ABACUS/src/HEIS/XXZ_Bethe_State.cc

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

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

Copyright (c)

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

File:  src/HEIS/XXZ_Bethe_State.cc

Purpose:  Defines all functions for XXZ_Bethe_State

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

#include "ABACUS.h"

using namespace std;

namespace ABACUS {

  // Function prototypes

  inline DP fbar_XXZ (DP lambda, int par, DP tannzetaover2);
  DP Theta_XXZ (DP lambda, int nj, int nk, int parj, int park, DP* tannzetaover2);
  DP hbar_XXZ (DP lambda, int n, int par, DP* si_n_anis_over_2);
  DP ddlambda_Theta_XXZ (DP lambda, int nj, int nk, int parj, int park, DP* si_n_anis_over_2);


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

  // Function definitions:  class XXZ_Bethe_State

  XXZ_Bethe_State::XXZ_Bethe_State ()
    : Heis_Bethe_State(), sinhlambda(Lambda(chain, 1)), coshlambda(Lambda(chain, 1)), tanhlambda(Lambda(chain, 1))
  {};

  XXZ_Bethe_State::XXZ_Bethe_State (const XXZ_Bethe_State& RefState) // copy constructor
    : Heis_Bethe_State(RefState), sinhlambda(Lambda(RefState.chain, RefState.base)), coshlambda(Lambda(RefState.chain, RefState.base)),
      tanhlambda(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;
  }

  XXZ_Bethe_State::XXZ_Bethe_State (const Heis_Chain& RefChain, int M)
    : Heis_Bethe_State(RefChain, M),
      sinhlambda(Lambda(RefChain, M)), coshlambda(Lambda(RefChain, M)), tanhlambda(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 XXZ_Bethe_State constructor");
  }

  XXZ_Bethe_State::XXZ_Bethe_State (const Heis_Chain& RefChain, const Heis_Base& RefBase)
    : Heis_Bethe_State(RefChain, RefBase),
      sinhlambda(Lambda(RefChain, RefBase)), coshlambda(Lambda(RefChain, RefBase)), tanhlambda(Lambda(RefChain, RefBase))
  {
    if ((RefChain.Delta <= -1.0) || (RefChain.Delta >= 1.0)) ABACUSerror("Delta out of range in XXZ_Bethe_State constructor");
  }
  /*
  XXZ_Bethe_State::XXZ_Bethe_State (const Heis_Chain& RefChain, long long int base_id_ref, long long int type_id_ref)
    : Heis_Bethe_State(RefChain, base_id_ref, type_id_ref),
      sinhlambda(Lambda(chain, base)), coshlambda(Lambda(chain, base)), tanhlambda(Lambda(chain, base))
  {
    if ((RefChain.Delta <= -1.0) || (RefChain.Delta >= 1.0)) ABACUSerror("Delta out of range in XXZ_Bethe_State constructor");
  }
  */

  XXZ_Bethe_State& XXZ_Bethe_State::operator= (const 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;
      label = RefState.label;

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

  // Member functions

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

    DP x = 0.0;
    for (int i = 0; i < chain.Nstrings; ++i) {

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

	if (chain.par[i] == 1) {
	  //lambda[i][alpha] = atanh(tan(chain.Str_L[i] * 0.5 * chain.anis) * tan(PI * 0.5 * Ix2[i][alpha]/chain.Nsites));
	  x = tan(chain.Str_L[i] * 0.5 * chain.anis) * tan(PI * 0.5 * Ix2[i][alpha]/chain.Nsites);
	  lambda[i][alpha] = atanh(x/sqrt(1.0 + x*x)); // lambda then always initiated real
	}

	else if (chain.par[i] == -1) {
	  //lambda[i][alpha] = atanh(-tan(PI * 0.5 * Ix2[i][alpha]/chain.Nsites)/tan(chain.Str_L[i] * 0.5 * chain.anis));
	  x = -tan(PI * 0.5 * Ix2[i][alpha]/chain.Nsites)/tan(chain.Str_L[i] * 0.5 * chain.anis);
	  lambda[i][alpha] = atanh(x/sqrt(1.0 + x*x)); // lambda then always initiated real
	}

	else ABACUSerror("Invalid parities in Set_Free_lambdas.");
	//cout << tan(chain.Str_L[i] * 0.5 * chain.anis) << endl;
	//cout << "Set_Free_lambdas: " << i << "\t" << alpha << "\t" << lambda[i][alpha] << "\t" << tan(chain.Str_L[i] * 0.5 * chain.anis) * tan(PI * 0.5 * Ix2[i][alpha]/chain.Nsites) << endl;

      }
    }

    return;
  }

  void 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 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 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 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) /*&& !(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 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 * 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 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 * 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;

	//if (is_nan(BE[j][alpha])) cout << "BE nan: " << j << "\t" << alpha << "\t" << lambda[j][alpha] << "\t" << tanhlambda[j][alpha] << endl;
      }
  }

  DP 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 *
					    //(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]))
      /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 * 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.");

      }

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

      //else cout << "Rapidity blows up !\t" << lambda[j][alpha] << "\t" << new_lambda << endl;
    } // else

    return(new_lambda);
  }

  bool 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;
  }

  DP XXZ_Bethe_State::String_delta ()
  {
    // Computes the sum of absolute value of \delta^{a, a+1} in string hypothesis, for a given bethe eigenstate

    DP delta = 0.0;

    int occupied_strings = 0;
    for (int i = 0; i < (*this).chain.Nstrings; ++i) if ((*this).chain.Str_L[i] > 1) occupied_strings += (*this).base.Nrap[i];

    //if ((*this).conv == 0) delta = 1.0;

    if (occupied_strings == 0) delta = 0.0;

    else {

      Vect_DP ln_deltadiff(0.0, 1000);  // contains ln |delta^{a, a+1}|
      Vect_DP deltadiff(0.0, 1000);  // contains |delta^{a, a+1}|

      complex<DP> log_BAE_reg = 0.0;

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

	  ln_deltadiff = 0.0;

	  for (int a = 1; a <= (*this).chain.Str_L[j]; ++a) {

	    if ((*this).chain.Str_L[j] > 1) {  // else the BAE are already 1

	      log_BAE_reg = DP((*this).chain.Nsites) * log(sinh((*this).lambda[j][alpha]
							   + 0.5 * II * (*this).chain.anis * ((*this).chain.Str_L[j] + 1.0 - 2.0 * a + 1.0)
				 + 0.25 * II * PI * (1.0 - (*this).chain.par[j]))
			    /sinh((*this).lambda[j][alpha] + 0.5 * II * (*this).chain.anis * ((*this).chain.Str_L[j] + 1.0 - 2.0 * a - 1.0)
				  + 0.25 * II * PI * (1.0 - (*this).chain.par[j])));

	      for (int k = 0; k < (*this).chain.Nstrings; ++k)
		for (int beta = 0; beta < (*this).base[k]; ++beta)
		  for (int b = 1; b <= (*this).chain.Str_L[k]; ++b) {
		    if ((j != k) || (alpha != beta) || (a != b - 1))

		      log_BAE_reg += log(sinh(((*this).lambda[j][alpha] + 0.5 * II * (*this).chain.anis * ((*this).chain.Str_L[j] + 1.0 - 2.0 * a )
				       + 0.25 * II * PI * (1.0 - (*this).chain.par[j]))
				      - ((*this).lambda[k][beta] + 0.5 * II * (*this).chain.anis * ((*this).chain.Str_L[k] + 1.0 - 2.0 * b )
					 + 0.25 * II * PI * (1.0 - (*this).chain.par[k])) - II * (*this).chain.anis));

		    if ((j != k) || (alpha != beta) || (a != b + 1))

		      log_BAE_reg -= log(sinh(((*this).lambda[j][alpha] + 0.5 * II * (*this).chain.anis * ((*this).chain.Str_L[j] + 1.0 - 2.0 * a )
				       + 0.25 * II * PI * (1.0 - (*this).chain.par[j]))
				      - ((*this).lambda[k][beta] + 0.5 * II * (*this).chain.anis * ((*this).chain.Str_L[k] + 1.0 - 2.0 * b )
					 + 0.25 * II * PI * (1.0 - (*this).chain.par[k])) + II * (*this).chain.anis));
		  }

	      // The regular LHS of BAE is now defined.  Now sum up the deltas...

	      if (a == 1) ln_deltadiff[0] = - real(log_BAE_reg);

	      else if (a < (*this).chain.Str_L[j]) ln_deltadiff[a - 1] = ln_deltadiff[a-2] - real(log_BAE_reg);

	      else if (a == (*this).chain.Str_L[j]) ln_deltadiff[a-1] = real(log_BAE_reg);

	    } // if ((*this).chain.Str_L[j] > 1)

	  } // for (int a = 1; ...

	  for (int a = 0; a < (*this).chain.Str_L[j]; ++a) {
	    deltadiff[a] = ln_deltadiff[a] != 0.0 ? exp(ln_deltadiff[a]) : 0.0;
	    delta += fabs(deltadiff[a]);
	  }

	} // alpha sum
      } // j sum

      if (is_nan(delta)) delta = 1.0; // sentinel

    } // else

    return delta;
  }


  void 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 XXZ_Bethe_State::Compute_Momentum ()
  {
    int sum_Ix2 = 0;
    DP sum_M = 0.0;

    for (int j = 0; j < chain.Nstrings; ++j) {
      sum_M += 0.5 * (1.0 + chain.par[j]) * base[j];
      for (int alpha = 0; alpha < base[j]; ++alpha) {
	sum_Ix2 += Ix2[j][alpha];
      }
    }

    iK = (chain.Nsites/2) * int(sum_M + 0.1) - (sum_Ix2/2);  // + 0.1:  for safety...

    while (iK >= chain.Nsites) iK -= chain.Nsites;
    while (iK < 0) iK += chain.Nsites;

    K = PI * sum_M - PI * sum_Ix2/chain.Nsites;

    while (K >= 2.0*PI) K -= 2.0*PI;
    while (K < 0.0) K += 2.0*PI;

    return;
  }
  */
  void 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
		      += 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 * 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> (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 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 fbar_XXZ.");

    return (result);
  }

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

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

    else {

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

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

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

    return (result);
  }

  DP 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 hbar_XXZ.");

    return (result);
  }

  DP 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 : 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 * hbar_XXZ(lambda, fabs(nj - nk) + 2*a, parj*park, si_n_anis_over_2);

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

    return (result);
  }


  XXZ_Bethe_State Add_Particle_at_Center (const XXZ_Bethe_State& RefState)
  {
    if (2*RefState.base.Mdown == RefState.chain.Nsites)
      ABACUSerror("Trying to add a down spin to a zero-magnetized chain in Add_Particle_at_Center.");

    Vect<int> newM = RefState.base.Nrap;
    newM[0] = newM[0] + 1;

    Heis_Base newBase (RefState.chain, newM);

    XXZ_Bethe_State ReturnState (RefState.chain, newBase);

    for (int il = 1; il < RefState.chain.Nstrings; ++il)
      for (int alpha = 0; alpha < RefState.base.Nrap[il]; ++alpha)
	ReturnState.Ix2[il][alpha] = RefState.Ix2[il][alpha];

    // Add a quantum number in middle (explicitly: to right of index M[0]/2)
    // and shift quantum numbers by half-integer away from added one:
    ReturnState.Ix2[0][RefState.base.Nrap[0]/2] = RefState.Ix2[0][RefState.base.Nrap[0]/2] - 1;
    for (int i = 0; i < RefState.base.Nrap[0] + 1; ++i)
      ReturnState.Ix2[0][i + (i >= RefState.base.Nrap[0]/2)] = RefState.Ix2[0][i] - 1 + 2*(i >= RefState.base.Nrap[0]/2);

    return(ReturnState);
  }


  XXZ_Bethe_State Remove_Particle_at_Center (const XXZ_Bethe_State& RefState)
  {
    if (RefState.base.Nrap[0] == 0)
      ABACUSerror("Trying to remove a down spin in an empty Nrap[0] state.");

    Vect<int> newM = RefState.base.Nrap;
    newM[0] = newM[0] - 1;

    Heis_Base newBase (RefState.chain, newM);

    XXZ_Bethe_State ReturnState (RefState.chain, newBase);

    for (int il = 1; il < RefState.chain.Nstrings; ++il)
      for (int alpha = 0; alpha < RefState.base.Nrap[il]; ++alpha)
	ReturnState.Ix2[il][alpha] = RefState.Ix2[il][alpha];

    // Remove midmost and shift quantum numbers by half-integer towards removed one:
    for (int i = 0; i < RefState.base.Nrap[0]-1; ++i)
      ReturnState.Ix2[0][i] = RefState.Ix2[0][i + (i >= RefState.base.Nrap[0]/2)] + 1 - 2*(i >= RefState.base.Nrap[0]/2);

    return(ReturnState);
    }

} // namespace ABACUS