Topological Order and Protected Semimetals in Non-Symmorphic CrystalsSiddharth A. Parameswaran

University of California, Irvine

Motivation: A question of ‘oddness’

Compare even vs. odd number of electrons per unit cell:

⌫" + ⌫# = 2⌫" + ⌫# = 1

k�⇡ ⇡ k�⇡ ⇡

•Odd filling: (Hastings/Oshikawa/Lieb-Schulz-Mattis [1])• cannot continue to translation-invariant band insulator• symmetry-preserving gapped phases ⇐⇒ non-unique g.s.• topological order =⇒ gapped degenerate symmetric g.s.• intuition: unfilled bands =⇒ ‘band insulator’ forbidden!

•Even filling:• can continue to band insulator w/ translation invariance(tight-binding ∼ more restrictive)

• apparently admits unique g.s.• symmetric g.s. w/o topological order• naïve expectation: filled bands =⇒ band insulator allowed

•Can we say more for even fillings?• Point-group symmetries: some known results

• e.g. fragile Mott insulators ∼ interacting g.s. carries nontrivial PGrepresentation [2]; unique g.s. =⇒ ‘featureless’ w/o topological order

•Goal: identify cases where symmetric gapped phase must bedegenerate, so band insulators are forbidden at even filling.

‘Laughlin’ Argument

•U(1) charge conservation + no spin-orbit• effective filling: ν = ν↑+ν↓

2 (even/odd ↔ integer/fraction)•Adiabatic cycle: given insulating g.s. |Ψ0〉, producedegenerate state |Ψ0〉 (flux insertion + large gauge transf.)

A

�0

| 0i

| 00i

fluxinsertion

gauge transformation

| 0i = U(�0)| 00i

U(�0)

�(t)

•Can we distinguish |Ψ0〉, |Ψ0〉 by symmetry quantumnumbers? (compute formally [1, 3] or graphically [3, 4])

Acknowledgements: Part of this work was performed with AshvinVishwanath, Ari Turner & Dan Arovas or grew out of discussions withMike Zaletel. I acknowledge support from the Simons Foundation and theNational Science Foundation via Grant No. DMR-1455366.

Flux Insertion & Symmetry: Graphical Calculation

T a12

�2A

A

T a12

�2A

⌫ = 1

�P �P �P

⌫ = 1/2

symmorphic non-symmorphic

glide is not a symmetry

glide is a symmetry� =

ZA · dr •Pure gauge flux ⇐⇒ A = k/N , k ∈ reciprocal lattice.

•Change in ground-state momentum: F = A (∼ Faraday’s law),

∆P =Q∑

i=1

∫F idt = Qk

N= N 2νk

where Q = νNd is the total charge in N × . . . N unit cells.•Translational symmetry (usual HOLSM):

•Fractional ν: ∆P 6∈ reciprocal lattice =⇒ |Ψ0〉, |Ψ0〉 havedifferent crystal momentum =⇒ ground state degeneracy.

• Integer ν: ∆P ∈ reciprocal lattice =⇒ |Ψ0〉, |Ψ0〉 havesame crystal momentum =⇒ no further insight.

•At integer ν, use non-symmorphic symmetries thatproduce ‘extinctions’ in reciprocal lattice. If ∆P = extinctlattice vector, |Ψ0〉, |Ψ0〉 have different quantum numbers [3]

Classifying Crystals: ‘Non-Symmorphic Rank’

•When is space group G non-symmorphic?• formally: G 6= (translation) o (point group)• informally: G 3 screw rotations/glide mirrors (any origin [5])

•Screw/glide: rotation/reflection + fractionaltranslation along rotation axis/mirror plane

• 157/230 space groups are non-symmorphic• Includes common examples, e.g. diamond, hcp, ...

•For each G compute ‘non-symmorphic rank’ S [3]:• ∆P ≡ extinction unless ν ≡ 0 (mod S) =⇒ |Ψ0〉, |Ψ0〉have different PG quantum numbers

•S > 1 ⇐⇒ G is non-symmorphic

diamond (G = Fd3m)Tz/4

R⇡/2

T c2

R⇡3

h.c.p. (G = P63/mmc)

T a12

�2

Shastry-Sutherland (G = p4g)

Lattices with S = 2 Space Groups

Applications

•Band insulator has unique g.s. =⇒ forbidden if ν 6= pS (forelectrons: ν↑ + ν↓ 6= 2pS)• Bands ‘stick together’ in multiplets of S [6]• Multiplets can be detached ⇐⇒ symmetry is broken!

• Integer filling =⇒ Fermi volume is zero =⇒ semimetal• To open gap: break U(1)/glide/screw symmetry or triggertopological order

• Constrains fractionalization: e.g., anyons in Z2 spin liquidcarry fractionalized U(1)/glide/screw quantum numbers [7]

•Potentially identifies new ‘Landau-forbidden’ critical points [8]

•Can apply to other U(1)-conserving systems, e.g. magnetizationplateaus, Bose-Hubbard models, Kondo lattice, etc.

glide/screw preserved, bands ‘stick’

glide/screw broken, bands detach

h.c.p., S = 2

‘Luttinger Invariants’ & Spectral Flow

•Fractional filling: partially filled bands =⇒ metal;momentum balance sets Fermi surface volume (Oshikawa’sproof of Luttinger’s theorem [9]): (2π)−d VFS ≡ ν (mod 1)

• Integer filling: total FS volume vanishes, but non-symmorphiccrystals must be gapless if ν 6≡ 0 (mod S) =⇒ semimetal

• New ‘Luttinger invariant’ from non-symmorphic rank S [10]:χF ≡ ν (mod S)

• Intuition: χF counts ‘leftover’ charge not in filled sets of Sbands (cf. VFS versus filled bands for S = 1)

• Characterizes spectral flow required by symmetry balance [10]

kx0 2⇡ kx0 2⇡

m�

m+

m�

m+

kx0 2⇡kx0 2⇡

m�

m+

before cycle after cycle

one electron transferred from

conduction to valence band

kx ! kx +2⇡

L

⌫ = 1 semimetal: non-trivial

⌫ = 2: trivial ⌫ = 1 insulator: trivial

SPECTRAL FLOWS FOR S = 2

Extensions & Future Directions

•Can connect to quantum theory of polarization [3]• Spin-orbit can be included via entanglement methods [11]• Symmetry counting/balance arguments are relevant tounderstanding quantum numbers of excitations in topologicallyordered phases [4, 7], possible unconventional critical points [8],and ‘fractionalized Fermi liquids’ [12, 13]

•Non-symmorphic symmetries also relevant to crystalline TIs [14]

References

[1] E.H. Lieb, T. Schulz, & D. Mattis, Ann. Phys. 61, 407 (1961); M. Oshikawa, Phys. Rev. Lett. 84, 1535(2000); M. Hastings, Phys. Rev. B 69 104431 (2004) and Europhys. Lett. 70, 824 (2005).

[2] H. Yao and S. Kivelson, Phys. Rev. Lett. 105, 166402 (2010).[3] S.A. Parameswaran, A. M. Turner, D. P. Arovas & A. Vishwanath, Nat. Phys. 9, 299 (2013); see also R.

Roy, arXiv:1212.2944.[4] A. Paramekanti & A. Vishwanath, Phys. Rev. B 70, 245118 (2004).[5] A. König & N.D. Mermin Proc. Nat. Acad. Sci. 96, 3502 (1999).[6] L. Michel & J. Zak, Phys. Rev. B 59, 5998 (1999).[7] S.B. Lee, M. Hermele, & S.A. Parameswaran, in preparation.[8] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, & M.P. A. Fisher, Science 303, 1490 (2004).[9] M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000).[10] S.A. Parameswaran, arXiv:1508.01546.[11] H. Watanabe, H. Po, A. Vishwanath, & M. Zaletel, Proc. Nat. Acad. Sci. 112, 14551 (2015).[12] T. Senthil, S. Sachdev, & M. Vojta, Phys. Rev. Lett. 90, 216403 (2003); T. Senthil, M. Vojta & S. Sachdev,

Phys. Rev. B 69, 035111 (2004).[13] B. Brandom, S.B. Lee, & S.A. Parameswaran, in preparation.[14] C-X Liu, R-X Zhang, & B. K. VanLeeuwen, Phys. Rev. B 90, 085304 (2014).