Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ(3) and χ(4)

Abstract

We present a simple, but efficient, way to calculate connection matrices between sets of independent local solutions, defined at two neighbouring singular points, of Fuchsian differential equations of quite large orders, such as those found for the third and fourth contribution (χ(3) and χ(4)) to the magnetic susceptibility of the square lattice Ising model. We deduce all the critical behaviours of the solutions χ(3) and χ(4), as well as the asymptotic behaviour of the coefficients in the corresponding series expansions. We confirm that the newly found quadratic singularities of the Fuchsian ODE associated with χ(3) are not singularities of the particular solution χ(3) itself. We use the previous connection matrices to get the exact expressions of all the monodromy matrices of the Fuchsian differential equation for χ(3) (and χ(4)) expressed in the same basis of solutions. These monodromy matrices are the generators of the differential Galois group of the Fuchsian differential equations for χ(3) (and χ(4)), whose analysis is just sketched here. As far as the physics implications of the solutions are concerned, we find challenging qualitative differences when comparing the corrections to scaling for the full susceptibility χ at high temperature (respectively low temperature) and the first two terms χ(1) and χ(3) (respectively χ(2) and χ(4)).

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arXiv:math-ph/0506065v3 19 Sep 2005
Square lattice Ising model susceptibility: connection
matrices and singular behavior of χ(3)and χ(4)
N. Zenine§, S. Boukraa†, S. Hassani§and J.-M. Maillard‡
§Centre de Recherche Nucl´ eaire d’Alger,
2 Bd. Frantz Fanon, BP 399, 16000 Alger, Algeria
†Universit´ e de Blida, Institut d’A´ eronautique, Blida, Algeria
‡ LPTMC, Universit´ e de Paris 6, Tour 24, 4` eme ´ etage, case 121,
4 Place Jussieu, 75252 Paris Cedex 05, France
E-mail: maillard@lptmc.jussieu.fr, maillard@lptl.jussieu.fr,
sboukraa@wissal.dz, njzenine@yahoo.com
Abstract.
matrices between sets of independent local solutions, defined at two neighboring
singular points, of Fuchsian differential equations of quite large orders, such
as those found for the third and fourth contribution (χ(3)and χ(4)) to the
magnetic susceptibility of the square lattice Ising model.
critical behavior of the solutions χ(3)and χ(4), as well as the asymptotic behavior
of the coefficients in the corresponding series expansions. We confirm that the
newly found quadratic singularities of the Fuchsian ODE associated with χ(3)
are not singularities of the particular solution χ(3)itself. We use the previous
connection matrices to get the exact expressions of all the monodromy matrices of
the Fuchsian differential equation for χ(3)(and χ(4)) expressed in the same basis
of solutions. These monodromy matrices are the generators of the differential
Galois group of the Fuchsian differential equations for χ(3)(and χ(4)), whose
analysis is just sketched here. As far as the physics implications of the solutions
are concerned, we find challenging qualitative differences when comparing the
corrections to scaling for the full susceptibillity χ at high temperature (resp. low
temperature) and the first two terms χ(1)and χ(3)(resp. χ(2)and χ(4)) .
We present a simple, but efficient, way to calculate connection
We deduce all the
PACS: 05.50.+q, 05.10.-a, 02.30.Hq, 02.30.Gp, 02.40.Xx
AMS Classification scheme numbers: 34M55, 47E05, 81Qxx, 32G34, 34Lxx,
34Mxx, 14Kxx
Key-words:
monodromy group, connection matrices, singular behavior, asymptotics, Fuchsian
differential equations, factorization of linear differential operators,
singularities,critical behaviors,confluent singularities,
constants.
Susceptibility of the Ising model, differential Galois group,
apparent
Euler’s and Catalan’s
1. Introduction
Since a pioneering, and quite monumental, paper [1] on the two-dimensional Ising
models, it has been known that the magnetic susceptibility of square lattice Ising

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Galois group for the Ising model2
model, can be written [1] as an infinite sum of (n − 1)-dimensional integrals [2, 3, 4,
5, 6, 7] contributions:
χ(T) =
∞
?
n=1
χ(n)(T) (1)
The odd (respectively even) n correspond to the high (respectively low) temperature
domain. These (n−1)-dimensional integrals are known to be holonomic, since they are
integrals of holonomic (actually algebraic) integrands. Besides the known χ(1)and
χ(2)terms, which can be expressed in terms of simple algebraic or hypergeometric
functions, it is only recently that the Fuchsian differential equations satisfied by
the χ(3)and χ(4)terms have been found [8, 9, 10]. These two exact differential
equations of quite large orders (seven and ten) can be used to find answers to a
set of problems traditionally known to be subtle, and difficult, for functions with
confluent singularities, like the fine-tuning of the singular behaviors for all the
singularities (dominant singular behavior, sub-dominant, etc.), accurate calculations
of the asymptotic behavior of the coefficients, etc.
Recall that the third, and fourth, contribution to the magnetic susceptibility χ(3),
and χ(4), are given by multi-integrals and each is, thus, a particular solution of the
corresponding differential equation. These differential equations exhibit a finite set of
regular singular points that may (or may not) appear in the physical solutions χ(3)
and χ(4). Besides the physical singularities and the non physical singularities s = ±i
(where s = sinh(2K), K being the usual Ising model coupling constant, K = β J),
it is commonly believed that the χ(n)’s have, at least, other non physical singularities
given by B. Nickel [6, 7]. The dominant singular behaviors at all these (non physical)
singularities (χ(3)and χ(4)) have also been given by B. Nickel.
equations of the χ(n)’s, which “encode” all the information on the solutions and their
singular behavior, in fact, allow us to obtain not only the dominant, but also all the
subdominant singular behavior, hardly detectable from straight series analysis. It is
thus of interest to get (or confirm) these singular behaviors from the exact Fuchsian
differential equations that we have actually obtained for χ(3)and χ(4)and, especially,
the singular behavior at the two new quadratic singularities, 1 + 3w + 4w2= 0,
(where w = s/(1 + s2)/2) found for χ(3)[8].
The physical solution χ(3)is defined by a double integral on two angles and is
known as a series obtained by expansion (then integration) of the double integral at
w = 0 (or s = 0). It is certainly not simple to obtain the χ(3)expansion around
(say) the ferromagnetic critical point w = 1/4, due to a singular logarithmic behavior.
However, one can overcome this difficulty since, with a differential equation, it is
straightforward to obtain the formal series solutions at each regular singular point
(i.e., a local basis of series solutions). By connecting the formal solutions around
w = 0 and the formal series solutions around another regular singular point like
w = 1/4, one will be able to express the particular solution χ(3)(and also all the other
formal solutions) as a linear combination of solutions valid at w = 1/4. The seven local
solutions at w = 0 will, then, be given by the product of a 7×7 matrix with the vector
having the seven local solutions at w = 1/4 as entries. In other words, succeeding in
obtaining these connection matrices amounts to building a common (global) basis of
solutions valid for all the regular singular points. Furthermore, with these connection
matrices, we obtain, in fact, the analytic continuation in the whole complex plane of
the variable w, of χ(3)and χ(4), which are known as integral representations.
Note that, remarkably, the Fuchsian differential equation for χ(3)has simple
The differential

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Galois group for the Ising model3
rational, and algebraic, solutions. These rational or algebraic solutions, known in
closed form, can be understood globally. One can easily expand such globally defined
solutions around any singular point of the ODE, and follow these solutions through
any “jump” from one regular singularity to another one, and, therefore, from one well-
suited basis to another well-suited basis. For a function not known in closed form,
like the “physical” solution χ(3), the decomposition on each well-suited local basis
associated with every singular point of the ODE, is far from clear. The correspondence
between these various (well-suited) local bases associated with each singular point
of the ODE, is typically a global problem and, thus, a quite difficult one.
clearly needs to build effective methods to find such connection matrices in the case of
Fuchsian differential equations of order seven, or ten (χ(3)and χ(4)), or of much higher
orders (χ(5), χ(6), etc.). With a method of matching of series, we will show that the
connection matrices matching these various well-suited bases of series-solutions can
be obtained explicitly. The entries of these matrices can be calculated with as many
digits as we want. We will show that we can actually find the exact expressions of
these entries as simple algebraic expressions of (in the case of the Fuchsian ODE’s of
χ(3)and χ(4)) powers of π, ln(2), ln(3) and various algebraic numbers or integers,
together with more “transcendental” numbers like the ”ferromagnetic constant” I+
introduced in equation (7.12) of [1]:
?∞
= 0.0008144625656625044393912171285627219978 ···
Y =
(y1+ y2)(y2 + y3)(y1 + y2+ y3)
Focusing on χ(3), and since this physical solution is known as a series expansion at
w = 0 (low or high temperature expansions), we will give all the connection matrices
between this w = 0 regular singular point and all the other regular singularities of
the differential equation including the two new complex regular singularities [8, 9]
which are roots of 1 + 3w + 4w2= 0. We will comment on the occurrence of the
”ferromagnetic constant” I+
3in the various blocks of the connection matrices. The
decomposition of χ(3)in the well-suited basis for each regular singular point allows
us to find all the singular behavior of the physical solution.
we will deduce the asymptotic behavior of the coefficients of the series expansion
of χ(3). These last problems are interesting, per se, for series expansions analysis
of lattice statistical mechanics, since they correspond to subtle analysis of confluent
singularities. Actually, we will see that even the last asymptotic evaluation problem is
a (global) connection problem since the physical solution like χ(3)does not correspond
to the obvious dominant singular behavior one might have imagined from the indicial
equation.
Focusing on the two new singularities, the roots of 1 + 3w + 4w2= 0, we will
show that the physical solution χ(3)is not singular at these points. The factor of
the logarithmic term, in the decomposition of χ(3)at these singular points, is known
exactly and vanishes identically.
Note that a fundamental concept to understand (the symmetries, the solutions of)
these exact Fuchsian differential equations is the so-called differential Galois group [11].
Differential Galois groups have been calculated for simple enough second order, or even
third order, ODE’s (see for instance [12]). However, finding the differential Galois
group of such higher order Fuchsian differential equations (order seven for χ(3), order
One
3
I+
3=
1
2π2·
1
?∞
1
?∞
1
dy1dy2dy3·
?
y2
2− 1
(y2
1− 1)(y2
3− 1)
?1/2
· Y2
(2)
y1 − y3
From these results,

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Galois group for the Ising model4
ten for χ(4)) with eight regular singular points (for χ(3)) is not an easy task [12] and
requires the computation of all the monodromy matrices associated with each (non
apparent) regular singular point, considered in the same basis‡.
We will give the exact expression of all the monodromy matrices expressed in
the same (w = 0) basis of solutions, these eight matrices being the generators of the
differential Galois group, which will be given in a forthcoming publication [14].
This method can be generalized, mutatis mutandis, to the Fuchsian differential
equation of χ(4). Here, we give the connection matrix between w = 0 and, both,
the ferromagnetic, and anti-ferromagnetic, critical points. The singular behavior is
straightforwardly obtained with the asymptotic behavior of the series coefficients of
the physical solution χ(4). The monodromy matrices, expressed in the same basis of
solutions are also obtained.
The paper is organized as follows. We recall, in Section 2, some results on the
Fuchsian differential equation satisfied by χ(3), and give a new factorization for the
corresponding order seven differential operator, yielding the emergence of an order
two, and an order three, differential operator (denoted Z2 and Y3 below).
give, in Section 3, the connection matrices matching the (series) solutions around the
regular singular point w = 0 and around all the other regular singular points. With
these connection matrices we deduce the singularity behavior and the asymptotics
on the physical solution of this ODE (Section 4).
exact expressions of the monodromy matrices expressed in the same basis. Section 6
generalizes these results to the Fuchsian differential equation satisfied by χ(4). Some
physics implications of our results at scaling are discussed in Section 7. Our conclusion
is given in Section 8.
We
In Section 5, we deduce the
2. The order seven operator L7
Let us first recall, with the same notations as in [8, 9], the seven linearly independent
solutions given in [8, 9] for the order seven differential operator L7, associated with§
˜ χ(3).
One finds two remarkable rational, and algebraic, solutions of the order seven
differential equation associated with ˜ χ(3), namely:
S(L1) =
w
1 − 4w,
S(N1) =
w2
(1 − 4w)√1 − 16w2
(3)
associated with the two order 1 differential operators given in [8]:
L1 =
d
dw−
1
w (1 − 4w),N1 =
d
dw−
2(1 + 2w)
w(1 − 16w2)
(4)
There is a solution behaving like w3, that we denote S3:
S3 = w3+ 3w4+ 22w5+ 74w6+ 417w7+ 1465w8
+ 7479w9+ 26839w10+ ···
and three solutions with logarithmic terms given by equation (17) in [8]. Note the
singled-out series expansion starting with w9, corresponding to the physical solution
(5)
‡ These monodromy matrices are the generators of the monodromy group which identifies with the
differential Galois group when there are no irregular singularities, and, thus, no Stokes matrices [13].
§ ˜ χ(n)is defined as χ(n)= (1 − s4)1/4/s · ˜ χ(n), for n odd.

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Galois group for the Ising model5
˜ χ(3):
S9 =
˜ χ(3)(w)
8
= w9+ 36w11+ 4w12+ 884w13+ 196w14+ ··· (6)
The choice of this set of linearly independent solutions (and of these series) is,
in fact, arbitrary since any linear combination of solutions is also a solution of the
differential equation.Three of the above solutions are however singled out: the
solutions S(L1) and S(N1) which are global (since they have closed expression), and
the series S9 associated with the highest critical exponent in the indicial equation
(w9+···), which has a unique (well-defined) expression and happens to correspond to
the “physical” solution ˜ χ(3). Linear combinations, like S3 − α · S9, are, at first sight,
on the same footing.
Nevertheless, introducing such a specific linear combination, B. Nickel? has been
able to show that the resulting series for the particular value α = 16 is, also, the
solution of a linear differential equation of lower order, namely order four. With this
result, the factorization scheme of L7becomes‡ :
L7 = M1· Y3· Z2· N1 = B3· X1· Z2· N1
= B3· B2· O1· N1 = B3· B2· T1· L1
where the indices correspond to the order of the differential operators (B3, Y3 are
order three, B2, Z2 order two, ...). The differential operators L7, M1 and T1 have
been given in [8]. We give in Appendix A, the differential operators X1, Z2and Y3.
With these differential operators, all the factorizations (7) can be found by left and
right division.
From these factorizations of L7, one can see that the general solution of the
corresponding differential equation is the direct sum of the solution of L1and of the
general solution of the differential operator L6= Y3· Z2· N1. The operator L7has
the following decomposition:
(7)
L7 = L6⊕ L1.(8)
We thus consider, from now on, the differential operator L6.
The formal solutions of L6 (at the singular point w = 0) show the occurrence
of three Frobenius series and three solutions carrying logarithmic terms. With the
factorizations (7), it is interesting to see which operator brings with it a singular
behavior for a given regular singular point. Table 1 shows the critical exponents at
each regular singular point for both differential operators Z2· N1 and Y3· Z2· N1.
In the third and sixth column the number of independent solutions with logarithmic
terms is shown.
At the singular points w = 1, w = −1/2, and at the two roots w1, w2 of
1 + 3w + 4w2= 0, we remark that the solution carrying a logarithmic term is in
fact a solution of Z2· N1. Therefore, the three solutions of the differential operator
Y3· Z2· N1, emerging from Y3, are analytical at the non physical singular points
w = 1, w = −1/2, and at the quadratic roots of 1 + 3w + 4w2= 0. At the singular
point w = 1/4, we also note that the differential operator Z2· N1 is responsible of
the (1−4w)−1behavior. We will then expect the ”ferromagnetic constant” I+
localized in the blocks of the connection matrix corresponding to the solutions of the
order three differential operator Z2· N1at the point w = 1/4.
? We thank B. Nickel for kindly communicating this result.
‡ The order four differential operator found by B. Nickel corresponds to B2·T1·L1= B2·O1·N1=
X1· Z2· N1.
3to be