Topological Obstructions to Band Insulator Behavior in Non-symmorphic Crystals

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2 Topological Obstructions to Band Insulator Behavior in Non-symmorphic Crystals D. P. Arovas, UCSD Siddharth Parameswaran UC Berkeley UC Irvine Ari Turner Univ. Amsterdam JHU Ashvin Vishwanath UC Berkeley this is how I draw myself Article : Nature Physics 9, 299 (2013) Support : NSF, Simons Foundation, KITP, Aspen Center for Physics

3 Phases of Matter 1. Classification by response functions: METAL INSULATOR 2. Classification by crystalline symmetries: DIAMOND HCP

4 Band theory establishes a connection: Bloch states: Energy bands: nk (r) =u nk (r) eik r " n (k) =" n (k + K) =" n (gk) METAL some bands partially filled INSULATOR all bands either filled or empty noninteracting systems only!

5 Effects of interactions 1) Benign : interacting ground state is adiabatically connected to noninteracting ground state from band theory. 2) Essential : interacting ground state exhibits phenomenon of spontaneous symmetry breaking : B C Q = π a ˆx + π a ŷ g 2 N 0 0 FEATURELESS g 1 A MAGNETS DENSITY WAVES SUPERFLUIDS PHASE DIAGRAM 3) Any other possibilities?

6 Can interactions produce a distinct symmetric phase not adiabatically connected to the noninteracting ground state? Examples of trivial phases without SSB: band insulators both topological and nontopological quantum paramagnets e.g. AKLT states k F Fermi liquids metals, semimetals if gapped : unique ground state if gapless : no fractionalization

7 Examples of nontrivial phases without SSB: Luttinger liquids separation of spin and charge DOF spin liquids emergent gauge fields fractionalization FQHE fractional charge and statistics if gapped : degenerate vacua all cases : fractionalization

8 Half-filled Hubbard model in d=1 dimension U t U t : Umklapp scattering relevant (g-ology) : strong coupling, / U /t With t =1: = 4+U +8 Z 1 0 d!! J 1 (!) 1 + exp(u!/2) U/t Spin-charge separation : spin metal + charge insulator This state is adiabatically disconnected from the band insulator. Filling factor : # electrons per cell per spin polarization 2 Z : adiabatic connection to band insulator 2 Z : Mott phase preserves all symmetries, but with no adiabatic connection to band insulator

9 At fractional filling, not even interactions can induce a unique symmetry-unbroken ground state. This prompts the following question: Q: If 2 Z is there always a band insulator? A: No! For nonsymmorphic crystals, there is a topological obstruction to band insulator behavior unless the filling is ν is an integer multiple of the non-symmorphic rank, S 2 Z For 6= k S, the ground state must either be either (i) gapless or (ii) topologically degenerate

10 Non-symmorphic crystallographic space groups Basically, a space group S is nonsymmorphic if it includes screw axes or glide planes. Both of these operations involve fractional translations that are not within the Bravais lattice. Formally, a crystallographic space group element R, acts as R, r = Rr + translation vector point group matrix operation This forms a group under multiplication: R, R 0, 0 = RR 0,R 0 + A space group is symmorphic when it is a semidirect product of the point group P and Bravais lattice translations T : S = P o T A space group is nonsymmorphic when there is no possible choice of origin about which all its elements can be decomposed into a product of a lattice translation and a point group element.

11 Example : a lattice with a 2D glide line space group p4g (S. Parameswaran)

12 Example : a lattice with a 2D glide line space group p4g (S. Parameswaran) glide mirror

13 Example : a lattice with a 2D glide line space group p4g (S. Parameswaran) a 2 glide mirror a 1

14 Example : a lattice with a 2D glide line space group p4g (S. Parameswaran) a 2 glide mirror a 1

15 Example : a lattice with a 2D glide line space group p4g (S. Parameswaran) a 2 glide mirror a 1 Glide operation : reflection in x-axis followed by translation by 1 2 a 1 g = I x, 1 2 a 1

16 Essential vs. inessential nonsymmorphic operations Graphene has a glide plane, yet its space group p6m is symmorphic! B1 A5 B5 B6 B7 B8 A1 A2 A3 A4 B2 B3 B4 A6 A7 A8 a 2 a 1 B5 B6 B7 B8 A1 A2 A3 A4 B1 B1 B2 B3 B4 The honeycomb glide is inessential, and may be written as an ordinary point group operation followed by a direct lattice vector translation. A5 A6 A7 A8

17 Essential vs. inessential nonsymmorphic operations Graphene has a glide plane, yet its space group p6m is symmorphic! B1 A5 B5 B6 B7 B8 A1 A2 A3 A4 B2 B3 B4 A6 A7 A8 a 2 a 1 B5 B6 B7 B8 A1 A2 A3 A4 B1 B1 B2 B3 B4 The honeycomb glide is inessential, and may be written as an ordinary point group operation followed by a direct lattice vector translation. A5 A6 A7 A8

18 Essential vs. inessential nonsymmorphic operations Graphene has a glide plane, yet its space group p6m is symmorphic! B5 B1 B1 B5 A6 B6 A7 B7 A8 B8 A1 B2 A2 B3 A3 B4 A4 A1 B2 A2 B3 A3 B4 A4 A5 B5 A6 B6 A7 B7 A8 B8 a 2 a 1 B5 B6 B7 B8 A1 A2 A3 A4 B1 B1 B2 B3 B4 The honeycomb glide is inessential, and may be written as an ordinary point group operation followed by a direct lattice vector translation. A5 A6 A7 A8

19 Essential vs. inessential nonsymmorphic operations Graphene has a glide plane, yet its space group p6m is symmorphic! B5 B6 A6 B7 A7 B8 A8 B1 A1 A2 B2 B3 A3 A4 B4 B2 A1 A2 B3 B4 A3 A6 B5 B6 A7 A8 B7 A4 B8 a 2 a 1 B5 B6 B7 B8 A1 A2 A3 A4 B1 B1 B2 B3 B4 The honeycomb glide is inessential, and may be written as an ordinary point group operation followed by a direct lattice vector translation. A5 A6 A7 A8

20 Essential vs. inessential nonsymmorphic operations Graphene has a glide plane, yet its space group p6m is symmorphic! B5 B6 A6 B7 A7 B8 A8 B1 A1 A2 B2 B3 A3 A4 B4 B2 A1 A2 B3 B4 A3 A6 B5 B6 A7 A8 B7 A4 B8 a 2 a 1 B5 B6 B7 B8 A1 A2 A3 A4 B1 B1 B2 B3 B4 The honeycomb glide is inessential, and may be written as an ordinary point group operation followed by a direct lattice vector translation. A5 A6 A7 A8

21 Essential vs. inessential nonsymmorphic operations Graphene has a glide plane, yet its space group p6m is symmorphic! B5 B6 A6 B7 A7 B8 A8 B1 A1 A2 B2 B3 A3 A4 B4 B2 A1 A2 B3 B4 A3 A6 B5 B6 A7 A8 B7 A4 B8 a 2 a 1 B5 B6 B7 B8 A1 A2 A3 A4 B1 B1 B5 A6 B2 A7 B3 A8 B4 The honeycomb glide is inessential, and may be written as an ordinary point group operation followed by a direct lattice vector translation. A5 B1 A1 A6 B2 A7 B3 A8 B4 A2 A3 A4 B5 B6 B7 B8

22 Essential vs. inessential nonsymmorphic operations Graphene has a glide plane, yet its space group p6m is symmorphic! B5 B6 A6 B7 A7 B8 A8 B1 A1 A2 B2 B3 A3 A4 B4 B2 A1 A2 B3 B4 A3 A6 B5 B6 A7 A8 B7 A4 B8 a 2 a 1 B5 A6 A7 A8 B1 B2 B3 B4 A1 A2 A3 A4 The honeycomb glide is inessential, and may be written as an ordinary point group operation followed by a direct lattice vector translation. B5 B6 B7 B8

23 Ubiquity of nonsymmorphicity Table 1 Some non-symmorphic groups and their ranks, colloquial structure names and representative materials. d Name Examples Space group S 2D Shastry-Sutherland glide plane ; S=2 ˆR 3 ˆT c 2 3D hexagonal close packed screw axis ; S=2 2 Shastry Sutherland SrCu 2 (BO 3 ) 2 p4g 2 3 hcp Be, Mg, Zn P6 3 /mmc 2 3 Diamond C, Si Fd 3m 2 3 Pyrochlore Dy 2 Ti 2 O 7 (spin ice) Fd 3m 2 3 -SiO 2, GeO 2 P CrSi 2 P Pr 2 Si 2 O 7,La 2 Si 2 O 7 P Hex. perovskite CsCuCl 3 P6 1 6 CRYSTALLOGRAPHY d=2 d=3 LATTICES 5 14 POINT GROUPS SPACE GROUPS SYMMORPHIC NON-SYMMORPHIC Of the 157 nonsymmorphic three-dimensional space groups, 155 involve glide planes or screw axes, and two are exceptional cases.

24 Stickiness of nonsymmorphic energy bands Results known from band theory. Bloch bands stick together in groups of S. Impossible to detach without breaking the symmetry. ˆR 3 ˆT c 2 screw symmetry unbroken HCP screw symmetry broken

25 Lieb-Schultz-Mattis-Oshikawa-Hastings theorem Lieb, Schultz, Mattis (1961) / Altman and Auerbach (1998) / Oshikawa (2000) / Hastings (2004,2005) At fractional filling ν, a unique, gapped, featureless,! insulating ground state is impossible. /2 Z unique gapped featureless insulator EXAMPLE NOT POSSIBLE METALLIC DENSITY WAVE SPIN-CHARGE! SEPARATION TOPOLOGICAL ORDER BAND INSULATOR

26 Lieb-Schultz-Mattis argument Consider an XXZ chain with periodic boundary conditions, NX NX Ĥ = 1 2 J? S + n S n+1 + S n S + n+1 + J z S z n S z n+1 n=1 n=1 Suppose 0i is the ground state, with t 0i = e ik 0 0i. Now apply the spin twist operator U =exp Find tut = Ue 2 isz tot /N e 2 isz 1 2 i N NX j=1 js z j to 0i., so if 0i is a spin singlet, then t 1i = e ik 1 1i with 1i = U 0i K 1 = K 0 +2 S and U S + n S n+1 U = e 2 i/n S + n S n+1 h 1 Ĥ 1i = E 0 + O(1/N ) Thus, we have found an orthogonal state 1i which is degenerate with 0i in the thermodynamic limit.

27 Example: next-nearest neighbor Heisenberg chain Ĥ = X n hs n S n+1 + g S n S n+2 i GAPLESS SPIN-PEIERLS 0 g c 0.5 Heisenberg model (Bethe Ansatz) Majumdar-Ghosh point (dimers) g For For g<g c ' , the spectrum is gapless. g>g c, the system is in a spin-peierls phase (doubly degenerate ground state with excitation gap) Majumdar-Ghosh point (g = 1 2 ): Ai = Bi = Ai ± Bi has crystal momentum 0,

28 Oshikawa-Hastings extension of LSM theorem The LSM argument works only in d=1 because h 0 U 2 ĤU 0i = E 2 0 N h 0 Ĥ? 0i = E O(N d 2 ) Oshikawa (2000) extended this argument to higher dimensions by examining the consequences of adiabatic flux threading. Place the system on a d - dimensional torus, and thread a flux through Ĥ( ) one of Ĥ( its cycles, resulting in a translationally-invariant. Since ),t =0, the crystal momentum of 0( )i 1i = U 0(2 )i Ĥ( = 0) the adiabatic ground state is constant (0). Now define a ground state of, which must be, but with momentum K =2 N? ê Here = p is the filling, and N the number of sites in the q? transverse direction. If K not an RLV, then h 0 1i =0.

29 K ê =2 N? 6= 2 n N? q The condition requires and to be relatively prime, and does not require d=1. In this case, the ground state cannot be unique at fractional filling. At integer filling, K is a reciprocal lattice vector, hence the states 0i and 1i cannot be distinguished - or so it would seem! Some crystalline structures, however, exhibit systematic extinctions in their Bragg patterns. Such is the case with nonsymmorphic lattices. In such cases, 0i and 1i can be distinguished. This is the essential content of our observation. Fourier space crystallography and extinctions: Under a space group operation {R, }, the Fourier components of the density n k transform as n k! n Rk e ik. If k 6= 2 n, then n k =0.

30 Examples: For a S = 1 Kagome lattice model with one electron per site, 2 there are three sites per cell, hence = 3. With a featureless, 2 gapped, insulating ground state, the system must exhibit topological degeneracy and fractionalization. Yan, Huse, White (2011) Jiang, Wang, Balents (2012) chiral order? Capponi et al. (2013) For a S = 1 honeycomb lattice model with one electron per 2 site, there are two sites per cell, and the filling is =1. Is a featureless, gapped, insulating ground state necessarily a spin liquid? No. It could be a Mott insulator. Kimchi et al. (2013)

31 Gist of our argument (details in paper) First, we require a conserved U(1) charge, which could arise from spinless or spinful fermions, bosons, or magnets where S z is a good quantum number. We then define total U(1) charge number of unit cells Next, consider a nonsymmorphic SG operation Ĝ={R, }, where is not in the direct lattice, and R = (such as in a screw or glide). Now let b be the smallest reciprocal lattice vector for which Rb=b. Now adiabatically thread a flux with vector potential A=b/N, which corresponds to a pure gauge. Starting with a ground state which is presumed to be an eigenstate of all symmetry operations, we must have Ĝ 0i=e i 0i. Let 0i be the adiabatic image of 0i after flux insertion. Flux threading commutes with Ĝ. We pull back to the original flux-less Hamiltonian via where ˆ (r) Û b =exp i N Z d d r b r ˆ (r) is the density operator for the U(1) charge. This removes the flux.

32 Following Oshikawa (2000), define 1i=Ûb 0i. Now Ĝ 1 Û b Ĝ=Ûb, where Q = NN? is the total U(1) charge. Furthermore, is fractional, hence eib Q/N t S G b =2 t/s G h 0 1i=0 where and are relatively prime. We therefore conclude that whenever t N? /S G is fractional. We can always choose N? so that it is relatively prime to S G. We conclude that adiabatic flux insertion and removal generates a distinct ground state whenever the filling is not an integer multiple of S G. The least common multiple of the {S G } must divide the nonsymmorphic rank S. A necessary consequence: Both diamond and pyrochlore lattices have space group Fd 3m, for which S =2. For interacting electrons with one e - per site, one has =1 for diamond, and =2 for pyrochlore. Thus, a trivial insulator at this filling is impossible on the diamond lattice, but possible on the pyrochlore lattice.

33 magnetization Magnetization plateaux in SrCu2(BO3)2 CuBO3 layers form Shastry-Sutherland lattice In a field, SCBO exhibits a magnetization plateau at half the saturation value =1 Accordingly, since SSSL=2, the plateau state must be topologically ordered, or else break a symmetry. [See Momoi and Totsuka (2000)] ν=1 applied field (T) Jaime et al. (PNAS, 2012) Sebastian et al. (PNAS, 2008)

34 Summary Basic question: is there always a band insulator for 2 Z? NO! Discrete invariant of space group: nonsymmorphic rank S - quantum of filling for featureless insulators - Bloch bands stick in groups of S and can t be unstuck without breaking symmetry - implications for band theory, interacting Bose and Fermi systems, topological degeneracy, fractionalization of 230 three-dimensional space groups are nonsymmorphic (S > 1) ˆR 3 ˆT c 2 p4g Fd 3m P 6 3 /mmc Questions: time-reversal? spin-orbit? quasicrystals? defects?