Luca de’ Medici ESPCI - Paris
Winter School on Magnetism Vienna TU, February 20-24th 2017 Band Theory of electrons in a solid
2 2 2 2 ~ Z↵e 1 e Hˆ = ri + 2me R↵ ri 2 ri ri i i↵ i=i 0 X X | | X6 0 | |
2 2 ~ i Independent electron hˆ(ri) r + veff (ri) approximation ⌘ 2me
one-particle ˆ Schrödinger equation h(r) a(r)=✏a a(r)
In a translationally invariant system (crystal) ik r (r) (r)=e · u (r) Bloch Functions a ⌘ kn nk Many-body wave function
Band theory: factorized many-body wave function
(x1, x2)= a(x1) b(x2) E = ✏a + ✏b
Antisymmetrization 1 (x1, x2)= [ a(x1) b(x2) b(x1) a(x2)] p2
Slater determinant a1 (x1) a1 (x2) ... a1 (xN ) (x ) (x ) ... (x ) 1 a2 1 a2 2 a2 N (x1, x2,...,xN )= . . . . pN! . . . . (x ) (x ) ... (x ) aN 1 aN 2 a3 N E = ✏ai i X Band “filling”, excitations, DOS
DOS 10 12 14 16 18 20 0 2 4 6 8
2 -2 -1.5 1.5 band “filling” 1 -1 density of -0.5 0.5 states 0 0
0.5 (DOS) energy -0.5 1 -1
1.5 N(✏) -1.5 2 -2 -3 -2 -1 0 1 2 3
kx
2 paramagnetic susceptibility = µBN(✏F )
⇡2k2 specific heat “Sommerfeld” coefficient = B N(✏ ) 3 F C T V ⇠ Factorized wave function: a two-site example
a(x)=caL L(x)+caR R(x)
L R b(x)=cbL L(x)+cbR R(x)
There is no way, by changing the one-electron wave functions
( i.e. by changing caL, caR, cbL and cbR) to:
disfavour these contributions
a(x1) b(x2)=caLcbL L(x1) L(x2)+caRcbR R(x1) R(x2)
+caLcbR L(x1) R(x2)+caRcbL R(x1) L(x2))
favour these contributions
Correlated Materials
Band Fermi-liquid, enhanced masses, Mott theory instabilities, local magnetism,… insulator increasing electronic correlationsCorrelated Materials ...... typically contain partially filled d- or f-shells Unconventional phenomena: WebElements: the periodic table on the world-wide web - High-Tc superconductivity http://www.shef.ac.uk/chemistry/web-elements/
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 hydrogen helium - large thermoelectric responses 1 2 H He 1.00794(7) Key: 4.002602(2) lithium beryllium element name boron carbon nitrogen oxygen fluorine neon - colossal magnetoresistance, … 34 atomic number 5 6 7 8 9 10 Li Be element symbol B C N O F Ne 6.941(2) 9.012182(3) 1995 atomic weight (mean relative mass) 10.811(7) 12.0107(8) 14.00674(7) 15.9994(3) 18.9984032(5) 20.1797(6) sodium magnesium aluminium silicon phosphorus sulfur chlorine argon 11 12 13 14 15 16 17 18 Na Mg Al Si P S Cl Ar 22.989770(2) 24.3050(6) 26.981538(2) 28.0855(3) 30.973761(2) 32.066(6) 35.4527(9) 39.948(1) potassium calcium scandium titanium vanadium chromium manganese iron cobalt nickel copper zinc gallium germanium arsenic selenium bromine krypton 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr 39.0983(1) 40.078(4) 44.955910(8) 47.867(1) 50.9415(1) 51.9961(6) 54.938049(9) 55.845(2) 58.933200(9) 58.6934(2) 63.546(3) 65.39(2) 69.723(1) 72.61(2) 74.92160(2) 78.96(3) 79.904(1) 83.80(1) rubidium strontium yttrium zirconium niobium molybdenum technetium ruthenium rhodium palladium silver cadmium indium tin antimony tellurium iodine xenon 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Materials typically from 3d and 85.4678(3) 87.62(1) 88.90585(2) 91.224(2) 92.90638(2) 95.94(1) [98.9063] 101.07(2) 102.90550(2) 106.42(1) 107.8682(2) 112.411(8) 114.818(3) 118.710(7) 121.760(1) 127.60(3) 126.90447(3) 131.29(2) caesium barium lutetium hafnium tantalum tungsten rhenium osmium iridium platinum gold mercury thallium lead bismuth polonium astatine radon 55 56 57-70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 Cs Ba * Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn 4f open shells: Cuprates, Fe- 132.90545(2) 137.327(7) 174.967(1) 178.49(2) 180.9479(1) 183.84(1) 186.207(1) 190.23(3) 192.217(3) 195.078(2) 196.96655(2) 200.59(2) 204.3833(2) 207.2(1) 208.98038(2) [208.9824] [209.9871] [222.0176] francium radium lawrencium rutherfordium dubnium seaborgium bohrium hassium meitnerium ununnilium unununium ununbium 87 88 89-102 103 104 105 106 107 108 109 110 111 112 Fr Ra ** Lr Rf Db Sg Bh Hs Mt Uun Uuu Uub based superconductors, [223.0197] [226.0254] [262.110] [261.1089] [262.1144] [263.1186] [264.12] [265.1306] [268] [269] [272] [277]
lanthanum cerium praseodymium neodymium promethium samarium europium gadolinium terbium dysprosium holmium erbium thulium ytterbium Manganites, heavy-fermions, 57 58 59 60 61 62 63 64 65 66 67 68 69 70 *lanthanides La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb 138.9055(2) 140.116(1) 140.90765(2) 144.24(3) [144.9127] 150.36(3) 151.964(1) 157.25(3) 158.92534(2) 162.50(3) 164.93032(2) 167.26(3) 168.93421(2) 173.04(3) actinium thorium protactinium uranium neptunium plutonium americium curium berkelium californium einsteinium fermium mendelevium nobelium … 89 90 91 92 93 94 95 96 97 98 99 100 101 102 **actinides Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No [227.0277] 232.0381(1) 231.03588(2) 238.0289(1) [237.0482] [244.0642] [243.0614] [247.0703] [247.0703] [251.0796] [252.0830] [257.0951] [258.0984] [259.1011]
Symbols and names: the symbols of the elements, their names, and their spellings are those recommended by IUPAC. After some controversy, the names of elements 101-109 are now confirmed: see Pure & Appl. Chem., 1997, 69, 2471Ð2473. Names have not been proposed as yet for the most recently discovered elements 110Ð112 so those used here are IUPACÕs temporary systematic names: see Pure & Appl. Chem., 1979, 51, 381Ð384. In the USA and some other countries, the spellings aluminum and cesium are normal while in the UK and elsewhere the usual spelling is sulphur. Periodic table organisation: for a justification of the positions of the elements La, Ac, Lu, and Lr in the WebElements periodic table see W.B. Jensen, ÒThe positions of lanthanum (actinium) and lutetium (lawrencium) in the per iodic tableÓ, J. Chem. Ed., 1982, 59, 634Ð636. Group labels: the numeric system (1Ð18) used here is the current IUPAC convention. For a discussion of this and other common systems see: W.C. Fernelius and W.H. Powell, ÒConfusion in the periodic table of the elem entsÓ, J. Chem. Ed., 1982, 59, 504Ð508. Atomic weights (mean relative masses): see Pure & Appl. Chem., 1996, 68, 2339Ð2359. These are the IUPAC 1995 values. Elements for which the atomic weight is contained within square brackets have no stable nuclides and are represented by one of the elementÕs more importan t isotopes. However, the three elements thorium, protactinium, and uranium do have characteristic terrestrial abundances and these are the values quoted. The last significant figure of each value is considered reliable to ±1 except where a larger uncertainty is given in parentheses. ©1998 Dr Mark J Winter [University of Sheffield, [email protected]]. For updates to this table see http://www.shef.ac.uk/chemistry/web-elements/pdf/periodic-table.html. Version date: 1 March 1998.
→ transition metal oxides/sulfides, rare earth or actinide compounds (but also: low-dimensional systems, organics ...)
–p.33 Mott transition
Localization of conduction electrons by correlations
Relevant for transition metal oxides, Fullerenes, organic superconductors,…
V2O3 The proximity to a Mott state strongly affects the properties of a system:
• reduced metallicity (Fermi-liquid: quasiparticle effective mass) • transfer of spectral weight from low to high energy (e.g. in optical response) • tendency towards magnetism • … Limelette et al. Science 302, 89 (2003)
Fermi liquid theory
quasiparticles instead of particles
1 2 2 = a(✏ ✏ ) + bT infinite lifetime at T=0 at the Fermi level ⌧ F
✏ ✏⇤ renormalized dispersion nk ! nk
N(✏ ) N ⇤(✏ ) renormalized DOS F ! F 1 ~k vn⇤k = k✏n⇤k vn⇤k = effective mass ~r | | m⇤
2 N ⇤(✏F ) = µB s paramagnetic susceptibility 1+FA
2 2 ⇡ kB = N ⇤(✏ ) Sommerfeld coefficient 3 F Hubbard model 2 2 Z e2 ˆ ~ ↵ H0 = dr † (r) r + V (r) (r) V (r)= 2m R↵ r e ↵ X Z X | | 1 e2 Hˆint = drdr0 † (r) † (r0) (r0) (r) 0 2 r r 0 0 X X 0 Z | |
† (r)= w⇤ (r R )d† n i in in X
mm0 1 mm0nn0 Hˆ = t d† djm + V d† d† dkn dln ij im 0 2 ijkl im jm0 0 0 ijmmX0 Xijkl mmX0nn0 X 0 2 2 mm0 ~ t = dr w⇤ (r R ) r + V (r) w (r R ) ij m i 2m m0 j Z e 2 mm0nn0 e V = drdr0w⇤ (r Ri)w⇤ (r0 Rj) wn(r0 Rk)wn (r Rl) ijkl m m0 r r 0 Z | 0| see e.g. LdM and M. Capone, Vietri Lectures, ArXiv:1607.08468 Hubbard model
mm0 1 mm0nn0 Hˆ = t d† djm + V d† d† dkn dln ij im 0 2 ijkl im jm0 0 0 ijmmX0 Xijkl mmX0nn0 X 0 2 2 mm0 ~ t = dr w⇤ (r R ) r + V (r) w (r R ) ij m i 2m m0 j Z e 2 mm0nn0 e V = drdr0w⇤ (r Ri)w⇤ (r0 Rj) wn(r0 Rk)wn (r Rl) ijkl m m0 r r 0 Z | 0|
mm0nn0 mm0nn0 Vijkl = U ij ik il
ˆ H = tijdi† dj + U ni ni (nim d† dim ) " # im ij i ⌘ X X
see e.g. LdM and M. Capone, Vietri Lectures, ArXiv:1607.08468 Localization of the Wannier functions
1 ik R ik r w (r R)= e · (r) (r)=e · u (r) n p kn kn nk N k BZ X2
1 1 ik r R=0 : w (r)= (r)= e · u (r) n p kn p nk N k BZ N k BZ X2 X2
Evaluate w n ( r ) at large r = Ri
1 ik r w (r = R )= e · u (0) n i !1 p nk N k BZ X2
the phase factor oscillates faster and faster as r = R i !1 à the Wannier function decays in space i.e. is localized Maximally localized Wannier functions
gauges in the construction: phase factors 1 ik R i↵ (k) w (r R)= e · e n (r) n p kn N k BZ X2
gauges in the construction: unitary transformations at each k point
1 ik R (k) w (r R)= e · U (r) n p mn km N k BZ m X2 X
(k) Minimize (by respect to all possible U mn ) the following functional
⌦ = [ n r2 n n r n 2] h | | i h | | i n X “Maximally Localized Wannier Functions: Theory and applications” Marzari et al. RMP 84, 1419 (2012) Weiss mean-field on the Ising model
Lattice H = J SiSj h Si Ising Model model
Effective single spin in an local Heff = (h + zJm)S0 model effective Weiss field
Local Heff magnetization observable m = Tre S0
relation between Self- m = tgh [ (h + Jmz)] consistency Weiss field and magnetization
exact in the infinite coordination (“dimension”) limit Dynamical mean-field theory
Lattice ˆ H = tijdi† dj + U ni ni Hubbard Model model " # ij i X X Anderson Impurity Model Effective local HAIM = ✏lal† al + Vl(al† dl + h.c.) µd0† d0 +Hint model Xl Xl
1 0(i!n) “Weiss field” G (⌧)= T d (⌧)d† (0) { h ⌧ i G Local 2 1 Vl observable Green’s G (i!n) = i!n +µ ⌃ (i!n) i!n ✏l function l X { (i!n)
Self- 1 exact in the infinite consistency G (i!n)= coordination i!n + µ ✏k ⌃ (i!n) (“dimension”) limit Xk Choose the AIM such that G ( i ! n ) and ⌃ ( i ! n ) have the same functional relation than in the lattice Dynamical mean-field theory
DMFT: use of a self-consistent impurity model to calculate the local correlation functions of a lattice model
1 Impurity model S = d⌧d⌧ 0 d† (⌧) (⌧ ⌧ 0)d (⌧ 0)+Hint G0 0 0 Z Z X
Georges et al. RMP 68, 13 (1996) K. Held et al. Phys. Stat. Sol. 243, 2599 (2006) Kotliar et al. RMP 78, 865 (2006) Kotliar and Vollhardt, Physics Today 3, 53 (2004) Long-range ordered phases in DMFT
No symmetry breaking assumed thus far. Easy generalization for symmetry-broken phases
Ferromagnetism: spin-dependent bath
H = ✏ a† a + V (a† d + h.c.) µd† d +H AIM l l l l l l 0 0 int l l X X 1 S = d⌧d⌧ 0 d† (⌧) (⌧ ⌧ 0)d (⌧ 0)+H G0 int 0 0 Z Z X Antiferromagnetism: two sublattices A and B (where ) ! GA (i!n)=GB, (i!n) A in the bath of B
⌃A (i!n)=⌃B, (i!n) A " G = G ⌃ (i! ), ⌃ (i! ) k { A n B n } Xk Analogous extensions for superconductivity, etc. … Impurity solvers
The Anderson Impurity Model has to be solved numerically: 2 Vl given 0 ( i ! n ) that is ( i ! n ) , (because 0 ( i ! n )= i ! n + µ ) G G i!n ✏l Xl we have to calculate the Green’s function G ( i ! n ) (that is ⌃ ( i ! n ) ) Many impurity solvers: - Iterated perturbation theory (IPT) - Exact diagonalization (ED): AIM truncated bath (limited # of levels) - Quantum Montecarlo (QMC) method - Numerical Renormalization Group (NRG) - Density-matrix Renormalization Group (DMRG) - ….
Iterative solution: DMFT: features, advantages and limitations
868 Kotliar et al.: Electronic structure calculations with ¼ DMFT captures the lifetimes of excitations (coherence vs incoherence)
We can compute static and dynamic observables
We have access to spectral features and to omega- FIG. 1. ͑Color online͒ Examples of a spectrum ͑a͒ for a weakly dependent response correlated system and ͑b͒ for a strongly correlated system. The situationAdvantages: in ͑a͒ can be modeled by the Kohn-Sham spectra of functions: spectroscopy LDA-like - zero treatmentsand finite while temperature the description of ͑b͒ requires a many-body - non-perturbative treatment such asmethod DMFT. - exact for infinite coordination (dimension): sum-rules preserved ͑Dagotto, 1994͒, density-matrix renormalization-group methodsLimitations:͑White, 1992; Schollwöck, 2005͒, and so on. The development of LDA+U Anisimov, Aryasetiawan, - non-local quantum fluctuations͑ are treated at the static mean-field level et al., 1997͒ and self-interaction corrected ͑SIC͒͑Svane FIG. 2. Local spectral function of the fully frustrated Hubbard and - cannot Gunnarsson, treat 1990non-local; Szotek orderet al., parameters 1993͒ methods, model at T=0, for several values of the interaction strength U, many-body - state-of-the-art perturbative impurity approaches solvers based are on GW numericallyand obtained challenging with the iterated perturbation theory approximation its extensions ͑Aryasetiawan and Gunnarsson, 1998͒, as ͑Zhang et al., 1993͒. The evolution from a weakly correlated well as the time-dependent version of the density- regime at small U, where the density of states resemble the functional theory ͑Gross et al., 1996͒, have been carried band density of states, to a Mott insulator at large U is shown, out by the electronic structure community to address the characterized by two Hubbard bands separated by a gap of problem of strongly correlated materials. Some of these order U. For intermediate U the metal exhibits a complex techniques are already much more complicated and time spectral function with Hubbard bands at high energies and consuming compared to the standard LDA-based algo- quasiparticle bands at low energies. The Mott transition is rithms, and therefore the real exploration of materials is driven by transfer of spectral weight between these features. frequently performed by simplified versions utilizing ap- From Zhang et al., 1993. proximations such as the plasmon-pole form for the di- electric function ͑Hybertsen and Louie, 1986͒, omitting delocalization crossover is nonuniversal, system specific, the self-consistency within GW ͑Aryasetiawan and Gun- and sensitive to the lattice structure and orbital degen- narsson, 1998͒ or assuming locality of the GW self- eracy which is unique to each compound. We believe energy ͑Zein and Antropov, 2002͒. that incorporating this information into the many-body To motivate the dynamical mean-field theory ap- treatment of this system is a necessary first step before proach we recall the nature of the one-electron ͑or one- more general lessons about strong-correlation phenom- particle͒ density of states of strongly correlated systems ena can be drawn. In this respect, we recall that DFT in may display both renormalized quasiparticles and atom- its common approximations, such as LDA or GGA, iclike states simultaneously ͑Georges and Kotliar, 1992; brings a system specific description into calculations. Zhang et al., 1993͒. To describe this method one needs a Despite the great success of DFT for studying weakly technique which is able to treat quasiparticle bands and correlated solids, it has not been able thus far to address Hubbard bands on equal footing, and which is able to strongly correlated phenomena. So, we see that both interpolate between atomic and band limits. Dynamical density-functional-based and many-body model Hamil- mean-field theory ͑Georges et al., 1996͒ is the simplest tonian approaches are to a large extent complementary approach which captures these features; it has been ex- to each other and hence can be merged. One-electron tensively developed to study model Hamiltonians. Fig- Hamiltonians, which are necessarily generated within ure 2 shows the development of the spectrum while in- density-functional approaches ͑i.e., the hopping terms͒, creasing the strength of Coulomb interaction U as can be used as input for more challenging many-body obtained by DMFT solution of the Hubbard model. It calculations. This path was undertaken by Anisimov et illustrates the necessity to go beyond static mean-field al. ͑Anisimov, Poteryaev, et al., 1997͒ who introduced the treatments in situations when the on-site Hubbard U be- LDA+DMFT method of electronic structure for comes comparable with the bandwidth W. strongly correlated systems and applied it to the photo- Model Hamiltonian-based DMFT methods have suc- emission spectrum of La1−xSrxTiO3. Near the Mott tran- cessfully described regimes U/Wտ1. However, to de- sition, this system shows a number of features incompat- scribe strongly correlated materials we need to incorpo- ible with the one-electron description ͑Fujimori, Hase, rate realistic electronic structure calculations. The low- Nakamura, et al., 1992͒. The electronic structure of Fe temperature physics of systems near localization- has been shown to be in better agreement with experi-
Rev. Mod. Phys., Vol. 78, No. 3, July–September 2006 Fermi liquid (Green functions)
@⌃0(k, !) Fermi-liquid self-energy ⌃(k, !) ⌃0(k, 0) + ! 2 ' @! ( ⌃ 00 ( ! ) ! ) 0 ⇠ 1 1 G(k, !) = ' @⌃0(k,!) @⌃0(k,!) ! ✏k + µ ⌃0(0) ! @! ! 1 @! ✏k0 + µ 0 0 1 Z ⇣ ⌘ = = k !/Z ✏ + µ ! Z (✏ µ ) k k0 k k0 1 (where ✏ k0 = ✏ k + ⌃ 0 ( k , 0) and Z k = ) @⌃0(k,!) 1 @! 0 1 @⌃0(k,0) m @k mass enhancement e kF = m @⌃0(kF ,!) 1 @! ⇤ 0 m 1 if ⌃ ( k , ! )= ⌃ ( ! ) then ⇤ = me Z
AIM at T=0 generates a ⌃ ( ! ) of this form: DMFT captures Fermi liquids Nozières, J. Low Temp. Phys. 17, 31 (1974) Fermi-liquid renormalization and Mott transition
1-band Hubbard model (semicircular DOS of width W=2D)
1
0.8
0.6 Z 0.4
0.2
0 0 0.5 1 1.5 2 2.5 3 3.5 4 U/D
Mott transition Mott transition: the atomic Mott gap
1 1 Single band Hubbard model U(ni )(ni ) " 2 # 2 in the atomic limit i X
U U U U U U + + 4 4 4 4 4 4
Potential energy barrier to overcome to establish conduction (ionization energy - electron affinity) =E (n +1) E (n)+E (n 1) E (n) at at at at at =E (n +1)+E (n 1) 2E (n) at at at
at = U Mott transition: Hubbard bands However these excited states are mobile (high degeneracy removed by hopping à energy dispersion ~W) particle excitation
hole excitation
Formation of two “Hubbard bands” in the spectrum of excitations
Δat
W W Hubbard criterion for the = U W U = W c Mott transition DMFT: spectral density 1-band Hubbard model: DMFT phase diagram Multi-Orbital Hubbard model
mm0 ˆ † H0 = tij dim djm0 ijmmX0 mmmm 2 2 V U = drdr0 w (r) W (r, r0) w (r0) iiii ⌘ | m | | m | Z mm0m0m 2 2 V U 0 = drdr0 w (r) W (r, r0) w (r0) iiii ⌘ | m | | m0 | Z mm0mm0 V J = drdr0 w⇤ (r)w⇤ (r0) W (r, r0) wm(r0)wm (r) iiii ⌘ m m0 0 Z
Hˆint = U nm nm + U 0 nm nm +(U 0 J) nm nm " # " 0# 0 m m=m m In correlated solids typically J=0.1÷0.2U Multi-Orbital Hubbard model mm0 ˆ † H0 = tij dim djm0 ijmmX0 mmmm 2 2 V U = drdr0 w (r) W (r, r0) w (r0) iiii ⌘ | m | | m | Z mm0m0m 2 2 V U 0 = drdr0 w (r) W (r, r0) w (r0) iiii ⌘ | m | | m0 | Z mm0mm0 V J = drdr0 w⇤ (r)w⇤ (r0) W (r, r0) wm(r0)wm (r) iiii ⌘ m m0 0 Z Hˆint = U nm nm + U 0 nm nm +(U 0 J) nm nm " # " 0# 0 m m=m m In correlated solids typically J=0.1÷0.2U Hund’s rules: atoms Aufbau 3d# 4s# Hund’s Rules 21Sc 22Ti 23V In open shells: 24Cr 25Mn 1. Maximize total spin S 26Fe 2. Maximize total angular 27Co momentum T 28Ni (3. Dependence on J=T+S, 29Cu Spin-orbit effects) 30Zn Mott transition in degenerate systems vs J: DMFT results HUND’s COUPLING AND ITS KEY ROLE IN TUNING ... 2 orbital Hubbard PHYSICALmodel half-filling REVIEW B (n=2)83,205112(2011) 8 1 7 0.8 DMFT- 2 electrons in 2 bands 6 5 0.6 2 /D =Z c 4 1 Z U 0.4 3 N=2, ntot=2 J/U J/U 0 =0 2 N=2, ntot=1 .05 0.2 0.1 =0 0.15 N=3, ntot=3 0.2 1 N=3, ntot=2 .25 N=3, ntot=1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 J/D U/D 1 FIG. 5. (Color online) Same phase diagram as in Fig. 1,plotted here as a function of J. The dashed lines indicate the strong J behavior 0.8 Uc diminishes with J calculated analytically in the atomic limit, Eq. (8) (from top to bottom DMFT - 1 electron in 2 bands δUc 3J, J, 2J ). 0.6 LdM, PRB 2011 ∝ − − 2 =Z 1 Z 0.4 half-filling this effect is controlled by n/N (for n 205112-5 Mott gap for multiorbital interaction at J=0 At J=0, the interaction is U 1 2 U tot 2 proportional to the total Hint = ( (nim )) = (n M) 2 2 2 i i m i charge on the site X X X squared M number of orbitals all configurations with the same number of electrons on a site have the same interaction energy. orb1 orb2 U U E =0 0 0 + + 0 2 2 = E (n +1)+E (n 1) 2E (n) at at at at The atomic potential barrier is still U at = U Hubbard bands at J=0 At J=0, the interaction is U 1 2 U tot 2 proportional to the total Hint = ( (nim )) = (n M) 2 2 2 i i m i charge on the site X X X M number of orbitals all configurations with the same number of electrons on a site have the same interaction energy. orb1 orb2 More hopping channels, larger degeneracy-removal energy Wider Hubbard bands Gunnarsson et al. Δ at W˜ = pMW PRB 56, 1146 (1997) ~ ~ W" W" Mott transition in multi-orbital models Δat ~ ~ W" W" N orbitals = U W˜ U = W˜ = pMW c Mott transitions at J=0 • Mott transition at every integer filling, Uc grows with M • Uc is maximum at half-filling (N orbitals) Lu,PRB 49, 5687 (1994), Rozenberg, PRB 55, R4855 (1997), Florens et al. 66, 205102 (2002) M-orbital degenerate Hubbard model: DMFT results HUND’s COUPLING AND ITS KEY ROLE IN TUNING ... PHYSICAL REVIEW B 83,205112(2011) effect of Hund’s J Quasiparticle renormalization factor Z=(m*/m)-1 8 1 7 0.8 DMFT-2 el 2 electrons2 bands in 2 bands 6 1 el 2 bands 5 0.6 2 /D =Z c 4 1 Z U 0.4 3 N=2, ntot=2 J/U J/U 0 =0 2 N=2, ntot=1 .05 0.2 0.1 =0 0.15 N=3, ntot=3 0.2 1 N=3, ntot=2 .25 N=3, ntot=1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 J/D U/D 1 FIG. 5. (Color online) Same phase diagram as in Fig. 1,plotted 16 DMFT - M=2N=2, ntot=1 here as a function of J. The dashed lines indicate the strong J behavior SSMF - N=2, n =1 0.8 Uc is lowered by J 14 M=2 tot calculated analytically in the atomic limit, Eq. (8) (from top to bottom DMFT - M=2N=2, ntot=2 DMFT - 1 electron in 2 bands 12 SSMF - M=2N=2, ntot=2 δUc 3J, J, 2J ). 0.6 at half-filling. ∝ − − 2 10 =Z 1 /D Z 0.4 For other fillings is c 8 U half-filling this effect is controlled by n/N (for n 205112-5 Hund’s coupling changes the Mott gap: increased at ½ filling n=0 Mott gap (atomic limit): n=4 Δ=E(n+1)-E(n) - [E(n)-E(n-1)] n=1 n=3 n=2 At half-filling Δ=U+J à The gap increases with J Needs a smaller U for the Mott transition: Uc decreases Hund’s coupling changes the Mott gap: decreased for a generic filling For Needs a larger U for the Mott n=1 transition: Uc increases Δ=U-3J M=2 M=2 (slave-spin M=3 mean-field M=3 M=3 results) Δ=U+J Suppression of orbital fluctuations by J In all cases different from Δ=U-3J one electron (n=1) or one hole (n=2M-1) filling, there is a fast M=2 M=2 suppression of Uc at M=3 M=3 small J M=3 Δ=U+J ✗ ✔ ✔ orb1 orb2 J limits again the hopping channels, smaller dispersion of excitations W˜ = pMW J Δat Δat ~ ~ W" W" W W Ground-state degeneracy and hopping processes Both the Hubbard band widths and the quasiparticle mass are governed by the same hopping processes which are the less effective the lower the ground-state degeneracy Hund’s coupling lowers the ground-state degeneracy - reduces the width of the Hubbard band - reduces the quasiparticle weight (enhances the effective mass) 3"orbitals 2"orbitals ground'state'degeneracy ground'state'degeneracy n n J=0 finite'J J=0 finite'J 0'(or'6) 1 1 0'(or'4) 1 1 1'(or'5) 6 6 1'(or'3) 4 4 2'(or'4) 15 9 2 6 3 3 20 4 3-band Hubbard model (semicircular DOS) in DMFT LdM, J. Mravlje, A. Georges, M=3 orbitals (relevant for t2g materials) PRL 107, 256401 (2011) n=1 (n=5) n=2 (n=4) n=3 Favored: metal bad metal Mott insulator ‘Janus’ case: two contrasting effects Strong correlations far from a Mott insulator! Janus Bifrons In ancient Roman religion and mythology, Janus is the god of beginnings and transitions, then also of gates, doors, doorways, endings and time. Most often he is depicted as having two heads […] 3-band Hubbard model (semicircular DOS) quasiparticle J/U=0.15 weight Z interactionstrength U/D LdM, J. Mravlje, A. Georges, PRL 107, 256401 (2011) number of electrons per site (n) 1142 Imada, Fujimori, and Tokura: Metal-insulator transitions FIG.3-orbital 64. A guide map forHubbard the synthesis of filling-controlled model (FC) 3d transition-metalwith t2g oxides density with perovskite and of layered states perovskite (K2NiF4-type) structures. and CuO chains. The nominal valence of Cu on the Filling control by use of nonstoichiometry (offstoichi- sheet and chain is estimated to be approximately ⇤2.25 ometry) has also been carried out for other systems, for J/U=0.15 and 2.50, respectively (Tokura et al., 1988). In other example V2⌅yO3 (see Sec. IV.A.1) and LaTiO3⇤y (Sec. words, the holes are almost optimally doped into the IV.B.1), which both show the Mott-insulator-to-metal sheet to produce superconductivity in this stoichiometric transition with such slight offstoichiometry as y⇥0.03. compound. The oxygen content on the chain site can be The advantage of utilizing oxygen nonstoichiometry is reduced, either partially (0 y 1) or totally (y⇥0), that one can accurately vary the filling on the same which results in a decrease in the nominal valence of Cu specimen by a post-annealing procedure under oxidizing on the sheet, lowering the superconducting transition or reducing atmosphere. Since vacancies or interstitials temperature Tc and finally (y 0.4) driving the system may cause an additional random potential, the above• 103 cubic to a Mott insulator (or more rigorously, a CT insulator). method is not appropriate for covering a broad range of The3d nominal valence of the chain-site Cu in the y⇥0 fillings. • 214 tetragonal compound is ⇤1 due to the twofold coordination, while (further splitting) that for the y⇥1 compounds is ⇤2.5. Thus the nomi- 3. Dimensionality control nal hole concentration or Cu valence in the sheet can apparently4d be controlled by oxygen nonstoichiometry on Anisotropic electronic structure and the resultant an- the chain site, yet it bears a complicated relation to the isotropy in the electrical and magnetic properties of d oxygen content (y) and furthermore depends on the de- electron systems arises in general from anisotropic net- tailed ordering pattern of the oxygen on the chain sites. work patterns of covalent bondings in the compounds. A. Georges, LdM, J. Mravlje, Annual Reviews Cond. Mat. 4, 137 (2013) Imada, Fujimori, Tokura, RMP 70, 1039(1998) FIG. 65. A schematic metal-insulator diagram for the filling-control (FC) and bandwidth-control (BC) 3d transition-metal oxides with perovskite structure. From Fujimori, 1992. Rev. Mod. Phys., Vol. 70, No. 4, October 1998 Multi-orbital AF insulator Static vs dynamic mean-field in magnetic phases PHYSICAL REVIEW B 85,085124(2012) Signature of antiferromagnetic long-range order in the optical spectrum of strongly correlated electron systems C. Taranto,1 G. Sangiovanni,1 K. Held,1 M. Capone,2 A. Georges,3,4,5 and A. Toschi1 1Institute for Solid State Physics, Vienna University of Technology, A-1040 Vienna, Austria 2Democritos National Simulation Center, Consiglio Nazionale delle Ricerche, Istituto Officina dei Materiali (IOM) and Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, I-34136 Trieste, Italy Structures in3Centre the de Physique Hubbard Theorique,´ Ecole bands: Polytechnique, CNRS, F-91128 Palaiseau Cedex, France theory (Hubbard4 Collmodel)ege` de France, 11 place Marcelin Berthelot, F-75005 Paris, FranceExperiments: Optical SIGNATURE OF ANTIFERROMAGNETIC5DPMC, Universit LONG-RANGEedeGen´ eve,` ... 24 quai Ernest Ansermet, CH-1211PHYSICAL Geneve,` Switzerland REVIEW B 85,085124(2012) C. TARANTO et al. (Received 31 December 2011; published 23 February 2012) conductivityPHYSICAL of REVIEW LaSrMnO4 B 85,085124(2012) 3 We show3 how the onset of a non-Slater3 antiferromagnetic ordering in a correlated material can be 9 U=D detectedthe Hartree-Fock by opticalU=3D spectroscopy. calculations. UsingThis dynamical is why the mean-field gapU=5D in theoryFig. 1 we identify two distinctiveT=15K features: Theis antiferromagnetic of order U 2D ordering, whileis it wouldassociated be exactly with anU enhancedwithin spectralstatic weight above theT=52K optical gap, and − T=100K well-separatedmean-field spin-polaron theory. peaks emerge in the optical spectrum. Both features are indeedFIG. observed2. (ColorT=120K in online) LaSrMnO Lower4 panels: 1.5 1.5 1.5 T=150K [A. Gossling¨In theett- al.J ,modelPhys. Rev. in infinite B 77,035109(2008) dimensions]. (as well as in the Evolution ofT=200K the optical conductivity of HF Hubbard model within AF-DMFT, up to higher-order correc- Hubbard modelT=250K from weak (left) to strong DMFT DOI: 10.1103/PhysRevB.85.0851242 PACS number(s): 71.27. a, 71.10.Fd, 75T=295K.30. m, 78.20.Bh tions in t /U)thepositionofthepeakscanbeanalytically + coupling (right).− Shown are both AF 6 ) ω local spectral function 0 0 0 expressed through the zeros of the Airy functions and the phase( and PM phase. In the figure the -3 -1.5 0 -3 -2 -1 0 -3 -2 -1 0 σ D1/3J 2/3 ω/D spacingI. INTRODUCTION between themω/D is proportional to ωas/D one.Aswe cools the systemcorresponding from above values the of the Neel´ optical temperature inte- will also discuss in Sec. V these analytic expressionsdown to hold low temperature.grals ( kinetic This energy) transfer are actually also reported, provides This paper deals with the following issue: Is there a dis- strictly only in infinite dimensions. Hence the applicabilityadiagnosticsofthestrong-coupling(superexchange)or of showing= a change of the “hierarchy” be- 1 tinctive signature of the 0.10 onset of antiferromagnetic0.10 (AF) long- U=D our results to real materials,U=3D and in particularU=5D theweak-coupling emergence (spin-densitytween the kinetic wave) energy nature scales of the of the antiferro- two range order in the optical spectrum ofAF a strongly correlated AF KAF=-0.366Dof spin-polaron peaks,K =-0.163D strongly relies on the assumptionK magnetism.=-0.097D that In thephases former at case, intermediate spectral coupling. weight increase Upper is insulating antiferromagnet, for examplePM an antiferromagnetic PM KPM=-0.378Dit is possible to neglectK =-0.084D the quantum spin-flipK processes.observed,=-0.046D corresponding In panels: to Corresponding a kinetic energy spectral gain function associated ) transition-metal oxide? ω with a multipeak (spin-polaron) structure ( 0.5 principle0.05 this is not guaranteed for0.05 finite-dimensionalwith systems; the ordering. In the0 latter case,1 a spectral2 weight decrease3 4 5 σ AF ordering is a characteristic feature of strongly correlated takes place, corresponding(Ref. 21). to a kinetic energy loss (potential Mott insulatorstherefore where superexchange one may expect drives some the ordering of the peaks of to broaden or ω(eV) energy gain). For superconducting long-range order, similar localizedAF phase, T=0 magneticeven moments. to be washed On out, the otherdue to hand, the coupling AF can to dispersive spin PM phase, T=0 5,8,9 diagnostics of weak vs strong-coupling pairing have been also arise from Fermi-surfacewaves. On nestingthe other in hand, a weakly recent correlated sophisticated calculations FIG. 4. (Color online) Optical conductivity of LaSrMnO4 be- 0 0 0 discussed and probed experimentally in the context of cuprate 0material.2.5 From5 for7.5 a theoreticalthe two-dimensional0 point2.5 of5t view,-J model7.5 a continuous give0 clear2.5 evidence5 for7.5 the tweenGössling 0.75 and 5et.50 al. eV forPRB different 77, temperatures. 035109 The (2008) data have ω/D ω/D ωsuperconductors/D 2 and for models with attractive interaction.3,4 crossover drivensurvival by the interaction of the first strength one or connects two spin-polaron weak- peaks with a been kindly provided by the authors of Ref. 10. Second, a multipeak23 structure develops above the gap in the and strong-couplingsimilar antiferromagnets, separation as in and the infinite-dimensional an unambiguous calculation. ordered phase at strong coupling. These peaks correspond to seefavored also:distinction if the G. spin Sangiovanni between patternFurthermore is local-moment as close the et as case possibleal. and of Fermi-surfacePRB a todouble-layer the 73, Neel´ AF205121 antiferromagnet isin potential (2006) energy. was At intermediateSince the separation values of the between interaction the multiplet peaks is larger the spin-polaron excitations associated with the motion of a one.3,9 Thuslacking. the lower therecently temperature, considered the larger in Ref. the staggered24. There, itstrength, is shown corresponding that the than to the the crossover energy scale between associated weak andwith the spin-polaron ex- hole in an antiferromagnetic background, a problem which magnetizationIn this and work in turninterlayer we the propose larger exciton the that, spectral remarkably, made weight. of one optical hole spec- andstrong one coupling, doublon wein havecitations, a mixed we regime expect in which to be able both to energy detect the above-mentioned has been addressed since the late sixties in a series of pivotal If wetroscopy consider can instead beeach used the weak-couplinglayer to infer displays the correlatedconfinement regime U nature effectsD of,a an AF whendifferences the interlayer are negative,increase3,4 i.e., of where spectral the weight onset of around the AF each is of the main absorption works.5 In the context of infinite dimensions the physics totally differentstate. Optical mechanism spectroscopycoupling for the is is smaller stabilization an invaluable than the of experimental the intralayer ordered≪ one. probe Instabilized such a situation, by both kineticpeaks and and potential the consequent energy. appearance of spin-polaron struc- of the spin polarons has also been extensively analyzed6–9 phase takesof correlated place. The materials,the AF spin-polaron phase providing is stabilized excitations key physical by a discussed (small) information here(ii) maySpin-polaron become peakstures. when To clarify going below the origin the Neel´ of temperature. the and this work identifies their signature in the optical potentialsuch energy as thegain, optical asparticularly the onset gap, theof relevant. a relativenonzero weight order parameter of low-energymultipeak structure, we compareAll curves in Fig. below2 theTN evolutiondisplay, from in the same region, a very conductivity. Drude excitations in metallic systems, and quite importantly similar low-frequency tail (below 3eV)whileinthatregion is correctly described byWhen the static considering mean-field three-dimensional theory in this cases,weak where toFinally, strong DMFT coupling we consider of the the optical possibility conductivity of detecting to that these the transfers of spectral weights often observed in correlated the higher-temperature measurements∼ show a gradual increase regime. These theoreticalis typically considerations more accurate are well and confirmed finite-dimensionalityof thesignatures one-particle effects are experimentally spectral function. and discuss It is in evident detail that the case the of materials when temperature or composition is varied (for 25 of spectral weight that6,9 we attribute to a more standard thermal by our numerical datadefinitively shown in the weaker bottom than panel in twoof Fig. dimensions,2 multipeakLaSrMnOspin-polaron structure, where of the we spectral argue that function some ofsurvives these effects in the may arecentreviewseeRef.AF 1PM). Spin degrees of freedom are 4 where we compare Kfeaturesand areK expected,thekineticenergyof to be even more visible.optical In thealready conductivity, following have been asbroadening alreadyobserved. observed10 effect. for It the is likelypure t- thatJ case such thermal effects are however much less coupled to light than the charge so that 22 the ordered and disorderedsection, phase, we respectively. discuss indeed The the complete relevance ofby our Cappelluti resultsThe for outlineet the al. ofThevisible the peculiar paper also in is behavior the as low-energy follows: of the In multiplet spectral Sec. II we peak at 3.5 eV, which the signatures of magnetism are expected to be comparatively evolution of the kineticthree-dimensional and potential energy manganite as a function LaSrMnO of 4. functionsummarize in infinite the dimensions mainlies aspects closer was of to first our this understood calculation tail. This in ofmakes the thet optical- thisJ spectral feature less weak. Zone-center magnons, for example, do not show up in 6 U is shown in Fig. 3.Atstrongcouplingtheantiferromagnetic modelconductivity in a pioneering for workthevisible Hubbard by Strack with and a and firstt-J Vollhardtmodel. distinguishable Detailsand more on excitation the shifting with the optical conductivity unless inversion symmetry is broken.1 temperature unlike what happens for the second multiplet order is clearly stabilized by theIV. kinetic COMPARISON energy while WITH at weak EXPERIMENTSrecentlydynamical reanalyzed mean-field for the case theory of the calculations Hubbard modelfor the by two some models Furthermore, the key energy scale associated with magnetic 9 structure at 4.5 eV. coupling (U<2D)theorderedphaseisdrivenbythegain2 of us.areThe provided multiple in Appendix peaks arise 1 and from 2, respectively. stringlike excitations In Sec. III we ordering, the superexchange J ( D /U with D being half the 5 On the basis of the previous discussion,associated the idealdiscuss material with the maina hole theoretical movingWe focus inresults a therefore N beforeeel´ background. we on compare the high-energyIn them spectral struc- bandwidth), is much smaller than∼ the scale U corresponding to for the observation of the temperature evolutioninfiniteto experimentdimensions predicted in the Sec.ture lifetimeIV around.Asummaryandconclusionaregiven of such 4.5 eV, string which excitations seems to is be unaffected by the the local matrix element of the screened Coulomb interaction, 0.05 above would be a completely isotropicinfinite three-dimensionalin due Sec. toV. the absencetemperature of quantum broadening spin-flip processesof the low-frequency6,7,9 data, and we and hence generally significantly smaller than the optical gap Mott antiferromagnet. To our knowledge(which the most would relevant act as “damage-repairing”zoom the plot tofocus processes), on this meaning feature. In order to highlight itself ( U 2D). ∼ − set of measurements in this respect isthat that the by Heisenberg Gossling¨ interactionthe temperature reduces effect, to the Ising we display one (see in Fig. 5 the difference Despite these shortcomings,10 we show that clear signatures et al., on LaSrMnO4,athree-dimensionalAppendixC-type 1 layered for furtherII.between MODELSdetails). the This AND optical results METHODS conductivity in the formation at a given temperature and associated with antiferromagnetismantiferromagnetic material are expected with inT the optical133 K, while we are not the conductivity at the highest measured temperature (300 K). 0conductivity below the Neel´ temperature (T ) whenN the systemof coherent but heavy quasiparticles, “dressed” by a spinon aware of similar studies asN a function= of temperatureIn on G order-type to studyFigure the optical5 and conductivityits theoretical of counterpart strongly (Fig. 6)endorse is in the strong-coupling (Mott insulating) regime. Thesecloud which is commonly called a “spin polaron.” antiferromagnets, which would be the closest realizationcorrelated antiferromagnets,of our physical we interpretation perform dynamical of the peaks mean- in the optical signal. signatures are twofold. First, there is a spectral weight transfer Thefield formation theory (DMFT) of the spin calculations polaron can for be a half-filled described Hubbard only our large-coordination results. In Fig. 4 weusing show methods the data (such of asAlthough DMFT) that the go frequency beyond the grid static in mean the experimental data is Ref. 10 for the optical conductivity of LaSrMnOfield (Hartree-Fock)4 along the a approximation0.05 eV, at least of a two spin-density well-spaced wave. peak structures emerge -0.051098-0121/2012/85(8)/085124(10)085124-1 ©2012 American Physical Society axis at different temperatures. The two mainThis structures is illustrated around in thebelow upperTN panelsand, as expectedof Fig. 2 within, where our we spin-polaron picture, the Energy difference/D 3.5 eV and 4.5 eV have been explained incompare terms of the multiplet evolution withlowerU theof temperature the dynamical the (red weaker line) the and temperature dependence Kinetic energysplit difference transitions from a d4 to a d5 configurationstatic of mean-field the Mn. Our (greenbecomes. line) results Indeed for the between electron-removal 52 and 15 K there is hardly Total energy difference Potential energymain difference interest instead is focused on the finespectrum. structure One arising clearlyany sees difference that the only in thefeature optical of the conductivity static in this frequency -0.1 0.5 1 1.5upon2 cooling2.5 3 on top3.5 of4 both4.5 of these5 main structures.mean-field (Hartree-Fock)region, calculation as one can is aexpect Slater-like from the single behavior of the staggered In the light of our theoretical analysis, wepeak can directly centered relate around magnetizationU/2. The peak (see has inset weight to Fig.Z 1),1and which is almost saturated far U/D − = this observation with the physics of the spin polaronsits width describedis given by J ; i.e.,below theT widthN .Theappearanceofthespin-polaronfinestructures shrinks to 0 as 1/U upon in the previous section. Since the two structuresincreasing areU. In well the dynamicalbecome mean-field clear below calculation,TN 133 instead, K. Yet, the polaronic features FIG. 3. (Color online) Difference of kinetic, potential, and total = energy between AF andseparated, PM phase fromwe shall weak make to strong a simplifying coupling assumptionthe weight (not of meant the loweststart lying to excitation be visible is already much smaller in the (Z 150is K curve. We attribute (computed with Lanczosto at beT quantitative,0). At strong but sufficient coupling to AF reveal is theproportional main qualitative to J/D)andtheremainingweightistransferredthis to fluctuation effects above TN : Even if long-range order stabilized by the kinetic energyaspect and of= at the weak phenomenon). coupling by the Namely, potential we shallto the assume higher-energy that we peaks.is not This yet makes fully a developed, total bandwidth a sufficiently of long coherence 4 5 energy. can adopt for each d d transition a simpleorder 2 single-bandD rather then J ,resultinginasmallergapcomparedtolength of the spin-spin correlations is enough to sustain a Hubbard model description.→ non-fully-developed spin-polaron mechanism. 085124-3 085124-4 Correlated metallic-magnetic phases: BaCr2As2 ArXiv:1610.10054 BaCr2As2 Wannier functions bare bandwidth of Cr d bands W~5eV GGA+DMFT vs GGA+U Maximum Entropy analytical continuation of G(iωn) (green curve) Z and shifts from Σ(iωn) (quasiparticles only, N*(εF) 20 total d 15 majority/minority spin 10 5 0 -5 -10 -15 -20 3.5 4 4.5 5 5.5 6 6.5 7 References DMFT Intro G. Kotliar and D. Vollhardt, Physics Today 3, 53 (2004) DMFT A. Georges et al. RMP 68, 13 (1996) Reviews K. Held et al. Phys. Stat. Sol. 243, 2599 (2006) G. Kotliar et al. RMP 78, 865 (2006) A. Georges, AIP Conference Proceedings 715, 3 (2004) Lectures D. Vollhardt, AIP Conference Proceedings 1297, 339 (2010) B. Amadon, ISTPC school 2015 (webpage at u-strasbg.fr) on DMFT Juelich autumn school on correlated electrons www.cond-mat.de/events/correl.html lectures from: A. Georges, M. Kollar, G. Kotliar, A. Lichtenstein, E. Pavarini, D. Vollhardt and many others… main LdM, PRB 83, 205112 (2011) references LdM et al., PRL 107, 256401 (2011) in the talk A. Georges, LdM, J. Mravlje, Annual Reviews Cond. Mat. 4, 137 (2013) • static vs dynamic MF x AF: C. Taranto et al. PRB 85, 085124 (2012) • Néèl temp SrTcO3: J. Mravlje, M. Aichhorn et al., PRL 108, 197202 (2012) • BaCr2As2: M. Edelmann et al. ArXiv:1610.10054