Quantum phase transition from a Antiferromagnetic-Insulator to a Paramagnetic-Metal laying beneath the superconducting dome
Victor M. Martinez Alvarez, Alejandro Cabo-Bizet, Alejandro Cabo Montes de Oca
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Quantum phase transition from a Antiferromagnetic-Insulator to aParamagnetic-Metal laying beneath the superconducting dome
Victor M. Martinez Alvarez ∗ , ∗∗∗ , Alejandro Cabo-Bizet ∗∗ and Alejandro Cabo ∗ ∗ Department de F´ısica Te´orica, Instituto de Cibern´etica,Matem´atica y F´ısica, Calle E, No. 309, Vedado, La Habana, Cuba. ∗∗ Department de F´ısica, Centro de Aplicaciones Tecnol´ogicas y Desarrollo Nuclear,Calle 30, No. 502 e/ 5ta y 7ma Avenida, Playa, La Habana, Cuba. and ∗∗∗
Department de F´ısica, Universidad de Camag¨uey, Circunvalaci´on Norte, km 5, Camag¨uey, Cuba.
The effect of hole doping on the Tight-Binding (
T B ) model of the Cu - O planes in the La CuO constructed in previous works is investigated here. Firstly, it is pointed out that the model employedconstitutes a generalization of the Hubbard one for the same system. Thus, the former predictions ofthe insulator gap, antiferromagnetic ( AF ) character and the existence of a paramagnetic-pseudogap( P P G ) state at half-filling, become natural ones to be expected from this more general picture.The effect of hole doping on the antiferromagnetic-insulator state (
AF I ) and the paramagnetic-pseudogap one at half-filling, is investigated here at T = 0 K . The results predict the occurrenceof a quantum phase transition ( QP T ) from the antiferromagnetic- insulator state at low dopingto a paramagnetic-metallic state (
P M ) at higher hole densities. Therefore, a clear description ofthe hidden
QP T laying beneath the “dome” in high critical temperature (
HT c ) superconductingmaterials is found. At low doping, the systems prefers the
AF I state, and at the critical value ofthe doping density δ c = 0 .
2, the energy of a metallic state starts becoming lower. The evolutionwith small doping values of the band spectrum of the
AF I state, shows that the holes tend tobecome localized at the middle of the sides of the reduced Brillouin zone ( BZ ). Then, when δ passes through the critical value, the holes of the AF I state move to become situated at the cornersof the same reduced BZ , showing in this way a structural change at the phase transition point.Thus, the P M state which appears at the transition point acquires the same behavior with respectto the localization of holes as the pseudogap state. In the small doping limit a clear differencebetween the degree of convergence of the iterative self-consistent solution is associated to an evenor odd number of electrons. It suggests that the Kramers degeneration in combination with thespin-spatial entangled nature of the hole states, leads to a new kind of pair interaction between twoholes. The binding energy value is estimated as a function of the screening.
PACS numbers: 71.10.-w,74.72.-h,74.72.Gh,74.72.Kf,75.10.-b,71.30.+h
I. INTRODUCTION
Since the discovery of the high critical temperature superconductivity in the called “cuprates” (made up of copperoxides) [1], the search about the nature of their behavior has been and it continues being, one of the most defiantand investigated topics in Condensed Matter Physics. Among these compounds is the La CuO , that is a classicexample of the so-called strongly correlated electron systems ( SCES ). However, due to the inherent difficulties ofthe many body problem for this class of materials, numerous properties continue without being clearing up, in spiteof the efforts done by the investigators in this subject. The La CuO superconductor is characterized by a drasticchange in its behavior depending on the density of electrons in the two-dimensional planes CuO [2], and it happensin general in the HT c superconductivity materials. All the cuprate superconductors have as a common element, theexistence in their crystal structure of CuO planes.The standard band calculations predict a character of paramagnetic-conductor to this material [3], in drastic contrastwith its experimentally observed, insulating and antiferromagnetic nature. These two properties are associated withthe presence of strong correlations, and are not derivable starting from an independent particle scheme as the Hartree-Fock ( HF ) one, when the whole many electron crystal system is considered [2]. However, in Refs. [4–6], on the base ofa single band model solved in the HF approximation, it was possible to predict the mentioned SCES properties of thismaterial. At first sight, it might seem that to obtain the AF and insulating properties of La CuO from a HF scheme,could be a contradictory outcome, since those properties are thought as essentially indicating the strong correlationnature of this material. However, those results become natural ones, when considering the following circumstances. Infirst place, the HF procedure was not applied in its quality of a F irst P rinciple method to solve the full Hamiltonianproblem associated to the total electronic structure of La CuO . Alternatively, the procedure was employed for solvingthe simple model of the CuO layers built in Refs. [4–6]. It is well-known that the impossibility of describing strongcorrelation properties by employing the HF scheme, appears when the scheme is applied to find the solution of theexact many electron problem defined by all the electrons and nuclei constituting a crystalline solid. For example, thedirect HF calculation of the band structure of La CuO , predicts an enormous electronic gap of nearly of 17 eV [7].In addition, the model introduced in Refs. [4–6], only considers the electrons that half-filled the single band crossingthe Fermi level appearing in the early electronic band calculation [3]. In fact, the proposed simple Hamiltonian canbe viewed as a preliminary stage within the definition of one among the variety of Hubbard’s Hamiltonians. One,in which the last step of imposing the nearest neighbor approximations leading to the Hubbard theories had notbeen implemented. Therefore, the proposed model could be considered as an improvement of a Hubbard one, whichretains the full Coulomb repulsion operator intact [8]. This last point can be understood by keeping in mind thatits free Hamiltonian is basically a T B one in which the full Coulomb interaction operator between the electronshad been retained. Thus, after performing the nearest neighbor approximations, a typical Hubbard scheme arises.Henceforth, since the Hubbard model is recognized to convey strong correlation effects, and moreover, since it isalso known that its mean field solutions also can exhibit such effects [8], the appearance of the insulator gap andthe antiferromagnetic structure in Refs. [4–6], becomes a reasonable conclusion. However, a new physical predictioncoming from the analysis is the existence of pseudogap states, which are not following in the Hubbard approximation.These mentioned results emerged as a consequence of a combination of lattice symmetry breaking with a spin-spaceentanglement structure in the single particle solutions of the HF problem. The treatment gave a clear explanationabout the nature of the antiferromagnetism and the insulating structure of this high temperature superconductor.The process through which the electronic structure evolves with the hole doping, from the AF I state at low dopingto the superconductor state, and then to a normal metal phase at large hole concentrations, has been the subject ofa strong interest in the literature. It is known that the magnitude of the gap in the normal state is of the same orderthat the superconductor gap [9]. With the objective of studying the hole doping dependence of the band structure,the La − x Sr x CuO ( LSCO ) system is for several reasons appropriate among the family of the
HT c superconductors.In first place, the
LSCO has a simple crystalline structure of
CuO layers. Second, the concentration of holes inthese planes can be controlled in a wide range and is uniquely determined by the concentration x of Sr [9]. For thisreason, one can develop samples of the material by varying doping continuously from the insulating state withoutdoping ( x = 0) up to the high doping limit ( x = 0 .
3) in the same system.Numerous experiments made in copper oxide compounds, suggest the existence of a quantum phase transition at T = 0 K , which lays inside the doping interval of the superconducting dome (this is colloquially referred as “layingbeneath” the dome). It is believed that this QP T is the key to understand the high temperature superconductivityand also to explain the properties of the normal state in those materials [10, 11]. The present work presents resultsthat predict the existence of this sort of
QP T in the context of the model for the La CuO constructed in Refs. [4–6]when the hole doping is incorporated. The HF system of equations is solved here by employing the same methodused in Refs. [4–6]. In the present case, again the parameters of the model are determined of imposing the conditionthat the HF solution without crystalline symmetry breaking, reproduces the form of the single band crossing theFermi level in the band calculations of the material [3]. Then, we investigated the changes that occur in the bandstructure when the system is doped with holes. As in Refs. [4–6] at exact half-filling, the HF solution predictsthe existence of the AF I and
P P G states. This previous result motivated the idea of investigating the hole dopingconsequences, after conceiving the possibility of predicting the quantum phase transition at T = 0 K . The sameexistence of this transition, is currently one of the most fundamental questions in high temperature superconductivityresearch [10, 11]. A transition like this is now considered that should determine the properties of the normal statein the diverse regions of the phase diagram. The present investigation aim consists in studying the existence of thisquantum phase transition beneath the superconducting dome.The Hartree-Fock description of the simple model for the Cu - O planes of La CuO including symmetry breakingeffects and spin-spatial “entanglement” of the single particle states at half-filling, is here extended to consider theeffects of hole doping. It is observed how the effect of doping is able to predict a variety of the most interestingproperties of this material a T = 0 K . The evolution of the band spectrum of the antiferromagnetic-insulator andthe paramagnetic-pseudogap states as functions of the hole doping parameter are determined, for a wide range of thehole concentration 0 ≤ x ≤ .
3. Around the critical doping x c = 0 .
2, the results show that for the
AF I state, theband spectrum suffers a gradual change, in which the insulator gap diminishes to completely closing. Surprisingly,the same behavior occurs for the
P P G state for which the pseudogap also collapses. For higher values than thementioned critical doping one, the
AF I and
P P G states coalesce in a single HF solution. The magnetization of thishigh doping state vanishes as well as its gap, thus, it describes a paramagnetic-metallic phase. The results also reveala drastic change in the Fermi surface, which goes from a hole-like Fermi surface centered at ( π, π ) for 0 < x < . ,
0) for 0 . < x ≤ .
3. These results evidence the existence of a quantum phasetransition laying beneath the superconductor dome, in which an insulator state with antiferromagnetic correlationstransits to a paramagnetic-metallic phase. The work also identifies a new mechanism of hole pairing which could giverise to the superconductivity. The effect results from a combination of the Kramers degeneration with the spin-spatialentanglement of the single particle HF states. The discussion helps to clarify the relation between: the pair of states AF I and
P P G in the Physics of the strongly correlated electron systems with the superconductor state and thequantum critical point, around which the phase transition occurs [10–13]. An evaluation of the hole pairs bindingenergy as a function of the dielectric constant is obtained, which predicts binding of the pairs at the experimentallymeasured values of the dielectric constant of La CuO [14]. However, the thermodynamic limit for this calculationhad not been attained yet in this work. In addition, the parameters of the model should be optimized yet to matchthe measured parameters of the material as the insulator gap of 2 eV [15], and the dielectric constant having a valueof the order of 25 [14]. Here, it is pointed out that this binding effect could be the acting mechanism determining thephysical relevance of the doubly charged bosonic fields argued in Ref. [16].The papers is organized as follows. In section 2 we review the one band model of the Cu - O planes in La CuO introduced in Refs. [4–6], the procedure for the determination of its parameters and the Hartree-Fock solution for thehere considered situation: the study of the hole doping effects. In section 3 the results are presented for the evolutionof the band spectrum of the P P G state in the range of doping 0 ≤ x ≤ .
3. Next, the section 4 considers the samestudy for the
AF I state. Section 5 presents the results for the changes of the Fermi surfaces as the doping is increased.The quantum phase transition properties are discussed in section 6. At the section 7 we exposes the identificationof a possible hole pairing mechanisms which could give rise to the superconductivity and estimate the values of theenergy for pair binding. Finally, the conclusion are presented in section 8.
II. THE
CuO MODEL INCLUDING HOLE DOPING
In this section we will review the main ideas and elements defining the model for the La CuO introduced inRefs. [4–6]. In the first subsection, the structure and notation of the unrestricted Hartree-Fock scheme employed isdescribed. Next, the model is presented. A. Fully unrestricted Hartree-Fock scheme
The N electron system considered in the T B model discussed in Refs. [4–6] was described by a fully unrestrictedSlater determinant f n ( x ; s , ..., x N ; s N ) state constructed with single particle orbitals φ k i ( x i , s i ) with i = 1 , ..., N ,which arbitrarily depends on the spin variables at any point of the space. The index n represents the set of quantumnumbers of the many electron system. In this subsection, as usual, generalized coordinates will be assumed toincorporate the spatial in common with the spin ones. For the electron case under study, the Slater determinant getsthe expression f ( x ; s , ..., x N ; s N ) = √ N ! P η ,...,η N ǫ η ,...,η N φ η ( x , s ) ...φ η N ( x N , s N ) , ∀ i η i = k , ..., k N , (1)where ǫ η ,...,η N is the Levi-Civita tensor. The single particle HF orbitals satisfy a set of coupled integro-differentialequations of the Pauli kind. This set is derived from the minimization of the HF energy functional of the systemunder the conditions of normalization of f and the normalization of all the single particle orbitals φ k i . For moredetails see Refs. [4–6]. In general, the Hamiltonian of usual electronic systems has a free term P i ˆ H ( x i ) (a kineticenergy plus an interaction with an external field one) with an addition associated to the pair Coulomb interactionbetween the electrons in the form ˆ H ( x , ..., x N ) = X i ˆ H ( x i ) + 12 X j = i V ( x i , x j ) . (2)After performing the above referred minimization process by using the Lagrange multipliers scheme, a “fully unre-stricted” set of HF equations for the orbitals φ k i is obtained in the form[ ˆ H ( x ) + X η X s ′ Z d x ′ φ ∗ η ( x ′ , s ′ ) V ( x, x ′ ) φ η ( x ′ , s ′ )] φ η ( x, s ) − X η [ X s ′ Z d x ′ φ ∗ η ( x ′ , s ′ ) V ( x, x ′ ) φ η ( x ′ , s ′ )] φ η ( x, s ) = ε η φ η ( x, s ) , (3)where η, η = k , ..., k N . In this completely unrestricted way, the HF set of equations was wrote by Dirac [17]. In thisform, the orbitals are allowed to show arbitrary spin projection at any point of the space, and this determines the useof the terms “fully unrestricted” for this set of equations. The total HF energy of the N electron system E HF andthe HF single particle energies have the forms E HF = P η h η | ˆ H | η i + P η,η h η, η | V | η , η i − P η,η h η, η | V | η, η i ,ε η = h η | ˆ H | η i + P η h η, η | V | η , η i − P η h η, η | V | η, η i . (4)The definitions of the basis states of the one electron band model and their scalar products appearing in the aboveequations are specified in Refs. [4–6].
1. The α and β spin constraints. The analysis in Refs. [4–6] was initially motivated by the aim of examining the restrictions that could be introducedin the description of many particle systems, by the frequently employed assumption about that the HF single particlestates should have a definite +1 / α ) or − / β ) projection of their spin at all the spatial points. That is, to showthe structure: φ k ( x, s ) = (cid:26) φ αk ( x ) u ↑ ( s ) type α,φ βk ( x ) u ↓ ( s ) type β. (5)Whenever the spatial functions φ αk ( x ) are identical, the HF evaluation is called a restricted one, and when they canbe different it is described as an unrestricted one [18]. However, both of these cases are in fact restrictive ones forthe allowed spin orientations of the HF orbitals. It was emphasized in references [4–6] that these assumptions aboutthe spin structure of the orbitals can be characterized as definite constraints which drastically limits the generalityof the space of function in which the HF single particle orbitals are searched. It can be noted that the HF schemeintroduced by Dirac [17] does not include any restriction on the spin structure of the searched orbitals. B. The model for the
CuO planes FIG. 1: Band structure of the La CuO , calculated by Matheiss 1987. The single half-filled band suggests theconsideration of the here investigated model, which is based in a set of interacting electrons subject to a crystalline T B potential, tightly binding them to the points of the lattice formed by the Cu atoms in the material.In figure 1, a calculated band diagram of the La CuO is illustrated. The band structure was obtained by the useof the LAP W (Linear Augmented Plane Waves) method [3]. Let us describe below the formulation of the model forthe
CuO planes introduced in Refs. [4–6], which is the starting point of the present work. It can be noted that thelast band occupied by electrons is half-filled, a fact that indicates a metallic character for the material. The form ofthis band suggested the validity of an approximate T B description for the electrons populating it. The less boundkind of electron in the compound La CuO , is the unpaired one in the Cu , which at variance with the O − in theplane, does not has its last shell (3 d ) closed. These electrons, in a qualitative picture can be estimated to be the onesconstituting the mentioned unique band crossing the Fermi level in the band calculations done in the Ref. [3]. Theseelectrons can be reasonably considered as tightly bound to the Cu atoms in the CuO planes, assumed the completionof the O − atoms in those planes. The last remarks, support the idea of taking as the lattice defining the T B model,that one giving rise to the half-filled band in Ref. [3]. That is, the planar squared lattice of points coinciding withthe sites of the Cu atoms in the CuO plane (see figure 2).The presence of all the other electrons filling the rest of the bands, in common with all the neutralizing nuclearcharges, were taken to play a double role in the model defined in Refs. [4–6]. In one sense, this system is assumedas a polarizable effective medium which screens the Coulomb repulsion by means of effective dielectric constant ǫ . Insecond place, this system is assumed to generate a mean periodic Tight Binding potential W γ , which strongly confinesthe electrons of the single partially filled band to be close to the Cu atoms in the CuO planes. The model is assumedto be purely 2 D , that is, the space coordinates are assumed to have only two planar components x = ( x , x ). Itis completed by assuming that the electrons in the partially filled band are also interacting with the potential F b generated by a jellium charge density which neutralizes the net charge of the electrons of the model. The jelliumcharges were assumed as periodically concentrated and having a planar gaussian distribution with a radial widthdefined by the parameter b . In resume, the free Hamiltonian for the T B model for the electrons in the half-filled bandhas the form ˆ H ( x ) = p m + W γ ( x ) + F b ( x ) , (6)in which the periodic T B potential is satisfying the periodicity condition W γ ( x ) = W γ ( x + R ) , (7)and the jellium potential is defined by F b ( x ) = e πε ε X R Z d y exp( − ( y − R ) b ) /πb | x − y | , b ≪ p, (8)in which the coordinate vectors of the planar Cu atoms are R = n x p e x + n x p e x where n x , n x ∈ Z , (9)where the unit vectors e x and e x lay on directions defined by the vectors joining the nearest neighbors of the Cu atoms in the lattice in figure 2. It is known that the distance between a Cu atoms and its nearest neighbor is p = 3 . A [19]. Further, the Coulomb interaction between two electrons in the partially filled band of the model is assumedin the form V = e πǫ ǫ | x − y | , (10)including a dielectric constant which is supposed to be generated by the polarization of the set of electrons filling theother bands and all the nuclei in the La CuO , through which the electrons of the partially filled band of the modelare assumed to move.The model was started to be constructed with the initial idea of searching for HF single particle states of theelectrons in the partially filled band, not being of an α or β types, that is, not being separable in their spatial andspin dependence. In other words, allowing that in those HF orbitals the spin projection could depend on the spatialposition (spin-space entanglement or non collinear spin structure). Materials showing AF structure were suspectedto lead to HF solutions of this kind, a fact that also suggested the possibility of improving the understanding of thetroublesome state of knowledge of the band structure of such materials. In the particular case of the La CuO , it isknown that it is an antiferromagnetic compound. In the AF systems, normally the crystal symmetry of the substanceis directly broken by the AF order. Therefore the description of the model was also designed to incorporate thepossibility of the breaking of the crystalline translation invariance. The AF order has translation symmetry in eachone of the two interpenetrating squared sublattices in which the planar Cu lattices can be decomposed. Thus, the HF single particle orbitals in the model were assumed to retain the Bloch functions character under the commonreduced translation symmetry group of these sublattices. Then, the physical state should be equivalent under thetranslations leaving invariant the sublattices, but not under the one which transforms a sublattice into the other. Thisreduced symmetry is associated to a subgroup of the total translation group of the crystal and therefore its spacerepresentation should be more reduced in its number of states as classified by the momenta k in the reciprocal space. (a) (b) FIG. 2: (a) The picture illustrates the
CuO planes in the material. (b) Point lattice associated to those planes. Inthe search of the possibility of describing the AF properties of the material through the elimination of the symmetryrestrictions (over the space of functions in which the single particle states were searched) it was helpful todecompose the full lattice in the two sublattices which are differentiated in the picture.The two sublattices of point depicted in figure 2, that will be described below by the indices r = 1 or 2, areanalytically defined in the form R ( r ) = √ n p q + √ n p q + q ( r ) , (11) q ( r ) = (cid:26) if r = 1 p e x if r = 2 , where q and q are the two unit vectors defining the directions of the unit cell vector of the sublattices.
1. Sublattice translations
Then, the searched HF single particle wave functions should be eigenfunctions of the discrete translation operatorˆ T R (1) , which transform a sublattice in itself ˆ T R (1) φ k ,l = exp( i k · R (1) ) φ k ,l . (12)When the point lattice is infinite, the Brillouin cell formed by the set of all the momenta k indexing the eigenfunctionsof ˆ T R (1) is the shadowed zone in figure 3, while the large square represents the set of momenta associated to theeigenfunctions of the group of translations in the original lattice. Since it is impossible to numerically treat theinfinite lattices because it has a continuum of states, periodical boundary conditions were chosen in order to makefinite the number of eigenfunctions in the Brillouin zone. The boundary conditions were imposed fixing the periodicityof the eigenfunctions φ k ,l in the boundaries defined by x = − Lp and x = Lp along the x axis and by x = − Lp and x = Lp along the x axis. The finite set of momenta vectors obeying the boundary conditions are k = πLp ( n x e x + n x e x ) n x , n x ∈ Z − L ≤ n x ± n x < L . The number of electron eigenfunctions obeying the conditions is N , that is, a half of the number of functions satisfyingthe same boundary periodicity conditions when the group is the group of translations in the full point lattice.
2. Tight-Binding single band basis
The
T B basis that was employed for describing the one band model had the form ϕ ( r,σ z ) k ( x , s ) = r N u σ z ( s ) X R ( r ) exp( i k · R ( r ) ) ϕ ( x − R ( r ) ) , (13)FIG. 3: The darker region shows the reduced Brillouin zone associated to the Bloch functions defined on thesublattices. The length of its sides is √ πp .ˆ σ z u σ z = σ z u σ z ,ϕ ( x ) = 1 √ πa exp( − x a ) , a ≪ p. (14)where N is the number of electrons in the model electron gas filling part of the band crossing the Fermi level and ˆ σ z is the spin projection operator in the z axis, which is assumed to be orthogonal to the CuO planes ; σ z = − , r = 1 and 2, is the above defined index for each of the two sublattices. Note that in thesmall overlapping approximation between nearest neighborhood points (which should be in different sublattices), theorthogonality between the members of this basis is only lost between functions belonging to different sublattices havingthe same spin polarization and momentum. The orthogonality between distinct elements corresponding to the samesublattice is valid by construction.The chosen Wannier orbital of the model ϕ ( x − R ( r ) ) were fixed for simplicity. Their gaussian form implies that itwas supposed that the net T B potential in the close neighborhood of each Cu atom had been assumed to be a simple2D harmonic potential. This simplifying assumption was adopted following the idea that the main forces in definingthe AF and insulator properties of the CuO planes were in fact determined by the spontaneous breaking of thecrystalline symmetry plus the spin entangled structure of the HF electron orbitals. Thus, simplifying considerationsled to fix the explicit form of equation (13) for the starting basis states of the model. However, taking into accountimproved definitions for the Wannier’s orbitals defining the single band model, might be of help for the description ofother effects, by example the magnetic anisotropy of the AF order in La CuO . For this purpose, the model couldbe generalized by considering atomic 3 D representation, employing Wannier’s orbitals like the d ones of the Cu andthe inclusion of spin-orbit interactions. This study is expected to be considered elsewhere. C. Hartree-Fock solution with hole doping
In this subsection we present the matrix problem to which the HF set of equations of the model was reduced inRefs. [4–6], after its projection on the defined basis. In the present work, these equations will be solved for the moregeneral situation in which the system is doped with holes. In the coming subsection it will be reviewed how themodel reproduces the main characteristics of the dispersion of the single band crossing the Fermi level in Ref. [3](predicting a metallic and paramagnetic state) when full crystalline symmetry and the HF orbitals of the type α and β are assumed. Afterwards, the next sections will illustrate the consequences of the elimination of the symmetryrestrictions in the space of functions in which the HF energy functional is minimized. It was performed in a similarway as it was done in Refs. [4–6], but for the important situation in which the systems is doped with holes. At verysmall doping values, the solutions are the ones obtained in Refs. [4–6] for the exact half-filling situation: the groundstate is an antiferromagnetic-insulator and the excited phase corresponds to the paramagnetic-pseudogap state. Whenthe hole doping is augmented the results indicate the appearance of a quantum phase transition at the doping value x = 0 .
2, in which the
AF I and
P P G states both coalesce in one single metallic state for higher doping values.As in Refs. [4–6] the searched HF single particle states are expressed as a linear combination of the before definedbasis functions in the form φ k ,l ( x , s ) = X r,σ z B k ,lr,σ z ϕ ( r,σ z ) k ( x , s ) , (15)where l is the index of the quantum numbers required to uniquely define the HF single particle states. Aftersubstituting the above expression for the orbitals in the HF equations and taking their scalar product with anarbitrary state, the equivalent self-consistent matrix problem can be written in the form (cid:2) E k + ˜ χ ( G Co k − G int k − F k ) (cid:3) · B k ,l = ˜ ε l ( k ) I k · B k ,l , (16)with the definitions for the constants ˜ χ = me a π ~ ǫǫ p , (17)˜ ε l ( k ) = ma ~ ε l ( k ) , (18)which are dimensionless, as also are all the implicit parameters in the definitions of all the entering matrices E k = (cid:13)(cid:13)(cid:13) E k , ( t,r,α z ,σ z ) (cid:13)(cid:13)(cid:13) × ,G C k = (cid:13)(cid:13)(cid:13) G C k , ( t,r,α z ,σ z ) (cid:13)(cid:13)(cid:13) × ,G i k = (cid:13)(cid:13)(cid:13) G i k , ( t,r,α z ,σ z ) (cid:13)(cid:13)(cid:13) × ,F k = (cid:13)(cid:13) F k , ( t,r,α z ,σ z ) (cid:13)(cid:13) × ,I k = (cid:13)(cid:13) I k , ( t,r,α z ,σ z ) (cid:13)(cid:13) × . They are respectively associated to the kinetic term plus the periodic
T B potential, the Coulomb direct and exchangepotential terms, the potential generated by the compensating jellium charge density and the overlapping matrixbetween the basis functions. The formulae for the matrix elements are shown in the Appendices of the Refs. [5, 6].In the employed representation the normalization condition of the HF single particle states and the formula for the HF energy take the forms 1 = B k ,l ∗ .I k .B k ,l , (19) E HF k ,l = X k ,l Θ (˜ ε F − ˜ ε l ( k )) [˜ ε l ( k ) − ˜ χ B k ,l ∗ . ( G C k − G i k ) .B k ,l ] . (20)The HF matrix system of equations (16) is a non linear one in the variables B k ,lr,σ z , which are the four componentsof the vector B k ,l for each value of k . The B constants can be interpreted as determining the probability amplitudesto find the electron in the states ( k , l ), of the sublattice r , and spin along the z axis. Note that for each k value, fourvalues of the quantum numbers of the HF single particle states l = 1 , , , BZ . From the equations (16) it can be observed that the full matrix representation of theFock operator is block diagonal in the momenta indices k , which is a direct consequence of the symmetry with respectto the reduced translation invariance of the system. D. Maximal symmetry solutions: fixing the model parameters
In first place the HF problem was solved by assuming that the HF single particle states satisfied the translationinvariance in the whole lattice and also were showing α or β spin dependence. After this assumptions, as it wassuspected, the HF solution of the model produced the single particle spectrum depicted in the figure 4, in a similarway as in Refs. [4–6]. As before we will adjust the free parameters ǫ : the dielectric constant of the effective medium;˜ a : the radial distance for which the gaussian Wannier’s orbitals are appreciably different from zero; ˜ γ : the hoppingprobability between nearest neighbor sites fixed by the effective medium and ˜ b : radial distances inside which thejellium charges are concentrated.The Bloch Tight-Binding single band basis for this problem had the form ϕ σ z Q ( x , s ) = r N u σ z ( s ) X R exp( i Q · R ) ϕ ( x − R ) , (21)where the appearing momenta Q are given by Q = πLp ( n x e x + n x e x ) n x , n x ∈ Z − L ≤ n x , n x < L These functions define the Bloch states forming the basis of the maximal translation group with periodic boundaryconditions in the same region defined before. In addition N = L × L , and R are the number of cells in the total latticeof points and the vectors defining the positions of the lattice. Let the HF single particle states expressed in the form φ Q ,l ( x , s ) = X σ z B Q ,lσ z ϕ σ z Q ( x , s ) , (22)in the just defined basis. In this case, the equivalent HF matrix problem obtained after substituting these functionsin the set of HF equations, is now of dimension two. That is, there are two components l = 1 , Q . In this way, in an analogous form as in (16) is possible to obtain the HF matrix problem in the form (cid:2) E Q + ˜ χ ( G C Q − G i Q − F Q ) (cid:3) · B Q ,l = ˜ ε l ( Q ) I Q · B Q ,l , (23)which is a system of non linear matrix equations to be solved by iterations.For starting the iterative process an initial paramagnetic state is employed as an ansatz. The figure 4 showsthe doubly degenerated metallic and paramagnetic band which is obtained at half-filling conditions. That is, with N = 20 ×
20 electrons. The chosen parameters were: ǫ = 12 .
5, which is a typical value in semiconductor systems. Inwhat follows we will explain the reasons for selecting the values ˜ a = 0 .
25, ˜ b = 0 .
05 and ˜ γ = 0 .
03, always followingthe criterium of obtaining a band width of 3 . eV as it is suggested by the band diagram in figure 1.FIG. 4: The doubly degenerated paramagnetic and metallic band arising from the HF solution with full translationsymmetry and α and β spin structure of the single particle states.Observe the coincidence in form between the calculated band and the conduction one appearing in figure 1. In bothof them, the Fermi level is crossed by the dispersion curve at the mid point between the top and the bottom of thebands, in the direction Γ- X , while in the direction laying at 45 o respect to Γ- X the Fermi level becomes tangent tothe dispersion curve in its maximum.0 III. EVOLUTION OF THE PARAMAGNETIC-PSEUDOGAP BAND SPECTRUM
In this section we will present the results of solving the HF matrix set of equations for a variable hole doping in thecase of the excited solution which evolves from the pseudogap state obtained in Refs. [4–6] for the exact half-fillingsituation. It is important to recall that this state emerges as a solution after only eliminating the restriction on thesingle particle states of being a Bloch function in the full lattice formed by all the planar Cu atoms within the CuO planes. The constraints of being single particle states of α or β types were yet maintained. This process allows toobtaining of a doubly degenerated HF solution which shows a pseudogap resembling the one observed in the normalstate of the HT c superconductor materials.Figure 5 shows the evolution with increasing hole concentration of the bands of the pseudogap state, obtained for apoint lattice of 20 ×
20 points. All the graphics are plotted in the Brillouin zone of the sublattices, that is the darkerzone in the figure 3. Note the existence of a pseudogap which attains its maximum of the order ∼ meV . Theparameters fixed in the previous section were employed for this evaluation. The maximal value for the gap appearsat the mid points of the sides of the Brillouin zone of the sublattices and furnishes an estimate of T ≃ K for thetemperature at which the pseudogap starts to be observed in the experiments [12, 13, 20, 21]. (a) (b)(c) (d) FIG. 5: The evolution of the band spectrum of the
P P G as the hole concentration grows. The Fermi level lays inthe zero of the graphics. It can be observed how the pseudogap starts closing at the corners of the Brillouin zone asthe doping increases.The evaluated result for the pseudogap depends from the effective dielectric constant ǫ and the set of parameterswhich were fixed in order to reproduce the single band crossing the Fermi level in the band calculation done byMatheiss [3]. Then, the value obtained here should not be taken as precise prediction for the temperature of theobserved pseudogap. However, the ARPES experimental results for the doped La CuO indicates a pseudogap1temperature in the region 100-200 meV , that is value close to 1000 K . Thus, the P P G
Hartree-Fock solution offers areasonable estimate of the pseudogap temperature T ∗ [12, 13, 20, 21].One interesting results which can be observed from figure 5 is that by augmenting the hole concentration, theso-called “pockets” [22–24] in the corners of the Brillouin zone are formed. In addition it can observed that thepseudogap starts to diminish first in the corners until it fully collapses at the critical doping x c = 0 .
2. After thiscritical concentration of holes, the state starts behaving as a paramagnetic metal.
IV. EVOLUTION OF THE ANTIFERROMAGNETIC-INSULATOR BAND SPECTRUM
In this section we will consider the solution of the HF system of equations (16) by again using the successiveiterations method, but in this case, starting from an antiferromagnetic initial ansatz for the Slater determinant state.The results presented below, were found for the values of the set of free parameters: ǫ = 12 .
5, ˜ a = 0 .
25, ˜ γ = 0 .
03 and˜ b = 0 .
05, which were also employed in the previous section.The figure 6 (a) shows the band obtained for the point lattice of 20 ×
20 points. The bands are clearly associatedto an insulator system, which resulted as the ground state with respect to the pseudogap phase. (a) (b)(c) (d)
FIG. 6: Evolution of the band spectrum
AF I as the hole doping increases. The Fermi level is laying in the zero ofthe plots. The hole states form the the so-called Fermi arcs at the middle of the sides of the reduced Brillouin celland the gap diminishes until it is fully closed at the critical doping x c = 0 . δ c = 0 .
2, in which the form of thebands becomes different before and after the states passes through this particular hole concentration. In other words aquantum critical phase transition occurs at this value of the hole doping. Thus, the model predicts a phase transitionwhich had being argued to exist beneath the superconductor dome [10, 11].The results state that at the moment in which the
AF I and
P P G become degenerated at the quantum criticalpoint, the obtained HF solution of the problem becomes a unique metallic and paramagnetic phase. An importantpoint to underline, is that, as it is shown in figure 7, around the critical doping x c = 0 .
2, the results predict that theinsulator gap of the
AF I state diminishes until it completely collapses. The same occurs to the state
P P G . Thisproduces the metallic behavior which coincides with the one observed in the material in the high doping region. (a) (b)
FIG. 7: Diminishing of the gap with the increase of the hole concentration. a) For the antiferromagnetic-insulatorspectrum. b) For the paramagnetic-pseudogap state. Note that in both cases the gaps are completely closed whenthe concentration of holes is x c ∼ .
2, predicting a metallic behavior.
V. EVOLUTION OF THE FERMI SURFACE
Let us expose now the results for the modifications of the Fermi surface induced by the hole doping. The Fermisurface, as separating in the reciprocal space, the occupied from the empty orbitals [25], allows to define many physicalproperties of the materials. It is known that in the largely non understood normal state of the
HT c superconductors,the Fermi surface becomes truncated in parts called Fermi arcs. One of the most important open questions insuperconductivity theory is how the Fermi arcs and the superconductivity are mutually related [26, 27]. We haveinvestigated the Fermi surface predicted by the model and its dependence on the hole concentration for a widerange of concentrations 0 ≤ x ≤ .
3, defining the evolution with doping from the AF -insulator region up to theparamagnetic-metallic one.The figure 8 shows the evolution of the Fermi surface as the hole doping grows. It can be observed that the modelpredicts that in the low doping region ( x ≤ .
15) the Fermi surface is composed of the above mentioned Fermi arcsaround the nodal region ( π, π ). In this direction is where the doped holes establish as it was mentioned before (thegrey zone in figure 8). It was possible to evidence that the length of these arcs grows in proportion with the doping,up to the attainment of the critical doping at x c = 0 . , where a sort of hole pockets form in the corners of the reducedBrillouin zone.In accordance with our calculations shown in figure 8, the Fermi surface for x = 0 . π , π ). The areas closed by the Fermi surfaces are 70, 80, 85 and 95% of thehalf Brillouin zone area and corresponds to x = 0 .
3, 0 .
2, 0 .
15 and 0 .
05, respectively. These proportions are consistentwith the Luttinger sum rule, if the electron density is 1 − x = (70, 80, 85 and 95%, respectively). It can be concludedthat our results have a close qualitative agreement with the ones shown in figure 9, that were obtained by means ofthe ARPES in Ref. [9].Therefore, it follows that the considered model predicts that the Fermi surface of the LSCO undergoes a transition,which goes from a hole-like Fermi surface centered at ( π, π ) for 0 < x < . x = 0 .
3, 0 .
2, 0 .
15 and 0 .
05, respectively. Theseproportions are consistent with the Luttinger sum rule, if the electron density is 1 − x = (70, 80, 85 and 95%,respectively).(0 ,
0) for 0 . < x ≤ .
3. It is believed that the drastic change of the Fermi surface can correspond with the fact thatthe sign of the Hall coefficient changes from positive to negative values around x = 0 .
25 in the
LSCO [28, 29]. Inthe framework of the investigated model, the mentioned change is associated to the occurrence of a quantum phasetransition around the critical point x c = 0 . VI. THE QUANTUM PHASE TRANSITION
One of the current questions in
HT c superconductor theory is the occurrence of the superconductivity as determinedby the existence of a quantum critical point laying within the hole doping interval at which the superconductivityoccurs. Today is widely debated the question about what is the detailed connection between the critical point andthe superconductivity effect. Experimental indications about the existence of a critical quantum point comes fromthe transport and thermodynamic measurements [30–33]. In the framework of the present study, the HF energy perparticle for the AF I and
P P G states were evaluated by varying the hole concentration in the range 0 ≤ x ≤ .
3, byusing the formula E HF k ,l = X k ,l Θ (˜ ε F − ˜ ε l ( k )) [˜ ε l ( k ) − ˜ χ B k ,l ∗ . ( G C k − G i k ) .B k ,l ] . La − x Sr x CuO , obtained by theARPES experiment. Taken from: Ino A 1999. The dotted line in the first figure represent the so-called Fermi arc.That is a discontinuous Fermi surface.The figure 10 (a) show how the AF I state, which have the lowest energy at half-filling, evolves and becomesdegenerate with the
P P G state at the critical doping x c = 0 .
2. At this point, as it can be seen in the figure 6,the results predict for the
AF I state, that the insulator gap diminishes to be completely closed. The same behavioris shown by the
P P G state for which the momentum dependent pseudogap collapses. The solution of the problemdetermines a metallic behavior which is the observed nature of La CuO for high hole concentrations. (a) (b) FIG. 10: (a) Dependence of the HF energy per particle with the hole doping for the AF I and
P P G states. (b)Generic phase diagram where it can be observed the quantum phase transition from the
AF I state to aparamagnetic-conductor.The magnetization of the
AF I state was also evaluated by its defining formula m ( x ) = X k ,l X s,s ′ φ ∗ k ,l ( x , s ) σ ( s, s ′ ) φ k ,l ( x , s ′ ) , (24)where σ ( s, s ′ ) = σ x ( s, s ′ ) e x + σ x ( s, s ′ ) e x + σ z ( s, s ′ ) e z , (25)and σ x , σ x and σ z are the Pauli matrices. One important conclusion in this study is that in approaching the criticaldoping value the local magnetization of the AF I state tends to vanish (see figure 11), a property that the
P P G showed along its whole hole doping evolution from the initial half-filled state.
VII. A NEW HOLE BINDING MECHANISM: SPIN-SPACE ENTANGLEMENT
The evaluation procedure of the energy for the
AF I state in the region of low doping showed a clear dependenceon the odd or even character of the number of holes added. That is, it was needed a very much larger number ofiterations for attaining convergence when the number of holes was odd.5FIG. 11: Reduction of the magnetic moment of the
AF I state under the doping increasing.We estimate that a reason for such a behavior might be that adding an odd number of holes, could be expectedto show a larger complexity of the HF single particle states, since the Kramers degeneracy should be implemented.That is, it could be expected that the double degeneracy implied by the time inversion invariance of the system, whenthe odd number of holes does not allow to fill an integer number of degenerate pairs of states, should make such singleparticle states to show a very much complex structure.A perhaps related circumstance is that the spin-spatial entangled structure of the HF single particle states leadsto imagine a mechanism of hole pairing that could be acting in the considered system. The idea is that this entangledstructure of the single particle states might produce that two, well separated in space, hole wave packets could show,each of them, their relatively complex spin-spatial entangled constitution. However, when the two holes are allowedto be close in space, it is possible that they could tend to compensate their spatially dependent spin and magneticmoment structures and show a lower energy than the other pair of well separated holes. In the case of the paramagneticsystems such an effect seem to be very much weaker, due to the simplicity of the spin structure.In order to check for this possibility we defined a quantitative measure of the binding between two holes in theantiferromagnetic state. This definition was based in the energy of the ground state of the system E at half-fillingand the ground state energies after doped with one hole E and with two holes E in the form △ B = e − e , where e = E − E and e = E − E . Whenever, two holes minimize their energy by producing a bound state,then △ B becomes negative. When △ B vanishes the holes may not form a bound state since then e = 2 e and it isexpected that they behave as independent excitations.The binding energy of two holes △ B as a function of the dielectric constant ǫ for point lattices of 16 ×
16 sites(squares) and 20 ×
20 sites (circles) in the case of the
AF I state is shown in figure 12. Note that the bound states ofthe two holes form at low doping value as helped by the amount of dielectric screening of the Coulomb interaction.The figure 12 indicates that in general, when the dielectric constant ǫ increases, the binding energy reduces until theyform a bound state. The critical value for the appearance of binding for the holes slowly grow with the increasing ofthe size of the sample (the region in which periodic boundary conditions were imposed).The results indicate that although the thermodynamical limit is not yet attained for the evaluation of this quantity △ B , it seemingly exists. An interesting result is that in La CuO , it has experimentally determined that a supercon-ductor gap △ SC is the order of the 10 meV [9], which is of the similar magnitude of the shown values in figure 12, forthe two holes binding energy. That is, the measured superconductor gap approximately coincides with the energies △ B required to break the two bound holes. This outcome suggests the possibility that the HF solution is able to alsoconvey a pairing effect determining the presence of HT c superconductivity and the constitution of the Cooper pairs.A possibly acting binding mechanism could be as follows. Firstly, recall that the HF single particle states showthe more complicate than the standard spin-spatial entangled structure. This complex composition can be expectto provoke, if the screening is assumed to be strongly enough, a diminishing energy effect, in which the respectiveentangled magnetic moment structure of the combined two states could tend to compensate one to another. Suchan effect should appear in the bound state Bethe-Salpeter equations for two holes. In the described picture, thebound state of two holes at low doping in the AF I phase will be formed in this picture thanks to the screening of theCoulomb interaction. The fact that the measured static dielectric constant of the La CuO is around the value 25,indicates that the amount of screening is high.In the more investigated models, like the t − J one, it is argued that the existence of each hole breaks fourantiferromagnetic bounds [2], an effect that have an energy cost of the order of the coupling energy (0 . eV ). Therefore,at least in the low hole density limit, like the case under consideration, two holes minimize the energy to create themby sharing a common bound. In this way they minimize the number of broken antiferromagnetic bounds. Note that6FIG. 12: Binding energy △ B of two holes as a function of the dielectric constant ǫ for point lattices of 16 ×
16 (bluesquares) and 20 ×
20 (red circles).this picture seems to be compatible with the one previously exposed on the basis of the spin-spatial entangled natureof the hole states. It is also known that the size of the Cooper pairs of the material is a small quantity (around twolattice constants) [2].It had been also argued that the existence of preformed hole pairs could not be sufficient evidence for the appearanceof the superconductivity. In the context of the present model, it has been argued that for large values the dielectricconstant and the presence of spin-spatial entanglement, the pairs could be formed. Then, let us assume that theirpair wave function can be formed by superposing products of single particle waves having momenta values close to thecenters of the four reduced Brillouin zone sides in figure 6. Then, the amount of the momentum transfer associatedto these functions in the Bethe-Salpeter equation could be expected to produce pair wave functions showing sizes ofthe order of few lattice cells. If such results to be the situation, then, the hole pairs could be expected to condenseat T = 0 K giving rise to a Bose condensate of Cooper pairs showing superconductivity. These possibilities will beinvestigated elsewhere. In ending this section, we also want to express that the results also suggests that the hereargued binding effects between two added holes, could constitute a dynamical foundation of the physical meaning ofthe charge two boson excitations identified in Ref. [16], as being relevant for the description of Hubbard models. VIII. CONCLUSIONS
We have investigated the model for the La CuO defined in Refs. [4–6] applied to the situation in which the materialis doped with holes. The main elements of the model were reviewed by fixing its free parameters and solving theHartree-Fock equations. The evolution with doping of the band spectrum of the antiferromagnetic-insulator state andthe metallic-paramagnetic pseudogap one are presented for a wide range of hole concentrations 0 ≤ x ≤ .
3. Aroundthe critical doping density x c = 0 .
2, the results show that for the
AF I state, the band spectrum suffers a change instructure of the bands and the insulating band gap diminishes until its complete closing. A similar process also occursfor the
P P G state where the pseudogap collapses at the same special hole density. Over this critical density thesystem only shows a single paramagnetic-metallic state. The magnetization of the
AF I state becomes zero exactly atthe critical point. The solutions evidence that the destruction of the antiferromagnetic order is produced by the factthat added holes tend to occupy the states which show a more intense AF order: that ones which are closer to theboundary of the reduced Brillouin zone. The results show a drastic change of the nature of the Fermi surface, whichgoes from a hole-like Fermi surface centered at ( π, π ) for 0 < x < . ,
0) for0 . < x ≤ .
3. Henceforth, the whole discussion had shown that the investigated model predicts the existence of aquantum phase transition at critical doping value which is beneath the superconducting dome in La CuO . It wasalso identified a possibility for the pairing of holes which could give rise to the superconductivity and the boundingenergy was estimated. The effect seems to be closely linked to the spin-space “entanglement”effect of the HF singleparticle states. Possible connections with the existence of bosonic doubly charged excitations investigated in Ref. [16]were also pointed out.In ending, let us comment on some possible extensions of the present work. It seems convenient to consider a newparameter for to be fixed in the Hamiltonian of the model. It is related with the effective mass defining the kineticterm of the free part of Hamiltonian. This constant was assumed to be equal to the electron mass in free space, which,possibly, is a somewhat rude choice, since such electrons are assumed to move in the crystalline potential generatedby all the other particles, filling the rest of the many bands. This new freedom in the parameters might be relevant7when the spin-orbit effects will be taken into account in order to describe the magnetic anisotropy of the AF orderin further discussions. It would be also of help in allowing to more precisely fix the value of the insulator gap of the AF I state which is known to be close to 2 eV . One important point in this respect is that the dielectric constant ǫ hasbeen measured for La CuO . Therefore, it will be needed to fix the free parameters in order to define the observedvalue of the gap. At this point it can be noticed that electron effective mass will be an influential value which cancontrol the energy scale of the bands and could be phenomenologically fixed to define the 2 eV insulator gap. It canbe observed from equations (17) and (18).It seems also of interest to perform calculations in order to attain the thermodynamic limit in the hole pairing effect,and also to attempt deriving this binding effect but in the framework of the Bethe-Salpeter equation for two holesmoving in the medium. This would confirm the presence of preformed Cooper pairs in the AF I state of the model.Lastly, it will seem convenient to improve the formulation of the model by employing 3D Wannier states resemblingthe incomplete 3D shell of the Cu atoms within the CuO planes. Upon this, the incorporation of the spin-orbitinteractions will be needed in order to further describe the magnetic anisotropy of the AF order in La CuO . Acknowledgments
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