quantum exotic states in correlated topological insulators su-peng kou ( 寇谡鹏 ) beijing normal...
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Quantum exotic states in Quantum exotic states in correlated correlated topological insulatorstopological insulators
Su-Peng Kou (寇谡鹏 )
Beijing Normal University
OutlineOutline MotivationMotivation Topological spin density waves in correlated Topological spin density waves in correlated
topological insulatorstopological insulators Quantum spin liquid states in correlated Quantum spin liquid states in correlated
topological insulatorstopological insulators ConclusionConclusion
[1] Kou SP , PHYS. REV. B 78, 233104 ( 2008 ) .
[2] Sun GY and Kou SP, EPL, 87 67002 (2009).
[3] Kou SP, and Liu LF , EUR. PHYS. J. B. 81, 165 (2011) .
[4] Sun GY and Kou SP, J. Phys. C 23 (2011) 045603.
[5] He J, Kou SP, Liang Y, Feng SP, PHYS. REV. B 83, 205116 (2011) .
[6] He J, Zong YH, Kou SP, Liang Y, Feng SP, PHYS. REV. B 84, 035127 (2011) .
[7] He J, Liang Y, Kou SP, PHYS. REV. B. 85, 205107 (2012).
[8] He J, Wang B, Kou SP, PHYS. REV. B. submitted, arXiv:1204.4766.
[9] Kou SP, “Insulators: Types, Properties and Uses” (Nova Science Publishers).
I. Motivation: LookLook for quantum exotic states in for quantum exotic states in
correlated topological insulatorcorrelated topological insulator
X. G. Wen, Quantum Field Theory of Many-Body Systems
Spin liquid – emergent in physicsSpin liquid – emergent in physics
No broken symmetry
Deconfined spinons+
Spin liquid
Emergent gauge field+
Spin orders in strongly correlated electron systems
G. Misguich, arXiv:cond-mat/0310405
II. Topological spin density wave states II. Topological spin density wave states in correlated topological insulatorsin correlated topological insulators
Instability of an interacting fermion system with topologically nontrivial band structure
1.1. Interacting spinful Haldane modelInteracting spinful Haldane model2. Interacting Kane-Mele model
The spinful Haldane model – spin rotation symmetry, no T symmetry
The Kane-Mele model – T symmetry, no spin
rotation symmetry Kane and Mele, Phys. Rev. Lett.
95, 146802 (2005)
http://www.physics.upenn.edu/~kane/
Possible quantum spin liquid in the interacting Kane-Mele model – T symmetry,
no spin rotation symmetry
Slave-rotor theory: Stephan Rachel and Karyn Le Hury, Phys. Rev. B 82. 075106 (2010)
QMC: M. Hohenadler, T. C. Lang, F. F. Assaad, Phys. Rev. Lett. 106, 100403 (2011) Dong Zheng, Congjun Wu and Guang-Ming Zhang, Phys. Rev. B 84, 205121 (2011)
VCA : Shun-Li Yu, X.C. Xie, Jian-Xin Li, Phys. Rev. Lett. 107, 010401 (2011)
DMF: Wei Wu, S. Rachel, Wu-Ming Liu, K. Le Hur, Phys. Rev. B 85, 205102 (2012)
1. Topological spin-density-wave states Topological spin-density-wave states in interacting spinful Haldane model in interacting spinful Haldane model - spin rotation symmetry, no T symmetry
He J, Zong YH, Kou SP, Liang Y, Feng SP, PHYS. REV. B 84, 035127 (2011)
What is the ground state for the spinful Haldane model with the on-site interaction?
Mean field equation
where
Mean field approach
M is the staggered magnetization.
Phase diagramPhase diagram
C=2 topological insulator - QAH
Band insulator
Trivial AF-SDW order
B-type topologicalSDW order
A-type topologicalSDW order
3.0 3.1 3.2 3.3 3.40.0
0.1
0.2
0.3
0.4
0.5
U0.0
0.5
1.0
AF
A-TSDW
QAHM
B-TSDW
Low energy effective modelLow energy effective model
K-matrix formulationK-matrix formulation
Spin-charge separated charge-Spin-charge separated charge-flux binding effect in A-TSDWflux binding effect in A-TSDW
spin-charge synchronization charge-spin-charge synchronization charge-flux binding effect in B-TSDWflux binding effect in B-TSDW
Different spin-density-wave states in correlated topological insulators with the same local order parameter may have different topological properties, including the induced quantum numbers on topological objects, the edge states, the quantum Hall effects.
2. Quantum spin orders in 2. Quantum spin orders in correlated topological insulator with correlated topological insulator with
flat-bandflat-band
Possible fractional quanum hall states
1. What is the ground state for the correlated topological insulators in the flat-band limit?
2. What’s the dispersion of electrons and spin waves for correlated topological insulators in the flat-band limit?
Phase diagram : electrons on TFBPhase diagram : electrons on TFB
d is the hole concentration.
FM (topological) spin-density-wave
Dispersion of electrons in A-TSDW
Dispersion of spin-waves in
A-TSDW
A-TSDW : Half filling caseA-TSDW : Half filling case
qq )(
A-TSDW
AF-SDW
TFB
TFB
FM (topological) spin-density-wave: quarter filling caseFM (topological) spin-density-wave: quarter filling case
Dispersion of electrons in FM order
Dispersion of spin wave in FM order
2)( qq
FM order and AF order :FM order and AF order :d=0.3 filling cased=0.3 filling case
Dispersion of electrons
Order parameters
III. Quantum spin liquids in III. Quantum spin liquids in interacting spinful Haldane modelinteracting spinful Haldane model
Short range A-type topological spin Short range A-type topological spin
density wave state: density wave state: chiral spin liquidchiral spin liquid
Short range B-type topological spin Short range B-type topological spin
density wave state : density wave state : composite spin composite spin
liquid liquid
Quantum spin-fluctuations in topological spin density wave states
Transverse spin susceptibility is
Spin coupling constant
Spin wave velocity
X. G. Wen, Quantum Field Theory of Many-Body Systems,(Oxford Univ. Press, Oxford, 2004)
One obtains spin stiffness and the transverse spin susceptibility:
nz
H.J. Schulz, in The hubbard Model, edited by D. Baeriswyl(Plenum, New York, 1995).Z. Y. Weng, C. S. Ting, and T. K. Lee, Phys. Rev. B43, 3790 (1991).K. Borejsza, N. Dupuis, Euro Phys. Lett. 63, 722 (2003); Phys. Rev. B 69, 085119 (2004).
Spin coupling constant Spin coupling constant
t’=0.0228tt’=0.1t
? ?
?
What is the nature of the quantum What is the nature of the quantum disordered states for TSDWs? disordered states for TSDWs?
S. Chakravarty, et al., Phys. Rev. B 39, 2344 (1989).
He J, Liang Y, Kou SP, PHYS. REV. B. 85, 205107 (2012).
Properties of chiral spin liquidProperties of chiral spin liquid
Spinon is semion with fractional statisticsSpinon is semion with fractional statistics Ground state degeneracy : 2 on torusGround state degeneracy : 2 on torus Chiral gapless edge states Chiral gapless edge states
He J, Liang Y, Kou SP, PHYS. REV. B. 85, 205107 (2012).
X. G.Wen, F.Wilczek, and A. Zee, Phys. Rev. B 39, 11413 (1989).
Slave-rotor approachSlave-rotor approach
Mean field approachMean field approach
C=2 topological insulator Chiral spin liquid
Trvial AF order
A-TSDW
Chiral spin order parameterChiral spin order parameter
π- vortex is semion
Statistics angle θ = π/2
With induced fermion numbWith induced fermion number , er , ππ-vortex beco-vortex becomes mes semionsemion. .
1fN
Effective Lagrangian from Effective Lagrangian from slave-rotor approach slave-rotor approach
N = 4
?
S=1/2, charge S=1/2, charge e fermione fermion
Composite spin liquid spin liquid
2.60 2.650.00
0.05
0.10E
nerg
y ga
p 2.55 2.60 2.65
0.0
0.2
0.4Electron's energy gap
U/t
E/t
U/t
B-TSDW
Charged skyrmion gap
Spin gap
g > gc g < gc
S=1/2, charge e fermionS=1/2, charge e fermion
?
To be confirmed by QMC, …
IV. ConclusionIV. Conclusion
?
Thanks for your attention!
Spin susceptility of spin order Spin susceptility of spin order in metallic spin orderin metallic spin order
1. Spin liquid 1. Spin liquid in the π-flux Hubbard model and the Hubbard model on honeycomb lattice
tt ji ,ittttyx eiieii ,, ,
Quantum spin liquid near Mott transition of Quantum spin liquid near Mott transition of ππ--flux Hubbard modelflux Hubbard model
Sun GY and Kou SP, EPL, 87 67002 (2009).Kou SP, Liu LF, He J, Wu YJ , EUR. PHYS. J. B. 81, 165 (2011).
Gapless Gapless Z2 topological spin liquid Z2 topological spin liquid
There are three types of quasi-particles : gapped fermionic spinons, gapped bosonic spinons and the gapped gauge field.
Nodal spin liquidNodal spin liquid
There are three types of quasi-particles : gapless fermionic spinons, gapped bosonic spinons and the roton-like gauge fie
ld.
Results from QMCResults from QMC
Chia-Chen Chang and Richard T. Scalettar, Phys. Rev. Lett. 109,
026404 (2012)
Global Phase diagram by spin-fluctuation theory
Sun GY and Kou SP, J. Phys. C. 23 (2011)
045603
Quantum spin liquid from QMCQuantum spin liquid from QMC
Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad & A. Muramatsu, Nature 464, 847 (2010)
Results of the Hubbard Model on the Results of the Hubbard Model on the Honeycomb Lattice from QMC of bigger sizeHoneycomb Lattice from QMC of bigger size
Sandro Sorella, Yuichi Otsuka, Seiji Yunoki, arXiv:1207.1783.
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