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

Transcription

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

2 Motivation - unconventional long-range order incommensurate spin spirals complex order parameters spin-orbit multipoles Typical (conventional) long-range order involves charge- or spin-density modulations in inter-atomic scale. Unconventional order involves intra-atomic density modulations (e.g. magnetic multipoles) or no density modulations (e.g. ordered current patterns).

3 Outline Particle-hole instabilities of the two-band Hubbard model Excitonic condensation (EC) with DMFT EC in strong and weak coupling limits EC in real materials Dynamical Mean-Field Theory

4 Spin-state transition Spin-state transition is a rapid change of magnitude or disappearance of the fluctuating or ordered local moment. Mn, Fe, Co compounds under high pressures (~ 10 GPa) low spin S=0 high spin S=1 Local degeneracy at the spin-state transition may lead to long-range ordering.

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 ( 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) orbital diagonal orb. off-diagonal spin longitudinal Excitonic instability close to spin-state transition Two-band Hubbard model at n=2 (half filling) χ λ 4 2 [0,0] [0,π] [π,π] [0,0] (1) [0,π] [π,π] [0,0] [0,π] [π,π] U/(t a +t b ) 4 2 HS Mott insulator metal LS band insulator /(t a +t b ) (b) χ -1 (arb. units) Temperature (K)

6 Excitonic instability close to spin state transition U=4, J=1 t a2 +t b2 =const Δ=3.40+δ V=0 Temperature (K) solid order normal phase excitonic insulator (superfluid) gap closing Solid HS-LS order: non-degenerate mode (staggered crystal-field) no weak coupling analog checker board order ζ ζ = 2t at b. t 2 a +t2 b cussion of t excitonic insulator (superfluid): degenerate mode (magnetic channel) exists also at weak coupling (BCS-BEC crossover) possible at various q JK and Augustinsky, 2014

7 Excitonic instability close to spin state transition U=4, J=1 t a2 +t b2 =const Δ=3.40+δ V=0 Temperature (K) solid order normal phase excitonic insulator (superfluid) gap closing Solid HS-LS order: non-degenerate mode (staggered crystal-field) no weak coupling analog checker board order ζ ζ = 2t at b. t 2 a +t2 b cussion of t excitonic insulator (superfluid): degenerate mode (magnetic channel) exists also at weak coupling (BCS-BEC crossover) possible at various q JK and Augustinsky, 2014

8 Solid order - strongly asymmetric bands bipartite lattice ordered phase Local spin susceptibility Asymmetric bands X JK and Krapek, 2011 HS are immobile (classical BEG model). The physics is dominated by HS-HS repulsion and HS-HS anti-ferromagnetic exchange.

9 Excitonic condensation U=4, J=1 t a2 +t b2 =const Δ=3.40+δ V=0 Temperature (K) solid order normal phase excitonic insulator (superfluid) gap closing Solid HS-LS order: non-degenerate mode (staggered crystal-field) no weak coupling analog checker board order ζ ζ = 2t at b. t 2 a +t2 b cussion of t excitonic insulator (superfluid): degenerate mode (magnetic channel) exists also at weak coupling (BCS-BEC crossover) possible at various q JK and Augustinsky, 2014

10 Excitonic condensation order parameter r φ = a b + a b Spectral density (diagonal elements) φ Temperature (K) A bb (ev -1 ) A aa (ev -1 ) Temperature Energy (ev)

11 Excitonic condensation order parameter r φ = a b + a b Spectral density (diagonal elements) φ Temperature (K) Σ ij Self-energy A bb (ev -1 ) A aa (ev -1 ) 0 20 Energy (ev) Energy (ev) τ Temperature Green s function off-diagonal diagonal 0 β G ij (τ)

12 Excitonic condensation order parameter r φ = a b + a b Optical conductivity (dc resistivity) φ Temperature (K) Uniform spin susceptibility Conductivity (arb. units) Resitivity (arb. units) Temperature T (K) Energy (ev)

13 Strong coupling picture of excitonic condensation Strong coupling: HS states behave as hard-core bosons with the vacuum state vac LS Bose-Einstein condensation = spontanous hybridization between HS and LS states on the same site (breaks spin rotational symmetry)

14 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

15 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 (10 x 10): spin structure: orbital structure: D = D 0 + φ z φ x + iφ y (φ x + iφ y ) D 0 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.

16 Examples of LDA+U EC solutions, φ α β LaCoO 3 AF-EC order local spin density residual group D 4h U(1), y D 3d U(1) X X 0 O h

17 Examples of EC solutions, φ α β LaCoO 3 AF-EC order local spin density residual group D 4h U(1), y D 3d U(1) X X 0 O h

18 Examples of LDA+U EC solutions LaCoO 3 AF-EC order, φ α β X X 0 i X E (i) [mev/f.u.]

19 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, (2014) arxiv:

20 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:

21 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=0 in the absence of spin-orbit coupling in Pr 4f shell. With SOC splitting on 10 mev scale is obtained.