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ABACUS/src/HEIS/XXX_Bethe_State.cc

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/**********************************************************
This software is part of J.-S. Caux's ABACUS++ library.
Copyright (c)
-----------------------------------------------------------
File: src/HEIS/XXX_Bethe_State.cc
Purpose: Defines all functions for XXX_Bethe_State
******************************************************************/
#include "JSC.h"
using namespace std;
namespace JSC {
// Function prototypes
DP Theta_XXX (DP lambda, int nj, int nk);
DP ddlambda_Theta_XXX (DP lambda, int nj, int nk);
//***************************************************************************************************
// Function definitions: class XXX_Bethe_State
XXX_Bethe_State::XXX_Bethe_State ()
: Heis_Bethe_State()
{};
XXX_Bethe_State::XXX_Bethe_State (const XXX_Bethe_State& RefState) // copy constructor
: Heis_Bethe_State(RefState)
{
}
XXX_Bethe_State::XXX_Bethe_State (const Heis_Chain& RefChain, int M)
: Heis_Bethe_State(RefChain, M)
{
if (RefChain.Delta != 1.0) {
cout << setprecision(16) << RefChain.Delta << endl;
JSCerror("Delta != 1.0 in XXX_Bethe_State constructor");
}
}
XXX_Bethe_State::XXX_Bethe_State (const Heis_Chain& RefChain, const Heis_Base& RefBase)
: Heis_Bethe_State(RefChain, RefBase)
{
if (RefChain.Delta != 1.0) {
cout << setprecision(16) << RefChain.Delta << endl;
JSCerror("Delta != 1.0 in XXX_Bethe_State constructor");
}
}
/*
XXX_Bethe_State::XXX_Bethe_State (const Heis_Chain& RefChain, long long int base_id_ref, long long int type_id_ref)
: Heis_Bethe_State(RefChain, base_id_ref, type_id_ref)
{
if (RefChain.Delta != 1.0) {
cout << setprecision(16) << RefChain.Delta << endl;
JSCerror("Delta != 1.0 in XXX_Bethe_State constructor");
}
}
*/
XXX_Bethe_State& XXX_Bethe_State::operator= (const XXX_Bethe_State& RefState)
{
if (this != &RefState) {
chain = RefState.chain;
base = RefState.base;
//offsets = RefState.offsets;
Ix2 = RefState.Ix2;
lambda = RefState.lambda;
BE = RefState.BE;
diffsq = RefState.diffsq;
conv = RefState.conv;
iter = RefState.iter;
iter_Newton = RefState.iter_Newton;
E = RefState.E;
iK = RefState.iK;
K = RefState.K;
lnnorm = RefState.lnnorm;
//base_id = RefState.base_id;
//type_id = RefState.type_id;
//id = RefState.id;
//maxid = RefState.maxid;
//nparticles = RefState.nparticles;
label = RefState.label;
}
return(*this);
}
// Member functions
void XXX_Bethe_State::Set_Free_lambdas()
{
// Sets all the rapidities to the solutions of the BAEs without scattering terms
for (int i = 0; i < chain.Nstrings; ++i) {
for (int alpha = 0; alpha < base[i]; ++alpha) {
lambda[i][alpha] = chain.Str_L[i] * 0.5 * tan(PI * 0.5 * Ix2[i][alpha]/chain.Nsites);
}
}
return;
}
bool XXX_Bethe_State::Check_Admissibility(char option)
{
// This function checks the admissibility of the Ix2's of a state:
// returns false if there are higher strings with Ix2 = 0, a totally symmetric distribution of I's at each level,
// and strings of equal length modulo 2 and parity with Ix2 = 0, meaning at least two equal roots in BAE.
bool answer = true;
//int test1, test3;
Vect<bool> Zero_at_level(false, chain.Nstrings); // whether there exists an Ix2 == 0 at a given level
bool higher_string_on_zero = false;
for (int j = 0; j < chain.Nstrings; ++j) {
// The following line puts answer to true if there is at least one higher string with zero Ix2
for (int alpha = 0; alpha < base[j]; ++alpha) if ((Ix2[j][alpha] == 0) && (chain.Str_L[j] > 2) && !(chain.Str_L[j] % 2))
higher_string_on_zero = true;
for (int alpha = 0; alpha < base[j]; ++alpha) if (Ix2[j][alpha] == 0) Zero_at_level[j] = true;
// NOTE: if base[j] == 0, Zero_at_level[j] remains false.
}
bool symmetric_state = (*this).Check_Symmetry();
bool string_coincidence = false;
// Checks that we have strings of equal length modulo 2 with Ix2 == 0, so equal rapidities, and inadmissibility
for (int j1 = 0; j1 < chain.Nstrings; ++j1) {
for (int j2 = j1 + 1; j2 < chain.Nstrings; ++j2)
if (Zero_at_level[j1] && Zero_at_level[j2] && (!((chain.Str_L[j1] + chain.Str_L[j2])%2)))
string_coincidence = true;
}
/*
bool onep_onem_on_zero = false;
if (option == 'm') { // for Smin, we also exclude symmetric states with 1+ and 1- strings on zero
if (Zero_at_level[0] && Zero_at_level[1]) onep_onem_on_zero = true;
}
*/
answer = !(symmetric_state && (higher_string_on_zero || string_coincidence /*|| onep_onem_on_zero*/));
// Now check that no Ix2 is equal to +N (since we take -N into account, and I + N == I by periodicity of exp)
for (int j = 0; j < chain.Nstrings; ++j)
for (int alpha = 0; alpha < base[j]; ++alpha) if ((Ix2[j][alpha] < -chain.Nsites) || (Ix2[j][alpha] >= chain.Nsites)) answer = false;
if (!answer) {
E = 0.0;
K = 0.0;
conv = 0;
iter = 0;
iter_Newton = 0;
lnnorm = -100.0;
}
return(answer); // answer == true: nothing wrong with this Ix2_config
}
void XXX_Bethe_State::Compute_BE (int j, int alpha)
{
// Fills in the BE members with the value of the Bethe equations.
DP sumtheta = 0.0;
sumtheta = 0.0;
for (int k = 0; k < chain.Nstrings; ++k) {
for (int beta = 0; beta < base[k]; ++beta)
if ((chain.Str_L[j] == 1) && (chain.Str_L[k] == 1))
sumtheta += atan(lambda[j][alpha] - lambda[k][beta]);
else sumtheta += 0.5 * Theta_XXX((lambda[j][alpha] - lambda[k][beta]), chain.Str_L[j], chain.Str_L[k]);
}
sumtheta *= 2.0;
BE[j][alpha] = 2.0 * atan(2.0 * lambda[j][alpha]/chain.Str_L[j]) - (sumtheta + PI*Ix2[j][alpha])/chain.Nsites;
}
void XXX_Bethe_State::Compute_BE ()
{
// Fills in the BE members with the value of the Bethe equations.
DP sumtheta = 0.0;
for (int j = 0; j < chain.Nstrings; ++j) {
for (int alpha = 0; alpha < base[j]; ++alpha) {
sumtheta = 0.0;
for (int k = 0; k < chain.Nstrings; ++k) {
for (int beta = 0; beta < base[k]; ++beta)
if ((chain.Str_L[j] == 1) && (chain.Str_L[k] == 1))
sumtheta += atan(lambda[j][alpha] - lambda[k][beta]);
else sumtheta += 0.5 * Theta_XXX((lambda[j][alpha] - lambda[k][beta]), chain.Str_L[j], chain.Str_L[k]);
}
sumtheta *= 2.0;
BE[j][alpha] = 2.0 * atan(2.0 * lambda[j][alpha]/chain.Str_L[j]) - (sumtheta + PI*Ix2[j][alpha])/chain.Nsites;
}
}
}
DP XXX_Bethe_State::Iterate_BAE (int j, int alpha)
{
// Returns a new iteration value for lambda[j][alpha] given BE[j][alpha]
return(0.5 * chain.Str_L[j] * tan(0.5 *
//(PI * Ix2[j][alpha] + sumtheta)/chain.Nsites
(2.0 * atan(2.0 * lambda[j][alpha]/chain.Str_L[j]) - BE[j][alpha])
));
}
/*
void XXX_Bethe_State::Iterate_BAE ()
{
// Recalculates the rapidities by iterating Bethe equations
Lambda New_lambda(chain, base);
DP sumtheta = 0.0;
DP arg = 0.0;
for (int j = 0; j < chain.Nstrings; ++j) {
for (int alpha = 0; alpha < base[j]; ++alpha) {
sumtheta = 0.0;
for (int k = 0; k < chain.Nstrings; ++k) {
for (int beta = 0; beta < base[k]; ++beta)
if ((chain.Str_L[j] == 1) && (chain.Str_L[k] == 1))
sumtheta += atan(lambda[j][alpha] - lambda[k][beta]);
else sumtheta += 0.5 * Theta_XXX(lambda[j][alpha] - lambda[k][beta], chain.Str_L[j], chain.Str_L[k]);
}
sumtheta *= 2.0;
New_lambda[j][alpha] = 0.5 * chain.Str_L[j] * tan((PI * 0.5 * Ix2[j][alpha] + 0.5 * sumtheta)/chain.Nsites);
}
}
DP New_diffsq = 0.0;
for (int j = 0; j < chain.Nstrings; ++j) {
for (int alpha = 0; alpha < base[j]; ++alpha) {
New_diffsq += pow(New_lambda[j][alpha] - lambda[j][alpha], 2.0);
lambda[j][alpha] = 1.0 * New_lambda[j][alpha] + 0.0 * lambda[j][alpha];
}
}
diffsq = New_diffsq;
iter++;
return;
}
*/
/*
void XXX_Bethe_State::Iterate_BAE_Newton ()
{
// does one step of a Newton method on the rapidities...
Vect_DP RHSBAE (0.0, base.Nraptot); // contains RHS of BAEs
Vect_CX dlambda (0.0, base.Nraptot); // contains delta lambda computed from Newton's method
SQMat_CX Gaudin (0.0, base.Nraptot);
Vect_INT indx (base.Nraptot);
DP sumtheta = 0.0;
DP arg = 0.0;
DP fn_arg = 0.0;
DP olddiffsq = diffsq;
// Compute the RHS of the BAEs:
int index = 0;
for (int j = 0; j < chain.Nstrings; ++j) {
for (int alpha = 0; alpha < base[j]; ++alpha) {
sumtheta = 0.0;
for (int k = 0; k < chain.Nstrings; ++k) {
for (int beta = 0; beta < base[k]; ++beta)
if ((chain.Str_L[j] == 1) && (chain.Str_L[k] == 1))
sumtheta += atan(lambda[j][alpha] - lambda[k][beta]);
else sumtheta += 0.5 * Theta_XXX((lambda[j][alpha] - lambda[k][beta]), chain.Str_L[j], chain.Str_L[k]);
}
sumtheta *= 2.0;
RHSBAE[index] = chain.Nsites * 2.0 * atan(2.0 * lambda[j][alpha]/chain.Str_L[j]) - sumtheta - PI*Ix2[j][alpha];
index++;
}
}
(*this).Build_Reduced_Gaudin_Matrix (Gaudin);
for (int i = 0; i < base.Nraptot; ++i) dlambda[i] = - RHSBAE[i];
DP d;
ludcmp_CX (Gaudin, indx, d);
lubksb_CX (Gaudin, indx, dlambda);
diffsq = 0.0;
for (int i = 0; i < base.Nraptot; ++i) diffsq += norm(dlambda[i]);
// if we've converged, calculate the norm here, since the work has been done...
if (diffsq < chain.prec) {
lnnorm = 0.0;
for (int j = 0; j < base.Nraptot; j++) lnnorm += log(abs(Gaudin[j][j]));
}
index = 0;
for (int j = 0; j < chain.Nstrings; ++j) {
for (int alpha = 0; alpha < base[j]; ++alpha) {
lambda[j][alpha] += real(dlambda[index]);
index++;
}
}
iter_Newton++;
return;
}
*/
bool XXX_Bethe_State::Check_Rapidities()
{
bool nonan = true;
for (int j = 0; j < chain.Nstrings; ++j)
for (int alpha = 0; alpha < base[j]; ++alpha) nonan *= !is_nan(lambda[j][alpha]);
return nonan;
}
DP XXX_Bethe_State::String_delta ()
{
// Computes the sum of absolute value of \delta^{a, a+1} in string hypothesis, for a given Bethe eigenstate
DP delta = 0.0;
int occupied_strings = 0;
for (int i = 0; i < (*this).chain.Nstrings; ++i) if ((*this).chain.Str_L[i] > 1) occupied_strings += (*this).base.Nrap[i];
//if ((*this).conv == 0) delta = 1.0;
if (occupied_strings == 0) delta = 0.0;
else {
Vect_DP ln_deltadiff(0.0, 1000); // contains ln |delta^{a, a+1}|
Vect_DP deltadiff(0.0, 1000); // contains |delta^{a, a+1}|
complex<DP> log_BAE_reg = 0.0;
for (int j = 0; j < (*this).chain.Nstrings; ++j) {
for (int alpha = 0; alpha < (*this).base[j]; ++alpha) {
ln_deltadiff = 0.0;
for (int a = 1; a <= (*this).chain.Str_L[j]; ++a) {
if ((*this).chain.Str_L[j] > 1) { // else the BAE are already 1
log_BAE_reg = DP((*this).chain.Nsites) * log(((*this).lambda[j][alpha] + 0.5 * II * ((*this).chain.Str_L[j] + 1.0 - 2.0 * a + 1.0))
/((*this).lambda[j][alpha] + 0.5 * II * ((*this).chain.Str_L[j] + 1.0 - 2.0 * a - 1.0)));
for (int k = 0; k < (*this).chain.Nstrings; ++k)
for (int beta = 0; beta < (*this).base[k]; ++beta)
for (int b = 1; b <= (*this).chain.Str_L[k]; ++b) {
if ((j != k) || (alpha != beta) || (a != b - 1))
log_BAE_reg += log((*this).lambda[j][alpha] + 0.5 * II * ((*this).chain.Str_L[j] + 1.0 - 2.0 * a )
- ((*this).lambda[k][beta] + 0.5 * II * ((*this).chain.Str_L[k] + 1.0 - 2.0 * b )
) - II );
if ((j != k) || (alpha != beta) || (a != b + 1))
log_BAE_reg -= log(((*this).lambda[j][alpha] + 0.5 * II * ((*this).chain.Str_L[j] + 1.0 - 2.0 * a )
)
- ((*this).lambda[k][beta] + 0.5 * II * ((*this).chain.Str_L[k] + 1.0 - 2.0 * b )
) + II );
}
// The regular LHS of BAE is now defined. Now sum up the deltas...
if (a == 1) ln_deltadiff[0] = - real(log_BAE_reg);
else if (a < (*this).chain.Str_L[j]) ln_deltadiff[a - 1] = ln_deltadiff[a-2] - real(log_BAE_reg);
else if (a == (*this).chain.Str_L[j]) ln_deltadiff[a-1] = real(log_BAE_reg);
} // if ((*this).chain.Str_L[j] > 1)
} // for (int a = 1; ...
for (int a = 0; a < (*this).chain.Str_L[j]; ++a) {
deltadiff[a] = ln_deltadiff[a] != 0.0 ? exp(ln_deltadiff[a]) : 0.0;
delta += fabs(deltadiff[a]);
}
} // alpha sum
} // j sum
if (is_nan(delta)) delta = 1.0; // sentinel
} // else
return delta;
}
void XXX_Bethe_State::Compute_Energy ()
{
DP sum = 0.0;
for (int j = 0; j < chain.Nstrings; ++j) {
for (int alpha = 0; alpha < base[j]; ++alpha) {
sum += chain.Str_L[j] / ( 4.0 * lambda[j][alpha] * lambda[j][alpha] + chain.Str_L[j] * chain.Str_L[j]);
}
}
sum *= - chain.J * 2.0;
E = sum;
return;
}
/*
void XXX_Bethe_State::Compute_Momentum ()
{
int sum_Ix2 = 0;
DP sum_M = 0.0;
for (int j = 0; j < chain.Nstrings; ++j) {
sum_M += base[j];
for (int alpha = 0; alpha < base[j]; ++alpha) {
sum_Ix2 += Ix2[j][alpha];
}
}
iK = (chain.Nsites/2) * int(sum_M) - (sum_Ix2/2);
while (iK >= chain.Nsites) iK -= chain.Nsites;
while (iK < 0) iK += chain.Nsites;
K = PI * sum_M - PI * sum_Ix2/chain.Nsites;
while (K >= 2.0*PI) K -= 2.0*PI;
while (K < 0.0) K += 2.0*PI;
return;
}
*/
void XXX_Bethe_State::Build_Reduced_Gaudin_Matrix (SQMat<complex<DP> >& Gaudin_Red)
{
if (Gaudin_Red.size() != base.Nraptot) JSCerror("Passing matrix of wrong size in Build_Reduced_Gaudin_Matrix.");
int index_jalpha;
int index_kbeta;
DP sum_hbar_XXX = 0.0;
index_jalpha = 0;
for (int j = 0; j < chain.Nstrings; ++j) {
for (int alpha = 0; alpha < base[j]; ++alpha) {
index_kbeta = 0;
for (int k = 0; k < chain.Nstrings; ++k) {
for (int beta = 0; beta < base[k]; ++beta) {
if ((j == k) && (alpha == beta)) {
sum_hbar_XXX = 0.0;
for (int kp = 0; kp < chain.Nstrings; ++kp) {
for (int betap = 0; betap < base[kp]; ++betap) {
if (!((j == kp) && (alpha == betap)))
sum_hbar_XXX
+= ddlambda_Theta_XXX (lambda[j][alpha] - lambda[kp][betap], chain.Str_L[j], chain.Str_L[kp]);
}
}
Gaudin_Red[index_jalpha][index_kbeta]
= complex<DP> ( chain.Nsites * chain.Str_L[j]/(lambda[j][alpha] * lambda[j][alpha] + 0.25 * chain.Str_L[j] * chain.Str_L[j])
- sum_hbar_XXX);
}
else {
if ((chain.Str_L[j] == 1) && (chain.Str_L[k] == 1))
Gaudin_Red[index_jalpha][index_kbeta] =
complex<DP> ( 2.0/(pow(lambda[j][alpha] - lambda[k][beta], 2.0) + 1.0));
else
Gaudin_Red[index_jalpha][index_kbeta] =
complex<DP> (ddlambda_Theta_XXX (lambda[j][alpha] - lambda[k][beta], chain.Str_L[j], chain.Str_L[k]));
}
index_kbeta++;
}
}
index_jalpha++;
}
}
return;
}
bool XXX_Bethe_State::Check_Finite_rap ()
{
bool answer = true;
for (int j = 0; j < chain.Nstrings; ++j) {
for (int alpha = 0; alpha < base[j]; ++alpha) {
if (fabs(lambda[j][alpha]) > 1.0e6) answer = false;
}
}
return(answer);
}
// ****************************************************************************************************
// non-member functions
DP Theta_XXX (DP lambda, int nj, int nk)
{
DP result;
if ((nj == 1) && (nk == 1)) result = 2.0 * atan(lambda);
else {
result = (nj == nk) ? 0.0 : 2.0 * atan(2.0 * lambda/fabs(nj - nk));
for (int a = 1; a < JSC::min(nj, nk); ++a) result += 4.0 * atan(2.0 * lambda/(fabs(nj - nk) + 2*a));
result += 2.0 * atan(2.0 * lambda/(nj + nk));
}
return (result);
}
DP ddlambda_Theta_XXX (DP lambda, int nj, int nk)
{
int n = abs(nj - nk);
DP result = (nj == nk) ? 0.0 : DP(n)/(lambda * lambda + 0.25 * n * n);
for (int a = 1; a < JSC::min(nj, nk); ++a) result += 2.0 * (n + 2.0*a)
/ (lambda * lambda + 0.25 * (n + 2.0*a) * (n + 2.0*a));
result += DP(nj + nk)/(lambda * lambda + 0.25 * (nj + nk) * (nj + nk));
return (result);
}
/*
DP ddlambda_Theta_XXX (DP lambda, int nj, int nk)
{
DP result = (nj == nk) ? 0.0 : DP(nj - nk)/(lambda * lambda + 0.25 * (nj - nk) * (nj - nk));
for (int a = 1; a < JSC::min(nj, nk); ++a) result += 2.0 * (nj - nk + 2.0*a) * (nj - nk + 2.0*a)
/ (lambda * lambda + 0.25 * (nj - nk + 2.0*a) * (nj - nk + 2.0*a));
result += DP(nj + nk)/(lambda * lambda + 0.25 * (nj + nk) * (nj + nk));
return (result);
}
*/
XXX_Bethe_State Add_Particle_at_Center (const XXX_Bethe_State& RefState)
{
if (2*RefState.base.Mdown == RefState.chain.Nsites)
JSCerror("Trying to add a down spin to a zero-magnetized chain in Add_Particle_at_Center.");
Vect<int> newM = RefState.base.Nrap;
newM[0] = newM[0] + 1;
Heis_Base newBase (RefState.chain, newM);
XXX_Bethe_State ReturnState (RefState.chain, newBase);
for (int il = 1; il < RefState.chain.Nstrings; ++il)
for (int alpha = 0; alpha < RefState.base.Nrap[il]; ++alpha)
ReturnState.Ix2[il][alpha] = RefState.Ix2[il][alpha];
// Add a quantum number in middle (explicitly: to right of index M[0]/2)
// and shift quantum numbers by half-integer away from added one:
ReturnState.Ix2[0][RefState.base.Nrap[0]/2] = RefState.Ix2[0][RefState.base.Nrap[0]/2] - 1;
for (int i = 0; i < RefState.base.Nrap[0] + 1; ++i)
ReturnState.Ix2[0][i + (i >= RefState.base.Nrap[0]/2)] = RefState.Ix2[0][i] - 1 + 2*(i >= RefState.base.Nrap[0]/2);
return(ReturnState);
}
XXX_Bethe_State Remove_Particle_at_Center (const XXX_Bethe_State& RefState)
{
if (RefState.base.Nrap[0] == 0)
JSCerror("Trying to remove a down spin in an empty Nrap[0] state.");
Vect<int> newM = RefState.base.Nrap;
newM[0] = newM[0] - 1;
Heis_Base newBase (RefState.chain, newM);
XXX_Bethe_State ReturnState (RefState.chain, newBase);
for (int il = 1; il < RefState.chain.Nstrings; ++il)
for (int alpha = 0; alpha < RefState.base.Nrap[il]; ++alpha)
ReturnState.Ix2[il][alpha] = RefState.Ix2[il][alpha];
// Remove midmost and shift quantum numbers by half-integer towards removed one:
for (int i = 0; i < RefState.base.Nrap[0]-1; ++i)
ReturnState.Ix2[0][i] = RefState.Ix2[0][i + (i >= RefState.base.Nrap[0]/2)] + 1 - 2*(i >= RefState.base.Nrap[0]/2);
return(ReturnState);
}
} // namespace JSC
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