limit. However, as soon as q is slightly increased from 2 the
regular shape vanishes and the structure of fig. 3.41 d) (q = 3) emerges. This
last case has been investigated in [11] and the features of the zeros in fig.
66
-2
-1
1
2
Re x
-1.5
-1
-0.5
0.5
1
1.5
Im x
-2
-1
1
2
Re x
-1.5
-1
-0.5
0.5
1
1.5
Im x
-1.5
-1
-0.5
0.5
1
1.5
2
Re x
-1.5
-1
-0.5
0.5
1
1.5
Im x
-2
-1
1
2
Re x
-1.5
-1
-0.5
0.5
1
1.5
Im x
Figure 3.43: honeycomb lattice: C
x
= 5 and C
y
= 3, free bound-
aries in x and periodic in ydirection; zeros in the x-plane for
q = 2, 2.1, 2.5, 3 (from top left).
67
3.41 d) agree with their fig. 1. Note that in this reference the zeros are plotted
in the aplane. The circles are the curves that would be formed by the zeros
in the thermodynamic limit for q = 2.
The zerodistributions for the triangular, honeycomb and the kagom´
e lat-
tice (cf. figs. 3.42, 3.43 and 3.44) show the same behavior. As soon as q is
increased from 2 the zeros leave the curves that are given by eqs. 3.15 and
3.17 and determine the locus
B. The CTzeros of the triangular and honey-
comb lattice are connected via the duality relation eq. (1.16). Further plots
are shown in appendix C.2.
3.4.2
CTzeros for large values of q
For very large values of q there are some exact results. In the many
component limit meanfieldtheory is exact and the Potts model is solvable
[47]. It has been conjectured on the basis of numerical results [48] that the
zeros of the square lattice Pottsmodel lie on the unitcircle in the xplane for
the infinite q limit [48]. This conjecture has been proved by the same authors
[49] using known solutions for latticeanimal and enumeration problems.
This result is reflected in the numerical data shown in figs. 3.45 d) and
3.46. To show the approach of this limit the zeros in the aplane for smaller
values of q are plotted, too (fig. 3.45, a) to c)).
The partition function in the many componentlimit (the zeros for q = 100
are plotted fig. 3.45 d) is determined by the leading coefficients in q. For the
68
-2.5
-2
-1.5
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
-2.5
-2
-1.5
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
-2.5
-2
-1.5
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
-2.5
-2
-1.5
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
Figure 3.44: kagom´
e lattice: C
x
= 3 and C
y
= 2, free boundaries
in x and periodic in y direction; zeros in the x-plane for q =
2, 2.1, 2.5, 3 (from top left).
69
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
-1.5
-1
-0.5
0.5
Re x
-1
-0.5
0.5
1
Im x
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
Figure 3.45: square lattice: L
x
= 10 and L
y
= 4, free boundaries
in x and periodic in ydirection; zeros in the x-plane for q =
4, 10, 25, 100 (from top left).
70
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
Figure 3.46: CTzeros for the leading coefficient of Z
sq
(4,10),fp
(x, q)
in q (cf. eq. (3.18). Further details are explained in the text.
4x10square lattice with P BC
y
(sq
(4,10),fp
) they are
Z
sq
(4,10),fp
= q
40
+ 76 x q
79/2
+ (2850 x
2
+ 46 x
4
+ 9 x
12
+ 8 x
20
+ 7 x
28
+ 6 x
36
+ 5 x
44
+ 4 x
52
+ 3 x
60
+ 2 x
68
+ x
76
) q
39
+ . . .
(3.18)
". . ." represents terms of order lower than q
39
; the degree of Z
sq
(4,10),fp
in x
is N
B
= 76. The zeros of the third term in eq. (3.18) are shown in fig. 3.46
and are basically the same as those in fig. 3.45 d). Hence the limit q
can be understood from the coefficients of the leading powers of q in Z
G
(x, q),
which are themselves polynomials in x.
From eqs. (3.13) and (3.14) the behavior of the critical point in the many
71
-0.6
-0.4
-0.2
0.2
0.4
Re x
-0.6
-0.4
-0.2
0.2
0.4
0.6
Im x
-0.4
-0.2
0.2
0.4
Re x
-0.4
-0.2
0.2
0.4
Im x
-1.25
-1
-0.75 -0.5 -0.25
0.25
0.5
Re x
-0.75
-0.5
-0.25
0.25
0.5
0.75
Im x
-0.8 -0.6 -0.4 -0.2
0.2
0.4
Re x
-0.6
-0.4
-0.2
0.2
0.4
0.6
Im x
Figure 3.47: triangular lattice: L
x
= 8 and L
y
= 4, free boundaries
in x and periodic in ydirection; zeros in the x-plane for q =
4, 10, 25, 100 (from top left).
72
component limit for the honeycomb and triangular lattice is determined by
x
q
1/6
honeycomb
(3.19)
x
q
-1/6
triangular
(3.20)
The same calculations as for the square lattice have been carried out for the
triangular and honeycomb lattice (figs. 3.47 and 3.48). The zero distributions
are again connected by the duality relation eq. (1.16) and the numerical results
are very close to this. The lattices are not exactly dual to each other.
The exact values for the critical point for the honeycomb lattice and q =
100 is x = 2.613, for the triangular it is x = .383. The numerical values are
in good agreement with this. The leading terms of the partition functions for
these lattices are given in appendix D.
The remarkable result is that the zeros in the many componentlimit are
close to circles
|x| = q
1/6
and
|x| = q
-1/6
for the honeycomb and its dual the
triangular lattice, respectively. Hence it can be conjectured that the cuve
B
forms a circle as q
not only for the square but also for the honeycomb
and triangular lattice.
73
-1
1
2
Re x
-2
-1
1
2
Im x
-1
1
2
Re x
-2
-1
1
2
Im x
-2
-1.5
-1
-0.5
0.5
1
1.5
Re x
-1.5
-1
-0.5
0.5
1
1.5
Im x
-1
1
2
Re x
-1.5
-1
-0.5
0.5
1
1.5
Im x
Figure 3.48: honeycomb lattice: C
x
= 5 and C
y
= 3, free bound-
aries in x and periodic in ydirection; zeros in the x-plane for
q = 4, 10, 25, 100 (from top left).
74
-2
-1.5
-1
-0.5
0.5
Re x
-1.5
-1
-0.5
0.5
1
1.5
Im x
-1.5
-1
-0.5
0.5
Re x
-1.5
-1
-0.5
0.5
1
1.5
Im x
-1
-0.5
0.5
Re x
-1
-0.5
0.5
1
Im x
-4
-3
-2
-1
Re x
-3
-2
-1
1
2
3
Im x
-3
-2.5
-2
-1.5
-1
-0.5
Re x
-2
-1
1
2
Im x
-2.5
-2
-1.5
-1
-0.5
Re x
-2
-1
1
2
Im x
Figure 3.49: square lattice: L
x
= 10 and L
y
= 4, free boundaries
in x and periodic in ydirection; zeros in the x-plane for q =
-1, -2, -3, -4, -25, -100 (from top left).
3.4.3
CTzeros for negative q
The remaining possible values of q that have not been covered yet are
negative numbers. Although there is no physical significance for negative q
and this case has not been the interest of research until now, it is of course
possible to look at Z
G
(a, q) just as a polynomial and plot its zeros for negative
q. This might be at least of some mathematical interest. The representation
is restricted the square lattice (fig. 3.49).
75
Chapter 4
Conclusions
With the interfacemethod it is possible to calculate the partition function
Z
G
(q, a) as a function of both the spindimensionality and the temperature
for finite lattices. This contains the chromatic polynomial: P
G
(q)
Z
G
(0, q).
The main results can be divided into two parts. First the zeros in the
complex qplane, where the starting point is the chromatic polynomial and its
connection to the the ground state entropy and second the complex tempera-
ture zeros. For the first case the most important results are the following.
· The chromatic zeros (a = 0) shrink to the origin for 0 < a < 1. For
a = 1 there is only one zero q = 0 left and for a > 1 they lie basically in
the left halfplane.
· With decreasing a < 0 the zeros move outwards compared to the chro-
matic zeros.
· The same holds for a > 1: the zeros are shifted further away from the
origin for increasing a.
76
· For free boundaries in both directions there are no qzeros with negative
imaginary part for any value of a.
· For the honeycomb lattice and F BC
x
, F BC
y
some zeros lie on the
axis for values of a close to one.
· Zeros can lie in the left halfplane ( (q) < 0) for the square and tri-
angular lattice if boundary conditions in ydirection are periodic, P
G
(q)
does not factorize and a
(0, 1).
· The only case where zeros with (q) < 0 occurred for P
G
(q) was the
honeycomb lattice with periodic boundary conditions in vertical direc-
tion.
· The zeros with (q) 0 have to be treated cautiously since they could
well be transient, e. g. disappear again for infinite strips.
For the second aspect of the problem, namely critical properties and complex
temperature zeros, the results can be summarized as follows.
· There is no region around q = 2 where the CTzeros for the square
lattice stay on the wellknown Fishercircles. Hence there is no hint
that the model could be solvable for some nonintegral values of q near
two. The same holds for the honeycomb and triangular lattice (with the
appropriate loci
B).
· In the many component limit (q ) the zerodistribution for the
square lattice agrees with the known unit circle in the xplane (x
e
K
-1
q
even for moderate lattice sizes.
77
· The zeros for the honeycomb and triangular lattice in the q limit
seem to lie on circles
|x| = q
1/6
and
|x| = q
-1/6
, respectively.
As shown in section 2.4 and appendix B.4, the calculation can be optimized
for retrieving P
G
(q).
The requirement for memory and computation time
would be drastically reduced. Thus one could calculate larger lattices. This
could be one aspect rewarding to pursue further.
The same optimization argument holds for the calculation of partition
functionzeros for fixed q. It is important to remember that for Z
G
(a, q) all
the Z
G,q
l
(a) for q
l
= 1, . . . , N
S
have to be known and calculating Z
G,q
l
(a) is by
far easier than calculating Z
G
(a, q). This aspect of the Pottsmodel is already
quite understood, however [40, 41], at least for small values of q. Once one
wants to get to high values of q above N
S
it is certainly better to calculate
Z
G
(a, q).
Finally the properties of the partition functions that have already been
calculated could be investigated further. There might also be some features of
the partitionfunctions that are not revealed by plotting the zeros in the q or
CTplane, even if this seems to be the best way to visualize their properties.
78
Bibliography
[1] E. Ising, Z. Phys. 21, 613 (1925).
[2] L. Onsager, Phys. Rev. 65, 117 (1944).
[3] C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952); ibid., 410 (1952).
[4] M. E. Fisher, in Lecture Notes in Theoretical Physics, edited by W. E.
Brittin (Univ. of Colorado Press, Boulder, 1965), vol. 7c.
[5] R. E. Shrock and S.-H. Tsai, Phys. Rev. E 55, 5165 (1997).
[6] R. B. Potts, Ph.D. thesis, University of Oxford (1951); Proc. Cambridge
Philos. Soc. 48, 106 (1952).
[7] F. Y. Wu, The Potts Model. Rev. of Mod. Ph. 54, 235 (1982).
[8] Baxter, R.J., J. Phys. C 6, L445 (1973).
[9] R. Martin, Potts Models and Related Problems in Statistical Mechanics
(World Scientific, Singapore, 1991).
[10] R. E. Shrock and V. Matveev, J. Phys. A 28, 1557 (1995).
[11] R. E. Shrock and V. Matveev, Phys. Rev. E 54, 6174 (1996).
79
[12] T. Kihara, Y. Midzuno and T. Shizume, J. Phys. Soc. Japan 9, 681 (1954).
[13] W. T. Tutte, Graph Theory, Vol. 21 of Encyclopedia of Mathematics
and its Applications, edited by G. C. Rotta (AddisonWesley, New York,
1984).
[14] S. Fiorini and R.J. Wilson, Edge Colorings of Graphs (Pitman, London,
1997).
[15] R. C. Read and W. T. Tutte, Chromatic Polynomials, in Selected Topics
in Graph Theory, 3, edited by L. W. Beinecke and R. J. Wilson (Academic
Press, New York, 1988).
[16] R. C. Read, J. Comb. Theor. 4 52 (1968).
[17] G. D. Birkhoff, Ann. Math 14, 42 (1912).
[18] H. Whitney, Ann. of Math. 33 688 (1932). "On the coloring of maps"
[19] F. Harary, Graph Theory and Applications, (AddisonWesley, Reading,
MA, 1969).
[20] B. Carr´
e, Graphs and Networks (Clarendon Press, Oxford, 1979).
[21] K. Appel, W. Haken and J. Koch, Illinois J. Math. 21 491 (1977).
[22] N. Biggs, Algebraic Graph Theory (Cambridge University Press, Cam-
bridge, 1996).
80
[23] The software is written and maintained by Victor Shoup. We used NTL
version 3.1 obtained from the authors homepage. The current address is
http://www.shoup.net
[24] H. Cohen, A course in computational algebraic number theory (Springer,
Berlin, 1993).
[25] C. N. Chen and C. K. Hu, Phys. Rev. B 43, 11519 (1991).
[26] H. Kl¨
upfel and R. Shrock, ITP-SB-99-32 (1999).
[27] M. Ro
cek, R. Shrock and S.-H. Tsai, Physica A 259, 367 (1998).
[28] M. Ro
cek, R. Shrock and S.-H. Tsai, Physica A 252, 505 (1998).
[29] R. Shrock and S.-H. Tsai, Phys. Rev. E 58, 4332 (1998).
[30] R. Shrock, ITP-SB-99-31 (1999).
[31] R. J. Baxter, J. Phys. A 20, 5241 (1987).
[32] Kenna R, Journal Of Physics A 31 9419 (1998).
[33] R. Abe, Prog. Theor. Phys. 38, 322 (1967).
[34] V. Matveev and R. Shrock, J. Phys. A 28, 1557 (1995).
[35] V. Matveev and R. Shrock, Phys. Rev. E53, 254 (1996).
[36] R. Shrock and V. Matveev, J. Phys. A 28, 5235 (1995).
[37] R. Shrock and V. Matveev, J. Phys. A 28, L533 (1995).
81
[38] D. W. Wood, J. Phys. A 20, 3471 (1987); D. W. Wood, R. W. Turnbull,
and J. K. Ball, ibid. 3495 (1987). Phys. Rev. E. 57, 1335 (1998).
[39] D. Kim and R. J. Joseph, J. Phys. C 7, L167 (1974); T. W. Burkhardt
and B. W. Southern, J. Phys. A 11, L247 (1978.
[40] H. Feldmann, R. Shrock and S.-H. Tsai,
[41] H. Feldmann, A. J. Guttmann, I. Jensen, R. Shrock and S.-H. Tsai, J.
Phys. A 31, 2287 (1998).
[42] A. Hintermann, H. Kunz and F. Y. Wu, J. Stat. Phys. 19, 623 (1978).
[43] H. Kl¨
upfel and R. Shrock, ITP-SB-99-33 (1999).
[44] S. Katsura, Prog. Theor. Phys. 38, 1415 (1967).
[45] S. Ono, Y. Karaki, M. Suzuki, and C. Kawabata, J. Phys. Soc. Jpn. 25,
54 (1968).
[46] R. Shrock and V. Matveev, J. Phys. A 29, 803 (1996).
[47] P. A. Pearce and B. Griffiths, J. Phys. A 13 2143 (1980).
[48] C. N. Chen, C. K. Hu, and F. Y. Wu, Phys. Rev. Lett. 76, 169 (1996).
[49] F. Y. Wu, G. Rollet, H. Y. Huang, J. M. Maillard, C. K. Hu, and C. N.
Chen, Phys. Rev. Lett. 76, 173 (1996).
[50] R. Shrock, ITP-SB-99-23 (1999).
[51] N. L. Biggs and R. Shrock, ITP-SB-99-30 (1999).
82
Appendix A
Lattice size and calculation time
The calculation of the chromatic polynomial is known as a NPcomplete
problem (cf. section 2.1), e.g. there is no upper limit for the computation
time that is a polynomial function of some characteristic parameters of the
problem. For the Potts model partition function the characteristic parameter
is the lattice size or more specific the number of sites N
S
. The number of bonds
N
B
grows in the same way as N
S
with the extent of the lattice in horizontal
and vertical direction L
x
and L
y
, viz. linearly (N
B
=
2
·N
S
, = coordination
number).
The following tables show the different lattices that were used in this work.
They also contain the characteristic parameters of those lattices N
S
, N
B
, L
x
and L
y
, respectively C
x
and C
y
, where C
x
and C
y
denote the number of poly-
gons in the x (horizontal) and ydirection (vertical).
The computation time given here is of course only a rough measure for
the complexity of the problem, as it depends also on a lot of other facts. The
computations have been carried out on different machines and usually there
83
Table A.1: Time required for the calculation of the partition function Z
sq
(q, a)
for square lattices of different sizes. The different symbols are explained in the
main text. For further details concerning the denotation of the lattices cf.
figure 3.1.
L
y
L
x
bc
N
S
N
B
time
token
3
18
fbc/fbc
54
87
01:01:16
sq3x18ff
3
18
fbc/pbc
54
105
0:45:08
sq3x18fp
4
8
fbc/fbc
32
52
16:21:39
sq4x8ff
4
8
fbc/pbc
32
60
18:45:54
sq4x8fp
4
10
fbc/fbc
40
66
35:10:16
sq4x10ff
4
10
fbc/pbc
40
76
20:31:58
sq4x10fp
5
5
fbc/fbc
25
40
353:8:13
sq5x5ff
84
Table A.2: Time required for the calculation of the partition function Z
tri
(q, a)
for triangular lattices of different sizes. N
x
and N
y
are the number of sites in
x- and y-direction, respectively. For further details concerning the denotation
of the lattices cf. figure 3.1.
L
y
L
x
bc
N
S
N
B
time
token
3
14
fbc/fbc
42
93
0:16:20
tri3x14ff
3
14
fbc/pbc
42
120
0:34:08
tri3x14fp
4
8
fbc/fbc
32
73
14:13:38
tri4x8ff
4
8
fbc/pbc
32
88
103:31:18
tri4x8fp
4
10
fbc/fbc
40
93
tri4x10ff
4
10
fbc/pbc
40
112
40:50:22
tri4x10fp
4
14
fbc/fbc
56
133
tri4x14ff
85
Table A.3: Time required for the calculation of the partition function Z
hc
(q, a)
for honeycomb lattices of different sizes. C
x
and C
y
are the number of hexagons
in x- and y-direction, respectively. For further details concerning the denota-
tion of the lattices cf. figure 3.1.
C
y
C
x
bc
N
S
N
B
time
token
2
7
fbc/fbc
46
59
0:26:55
hc2x7ff
2
9
fbc/fbc
58
75
1:44:04
hc2x9ff
3
3
fbc/fbc
30
38
7:55:00
hc3x3ff
3
5
fbc/fbc
46
60
70:11:49
hc3x5ff
3
5
fbc/pbc
46
68
84:01:00
hc3x5fp
86
Table A.4: Time required for the calculation of the partition function Z
kag
(q, a)
for kagome lattices of different sizes. C
x
and C
y
are the number of hexagons in
x- and y-direction, respectively. For further details concerning the denotation
of the lattices cf. figure 3.1.
C
y
C
x
bc
N
S
N
B
time
token
1
10
fbc/fbc
53
82
0:44:56
kag1x10ff
1
10
fbc/pbc
53
103
0:42:43
kag1x10fp
2
3
fbc/pbc
29
53
30 days kag2x3fp
Table A.5: Time required for the calculation of the partition function Z
hc,q
(a)
for different honeycomb lattices and a fixed value of q. C
x
and C
y
are the
number of polygons in x- and y-direction, respectively. For further details
concerning the denotation of the lattices cf. figure 3.1.
C
x
C
y
type
bc
q
N
S
N
B
time
token
5
5
hc
fbc/pbc
4
70
94
< 1h
hc5x5ffq3
5
5
hc
fbc/fbc
5
70
94
1:30:04
hc5x5ffq5
87
have been run other jobs at the same time. Anyway, one can still get a feeling
for the very fast growth of the required resources with the size of the lattice.
88
Appendix B
Implementation of the algorithm
B.1
Tensor products and direct products of
matrices
The tensorproducts in sections 2.2 and 2.3 can be written as direct prod-
ucts of matrices in the following way:
ii
|A × B|jj i|A|j i |B|j
(B.1)
The generalization to multiple products is straightforward:
ii . . . i
|A × B × · · · × C|jj . . . j i|A|j i |B|j . . . i |C|j
(B.2)
For the case of the local transfer matrices eqs. (2.7) and (2.8) these formulas
yield:
89
k
|t
i.
|l = k
0
. . . k
W
-1
|t
i.
|l
0
. . . l
W
-1
= k
0
|1
q
|l
0
. . . k
i
|M|l
i
. . . k
W
-1
|1
q
|l
W
-1
= (a
- 1) +
k
0
l
0
· · · 1 · · ·
k
W -1
l
W -1
(B.3)
k
i
k
i
|D|k
i
+1
k
i
+1
=
k
i
k
i+1
i
|1
q
|j =
ij
i, j = 0, . . . , q
- 1
(B.4)
i
|1
q
W
|j =
ij
i, j = 0, . . . , q
- 1
(B.5)
ii
|D|jj =
ii
i j
jj
i, i , j, j = 0, . . . , q
- 1
(B.6)
i
|M|j = 1 i, j = 0, . . . , q - 1
(B.7)
B.2
Adding sites and bonds
Since they are rather short, the key algorithms will be shown in this section.
They are of course not necessary for an understanding of the results. Adding
one bond at site i:
for i = 0 to W
- 1
for k = 0 to q
W
(for k = 0 to ds)
sum = 0
for l = 0 to k
- 1 and l = k + 1 to q
W
-1
if compare(k, l, i) : sum+ = Z
l
(if compare(d[k], l, i))
Z
k
= a
· Z
k
+ sum
(B.8)
The function compare(k, l, i) is defined via the r.h.s. of eq. (2.7). The parts
in parentheses are valid if the S
q
symmetry of the Hamiltonian is exploited to
90
speed up the calculation. This was of course done and is necessary to make
the task feasible at all. This is explained more thoroughly in the next section,
where also the vector d is introduced.
Connecting two new sites i and i + 1:
for i = 0 to W
- 2
for k = 0 to ds
if compare(k, i) : Z
k
= a
· Z
k
(if compare(d[k], i))
else Z
k
= Z
k
(B.9)
The function compare(k, i) is defined via the r.h.s. of eq. (2.8). d is defined
in the following section.
To show how the honeybomb and kagom´
e lattices are build up by this
method (adding new bonds at a certain position i in the rightmost column
and then connecting those new sites) they are shown in figures B.1 and B.2.
Both have to be looked at as brick lattices to allow a efficient algorithmic
implementation.
The method for the calculation of the transfer matrices can be further
improved by considering the S
q
symmetry of the PMHamiltonian.
B.3
Using the S
q
symmetry of the Hamilto-
nian
Any permutation of the spindirections does not change the partition func-
tion, because it depends only on the number of "filled bonds", e.g. the number
91
1
2
3
4
5
6
7
1
2
3
4
5
6
7
Figure B.1: Honeycomb lattice drawn as a brick lattice to illus-
trate the way its partition function is computed with the interface
method.
1
2
3
4
5
6
7
1
2
3
4
5
6
7
Figure B.2: Kag´
ome lattice drawn in a different way to illustrate
the how its partition function is calculated by the interface method.
92
of bonds with the two adjacent spins in the same direction. The q
W
possible
configurations of one row are labelled by k and the according partition func-
tions stored in a vector Z
W
V
q
indexed by k (cf. eqs. (2.5) and (2.6)). The
permutationsymmetry can be exploited by transforming k into the smallest
number k that has an equivalent representation in the qbasis. Equivalent
representation means that there exists a permutation of the digits that trans-
forms k into k . To make this point clearer an example for q = 3 is given in
eq. (B.10). "
" connects the same number represented in different basises,
here decimal and with basis number 3.
25
221
permutation
----------
001
1
(B.10)
Hence 25 is mapped onto 1. The whole mapping for all numbers from 0 to
q
W
- 1 (W is the height of one layer or column) can be stored in a vector d of
length 012 . . . q
- 1 . . . q - 1 (representation to the basis q), where the entries k
are all the different (in the sense of the beforementioned equivalence) possible
numbers between 0 and q
W
- 1 and the index of d gives k .
The method outlined in this section is crucial for an effective implementa-
tion of the algorithm and reduces the computation time certainly by several
orders of magnitude.
B.4
Simplifications for zero temperature
Considerable simplification is possible for the case a = 0 (zero temperature
and antiferromagnetic coupling), e.g. the calculation of the chromatic poly-
nomial. Instead of a vector of polynomials, Z is now a vector of numbers. The
93
algorithm is also distinctively easier. However, these possible simplifications
were not used because the main interest was the calculation of the complete
partition function.
For a = 0 the algorithms (B.8) and (B.9) simplify as following: Adding a
new site at position i:
for i = 0 to W
- 1
Z
k
=
q
W
-1
l
=0
Z
l
· (-
kl
+ compare(k, l, i)) =
-Z
k
+
q
W
-1
l
=0
Z
l
(B.11)
Connecting two new sites i and i + 1:
for i = 0 to W
- 2
for k = 0 to ds
if compare(k, i) : Z
k
= 0
(if compare(d[k], i))
(B.12)
Of course the improvements outlined in section B.3 can still be used.
94
Appendix C
Zeros for further lattices of different sizes
C.1
Complex qplane
As can be seen from fig. C.1 smaller lattices with fewer sites N
S
do not show
as a distinct qzeros distribution than e. g. stripgraphs with more sites. It is
therefore more difficult to infere the form of
B and its behavior as a is increased
from zero. On the other hand, those lattice are closer to the twodimensional
thermodynamic limit than stripgraphs.
Fig. C.4 shows the qzeros for a strip of the honeycomb lattice. The
reference for the chromatic zeros is fig. 6 in [28].
95
0.5
1
1.5
2
2.5
Re q
-1.5
-1
-0.5
0.5
1
1.5
Im q
Figure C.1: square lattice: L
x
= 5 and L
y
= 5, free boundaries
in both directions; zeros in the q-plane for a = 0 (larger dots),
a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
0.5
1
1.5
2
2.5
3
Re q
-2
-1
1
2
Im q
0.5
1
1.5
2
2.5
3
3.5
Re q
-3
-2
-1
1
2
3
Im q
Figure C.2: sq
(5,5),ff
(cf. fig. C.1) zeros in the q-plane for a) a = 0
(stars) and a =
-0.25 (triangles); b) a = 0 (stars) and a = -0.5
(triangles)
96
-0.03 -0.02 -0.01
0.01
0.02
0.03
Re q
-0.02
-0.01
0.01
0.02
Im q
0.1
0.15 0.2
0.3
0.5 0.7
1
1.5
1-a
1
1.5
2
3
q_max
Figure C.3: sq
(5,5),ff
(cf. fig. C.1); a) zeros in the q-plane for a =
0.99 (stars) and a = 1.01 (triangles), b) plot of the realpart (larger
dots) of the rightmost and the imaginary part of the uppermost zero
against 1
- a.
0.5
1
1.5
2
Re q
-1
-0.5
0.5
1
Im q
Figure C.4: honeycomb lattice: C
x
= 7 and C
y
= 2, free boundaries
in both directions; zeros in the q-plane for a = 0 (larger dots),
a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
97
0.5
1
1.5
2
2.5
Re q
-1.5
-1
-0.5
0.5
1
1.5
Im q
0.5
1
1.5
2
2.5
3
3.5
Re q
-2
-1
1
2
Im q
Figure C.5: hc
(2,7),ff
(cf. fig. C.4) zeros in the q-plane for a) a = 0
(stars) and a =
-0.25 (triangles); b) a = 0 (stars) and a = -0.5
(triangles)
-0.02
-0.01
0.01
0.02
Re q
-0.01
-0.005
0.005
0.01
Im q
0.1
0.15 0.2
0.3
0.5 0.7
1
1.5
1-a
0.5
1
2
q_max
Figure C.6: hc
(2,7),ff
(cf. fig. C.4); a) zeros in the q-plane for a =
0.99 (stars) and a = 1.01 (triangles), b) plot of the realpart (larger
dots) of the rightmost and the imaginary part of the uppermost zero
against 1
- a.
98
0.5
1
1.5
2
Re q
-1
-0.5
0.5
1
Im q
Figure C.7: honeycomb lattice: C
x
= 5 and C
y
= 3, free boundaries
in both directions; zeros in the q-plane for a = 0 (larger dots),
a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
0.5
1
1.5
2
2.5
Re q
-1.5
-1
-0.5
0.5
1
1.5
Im q
0.5
1
1.5
2
2.5
3
3.5
Re q
-2
-1
1
2
Im q
Figure C.8: hc
(3,5),ff
(cf. fig. C.7) zeros in the q-plane for a) a = 0
(stars) and a =
-0.25 (triangles); b) a = 0 (stars) and a = -0.5
(triangles)
99
-0.02
-0.01
0.01
0.02
Re q
-0.015
-0.01
-0.005
0.005
0.01
0.015
Im q
0.1
0.15 0.2
0.3
0.5 0.7
1
1.5
1-a
0.5
1
2
q_max
Figure C.9: hc
(3,5),ff
(cf. fig. C.7); a) zeros in the q-plane for a =
0.99 (stars) and a = 1.01 (triangles), b) plot of the realpart (larger
dots) of the rightmost and the imaginary part of the uppermost zero
against 1
- a.
0.5
1
1.5
2
2.5
Re q
-1.5
-1
-0.5
0.5
1
1.5
Im q
Figure C.10: kag´
ome lattice: C
x
= 2 and C
y
= 3, free boundaries in
x and in ydirection; zeros in the q-plane for a = 0 (larger dots),
a = 0.25, a = 0.5, a = 0.75 and a = 0.9.
100
0.5
1
1.5
2
2.5
3
3.5
Re q
-2
-1
1
2
Im q
1
2
3
4
Re q
-2
-1
1
2
Im q
Figure C.11: kag
(2,3),fp
(cf. fig. C.10) zeros in the q-plane for a)
a = 0 (stars) and a =
-0.25 (triangles); b) a = 0 (stars) and
a =
-0.5 (triangles)
-0.03 -0.02 -0.01
0.01
0.02
0.03
Re q
-0.02
-0.01
0.01
0.02
Im q
0.1
0.15 0.2
0.3
0.5 0.7
1
1.5
1-a
0.5
1
2
5
q_max
Figure C.12: kag
(2,3),fp
(cf. fig. C.10); a) zeros in the q-plane for
a = 0.99 (stars) and a = 1.01 (triangles), b) plot of the real
part (larger dots) of the rightmost and the imaginary part of the
uppermost zero against 1
- a.
101
-0.4
-0.2
0.2
Re q
-0.2
-0.1
0.1
0.2
Im q
-1
-0.5
0.5
Re q
-0.6
-0.4
-0.2
0.2
0.4
0.6
Im q
Figure C.13: sq
(4,10),fp
a) zeros in the q-plane for a = 0.9 (stars) and
a = 1.1 (triangles), b) for a = 0.75 (stars) and a = 1.25 (triangles).
-0.4
-0.2
0.2
0.4
Re q
-0.4
-0.2
0.2
0.4
Im q
-1
-0.5
0.5
1
Re q
-1
-0.5
0.5
1
Im q
Figure C.14: tri
(4,8),fp
a) zeros in the q-plane for a = 0.9 (stars) and
a = 1.1 (triangles), b) for a = 0.75 (stars) and a = 1.25 (triangles).
102
-0.3
-0.2
-0.1
0.1
0.2
0.3
Re q
-0.15
-0.1
-0.05
0.05
0.1
0.15
Im q
-0.8 -0.6 -0.4 -0.2
0.2
0.4
0.6
Re q
-0.4
-0.2
0.2
0.4
Im q
Figure C.15: hc
(3,5),fp
a) zeros in the q-plane for a = 0.9 (stars) and
a = 1.1 (triangles), b) for a = 0.75 (stars) and a = 1.25 (triangles).
-0.3
-0.2
-0.1
0.1
0.2
0.3
Re q
-0.2
-0.1
0.1
0.2
Im q
-0.75 -0.5 -0.25
0.25
0.5
0.75
Re q
-0.6
-0.4
-0.2
0.2
0.4
0.6
Im q
Figure C.16: kag
(2,3),fp
a) zeros in the q-plane for a = 0.9 (stars) and
a = 1.1 (triangles), b) for a = 0.75 (stars) and a = 1.25 (triangles).
103
-2.5
-2
-1.5
-1
-0.5
Re x
-1.5
-1
-0.5
0.5
1
1.5
Im x
-2
-1.5
-1
-0.5
Re x
-1
-0.5
0.5
1
Im x
-1.5
-1
-0.5
0.5
Re x
-1
-0.5
0.5
1
Im x
-8
-6
-4
-2
Re x
-6
-4
-2
2
4
6
Im x
-4
-3
-2
-1
Re x
-2
-1
1
2
Im x
-2.5
-2
-1.5
-1
-0.5
Re x
-1.5
-1
-0.5
0.5
1
1.5
Im x
Figure C.17: triangular lattice: L
x
= 8 and L
y
= 4, free boundaries
in x and periodic in ydirection; zeros in the x-plane for q =
0.1, 0.5, 0.9, 1.1, 1.5, 1.8 (from top left).
C.2
Complex xplane
The following figures show further plots of CTzeros. The lattice types have
already been treated in earlier sections. The differences are the lattice size and
in some cases the boundary conditions. Since all the important features have
already been pointed out, I will not comment any further on them. Hence this
section is mostly meant as a reference.
104
-4
-3
-2
-1
Re x
-2
-1
1
2
Im x
-3
-2
-1
1
Re x
-2
-1
1
2
Im x
-2
-1
1
2
Re x
-1.5
-1
-0.5
0.5
1
1.5
Im x
-15
-12.5
-10
-7.5
-5
-2.5
Re x
-10
-5
5
10
Im x
-6
-5
-4
-3
-2
-1
Re x
-4
-2
2
4
Im x
-4
-3
-2
-1
Re x
-3
-2
-1
1
2
3
Im x
Figure C.18: honeycomb lattice: C
x
= 5 and C
y
= 3, free bound-
aries in x and periodic in ydirection; zeros in the x-plane for
q = 0.1, 0.5, 0.9, 1.1, 1.5, 1.8 (from top left).
105
-3
-2
-1
1
2
Re x
-3
-2
-1
1
2
3
Im x
-1.5
-1
-0.5
0.5
1
Re x
-1.5
-1
-0.5
0.5
1
1.5
Im x
-7.5
-5
-2.5
2.5
5
Re x
-7.5
-5
-2.5
2.5
5
7.5
Im x
-6
-4
-2
2
4
Re x
-4
-2
2
4
Im x
Figure C.19: square lattice: L
x
= 10 and L
y
= 4, free boundaries
in both directions; zeros in the x-plane for q = 4, 10, 25, 100 (from
top left).
106
-3
-2
-1
Re x
-3
-2
-1
1
2
3
Im x
-3
-2
-1
Re x
-2
-1
1
2
Im x
-2
-1
1
Re x
-2
-1
1
2
Im x
-5
-4
-3
-2
-1
Re x
-4
-2
2
4
Im x
-4
-3
-2
-1
Re x
-3
-2
-1
1
2
3
Im x
-3
-2
-1
Re x
-3
-2
-1
1
2
3
Im x
Figure C.20: honeycomb lattice: C
x
= 5 and C
y
= 3, free bound-
aries in x and periodic in ydirection; zeros in the x-plane for
q =
-1, -2, -3, -4, -25, -100 (from top left).
107
-1.5
-1
-0.5
Re x
-1
-0.5
0.5
1
Im x
-1
-0.75 -0.5 -0.25
0.25
Re x
-0.75
-0.5
-0.25
0.25
0.5
0.75
Im x
-0.6
-0.4
-0.2
0.2
0.4
Re x
-0.4
-0.2
0.2
0.4
Im x
-3
-2.5
-2
-1.5
-1
-0.5
Re x
-2
-1
1
2
Im x
-2
-1.5
-1
-0.5
Re x
-1.5
-1
-0.5
0.5
1
1.5
Im x
-1.5
-1
-0.5
Re x
-1
-0.5
0.5
1
Im x
Figure C.21: triangular lattice: L
x
= 8 and L
y
= 4, free boundaries
in x and periodic in ydirection; zeros in the x-plane for q =
-1, -2, -3, -4, -25, -100 (from top left).
108
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
-1.5
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
-1
-0.5
0.5
1
Re x
-1
-0.5
0.5
1
Im x
Figure C.22: kagom´
e lattice: C
x
= 3 and C
y
= 2, free boundaries
in x and periodic in y direction; zeros in the x-plane for q =
4, 10, 25, 100 (from top left).
109
Appendix D
Leading terms of some partition functions
For further reference some of the partition functions Z
G
(x, q) are listed in
this appendix. Since they are polynomials of high order in both x (N
B
) and
q and the coefficients are very large integers it is not possible to show them
in full form. Instead only the leading and trailing terms are shown. Here the
variable x =
a
-1
q
e
K
-1
q
is used.
Z
hc
(3,5),fp
= (1 + x
2
) q
46
+ (68 x + 63 x
3
) q
91/2
+ (2227 x
2
+ 2147 x
4
+ 2 x
6
) q
45
+ (50050 x
3
+ 45888 x
5
+ 126 x
7
) q
89/2
+ . . .
+ (5951382871517013 x
44
+ 1822201598813250 x
46
) q
24
+
+ 558945176147820 x
45
) q
47/2
(D.1)
110
Z
tri
(4,8),fp
= x
88
q
45
+ (2 x
85
+ 88 x
87
) q
89/2
+ (3 x
82
+ 174 x
84
+ 3828 x
86
) q
44
+ . . .
+ (175364862886905376372 x
30
+ 518150275722326512581 x
32
) q
17
+
+ 41732358981384296680 x
31
q
33/2
(D.2)
Another representation focusses on the variable q and hence shows the
powers of q with coefficients that are polynomials in a. Again only the leading
and trailing coefficients are given.
Z
hc
(3,5),ff
= (-1176499224 q + 2726969683 q
2
- 56156314 q
3
- 4743732618 q
4
+ 4591704419 q
5
- 1184832527 q
6
- 275831602 q
7
+ 107409110 q
8
+ 10818930 q
9
+ 150027 q
10
+ 116 q
11
) a
46
+ (2144463440 q - 18008085422 q
2
+ 42922940168 q
3
- 41851952876 q
4
+ 13028296156 q
5
+ 5353728522 q
6
- 4012891751 q
7
+ 268193952 q
8
+ 148777551 q
9
+ 6491012 q
10
+ 39242 q
11
+ 6q
12
) a
45
+ . . .
+ (903797185376 q - 15127351294804 q
1
+ 128351603127992 q
2
-
+ 734347948656301 q
4
+ 3179394760306945 q
5
- · · · + 1950540 q
42
+ 102660 q
43
- 3540 q
44
+ 60q
45
) a
- 20960437373 q + 357284253787 q
2
- 3090267495200 q
3
+ . . .
- 5461497 q
41
+ 487635 q
42
- 34220 q
43
+ 1770q
44
- 60q
45
+ q
46
(D.3)
Excerpt out of 122 pages
- Quote paper
- Hubert Klüpfel (Author), 1999, The q –state Potts Model: Partition functions and their zeros in the complex temperature– and q–plane, Munich, GRIN Verlag, https://www.grin.com/document/201506
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