The uses of Instantons for classifying Topological Phases

Transcription

1 The uses of Instantons for classifying Topological Phases - anomaly-free and chiral fermions Juven Wang, Xiao-Gang Wen (arxiv: , arxiv:140?.????) MIT/Perimeter Inst. APS March

2 A Lattice Non-Perturbative Definition of 1+1D Anomaly-Free Chiral Fermions and Bosons. arxiv: The uses of Instantons... work to appear. arxiv: 140?.???? w/ Prof. X-G Wen (Perimeter and MIT) uven Wang, Xiao-Gang Wen The(arXiv: , uses of InstantonsarXiv:140?.????)

3 Outline - Motivations Three Key Questions : Q1: Topological phases with/out symmetry may have edge states, how to gap out (i.e. fully open mass gaps) edge states? Q2: How to design a bulk lattice model with one mirrored sector fully gapped) of edge states? - solve Lattice Chiral Fermion Q3: How to detect non-trivialness of bulk SPT or Topological Order?

4 In this talk, just Three Key Messages : 1st. Anomaly and Topological States Boundary Anomaly Quantum Hall States y x J J J d-dim t Hooft anomaly matching conditions d-dim Boundary Fully Gapping Rules (d + 1-Dim bulk)

5 In this talk, just Three Key Messages : 2nd. Lattice Chiral Fermions Ginsparg-Wilson Fermion Our approach ql ql Nontrivial SPT (Topological insulator) Trivial SPT (Trivial insulator) qr qr

6 In this talk, just Three Key Messages : 3rd. The uses of topological defects (eg. Instanton, soliton, domain wall, vortex line) to detect gauge or gravitational anomaly, thus detect SPT and topological order. 0D 1D 1D or 2D x + + x

7 Review on the Basics about: Symmetric Protected Trivial(SPT) Order and intrinsic Topological Order

8 Symmetric Protected Trivial(SPT) and intrinsic Topological Order Symmetric Protected Trivial(SPT) order : Topological Insulator -Time reversal invariant with U(1) charge Z2 T Z2 f symmetry, giving a Z 2 class Haldane spin-1 chain - with SO(3) symmetry, H 2 (SO(3), U(1)) = Z 2 class. Classified by Group Cohomology (arxiv , Science 338, 1604 (2012), Chen-Gu-Liu-Wen) intrinsic Topological Order : fractional quantum hall states - such as Chern-Simons theory level k.

9 Symmetric Protected Trivial(SPT) and intrinsic Topological Order SPT order : Short ranged entangled states. No topological entanglement entropy. No fractionalized charge, No fractionalized statistics (no anyons) The bulk realizes the symmetry G onsite. U(g) = i U i (g), g G,

10 Symmetric Protected Trivial(SPT) and intrinsic Topological Order SPT order : Short ranged entangled states. No topological entanglement entropy. No fractionalized charge, No fractionalized statistics (no anyons) The bulk realizes the symmetry G onsite. U(g) = i U i (g), g G, The boundary realizes the symmetry G non-onsite, with (1) gapless edge states, or (2) Ground state degeneracy(gsd) from symmetry breaking, or (3) GSD from boundary topological ordered.

11 Symmetric Protected Trivial(SPT) and intrinsic Topological Order intrinsic Topological Order : Long ranged entangled states. (usually) with topological entanglement entropy. (usually) fractionalized charge. (usually) fractionalized statistics.

12 Symmetric Protected Trivial(SPT) and intrinsic Topological Order intrinsic Topological Order : Long ranged entangled states. (usually) with topological entanglement entropy. (usually) fractionalized charge. (usually) fractionalized statistics. (1) GSD on the nontrivial topology space. (2) non-abelian Berry s phases on coupling const moduli space

13 Anomaly and Quantum Hall (Laughlin s flux insertion) y x J J J In 1+1D boundary/2+1d bulk with a bulk Chern-Simons theory ( K 4π a da + q σh 2π A da 2 A da): In 1+1D its own: µ j µ V = µ (j µ L + j µ R ) = 0, µ j µ b = σ H 2 εµν F µν = σ H E x = J y µ j L = = +J y, µ j R = = J y. µ j µ A = µ (j µ L j µ R ) = σ Hε µν F µν 0... U(1) V current conserved... ABJ U(1) A chiral anomaly

14 Anomaly and Quantum Hall (2+1D bulk/1+1d edge) Quantum Hall States y A d-dim anomaly free edge theory there is NO need for an extended d + 1-Dim bulk theory to carry the anomalous current leaking from its d- dimensional boundary theory. By J y = σ H E x, so J y = 0 σ H = 0 A = (ql 2 qr) 2 = 0 σ H,xy = 1 (q 2 2π L qr) 2 = 0 A (anomaly factor) = 2πσ H,xy (effective Hall conductance) Juven Wang, Xiao-Gang Wen (arxiv: , arxiv:140?.????) The uses of Instantons J x J J

15 Question 1: How to gap out (i.e. fully open all mass gaps) of edge states? Quantum Hall States y x J J J One Way (chiral) or Multiple Ways (chiral/non-chiral)

16 Question 1: How to gap out (i.e. fully open all mass gaps) of edge states? How to find proper gapping or interaction terms?

17 Question 1: How to gap out (i.e. fully open all mass gaps) of edge states? How to find proper gapping or interaction terms?

18 Question 1: How to gap out (i.e. fully open all mass gaps) of edge states? How to find possible gapping or interaction terms? Analogy between lane traffics quantum system. To gap out, i.e. fully open mass gaps, it requires: lane traffics equal number of lanes momentum conserved=0 traffic accidents quantum system equal number of Left/Right modes quantum-fluctuation(a-b) phase=0 gapping terms

19 uven Wang, Xiao-Gang Wen The(arXiv: , uses of InstantonsarXiv:140?.????) A conjecture and a proof: Anomaly-Matching Conditions = Boundary Fully Gapping Rules

20 uven Wang, Xiao-Gang Wen The(arXiv: , uses of InstantonsarXiv:140?.????) A conjecture and a proof: Anomaly-Matching Conditions = Boundary Fully Gapping Rules Anomaly-Matching conditions Hall conductance: A = q I K 1 IJ q J = 0 (t Hooft s Adler-Bell-Jackiw U(1)) in 1+1D

21 A conjecture and a proof: Anomaly-Matching Conditions = Boundary Fully Gapping Rules Anomaly-Matching conditions Hall conductance: A = q I K 1 IJ q J = 0 (t Hooft s Adler-Bell-Jackiw U(1)) in 1+1D Boundary Fully Gapping Rules l a,i K 1 IJ l b,j = 0, so Aharanov-Bohm phase = 1 l Lagrangian subgroup. Find N independent sets of l for 2N modes. Here l is Wilson-line operator. (Proof for U(1) N sym case, arxiv: , JW and X-G Wen)

22 Question 1: How to gap out (i.e. fully open all mass gaps) of edge states? 1st. Key Message : Boundary Anomaly Quantum Hall States y d-dim t Hooft anomaly matching conditions d-dim Boundary Fully Gapping Rules (d + 1-Dim bulk) J x J J

23 Question 2: How to design a bulk lattice model with one mirrored sector fully gapped) of edge states? - solve Lattice Chiral Fermion x y A B y A B a! x b! (a) (b) (c) chiral π-flux square lattice on the cylinder ladder Use copies of Chern insulator: Chern-band with nontrivial Chern number: C 1 = 1 d 2 k ɛ µν 2π k BZ kµ ψ(k) i kν ψ(k)

24 Question 2: How to design a bulk lattice model with one mirrored sector fully gapped) of edge states? - Lattice Chiral Fermion problem 2nd. Key Message : Ginsparg-Wilson Fermion Our approach ql ql Nontrivial SPT (Topological insulator) Trivial SPT (Trivial insulator) p.s. DMRG or tensor-networks methods to simulate 1+1D Lattice Chiral Fermion. qr qr

25 Question 3: How to detect non-trivialness of bulk SPT or Topological Order? Design proper interactions and geometry, ask whether the edge states can be gapped. 1+1D example: Boson sign-gordon: S = dt dx 1 4π (K IJ t Φ I V IJ x Φ I ) x Φ J + a g a cos(l a,i Φ I ) Multi-fermion interactions: S Ψ = dt dx (i ΨΓ µ µ Ψ + U interaction,b ( ψ q,..., N x ψ q,... ) ) Yukawa-Higgs (induce instantons or defects): S Ψ,Φ = dt d D x (i Ψ n Γ µ µ Ψ n + Ψ pn n Φ n,m Ψ pm m )

26 Question 3: How to detect non-trivialness of bulk SPT or Topological Order? 3rd. Key Message : Use topological defects (eg. Instanton). By gauge or gravitational anomaly. 0D 1D 1D or 2D x + + x

27 Q 1: How to gap out edge states? 1st. Key: Anomaly-free edge Edge states can be gapped Quantum Hall States y Boundary Anomaly x J J Q2: How to solve Lattice Chiral Fermion problem 2nd. Key: On-site symm. Chern insulators w/ strong interactions J Ginsparg-Wilson Fermion Our approach ql ql Nontrivial SPT (Topological insulator) Trivial SPT (Trivial insulator) qr qr Q3: Detect non-trivialness of SPT or Topological Order? 3rd. Key: Use topological defects (eg. Instanton, soliton, domain wall, vortex line). By gauge or gravitational anomaly. Juven Wang, Xiao-Gang Wen (arxiv: , arxiv:140?.????) The uses of Instantons

28 Work to appear For people like to know more INFO or ask for a copy of talk slides, please do feel free send me an JUVEN@MIT.EDU Juven Wang Thank you.

29 Symmetric Protected Trivial(SPT) and intrinsic Topological Order SPT order : Short ranged entangled states. No topological entanglement entropy. No fractionalized charge, no fractionalized statistics. The bulk realizes the symmetry G on-site. U(g) = i U i (g), g G,

30 Symmetric Protected Trivial(SPT) and intrinsic Topological Order SPT order : Short ranged entangled states. No topological entanglement entropy. No fractionalized charge, no fractionalized statistics. The bulk realizes the symmetry G on-site. U(g) = i U i (g), g G, The boundary realizes the symmetry G non-on-site, with (1) gapless edge states, or (2) Ground state degeneracy(gsd) from symmetry breaking, or (3) GSD from boundary topological ordered.

31 Symmetric Protected Trivial(SPT) and intrinsic Topological Order SPT order : Short ranged entangled states. No topological entanglement entropy. No fractionalized charge, no fractionalized statistics. The bulk realizes the symmetry G on-site. U(g) = i U i (g), g G, The boundary realizes the symmetry G non-on-site, with (1) gapless edge states, or (2) Ground state degeneracy(gsd) from symmetry breaking, or (3) GSD from boundary topological ordered. intrinsic Topological Order : Long ranged entangled states. (usually) with topological entanglement entropy. (usually) fractionalized charge, fractionalized statistics.

32 Symmetric Protected Trivial(SPT) and intrinsic Topological Order SPT order : Short ranged entangled states. No topological entanglement entropy. No fractionalized charge, no fractionalized statistics. The bulk realizes the symmetry G on-site. U(g) = i U i (g), g G, The boundary realizes the symmetry G non-on-site, with (1) gapless edge states, or (2) Ground state degeneracy(gsd) from symmetry breaking, or (3) GSD from boundary topological ordered. intrinsic Topological Order : Long ranged entangled states. (usually) with topological entanglement entropy. (usually) fractionalized charge, fractionalized statistics. (1) GSD on the nontrivial topology space. (2) non-abelian Berry s phases on coupling const moduli space

33 Anomaly and Quantum Hall (Laughlin s flux insertion) y x J J J In 1+1D boundary/2+1d bulk with a bulk Chern-Simons theory ( K 4π a da + q 2π A da σ H 2 A da): In 1+1D its own: µ j µ b = σ H 2 εµν F µν = σ H E x = J y µ j L = µ ( q 2π ɛµν ν Φ) = µ (q ψγ µ P L ψ) = +J y, µ j R = µ ( q 2π ɛµν ν Φ) = µ (q ψγ µ P R ψ) = J y. µ j µ V = µ (j µ L + j µ R ) = 0, µ j µ A = µ (j µ L j µ R ) = σ Hε µν F µν 0... U(1) V current conserved... ABJ U(1) A chiral anomaly

34 Question 1: How to gap out (i.e. fully open all mass gaps) of edge states? How to find proper gapping or interaction terms?

35 Question 1: How to gap out (i.e. fully open all mass gaps) of edge states? How to find proper gapping or interaction terms?