Wiring Topological Phases - PDF Free Download

1 Wiring Topological Phases Quantum Condensed Matter Journal Club Adhip Agarwala Department of Physics Indian Institute of Science adhip@physics.iisc.ernet.in February 4, 2016

So you are interested in topological stuff? 2

4 References E. Sagi, Yuval Oreg, Stern PRB 91, 245144 (2015) (Editor s Suggestion) E. Sagi, Yuval Oreg, PRB 92, 195137 (2015) E. Sagi, Yuval Oreg, PRB 90, 201102(R) (2014)(Editor s Suggestion)

4 References E. Sagi, Yuval Oreg, Stern PRB 91, 245144 (2015) (Editor s Suggestion) E. Sagi, Yuval Oreg, PRB 92, 195137 (2015) E. Sagi, Yuval Oreg, PRB 90, 201102(R) (2014)(Editor s Suggestion) Jefferey Teo, Kane PRB 89, 085101 (2014) (Editor s Suggestion) Ranjan Mukhopadhay, C. Kane, Lubensky PRB 64, 045120 (2001) C. Kane, Ranjan Mukhpadhay, Lubensky PRL 88, 036401 (2002) Commentary Anton Akhmerov: Topological Quantum Lego Condensed Matter Journal Club 2015. Related work: Classifying Topological Phases (Not to be discussed) Neupert, Chamon, Mudry, Thomale PRB 90, 205101 (2014)

5 Mandate Introduction Non-Interacting Systems Integer Quantum Hall Effect Quantum Spin Hall Effect Bosonization by Assertion Fractional Stuff From 1D Wire to Fractional Majorana

Introduction 6

6 Introduction Bottomline The idea is to construct various topological phases using 1D wires.

6 Introduction Bottomline The idea is to construct various topological phases using 1D wires. Why? Simple construction. Simple way to include interactions.

6 Introduction Bottomline The idea is to construct various topological phases using 1D wires. Why? Simple construction. Simple way to include interactions. So? Apart from producing the known things in a different context, also produces newer things.

7 Mandate Motivation and Introduction Non-Interacting Systems Integer Quantum Hall Effect Quantum Spin Hall Effect Bosonization by Assertion Fractional Stuff From 1D Wire to Fractional Majorana

8 1D wire Consider a particle moving in 1D(spinless) ɛ k = k2 2 (1)

9 1D wires Consider some set of wires.

10 Apply Magnetic Field A = By ˆx (2) E j (k) = 1 (k bj)2 b 2 = eb (3)

11 Apply Magnetic Field A = By ˆx (4) E j (k) = 1 (k bj)2 b 2 = eb (5)

Allow hybridization between wires. 12

13 IQHE Kane et. al. PRL 036401 2002, Teo et al. PRB 89, 085101 2014

14 IQHE at ν = 1

15 IQHE Crossing point with the j th wire k 2 2 = (k jb)2 2 (6)

IQHE Crossing point with the j th wire k 2 2 = (k jb)2 2 (6) k 2 F = (k F jb) 2 (7) Filling :ν = 2k F /b (8) Kane et. al. PRL 036401 2002, Teo et al. PRB 89, 085101 2014 15

We have seen Integer Quantum Hall. Lets see if we can get spin hall state? 16

Two wires at each j for each spin. E j (k) = 1 2 (k + k SO(2j 1)σ z ) 2 (9) Sagi PRB 91, 255144 (2015) 17

Apply SO coupling, changing with wire index and opposite sign for the spins. E j (k) = 1 2 (k + k SO(2j 1)σ z ) 2 (10) Sagi PRB 91, 255144 (2015) 18

19 Allowing the hybridization with the same spins. Sagi PRB 91, 255144 (2015)

20 Introducing pictorial representation. This will be used later as well. Sagi PRB 91, 255144 (2015) : Left : Right (11) red : blue : (12)

21 Mandate Introduction Non-Interacting Systems Integer Quantum Hall Effect Quantum Spin Hall Effect Bosonization by Assertion Fractional Stuff From 1D Wire to Fractional Majorana

Bosonization Ψn,σ R/L Kane, Lectures in Boulder School 2005 22 e i(φr/l n,σ +kn,σ R/L x) (13)

23 Going back to the Integer Quantum Hall, look at the hybridization N 1 H = t = t ko F π j=1 j=1 dx(ψ L j+1 ψr j + h.c.) (14) N 1 dx cos(φ L j+1 φ R j ) (15)

For SO coupling case 24

For SO coupling case N 1 H = t = t ko F π j=1 N 1 j=1 N 1 H = t = t ko F π j=1 N 1 j=1 Sagi PRB 91, 255144 (2015) dx(ψ L j+1, ψr j, + h.c.) (16) dx cos(φ L j+1, φ R j, ) (17) dx(ψ L j+1, ψr j, + h.c.) (18) dx cos(φ L j+1, φ R j, ) (19) 24

25 Mandate Introduction Non-Interacting Systems Integer Quantum Hall Effect Quantum Spin Hall Effect Topological Insulator Bosonization by Assertion Fractional Stuff From 1D Wire to Fractional Majorana

26 Filling should be < 1 Fractional Quantum Hall

Filling should be < 1 Fractional Quantum Hall R. Mukhopadhay, C. Kane, Lubensky PRB 64, 045120 (2001) 26

Fractional Quantum Hall 27

27 Fractional Quantum Hall momentum conserving scattering, multiple electron operators are involved. Ranjan Mukhopadhay, C. Kane, Lubensky PRB 64, 045120 (2001) Teo et al. PRB 89, 085101 (2014)

28 Define new operators, ψ R/L j = (ψ R/L j ) 2 (ψ L/R j ) e i(ηr/l j +p R/L j ) (20) η R/L j p R/L j = 2φ R/L j = 2k R/L j φ L/R j (21) k L/R j (22) Ranjan Mukhopadhay, C. Kane, Lubensky PRB 64, 045120 (2001) Teo et al. PRB 89, 085101 (2014)

29 Transform to a Non-Interacting problem ψ R/L j = (ψ R/L j ) 2 (ψ L/R j ) e i(ηr/l j +p R/L j ) (23)

29 Transform to a Non-Interacting problem ψ R/L j = (ψ R/L j ) 2 (ψ L/R j ) e i(ηr/l j +p R/L j ) (23) ψ R/L operators are fermionic operators. N 1 L H = t dx( ψ ψ R j+1 j + h.c.) (24) j=1 = t N 1 4 (ko F π )3 dx cos(ηj+1 L ηj R ) (25) j=1 Looks like the non-interacting problem!

Transform to a Non-Interacting problem ψ R/L j = (ψ R/L j ) 2 (ψ L/R j ) e i(ηr/l j +p R/L j ) (23) ψ R/L operators are fermionic operators. N 1 L H = t dx( ψ ψ R j+1 j + h.c.) (24) j=1 = t N 1 4 (ko F π )3 dx cos(ηj+1 L ηj R ) (25) j=1 Looks like the non-interacting problem! Sagi PRB 91, 255144 (2015) Teo et al. PRB 89, 085101 (2014) 29

30 Why fractional? θ l = ηr j η L j+1 2 (26)

30 Why fractional? θ l = ηr j η L j+1 2 (26) ρ l = x θ l /mπ (27)

30 Why fractional? θ l = ηr j η L j+1 2 (26) ρ l = x θ l /mπ (27) This means the kink in the bosonic field is associated with charge e/m.

30 Why fractional? θ l = ηr j η L j+1 2 (26) ρ l = x θ l /mπ (27) This means the kink in the bosonic field is associated with charge e/m. The phase gathered by exchange of these quasiparticles, = e 2πiN QP /m (28) Teo et al. PRB 89, 085101 (2014)

How about the same in Spin Hall effect at fractional filling? 31

Fractional Spin Hall Effect 32

32 Fractional Spin Hall Effect N 1 H = t j=1 L dx( ψ ψ R j+1, j + h.c.)

32 Fractional Spin Hall Effect N 1 H = t j=1 L dx( ψ ψ R j+1, j + h.c.) N 1 j=1 dx cos(η L j+1, η R j, ) (29)

32 Fractional Spin Hall Effect N 1 H = t j=1 N 1 H = t j=1 L dx( ψ ψ R j+1, j + h.c.) L dx( ψ ψ R j+1, j + h.c.) E. Sagi, PRB 91, 245144 (2015) N 1 j=1 N 1 j=1 dx cos(η L j+1, η R j, ) (29) dx cos(η L j+1, η R j, ) (30)

33 Mandate Introduction Non-Interacting Systems Integer Quantum Hall Effect Quantum Spin Hall Effect Topological Insulator Bosonization by Assertion Fractional Stuff From 1D Wire to Fractional Majorana

We will use whatever we have seen to build another model which will start from a a set of wires and reach fractional Majoranas(!). 34

35 Starting from 1D 4 wire unit cell. σ z : 1 1 1 1 (31) τ z : 1 1 1 1 (32)

Starting from 1D 4 wire unit cell. σ z : 1 1 1 1 (31) τ z : 1 1 1 1 (32) H o = (m 2t x cos(k x a))σ z τ z (33) Sagi et al. PRB 92, 195137(2015) 35

36 Add Spin-Orbit Coupling σ z : 1 1 1 1 (34) τ z : 1 1 1 1 (35) H = (m 2t x cos(k x a))σ z τ z (36) λ SO s z τ z sin(k x a) (37) s z is the spin label on each wire. Sagi et al. PRB 92, 195137(2015)

t x = t x cos(k so a) λ so = 2 t x sin(k so a) m = 2 t x cos(k o F a) (38) 37

37 t x = t x cos(k so a) λ so = 2 t x sin(k so a) m = 2 t x cos(k o F a) (38) E = σ z τ z 2 t x (cos(k o F a) cos(k x s z σ z k so a)) (39)

37 t x = t x cos(k so a) λ so = 2 t x sin(k so a) m = 2 t x cos(k o F a) (38) E = σ z τ z 2 t x (cos(k o F a) cos(k x s z σ z k so a)) (39) ν = ko F k so (40)

t x = t x cos(k so a) λ so = 2 t x sin(k so a) m = 2 t x cos(k o F a) (38) E = σ z τ z 2 t x (cos(k o F a) cos(k x s z σ z k so a)) (39) ν = ko F k so (40) Sagi et al. PRB 92, 195137(2015) 37

38 From 1D to 2D σ z : 1 1 1 1 (41) τ z : 1 1 1 1 (42)

38 From 1D to 2D σ z : 1 1 1 1 (41) τ z : 1 1 1 1 (42) H = E 1D t y τ x t y ((τ + σ + τ + σ + e i4ky ) + h.c) (43) Sagi et al. PRB 92, 195137(2015)

From 1D to 2D Treat t y s perturbatively Sagi et al. PRB 92, 195137(2015) : Left : Right (44) 39

40 From 1D to 2D Treat t y s perturbatively t y = 0, t y 0 : : Left : Right (45)

40 From 1D to 2D Treat t y s perturbatively t y = 0, t y 0 : Trivial : Left : Right (45)

40 From 1D to 2D Treat t y s perturbatively : Left : Right (45) t y = 0, t y 0 : Trivial t y 0, t y = 0 : Topological (46)

40 From 1D to 2D Treat t y s perturbatively : Left : Right (45) t y = 0, t y 0 : Trivial t y 0, t y = 0 : Topological (46) Critical point : t y = t y

40 From 1D to 2D Treat t y s perturbatively : Left : Right (45) t y = 0, t y 0 : Trivial t y 0, t y = 0 : Topological (46) Critical point : t y = t y Sagi et al. PRB 92, 195137(2015)

Fractionalize! 41

41 Fractionalize! Multi-electron processes needed to gap the system.

41 Fractionalize! Multi-electron processes needed to gap the system. operators look non-interacting. All hopping terms become cosines in bosonized language.

41 Fractionalize! Multi-electron processes needed to gap the system. operators look non-interacting. All hopping terms become cosines in bosonized language. This is the two-dimensional fractional topological insulator. Sagi et al. PRB 92, 195137(2015)

42 Integer Case: From 2D to 3D set ν = 1.

42 Integer Case: From 2D to 3D set ν = 1. H z = 1 2 [(m 2t z cos(k z ))τ x + 2t z sin(k z )s y τ z ](1 σ x ) (47) Sagi et al. PRB 92, 195137(2015)

43 Looking for Dirac Cones Set t y = t y, ν = 1 setup. Look at the low energy subspace at k x = k y = 0. These are, σ y = 1; τ x = ±1; s.

43 Looking for Dirac Cones Set t y = t y, ν = 1 setup. Look at the low energy subspace at k x = k y = 0. These are, σ y = 1; τ x = ±1; s. Linearizing, H xy = 2a[k y t y σ y + k x t x sin(k so σ x )] (48)

43 Looking for Dirac Cones Set t y = t y, ν = 1 setup. Look at the low energy subspace at k x = k y = 0. These are, σ y = 1; τ x = ±1; s. Linearizing, H xy = 2a[k y t y σ y + k x t x sin(k so σ x )] (48) The z-direction problem becomes, H z = 2t z sin(k z a)σ y + (m 2t z cos(k z a))σ z (49) SSH Deja-vu in z direction and indeed is the condition for strong topological insulator. Explicit solutions for the edge modes can be found which are exponentially decaying into the bulk. Sagi et al. PRB 92, 195137(2015)

44 Fractionalize! Prescription: Write the same Hamiltonian in the fields.

44 Fractionalize! Prescription: Write the same Hamiltonian in the fields. Exactly the same Hamiltonian in the y and z as we saw previously, but now for the fields. This is the reduced model and look at the edge modes.

This is the reduced model and look at the edge modes. This edge mode cannot be described by a free Dirac theory. This is what is termed fractional Dirac liquid. Its a surface state, but of these new quasiparticles. Sagi et al. PRB 92, 195137(2015) 44 Fractionalize! Prescription: Write the same Hamiltonian in the fields. Exactly the same Hamiltonian in the y and z as we saw previously, but now for the fields.

45 Integer Case: Divide and Conquer! Usual 3D TIs are protected by time reversal and charge conservation and support exotic states at boundaries of symmetry breaking fields.

45 Integer Case: Divide and Conquer! Usual 3D TIs are protected by time reversal and charge conservation and support exotic states at boundaries of symmetry breaking fields. Case 1: Integer Quantum Hall Effect Case II: Counter Propagating Majorana Case III: Chiral Majorana Taylor Hughes in Topological Insulators and Topological Superconductors, Princeton Press

46 Fractionalize, then Divide and Conquer! Fractionalized modes are still protected by time reversal and charge conservation. Breaking these symmetries bring edge modes which are called the halved fractional quantum hall state and fractional Majorana state. Sagi et al. PRB 92, 195137(2015)

47 Mandate Introduction Non-Interacting Systems Integer Quantum Hall Effect Quantum Spin Hall Effect Bosonization by Assertion Fractional Stuff From 1D Wire to Fractional Majorana

Is that it? 48

48 Is that it? Electron-hole double layer constructions to simulate a torus. Can reproduce topological degeneracy of these states. Predictions for an experiment. E. Sagi, Yuval Oreg, Stern PRB 91, 245144 (2015) (Editor s Suggestion) Moore-Read state. Bosonic systems. Non-uniform magnetic flux in a unit cell: q-pfaffian state. Read-Rezayi Sequence. Teo, Kane PRB 89, 0851101(2014) (Editor s Suggestion) Scattering calculations for wires. Two-terminal conductances for fractional cases. Oreg, Sela, Stern PRB 89, 115402 (2014)

49 References E. Sagi, Yuval Oreg, Stern PRB 91, 245144 (2015) (Editor s Suggestion) E. Sagi, Yuval Oreg, PRB 92, 195137 (2015) E. Sagi, Yuval Oreg, PRB 90, 201102(R) (2014)(Editor s Suggestion) Jefferey Teo, Kane PRB 89, 085101 (2014) (Editor s Suggestion) Ranjan Mukhopadhay, C. Kane, Lubensky PRB 64, 045120 (2001) C. Kane, Ranjan Mukhpadhay, Lubensky PRL 88, 036401 (2002) Commentary Anton Akhmerov: Topological Quantum Lego Condensed Matter Journal Club 2015. Related work: Classifying Topological Phases Neupert, Chamon, Mudry, Thomale PRB 90, 205101 (2014)

And so if you thought you are going to understand all of topology by attending this talk 50

And so if you thought you are going to understand all of topology by attending this talk or by preparing for it. Well.. 50

50 And so if you thought you are going to understand all of topology by attending this talk or by preparing for it. Well.. Thanks.

50 Weak TIs H = 2tσ y sin(k z a) + vk x σ z (50) Two Dirac cones at (k x = 0, k z = 0) and (k x = 0, k π = 0)

50 References Link E. Sagi, Yuval Oreg, Stern PRB 91, 245144 (2015) (Editor s Suggestion) Link E. Sagi, Yuval Oreg, PRB 92, 195137 (2015) Link E. Sagi, Yuval Oreg, PRB 90, 201102(R) (2014)(Editor s Suggestion) Link Jefferey Teo, Kane PRB 89, 085101 (2014) (Editor s Suggestion) Link Ranjan Mukhopadhay, C. Kane, Lubensky PRB 64, 045120 (2001) Link C. Kane, Ranjan Mukhpadhay, Lubensky PRL 88, 036401 (2002)Link Commentary Anton Akhmerov: Topological Quantum Lego Condensed Matter Journal Club 2015. Related work: Classifying Topological Phases Neupert, Chamon, Mudry, Thomale PRB 90, 205101 (2014) Link 1D SSH Model Link