# Hartree-Fock Computation of the High-T Cuprate Phase Diagram

###### Abstract

A computation of the cuprate phase diagram is presented. Adiabatic deformability back to the density function band structure plus symmetry constraints lead to a Fermi liquid theory with five interaction parameters. Two of these are forced to zero by experiment. The remaining three are fit to the moment of the antiferromagnetic state at half filling, the superconducting gap at optimal doping, and the maximum pseudogap. The latter is identified as orbital antiferromagnetism. Solution of the Hartree-Fock equations gives, in quantitative agreement with experiment, (1) quantum phase transitions at 5% and 16% -type doping, (2) insulation below 5%, (3) a -wave pseudogap quasiparticle spectrum, (4) pseudogap and superconducting gap values as a function of doping, (5) superconducting versus doping, (6) London penetration depth versus doping, and (7) spin wave velocity. The fit points to superexchange mediated by the bonding O atom in the Cu-O plane as the causative agent of all three ordering phenomena.

R. B. Laughlin: ]http://large.stanford.edu

## I Introduction

The purpose of this paper is to discuss the theoretical phase diagram for the high- cuprate superconductors shown in Fig. 1 bonn ; copper ; broun ; orenstein ; mind ; manifesto ; wahl ; rblprl . It is generated using standard Hartree-Fock methods starting from a Fermi liquid theory with three interaction parameters agd ; fetter ; pines . It is characterized by three interpenetrating order parameters: spin antiferromagnetism, or spin density wave (SDW), -wave superconductivity (DWS) and orbital current antiferromagnetism, or -density wave (DDW) ddw ; nayak ; dimov .

However, the central issue of the paper is not building better models for the cuprates but the application of elementary quantum mechanics to them. The equations that generate Fig. 1 are not just made up. They are the only equations one can write down that are compatible with adiabatic evolution out of a fictitious metallic parent, plus a handful of experimental fiducials. This evolution, which strictly enforces the Feynman rules, is the starting point of all conventional solid state physics. The ability of these equations to account broadly and well for all key aspects of high- phenomenology thus indicates that high- cuprates are not qualitatively different from other solids, as has often been suggested might be the case, but are simply materials with unusually complex low-energy spectroscopy. This complexity, which results from delicate interplay of multiple order parameters, has prevented the problem from being solved empirically. But elementary quantum mechanics so circumscribes the mathematics that one can say with confidence that the phase diagram in Fig. 1 is correct, even though one of its features, the identification of DDW with the cuprate pseudogap, is still in doubt phenomenologically ckn ; stock ; macridin ; kee ; macdougall .

The results reported in this paper therefore have significance far greater than simply accounting for the behavior of a particular class of materials. The 25-year history of the cuprates has demonstrated rather brutally that first-principles theoretical control of solids at the energy scales relevant to electronic transport does not exist. This is so even though the underlying Hamiltonian of conventional matter is known exactly. Computers were not able to solve this problem. It was too hard for them. One obtains control, if at all, only by exploiting the universal low-energy properties of quantum phases. The simple equations that describe these properties are the starting point for predictive computation. The practice of adiabatically evolving from fictional parent vacua is precisely what distinguishes solid state physics from materials chemistry, and is what makes it so much more powerful than the latter as a basis of engineering.

### Historical Background

The discovery of high- cuprate superconductivity revealed that standard methods for computing the properties of solids were more seriously flawed than previously thought bednorz ; narrows ; wu ; maeda ; schilling . On the one hand, the materials in question were sufficiently conventional chemically that they should have yielded to ordinary self-consistent band structure analysis mattheiss . The latter requires them to be metals in the absence of translational symmetry breaking. On the other, their phenomenology was totally incompatible with the conventional theory of metals metals . Not only were their superconducting transition temperatures higher than existing theory said was possible without structural instability, their transport phenomenology was wildly irregular, and the violent variation of the superconducting transition temperature and superfluid density had no precedent cohen ; marginal ; takagi .

Accordingly, Anderson and others suggested at the time that cuprate superconductivity might be an important new aspect of Mott insulation, an equally serious conceptual issue that had emerged 40 years earlier rvb ; cyrot ; mott . This proposition was, and still is, extremely radical. Its central premise is that standard practices of solid state physics based on the adiabatic principle are irrelevant to these materials brainwash . Nonetheless it has now become mainstream and central to the field, in part because so many experiments have defied conventional explanation. It has also given rise to a number of related non-adiabatic theoretical ideas such as the non-Fermi liquid state, the holographic metal and the orthogonal metal schofield ; stewart ; kirkpatrick ; jiang ; holographic ; orthogonal .

Unfortunately, the phenomenological definition of a Mott insulator has always been somewhat difficult to state and is sometimes expanded to include the entire class of ordinary magnetic insulators brandow . The underlying idea is of a system that insulates when it ought not to. Thus the spin-unpolarized band structures of the transition metal monoxides FeO, CoO, and NiO are metallic but the materials themselves are all good insulators norman ; ohta . CoO has an odd number of electrons per unit cell. All three oxides possess antiferromagnetic order at zero temperature, which doubles the unit cell and thus formally allows them to insulate by the usual rules of band structure slater ; terakura ; anisimov . However, the strength of conventional exchange is inadequate to cause insulation in this way except in NiO, and all three materials continue to insulate above their Neèl temperatures. Other materials typically (but not always) categorized as Mott insulators include MnO, VO, FeO, LaTiO, YRuO, YTiO, YVO, and SrVO mcwhan ; nakotte ; pasternak ; lee ; park ; patterson ; ulrich ; zhou . The majority of identified Mott insulators are transition metal oxides.

The enormous amount of theoretical work stimulated by the cuprate discovery has now built up a strong case that the Mott insulator does not exist as a distinct zero-temperature state of matter. This was arguably unclear when the cuprates were first discovered, but it is no more. Two decades of intense focus on the problem have not led to a single instance of an actual wavefunction for a Mott insulator written down in terms of the underlying electron coordinates imada ; sachdev ; pal ; senthil ; gebhard . The resonating valence bond state of Anderson appears to be a counterexample, but this is not so vanilla ; jain . It is actually a -wave superconductor. It is made by adding a short-range Coulomb repulsion to a superconducting Hamiltonian and then taking the strength of this repulsion to large values while legislating that no phase transition occurs gossamer . Were such a perturbation actually applied without the unphysical constraint it would cause a phase transition to spin antiferromagnetism. No numerical calculation based on a conventional Hamiltonian finds a resonating valence bond state dagotto ; maier ; edegger ; leung .

## Experimental Constraints

As materials and experimental techniques improved over time, the purely empirical case for a new quantum state incompatible with the theory of metals became progressively weaker. After several years of failure Josephson tunneling was finally observed between YBaCO and Pb, thus dispelling concerns that cuprate superconductivity might not be a traditional Cooper pair condensate sun ; notoe . Ideas about the non-fermionic nature of the superconductor’s excitation spectrum were laid to rest by observation sharp fermionic quasiparticles in photoemission damascelli . Controversy over the symmetry of the superconducting order parameter was settled in favor of -wave pairing by observation of the node in photoemission, a sign change in Josephson tunneling, and half-integral trapped flux in magnetometery monthoux ; littlewood ; wollman ; scalapino ; tsuei . Ideas about the non-existence of a Fermi surface were disproved by photoemission observation of Fermi surfaces in overdoped samples agreeing in detail with band structure and the Luttinger sum rule plate ; luttinger . And, finally, superconductors placed in magnetic fields strong enough to crush their superconductivity were found to exhibit quantum oscillations, thus demonstrating the presence of a Fermi surface at low-energy scales in the zero-temperature normal state hussey ; sebastian ; vignolle .

The finite-temperature properties of the cuprates continued to be problematic, especially above the superconducting transition temperature near optimal doping batlogg . However, as the temperatures were lowered to zero the behavior inevitably evolved into something simple and conventional, most notably when the superconductivity was suppressed with a magnetic field ando . Repeated and consistent failure of the strange metal behavior to persist to low temperature has now demonstrated that it has nothing to do with quantum states of matter but is rather a critical phenomenon associated with a zero-temperature phase transition beneath the superconducting dome zaanen ; kallin ; mielke ; butch . The possibility that this phase transition is pseudogap development remains controversial varma ; where .

The occurrence of the pseudogap below optimal doping is associated with the reconstruction of the Fermi surface into pockets, as would be expected if a density wave had formed chakravarty . The pseudogap, first discovered in magnetic resonance and optical conductivity, was later identified in photoemission as a -wave quasiparticle dispersion that persisted above the superconducting transition temperature timusk ; loeser . Subsequent experiments revealed the existence of two -wave gaps, one associated with the superconductivity and another antagonistic to it tacon ; doiron ; valenzuela ; trisect . Scanning tunneling microscopy has now shown that pseudogap has complex position-dependent structure that is inherently glassy rosch ; kohsaka . It has also shown that conventional fermionic quasiparticles exist in the presence of the pseudogap and that they have the ability to propagate coherently large distances through it and interfere mcelroy .

### Orbital Antiferromagnetism

Several years before the quantum oscillation discovery, a group of us predicted that a reconstructed Fermi surface would appear when the superconductivity was destroyed by a strong magnetic field ddw . We argued that conventional translational symmetry breaking had to be the cause of the pseudogap because nothing else could be written down as actual equations. We proposed specifically that the pseudogap was the signature of orbital antiferromagnetism. Our grounds were (1) that there was no other way to account for a -wave pseudogap that was compatible with the adiabatic principle and (2) that instability to such order was an unavoidable consequence of antiferromagnetic exchange stabilization of -wave superconductivity out of a metallic parent. We named this order -density wave (DDW) to distinguish it from the gauge theory flux vacua, which were mathematically similar but conceptually different affleck ; liang . However, after much searching the predicted magnetic Bragg peaks were not found, so the purely empirical case for the order could not be made dai .

The subsequent discovery of quantum oscillations changed this situation. The original theoretical grounds for anticipating Fermi surface reconstruction had not changed, and attempts to reconcile it with the underlying quantum mechanics without doubling the unit cell proved impossible rice . The only explanation compatible with the adiabatic principle is that DDW order is, in fact, present in the cuprates, and that failure to detect clean magnetic Bragg peaks from it has been a consequence of pseudogap glassiness.

There is nothing extraordinary about orbital magnetism from the point of view of quantum mechanics. The ground state of the neutral O atom has a total magnetic moment of , of which comes from the spin and of which comes from the orbit. Orbital antiferromagnetism is normally quenched in solids, but is it conceptually no different from spin antiferromagnetism schroeter .

The failure to find signature magnetic Bragg peaks contributed materially to the development of the Varma current-loop theory, which has many similar features but does not break translation symmetry loop ; loopmodes . There is now some experimental support for this theory, although it is controversial fauque ; mook ; sonier ; li ; bourges ; strassle . However, the loop current insulating state has the same problem the Mott insulator does: It cannot be written down in electron coordinates.

Both kinds of spontaneous current would be difficult to detect in conventional Cu or O magnetic resonance in any cuprate because of symmetry lederer .

## Ii Adiabatic Evolution

Let us now briefly review the idea of adiabatic evolution. We imagine a fictitious Hamiltonian , usually noninteracting electrons moving in a periodic potential, that is easy to diagonalize. We then add a “perturbation” , where is the true Hamiltonian, and slowly increase from 0 to 1. Each time we infinitesimally increment , one of two things happens: either (1) the ground state and low-lying excitations remain in one-to-one correspondence, or (2) they do not. If the former is the case, we say that the system has remained in the same quantum phase. If the latter is the case, we say that it has undergone a quantum phase transition shankar ; sondhi .

Adiabatic mapping is what enables universal characterization of low-energy excitations inside a given phase. Thus, for example, when we say that electrons in a conventional metal behave as though they do not interact, we really mean that there is an adiabatic path back to that does not encounter any phase transitions. Were there no such path, it would make no sense to talk about a metal’s electrons and holes as abstractions, or to write down equations for them that involve only small effective interactions. The electrons in a conventional metal interact extremely strongly, as do electrons in solids generally. The correct Hamiltonian is always

(1) |

where denotes the location of the th electron and denotes the location of an ion of mass . The kinetic and potential energies of the valence electrons are therefore always comparable by virtue of the virial theorem. It is not true that the Coulomb interactions in the cuprates are enormously bigger than they are in other solids, such as elemental Si or Na metal. They are just the same.

Whether the interactions of Eq. (1) are strong enough to destabilize the metallic state necessarily and always is an interesting question, but an academic one in light of the enormous body of experimental precedent in metals catastrophe ; trash . Moreover the idea that metals might be inherently unstable at low-energy scales does not in any way invalidate computational procedures based on adiabatic evolution from a fictitious noninteracting parent state, which is to say, sums of conventional metallic Feynman graphs schrieffer .

It is not necessary that phase transitions should occur at isolated points in the interval , but this is usually the case. The reason is renormalization belitz . As a system is made larger and larger its measured properties eventually begin to change in a deterministic way. A renormalization fixed point is a Hamiltonian whose low-energy excitation spectrum stays the same when the system size is made larger. If all the perturbations to this Hamiltonian diminish under renormalization, we say the fixed point is attractive, and we associate it with a stable state of matter. If at least one perturbation grows with renormalization, we say the fixed point is repulsive, and we associate it with a continuous phase transition. The difference between repulsion and attraction is why phases of matter occupy open sets of values, while the transitions between them tend to occur at points.

The logical inconsistency of the Mott insulator concept is now easy to spot: It is perfectly reasonable that a system should pass through a new phase of matter on the way from (a metal) to (a -wave superconductor) but one is obligated to say what it is.

## Iii Fundamental Equations

Correct equations for the cuprate problem are easy to write down once one accepts that they must describe adiabatic evolution out of the density functional band structure. The latter is described adequately by mattheiss

(2) |

where denotes the set of near-neighbor pairs of sites on a planar square lattice of lattice constant and denotes the set of second-neighbor pairs. This description is inaccurate far from the fermi surface, but the high-energy excitations poorly described are not important.

### Fermi Liquid Parameters

Each time one increments the small perturbations renormalize to an effective Hamiltonian of the form

(3) |

This represents the most general set of fermi liquid parameters allowed by symmetry. Only pairwise interactions are relevant because the perturbation excites a quantum-mechanical gas of quasiparticles that is dilute. Only lattice terms closer than second neighbors are relevant because these exhaust the low angular momentum scattering channels. All terms associated with bonds must be rotationally invariant about the bond axis and reflection symmetric. All terms must be spin-rotationally invariant and time-reversal syymmetric. The parameters can, in principle, be energy-dependent, as they are, for example, if they are mediated by phonons, but this is irrelevant at the lowest energy scales. The difference between phonon-mediated pairing and purely electronic pairing is retardation, and this shows up only in high-energy spectroscopy schrieffer . The parameters can also be doping dependent, but I find that they are not.

All the various parts of added by incrementing renormalize into fermi liquid parameters by definition if the system has not yet made a phase transition out of the metallic state abrikosov ; nozieres ; wolfle ; depuis . But they also do if a phase transition has occurred along the way that is mild. This is why conventional superconductors may be described simply with Feynman graph sums schrieffer . The new state is still a quantum-mechanical combination of electron and holes of the parent metal.

Since is the most general effective Hamiltonian possible, all of the behaviors of the cuprate superconductors must be contained in it. There is no other possibility.

### Band Rigidity

We may immediately set to zero. Were it present its main effect would be to renormalize the kinetic energy in the doping-dependent way

(4) |

where denotes ground state expectation value. However, angle-resolved photoemission measurements find the asymptotic nodal fermi velocity to be eV Å for both -type and -type materials, regardless of doping vishik ; johnson ; armitage .

The absence of such a term makes sense physically. Doping dependence of simply means that the potential barrier through which the electrons tunnel depends on electron density. Such dependence is already taken care of in the band structure through self-consistency.

## Iv Hartree-Fock Solution

The ground state is characterized by the expectation values

(5) |

with the signs as in Fig. 1. The order parameter describes -wave superconductivity (DWS). The order parameter describes orbital antiferromagnetism (DDW). The order parameter describes spin antiferromagnetism (SDW). and are not order parameters but measures of the ground state kinetic energy. is the site occupancy.

To simplify the calculation we constrain both and to be periodic in the doubled unit cell. This creates a mild artifact of allowing the system to conduct electricity at all nonzero dopings. If this constraint is relaxed, SDW domain walls form, trapping carriers and causing the system to insulate everywhere SDW order is developed schulz ; inui ; sarkar ; chubukov ; gong . The similar problem with DDW, while important, is less severe because (1) the DDW quasiparticle spectrum is gapless at half-filling and (2) the DDW symmetry breaking is discrete.

The Hartree-Fock approximation is expressed either as a sum of rainbow Feynman graphs or as a single Slater determinant variational ground state fetter . In either case, each electron effectively moves in an average field generated by all the others. The variational ground state takes the form

(6) |

where

(7) |

is the creation operator for an electron of crystal momentum and spin in band . The index is required because both kinds of antiferromagnetism double the unit cell. As usual, the vector denotes the location of the th site. The coefficients are the same in every unit cell and satisfy

(8) |

(9) |

where

(10) |

(11) |

(12) |

(13) |

(14) |

in units for which the bond length equals 1.

The absence of band mixing in Eq. (6) is a consequence of the system’s special symmetries. Combining Eq. (8) with the complex conjugate of Eq. (9) in the presence of -wave superconductivity, we obtain the Nambu matrix

(15) |

where

(16) |

(17) |

with as the chemical potential. However since

(18) |

the eigenvalues of are , where

(19) |

The bands therefore do not mix, and we have

(20) |

(21) |

### Quasiparticle Energies

The operators

(22) |

(23) |

destroy . Their adjoints create quasiparticles of energy .

### Pairwise Contractions

With a variational ground state of the form of Eq. (6), the expected interaction energy is the sum of all pairwise contractions. Thus, for example, the second term of Eq. (3) gives a ground state expectation value of

(24) |

Thus this term stabilizes all three kinds of order. Here we have used the relations

(25) |

(26) |

implicit in Eq. (6).

The last term in Eq. (3) gives

(27) |

It thus has no effect on either DWS or SDW but stabilizes DDW.

The remaining two “coulombic” terms in Eq. (3) give

(28) |

per Eq. (24) and

(29) |

### Variational Energy

Thus the total expected ground state energy is

(30) |

### Self-Consistency Equations

The extremal condition

(31) |

then gives the equations

(32) |

(33) |

(34) |

(35) |

(36) |

Iterating these to convergence locates the minimum variational ground state energy

## V Limiting Cases

As a preliminary to fitting the parameters , , and to experiment we shall consider a handful of limiting cases.

### Half-Filling Antiferromagnetism

Let us first set all the parameters except , , and to zero and adjust the chemical potential to half-filling (). The superconductivity is problematic in this limit, so let us also force by hand. Iterating Eqs. (32) - (36) to self-consistency, we obtain the result shown in Fig. 2. There are two second-order phase transitions bounding a region of coexistence between spin antiferromagnetism (spin density wave SDW) and orbital antiferromagnetism (-density wave DDW).

It is immediately clear from this figure that pure exchange characterized by tends to stabilize SDW and DDW equally. For one needs , which is unphysical, to achieve coexistence. But the more important observation is that the requisite is relatively small. The boundary in question is also actually multicritical. DDW and DWS are exactly degenerate in this limit. This may be seen by comparing Eqs. (35) and (36), but the actual cause is a special symmetry described in Appendix A.

This result thus shows that strongly correlated superexchange () cannot account quantitatively for DWS in the cuprates superex . One of the key experimental features of these materials is that small amounts of doping, a delicate perturbation, can violently disrupt the SDW and completely replace it with DWS. This implies that the system lies near a phase boundary. But Fig. 2 shows that the purely magnetic system cannot be near the phase boundary between SDW and DWS unless is small. This observation is backed up by the large body of numerical work on the Hubbard model, which shows that it does not account well for properties of the cuprates hubbard .

### SDW vs. DDW

Let us next relax the condition that and repeat the calculation of Fig. 2, again artificially forcing , with fixed values of and . These parameters are unphysically large, in part because , per Eq. (11). They are fit to the conditions that (1) the system lie near the upper edge of the coexistence region in Fig. 2 and (2) the spin moment at half-filling be , half the maximum classically allowed value. A moment of this size is characteristic of the insulating cuprates yamada ; vaknin ; regnault ; tranquada . The result, shown in Fig. 3, reveals that the SDW is destroyed by a doping of 4%, a number characteristic of spin antiferromagnetism disappearance of the cuprates. SDW is supplanted at this density by DDW, a pseudogap candidate with a -wave quasiparticle spectrum. DDW itself then becomes unstable at 14% doping, exactly where the optimal superconducting is observed in the cuprates and where the pseudogap is observed to disappear. There is a slight first-order re-entrance of SDW at 14% when the DDW vanishes, indicating an intense struggle for dominance between the two kinds of order. Neither the specific transition doping densities nor the order parameter functional forms are fit.

Why this sequence of phase transitions takes place is easy to understand physically. Figure 4 shows the evolution of density of states

(37) |

as doping is increased. The rough equivalence of the two orders at half-filling, reflected in equality of their energy gaps, becomes unbalanced in favor of DDW when carriers are added because DDW, which has a node, allows them to be added at zero energy. Deeper doping then destabilizes DDW because it causes the Fermi surface to contract around the lines , where the state’s node prevents it from extracting condensation energy. SDW order, which can extract condensation energy from this region, is then briefly resurrected, but it shortly falls victim to the Fermi surface shrinkage that occurs as doping is increased further.

The larger implication of Fig. 3 is that any system with a moment of and rough balance between SDW and DDW at half filling will have phase transitions when doped at densities consistent with those observed in the cuprates.

### SDW vs. DWS

Let us now repeat the calculation of Fig. 3 with the constraint removed. The result, shown in Fig. 5, reveals that DWS now overwhelms DDW completely. The effect may be understood as an analog of a spin flop demler . The two kinds of order are exactly degenerate at , the way the , and components of an ideal antiferromagnet are, as described in Appendix A. But the nesting condition required to stabilize DDW becomes degraded at any finite doping while DWS, which does not require nesting, remains robust. Adding carriers is thus analogous to adding an anisotropy field to the antiferromagnet, and the system responds by “flopping” to DWS.

The persistence of SDW to 12% doping in Fig. 5 also shows that DWS is less effective at crushing the SDW at low dopings than DDW is, even though it is more stable. The reason is that DWS does not exploit any special degeneracies of the band nesting and thus does not use them up and make them unavailable to SDW formation the way DDW would. Reentrant spin antiferromagnetism is also absent in Fig. 5, but this is a simple consequence of the persistence of nonzero to high dopings.

The coexistence of DWS with SDW everywhere the latter is developed in Fig. 5 is allowed by the special (imposed) symmetries of the problem, which guarantee that spin antiferromagnetism fights superconductivity through gap formation only, not through pair breaking. The system can then become superconducting by exciting electrons across the SDW gap quantum mechanically and binding them into Cooper pairs there. A system nominally an insulator in this way becomes a superconductor. The effect is shown more explicitly in Fig. 6, where the superconducting spectral function

(38) |

is plotted. As doping increases the SDW gap eventually collapses, restoring the -wave node.

### DWS vs. DDW

Let us now get all three order parameters to appear in the phase diagram by repeating the calculation of Fig. 5 in the presence of . This parameter breaks the degeneracy between DDW and DWS at half filling, encouraging the former over the latter. The result is shown in Fig. 7. In order to maintain the half-filling conditions implicit in Figs. 3 and 5, we accompany the increase of with an adjustment of the parameters and that keeps and constant, per Eqs. (11) and (33) - (35). When is increased to the value used in Fig. 7, this causes the interaction parameters , and all to become reasonably sized (i.e. comparable to ), and, even more importantly, causes to switch from negative to positive. The behavior of Fig. 3 is now restored, this time legitimately, but DDW is accompanied by doping-dependent DWS, with which it coexists. The latter first acquires significant magnitude when the SDW is destroyed at 5% and then rises up as the DDW dies away, peaking at 15% where the latter disappears, and then declining rapidly. The sequence of events is identical to that observed in -type cuprates.

The result of Fig. 7 cannot be achieved using . The half-filling degeneracy of DDW and DWS is broken the same way by both parameters, but has the effect of suppressing DWS, per Eq. (36). This suppression can be counteracted by increasing the value of , but this then requires making more negative, per Eq. (33). Negative is highly unphysical. Were it present, it would stabilize -wave superconductivity, a phenomenon not observed in the cuprates.

### Particle-Hole Asymmetry

Let us now introduce particle-hole asymmetry by repeating the calculation of Fig. 7 with an added . The result is shown in Fig. 8. The pseudogap is now completely absent on the -type side, and the re-entrant spin antiferromagnetism has disappeared from both sides.

The ability of such a small change in the underlying band structure to violently rearrange the phase diagram is consistent with the variability of cuprate experiments, both among different materials and among samples of the same material prepared different ways. It occurs because small perturbations tip the fine balance among orderings. Defects and crystal boundaries would be expected to rearrange the order parameters locally in a similar fashion, thus causing large effects that have no analog in semiconductors. This result is also consistent with the observed complexity of lattice instabilities in these materials tranquada . Describing the latter requires addition of an electron-phonon interaction to , but doing so is straightforward given that the only effect of moving atoms around is to change the underlying band structure.

## Vi Parametric Constraints

Figures 7 and 8 show that the interaction parameters , , and are fixed by the magnitudes of , , and , and that is fixed by the overall shape of the phase diagram. The conclusion that resolves the important controversy of whether antiferromagnetic exchange or near-neighbor attraction, possibly phonon-mediated, causes high- superconductivity. Only the former is both physically reasonable and compatible with experiment.

The reason that one cannot have is subtle and requires some discussion. Only and have the ability to stabilize -wave superconductivity, so at least one of them must be present and sizable. However, a of the requisite magnitude is already present, as demonstrated by the antiferromagnetism of the undoped insulator. Adding would further stabilize -wave superconductivity, but unfortunately it would do so near half-filling and destabilize the pseudogap. To restore the pseudogap, one would then have to compensate by making either or more positive, per Eq. (35). But, actually has to decrease if the superconducting aspects of Fig. 7 are to remain the same, per Eq. (36). The required increase in would then severely narrow the quasiparticle bandwidth, per Eq. (11), an effect not observed experimentally. Accordingly, can be neither negative nor positive but must be zero.

The conclusion that also makes sense physically. The parameter is a Coulomb interaction. There is fundamentally no reason for it to be negative, just as there is no reason for to be negative. Indeed one’s first guess would be that both parameters had been mostly accounted for in the generation of the band structure. A parameter would also tend to stabilize -wave superconductivity unless prevented from doing so by a sufficiently large . The latter would have to be large enough to prevent mixed superconductivity, a phenomenon not observed in any part of the cuprate phase diagram. The parameter by contrast is not only demonstrably present but something unique to the cuprates.

In addition to properly balancing the phase diagram, the parameter has three important effects that indicate it is not only useful but actually physically necessary: (1) it enables to be positive, (2) it reduces to a reasonable size, and (3) it enables to be less than . The latter is particularly important. The parameter fixes physical quasiparticle bandwidth, a quantity observed experimentally not to be broadened, but it also sets quasiparticle Fermi velocity. Were , the oscillator strength of the Fermi sea conductivity pole would exceed the total -sum rule, which is fixed by solely, thus indicating that the system was not in its ground state. This sum rule problem is discussed further Section VII in the context of the superfluid density.

Given the insight that postulating makes no sense without also postulating of roughly the same magnitude, the source of the latter is easily identified: It is a secondary aspect of superexchange. This is illustrated in Fig. 9. Spin antiferromagnetism is due to blocking superex . One knows this because the spin-orbit and magnetic dipole interactions in typical antiferromagnets are too small to account for the observed spin stiffness. The bond quantum mechanics may therefore be adiabatically deformed to a Hubbard model characterized by parameters and . In terms of the four configurational states

(39) |

we then have the Hamiltonian

(40) |

Second-order perturbation theory then gives a spin-exchange matrix element of and a Cooper pair tunneling matrix element of .

Correlation corrections to the properties of metals are notoriously difficult to calculate from first principles, especially if they are performed by summing Feynman graphs. In graphical sums, the negative-energy denominators required to obtain show up in the reverse time orderings fetter . Such calculations are beyond the scope of this work. The purpose of Eq. (40) is only to show that it is physically reasonable for to appear whenever does.

## Vii Superconducting Properties

In what follows we shall assume the parameters used to generate Fig. 1: , , , , , and eV. These amount to a small fine-tuning of the parameters of Fig. 8.

### Energy Gap

Figure 10 compares the maximum gaps computed using Eq. (32) - (36) with pseudogap and superconducting gap value estimated by Hüfner et al from a variety of experimental sources hufner . The latter include scanning tunneling microscopy, photoemission, Raman scattering, break junction tunneling, magnetic resonance, inelastic neutron scattering, thermal transport and Andreev scattering ding ; peets ; venturini ; sugai ; zasadzinski ; sidis ; he ; hawthorn ; sutherland ; deutscher . The experimental plot is qualitatively similar to that of Le Tacon et al. and of Valenzuela and Bascones but is in absolute units tacon ; valenzuela . The assignment of specific values to the experimental gaps is somewhat subjective because they are not sharply defined. A reasonable estimate of the error bar is 30%. Particularly important are the Andreev reflection experiments, which show that particle-hole mixing characteristic of superconducting order vanishes at an energy scale much lower than that of the pseudogap deutscher ; gonnelli .

Figure 10 severely constrains the choice of , , and . The scale of superconducting gap fixes . The scale of the pseudogap fixes . The condition that at half filling fixes . These parameters then place the system at the edge of the half-filling coexistence region of Fig. 2 (with the substitution keeping fixed), thus causing the two phase transitions take place at their experimentally observed doping values of 5% and 16%.

### London Penetration Depth

The zero-temperature London penetration depth is given by

(41) |

where Å is an interlayer spacing appropriate for YBCO and in the effective superfluid density per Cu atom

(42) |

The superfluid density is not a well-defined quantity, so the choice of as the conversion factor between and is somewhat arbitrary. It corresponds to the effective mass formula

(43) |

with a bond length Å.

The formal justification of Eq. (42) when DDW order is developed is complicated by the fact that the DDW order parameter is gauge-variant, while the Hamiltonian parameter that gives rise to it is not. Handling this properly requires executing a vertex correction, which is technically beyond the scope of this work. However, it is easy to see on physical grounds that Eq. (42) must be correct. It says that each quasiparticle carries electric charge and moves with a speed that is the momentum derivative of its energy, just as occurs in SDW. The quasiparticles of DDW and SDW must behave similarly because orbital antiferromagnetism and spin antiferromagnetism are aptly analogous.

Figure 11 shows the superfluid density calculated using Eq. (42) and the parameters of Fig. 1 as a function of doping. It is free-electron-like except when the DDW and SDW orders develop near , when it is strongly suppressed. This reflects the reorganization of the Fermi surface into pockets. Loss of superfluid density due to an elementary Fermi surface reconstruction is fully consistent with the finding of Tallon et al. that the ratio of the superconducting temperature and the specific heat jump remains constant in this limit while both quantities individually vanish tallon .

The values of implicit in Fig. 11 through Eq. (41) are compared with experiment in Fig. 10 tallon ; bernhard ; pereg . The large variability among them is symptomatic of disorder degradation. The proper comparison to make is thus with the best samples at optimal doping. The good agreement in these cases confirms .

The nonzero superfluid density for in Fig. 11 ( in Fig. 10) is an artifact of having forced the SDW to periodic in the doubled unit cell. Relaxing this condition will cause the quasiparticles to trap in domain walls, thus forcing the superfluid density to zero, in agreement with experiment schulz ; inui ; sarkar ; chubukov ; gong . The system is an insulator over the entire range in which SDW is developed—with the possible exception of , where coexistence of SDW and DWS has been reported yu ; luo . The superfluid oscillator strength in question, however, is not lost but is simply transported upward to a small, but finite frequency characteristic of the trapping. This is seen experimentally in the lightly-doped cuprates as mid-infrared absorption uchida ; padilla . It is centered at about 0.6 eV in LaSrCuO.

Figure 11 also shows that the total -sum rule

derived in Appendix C, where

(44) |

per Eqs. (5), exceeds . This is an important stability test. is the sum of plus all the additional optical oscillator strength at higher frequencies. If one had at any doping value, it would imply that the system had negative oscillator strength (laser gain) at higher frequencies, and thus that it was not in its ground state. But Eqs. (32) - (36) show that is enforced by . Thus Fig. 11 also shows that the presence of a is required for to make sense physically. Moreover, the particular values of these parameters used to generate Fig. 11 give comparable to