1. La 2 CuO 4 insulator gap, AF structure and pseudogaps from spin-space entangled orbitals in the HF scheme Alejandro Cabo-Bizet, CEADEN, Havana, Cuba acbizet@gmail.com Alejandro Cabo Montes de Oca, ICIMAF, Havana, Cuba cabo@icmf.inf.cu Phys. Lett. A 373, (2009) 1865 Symmetry 2010, 2, 388

2. Overview of the results It is argued that La 2 CuO 4, which is conventionally considered as a primer of a Mott Insulator, can be also described as a Slater Insulator in the framework of the HF scheme. Then, it follows that its assumed strong correlations properties are in fact described by optimizing the mean field solutions. The results are natural ones after bearing in mind that correlation effects should be properly defined as the ones which are not taken into account by the “best” lower energy HF solution. The discussion also furnishes a clear picture of the sources of the AF order in this HTc superconductor materials, as following from a spin-orbit entanglement effect. HF pseudogap states are also determined. The analysis indicates hints for clarifying the Mott-First Principles debate for the transition metal oxides; and suggests a path for quantitatively describing the phase diagram of La 2 CuO 4 in the temperature and doping plane.

3. Overview of the Model The first band calculations indicated a paramegnetic metal nature of the material Band structure of La 2 CuO 4 calculated (Matheiss,PRL 58 (1987)1028) for orbitals of type  or  and showing Bloch property in the lattice of half the size of the AF order one. The N Coulomb interacting electrons of the model will be the ones associated to the partially filled band shown in red. They will be assumed to have a Tight Binding free hamiltonian associated to an effective mean field W  created by the rest of the electrons in the multiple filled bands. The Coulomb interacting potential between the N electrons is assumed to be screened by the mascroscopic dielectric  constant of the cloud of the electrons in the filled bands and the nuclei.

4. HF method deleting usual symmetry constraints N electron HF Slater determinant state The HF equations, without any restriction of the spin structure of the single particle orbitals Full Hamiltonian of the N electron system The single particle orbitals can show a kind of spin spatial entanglement The quantum numbers of the HF orbitals

5. The Tight Binding single particle free Hamiltonian of the metallic half filled band of the N electron system The Tight Binding effective potential created by the electrons in the completely filled electron bands, The potential have the symmetrry under the spatial shifts R of the Cu atoms in the CuO planes Cu atoms lattice in the CuO planes: p is the smallest distance between Cu atoms F b is the jellium potential generated by Gaussian distributed densities of charges centered around any point of the lattice R The interacting Coulomb potential is assumed to be screened by the dielectric response of the set of filled electron bands and the nuclei. p

6. Sublatt. r=1Sublatt. r=2 The lattice R is decomposed in two sublattices R (1) and R (2) for allowing the breaking the translational symmetry of the orbitals by only imposing on them to obey the Bloch condition for symmetry translations limited to the reduced invariance group of the sublattices. The vectors q relate the two sublattices by shifting one of them into the other.

7. Bloch functions are constructed starting from Gaussian Tight binding functions of typical width a centered in the points of each of the two sublattices r=1 and r=2. They are indexed by a quasimomentum quantum number k pertaining to the common Brillouin cell of each sublattice. The quasimomenta are discretized by imposing periodic boundary conditions in a large squared box of lateral size L, of the original lattice R. The spins are assumed to be of types  or   The HF orbitals breaking the translation and spin structures in the Matheiss etal band calculations. The linear combination of the Bloch functions with the coefficients B allows for spin projections depending on the position and translation symmetries smaller than the ones in the original lattice R.

8. HF equations reduced to a set of 4x4 matrix equations, one for each quasimomenta k. As mentioned before the orbitals are proposed in the form showing the spin spatial entanglement The HF system of equations can be reduced to set of 4x4 matrix equations: one for each quasimomenta value k The systems is solved by an iterative method of solution starting from a state ansatz. For each value of k, four eigenvalues are obtained corresponding to four electorn bands.The Kramers degeneracy explains why only two energy bands follows. Free T.B. Hamiltonian matrix Jellium potential matrix Overlapping matrix, reflecting that the Bloch functions for different sublattices may not be orthogonal Direct and Exchange Potential matrix

10. Paramagnetic Metallic higher energy solution The first solution that was searched was one having the usual symmetry restrictions. That is, being a Bloch function in the original lattice of the Cu atoms and showing the  or  restricted spin orientations. The same before defined 2D Gaussian orbitals were used for this construction

11. Dispersion relation obtained from the iterative solutions of the HF equations. It very well reproduces the Matheiss etal single band crossing the Fermi level in their band calculations. This result allowed to select the parameters of the model by reproducing the calculated bandwith.

12. AF insulator lowest energy solution Afterwards, the symmetry restrictions were deleted. The results for the HF single electron spectrum obtained from the iterative solution of the HF equations without any of the two restrictions gave the following band structure: The N electrons fill the two degenerated lower bands The resulting HF state corresponds to an antiferromagnetic insulator, showing that La 2 CuO 4 can be also described as a Slater insulator even in a mean field description.

13. AF order of the isolator solution An AF magnetic moment of 0.68 mB for any base of CuO2 is concentrated around the Cu sites. Its direction rests at 45 degree respect to the CuO bonds. However, this is an artifact of the way of solution, because spin–orbit effects were not taken into account. Local spin density spin operator

14. The single particle states exhibit a sharp antiferromagnetism in the proximities of the Brillouin zone boundaries. The figue shows the k dependence of the angle between two magnetic moment vectors associated to each of the sublattices for a given orbital. These vectors are defined as the: integrals of the magnetic moment density of the given state, over all the units cells of the original lattice R centered in the corresponding sublattice points. Note that for the orbitals being closer to the Birllouin zone border, the AF order is more intense. This furnishes a clear explanation of the strong effect of doping in the destruction of the AF order. Electrons show a spin-orbit entanglemenet x s

15. The occupied bands of the AF-Insulator and Paramgetic metallic HF solutions in the same plot. Note that the contribution of each orbital to the energy difference is higher in approaching the border. Thus, doping of holes also should tend to reduce the total energy difference between the two states

16. Paramagnetic Pseudogap higher energy solution Next, only the symmetry constraint of being Bloch functions in the original Cu lattice R was removed. The spin orientation were yet fixed to be of  (+1/2) or  (- 1/2) type. Then, the iterative solution of the HF problem gave a state which numerically coincides with the found metallic paramagnetic solution for the filled electron states. However, the excited single particle states showed a gap which depends of the momenta values along the Fermi surface. Thus, a pseudogap state is also following from the given HF discussion. The graphic at the left shows the dependence of the pseudogap along one of the sides of the Fermi surface (curve). The maximal value of the gap at the mid point of the side is ~ 100 meV. Similar values had been estimated through ARPES for the higher pseudogap in La 2 CuO 4. ~100 meV

17. If the energy reference is at the Fermi level of the AF filled band, then: a)The doping with holes of the excited pseudogap state should reduce its energy. b) On the contrary the, same doping should increase the energy of the AF state. c) Due to the adopted energy reference, the slope of the energy decaying with doping of the excited state seems that should be larger than the growing slope with doping of the AF state. Then, a phase transition between the AF state and the pseudogap one is expected to occurs after doping. This possibility suggest a following picture which is only starting being explored.

18. Possible description of the SC transition with varying doping The large screening at the corners of the Brillouin cell suggest the possibility that the strong Coulomb repulsion at sizes of the Cooper pairs (few lattice cells) could be strongly reduced by the combination of the large homogeneous dielectric constant of the material (  and the momentum dependent 2D screening. However, it has been suggested that Coulomb interaction can lead to binding, but in any case at zero doping the ground state is the AF one.

19. Assuming that under sufficient large doping the pseudogap state gets lower energy than the AF one. Then, it could be possible that the single particle of higher energy HF state can develope a spin dependent kernel of the Bethe Salpeter equation which could bind preformed hole-hole pairs, thanks to the before mentioned strong screening of the Coulomb interaction kernel. Then, after increasing the doping the preformed pairs could condense to produce the superconductor state.

20. Conclusions Its is argued that a primer of a Mott insulator as La 2 CuO 4 can be also described as a Slater insulator, within a mean field formulation. Then, the connection between the Mott and First Principles pictures is clarified in this material. Therefore, the usually assumed strong correlation properties of this superconductor, as its insulator character and the AF order, are shown to be described as mean field properties. This conclusion is not strange if we define the correlation properties as the ones which are not predicted by the “best” (lower energy) HF solution of the problem. A natural explanation of the nature of the antiferromagnetism in HTSC materials is also given: the single particle HF orbitals show an entangled spatial and spin structure, in a way that every particle exhibits a kind of AF or ferrimagnetic order. The work also predicts Hartree Fock states showing pseudogaps having a Coulomb interaction origin. The energies of such states are higher than the ones of the AF Insulator states, but the doping with holes seems that can bring their energies to coincidence. The maximal energy gap appears for momenta along the CuO bonds, as experiments indicate. It has a magnitude of nearly 100 meV, which is close to the ARPES result for La 2 CuO 4 in the zero doping limit. The results suggest a path for clarifying the connections between the Mott and Slater (Band or First Pinciples) pictures in the Physics of the transition metal oxides, a long standing problem of Solid State Physics.