Density matrix renormalization group study of a three- orbital Hubbard model with spin- orbit coupling in one dimension

Transcription

1 Density matrix renormalization group study of a three- orbital Hubbard model with spin- orbit coupling in one dimension Nitin Kaushal, Jacek Herbrych, Alberto Nocera, Gonzalo Alvarez, Adriana Moreo and Elbio Dagotto University of Tennessee, Knoxville, USA Oak Ridge National Laboratory

2 Introduction and Motivation Iron Based Superconductors, Cuprates etc. Sr 2 RhO 4 [d 5 ], Sr 2 RuO 4 [d 4 ], etc. Sr 2 IrO 4 [d 5 ], Ba 2 YIrO 6 [d 4 ], etc. Spin- orbit coupling = λ(l.s) ; grows as Z # for outer shell electrons. Gang Cao, Pedro Schlotman, 2018 Rep. Prog. Phys.

3 Introduction and Motivation Iron Based Superconductors, Cuprates etc. Sr 2 RhO 4 [d 5 ], Sr 2 RuO 4 [d 4 ], etc. Sr 2 IrO 4 [d 5 ], Ba 2 YIrO 6 [d 4 ], etc. CF(Crystal Field) + Spin orbit coupling: d 2 3, d (e :) l =>> = 0 d orbitals l = 2 Δ BCD j =>> = 1/2,m = ±1/2 d 57, d 72, d 52 (t #< ) l =>> = 1 ~3λ/2 j =>> = 3/2,m = ±1/2 j =>> = 3/2,m = ±3/2

4 Introduction and Motivation Case: d L [Relevant for materials like Sr 2 YIrO 4, Ba 2 YIrO 6 ] Band Insulator j =>> = 1/2 E(k) Turn on the hopping µμ j =>> = 3/2 k Turn on the hopping+ Correlation [U] Very limited information?

5 Introduction and Motivation Case: d L [Relevant for materials like Sr 2 YIrO 4, Ba 2 YIrO 6 ] Band Insulator j =>> = 1/2 E(k) Turn on the hopping µμ j =>> = 3/2 k Turn on the hopping+ Correlation [U] AFM predicted by theory in strong coupling. Experiments on Sr 2 YIrO 6, Ba 2 YIrO 6 have also show AFM at T Q ~2K. 1.*G. Khaliullin, Phys. Rev. Lett. 111, (2013) 3. G.Cao et al., Phys. Rev. Lett. 112, (2014) 2.*C. Svoboda et al., Phys. Rev. B 95, (2017) 4. Q. Chen et al., Phys. Rev. B 96, (2017)

6 Focus of the work: Creating a λ vs U phase diagram for n=4 case. We aim to understand the phases emerging from SOC and U competition. Why is this important? Because ours is the first numerically exact study of SOC+U effects. Previous work relied on static and dynamical mean field theory*. Numerical Technique used: Density Matrix Renormalization Group technique is used to solve one- dimensional multi(3)- orbital Hubbard model in presence of spin- orbit coupling. *T. Sato et al., Phys. Rev. B 91, (2015) T. Sato et al., arxiv:

7 Model Increasing λ, for Non- interacting case Metal Band- insulator

8 Model Increasing λ, for Non- interacting case Metal Band- insulator

9 Phase Diagram [U vs λ] for n=4 Calculations are performed for L=16 [upto L=32] sites. J Z U = 1 4 N. Kaushal et al., Phys. Rev. B 96, (2017).

10 Phase Diagram [U vs λ] for n=4 Increasing λ, for Non- interacting case Weak coupling limit: Metal to Relativistic Band Insulator transition

11 Phase Diagram [U vs λ] for n=4 Block Phase Ferromagnetic + Orbital Ordering Both phases are well studied in literature on the same model: J. Rincon et al., Phys. Rev. Lett. 112, (2014) G. Liu et al., Phys. Rev. E 93, (2016) S. Li et al., Phys. Rev. B. 94, (2016)

12 site - i i + 1 i + 2 ȷ = 1/2, j = 3/2 m = ±1/2 Phase Diagram [U vs λ] for n=4 Antiferromagnetism + Excitons Coherent state with π phase + +

13 Phase Diagram [U vs λ] for n=4 Antiferromagnetism + Excitons Present for U/W > 0.2 Intermediate coupling is sufficient for Excitoniccondensate. 4d, 5d materials have small U~W Sr # YIrO L (d L ) U λ ~1.0, W ~0.23, W D3h W D3h ~ 2eV D3h S. Bhowal et al., Phys. Rev. B 92, (R)

14 Conclusion and importance of this work: First DMRG study on multiorbital Hubbard model with spin- orbit coupling. Full U vs λ phase diagram presented. We found excitonic magnetism in intermediate coupling region, within the physical region for 5d materials. Future Directions: Although doping 4d/5d materials is difficult experimentally, in numerical studies we can predict phases that may emerge from doping with both U and SOC.

15 THANK YOU!!

16 EXTRA SLIDES:

17 CF + hopping in (j eff, m) basis:

18 To have

19 Current Work Effect of different filling (n)?

20 Phase Diagram [n vs λ] for U/W= Excitons? Model: Degenerate t #< bands are used. Relevant for materials with nearly octahedral crystal field splitting. /W 0.25 AF M IC SDW n Incommensurate Spin density wave coming from nesting in bands. N. Kaushal et al. Unpublished (Under Progress)

21 Phase Diagram [n vs λ] for U/W= Excitons? Model: Degenerate t #< bands are used. Relevant for materials with nearly octahedral crystal field splitting. /W 0.25 AF M 0 IC SDW n Antiferromagnetic ordering is found for n < 4 as well. Promising region to look for Excitonic condensate? Incommensurate Spin density wave coming from nesting in bands. N. Kaushal et al. Unpublished (Under Progress)

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23 Excitons at Metal Insulator transition [In Intermediate coupling limit]: Semi- metal Semiconductor Binding energy of exciton E = μ/m = ε # µμ E µμ k Small no of carriers à Unscreened coloumb replusion à electron- hole pairs Energy to create Exciton = E E k If E < E à Spontaneous formation of Excitons B. Halperin and T. M. Rice, REVIEWS OF MODERN PHYSICS, VOLUME 40, No. 4, 1968

24 Excitons as Singlet- Triplet Excitations [In Strong coupling limit]: 1 state J = 0 6 states J = 1 8 states J = 2 In strong coupling limit ~J =5C ( ) Giniyat Khaliullin, PRL 111, (2013) Christopher Svoboda et al., PHYSICAL REVIEW B 95, (2017)

25 L # =>> in t #< and e < sectors: In t 2g : and

26 SOC term:

27 Introduction and Motivation Case: d Half- filled; In presence of e- e correlations and hopping, a Mott insulator is stabilized [opens a gap in j =>> = 1/2 band] Explain why materials like Sr 2 IrO 4 [d 5]*, Ba 2 IrO 4 [d 5 ] shows Mott insulating behavior. *B J Kim et al., Phys. Rev. Lett. 101, (2008)

28 Site - i Site - i+1 j =>> = 1/2, m = ±1/2 K. E. ~0 E (U, t, λ ) j =>> = 3/2, m = ±1/2 j =>> = 3/2, m = ±3/2 If Δ K. E + Int Energy + E < 0 Cost ~ E K. E < 0 Δ(Int. Energy) < 0 or