ABACUS/dev/ODSLF/ODSLF_XXZ_Bethe_State.cc

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

This software is part of J.-S. Caux's ABACUS library.

Copyright (c) J.-S. Caux.

-----------------------------------------------------------

File:  src/ODSLF/ODSLF_XXZ_Bethe_State.cc

Purpose:  defines Bethe states for 1d spinless fermions.

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

#include "ABACUS.h"

using namespace std;

namespace ABACUS {

  // Function prototypes

  inline DP ODSLF_fbar_XXZ (DP lambda, int par, DP tannzetaover2);
  DP ODSLF_Theta_XXZ (DP lambda, int nj, int nk, int parj, int park, DP* tannzetaover2);
  DP ODSLF_hbar_XXZ (DP lambda, int n, int par, DP* si_n_anis_over_2);
  DP ODSLF_ddlambda_Theta_XXZ (DP lambda, int nj, int nk, int parj, int park, DP* si_n_anis_over_2);


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

  // Function definitions:  class ODSLF_XXZ_Bethe_State

  ODSLF_XXZ_Bethe_State::ODSLF_XXZ_Bethe_State ()
    : ODSLF_Bethe_State(), sinhlambda(ODSLF_Lambda(chain, 1)), coshlambda(ODSLF_Lambda(chain, 1)),
      tanhlambda(ODSLF_Lambda(chain, 1))
  {};

  ODSLF_XXZ_Bethe_State::ODSLF_XXZ_Bethe_State (const ODSLF_XXZ_Bethe_State& RefState) // copy constructor
    : ODSLF_Bethe_State(RefState), sinhlambda(ODSLF_Lambda(RefState.chain, RefState.base)),
      coshlambda(ODSLF_Lambda(RefState.chain, RefState.base)),
      tanhlambda(ODSLF_Lambda(RefState.chain, RefState.base))
  {
    // copy arrays into new ones

    //cout << "Calling XXZ state copy constructor." << endl;
    for (int j = 0; j < RefState.chain.Nstrings; ++j) {
      for (int alpha = 0; alpha < RefState.base[j]; ++j) {
	sinhlambda[j][alpha] = RefState.sinhlambda[j][alpha];
	coshlambda[j][alpha] = RefState.coshlambda[j][alpha];
	tanhlambda[j][alpha] = RefState.tanhlambda[j][alpha];
      }
    }
    //cout << "Done calling XXZ state copy constructor." << endl;
  }

  ODSLF_XXZ_Bethe_State::ODSLF_XXZ_Bethe_State (const Heis_Chain& RefChain, int M)
    : ODSLF_Bethe_State(RefChain, M),
      sinhlambda(ODSLF_Lambda(RefChain, M)), coshlambda(ODSLF_Lambda(RefChain, M)), tanhlambda(ODSLF_Lambda(RefChain, M))
  {
    //cout << "Here in XXZ BS constructor." << endl;
    //cout << (*this).lambda[0][0] << endl;
    //cout << "OK" << endl;
    if ((RefChain.Delta <= -1.0) || (RefChain.Delta >= 1.0))
      ABACUSerror("Delta out of range in ODSLF_XXZ_Bethe_State constructor");
  }

  ODSLF_XXZ_Bethe_State::ODSLF_XXZ_Bethe_State (const Heis_Chain& RefChain, const ODSLF_Base& RefBase)
    : ODSLF_Bethe_State(RefChain, RefBase),
      sinhlambda(ODSLF_Lambda(RefChain, RefBase)), coshlambda(ODSLF_Lambda(RefChain, RefBase)),
      tanhlambda(ODSLF_Lambda(RefChain, RefBase))
  {
    if ((RefChain.Delta <= -1.0) || (RefChain.Delta >= 1.0))
      ABACUSerror("Delta out of range in ODSLF_XXZ_Bethe_State constructor");
  }

  ODSLF_XXZ_Bethe_State::ODSLF_XXZ_Bethe_State (const Heis_Chain& RefChain,
						long long int base_id_ref, long long int type_id_ref)
    : ODSLF_Bethe_State(RefChain, base_id_ref, type_id_ref),
      sinhlambda(ODSLF_Lambda(chain, base)), coshlambda(ODSLF_Lambda(chain, base)), tanhlambda(ODSLF_Lambda(chain, base))
  {
    if ((RefChain.Delta <= -1.0) || (RefChain.Delta >= 1.0))
      ABACUSerror("Delta out of range in ODSLF_XXZ_Bethe_State constructor");
  }


  ODSLF_XXZ_Bethe_State& ODSLF_XXZ_Bethe_State::operator= (const ODSLF_XXZ_Bethe_State& RefState)
  {
    if (this != &RefState) {
      chain = RefState.chain;
      base = RefState.base;
      offsets = RefState.offsets;
      Ix2 = RefState.Ix2;
      lambda = RefState.lambda;
      BE = RefState.BE;
      diffsq = RefState.diffsq;
      conv = RefState.conv;
      iter = RefState.iter;
      iter_Newton = RefState.iter_Newton;
      E = RefState.E;
      iK = RefState.iK;
      K = RefState.K;
      lnnorm = RefState.lnnorm;
      base_id = RefState.base_id;
      type_id = RefState.type_id;
      id = RefState.id;
      maxid = RefState.maxid;
      nparticles = RefState.nparticles;

      sinhlambda = RefState.sinhlambda;
      coshlambda = RefState.coshlambda;
      tanhlambda = RefState.tanhlambda;
    }
    return(*this);
  }

  // Member functions

  void ODSLF_XXZ_Bethe_State::Set_Free_lambdas()
  {
    // Sets all the rapidities to the solutions of the BAEs without scattering terms

    for (int i = 0; i < chain.Nstrings; ++i) {

      for (int alpha = 0; alpha < base[i]; ++alpha) {

	if(chain.par[i] == 1)
	  lambda[i][alpha] = (tan(chain.Str_L[i] * 0.5 * chain.anis) * tan(PI * 0.5 * Ix2[i][alpha]/chain.Nsites));
	else if (chain.par[i] == -1)
	  lambda[i][alpha] = (-tan((PI * 0.5 * Ix2[i][alpha])/chain.Nsites)/tan(chain.Str_L[i] * 0.5 * chain.anis));

	else ABACUSerror("Invalid parities in Set_Free_lambdas.");

      }
    }

    return;
  }

  void ODSLF_XXZ_Bethe_State::Compute_sinhlambda()
  {
    for (int j = 0; j < chain.Nstrings; ++j) {

      for (int alpha = 0; alpha < base[j]; ++alpha) sinhlambda[j][alpha] = sinh(lambda[j][alpha]);
    }
    return;
  }

  void ODSLF_XXZ_Bethe_State::Compute_coshlambda()
  {
    for (int j = 0; j < chain.Nstrings; ++j) {

      for (int alpha = 0; alpha < base[j]; ++alpha) coshlambda[j][alpha] = cosh(lambda[j][alpha]);
    }
    return;
  }

  void ODSLF_XXZ_Bethe_State::Compute_tanhlambda()
  {
    for (int j = 0; j < chain.Nstrings; ++j) {

      for (int alpha = 0; alpha < base[j]; ++alpha) tanhlambda[j][alpha] = tanh(lambda[j][alpha]);
    }
    return;
  }

  bool ODSLF_XXZ_Bethe_State::Check_Admissibility(char option)
  {
    // This function checks the admissibility of the Ix2's of a state:
    // returns false if there are higher strings with Ix2 = 0, a totally symmetric distribution of I's at each level,
    // and strings of equal length modulo 2 and parity with Ix2 = 0, meaning at least two equal roots in BAE.

    bool answer = true;
    Vect<bool> Zero_at_level(false, chain.Nstrings); // whether there exists an Ix2 == 0 at a given level

    bool higher_string_on_zero = false;
    for (int j = 0; j < chain.Nstrings; ++j) {
      // The following line puts answer to true if there is at least one higher string with zero Ix2
      for (int alpha = 0; alpha < base[j]; ++alpha) if ((Ix2[j][alpha] == 0) && (chain.Str_L[j] >= 2))
						      higher_string_on_zero = true;
      for (int alpha = 0; alpha < base[j]; ++alpha) if (Ix2[j][alpha] == 0) Zero_at_level[j] = true;
      // NOTE:  if base[j] == 0, Zero_at_level[j] remains false.
    }

    // check symmetry of Ix2 at each level, if there exists a potentially risky Ix2...

    bool symmetric_state = (*this).Check_Symmetry();

    bool string_coincidence = false;
    for (int j1 = 0; j1 < chain.Nstrings; ++j1) {
      for (int j2 = j1 + 1; j2 < chain.Nstrings; ++j2)
	if (Zero_at_level[j1] && Zero_at_level[j2] && (chain.par[j1] == chain.par[j2])
	    && (!((chain.Str_L[j1] + chain.Str_L[j2])%2)))
	  string_coincidence = true;
    }

    bool M_odd_and_onep_on_zero = false;
    if (option == 'z') { // for Sz, if M is odd, exclude symmetric states with a 1+ on zero
                         // (zero rapidities in left and right states, so FF det not defined).
      bool is_ground_state = base.Nrap[0] == base.Mdown && Ix2[0][0] == -(base.Mdown - 1)
	&& Ix2[0][base.Mdown-1] == base.Mdown - 1;
      if (Zero_at_level[0] && (base.Mdown % 2) && !is_ground_state) M_odd_and_onep_on_zero = true;
    }

    bool onep_onem_on_zero = false;
    if (option == 'm' || option == 'p') { // for Smin, we also exclude symmetric states with 1+ and 1- strings on zero
      if (Zero_at_level[0] && Zero_at_level[1]) onep_onem_on_zero = true;
    }

    answer = !(symmetric_state && (higher_string_on_zero || string_coincidence
				   || onep_onem_on_zero || M_odd_and_onep_on_zero));

    // Now check that no Ix2 is equal to +N (since we take -N into account, and I + N == I by periodicity of exp)

    for (int j = 0; j < chain.Nstrings; ++j)
      for (int alpha = 0; alpha < base[j]; ++alpha)
	if ((Ix2[j][alpha] < -chain.Nsites) || (Ix2[j][alpha] >= chain.Nsites)) answer = false;

    if (!answer) {
      E = 0.0;
      K = 0.0;
      conv = 0;
      iter = 0;
      iter_Newton = 0;
      lnnorm = -100.0;
    }

    return(answer);  // answer == true:  nothing wrong with this Ix2_config
  }

  void ODSLF_XXZ_Bethe_State::Compute_BE (int j, int alpha)
  {
    tanhlambda[j][alpha] = tanh(lambda[j][alpha]);

    DP sumtheta = 0.0;

    for (int k = 0; k < chain.Nstrings; ++k)
      for (int beta = 0; beta < base[k]; ++beta) {
	if ((chain.Str_L[j] == 1) && (chain.Str_L[k] == 1))
	  sumtheta += (chain.par[j] == chain.par[k])
	    ? atan((tanhlambda[j][alpha] - tanhlambda[k][beta])
		   /((1.0 - tanhlambda[j][alpha] * tanhlambda[k][beta]) * chain.ta_n_anis_over_2[2]))
	    : - atan(((tanhlambda[j][alpha] - tanhlambda[k][beta])
		      /(1.0 - tanhlambda[j][alpha] * tanhlambda[k][beta])) * chain.ta_n_anis_over_2[2]) ;
	else sumtheta += 0.5 * ODSLF_Theta_XXZ((tanhlambda[j][alpha] - tanhlambda[k][beta])
					       /(1.0 - tanhlambda[j][alpha] * tanhlambda[k][beta]),
					       chain.Str_L[j], chain.Str_L[k], chain.par[j], chain.par[k],
					       chain.ta_n_anis_over_2);
      }
    sumtheta *= 2.0;

    BE[j][alpha] =
      ((chain.par[j] == 1) ? 2.0 * atan(tanhlambda[j][alpha]/chain.ta_n_anis_over_2[chain.Str_L[j]])
       : -2.0 * atan(tanhlambda[j][alpha] * chain.ta_n_anis_over_2[chain.Str_L[j]]))
      - (sumtheta + PI*Ix2[j][alpha])/chain.Nsites;

  }

  void ODSLF_XXZ_Bethe_State::Compute_BE ()
  {
    // Fills in the BE members with the value of the Bethe equations.

    (*this).Compute_tanhlambda();

    DP sumtheta = 0.0;

    for (int j = 0; j < chain.Nstrings; ++j)
      for (int alpha = 0; alpha < base[j]; ++alpha) {

	sumtheta = 0.0;
	for (int k = 0; k < chain.Nstrings; ++k)
	  for (int beta = 0; beta < base[k]; ++beta) {
	    if ((chain.Str_L[j] == 1) && (chain.Str_L[k] == 1))
	      sumtheta += (chain.par[j] == chain.par[k])
		? atan((tanhlambda[j][alpha] - tanhlambda[k][beta])
		       /((1.0 - tanhlambda[j][alpha] * tanhlambda[k][beta]) * chain.ta_n_anis_over_2[2]))
		: - atan(((tanhlambda[j][alpha] - tanhlambda[k][beta])
			  /(1.0 - tanhlambda[j][alpha] * tanhlambda[k][beta])) * chain.ta_n_anis_over_2[2]) ;
	    else sumtheta += 0.5 * ODSLF_Theta_XXZ((tanhlambda[j][alpha] - tanhlambda[k][beta])
						   /(1.0 - tanhlambda[j][alpha] * tanhlambda[k][beta]),
						   chain.Str_L[j], chain.Str_L[k], chain.par[j], chain.par[k],
						   chain.ta_n_anis_over_2);
	  }
	sumtheta *= 2.0;

	BE[j][alpha] =
	  ((chain.par[j] == 1) ? 2.0 * atan(tanhlambda[j][alpha]/chain.ta_n_anis_over_2[chain.Str_L[j]])
	   : -2.0 * atan(tanhlambda[j][alpha] * chain.ta_n_anis_over_2[chain.Str_L[j]]))
	  - (sumtheta + PI*Ix2[j][alpha])/chain.Nsites;

      }
  }

  DP ODSLF_XXZ_Bethe_State::Iterate_BAE (int j, int alpha)
  {
    // Returns a new iteration value for lambda[j][alpha] given tanhlambda and BE Lambdas
    // Assumes that tanhlambda[][] and BE[][] have been computed.

    DP new_lambda = 0.0;
    DP arg = 0.0;

    if (chain.par[j] == 1) arg = chain.ta_n_anis_over_2[chain.Str_L[j]]
			     * tan(0.5 *
				   //(PI * Ix2[j][alpha] + sumtheta)/chain.Nsites
				   (2.0 * atan(tanhlambda[j][alpha]/chain.ta_n_anis_over_2[chain.Str_L[j]]) - BE[j][alpha])
				   );

    else if (chain.par[j] == -1)
      arg = -tan(0.5 *
		 (-2.0 * atan(tanhlambda[j][alpha] * chain.ta_n_anis_over_2[chain.Str_L[j]]) - BE[j][alpha]))
	/chain.ta_n_anis_over_2[chain.Str_L[j]];

    if (fabs(arg) < 1.0) {
      new_lambda = atanh(arg);
    }

    else {
      new_lambda = lambda[j][alpha];  // back to drawing board...
      int block = 0;  // counter to prevent runaway while loop
      DP new_tanhlambda = 0.0;
      DP sumtheta = 0.0;
      arg = 10.0; // reset value to start while loop
      while ((fabs(arg) > 1.0) && (block++ < 100)) {  // recompute the diverging root on its own...
	new_lambda *= 1.01; // try to go slowly towards infinity...
	new_tanhlambda = tanh(new_lambda);
	sumtheta = 0.0;
	for (int k = 0; k < chain.Nstrings; ++k) {
	  for (int beta = 0; beta < base[k]; ++beta)
	    if ((chain.Str_L[j] == 1) && (chain.Str_L[k] == 1))
	      sumtheta += (chain.par[j] == chain.par[k])
		? atan((new_tanhlambda - tanhlambda[k][beta])
		       /((1.0 - new_tanhlambda * tanhlambda[k][beta]) * chain.ta_n_anis_over_2[2]))
		: - atan(((new_tanhlambda - tanhlambda[k][beta])
			  /(1.0 - new_tanhlambda * tanhlambda[k][beta])) * chain.ta_n_anis_over_2[2]) ;
	    else sumtheta += 0.5 * ODSLF_Theta_XXZ((new_tanhlambda - tanhlambda[k][beta])
						   /(1.0 - new_tanhlambda * tanhlambda[k][beta]),
						   chain.Str_L[j], chain.Str_L[k], chain.par[j], chain.par[k],
						   chain.ta_n_anis_over_2);
	}
	sumtheta *= 2.0;
	if (chain.par[j] == 1)
	  arg = chain.ta_n_anis_over_2[chain.Str_L[j]] * tan(0.5 * (PI * Ix2[j][alpha] + sumtheta)/chain.Nsites);

	else if (chain.par[j] == -1)
	  arg = -tan(0.5 * (PI * Ix2[j][alpha] + sumtheta)/chain.Nsites)/chain.ta_n_anis_over_2[chain.Str_L[j]];

	else ABACUSerror("Invalid parities in Iterate_BAE.");

      }

    }

    return(new_lambda);
  }

  bool ODSLF_XXZ_Bethe_State::Check_Rapidities()
  {
    bool nonan = true;

    for (int j = 0; j < chain.Nstrings; ++j)
      for (int alpha = 0; alpha < base[j]; ++alpha) nonan *= !is_nan(lambda[j][alpha]);

    return nonan;
  }

  void ODSLF_XXZ_Bethe_State::Compute_Energy ()
  {
    DP sum = 0.0;

    for (int j = 0; j < chain.Nstrings; ++j) {
      for (int alpha = 0; alpha < base[j]; ++alpha) {
	sum += sin(chain.Str_L[j] * chain.anis)
	  / (chain.par[j] * cosh(2.0 * lambda[j][alpha]) - cos(chain.Str_L[j] * chain.anis));
      }
    }

    sum *= - chain.J * sin(chain.anis);

    E = sum;

    return;
  }

  void ODSLF_XXZ_Bethe_State::Build_Reduced_Gaudin_Matrix (SQMat<complex<DP> >& Gaudin_Red)
  {

    if (Gaudin_Red.size() != base.Nraptot) ABACUSerror("Passing matrix of wrong size in Build_Reduced_Gaudin_Matrix.");

    int index_jalpha;
    int index_kbeta;

    DP sum_hbar_XXZ = 0.0;

    DP sinzetasq = pow(sin(chain.anis), 2.0);

    (*this).Compute_sinhlambda();
    (*this).Compute_coshlambda();

    index_jalpha = 0;
    for (int j = 0; j < chain.Nstrings; ++j) {
      for (int alpha = 0; alpha < base[j]; ++alpha) {
	index_kbeta = 0;
	for (int k = 0; k < chain.Nstrings; ++k) {
	  for (int beta = 0; beta < base[k]; ++beta) {

	    if ((j == k) && (alpha == beta)) {

	      sum_hbar_XXZ = 0.0;

	      for (int kp = 0; kp < chain.Nstrings; ++kp) {
		for (int betap = 0; betap < base[kp]; ++betap) {
		  if (!((j == kp) && (alpha == betap)))
		    sum_hbar_XXZ
		      += ODSLF_ddlambda_Theta_XXZ (lambda[j][alpha] - lambda[kp][betap], chain.Str_L[j], chain.Str_L[kp],
						   chain.par[j], chain.par[kp], chain.si_n_anis_over_2);
		}
	      }

	      Gaudin_Red[index_jalpha][index_kbeta]
		= complex<DP> ( chain.Nsites * ODSLF_hbar_XXZ (lambda[j][alpha], chain.Str_L[j], chain.par[j],
							       chain.si_n_anis_over_2) - sum_hbar_XXZ);
	    }

	    else {
	      if ((chain.Str_L[j] == 1) && (chain.Str_L[k] == 1))
		Gaudin_Red[index_jalpha][index_kbeta] =
		  complex<DP> ((chain.par[j] * chain.par[k] == 1)
			       ? chain.si_n_anis_over_2[4]
			       /(pow(sinhlambda[j][alpha] * coshlambda[k][beta]
				     - coshlambda[j][alpha] * sinhlambda[k][beta], 2.0) + sinzetasq)
			       : chain.si_n_anis_over_2[4]
			       /(-pow(coshlambda[j][alpha] * coshlambda[k][beta]
				      - sinhlambda[j][alpha] * sinhlambda[k][beta], 2.0) + sinzetasq) );
	      else
		Gaudin_Red[index_jalpha][index_kbeta] =
		  complex<DP> (ODSLF_ddlambda_Theta_XXZ (lambda[j][alpha] - lambda[k][beta], chain.Str_L[j], chain.Str_L[k],
							 chain.par[j], chain.par[k], chain.si_n_anis_over_2));
	    }
	    index_kbeta++;
	  }
	}
	index_jalpha++;
      }
    }
    return;
  }

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

  // non-member functions

  inline DP ODSLF_fbar_XXZ (DP tanhlambda, int par, DP tannzetaover2)
  {
    DP result = 0.0;

    if (par == 1) result = 2.0 * atan(tanhlambda/tannzetaover2);

    else if (par == -1) result = -2.0 * atan(tanhlambda * tannzetaover2);

    else ABACUSerror("Faulty parity in ODSLF_fbar_XXZ.");

    return (result);
  }

  DP ODSLF_Theta_XXZ (DP tanhlambda, int nj, int nk, int parj, int park, DP* tannzetaover2)
  {
    DP result = 0.0;

    if ((nj == 1) && (nk == 1)) result = ODSLF_fbar_XXZ(tanhlambda, parj*park, tannzetaover2[2]);

    else {

      result = (nj == nk) ? 0.0 : ODSLF_fbar_XXZ(tanhlambda, parj*park, tannzetaover2[fabs(nj - nk)]);

      for (int a = 1; a < ABACUS::min(nj, nk); ++a)
	result += 2.0 * ODSLF_fbar_XXZ(tanhlambda, parj*park, tannzetaover2[fabs(nj - nk) + 2*a]);

      result += ODSLF_fbar_XXZ(tanhlambda, parj*park, tannzetaover2[nj + nk]);
    }

    return (result);
  }

  DP ODSLF_hbar_XXZ (DP lambda, int n, int par, DP* si_n_anis_over_2)
  {
    DP result = 0.0;

    if (par == 1) result = si_n_anis_over_2[2*n]/(pow(sinh(lambda), 2.0) + pow(si_n_anis_over_2[n], 2.0));

    else if (par == -1) result = si_n_anis_over_2[2*n]/(-pow(cosh(lambda), 2.0) + pow(si_n_anis_over_2[n], 2.0));

    else ABACUSerror("Faulty parity in ODSLF_hbar_XXZ.");

    return (result);
  }

  DP ODSLF_ddlambda_Theta_XXZ (DP lambda, int nj, int nk, int parj, int park, DP* si_n_anis_over_2)
  {
    DP result = (nj == nk) ? 0.0 : ODSLF_hbar_XXZ(lambda, fabs(nj - nk), parj*park, si_n_anis_over_2);

    for (int a = 1; a < ABACUS::min(nj, nk); ++a)
      result += 2.0 * ODSLF_hbar_XXZ(lambda, fabs(nj - nk) + 2*a, parj*park, si_n_anis_over_2);

    result += ODSLF_hbar_XXZ(lambda, nj + nk, parj*park, si_n_anis_over_2);

    return (result);
  }


} // namespace ABACUS