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https://github.com/paboyle/Grid.git
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230 lines
7.3 KiB
C++
230 lines
7.3 KiB
C++
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#include <Grid.h>
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#include <qcd/utils/CovariantCshift.h>
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#include <qcd/utils/WilsonLoops.h>
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using namespace std;
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using namespace Grid;
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using namespace Grid::QCD;
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class suN {
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public:
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static int generators(int ncolour) { return ncolour*ncolour-1; }
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static int su2subgroups(int ncolour) { return (ncolour*(ncolour-1))/2; }
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template<typename CComplex,int N> using suNmatrix = iScalar<iScalar<iMatrix<CComplex,N> > > ;
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////////////////////////////////////////////////////////////////////////
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// There are N^2-1 generators for SU(N).
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//
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// We take a traceless hermitian generator basis as follows
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//
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// * Normalisation: trace ta tb = 1/2 delta_ab
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//
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// * Off diagonal
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// - pairs of rows i1,i2 behaving like pauli matrices signma_x, sigma_y
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//
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// - there are (Nc-1-i1) slots for i2 on each row [ x 0 x ]
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// direct count off each row
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//
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// - Sum of all pairs is Nc(Nc-1)/2: proof arithmetic series
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//
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// (Nc-1) + (Nc-2)+... 1 ==> Nc*(Nc-1)/2
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// 1+ 2+ + + Nc-1
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//
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// - There are 2 x Nc (Nc-1)/ 2 of these = Nc^2 - Nc
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//
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// - We enumerate the row-col pairs.
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// - for each row col pair there is a (sigma_x) and a (sigma_y) like generator
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//
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//
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// t^a_ij = { in 0.. Nc(Nc-1)/2 -1} => delta_{i,i1} delta_{j,i2} + delta_{i,i1} delta_{j,i2}
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// t^a_ij = { in Nc(Nc-1)/2 ... Nc^(Nc-1) -1} => i delta_{i,i1} delta_{j,i2} - i delta_{i,i1} delta_{j,i2}
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//
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// * Diagonal; must be traceless and normalised
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// - Sequence is
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// N (1,-1,0,0...)
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// N (1, 1,-2,0...)
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// N (1, 1, 1,-3,0...)
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// N (1, 1, 1, 1,-4,0...)
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//
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// where 1/2 = N^2 (1+.. m^2)etc.... for the m-th diagonal generator
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// NB this gives the famous SU3 result for su2 index 8
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//
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// N= sqrt(1/2 . 1/6 ) = 1/2 . 1/sqrt(3)
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//
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// ( 1 )
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// ( 1 ) / sqrt(3) /2 = 1/2 lambda_8
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// ( -2)
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////////////////////////////////////////////////////////////////////////
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template<class CComplex,int Ncolour>
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static void suNgenerator(int lieIndex,suNmatrix<CComplex,Ncolour> &ta){
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// map lie index to which type of generator
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int diagIndex;
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int su2Index;
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int sigxy;
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int NNm1 = Ncolour*(Ncolour-1);
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if ( lieIndex>= NNm1 ) {
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diagIndex = lieIndex -NNm1;
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suNgeneratorDiagonal(diagIndex,ta);
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return;
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}
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sigxy = lieIndex&0x1;
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su2Index= lieIndex>>1;
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if ( sigxy ) suNgeneratorSigmaY(su2Index,ta);
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else suNgeneratorSigmaX(su2Index,ta);
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}
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template<class CComplex,int Ncolour>
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static void suNgeneratorSigmaX(int su2Index,suNmatrix<CComplex,Ncolour> &ta){
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ta=zero;
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int i1,i2;
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su2SubGroupIndex<Ncolour>(i1,i2,su2Index);
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ta()()(i1,i2)=1.0;
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ta()()(i2,i1)=1.0;
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ta= ta *0.5;
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}
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template<class CComplex,int Ncolour>
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static void suNgeneratorSigmaY(int su2Index,suNmatrix<CComplex,Ncolour> &ta){
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ta=zero;
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Complex i(0.0,1.0);
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int i1,i2;
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su2SubGroupIndex<Ncolour>(i1,i2,su2Index);
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ta()()(i1,i2)=-i;
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ta()()(i2,i1)= i;
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ta= ta *0.5;
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}
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template<class CComplex,int Ncolour>
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static void suNgeneratorDiagonal(int diagIndex,suNmatrix<CComplex,Ncolour> &ta){
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ta=zero;
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int trsq=0;
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int last=diagIndex+1;
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for(int i=0;i<=diagIndex;i++){
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ta()()(i,i) = 1.0;
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trsq++;
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}
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ta()()(last,last) = -last;
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trsq+=last*last;
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RealD nrm = 1.0/std::sqrt(2.0*trsq);
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ta = ta *nrm;
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}
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////////////////////////////////////////////////////////////////////////
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// Map a
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//
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////////////////////////////////////////////////////////////////////////
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template<int Ncolour>
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static void su2SubGroupIndex(int &i1,int &i2,int su2_index){
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assert( (su2_index>=0) && (su2_index< (Ncolour*(Ncolour-1))/2) );
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int spare=su2_index;
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for(i1=0;spare >= (Ncolour-1-i1);i1++ ){
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spare = spare - (Ncolour-1-i1); // remove the Nc-1-i1 terms
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}
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i2=i1+1+spare;
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}
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template<class CComplex,int Ncolour>
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static void su2Extract(std::vector<LatticeComplex> &r,const Lattice<suNmatrix<CComplex,Ncolour> > &source, int su2_index)
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{
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assert(r.size() == 4); // store in 4 real parts
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for(int i=0;i<4;i++){
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conformable(r[i],source);
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}
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int i1,i2;
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su2SubGroupIndex<Ncolour>(i1,i2,su2_index);
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/* Compute the b(k) of A_SU(2) = b0 + i sum_k bk sigma_k */
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r[0] = real(source()()(i1,i1) + source()()(i2,i2));
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r[1] = imag(source()()(i1,i2) + source()()(i2,i1));
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r[2] = real(source()()(i1,i2) - source()()(i2,i1));
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r[3] = imag(source()()(i1,i1) - source()()(i2,i2));
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}
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template<int Ncolour> static void printGenerators(void)
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{
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for(int gen=0;gen<suN::generators(Ncolour);gen++){
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suN::suNmatrix<Complex,Ncolour> ta;
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suN::suNgenerator(gen,ta);
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std::cout<< "Nc = "<<Ncolour<<" t_"<<gen<<std::endl;
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std::cout<<ta<<std::endl;
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}
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}
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template<int Ncolour> static void testGenerators(void){
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suNmatrix<Complex,Ncolour> ta;
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suNmatrix<Complex,Ncolour> tb;
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std::cout<<"Checking trace ta tb is 0.5 delta_ab"<<std::endl;
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for(int a=0;a<generators(Ncolour);a++){
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for(int b=0;b<generators(Ncolour);b++){
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suNgenerator(a,ta);
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suNgenerator(b,tb);
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Complex tr =TensorRemove(trace(ta*tb));
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std::cout<<tr<<" ";
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if(a==b) assert(abs(tr-Complex(0.5))<1.0e-6);
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if(a!=b) assert(abs(tr)<1.0e-6);
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}
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std::cout<<std::endl;
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}
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std::cout<<"Checking hermitian"<<std::endl;
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for(int a=0;a<generators(Ncolour);a++){
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suNgenerator(a,ta);
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std::cout<<a<<" ";
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assert(norm2(ta-adj(ta))<1.0e-6);
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}
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std::cout<<std::endl;
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std::cout<<"Checking traceless"<<std::endl;
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for(int a=0;a<generators(Ncolour);a++){
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suNgenerator(a,ta);
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Complex tr =TensorRemove(trace(ta));
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std::cout<<a<<" ";
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assert(abs(tr)<1.0e-6);
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}
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std::cout<<std::endl;
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}
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};
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int main (int argc, char ** argv)
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{
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Grid_init(&argc,&argv);
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std::vector<int> simd_layout = GridDefaultSimd(4,vComplexF::Nsimd());
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std::vector<int> mpi_layout = GridDefaultMpi();
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std::vector<int> latt_size ({4,4,4,4});
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GridCartesian Fine(latt_size,simd_layout,mpi_layout);
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LatticeGaugeField Umu(&Fine);
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std::cout<<"*********************************************"<<std::endl;
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std::cout<<"* Generators for SU(2)"<<std::endl;
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std::cout<<"*********************************************"<<std::endl;
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suN::printGenerators<2>();
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suN::testGenerators<2>();
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std::cout<<"*********************************************"<<std::endl;
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std::cout<<"* Generators for SU(3)"<<std::endl;
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std::cout<<"*********************************************"<<std::endl;
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suN::printGenerators<3>();
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suN::testGenerators<3>();
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std::cout<<"*********************************************"<<std::endl;
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std::cout<<"* Generators for SU(4)"<<std::endl;
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std::cout<<"*********************************************"<<std::endl;
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suN::printGenerators<4>();
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suN::testGenerators<4>();
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std::cout<<"*********************************************"<<std::endl;
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std::cout<<"* Generators for SU(5)"<<std::endl;
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std::cout<<"*********************************************"<<std::endl;
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suN::printGenerators<5>();
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suN::testGenerators<5>();
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Grid_finalize();
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}
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