ABACUS/src/LIEBLIN/ln_g2_ME.cc

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

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

Copyright (c) J.-S. Caux.

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

File:  src/LIEBLIN/ln_g2_ME.cc

Purpose:  provides the matrix element of the second density moment g2(x = 0)

NOTE:  based on Lorenzo Piroli's expression of the matrix element

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

#include "ABACUS.h"

using namespace std;
using namespace ABACUS;

namespace ABACUS {

  complex<DP> Fn_V_g2 (int j, LiebLin_Bethe_State& lstate, LiebLin_Bethe_State& rstate)
  {
    complex<DP> result = 1.0;

    for (int m = 0; m < lstate.N; ++m) {
      result *= (lstate.lambdaoc[m] - rstate.lambdaoc[j] + II)/(rstate.lambdaoc[m] - rstate.lambdaoc[j] + II);
    }

    return(result);
  }


  complex<DP> ln_g2_ME (LiebLin_Bethe_State& lstate, LiebLin_Bethe_State& rstate)
  {
    // Computes the log of the density operator \rho(x = 0) matrix element between lstate and rstate.

    // If we have lstate == rstate, density matrix element = 0:
    if (lstate.Ix2 == rstate.Ix2) return(-200.);

    SQMat_DP one_plus_U (0.0, lstate.N);

    Vect_CX Vplus (0.0, lstate.N);  // contains V^+_j;  V^-_j is the conjugate
    Vect_CX ln_Fn_Prod (0.0, lstate.N);  // product_{m\neq j} (\mu_m - \lambdaoc_j)/(\lambdaoc_m - \lambdaoc_j)
    Vect_DP rKern (0.0, lstate.N);   // K(lambdaoc_j - lambdaoc_(p == arbitrary))

    int p = rstate.N/2-1; // choice doesn't matter, see 1990_Slavnov_TMP_82 after (3.8). Choose rapidity around the middle.

    DP c_int = rstate.c_int;

    DP Eout = log(pow(lstate.E - rstate.E,2.)) - (2)*log(c_int)  - log(2*lstate.N);

    for (int a = 0; a < lstate.N; ++a) {
      Vplus[a] = Fn_V_g2 (a, lstate, rstate);
      ln_Fn_Prod[a] = 0.0;;
      for (int m = 0; m < lstate.N; ++m)
	if (m != a) ln_Fn_Prod[a] += log(complex<DP>(lstate.lambdaoc[m] - rstate.lambdaoc[a])
					 /(rstate.lambdaoc[m] - rstate.lambdaoc[a]));
      rKern[a] = rstate.Kernel (a, p);
    }

    for (int a = 0; a < lstate.N; ++a)
      for (int b = 0; b < lstate.N; ++b)
	one_plus_U[a][b] = (a == b ? 1.0 : 0.0) + 0.5 * 1./imag(Vplus[a])
	  * ((lstate.lambdaoc[a] - rstate.lambdaoc[a])*real(exp(ln_Fn_Prod[a]))
	     * (rstate.Kernel(a,b) - rKern[b]) + rKern[b]);

    complex<DP> ln_ddalpha_sigma = lndet_LU_dstry(one_plus_U);

    complex<DP> ln_prod_V = 0.0;
    for (int a = 0; a < lstate.N; ++a) ln_prod_V += log(2.0 * II * imag(Vplus[a]));

    complex<DP> ln_prod_2 = 0.0;
    for (int a = 0; a < lstate.N; ++a)
      for (int b = 0; b < lstate.N; ++b)
	ln_prod_2 += log((rstate.lambdaoc[a] - rstate.lambdaoc[b] + II)/(lstate.lambdaoc[a] - rstate.lambdaoc[b]));

    ln_ddalpha_sigma += ln_prod_V + ln_prod_2 - log(2.0 * II * imag(Vplus[p]));

    return (log(II) + Eout + ln_ddalpha_sigma - 0.5 * (lstate.lnnorm + rstate.lnnorm));
  }

} // namespace ABACUS