Lattice Green functions: the d-dimensional face-centred cubic lattice,
d = 8, 9, 10, 11, 12

Saoud Hassani, Christoph Koutschan, Jean-Marie Maillard, Nadjah Zenine

Abstract

We previously reported on a recursive method to generate the expansion of the lattice Green function of the d-dimensional face-centred cubic lattice. The method was used to generate many coefficients for d = 7 and the corresponding linear differential equation has been obtained. We show the strength and the limit of the method by producing the series and the corresponding differential equations for d = 8, 9, 10, 11. The differential Galois groups of these differential equations are shown to be symplectic for d = 8, 10 and orthogonal for d = 9, 11.

Paper

The full paper is available here.

Supplementary electronic material

Here we give the minimal-order differential operators for the lattice Green functions of the face-centred cubic lattice, for different dimensions d. The second column displays the order of the operator, the third column the maximal degree of its coefficients, and the fourth column shows the number of terms of the Taylor series that are necessary to guess some non-minimal differential operators which allow to obtain the minimal-order one as their greatest common right divisor. In the sixth column, we give the recurrence equation for the number a(n) of returning walks in the d-dimensional f.c.c. lattice; the n-th Taylor coefficient of the lattice Green function is equal to a(n)/cⁿ, where c = 2d(d-1). Note that the operator for d = 12 is only available modulo the prime p = 2³¹ - 1.

d	order	degree	terms	operator	recurrence
3	3	5	20	fcc3_mop	fcc3_rec
4	4	10	40	fcc4_mop	fcc4_rec
5	6	17	88	fcc5_mop	fcc5_rec
6	8	43	228	fcc6_mop	fcc6_rec
7	11	68	391	fcc7_mop	fcc7_rec
8	14	126	714	fcc8_mop	fcc8_rec
9	18	169	999	fcc9_mop	fcc9_rec
10	22	300	1739	fcc10_mop	fcc10_rec
11	27	409	2464	fcc11_mop
12	32	617	3618	fcc12_mop_mod

In the following we provide the ODEs and PDEs (in operator notation) for T₂(z,y) and T₃(z,y):

• ODE L₅ with respect to z for T₂(z,y): T2_ode_z.txt
• ODE N₅ with respect to y for T₂(z,y): T2_ode_y.txt
• The operator PDE₃⁽¹⁾ for T₂(z,y): T2_pde_3_1.txt
• The operator PDE₃⁽²⁾ for T₂(z,y): T2_pde_3_2.txt
• The operator PDE₂ for T₂(z,y): T2_pde_2.txt
• Gröbner basis of PDE for ann(T₂): T2_pde_gb.txt
• ODE L₉ with respect to z for T₃(z,y): T3_ode_z.txt
• ODE N₉ with respect to y for T₃(z,y): T3_ode_y.txt
• The operator PDE₄⁽¹⁾ for T₃(z,y): T3_pde_4_1.txt
• The operator PDE₄⁽²⁾ for T₃(z,y): T3_pde_4_2.txt
• The operator PDE₃ for T₃(z,y): T3_pde_3.txt
• Gröbner basis of PDE for ann(T₃): T3_pde_gb.txt

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