Edge universality in interacting 2 d topological insulators Marcello - - PowerPoint PPT Presentation

Edge universality in interacting 2d topological insulators

Marcello Porta Joint with: G. Antinucci (UZH) and V. Mastropietro (Milan)

Summary

• Introduction: edge transport in noninteracting quantum Hall systems

and time-reversal invariant systems. Bulk-edge duality.

• Many-body quantum systems. Results:

Edge transport coefficients for quantum Hall and TRI systems. Interacting bulk-edge correspondence, Haldane relations.

• Sketch of the proof: Renormalization group and Ward identities.
• Conclusions.

Marcello Porta Edge universality October 3, 2018 1 / 22

Introduction

Introduction: noninteracting systems

Marcello Porta Edge universality October 3, 2018 1 / 22

Introduction

Integer quantum Hall effect

• Bulk topological order in condensed matter systems is deeply related to

the emergence of gapless edge modes.

• Example. Integer quantum Hall effect [von Klitzing et al. ’80]

2d insulators exposed to strong magnetic field and in-plane electric field.

Marcello Porta Edge universality October 3, 2018 2 / 22

Introduction

Integer quantum Hall effect

• Bulk topological order in condensed matter systems is deeply related to

the emergence of gapless edge modes.

• Example. Integer quantum Hall effect [von Klitzing et al. ’80]

2d insulators exposed to strong magnetic field and in-plane electric field. Linear response: J = σE + o(E) with σ = conductivity matrix: σ =

• n

2π

− n

2π

• ,

n ∈ Z.

Marcello Porta Edge universality October 3, 2018 2 / 22

Introduction

Integer quantum Hall effect: theory

• Noninteracting fermions. H = 1-particle Hamiltonian, on ℓ2(Z2; CM).

Suppose that σ(H) is gapped, µ = Fermi level ∈ gap(H). µ R

Marcello Porta Edge universality October 3, 2018 3 / 22

Introduction

Integer quantum Hall effect: theory

• Noninteracting fermions. H = 1-particle Hamiltonian, on ℓ2(Z2; CM).

Suppose that σ(H) is gapped, µ = Fermi level ∈ gap(H).

• For simplicity, H(x; y) ≡ H(x − y). Bloch decomp.: H =

⊕

T2 dk ˆ

H(k) Let ˆ Pµ(k) = χ( ˆ H(k) ≤ µ) = Fermi projector. Thouless et al. ’82: σ12 = i

• T2

dk (2π)2 TrCM ˆ Pµ(k)[∂k1 ˆ Pµ(k), ∂k2 ˆ Pµ(k)]∈ 1 2π Z σ12 = Chern number of Bloch bundle: EB = {(k, u) ∈ T2 × CM | u ∈ Ran ˆ Pµ(k)}

Marcello Porta Edge universality October 3, 2018 3 / 22

Introduction

Integer quantum Hall effect: theory

• Noninteracting fermions. H = 1-particle Hamiltonian, on ℓ2(Z2; CM).

Suppose that σ(H) is gapped, µ = Fermi level ∈ gap(H).

• For simplicity, H(x; y) ≡ H(x − y). Bloch decomp.: H =

⊕

T2 dk ˆ

H(k) Let ˆ Pµ(k) = χ( ˆ H(k) ≤ µ) = Fermi projector. Thouless et al. ’82: σ12 = i

• T2

dk (2π)2 TrCM ˆ Pµ(k)[∂k1 ˆ Pµ(k), ∂k2 ˆ Pµ(k)]∈ 1 2π Z σ12 = Chern number of Bloch bundle: EB = {(k, u) ∈ T2 × CM | u ∈ Ran ˆ Pµ(k)}

• IQHE for general (disordered) systems:

Bellissard et al. ’94. σ12 = Noncommutative Chern number. Avron-Seiler-Simon ’94. σ12 = index of a pair of projections. Aizenman-Graf ’98. Strong disorder ⇒ Hall plateaux.

Marcello Porta Edge universality October 3, 2018 3 / 22

Introduction

Edge states in quantum Hall systems

• Halperin ’82. Hall phases must come with robust edge currents.
• Intuition. For a weak, slowly varying vector potential A,

Z(A) Z(0) = eiσ12

• Ω A∧dA+irr.

(gap assumption)

= eiσ12

• Ω(A+dα)∧d(A+dα)+irr.

(gauge inv.)

= Z(A) Z(0) eiσ12

• ∂Ω dα∧A+irr.

(Stokes)

σ12 = 0 ⇒ The gap assumption cannot be true!

Marcello Porta Edge universality October 3, 2018 4 / 22

Introduction

Edge states in quantum Hall systems: more precise

• Let H be a lattice Schr¨
• dinger operator on the cylinder:

(0, 0) (L, 0) (L, L) (0, L) Figure: Dotted lines: Dirichlet boundary conditions. Identify vertical sides.

Marcello Porta Edge universality October 3, 2018 5 / 22

Introduction

Edge states in quantum Hall systems: more precise

• Let H be a lattice Schr¨
• dinger operator on the cylinder:
• Let Hp the counterpart of H with periodic b.c.. Hyp.: Hp is gapped.

Marcello Porta Edge universality October 3, 2018 5 / 22

Introduction

Edge states in quantum Hall systems: more precise

• Let H be a lattice Schr¨
• dinger operator on the cylinder:
• Let Hp the counterpart of H with periodic b.c.. Hyp.: Hp is gapped.

σ(H) might differ from σ(Hp) by the presence of edge states. Figure: H = ⊕

T1 dk1 ˆ

H(k1), ˆ H(k1) = 1d Hamiltonian. Spectrum of ˆ H(k1).

• Red curve: eigenvalue branch ε(k1), with eigenstates (edge modes)

ϕx(k1) = eik1x1ξx2(k1) , with ξx2(k1) ∼ e−cx2 .

Marcello Porta Edge universality October 3, 2018 5 / 22

Introduction

The bulk-edge correspondence

• Bulk-edge duality: relation between σ12 of Hp and the edge states of H.

σ12 =

• e

ωe 2π with ωe = ±1 (chirality of the edge state.) Figure: (a) : σ12 =

1 2π,

(b) : σ12 = − 1

2π,

(c) : σ12 = 0.

Marcello Porta Edge universality October 3, 2018 6 / 22

Introduction

The bulk-edge correspondence

• Bulk-edge duality: relation between σ12 of Hp and the edge states of H.

σ12 =

• e

ωe 2π with ωe = ±1 (chirality of the edge state.) Figure: (a) : σ12 =

1 2π,

(b) : σ12 = − 1

2π,

(c) : σ12 = 0.

• Rigorous results for noninteracting systems:

Hatsugai, ’93: Translation invariant systems. Schulz-Baldes et al. ’00: Disordered systems (with bulk gap). Graf et al. ’02: Anderson localization regime.

Marcello Porta Edge universality October 3, 2018 6 / 22

Introduction

Time-reversal invariant systems

• Quantum Hall systems are an example of topological insulators.

Necessary condition for σ12 = 0: breaking of TRS (magnetic field).

• Unbroken TRS: charge transport is trivial but spin transport is possible.

spin down Edge void Bulk spin up Edge void

• Spin Hall effect: Murakami-Nagaosa-Zhang ’03, ... (Fr¨
• hlich et al. ’93.) Model:

Kane-Mele ’05. Discovery: Bernevig-Hughes-Zhang ’06 (theory), K¨

• nig et al. ’07.

Marcello Porta Edge universality October 3, 2018 7 / 22

Introduction

Edge spin transport

• Gapped TRI model on a cylinder, Hamiltonian H =

⊕

T1 dk1 ˆ

H(k1). TRS: ˆ H(k1) = Θ−1 ˆ H(−k1)Θ, with Θ2 = −1, Θ antiunitary. Figure: σ( ˆ H(k1)) for the Kane-Mele model. σ( ˆ H(k1)) = σ( ˆ H(−k1)).

• Eigenvalues at k1 = −k1 are even degenerate (Kramers degeneracy).

⇒ (edge) Z2 classification of H: parity of # of pairs of edge modes at µ.

Marcello Porta Edge universality October 3, 2018 8 / 22

Introduction

Edge spin transport

• Gapped TRI model on a cylinder, Hamiltonian H =

⊕

T1 dk1 ˆ

H(k1). TRS: ˆ H(k1) = Θ−1 ˆ H(−k1)Θ, with Θ2 = −1, Θ antiunitary. π k

• Eigenvalues at k1 = −k1 are even degenerate (Kramers degeneracy).

⇒ (edge) Z2 classification of H: parity of # of pairs of edge modes at µ.

Marcello Porta Edge universality October 3, 2018 8 / 22

Introduction

Edge spin transport

• Gapped TRI model on a cylinder, Hamiltonian H =

⊕

T1 dk1 ˆ

H(k1). TRS: ˆ H(k1) = Θ−1 ˆ H(−k1)Θ, with Θ2 = −1, Θ antiunitary. π k

• Eigenvalues at k1 = −k1 are even degenerate (Kramers degeneracy).

⇒ (edge) Z2 classification of H: parity of # of pairs of edge modes at µ.

Marcello Porta Edge universality October 3, 2018 8 / 22

Introduction

Edge spin transport

• Gapped TRI model on a cylinder, Hamiltonian H =

⊕

T1 dk1 ˆ

H(k1). TRS: ˆ H(k1) = Θ−1 ˆ H(−k1)Θ, with Θ2 = −1, Θ antiunitary. π k

• Eigenvalues at k1 = −k1 are even degenerate (Kramers degeneracy).

⇒ (edge) Z2 classification of H: parity of # of pairs of edge modes at µ.

Marcello Porta Edge universality October 3, 2018 8 / 22

Introduction

Edge spin transport

• Gapped TRI model on a cylinder, Hamiltonian H =

⊕

T1 dk1 ˆ

H(k1). TRS: ˆ H(k1) = Θ−1 ˆ H(−k1)Θ, with Θ2 = −1, Θ antiunitary. π k

• Eigenvalues at k1 = −k1 are even degenerate (Kramers degeneracy).

⇒ (edge) Z2 classification of H: parity of # of pairs of edge modes at µ.

Marcello Porta Edge universality October 3, 2018 8 / 22

Introduction

Edge spin transport

• Gapped TRI model on a cylinder, Hamiltonian H =

⊕

T1 dk1 ˆ

H(k1). TRS: ˆ H(k1) = Θ−1 ˆ H(−k1)Θ, with Θ2 = −1, Θ antiunitary. π k

• Eigenvalues at k1 = −k1 are even degenerate (Kramers degeneracy).

⇒ (edge) Z2 classification of H: parity of # of pairs of edge modes at µ.

• Bulk Z2 classif. is also possible (no direct connection with transport).
• Graf-P. ’13: bulk-edge duality for TRI systems.

Marcello Porta Edge universality October 3, 2018 8 / 22

Many-body quantum systems

Many-body quantum systems

Marcello Porta Edge universality October 3, 2018 8 / 22

Many-body quantum systems

Many-body quantum systems

• Interacting many-body Fermi system on ΛL ⊂ Z2.
• Fock space Hamiltonian: H = H0 + λV with

H0 =

• x,y
• σ,σ′

a+

x,σH(x, y)a− y,σ ,

V =

• x,y
• σ,σ′

v(x − y)a+

x,σa+ y,σ′a− y,σ′a− x,σ

with {a+

x,σ, a− y,σ′} = δx,yδσ,σ′,

{a+

x,σ, a+ y,σ′} = {a− x,σ, a− y,σ′} = 0,

and H, v short ranged.

• Finite volume, finite temperature Gibbs state:

·β,L = Tr · e−β(H−µN ) Zβ,L , Zβ,L = Tr e−β(H−µN ) , β = 1/T with µ = chemical potential and: N =

• x
• σ

a+

x,σa− x,σ ≡

• x

nx .

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Many-body quantum systems

Interacting bulk transport

• Periodic boundary conditions. Many-body Kubo formula:

σij = lim

η→0+

lim

β,L→∞

i ηL2

−∞

dt eηt

• Ji(t), Jj
• β,L −
• Ji, Xj
• β,L
• X =

x xnx, J = i[H , X] = current operator and Ji(t) = eiHtJie−iHt.

• Hastings-Michalakis ’15. Quantization of σ12. (Quasi-adiabatic methods)

Hyp.: the ground state of H is gapped.

• Giuliani-Mastropietro-P. ’16. Universality of σij. (RG & Ward identities)

Hyp.: fast enough algebraic decay of corr.. E.g.: gapped ground states; graphene-like models (+Jauslin ’16: critical Haldane model).

• Bachmann-de Roeck-Fraas ’17. Validity of Kubo formula.

Hyp.: the ground state of H(t) is gapped for all times.

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Many-body quantum systems

Interacting edge transport

• Edge transport. Localize observables at distance ≤ ℓ from x2 = 0.

(Cylindric boundary conditions & transl. inv. in direction x1.)

(0, 0) (L, 0) (L, L) (0, L) (0, ℓ)

• Interesting quantities: charge density nx and current density

jx, nx = a+

x a− x ,

∂tnx(t) + divx jx(t) = 0 . Let: ˆ np1 =

x=(x1,x2) eip1x1nx

and ˆ nℓ

p1 = x1 eip1x1 x2≤ℓ nx.

Marcello Porta Edge universality October 3, 2018 11 / 22

Many-body quantum systems

Interacting edge transport

• Edge transport. Localize observables at distance ≤ ℓ from x2 = 0.

(Cylindric boundary conditions & transl. inv. in direction x1.)

(0, 0) (L, 0) (L, L) (0, L) (0, ℓ)

• Interesting quantities: charge density nx and current density

jx, nx = a+

x a− x ,

∂tnx(t) + divx jx(t) = 0 . Let: ˆ np1 =

x=(x1,x2) eip1x1nx

and ˆ nℓ

p1 = x1 eip1x1 x2≤ℓ nx.

• Spin transport (current well defined if [H, S3] = 0, with S3 =

σ σnx,σ):

nx → nx,↑ − nx,↓ , j1,x → j1,x,↑ − j1,x,↓

Marcello Porta Edge universality October 3, 2018 11 / 22

Many-body quantum systems

Edge transport coefficients

• Edge charge susceptibility:

κℓ(η, p1) := − lim

β,L→∞

i L

−∞

dt etη [ˆ np1(t) , ˆ nℓ

−p1]β,L

(response of the edge density to a density perturbation)

• Edge charge conductance:

Gℓ(η, p1) := − lim

β,L→∞

i L

−∞

dt etη [ˆ np1(t) , ˆ jℓ

1,−p1]β,L

(response of the edge current to a density perturbation)

• Edge Drude weight:

Dℓ(η, p1) := lim

β,L→∞

i L

−∞

dt etη [ˆ j1,p1(t) , ˆ jℓ

1,−p1]β,L

(response of the edge current to an electric field)

Marcello Porta Edge universality October 3, 2018 11 / 22

Many-body quantum systems

Effective description of the edge modes

• Effective 1d theory for a single edge mode: chiral Luttinger model.

HχL =

• σ=↑↓
• dk vek ˆ

a+

k,σˆ

a−

k,σ + λ

• dpdkdk′ ˆ

a+

k+p,↑ˆ

a+

k′−p,↓ˆ

a−

k,↓ˆ

a−

k′,↑

• Wen ’90. Theory of interacting Hall edge currents based on χL.

Advantage: χL exactly solvable by bosonization [Mattis-Lieb ’65.]

• Effective 1d theory for TRI systems: helical Luttinger model.

HHL =

• σ=↑↓
• dk σvek ˆ

a+

k,σˆ

a−

k,σ + λ

• dpdkdk′ ˆ

a+

k+p,↑ˆ

a+

k′−p,↓ˆ

a−

k,↓ˆ

a−

k′,↑

Marcello Porta Edge universality October 3, 2018 12 / 22

Many-body quantum systems

Effective description of the edge modes

• Effective 1d theory for a single edge mode: chiral Luttinger model.

HχL =

• σ=↑↓
• dk vek ˆ

a+

k,σˆ

a−

k,σ + λ

• dpdkdk′ ˆ

a+

k+p,↑ˆ

a+

k′−p,↓ˆ

a−

k,↓ˆ

a−

k′,↑

• Wen ’90. Theory of interacting Hall edge currents based on χL.

Advantage: χL exactly solvable by bosonization [Mattis-Lieb ’65.]

• Effective 1d theory for TRI systems: helical Luttinger model.

HHL =

• σ=↑↓
• dk σvek ˆ

a+

k,σˆ

a−

k,σ + λ

• dpdkdk′ ˆ

a+

k+p,↑ˆ

a+

k′−p,↓ˆ

a−

k,↓ˆ

a−

k′,↑

• Remark. Integrability is nongeneric: broken by, e.g., nonlinearities of the

dispersion relation or by the bulk degrees of freedom.

Marcello Porta Edge universality October 3, 2018 12 / 22

Many-body quantum systems

Interacting edge transport: single mode edge currents

Theorem (Antinucci-Mastropietro-P., Comm. Math. Phys. ’18) Suppose that H has one edge mode per edge. Then, ∃ λ0 > 0 s.t. for |λ| < λ0 the Gibbs state is analytic in λ. Moreover, the edge transport coefficients are: κℓ(η, p1) = 1 π|v| vp1 −iη + vp1 + Rℓ

κ(η, p1)

Gℓ(η, p1) = ω π vp1 −iη + vp1 + Rℓ

G(η, p1) ,

(ω = sgn(v)) Dℓ(η, p1) = |v| π −iη −iη + vp1 + Rℓ

D(η, p1)

v ≡ v(λ) = dressed Fermi velocity, limℓ→∞ limη,p1→0 Rℓ

♯(η, p1) = 0.

• The results agrees with the predictions based on bosonization:

“edge states ≃ noninteracting 1d Bose gas”.

• Proof based on renormalization group methods and on a rigorous

comparison with χL [Benfatto-Falco-Mastropietro ’10+].

Marcello Porta Edge universality October 3, 2018 13 / 22

Many-body quantum systems

Bulk-edge correspondence and Haldane relations

• Interacting bulk-edge duality:

G = lim

ℓ→∞

lim

p1,η→0+ Gℓ(η, p1)

= ω π =

• bulk-edge corresp.

σ12(λ = 0) The bulk-edge duality follows from bulk universality: σ12(0) = σ12(λ).

• In contrast, the Drude weight and the susceptibility are nonuniversal:

κ = lim

ℓ→∞

lim

p1,η→0+ κℓ(η, p1) =

1 π|v| D = lim

ℓ→∞

lim

η,p1→0+ Dℓ(η, p1) = |v|

π . (v ≡ v(λ)) Nevertheless, they satisfy the Haldane relation: D κ = v2 first predicted to hold for 1d systems by [Haldane ’80].

Marcello Porta Edge universality October 3, 2018 14 / 22

Many-body quantum systems

Interacting edge transport: TRI systems

Theorem (Mastropietro-P., Phys. Rev. B ’17) Suppose that H is TRS, and that H has one pair of edge states per edge. Also, suppose that [H, S3] = 0. Then, ∃ λ0 > 0 s.t. for |λ| < λ0: Gs = ω π , ω = sgn(v) . Moreover the charge and spin edge Drude weights and susceptibilities are: κc = K πv , Dc = vK π , κs = 1 πvK , Ds = v πK with K ≡ K(λ) = 1 + O(λ) = 1, v ≡ v(λ) = v↑ + O(λ). Finally, the 2-point function decays with anomalous exponent η = (K + K−1 − 2)/2.

• Remark. In the single edge mode case, K = 1 (no anomalous exponents.)

Marcello Porta Edge universality October 3, 2018 15 / 22

Sketch of the proof

Sketch of the proof

(one edge mode)

Marcello Porta Edge universality October 3, 2018 15 / 22

Sketch of the proof

Perturbation theory

• Wick rotation. Transport coefficients can be expressed via imaginary

time correlations (T = time ordering): Gℓ(η, p1) = lim

β,L→∞

β/2

−β/2

e−iηt 1 LT ˆ np1(−it) ; ˆ jℓ

−p1β,L .

• Let At ≡ A(−it). Perturbative expansion of Euclidean correlations:

T At ; Bβ,L =

• n≥0

λn n!

• [0,β)n dt1 . . . dtnT At ; B ; Vt1 ; · · · ; Vtnβ,L
• λ=0

Expansion in terms of Feynman diagrams. Covariance (β, L → ∞): g(t1, x; t2, y) = T a−

(t1,x)a+ (t2,y)

• λ=0 =

θ(t1 − t2)e(t2−t1)(H−µ)P ⊥

µ (H) − θ(t2 − t1)e(t2−t1)(H−µ)Pµ(H)

• Problems. 1) (2n)! diagrams;

2) gapless modes: slow space-time decay.

Marcello Porta Edge universality October 3, 2018 16 / 22

Sketch of the proof

Grassmann integral formulation

Tr e−βH Tr e−βH0 =

• µ(dψ)eV (ψ)
• ψ±

x = Grassmann field,

V (ψ) = “λψ4”, µ(dψ) = N −1e−(ψ+,g−1ψ−)dψ

• ψ = ψe + ψb, where ψb has gapped covariance g(bulk) ≡ gχ(|H − µ| > δ).
• µ(dψ)eV (ψ) =
• µe(dψe)µb(dψb)eV (ψe+ψb) ≡ eF (b)

β,L

• µe(dψe)eV (e)(ψe)

Marcello Porta Edge universality October 3, 2018 17 / 22

Sketch of the proof

Grassmann integral formulation

Tr e−βH Tr e−βH0 =

• µ(dψ)eV (ψ)
• ψ±

x = Grassmann field,

V (ψ) = “λψ4”, µ(dψ) = N −1e−(ψ+,g−1ψ−)dψ

• ψ = ψe + ψb, where ψb has gapped covariance g(bulk) ≡ gχ(|H − µ| > δ).
• µ(dψ)eV (ψ) =
• µe(dψe)µb(dψb)eV (ψe+ψb) ≡ eF (b)

β,L

• µe(dψe)eV (e)(ψe)

Brydges-Battle-Federbush formula. Solution of 2n! problem. ψP1

b ; . . . ; ψPn b µb =

• T ∈T

αT

• ℓ∈T

g(bulk)

ℓ

• νT (dt) det G(bulk)

T

(t) T = spanning tree of {Pi}, #{T} ≤ Cn!, det G(bulk)

T

(t)∞ ≤ Cn , |g(bulk)

ℓ

| ≤ (C/δ)e−cδℓ .

Marcello Porta Edge universality October 3, 2018 17 / 22

Sketch of the proof

Effective 1d model: RG analysis

• µe(dψe)eV (e)(ψe) =
• µ1d(dϕ)eV (e)(ϕ∗ξ(e)) ≡
• µ1d(dϕ)eV (1d)(ϕ)
• (ϕ ∗ ξ(e))(ω, k1, x2) = ˆ

ξe

x2(k1) ˆ

ϕ(ω,k1) with ξ(e) = edge state and: ˆ ϕ−

(ω,k1) ˆ

ϕ+

(ω,k1) =

χe(k1) −iω + ε(k1) − µ ≃ 1 −iω + v(k1 − kF ) χL model

• Multiscale evaluation of the Grassmann integral:

[Gawedzki, Kupiainen, Feldman, Magnen, Rivasseau, S´ en´ eor, Lesniewski, Benfatto, Gallavotti, Mastropietro, Balaban, Kn¨

• rrer, Salmhofer, Trubowitz, Brydges, Slade...]

Write ϕ = 0

h=hβ ϕ(0) and integrate ϕ(h) progressively:

• µ1d(dϕ)eV (1d)(ϕ) =
• µhβ(dϕ(hβ)) · · · µh(dϕ(h))eV (h)(ϕ(hβ )+...+ϕ(h))

ˆ ϕ(h)

(ω,k1) supported for |ω|2 + |k1 − kF |2 ∼ 22h,

covariance ˆ g(h).

Marcello Porta Edge universality October 3, 2018 18 / 22

Sketch of the proof

The flow of the beta function

• Goal: control the map (µh, V (h)) → (µh−1, V (h−1)). Morally,

V (h)(ϕ(h)) =

• dt
• x1

λhϕ(h)+

x,↑ ϕ(h)− x,↑ ϕ(h)+ x,↓ ϕ(h)− x,↓

+ irrelevant terms λh = λh+1 + βh+1(λh+1, . . . , λ0) , λ0 ≡ λ .

• In general, |βh+1| ≤ C maxk≥h |λk|2. Not summable.

Marcello Porta Edge universality October 3, 2018 19 / 22

Sketch of the proof

The flow of the beta function

• Goal: control the map (µh, V (h)) → (µh−1, V (h−1)). Morally,

V (h)(ϕ(h)) =

• dt
• x1

λhϕ(h)+

x,↑ ϕ(h)− x,↑ ϕ(h)+ x,↓ ϕ(h)− x,↓

+ irrelevant terms λh = λh+1 + βh+1(λh+1, . . . , λ0) , λ0 ≡ λ .

• In general, |βh+1| ≤ C maxk≥h |λk|2. Not summable.
• Crucial remark: βh+1 = βχL

h+1 + δβh+1, with [Falco-Mastropietro ’08]:

βχL

h+1= 0 ,

|δβh+1| ≤ C2h max

k≥h |λk|2 .

Summable iteration! Analyticity of V (hβ), unif. in β, L, follows.

Marcello Porta Edge universality October 3, 2018 19 / 22

Sketch of the proof

Comparison with the effective 1d theory

• RG allows to express the lattice correlations via the χL:

T jµ,(t,x) ; jν,y = Zµ(x2)Zν(y2)T n(t,x1) ; ny1χL + “small errors” where |Zµ(x2)| ≤ Ce−c|x2| (from the decay of edge modes), and: (FTT n(t,x1) ; ny1χL)(ω, p1) = − 1 2πv 1 Z2(1 − τ) −iω − vp1 −iω + ˜ vp1 , ˜ v = 1−τ

1+τ

• v,

τ =

λ 2πv = anomaly,

v = ve + O(λ), Z = 1 + O(λ2).

Marcello Porta Edge universality October 3, 2018 20 / 22

Sketch of the proof

Comparison with the effective 1d theory

• RG allows to express the lattice correlations via the χL:

T jµ,(t,x) ; jν,y = Zµ(x2)Zν(y2)T n(t,x1) ; ny1χL + “small errors” where |Zµ(x2)| ≤ Ce−c|x2| (from the decay of edge modes), and: (FTT n(t,x1) ; ny1χL)(ω, p1) = − 1 2πv 1 Z2(1 − τ) −iω − vp1 −iω + ˜ vp1 , ˜ v = 1−τ

1+τ

• v,

τ =

λ 2πv = anomaly,

v = ve + O(λ), Z = 1 + O(λ2). (empty bubble) = τ

λ(iω + vp1)

(i) D(p) = −iω + vp1; (ii) the circle localizes the lines on the UV cutoff scale; (iii) the last term vanishes as the UV cutoff is removed.

Marcello Porta Edge universality October 3, 2018 20 / 22

Sketch of the proof

Comparison with the effective 1d theory

• RG allows to express the lattice correlations via the χL:

T jµ,(t,x) ; jν,y = Zµ(x2)Zν(y2)T n(t,x1) ; ny1χL + “small errors” where |Zµ(x2)| ≤ Ce−c|x2| (from the decay of edge modes), and: (FTT n(t,x1) ; ny1χL)(ω, p1) = − 1 2πv 1 Z2(1 − τ) −iω − vp1 −iω + ˜ vp1 , ˜ v = 1−τ

1+τ

• v,

τ =

λ 2πv = anomaly,

v = ve + O(λ), Z = 1 + O(λ2).

• To prove universality, need to find a cancellation between Z, τ, v and the

vertex renormalization: Zµ =

• x2

Zµ(x2) The cancellation follows from Ward identities: consequences of the continuity equation ∂µjµ,x = 0 for the correlations.

Marcello Porta Edge universality October 3, 2018 20 / 22

Sketch of the proof

Ward identities

• Both ·β,L and ·χL satisfy vertex WIs:

(x = (t, x1, x2) = (x, x2))

∂µTjµ,z ; a−

y a+ x β,L

=

• δx,zTa−

y a+ x β,L − δy,zTa− y a+ x β,L

• (∂0 + v∂1)Tnz ; ϕ−

y ϕ+ x χL

= 1 Z(1 − τ)

• δx,zTϕ−

y ϕ+ x χL − δy,zTϕ− y ϕ+ x χL

• Marcello Porta

Edge universality October 3, 2018 21 / 22

Sketch of the proof

Ward identities

• Both ·β,L and ·χL satisfy vertex WIs:

(x = (t, x1, x2) = (x, x2))

∂µTjµ,z ; a−

y a+ x β,L

=

• δx,zTa−

y a+ x β,L − δy,zTa− y a+ x β,L

• (∂0 + v∂1)Tnz ; ϕ−

y ϕ+ x χL

= 1 Z(1 − τ)

• δx,zTϕ−

y ϕ+ x χL − δy,zTϕ− y ϕ+ x χL

• For large space-time distances:
• T jµ,(t1,x) ; a−

(t2,y)a+ z

• β,L

≃ Zµ(x2)ξy2ξz2

• T n(t1,x1) ; ϕ−

(t2,y1)ϕ+ z1

• χL

T a−

(t,x) ; a+ y β,L

≃ ξx2ξy2T ϕ−

(t,x1) ; ϕ+ y1χL

(∗)

• Plugging (∗) in the WIs, we get relations between Zµ, Z, τ and v:

Z0 = Z(1 − τ) , Z1 = Zv(1 − τ) . These identities imply the universality of the edge conductance.

Marcello Porta Edge universality October 3, 2018 21 / 22

Conclusions

Conclusions

• Today: Edge transport coefficients for 2d topological insulators with:

(i) single-mode edge currents, or (ii) one pair of counterpropagating edge modes. Consequences: bulk-edge duality, Haldane relation.

• Based on RG, and on Ward identities for relativistic & lattice model.
• Open problems:

(i) Multi-edge states topological insulators? (edge states scattering?) (ii) Validity of edge linear response theory? (already for λ = 0!) (iii) Fractional Quantum Hall effect...?

Marcello Porta Edge universality October 3, 2018 22 / 22

Conclusions

Thank you!

Marcello Porta Edge universality October 3, 2018 22 / 22