Interpretations of ground-state symmetry breaking and strong correlation in wavefunction and density functional theories

Significance

The ground state of a quantum mechanical system is the lowest-energy eigenstate of the Hamiltonian. In isolation, it persists unchanged forever, with symmetries dictated by those of the Hamiltonian. But near-eigenstates of broken symmetry can persist for long times, even on the scale of human measurement. The appearance of broken symmetries of the electron density or spin density in a density functional calculation can reveal strong correlations among the electrons that are present in a symmetry-unbroken wavefunction. Symmetry breaking can arise when a wave-like fluctuation drops to zero frequency. The presented examples are the stretched hydrogen molecule, antiferromagnetism in solids, and the static charge-density wave in a low-density jellium, which is shown quantitatively to be a zero-frequency plasma wave.

Abstract

Strong correlations within a symmetry-unbroken ground-state wavefunction can show up in approximate density functional theory as symmetry-broken spin densities or total densities, which are sometimes observable. They can arise from soft modes of fluctuations (sometimes collective excitations) such as spin-density or charge-density waves at nonzero wavevector. In this sense, an approximate density functional for exchange and correlation that breaks symmetry can be more revealing (albeit less accurate) than an exact functional that does not. The examples discussed here include the stretched H₂ molecule, antiferromagnetic solids, and the static charge-density wave/Wigner crystal phase of a low-density jellium. Time-dependent density functional theory is used to show quantitatively that the static charge-density wave is a soft plasmon. More precisely, the frequency of a related density fluctuation drops to zero, as found from the frequency moments of the spectral function, calculated from a recent constraint-based wavevector- and frequency-dependent jellium exchange-correlation kernel.

This article presents examples and interpretations of broken symmetry and strong correlation, complexities that sometimes show up in quantum mechanical systems of many particles, including molecules and solids. Fig. 1 suggests why several different perspectives can be helpful to understand complex phenomena.

Fig. 1.

The parable of the blind men and the elephant suggests that scientists who study the same complex problem from different perspectives should pool their insights. This article touches four perspectives on broken symmetry and strong correlation in many-electron systems: ground-state and time-dependent density functional, wavefunction, and model Hamiltonian. The DFT+U, dynamical mean field, and Green’s functions approaches are other important perspectives that are mentioned briefly in Conclusions. Image credit: Liyu Ye (artist).

The density functional theory (DFT) of Kohn and Sham (1) is an exact-in-principle mean-field-like theory for the ground-state (lowest) energy and electron density (or spin densities) of any N-electron system with a Hamiltonian or energy operator of the form

$$Ĥ=\sum _{i=1}^N\left[-\frac{1}{2}{\nabla }_i^2+v\left({\mathit{r}}_i\right)+\sum _{i<j}{\left|{\mathit{r}}_i-{\mathit{r}}_j\right|}^{-1}\right],$$ [1]

(taking fundamental constants to be one). $v\left(\mathit{r}\right)$ is an external scalar potential, such as the Coulomb attraction to the nuclei, and the electron–electron Coulomb repulsion is explicitly included. A ground-state $N$-electron wavefunction is an eigenstate of the Hamiltonian for the lowest-energy eigenvalue. Since electrons are spin-1/2 fermions, the physical wavefunctions must be antisymmetric, changing sign under the exchange of any two electrons. Squaring a wavefunction yields a positive probability distribution. Integrating the square of the wavefunction over $N-1$ electron position vectors and $N-1$ spin coordinates yields a one-electron spin density $n_{\sigma }\left(\mathit{r}\right)$ (where $\sigma \hspace{0.17em}=\uparrow$ or $\downarrow$ along the $z$ axis of spin quantization) and total electron density $n\left(\mathit{r}\right)=n_{\uparrow }\left(\mathit{r}\right)+n_{\downarrow }\left(\mathit{r}\right)$ that define the starting point for DFT. Integrating over all but two electron positions $\mathit{r}$ and spins $\sigma$ yields an electron-pair density ${\rho }_{\sigma {\sigma }^{\prime }}\left(\mathit{r},{\mathit{r}}^{\prime }\right)$ that shows explicitly how the electrons avoid one another due to wavefunction antisymmetry (which makes ${\rho }_{\sigma \sigma }\left(\mathit{r},\mathit{r}\right)=0$) and Coulomb repulsion. Although the Hamiltonian of Eq. 1 does not depend explicitly on spin, the spin state importantly affects the ground-state energy through the antisymmetry of the wavefunction.

There is an exact ground-state density functional for the energy, including noninteracting kinetic, external, Hartree, and exchange-correlation terms, whose minimization leads to exact one-electron Schrödinger equations for the orbitals or one-electron wavefunctions used to construct the noninteracting kinetic energy and the electron density. A practically useful method is attained by approximating only the negative exchange-correlation energy functional $E_{\mathrm{x}\mathrm{c}}\left[n_{\uparrow },n_{\downarrow }\right]$, sometimes based on exact constraints and on appropriate norms, such as the uniform electron gas, for which all functionals discussed here are exact.

Coulomb correlation means all effects beyond the symmetry-unbroken self-consistent mean-field Hartree–Fock approximation (2). The mean-field appearance of Kohn–Sham theory too often creates the impression that this theory lacks correlation, or at least strong correlation. But Coulomb correlation is important, even in ordinary sp-bonded molecules, and this correlation is well described by nonempirical approximate functionals (1, 3–5), as Table 1 shows. Kohn–Sham DFT achieves computational efficiency by not calculating a correlated many-electron wavefunction, but it still includes the Coulomb-repulsion-driven correlation among electrons in its functional or rule for finding the exchange-correlation term of the total energy from the electron up- and down-spin densities and in its related exchange-correlation potential in the one-electron Schrödinger equation that shapes the orbitals and the spin densities in a self-consistent calculation.

Table 1.

Mean absolute errors (MAEs) in electronvolts for the atomization energies of the six representative AE6 (6) sp-bonded molecules, for Hartree–Fock exchange (x) and for exchange-correlation (xc) functionals on the first three rungs of a ladder of approximations (none of them fitted to bonded systems)

Approximation	AE6 MAE, eV
Hartree–Fock x	6.3
LSDA xc	3.3
PBE GGA xc	0.6
SCAN meta-GGA xc	0.1

Hartree–Fock results are from Lynch and Truhlar (6). LSDA (1, 3), PBE GGA (7), and SCAN meta-GGA (4) results are from Bhattarai et al. (5). Note that the semilocal approximations LSDA, PBE, and SCAN are much better here for xc together than for x or c separately, due to an understood cancellation between the full nonlocalities of exact x and exact c. The LSDA exchange-correlation energy density depends only on the local spin densities; the GGA further includes the density gradients; and the meta-GGA still further includes the orbital kinetic energy densities.

“Strong correlation” is sometimes used to mean “everything that DFT gets wrong.” Yet, hybrid functionals (including part of the Hartree–Fock exchange) like HSE06 (8–10) (for nonmetallic states) and meta-generalized gradient approximations (meta-GGAs) like the strongly constrained and appropriately normed (SCAN) functional (4, 11–18) are yielding quantitatively correct ground-state (and we emphasize “ground-state”) results by symmetry breaking for some systems that have long been regarded as strongly correlated. In the cuprate high-temperature superconducting materials, for example, the SCAN meta-GGA (4) is able to do what simpler density functionals (local spin density approximation [LSDA] and generalized gradient approximation [GGA]) cannot (13), by creating the correct spin moments on the copper atoms, their antiferromagnetic order, and a correctly nonzero band gap in the undoped material that correctly disappears under the doping that also leads to superconductivity (12–14). Full, but better-designed, self-interaction corrections (5) might eventually further improve approximate DFT for other strongly correlated systems. And, as will be argued here, even the simplest DFT approximations yield a qualitative insight into some such systems.

The first interpretation to be expressed here is that certain strong correlations that are present as fluctuations in the exact symmetry-unbroken ground-state wavefunction are “frozen” in symmetry-broken electron densities or spin densities of approximate DFT. For finite systems, this would not happen with the exact functional. So, while the exact functional would always be exact for the ground-state energy and density, it would not always be as qualitatively revealing as the approximate one.

This first interpretation (but without the term “strong correlations”) is eloquently expressed in P. W. Anderson’s famous essay “More Is Different” (19), which explains that, while a ground-state wavefunction is necessarily static, it must describe fluctuations in the expectation values of operators that are not diagonalized along with the Hamiltonian, and that these fluctuations can freeze as the number of particles in the system grows large, leading to an observable symmetry breaking that would not be possible in a few-particle system. Symmetry breaking in approximate DFT can thus be more correct for solids than it is for small molecules, but, in either case, it can reveal a strong correlation in a symmetry-unbroken wavefunction. The possibility of observable symmetry breaking is compatible (20) with the continued existence of a symmetry-unbroken exact ground-state wavefunction. Symmetry breaking is surprising only in quantum physics; in classical physics, it is familiar and intuitive (21). For an eloquent exposition of symmetry breaking, and more generally of theoretical physics, the interested reader is referred to ref. 22.

The second interpretation involves the wave-like fluctuations and collective excitations of a solid. A wave in the total electron density $n\left(\mathit{r}\right)=n_{\uparrow }\left(\mathit{r}\right)+n_{\downarrow }\left(\mathit{r}\right)$ is a plasma wave (quantized as plasmons), and a wave in the net electron spin density $m\left(\mathit{r}\right)=n_{\uparrow }\left(\mathit{r}\right)-n_{\downarrow }\left(\mathit{r}\right)$ is a spin-density wave. Like other waves, these have amplitude, wavelength $\lambda$ or wavevector $q=2\pi /\lambda$, and frequency $\omega$, and at small amplitudes can be simply superposed or added together. However, their frequencies are not sharply defined. By the uncertainty principle, a smaller frequency width leads to a longer lifetime. Collective excitation modes are distinguished from other fluctuation modes by much smaller frequency widths. The second interpretation explains how some symmetry breakings and strong correlations can arise: Under a variation of the external potential, a collective excitation or fluctuation of the electrons of nonzero wavevector, such as a charge-density wave or a spin-density wave, can soften, with an excitation energy or frequency tending to zero, until it appears as a static wave in the symmetry-broken density or spin density of an approximate functional. In small systems, where symmetry breaking is unobservable, the exact frequency might only approach, but not reach, zero. Since this frequency is nonnegative, the frequency width must go to zero when the frequency does. This mechanism is analogous to the softening of a phonon mode of nonzero wavevector that can lead to a distortion or structural phase transition of an ionic lattice. The ground-state energy of a quantum lattice nearly equals that of the classical lattice plus the zero-point energy of its lattice vibrational modes. Via the fluctuation-dissipation theorem (23, 24), the ground-state total energy of a system of interacting electrons, including the Coulomb correlation energy, has contributions from fluctuations of various wavevectors, including the nonnegative zero-point energies $\hslash \omega /2$ of its collective excitations. Collective excitations, like observable or physical symmetry breakings, are emergent phenomena, arising only in systems with large numbers of particles (25). Fluctuations, even when they are not true collective excitations, are described by the dynamic structure factor (24) or spectral function of time-dependent DFT (25).

Symmetry in Quantum Mechanics

The Hamiltonian of Eq. 1 can break spatial symmetries through its external potential, which violates the homogeneity of space, but that is not the subject of the discussion. We also note that this Hamiltonian, while a good starting point, is not exact. It omits relativistic effects, including the spin–orbit interaction, and it neglects the fact that the external potential comes from nuclei that can move. These physical effects will be ignored here for the sake of clarity and simplicity.

We know that it is possible to find the eigenstates of a complete set of commuting observables, including the Hamiltonian. The states with the lowest eigenvalue of the Hamiltonian are the ground states. If the ground state is nondegenerate, the ground-state density will have all of the symmetries of the external potential. For example, the ground state of the Ne atom is nondegenerate, and its electron density has the spherical symmetry of the $v\left(r\right)=-10/r$ Coulomb attraction to the nucleus. If the ground state is degenerate, an equally weighted average (equi-ensemble) of all ground-state densities or spin densities will have the symmetries of the external potential: “The symmetry of the Hamiltonian is the symmetry of the ground state” (20).

The Hamiltonian of Eq. 1 is independent of electron spin. Then, the complete set of commuting observables can include the square and $z$-component of the total electron spin. When the ground state is nondegenerate, as in a closed-shell atom or molecule, or presumably in a finite nonferromagnetic and nonferrimagnetic crystal, the square of the total spin must be zero ($S=0$ or spin singlet state), and, thus, the local net spin density must be zero, and the exact ground-state density is expected to have all of the symmetries of the external potential. When there is a net spin moment or a net current density, of course, the ground state must be degenerate, and a single ground state from a degenerate set need not have the symmetry of the external potential, although the state of thermal equilibrium in the zero-temperature limit still should have it.

Stretched Hydrogen Molecule

A closed-shell molecule like ${\mathrm{H}}_2$ has a nondegenerate ground state whose exact electron density must remain spin-unpolarized at all bond lengths. In 1976, Gunnarsson and Lundqvist (26) made an LSDA calculation for the ground-state ${\mathrm{H}}_2$ molecule at various bond lengths. For bond lengths less than a critical value (1.7 Å), they found a spin-unpolarized density and a realistic binding-energy curve. Above the critical bond length, they found an unobservable spin-symmetry breaking, with the net spin up near one nuclear center and down near the other (a kind of molecular precursor of the antiferromagnetism to be discussed in Antiferromagnetic Solids). The whole binding-energy curve was realistic only when the symmetry of the spin density was allowed to break under bond stretching. A constraint that prevented spin-symmetry breaking led in the limit of infinite bond length to an LSDA total energy much higher than the LSDA energy of two individual hydrogen atoms. Ref. 26 also mentioned a fluctuation in which the spins on the two nuclei are interchanged, with a frequency that drops toward zero as the bond length tends toward infinity.

The exact wavefunction tells us that, at a stretched, but finite, bond length, when an electron with a given spin direction is near one nucleus, an electron with the opposite spin direction is highly likely to be near the other. This is a strong correlation within a symmetry-unbroken wavefunction of zero total spin, which is revealed to us by the breaking of the symmetry of the spin density in the LSDA calculation. As the bond length tends to infinity, the energy of stretched ${\mathrm{H}}_2$ becomes identical to the energy of two separate H atoms, one with spin up and the other with spin down. Then, even an infinitesimal, staggered magnetic field can break the symmetry and select out the spin-polarized atomic densities. From the (classical) viewpoint of human intuition, the strongly correlated limit of infinite bond length is just the limit of independent atoms. Greatly improved binding-energy curves from approximate functionals are achieved by spin-symmetry breaking not only for ${\mathrm{H}}_2$, but also for LiH (27). For LiH, the spin-symmetry breaking even prevents (27) an incorrect dissociation to fractionally charged fragments.

Antiferromagnetic Solids

Similarly, symmetry suggests that the exact ground-state density of any finite system with zero net spin moment should be spin-unpolarized. But approximate spin-density functional calculations, e.g., refs. 11–15 and 17, find that many materials, and especially transition-metal oxides, are antiferromagnetic, with alternating net spin moments on alternating atomic sites. This is presumably a strong correlation within the exact symmetry-unbroken ground-state wavefunction that freezes into the symmetry-broken ground-state spin densities of the approximate density functional. Strong correlations within an exact symmetry-unbroken wavefunction might tell us that there are local spin moments on the transition-metal atoms that are correlated, so that, for example, a spin-up moment on a given atom has spin-down neighbors on the nearest-neighbor atoms, although the direction of the spin moment on a given atom is not fixed.

A connection between spin-density waves and antiferromagnetism was proposed by Overhauser (28). Our interpretation is that a spin-density wave drops down (as a function of lattice constant) in energy and frequency to create the broken-symmetry static spin density. Under extreme compression of a given lattice, any material will be a nonmagnetic metal with zero energy gap and no local magnetic moment. As this compression is relaxed, we imagine that a spin-density wave will soften. To create an ordered antiferromagnetic state, the nonzero wavevector of the soft spin-density wave must extend from the center of the first Brillouin zone of the underlying ionic lattice to the center of one of the Brillouin-zone faces, typically doubling the size of the unit cell in real space and opening an energy gap that makes the material an insulator. In approximate Kohn–Sham DFT, the metallic, nonmagnetic state of unbroken symmetry remains as a solution of the Kohn–Sham equations, but not as the solution of lowest energy. Many correlations become stronger, and symmetries tend to break as the electron density and electron kinetic energy decrease.

Anderson (29) found an accurate estimate of the ground-state energy for an antiferromagnetic Heisenberg model Hamiltonian describing the effective interaction between localized spin moments on a simple cubic lattice. He started from the energy of a classical antiferromagnetic ground state, then included the zero-point energy of the transverse spin-wave excitations. Here, the transverse directions are perpendicular to the $z$ direction—i.e., to the spin direction on the first of the two spin sublattices. The frequency is, of course, zero for any long-wavelength mode (Goldstone mode) that simply rotates the spin direction, since there is no restoring force. The expectation value of the spin-moment vector on each site of a sublattice can point in any direction, as long as the expectation value on each site of the other sublattice points in the opposite direction. Anderson estimated the time (proportional to the number of atoms present) for the spin direction to rotate by 90° as 3 years, much longer than the time scale of a neutron-diffraction experiment. The static spin-density wave that creates antiferromagnetism must be distinguished from the Goldstone mode of this paragraph.

Interestingly, the local spin moment on a transition-metal atom can persist even above the magnetic disordering temperature, both in ferromagnetic iron (30) and in transition-metal monoxides and other strongly correlated materials (where it can still give rise to an insulating gap in the Kohn–Sham density of states) (11–15, 31). Refs. 15, 18, and 31 use supercells to describe disordered local moments. Surprisingly, different symmetry-broken magnetic configurations very close in energy have been found in the Kondo topological insulator ${\mathrm{S}\mathrm{m}\mathrm{B}}_6$ (18), revealing spin fluctuations in the system.

Local spin-magnetic moments also appear in $\alpha$-Ce, where they are disordered experimentally (32), although they have been treated as ordered computationally (33, 34). The total energy of face-centered-cubic cerium exhibits two minima as a function of unit-cell volume and an electronic phase transition between them. The smaller-volume phase, $\alpha$-Ce, has little or no local magnetic moment, while the larger-volume phase, $\gamma$-Ce, has a substantial local magnetic moment on each atom (34). Symmetry-broken ground-state spin-density functional calculations confirm this picture (33).

Static Charge-Density Wave in Jellium

Now, consider a standard model for a simple metal, an infinite jellium in its ground state. The positive background charge is uniform and rigid, and there should always be a symmetry-unbroken ground-state electronic wavefunction with a uniform negative charge density to neutralize the positive density. But, at low density ($r_s\approx 69$, where the density is $n=3/\left[4\pi r_s^3\right]=k_F^3/\left(3{\pi }^2\right)$, and thus $r_s$ is the radius of a sphere containing, on average, one electron), where the jellium is greatly expanded from its equilibrium density ($r_s\cong 4$), an exchange-correlation kernel-corrected linear-response density functional calculation (35) finds a static charge-density wave. Approximate density functionals like LSDA also predict (36) a charge-density wave instability of the uniform electron density, but often at the wrong density. The charge-density wave is an incipient Wigner lattice, in the sense that it has a wavevector $q\approx 2.28k_F$ close to the first reciprocal lattice vector of a body-centered cubic (bcc) lattice, so that a bcc Wigner lattice can be constructed by superposing charge-density waves with wavevectors along the 12 (110) directions. At the density where the Wigner lattice first appears [$r_s\approx 85\pm 20$ from a quantum Monte Carlo calculation (37)], each electron is spread out over its whole Wigner–Seitz cell, but in the $r_s\to \infty$ limit, each electron further localizes to the center of its cell. The Wigner lattice is a strong correlation within the exact symmetry-unbroken ground-state wavefunction at low density and is justified by the theory of strictly correlated electrons (38).

The charge-density wave in jellium is a case in which we can computationally identify the collective mode or fluctuation and see how its excitation energy or frequency approaches zero as the external potential is varied. The charge-density wave appears (35) to arise from a soft plasmon: a collective density-wave excitation of the system that drops down in excitation energy or frequency and eventually freezes in the broken-symmetry density of an approximate density functional calculation. The argument for this in ref. 35 was based on extrapolation of the plasmon frequency ${\omega }_p\left(q\right)$ into the range of wavevectors $q$, where the plasmon frequency is mixed with the continuum of single particle-hole excitations, and the plasmon is no longer a true collective excitation. Figs. 2 and 3, however, show that this extrapolation is well justified. Fig. 2 shows the average frequency of a density fluctuation of wavevector $q$, and Fig. 3 shows the root-mean-square deviation from this average, both computed from the dynamic structure factor (23, 24) or spectral function (25) $S\left(q,\omega \right)$ of time-dependent DFT (25).

Fig. 2.

Charge-density wave in jellium as a soft plasmon. The frequency of a density fluctuation as a function of its wavevector $q=2\pi /\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}$ in a uniform electron gas at its equilibrium density ($n=3/\left[4\pi r_s^3\right]=k_F^3/\left(3{\pi }^2\right)$; $r_s=4$) and at a much lower density ($r_s=69$) where a static charge-density wave appears and breaks translational symmetry. At $r_s=4$, the frequency increases with $q$, but at $r_s=69$, the frequency softens, dropping to zero around $q=2.28k_F$, the critical wavevector of the static charge-density wave. The solid part of the $r_s=69$ curve was computed as in ref. 35, at wavevectors where the plasmon has not yet penetrated into the continuum of single particle-hole excitations. For $q/k_F\gtrsim 1.4$, the plasmon or collective excitation is replaced by a density fluctuation. The dashed parts of the curves were computed here at all wavevectors by using the frequency distribution [structure factor (23, 24) or spectral function (25, 39)] $S\left(q,\omega \right)=\left[-1/\left(\pi n\right)\right]\mathrm{I}\mathrm{m}\chi \left(q,\omega \right)>0$; $\chi$ is the density response function at full coupling strength defined in ref. 35. The dashes show the average frequency $⟨{\omega }_p\left(q\right)⟩={\int }_0^{\infty }\omega S\left(q,\omega \right)d\omega /{\int }_0^{\infty }S\left(q,\omega \right)d\omega$. The bulk plasma frequency is ${\omega }_p\left(0\right)={\left[4\pi n\right]}^{1/2}$. For plots of $S\left(q\right)={\int }_0^{\infty }S\left(q,\omega \right)d\omega$, refer to SI Appendix, Figs. S6 and S8.

Fig. 3.

An analogous plot to Fig. 2, presenting the standard deviation in the frequency of a density fluctuation, with the variance defined as ${⟨\mathrm{\Delta }{\omega }_p\left(q\right)⟩}^2={\int }_0^{\infty }{\omega }^{2}S\left(q,\omega \right)d\omega /{\int }_0^{\infty }S\left(q,\omega \right)d\omega -{⟨{\omega }_p\left(q\right)⟩}^2$, as a function of its wavevector $q$. At $r_s=4$ the variance is monotonic, but at $r_s=69$, the variance begins decreasing as $q$ approaches its critical value ($2.28k_F$) for the static charge-density wave. There is a rather sudden increase in the width of the spectral function as the plasmon enters the continuum of single particle-hole excitations (at $q/k_F>0.9$ and 1.4 for $r_s=4$ and 69, respectively) that is not reflected in the variance. The variance is controlled by a much higher and narrower central peak. See the contour plots of $S\left(q,\omega \right)$ (and the dielectric function) in SI Appendix.

Although $S\left(q,\omega \right)$ is often plotted for qualitative interpretation, we have not seen it used before as it is used in Figs. 2 and 3. The development (35) for jellium of a constraint-based exchange-correlation kernel $f_{\mathrm{x}\mathrm{c}}\left(q,\omega \right)$ motivated and enabled this study. Additional figures and formulas are presented in SI Appendix. Numeric values of the integrals displayed in Figs. 2 and 3 are presented in SI Appendix, with requisite parameters needed to reproduce them.

SI Appendix, Fig. S8 shows that the kernel of ref. 35 yields a static structure factor $S\left(q\right)$ for $r_s=4$ jellium in close agreement with that of a theoretically constrained fit (40) to quantum Monte Carlo data for $r_s\le 10$. SI Appendix, Fig. S9 shows that the spectral function for $r_s=4$ satisfies the third-frequency-moment sum rule (25, 41). But SI Appendix, Figs. S8 and S9 also show that the model kernel is less satisfactory for $r_s=69$. SI Appendix, Fig. S10 shows that the model kernel reasonably describes ground-state jellium correlation energies per particle at a wide range of densities, but not for $r_s\gtrsim 69$.

The kernel $f_{\mathrm{x}\mathrm{c}}\left(q,\omega \right)$ from ref. 35 satisfies many exact constraints on the static limit $f_{\mathrm{x}\mathrm{c}}\left(q,0\right)$ and the long-wavelength limit $f_{\mathrm{x}\mathrm{c}}\left(0,\omega \right)$. The way in which these two pieces are combined, equation 24 of ref. 35, employs a factor $e^{-k{q}^2}$ to damp out the frequency dependence at large $q$, but the transfer of $k$ and its $r_s$ dependence from $f_{\mathrm{x}\mathrm{c}}\left(q,0\right)$ is not rigorous. The data in SI Appendix suggest that a stronger damping is needed at $r_s=4$ and a weaker damping at $r_s=69$. An improvement will be explored in future work, but should not qualitatively change Figs. 2 and 3, because the static $f_{\mathrm{x}\mathrm{c}}\left(q,0\right)$ already leads (35) to a static charge-density wave at $r_s=69$ and $q/k_F\approx 2.2$.

Conclusions

The refinement of approximate density functionals toward the exact functional offers the possibility of suppressing symmetry breaking (42), but the result would not be an unmixed blessing, since qualitative insights into strong correlation could be lost.

We can now propose a definition of strong correlation: Strong correlation in a symmetry-constrained wavefunction is any correlation between electrons that results in an exceptionally structured electron-pair density, or is otherwise qualitatively different from the “normal” Coulomb correlation found in simple sp-bonded materials in their ground states near equilibrium nuclear geometries and having Hamiltonians of the form of Eq. 1. This definition has little in common with “everything that DFT gets wrong.” Indeed, DFT with standard approximate functionals (e.g., LSDA, Perdew–Burke–Ernzerhof [PBE], or SCAN) and symmetry-unbroken (uniform) densities perfectly captures the energetic consequences of strict correlation (38) in a jellium of very low density. Strong correlation by this definition is sometimes displayed by symmetry breaking in a symmetry-unconstrained wavefunction or density. DFT increasingly describes the ground-state energies and densities of systems with d or f electrons that are widely regarded as strongly correlated.

There is another valid interpretation (43) of spin symmetry breaking in spin-density functional theory: Approximate spin-density functionals reliably predict the total electron density, but predict the on-top electron-pair density ${\sum }_{\sigma ,{\sigma }^{\prime }}{\rho }_{\sigma {\sigma }^{\prime }}\left(\mathit{r},\mathit{r}\right)$ (yielding the probability to find two electrons together at the same position in space) more reliably than they predict the net spin density; such functionals predict both reliably when the correlation is not strong. This interpretation is now being used to merge the wavefunction and density functional approaches (44).

The jellium exchange-correlation kernel of ref. 35 is, like the PBE and SCAN exchange-correlation energy functionals, based upon the satisfaction of known exact constraints. Its accuracy and predictiveness suffice for this work and can be improved in future work.

This article has emphasized the perspectives of ground-state and time-dependent density functional, wavefunction, and model Hamiltonian theories, but will close by mentioning some other perspectives that are also important for understanding strong correlation in many-electron systems. The DFT+U (44, 45) method adds an on-site Coulomb repulsion, $U$, to a density functional approximation for exchange and correlation. The dynamical mean field theory (DMFT), specifically designed for strong correlation, treats a small fragment (e.g., an active metallic site) of a solid as an impurity whose frequency-dependent Green’s function is coupled to an extended environment (46, 47). DMFT has been used to study antiferromagnetism (48) and spin-density waves (49) within the context of the idealized Hubbard model. This and other Green’s function-based approaches (50, 51) can describe the quasiparticle excited-state spectrum.

These latter methods are more numerically intensive than ground-state DFT. However, they may be performed in conjunction with an approximate ground-state DFT calculation to achieve numerically efficient, and often highly accurate, descriptions of quasiparticle spectra in real materials. Established methods include, but are not limited to, LSDA+DMFT, as used to study ${\mathrm{C}\mathrm{r}\mathrm{O}}_2$ (52) and the volume-collapse transition in Ce (53); and DFT+GW (where “G” stands for Green’s function and “W” for the screened Coulomb interaction), as used for ZnO (54) and antiferromagnetic NiO and MnO (55).

The broad successes of density functional approaches, either on their own or in combination with excited-state methods, should indicate the need to collaborate and gather insights from all perspectives.

Data Availability.

All relevant data are available within the manuscript and SI Appendix. All raw data are publicly available at the Materials Cloud Archive (DOI: 10.24435/materialscloud:vh-wc) (56). The code used was written for this project and is publicly available in GitLab at https://gitlab.com/dhamil/mcp07-kernel-testing.