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5028969d4b
Grid/lib/qcd/utils/SUn.h
Guido Cossu 5028969d4b Added generators for the adjoint representation
2016-07-08 15:40:11 +01:00

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/*************************************************************************************
Grid physics library, www.github.com/paboyle/Grid
Source file: ./lib/qcd/utils/SUn.h
Copyright (C) 2015
Author: Azusa Yamaguchi <ayamaguc@staffmail.ed.ac.uk>
Author: Peter Boyle <paboyle@ph.ed.ac.uk>
Author: neo <cossu@post.kek.jp>
Author: paboyle <paboyle@ph.ed.ac.uk>
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License along
with this program; if not, write to the Free Software Foundation, Inc.,
51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
See the full license in the file "LICENSE" in the top level distribution
directory
*************************************************************************************/
/* END LEGAL */
#ifndef QCD_UTIL_SUN_H
#define QCD_UTIL_SUN_H
namespace Grid {
namespace QCD {
template <int ncolour>
class SU {
public:
static int generators(void) { return ncolour * ncolour - 1; }
static int su2subgroups(void) { return (ncolour * (ncolour - 1)) / 2; }
template <typename vtype>
using iSUnMatrix = iScalar<iScalar<iMatrix<vtype, ncolour> > >;
template <typename vtype>
using iSU2Matrix = iScalar<iScalar<iMatrix<vtype, 2> > >;
template <typename vtype>
using iSUnAdjointMatrix = iScalar<iScalar<iMatrix<vtype, (ncolour*ncolour - 1)> > >;
//////////////////////////////////////////////////////////////////////////////////////////////////
// Types can be accessed as SU<2>::Matrix , SU<2>::vSUnMatrix,
// SU<2>::LatticeMatrix etc...
//////////////////////////////////////////////////////////////////////////////////////////////////
typedef iSUnMatrix<Complex> Matrix;
typedef iSUnMatrix<ComplexF> MatrixF;
typedef iSUnMatrix<ComplexD> MatrixD;
typedef iSUnMatrix<vComplex> vMatrix;
typedef iSUnMatrix<vComplexF> vMatrixF;
typedef iSUnMatrix<vComplexD> vMatrixD;
// Actually the adjoint matrices are real...
// Consider this overhead... FIXME
typedef iSUnAdjointMatrix<Complex> AMatrix;
typedef iSUnAdjointMatrix<ComplexF> AMatrixF;
typedef iSUnAdjointMatrix<ComplexD> AMatrixD;
typedef iSUnAdjointMatrix<vComplex> vAMatrix;
typedef iSUnAdjointMatrix<vComplexF> vAMatrixF;
typedef iSUnAdjointMatrix<vComplexD> vAMatrixD;
typedef Lattice<vMatrix> LatticeMatrix;
typedef Lattice<vMatrixF> LatticeMatrixF;
typedef Lattice<vMatrixD> LatticeMatrixD;
typedef iSU2Matrix<Complex> SU2Matrix;
typedef iSU2Matrix<ComplexF> SU2MatrixF;
typedef iSU2Matrix<ComplexD> SU2MatrixD;
typedef iSU2Matrix<vComplex> vSU2Matrix;
typedef iSU2Matrix<vComplexF> vSU2MatrixF;
typedef iSU2Matrix<vComplexD> vSU2MatrixD;
typedef Lattice<vSU2Matrix> LatticeSU2Matrix;
typedef Lattice<vSU2MatrixF> LatticeSU2MatrixF;
typedef Lattice<vSU2MatrixD> LatticeSU2MatrixD;
////////////////////////////////////////////////////////////////////////
// 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)
//
//
// * Adjoint representation generators
//
// base for NxN hermitian traceless matrices
// normalized to 1:
//
// (e_Adj)^a = t^a / sqrt(T_F)
//
// then the real, antisymmetric generators for the adjoint representations
// are computed ( shortcut: e^a == (e_Adj)^a )
//
// (iT_adj)^d_ab = i tr[e^a t^d e^b - t^d e^a e^b]
//
////////////////////////////////////////////////////////////////////////
template <class cplx>
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>
static void generatorSigmaX(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>
static void generatorSigmaY(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>
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;
}
template <class cplx>
static void generatorAdjoint(int Index, iSUnAdjointMatrix<cplx> &iAdjTa){
// returns i(T_Adj)^index necessary for the projectors
// see definitions above
iAdjTa = zero;
Vector< iSUnMatrix<cplx> > ta(ncolour*ncolour -1);
iSUnMatrix<cplx> tmp;
// FIXME not very efficient to get all the generators everytime
for (int a = 0; a < (ncolour * ncolour - 1); a++)
generator(a, ta[a]);
for (int a = 0; a < (ncolour*ncolour - 1); a++){
tmp = ta[a] * ta[Index] - ta[Index] * ta[a];
for (int b = 0; b < (ncolour*ncolour - 1); b++){
iSUnMatrix<cplx> tmp1 = 2.0 * tmp * ta[b]; // 2.0 from the normalization
Complex iTr = TensorRemove(timesI(trace(tmp1)));
iAdjTa()()(b,a) = iTr;
}
}
}
////////////////////////////////////////////////////////////////////////
// Map a su2 subgroup number to the pair of rows that are non zero
////////////////////////////////////////////////////////////////////////
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>
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);
PARALLEL_FOR_LOOP
for (int ss = 0; ss < grid->oSites(); ss++) {
subgroup._odata[ss]()()(0, 0) = source._odata[ss]()()(i0, i0);
subgroup._odata[ss]()()(0, 1) = source._odata[ss]()()(i0, i1);
subgroup._odata[ss]()()(1, 0) = source._odata[ss]()()(i1, i0);
subgroup._odata[ss]()()(1, 1) = source._odata[ss]()()(i1, i1);
iSU2Matrix<vcplx> Sigma = subgroup._odata[ss];
Sigma = Sigma - adj(Sigma) + trace(adj(Sigma));
subgroup._odata[ss] = Sigma;
// this should be purely real
Determinant._odata[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>
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
PARALLEL_FOR_LOOP
for (int ss = 0; ss < grid->oSites(); ss++) {
dest._odata[ss]()()(i0, i0) = subgroup._odata[ss]()()(0, 0);
dest._odata[ss]()()(i0, i1) = subgroup._odata[ss]()()(0, 1);
dest._odata[ss]()()(i1, i0) = subgroup._odata[ss]()()(1, 0);
dest._odata[ss]()()(i1, i1) = subgroup._odata[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.
//
///////////////////////////////////////////////
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;
int ntrials = 0;
int nfails = 0;
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;
alpha = toReal(2.0 * 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) {
for (int gen = 0; gen < generators(); gen++) {
Matrix ta;
generator(gen, ta);
std::cout << GridLogMessage << "Nc = " << ncolour << " t_" << gen
<< std::endl;
std::cout << GridLogMessage << ta << std::endl;
}
}
static void printAdjointGenerators(void) {
for (int gen = 0; gen < generators(); gen++) {
AMatrix ta;
generatorAdjoint(gen, ta);
std::cout << GridLogMessage << "Nc = " << ncolour << " t_" << gen
<< std::endl;
std::cout << GridLogMessage << ta << std::endl;
}
}
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 < generators(); a++) {
for (int b = 0; b < generators(); 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 < generators(); 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 < generators(); 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;
AMatrix adjTa;
std::cout << GridLogMessage << "Adjoint - Checking if real" << std::endl;
for (int a = 0; a < generators(); a++) {
generatorAdjoint(a, adjTa);
std::cout << GridLogMessage << a << std::endl;
assert(norm2(adjTa - conjugate(adjTa)) < 1.0e-6);
}
std::cout << GridLogMessage << std::endl;
std::cout << GridLogMessage << "Adjoint - Checking if antisymmetric" << std::endl;
for (int a = 0; a < generators(); a++) {
generatorAdjoint(a, adjTa);
std::cout << GridLogMessage << a << std::endl;
assert(norm2(adjTa + transpose(adjTa)) < 1.0e-6);
}
std::cout << GridLogMessage << std::endl;
}
// reunitarise??
template <typename LatticeMatrixType>
static void LieRandomize(GridParallelRNG &pRNG, LatticeMatrixType &out,
double scale = 1.0) {
GridBase *grid = out._grid;
typedef typename LatticeMatrixType::vector_type vector_type;
typedef typename LatticeMatrixType::scalar_type scalar_type;
typedef iSinglet<vector_type> vTComplexType;
typedef Lattice<vTComplexType> LatticeComplexType;
typedef typename GridTypeMapper<
typename LatticeMatrixType::vector_object>::scalar_object MatrixType;
LatticeComplexType ca(grid);
LatticeMatrixType lie(grid);
LatticeMatrixType la(grid);
ComplexD ci(0.0, scale);
ComplexD cone(1.0, 0.0);
MatrixType ta;
lie = zero;
for (int a = 0; a < generators(); a++) {
random(pRNG, ca);
ca = (ca + conjugate(ca)) * 0.5;
ca = ca - 0.5;
generator(a, ta);
la = ci * ca * ta;
lie = lie + la; // e^{i la ta}
}
taExp(lie, out);
}
static void GaussianFundamentalLieAlgebraMatrix(GridParallelRNG &pRNG,
LatticeMatrix &out,
Real scale = 1.0) {
GridBase *grid = out._grid;
LatticeReal ca(grid);
LatticeMatrix la(grid);
Complex ci(0.0, scale);
Matrix ta;
out = zero;
for (int a = 0; a < generators(); a++) {
gaussian(pRNG, ca);
generator(a, ta);
la = toComplex(ca) * ci * ta;
out += la;
}
}
static void FundamentalLieAlgebraMatrix(Vector<Real> &h, LatticeMatrix &out,
Real scale = 1.0) {
GridBase *grid = out._grid;
LatticeMatrix la(grid);
Matrix ta;
out = zero;
for (int a = 0; a < generators(); a++) {
generator(a, ta);
la = Complex(0.0, h[a]) * scale * ta;
out += la;
}
}
template <typename GaugeField>
static void HotConfiguration(GridParallelRNG &pRNG, GaugeField &out) {
typedef typename GaugeField::vector_type vector_type;
typedef iSUnMatrix<vector_type> vMatrixType;
typedef Lattice<vMatrixType> LatticeMatrixType;
LatticeMatrixType Umu(out._grid);
for (int mu = 0; mu < Nd; mu++) {
LieRandomize(pRNG, Umu, 1.0);
PokeIndex<LorentzIndex>(out, Umu, mu);
}
}
static void TepidConfiguration(GridParallelRNG &pRNG,
LatticeGaugeField &out) {
LatticeMatrix Umu(out._grid);
for (int mu = 0; mu < Nd; mu++) {
LieRandomize(pRNG, Umu, 0.01);
PokeIndex<LorentzIndex>(out, Umu, mu);
}
}
static void ColdConfiguration(GridParallelRNG &pRNG, LatticeGaugeField &out) {
LatticeMatrix Umu(out._grid);
Umu = 1.0;
for (int mu = 0; mu < Nd; mu++) {
PokeIndex<LorentzIndex>(out, Umu, mu);
}
}
static void taProj(const LatticeMatrix &in, LatticeMatrix &out) {
out = Ta(in);
}
template <typename LatticeMatrixType>
static void taExp(const LatticeMatrixType &x, LatticeMatrixType &ex) {
typedef typename LatticeMatrixType::scalar_type ComplexType;
LatticeMatrixType xn(x._grid);
RealD nfac = 1.0;
xn = x;
ex = xn + ComplexType(1.0); // 1+x
// Do a 12th order exponentiation
for (int i = 2; i <= 12; ++i) {
nfac = nfac / RealD(i); // 1/2, 1/2.3 ...
xn = xn * x; // x2, x3,x4....
ex = ex + xn * nfac; // x2/2!, x3/3!....
}
}
};
typedef SU<2> SU2;
typedef SU<3> SU3;
typedef SU<4> SU4;
typedef SU<5> SU5;
}
}
#endif
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