mirror of
https://github.com/paboyle/Grid.git
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277 lines
8.8 KiB
C++
277 lines
8.8 KiB
C++
////////////////////////////////////////////////////////////////////////
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//
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// * Two index representation generators
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//
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// * Normalisation for the fundamental generators:
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// trace ta tb = 1/2 delta_ab = T_F delta_ab
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// T_F = 1/2 for SU(N) groups
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//
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//
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// base for NxN two index (anti-symmetric) matrices
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// normalized to 1 (d_ij is the kroenecker delta)
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//
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// (e^(ij)_{kl} = 1 / sqrt(2) (d_ik d_jl +/- d_jk d_il)
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//
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// Then the generators are written as
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//
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// (iT_a)^(ij)(lk) = i * ( tr[e^(ij)^dag e^(lk) T^trasp_a] +
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// tr[e^(lk)e^(ij)^dag T_a] ) //
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//
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//
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////////////////////////////////////////////////////////////////////////
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// Authors: David Preti, Guido Cossu
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#ifndef QCD_UTIL_SUN2INDEX_H
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#define QCD_UTIL_SUN2INDEX_H
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namespace Grid {
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namespace QCD {
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enum TwoIndexSymmetry { Symmetric = 1, AntiSymmetric = -1 };
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inline Real delta(int a, int b) { return (a == b) ? 1.0 : 0.0; }
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template <int ncolour, TwoIndexSymmetry S>
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class SU_TwoIndex : public SU<ncolour> {
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public:
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static const int Dimension = ncolour * (ncolour + S) / 2;
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static const int NumGenerators = SU<ncolour>::AdjointDimension;
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template <typename vtype>
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using iSUnTwoIndexMatrix = iScalar<iScalar<iMatrix<vtype, Dimension> > >;
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typedef iSUnTwoIndexMatrix<Complex> TIMatrix;
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typedef iSUnTwoIndexMatrix<ComplexF> TIMatrixF;
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typedef iSUnTwoIndexMatrix<ComplexD> TIMatrixD;
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typedef iSUnTwoIndexMatrix<vComplex> vTIMatrix;
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typedef iSUnTwoIndexMatrix<vComplexF> vTIMatrixF;
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typedef iSUnTwoIndexMatrix<vComplexD> vTIMatrixD;
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typedef Lattice<vTIMatrix> LatticeTwoIndexMatrix;
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typedef Lattice<vTIMatrixF> LatticeTwoIndexMatrixF;
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typedef Lattice<vTIMatrixD> LatticeTwoIndexMatrixD;
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typedef Lattice<iVector<iScalar<iMatrix<vComplex, Dimension> >, Nd> >
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LatticeTwoIndexField;
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typedef Lattice<iVector<iScalar<iMatrix<vComplexF, Dimension> >, Nd> >
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LatticeTwoIndexFieldF;
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typedef Lattice<iVector<iScalar<iMatrix<vComplexD, Dimension> >, Nd> >
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LatticeTwoIndexFieldD;
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template <typename vtype>
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using iSUnMatrix = iScalar<iScalar<iMatrix<vtype, ncolour> > >;
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typedef iSUnMatrix<Complex> Matrix;
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typedef iSUnMatrix<ComplexF> MatrixF;
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typedef iSUnMatrix<ComplexD> MatrixD;
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template <class cplx>
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static void base(int Index, iSUnMatrix<cplx> &eij) {
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// returns (e)^(ij)_{kl} necessary for change of base U_F -> U_R
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assert(Index < NumGenerators);
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eij = zero;
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// for the linearisation of the 2 indexes
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static int a[ncolour * (ncolour - 1) / 2][2]; // store the a <-> i,j
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static bool filled = false;
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if (!filled) {
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int counter = 0;
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for (int i = 1; i < ncolour; i++) {
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for (int j = 0; j < i; j++) {
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a[counter][0] = i;
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a[counter][1] = j;
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counter++;
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}
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}
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filled = true;
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}
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if (Index < ncolour * (ncolour - 1) / 2) {
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baseOffDiagonal(a[Index][0], a[Index][1], eij);
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} else {
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baseDiagonal(Index, eij);
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}
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}
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template <class cplx>
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static void baseDiagonal(int Index, iSUnMatrix<cplx> &eij) {
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eij = zero;
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eij()()(Index - ncolour * (ncolour - 1) / 2,
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Index - ncolour * (ncolour - 1) / 2) = 1.0;
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}
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template <class cplx>
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static void baseOffDiagonal(int i, int j, iSUnMatrix<cplx> &eij) {
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eij = zero;
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for (int k = 0; k < ncolour; k++)
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for (int l = 0; l < ncolour; l++)
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eij()()(l, k) = delta(i, k) * delta(j, l) +
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S * delta(j, k) * delta(i, l);
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RealD nrm = 1. / std::sqrt(2.0);
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eij = eij * nrm;
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}
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static void printBase(void) {
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for (int gen = 0; gen < Dimension; gen++) {
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Matrix tmp;
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base(gen, tmp);
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std::cout << GridLogMessage << "Nc = " << ncolour << " t_" << gen
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<< std::endl;
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std::cout << GridLogMessage << tmp << std::endl;
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}
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}
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template <class cplx>
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static void generator(int Index, iSUnTwoIndexMatrix<cplx> &i2indTa) {
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Vector<typename SU<ncolour>::template iSUnMatrix<cplx> > ta(
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ncolour * ncolour - 1);
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Vector<typename SU<ncolour>::template iSUnMatrix<cplx> > eij(Dimension);
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typename SU<ncolour>::template iSUnMatrix<cplx> tmp;
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i2indTa = zero;
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for (int a = 0; a < ncolour * ncolour - 1; a++)
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SU<ncolour>::generator(a, ta[a]);
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for (int a = 0; a < Dimension; a++) base(a, eij[a]);
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for (int a = 0; a < Dimension; a++) {
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tmp = transpose(ta[Index]) * adj(eij[a]) + adj(eij[a]) * ta[Index];
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for (int b = 0; b < Dimension; b++) {
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typename SU<ncolour>::template iSUnMatrix<cplx> tmp1 =
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tmp * eij[b];
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Complex iTr = TensorRemove(timesI(trace(tmp1)));
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i2indTa()()(a, b) = iTr;
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}
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}
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}
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static void printGenerators(void) {
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for (int gen = 0; gen < ncolour * ncolour - 1; gen++) {
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TIMatrix i2indTa;
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generator(gen, i2indTa);
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std::cout << GridLogMessage << "Nc = " << ncolour << " t_" << gen
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<< std::endl;
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std::cout << GridLogMessage << i2indTa << std::endl;
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}
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}
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static void testGenerators(void) {
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TIMatrix i2indTa, i2indTb;
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std::cout << GridLogMessage << "2IndexRep - Checking if traceless"
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<< std::endl;
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for (int a = 0; a < ncolour * ncolour - 1; a++) {
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generator(a, i2indTa);
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std::cout << GridLogMessage << a << std::endl;
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assert(norm2(trace(i2indTa)) < 1.0e-6);
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}
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std::cout << GridLogMessage << std::endl;
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std::cout << GridLogMessage << "2IndexRep - Checking if antihermitean"
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<< std::endl;
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for (int a = 0; a < ncolour * ncolour - 1; a++) {
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generator(a, i2indTa);
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std::cout << GridLogMessage << a << std::endl;
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assert(norm2(adj(i2indTa) + i2indTa) < 1.0e-6);
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}
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std::cout << GridLogMessage << std::endl;
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std::cout << GridLogMessage
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<< "2IndexRep - Checking Tr[Ta*Tb]=delta(a,b)*(N +- 2)/2"
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<< std::endl;
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for (int a = 0; a < ncolour * ncolour - 1; a++) {
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for (int b = 0; b < ncolour * ncolour - 1; b++) {
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generator(a, i2indTa);
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generator(b, i2indTb);
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// generator returns iTa, so we need a minus sign here
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Complex Tr = -TensorRemove(trace(i2indTa * i2indTb));
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std::cout << GridLogMessage << "a=" << a << "b=" << b << "Tr=" << Tr
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<< std::endl;
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}
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}
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std::cout << GridLogMessage << std::endl;
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}
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static void TwoIndexLieAlgebraMatrix(
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const typename SU<ncolour>::LatticeAlgebraVector &h,
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LatticeTwoIndexMatrix &out, Real scale = 1.0) {
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conformable(h, out);
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GridBase *grid = out._grid;
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LatticeTwoIndexMatrix la(grid);
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TIMatrix i2indTa;
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out = zero;
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for (int a = 0; a < ncolour * ncolour - 1; a++) {
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generator(a, i2indTa);
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la = peekColour(h, a) * i2indTa;
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out += la;
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}
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out *= scale;
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}
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// Projects the algebra components
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// of a lattice matrix ( of dimension ncol*ncol -1 )
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static void projectOnAlgebra(
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typename SU<ncolour>::LatticeAlgebraVector &h_out,
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const LatticeTwoIndexMatrix &in, Real scale = 1.0) {
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conformable(h_out, in);
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h_out = zero;
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TIMatrix i2indTa;
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Real coefficient = -2.0 / (ncolour + 2 * S) * scale;
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// 2/(Nc +/- 2) for the normalization of the trace in the two index rep
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for (int a = 0; a < ncolour * ncolour - 1; a++) {
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generator(a, i2indTa);
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auto tmp = real(trace(i2indTa * in)) * coefficient;
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pokeColour(h_out, tmp, a);
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}
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}
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// a projector that keeps the generators stored to avoid the overhead of
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// recomputing them
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static void projector(typename SU<ncolour>::LatticeAlgebraVector &h_out,
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const LatticeTwoIndexMatrix &in, Real scale = 1.0) {
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conformable(h_out, in);
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// to store the generators
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static std::vector<TIMatrix> i2indTa(ncolour * ncolour -1);
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h_out = zero;
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static bool precalculated = false;
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if (!precalculated) {
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precalculated = true;
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for (int a = 0; a < ncolour * ncolour - 1; a++) generator(a, i2indTa[a]);
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}
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Real coefficient =
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-2.0 / (ncolour + 2 * S) * scale; // 2/(Nc +/- 2) for the normalization
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// of the trace in the two index rep
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for (int a = 0; a < ncolour * ncolour - 1; a++) {
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auto tmp = real(trace(i2indTa[a] * in)) * coefficient;
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pokeColour(h_out, tmp, a);
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}
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}
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};
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// Some useful type names
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typedef SU_TwoIndex<Nc, Symmetric> TwoIndexSymmMatrices;
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typedef SU_TwoIndex<Nc, AntiSymmetric> TwoIndexAntiSymmMatrices;
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typedef SU_TwoIndex<2, Symmetric> SU2TwoIndexSymm;
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typedef SU_TwoIndex<3, Symmetric> SU3TwoIndexSymm;
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typedef SU_TwoIndex<4, Symmetric> SU4TwoIndexSymm;
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typedef SU_TwoIndex<5, Symmetric> SU5TwoIndexSymm;
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typedef SU_TwoIndex<2, AntiSymmetric> SU2TwoIndexAntiSymm;
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typedef SU_TwoIndex<3, AntiSymmetric> SU3TwoIndexAntiSymm;
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typedef SU_TwoIndex<4, AntiSymmetric> SU4TwoIndexAntiSymm;
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typedef SU_TwoIndex<5, AntiSymmetric> SU5TwoIndexAntiSymm;
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}
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}
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#endif
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