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ABACUS/src/HEIS/ln_Smm_ME_XXX.cc

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/**********************************************************
This software is part of J.-S. Caux's ABACUS library.
Copyright (c) J.-S. Caux.
-----------------------------------------------------------
File: ln_Smm_ME_XXX.cc
Purpose: compute the S^-_j S^-_{j+1} matrix elemment for XXX
***********************************************************/
#include "ABACUS.h"
using namespace std;
using namespace ABACUS;
namespace ABACUS {
inline complex<DP> phi(complex<DP> x){return x;}
inline complex<DP> a(complex<DP> x){return 1;}
inline complex<DP> b(complex<DP> x,complex<DP> y, complex<DP> eta)
{ return phi(x-y)/phi(x-y+complex<DP>(0.0,1.0)*eta);}
inline complex<DP> d(complex<DP> x, complex<DP> xi, complex<DP> eta, int N)
{return pow(b(x,xi,eta),N);}
inline complex<DP> ln_Fn_F (XXX_Bethe_State& B, int k, int beta, int b)
{
complex<DP> ans = 0.0;
for (int j = 0; j < B.chain.Nstrings; ++j) {
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)))
ans += log(B.lambda[j][alpha] - B.lambda[k][beta]
+ 0.5 * II * (B.chain.Str_L[j] - B.chain.Str_L[k] - 2.0 * (a - b)));
}
}
}
return(ans);
}
inline complex<DP> ln_Fn_G (XXX_Bethe_State& A, XXX_Bethe_State& B, int k, int beta, int b)
{
complex<DP> ans = 0.0;
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) {
ans += log(A.lambda[j][alpha] - B.lambda[k][beta]
+ 0.5 * II * (A.chain.Str_L[j] - B.chain.Str_L[k] - 2.0 * (a - b)));
}
}
}
return(ans);
}
inline complex<DP> Fn_K (XXX_Bethe_State& A, int j, int alpha, int a, XXX_Bethe_State& B, int k, int beta, int b)
{
return(1.0/((A.lambda[j][alpha] - B.lambda[k][beta]
+ 0.5 * II * (A.chain.Str_L[j] - B.chain.Str_L[k] - 2.0 * (a - b)))
* (A.lambda[j][alpha] - B.lambda[k][beta]
+ 0.5 * II * (A.chain.Str_L[j] - B.chain.Str_L[k] - 2.0 * (a - b - 1.0)) )));
}
inline complex<DP> Fn_L (XXX_Bethe_State& A, int j, int alpha, int a, XXX_Bethe_State& B, int k, int beta, int b)
{
return ((2.0 * (A.lambda[j][alpha] - B.lambda[k][beta]
+ 0.5 * II * (A.chain.Str_L[j] - B.chain.Str_L[k] - 2.0 * (a - b - 0.5))
))
* pow(Fn_K (A, j, alpha, a, B, k, beta, b), 2.0));
}
complex<DP> ln_Smm_ME (XXX_Bethe_State& A, XXX_Bethe_State& B)
{
const DP Zero_Center_Thres=1.0e-5;
const DP real_dev=1.0e-14;
const DP Diff_ME_Thres=1.e-6;
// This function returns the natural log of the S^z operator matrix element.
// The A and B states can contain strings.
// A is the averaging state.
// Check that the two states refer to the same XXX_Chain
if (A.chain != B.chain)
ABACUSerror("Incompatible XXX_Chains in Smm matrix element.");
// Check that A and B are compatible: same Mdown
if (A.base.Mdown != B.base.Mdown + 2)
ABACUSerror("Incompatible Mdown between the two states in Smm matrix element!");
// Some convenient arrays
complex<DP> eta=-II;
complex<DP> ln_prod = complex<DP>(0.0,0.0);
complex<DP> result=-300;
complex<DP> prev_result=-300;
XXX_Bethe_State B_origin; B_origin=B;
bool zero_string=false;
for (int j = 0; j < B_origin.chain.Nstrings; ++j)
for (int alpha = 0; alpha < B_origin.base.Nrap[j]; ++alpha)
if(abs(B_origin.lambda[j][alpha])<Zero_Center_Thres) zero_string=true;
// Some convenient arrays
bool real_dev_conv=false;
int dev=-1;
while(!real_dev_conv){
real_dev_conv=true;
dev++;
//add a delta to the origin of the centered strings
if(zero_string){
real_dev_conv=false;
for (int j = 0; j < B.chain.Nstrings; ++j)
for (int alpha = 0; alpha < B.base.Nrap[j]; ++alpha)
if(abs(B_origin.lambda[j][alpha])<Zero_Center_Thres)
B.lambda[j][alpha]=real_dev*pow(10.0,dev);
}
prev_result=result;
result=log(B.chain.Nsites*1.0);
int sizeA=0;
int sizeB=0;
for (int i = 0; i < A.chain.Nstrings; ++i)
sizeA+=A.base.Nrap[i]*A.chain.Str_L[i];
for (int i = 0; i < B.chain.Nstrings; ++i)
sizeB+=B.base.Nrap[i]*B.chain.Str_L[i];
complex<DP>* mu = new complex<DP>[sizeA];
complex<DP>* lam = new complex<DP>[sizeB];
int index=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)
{
mu[index]=(A.lambda[i][alpha] + 0.5 * -eta * (A.chain.Str_L[i] + 1.0 - 2.0 * a));
index++;
}
index=0;
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)
{
lam[index]=(B.lambda[i][alpha] + 0.5 * -eta * (B.chain.Str_L[i] + 1.0 - 2.0 * a));
index++;
}
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);
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));
}
}
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(A.lambda[i][alpha] + 0.5 * II * (A.chain.Str_L[i] + 1.0 - 2.0 * a - 1.0)));
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(B.lambda[i][alpha] + 0.5 * II * (B.chain.Str_L[i] + 1.0 - 2.0 * a - 1.0)) > 100.0 * MACHINE_EPS_SQ)
ln_prod2 += log(norm(B.lambda[i][alpha] + 0.5 * II * (B.chain.Str_L[i] + 1.0 - 2.0 * a - 1.0)));
// 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);
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);
}
result += 2.0*real(ln_prod1 - ln_prod2) - real(ln_prod3) + real(ln_prod4) - A.lnnorm - B.lnnorm;
// Define the F ones earlier...
int index_a = 0;
int index_b = 0;
complex<DP> Prod_powerN;
//mu is the ground state!
//A -> mu, B -> lam
SQMat_CX H(0.0, A.base.Mdown);
index_a = 0;
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;
complex<DP> Da;
complex<DP> Ca;
Da=eta/((mu[index_a]-eta*0.5)*(mu[index_a]+eta*0.5));
Ca=eta*((mu[index_a]-eta*0.5)+(mu[index_a]+eta*0.5))/pow(((mu[index_a]-eta*0.5)*(mu[index_a]+eta*0.5)),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) {
complex<DP> prodplus= complex<DP>(1.0,0.0);
complex<DP> prodminus= complex<DP>(1.0,0.0);
// use simplified code for one-string here: original form of Hm2P matrix
prodplus = Fn_K (A, j, alpha, a, B, k, beta, 0);// 1.0/ (phi(mu[l]-lam[k]) phi(mu[l]-lam[k]+eta) )
prodplus*= exp(re_ln_Fn_G_0[k][beta] + II * im_ln_Fn_G_0[k][beta]
- re_ln_Fn_F_B_0[k][beta]);//Prod phi(mu[l]-lam[k]+eta) / prod_l!=k |phi(lam[l]-lam[k]) |;
prodminus = Fn_K (A, j, alpha, a, B, k, beta, 1);// 1.0/ (phi(mu[l]-lam[k]) phi(mu[l]-lam[k]-eta) )
prodminus*= exp(re_ln_Fn_G_2[k][beta] + II * im_ln_Fn_G_2[k][beta]
- re_ln_Fn_F_B_0[k][beta]);//Prod phi(mu[l]-lam[k]-eta)/ prod_l!=k | phi(lam[l]-lam[k]) |;
Prod_powerN = pow((B.lambda[k][beta] - eta*0.5) /(B.lambda[k][beta] + eta*0.5), complex<DP> (B.chain.Nsites));
H[index_a][index_b] =eta*(prodplus-prodminus*Prod_powerN);
} // if (B.chain.Str_L == 1)
else {
// */{
if (b > 1){
H[index_a][index_b] = eta* Fn_K(A, j, alpha, a, B, k, beta, b-1)
*exp(ln_Fn_G(A,B,k,beta,b-1))
*exp(-real(ln_Fn_F(B, k, beta, b - 1)));//.../ prod_l!=k | phi(lam[l]-lam[k]) |
}
else if (b == 1) {
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);
complex<DP> sum1 = complex<DP>(0.0,0.0);
sum1 += Fn_K (A, j, alpha, a, B, k, beta, 0) *exp(ln_FunctionG[0]+ ln_FunctionG[1]
- ln_FunctionF[0] - ln_FunctionF[1]);//sum term when i=0
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]); //sum term when i=n
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]);
}
H[index_a][index_b] = eta * exp(ln_FunctionF[0]+ ln_FunctionF[1] - ln_FunctionG[1])
* sum1 * exp( - real(ln_Fn_F(B, k, beta, b - 1)));
//the absolute value prod_l!=k phi(lam[l]-lam[k]) : real(ln_...)
} // else if (b == B.chain.Str_L[k])
} // else
index_b++;
}}} // sums over k, beta, b
// now define the elements H[a][M] & H[a][M-1]
H[index_a][B.base.Mdown]=Da;
H[index_a][B.base.Mdown+1]=Ca;
index_a++;
}}} // sums over j, alpha, a
complex<DP> logF=lndet_LU_CX_dstry(H);
result+=2.0*real(logF);
result+=log(2.0)-log((A.chain.Nsites-A.base.Mdown*2+3.0)*((A.chain.Nsites-A.base.Mdown*2+4.0)));
if (!(real_dev_conv) && abs(exp(result)-exp(prev_result))<abs( Diff_ME_Thres*exp(result)))
real_dev_conv=true;
if (!(real_dev_conv) && dev >20){
result=-300;
real_dev_conv=true;
}
delete[] mu;
delete[] lam;
}
return(0.5 * result); // Return ME, not MEsq
}
} // namespace ABACUS
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