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Separated out everything SU specific
This commit is contained in:
Julian Lenz
parent
6e750ecb0e
commit
921e23e83c
@ -32,11 +32,25 @@ directory
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#ifndef QCD_UTIL_SUN_H
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#ifndef QCD_UTIL_SUN_H
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#define QCD_UTIL_SUN_H
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#define QCD_UTIL_SUN_H
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#define ONLY_IF_SU \
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typename dummy_name = group_name, \
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typename = std::enable_if_t<is_su<dummy_name>::value>
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NAMESPACE_BEGIN(Grid);
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NAMESPACE_BEGIN(Grid);
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namespace GroupName {
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namespace GroupName {
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class SU {};
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class SU {};
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} // namespace GroupName
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} // namespace GroupName
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template <typename group_name>
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struct is_su {
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static const bool value = false;
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};
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template <>
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struct is_su<GroupName::SU> {
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static const bool value = true;
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};
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template <int ncolour, class group_name = GroupName::SU>
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template <int ncolour, class group_name = GroupName::SU>
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class GaugeGroup;
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class GaugeGroup;
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@ -50,36 +64,35 @@ class GaugeGroup {
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static const int AdjointDimension = ncolour * ncolour - 1;
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static const int AdjointDimension = ncolour * ncolour - 1;
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static int su2subgroups(void) { return (ncolour * (ncolour - 1)) / 2; }
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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 iSUnMatrix = iScalar<iScalar<iMatrix<vtype, ncolour> > >;
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template <typename vtype>
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template <typename vtype>
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using iSU2Matrix = iScalar<iScalar<iMatrix<vtype, 2> > >;
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using iSU2Matrix = iScalar<iScalar<iMatrix<vtype, 2> > >;
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template <typename vtype>
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template <typename vtype>
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using iSUnAlgebraVector =
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using iGroupMatrix = iScalar<iScalar<iMatrix<vtype, ncolour> > >;
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iScalar<iScalar<iVector<vtype, AdjointDimension> > >;
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template <typename vtype>
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using iAlgebraVector = iScalar<iScalar<iVector<vtype, AdjointDimension> > >;
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//////////////////////////////////////////////////////////////////////////////////////////////////
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//////////////////////////////////////////////////////////////////////////////////////////////////
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// Types can be accessed as SU<2>::Matrix , SU<2>::vSUnMatrix,
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// Types can be accessed as SU<2>::Matrix , SU<2>::vSUnMatrix,
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// SU<2>::LatticeMatrix etc...
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// SU<2>::LatticeMatrix etc...
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//////////////////////////////////////////////////////////////////////////////////////////////////
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//////////////////////////////////////////////////////////////////////////////////////////////////
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typedef iSUnMatrix<Complex> Matrix;
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typedef iGroupMatrix<Complex> Matrix;
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typedef iSUnMatrix<ComplexF> MatrixF;
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typedef iGroupMatrix<ComplexF> MatrixF;
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typedef iSUnMatrix<ComplexD> MatrixD;
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typedef iGroupMatrix<ComplexD> MatrixD;
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typedef iSUnMatrix<vComplex> vMatrix;
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typedef iGroupMatrix<vComplex> vMatrix;
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typedef iSUnMatrix<vComplexF> vMatrixF;
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typedef iGroupMatrix<vComplexF> vMatrixF;
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typedef iSUnMatrix<vComplexD> vMatrixD;
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typedef iGroupMatrix<vComplexD> vMatrixD;
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// For the projectors to the algebra
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// For the projectors to the algebra
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// these should be real...
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// these should be real...
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// keeping complex for consistency with the SIMD vector types
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// keeping complex for consistency with the SIMD vector types
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typedef iSUnAlgebraVector<Complex> AlgebraVector;
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typedef iAlgebraVector<Complex> AlgebraVector;
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typedef iSUnAlgebraVector<ComplexF> AlgebraVectorF;
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typedef iAlgebraVector<ComplexF> AlgebraVectorF;
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typedef iSUnAlgebraVector<ComplexD> AlgebraVectorD;
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typedef iAlgebraVector<ComplexD> AlgebraVectorD;
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typedef iSUnAlgebraVector<vComplex> vAlgebraVector;
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typedef iAlgebraVector<vComplex> vAlgebraVector;
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typedef iSUnAlgebraVector<vComplexF> vAlgebraVectorF;
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typedef iAlgebraVector<vComplexF> vAlgebraVectorF;
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typedef iSUnAlgebraVector<vComplexD> vAlgebraVectorD;
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typedef iAlgebraVector<vComplexD> vAlgebraVectorD;
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typedef Lattice<vMatrix> LatticeMatrix;
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typedef Lattice<vMatrix> LatticeMatrix;
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typedef Lattice<vMatrixF> LatticeMatrixF;
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typedef Lattice<vMatrixF> LatticeMatrixF;
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@ -101,462 +114,7 @@ class GaugeGroup {
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typedef Lattice<vSU2MatrixF> LatticeSU2MatrixF;
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typedef Lattice<vSU2MatrixF> LatticeSU2MatrixF;
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typedef Lattice<vSU2MatrixD> LatticeSU2MatrixD;
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typedef Lattice<vSU2MatrixD> LatticeSU2MatrixD;
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////////////////////////////////////////////////////////////////////////
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#include "Grid/qcd/utils/SUn_impl.h"
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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 = T_F delta_ab
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// T_F = 1/2 for SU(N) groups
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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
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// generator
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//
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//
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// t^a_ij = { in 0.. Nc(Nc-1)/2 -1} => 1/2(delta_{i,i1} delta_{j,i2} +
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// delta_{i,i1} delta_{j,i2})
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// t^a_ij = { in Nc(Nc-1)/2 ... Nc(Nc-1) - 1} => i/2( delta_{i,i1}
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// 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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////////////////////////////////////////////////////////////////////////
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template <class cplx>
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static void generator(int lieIndex, iSUnMatrix<cplx> &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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generatorDiagonal(diagIndex, ta);
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return;
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}
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sigxy = lieIndex & 0x1; // even or odd
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su2Index = lieIndex >> 1;
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if (sigxy)
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generatorSigmaY(su2Index, ta);
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else
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generatorSigmaX(su2Index, ta);
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}
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template <class cplx>
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static void generatorSigmaY(int su2Index, iSUnMatrix<cplx> &ta) {
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ta = Zero();
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int i1, i2;
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su2SubGroupIndex(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 cplx>
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static void generatorSigmaX(int su2Index, iSUnMatrix<cplx> &ta) {
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ta = Zero();
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cplx i(0.0, 1.0);
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int i1, i2;
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su2SubGroupIndex(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 cplx>
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static void generatorDiagonal(int diagIndex, iSUnMatrix<cplx> &ta) {
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// diag ({1, 1, ..., 1}(k-times), -k, 0, 0, ...)
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ta = Zero();
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int k = diagIndex + 1; // diagIndex starts from 0
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for (int i = 0; i <= diagIndex; i++) { // k iterations
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ta()()(i, i) = 1.0;
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}
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ta()()(k, k) = -k; // indexing starts from 0
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RealD nrm = 1.0 / std::sqrt(2.0 * k * (k + 1));
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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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//////////////////////////////////////////////////////////////////////////////////////////
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// Pull out a subgroup and project on to real coeffs x pauli basis
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//////////////////////////////////////////////////////////////////////////////////////////
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template <class vcplx>
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static void su2Extract(Lattice<iSinglet<vcplx> > &Determinant,
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Lattice<iSU2Matrix<vcplx> > &subgroup,
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const Lattice<iSUnMatrix<vcplx> > &source,
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int su2_index) {
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GridBase *grid(source.Grid());
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conformable(subgroup, source);
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conformable(subgroup, Determinant);
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int i0, i1;
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su2SubGroupIndex(i0, i1, su2_index);
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autoView(subgroup_v, subgroup, AcceleratorWrite);
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autoView(source_v, source, AcceleratorRead);
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autoView(Determinant_v, Determinant, AcceleratorWrite);
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accelerator_for(ss, grid->oSites(), 1, {
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subgroup_v[ss]()()(0, 0) = source_v[ss]()()(i0, i0);
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subgroup_v[ss]()()(0, 1) = source_v[ss]()()(i0, i1);
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subgroup_v[ss]()()(1, 0) = source_v[ss]()()(i1, i0);
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subgroup_v[ss]()()(1, 1) = source_v[ss]()()(i1, i1);
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iSU2Matrix<vcplx> Sigma = subgroup_v[ss];
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Sigma = Sigma - adj(Sigma) + trace(adj(Sigma));
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subgroup_v[ss] = Sigma;
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// this should be purely real
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Determinant_v[ss] =
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Sigma()()(0, 0) * Sigma()()(1, 1) - Sigma()()(0, 1) * Sigma()()(1, 0);
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});
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}
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//////////////////////////////////////////////////////////////////////////////////////////
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// Set matrix to one and insert a pauli subgroup
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//////////////////////////////////////////////////////////////////////////////////////////
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template <class vcplx>
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static void su2Insert(const Lattice<iSU2Matrix<vcplx> > &subgroup,
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Lattice<iSUnMatrix<vcplx> > &dest, int su2_index) {
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GridBase *grid(dest.Grid());
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conformable(subgroup, dest);
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int i0, i1;
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su2SubGroupIndex(i0, i1, su2_index);
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dest = 1.0; // start out with identity
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autoView(dest_v, dest, AcceleratorWrite);
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autoView(subgroup_v, subgroup, AcceleratorRead);
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accelerator_for(ss, grid->oSites(), 1, {
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dest_v[ss]()()(i0, i0) = subgroup_v[ss]()()(0, 0);
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dest_v[ss]()()(i0, i1) = subgroup_v[ss]()()(0, 1);
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dest_v[ss]()()(i1, i0) = subgroup_v[ss]()()(1, 0);
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dest_v[ss]()()(i1, i1) = subgroup_v[ss]()()(1, 1);
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});
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}
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///////////////////////////////////////////////
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// Generate e^{ Re Tr Staple Link} dlink
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//
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// *** Note Staple should be appropriate linear compbination between all
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// staples.
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// *** If already by beta pass coefficient 1.0.
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// *** This routine applies the additional 1/Nc factor that comes after trace
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// in action.
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//
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///////////////////////////////////////////////
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static void SubGroupHeatBath(
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GridSerialRNG &sRNG, GridParallelRNG &pRNG,
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RealD beta, // coeff multiplying staple in action (with no 1/Nc)
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LatticeMatrix &link,
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const LatticeMatrix &barestaple, // multiplied by action coeffs so th
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int su2_subgroup, int nheatbath, LatticeInteger &wheremask) {
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GridBase *grid = link.Grid();
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const RealD twopi = 2.0 * M_PI;
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LatticeMatrix staple(grid);
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staple = barestaple * (beta / ncolour);
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LatticeMatrix V(grid);
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V = link * staple;
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// Subgroup manipulation in the lie algebra space
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LatticeSU2Matrix u(
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grid); // Kennedy pendleton "u" real projected normalised Sigma
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LatticeSU2Matrix uinv(grid);
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LatticeSU2Matrix ua(grid); // a in pauli form
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LatticeSU2Matrix b(grid); // rotated matrix after hb
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// Some handy constant fields
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LatticeComplex ones(grid);
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ones = 1.0;
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LatticeComplex zeros(grid);
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zeros = Zero();
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LatticeReal rones(grid);
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rones = 1.0;
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LatticeReal rzeros(grid);
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rzeros = Zero();
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LatticeComplex udet(grid); // determinant of real(staple)
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LatticeInteger mask_true(grid);
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mask_true = 1;
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LatticeInteger mask_false(grid);
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mask_false = 0;
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/*
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PLB 156 P393 (1985) (Kennedy and Pendleton)
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Note: absorb "beta" into the def of sigma compared to KP paper; staple
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passed to this routine has "beta" already multiplied in
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Action linear in links h and of form:
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beta S = beta Sum_p (1 - 1/Nc Re Tr Plaq )
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Writing Sigma = 1/Nc (beta Sigma') where sum over staples is "Sigma' "
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beta S = const - beta/Nc Re Tr h Sigma'
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= const - Re Tr h Sigma
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Decompose h and Sigma into (1, sigma_j) ; h_i real, h^2=1, Sigma_i complex
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arbitrary.
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Tr h Sigma = h_i Sigma_j Tr (sigma_i sigma_j) = h_i Sigma_j 2 delta_ij
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Re Tr h Sigma = 2 h_j Re Sigma_j
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Normalised re Sigma_j = xi u_j
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With u_j a unit vector and U can be in SU(2);
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Re Tr h Sigma = 2 h_j Re Sigma_j = 2 xi (h.u)
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4xi^2 = Det [ Sig - Sig^dag + 1 Tr Sigdag]
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u = 1/2xi [ Sig - Sig^dag + 1 Tr Sigdag]
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xi = sqrt(Det)/2;
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Write a= u h in SU(2); a has pauli decomp a_j;
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Note: Product b' xi is unvariant because scaling Sigma leaves
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normalised vector "u" fixed; Can rescale Sigma so b' = 1.
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*/
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////////////////////////////////////////////////////////
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// Real part of Pauli decomposition
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// Note a subgroup can project to zero in cold start
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////////////////////////////////////////////////////////
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su2Extract(udet, u, V, su2_subgroup);
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//////////////////////////////////////////////////////
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// Normalising this vector if possible; else identity
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|
||||||
//////////////////////////////////////////////////////
|
|
||||||
LatticeComplex xi(grid);
|
|
||||||
|
|
||||||
LatticeSU2Matrix lident(grid);
|
|
||||||
|
|
||||||
SU2Matrix ident = Complex(1.0);
|
|
||||||
SU2Matrix pauli1;
|
|
||||||
SU<2>::generator(0, pauli1);
|
|
||||||
SU2Matrix pauli2;
|
|
||||||
SU<2>::generator(1, pauli2);
|
|
||||||
SU2Matrix pauli3;
|
|
||||||
SU<2>::generator(2, pauli3);
|
|
||||||
pauli1 = timesI(pauli1) * 2.0;
|
|
||||||
pauli2 = timesI(pauli2) * 2.0;
|
|
||||||
pauli3 = timesI(pauli3) * 2.0;
|
|
||||||
|
|
||||||
LatticeComplex cone(grid);
|
|
||||||
LatticeReal adet(grid);
|
|
||||||
adet = abs(toReal(udet));
|
|
||||||
lident = Complex(1.0);
|
|
||||||
cone = Complex(1.0);
|
|
||||||
Real machine_epsilon = 1.0e-7;
|
|
||||||
u = where(adet > machine_epsilon, u, lident);
|
|
||||||
udet = where(adet > machine_epsilon, udet, cone);
|
|
||||||
|
|
||||||
xi = 0.5 * sqrt(udet); // 4xi^2 = Det [ Sig - Sig^dag + 1 Tr Sigdag]
|
|
||||||
u = 0.5 * u *
|
|
||||||
pow(xi, -1.0); // u = 1/2xi [ Sig - Sig^dag + 1 Tr Sigdag]
|
|
||||||
|
|
||||||
// Debug test for sanity
|
|
||||||
uinv = adj(u);
|
|
||||||
b = u * uinv - 1.0;
|
|
||||||
assert(norm2(b) < 1.0e-4);
|
|
||||||
|
|
||||||
/*
|
|
||||||
Measure: Haar measure dh has d^4a delta(1-|a^2|)
|
|
||||||
In polars:
|
|
||||||
da = da0 r^2 sin theta dr dtheta dphi delta( 1 - r^2 -a0^2)
|
|
||||||
= da0 r^2 sin theta dr dtheta dphi delta( (sqrt(1-a0^) - r)(sqrt(1-a0^) +
|
|
||||||
r) )
|
|
||||||
= da0 r/2 sin theta dr dtheta dphi delta( (sqrt(1-a0^) - r) )
|
|
||||||
|
|
||||||
Action factor Q(h) dh = e^-S[h] dh = e^{ xi Tr uh} dh // beta
|
|
||||||
enters through xi = e^{2 xi (h.u)} dh = e^{2 xi h0u0}.e^{2 xi h1u1}.e^{2
|
|
||||||
xi h2u2}.e^{2 xi h3u3} dh
|
|
||||||
|
|
||||||
Therefore for each site, take xi for that site
|
|
||||||
i) generate |a0|<1 with dist
|
|
||||||
(1-a0^2)^0.5 e^{2 xi a0 } da0
|
|
||||||
|
|
||||||
Take alpha = 2 xi = 2 xi [ recall 2 beta/Nc unmod staple norm];
|
|
||||||
hence 2.0/Nc factor in Chroma ] A. Generate two uniformly distributed
|
|
||||||
pseudo-random numbers R and R', R'', R''' in the unit interval; B. Set X =
|
|
||||||
-(ln R)/alpha, X' =-(ln R')/alpha; C. Set C = cos^2(2pi R"), with R"
|
|
||||||
another uniform random number in [0,1] ; D. Set A = XC; E. Let d = X'+A;
|
|
||||||
F. If R'''^2 :> 1 - 0.5 d, go back to A;
|
|
||||||
G. Set a0 = 1 - d;
|
|
||||||
|
|
||||||
Note that in step D setting B ~ X - A and using B in place of A in step E
|
|
||||||
will generate a second independent a 0 value.
|
|
||||||
*/
|
|
||||||
|
|
||||||
/////////////////////////////////////////////////////////
|
|
||||||
// count the number of sites by picking "1"'s out of hat
|
|
||||||
/////////////////////////////////////////////////////////
|
|
||||||
Integer hit = 0;
|
|
||||||
LatticeReal rtmp(grid);
|
|
||||||
rtmp = where(wheremask, rones, rzeros);
|
|
||||||
RealD numSites = sum(rtmp);
|
|
||||||
RealD numAccepted;
|
|
||||||
LatticeInteger Accepted(grid);
|
|
||||||
Accepted = Zero();
|
|
||||||
LatticeInteger newlyAccepted(grid);
|
|
||||||
|
|
||||||
std::vector<LatticeReal> xr(4, grid);
|
|
||||||
std::vector<LatticeReal> a(4, grid);
|
|
||||||
LatticeReal d(grid);
|
|
||||||
d = Zero();
|
|
||||||
LatticeReal alpha(grid);
|
|
||||||
|
|
||||||
// std::cout<<GridLogMessage<<"xi "<<xi <<std::endl;
|
|
||||||
xi = 2.0 * xi;
|
|
||||||
alpha = toReal(xi);
|
|
||||||
|
|
||||||
do {
|
|
||||||
// A. Generate two uniformly distributed pseudo-random numbers R and R',
|
|
||||||
// R'', R''' in the unit interval;
|
|
||||||
random(pRNG, xr[0]);
|
|
||||||
random(pRNG, xr[1]);
|
|
||||||
random(pRNG, xr[2]);
|
|
||||||
random(pRNG, xr[3]);
|
|
||||||
|
|
||||||
// B. Set X = - ln R/alpha, X' = -ln R'/alpha
|
|
||||||
xr[1] = -log(xr[1]) / alpha;
|
|
||||||
xr[2] = -log(xr[2]) / alpha;
|
|
||||||
|
|
||||||
// C. Set C = cos^2(2piR'')
|
|
||||||
xr[3] = cos(xr[3] * twopi);
|
|
||||||
xr[3] = xr[3] * xr[3];
|
|
||||||
|
|
||||||
LatticeReal xrsq(grid);
|
|
||||||
|
|
||||||
// D. Set A = XC;
|
|
||||||
// E. Let d = X'+A;
|
|
||||||
xrsq = xr[2] + xr[1] * xr[3];
|
|
||||||
|
|
||||||
d = where(Accepted, d, xr[2] + xr[1] * xr[3]);
|
|
||||||
|
|
||||||
// F. If R'''^2 :> 1 - 0.5 d, go back to A;
|
|
||||||
LatticeReal thresh(grid);
|
|
||||||
thresh = 1.0 - d * 0.5;
|
|
||||||
xrsq = xr[0] * xr[0];
|
|
||||||
LatticeInteger ione(grid);
|
|
||||||
ione = 1;
|
|
||||||
LatticeInteger izero(grid);
|
|
||||||
izero = Zero();
|
|
||||||
|
|
||||||
newlyAccepted = where(xrsq < thresh, ione, izero);
|
|
||||||
Accepted = where(newlyAccepted, newlyAccepted, Accepted);
|
|
||||||
Accepted = where(wheremask, Accepted, izero);
|
|
||||||
|
|
||||||
// FIXME need an iSum for integer to avoid overload on return type??
|
|
||||||
rtmp = where(Accepted, rones, rzeros);
|
|
||||||
numAccepted = sum(rtmp);
|
|
||||||
|
|
||||||
hit++;
|
|
||||||
|
|
||||||
} while ((numAccepted < numSites) && (hit < nheatbath));
|
|
||||||
|
|
||||||
// G. Set a0 = 1 - d;
|
|
||||||
a[0] = Zero();
|
|
||||||
a[0] = where(wheremask, 1.0 - d, a[0]);
|
|
||||||
|
|
||||||
//////////////////////////////////////////
|
|
||||||
// ii) generate a_i uniform on two sphere radius (1-a0^2)^0.5
|
|
||||||
//////////////////////////////////////////
|
|
||||||
|
|
||||||
LatticeReal a123mag(grid);
|
|
||||||
a123mag = sqrt(abs(1.0 - a[0] * a[0]));
|
|
||||||
|
|
||||||
LatticeReal cos_theta(grid);
|
|
||||||
LatticeReal sin_theta(grid);
|
|
||||||
LatticeReal phi(grid);
|
|
||||||
|
|
||||||
random(pRNG, phi);
|
|
||||||
phi = phi * twopi; // uniform in [0,2pi]
|
|
||||||
random(pRNG, cos_theta);
|
|
||||||
cos_theta = (cos_theta * 2.0) - 1.0; // uniform in [-1,1]
|
|
||||||
sin_theta = sqrt(abs(1.0 - cos_theta * cos_theta));
|
|
||||||
|
|
||||||
a[1] = a123mag * sin_theta * cos(phi);
|
|
||||||
a[2] = a123mag * sin_theta * sin(phi);
|
|
||||||
a[3] = a123mag * cos_theta;
|
|
||||||
|
|
||||||
ua = toComplex(a[0]) * ident + toComplex(a[1]) * pauli1 +
|
|
||||||
toComplex(a[2]) * pauli2 + toComplex(a[3]) * pauli3;
|
|
||||||
|
|
||||||
b = 1.0;
|
|
||||||
b = where(wheremask, uinv * ua, b);
|
|
||||||
su2Insert(b, V, su2_subgroup);
|
|
||||||
|
|
||||||
// mask the assignment back based on Accptance
|
|
||||||
link = where(Accepted, V * link, link);
|
|
||||||
|
|
||||||
//////////////////////////////
|
|
||||||
// Debug Checks
|
|
||||||
// SU2 check
|
|
||||||
LatticeSU2Matrix check(grid); // rotated matrix after hb
|
|
||||||
u = Zero();
|
|
||||||
check = ua * adj(ua) - 1.0;
|
|
||||||
check = where(Accepted, check, u);
|
|
||||||
assert(norm2(check) < 1.0e-4);
|
|
||||||
|
|
||||||
check = b * adj(b) - 1.0;
|
|
||||||
check = where(Accepted, check, u);
|
|
||||||
assert(norm2(check) < 1.0e-4);
|
|
||||||
|
|
||||||
LatticeMatrix Vcheck(grid);
|
|
||||||
Vcheck = Zero();
|
|
||||||
Vcheck = where(Accepted, V * adj(V) - 1.0, Vcheck);
|
|
||||||
// std::cout<<GridLogMessage << "SU3 check " <<norm2(Vcheck)<<std::endl;
|
|
||||||
assert(norm2(Vcheck) < 1.0e-4);
|
|
||||||
|
|
||||||
// Verify the link stays in SU(3)
|
|
||||||
// std::cout<<GridLogMessage <<"Checking the modified link"<<std::endl;
|
|
||||||
Vcheck = link * adj(link) - 1.0;
|
|
||||||
assert(norm2(Vcheck) < 1.0e-4);
|
|
||||||
/////////////////////////////////
|
|
||||||
}
|
|
||||||
|
|
||||||
static void printGenerators(void) {
|
static void printGenerators(void) {
|
||||||
for (int gen = 0; gen < AdjointDimension; gen++) {
|
for (int gen = 0; gen < AdjointDimension; gen++) {
|
||||||
@ -568,44 +126,6 @@ class GaugeGroup {
|
|||||||
}
|
}
|
||||||
}
|
}
|
||||||
|
|
||||||
static void testGenerators(void) {
|
|
||||||
Matrix ta;
|
|
||||||
Matrix tb;
|
|
||||||
std::cout << GridLogMessage
|
|
||||||
<< "Fundamental - Checking trace ta tb is 0.5 delta_ab"
|
|
||||||
<< std::endl;
|
|
||||||
for (int a = 0; a < AdjointDimension; a++) {
|
|
||||||
for (int b = 0; b < AdjointDimension; b++) {
|
|
||||||
generator(a, ta);
|
|
||||||
generator(b, tb);
|
|
||||||
Complex tr = TensorRemove(trace(ta * tb));
|
|
||||||
std::cout << GridLogMessage << "(" << a << "," << b << ") = " << tr
|
|
||||||
<< std::endl;
|
|
||||||
if (a == b) assert(abs(tr - Complex(0.5)) < 1.0e-6);
|
|
||||||
if (a != b) assert(abs(tr) < 1.0e-6);
|
|
||||||
}
|
|
||||||
std::cout << GridLogMessage << std::endl;
|
|
||||||
}
|
|
||||||
std::cout << GridLogMessage << "Fundamental - Checking if hermitian"
|
|
||||||
<< std::endl;
|
|
||||||
for (int a = 0; a < AdjointDimension; a++) {
|
|
||||||
generator(a, ta);
|
|
||||||
std::cout << GridLogMessage << a << std::endl;
|
|
||||||
assert(norm2(ta - adj(ta)) < 1.0e-6);
|
|
||||||
}
|
|
||||||
std::cout << GridLogMessage << std::endl;
|
|
||||||
|
|
||||||
std::cout << GridLogMessage << "Fundamental - Checking if traceless"
|
|
||||||
<< std::endl;
|
|
||||||
for (int a = 0; a < AdjointDimension; a++) {
|
|
||||||
generator(a, ta);
|
|
||||||
Complex tr = TensorRemove(trace(ta));
|
|
||||||
std::cout << GridLogMessage << a << " " << std::endl;
|
|
||||||
assert(abs(tr) < 1.0e-6);
|
|
||||||
}
|
|
||||||
std::cout << GridLogMessage << std::endl;
|
|
||||||
}
|
|
||||||
|
|
||||||
// reunitarise??
|
// reunitarise??
|
||||||
template <typename LatticeMatrixType>
|
template <typename LatticeMatrixType>
|
||||||
static void LieRandomize(GridParallelRNG &pRNG, LatticeMatrixType &out,
|
static void LieRandomize(GridParallelRNG &pRNG, LatticeMatrixType &out,
|
||||||
@ -679,49 +199,6 @@ class GaugeGroup {
|
|||||||
out += la;
|
out += la;
|
||||||
}
|
}
|
||||||
}
|
}
|
||||||
/*
|
|
||||||
* Fundamental rep gauge xform
|
|
||||||
*/
|
|
||||||
template <typename Fundamental, typename GaugeMat>
|
|
||||||
static void GaugeTransformFundamental(Fundamental &ferm, GaugeMat &g) {
|
|
||||||
GridBase *grid = ferm._grid;
|
|
||||||
conformable(grid, g._grid);
|
|
||||||
ferm = g * ferm;
|
|
||||||
}
|
|
||||||
/*
|
|
||||||
* Adjoint rep gauge xform
|
|
||||||
*/
|
|
||||||
|
|
||||||
template <typename GaugeField, typename GaugeMat>
|
|
||||||
static void GaugeTransform(GaugeField &Umu, GaugeMat &g) {
|
|
||||||
GridBase *grid = Umu.Grid();
|
|
||||||
conformable(grid, g.Grid());
|
|
||||||
|
|
||||||
GaugeMat U(grid);
|
|
||||||
GaugeMat ag(grid);
|
|
||||||
ag = adj(g);
|
|
||||||
|
|
||||||
for (int mu = 0; mu < Nd; mu++) {
|
|
||||||
U = PeekIndex<LorentzIndex>(Umu, mu);
|
|
||||||
U = g * U * Cshift(ag, mu, 1);
|
|
||||||
PokeIndex<LorentzIndex>(Umu, U, mu);
|
|
||||||
}
|
|
||||||
}
|
|
||||||
template <typename GaugeMat>
|
|
||||||
static void GaugeTransform(std::vector<GaugeMat> &U, GaugeMat &g) {
|
|
||||||
GridBase *grid = g.Grid();
|
|
||||||
GaugeMat ag(grid);
|
|
||||||
ag = adj(g);
|
|
||||||
for (int mu = 0; mu < Nd; mu++) {
|
|
||||||
U[mu] = g * U[mu] * Cshift(ag, mu, 1);
|
|
||||||
}
|
|
||||||
}
|
|
||||||
template <typename GaugeField, typename GaugeMat>
|
|
||||||
static void RandomGaugeTransform(GridParallelRNG &pRNG, GaugeField &Umu,
|
|
||||||
GaugeMat &g) {
|
|
||||||
LieRandomize(pRNG, g, 1.0);
|
|
||||||
GaugeTransform(Umu, g);
|
|
||||||
}
|
|
||||||
|
|
||||||
// Projects the algebra components a lattice matrix (of dimension ncol*ncol -1
|
// Projects the algebra components a lattice matrix (of dimension ncol*ncol -1
|
||||||
// ) inverse operation: FundamentalLieAlgebraMatrix
|
// ) inverse operation: FundamentalLieAlgebraMatrix
|
||||||
@ -740,7 +217,7 @@ class GaugeGroup {
|
|||||||
template <typename GaugeField>
|
template <typename GaugeField>
|
||||||
static void HotConfiguration(GridParallelRNG &pRNG, GaugeField &out) {
|
static void HotConfiguration(GridParallelRNG &pRNG, GaugeField &out) {
|
||||||
typedef typename GaugeField::vector_type vector_type;
|
typedef typename GaugeField::vector_type vector_type;
|
||||||
typedef iSUnMatrix<vector_type> vMatrixType;
|
typedef iGroupMatrix<vector_type> vMatrixType;
|
||||||
typedef Lattice<vMatrixType> LatticeMatrixType;
|
typedef Lattice<vMatrixType> LatticeMatrixType;
|
||||||
|
|
||||||
LatticeMatrixType Umu(out.Grid());
|
LatticeMatrixType Umu(out.Grid());
|
||||||
@ -752,7 +229,7 @@ class GaugeGroup {
|
|||||||
template <typename GaugeField>
|
template <typename GaugeField>
|
||||||
static void TepidConfiguration(GridParallelRNG &pRNG, GaugeField &out) {
|
static void TepidConfiguration(GridParallelRNG &pRNG, GaugeField &out) {
|
||||||
typedef typename GaugeField::vector_type vector_type;
|
typedef typename GaugeField::vector_type vector_type;
|
||||||
typedef iSUnMatrix<vector_type> vMatrixType;
|
typedef iGroupMatrix<vector_type> vMatrixType;
|
||||||
typedef Lattice<vMatrixType> LatticeMatrixType;
|
typedef Lattice<vMatrixType> LatticeMatrixType;
|
||||||
|
|
||||||
LatticeMatrixType Umu(out.Grid());
|
LatticeMatrixType Umu(out.Grid());
|
||||||
@ -764,7 +241,7 @@ class GaugeGroup {
|
|||||||
template <typename GaugeField>
|
template <typename GaugeField>
|
||||||
static void ColdConfiguration(GaugeField &out) {
|
static void ColdConfiguration(GaugeField &out) {
|
||||||
typedef typename GaugeField::vector_type vector_type;
|
typedef typename GaugeField::vector_type vector_type;
|
||||||
typedef iSUnMatrix<vector_type> vMatrixType;
|
typedef iGroupMatrix<vector_type> vMatrixType;
|
||||||
typedef Lattice<vMatrixType> LatticeMatrixType;
|
typedef Lattice<vMatrixType> LatticeMatrixType;
|
||||||
|
|
||||||
LatticeMatrixType Umu(out.Grid());
|
LatticeMatrixType Umu(out.Grid());
|
||||||
@ -778,7 +255,7 @@ class GaugeGroup {
|
|||||||
ColdConfiguration(out);
|
ColdConfiguration(out);
|
||||||
}
|
}
|
||||||
|
|
||||||
template <typename LatticeMatrixType>
|
template <typename LatticeMatrixType, ONLY_IF_SU>
|
||||||
static void taProj(const LatticeMatrixType &in, LatticeMatrixType &out) {
|
static void taProj(const LatticeMatrixType &in, LatticeMatrixType &out) {
|
||||||
out = Ta(in);
|
out = Ta(in);
|
||||||
}
|
}
|
||||||
|
|||||||
552
Grid/qcd/utils/SUn_impl.h
Normal file
552
Grid/qcd/utils/SUn_impl.h
Normal file
@ -0,0 +1,552 @@
|
|||||||
|
// This file is #included into the body of the class template definition of
|
||||||
|
// GaugeGroup. So, image there to be
|
||||||
|
//
|
||||||
|
// template <int ncolour, class group_name>
|
||||||
|
// class GaugeGroup {
|
||||||
|
//
|
||||||
|
// around it.
|
||||||
|
|
||||||
|
template <typename vtype, ONLY_IF_SU>
|
||||||
|
using iSUnMatrix = iGroupMatrix<vtype>;
|
||||||
|
template <typename vtype, ONLY_IF_SU>
|
||||||
|
using iSUnAlgebraVector = iAlgebraVector<vtype>;
|
||||||
|
////////////////////////////////////////////////////////////////////////
|
||||||
|
// There are N^2-1 generators for SU(N).
|
||||||
|
//
|
||||||
|
// We take a traceless hermitian generator basis as follows
|
||||||
|
//
|
||||||
|
// * Normalisation: trace ta tb = 1/2 delta_ab = T_F delta_ab
|
||||||
|
// T_F = 1/2 for SU(N) groups
|
||||||
|
//
|
||||||
|
// * Off diagonal
|
||||||
|
// - pairs of rows i1,i2 behaving like pauli matrices signma_x, sigma_y
|
||||||
|
//
|
||||||
|
// - there are (Nc-1-i1) slots for i2 on each row [ x 0 x ]
|
||||||
|
// direct count off each row
|
||||||
|
//
|
||||||
|
// - Sum of all pairs is Nc(Nc-1)/2: proof arithmetic series
|
||||||
|
//
|
||||||
|
// (Nc-1) + (Nc-2)+... 1 ==> Nc*(Nc-1)/2
|
||||||
|
// 1+ 2+ + + Nc-1
|
||||||
|
//
|
||||||
|
// - There are 2 x Nc (Nc-1)/ 2 of these = Nc^2 - Nc
|
||||||
|
//
|
||||||
|
// - We enumerate the row-col pairs.
|
||||||
|
// - for each row col pair there is a (sigma_x) and a (sigma_y) like
|
||||||
|
// generator
|
||||||
|
//
|
||||||
|
//
|
||||||
|
// t^a_ij = { in 0.. Nc(Nc-1)/2 -1} => 1/2(delta_{i,i1} delta_{j,i2} +
|
||||||
|
// delta_{i,i1} delta_{j,i2})
|
||||||
|
// t^a_ij = { in Nc(Nc-1)/2 ... Nc(Nc-1) - 1} => i/2( delta_{i,i1}
|
||||||
|
// delta_{j,i2} - i delta_{i,i1} delta_{j,i2})
|
||||||
|
//
|
||||||
|
// * Diagonal; must be traceless and normalised
|
||||||
|
// - Sequence is
|
||||||
|
// N (1,-1,0,0...)
|
||||||
|
// N (1, 1,-2,0...)
|
||||||
|
// N (1, 1, 1,-3,0...)
|
||||||
|
// N (1, 1, 1, 1,-4,0...)
|
||||||
|
//
|
||||||
|
// where 1/2 = N^2 (1+.. m^2)etc.... for the m-th diagonal generator
|
||||||
|
// NB this gives the famous SU3 result for su2 index 8
|
||||||
|
//
|
||||||
|
// N= sqrt(1/2 . 1/6 ) = 1/2 . 1/sqrt(3)
|
||||||
|
//
|
||||||
|
// ( 1 )
|
||||||
|
// ( 1 ) / sqrt(3) /2 = 1/2 lambda_8
|
||||||
|
// ( -2)
|
||||||
|
//
|
||||||
|
////////////////////////////////////////////////////////////////////////
|
||||||
|
template <class cplx, ONLY_IF_SU>
|
||||||
|
static void generator(int lieIndex, iSUnMatrix<cplx> &ta) {
|
||||||
|
// map lie index to which type of generator
|
||||||
|
int diagIndex;
|
||||||
|
int su2Index;
|
||||||
|
int sigxy;
|
||||||
|
int NNm1 = ncolour * (ncolour - 1);
|
||||||
|
if (lieIndex >= NNm1) {
|
||||||
|
diagIndex = lieIndex - NNm1;
|
||||||
|
generatorDiagonal(diagIndex, ta);
|
||||||
|
return;
|
||||||
|
}
|
||||||
|
sigxy = lieIndex & 0x1; // even or odd
|
||||||
|
su2Index = lieIndex >> 1;
|
||||||
|
if (sigxy)
|
||||||
|
generatorSigmaY(su2Index, ta);
|
||||||
|
else
|
||||||
|
generatorSigmaX(su2Index, ta);
|
||||||
|
}
|
||||||
|
|
||||||
|
template <class cplx, ONLY_IF_SU>
|
||||||
|
static void generatorSigmaY(int su2Index, iSUnMatrix<cplx> &ta) {
|
||||||
|
ta = Zero();
|
||||||
|
int i1, i2;
|
||||||
|
su2SubGroupIndex(i1, i2, su2Index);
|
||||||
|
ta()()(i1, i2) = 1.0;
|
||||||
|
ta()()(i2, i1) = 1.0;
|
||||||
|
ta = ta * 0.5;
|
||||||
|
}
|
||||||
|
|
||||||
|
template <class cplx, ONLY_IF_SU>
|
||||||
|
static void generatorSigmaX(int su2Index, iSUnMatrix<cplx> &ta) {
|
||||||
|
ta = Zero();
|
||||||
|
cplx i(0.0, 1.0);
|
||||||
|
int i1, i2;
|
||||||
|
su2SubGroupIndex(i1, i2, su2Index);
|
||||||
|
ta()()(i1, i2) = i;
|
||||||
|
ta()()(i2, i1) = -i;
|
||||||
|
ta = ta * 0.5;
|
||||||
|
}
|
||||||
|
|
||||||
|
template <class cplx, ONLY_IF_SU>
|
||||||
|
static void generatorDiagonal(int diagIndex, iSUnMatrix<cplx> &ta) {
|
||||||
|
// diag ({1, 1, ..., 1}(k-times), -k, 0, 0, ...)
|
||||||
|
ta = Zero();
|
||||||
|
int k = diagIndex + 1; // diagIndex starts from 0
|
||||||
|
for (int i = 0; i <= diagIndex; i++) { // k iterations
|
||||||
|
ta()()(i, i) = 1.0;
|
||||||
|
}
|
||||||
|
ta()()(k, k) = -k; // indexing starts from 0
|
||||||
|
RealD nrm = 1.0 / std::sqrt(2.0 * k * (k + 1));
|
||||||
|
ta = ta * nrm;
|
||||||
|
}
|
||||||
|
|
||||||
|
////////////////////////////////////////////////////////////////////////
|
||||||
|
// Map a su2 subgroup number to the pair of rows that are non zero
|
||||||
|
////////////////////////////////////////////////////////////////////////
|
||||||
|
template <ONLY_IF_SU>
|
||||||
|
static void su2SubGroupIndex(int &i1, int &i2, int su2_index) {
|
||||||
|
assert((su2_index >= 0) && (su2_index < (ncolour * (ncolour - 1)) / 2));
|
||||||
|
|
||||||
|
int spare = su2_index;
|
||||||
|
for (i1 = 0; spare >= (ncolour - 1 - i1); i1++) {
|
||||||
|
spare = spare - (ncolour - 1 - i1); // remove the Nc-1-i1 terms
|
||||||
|
}
|
||||||
|
i2 = i1 + 1 + spare;
|
||||||
|
}
|
||||||
|
|
||||||
|
//////////////////////////////////////////////////////////////////////////////////////////
|
||||||
|
// Pull out a subgroup and project on to real coeffs x pauli basis
|
||||||
|
//////////////////////////////////////////////////////////////////////////////////////////
|
||||||
|
template <class vcplx, ONLY_IF_SU>
|
||||||
|
static void su2Extract(Lattice<iSinglet<vcplx> > &Determinant,
|
||||||
|
Lattice<iSU2Matrix<vcplx> > &subgroup,
|
||||||
|
const Lattice<iSUnMatrix<vcplx> > &source,
|
||||||
|
int su2_index) {
|
||||||
|
GridBase *grid(source.Grid());
|
||||||
|
conformable(subgroup, source);
|
||||||
|
conformable(subgroup, Determinant);
|
||||||
|
int i0, i1;
|
||||||
|
su2SubGroupIndex(i0, i1, su2_index);
|
||||||
|
|
||||||
|
autoView(subgroup_v, subgroup, AcceleratorWrite);
|
||||||
|
autoView(source_v, source, AcceleratorRead);
|
||||||
|
autoView(Determinant_v, Determinant, AcceleratorWrite);
|
||||||
|
accelerator_for(ss, grid->oSites(), 1, {
|
||||||
|
subgroup_v[ss]()()(0, 0) = source_v[ss]()()(i0, i0);
|
||||||
|
subgroup_v[ss]()()(0, 1) = source_v[ss]()()(i0, i1);
|
||||||
|
subgroup_v[ss]()()(1, 0) = source_v[ss]()()(i1, i0);
|
||||||
|
subgroup_v[ss]()()(1, 1) = source_v[ss]()()(i1, i1);
|
||||||
|
|
||||||
|
iSU2Matrix<vcplx> Sigma = subgroup_v[ss];
|
||||||
|
|
||||||
|
Sigma = Sigma - adj(Sigma) + trace(adj(Sigma));
|
||||||
|
|
||||||
|
subgroup_v[ss] = Sigma;
|
||||||
|
|
||||||
|
// this should be purely real
|
||||||
|
Determinant_v[ss] =
|
||||||
|
Sigma()()(0, 0) * Sigma()()(1, 1) - Sigma()()(0, 1) * Sigma()()(1, 0);
|
||||||
|
});
|
||||||
|
}
|
||||||
|
|
||||||
|
//////////////////////////////////////////////////////////////////////////////////////////
|
||||||
|
// Set matrix to one and insert a pauli subgroup
|
||||||
|
//////////////////////////////////////////////////////////////////////////////////////////
|
||||||
|
template <class vcplx, ONLY_IF_SU>
|
||||||
|
static void su2Insert(const Lattice<iSU2Matrix<vcplx> > &subgroup,
|
||||||
|
Lattice<iSUnMatrix<vcplx> > &dest, int su2_index) {
|
||||||
|
GridBase *grid(dest.Grid());
|
||||||
|
conformable(subgroup, dest);
|
||||||
|
int i0, i1;
|
||||||
|
su2SubGroupIndex(i0, i1, su2_index);
|
||||||
|
|
||||||
|
dest = 1.0; // start out with identity
|
||||||
|
autoView(dest_v, dest, AcceleratorWrite);
|
||||||
|
autoView(subgroup_v, subgroup, AcceleratorRead);
|
||||||
|
accelerator_for(ss, grid->oSites(), 1, {
|
||||||
|
dest_v[ss]()()(i0, i0) = subgroup_v[ss]()()(0, 0);
|
||||||
|
dest_v[ss]()()(i0, i1) = subgroup_v[ss]()()(0, 1);
|
||||||
|
dest_v[ss]()()(i1, i0) = subgroup_v[ss]()()(1, 0);
|
||||||
|
dest_v[ss]()()(i1, i1) = subgroup_v[ss]()()(1, 1);
|
||||||
|
});
|
||||||
|
}
|
||||||
|
|
||||||
|
///////////////////////////////////////////////
|
||||||
|
// Generate e^{ Re Tr Staple Link} dlink
|
||||||
|
//
|
||||||
|
// *** Note Staple should be appropriate linear compbination between all
|
||||||
|
// staples.
|
||||||
|
// *** If already by beta pass coefficient 1.0.
|
||||||
|
// *** This routine applies the additional 1/Nc factor that comes after trace
|
||||||
|
// in action.
|
||||||
|
//
|
||||||
|
///////////////////////////////////////////////
|
||||||
|
template <ONLY_IF_SU>
|
||||||
|
static void SubGroupHeatBath(
|
||||||
|
GridSerialRNG &sRNG, GridParallelRNG &pRNG,
|
||||||
|
RealD beta, // coeff multiplying staple in action (with no 1/Nc)
|
||||||
|
LatticeMatrix &link,
|
||||||
|
const LatticeMatrix &barestaple, // multiplied by action coeffs so th
|
||||||
|
int su2_subgroup, int nheatbath, LatticeInteger &wheremask) {
|
||||||
|
GridBase *grid = link.Grid();
|
||||||
|
|
||||||
|
const RealD twopi = 2.0 * M_PI;
|
||||||
|
|
||||||
|
LatticeMatrix staple(grid);
|
||||||
|
|
||||||
|
staple = barestaple * (beta / ncolour);
|
||||||
|
|
||||||
|
LatticeMatrix V(grid);
|
||||||
|
V = link * staple;
|
||||||
|
|
||||||
|
// Subgroup manipulation in the lie algebra space
|
||||||
|
LatticeSU2Matrix u(
|
||||||
|
grid); // Kennedy pendleton "u" real projected normalised Sigma
|
||||||
|
LatticeSU2Matrix uinv(grid);
|
||||||
|
LatticeSU2Matrix ua(grid); // a in pauli form
|
||||||
|
LatticeSU2Matrix b(grid); // rotated matrix after hb
|
||||||
|
|
||||||
|
// Some handy constant fields
|
||||||
|
LatticeComplex ones(grid);
|
||||||
|
ones = 1.0;
|
||||||
|
LatticeComplex zeros(grid);
|
||||||
|
zeros = Zero();
|
||||||
|
LatticeReal rones(grid);
|
||||||
|
rones = 1.0;
|
||||||
|
LatticeReal rzeros(grid);
|
||||||
|
rzeros = Zero();
|
||||||
|
LatticeComplex udet(grid); // determinant of real(staple)
|
||||||
|
LatticeInteger mask_true(grid);
|
||||||
|
mask_true = 1;
|
||||||
|
LatticeInteger mask_false(grid);
|
||||||
|
mask_false = 0;
|
||||||
|
|
||||||
|
/*
|
||||||
|
PLB 156 P393 (1985) (Kennedy and Pendleton)
|
||||||
|
|
||||||
|
Note: absorb "beta" into the def of sigma compared to KP paper; staple
|
||||||
|
passed to this routine has "beta" already multiplied in
|
||||||
|
|
||||||
|
Action linear in links h and of form:
|
||||||
|
|
||||||
|
beta S = beta Sum_p (1 - 1/Nc Re Tr Plaq )
|
||||||
|
|
||||||
|
Writing Sigma = 1/Nc (beta Sigma') where sum over staples is "Sigma' "
|
||||||
|
|
||||||
|
beta S = const - beta/Nc Re Tr h Sigma'
|
||||||
|
= const - Re Tr h Sigma
|
||||||
|
|
||||||
|
Decompose h and Sigma into (1, sigma_j) ; h_i real, h^2=1, Sigma_i complex
|
||||||
|
arbitrary.
|
||||||
|
|
||||||
|
Tr h Sigma = h_i Sigma_j Tr (sigma_i sigma_j) = h_i Sigma_j 2 delta_ij
|
||||||
|
Re Tr h Sigma = 2 h_j Re Sigma_j
|
||||||
|
|
||||||
|
Normalised re Sigma_j = xi u_j
|
||||||
|
|
||||||
|
With u_j a unit vector and U can be in SU(2);
|
||||||
|
|
||||||
|
Re Tr h Sigma = 2 h_j Re Sigma_j = 2 xi (h.u)
|
||||||
|
|
||||||
|
4xi^2 = Det [ Sig - Sig^dag + 1 Tr Sigdag]
|
||||||
|
u = 1/2xi [ Sig - Sig^dag + 1 Tr Sigdag]
|
||||||
|
|
||||||
|
xi = sqrt(Det)/2;
|
||||||
|
|
||||||
|
Write a= u h in SU(2); a has pauli decomp a_j;
|
||||||
|
|
||||||
|
Note: Product b' xi is unvariant because scaling Sigma leaves
|
||||||
|
normalised vector "u" fixed; Can rescale Sigma so b' = 1.
|
||||||
|
*/
|
||||||
|
|
||||||
|
////////////////////////////////////////////////////////
|
||||||
|
// Real part of Pauli decomposition
|
||||||
|
// Note a subgroup can project to zero in cold start
|
||||||
|
////////////////////////////////////////////////////////
|
||||||
|
su2Extract(udet, u, V, su2_subgroup);
|
||||||
|
|
||||||
|
//////////////////////////////////////////////////////
|
||||||
|
// Normalising this vector if possible; else identity
|
||||||
|
//////////////////////////////////////////////////////
|
||||||
|
LatticeComplex xi(grid);
|
||||||
|
|
||||||
|
LatticeSU2Matrix lident(grid);
|
||||||
|
|
||||||
|
SU2Matrix ident = Complex(1.0);
|
||||||
|
SU2Matrix pauli1;
|
||||||
|
SU<2>::generator(0, pauli1);
|
||||||
|
SU2Matrix pauli2;
|
||||||
|
SU<2>::generator(1, pauli2);
|
||||||
|
SU2Matrix pauli3;
|
||||||
|
SU<2>::generator(2, pauli3);
|
||||||
|
pauli1 = timesI(pauli1) * 2.0;
|
||||||
|
pauli2 = timesI(pauli2) * 2.0;
|
||||||
|
pauli3 = timesI(pauli3) * 2.0;
|
||||||
|
|
||||||
|
LatticeComplex cone(grid);
|
||||||
|
LatticeReal adet(grid);
|
||||||
|
adet = abs(toReal(udet));
|
||||||
|
lident = Complex(1.0);
|
||||||
|
cone = Complex(1.0);
|
||||||
|
Real machine_epsilon = 1.0e-7;
|
||||||
|
u = where(adet > machine_epsilon, u, lident);
|
||||||
|
udet = where(adet > machine_epsilon, udet, cone);
|
||||||
|
|
||||||
|
xi = 0.5 * sqrt(udet); // 4xi^2 = Det [ Sig - Sig^dag + 1 Tr Sigdag]
|
||||||
|
u = 0.5 * u * pow(xi, -1.0); // u = 1/2xi [ Sig - Sig^dag + 1 Tr Sigdag]
|
||||||
|
|
||||||
|
// Debug test for sanity
|
||||||
|
uinv = adj(u);
|
||||||
|
b = u * uinv - 1.0;
|
||||||
|
assert(norm2(b) < 1.0e-4);
|
||||||
|
|
||||||
|
/*
|
||||||
|
Measure: Haar measure dh has d^4a delta(1-|a^2|)
|
||||||
|
In polars:
|
||||||
|
da = da0 r^2 sin theta dr dtheta dphi delta( 1 - r^2 -a0^2)
|
||||||
|
= da0 r^2 sin theta dr dtheta dphi delta( (sqrt(1-a0^) - r)(sqrt(1-a0^) +
|
||||||
|
r) )
|
||||||
|
= da0 r/2 sin theta dr dtheta dphi delta( (sqrt(1-a0^) - r) )
|
||||||
|
|
||||||
|
Action factor Q(h) dh = e^-S[h] dh = e^{ xi Tr uh} dh // beta
|
||||||
|
enters through xi = e^{2 xi (h.u)} dh = e^{2 xi h0u0}.e^{2 xi h1u1}.e^{2
|
||||||
|
xi h2u2}.e^{2 xi h3u3} dh
|
||||||
|
|
||||||
|
Therefore for each site, take xi for that site
|
||||||
|
i) generate |a0|<1 with dist
|
||||||
|
(1-a0^2)^0.5 e^{2 xi a0 } da0
|
||||||
|
|
||||||
|
Take alpha = 2 xi = 2 xi [ recall 2 beta/Nc unmod staple norm];
|
||||||
|
hence 2.0/Nc factor in Chroma ] A. Generate two uniformly distributed
|
||||||
|
pseudo-random numbers R and R', R'', R''' in the unit interval; B. Set X =
|
||||||
|
-(ln R)/alpha, X' =-(ln R')/alpha; C. Set C = cos^2(2pi R"), with R"
|
||||||
|
another uniform random number in [0,1] ; D. Set A = XC; E. Let d = X'+A;
|
||||||
|
F. If R'''^2 :> 1 - 0.5 d, go back to A;
|
||||||
|
G. Set a0 = 1 - d;
|
||||||
|
|
||||||
|
Note that in step D setting B ~ X - A and using B in place of A in step E
|
||||||
|
will generate a second independent a 0 value.
|
||||||
|
*/
|
||||||
|
|
||||||
|
/////////////////////////////////////////////////////////
|
||||||
|
// count the number of sites by picking "1"'s out of hat
|
||||||
|
/////////////////////////////////////////////////////////
|
||||||
|
Integer hit = 0;
|
||||||
|
LatticeReal rtmp(grid);
|
||||||
|
rtmp = where(wheremask, rones, rzeros);
|
||||||
|
RealD numSites = sum(rtmp);
|
||||||
|
RealD numAccepted;
|
||||||
|
LatticeInteger Accepted(grid);
|
||||||
|
Accepted = Zero();
|
||||||
|
LatticeInteger newlyAccepted(grid);
|
||||||
|
|
||||||
|
std::vector<LatticeReal> xr(4, grid);
|
||||||
|
std::vector<LatticeReal> a(4, grid);
|
||||||
|
LatticeReal d(grid);
|
||||||
|
d = Zero();
|
||||||
|
LatticeReal alpha(grid);
|
||||||
|
|
||||||
|
// std::cout<<GridLogMessage<<"xi "<<xi <<std::endl;
|
||||||
|
xi = 2.0 * xi;
|
||||||
|
alpha = toReal(xi);
|
||||||
|
|
||||||
|
do {
|
||||||
|
// A. Generate two uniformly distributed pseudo-random numbers R and R',
|
||||||
|
// R'', R''' in the unit interval;
|
||||||
|
random(pRNG, xr[0]);
|
||||||
|
random(pRNG, xr[1]);
|
||||||
|
random(pRNG, xr[2]);
|
||||||
|
random(pRNG, xr[3]);
|
||||||
|
|
||||||
|
// B. Set X = - ln R/alpha, X' = -ln R'/alpha
|
||||||
|
xr[1] = -log(xr[1]) / alpha;
|
||||||
|
xr[2] = -log(xr[2]) / alpha;
|
||||||
|
|
||||||
|
// C. Set C = cos^2(2piR'')
|
||||||
|
xr[3] = cos(xr[3] * twopi);
|
||||||
|
xr[3] = xr[3] * xr[3];
|
||||||
|
|
||||||
|
LatticeReal xrsq(grid);
|
||||||
|
|
||||||
|
// D. Set A = XC;
|
||||||
|
// E. Let d = X'+A;
|
||||||
|
xrsq = xr[2] + xr[1] * xr[3];
|
||||||
|
|
||||||
|
d = where(Accepted, d, xr[2] + xr[1] * xr[3]);
|
||||||
|
|
||||||
|
// F. If R'''^2 :> 1 - 0.5 d, go back to A;
|
||||||
|
LatticeReal thresh(grid);
|
||||||
|
thresh = 1.0 - d * 0.5;
|
||||||
|
xrsq = xr[0] * xr[0];
|
||||||
|
LatticeInteger ione(grid);
|
||||||
|
ione = 1;
|
||||||
|
LatticeInteger izero(grid);
|
||||||
|
izero = Zero();
|
||||||
|
|
||||||
|
newlyAccepted = where(xrsq < thresh, ione, izero);
|
||||||
|
Accepted = where(newlyAccepted, newlyAccepted, Accepted);
|
||||||
|
Accepted = where(wheremask, Accepted, izero);
|
||||||
|
|
||||||
|
// FIXME need an iSum for integer to avoid overload on return type??
|
||||||
|
rtmp = where(Accepted, rones, rzeros);
|
||||||
|
numAccepted = sum(rtmp);
|
||||||
|
|
||||||
|
hit++;
|
||||||
|
|
||||||
|
} while ((numAccepted < numSites) && (hit < nheatbath));
|
||||||
|
|
||||||
|
// G. Set a0 = 1 - d;
|
||||||
|
a[0] = Zero();
|
||||||
|
a[0] = where(wheremask, 1.0 - d, a[0]);
|
||||||
|
|
||||||
|
//////////////////////////////////////////
|
||||||
|
// ii) generate a_i uniform on two sphere radius (1-a0^2)^0.5
|
||||||
|
//////////////////////////////////////////
|
||||||
|
|
||||||
|
LatticeReal a123mag(grid);
|
||||||
|
a123mag = sqrt(abs(1.0 - a[0] * a[0]));
|
||||||
|
|
||||||
|
LatticeReal cos_theta(grid);
|
||||||
|
LatticeReal sin_theta(grid);
|
||||||
|
LatticeReal phi(grid);
|
||||||
|
|
||||||
|
random(pRNG, phi);
|
||||||
|
phi = phi * twopi; // uniform in [0,2pi]
|
||||||
|
random(pRNG, cos_theta);
|
||||||
|
cos_theta = (cos_theta * 2.0) - 1.0; // uniform in [-1,1]
|
||||||
|
sin_theta = sqrt(abs(1.0 - cos_theta * cos_theta));
|
||||||
|
|
||||||
|
a[1] = a123mag * sin_theta * cos(phi);
|
||||||
|
a[2] = a123mag * sin_theta * sin(phi);
|
||||||
|
a[3] = a123mag * cos_theta;
|
||||||
|
|
||||||
|
ua = toComplex(a[0]) * ident + toComplex(a[1]) * pauli1 +
|
||||||
|
toComplex(a[2]) * pauli2 + toComplex(a[3]) * pauli3;
|
||||||
|
|
||||||
|
b = 1.0;
|
||||||
|
b = where(wheremask, uinv * ua, b);
|
||||||
|
su2Insert(b, V, su2_subgroup);
|
||||||
|
|
||||||
|
// mask the assignment back based on Accptance
|
||||||
|
link = where(Accepted, V * link, link);
|
||||||
|
|
||||||
|
//////////////////////////////
|
||||||
|
// Debug Checks
|
||||||
|
// SU2 check
|
||||||
|
LatticeSU2Matrix check(grid); // rotated matrix after hb
|
||||||
|
u = Zero();
|
||||||
|
check = ua * adj(ua) - 1.0;
|
||||||
|
check = where(Accepted, check, u);
|
||||||
|
assert(norm2(check) < 1.0e-4);
|
||||||
|
|
||||||
|
check = b * adj(b) - 1.0;
|
||||||
|
check = where(Accepted, check, u);
|
||||||
|
assert(norm2(check) < 1.0e-4);
|
||||||
|
|
||||||
|
LatticeMatrix Vcheck(grid);
|
||||||
|
Vcheck = Zero();
|
||||||
|
Vcheck = where(Accepted, V * adj(V) - 1.0, Vcheck);
|
||||||
|
// std::cout<<GridLogMessage << "SU3 check " <<norm2(Vcheck)<<std::endl;
|
||||||
|
assert(norm2(Vcheck) < 1.0e-4);
|
||||||
|
|
||||||
|
// Verify the link stays in SU(3)
|
||||||
|
// std::cout<<GridLogMessage <<"Checking the modified link"<<std::endl;
|
||||||
|
Vcheck = link * adj(link) - 1.0;
|
||||||
|
assert(norm2(Vcheck) < 1.0e-4);
|
||||||
|
/////////////////////////////////
|
||||||
|
}
|
||||||
|
|
||||||
|
template <ONLY_IF_SU>
|
||||||
|
static void testGenerators(void) {
|
||||||
|
Matrix ta;
|
||||||
|
Matrix tb;
|
||||||
|
std::cout << GridLogMessage
|
||||||
|
<< "Fundamental - Checking trace ta tb is 0.5 delta_ab"
|
||||||
|
<< std::endl;
|
||||||
|
for (int a = 0; a < AdjointDimension; a++) {
|
||||||
|
for (int b = 0; b < AdjointDimension; b++) {
|
||||||
|
generator(a, ta);
|
||||||
|
generator(b, tb);
|
||||||
|
Complex tr = TensorRemove(trace(ta * tb));
|
||||||
|
std::cout << GridLogMessage << "(" << a << "," << b << ") = " << tr
|
||||||
|
<< std::endl;
|
||||||
|
if (a == b) assert(abs(tr - Complex(0.5)) < 1.0e-6);
|
||||||
|
if (a != b) assert(abs(tr) < 1.0e-6);
|
||||||
|
}
|
||||||
|
std::cout << GridLogMessage << std::endl;
|
||||||
|
}
|
||||||
|
std::cout << GridLogMessage << "Fundamental - Checking if hermitian"
|
||||||
|
<< std::endl;
|
||||||
|
for (int a = 0; a < AdjointDimension; a++) {
|
||||||
|
generator(a, ta);
|
||||||
|
std::cout << GridLogMessage << a << std::endl;
|
||||||
|
assert(norm2(ta - adj(ta)) < 1.0e-6);
|
||||||
|
}
|
||||||
|
std::cout << GridLogMessage << std::endl;
|
||||||
|
|
||||||
|
std::cout << GridLogMessage << "Fundamental - Checking if traceless"
|
||||||
|
<< std::endl;
|
||||||
|
for (int a = 0; a < AdjointDimension; a++) {
|
||||||
|
generator(a, ta);
|
||||||
|
Complex tr = TensorRemove(trace(ta));
|
||||||
|
std::cout << GridLogMessage << a << " " << std::endl;
|
||||||
|
assert(abs(tr) < 1.0e-6);
|
||||||
|
}
|
||||||
|
std::cout << GridLogMessage << std::endl;
|
||||||
|
}
|
||||||
|
|
||||||
|
/*
|
||||||
|
* Fundamental rep gauge xform
|
||||||
|
*/
|
||||||
|
template <typename Fundamental, typename GaugeMat, ONLY_IF_SU>
|
||||||
|
static void GaugeTransformFundamental(Fundamental &ferm, GaugeMat &g) {
|
||||||
|
GridBase *grid = ferm._grid;
|
||||||
|
conformable(grid, g._grid);
|
||||||
|
ferm = g * ferm;
|
||||||
|
}
|
||||||
|
/*
|
||||||
|
* Adjoint rep gauge xform
|
||||||
|
*/
|
||||||
|
|
||||||
|
template <typename GaugeField, typename GaugeMat, ONLY_IF_SU>
|
||||||
|
static void GaugeTransform(GaugeField &Umu, GaugeMat &g) {
|
||||||
|
GridBase *grid = Umu.Grid();
|
||||||
|
conformable(grid, g.Grid());
|
||||||
|
|
||||||
|
GaugeMat U(grid);
|
||||||
|
GaugeMat ag(grid);
|
||||||
|
ag = adj(g);
|
||||||
|
|
||||||
|
for (int mu = 0; mu < Nd; mu++) {
|
||||||
|
U = PeekIndex<LorentzIndex>(Umu, mu);
|
||||||
|
U = g * U * Cshift(ag, mu, 1);
|
||||||
|
PokeIndex<LorentzIndex>(Umu, U, mu);
|
||||||
|
}
|
||||||
|
}
|
||||||
|
template <typename GaugeMat, ONLY_IF_SU>
|
||||||
|
static void GaugeTransform(std::vector<GaugeMat> &U, GaugeMat &g) {
|
||||||
|
GridBase *grid = g.Grid();
|
||||||
|
GaugeMat ag(grid);
|
||||||
|
ag = adj(g);
|
||||||
|
for (int mu = 0; mu < Nd; mu++) {
|
||||||
|
U[mu] = g * U[mu] * Cshift(ag, mu, 1);
|
||||||
|
}
|
||||||
|
}
|
||||||
|
template <typename GaugeField, typename GaugeMat, ONLY_IF_SU>
|
||||||
|
static void RandomGaugeTransform(GridParallelRNG &pRNG, GaugeField &Umu,
|
||||||
|
GaugeMat &g) {
|
||||||
|
LieRandomize(pRNG, g, 1.0);
|
||||||
|
GaugeTransform(Umu, g);
|
||||||
|
}
|
||||||
Loading…
Reference in New Issue
Block a user