fermionic symmetry protected topological phase induced by interaction shangqiang ning first year phd...
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Fermionic Symmetry Protected Topological Phase
Induced by Interaction
Shangqiang NINGFirst year PHD student
Institute For Advanced Study, Tsinghua University Email: [email protected]
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Collaborators
Hongchen Jiang Slac, Stanford
Zhengxin LiuIASTU, Beijing
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Outline
1. Background and motivation 2. Study of a specific model 3. Conclusion
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Background and Motivation
2D topological insulator
Key properties of TI:
Bulk gapGapless edge states protected by time reversal symmetry and U(1)No symmetry breaking
Band insulator
Mass terms
breaks T breaks U(1)
)B(
)(
X. Qi, S.C. Zhang RevModPhys.83.1057
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Symmetry Protected Topological phase (SPT phase)
Key points of SPT No symmetry breaking bulk is gapped boundary excitation is gapless protected by symmetry
gapped
gapless
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Classification of SPT
Free fermion systems are classified by K-theory
For more symmetry group in
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β SPT phases also exist in Bosonic systems
β’ S=1 Haldane phase (Bosonic)
β’ Finite excitation gapβ’ No symmetry breakingβ’ SO(3) symmetry protected
spin-1/2 edge states β’ Degeneracy of entanglement spectrum
Spin-1/2Spin-1
Haldane 83; Affleck, Kennedy, Lieb, Tasaki 87Gu, Wen 09; Pollmann et.al 09; Chen,Gu,Wen 11
bulk gap
SO(3) symmetry
π―=βπ
π±ππΊπ βπΊπ+π
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entanglement spectrum for 1D
A B
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Entanglement spectrum
Entanglement spectrum
0 2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
Ent
angl
emen
t S
pect
rum
0 2 4 6 8 10 120
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Ent
angl
emen
t S
pect
rum
Entanglement spectrum
J2/J1=0.9
SO(3) symmetry
Non trivial Trivial
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Classification of Bosonic SPT
1D by the projective representation SO(3) : 2 projective representation
2D or higher by group cohomology
SO(3) symmetry: 2 phases
Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen,2011
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1D by projective representation 2D or higher partially by super-cohomology effect of interaction
Classification of Interacting Fermionic SPT
1D superconductor
2D superconductorTime reversal and mirror reflect Z Z8
It seems that interactions reduce the classification.However, it maybe not true!
Hong Yao and Shinsei Ryu, arXiv :1202.5805 Evelyn Tang and Xiao-Gang Wen arXiv:1204.0520v2
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Model
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Symmetry and classification
The model : U(1) and time reversal symmetry T
Classification
free case, only the trivial phase.
2 projective representation two SPT phases.
π 2=1
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Non-interacting limit U=0, J=0
=
2-fold 2-fold 2-fold
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Non-interacting limit U=0, J=0
SPT only focus on gapped phases.
π‘1>83π‘
0
πΌπππππππππ π‘1>83π‘
0
,ππππ¦π‘πππ£πππ hπ ππ π ,π .πππππ πππ π’πππ‘ππ
Open boundary, no zero mode,indicating the trivialness of phase
πΌπ13ππππππππππ π
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Interacting case
Add Hubbard interaction π»π’=πβπ
(π πβ1)2
the band insulator and the Mott insulator are adiabatically connected
the charge are localized at each rung in ground state.
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Interacting case
Add Heisenberg-like interaction ,
This term competes to exhibit the Haldane phaseοΌ so in the limit , the system would fall into Haldane phase.
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Interacting case
metal phase, trivial insulator phase, nontrivial Haldane phase.
For finite-parameters region, the system maybe exist the trivial and nontrivial gapped phases.
Competition
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Interacting caseπππ‘ π‘0=1 , π½=0.5 πππ·ππ πΊπ πππ’πππ‘πππ
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Interacting caseπππ‘ π‘0=1 , π½=0.5 πππ·ππ πΊπ πππ’πππ‘πππ
β
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Interacting caseπππ‘ π‘0=1 , π½=0.5 πππ·ππ πΊπ πππ’πππ‘πππ
β
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Interacting caseπππ‘ π‘0=1 , π½=0.5 πππ·ππ πΊπ πππ’πππ‘πππ
β
β
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Interacting caseπππ‘ π‘0=1 , π½=0.5 πππ·ππ πΊπ πππ’πππ‘πππ
β
β
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Interacting caseπππ‘ π‘0=1 , π½=0.5 πππ·ππ πΊπ πππ’πππ‘πππ
β β
β
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Interacting caseπππ‘ π‘0=1 , π½=0.5 πππ·ππ πΊπ πππ’πππ‘πππ
β β
β
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We find two phases in the interacting caseοΌ trivial insulator and Haldane insulator
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Conclusion
introduction of SPT
Interaction can induce new SPT phases for fermionic systems.
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Thanks for your attention!