mirror of
https://github.com/paboyle/Grid.git
synced 2024-11-10 15:55:37 +00:00
274 lines
8.7 KiB
C++
274 lines
8.7 KiB
C++
////////////////////////////////////////////////////////////////////////
|
|
//
|
|
// * Two index representation generators
|
|
//
|
|
// * Normalisation for the fundamental generators:
|
|
// trace ta tb = 1/2 delta_ab = T_F delta_ab
|
|
// T_F = 1/2 for SU(N) groups
|
|
//
|
|
//
|
|
// base for NxN two index (anti-symmetric) matrices
|
|
// normalized to 1 (d_ij is the kroenecker delta)
|
|
//
|
|
// (e^(ij)_{kl} = 1 / sqrt(2) (d_ik d_jl +/- d_jk d_il)
|
|
//
|
|
// Then the generators are written as
|
|
//
|
|
// (iT_a)^(ij)(lk) = i * ( tr[e^(ij)^dag e^(lk) T^trasp_a] +
|
|
// tr[e^(lk)e^(ij)^dag T_a] ) //
|
|
//
|
|
//
|
|
////////////////////////////////////////////////////////////////////////
|
|
|
|
// Authors: David Preti, Guido Cossu
|
|
|
|
#ifndef QCD_UTIL_SUN2INDEX_H
|
|
#define QCD_UTIL_SUN2INDEX_H
|
|
|
|
|
|
NAMESPACE_BEGIN(Grid);
|
|
|
|
enum TwoIndexSymmetry { Symmetric = 1, AntiSymmetric = -1 };
|
|
|
|
inline Real delta(int a, int b) { return (a == b) ? 1.0 : 0.0; }
|
|
|
|
template <int ncolour, TwoIndexSymmetry S>
|
|
class SU_TwoIndex : public SU<ncolour> {
|
|
public:
|
|
static const int Dimension = ncolour * (ncolour + S) / 2;
|
|
static const int NumGenerators = SU<ncolour>::AdjointDimension;
|
|
|
|
template <typename vtype>
|
|
using iSUnTwoIndexMatrix = iScalar<iScalar<iMatrix<vtype, Dimension> > >;
|
|
|
|
typedef iSUnTwoIndexMatrix<Complex> TIMatrix;
|
|
typedef iSUnTwoIndexMatrix<ComplexF> TIMatrixF;
|
|
typedef iSUnTwoIndexMatrix<ComplexD> TIMatrixD;
|
|
|
|
typedef iSUnTwoIndexMatrix<vComplex> vTIMatrix;
|
|
typedef iSUnTwoIndexMatrix<vComplexF> vTIMatrixF;
|
|
typedef iSUnTwoIndexMatrix<vComplexD> vTIMatrixD;
|
|
|
|
typedef Lattice<vTIMatrix> LatticeTwoIndexMatrix;
|
|
typedef Lattice<vTIMatrixF> LatticeTwoIndexMatrixF;
|
|
typedef Lattice<vTIMatrixD> LatticeTwoIndexMatrixD;
|
|
|
|
typedef Lattice<iVector<iScalar<iMatrix<vComplex, Dimension> >, Nd> >
|
|
LatticeTwoIndexField;
|
|
typedef Lattice<iVector<iScalar<iMatrix<vComplexF, Dimension> >, Nd> >
|
|
LatticeTwoIndexFieldF;
|
|
typedef Lattice<iVector<iScalar<iMatrix<vComplexD, Dimension> >, Nd> >
|
|
LatticeTwoIndexFieldD;
|
|
|
|
template <typename vtype>
|
|
using iSUnMatrix = iScalar<iScalar<iMatrix<vtype, ncolour> > >;
|
|
|
|
typedef iSUnMatrix<Complex> Matrix;
|
|
typedef iSUnMatrix<ComplexF> MatrixF;
|
|
typedef iSUnMatrix<ComplexD> MatrixD;
|
|
|
|
template <class cplx>
|
|
static void base(int Index, iSUnMatrix<cplx> &eij) {
|
|
// returns (e)^(ij)_{kl} necessary for change of base U_F -> U_R
|
|
assert(Index < NumGenerators);
|
|
eij = zero;
|
|
|
|
// for the linearisation of the 2 indexes
|
|
static int a[ncolour * (ncolour - 1) / 2][2]; // store the a <-> i,j
|
|
static bool filled = false;
|
|
if (!filled) {
|
|
int counter = 0;
|
|
for (int i = 1; i < ncolour; i++) {
|
|
for (int j = 0; j < i; j++) {
|
|
a[counter][0] = i;
|
|
a[counter][1] = j;
|
|
counter++;
|
|
}
|
|
}
|
|
filled = true;
|
|
}
|
|
|
|
if (Index < ncolour * (ncolour - 1) / 2) {
|
|
baseOffDiagonal(a[Index][0], a[Index][1], eij);
|
|
} else {
|
|
baseDiagonal(Index, eij);
|
|
}
|
|
}
|
|
|
|
template <class cplx>
|
|
static void baseDiagonal(int Index, iSUnMatrix<cplx> &eij) {
|
|
eij = zero;
|
|
eij()()(Index - ncolour * (ncolour - 1) / 2,
|
|
Index - ncolour * (ncolour - 1) / 2) = 1.0;
|
|
}
|
|
|
|
template <class cplx>
|
|
static void baseOffDiagonal(int i, int j, iSUnMatrix<cplx> &eij) {
|
|
eij = zero;
|
|
for (int k = 0; k < ncolour; k++)
|
|
for (int l = 0; l < ncolour; l++)
|
|
eij()()(l, k) = delta(i, k) * delta(j, l) +
|
|
S * delta(j, k) * delta(i, l);
|
|
|
|
RealD nrm = 1. / std::sqrt(2.0);
|
|
eij = eij * nrm;
|
|
}
|
|
|
|
static void printBase(void) {
|
|
for (int gen = 0; gen < Dimension; gen++) {
|
|
Matrix tmp;
|
|
base(gen, tmp);
|
|
std::cout << GridLogMessage << "Nc = " << ncolour << " t_" << gen
|
|
<< std::endl;
|
|
std::cout << GridLogMessage << tmp << std::endl;
|
|
}
|
|
}
|
|
|
|
template <class cplx>
|
|
static void generator(int Index, iSUnTwoIndexMatrix<cplx> &i2indTa) {
|
|
Vector<typename SU<ncolour>::template iSUnMatrix<cplx> > ta(
|
|
ncolour * ncolour - 1);
|
|
Vector<typename SU<ncolour>::template iSUnMatrix<cplx> > eij(Dimension);
|
|
typename SU<ncolour>::template iSUnMatrix<cplx> tmp;
|
|
i2indTa = zero;
|
|
|
|
for (int a = 0; a < ncolour * ncolour - 1; a++)
|
|
SU<ncolour>::generator(a, ta[a]);
|
|
|
|
for (int a = 0; a < Dimension; a++) base(a, eij[a]);
|
|
|
|
for (int a = 0; a < Dimension; a++) {
|
|
tmp = transpose(ta[Index]) * adj(eij[a]) + adj(eij[a]) * ta[Index];
|
|
for (int b = 0; b < Dimension; b++) {
|
|
typename SU<ncolour>::template iSUnMatrix<cplx> tmp1 =
|
|
tmp * eij[b];
|
|
Complex iTr = TensorRemove(timesI(trace(tmp1)));
|
|
i2indTa()()(a, b) = iTr;
|
|
}
|
|
}
|
|
}
|
|
|
|
static void printGenerators(void) {
|
|
for (int gen = 0; gen < ncolour * ncolour - 1; gen++) {
|
|
TIMatrix i2indTa;
|
|
generator(gen, i2indTa);
|
|
std::cout << GridLogMessage << "Nc = " << ncolour << " t_" << gen
|
|
<< std::endl;
|
|
std::cout << GridLogMessage << i2indTa << std::endl;
|
|
}
|
|
}
|
|
|
|
static void testGenerators(void) {
|
|
TIMatrix i2indTa, i2indTb;
|
|
std::cout << GridLogMessage << "2IndexRep - Checking if traceless"
|
|
<< std::endl;
|
|
for (int a = 0; a < ncolour * ncolour - 1; a++) {
|
|
generator(a, i2indTa);
|
|
std::cout << GridLogMessage << a << std::endl;
|
|
assert(norm2(trace(i2indTa)) < 1.0e-6);
|
|
}
|
|
std::cout << GridLogMessage << std::endl;
|
|
|
|
std::cout << GridLogMessage << "2IndexRep - Checking if antihermitean"
|
|
<< std::endl;
|
|
for (int a = 0; a < ncolour * ncolour - 1; a++) {
|
|
generator(a, i2indTa);
|
|
std::cout << GridLogMessage << a << std::endl;
|
|
assert(norm2(adj(i2indTa) + i2indTa) < 1.0e-6);
|
|
}
|
|
|
|
std::cout << GridLogMessage << std::endl;
|
|
std::cout << GridLogMessage
|
|
<< "2IndexRep - Checking Tr[Ta*Tb]=delta(a,b)*(N +- 2)/2"
|
|
<< std::endl;
|
|
for (int a = 0; a < ncolour * ncolour - 1; a++) {
|
|
for (int b = 0; b < ncolour * ncolour - 1; b++) {
|
|
generator(a, i2indTa);
|
|
generator(b, i2indTb);
|
|
|
|
// generator returns iTa, so we need a minus sign here
|
|
Complex Tr = -TensorRemove(trace(i2indTa * i2indTb));
|
|
std::cout << GridLogMessage << "a=" << a << "b=" << b << "Tr=" << Tr
|
|
<< std::endl;
|
|
}
|
|
}
|
|
std::cout << GridLogMessage << std::endl;
|
|
}
|
|
|
|
static void TwoIndexLieAlgebraMatrix(
|
|
const typename SU<ncolour>::LatticeAlgebraVector &h,
|
|
LatticeTwoIndexMatrix &out, Real scale = 1.0) {
|
|
conformable(h, out);
|
|
GridBase *grid = out._grid;
|
|
LatticeTwoIndexMatrix la(grid);
|
|
TIMatrix i2indTa;
|
|
|
|
out = zero;
|
|
for (int a = 0; a < ncolour * ncolour - 1; a++) {
|
|
generator(a, i2indTa);
|
|
la = peekColour(h, a) * i2indTa;
|
|
out += la;
|
|
}
|
|
out *= scale;
|
|
}
|
|
|
|
// Projects the algebra components
|
|
// of a lattice matrix ( of dimension ncol*ncol -1 )
|
|
static void projectOnAlgebra(
|
|
typename SU<ncolour>::LatticeAlgebraVector &h_out,
|
|
const LatticeTwoIndexMatrix &in, Real scale = 1.0) {
|
|
conformable(h_out, in);
|
|
h_out = zero;
|
|
TIMatrix i2indTa;
|
|
Real coefficient = -2.0 / (ncolour + 2 * S) * scale;
|
|
// 2/(Nc +/- 2) for the normalization of the trace in the two index rep
|
|
for (int a = 0; a < ncolour * ncolour - 1; a++) {
|
|
generator(a, i2indTa);
|
|
auto tmp = real(trace(i2indTa * in)) * coefficient;
|
|
pokeColour(h_out, tmp, a);
|
|
}
|
|
}
|
|
|
|
// a projector that keeps the generators stored to avoid the overhead of
|
|
// recomputing them
|
|
static void projector(typename SU<ncolour>::LatticeAlgebraVector &h_out,
|
|
const LatticeTwoIndexMatrix &in, Real scale = 1.0) {
|
|
conformable(h_out, in);
|
|
// to store the generators
|
|
static std::vector<TIMatrix> i2indTa(ncolour * ncolour -1);
|
|
h_out = zero;
|
|
static bool precalculated = false;
|
|
if (!precalculated) {
|
|
precalculated = true;
|
|
for (int a = 0; a < ncolour * ncolour - 1; a++) generator(a, i2indTa[a]);
|
|
}
|
|
|
|
Real coefficient =
|
|
-2.0 / (ncolour + 2 * S) * scale; // 2/(Nc +/- 2) for the normalization
|
|
// of the trace in the two index rep
|
|
|
|
for (int a = 0; a < ncolour * ncolour - 1; a++) {
|
|
auto tmp = real(trace(i2indTa[a] * in)) * coefficient;
|
|
pokeColour(h_out, tmp, a);
|
|
}
|
|
}
|
|
};
|
|
|
|
// Some useful type names
|
|
typedef SU_TwoIndex<Nc, Symmetric> TwoIndexSymmMatrices;
|
|
typedef SU_TwoIndex<Nc, AntiSymmetric> TwoIndexAntiSymmMatrices;
|
|
|
|
typedef SU_TwoIndex<2, Symmetric> SU2TwoIndexSymm;
|
|
typedef SU_TwoIndex<3, Symmetric> SU3TwoIndexSymm;
|
|
typedef SU_TwoIndex<4, Symmetric> SU4TwoIndexSymm;
|
|
typedef SU_TwoIndex<5, Symmetric> SU5TwoIndexSymm;
|
|
|
|
typedef SU_TwoIndex<2, AntiSymmetric> SU2TwoIndexAntiSymm;
|
|
typedef SU_TwoIndex<3, AntiSymmetric> SU3TwoIndexAntiSymm;
|
|
typedef SU_TwoIndex<4, AntiSymmetric> SU4TwoIndexAntiSymm;
|
|
typedef SU_TwoIndex<5, AntiSymmetric> SU5TwoIndexAntiSymm;
|
|
|
|
NAMESPACE_END(Grid);
|
|
|
|
#endif
|