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

15

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

17

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

22

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)

26

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

29

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

30

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, …

38