Unfolding of bandstructres Image Image An alloy on triangular lattice generate random binary potential in the supercell How to average over different realizations of the disorder? How to get a 'bandstructure' in the elementary (1-atom) unit cell? Brute force approach - many realizations of disorder Image Image Image no disorder weak disorder strong disorder Calculate the band dispersion and density of states for a triangular lattices with 120 def spin order. Consider non-interacting electrons on triangular lattice (calculate the band dispersion and density of states). Add a local exchange field which has a direction as indicated in the picture. Hint: Use the enlarged unit cell indicated in the figure. Note that the local term depend on the lattice site (sublattice) and mixes the up and down spin directions (i.e. spin is not a good quantum number). Use t=1 and several different values of b (starting from 0). 120 deg order on triangular lattice H_0=t_sum_langle.pdf H_i(b)=_&_textco.pdf Tight-binding Hamiltonian Type to enter text (0,0) (0,1) (1,1) a1 a2 (1,0) b^dagger_r_&_qua.pdf X=_1_&_0_&_0_0_&.pdf Image H_i(b)=_&_textco.pdf H_mathbf_k_(b)=_.pdf T_mathbf_k_=_0_&.pdf hopping local fields in xy-plane total 6x6 structure b1 b2 (1,0) (1,1) (0,1) (0,0) Bandstructure (3-site unit cell) Image Image Image Image b=0 b=1 Image Image Density of states b1 b2 k=(x,0) k=(x,-x) Bandstructure (3-site unit cell) Image Image Image Image b=0 b=1 Image Image Density of states b1 b2 k=(x,0) k=(x,-x) How to go back to 1-band (= 1 atom unit cell)? Bandstructure (3-site unit cell) Image Image Image Image b=0 b=1 Image Image Density of states b1 b2 k=(x,0) k=(x,-x) How to go back to 1-band (= 1 atom unit cell)? Chose the right quantity! Dispersion is too 'narrow'. Spectral function epsilon(_mathbf_.pdf A(_mathbf_k_,_om.pdf A(_mathbf_k_,_om.pdf Spectral function langle_b;a^dagge.pdf langle_c^phantom.pdf langle_c^phantom.pdf many-body (general) formalism non-interacting electrons (1p functions) k-diagonal elements of object, which has also off-diagonal (kk') elements Going between different cells Image Type to enter text (0,0) (0,1) (1,1) a1 a2 (1,0) b1 b2 mathbf_b_1&=_mat.pdf m_n_end_pmatrix_.pdf tilde_m_tilde_n_.pdf mathbf_tilde_R_&.pdf tilde_m_tilde_n_.pdf text_Flavor_=_op.pdf Going between different cells Image Type to enter text (0,0) (0,1) (1,1) a1 a2 (1,0) b1 b2 mathbf_b_1&=_mat.pdf m_n_end_pmatrix_.pdf tilde_m_tilde_n_.pdf mathbf_tilde_R_&.pdf tilde_m_tilde_n_.pdf text_Flavor_=_op.pdf operatorname_Div.pdf operatorname_Mod.pdf Going between different cells Image Type to enter text (0,0) (0,1) (1,1) a1 a2 (1,0) b1 b2 mathbf_b_1&=_mat.pdf m_n_end_pmatrix_.pdf tilde_m_tilde_n_.pdf mathbf_tilde_R_&.pdf tilde_m_tilde_n_.pdf text_Flavor_=_op.pdf operatorname_Mod.pdf operatorname_Div.pdf Going between different cells Image Type to enter text (0,0) (0,1) (1,1) a1 a2 (1,0) b1 b2 mathbf_b_1&=_mat.pdf m_n_end_pmatrix_.pdf tilde_m_tilde_n_.pdf mathbf_tilde_R_&.pdf tilde_m_tilde_n_.pdf text_Flavor_=_op.pdf operatorname_Div.pdf operatorname_Mod.pdf Going between different cells Image Type to enter text (0,0) (0,1) (1,1) a1 a2 (1,0) b1 b2 mathbf_b_1&=_mat.pdf c^dagger_mathbf_.pdf r^dagger_mathbf_.pdf sum_mathbf_R_f(_.pdf Going between different cells Image Type to enter text (0,0) (0,1) (1,1) a1 a2 (1,0) b1 b2 mathbf_b_1&=_mat.pdf c^dagger_mathbf_.pdf r^dagger_mathbf_.pdf sum_mathbf_R_f(_.pdf No reference to Hamiltonian (lattice dimension) needed in principle Type to enter text (0,0) (0,1) (1,1) a1 a2 (1,0) b1 b2 mathbf_b_1&=_mat.pdf c^dagger_mathbf_.pdf r^dagger_mathbf_.pdf sum_mathbf_R_f(_.pdf sum_mathbf_R_e^i.pdf mathbf_tilde_R_&.pdf mathbf_k_cdot_ma.pdf Going between different cells b1 b2 Type to enter text (0,0) (0,1) (1,1) a1 a2 (1,0) mathbf_b_1&=_mat.pdf c^dagger_mathbf_.pdf r^dagger_mathbf_.pdf sum_mathbf_R_f(_.pdf sum_mathbf_R_e^i.pdf mathbf_tilde_R_&.pdf sum_mathbf_R_e^i.pdf tilde_k_1&=k_1-k.pdf mathbf_k_cdot_ma.pdf Going between different cells k^alpha_R_alpha.pdf Strictly speaking we are working with covariant and contravariant tensors: c^dagger_mathbf_.pdf Spectral function langle_b;a^dagge.pdf langle_c^phantom.pdf langle_c^phantom.pdf many-body (general) formalism non-interacting electrons (1p functions) k-diagonal elements of object, which has also off-diagonal (kk') elements Spectral function langle_b;a^dagge.pdf langle_c^phantom.pdf langle_c^phantom.pdf many-body (general) formalism non-interacting electrons (1p functions) k-diagonal elements of object, which has also off-diagonal (kk') elements Image Image Image Image b=0 b=1 Image Image Density of states Bandstructure (3-site unit cell) k=(x,0) k=(x,-x) Image Image 1-site cell 3-site cell unfolded Image Image b=0 langle_c^phantom.pdf Image Image Density of states Bandstructure (1-site unit cell) Type to enter text a1 a2 k=(x,0) k=(x,x) Type to enter text a1 a2 Image Bandstructure (1-site unit cell) Image b=0 langle_c^phantom.pdf Image b=1 Image Image Image Density of states Type to enter text a1 a2 Image Bandstructure (1-site unit cell) Image b=0 langle_c^phantom.pdf b=2.5 Image Density of states Image Image Image Symmetry and asymmetry (1-site unit cell) langle_c^phantom.pdf Type to enter text b=2.5 k-space: Type to enter text a1 a2 mathbf_b_1&=_mat.pdf tilde_k_1&=k_1-k.pdf Brillouin zone folding basis vectors: k-vectors coordinates: 1-atom unit cell: The reciprocal lattice is also triangular Type to enter text a1 a2 mathbf_b_1&=_mat.pdf tilde_k_1&=k_1-k.pdf Brillouin zone folding basis vectors: k-vectors coordinates: 1-atom unit cell: (k-space) Brillouin zone Type to enter text Type to enter text a1 a2 mathbf_b_1&=_mat.pdf tilde_k_1&=k_1-k.pdf Brillouin zone folding basis vectors: k-vectors coordinates: Brillouin zone Type to enter text 3-atom unit cell (k-space): 3-atom unit cell (lattice): 1-atom unit cell: (k-space) The small BZ is 1/3 of the large BZ. Type to enter text a1 a2 mathbf_b_1&=_mat.pdf tilde_k_1&=k_1-k.pdf Brillouin zone folding basis vectors: k-vectors coordinates: Brillouin zone Type to enter text 3-atom unit cell (k-space): For each point in the small (blue) BZ there are 3 points in the large BZ. Band folding: 1 -> 3 bands Band unfolding: 3 bands (+ off diagonal elements) @ k-point in small BZ -> 3 k-points in large BZ 1-atom unit cell: (k-space) An alloy on triangular lattice generate random binary potential in the supercell How to average over different realizations of the disorder? How to get a 'bandstructure' in the elementary (1-atom) unit cell? Brute force approach - many realizations of disorder Image Image Image no disorder weak disorder strong disorder (0,0) (1,0) (0,1) (-1,0) a b (0,-1) A B A: s=(2/3,1/3) B: s=(1/3,2/3) m(r)&=_sum_R,s_S.pdf langle_m(k);_m(-.pdf Lattice <-> continuum (theo <-> exp) Lattice models 'live' on k-space torus <-> materials live in non-compact k-space ? Bring in the structure of the underlying orbitals. (0,0) (1,0) (0,1) (-1,0) a b (0,-1) A B A: s=(2/3,1/3) B: s=(1/3,2/3) m(r)&=_sum_R,s_S.pdf langle_m(k);_m(-.pdf Lattice <-> continuum (theo <-> exp) Lattice models 'live' on k-space torus <-> materials live in non-compact k-space ? The model correlation functions capture long wavelength behavior (k inside the 1st BZ). The matrix elements encode the short wavelength behavior (variation between BZs).