quantum phase transitions in correlated topological insulators · 2013. 6. 16. · correlation...
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EQPCM 2013 ISSP, June 12-14, 2013
Norio Kawakami
Quantum phase transitions in
correlated topological insulators
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Kyoto University
京都 M. Sigrist
R. Peters (PD) S. Fujimoto
Kyoto
ETH ISSP
S. Ueda (D2)
Y. Tada M. Oshikawa
Collaborators: Correlated Topological Insulators
T. Yoshida (D3)
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~Topological phases in correlated electron systems ~
LuPtBi (Heusler compound)
Lu
Pt
Sb
Chadov et al. Nature Materials (2010)
Lin et al. Nat. Mat. (2010)
Ir
O
Shitade et al. PRL (2009)
Na2IrO3
Electron correlations + SO coupling
SmB6 (Kondo insulator)
Dzero et al. PRL (2010)
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◇ Interaction-driven topological insulators
◇ Competing phases : [Topological phase] v.s. [ordered phases] magnetic phase, charge density wave phase
Coulomb interaction + Topological nature
Exotic phenomena are expected.
etc…
Correlation effects on Topological phases
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Interaction-driven TI ・S. Raghu, Xl. L. Qi, C. Henerkampp, and S. C. Zhang, PRL 100, 156401 (2008).
・Y. Zhang, Y. Rau, and A. Vishwanath, PRB 79, 245331 (2009).
・M. Kurita, Y. Yamaji, and M. Imada, J. Phys. Soc. Jpn. 80 044708 (2011).
・G. A. Fiete, V. Chua, X. Hu, M. Kargarian, R. Lundgren, A. Ruegg, J. Wen and V. Zyuzin, arXiv:1106.0013
Correlation Effects on TI ・Y. Yamaji and M. Imada, PRB 83, 205122 (2011).
・ M. Hohenadler, T. C. Lang, and F. F. Assad. PRL 106, 100403 (2011).
・ D. Zheng, C. Wu, and G-M Zhang, arXiv: 1011.5858v2 (2010).
・ S. L. Yu, X. C. Xie, and J. X. Li, PRL 107, 010401 (2011).
・ W. Wu , S. Rachel, W-M Liu, and K. L. Hur, arXiv:1106.0943v1. (2011)
・ S. Rachel and K. L. Hur, PRB 82 075106 (2010).
・ C. N. Varney, K. Sun, M. Rigol, and V. Galitski, PRB 82, 115125 (2010).
・ L. Wang, H. Shi, S. Zhang, X. Wang, X Dai and X. C. Xie, arXiv. 1012.5163v1 (2011).
・ T. Yoshida, S. Fujimoto, and N. Kawakami, PRB 85, 125113 (2012).
・ Y. Tada, R. Peters, M. Oshikawa, A. Koga, N. Kawakami, and S. Fujimoto, PRB 85, 165138 (2012).
Topological AF ・ R. S. K. Mong, A. M. Essin, and J. E. Moore, PRB 81, 245209 (2010)
・ A. M. Essin and V. Gurarie, arXiv:1112.6013v1 (2011)
・ J. He, Y-H, Zong, S-P Kou, Y Liang, and S. Feng, PRB 84, 035127 (2011)
・ J. He, B. Wang, and S-P Kou, arXiv:1204.4766 (2012)
・ H. Guo, S. Feng, and S-Q Shen, PRB 83 045114 (2011)
・ T. Yoshida, R. Peters, S. Fujimoto and N. Kawakami, PRB 87, 085134 (2013)
Topological Kondo Insulator ・ M. Dzero, K. Sun, V. Galitski, P. Coleman, PRL 104, 106408 (2010).
・ T. Takimoto, J. Phys. Soc. Jpn 80, 123710 (2011).
・ F. Lu, J. Zhao, H. Weng, Z. Fang, and X. Dai, arXiv:1211.5863
・ T.Yoshida, R. Peters, S. Fujimoto and N. Kawakami, PRB 87, 165109 (2013)
etc., etc
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Contents
Correlation Effects on Topological Insulators
1. Correlated TI at Finite T
2. Topological Kondo Insulator in a Metal
Collaboration, topology, ferromag, Kondo effect
Electron correlation
Strong renormalization effects
Edge states
Nontrivial phase in a metal
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Correlated Topological Insulators at Finite Temperatures
T. Yoshida, S. Fujimoto, NK
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Bernevig-Hughes-Zhang model +U
Orbital 2
Orbital 1
gap-closing
k
Trivial Large Δ SmallΔ Nontrivial
BHZ+U DMFT
Non-interacting case
Two-orbital system
Y
X
SO-coupling
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Spin Hall conductivity
Mott
[gap size]
Trivial Mott
TBI
Gap narrowing
Increase of effective temperature ([Temp.]/[Gap size])
: Quantized at T=0
jump
hysteresis
1st order transition
Gap narrowing
Δ = t
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Spectral function
:TBI
:MI
Mott transition
gap
(1st order transition)
Change of Topological structure without gap closing
Non-trivial trivial
TBI
MI
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TBI
MI
Gap renormalization
In generic band insulators
gap renormalization depends on the origin of the gap at U=0.
U≠0 case:
gets wider or narrower ?
Renormalization due to correlations => at T≠0
Gap Narrowing is generic behavior of TBIs.
Gap Spin orbit interaction (contributes to kinetic term)
(narrow the gap)
Important ! QPT
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Finite-size effects on correlated TBI
Y. Tada, R. Peters, M. Oshikawa, A. Koga, NK, S. Fujimoto
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Finite-Size Effects
y
x
10yL
Hubbard interaction
Spin-diagonal
0.1,25.0 0 Mt
Bernevig-Hughes-Zhang model
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Inhomogeneous DMFT
y
x
Site dependent LDOS
Advantage: Applicable for geometries with edges Extension to higher dimension is easy
Disadvantage: Spatial correlation is not incorporated cluster extensions will improve
Successful application to heterostructures, optical lattices
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Finite size effects
Edge states = massive renormalized massive Dirac Bulk = gapped Fermi liquid
◆Oscillations due to lattices ◆Ly = 10, Δf ~ 0.1ΔTI
]exp[~ 0 Lf
tTI 4~
M0
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Finite size effects with interaction DMFT + IPT
Bulk gap
Finite size gap
Z-factor
Finite size gap is simply renormalized Consistent with gapped Fermi liquid picture
Renormalization factor
M0
scaled finite size gap δf
Z-factor
U M
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Site-dependence
Strong renormalization at the edges due to reduction of coordinate number
✔Strong site dependence: z (edge) << z (bulk)
Renormalization factor
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Local density of states
Edge site Bulk site
incr
eas
ing
U
✔U<Uc: renormalized gap around ω~0
✔U>Uc: Mott insulating gap ΔMI~ U for all the sites
Generally, discontinuous transitions Gap closing is not required
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Competition between TBI and MI
BHZ model + U
Topological insulator
(Topologically trivial) Mott phase
1st order
no gap-closing
Strong renormalization: near Mott transition
Mott transition
✔renormalization of finite size gap
✔simple Mott transition
Finite size effects at T=0
Summary of Part I
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hidden in a metallic phase
Spin-selective Topological insulator
T.Yoshida, R. Peters, S. Fujimoto, NK
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● Topological Kondo Insulator at half filling
Coulomb interaction + Topological nature
Exotic phenomena
Correlation effects on Topological phases
M. Dzero et al. PRL (2010), PRB (2012).
M.T. Tran et al. PRB (2012), T. Takimoto, JSPS (2011)
Topological Kondo Insulator in a metallic phase !?
● SmB6 midgap state -> edges state
S. Wolgast et al, cond-mat 1211.5104 (2012) resistivity
J. Botimer et al, cond-mat 1211.6769 (2012) Hall effect
X. Zhang et al, cond-mat-1211.5532 (2012) tunneling
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Correlated f-electrons
+ c-electrons
At half-filling
Kondo insulator
Antiferromagnetic insulator
Away from half-filling
Paramagnetic metal
Ferromagnetic metal
Unconventional superconductor
Intriguing phenomena
Topological
Kondo insulator
SmB6 ? SO coupling
????
SO coupling
Spin-selective
Topological
Kondo insulator
f
c
Heavy fermion systems Ce, Yb, Sm…
New phase
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Y
X
Ferromagnetic metallic phase: spin-selective Kondo insulator
(half-metallic, half-insulating)
Topologically nontrivial phase
Spin-selective topological Kondo insulator
~Topological ins. in a metallic phase ~
c-f mixing (SO coupling)
DMFT calculation
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Phase Diagram: Topological PAM
RKKY
Kondo
Antiferro.
Topological Kondo insulator (spin Chern #=1) (M. Dzero et al.) possibly SmB6
Topological Chern # =1
Spin-selective Topological Kondo insulator
[conduction band width] =8
Ferromagnetic
metal
Half-filling
hole dope
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up: gap down: metallic
energy
DOS( ) DOS( )
energy
DOS( ) DOS( )
energy
DOS( ) DOS( ) topological insulator hidden in metallic phase!
RKKY+Kondo
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DOS (up-spin)
RKKY and Kondo effect reconstruct the gap and
restore topological properties! Only is changed
electron dope
Kondo gap
Electron filling
filling
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Phase Diagram: Topological PAM
RKKY
Kondo
Topological Chern # =1
Spin-selective Topological Kondo insulator
Ferromagnetic
metal
Half-filling
energy
DOS( ) DOS( )
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Chiral edge mode (up spin)
+ 2D ferromag. fluctuations
Edge states
Tomonaga boson
Bulk spin- fluctuations
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cf 2D systems
Spin fluctuations become stronger
Edge contribution becomes dominant
Non-Tomonaga-Luttinger (dissipative behavior)
NMR relaxation rate
3
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bulk gap is induced by interaction.
⇒ Kondo insulator in Ferromagnetic metal
Even in a metallic phase
Spin-selective Topological Kondo insulator half-metallic, half insulating
Collaboration: Topology & Correlation
Correlated Topological insulator
Topological Kondo insulator
Summary of Part II
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Summary
Correlation Effects on Topological Insulators
1. Correlated TI: Mott transition
2. Topological Kondo Insulator in a Metal
Collaboration, topology, ferromag, Kondo effect
Electron correlation
Strong renormalization effects
Edge states
Nontrivial phase in a metal