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Register a new userSquare lattice Ising model $tilde{chi}^{(5)}$ ODE in exact arithmetic
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We obtain in exact arithmetic the order 24 linear differential operator $L_{24}$ and right hand side $E^{(5)}$ of the inhomogeneous equation$L_{24}(Phi^{(5)}) = E^{(5)}$, where $Phi^{(5)} =tilde{chi}^{(5)}-tilde{chi}^{(3)}/2+tilde{chi}^{(1)}/120$ is a linear combination of $n$-particle contributions to the susceptibility of the square lattice Ising model. In Bostan, et al. (J. Phys. A: Math. Theor. {bf 42}, 275209 (2009)) the operator $L_{24}$ (modulo a prime) was shown to factorize into $L_{12}^{(rm left)} cdot L_{12}^{(rm right)}$; here we prove that no further factorization of the order 12 operator $L_{12}^{(rm left)}$ is possible. We use the exact ODE to obtain the behaviour of $tilde{chi}^{(5)}$ at the ferromagnetic critical point and to obtain a limited number of analytic continuations of $tilde{chi}^{(5)}$ beyond the principal disk defined by its high temperature series. Contrary to a speculation in Boukraa, et al (J. Phys. A: Math. Theor. {bf 41} 455202 (2008)), we find that $tilde{chi}^{(5)}$ is singular at $w=1/2$ on an infinite number of branches.
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We consider the Fuchsian linear differential equation obtained (modulo a prime) for $tilde{chi}^{(5)}$, the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination of $tilde{chi}^{(1)}$ and $tilde{chi}^{(3)}$ can be removed from $tilde{chi}^{(5)}$ and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth order linear differential operator occurs as the left-most factor of the depleted differential operator and it is shown to be equivalent to the symmetric fourth power of $L_E$, the linear differential operator corresponding to the elliptic integral $E$. This result generalizes what we have found for the lower order terms $tilde{chi}^{(3)}$ and $tilde{chi}^{(4)}$. We conjecture that a linear differential operator equivalent to a symmetric $(n-1)$-th power of $L_E$ occurs as a left-most factor in the minimal order linear differential operators for all $tilde{chi}^{(n)}$s.
This paper deals with $tilde{chi}^{(6)}$, the six-particle contribution to the magnetic susceptibility of the square lattice Ising model. We have generated, modulo a prime, series coefficients for $tilde{chi}^{(6)}$. The length of the series is sufficient to produce the corresponding Fuchsian linear differential equation (modulo a prime). We obtain the Fuchsian linear differential equation that annihilates the depleted series $Phi^{(6)}=tilde{chi}^{(6)} - {2 over 3} tilde{chi}^{(4)} + {2 over 45} tilde{chi}^{(2)}$. The factorization of the corresponding differential operator is performed using a method of factorization modulo a prime introduced in a previous paper. The depleted differential operator is shown to have a structure similar to the corresponding operator for $tilde{chi}^{(5)}$. It splits into factors of smaller orders, with the left-most factor of order six being equivalent to the symmetric fifth power of the linear differential operator corresponding to the elliptic integral $E$. The right-most factor has a direct sum structure, and using series calculated modulo several primes, all the factors in the direct sum have been reconstructed in exact arithmetics.
We study the class of non-holonomic power series with integer coefficients that reduce, modulo primes, or powers of primes, to algebraic functions. In particular we try to determine whether the susceptibility of the square-lattice Ising model belongs to this class, and more broadly whether the susceptibility is a solution of a differentially algebraic equation. Initial results on Tuttes non-linear ordinary differential equation (ODE) and other simple quadratic non-linear ODEs suggest that a large set of differentially algebraic power series solutions with integer coefficients might reduce to algebraic functions modulo primes, or powers of primes. Here we give several examples of series with integer coefficients and non-zero radius of convergence that reduce to algebraic functions modulo (almost) every prime (or power of a prime). These examples satisfy differentially algebraic equations with the encoding polynomial occasionally possessing quite high degree (and thus difficult to identify even with long series). Additionally, we have extended both the high- and low-temperature Ising square-lattice susceptibility series to 5043 coefficients. We find that even this long series is insufficient to determine whether it reduces to algebraic functions modulo $3$, $5$, etc. This negative result is in contrast to the comparatively easy confirmation that the corresponding series reduce to algebraic functions modulo powers of $2$.
The partition function of the square lattice Ising model on the rectangle with open boundary conditions in both directions is calculated exactly for arbitrary system size $Ltimes M$ and temperature. We start with the dimer method of Kasteleyn, McCoy & Wu, construct a highly symmetric block transfer matrix and derive a factorization of the involved determinant, effectively decomposing the free energy of the system into two parts, $F(L,M)=F_mathrm{strip}(L,M)+F_mathrm{strip}^mathrm{res}(L,M)$, where the residual part $F_mathrm{strip}^mathrm{res}(L,M)$ contains the nontrivial finite-$L$ contributions for fixed $M$. It is given by the determinant of a $frac{M}{2}times frac{M}{2}$ matrix and can be mapped onto an effective spin model with $M$ Ising spins and long-range interactions. While $F_mathrm{strip}^mathrm{res}(L,M)$ becomes exponentially small for large $L/M$ or off-critical temperatures, it leads to important finite-size effects such as the critical Casimir force near criticality. The relations to the Casimir potential and the Casimir force are discussed.
We calculate very long low- and high-temperature series for the susceptibility $chi$ of the square lattice Ising model as well as very long series for the five-particle contribution $chi^{(5)}$ and six-particle contribution $chi^{(6)}$. These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150000 CPU hours on computer clusters. For $chi^{(5)}$ 10000 terms of the series are calculated {it modulo} a single prime, and have been used to find the linear ODE satisfied by $chi^{(5)}$ {it modulo} a prime. A diff-Pade analysis of 2000 terms series for $chi^{(5)}$ and $chi^{(6)}$ confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the $n$-particle components of the susceptibility, up to a small set of ``additional singularities. We find the presence of singularities at $w=1/2$ for the linear ODE of $chi^{(5)}$, and $w^2= 1/8$ for the ODE of $chi^{(6)}$, which are {it not} singularities of the ``physical $chi^{(5)}$ and $chi^{(6)},$ that is to say the series-solutions of the ODEs which are analytic at $w =0$. Furthermore, analysis of the long series for $chi^{(5)}$ (and $chi^{(6)}$) combined with the corresponding long series for the full susceptibility $chi$ yields previously conjectured singularities in some $chi^{(n)}$, $n ge 7$. We also present a mechanism of resummation of the logarithmic singularities of the $chi^{(n)}$ leading to the known power-law critical behaviour occurring in the full $chi$, and perform a power spectrum analysis giving strong arguments in favor of the existence of a natural boundary for the full susceptibility $chi$.
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