int main(int argc, char* argv[]) {
boost::mpi::environment env(argc,argv);
boost::mpi::communicator comm;
print_section("Hubbard nxn");
int size_x, size_y, wn;
RealType t, mu, U, beta, reduce_tol, coeff_tol;
bool calc_gf, calc_2pgf;
int wf_max, wb_max;
double eta, hbw, step; // for evaluation of GF on real axis
try { // command line parser
TCLAP::CmdLine cmd("Hubbard nxn diag", ' ', "");
TCLAP::ValueArg<RealType> U_arg("U","U","Value of U",true,10.0,"RealType",cmd);
TCLAP::ValueArg<RealType> mu_arg("","mu","Global chemical potential",false,0.0,"RealType",cmd);
TCLAP::ValueArg<RealType> t_arg("t","t","Value of t",false,1.0,"RealType",cmd);
TCLAP::ValueArg<RealType> beta_arg("b","beta","Inverse temperature",true,100.,"RealType");
TCLAP::ValueArg<RealType> T_arg("T","T","Temperature",true,0.01,"RealType");
cmd.xorAdd(beta_arg,T_arg);
TCLAP::ValueArg<size_t> x_arg("x","x","Size over x",false,2,"int",cmd);
TCLAP::ValueArg<size_t> y_arg("y","y","Size over y",false,2,"int",cmd);
TCLAP::ValueArg<size_t> wn_arg("","wf","Number of positive fermionic Matsubara Freqs",false,64,"int",cmd);
TCLAP::ValueArg<size_t> wb_arg("","wb","Number of positive bosonic Matsubara Freqs",false,1,"int",cmd);
TCLAP::SwitchArg gf_arg("","calcgf","Calculate Green's functions",cmd, false);
TCLAP::SwitchArg twopgf_arg("","calc2pgf","Calculate 2-particle Green's functions",cmd, false);
TCLAP::ValueArg<RealType> reduce_tol_arg("","reducetol","Energy resonance resolution in 2pgf",false,1e-5,"RealType",cmd);
TCLAP::ValueArg<RealType> coeff_tol_arg("","coefftol","Total weight tolerance",false,1e-12,"RealType",cmd);
TCLAP::ValueArg<RealType> eta_arg("","eta","Offset from the real axis for Green's function calculation",false,0.05,"RealType",cmd);
TCLAP::ValueArg<RealType> hbw_arg("D","hbw","Half-bandwidth. Default = U",false,0.0,"RealType",cmd);
TCLAP::ValueArg<RealType> step_arg("","step","Step on a real axis. Default : 0.01",false,0.01,"RealType",cmd);
cmd.parse( argc, argv );
U = U_arg.getValue();
mu = (mu_arg.isSet()?mu_arg.getValue():U/2);
boost::tie(t, beta, calc_gf, calc_2pgf, reduce_tol, coeff_tol) = boost::make_tuple( t_arg.getValue(), beta_arg.getValue(),
gf_arg.getValue(), twopgf_arg.getValue(), reduce_tol_arg.getValue(), coeff_tol_arg.getValue());
boost::tie(size_x, size_y) = boost::make_tuple(x_arg.getValue(), y_arg.getValue());
boost::tie(wf_max, wb_max) = boost::make_tuple(wn_arg.getValue(), wb_arg.getValue());
boost::tie(eta, hbw, step) = boost::make_tuple(eta_arg.getValue(), (hbw_arg.isSet()?hbw_arg.getValue():2.*U), step_arg.getValue());
calc_gf = calc_gf || calc_2pgf;
calc_gf = calc_gf || calc_2pgf;
}
catch (TCLAP::ArgException &e) // catch parsing exceptions
{ std::cerr << "error: " << e.error() << " for arg " << e.argId() << std::endl; exit(1);}
int L = size_x*size_y;
INFO("Diagonalization of " << L << "=" << size_x << "*" << size_y << " sites");
Lattice Lat;
/* Add sites */
std::vector<std::string> names(L);
for (size_t y=0; y<size_y; y++)
for (size_t x=0; x<size_x; x++)
{
size_t i = SiteIndexF(size_x, x, y);
std::stringstream s; s << i;
names[i]="S"+s.str();
Lat.addSite(new Lattice::Site(names[i],1,2));
};
INFO("Sites");
Lat.printSites();
/* Add interaction on each site*/
MelemType U_complex;
U_complex = std::complex<double> (U,0);
MelemType mu_complex;
mu_complex = std::complex<double> (mu,0);
for (size_t i=0; i<L; i++) LatticePresets::addCoulombS(&Lat, names[i], U_complex, -mu_complex);
/* Add hopping */
for (size_t y=0; y<size_y; y++) {
for (size_t x=0; x<size_x; x++) {
size_t pos = SiteIndexF(size_x, x,y);
size_t pos_right = SiteIndexF(size_x, (x+1)%size_x,y); /*if (x == size_x - 1) pos_right = SiteIndexF(0,y); */
size_t pos_up = SiteIndexF(size_x, x,(y+1)%size_y);
if (size_x > 1) LatticePresets::addHopping(&Lat, std::min(names[pos], names[pos_right]), std::max(names[pos], names[pos_right]), -t);
if (size_y > 1) LatticePresets::addHopping(&Lat, std::min(names[pos], names[pos_up]), std::max(names[pos], names[pos_up]), -t);
};
};
int rank = comm.rank();
if (!rank) {
INFO("Terms with 2 operators");
Lat.printTerms(2);
INFO("Terms with 4 operators");
Lat.printTerms(4);
};
IndexClassification IndexInfo(Lat.getSiteMap());
IndexInfo.prepare(false); // Create index space
if (!rank) { print_section("Indices"); IndexInfo.printIndices(); };
int index_size = IndexInfo.getIndexSize();
print_section("Matrix element storage");
IndexHamiltonian Storage(&Lat,IndexInfo);
Storage.prepare(); // Write down the Hamiltonian as a symbolic formula
print_section("Terms");
if (!rank) INFO(Storage);
Symmetrizer Symm(IndexInfo, Storage);
Symm.compute(); // Find symmetries of the problem
StatesClassification S(IndexInfo,Symm); // Introduce Fock space and classify states to blocks
S.compute();
Hamiltonian H(IndexInfo, Storage, S); // Hamiltonian in the basis of Fock Space
H.prepare(); // enter the Hamiltonian matrices
H.compute(); // compute eigenvalues and eigenvectors
RealVectorType evals (H.getEigenValues());
std::sort(evals.data(), evals.data() + H.getEigenValues().size());
savetxt("spectrum.dat", evals); // dump eigenvalues
DensityMatrix rho(S,H,beta); // create Density Matrix
rho.prepare();
rho.compute(); // evaluate thermal weights with respect to ground energy, i.e exp(-beta(e-e_0))/Z
INFO("<N> = " << rho.getAverageOccupancy()); // get average total particle number
savetxt("N_T.dat",rho.getAverageOccupancy());
// Green's function calculation starts here
FieldOperatorContainer Operators(IndexInfo, S, H); // Create a container for c and c^+ in the eigenstate basis
if (calc_gf) {
INFO("1-particle Green's functions calc");
std::set<ParticleIndex> f;
std::set<IndexCombination2> indices2;
ParticleIndex d0 = IndexInfo.getIndex("S0", 0, down);
ParticleIndex u0 = IndexInfo.getIndex("S0", 0, up);
f.insert(u0);
f.insert(d0);
for (size_t x = 0; x < size_x; x++) {
ParticleIndex ind = IndexInfo.getIndex(names[SiteIndexF(size_x, x, 0)], 0, down);
f.insert(ind);
indices2.insert(IndexCombination2(d0, ind));
};
Operators.prepareAll(f);
Operators.computeAll(); // evaluate c, c^+ for chosen indices
GFContainer G(IndexInfo, S, H, rho, Operators);
G.prepareAll(indices2); // identify all non-vanishing block connections in the Green's function
G.computeAll(); // Evaluate all GF terms, i.e. resonances and weights of expressions in Lehmans representation of the Green's function
if (!comm.rank()) { // dump gf into a file
std::set<IndexCombination2>::iterator ind2;
for (ind2 = indices2.begin(); ind2 !=
indices2.end(); ++ind2) { // loops over all components (pairs of indices) of the Green's function
// Save Matsubara GF from pi/beta to pi/beta*(4*wf_max + 1)
std::cout << "Saving imfreq G" << *ind2 << " on " << 4 * wf_max << " Matsubara freqs. " << std::endl;
std::stringstream fname;
fname << "gw_imag" << (*ind2).Index1 << (*ind2).Index2 << ".dat";
std::ofstream gw_im(fname.str().c_str());
const GreensFunction &GF = G(*ind2);
for (int wn = 0; wn < wf_max * 4; wn++) {
ComplexType val = GF(
I * FMatsubara(wn, beta)); // this comes from Pomerol - see GreensFunction::operator()
gw_im << std::scientific << std::setprecision(12) << FMatsubara(wn, beta) << " " << real(val) <<
" " << imag(val) << std::endl;
};
gw_im.close();
}
}
}
return 0;
}
int main(int argc, char* argv[])
{
boost::mpi::environment env(argc,argv);
boost::mpi::communicator world;
Lattice L;
L.addSite(new Lattice::Site("A",1,2));
LatticePresets::addCoulombS(&L, "A", U, -mu);
L.addSite(new Lattice::Site("B",1,2));
LatticePresets::addCoulombS(&L, "B", U, -mu);
LatticePresets::addHopping(&L, "A","B", -1.0);
INFO("Sites");
L.printSites();
INFO("Terms");
L.printTerms(2);
INFO("Terms with 4 operators");
L.printTerms(4);
IndexClassification IndexInfo(L.getSiteMap());
IndexInfo.prepare();
print_section("Indices");
IndexInfo.printIndices();
ParticleIndex NModes = IndexInfo.getIndexSize();
std::vector<ParticleIndex> SpinUpIndices;
for (ParticleIndex i=0; i<NModes; i++) if (IndexInfo.getInfo(i).Spin) SpinUpIndices.push_back(i);
print_section("Matrix element storage");
IndexHamiltonian Storage(&L,IndexInfo);
Storage.prepare();
INFO("Terms");
INFO(Storage);
ParticleIndex IndexSize = IndexInfo.getIndexSize();
OperatorPresets::Sz Sz(IndexSize, SpinUpIndices);
INFO("Sz terms");
Sz.printAllTerms();
OperatorPresets::N N(IndexSize);
INFO("N terms");
N.printAllTerms();
if (!(Sz.commutes(N))) return EXIT_FAILURE;
if (!(Storage.commutes(N))) return EXIT_FAILURE;
INFO("H commutes with N");
if (!(Storage.commutes(Sz))) return EXIT_FAILURE;
INFO("H commutes with Sz");
Symmetrizer Symm(IndexInfo, Storage);
Symm.compute();
StatesClassification S(IndexInfo,Symm);
S.compute();
Hamiltonian H(IndexInfo, Storage, S);
H.prepare();
H.getPart(BlockNumber(4)).print_to_screen();
H.getPart(BlockNumber(5)).print_to_screen();
H.compute(world);
H.getPart(BlockNumber(4)).print_to_screen();
H.getPart(BlockNumber(5)).print_to_screen();
INFO("The value of ground energy is " << H.getGroundEnergy());
RealType beta = 10.0;
DensityMatrix rho(S,H,beta);
rho.prepare();
rho.compute();
for (QuantumState i=0; i<S.getNumberOfStates(); ++i) INFO(rho.getWeight(i));
FieldOperatorContainer Operators(IndexInfo, S, H);
Operators.prepareAll();
Operators.computeAll();
ParticleIndex down_index = IndexInfo.getIndex("A",0,down);
FieldOperator::BlocksBimap c_map = Operators.getCreationOperator(0).getBlockMapping();
for (FieldOperator::BlocksBimap::right_const_iterator c_map_it=c_map.right.begin(); c_map_it!=c_map.right.end(); c_map_it++)
{
INFO(c_map_it->first << "->" << c_map_it->second);
}
GreensFunction GF(S,H,Operators.getAnnihilationOperator(0), Operators.getCreationOperator(0), rho);
GF.prepare();
GF.compute();
ComplexVectorType G_ref(10);
G_ref << -2.53021005e-01*I,
-4.62090702e-01*I,
-4.32482782e-01*I,
-3.65598615e-01*I,
-3.07785174e-01*I,
-2.62894141e-01*I,
-2.28274316e-01*I,
-2.01170772e-01*I,
-1.79539602e-01*I,
-1.61950993e-01*I;
bool result = true;
for(int n = 0; n<10; ++n) {
INFO(GF(n) << " == " << G_ref(n));
result = (result && compare(GF(n),G_ref(n)));
}
if (!result) return EXIT_FAILURE;
return EXIT_SUCCESS;
}
int main(int argc, char* argv[])
{
boost::mpi::environment MpiEnv(argc, argv);
world = boost::mpi::communicator();
/* As pomerol is an ED code, it requires a finite-size lattice to be
* provided. Here is an example of a lattice of 2 sites. */
// First, we construct an empty lattice
Lattice L;
// Add a site with a name "A", that has 1 orbitals and 2 spins.
L.addSite(new Lattice::Site("A",1,2));
// Add one more site with a name "B". It also has 1 orbitals and 2 spins.
L.addSite(new Lattice::Site("B",1,2));
// Let us now connect two sites with a hopping term, with matrix element -1.
RealType t=1.0;
LatticePresets::addHopping(&L, "A","B", -t);
/* Now add interaction. In order to provide some custom interaction, one can
* give any custom term of 4,6 operators. This is done via
* Lattice.addTerm(Lattice::Term) method.
* We will use Hubbard-type n_{up}n_{down} interaction. For this and some
* other typical interactions, such as SzSz or SS couplings a shortcut is
* provided in the LatticePresets class.
*/
RealType U = 2.0;
RealType mu = 1.0;
// LatticePresets::addCoulombS adds Un_{up}n_{down} for 1 orbital and 2 spins.
LatticePresets::addCoulombS(&L, "A", U, -mu);
LatticePresets::addCoulombS(&L, "B", U, -mu);
// Let us now print which sites and terms are defined.
if (!world.rank()) {
INFO("Sites"); // equivalent to std::cout << "Sites" << std::endl;
L.printSites();
INFO("Terms");
L.printTerms(2);
INFO("Terms with 4 operators");
L.printTerms(4);
print_section("Indices");
};
/* In order to go further, we need to introduce the index space. An index
* is a number that uniquely identifies a combination of (site,orbital,spin).
* The object that takes care of handling indices is called
* IndexClassification.
*/
// Construct IndexClassification
IndexClassification IndexInfo(L.getSiteMap());
/* Important remark 1!
* Most of the objects that are defined within Pomerol library have the
* following semantics. They can be constructed, prepared and computed.
* This means
* - constructed: No operations are done except from initializing links to
* other objects that current class depends on.
* - prepared: Typically, this is when all memory allocation takes place.
* - computed: The actual computation. This is the most costly operation.
* When no heavy computation is done, it can be done during
* preparation stage.
*/
// IndexClassification does not require much computing time, so everything
// is calculated during prepare() method.
IndexInfo.prepare();
// Print which indices we have
IndexInfo.printIndices();
// Save the total number of indices.
ParticleIndex IndexSize = IndexInfo.getIndexSize();
print_section("Matrix element storage");
/* We now need to construct the actual Hamiltonian. First, we need to create
* it in the index space, e.g. write down a formula with all terms. It is
* useful for a subsequent symmetry analysis. The corresponding class is called
* an IndexHamiltonian. It is inherited from Operator class, which is designed
* to represent a generic second-quantized fermionic operator and fully model
* the algebra of fermionic operators. This means one can multiply, add,
* subtract it, and also calculate the commutator of two operators, etc.
*/
// First construct the IndexHamiltonian object.
IndexHamiltonian Storage(&L,IndexInfo);
// Then prepare it. As previously no long computation required, so everything
// is done in the prepare() method.
Storage.prepare();
// Print out the Hamiltonian.
INFO("Terms");
INFO(Storage);
// Let us make a test, that our Hamiltonian commutes with an operator, that
// represents the total number of particles. The preset for such operator is
// defined in OperatorPresets.h and .cpp files.
OperatorPresets::N N(IndexSize);
INFO("N terms");
N.printAllTerms();
if ((Storage.commutes(N))) INFO("H commutes with N");
/* The Hamiltonian that is set now, has a set of symmetries. One can identify
* them by checking that the IndexHamiltonian commutes with the corresponding
* symmetry operator. The class that does it is called Symmetrizer.
*/
// Construct Symmetrizer
Symmetrizer Symm(IndexInfo, Storage);
/* At this stage Symmetrizer checks symmetries with some predefined operators,
* such as number-of-particles and sz operators.
*/
Symm.compute();
/* A custom check for a symmetry can be done using
* Symmetrizer.checkSymmetry(Operator).
* After checking all the symmetry operations, Symmetrizer defines a set of
* quantum numbers, which uniquely identifies the closed region in the
* phase space.
* Calling Symmetrizer::compute(true) will skip all symmetry operations and
* result in a single Hamiltonian block in the end.
*/
/* We shall proceed now with obtaining the spectrum of the Hamiltonian.
* First, introduce a basis of Fock states and classify them into blocks
* that correspond to a set of quantum numbers. This is done in the
* StatesClassification class. It provides all the information about Blocks
* and FockStates.
*/
StatesClassification S(IndexInfo,Symm);
S.compute();
/* We now convert the IndexHamiltonian into the basis of Fock States. The
* Hamiltonian class is the one, which does it.
*/
Hamiltonian H(IndexInfo, Storage, S);
// Enter all blocks of the Hamiltonian.
H.prepare();
// Diagonalize them.
H.compute(world);
// Get ground energy.
INFO("The value of ground energy is " << H.getGroundEnergy());
/* Important remark 2!
* All the calculations done in the pomerol code are done with respect to the
* block structure of the Hamiltonian. All objects that operate in Fock
* space and all thermal objects, such as Green's functions are in fact a
* set of pieces (called "parts") that operate on a certain block or set of
* blocks. As such all actual computations are done within this parts and
* grand objects like Green's functions or Hamiltonian basically just loops
* over parts and tells them to run prepare() or compute() methods.
*/
/* At this stage the Hamiltonian, defined in Fock Space is
* entered and diagonalized and it's spectrum and eigenfunctions can be
* directly accessed to calculate some observables.
*
* We shall now proceed to the calculations of thermal quantities, e.g.
* assume that our finite-size system was adiabatically connected to a thermal
* reservoir, that sets certain temperature (in fact, inverse temperature
* beta). This means, that the observables in the systems should be calculated
* with a density-matrix exp(-\beta H), rather than by averaging with the
* ground state. In the eigenbasis of the Hamiltonian the calculation of
* a density matrix is straightforward - it is just \exp(-\beta (E_i - E_0)),
* where E_i is an energy of the excited state, and E_0 is the ground energy.
* The procedure is done as following:
*/
// Define inverse temperature
RealType beta = 10.0;
// Create the Density Matrix.
DensityMatrix rho(S,H,beta);
// Allocate all internal parts of density matrix.
rho.prepare();
// Actually compute the density matrix.
rho.compute();
/* Lehmanns representation of the Green's function required creation and
* annihilation operators, calculated in the basis of eigenstates of the
* Hamiltonian. Creation/AnnihilationOperator are the classes that do it.
*/
// Let us create c^+_{"A",up} and c_{"A",up}
ParticleIndex up_index = IndexInfo.getIndex("A",0,up);
CreationOperator CX(IndexInfo,S,H,up_index);
CX.prepare();
CX.compute();
AnnihilationOperator C(IndexInfo,S,H,up_index);
C.prepare();
C.compute();
// The local Greens function in the Matsubara domain G_{"A",up}(i\omega_n)
GreensFunction GF(S,H,C,CX, rho);
GF.prepare();
/* Calculate the GF and cache values for 10 Matsubaras.
* These values will be fast to retrieve. The other values are also
* accessible, but will require a short calculation to be done.
*/
GF.compute();
for(int n = 0; n<10; ++n) {
INFO(n << " | " << GF(n));
}
/* The two particle GF is constructed in analogy to the single-particle GF,
* it requires 4 operators to be provided though.
*/
TwoParticleGF Chi(S,H,C,C,CX,CX,rho);
/* Some knobs to make calc faster - the larger the values of tolerances, the faster is calc, but rounding errors may show.
Typically these errors are less then 10^{-3} of the value.
Here are some knobs that give very high-precision. If you want to make things faster, and when many values on
different frequencies required - change ReduceResonanceTolerance to something like 10^-4. */
/** A difference in energies with magnitude less than this value is treated as zero. */
Chi.ReduceResonanceTolerance = 1e-8;
/** Minimal magnitude of the coefficient of a term to take it into account. */
Chi.CoefficientTolerance = 1e-16;
/** Knob that controls the caching frequency. */
Chi.ReduceInvocationThreshold = 1e5;
/** Minimal magnitude of the coefficient of a term to take it into account with respect to amount of terms. */
Chi.MultiTermCoefficientTolerance = 1e-6;
Chi.prepare();
Chi.compute(world);
if (world.rank()==0) {
int nm = 2;
for(int n1 = -nm; n1<nm; ++n1)
for(int n2 = -nm; n2<nm; ++n2)
for(int n3 = -nm; n3<nm; ++n3){
INFO(n1 << " " << n2 << " " << n3 << "|" << Chi(n1,n2,n3));
};
};
}