ABACUS/src/HEIS/XXZ_gpd_Bethe_State.cc

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

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

Copyright (c)

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

File:  src/HEIS/XXZ_gpd_Bethe_State.cc

Purpose:  Defines all functions for XXZ_gpd_Bethe_State

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

#include "ABACUS.h"

using namespace std;

namespace ABACUS {

  // Function prototypes

  inline DP fbar_XXZ_gpd (DP lambda, int par, DP tanhnetaover2);
  DP Theta_XXZ_gpd (DP lambda, int nj, int nk, DP* tanhnetaover2);
  DP hbar_XXZ_gpd (DP lambda, int n, DP* si_n_anis_over_2);
  DP ddlambda_Theta_XXZ_gpd (DP lambda, int nj, int nk, DP* si_n_anis_over_2);


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

  // Function definitions:  class XXZ_gpd_Bethe_State

  XXZ_gpd_Bethe_State::XXZ_gpd_Bethe_State ()
    : Heis_Bethe_State(), sinlambda(Lambda(chain, 1)), coslambda(Lambda(chain, 1)), tanlambda(Lambda(chain, 1))
  {};

  XXZ_gpd_Bethe_State::XXZ_gpd_Bethe_State (const XXZ_gpd_Bethe_State& RefState) // copy constructor
    : Heis_Bethe_State(RefState),
      sinlambda(Lambda(RefState.chain, RefState.base)), coslambda(Lambda(RefState.chain, RefState.base)),
      tanlambda(Lambda(RefState.chain, RefState.base))
  {
    // copy arrays into new ones

    for (int j = 0; j < RefState.chain.Nstrings; ++j) {
      for (int alpha = 0; alpha < RefState.base[j]; ++j) {
	sinlambda[j][alpha] = RefState.sinlambda[j][alpha];
	coslambda[j][alpha] = RefState.coslambda[j][alpha];
	tanlambda[j][alpha] = RefState.tanlambda[j][alpha];
      }
    }
  }

  XXZ_gpd_Bethe_State::XXZ_gpd_Bethe_State (const Heis_Chain& RefChain, int M)
    : Heis_Bethe_State(RefChain, M),
      sinlambda(Lambda(RefChain, M)), coslambda(Lambda(RefChain, M)), tanlambda(Lambda(RefChain, M))
  {
    if (RefChain.Delta <= 1.0) ABACUSerror("Delta too low in XXZ_gpd_Bethe_State constructor");
  }

  XXZ_gpd_Bethe_State::XXZ_gpd_Bethe_State (const Heis_Chain& RefChain, const Heis_Base& RefBase)
    : Heis_Bethe_State(RefChain, RefBase),
      sinlambda(Lambda(RefChain, RefBase)), coslambda(Lambda(RefChain, RefBase)),
      tanlambda(Lambda(RefChain, RefBase))
  {
    if (RefChain.Delta <= 1.0) ABACUSerror("Delta too low in XXZ_gpd_Bethe_State constructor");
  }
  /*
  XXZ_gpd_Bethe_State::XXZ_gpd_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),
      sinlambda(Lambda(chain, base)), coslambda(Lambda(chain, base)), tanlambda(Lambda(chain, base))
  {
    if (RefChain.Delta <= 1.0) ABACUSerror("Delta too low in XXZ_gpd_Bethe_State constructor");
  }
  */
  XXZ_gpd_Bethe_State& XXZ_gpd_Bethe_State::operator= (const XXZ_gpd_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;

      sinlambda = RefState.sinlambda;
      coslambda = RefState.coslambda;
      tanlambda = RefState.tanlambda;
    }
    return(*this);
  }


  // Member functions

  void XXZ_gpd_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) {

	lambda[i][alpha] = atan((tanh(chain.Str_L[i] * 0.5 * chain.anis) * tan(PI * 0.5 * Ix2[i][alpha]/chain.Nsites)));

      }
    }

    return;
  }

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

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

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

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

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

      for (int alpha = 0; alpha < base[j]; ++alpha) {
	tanlambda[j][alpha] = tan(lambda[j][alpha]);
	//if (lambda[j][alpha] > 0.5*PI) cout << "Rapidity higher than 0.5*PI:  j = " << j << "\talpha = " << alpha << "\trap = " << lambda[j][alpha] << "\ttan = " << tanlambda[j][alpha] << endl;
      }
    }
    return;
  }

  bool XXZ_gpd_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

    /*
    Vect<bool> min_Ix2_max_busy(false, chain.Nstrings);
    Vect<bool> plus_Ix2_max_busy(false, chain.Nstrings);
    for (int j = 0; j < chain.Nstrings; ++j)
      for (int alpha = 0; alpha < base[j]; ++alpha) {
	if (Ix2[j][alpha] == -base.Ix2_max[j]) min_Ix2_max_busy[j] = true;
	if (Ix2[j][alpha] == base.Ix2_max[j]) plus_Ix2_max_busy[j] = true;
      }
    */
    /*
    // State is not admissible if this is false:  -N/2 + 1 \leq \sum I^j_{\alpha} \leq N
    int sum_all_Ix2 = 0;
    for (int j = 0; j < chain.Nstrings; ++j)
      for (int alpha = 0; alpha < base[j]; ++alpha) {
	sum_all_Ix2 += Ix2[j][alpha];
      }
    if (sum_all_Ix2 > chain.Nsites || sum_all_Ix2 <= -chain.Nsites) {
      cout << "\tSum Ix2 out of fundamental interval: sum_all_Ix2 = " << sum_all_Ix2 << "\tfor N = " << chain.Nsites << endl;
      return(false);
    }
    */
    //Deactivated 2014 06 11, put in Heis_Form_Factor_Entry.cc
    /*
    // State is not admissible if I_max > 1/2 (N - \sum_k t_{jk} M_k) or I_min < -1/2 (N - sum_k...) + 1 at any level:
    int sum1 = 0;
    for (int j = 0; j < chain.Nstrings; ++j) {
      sum1 = 0;
      for (int k = 0; k < chain.Nstrings; ++k) {
	sum1 += base[k] * (2 * ABACUS::min(chain.Str_L[j], chain.Str_L[k]) - ((j == k) ? 1 : 0));
      }
      // Define limits...
      //if (!((Nrap[j] + Ix2_max[j]) % 2)) Ix2_max[j] -= 1;

      // This almost does it:  only missing are the states with one on -PI/2 and one on PI/2
      if (base[j] >= 1 && (Ix2[j][0] <= -(chain.Nsites - sum1) ||
			   (Ix2[j][base[j] - 1] - Ix2[j][0]) > 2*(chain.Nsites - sum1))) {
	//cout << "\tAn Ix2 is out of interval at level " << j << endl;
	//cout << Ix2[j][base[j] - 1] << "\t" << Ix2[j][0] << "\t" << chain.Nsites << "\t" << sum1 << endl;
	return(false);
      }
    }
    */

    /*
    // State is not admissible if both a min_ and plus_Ix2_max are busy simultaneously:
    bool is_a_min_Ix2_max_busy = false;
    for (int j = 0; j < chain.Nstrings; ++j) if (min_Ix2_max_busy[j]) is_a_min_Ix2_max_busy = true;
    bool is_a_plus_Ix2_max_busy = false;
    for (int j = 0; j < chain.Nstrings; ++j) if (plus_Ix2_max_busy[j]) is_a_plus_Ix2_max_busy = true;
    */
    /*
    // State is not admissible if all min_Ix2_max are busy simultaneously:
    bool any_min_Ix2_max_free = false;
    for (int j = 0; j < chain.Nstrings; ++j)
      if (base[j] > 0 && !min_Ix2_max_busy[j]) any_min_Ix2_max_free = true;
    if (!any_min_Ix2_max_free) return(false);
    */
    /*
    // State is not admissible if -Ix2_max, -Ix2_max + 2, ..., -Ix2_max + 2*(Str_L - 1) are busy:
    for (int j = 0; j < chain.Nstrings; ++j)
      if (base[j] > 0 && Ix2[j][0] <= -base.Ix2_max[j] + 2*(chain.Str_L[j] - 1))
	return(false);
    // Almost correct with above !
    // State is not admissible if Ix2_max - 2, ..., Ix2_max - 2*(Str_L - 2) are busy (NB:  one slot more than on left):
    for (int j = 0; j < chain.Nstrings; ++j)
      if (base[j] > 0 && Ix2[j][base[j] - 1] >= base.Ix2_max[j] - 2*(chain.Str_L[j] - 2))
	return(false);
    */

    // Check that at all at least doubly occupied levels, the difference between max and min quantum numbers
    // is strictly smaller than 2*Ix2_max - 2, so that lambda_max - lambda_min < PI at each level:
    //for (int j = 0; j < chain.Nstrings; ++j)
    //if (base[j] >= 2 && Ix2[j][base[j] - 1] - Ix2[j][0] >= 2* base.Ix2_max[j] - 2) answer = false;

    //if (is_a_min_Ix2_max_busy && is_a_plus_Ix2_max_busy) return(false);

    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.
    }

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

    bool string_coincidence = false;
    // Checks that we have strings of equal length modulo 2 and same parity with Ix2 == 0, so equal rapidities, and inadmissibility
    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.Str_L[j1] + chain.Str_L[j2])%2)))
	  string_coincidence = 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;
    //}

    //cout << min_Ix2_max_busy << "\t" << symmetric_state  << "\t" << higher_string_on_zero << "\t" << string_coincidence << "\t" << onep_onem_on_zero << endl;

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

    // Explicitly exclude state with min_Ix2_max_busy && iK == 0
    //Compute_Momentum();
    //answer = !((min_Ix2_max_busy && iK == 0) || (symmetric_state && (higher_string_on_zero || string_coincidence || onep_onem_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_gpd_Bethe_State::Compute_BE (int j, int alpha)
  {
    // Fills in the BE members with the value of the Bethe equations.

    tanlambda[j][alpha] = tan(lambda[j][alpha]);

    DP sumtheta = 0.0;

    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 += atan ((tanlambda[j][alpha] - tanlambda[k][beta])/((1.0 + tanlambda[j][alpha] * tanlambda[k][beta])
									* chain.ta_n_anis_over_2[2]))
	    + PI * floor(0.5 + (lambda[j][alpha] - lambda[k][beta])/PI);
	}
	else sumtheta += 0.5 * Theta_XXZ_gpd((tanlambda[j][alpha] - tanlambda[k][beta])/(1.0 + tanlambda[j][alpha] * tanlambda[k][beta]),
					     chain.Str_L[j], chain.Str_L[k], chain.ta_n_anis_over_2)
	  + PI * (2.0 * ABACUS::min(chain.Str_L[j], chain.Str_L[k]) - ((j == k) ? 1.0 : 0))
	  * floor(0.5 + (lambda[j][alpha] - lambda[k][beta])/PI);
    }
    sumtheta *= 2.0;

    BE[j][alpha] = 2.0 * (atan(tanlambda[j][alpha]/chain.ta_n_anis_over_2[chain.Str_L[j]])
			  + PI * floor(0.5 + lambda[j][alpha]/PI))
      - (sumtheta + PI*Ix2[j][alpha])/chain.Nsites;
  }

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

    (*this).Compute_tanlambda();

    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 += atan ((tanlambda[j][alpha] - tanlambda[k][beta])/((1.0 + tanlambda[j][alpha] * tanlambda[k][beta])
									    * chain.ta_n_anis_over_2[2]))
		+ PI * floor(0.5 + (lambda[j][alpha] - lambda[k][beta])/PI);
	    }
	    else sumtheta += 0.5 * Theta_XXZ_gpd((tanlambda[j][alpha] - tanlambda[k][beta])/(1.0 + tanlambda[j][alpha] * tanlambda[k][beta]),
						 chain.Str_L[j], chain.Str_L[k], chain.ta_n_anis_over_2)
	      + PI * (2.0 * ABACUS::min(chain.Str_L[j], chain.Str_L[k]) - ((j == k) ? 1.0 : 0))
	      * floor(0.5 + (lambda[j][alpha] - lambda[k][beta])/PI);
	}
	sumtheta *= 2.0;

	BE[j][alpha] = 2.0 * (atan(tanlambda[j][alpha]/chain.ta_n_anis_over_2[chain.Str_L[j]])
			      + PI * floor(0.5 + lambda[j][alpha]/PI))
	  - (sumtheta + PI*Ix2[j][alpha])/chain.Nsites;
      }
  }

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

    DP arg0 = 0.5 * (2.0 * (atan(tanlambda[j][alpha]/chain.ta_n_anis_over_2[chain.Str_L[j]])
			    + PI * floor(0.5 + lambda[j][alpha]/PI)) - BE[j][alpha]);
    DP arg = chain.ta_n_anis_over_2[chain.Str_L[j]] * tan(
							  arg0
							  //0.5 *
							  //(PI * Ix2[j][alpha] + sumtheta)/chain.Nsites
							  //(2.0 * (atan(tanlambda[j][alpha]/chain.ta_n_anis_over_2[chain.Str_L[j]])
							  //  + PI * floor(0.5 + lambda[j][alpha]/PI)) - BE[j][alpha])
							  );

    return(atan(arg)
	   //+ PI * floor(0.5 + arg0)
				  //0.5 * (Ix2[j][alpha] + sumtheta/PI)/(chain.Nsites)
	   + PI * floor(0.5 + arg0/PI)
				  );

  }
  /*
  void XXZ_gpd_Bethe_State::Iterate_BAE ()
  {
    // Recalculates the rapidities by iterating Bethe equations

    Lambda New_lambda(chain, base);
    DP sumtheta = 0.0;
    DP arg = 0.0;

    // First, compute the tan of rapidities:
    (*this).Compute_tanlambda();

    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 += atan((tanlambda[j][alpha] - tanlambda[k][beta])/((1.0 + tanlambda[j][alpha] * tanlambda[k][beta])
									   * chain.ta_n_anis_over_2[2]))
		+ PI * floor(0.5 + (lambda[j][alpha] - lambda[k][beta])/PI);

	    else sumtheta += 0.5 * Theta_XXZ_gpd((tanlambda[j][alpha] - tanlambda[k][beta])/(1.0 + tanlambda[j][alpha] * tanlambda[k][beta]),
					     chain.Str_L[j], chain.Str_L[k], chain.ta_n_anis_over_2)
	      + PI * (2.0 * ABACUS::min(chain.Str_L[j], chain.Str_L[k]) - ((j == k) ? 1.0 : 0))
	      * floor(0.5 + (lambda[j][alpha] - lambda[k][beta])/PI);
	}
	sumtheta *= 2.0;

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

	New_lambda[j][alpha] = atan(arg) + PI * floor(0.5 + (0.5 * Ix2[j][alpha] + 0.5 * sumtheta/PI)/(chain.Nsites));

      }
    }

    DP New_diffsq = 0.0;

    for (int j = 0; j < chain.Nstrings; ++j) {
      for (int alpha = 0; alpha < base[j]; ++alpha) {
	//New_diffsq += pow(tan(New_lambda[j][alpha]) - tanlambda[j][alpha], 2.0);
	New_diffsq += pow(New_lambda[j][alpha] - lambda[j][alpha], 2.0);
	lambda[j][alpha] = 1.0 * New_lambda[j][alpha] + 0.0 * lambda[j][alpha];
      }
    }

    diffsq = New_diffsq;
    iter++;

    return;
  }
  */
  /*
  void XXZ_gpd_Bethe_State::Iterate_BAE_Newton ()
  {
    // does one step of a Newton method on the rapidities...

    Vect_DP RHSBAE (0.0, base.Nraptot);  // contains RHS of BAEs
    Vect_CX dlambda (0.0, base.Nraptot);  // contains delta lambda computed from Newton's method
    SQMat_CX Gaudin (0.0, base.Nraptot);
    Vect_INT indx (base.Nraptot);
    DP sumtheta = 0.0;
    DP arg = 0.0;
    DP fn_arg = 0.0;
    DP olddiffsq = diffsq;

    // Compute the RHS of the BAEs:

    int index = 0;

    (*this).Compute_tanlambda();

    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 += atan ((tanlambda[j][alpha] - tanlambda[k][beta])/((1.0 + tanlambda[j][alpha] * tanlambda[k][beta])
									    * chain.ta_n_anis_over_2[2]))
		+ PI * floor(0.5 + (lambda[j][alpha] - lambda[k][beta])/PI);
	    }
	    else sumtheta += 0.5 * Theta_XXZ_gpd((tanlambda[j][alpha] - tanlambda[k][beta])/(1.0 + tanlambda[j][alpha] * tanlambda[k][beta]),
					     chain.Str_L[j], chain.Str_L[k], chain.ta_n_anis_over_2)
	      + PI * (2.0 * ABACUS::min(chain.Str_L[j], chain.Str_L[k]) - ((j == k) ? 1.0 : 0))
	      * floor(0.5 + (lambda[j][alpha] - lambda[k][beta])/PI);
	}
	sumtheta *= 2.0;

	RHSBAE[index] = chain.Nsites * 2.0 * (atan(tanlambda[j][alpha]/chain.ta_n_anis_over_2[chain.Str_L[j]])
       					      + PI * floor(0.5 + lambda[j][alpha]/PI))
					      //					      )
	  - sumtheta - PI*Ix2[j][alpha];
	index++;
      }
    }

    (*this).Build_Reduced_Gaudin_Matrix (Gaudin);

    for (int i = 0; i < base.Nraptot; ++i) dlambda[i] = - RHSBAE[i];

    DP d;
    ludcmp_CX (Gaudin, indx, d);
    lubksb_CX (Gaudin, indx, dlambda);

    diffsq = 0.0;
    //    for (int i = 0; i < base.Nraptot; ++i) diffsq += norm(dlambda[i]);
    int ctr = 0;
    for (int j = 0; j < chain.Nstrings; ++j) {
      for (int alpha = 0; alpha < base[j]; ++alpha) {
	diffsq += norm(tan(lambda[j][alpha] + dlambda[ctr]) - tanlambda[j][alpha]);
	//	cout << "lambda = " << lambda[j][alpha] << "\tdlambda = " << dlambda[ctr] << endl;
	ctr++;
      }
    }

    // if we've converged, calculate the norm here, since the work has been done...

    if (diffsq < chain.prec) {
      lnnorm = 0.0;
      for (int j = 0; j < base.Nraptot; j++) lnnorm += log(abs(Gaudin[j][j]));
    }

    index = 0;
    for (int j = 0; j < chain.Nstrings; ++j) {
      for (int alpha = 0; alpha < base[j]; ++alpha) {
	lambda[j][alpha] += real(dlambda[index]);
	index++;
      }
    }

    iter_Newton++;

    //    cout << "iter_N = " << iter_Newton << "\t" << diffsq << endl;

    return;
  }
  */
  bool XXZ_gpd_Bethe_State::Check_Rapidities()
  {
    bool nonan = true;

    for (int j = 0; j < chain.Nstrings; ++j)
      for (int alpha = 0; alpha < base[j]; ++alpha) if (nonan) nonan = ((!is_nan(lambda[j][alpha]))
									//&& (lambda[j][alpha] > -0.5*PI*chain.Str_L[j])
									//&& (lambda[j][alpha] <= 0.5*PI*chain.Str_L[j])
									);

    bool all_within_pi_interval = true;
    DP min_lambda = 10.0;
    DP max_lambda = -10.0;
    for (int j = 0; j < chain.Nstrings; ++j)
      for (int alpha = 0; alpha < base[j]; ++alpha) {
	if (lambda[j][alpha] < min_lambda) min_lambda = lambda[j][alpha];
	if (lambda[j][alpha] > max_lambda) max_lambda = lambda[j][alpha];
      }

    //all_within_pi_interval = (fabs(max_lambda - min_lambda) < PI - 0.0 * 1.0e-12);
    // Use pi interval, but centered on 0:
    all_within_pi_interval = max_lambda < 0.5* PI + 1.0e-8 && min_lambda > -0.5* PI - 1.0e-8;
    //nonan *= all_within_pi_interval;

    return nonan;
  }

  DP XXZ_gpd_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(sin((*this).lambda[j][alpha] + 0.5 * II * (*this).chain.anis
							     * ((*this).chain.Str_L[j] + 1.0 - 2.0 * a + 1.0))
			    /sin((*this).lambda[j][alpha] + 0.5 * II * (*this).chain.anis * ((*this).chain.Str_L[j] + 1.0 - 2.0 * a - 1.0)));

	      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(sin(((*this).lambda[j][alpha] + 0.5 * II * (*this).chain.anis * ((*this).chain.Str_L[j] + 1.0 - 2.0 * a ))
				      - ((*this).lambda[k][beta] + 0.5 * II * (*this).chain.anis * ((*this).chain.Str_L[k] + 1.0 - 2.0 * b ))
					     - II * (*this).chain.anis));

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

		      log_BAE_reg -= log(sin(((*this).lambda[j][alpha] + 0.5 * II * (*this).chain.anis * ((*this).chain.Str_L[j] + 1.0 - 2.0 * a ))
				      - ((*this).lambda[k][beta] + 0.5 * II * (*this).chain.anis * ((*this).chain.Str_L[k] + 1.0 - 2.0 * b ))
					     + 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_gpd_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 += sinh(chain.Str_L[j] * chain.anis) / (cos(2.0 * lambda[j][alpha]) - cosh(chain.Str_L[j] * chain.anis));
      }
    }

    sum *= chain.J * sinh(chain.anis);

    E = sum;

    return;
  }

  /*
  void XXZ_gpd_Bethe_State::Compute_Momentum ()
  {
    int sum_Ix2 = 0;
    DP sum_M = 0.0;

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

    iK = (chain.Nsites/2) * int(sum_M) - (sum_Ix2/2);

    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_gpd_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 sinhetasq = pow(sinh(chain.anis), 2.0);

    (*this).Compute_sinlambda();
    (*this).Compute_coslambda();

    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_gpd (lambda[j][alpha] - lambda[kp][betap], chain.Str_L[j], chain.Str_L[kp],
					     chain.si_n_anis_over_2);
		}
	      }

	      Gaudin_Red[index_jalpha][index_kbeta]
		= complex<DP> ( chain.Nsites * hbar_XXZ_gpd (lambda[j][alpha], chain.Str_L[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.si_n_anis_over_2[4]/(pow(sinlambda[j][alpha] * coslambda[k][beta]
							       - coslambda[j][alpha] * sinlambda[k][beta], 2.0) + sinhetasq));
	      else
		Gaudin_Red[index_jalpha][index_kbeta] = complex<DP> (ddlambda_Theta_XXZ_gpd (lambda[j][alpha] - lambda[k][beta], chain.Str_L[j],
											     chain.Str_L[k], chain.si_n_anis_over_2));
	    }
	    index_kbeta++;
	  }
	}
	index_jalpha++;
      }
    }

    return;
  }


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

  // non-member functions

  inline DP fbar_XXZ_gpd (DP tanlambda, DP tanhnetaover2)
  {
    return (2.0 * atan(tanlambda/tanhnetaover2));
  }

  DP Theta_XXZ_gpd (DP tanlambda, int nj, int nk, DP* tanhnetaover2)
  {
    DP result;

    if ((nj == 1) && (nk == 1)) result = fbar_XXZ_gpd(tanlambda, tanhnetaover2[2]);

    else {

      result = (nj == nk) ? 0.0 : fbar_XXZ_gpd(tanlambda, tanhnetaover2[fabs(nj - nk)]);

      for (int a = 1; a < ABACUS::min(nj, nk); ++a) result += 2.0 * fbar_XXZ_gpd(tanlambda, tanhnetaover2[fabs(nj - nk) + 2*a]);

      result += fbar_XXZ_gpd(tanlambda, tanhnetaover2[nj + nk]);
    }

    return (result);
  }

  DP hbar_XXZ_gpd (DP lambda, int n, DP* si_n_anis_over_2)
  {
    return (si_n_anis_over_2[2*n]/(pow(sin(lambda), 2.0) + pow(si_n_anis_over_2[n], 2.0)));
  }

  DP ddlambda_Theta_XXZ_gpd (DP lambda, int nj, int nk, DP* si_n_anis_over_2)
  {
    DP result = (nj == nk) ? 0.0 : hbar_XXZ_gpd(lambda, fabs(nj - nk), si_n_anis_over_2);

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

    result += hbar_XXZ_gpd(lambda, nj + nk, si_n_anis_over_2);

    return (result);
  }


  XXZ_gpd_Bethe_State Add_Particle_at_Center (const XXZ_gpd_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_gpd_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_gpd_Bethe_State Remove_Particle_at_Center (const XXZ_gpd_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_gpd_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