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Downloaded by guest on September 28, 2021 lws)eeg n lcrndniy(rsi este)o any of densities) form (or density N and ground-state energy the (lowest) for theory mean-field-like complex exact-in-principle an understand to helpful several be why parti- phenomena. suggests can many 1 of perspectives Fig. systems . different and mechanical molecules quantum including in cles, up show times T (DFT) theory functional density a jellium of kernel. frequency-dependent exchange-correlation frequency fre- and recent the a wavevector- the from from calculated constraint-based precisely, found function, charge-density spectral as the More zero, of static moments to quency . drops the fluctuation soft that density related a quantitatively is show a theory wave to of functional phase used density crystal is Time-dependent wave/Wigner jellium. charge-density low-density static discussed examples the The and H not. stretched does the include that here accurate) functional less exact (albeit an revealing more correlation than sense, be and can this exchange symmetry breaks In for that wavevector. functional density nonzero approximate spin-density at an as waves such charge-density excitations) or fluc- collective of which modes (sometimes soft densities, from tuations total arise can or They observable. densities sometimes are spin symmetry-broken the- as functional ground-state Vignale) density ory Giovanni approximate and symmetry-unbroken in Maitra up Neepa show a by can reviewed wavefunction within 2020; 24, correlations August review for Strong (sent 2020 23, November Perdew, P. John 70118 by LA Contributed Orleans, New University, Tulane Physics, Engineering and Physics of a Perdew P. John density theories and functional wavefunction in correlation strong and breaking symmetry ground-state of Interpretations PNAS Eq. of Hamiltonian antisymmetry the wavefunction to makes due (which another one avoid positions electron density two electron-pair an but all over grating n density the spin along one-electron a wavefunction yields the nates any prob- of positive of square a the over exchange yields Integrating the wavefunction distribution. a ability under Squaring sign electrons. two changing must antisymmetric, wavefunctions physical be the Since fermions, eigenvalue. the spin-1/2 lowest-energy are the electrons to for Hamiltonian attraction the of state Coulomb ground-state the explicitly A is repulsion included. as Coulomb electron–electron such the one). and nuclei, potential, be scalar to nal constants fundamental (taking eateto hsc,Tml nvriy hldlha A19122; PA Philadelphia, University, Temple Physics, of Department = (r) eeto ytmwt aitna reeg prtro the of operator energy or Hamiltonian a with system -electron h est ucinlter DT fKh n hm()is (1) Sham and Kohn of (DFT) theory functional density The ymtyadsrn orlto,cmlxte htsome- that complexities correlation, broken of strong interpretations and and symmetry examples presents article his 01Vl 1 o e2017850118 4 No. 118 Vol. 2021 N n − ↑ + (r) z H 1 ˆ xso pnqatzto)adttleeto density electron total and quantization) spin of axis lcrnpsto etr and vectors position electron = n ρ X i =1 N σσ ↓ a,b,1 (r) (r, " − htdfietesatn on o F.Inte- DFT. for point starting the define that 0 = r) din Ruzsinszky Adrienn , 1 2 1 ∇ ρ osntdpn xlctyo pn the spin, on explicitly depend not does σσ i 2 N n olm euso.Although repulsion. Coulomb and ) | + 0 2 ymtybreaking symmetry eeto aeucini neigen- an is wavefunction -electron (r, oeue niermgei solids, antiferromagnetic molecule, v (r r 0 i ) + ) htsosepiil o the how explicitly shows that X i

PHYSICS 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 possibil- ity 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 . A wave in the total elec- tron density n(r) = n↑(r) + n↓(r) is a plasma wave (quantized as ), and a wave in the net electron spin density m(r) = n↑(r) − n↓(r) is a spin-density wave. Like other waves, these have amplitude, wavelength λ or wavevector q = 2π/λ, and frequency ω, and at small amplitudes can be simply superposed or added together. However, their frequencies are not sharply defined. Fig. 1. The parable of the blind men and the elephant suggests that By the uncertainty principle, a smaller frequency width leads to scientists who study the same complex problem from different perspec- a longer lifetime. Collective excitation modes are distinguished tives should pool their insights. This article touches four perspectives on from other fluctuation modes by much smaller frequency widths. broken symmetry and strong correlation in many-electron systems: ground- The second interpretation explains how some symmetry break- state and time-dependent density functional, wavefunction, and model ings and strong correlations can arise: Under a variation of the Hamiltonian. The DFT+U, dynamical mean field, and Green’s functions external potential, a collective excitation or fluctuation of the approaches are other important perspectives that are mentioned briefly in electrons of nonzero wavevector, such as a charge-density wave Conclusions. Image credit: Liyu Ye (artist). 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 approx- exchange-correlation potential in the one-electron Schrodinger¨ imate functional. In small systems, where symmetry breaking equation that shapes the orbitals and the spin densities in a is unobservable, the exact frequency might only approach, but self-consistent calculation. not reach, zero. Since this frequency is nonnegative, the fre- “Strong correlation” is sometimes used to mean “everything quency width must go to zero when the frequency does. This that DFT gets wrong.” Yet, hybrid functionals (including part of mechanism is analogous to the softening of a mode of the Hartree–Fock exchange) like HSE06 (8–10) (for nonmetallic nonzero wavevector that can lead to a distortion or structural states) and meta-generalized gradient approximations (meta- of an ionic lattice. The ground-state energy GGAs) like the strongly constrained and appropriately normed of a quantum lattice nearly equals that of the classical lattice (SCAN) functional (4, 11–18) are yielding quantitatively correct plus the zero-point energy of its lattice vibrational modes. Via ground-state (and we emphasize “ground-state”) results by sym- the fluctuation-dissipation theorem (23, 24), the ground-state metry breaking for some systems that have long been regarded total energy of a system of interacting electrons, including the as strongly correlated. In the cuprate high-temperature super- Coulomb correlation energy, has contributions from fluctuations conducting materials, for example, the SCAN meta-GGA (4) is of various wavevectors, including the nonnegative zero-point able to do what simpler density functionals (local spin density energies ~ω/2 of its collective excitations. Collective excitations, approximation [LSDA] and generalized gradient approximation like observable or physical symmetry breakings, are emergent [GGA]) cannot (13), by creating the correct spin moments on phenomena, arising only in systems with large numbers of par- the copper atoms, their antiferromagnetic order, and a correctly ticles (25). Fluctuations, even when they are not true collective nonzero band gap in the undoped material that correctly dis- appears under the doping that also leads to (12–14). Full, but better-designed, self-interaction corrections Table 1. Mean absolute errors (MAEs) in electronvolts for the (5) might eventually further improve approximate DFT for other atomization energies of the six representative AE6 (6) sp-bonded strongly correlated systems. And, as will be argued here, even molecules, for Hartree–Fock exchange (x) and for the simplest DFT approximations yield a qualitative insight into exchange-correlation (xc) functionals on the first three rungs some such systems. of a ladder of approximations (none of them fitted to The first interpretation to be expressed here is that certain bonded systems) strong correlations that are present as fluctuations in the exact symmetry-unbroken ground-state wavefunction are “frozen” in Approximation AE6 MAE, eV symmetry-broken electron densities or spin densities of approx- Hartree–Fock x 6.3 imate DFT. For finite systems, this would not happen with the LSDA xc 3.3 exact functional. So, while the exact functional would always be PBE GGA xc 0.6 exact for the ground-state energy and density, it would not always SCAN meta-GGA xc 0.1 be as qualitatively revealing as the approximate one. This first interpretation (but without the term “strong corre- Hartree–Fock results are from Lynch and Truhlar (6). LSDA (1, 3), PBE GGA lations”) is eloquently expressed in P. W. Anderson’s famous (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 essay “More Is Different” (19), which explains that, while a for xc together than for x or c separately, due to an understood cancellation ground-state wavefunction is necessarily static, it must describe between the full nonlocalities of exact x and exact c. The LSDA exchange- fluctuations in the expectation values of operators that are not correlation energy density depends only on the local spin densities; the diagonalized along with the Hamiltonian, and that these fluctua- GGA further includes the density gradients; and the meta-GGA still further tions can freeze as the number of particles in the system grows includes the orbital kinetic energy densities.

2 of 6 | PNAS Perdew et al. https://doi.org/10.1073/pnas.2017850118 Interpretations of ground-state symmetry breaking and strong correlation in wavefunction and density functional theories Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 nepeain fgon-tt ymtybekn n togcreaini aeucinand wavefunction in correlation theories strong functional and density breaking symmetry ground-state of Interpretations spin al. with et other Perdew the and up spin with one atoms, H separate den- infinity, stretched two spin to tends of the length of energy bond symmetry the the As within the calculation. of LSDA correlation is the breaking in which strong the sity spin, a by total us is to zero This revealed of other. wavefunction the symmetry-unbroken near a highly be is direction near to spin is opposite likely direction the spin with electron given an a nucleus, with one electron an when length, bond toward tends length bond the fre- a as with zero infinity. interchanged, toward are drops nuclei two that the two quency on fluctuation of spins a the energy mentioned which also LSDA an in 26 the to Ref. than length atoms. hydrogen higher bond individual infinite much of energy limit total the LSDA in spin- led prevented allowed breaking that was constraint symmetry A density stretching. spin bond the under break of to symmetry the when of only discussed kind realistic be (a to other the antiferromagnetism in near the down of and precursor center molecular nuclear one spin they near net length, the up with bond critical breaking, spin-symmetry critical a unobservable the an than found Above less curve. binding-energy lengths istic bond (26) For (1.7 ground-state Lundqvist lengths. value the and bond for Gunnarsson various calculation 1976, at LSDA In an lengths. made bond spin-unpolarized all remain must at density electron exact whose like state molecule closed-shell A Molecule the Hydrogen in Stretched equilibrium it. thermal have exter- of should the still state of limit symmetry the zero-temperature state the course, although ground have of potential, not single density, need nal a current set and When degenerate net degenerate, potential. a a be from external or must state moment the ground spin of the net symmetries a the expected is is of there density all ground-state exact have the zero to and be zero, must be must spin sity total the non- of and (S or square nonferromagnetic atom the finite closed-shell crystal, a a ferrimagnetic in in presumably as or nondegenerate, molecule, is include state can ground observables the commuting and of square set the complete the Then, (20). state” the ground of the symmetry of “The symmetry the potential: is external Hamiltonian the have will of (equi- densities symmetries spin average the or weighted densities ground-state equally all an of ensemble) degenerate, is is state atom ground Ne symme- the the spherical of of the try the has state density electron ground nondegenerate, its exter- the and the is nondegenerate, example, of state symmetries For the are potential. ground of Hamiltonian nal all the have the will If of density ground-state states. eigenvalue lowest ground the the Hamiltonian. with the states including observables, The commuting of set plete the for simplicity. here and ignored clarity be nuclei of will from effects sake comes It physical potential These exact. external move. can the not that that that is fact the note point, and neglects also interaction, it starting spin–orbit We the including good discussion. effects, the relativistic a omits of while subject space, Hamiltonian, the of this not homogeneity is the that violates but which potential, external its Eq. of Hamiltonian The Mechanics Quantum in Symmetry (25). or DFT (24) time-dependent factor of structure function dynamic spectral the by described are excitations, h xc aeucintlsu ht tasrthd u finite, but stretched, a at that, us tells wavefunction exact The Eq. of Hamiltonian The com- a of eigenstates the find to possible is it that know We .Tewoebnigeeg uv was curve binding-energy whole The Solids). Antiferromagnetic 0 = rsi ige tt) n,tu,telclntsi den- spin net local the thus, and, state), singlet spin or v ) hyfudasi-noaie est n real- a and density spin-unpolarized a found they A), (r ˚ = ) z −10/r cmoeto h oa lcrnsi.When spin. electron total the of -component olm trcint h ulu.I the If nucleus. the to attraction Coulomb H 1 2 a ra pta ymtisthrough symmetries spatial break can eoe dnia oteeeg of energy the to identical becomes 1 H sidpneto lcrnspin. electron of independent is 2 a odgnrt ground nondegenerate a has H 2 molecule rmapoiaefntoasaeahee yspin-symmetry by achieved for only are not limit breaking functionals curves the approximate binding-energy just improved is from Greatly length atoms. bond atomic independent infinite of the of spin-polarized intuition, limit the human correlated of out strongly viewpoint (classical) select the From and can densities. field symmetry magnetic the staggered infinitesimal, break an even Then, down. ooie n te togycreae aeil weeit (where materials transition-metal correlated in strongly and other (30) temper- iron and disordering ferromagnetic monoxides magnetic in the both above ature, even persist can atom paragraph. this The of experiment. mode Goldstone neutron-diffraction be the must from a antiferromagnetism distinguished creates of that wave 90 scale spin-density static by time present) rotate the atoms to of than direction the number spin any of the the to in site esti- for (proportional Anderson each point direction. time on can opposite the value the mated sublattice in expectation points a spin- the sublattice the of as other of long site value as each expectation direction, on The since vector force. direction, restoring spin moment no the is mode rotates sublattices. long-wavelength simply there spin any two that for the mode) zero of course, (Goldstone first of the is, on frequency direction The spin the the to to perpendicular i.e., Here, are excitations. directions the spin-wave transverse transverse included energy the the then the of state, from energy ground started zero-point antiferromagnetic He classical lattice. a cubic spin of simple localized Hamilto- a between on model interaction moments effective Heisenberg the antiferromagnetic describing nian an for energy energy kinetic electron low- and of density electron solution decrease. the symmetries as the and break stronger, as to become tend not correlations but Many energy. solution equations, est a Kohn–Sham as remains the symmetry of unbroken metallic, of the DFT, state mate- Kohn–Sham nonmagnetic the approximate makes In that . gap energy an unit an rial the opening of and space size one real the first in of doubling cell center the typically the faces, of to Brillouin-zone lattice center the ionic of the underlying an the from of create extend zone Brillouin To must the of wave soften. wavevector spin-density will nonzero soft wave the relaxed, state, spin-density is antiferromagnetic a compression ordered gap that this energy As imagine zero moment. with we magnetic metal local nonmagnetic no a and be lattice, broken-symmetry will given the a material of create any compression extreme to Under density. frequency con- spin lattice and static of energy function a is in (as interpretation down stant) drops Our wave (28). spin-density a Overhauser that by fixed. proposed not was is atom netism given a on although moment spin atoms, the nearest-neighbor of direction has the the atom given on a on neighbors moment spin-up spin-down corre- a are example, are for that that, there atoms exact so transition-metal lated, that the an on us moments within spin tell local correlations might wavefunction Strong symmetry-unbroken functional. approx- the density exact of densities imate the spin within ground-state symmetry-broken correlation into the freezes strong that atomic a wavefunction alternating ground-state presumably symmetry-unbroken materi- on antiferromag- is moments many are This spin that sites. net oxides, should alternating find transition-metal with moment 17, netic, especially spin and and net 11–15 functional als, refs. zero spin-density e.g., with approximate calculations, But system den- spin-unpolarized. finite ground-state be exact any the of that sity suggests symmetry Similarly, Solids Antiferromagnetic dissociation incorrect fragments. charged an fractionally (27) to prevents even breaking symmetry neetnl,telclsi oeto transition-metal a on moment spin local the Interestingly, ground-state the of estimate accurate an found (29) Anderson antiferromag- and waves spin-density between connection A H 2 u lofrLH(7.FrLH h spin- the LiH, For (27). 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PHYSICS 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 ≈ 2.28kF close to the first reciprocal lattice vector of a body-centered cubic (bcc) lattice, so that a bcc Wigner lattice can be constructed by superpos- ing charge-density waves with wavevectors along the 12 (110) directions. At the density where the Wigner lattice first appears [rs ≈ 85 ± 20 from a quantum Monte Carlo calculation (37)], each electron is spread out over its whole Wigner–Seitz cell, but in the rs → ∞ 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 exci- Fig. 2. Charge-density wave in jellium as a soft plasmon. The frequency of tation of the system that drops down in excitation energy or a density fluctuation as a function of its wavevector q = 2π/wavelength in frequency and eventually freezes in the broken-symmetry density 3 3 2 a uniform electron gas at its equilibrium density (n = 3/[4πrs ] = kF /(3π ); of an approximate density functional calculation. The argument rs = 4) and at a much lower density (rs = 69) where a static charge-density for this in ref. 35 was based on extrapolation of the plasmon fre- wave appears and breaks translational symmetry. At rs = 4, the frequency quency ωp (q) into the range of wavevectors q, where the plasmon increases with q, but at rs = 69, the frequency softens, dropping to zero frequency is mixed with the continuum of single particle-hole around q = 2.28kF , the critical wavevector of the static charge-density excitations, and the plasmon is no longer a true collective exci- = wave. The solid part of the rs 69 curve was computed as in ref. 35, tation. Figs. 2 and 3, however, show that this extrapolation is well at wavevectors where the plasmon has not yet penetrated into the con- justified. Fig. 2 shows the average frequency of a density fluctu- tinuum of single particle-hole excitations. For q/kF & 1.4, the plasmon or collective excitation is replaced by a density fluctuation. The dashed parts ation of wavevector q, and Fig. 3 shows the root-mean-square of the curves were computed here at all wavevectors by using the fre- deviation from this average, both computed from the dynamic quency distribution [structure factor (23, 24) or spectral function (25, 39)] structure factor (23, 24) or spectral function (25) S(q, ω) of S(q, ω) = [−1/(πn)]Imχ(q, ω) > 0; χ is the density response function at full time-dependent DFT (25). coupling strength defined in ref. 35. The dashes show the average fre- Although S(q, ω) is often plotted for qualitative interpreta- quency hω (q)i = R ∞ ωS(q, ω)dω/ R ∞ S(q, ω)dω. The bulk plasma frequency p 0 0 tion, we have not seen it used before as it is used in Figs. 2 is ω (0) = [4π ]1/2. For plots of ( ) = R ∞ ( , ω) ω, refer to , p n S q 0 S q d SI Appendix and 3. The development (35) for jellium of a constraint-based Figs. S6 and S8. exchange-correlation kernel fxc(q, ω) motivated and enabled

can still give rise to an insulating gap in the Kohn–Sham den- sity 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 SmB6 (18), revealing spin fluctuations in the system. Local spin-magnetic moments also appear in α-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, α-Ce, has little or no local mag- netic moment, while the larger-volume phase, γ-Ce, has a sub- stantial 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 infi- nite jellium in its ground state. The positive background charge is uniform and rigid, and there should always be a symmetry- Fig. 3. An analogous plot to Fig. 2, presenting the standard deviation unbroken ground-state electronic wavefunction with a uniform in the frequency of a density fluctuation, with the variance defined as h∆ω ( )i2 = R ∞ ω2 ( , ω) ω/ R ∞ ( , ω) ω − hω ( )i2, as a function of its negative charge density to neutralize the positive density. But, p q 0 S q d 0 S q d p q 3 wavevector q. At rs = 4 the variance is monotonic, but at rs = 69, the vari- at low density (rs ≈ 69, where the density is n = 3/[4πrs ] = 3 2 ance begins decreasing as q approaches its critical value (2.28kF ) for the k /(3π ), and thus rs is the radius of a sphere containing, on F static charge-density wave. There is a rather sudden increase in the width average, one electron), where the jellium is greatly expanded of the spectral function as the plasmon enters the continuum of single from its equilibrium density (rs =∼ 4), an exchange-correlation particle-hole excitations (at q/kF > 0.9 and 1.4 for rs = 4 and 69, respectively) kernel-corrected linear-response density functional calculation that is not reflected in the variance. The variance is controlled by a much (35) finds a static charge-density wave. Approximate density higher and narrower central peak. See the contour plots of S(q, ω) (and the functionals like LSDA also predict (36) a charge-density wave dielectric function) in SI Appendix.

4 of 6 | PNAS Perdew et al. https://doi.org/10.1073/pnas.2017850118 Interpretations of ground-state symmetry breaking and strong correlation in wavefunction and density functional theories Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 h oe enli esstsatr for satisfactory S10 less is kernel model But the 41). (25, rule factor structure static them. in reproduce 2 in to presented Figs. presented needed in are are displayed integrals 3 formulas the and of and values Numeric figures Appendix. SI Additional study. this rlfnto for function tral Monte quantum for to data (40) fit Carlo constrained theoretically a of that with nepeain fgon-tt ymtybekn n togcreaini aeucinand wavefunction in correlation theories strong functional and density breaking symmetry ground-state of Interpretations al. et Perdew iy u rdc h ntpeeto-ardensity electron-pair den- Approximate on-top electron the total theory: predict the but functional predict sity, reliably spin-density functionals in spin-density breaking try sys- of densities this correlated. and by energies a with ground-state in correlation tems the breaking Strong describes increas- symmetry DFT density. ingly density. by or low displayed wavefunction very symmetry-unconstrained sometimes (38) of is correlation jellium definition strict of a consequences in energetic approxi- the perfectly standard densities captures (uniform) with symmetry-unbroken and DFT SCAN) [PBE], or Perdew–Burke–Ernzerhof Indeed, LSDA, of (e.g., wrong.” functionals form mate gets the DFT equilibrium of that near Hamiltonians states having simple Eq. ground in and found their geometries correlation nuclear in Coulomb materials differ- “normal” qualitatively sp-bonded the otherwise from is or ent struc- exceptionally density, an electron-pair in cor- tured results any that is electrons between wavefunction relation symmetry-constrained a in correlation lost. blessing, be unmixed could an correlation be strong into not symmetry insights would qualitative suppressing result since the of but possibility the (42), the toward breaking functionals offers density functional approximate exact of refinement The Conclusions at static change wave the qualitatively charge-density because not static 3, should and but 2 work, Figs. future in explored be r in factor large a at employs dependence 35, frequency ref. the of out 24 equation bined, limit static limit the on straints of range wide a for at not particle but per densities, energies correlation jellium state n its and .J .Pre,K uk,M rzro,Gnrlzdgain prxmto made approximation gradient Generalized thermochemical Ernzerhof, M. for Burke, benchmarks K. representative Perdew, Small P. J. Truhlar, 7. G. D. Lynch, J. B. 6. normed appropriately and constrained Bhattarai P. Strongly Perdew, 5. P. J. Ruzsinszky, A. Sun, J. electron-gas the 4. of representation analytic simple and Accurate Wang, Y. Perdew, P. J. 3. Ostlund, S. correlation. N. and Szabo, exchange A. including equations 2. Self-consistent Sham, J. L. Kohn, W. 1. s i.S8 Fig. Appendix, SI hr saohrvlditrrtto 4)o pnsymme- spin of (43) interpretation valid another is There Strong correlation: strong of definition a propose now can We h kernel The 4 = simple. approximation. calculations. of levels three at Phys. consistency Chem. Gauge J. II. correction. self-interaction functional. density semilocal energy. correlation 1982). Rev. Phys. IAppendix SI hw httemdlkre esnbydsrbsground- describes reasonably kernel model the that shows hsdfiiinhsltl ncmo ih“everything with common in little has definition This 1. f xc n ekrdmigat damping weaker a and r hs e.Lett. Rev. Phys. (0, s 13–13 (1965). A1133–A1138 140, d eednefrom dependence tpi h ieto frsligteprdxo Perdew-Zunger of paradox the resolving of direction the in step A al., et ω .Py.Ce.A Chem. Phys. J. or 119(2020). 214109 152, h a nwihteetopee r com- are pieces two these which in way The ). f r xc f s r ugs htasrne apn snee at needed is damping stronger a that suggest hs e.B Rev. Phys. ≤ s lcrn htaewdl eadda strongly as regarded widely are that electrons (q 4 = is 8adS9 and S8 Figs. Appendix, SI 8536 (1996). 3865–3868 77, 10. , ω r oenQatmChemistry Quantum Modern S s ) aifistetidfeunymmn sum third-frequency-moment the satisfies i.S9 Fig. Appendix, SI hs e.Lett. Rev. Phys. hw httekre frf 5yed a yields 35 ref. of kernel the that shows & (q 9689 (2003). 8996–8999 107, ihrqiieparameters requisite with Appendix, SI rmrf 5stse ayeatcon- exact many satisfies 35 ref. from 34–34 (1992). 13244–13249 45, 69. ) for f r f xc s xc r 69 = (q (q s f xc 4 = r 0) , 0) , s 342(2015). 036402 115, (q nipoeetwill improvement An 69. = and 0) , elu ncoeagreement close in jellium n h long-wavelength the and sntrgru.Tedata The rigorous. not is r s q q 69. = led ed 3)t a to (35) leads already u h rnfrof transfer the but , /k Mciln e ok NY, York, New (Macmillan, hw httespec- the that shows F ≈ Fig. Appendix, SI 2. e P loso that show also −kq σ ,σ 2 0 ρ odamp to σσ 0 (r, r) k 3 .Pokharel K. 13. in corrections self-interaction and Waals Furness der W. J. van of 12. Synergy Perdew, P. theory J. functional Peng, Density H. Scuseria, 11. E. G. Henderson, M. T. Martin, L. R. Wen, X.-D. 10. e o hspoetadi ulcyaalbei iLbat GitLab in available publicly is and project this for ten rhv (DOI: Archive Appendix SI Availability. methods, Data excited-state from insights with gather and combination perspectives. collaborate all to in need the or indicate should own their on and (55). (54) MnO ZnO and for NiO used antiferromagnetic as interaction), “W” Coulomb and screened function to the Green’s for for used stands “G” as (where LSDA+DMFT, DFT+GW and to, limited not study are but meth- Established include, materials. ods real in descrip- spectra accurate, to con- of highly tions calculation often in and DFT performed efficient, ground-state numerically be achieve approximate an may with they junction However, DFT. ground-state ideal- function-based excited-state quasiparticle the Green’s the of other describe spectrum. can context and 51) (48) the This (50, antiferromagnetism approaches model. within study Hubbard (49) to (46, ized waves used environment spin-density frequency-dependent extended been whose and an has impurity to DMFT an coupled 47). as is solid function a metal- Green’s active of an (e.g., site) fragment for lic The small designed a specifically correlation. treats correlation, (DMFT), strong and theory field exchange mean repulsion, dynamical for Coulomb approximation DFT+U on-site functional The an strong systems. adds understanding method many-electron for important other in some also correlation mentioning are by that close perspectives will but theories, model Hamiltonian and wavefunction, functional, density time-dependent and future in improved be can work. and work and this accuracy for Its suffice constraints. predictiveness exact known based of satisfaction functionals, the energy upon exchange-correlation SCAN and to PBE used (44). being approaches now functional is density and interpretation wavefunction This the merge cor- strong. net the not the when the predict reliably is both at they relation predict than together functionals reliably such electrons density; more spin two space) in find position to same probability the (yielding rnirRsac etrfrCmlxMtrasfo is rnilsGrant Energy Principles the First through from University. Materials Sciences, Temple and Complex Energy DE-SC0012575; for Basic supported Center Energy, was Research of of A.D.K. Frontier Office Department DE-SC0019350. Energy, Grant the of Sciences by Department Grant Energy the NSF Basic by by supported Science, supported were was N.K.N. J.S. DMR- and DMR-1553022. A.R. Compu- Grant Chemistry. and NSF of Modeling, Theory, Division by Chemical tation, from supported contribution was a J.P.P. with correlation. 1939528, strong of discussions ACKNOWLEDGMENTS. dhamil/mcp07-kernel-testing. .P .Hy .L atn .Udn .E csra hoeia td fCeO of a study on Theoretical Scuseria, based E. G. functionals Uddin, J. “Hybrid Martin, Erratum: L. R. Ernzerhof, Hay, J. P. M. 9. Scuseria, E. G. Heyd, J. 8. h ra ucse fdniyfntoa prahs either approaches, functional density of successes broad The than intensive numerically more are methods latter These ground-state of perspectives the emphasized has article This the like is, 35 ref. of kernel exchange-correlation jellium The eprtr urt uecnutr:Acmaaiedniyfntoa study. functional density comparative 2020). A April (17 arXiv:2004.08047 superconductors: cuprate temperature superconductors. (2018). cuprate high-temperature of structure tronic monoxides. metal transition oxides. actinide (2013). state solid 1096 of structure electronic the of studies sn cendhbi est functional. density hybrid screened a using potential”. Coulomb screened CrO 2 l a aaaepbil vial tteMtrasCloud Materials the at available publicly are data raw All . 5) h oeue a writ- was used code The (56). 10.24435/materialscloud:vh-wc) nacrt rtpicpe ramn fdpn-eedn elec- doping-dependent of treatment first-principles accurate An al., et 5)adtevlm-olpetasto nC (53); Ce in transition volume-collapse the and (52) tal. et l eeatdt r vial ihntemnsrp and manuscript the within available are data relevant All biii ecito fteeetoi tutr fhigh- of structure electronic the of description initio Ab , ...adJS hn lxZne o stimulating for Zunger Alex thank J.S. and J.P.P. hs e.B Rev. Phys. .Ce.Phys. Chem. J. https://doi.org/10.1073/pnas.2017850118 011R (2017). 100101(R) 96, .Ce.Phys. Chem. J. 196(2006). 219906 124, 372(2006). 034712 125, U hm Rev. Chem. https://gitlab.com/ omn Phys. Commun. oadensity a to , PNAS 2 4,45) (44, n Ce and 1063– 113, | f6 of 5 11 1, 2 O 3

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