ABACUS/src/TBA/TBA_XXZ.cc

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

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

Copyright (c) J.-S. Caux.

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

File:  TBA_XXZ.cc

Purpose:  TBA for the gapless XXZ antiferromagnet.

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

#include "ABACUS.h"

using namespace std;
using namespace ABACUS;

namespace ABACUS {

  // First, define some useful kernels:

  DP XXZ_phi1_kernel (DP zeta, DP lambda)
  {
    return(2.0 * atan(tanh(lambda)/tan(0.5*zeta)));
  }

  DP XXZ_phi2_kernel (DP zeta, DP lambda)
  {
    return(2.0 * atan(tanh(lambda)/tan(zeta)));
  }

  DP XXZ_a1_kernel (DP sinzeta, DP coszeta, DP lambda)
  {
    DP expmin2abslambda = exp(-2.0 * fabs(lambda));

    DP answer = sinzeta * expmin2abslambda
      /(PI * (0.5 * (1.0 + expmin2abslambda * expmin2abslambda) - coszeta * expmin2abslambda));

    return(answer);
  }

  DP XXZ_da1dlambda_kernel (DP sinzeta, DP coszeta, DP lambda)
  {
    DP expmin2abslambda = exp(-2.0 * fabs(lambda));
    DP signlambda = lambda >= 0.0 ? 1.0 : -1.0;
    DP answer = -sinzeta * signlambda * expmin2abslambda * (1.0 - expmin2abslambda * expmin2abslambda)
      /(PI * pow((0.5 * (1.0 + expmin2abslambda * expmin2abslambda)
		  - coszeta * expmin2abslambda), 2.0));

    return(answer);
  }

  DP XXZ_a2_kernel (DP sin2zeta, DP cos2zeta, DP lambda)
  {
    DP expmin2abslambda = exp(-2.0 * fabs(lambda));

    DP answer = sin2zeta * expmin2abslambda
      /(PI * (0.5 * (1.0 + expmin2abslambda * expmin2abslambda) - cos2zeta * expmin2abslambda));

    return(answer);
  }

  void Iterate_XXZ_rhotot_GS (DP zeta, DP B, Root_Density& rhotot_GS)
  {
    // Performs an iteration of the TBA equation for rhotot_GS:

    // First, copy the actual values into the previous ones:
    for (int i = 0; i < rhotot_GS.Npts; ++i) rhotot_GS.prev_value[i] = rhotot_GS.value[i];

    // Calculate the new values:
    DP conv = 0.0;
    DP sinzeta = sin(zeta);
    DP sin2zeta = sin(2.0*zeta);
    DP coszeta = cos(zeta);
    DP cos2zeta = cos(2.0*zeta);
    for (int i = 0; i < rhotot_GS.Npts; ++i) {
      // First, calculate the convolution
      conv = 0.0;
      for (int j = 0; j < rhotot_GS.Npts; ++j)
	conv += fabs(rhotot_GS.lambda[j]) > B ? 0.0 :
	  XXZ_a2_kernel (sin2zeta, cos2zeta, rhotot_GS.lambda[i] - rhotot_GS.lambda[j])
	  * rhotot_GS.prev_value[j] * rhotot_GS.dlambda[j];
      rhotot_GS.value[i] = XXZ_a1_kernel(sinzeta, coszeta, rhotot_GS.lambda[i]) - conv;
    }

    // Calculate the sum of differences:
    rhotot_GS.diff = 0.0;
    for (int i = 0; i < rhotot_GS.Npts; ++i)
      rhotot_GS.diff += fabs(rhotot_GS.value[i] - rhotot_GS.prev_value[i]);

    return;
  }

  Root_Density XXZ_rhotot_GS (DP Delta, DP B, DP lambdamax, int Npts, DP req_prec)
  {

    // This function returns a Root_Density object corresponding to the ground state
    // root distribution of the XXZ model.

    if (Delta >= 1.0 || Delta <= 0.0) ABACUSerror("Delta out of bounds in XXZ_rhotot_GS.");

    DP zeta = acos(Delta);

    Root_Density rhotot_GS(Npts, lambdamax);

    // We now attempt to find a solution...

    int niter = 0;
    do {
      Iterate_XXZ_rhotot_GS (zeta, B, rhotot_GS);
      niter ++;
    } while (rhotot_GS.diff > req_prec);

    return(rhotot_GS);
  }

  DP Return_GS_Sz_tot_value (DP B, Root_Density& rhotot_GS)
  {
    DP integral = 0.0;
    for (int i = 0; i < rhotot_GS.Npts; ++i)
      integral += fabs(rhotot_GS.lambda[i]) > B ? 0.0 : rhotot_GS.value[i] * rhotot_GS.dlambda[i];

    return(0.5 - integral);
  }

  void Iterate_XXZ_eps_GS (DP zeta, DP Hz, Root_Density& eps_GS)
  {
    // Performs an iteration of the TBA equation for eps_GS:

    // First, copy the actual values into the previous ones:
    for (int i = 0; i < eps_GS.Npts; ++i) eps_GS.prev_value[i] = eps_GS.value[i];

    // Calculate the new values:
    DP conv = 0.0;
    DP sinzeta = sin(zeta);
    DP sin2zeta = sin(2.0*zeta);
    DP coszeta = cos(zeta);
    DP cos2zeta = cos(2.0*zeta);
    for (int i = 0; i < eps_GS.Npts; ++i) {
      // First, calculate the convolution
      conv = 0.0;
      for (int j = 0; j < eps_GS.Npts; ++j)
	conv += eps_GS.prev_value[j] > 0.0 ? 0.0 :
	  XXZ_a2_kernel (sin2zeta, cos2zeta, eps_GS.lambda[i] - eps_GS.lambda[j])
	  * eps_GS.prev_value[j] * eps_GS.dlambda[j];
      eps_GS.value[i] = Hz - PI * sinzeta * XXZ_a1_kernel(sinzeta, coszeta, eps_GS.lambda[i]) - conv;
    }

    // Calculate the sum of differences:
    eps_GS.diff = 0.0;
    for (int i = 0; i < eps_GS.Npts; ++i)
      eps_GS.diff += fabs(eps_GS.value[i] - eps_GS.prev_value[i]);

    return;
  }

  Root_Density XXZ_eps_GS (DP Delta, DP Hz, DP lambdamax, int Npts, DP req_prec)
  {

    // This function returns a Root_Density object corresponding to the ground state
    // epsilon function of the XXZ model.

    if (Delta >= 1.0 || Delta <= 0.0) ABACUSerror("Delta out of bounds in XXZ_eps_GS.");

    DP zeta = acos(Delta);

    Root_Density eps_GS(Npts, lambdamax);

    // We now attempt to find a solution...

    int niter = 0;
    do {
      Iterate_XXZ_eps_GS (zeta, Hz, eps_GS);
      niter++;
    } while (eps_GS.diff > req_prec);

    return(eps_GS);
  }

  void Iterate_XXZ_depsdlambda_GS (DP zeta, DP B, Root_Density& depsdlambda_GS)
  {
    // Performs an iteration of the TBA equation for depsdlambda_GS:

    // First, copy the actual values into the previous ones:
    for (int i = 0; i < depsdlambda_GS.Npts; ++i) depsdlambda_GS.prev_value[i] = depsdlambda_GS.value[i];

    // Calculate the new values:
    DP conv = 0.0;
    DP sinzeta = sin(zeta);
    DP sin2zeta = sin(2.0*zeta);
    DP coszeta = cos(zeta);
    DP cos2zeta = cos(2.0*zeta);
    for (int i = 0; i < depsdlambda_GS.Npts; ++i) {
      // First, calculate the convolution
      conv = 0.0;
      for (int j = 0; j < depsdlambda_GS.Npts; ++j)
	conv += fabs(depsdlambda_GS.lambda[j]) > B ? 0.0 :
	  XXZ_a2_kernel (sin2zeta, cos2zeta, depsdlambda_GS.lambda[i] - depsdlambda_GS.lambda[j])
	  * depsdlambda_GS.prev_value[j] * depsdlambda_GS.dlambda[j];
      depsdlambda_GS.value[i] = -PI * sinzeta * XXZ_da1dlambda_kernel(sinzeta, coszeta, depsdlambda_GS.lambda[i]) - conv;
    }

    // Calculate the sum of differences:
    depsdlambda_GS.diff = 0.0;
    for (int i = 0; i < depsdlambda_GS.Npts; ++i)
      depsdlambda_GS.diff += fabs(depsdlambda_GS.value[i] - depsdlambda_GS.prev_value[i]);

    return;
  }

  Root_Density XXZ_depsdlambda_GS (DP Delta, DP B, DP lambdamax, int Npts, DP req_prec)
  {

    // This function returns a Root_Density object corresponding to the ground state
    // (d/d\lambda) epsilon function of the XXZ model.

    if (Delta >= 1.0 || Delta <= 0.0) ABACUSerror("Delta out of bounds in XXZ_rho_GS.");

    DP zeta = acos(Delta);

    Root_Density depsdlambda_GS(Npts, lambdamax);

    // We now attempt to find a solution...

    int niter = 0;
    do {
      Iterate_XXZ_depsdlambda_GS (zeta, B, depsdlambda_GS);
      niter++;
    } while (depsdlambda_GS.diff > req_prec);

    return(depsdlambda_GS);
  }

  void Iterate_XXZ_b2BB_lambda_B (DP zeta, DP B, Root_Density& b2BB)
  {
    // Calculates the vector corresponding to b2BB (lambda,B)

    // First, copy the actual values into the previous ones:
    for (int i = 0; i < b2BB.Npts; ++i) b2BB.prev_value[i] = b2BB.value[i];

    // Calculate the new values:
    DP conv = 0.0;
    DP sin2zeta = sin(2.0*zeta);
    DP cos2zeta = cos(2.0*zeta);
    for (int i = 0; i < b2BB.Npts; ++i) {
      if (fabs(b2BB.lambda[i]) > B) b2BB.value[i] = 0.0;
      else {
	// First, calculate the convolution
	conv = 0.0;
	for (int j = 0; j < b2BB.Npts; ++j)
	  conv += fabs(b2BB.lambda[j]) > B ? 0.0 :
	    XXZ_a2_kernel (sin2zeta, cos2zeta, b2BB.lambda[i] - b2BB.lambda[j])
	    * b2BB.prev_value[j] * b2BB.dlambda[j];
	b2BB.value[i] = -XXZ_a2_kernel(sin2zeta, cos2zeta, b2BB.lambda[i] - B)
	  - conv;
      }
    }
    // Calculate the sum of differences:
    b2BB.diff = 0.0;
    for (int i = 0; i < b2BB.Npts; ++i)
      b2BB.diff += fabs(b2BB.value[i] - b2BB.prev_value[i]);

    return;
  }

  Root_Density XXZ_b2BB_lambda_B (DP Delta, DP B, DP lambdamax, int Npts, DP req_prec)
  {

    // This function returns a Root_Density object corresponding to the inverse
    // kernel b2BB (lambda, B) used in Kbar function.

    if (Delta >= 1.0 || Delta <= 0.0) ABACUSerror("Delta out of bounds in XXZ_Kbackflow_GS.");

    DP zeta = acos(Delta);

    Root_Density b2BB_lambda_B(Npts, lambdamax);

    // We now attempt to find a solution...

    int niter = 0;
    do {
      Iterate_XXZ_b2BB_lambda_B (zeta, B, b2BB_lambda_B);
      niter++;
    } while (b2BB_lambda_B.diff > req_prec);

    return(b2BB_lambda_B);
  }

  void Iterate_XXZ_b2BB_lambda_lambdap (DP zeta, DP B, DP lambdap, Root_Density& b2BB)
  {
    // Calculates the vector corresponding to b2BB (lambda,lambdap)

    // First, copy the actual values into the previous ones:
    for (int i = 0; i < b2BB.Npts; ++i) b2BB.prev_value[i] = b2BB.value[i];

    // Calculate the new values:
    DP conv = 0.0;
    DP sin2zeta = sin(2.0*zeta);
    DP cos2zeta = cos(2.0*zeta);
    for (int i = 0; i < b2BB.Npts; ++i) {
      if (fabs(b2BB.lambda[i]) > B) b2BB.value[i] = 0.0;
      else {
	// First, calculate the convolution
	conv = 0.0;
	for (int j = 0; j < b2BB.Npts; ++j)
	  conv += fabs(b2BB.lambda[j]) > B ? 0.0 :
	    XXZ_a2_kernel (sin2zeta, cos2zeta, b2BB.lambda[i] - b2BB.lambda[j])
	    * b2BB.prev_value[j] * b2BB.dlambda[j];
	b2BB.value[i] = -XXZ_a2_kernel(sin2zeta, cos2zeta, b2BB.lambda[i] - lambdap)
	  - conv;
      }
    }
    // Calculate the sum of differences:
    b2BB.diff = 0.0;
    for (int i = 0; i < b2BB.Npts; ++i)
      b2BB.diff += fabs(b2BB.value[i] - b2BB.prev_value[i]);

    return;
  }

  Root_Density XXZ_b2BB_lambda_lambdap (DP Delta, DP B, DP lambdap, DP lambdamax, int Npts, DP req_prec)
  {

    // This function returns a Root_Density object corresponding to the inverse
    // kernel b2BB (lambda, lambdap) used in Kbar function.

    if (Delta >= 1.0 || Delta <= 0.0) ABACUSerror("Delta out of bounds in XXZ_Kbackflow_GS.");

    DP zeta = acos(Delta);

    Root_Density b2BB_lambda_lambdap(Npts, lambdamax);

    // We now attempt to find a solution...

    int niter = 0;
    do {
      Iterate_XXZ_b2BB_lambda_lambdap (zeta, B, lambdap, b2BB_lambda_lambdap);
      niter++;
    } while (b2BB_lambda_lambdap.diff > req_prec);

    return(b2BB_lambda_lambdap);
  }

  void Iterate_XXZ_Kbackflow_GS (DP zeta, DP B, DP lambda_p, DP lambda_h, Root_Density& Kbackflow_GS)
  {
    // Performs an iteration of the TBA equation for Kbackflow_GS:

    // First, copy the actual values into the previous ones:
    for (int i = 0; i < Kbackflow_GS.Npts; ++i) Kbackflow_GS.prev_value[i] = Kbackflow_GS.value[i];

    // Calculate the new values:
    DP conv = 0.0;
    DP sin2zeta = sin(2.0*zeta);
    DP cos2zeta = cos(2.0*zeta);
    for (int i = 0; i < Kbackflow_GS.Npts; ++i) {
      if (false && fabs(Kbackflow_GS.lambda[i]) > B) Kbackflow_GS.value[i] = 0.0;
      else {
	// First, calculate the convolution
	conv = 0.0;
	for (int j = 0; j < Kbackflow_GS.Npts; ++j)
	  conv += fabs(Kbackflow_GS.lambda[j]) > B ? 0.0 :
	    XXZ_a2_kernel (sin2zeta, cos2zeta, Kbackflow_GS.lambda[i] - Kbackflow_GS.lambda[j])
	    * Kbackflow_GS.prev_value[j] * Kbackflow_GS.dlambda[j];
	Kbackflow_GS.value[i] = -XXZ_a2_kernel(sin2zeta, cos2zeta, Kbackflow_GS.lambda[i] - lambda_p)
	  + XXZ_a2_kernel(sin2zeta, cos2zeta, Kbackflow_GS.lambda[i] - lambda_h) - conv;
      }
    }
    // Calculate the sum of differences:
    Kbackflow_GS.diff = 0.0;
    for (int i = 0; i < Kbackflow_GS.Npts; ++i)
      Kbackflow_GS.diff += fabs(Kbackflow_GS.value[i] - Kbackflow_GS.prev_value[i]);

    return;
  }

  Root_Density XXZ_Kbackflow_GS (DP Delta, DP B, DP lambdamax, DP lambda_p, DP lambda_h, int Npts, DP req_prec)
  {

    // This function returns a Root_Density object corresponding to the ground state
    // backflow function of the XXZ model.

    if (Delta >= 1.0 || Delta <= 0.0) ABACUSerror("Delta out of bounds in XXZ_Kbackflow_GS.");

    DP zeta = acos(Delta);

    Root_Density Kbackflow_GS(Npts, lambdamax);

    // We now attempt to find a solution...

    int niter = 0;
    do {
      Iterate_XXZ_Kbackflow_GS (zeta, B, lambda_p, lambda_h, Kbackflow_GS);
      niter++;
    } while (Kbackflow_GS.diff > req_prec);

    return(Kbackflow_GS);
  }

  void Iterate_XXZ_Fbackflow_GS (DP zeta, DP B, DP lambda_p, DP lambda_h, Root_Density& Fbackflow_GS)
  {
    // Performs an iteration of the TBA equation for Fbackflow_GS:

    // First, copy the actual values into the previous ones:
    for (int i = 0; i < Fbackflow_GS.Npts; ++i) Fbackflow_GS.prev_value[i] = Fbackflow_GS.value[i];

    // Calculate the new values:
    DP conv = 0.0;
    DP sin2zeta = sin(2.0*zeta);
    DP cos2zeta = cos(2.0*zeta);
    for (int i = 0; i < Fbackflow_GS.Npts; ++i) {
      if (fabs(Fbackflow_GS.lambda[i]) > B) Fbackflow_GS.value[i] = 0.0;
      else {
	// First, calculate the convolution
	conv = 0.0;
	for (int j = 0; j < Fbackflow_GS.Npts; ++j)
	  conv += fabs(Fbackflow_GS.lambda[j]) > B ? 0.0 :
	    XXZ_a2_kernel (sin2zeta, cos2zeta, Fbackflow_GS.lambda[i] - Fbackflow_GS.lambda[j])
	    * Fbackflow_GS.prev_value[j] * Fbackflow_GS.dlambda[j];
	Fbackflow_GS.value[i] = (-XXZ_phi2_kernel(zeta, Fbackflow_GS.lambda[i] - lambda_p)
				 + XXZ_phi2_kernel(zeta, Fbackflow_GS.lambda[i] - lambda_h))/twoPI - conv;
      }
    }
    // Calculate the sum of differences:
    Fbackflow_GS.diff = 0.0;
    for (int i = 0; i < Fbackflow_GS.Npts; ++i)
      Fbackflow_GS.diff += fabs(Fbackflow_GS.value[i] - Fbackflow_GS.prev_value[i]);

    return;
  }

  Root_Density XXZ_Fbackflow_GS (DP Delta, DP B, DP lambdamax, DP lambda_p, DP lambda_h, int Npts, DP req_prec)
  {

    // This function returns a Root_Density object corresponding to the ground state
    // F backflow function of the XXZ model.

    if (Delta >= 1.0 || Delta <= 0.0) ABACUSerror("Delta out of bounds in XXZ_Fbackflow_GS.");

    DP zeta = acos(Delta);

    Root_Density Fbackflow_GS(Npts, lambdamax);

    // We now attempt to find a solution...

    int niter = 0;
    do {
      Iterate_XXZ_Fbackflow_GS (zeta, B, lambda_p, lambda_h, Fbackflow_GS);
      niter++;
    } while (Fbackflow_GS.diff > req_prec);

    return(Fbackflow_GS);
  }

  void Iterate_XXZ_Z_GS (DP zeta, DP B, Root_Density& Z_GS)
  {
    // Performs an iteration of the TBA equation for the dressed charge:

    // First, copy the actual values into the previous ones:
    for (int i = 0; i < Z_GS.Npts; ++i) Z_GS.prev_value[i] = Z_GS.value[i];

    // Calculate the new values:
    DP conv = 0.0;
    DP sin2zeta = sin(2.0*zeta);
    DP cos2zeta = cos(2.0*zeta);
    for (int i = 0; i < Z_GS.Npts; ++i) {
      // First, calculate the convolution
      conv = 0.0;
      for (int j = 0; j < Z_GS.Npts; ++j)
	conv += fabs(Z_GS.lambda[j]) > B ? 0.0 :
	  XXZ_a2_kernel (sin2zeta, cos2zeta, Z_GS.lambda[i] - Z_GS.lambda[j])
	  * Z_GS.prev_value[j] * Z_GS.dlambda[j];
      Z_GS.value[i] = 1.0 - conv;
    }

    // Calculate the sum of differences:
    Z_GS.diff = 0.0;
    for (int i = 0; i < Z_GS.Npts; ++i)
      Z_GS.diff += fabs(Z_GS.value[i] - Z_GS.prev_value[i]);

    return;
  }

  Root_Density XXZ_Z_GS (DP Delta, DP B, DP lambdamax, int Npts, DP req_prec)
  {

    // This function returns a Root_Density object corresponding to the ground state
    // dressed charge function of the XXZ model.

    if (Delta >= 1.0 || Delta <= 0.0) ABACUSerror("Delta out of bounds in XXZ_Z_GS.");

    DP zeta = acos(Delta);

    Root_Density Z_GS(Npts, lambdamax);

    // We now attempt to find a solution...

    do {
      Iterate_XXZ_Z_GS (zeta, B, Z_GS);
    } while (Z_GS.diff > req_prec);

    return(Z_GS);
  }


}  // namespace ABACUS