ABACUS/src/HEIS/ln_Smin_ME_XXZ.cc

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

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

Copyright (c) J.-S. Caux.

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

File: ln_Smin_ME_XXZ.cc

Purpose: compute the S^- matrix elemment for XXZ

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

#include "ABACUS.h"

using namespace std;
using namespace ABACUS;

namespace ABACUS {

inline complex<DP> ln_Fn_F (XXZ_Bethe_State& B, int k, int beta, int b)
{
  complex<DP> ans = 0.0;
  complex<DP> prod_temp = 1.0;
  int counter = 0;
  int arg = 0;
  int absarg = 0;
  int par_comb_1, par_comb_2;

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

    par_comb_1 = B.chain.par[j] == B.chain.par[k] ? 1 : 0;
    par_comb_2 = B.chain.par[k] == B.chain.par[j] ? 0 : B.chain.par[k];

    for (int alpha = 0; alpha < B.base.Nrap[j]; ++alpha) {
      for (int a = 1; a <= B.chain.Str_L[j]; ++a) {

	if (!((j == k) && (alpha == beta) && (a == b))) {

	  arg = B.chain.Str_L[j] - B.chain.Str_L[k] - 2 * (a - b);
	  absarg = abs(arg);
	  /*
	  prod_temp *= 0.5 * //done later...
	    ((B.sinhlambda[j][alpha] * B.coshlambda[k][beta] - B.coshlambda[j][alpha] * B.sinhlambda[k][beta])
	     * (B.chain.co_n_anis_over_2[absarg] * (1.0 + B.chain.par[j] * B.chain.par[k])
		- sgn_int(arg) * B.chain.si_n_anis_over_2[absarg] * (B.chain.par[k] - B.chain.par[j]))
			+ II * (B.coshlambda[j][alpha] * B.coshlambda[k][beta] - B.sinhlambda[j][alpha] * B.sinhlambda[k][beta])
			* (sgn_int(arg) * B.chain.si_n_anis_over_2[absarg] * (1.0 + B.chain.par[j] * B.chain.par[k])
			   + B.chain.co_n_anis_over_2[absarg] * (B.chain.par[k] - B.chain.par[j])) );
	  */

	  prod_temp *= ((B.sinhlambda[j][alpha] * B.coshlambda[k][beta] - B.coshlambda[j][alpha] * B.sinhlambda[k][beta])
			* (B.chain.co_n_anis_over_2[absarg] * par_comb_1 - sgn_int(arg) * B.chain.si_n_anis_over_2[absarg] * par_comb_2)
			+ II * (B.coshlambda[j][alpha] * B.coshlambda[k][beta] - B.sinhlambda[j][alpha] * B.sinhlambda[k][beta])
			* (sgn_int(arg) * B.chain.si_n_anis_over_2[absarg] * par_comb_1 + B.chain.co_n_anis_over_2[absarg] * par_comb_2));
	}

	if (counter++ > 100) {  // we do at most 100 products before taking a log
	  ans += log(prod_temp);
	  prod_temp = 1.0;
	  counter = 0;
	}

      }}}

  return(ans + log(prod_temp));
}

inline complex<DP> ln_Fn_G (XXZ_Bethe_State& A, XXZ_Bethe_State& B, int k, int beta, int b)
{
  complex<DP> ans = 0.0;
  complex<DP> prod_temp = 1.0;
  int counter = 0;
  int arg = 0;
  int absarg = 0;
  int par_comb_1, par_comb_2;

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

    par_comb_1 = A.chain.par[j] == B.chain.par[k] ? 1 : 0;
    par_comb_2 = B.chain.par[k] == A.chain.par[j] ? 0 : B.chain.par[k];

    for (int alpha = 0; alpha < A.base.Nrap[j]; ++alpha) {
      for (int a = 1; a <= A.chain.Str_L[j]; ++a) {

	arg = A.chain.Str_L[j] - B.chain.Str_L[k] - 2 * (a - b);
	absarg = abs(arg);
	/*
	prod_temp *= 0.5 * //done later...
	  ((A.sinhlambda[j][alpha] * B.coshlambda[k][beta] - A.coshlambda[j][alpha] * B.sinhlambda[k][beta])
		      * (A.chain.co_n_anis_over_2[absarg] * (1.0 + A.chain.par[j] * B.chain.par[k])
			 - sgn_int(arg) * A.chain.si_n_anis_over_2[absarg] * (B.chain.par[k] - A.chain.par[j]))
		      + II * (A.coshlambda[j][alpha] * B.coshlambda[k][beta] - A.sinhlambda[j][alpha] * B.sinhlambda[k][beta])
		      * (sgn_int(arg) * A.chain.si_n_anis_over_2[absarg] * (1.0 + A.chain.par[j] * B.chain.par[k])
			 + A.chain.co_n_anis_over_2[absarg] * (B.chain.par[k] - A.chain.par[j])) );
	*/
	prod_temp *= ((A.sinhlambda[j][alpha] * B.coshlambda[k][beta] - A.coshlambda[j][alpha] * B.sinhlambda[k][beta])
		      * (A.chain.co_n_anis_over_2[absarg] * par_comb_1 - sgn_int(arg) * A.chain.si_n_anis_over_2[absarg] * par_comb_2)
		      + II * (A.coshlambda[j][alpha] * B.coshlambda[k][beta] - A.sinhlambda[j][alpha] * B.sinhlambda[k][beta])
		      * (sgn_int(arg) * A.chain.si_n_anis_over_2[absarg] * par_comb_1 + A.chain.co_n_anis_over_2[absarg] * par_comb_2));

	if (counter++ > 100) {  // we do at most 100 products before taking a log
	  ans += log(prod_temp);
	  prod_temp = 1.0;
	  counter = 0;
	}
      }}}

  return(ans + log(prod_temp));
}

inline complex<DP> Fn_K (XXZ_Bethe_State& A, int j, int alpha, int a, XXZ_Bethe_State& B, int k, int beta, int b)
{
  int arg1 = A.chain.Str_L[j] - B.chain.Str_L[k] - 2 * (a - b);
  int absarg1 = abs(arg1);
  int arg2 = arg1 + 2;
  int absarg2 = abs(arg2);

  return(4.0/(
	      ((A.sinhlambda[j][alpha] * B.coshlambda[k][beta] - A.coshlambda[j][alpha] * B.sinhlambda[k][beta])
		      * (A.chain.co_n_anis_over_2[absarg1] * (1.0 + A.chain.par[j] * B.chain.par[k])
			 - sgn_int(arg1) * A.chain.si_n_anis_over_2[absarg1] * (B.chain.par[k] - A.chain.par[j]))
		      + II * (A.coshlambda[j][alpha] * B.coshlambda[k][beta] - A.sinhlambda[j][alpha] * B.sinhlambda[k][beta])
		      * (sgn_int(arg1) * A.chain.si_n_anis_over_2[absarg1] * (1.0 + A.chain.par[j] * B.chain.par[k])
			 + A.chain.co_n_anis_over_2[absarg1] * (B.chain.par[k] - A.chain.par[j])) )
	 *
	 ((A.sinhlambda[j][alpha] * B.coshlambda[k][beta] - A.coshlambda[j][alpha] * B.sinhlambda[k][beta])
		      * (A.chain.co_n_anis_over_2[absarg2] * (1.0 + A.chain.par[j] * B.chain.par[k])
			 - sgn_int(arg2) * A.chain.si_n_anis_over_2[absarg2] * (B.chain.par[k] - A.chain.par[j]))
		      + II * (A.coshlambda[j][alpha] * B.coshlambda[k][beta] - A.sinhlambda[j][alpha] * B.sinhlambda[k][beta])
		      * (sgn_int(arg2) * A.chain.si_n_anis_over_2[absarg2] * (1.0 + A.chain.par[j] * B.chain.par[k])
			 + A.chain.co_n_anis_over_2[absarg2] * (B.chain.par[k] - A.chain.par[j])) )
	      ));

}

inline complex<DP> Fn_L (XXZ_Bethe_State& A, int j, int alpha, int a, XXZ_Bethe_State& B, int k, int beta, int b)
{
  return (sinh(2.0 * (A.lambda[j][alpha] - B.lambda[k][beta]
		      + 0.5 * II * B.chain.anis * (A.chain.Str_L[j] - B.chain.Str_L[k] - 2.0 * (a - b - 0.5))
		      + 0.25 * II * PI * complex<DP>(-A.chain.par[j] + B.chain.par[k])))
	  * pow(Fn_K (A, j, alpha, a, B, k, beta, b), 2.0));
}

complex<DP> ln_Smin_ME (XXZ_Bethe_State& A, XXZ_Bethe_State& B)
{
  // This function returns the natural log of the S^- operator matrix element.
  // The A and B states can contain strings.

  // Check that the two states are compatible

  if (A.chain != B.chain) ABACUSerror("Incompatible XXZ_Chains in Smin matrix element.");

  // Check that A and B are Mdown-compatible:

  if (A.base.Mdown != B.base.Mdown + 1) {
    cout << "A.base.Mdown = " << A.base.Mdown << "\tB.base.Mdown = " << B.base.Mdown << endl;
    cout << "A: " << A << endl << "B: " << B << endl;
    ABACUSerror("Incompatible Mdown between the two states in Smin matrix element!");
  }

  // Compute the sinh and cosh of rapidities

  A.Compute_sinhlambda();
  A.Compute_coshlambda();
  B.Compute_sinhlambda();
  B.Compute_coshlambda();

  // Some convenient arrays

  Lambda re_ln_Fn_F_B_0(B.chain, B.base);
  Lambda im_ln_Fn_F_B_0(B.chain, B.base);
  Lambda re_ln_Fn_G_0(B.chain, B.base);
  Lambda im_ln_Fn_G_0(B.chain, B.base);
  Lambda re_ln_Fn_G_2(B.chain, B.base);
  Lambda im_ln_Fn_G_2(B.chain, B.base);

  complex<DP> ln_prod1 = 0.0;
  complex<DP> ln_prod2 = 0.0;
  complex<DP> ln_prod3 = 0.0;
  complex<DP> ln_prod4 = 0.0;

  for (int i = 0; i < A.chain.Nstrings; ++i)
    for (int alpha = 0; alpha < A.base.Nrap[i]; ++alpha)
      for (int a = 1; a <= A.chain.Str_L[i]; ++a)
	ln_prod1 += log(norm(sinh(A.lambda[i][alpha] + 0.5 * II * A.chain.anis * (A.chain.Str_L[i] + 1.0 - 2.0 * a - 1.0)
				  + 0.25 * II * PI * (1.0 - A.chain.par[i]))));

  for (int i = 0; i < B.chain.Nstrings; ++i)
    for (int alpha = 0; alpha < B.base.Nrap[i]; ++alpha)
      for (int a = 1; a <= B.chain.Str_L[i]; ++a)
	if (norm(sinh(B.lambda[i][alpha] + 0.5 * II * B.chain.anis * (B.chain.Str_L[i] + 1.0 - 2.0 * a - 1.0)
		      + 0.25 * II * PI * (1.0 - B.chain.par[i]))) > 100.0 * MACHINE_EPS_SQ)
	  ln_prod2 += log(norm(sinh(B.lambda[i][alpha] + 0.5 * II * B.chain.anis * (B.chain.Str_L[i] + 1.0 - 2.0 * a - 1.0)
				    + 0.25 * II * PI * (1.0 - B.chain.par[i]))));

  // Define the F ones earlier...

  for (int j = 0; j < B.chain.Nstrings; ++j) {
    for (int alpha = 0; alpha < B.base.Nrap[j]; ++alpha) {
      re_ln_Fn_F_B_0[j][alpha] = real(ln_Fn_F(B, j, alpha, 0));
      im_ln_Fn_F_B_0[j][alpha] = imag(ln_Fn_F(B, j, alpha, 0));
      re_ln_Fn_G_0[j][alpha] = real(ln_Fn_G(A, B, j, alpha, 0));
      im_ln_Fn_G_0[j][alpha] = imag(ln_Fn_G(A, B, j, alpha, 0));
      re_ln_Fn_G_2[j][alpha] = real(ln_Fn_G(A, B, j, alpha, 2));
      im_ln_Fn_G_2[j][alpha] = imag(ln_Fn_G(A, B, j, alpha, 2));
    }
  }

  DP logabssinzeta = log(abs(sin(A.chain.anis)));

  // Define regularized products in prefactors

  for (int j = 0; j < A.chain.Nstrings; ++j)
    for (int alpha = 0; alpha < A.base.Nrap[j]; ++alpha)
      for (int a = 1; a <= A.chain.Str_L[j]; ++a)
	ln_prod3 += ln_Fn_F(A, j, alpha, a - 1);  // assume only one-strings here

  ln_prod3 -= A.base.Mdown * log(abs(sin(A.chain.anis)));

  for (int k = 0; k < B.chain.Nstrings; ++k)
    for (int beta = 0; beta < B.base.Nrap[k]; ++beta)
      for (int b = 1; b <= B.chain.Str_L[k]; ++b) {
	if (b == 1) ln_prod4 += re_ln_Fn_F_B_0[k][beta];
	else if (b > 1) ln_prod4 += ln_Fn_F(B, k, beta, b - 1);
    }

  ln_prod4 -= B.base.Mdown * log(abs(sin(B.chain.anis)));

  // Now proceed to build the Hm matrix

  SQMat_CX Hm(0.0, A.base.Mdown);

  int index_a = 0;
  int index_b = 0;

  complex<DP> sum1 = 0.0;
  complex<DP> sum2 = 0.0;
  complex<DP> prod_num = 0.0;
  complex<DP> Fn_K_0_G_0 = 0.0;
  complex<DP> Prod_powerN = 0.0;
  complex<DP> Fn_K_1_G_2 = 0.0;
  complex<DP> one_over_A_sinhlambda_sq_plus_sinzetaover2sq;


  for (int j = 0; j < A.chain.Nstrings; ++j) {
    for (int alpha = 0; alpha < A.base.Nrap[j]; ++alpha) {
      for (int a = 1; a <= A.chain.Str_L[j]; ++a) {

	index_b = 0;

	one_over_A_sinhlambda_sq_plus_sinzetaover2sq = 1.0/((sinh(A.lambda[j][alpha] + 0.5 * II * A.chain.anis * (A.chain.Str_L[j] + 1.0 - 2.0 * a)
								  + 0.25 * II * PI * (1.0 - A.chain.par[j])))
							    * (sinh(A.lambda[j][alpha] + 0.5 * II * A.chain.anis * (A.chain.Str_L[j] + 1.0 - 2.0 * a)
								  + 0.25 * II * PI * (1.0 - A.chain.par[j])))
							    + pow(sin(0.5*A.chain.anis), 2.0));

	for (int k = 0; k < B.chain.Nstrings; ++k) {
	  for (int beta = 0; beta < B.base.Nrap[k]; ++beta) {
	    for (int b = 1; b <= B.chain.Str_L[k]; ++b) {

	      if (B.chain.Str_L[k] == 1) {

		// use simplified code for one-string here:  original form of Hm matrix

		Fn_K_0_G_0 = Fn_K (A, j, alpha, a, B, k, beta, 0) *
		  exp(re_ln_Fn_G_0[k][beta] + II * im_ln_Fn_G_0[k][beta] - re_ln_Fn_F_B_0[k][beta] + logabssinzeta);
		Fn_K_1_G_2 = Fn_K (A, j, alpha, a, B, k, beta, 1) *
		  exp(re_ln_Fn_G_2[k][beta] + II * im_ln_Fn_G_2[k][beta] - re_ln_Fn_F_B_0[k][beta] + logabssinzeta);

		Prod_powerN = pow( B.chain.par[k] == 1 ?
				   (B.sinhlambda[k][beta] * B.chain.co_n_anis_over_2[1] + II * B.coshlambda[k][beta] * B.chain.si_n_anis_over_2[1])
				   /(B.sinhlambda[k][beta] * B.chain.co_n_anis_over_2[1] - II * B.coshlambda[k][beta] * B.chain.si_n_anis_over_2[1])
				   :
				   (B.coshlambda[k][beta] * B.chain.co_n_anis_over_2[1] + II * B.sinhlambda[k][beta] * B.chain.si_n_anis_over_2[1])
				   /(B.coshlambda[k][beta] * B.chain.co_n_anis_over_2[1] - II * B.sinhlambda[k][beta] * B.chain.si_n_anis_over_2[1])
				   , complex<DP> (B.chain.Nsites));

		Hm[index_a][index_b] = Fn_K_0_G_0 - Prod_powerN * Fn_K_1_G_2;

	      } // if (B.chain.Str_L == 1)

	      else {

		if (b <= B.chain.Str_L[k] - 1) Hm[index_a][index_b] = Fn_K(A, j, alpha, a, B, k, beta, b);
		else if (b == B.chain.Str_L[k]) {

		  Vect_CX ln_FunctionF(B.chain.Str_L[k] + 2);
		  for (int i = 0; i < B.chain.Str_L[k] + 2; ++i) ln_FunctionF[i] = ln_Fn_F (B, k, beta, i);

		  Vect_CX ln_FunctionG(B.chain.Str_L[k] + 2);
		  for (int i = 0; i < B.chain.Str_L[k] + 2; ++i) ln_FunctionG[i] = ln_Fn_G (A, B, k, beta, i);

		  sum1 = 0.0;

		  sum1 += Fn_K (A, j, alpha, a, B, k, beta, 0) * exp(ln_FunctionG[0] + ln_FunctionG[1] - ln_FunctionF[0] - ln_FunctionF[1]);

		  sum1 += Fn_K (A, j, alpha, a, B, k, beta, B.chain.Str_L[k])
		    * exp(ln_FunctionG[B.chain.Str_L[k]] + ln_FunctionG[B.chain.Str_L[k] + 1]
			  - ln_FunctionF[B.chain.Str_L[k]] - ln_FunctionF[B.chain.Str_L[k] + 1]);

		  for (int jsum = 1; jsum < B.chain.Str_L[k]; ++jsum)

		    sum1 -= Fn_L (A, j, alpha, a, B, k, beta, jsum) *
		      exp(ln_FunctionG[jsum] + ln_FunctionG[jsum + 1] - ln_FunctionF[jsum] - ln_FunctionF[jsum + 1]);

		  /*
		  sum2 = 0.0;

		  for (int jsum = 1; jsum <= B.chain.Str_L[k]; ++jsum)  sum2 += exp(ln_FunctionG[jsum] - ln_FunctionF[jsum]);

		  */
		  prod_num = exp(II * im_ln_Fn_F_B_0[k][beta] + ln_FunctionF[1] - ln_FunctionG[B.chain.Str_L[k]] + logabssinzeta);

		  for (int jsum = 2; jsum <= B.chain.Str_L[k]; ++jsum)
		    prod_num *= exp(ln_FunctionG[jsum] - real(ln_Fn_F(B, k, beta, jsum - 1)) + logabssinzeta);
		  // include all string contributions F_B_0 in this term

		  Hm[index_a][index_b] = prod_num * sum1;

		} // else if (b == B.chain.Str_L[k])
	      } // else

	      index_b++;
	    }}} // sums over k, beta, b

	// now define the elements Hm[a][M]

	Hm[index_a][B.base.Mdown] = one_over_A_sinhlambda_sq_plus_sinzetaover2sq;

	index_a++;
      }}} // sums over j, alpha, a

  complex<DP> ln_ME_sq = log(1.0 * A.chain.Nsites) + real(ln_prod1 - ln_prod2) - real(ln_prod3) + real(ln_prod4)
    + 2.0 * real(lndet_LU_CX_dstry(Hm)) + logabssinzeta - A.lnnorm - B.lnnorm;

  //return(ln_ME_sq);
  return(0.5 * ln_ME_sq); // Return ME, not MEsq

}

} // namespace ABACUS