ABACUS/dev/ODSLF/ln_Smin_ME_ODSLF_XXZ.cc

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

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

Copyright (c) J.-S. Caux.

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

File:  ln_Smin_ME_ODSLF_XXZ.cc

Purpose:  S^- matrix element

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

#include "ABACUS.h"

using namespace ABACUS;

namespace ABACUS {

  inline complex<DP> ln_Fn_F (ODSLF_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 *= ((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 (ODSLF_XXZ_Bethe_State& A, ODSLF_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 *= ((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 (ODSLF_XXZ_Bethe_State& A, int j, int alpha, int a,
			   ODSLF_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 (ODSLF_XXZ_Bethe_State& A, int j, int alpha, int a,
			   ODSLF_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 (ODSLF_XXZ_Bethe_State& A, ODSLF_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 ODSLF_XXZ_Chains in Smin matrix element.");

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

    if (A.base.Mdown != B.base.Mdown + 1)
      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

    ODSLF_Lambda re_ln_Fn_F_B_0(B.chain, B.base);
    ODSLF_Lambda im_ln_Fn_F_B_0(B.chain, B.base);
    ODSLF_Lambda re_ln_Fn_G_0(B.chain, B.base);
    ODSLF_Lambda im_ln_Fn_G_0(B.chain, B.base);
    ODSLF_Lambda re_ln_Fn_G_2(B.chain, B.base);
    ODSLF_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 - (1.0 - 2.0 * (B.base.Mdown % 2)) * // MODIF from XXZ
		    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 += (1.0 - 2.0 * (B.base.Mdown % 2)) * // MODIF from XXZ
		      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]);

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

  }

} // namespace ABACUS