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sp2n generators
This commit is contained in:
Alessandro Lupo
396
Grid/qcd/utils/Sp2n.h
Normal file
396
Grid/qcd/utils/Sp2n.h
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#ifndef QCD_UTIL_Sp2n_H
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#define QCD_UTIL_Sp2n_H
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NAMESPACE_BEGIN(Grid);
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// Sp(2N)
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// ncolour = N
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template <int ncolour>
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class Sp {
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public:
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static const int Dimension = ncolour*2;
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static const int AlgebraDimension = ncolour*(2*ncolour +1);
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static int su2subgroups(void) { return (ncolour * (ncolour - 1)) / 2; }
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template <typename vtype>
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using iSp2nMatrix = iScalar<iScalar<iMatrix<vtype, Dimension> > >;
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template <typename vtype>
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using iSU2Matrix = iScalar<iScalar<iMatrix<vtype, 2> > >;
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template <typename vtype>
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using iSp2nAlgebraVector = iScalar<iScalar<iVector<vtype, AlgebraDimension> > >;
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typedef iSp2nMatrix<Complex> Matrix;
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typedef iSp2nMatrix<vComplex> vMatrix;
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typedef Lattice<vMatrix> LatticeMatrix;
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// Sp(2N) has N(2N+1) = 2N^2+N generators
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//
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// normalise the generators such that
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// Trace ( Ta Tb) = 1/2 delta_ab
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//
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// N generators in the cartan, 2N^2 off
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// off diagonal:
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// there are 6 types named a,b,c,d and w,z
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// abcd are N(N-1)/2 each while wz are N each
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template <class cplx>
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static void generator(int lieIndex, iSp2nMatrix<cplx> &ta) {
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// map lie index into type of generators: diagonal, abcd type, wz type
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int diagIndex;
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int aIndex, bIndex, cIndex, dIndex;
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int wIndex, zIndex; // a,b,c,d are N(N-1)/2 and w,z are N
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int mod = ncolour * (ncolour-1) * 0.5;
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int offdiag = 2*ncolour*ncolour; // number of generators not in the cartan subalgebra
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int wmod = 4*mod;
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int zmod = wmod+ncolour;
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if (lieIndex >= offdiag) {
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diagIndex = lieIndex - offdiag; // 0, ... ,N-1
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//std::cout << GridLogMessage << "diag type " << std::endl;
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generatorDiagtype(diagIndex, ta);
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return;
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}
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if ( (lieIndex >= wmod) && (lieIndex < zmod) ) {
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//std::cout << GridLogMessage << "w type " << std::endl;
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wIndex = lieIndex- wmod; // 0, ... ,N-1
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generatorWtype(wIndex,ta);
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return;
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}
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if ( (lieIndex >= zmod) && (lieIndex < offdiag) ) {
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//std::cout << GridLogMessage << "z type " << std::endl;
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//std::cout << GridLogMessage << "lie index " << lieIndex << std::endl;
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//std::cout << GridLogMessage << "z mod " << zmod << std::endl;
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zIndex = lieIndex - zmod; // 0, ... ,N-1
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generatorZtype(zIndex,ta);
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return;
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}
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if (lieIndex < mod) { // atype 0, ... , N(N-1)/2=mod
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//std::cout << GridLogMessage << "a type " << std::endl;
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aIndex = lieIndex;
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//std::cout << GridLogMessage << "a indx " << aIndex << std::endl;
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generatorAtype(aIndex, ta);
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return;
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}
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if ( (lieIndex >= mod) && lieIndex < 2*mod) { // btype mod, ... , 2mod-1
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//std::cout << GridLogMessage << "b type " << std::endl;
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bIndex = lieIndex - mod;
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generatorBtype(bIndex, ta);
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return;
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}
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if ( (lieIndex >= 2*mod) && lieIndex < 3*mod) { // ctype 2mod, ... , 3mod-1
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//std::cout << GridLogMessage << "c type " << std::endl;
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cIndex = lieIndex - 2*mod;
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generatorCtype(cIndex, ta);
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return;
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}
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if ( (lieIndex >= 3*mod) && lieIndex < wmod) { // ctype 3mod, ... , 4mod-1 = wmod-1
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//std::cout << GridLogMessage << "d type " << std::endl;
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dIndex = lieIndex - 3*mod;
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generatorDtype(dIndex, ta);
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return;
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}
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} //end of generator
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template <class cplx>
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static void generatorDiagtype(int diagIndex, iSp2nMatrix<cplx> &ta) {
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// ta(i,i) = - ta(i+N,i+N) = 1/2 for each i index of the cartan subalgebra
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ta = Zero();
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RealD nrm = 1.0 / 2;
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ta()()(diagIndex,diagIndex) = nrm;
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ta()()(diagIndex+ncolour,diagIndex+ncolour) = -nrm;
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}
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template <class cplx>
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static void generatorAtype(int aIndex, iSp2nMatrix<cplx> &ta) {
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// ta(i,j) = ta(j,i) = -ta(i+N,j+N) = -ta(j+N,i+N) = 1 / 2 sqrt(2)
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// with i<j and i=0,...,N-2
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// follows that j=i+1, ... , N
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int i1, i2;
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ta = Zero();
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RealD nrm = 1 / (2 * std::sqrt(2) );
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su2SubGroupIndex(i1, i2, aIndex);
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ta()()(i1,i2) = 1;
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ta()()(i2,i1) = 1;
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ta()()(i1+ncolour,i2+ncolour) = -1;
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ta()()(i2+ncolour,i1+ncolour) = -1;
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ta = ta * nrm;
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}
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template <class cplx>
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static void generatorBtype(int bIndex, iSp2nMatrix<cplx> &ta) {
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// ta(i,j) = -ta(j,i) = ta(i+N,j+N) = -ta(j+N,i+N) = i / 1/ 2 sqrt(2)
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// with i<j and i=0,...,N-2
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// follows that j=i+1, ... , N-1
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int i1, i2;
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ta = Zero();
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cplx i(0.0, 1.0);
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RealD nrm = 1 / (2 * std::sqrt(2));
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su2SubGroupIndex(i1, i2, bIndex);
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ta()()(i1,i2) = i;
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ta()()(i2,i1) = -i;
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ta()()(i1+ncolour,i2+ncolour) = i;
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ta()()(i2+ncolour,i1+ncolour) = -i;
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ta = ta * nrm;
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}
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template <class cplx>
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static void generatorCtype(int cIndex, iSp2nMatrix<cplx> &ta) {
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// ta(i,j+N) = ta(j,i+N) = ta(i+N,j) = ta(j+N,i) = 1 / 2 sqrt(2)
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int i1, i2;
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ta = Zero();
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RealD nrm = 1 / (2 * std::sqrt(2) );
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su2SubGroupIndex(i1, i2, cIndex);
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ta()()(i1,i2+ncolour) = 1;
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ta()()(i2,i1+ncolour) = 1;
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ta()()(i1+ncolour,i2) = 1;
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ta()()(i2+ncolour,i1) = 1;
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ta = ta * nrm;
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}
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template <class cplx>
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static void generatorDtype(int dIndex, iSp2nMatrix<cplx> &ta) {
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// ta(i,j+N) = ta(j,i+N) = -ta(i+N,j) = -ta(j+N,i) = i / 2 sqrt(2)
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int i1, i2;
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ta = Zero();
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cplx i(0.0, 1.0);
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RealD nrm = 1 / (2 * std::sqrt(2) );
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su2SubGroupIndex(i1, i2, dIndex);
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ta()()(i1,i2+ncolour) = i;
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ta()()(i2,i1+ncolour) = i;
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ta()()(i1+ncolour,i2) = -i;
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ta()()(i2+ncolour,i1) = -i;
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ta = ta * nrm;
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}
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template <class cplx>
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static void generatorWtype(int wIndex, iSp2nMatrix<cplx> &ta) {
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// ta(i,i+N) = ta(i+N,i) = 1/2
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ta = Zero();
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RealD nrm = 1.0 / 2; //check
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ta()()(wIndex,wIndex+ncolour) = 1;
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ta()()(wIndex+ncolour,wIndex) = 1;
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ta = ta * nrm;
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}
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template <class cplx>
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static void generatorZtype(int zIndex, iSp2nMatrix<cplx> &ta) {
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// ta(i,i+N) = - ta(i+N,i) = i/2
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ta = Zero();
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RealD nrm = 1.0 / 2; //check
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cplx i(0.0, 1.0);
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ta()()(zIndex,zIndex+ncolour) = i;
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ta()()(zIndex+ncolour,zIndex) = -i;
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ta = ta * nrm;
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}
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////////////////////////////////////////////////////////////////////////
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// Map a su2 subgroup number to the pair of rows that are non zero
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////////////////////////////////////////////////////////////////////////
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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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static void printGenerators(void) {
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for (int gen = 0; gen < AlgebraDimension; gen++) {
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Matrix ta;
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generator(gen, ta);
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std::cout << GridLogMessage << "Nc (2n) = " << 2*ncolour << std::endl;
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std::cout << GridLogMessage << " t_" << gen << std::endl;
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std::cout << GridLogMessage << ta << std::endl;
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}
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}
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static void testGenerators(void) {
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Matrix ta;
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Matrix tb;
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std::cout << GridLogMessage << "Fundamental - Checking trace ta tb is 0.5 delta_ab " << std::endl;
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for (int a = 0; a < AlgebraDimension; a++) {
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for (int b = 0; b < AlgebraDimension; b++) {
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generator(a,ta);
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generator(b,tb);
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Complex tr = TensorRemove(trace( ta * tb) );
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std::cout << GridLogMessage << "(" << a << "," << b << ") = " << tr
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<< std::endl;
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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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}
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std::cout << GridLogMessage << std::endl;
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std::cout << GridLogMessage << "Fundamental - Checking if hermitian" << std::endl;
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for (int a = 0; a < AlgebraDimension; a++) {
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generator(a,ta);
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std::cout << GridLogMessage << a << std::endl;
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assert(norm2(ta - adj(ta)) < 1.0e-6);
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}
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std::cout << GridLogMessage << std::endl;
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std::cout << GridLogMessage << "Fundamental - Checking if traceless" << std::endl;
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for (int a = 0; a < AlgebraDimension; a++) {
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generator(a, ta);
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Complex tr = TensorRemove(trace(ta));
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std::cout << GridLogMessage << a << std::endl;
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assert(abs(tr) < 1.0e-6);
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}
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}
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static void GaussianFundamentalLieAlgebraMatrix(GridParallelRNG &pRNG, //same as sun
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LatticeMatrix &out,
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Real scale = 1.0) {
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GridBase *grid = out.Grid();
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LatticeReal ca(grid);
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LatticeMatrix la(grid);
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Complex ci(0.0, scale);
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Matrix ta;
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out = Zero();
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for (int a = 0; a < AlgebraDimension; a++) {
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gaussian(pRNG, ca);
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generator(a, ta);
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la = toComplex(ca) * ta;
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out += la;
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}
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out *= ci;
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}
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template <typename LatticeMatrixType>
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static void taExp(const LatticeMatrixType &x, LatticeMatrixType &ex) { // same as sun
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typedef typename LatticeMatrixType::scalar_type ComplexType;
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LatticeMatrixType xn(x.Grid());
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RealD nfac = 1.0;
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xn = x;
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ex = xn + ComplexType(1.0); // 1+x
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// Do a 12th order exponentiation
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for (int i = 2; i <= 12; ++i) {
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nfac = nfac / RealD(i); // 1/2, 1/2.3 ...
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xn = xn * x; // x2, x3,x4....
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ex = ex + xn * nfac; // x2/2!, x3/3!....
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}
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}
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}; // end of class Sp
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typedef Sp<2> Sp4;
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typedef Sp<3> Sp6;
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typedef Sp<4> Sp8;
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NAMESPACE_END(Grid);
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#endif
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/* }
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sigxy = lieIndex & 0x1; // 1 if odd, 0 if even
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su2Index = lieIndex >> 1 ; // where to put the sigma_x(y)
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//for the even(odd) lieindex, sigmax(y)
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if (sigxy)
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//put sigmay at su2index
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else
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//put sigmax */
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@@ -9,6 +9,7 @@
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// Include representations
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#include <Grid/qcd/utils/SUn.h>
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#include <Grid/qcd/utils/Sp2n.h>
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#include <Grid/qcd/utils/SUnAdjoint.h>
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#include <Grid/qcd/utils/SUnTwoIndex.h>
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46
tests/core/test_sp2n_lie_gen.cc
Normal file
46
tests/core/test_sp2n_lie_gen.cc
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@@ -0,0 +1,46 @@
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#include <Grid/Grid.h>
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#include <Grid/qcd/utils/SUn.h>
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using namespace Grid;
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int main(int argc, char** argv) {
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Grid_init(&argc, &argv);
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//std::vector<int> latt({4, 4, 4, 8});
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//GridCartesian* grid = SpaceTimeGrid::makeFourDimGrid(
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//latt, GridDefaultSimd(Nd, vComplex::Nsimd()), GridDefaultMpi());
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//GridRedBlackCartesian* rbGrid = SpaceTimeGrid::makeFourDimRedBlackGrid(grid);
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std::cout << GridLogMessage << "*********************************************"
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<< std::endl;
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std::cout << GridLogMessage << "* Generators for Sp(4)" << std::endl;
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std::cout << GridLogMessage << "*********************************************"
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<< std::endl;
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Sp4::printGenerators();
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Sp4::testGenerators();
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std::cout << GridLogMessage << "*********************************************"
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<< std::endl;
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std::cout << GridLogMessage << "* Generators for Sp(6)" << std::endl;
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std::cout << GridLogMessage << "*********************************************"
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<< std::endl;
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Sp6::printGenerators();
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Sp6::testGenerators();
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std::cout << GridLogMessage << "*********************************************"
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<< std::endl;
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std::cout << GridLogMessage << "* Generators for Sp(8)" << std::endl;
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std::cout << GridLogMessage << "*********************************************"
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<< std::endl;
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Sp8::printGenerators();
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Sp8::testGenerators();
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Grid_finalize();
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}
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