Topological crystalline insulators and topological magnetic
crystalline insulators
Chaoxing Liu
Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
Collaborated with Ruixing Zhang, Xiaoyu Dong (Tsinghua), and Brian VanLeeuwen
cond-mat/1304.6455, Phys. Rev. B 90, 085304 cond-mat/1401.6922
Topological insulators and mathematical science Harvard University
Outline
• Introduction: topological insulators and topological crystalline insulators
• Symmetry induced degeneracy and surface states
• Topological crystalline insulators and topological magnetic crystalline insulators
• Summary and outlook
2
Topological states of matters
• Band theory of metals and insulators
Insulating Metallic
Ef Ef
Eg
Transport properties of materials are usually determined by bulk band gap and the position of Fermi energy in an electronic system.
3
Topological states of matters
• Topological phases of free fermions
4
Topological phases are usually characterized by insulating bulk states and metallic edge/surface states in a free fermion system. e.g. quantum Hall effect
Symmetry protected topological phases
• Time reversal (TR) invariant topological insulators (TIs) Topological phases due to the protection of time reversal symmetry, helical edge states
5
Topological insulators
• Experimental observation of helical edge states in TR invariant TIs
6
Bernevig, et al (2006), Konig, et al (2007)
Hsieh, et al, (2008), (2009); Roushan, et al, (2009), H.J. Zhang et al (2009); Xia et al (2009); Y. L. Chen (2009)
Topological insulators
• Kramers’ theorem
7
𝐸 𝑘, ↑ = 𝐸(−𝑘, ↓)
𝒌
𝑬
𝟎 𝝅 −𝝅 𝑘 = −𝑘 + �⃗�
• Non-trivial surface states protected by Kramers’ degeneracy
Topological crystalline insulators
• Can degenerate points be protected by other symmetries?
8
SnTe, mirror symmetry Fu (2011); Timothy, et al, (2012)
• Gapless edge/surface states protected by crystalline symmetry, topological crystalline insulators (TCIs)
Topological crystalline insulators
• Experimental observations of TCIs
9 SuYang Xu, et al (2012) Dziawa, et al (2012) Tanaka, et al (2012)
Topological crystalline insulators
• How to find a practical and systematical way to identify which types of crystalline symmetry group can protect non-trivial surface states?
10
Taylor, et al (2010), Fang, et al (2012), Slager, et al (2012), Jadaun, et al (2012)
Most of these works are constructing bulk topological invariants based on bulk symmetry directly. However, surfaces can break crystalline symmetry of a bulk system.
Topological crystalline insulators
11
• Our strategy: classification based on symmetry groups of surface states.
Look for gapless surface states in a semi-infinite system
𝒌𝒚
𝒌𝒙 𝚪� 𝐗�
𝒀� 𝑴�
Topological crystalline insulators
12
• Degeneracy and symmetry
Different irreducible representations (type I) E.g. Mirror topological insulators
Non-commutation relation, high dimensional irreducible representations (type II) E.g. non-symmorphic topological insulators
Anti-unitary symmetry operator (type III) E.g. Time reversal invariant topological insulators, magnetic topological insulators
Wigner, ...
• How to protect these degeneracies?
Summary of our approach
• Determine 2D space symmetry group of a semi-infinite bulk system with one surface
• Determine wave vector groups of each momentum in the surface Brillouin zone.
13
• Determine degeneracies of high symmetric momenta from representations of wave vector group.
• Determine possible non-trivial surface states and bulk topological invariants.
Γ
Bulk BZ
Surface BZ
Topological crystalline insulators
• Advantages of our approach
14
We can guarantee the existence of surface states. We show how to use the representation theory of
symmetry groups to classify different surface states.
There are only 17 2D space group (wall paper group). Therefore, it is possible to get a complete study of all the possible groups.
Topological crystalline insulators
• 17 2D space group, wall paper group
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Group Deg Group Deg P1 No Pmm No P2 No Cmm No P4 No Pmg I, II P3 No Pgg I, II P6 No P4m I, II Pm I P4g I, II Pg I P3m1 I, II Cm I P31m I, II
P6m I, II
Type I TCIs
• Degeneracy due to different representations of a symmetry group
16
𝒌𝒚
𝒌𝒙 𝚪�
𝜎𝑦
Pm group: a line in the Brillouin zone has symmetry 𝜎𝑦: 𝑥,𝑦, 𝑧 →(𝑥,−𝑦, 𝑧)
Type I TCIs
• Two surface states will not couple to each other if they belong to different representations
• Corresponding to mirror Chern number, mirror topological insulators
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𝒌𝒚
𝒌𝒙 𝚪�
𝚪� 𝑿� −𝑿�
𝑬
−𝑿� 𝑿�
𝐻 = 𝐻+ 00 𝐻−
SnTe system: Timothy, et al, (2012)
Topological crystalline insulators
• 17 2D space group, no anti-unitary operators
18
Group Deg Group Deg P1 No Pmm No P2 No Cmm No P4 No Pmg I, II P3 No Pgg I, II P6 No P4m I, II Pm I P4g I, II Pg I P3m1 I, II Cm I P31m I, II
P6m I, II
Type II TCIs
19
• Degeneracy due to high dimensional irreducible representations, non-commutation between symmetry operations
Eg. zinc-blende semiconductors
Type II TCIs
20
• A special case, anti-commutation relation
𝑅,𝐻 = 0, 𝑆,𝐻 = 0, 𝑅, 𝑆 = 0
If 𝐻 𝜙 = 𝐸|𝜙⟩ and 𝑅 𝜙 = 𝑟|𝜙⟩, 𝑆|𝜙⟩ and |𝜙⟩ are two orthogonal and degenerate eigen states.
𝐻𝑆 𝜙 = 𝑆𝐻 𝜙 = 𝐸𝑆 𝜙 → 𝑆|𝜙⟩ is an eigen-state
𝑅𝑆 𝜙 = −𝑆𝑅 𝜙 = −𝑟𝑆 𝜙 → 𝑆|𝜙⟩ is different from |𝜙⟩.
E.g. non-symmorphic symmetry; all the states are doubly degenerate at some special momenta; non-symmorphic topological insulators
Non-symmorphic symmetry
• Non-symmorphic symmetry group
21
𝐴
𝐵
𝐴
𝑐
𝑎
𝑏
Two symmetry operations, 𝜎𝑧: 𝑥,𝑦, 𝑧 → 𝑥,𝑦,−𝑧 𝑔𝑥 = 𝜎𝑥 𝜏 ∶ 𝑥,𝑦, 𝑧 → −𝑥,𝑦, 𝑧 +
𝑐2
, 𝜏 = 0,0,𝑐2
Non-symmorphic symmetry
• Non-symmorphic symmetry
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𝜎𝑧𝑔𝑥 = 𝐶2𝑦 −𝝉 , 𝑔𝑥𝜎𝑧 = 𝐶2𝑦 𝝉 , 𝝉 = 0,0,𝑐2
𝐶2𝑦 −𝜏 𝜙𝑘 = 𝑒−𝑖𝑘⋅𝜏𝐶2𝑦 𝜙𝑘= −𝑖𝐶2𝑦|𝜙𝑘⟩
𝐶2𝑦 𝜏 𝜙𝑘 = 𝑖𝐶2𝑦|𝜙𝑘⟩
When 𝑘 = 0,𝑘𝑦 , 𝜋𝑐
𝑜𝑟 𝜋𝑎
,𝑘𝑦 , 𝜋𝑐
,
{𝜎𝑧,𝑔𝑥} = 0
𝑔𝑥𝜎𝑧 = 𝜎𝑧𝑔𝑥 + 𝒕, 𝒕 = 0,0, 𝑐
𝒌𝒛
𝒌𝒙
𝒁�
𝚪� 𝐗�
𝑼�
Non-symmorphic symmetry
• Non-symmorphic symmetry
23
𝒌𝒛
𝒌𝒙
𝒁�
𝚪� 𝐗�
𝑼� Anti-commutation relation results in the degeneracy at �̅� and 𝑈�.
Wannier function center
• Surface states and topological invariant
CXL, RXZ and BV , Phys. Rev. B 90, 085304
R. Yu, PRB (2011)
Non-symmorphic symmetry
• Eg. Non-symmorphic symmetry
24
𝒌𝒛
𝒌𝒙
𝒁�
𝚪� 𝐗�
𝑼�
TI
Topological crystalline insulators
• 17 2D space group, no anti-unitary operators
25
Group Deg Group Deg P1 No Pmm No P2 No Cmm No P4 No Pmg I, II P3 No Pgg I, II P6 No P4m I, II Pm I P4g I, II Pg I P3m1 I, II Cm I P31m I, II
P6m I, II
P4m Group
• Double degeneracy due to mirror and rotation symmetry operation (C4v)
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𝒌𝒚
𝒌𝒙
𝒀�
𝚪� 𝐗�
𝑴�
Bernevig’s group (2014)
P4m Group
• Double degeneracy due to mirror and rotation symmetry operation (C4v)
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Bernevig’s group (2014)
Γ�, M� : 𝐶4𝑣 symmetry
Γ� − M� line has mirror symmetry
Halved mirror chirality 𝜒 can be defined
Type III magnetic TCIs
• Degeneracy due to anti-unitary symmetry operations
28
Kramers’ degeneracy, spinful fermions, Θ2 = −1, TR invariant TIs.
𝐸 𝑘, ↑ = 𝐸(−𝑘, ↓)
𝒌𝒙
𝒌𝒚 𝑘 = −𝑘 + �⃗�
Type III magnetic TCIs
• Other anti-unitary operator, magnetic group
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ℳ = 𝒢 + 𝐴𝒢 RMP, 40, 359, (1968)
𝒢 is the unitary sub-group 𝐴 = Θ𝑅 is an anti-unitary element, Θ time reversal and the unitary operator 𝑅 ∉ 𝒢.
� 𝜒(𝐵2)𝐵∈𝐴𝒢
= 𝐺 𝑐𝑎𝑐𝑒 𝑎 ;
= − 𝐺 𝑐𝑎𝑐𝑒 𝑏 ;
= 0 𝑐𝑎𝑐𝑒 𝑐 .
Δ real, reducible
Δ = 𝑃Δ∗𝑃−1, irreducible
Δ ≠ 𝑃Δ∗𝑃−1, irreducible
Herring rule Wigner (1932), Herring (1937)
RX Zhang and CXL (arxiv: 1401.6922)
Type III magnetic TCIs
• Magnetic topological insulators
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𝝉Θ with translation operation 𝝉 and time reversal symmetry Θ. 𝝉Θ 2 = −1 can exist for both spinless and spinful fermions.
CnΘ with rotation operation 𝐶𝑛 and time reversal symmetry Θ. 𝐶𝑛Θ 2 = −1 exists for both spinless and spinful fermions.
Mong, Essin, Moore (2010), Fang, et al (2013), Liu (2013)
Fu (2011), RX Zhang and CXL (arxiv: 1401.6922)
Type III magnetic TCIs
• 𝐶4Θ model
31
Eg. 𝐶4Θ with four-fold rotation symmetry 𝐶4 and time reversal symmetry Θ.
𝒌𝒚
𝒌𝒙
𝐶4Θ 2 = 𝜔𝐶2, 𝜔 = ±1
𝐶2
Type III magnetic TCIs
• 𝐶4Θ model
32 RX Zhang and CXL (arxiv: 1401.6922)
Type III magnetic TCIs
• 𝐶4Θ model, surface states and Z2 topological invariants.
33 RX Zhang and CXL (arxiv: 1401.6922)
Wannier function center
R. Yu, et al (2011)
Type III magnetic TCIs
• Generalization to other CnΘ symmetry
34 RX Zhang and CXL (arxiv: 1401.6922)
Value of n single group double group
mirror no no
C2Θ no no
C3Θ no no
C4Θ yes (Z2) yes (Z2)
C6Θ yes (Z2×Z2) yes (Z2×Z2)
Type III magnetic TCIs
• C6Θ symmetry
35 RX Zhang and CXL (arxiv: 1401.6922)
Δ1+Δ2+Δ3=0 mod 2 only two Δi s are indepedent
{(0,0),(1,0),(0,1),(1,1)}
Z2×Z2 topological invariant pair
Type III magnetic TCIs
• C6Θ symmetry
37 RX Zhang and CXL (arxiv: 1401.6922)
Summary and Outlook
• We have presented a theory to explore different topological crystalline insulators by constructing non-trivial surface states.
• Our approach makes it possible to get a complete table for possible topological phases of different crystalline structures in a free fermion system.
• Our approach might be generalized to other systems, such as Weyl semi-metal systems, boson systems, superconducting systems, …
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