Many-body topological invariants for topological superconductors (and insulators)

Transcription

1 Many-body topological invariants for topological superconductors (and insulators) Shinsei Ryu The University of Chicago July 5, 2017

2 Outline Motivations: the Kitaev chain with interactions The kitaev chain with reection symmetry The kitaev chain with time-reversal symmetry Summary Collaborators: Hassan Shapourian (UIUC) and Ken Shiozaki (UIUC)

3 Motivating example: The Kitaev chain The Kitaev chain H = j [ tc j c j+1 + c j+1 c j + h.c. ] µ j c j c j Phase diagram: there are only two phases: Topologically non-trivial phase is realized when 2 t µ.

4 The topological phase is characterized by (i) a bulk Z 2 topological invariant: [ π ] exp i dk A x(k) = ±1 where A(k) = i u(k) du(k) is the Berry connection. π (ii) Majorana end states: Characterized by the Z 2 topological invariant the even/odd Majorana end states.

5 The Kitaev chain with TR or reection symmetry The Kitaev chain can be studied in the presence of time-reversal or reection symmetry. T c j T 1 = c j (T it 1 = i) or Rc j R 1 = ic j. Once we impose more symmetries (T or R), we distinguish more phases. Phases are classied by an integer Z. There is a topological invariant written in terms of Bloch wave functions. ν = 1 2π π π dk A x(k) = (integer) Ex: For N f copies of the Kitaev chain H = N f a=1 Ha with µ < 2 t, the topological invariant is ν = N f.

6 The Kitaev chain with interactions The 1d Kitaev chain: H = N f a=1 Ha with TR or reection symmetry T HT 1 = H or RHR 1 = H Add interactions; Can we destropy the topological phase by interactions?; H H + wv I.e., Is it possible to go from topological to trivial? Interestingly, the answer is Yes! [Fidkowski-Kitaev(10)] You can show there is a (rather complicated) interaction V that destroys the topological case. Only possible when N f = 8 (N f 0 mod 8)

7 Issues and Goal Clearly, something is missing in non-interacting classication. We have not explored the phase diagram "hard enough". Various other examples in which the non-interacting classication breaks down. Non-interacting topological invariants are not enough/spurious. Goal: nd many-body invariants for fermionic symmetry-protected topological phases.

8 Main results [arxiv: and arxiv: ] We have succeeded in construction many-body topological invariants for many fermion SPT phases. These invariants do not refer to single particle wave functions (Bloch wave functions). They are written in terms of many-body ground states Ψ. C.f. Many-body Chern number. Strategy behind the construction (later). [Hsieh-Sule-Cho-SR-Leigh (14), Kapustin et al (14), Witten (15)]

9 The Kitaev chain with reection Consider Partial reection operation R part, which acts only a part of the system. We claim the phase of the overlap Z = Ψ R part Ψ is quantized, and serves as the many-body topological invariant. (Similar but somewhat more complicated invariant for TR symmetric case.)

10 Numerically check (blue lled circles). The phase of Z is the 8th root of unity.

11 Strategy behind the construction In the quantum Hall eect, the many-body Chern number is formulated by putting the system on the spatial torus: Introduce the twisting boundary condition by U(1), and measure the response: Ground state on a torus with ux Ψ(Φ x, Φ y) Berry connection in parameter space A i = i Ψ(Φ x, Φ y) Ψ(Φ x, Φ y) Φ i Many-body Chern number [Niu-Thouless-Wu (85)]: Ch = 1 2π 2π 0 2π dφ x dφ y( Φx A y Φy A x) 0

12 Strategy behind the construction I A famous saying I However, new phases of matter requires new kinds of manifolds, unoriented manifolds. E.g. Klein bottle

13 For topological phases with reection symmetry, we twist boundary conditions using the symmetry of the problem (reection)

14 Partial reection introduces a crosscap in space time: The spacetime is eectively the real projective plane, RP 2.

15 More general considerations For condensed matter systems with symmetries, we can consider Z(X, η, A) = D[φ i ] exp S(X, η, A, φ i ) (1) X: spacetime, η: structure (e.g., spin, pin, etc.), A: background gauge eld, φ i: matter eld. For orientation-reversing symmetries (time-reversal, reection, etc.), unoriented spacetime (with pin structure) plays the role of an external background eld. For gapped topological phases, we expect Z(X, η, A) consists of a topological term: Z(X, η, A) exp[is top(x, η, A, ) + ] (2) S top are our many-body topological invariants. For SPT phases (i.e., no topological order, unique ground state), S top are expected to be classied by cobordism theory. [Kapustin et al (14), Freed et al(14-16)]

16 Why does the real projective plane know 8? Interesting mathematics... [Kapustin et al (14), Witten (15), Freed et al(14-16)] Watch a Youtube video (thanks: Dennis Sullivan): Topology: The connected sum of 8 copies of Boys Surface is immersion-cobordant to zero. The addition and the cobordism are illustrated here.

17 Higher dimensions (3+1)d with inversion Consider 3 He-B: H = d 3 k Ψ (k)h(k)ψ(k), H(k) = [ k 2 Ψ(k) = (ψ (k), ψ (k), ψ ( k), ψ ( k))t Inversion symmetry: Iψ σ(r)i 1 = iψ σ( r) 2m µ σ k σ k k2 2m + µ Topologically protected surface Majorana cone (stable when the surface is inversion symmetric) ] Characterized by the integer topological invariant at non-interacting level.

18 Many-body topological invariant Previous studies indicate the non-interacting classication breaks down to Z 16. Surface topological order. [Fidkowski et al (13), Metlitski et al (14), Wang-Senthil (14)] We consider partial inversion I part on a ball D: Z = Ψ I part Ψ The spacetime is eectively four-dimensional projective plane, RP 4.

19 Calculations Numerics on a lattice: 0.5 Z/(π/4) Trivial Top. I Top. II Top. I 1.0 Trivial µ/t Matches with the analytical result [ Z = exp iπ ln(2) ζ(3) ( ) ] R 2 + ξ C.f. topologically ordered surfaces: [Wang-Levin, Tachikawa-Yonekura, Barkeshli et al (16)]

20 The Kitaev chain with Time-reversal partial time-reversal Start from the reduced density matrix for the interval I, ρ I := Tr Ī Ψ Ψ. I consists of two adjacent intervals, I = I 1 I 2. We consider partial time-reversal acting only for I 1; ρ I ρ T 1 I. Partial time reversal partial transpose has to be properly dened for fermionic systems [Shaporian-Shiozaki-Ryu 17]; (led to new entanglement measure, fermionic entanglement negativity.) The invariant: Z = Tr[ρ I ρ T 1 I ],

21 Z = Tr[ρ I ρ T 1 I ], The invariant "simulates" the path integral on RP 2 : = = = =

22 Calculations Analytical calculations in the zero correlation length limit: Tr(ρ I ρ T 1 I ) = 1 + i = eiπ/4 Numerically checked away from the zero correlation length (red triangles). (a) Z/(π/4) (b) Trivial Topological D + Refl. BDI Trivial µ/t Z µ/t

23 More on partial transpose The partial transpose for bosons; denition: for the density matrix ρ A1 A 2, e (1) i e (2) j ρ T 2 A 1 A 2 e (1) k e(2) l = e (1) where e (1,2) i is the basis of H A1,A 2. i e (2) l ρ A1 A 2 e (1) k e(2) j Can detect quantum correlation comes from o diagonal parts of density matrices. [Peres (96), Horodecki-Horodecki-Horodecki (96), Vidal-Werner (02), Plenio (05)...] Entanglement negativity and logarithmic negativity 1 2 (Tr ρt 2 A 1), EA = log Tr ρt 2 A

24 Issues in fermionic systems Fermionic version of the partial tranpose has been considered previously. E.g., [Eisler-Zimborás] Two issues: (i) Partial transpose of fermionic Gaussian states are not Gaussian Partial transpose of bosonic Gaussian states is still Gaussian; easy to compute by using the correlation matrix ρ T 1 can be written in terms of two Gaussian operators O ±: ρ T1 = 1 i 2 O i 2 O Negativity estimators/bounds using Tr [ O +O ] [Herzog-Y. Wang (16), Eisert-Eisler-Zimborás (16)] Spin structures: [Coser-Tonni-Calabrese, Herzog-Wang] (ii) Fails to capture Majorana dimers in the Kitaev chain Our new denition of partial time-reversal solves these issues

25 Issues in fermionic systems (2): The Kitaev chain Consider log negativity E for two adjacent intervals of equal length. (L = 4l = 8) Vertical axis: µ/t ranging from 0 to 6. (Blue circles) is computed for the bosonic many-particle density matrix in the Ising chain with periodic boundary condition. (Red crosses) curves are computed for the fermionic many-particle density matrix according to the rules by Eisler-Zimborás. Log negativity fails to capture Majorana dimers.

26 Partial transpose for fermions our denition In the Majorana representation, ρ R 1 A = κ,τ w κ,τ R(c κ 1 m 1 c κ 2k m 2k ) c τ 1 n 1 c τ 2l n 2l = κ,τ w κ,τ i κ c κ 1 m 1 c κ 2k m 2k c τ 1 n 1 c τ 2l n 2l where R satises: R(c) = ic, R(M 1M 2) = R(M 1)R(M 2) Furthermore, (ρ R 1 A )R 2 = ρ R A, (ρr A )R = ρ A, (ρ 1 A ρn A )R 1 = (ρ 1 A )R 1 (ρ n A )R 1 Fermions Gaussian states stay Gaussian after partial TR, ρ R 1 A = O +, ρ R 1 A = O, Possible to develop the free fermion formula. Dene log negativity as E := ln Tr ρ R 1 A = ln Tr O + O

27 Comparison Comparison: (Green triangles) are computed for the fermionic many-particle density matrix according to our rule. (Computed numerically) (Orange triangles) is computed using the free fermion formula: M E = ln ( λ i + Pf ( 1 λ i ) S iσ 2 ) Pf (S iσ 2 ) 2 i=1

28 Summary We have succeeded in constructing many-body topological invariants for SPT phases. These invariants do not refer to single particle wave functions (Bloch wave functions). They are written in terms of many-body ground states Ψ. Analogous to go from the single-particle TKNN formula to to the many-body Chern number. Many-body invariants in other cases (e.g., time-reversal symmetric topological insulators) can be constructed in a similar way.

29 Outlook Many future applications, in particular, in numerics. And...? NbSe 3 Möbius strip [Ningyuan-Owens-Sommer-Schuster-Simon (13)]