Lattice Green's functions

See examples in LGFhc.mw, LGFcp.mw, LGFbcc.mw

See mathematical details here

Notations

lat{name}{function}(args) where name is lattice name (Dsc,1,2sq,3sc,2tri) and function(args) is one of:

Basic functions

X(X)::X symmetry-unique site
O(X,sorted)::Xs site orbit, sorted means that X is already symmetry-unique
ONT(X,sorted)::[Osize,Otype::posint] orbit size and type (Wyckoff position)
CD(X,sorted)::nonnegint=|X|
ED2(X)::numeric=||X||^2
add(f::procedure(x1,...,xd),L) lattice summation of a full-symmetry function
p(t,X) propagator

Green's function

g(s,X) Green's function evaluated by best available method
h?(q,X)=g?(s,X) replace "g" by "h" to change argument
go(s) Green's function at origin
gos(s) derivative of the Green's function at origin
r(sigma) density of states
gF by Fourier integral
gL(s,X,{fulloutput}) by large-scale approximation, full output includes g,eta,kappa,Xsorted
gP Green's function evaluated by path expansion
gR by recurrence

Path expansion

gP(q,X,/N:=-1,output:=`if`(type(q,numeric),"","v"),digits:={8}) Green's function by path expansion up to order N or maximum available, where output is one of v=value without remainder, c=corrected by remainder value, r=[c,remainder], otherwise corrected value with checks
PR(q,X,N,shift:=0) series remainder, negative shift gives the upper bound, note that remainder to P(N) is PR(N+1)
N(n,X,{sorted,all}) path expansion numbers of order n≥0, sorted relates to X, all requests all numbers for given X
Nc(n,X,{sorted,all}) path expansion coefficients
N_init(d,L,N,{maxsize1,maxsize2,printout}) initialize path expansion numbers and coefficients, d<1 reset some or all tables and arrays (see examples)

Recurrent evaluation

hR(q,X,output:=`if`(type(q,numeric),"v","l")) Green's function by recurrence, where output is one of v=value, l=list of coefficients at basis functions
B(d,q)::[h0,h1,..,hd] natural basis for recurrence
M(X,{sorted}) recurrence polynomial coefficients

Recurrent evaluation

For d≤3 basis is available in terms of elliptic integrals, otherwise latDschP(q,X,_rest) is used

Series at zero

latDscSZ(d,p) singular function
latDscScB(d) coefficient at singular part
latDscSAJ2A(d,AJ,Am::list(Array),B::Array,extra:=0,{p0:=1})::list[d-1] coefficients at regular part from Ref.[Joyce03]
latDscSC(d,mv:=-1) get (mv::integer) or set (mv::list) coefficients at ODE basis, if mv<0 then all coefficients are returned, if mv=[] then default values are used
latDscSAB(a0,b0,N,dN:=1,IC:=[0=0],{printout}) series coefficients of regular A::Array(0..d,0..N) and singular B parts, a0,b0 can be either IC for recurrence or existing A,B to extend
latDscS(d,m,n,k,dN:=20,all) series coefficients BA[m,n,k], where m=0..d is for ODE basis, n=0..d is for recurrence basis, and k=0..N is at p^(2k)
latDschpS(p,X,N:=0,sorted,basis) function hp(p,X)=h(q,X)*q^(|X|-1) where p^2+q^2=1, if basis then basis functions are returned hp=add(C[m]*hb[m],m=1..d)+C[d]*hb[0]*Z+hb[1]-correction

Integral representations

Functions accepting special symbolic arguments for symbolic output

lat3scgos(s,/xi,K,E)
lat2sqB(q,/K,E)
lat3scB(q,/xi,K,L,/K1,K2) (the last two optional arguments are detected by _nrest)

Other dimension-specific functions

lat3scgo0 value at 0
lat3sc_q2xi(q),lat3sc_xi2q(xi) conversion between q and Joyce's xi variable
lat3sc_xi2k(xi) argument of the complete elliptic integrals used for evaluation of the Green's function
lat3scho_H0,1 two equivalent representations via Heun function

Triangular lattice

lat2trip(t,X,n) here n is number of terms in convolution
lat2tri_q2xi(q),lat2tri_xi2q(xi) conversion between q and Joyce's xi variable
lat2tri_xi2k(xi) argument of the complete elliptic integrals used for evaluation of the Green's function