A hierarchical supersymmetric model for weakly disordered 3 d - PowerPoint PPT Presentation

A hierarchical supersymmetric model for weakly disordered 3 d semimetals Marcello Porta Joint with: G. Antinucci (UZH) and L. Fresta (UZH)

Lattice Schr¨ odinger operators • Let H on ℓ 2 ( Z 3 ; C M ) be a translation invariant Schr¨ odinger operator: [ H, T e i ] = 0 , ( T e i ψ )( x ) = ψ ( x + e i ) . Suppose that H ( x, y ) ≡ H ( x − y ) is short-ranged. Bloch decomposition: � ⊕ T d dk ˆ H = H ( k ) , with ˆ H ( k ) ∈ C M × M , smooth in k . • Green’s function, � ∞ 1 dt e − ηt � δ x , e − i ( H − µ ) t δ y � G ( x, y ; µ + iη ) = H − µ − iη ( x, y ) = ( − i ) 0 Can be used to express the Fermi project P µ = χ ( H ≤ µ ). Marcello Porta SUSY semimetals August 19, 2019 1 / 15

Weyl semimetals • 3 d materials with pointlike Fermi surface. Figure: Energy bands of ˆ H ( k ). Marcello Porta SUSY semimetals August 19, 2019 2 / 15

Weyl semimetals • 3 d materials with pointlike Fermi surface. Discovery [Hasan et al. ’13], theoretical model [Delplace et al ’11] • Green’s function and conductivity, at the Fermi level µ = 0: 1 | G ( x, y ; 0) | ∼ � x − y � 2 , σ ( η ) ∼ η log η for small η > 0. The decay of the Green’s function implies that P ( x, y ) ∼ � x − y � − 3 . Rmk. For gapped H , G and P decay exponentially and σ ( η ) is smooth. Marcello Porta SUSY semimetals August 19, 2019 2 / 15

Weyl semimetals • 3 d materials with pointlike Fermi surface. Discovery [Hasan et al. ’13], theoretical model [Delplace et al ’11] • Green’s function and conductivity, at the Fermi level µ = 0: 1 | G ( x, y ; 0) | ∼ � x − y � 2 , σ ( η ) ∼ η log η for small η > 0. The decay of the Green’s function implies that P ( x, y ) ∼ � x − y � − 3 . Rmk. For gapped H , G and P decay exponentially and σ ( η ) is smooth. • Question: effect of disorder? H ω = H + γV ω , ( V ω ψ )( x ) = ω ( x ) ψ ( x ) , { ω ( x ) } i.i.d. | γ | ≫ 1: Anderson localization, well understood. What about | γ | ≪ 1? Marcello Porta SUSY semimetals August 19, 2019 2 / 15

SUSY Supersymmetric formulation Marcello Porta SUSY semimetals August 19, 2019 2 / 15

SUSY Supersymmetric formulation • Let Λ = [0 , L N ] 3 , { ω ( x ) } Gaussian i.i.d.. Let { ψ ± x,σ } = Grassmann vars: � � x,σ , ψ ε ′ { ψ ε dψ ε dψ ε x,σ ψ ε y,σ ′ } = 0 , x,σ = 0 , x,σ = 1 . Then, � [ � x,σ ] e − ( ψ + ,G − 1 ω ψ − ) ψ − x,σ dψ + x,σ dψ − x,σ ψ + y,σ G ω ( x, y ; µ + iη ) = � [ � x,σ ] e − ( ψ + ,G − 1 x,σ dψ + x,σ dψ − ω ψ − ) True for any invertible matrix. Choose G − 1 = ( H ω − µ ) − iη . ω Marcello Porta SUSY semimetals August 19, 2019 3 / 15

SUSY Supersymmetric formulation • Let Λ = [0 , L N ] 3 , { ω ( x ) } Gaussian i.i.d.. Let { ψ ± x,σ } = Grassmann vars: � � x,σ , ψ ε ′ { ψ ε dψ ε dψ ε x,σ ψ ε y,σ ′ } = 0 , x,σ = 0 , x,σ = 1 . Then, � [ � x,σ ] e − ( ψ + ,G − 1 ω ψ − ) ψ − x,σ dψ + x,σ dψ − x,σ ψ + y,σ G ω ( x, y ; µ + iη ) = � [ � x,σ ] e − ( ψ + ,G − 1 x,σ dψ + x,σ dψ − ω ψ − ) True for any invertible matrix. Choose G − 1 = ( H ω − µ ) − iη . ω • E ω G ω ? Difficulty: the denominator. Trick: � � ω ψ − ) = det iG − 1 x,σ ] e − ( ψ + ,iG − 1 dψ + x,σ dψ − [ ω x,σ � � ω φ − ) � − 1 � x,σ ] e − ( φ + ,iG − 1 dφ + x,σ dφ − = [ x,σ x,σ ∈ C and φ + = φ − . Integral makes sense for η > 0! where φ + Marcello Porta SUSY semimetals August 19, 2019 3 / 15

SUSY Supersymmetric formulation • Therefore: E ω iG ω ( x, y ; µ + iη ) � � � x ] e − ( ψ + ,iG − 1 ω ψ − ) e − ( φ + ,iG − 1 dψ − x dψ + dφ − x dφ + ω φ − ) ψ − x ψ + = E ω [ x ][ y x ∈ Λ x ∈ Λ � D Φ e − (Φ + ,iG − 1 ω Φ − ) ψ − x ψ + =: E ω y where Φ ± x = ( φ ± x , ψ ± x ) = Gaussian superfield. Using that: ω Φ − ) = e − (Φ + ,iG − 1 Φ − ) e − iγ � e − (Φ + ,iG − 1 x,σ ω x Φ + x,σ Φ − x,σ and taking the average with respect of { ω ( x ) } , we get, for λ = γ 2 / 2: � D Φ e − (Φ + ,iG − 1 Φ − ) e − λ � x (Φ + x ) 2 ψ − x · Φ − x ψ + E ω iG ω ( x, y ; µ + iη ) = y Marcello Porta SUSY semimetals August 19, 2019 4 / 15

SUSY Remarks � D Φ e − (Φ + ,iG − 1 Φ − ) e − λ � x (Φ + x ) 2 ψ − x · Φ − x ψ + E ω iG ω ( x, y ; µ + iη ) = y • Stat mech problem! [Wegner, Efetov, ’80s.] Difficulties: Perturbation theory for bosons does not work (large field problem) ˆ G ( k ) is singular if the Fermi surface is nonempty (infrared problem) Covariance of the bosonic Gaussian purely imaginary (as η → 0 + ). • Rigorous analysis of SUSY theories: Disertori-Spencer-Zirnbauer ’10 + : Effective model for loc/deloc trans. Shcherbina 2 ’08 + : Application to random matrix models. • Today. RG construction of the SUSY Gibbs state, in the hierarchical approximation. Marcello Porta SUSY semimetals August 19, 2019 5 / 15

Hierarchical model Hierarchical model for 3 d semimetals Marcello Porta SUSY semimetals August 19, 2019 5 / 15

Hierarchical model Hierarchical model • Hierarchical block spin decomposition: [Gawedzki-Kupiainen ’82] Hierarchical SUSY field: N � L − h A ⌊ L − h x ⌋ ζ ( h ) Φ x := ⌊ L − h − 1 x ⌋ h =0 where � � ζ − ( h ) ⌊ L − h − 1 x ⌋ ζ +( h ) A y = 0 , ⌊ L − h − 1 y ⌋ � = − iδ ⌊ L − h − 1 x ⌋ , ⌊ L − h − 1 y ⌋ . y ∈ � That is: 1 L Φ ( ≥ 1) A x ζ (0) Φ x ≡ Φ ( ≥ 0) = + x ⌊ x/L ⌋ ⌊ x/L ⌋ � �� � � �� � zero-sum fluctuation average “background spin” on � x • It mimics the block spin decomp. of the true Gaussian free field [GK85] Marcello Porta SUSY semimetals August 19, 2019 6 / 15

Hierarchical model Covariance of the hierarchical model • Hierarchical SUSY field: N � L − h A ⌊ L − h x ⌋ ζ ( h ) Φ x := ⌊ L − h − 1 x ⌋ h =0 Hierarchical covariance: N � L − 2 h A ⌊ L − h x ⌋ A ⌊ L − h y ⌋ � ζ − ( h ) ⌊ L − h − 1 x ⌋ ζ +( h ) C N ( x, y ) := ⌊ L − h − 1 y ⌋ � h =0 N A ⌊ L − h x ⌋ A ⌊ L − h y ⌋ − i � = d ( x, y ) 2 L 2( h − k ) h = k where d ( x, y ) is the hierarchical distance of x and y : d ( x, y ) := L k , k := min { j ∈ N | ⌊ x/L j ⌋ = ⌊ y/L j ⌋} • C N mimics the decay of the Green’s function of 3 d semimetals. Marcello Porta SUSY semimetals August 19, 2019 7 / 15

Hierarchical model Main result • Let: � � x ) 2 + iµ (Φ + x · Φ − (Φ + x · Φ − V (Φ) := λ x ) x ∈ Λ x ∈ Λ with µ playing the role of chemical potential. Define: N � 1 � dµ ( ζ ( h ) )] e − V ( h ) (Φ) P (Φ) . � P (Φ) � N := [ Z N h =0 Theorem (Antinucci, Fresta, P. 2019) For λ > 0 small enough, there exists µ ≡ µ ( λ ) = O ( λ ) such that: � φ + x,σ φ − −� ψ + x,σ ψ − y,σ ′ � N = y,σ ′ � N N A ⌊ L − h x ⌋ A ⌊ L − h y ⌋ − iδ σ,σ ′ � = + E N ( x, y ) d ( x, y ) 2 L 2( h − k ) h = k 1 2 − ε Kλ |E N ( x, y ) | ≤ 2 − ε , unif. in N . 5 d ( x, y ) Marcello Porta SUSY semimetals August 19, 2019 8 / 15

Hierarchical model Remarks • Theorem based on RG analysis of the model. Inspired by the analysis of [GK82] for the hierarchical φ 4 4 theory. Main differences wrt [GK82]: Imaginary covariance for the field. Oscillatory integrals! Bosons and fermions: need to exploit supersymmetry. Simplification: quartic interaction irrelevant instead of marginal. • Previous work on disordered hierarchical models: Bovier ’90: density of states, 1 d hier. Anderson with summable hopping. van Soosten-Warzel ’17: RG analysis for [B90]. Proof of localization. Marcello Porta SUSY semimetals August 19, 2019 9 / 15

Sketch of the proof Sketch of the proof Marcello Porta SUSY semimetals August 19, 2019 9 / 15

Sketch of the proof Multiscale analysis • Iterative scheme. Main ingredient: flow of effective interactions. • Let: x (Φ + x ) 2 − iµ � x (Φ + x · Φ − x · Φ − U (0) (Φ) := e − λ � x ) We define the effective interaction on scale h = 1 as: U (1) (Φ ( ≥ 1) ) T RG U (0) (Φ ( ≥ 0) ) := � dµ ( ζ (0) ) U (0) ( L − 1 Φ ( ≥ 1) + Aζ (0) ) . := • Nice thing about hier. models: T RG equivalent to a local map. Let Λ (1) := Z 3 ∩ L − 1 Λ. Then: U (1) (Φ ( ≥ 1) ) = � x ∈ Λ (1) U (1) (Φ ( ≥ 1) ), x � � U (1) (Φ ( ≥ 1) dµ ( ζ (0) U (0) ( L − 1 Φ ( ≥ 1) + A y ζ (0) ) = x ) x ) x x y ∈ � x Marcello Porta SUSY semimetals August 19, 2019 10 / 15

Sketch of the proof Multiscale analysis • Iteration to higher scales. For all x ∈ Λ ( h ) = Z (3) ∩ L − h Λ: � � U ( h +1) (Φ ( ≥ h +1) dµ ( ζ ( h ) U ( h ) ( L − 1 Φ ( ≥ h +1) + A y ζ ( h ) ) = x ) x ) x x y ∈ � x Using that � x ∈ � A x = 0: � L 3 U ( h +1) (Φ) = dµ ( ζ ) [ U ( h ) (Φ /L + ζ ) U ( h ) (Φ /L − ζ )] 2 • Simple power counting: Φ 4 → Φ 4 /L : the quartic interaction is RG-irrelevant Φ 2 → L Φ 2 : the chemical potential is RG-relevant. U ( h ) ≃ e − λ h (Φ + · Φ − ) 2 − iµ h (Φ + · Φ − ) , λ • Show: λ h ∼ L h , µ h ∼ λ h . Marcello Porta SUSY semimetals August 19, 2019 11 / 15