#
Hartee Fock Symmetry Breaking Effects in La_{2}CuO_{4}: Hints for connecting the Mott and Slater Pictures and Pseudogap Prediction

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## Abstract

**:**

## 1. Introduction

## 2. Rotational Invariant Hartree-Fock Method

#### 2.1. $\alpha $, $\beta $ and symmetry restrictions

**T**. Consider also the maximal subset U${}_{s}$ being invariant under the group of transformations

**T**${}_{s}$, which is a subgroup of

**T**. Then, the set U${}_{s}$ obtained from imposing less symmetry restrictions a priori, should contain the set U. Therefore, after finding the extremes of the same functional in U and U${}_{s}$, it could be possible to obtain different results. In this case, the solution in U${}_{s}$, in general, shall be the most stable of both. However, it could be also the case that looking for a solution in U${}_{s}$, an extremal function also pertaining to U arises as a solution. In such a situation, such a configuration could be found from the beginning by finding the extreme of the functional in U; that is: by imposing more symmetry restrictions. In terms of the HF scheme, this could mean that the states corresponding to both solutions have identical occupied single-particle states, but they curiously might show different sets of excited ones. Therefore, depending on the particular features of the material, removing a priori imposed symmetry restrictions on the set of allowed orbitals of the HF procedure can predict new properties for the excited single particle states of the system. Such one could be for instance the gap appearance. This effect can have physical relevance after noting that at finite temperatures the more stable state will be preferred by the system and then the state showing a gap should be expected to be selected at non zero temperature.

## 3. Tight Binding Electron Model: "Removing Symmetries"

#### 3.1. Model for the Cu-O planes

**AFM**, paramagnetic

**PM**and ferromagnetic

**FM**ones. For this purpose, let us define the points of the two sublattices with indices r = 1, 2, as follows

#### 3.2. Translations on the sublattices

**k**values

#### 3.3. Tight-Binding basis

## 4. Matrix Problem and Solutions

**IAF**) presented in Subsection 4.3.

#### 4.1. Tight-Binding representation

**k**. All the implicit parameters in the following 4×4 matrices are also dimensionless

**k**, l), in the sublattice r, with spin z-projection ${\sigma}_{z}$. In order to solve the equations numerically by the method of iterations, it is convenient to pre-multiply them by ${\mathrm{I}}_{\mathbf{k}}$ for each

**k**. Note that for each

**k**four eigenvalues (l = 1, 2, 3, 4.) will be obtained, or, equivalently, four bands on the Z.B. (Eq. 23). From Eq. (15), it can be observed that in the representation (13), the HF potentials and in general the total hamiltonian of the system, become block diagonal with respect the sets of states indexed by

**k**. This fact is a consequence of the commutation of each of them with every element of the reduced group of discrete translations; that is: the group of translations which leaves invariant a sublattice.

#### 4.2. Maximally translational symmetric solutions

**R**, which are the amount of cells in the absolute lattice and the corresponding parametrization (8), respectively. The functions ${\phi}_{0}(\mathbf{x})$ are the Gaussian orbitals defined in Section 3.3. The searched HF orbitals will have the form

#### 4.3. Insulating and antiferromagnetic solutions

#### 4.4. Paramagnetic solution showing a pseudogap

**PM**and superconductor

**SC**phases [14]. It should be noticed that the physical nature of the pseudogap phenomena is yet not fully understood. There exist various proposals for explaining its physical origins [15,38]. In this paper we adopt a definition of the pseudogap as a momentum dependent excitation gap existing along the Fermi surface of the system [15]. The presence of this gap clearly means a depletion of the density of states around the Fermi surface, which is another way to describe the concept [38]. As mentioned above, the experimentally detected pseudogaps are characterized by the fact that the measured gap is higher when the electrons travel parallel to the Cu-O bonds.

**PPG**Hartree-Fock solution gives a reasonable estimate for the upper pseudogap temperature parameter.

**PPG**wavefunction is exactly coincident with the one corresponding to the paramagnetic and metallic state presented in Subsection 4.2. Moreover, the lowest energy band in both solutions, also are identical, with an upper bound error of ${10}^{-6}$ in dimensionless energy units (that means ${10}^{-5}$ in eV). Thus, the occupied single particle states in both wavefunctions are identical and correspondingly, the momentum dependence of the filled energy band also coincides. Henceforth, the differences between the two paramagnetic solutions obtained only refer to the non occupied orbitals. Those single particle states can not be considered in the band model presented in Subsection 4.2 as a consequence of the crystal symmetry restriction imposed there. For instance, consider the expressions for two states, one occupied an another empty, which are associated to the same quasimomentum value

**PPG**state should be expected only in the zero doping limit, since after adding holes, magnetic correlations are expected to start acting as the energies of the

**PPG**and

**IAF**states approach coincidence.

**PM**), paramagnetic with pseudogap (

**PPG**) and insulator-antiferromagnetic (

**IAF**) ground states, are shown in Table 1; with the zero energy reference assumed on the last one:

**PM (PPG)-IAF**and the Néel temperature of this kind of materials are both of the order of various hundreds of Kelvin degrees at the vanishing doping limit under consideration. In figure 8 the

**PPG (PM)**and

**IAF**occupied bands are depicted in a common frame. Note that the main difference in their energies corresponds to the single particle states being closer to the Fermi surface. As we had noted before, the same behavior has the antiferromagnetic character of the single particle states of the

**IAF**solution. Thus, the obtained results suggest the possibility of having success in generalizing the HF discussion considered here to the description of the phase diagram of the La${}_{2}$CuO${}_{4}$.

**PPG**and

**IAF**, the added holes will tend to bunch near the Fermi surface (curve). Then, the figure indicates that the reduction of the energy in the

**IAF**state will be smaller than the one affecting the

**PPG**state. Assumed that this tendency is maintained, we can expect that the energies of both states will become equal at some critical doping parameter ${\delta}_{c}.$ Near this value of doping it can be expected that a real solution of the HF problem should show a kind of intermediate nature in which, although

**AF**correlations could yet remain, the long range

**AF**order has already disappeared. This occurrence will lead to an HF description of the disappearance of the

**AF**state when the doping starts increases from zero at vanishing temperature. In addition, the incorporation of the temperature to the HF procedure will also give access to describe the Neel transition for variable doping existing in the low doping region of the La${}_{2}$CuO${}_{4}$ phase diagram [34]. After the derivation of the evolution with doping and temperature of the HF energy per particle and the single particle energies, one could expect a decrease with doping of the maximal gap of the single particle energies. This behavior could further describe the decaying with doping of the measured upper pseudogap ${T}_{o}$ for La${}_{2}$CuO${}_{4}$ [35,36,37]. As for the lower pseudogap, it seems possible that after doping over the disappearance of the AF order, the HF solution could support bounded pairs of holes, glued by the new quasiparticles of this state. Those excitations should be expected to have some degree of magnetic order in their structures. If such is the case and the bounded pairs turn to be sufficiently small in size due to short ranged magnetic interactions, a finite doping threshold could be needed for the bounded pairs to condense in defining a superconducting transition. In this view, the $lower$ pseudogap temperature could related with the preformed pairs gap. We expect to explore these issues in the coming extension of the work.

**PM**and

**PPG**states. It can be estimated that, due to the presence of a pseudogap, the

**PPG**ground state should be more stable than the

**PM**at non vanishing temperatures. This might be the case because in order to create excitations on the

**PPG**ground state, temperatures of hundreds of Kelvin degrees are needed, while excitations in the

**PM**ground state can appear in any range of temperatures.

## 5. Conclusions

**PM**) was obtained. Its dispersion properties topologically coincide with the results given in Ref. [22] for the unique band crossing the Fermi level. Then, the free parameters of the effective model were fixed from the requirement of reproducing the bandwidth of the half filled band obtained in [22]. Employing those parameters and removing some symmetry restrictions, solutions were obtained showing new properties. In agreement with the experiment, the insulating-antiferromagnetic (

**IAF**) solution turns to be the most stable among all HF states found. Some of its properties are enumerated below:

- The isolator gap magnitude diminishes with the increasing of the screening constant $\u03f5$.
- The antiferromagnetic structure the HF orbitals increases when the states approach the Fermi level. In addition, the size of the outlying region in which antiferromagnetism persists depends on the screening created by the effective environment $\u03f5$. That is, by increasing screening, the size of the antiferromagnetic region reduces. Thus, the idea arises that after doping with holes (that is, solving for HF solutions not at half filling condition as it is done here) the antiferromagnetic zone, which is precisely concentrated near the Fermi level could be annihilated, producing in this way a phase transition to a non globally magnetically ordered ground state. This possibility indicates a way to describe the normal state properties of the HTc superconductors through a simple HF study.
- The magnetic moments per cell which are evaluated show a modular value of 0.67 ${\mu}_{B}$, which is close to the measured moments in La${}_{2}$CuO${}_{4}$ and interestingly almost coincide with the measured result of 0.68 ${\mu}_{B}$ for the Cu sites in the 3D solid CuO [34].

**PPG**). Some properties of this excited HF wavefunction are:

- The magnitude of the predicted maximal value of the pseudogap is of the order of 100 $meV$. This result is close to the range 100-200 $meV$ which is experimentally detected through ARPES for the $upper$ pseudogap ${T}_{o}$ in the zero doping limit for La${}_{2}$CuO${}_{4}$ [15,35,36,37]. The comparison of the filled band spectra of the
**IAF**and**PPG**states suggests that under a relatively small doping the energy of both states could evolve to coincidence. A decreasing evolution with doping can be expected from future temperature and doping dependent HF pseudogap evaluations. Thus, assuming that the maximal value of the obtained pseudogap describes the experimentally determined $upper$ pseudogap temperature ${T}_{o}$, a description of the observed decaying behavior of ${T}_{o}$ for increasing doping is suggested. [15]. - Similarly as it happens for the antiferromagnetic character of the
**IAF**ground state, the difference between the one particle energies of**IAF**and the**PPG (PM)**solutions is larger for orbitals closer to the Fermi surface. That is, it happens for the electrons with more energy and consequently the first to disappear under doping with holes. Therefore, this outcome further supports the possibility to describe a crossing of the energies of the**IAF**and**PPG**states under doping.k - The pseudogap magnitude diminishes with the increasing of the screening constant $\u03f5$. This property, and the fact that the set of parameters were non univocally fixed (to reproduce Matheiss results for the single band crossing the Fermi level in [22]) leads to the opportunity of a better determination of the parameters to match a larger set of observed physical properties of La${}_{2}$CuO${}_{4}$.
- At the $T=0$ limit considered here, the states
**PM**and**PPG**were identical. The difference between them only appears at the excitations of the system. Thus, the removal of some symmetry restrictions defines new properties for unoccupied single particle states. It seems feasible that it could be possible to obtain a gap instead of a pseudogap in other materials, even in the absence of magnetic order. Such an outcome could show the ability of a properly formulated HF description to describe general kinds of Mott insulators, being or not magnetically ordered.

**IAF**solution in which the HF single particle states have a neatly "entangled" non separable character in their spin and orbital dependence.

- To generalize the discussion in order to introduce the doping with holes and temperature as new parameters. This will allow to investigate the effects of these parameters on the determined HF states. Of particular interest in a first stage appears the study of the crossing of the energies per particle of the
**IAF**and**PPG**states under doping. As above remarked, this point could determine the**AF**destruction phase transition. Furthermore, the mean field study for even larger dopings could shed light on the superconducting phase change. - To compute the temperature and doping dependent electron Green function of the system, and use it to evaluate the effective polarization of the La${}_{2}$CuO${}_{4}$ in the obtained states.
- With the polarization results in hand, it could be possible to attempt solving the Bethe-Salpeter equation for two holes in the HF ground state, to find whether or not it is possible to decide about the existence of preformed Cooper Pairs in the HF model under finite doping and temperature. The possibility for their existence was suggested by the results of Ref. [46], in which it was argued that a strong 2D-screening of the Coulomb interaction is created by a half filled band of tight binding electrons.
- Finally, we intend to follow the hints given by the ideas exposed here to clarify the debate between the Mott and Slater pictures in connection with the electronic structure of transition metal oxides.

## Acknowledgements

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## A.. Matrix elements

#### A.1.. Brackets notation

#### A.2.. Dimensionless definitions

#### A.3.. Other definitions and properties

**R**${}^{(1)}$ and

**R**${}^{(2)}$ given in (10), and recall that the only Wannier orbitals which have non vanishing overlapping are those centered on the same site or those centered on nearest neighbors (the closest neighbors belong to different sublattices). Then, for fixed

**R**${}^{(r)}$ and

**R**${}^{(t)}$ the only non vanishing among all the quantities in the left hand side of (43) are

**p**${}_{i}$ are defined as

#### A.4.. Matrix Elements

#### A.5.. Reducing the order of some integrals

**Figure 1.**The figures shows: a) The point lattice associated with the Cu-O planes. For the search of the

**AFM**properties of the conduction electron, and more generally for removing the symmetry restrictions, it will be helpful to separate the lattice in the two represented sublattices; and b) the corresponding base of the Cu-O planes.

**Figure 2.**a) The Brillouin zone (B.Z.) associated with the infinite absolute point lattice. The vectors $\mathbf{k}$ label the eigenfunctions of the group of translations ${\widehat{T}}_{{\mathbf{R}}^{(r)}}$ in this infinite point lattice. The grey square indicates the corresponding B.Z. of the sublattices. The length of its side is $\sqrt{2}\pi /p$. b) The grid of points shows the discrete character of the B.Z. when the absolute point lattice is finite with periodic conditions fixed in its boundaries. The unit of the scale means $\frac{\pi}{p}$.

**Figure 3.**Figure a) shows the Brillouin zone associated to the absolute point lattice. The grey zone signals the occupied states in the paramagnetic metallic solution at half filling conditions. The unity of quasimomentum is $\frac{\pi}{p}$. Figure b) shows the doubly degenerated bands associated to the same paramagnetic and metallic state. Note the close correspondence between these results and those obtained by Matheiss in Ref. [22]. The zero energy level in all the band diagrams is the Fermi energy of the isolator-antiferromagnetic solution presented in subsection 4.3. The domain of the plot is the B.Z. of the sublattice shown in Fig. 2 a).

**Figure 4.**Energy bands obtained for: a) A sample of 20x20 cells, E${}_{gap}$ = 1.32 eV. b) A sample of 30x30 cells, E${}_{gap}$ = 1.32 eV. The parameter values chosen were $\tilde{a}$ = 0.25, $\tilde{b}$ = 0.05, $\tilde{\gamma}$ = -0.03 and $\u03f5$ = 10. The zero energy level is fixed on the Fermi level of the 20×20 system. Note that the difference in energy between the two bands is not appreciable in the employed energy scale. The domains of both plots is the B.Z. of the sublattice shown in Fig. 2 a)

**Figure 5.**The magnetization vector $\mathbf{m}$ of the more stable HF state determined here lies in the direction 1-2. a) This figure shows the projection ${\tilde{m}}_{12}$ of the dimensionless magnetization, in the 1-2 direction. The magnetization unit is $\frac{{\mu}_{B}}{{p}^{2}}$. b) The picture shows a scheme of the mean magnetic moment per site in the lattice. The modular value for the shown solution is 0.67${\mu}_{B}$.

**Figure 6.**The single particle states exhibit a sharp antiferromagnetism in the proximities of B.Z. boundaries. In the figure the angle between their magnetic moment components on each of both sublattices (after divided by $\pi $) is plotted against their Bloch states quasimomenta. Note that the states on the boundary have a perfect antiferromagnetism and that they become less antiferromagnetic as their quasimomenta move away from the boundary. The region of the plot is the B.Z. of the sublattice shown in Fig. 2 a)

**Figure 7.**The band structure associated to a paramagnetic HF solution presenting a pseudogap is shown. b) A frontal view of the plot evidences more clearly the existence of a momentum dependent gap along the Fermi surface, which is larger for momenta directions pointing along the Cu-O links on the plane. This properties indicate the presence of the pseudogap [15,38]. Note that the Fermi level of the

**PPG**solution is laying $\backsimeq 0.08$ dimensionless units of energy ($0.664\phantom{\rule{0.166667em}{0ex}}eV$) above the zero energy reference (the Fermi energy of the

**IAF**solution). Both graphics are plotted in the B. Z. of the sublattices, that is: the grey zone in figure 2 a).

**Figure 8.**The figure shows in the same plot the occupied bands corresponding to the states

**PPG**and

**IAF**. The difference in the energies of the orbitals is concentrated in the boundary of the Brillouin zone. The zero energy level coincides with the Fermi level of the

**IAF**solution. The domain of the plot is the B.Z. of the sublattices, given by the grey zone in figure 2 a).

State | IAF | PM | PPG |
---|---|---|---|

$\Delta $E (eV) | 0.0 | +0.076 | +0.076 |

© 2010 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license http://creativecommons.org/licenses/by/3.0/.

## Share and Cite

**MDPI and ACS Style**

Cabo-Bizet, A.; De Oca, A.C.M.
Hartee Fock Symmetry Breaking Effects in La_{2}CuO_{4}: Hints for connecting the Mott and Slater Pictures and Pseudogap Prediction. *Symmetry* **2010**, *2*, 388-417.
https://doi.org/10.3390/sym2010388

**AMA Style**

Cabo-Bizet A, De Oca ACM.
Hartee Fock Symmetry Breaking Effects in La_{2}CuO_{4}: Hints for connecting the Mott and Slater Pictures and Pseudogap Prediction. *Symmetry*. 2010; 2(1):388-417.
https://doi.org/10.3390/sym2010388

**Chicago/Turabian Style**

Cabo-Bizet, Alejandro, and Alejandro Cabo Montes De Oca.
2010. "Hartee Fock Symmetry Breaking Effects in La_{2}CuO_{4}: Hints for connecting the Mott and Slater Pictures and Pseudogap Prediction" *Symmetry* 2, no. 1: 388-417.
https://doi.org/10.3390/sym2010388