Density(matrix(renormalization(group(study( of ...
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Density matrix renormalization group study of a three-‐orbital Hubbard model
with spin-‐orbit coupling in one dimensionNitin Kaushal, Jacek Herbrych, Alberto Nocera, Gonzalo Alvarez,
Adriana Moreo and ElbioDagotto
University of Tennessee, Knoxville, USAOak Ridge National Laboratory
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Introduction and Motivation
Iron Based Superconductors, Cuprates etc.
Sr2RhO4 [d5], Sr2RuO4[d4], etc.
Sr2IrO4[d5], Ba2YIrO6[d4], etc.
Spin-‐orbit coupling = λ(L.S) ; grows as 𝑍# for outer shell electrons.
Gang Cao, Pedro Schlotman, 2018 Rep. Prog. Phys.
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Introduction and Motivation
Iron Based Superconductors, Cuprates etc.
Sr2RhO4 [d5], Sr2RuO4[d4], etc.
Sr2IrO4[d5], Ba2YIrO6[d4], etc.
CF(Crystal Field) + Spin orbit coupling:
“𝑑” 𝑜𝑟𝑏𝑖𝑡𝑎𝑙𝑠
𝑙 = 2
𝑑23 , 𝑑53673 (e:)
𝑑57, 𝑑72, 𝑑52(𝑡#<)
𝑙=>> = 0
𝑙=>> = 1
ΔBCD
~3𝜆/2
𝑗=>> = 3/2,𝑚 = ±1/2𝑗=>> = 3/2,𝑚 = ±3/2
𝑗=>> = 1/2,𝑚 = ±1/2
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Introduction and MotivationCase: 𝑑L [Relevant for materials like Sr2YIrO4 , Ba2YIrO6 ]
Turn on the hopping
Turn on the hopping+ Correlation [U]Very limited information ?
µμ
𝐸(𝑘)
𝑘
𝑗=>> = 1/2
𝑗=>> = 3/2
Band Insulator
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Introduction and MotivationCase: 𝑑L [Relevant for materials like Sr2YIrO4 , Ba2YIrO6 ]
Turn on the hopping
Turn on the hopping+ Correlation [U]AFM predicted by theory in strong coupling.Experiments on Sr2YIrO6, Ba2YIrO6 have also show AFM at 𝑇Q~2𝐾 .
µμ
𝐸(𝑘)
𝑘
𝑗=>> = 1/2
𝑗=>> = 3/2
Band Insulator
1.*G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013) 3. G.Cao et al., Phys. Rev. Lett. 112, 056402 (2014)2.*C. Svoboda et al., Phys. Rev. B 95, 014409 (2017) 4. Q. Chen et al., Phys. Rev. B 96, 144423 (2017)
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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, 125122 (2015) T. Sato et al., arXiv:1603.01800
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Model Increasing 𝜆, for Non-‐interacting case
Metal Band-‐insulator
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Model
Metal Band-‐insulator
Increasing 𝜆, for Non-‐interacting case
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Calculations are performed for L=16 [upto L=32] sites.
N. Kaushal et al., Phys. Rev. B 96, 155111 (2017).
Phase Diagram [𝑼 𝒗𝒔 𝝀] for n=4
𝐽Z𝑈 =
14
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Weak coupling limit:Metal to Relativistic Band Insulator transition
Increasing 𝜆, for Non-‐interacting case
Phase Diagram [𝑼 𝒗𝒔 𝝀] for n=4
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Block Phase
Ferromagnetic + Orbital Ordering
Both phases are well studied in literature on the same model:J. Rincon et al., Phys. Rev. Lett. 112, 106405 (2014)G. Liu et al., Phys. Rev. E 93, 063313 (2016)S. Li et al., Phys. Rev. B. 94, 235126 (2016)
Phase Diagram [𝑼 𝒗𝒔 𝝀] for n=4
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Phase Diagram [𝑼 𝒗𝒔 𝝀] for n=4Antiferromagnetism + Excitons
�̃� = 1/2, 𝑗 = 3/2𝑚 = ±1/2𝑠𝑖𝑡𝑒 -‐ 𝑖 𝑖 + 1 𝑖 + 2
Coherent state with 𝜋 phase
+ − +
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Antiferromagnetism + Excitons• Present for U/W > 0.2 • Intermediate coupling is sufficient for
Excitonic condensate.• “4d, 5d materials have small 𝑼~𝑾”
Phase Diagram [𝑼 𝒗𝒔 𝝀] for n=4
𝑆𝑟#𝑌𝐼𝑟𝑂L (𝑑L)𝑈𝑊D3h
~1.0, 𝜆𝑊D3h
~0.23,
S. Bhowal et al., Phys. Rev. B 92, 121113(R)
𝑊D3h~ 2𝑒𝑉
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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.
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THANK YOU!!
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EXTRA SLIDES:
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CF + hopping in (jeff , m) basis:
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To have
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Current WorkEffect of different filling (n)?
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0
0.25
0.5
3 3.5 4 4.5 5
�/W
n
AFM
Excitons?
IC � SDW
Phase Diagram [𝒏 𝒗𝒔 𝝀] for 𝑼/𝑾=𝟏. 𝟎
N. Kaushal et al. Unpublished (Under Progress)
Model:• Degenerate 𝑡#< bands are used.• Relevant for materials with nearly
octahedral crystal field splitting.
Incommensurate Spin density wave coming from nesting in bands.
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0
0.25
0.5
3 3.5 4 4.5 5
�/W
n
AFM
Excitons?
IC � SDW
Phase Diagram [𝒏 𝒗𝒔 𝝀] for 𝑼/𝑾=𝟏. 𝟎
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.
Model:• Degenerate 𝑡#< bands are used.• Relevant for materials with nearly
octahedral crystal field splitting.
N. Kaushal et al. Unpublished (Under Progress)
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Excitons at Metal –Insulator transition [In Intermediate coupling limit]:
µμ
𝑘
Semi-‐metal
Small no of carriersà Unscreened coloumbreplusionàelectron-‐hole pairs
µμ
𝑘
Semiconductor
𝐸�
𝐸� = 𝜇/𝑚=𝜖#Binding energy of exciton
Energy to create Exciton= 𝐸� − 𝐸�
If 𝐸� < 𝐸� à Spontaneous formation of Excitons
B. Halperin and T. M. Rice, REVIEWS OF MODERN PHYSICS, VOLUME 40, No. 4, 1968
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Excitons as Singlet-‐Triplet Excitations[In Strong coupling limit]:
𝐽 = 0 𝐽 = 1 𝐽 = 21 state 6 states 8 states
In strong coupling limit
~𝐽=5C( )
Giniyat Khaliullin, PRL 111, 197201 (2013)Christopher Svoboda et al., PHYSICAL REVIEW B 95, 014409 (2017)
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𝐿=>># 𝑖𝑛 𝑡#< 𝑎𝑛𝑑 𝑒< 𝑠𝑒𝑐𝑡𝑜𝑟𝑠:
and
In 𝒕𝟐𝒈:
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SOC term:
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Introduction and MotivationCase: 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 Sr2IrO4[d5]*, Ba2IrO4[d5] shows Mott insulating behavior.
*B J Kim et al., Phys. Rev. Lett. 101, 076402 (2008)
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𝑗=>> = 3/2, 𝑚 = ±3/2𝑗=>> = 3/2, 𝑚 = ±1/2
𝑗=>> = 1/2, 𝑚 = ±1/2
Site -‐ 𝑖 Site -‐ 𝑖+1
𝐾.𝐸. ~0
𝐾.𝐸 < 0 𝑜𝑟Δ(𝐼𝑛𝑡. 𝐸𝑛𝑒𝑟𝑔𝑦) < 0
𝐶𝑜𝑠𝑡 ~ 𝐸�
If Δ 𝐾.𝐸 + 𝐼𝑛𝑡 𝐸𝑛𝑒𝑟𝑔𝑦 + 𝐸� < 0
𝐸�(𝑈, 𝑡, 𝜆 )