energy_and_topo_charge_log.h

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/** @file energy_and_topo_charge_log.h */
 
#ifndef ENERGY_AND_TOPO_CHARGE_LOG_H_
#define ENERGY_AND_TOPO_CHARGE_LOG_H_
 
#include "hila.h"
 
template <typename group, typename atype = hila::arithmetic_type<group>>
void measure_topo_charge_and_energy_log(const GaugeField<group> &U, atype &qtopo_out,
                                        atype &energy_out) {
    // measure topological charge and field strength energy of the gauge field, using the
    // matrix logarithms of the plaquettes as components of the field strength tensor
 
    Reduction<double> qtopo = 0;
    Reduction<double> energy = 0;
    qtopo.allreduce(false).delayed(true);
    energy.allreduce(false).delayed(true);
 
#if NDIM == 4
    Field<group> F[6];
    // F[0]: F[0][1], F[1]: F[0][2], F[2]: F[0][3],
    // F[3]: F[1][2], F[4]: F[1][3], F[5]: F[2][3]
    Field<group> tF0, tF1;
 
    int k = 0;
    // (Note: by replacing log(...).expand() in the following by Lie-algebra projection,
    // one would end up computing the clover field strength)
    foralldir(dir1) foralldir(dir2) if (dir1 < dir2) {
        U[dir2].start_gather(dir1, ALL);
        U[dir1].start_gather(dir2, ALL);
 
        onsites(ALL) {
            // log of dir1-dir2-plaquette that starts and ends at X; corresponds to F[dir1][dir2]
            // at center location X+dir1/2+dir2/2 of plaquette:
            tF0[X] =
                log((U[dir1][X] * U[dir2][X + dir1] * (U[dir2][X] * U[dir1][X + dir2]).dagger()))
                    .expand();
            // parallel transport to X+dir1
            tF1[X] = U[dir1][X].dagger() * tF0[X] * U[dir1][X];
        }
 
        tF1.start_gather(-dir1, ALL);
        onsites(ALL) {
            tF0[X] += tF1[X - dir1];
        }
 
        U[dir2].start_gather(-dir2, ALL);
        tF0.start_gather(-dir2, ALL);
        onsites(ALL) {
            // get F[dir1][dir2] at X from average of the (parallel transported) F[dir1][dir2] from
            // the centers of all dir1-dir2-plaquettes that touch X :
            F[k][X] =
                (tF0[X] + U[dir2][X - dir2].dagger() * tF0[X - dir2] * U[dir2][X - dir2]) * 0.25;
        }
        ++k;
    }
    onsites(ALL) {
        qtopo += real(mul_trace(F[0][X], F[5][X]));
        qtopo += -real(mul_trace(F[1][X], F[4][X]));
        qtopo += real(mul_trace(F[2][X], F[3][X]));
 
        energy += F[0][X].squarenorm();
        energy += F[1][X].squarenorm();
        energy += F[2][X].squarenorm();
        energy += F[3][X].squarenorm();
        energy += F[4][X].squarenorm();
        energy += F[5][X].squarenorm();
    }
#endif
    qtopo_out = (atype)qtopo.value() / (4.0 * M_PI * M_PI);
    energy_out = (atype)energy.value();
}
 
 
#endif