def sorted_basis(basis_numbering=False): """Sorts the basis of states for the electrons in the molecule Enforces ordering in particle number an Sz projection""" ind = np.array([0, 1, 2, 4, 8, 5, 10, 6, 9, 12, 3, 7, 11, 13, 14, 15]) if basis_numbering: ind = basis_numbering basis = [fermion.destruct(4, sigma)[ind][:, ind] for sigma in range(4)] return basis
def test_gf_lehmann2(): """Verifies the lehmann representation of a random Hamiltonian""" w = np.linspace(-1.5, 1.5, 500) d_up = fermion.destruct(6, 0) H = np.random.rand(*d_up.shape) H += H.T e, v = operators.diagonalize(H) g_up_diag = operators.gf_lehmann(e, v, d_up.T, 400, w) g_up_cross = operators.gf_lehmann(e, v, d_up.T, 400, w, d_up) assert np.allclose(g_up_diag, g_up_cross)
def test_gf_lehmann2(): """Verifies the lehmann representation of a random Hamiltonian""" w = np.linspace(-1.5, 1.5, 500) d_up = fermion.destruct(6, 0) H = np.random.rand(*d_up.shape) H += H.T e, v = operators.diagonalize(H) g_up_diag = operators.gf_lehmann(e, v, d_up.T, 400, w) g_up_cross = operators.gf_lehmann(e, v, d_up.T, 400, w, d_up) assert np.allclose(g_up_diag, g_up_cross)
def __init__(self, beta, t): self.beta = beta self.t = t self.m2 = m2_weight(t) self.mu = 0. self.e_c = 0. self.eig_energies = None self.eig_states = None self.oper = [fermion.destruct(4, index) for index in range(4)] self.H_operators = self.hamiltonian() self.GF = {}
def __init__(self, beta, t): self.beta = beta self.t = t self.m2 = m2_weight(t) self.mu = 0. self.e_c = 0. self.eig_energies = None self.eig_states = None self.oper = [fermion.destruct(4, index) for index in range(4)] self.H_operators = self.hamiltonian() self.GF = {}
def hamiltonian(U, mu): r"""Generate a single orbital isolated atom Hamiltonian in particle-hole symmetry. Include chemical potential for grand Canonical calculations .. math:: \mathcal{H} - \mu N = -\frac{U}{2}(n_\uparrow - n_\downarrow)^2 - \mu(n_\uparrow + n_\downarrow) """ d_up, d_dw = [fermion.destruct(2, sigma) for sigma in range(2)] sigma_z = d_up.T * d_up - d_dw.T * d_dw H = -U / 2 * sigma_z * sigma_z - mu * (d_up.T * d_up + d_dw.T * d_dw) return H, d_up, d_dw
def test_fermion_anticommut(): """Verifies the fermion anticommutation relations""" particles = 3 c = [fermion.destruct(particles, index) for index in range(particles)] for i in range(particles): for j in range(i, particles): assert operators.anticommutator(c[i].T, c[j].T).nnz == 0 assert operators.anticommutator(c[i], c[j]).nnz == 0 k_delta = operators.anticommutator(c[i], c[j].T) if i == j: assert (k_delta - eye(2**particles)).nnz == 0 else: assert k_delta.nnz == 0
def test_fermion_anticommut(): """Verifies the fermion anticommutation relations""" particles = 3 c = [fermion.destruct(particles, index) for index in range(particles)] for i in range(particles): for j in range(i, particles): assert operators.anticommutator(c[i].T, c[j].T).nnz == 0 assert operators.anticommutator(c[i], c[j]).nnz == 0 k_delta = operators.anticommutator(c[i], c[j].T) if i == j: assert (k_delta - eye(2**particles)).nnz == 0 else: assert k_delta.nnz == 0
def test_gf_lehmann(U, mu, beta): """Verifies the lehmann representation of the atom""" w = np.linspace(-1.5*U, 1.5*U, 500) d_up, d_dw = [fermion.destruct(2, sigma) for sigma in range(2)] sigma_z = d_up.T*d_up - d_dw.T*d_dw H = - U/2 * sigma_z * sigma_z - mu * (d_up.T*d_up + d_dw.T*d_dw) e, v = operators.diagonalize(H.todense()) g_up = operators.gf_lehmann(e, v, d_up.T, beta, w) g_up_cross = operators.gf_lehmann(e, v, d_up.T, beta, w, d_up) Z = 1+2*np.exp(beta*(U/2+mu)) + np.exp(2*beta*mu) g_up_ref = ((1+np.exp(beta*(U/2+mu)))/(w + mu + U/2.) + (np.exp(beta*(U/2+mu)) + np.exp(2*beta*mu))/(w + mu - U/2.))/Z assert np.allclose(g_up, g_up_ref) assert np.allclose(g_up_cross, g_up_ref)
def test_gf_lehmann(U, mu, beta): """Verifies the lehmann representation of the atom""" w = np.linspace(-1.5 * U, 1.5 * U, 500) d_up, d_dw = [fermion.destruct(2, sigma) for sigma in range(2)] sigma_z = d_up.T * d_up - d_dw.T * d_dw H = -U / 2 * sigma_z * sigma_z - mu * (d_up.T * d_up + d_dw.T * d_dw) e, v = operators.diagonalize(H.todense()) g_up = operators.gf_lehmann(e, v, d_up.T, beta, w) g_up_cross = operators.gf_lehmann(e, v, d_up.T, beta, w, d_up) Z = 1 + 2 * np.exp(beta * (U / 2 + mu)) + np.exp(2 * beta * mu) g_up_ref = ((1 + np.exp(beta * (U / 2 + mu))) / (w + mu + U / 2.) + (np.exp(beta * (U / 2 + mu)) + np.exp(2 * beta * mu)) / (w + mu - U / 2.)) / Z assert np.allclose(g_up, g_up_ref) assert np.allclose(g_up_cross, g_up_ref)