Excitonic Condensation of Strongly Correlated Electrons. Jan Kuneš DFG FOR1346

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1 Excitonic Condensation of Strongly Correlated Electrons Jan Kuneš DFG FOR1346

2 Outline Excitonic condensation in fermion systems EC phase in the two-band Hubbard model (DMFT results) (PrxLn1-x)yCa1-yCoO3 (PCCO) LDA+U for PCCO excitonic condensation with orbital degeneracy

3 Excitonic insulator A band insulator with a very narrow gap (positive or negative) is unstable towards opening of a gap due to electron-hole attraction - condensation of excitons. The gap can have spin-singlet or spin-triplet symmetry and be real or imaginary. Which of these options is realised depends on the interaction term and details of the band structure. Mott, 1961 Halperin and Rice, 1968

4 Strong coupling picture of excitonic condensation Two-orbital atom with 2 electrons (crystal field and Hund s exchange determine the states of lowest energy): low spin S= high spin S=1 Strong coupling: HS states behave as hard-core bosons with the vacuum state vac LS Bose-Einstein condensation = spontaneous hybridization between HS and LS states on the same site (breaks spin rotational symmetry)

5 H t = ( n a 2 iσ n b ( iσ) + ta a iσ a jσ + t bb iσ b jσ) i,σ i,j,σ + ( V1 a iσ b jσ + V 2b iσ a jσ + c.c.) ij,σ H dd int = U i +(U 3J) iσ H int = J iσ Excitonic instability close to spin-state transition Two-band Hubbard model at n=2 (half filling) ( n a i n a i + nb i nb i ) +(U 2J) n a iσ nb iσ a iσ b i σ a i σ b iσ + J i i,σ n a iσ nb i σ ( a i a i b i b i + c.c.). Full p-h susceptibility tensor χ q (T) for various temperatures Investigate dominant eigenmodes of χ q (T) (1) U/(t a +t b ) 4 2 HS Mott insulator metal LS band insulator 2 4 /(t a +t b ) orbital diagonal orb. off-diagonal spin longitudinal X n a χ λ 4 n b a " b #,a # b ",b " a #,b # a " X n a + n b (b) χ -1 (arb. units) [,] [,π] [π,π] [,] [,π] [π,π] [,] [,π] [π,π] 4 6 8

6 H t = ( n a 2 iσ n b ( iσ) + ta a iσ a jσ + t bb iσ b jσ) i,σ i,j,σ + ( V1 a iσ b jσ + V 2b iσ a jσ + c.c.) ij,σ H dd int = U i +(U 3J) iσ H int = J iσ Excitonic instability close to spin-state transition Two-band Hubbard model at n=2 (half filling) ( n a i n a i + nb i nb i ) +(U 2J) n a iσ nb iσ a iσ b i σ a i σ b iσ + J i Full p-h susceptibility tensor χ q (T) for various temperatures Investigate dominant eigenmodes of χ q (T) i,σ n a iσ nb i σ ( a i a i b i b i + c.c.). (1) U/(t a +t b ) 4 2 HS Mott insulator LS band metal insulator ta 2 4 /(t a +t b ) tb V a b (b) χ λ 4 2 [,] [,π] [π,π] [,] [,π] [π,π] [,] [,π] [π,π] χ -1 (arb. units)

7 H t = ( n a 2 iσ n b ( iσ) + ta a iσ a jσ + t bb iσ b jσ) i,σ i,j,σ + ( V1 a iσ b jσ + V 2b iσ a jσ + c.c.) ij,σ H dd int = U i +(U 3J) iσ H int = J iσ Excitonic instability close to spin-state transition Two-band Hubbard model at n=2 (half filling) ( n a i n a i + nb i nb i ) +(U 2J) n a iσ nb iσ a iσ b i σ a i σ b iσ + J i i,σ n a iσ nb i σ ( a i a i b i b i + c.c.). Full p-h susceptibility tensor χ q (T) for various temperatures Investigate dominant eigenmodes of χ q (T) (1) U/(t a +t b ) 4 2 HS Mott insulator metal LS band insulator 2 4 /(t a +t b ) orbital diagonal orb. off-diagonal spin longitudinal X n a χ λ 4 n b a " b #,a # b ",b " a #,b # a " X n a + n b (b) χ -1 (arb. units) [,] [,π] [π,π] [,] [,π] [π,π] [,] [,π] [π,π] 4 6 8

8 Excitonic instability U=4, J=1 t a2 +t b2 =const Δ=3.4 V= ζ = 2t at b. t 2 a +t2 b cussion of t 1 normal phase 8 6 solid 4 order 2 excitonic insulator (superfluid) ζ JK and Augustinsky, 214

9 Excitonic instability U=4, J=1 t a2 +t b2 =const Δ=3.4 V= ζ = 2t at b. t 2 a +t2 b cussion of t 1 normal phase 8 6 solid 4 order 2 excitonic insulator (superfluid) ζ JK and Augustinsky, 214

10 U=4, J=1 t a2 +t b2 =const Δ=3.4 V= ζ = 2t at b. t 2 a +t2 b cussion of t Excitonic instability 1 normal phase 8 6 solid 4 order 2 excitonic insulator (superfluid) Divergent response to: a b ζ JK and Augustinsky, 214

11 Excitonic condensate - oder parameter Order parameter is a complex vector: z-axis parallel to: No cross hopping (V=) -> overall phase of does not matter -> phases are distinguished by and 3 possible phases: linear (L) elliptic (E) circular (C)

12 Excitonic condensate - oder parameter Order parameter is a complex vector: z-axis parallel to: No cross hopping (V=) -> overall phase of does not matter -> phases are distinguished by and 3 possible phases: linear (L) elliptic (E) circular (C)

13 Excitonic condensate - oder parameter Order parameter is a complex vector: z-axis parallel to: No cross hopping (V=) -> overall phase of does not matter -> phases are distinguished by and 3 possible phases: linear (L) elliptic (E) circular (C)

14 Excitonic instability U=4, J=1 t a2 +t b2 =const Δ=3.4 V= ζ = 2t at b. t 2 a +t2 b cussion of t 1 normal phase 8 6 solid 4 order 2 excitonic insulator (superfluid) ζ JK and Augustinsky, 214

15 Excitonic condensation (n=2) order parameter r φ = a b + a b Spectral density (diagonal elements) φ Internal energy (ev) Internal energy and specific heat 5 1 Specific heat A bb (ev -1 ) A aa (ev -1 ) Energy (ev) Temperature

16 Excitonic condensation (n=2) order parameter r φ = a b + a b Optical conductivity (dc resistivity) φ Conductivity (arb. units) Resitivity (arb. units) Temperature 5 1 T (K) Energy (ev)

17 Excitonic condensation (n=2) φ order parameter r φ = a b + a b Dynamical spin susceptibility (local) B(ω)/ω 4 2 Temperature.2.4 ω (ev)

18 order parameter Excitonic condensation (n=2) r φ = a b + a b 8 Dynamical spin susceptibility (local).4 6 φ Local picture Normal phase B(ω)/ω 4 2 Temperature.2.4 ω (ev) EC phase HS LS HS-LS hybridization

19 Excitonic condensation (doping) - all phases n-t phase diagram μ-t phase diagram T (K) 4 T (K) n h L phase µ (ev) E phase C phase nh - hole concentration ( N=2-nh)

20 specific heat: Pr.5 Ca.5 CoO 3 (PCCO) Fig. 1. Temperature dependence of the specific heat of (Pr 1 y Y y ).7 Ca.3 CoO 3 (y = ) and Pr.5 Ca.5 CoO 3. The critical temperatures of M-I transitions are marked. resistivity: Fig. 3. The t (y =..1 dent from the Pr 1 x Ca x CoO and the orig both a ther (Pr 1 y Y y ).7 shows itself a certainty, as electric powe of thermopow the charge c itinerancy is phase. In con subsystem at hole concentr Page 4 of 8 Eur. Phys. J. B (213) 86: ison 35 of the l Fig. inverse 1. Temperature susceptibility: dependence of the specific heat Fig. 2. The temperature dependence of electrical resistivity of (Pr 1 y Y y ).7 Ca.3 CoO 3 (y = ) and the Pr.9 Ca. of (Pr 1 y Y y ).7 Ca.3 CoO 3 (y = ) and cluded also in Pr Fig. 3. The thermoelectric power of (Pr Pr.5 Ca.5 CoO 3. The critical temperatures of M-I transitions are marked..5 Ca.5 CoO metallic 3. The ground transition state areis evi- glet.5 Cadue.5 CoO to 3 crystal, plottedfield in effects. log-log Our scale. experimentally 1 y TheY y data ).7 Cafor.3 CoO estimated value χ VV.3.6 emu mol 1 Oe 1 per Pr 3 Pr (y.7 Ca =.3..15) CoO 3 withand ferromagnetic Pr It is worth added dent for fromcomparison. the increase below the T MI,thedataforcobaltites and 3+ their hig matches the absolute value determined by Knížek et al. resistivity aft Pr 1 x Ca x CoO 3 of lower hole doping are given for comparison..3 CoO 2 [16]. Turning back to the open hysteresis loop value, and fu with of the y = Pr Cacontinuous.5 CoO 3 The,wenotethatitsoccurrencecan electrical phase resistivity transition of the increases its studied be related samples to the is presented fact that due Figure to some 2 together composition with the inhomogeneity and for theferromagnetic original 2. and/or metal-insulator oxygen one metallic ofnonstoichiometry Tsubouchi Pr.7 supposing th data transition Ca.3 CoO al. in 3 the [1], sample. not both Although allacothermal species character hysteresis are transformed of conduction at M-I intotransition. the can low be some- spin For state. the the origin sample, exhibit by high elasti what (Pr 1 y masked Y y ) 3..7 Ca drop by.3 the CoO of polycrystalline 3 samples, magnetic however, formsusceptibility AminorphaseofsupposedlyFMcharacteristhusdeveloped shows clearly itself below identify as non-hysteretic 7 two K and distinct interacts within behaviors: with the the experimental (i) the major realglassy un-the ma of the samples, transition sample when we macroscopically FM certainty, phase Tsubouchi as (T f documented insulating 6.5 K)[17]. et simultaneously state al., This with 22 complication steeply byincreas- ing in electric resistivity (Pr power at data low temperatures, in Figure 3. represented Here, the by sharp the jump Figures 4 an 1 y Y y ).7 Ca.3 CoO 3 samples and, in our opinion, theisthermo- absent (Pr 1 y Y y ).7 (Pr of 1 y thermopower Y y ).7 Hejtmanek Ca.3 CoO coefficient 3 sample et below al., y = 213 the.15t MI and indicates (ii) the that as 1/χ vs. te the observed Brillouin-like magnetism cannot be explained behavior the charge typical carrier for bad concentration or highlyis disordered decreased metals characterized by slowly increasing but finite resis- 4+ ions. and their T MI.Athigh without considering a contribution of magnetic Pr paramagnetic

21 Pr.5 Ca.5 CoO 3 (PCCO) Pr 3+ ->Pr 4+ valence transition: Pr 4+ Schottky peak: Eur. Phys. J. B (213) 86: 35 Page 5 of 8 4. Pr 3+ ->Pr 4+ valence transition 5. exchange splitting of Pr 4+ ground state, but no magnetic order detected Fig. 12. The temperature dependence of the normalized XANES spectra at the Co-K-edge for the Pr.63 Y.7 Ca.3 CoO 3 sample. The inset shows the magnification of the pre-edge peak including the data after subtraction of arctangent background (full lines). The arrows indicate the most marked changes of spectral weight between 3 and 8 K. which increases the main edge energy for the same valence of Co. The impact of the spin state is difficult to estimate quantitatively, but very roughly, based on the paper of Vankó etal.[21] onlacoo 3 the shift from LS to Fig. 6. The low-temperature specific heat of Pr.5 Ca.5 CoO 3 Fig. 7. The Schottky Pr 4+ contribution for Pr.5 Ca.5 CoO 3 and Pr.63 Y.7 Ca.3 CoO 3 measured in fields 9 T and HSplot- ted in log-log scale. In order to separate the Pr 4+ related The pre-edge leads to and increase Pr of the main edge energy of.5 ev..63 Y.7 Ca.3 CoO 3 plotted as C p /T vs. T together with the fit based peak, shown on anisotropic in more g-factor detail in as the described inset of in the text Schottky peaks, the background contribution comprising Figure com-12mon terms is marked by full line (see the text). perature. Compared to the Hejtmanek spectra of LnCoO et 3 (solid exhibits lines). alsolittlechangewithdecreasingtem- al.,,thatallow 213 adecompositionintotwocomponents and the temperature induced spin states can be readily probed [21 23], the pre-edge peak in mixed-valent cobaltites is much broader is characterized by coefficient α.6 J mol 1 K, which

22 Excitonic condensation (doping) - all phases n-t phase diagram 8.4 T (K) 6 4 φ n h L phase n (electrons per atom) E phase 5 1 C phase

23 EC in cubic d 6 perovskite Exciton = bound pair of e g electron and t 2g hole How do we detect the EC order? Local d-occupation matrix (1 x 1): spin structure: orbital structure: D = D + φ z φ x + iφ y (φ x + iφ y ) D φ z φ α xy d x 2 y 2 d xy + φ α zx d z 2 x 2 d zx + φ α yz d y 2 z 2 d yz. explicit form in spherical harmonic basis (real Φ): ( φ zx iφ yz) ( φ zx + iφ yz) 3 8 iφ xy 1 4 ( φ zx + iφ yz) ( φ zx iφ yz) ( φ zx + iφ yz) ( φ zx + iφ yz) 3 ( φ zx iφ yz) 1 4 ( φ zx iφ yz) iφ xy 1 ( 4 φ zx iφ yz) ( 8 φ zx + iφ yz) ( 1 4 φ zx + iφ yz) ( φ zx iφ yz)

24 EC in cubic d 6 perovskite Exciton = bound pair of e g electron and t 2g hole How do we detect the EC order? Local d-occupation matrix (1 x 1): spin structure: orbital structure: D = D + φ z φ x + iφ y (φ x + iφ y ) D φ z φ α xy d x 2 y 2 d xy + φ α zx d z 2 x 2 d zx + φ α yz d y 2 z 2 d yz. The order parameter has 9 components (or 18 real components), φ α β α = x, y, z transforms like a vector under spin rotations β = x, ŷ, ẑ transforms like a pseudovector under O h operations The spin and orbital symmetry does not specify the ordered phase uniquely, possible solutions can be classified by their residual symmetry.

25 Examples of LDA+U EC solutions, φ α β LaCoO 3 AF-EC order local spin density residual group X D 4h U(1), X X X y D 3d U(1) X X X O h

26 Examples of LDA+U EC solutions LaCoO 3 AF-EC order, φ α β X X X X X X X i X E (i) [mev/f.u.]

27 Examples of LDA+U for PCCO orthorombic structure: 4 Co atoms per f.u. two inequivalent Co positions Product solution: h φ yz φ zx φ xy Orbital pseudovectors on sym. related Co atoms:

28 Origin of exchange splitting on Pr Coupling of Pr 4f 1 spin to p-d orbitals: effective multi-channel Kondo Hamiltonian H (n) = αα mm Below T c effective exchange field appears: i S σ αα J (n) i,mm c imα c im α +c.c. h (n) γ = J (n) i,mm imm αα 2Re c imα σγ αα c im α The site symmetry of the EC order parameter with respect to the Pr site decides whether contributions of from different Co site interfere constructively or destructively. For the present EC solution h= in the absence of spin-orbit coupling in Pr 4f shell. With SOC splitting on 1 mev scale is obtained.

29 Conclusions Solids close to spin-state transition may be unstable towards condensation of spinful excitons. The excitonic condensate breaks the spin rotation symmetry. The excitonic condensation can lead to a long-range order of magnetic multipoles (but there are other possibilities as well). Phys. Rev. B 89, (214) Phys. Rev. B 9, (214) Phys. Rev. B 9, (214)

30 Solid order - strongly asymmetric bands bipartite lattice ordered phase Local spin susceptibility Asymmetric bands X JK and Krapek, 211 HS are immobile (classical BEG model). The physics is dominated by HS-HS repulsion and HS-HS anti-ferromagnetic exchange.

31 Excitonic condensation (n=2) order parameter r φ = a b + a b Spectral density (diagonal elements) φ Σ ij Self-energy A bb (ev -1 ) A aa (ev -1 ) 2 Energy (ev) Energy (ev) τ Temperature Green s function off-diagonal diagonal β G ij (τ)

32 Excitonic condensation (n=2) order parameter r φ = a b + a b Spectral density (diagonal elements) φ Internal energy (ev) Energy (ev) Internal energy and specific heat Specific heat Specific heat (R) ln2 Entropy Spec. heat TS E F A bb (ev -1 ) A aa (ev -1 ) Energy (ev) Temperature

33 Excitonic condensation (doping) - all phases n-t phase diagram μ-t phase diagram T (K) 4 T (K) n h L phase µ (ev) E phase C phase nh - hole concentration ( N=2-nh)

34 r φ = a b + a b Excitonic condensation (fixed μ) φ n (electrons per atom) A bb (ev -1 ) A aa (ev -1 ) (a) (b) Temperature T n(t) = const 2 T > T c T < T c Energy (ev)

35 Excitonic condensation (fixed μ) r φ = a b + a b Spin susceptibility.4 φ.2 n (electrons per atom) n(t) = const 2 T > T c T < T c

36 Examples of EC solutions, φ α β LaCoO 3 AF-EC order local spin density residual group X D 4h U(1), X X X y D 3d U(1) X X X O h

Excitonic Condensation in Systems of Strongly Correlated Electrons. Jan Kuneš and Pavel Augustinský DFG FOR1346

Excitonic Condensation in Systems of Strongly Correlated Electrons Jan Kuneš and Pavel Augustinský DFG FOR1346 Motivation - unconventional long-range order incommensurate spin spirals complex order parameters