Grid/lib/qcd/utils/WilsonLoops.h

/*************************************************************************************

    Grid physics library, www.github.com/paboyle/Grid

    Source file: ./lib/qcd/utils/WilsonLoops.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_UTILS_WILSON_LOOPS_H
#define QCD_UTILS_WILSON_LOOPS_H

NAMESPACE_BEGIN(Grid);

// Common wilson loop observables
template <class Gimpl> class WilsonLoops : public Gimpl {
public:
  INHERIT_GIMPL_TYPES(Gimpl);

  typedef typename Gimpl::GaugeLinkField GaugeMat;
  typedef typename Gimpl::GaugeField GaugeLorentz;

  //////////////////////////////////////////////////
  // directed plaquette oriented in mu,nu plane
  //////////////////////////////////////////////////
  static void dirPlaquette(GaugeMat &plaq, const std::vector<GaugeMat> &U,
                           const int mu, const int nu) {
    // Annoyingly, must use either scope resolution to find dependent base
    // class,
    // or this-> ; there is no "this" in a static method. This forces explicit
    // Gimpl scope
    // resolution throughout the usage in this file, and rather defeats the
    // purpose of deriving
    // from Gimpl.
    /*
      plaq = Gimpl::CovShiftBackward(
      U[mu], mu, Gimpl::CovShiftBackward(
      U[nu], nu, Gimpl::CovShiftForward(U[mu], mu, U[nu])));
    */
    // _
    //|< _|
    plaq = Gimpl::CovShiftForward(U[mu],mu,
				  Gimpl::CovShiftForward(U[nu],nu,
							 Gimpl::CovShiftBackward(U[mu],mu,
										 Gimpl::CovShiftIdentityBackward(U[nu], nu))));




  }
  //////////////////////////////////////////////////
  // trace of directed plaquette oriented in mu,nu plane
  //////////////////////////////////////////////////
  static void traceDirPlaquette(ComplexField &plaq,
                                const std::vector<GaugeMat> &U, const int mu,
                                const int nu) {
    GaugeMat sp(U[0]._grid);
    dirPlaquette(sp, U, mu, nu);
    plaq = trace(sp);
  }
  //////////////////////////////////////////////////
  // sum over all planes of plaquette
  //////////////////////////////////////////////////
  static void sitePlaquette(ComplexField &Plaq,
                            const std::vector<GaugeMat> &U) {
    ComplexField sitePlaq(U[0]._grid);
    Plaq = zero;
    for (int mu = 1; mu < Nd; mu++) {
      for (int nu = 0; nu < mu; nu++) {
        traceDirPlaquette(sitePlaq, U, mu, nu);
        Plaq = Plaq + sitePlaq;
      }
    }
  }
  //////////////////////////////////////////////////
  // sum over all x,y,z,t and over all planes of plaquette
  //////////////////////////////////////////////////
  static RealD sumPlaquette(const GaugeLorentz &Umu) {
    std::vector<GaugeMat> U(Nd, Umu._grid);
    // inefficient here
    for (int mu = 0; mu < Nd; mu++) {
      U[mu] = PeekIndex<LorentzIndex>(Umu, mu);
    }

    ComplexField Plaq(Umu._grid);

    sitePlaquette(Plaq, U);
    auto Tp = sum(Plaq);
    auto p = TensorRemove(Tp);
    return p.real();
  }


  //////////////////////////////////////////////////
  // average over all x,y,z,t and over all planes of plaquette
  //////////////////////////////////////////////////
  static RealD avgPlaquette(const GaugeLorentz &Umu) {
    RealD sumplaq = sumPlaquette(Umu);
    double vol = Umu._grid->gSites();
    double faces = (1.0 * Nd * (Nd - 1)) / 2.0;
    return sumplaq / vol / faces / Nc; // Nd , Nc dependent... FIXME
  }

  //////////////////////////////////////////////////
  // average over traced single links
  //////////////////////////////////////////////////
  static RealD linkTrace(const GaugeLorentz &Umu) {
    std::vector<GaugeMat> U(Nd, Umu._grid);

    ComplexField Tr(Umu._grid);
    Tr = zero;
    for (int mu = 0; mu < Nd; mu++) {
      U[mu] = PeekIndex<LorentzIndex>(Umu, mu);
      Tr = Tr + trace(U[mu]);
    }

    auto Tp = sum(Tr);
    auto p = TensorRemove(Tp);

    double vol = Umu._grid->gSites();

    return p.real() / vol / 4.0 / 3.0;
  };

  //////////////////////////////////////////////////
  // the sum over all staples on each site in direction mu,nu
  //////////////////////////////////////////////////
  static void Staple(GaugeMat &staple, const GaugeLorentz &Umu, int mu,
                     int nu) {

    GridBase *grid = Umu._grid;

    std::vector<GaugeMat> U(Nd, grid);
    for (int d = 0; d < Nd; d++) {
      U[d] = PeekIndex<LorentzIndex>(Umu, d);
    }
    staple = zero;

    if (nu != mu) {

      // mu
      // ^
      // |__>  nu

      //    __
      //      |
      //    __|
      //

      staple += Gimpl::ShiftStaple(
				   Gimpl::CovShiftForward(
							  U[nu], nu,
							  Gimpl::CovShiftBackward(
										  U[mu], mu, Gimpl::CovShiftIdentityBackward(U[nu], nu))),
				   mu);

      //  __
      // |
      // |__
      //
      //
      staple += Gimpl::ShiftStaple(
				   Gimpl::CovShiftBackward(U[nu], nu,
							   Gimpl::CovShiftBackward(U[mu], mu, U[nu])),
				   mu);
    }
  }


  // For the force term
  static void StapleMult(GaugeMat &staple, const GaugeLorentz &Umu, int mu) {
    GridBase *grid = Umu._grid;
    std::vector<GaugeMat> U(Nd, grid);
    for (int d = 0; d < Nd; d++) {
      // this operation is taking too much time
      U[d] = PeekIndex<LorentzIndex>(Umu, d);
    }
    staple = zero;
    GaugeMat tmp1(grid);
    GaugeMat tmp2(grid);

    for (int nu = 0; nu < Nd; nu++) {
      if (nu != mu) {
        // this is ~10% faster than the Staple
        tmp1 = Cshift(U[nu], mu, 1);
        tmp2 = Cshift(U[mu], nu, 1);
        staple += tmp1* adj(U[nu]*tmp2);
        tmp2 = adj(U[mu]*tmp1)*U[nu];
        staple += Cshift(tmp2, nu, -1);
      }
    }
    staple = U[mu]*staple;
  }

  //////////////////////////////////////////////////
  // the sum over all staples on each site
  //////////////////////////////////////////////////
  static void Staple(GaugeMat &staple, const GaugeLorentz &Umu, int mu) {

    GridBase *grid = Umu._grid;

    std::vector<GaugeMat> U(Nd, grid);
    for (int d = 0; d < Nd; d++) {
      U[d] = PeekIndex<LorentzIndex>(Umu, d);
    }
    staple = zero;

    for (int nu = 0; nu < Nd; nu++) {

      if (nu != mu) {

        // mu
        // ^
        // |__>  nu

        //    __
        //      |
        //    __|
        //
     
        staple += Gimpl::ShiftStaple(
				     Gimpl::CovShiftForward(
							    U[nu], nu,
							    Gimpl::CovShiftBackward(
										    U[mu], mu, Gimpl::CovShiftIdentityBackward(U[nu], nu))),
				     mu);

        //  __
        // |
        // |__
        //
        //

        staple += Gimpl::ShiftStaple(
				     Gimpl::CovShiftBackward(U[nu], nu,
							     Gimpl::CovShiftBackward(U[mu], mu, U[nu])), mu);
      }
    }
  }

  //////////////////////////////////////////////////
  // the sum over all staples on each site in direction mu,nu, upper part
  //////////////////////////////////////////////////
  static void StapleUpper(GaugeMat &staple, const GaugeLorentz &Umu, int mu,
                          int nu) {
    if (nu != mu) {
      GridBase *grid = Umu._grid;

      std::vector<GaugeMat> U(Nd, grid);
      for (int d = 0; d < Nd; d++) {
        U[d] = PeekIndex<LorentzIndex>(Umu, d);// some redundant copies
      }

      // mu
      // ^
      // |__>  nu

      //    __
      //      |
      //    __|
      //

      staple = Gimpl::ShiftStaple(
				  Gimpl::CovShiftForward(
							 U[nu], nu,
							 Gimpl::CovShiftBackward(
										 U[mu], mu, Gimpl::CovShiftIdentityBackward(U[nu], nu))),
				  mu);
    }
  }

  //////////////////////////////////////////////////
  // the sum over all staples on each site in direction mu,nu, lower part
  //////////////////////////////////////////////////
  static void StapleLower(GaugeMat &staple, const GaugeLorentz &Umu, int mu,
                          int nu) {
    if (nu != mu) {
      GridBase *grid = Umu._grid;

      std::vector<GaugeMat> U(Nd, grid);
      for (int d = 0; d < Nd; d++) {
        U[d] = PeekIndex<LorentzIndex>(Umu, d);// some redundant copies
      }

      // mu
      // ^
      // |__>  nu

      //  __
      // |
      // |__
      //
      //
      staple = Gimpl::ShiftStaple(
				  Gimpl::CovShiftBackward(U[nu], nu,
							  Gimpl::CovShiftBackward(U[mu], mu, U[nu])), mu);
    }
  }

  //////////////////////////////////////////////////////
  //  Field Strength
  //////////////////////////////////////////////////////
  static void FieldStrength(GaugeMat &FS, const GaugeLorentz &Umu, int mu, int nu){
    // Fmn +--<--+  Ut +--<--+
    //     |     |     |     |
    //  (x)+-->--+     +-->--+(x)
    //     |     |     |     |
    //     +--<--+     +--<--+

    GaugeMat Vup(Umu._grid), Vdn(Umu._grid);
    StapleUpper(Vup, Umu, mu, nu);
    StapleLower(Vdn, Umu, mu, nu);
    GaugeMat v = Vup - Vdn;
    GaugeMat u = PeekIndex<LorentzIndex>(Umu, mu);  // some redundant copies
    GaugeMat vu = v*u;
    FS = 0.25*Ta(u*v + Cshift(vu, mu, -1));
  }

  static Real TopologicalCharge(GaugeLorentz &U){
    // 4d topological charge
    assert(Nd==4);
    // Bx = -iF(y,z), By = -iF(z,y), Bz = -iF(x,y)
    GaugeMat Bx(U._grid), By(U._grid), Bz(U._grid);
    FieldStrength(Bx, U, Ydir, Zdir);
    FieldStrength(By, U, Zdir, Xdir);
    FieldStrength(Bz, U, Xdir, Ydir);

    // Ex = -iF(t,x), Ey = -iF(t,y), Ez = -iF(t,z)
    GaugeMat Ex(U._grid), Ey(U._grid), Ez(U._grid);
    FieldStrength(Ex, U, Tdir, Xdir);
    FieldStrength(Ey, U, Tdir, Ydir);
    FieldStrength(Ez, U, Tdir, Zdir);

    double coeff = 8.0/(32.0*M_PI*M_PI);

    ComplexField qfield = coeff*trace(Bx*Ex + By*Ey + Bz*Ez);
    auto Tq = sum(qfield);
    return TensorRemove(Tq).real();
  }

  //////////////////////////////////////////////////////
  // Similar to above for rectangle is required
  //////////////////////////////////////////////////////
  static void dirRectangle(GaugeMat &rect, const std::vector<GaugeMat> &U,
                           const int mu, const int nu) {
    rect = Gimpl::CovShiftForward(
				  U[mu], mu, Gimpl::CovShiftForward(U[mu], mu, U[nu])) * // ->->|
      adj(Gimpl::CovShiftForward(
				 U[nu], nu, Gimpl::CovShiftForward(U[mu], mu, U[mu])));
    rect = rect +
      Gimpl::CovShiftForward(
			     U[mu], mu, Gimpl::CovShiftForward(U[nu], nu, U[nu])) * // ->||
      adj(Gimpl::CovShiftForward(
				 U[nu], nu, Gimpl::CovShiftForward(U[nu], nu, U[mu])));
  }
  static void traceDirRectangle(ComplexField &rect,
                                const std::vector<GaugeMat> &U, const int mu,
                                const int nu) {
    GaugeMat sp(U[0]._grid);
    dirRectangle(sp, U, mu, nu);
    rect = trace(sp);
  }
  static void siteRectangle(ComplexField &Rect,
                            const std::vector<GaugeMat> &U) {
    ComplexField siteRect(U[0]._grid);
    Rect = zero;
    for (int mu = 1; mu < Nd; mu++) {
      for (int nu = 0; nu < mu; nu++) {
        traceDirRectangle(siteRect, U, mu, nu);
        Rect = Rect + siteRect;
      }
    }
  }

  //////////////////////////////////////////////////
  // sum over all x,y,z,t and over all planes of plaquette
  //////////////////////////////////////////////////
  static RealD sumRectangle(const GaugeLorentz &Umu) {
    std::vector<GaugeMat> U(Nd, Umu._grid);

    for (int mu = 0; mu < Nd; mu++) {
      U[mu] = PeekIndex<LorentzIndex>(Umu, mu);
    }

    ComplexField Rect(Umu._grid);

    siteRectangle(Rect, U);

    auto Tp = sum(Rect);
    auto p = TensorRemove(Tp);
    return p.real();
  }
  //////////////////////////////////////////////////
  // average over all x,y,z,t and over all planes of plaquette
  //////////////////////////////////////////////////
  static RealD avgRectangle(const GaugeLorentz &Umu) {

    RealD sumrect = sumRectangle(Umu);

    double vol = Umu._grid->gSites();

    double faces = (1.0 * Nd * (Nd - 1)); // 2 distinct orientations summed

    return sumrect / vol / faces / Nc; // Nd , Nc dependent... FIXME
  }

  //////////////////////////////////////////////////
  // the sum over all staples on each site
  //////////////////////////////////////////////////
  static void RectStapleDouble(GaugeMat &U2, const GaugeMat &U, int mu) {
    U2 = U * Cshift(U, mu, 1);
  }

  ////////////////////////////////////////////////////////////////////////////
  // Hop by two optimisation strategy does not work nicely with Gparity. (could
  // do,
  // but need to track two deep where cross boundary and apply a conjugation).
  // Must differentiate this in Gimpl, and use Gimpl::isPeriodicGaugeField to do
  // so .
  ////////////////////////////////////////////////////////////////////////////
  static void RectStapleOptimised(GaugeMat &Stap, std::vector<GaugeMat> &U2,
                                  std::vector<GaugeMat> &U, int mu) {

    Stap = zero;

    GridBase *grid = U[0]._grid;

    GaugeMat Staple2x1(grid);
    GaugeMat tmp(grid);

    for (int nu = 0; nu < Nd; nu++) {
      if (nu != mu) {

        // Up staple    ___ ___
        //             |       |
        tmp = Cshift(adj(U[nu]), nu, -1);
        tmp = adj(U2[mu]) * tmp;
        tmp = Cshift(tmp, mu, -2);

        Staple2x1 = Gimpl::CovShiftForward(U[nu], nu, tmp);

        // Down staple
        //             |___ ___|
        //
        tmp = adj(U2[mu]) * U[nu];
        Staple2x1 += Gimpl::CovShiftBackward(U[nu], nu, Cshift(tmp, mu, -2));

        //              ___ ___
        //             |    ___|
        //             |___ ___|
        //

        Stap += Cshift(Gimpl::CovShiftForward(U[mu], mu, Staple2x1), mu, 1);

        //              ___ ___
        //             |___    |
        //             |___ ___|
        //

        //  tmp= Staple2x1* Cshift(U[mu],mu,-2);
        //  Stap+= Cshift(tmp,mu,1) ;
        Stap += Cshift(Staple2x1, mu, 1) * Cshift(U[mu], mu, -1);
        ;

        //       --
        //      |  |
        //
        //      |  |

        tmp = Cshift(adj(U2[nu]), nu, -2);
        tmp = Gimpl::CovShiftBackward(U[mu], mu, tmp);
        tmp = U2[nu] * Cshift(tmp, nu, 2);
        Stap += Cshift(tmp, mu, 1);

        //      |  |
        //
        //      |  |
        //       --

        tmp = Gimpl::CovShiftBackward(U[mu], mu, U2[nu]);
        tmp = adj(U2[nu]) * tmp;
        tmp = Cshift(tmp, nu, -2);
        Stap += Cshift(tmp, mu, 1);
      }
    }
  }

  static void RectStaple(GaugeMat &Stap, const GaugeLorentz &Umu, int mu) {
    RectStapleUnoptimised(Stap, Umu, mu);
  }
  static void RectStaple(const GaugeLorentz &Umu, GaugeMat &Stap,
                         std::vector<GaugeMat> &U2, std::vector<GaugeMat> &U,
                         int mu) {
    if (Gimpl::isPeriodicGaugeField()) {
      RectStapleOptimised(Stap, U2, U, mu);
    } else {
      RectStapleUnoptimised(Stap, Umu, mu);
    }
  }

  static void RectStapleUnoptimised(GaugeMat &Stap, const GaugeLorentz &Umu,
                                    int mu) {
    GridBase *grid = Umu._grid;

    std::vector<GaugeMat> U(Nd, grid);
    for (int d = 0; d < Nd; d++) {
      U[d] = PeekIndex<LorentzIndex>(Umu, d);
    }

    Stap = zero;

    for (int nu = 0; nu < Nd; nu++) {
      if (nu != mu) {
        //           __ ___
        //          |    __ |
        //
        Stap += Gimpl::ShiftStaple(
				   Gimpl::CovShiftForward(
							  U[mu], mu,
							  Gimpl::CovShiftForward(
										 U[nu], nu,
										 Gimpl::CovShiftBackward(
													 U[mu], mu,
													 Gimpl::CovShiftBackward(
																 U[mu], mu,
																 Gimpl::CovShiftIdentityBackward(U[nu], nu))))),
				   mu);

        //              __
        //          |__ __ |

        Stap += Gimpl::ShiftStaple(
				   Gimpl::CovShiftForward(
							  U[mu], mu,
							  Gimpl::CovShiftBackward(
										  U[nu], nu,
										  Gimpl::CovShiftBackward(
													  U[mu], mu, Gimpl::CovShiftBackward(U[mu], mu, U[nu])))),
				   mu);

        //           __
        //          |__ __ |

        Stap += Gimpl::ShiftStaple(
				   Gimpl::CovShiftBackward(
							   U[nu], nu,
							   Gimpl::CovShiftBackward(
										   U[mu], mu,
										   Gimpl::CovShiftBackward(
													   U[mu], mu, Gimpl::CovShiftForward(U[nu], nu, U[mu])))),
				   mu);

        //           __ ___
        //          |__    |

        Stap += Gimpl::ShiftStaple(
				   Gimpl::CovShiftForward(
							  U[nu], nu,
							  Gimpl::CovShiftBackward(
										  U[mu], mu,
										  Gimpl::CovShiftBackward(
													  U[mu], mu, Gimpl::CovShiftBackward(U[nu], nu, U[mu])))),
				   mu);

        //       --
        //      |  |
        //
        //      |  |

        Stap += Gimpl::ShiftStaple(
				   Gimpl::CovShiftForward(
							  U[nu], nu,
							  Gimpl::CovShiftForward(
										 U[nu], nu,
										 Gimpl::CovShiftBackward(
													 U[mu], mu,
													 Gimpl::CovShiftBackward(
																 U[nu], nu,
																 Gimpl::CovShiftIdentityBackward(U[nu], nu))))),
				   mu);

        //      |  |
        //
        //      |  |
        //       --

        Stap += Gimpl::ShiftStaple(
				   Gimpl::CovShiftBackward(
							   U[nu], nu,
							   Gimpl::CovShiftBackward(
										   U[nu], nu,
										   Gimpl::CovShiftBackward(
													   U[mu], mu, Gimpl::CovShiftForward(U[nu], nu, U[nu])))),
				   mu);
      }
    }
  }
};

typedef WilsonLoops<PeriodicGimplR> ColourWilsonLoops;
typedef WilsonLoops<PeriodicGimplR> U1WilsonLoops;
typedef WilsonLoops<PeriodicGimplR> SU2WilsonLoops;
typedef WilsonLoops<PeriodicGimplR> SU3WilsonLoops;

NAMESPACE_END(Grid);

#endif