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

A hierarchical supersymmetric model for weakly disordered 3d semimetals

Marcello Porta Joint with: G. Antinucci (UZH) and L. Fresta (UZH)

Lattice Schr¨

• dinger operators
• Let H on ℓ2(Z3; CM) be a translation invariant Schr¨
• dinger operator:

[H, Tei] = 0 , (Teiψ)(x) = ψ(x + ei) . Suppose that H(x, y) ≡ H(x − y) is short-ranged. Bloch decomposition: H = ⊕

Td dk ˆ

H(k) , with ˆ H(k) ∈ CM×M, smooth in k.

• Green’s function,

G(x, y; µ + iη) = 1 H − µ − iη (x, y) = (−i) ∞ dt e−ηtδx, e−i(H−µ)tδy Can be used to express the Fermi project Pµ = χ(H ≤ µ).

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Weyl semimetals

• 3d materials with pointlike Fermi surface.

Figure: Energy bands of ˆ H(k).

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Weyl semimetals

• 3d 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:

|G(x, y; 0)| ∼ 1 x − y2 , 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.

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Weyl semimetals

• 3d 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:

|G(x, y; 0)| ∼ 1 x − y2 , 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?

Hω = H + γVω , (Vωψ)(x) = ω(x)ψ(x) , {ω(x)} i.i.d. |γ| ≫ 1: Anderson localization, well understood. What about |γ| ≪ 1?

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SUSY

Supersymmetric formulation

Marcello Porta SUSY semimetals August 19, 2019 2 / 15

SUSY

Supersymmetric formulation

• Let Λ = [0, LN]3, {ω(x)} Gaussian i.i.d.. Let {ψ±

x,σ} = Grassmann vars:

{ψε

x,σ , ψε′ y,σ′} = 0 ,

• dψε

x,σ = 0 ,

• dψε

x,σψε x,σ = 1 .

Then, Gω(x, y; µ + iη) =

• [

x,σ dψ+ x,σdψ− x,σ]e−(ψ+,G−1

ω ψ−)ψ−

x,σψ+ y,σ

• [

x,σ dψ+ x,σdψ− x,σ]e−(ψ+,G−1

ω ψ−)

True for any invertible matrix. Choose G−1

ω

= (Hω − µ) − iη.

Marcello Porta SUSY semimetals August 19, 2019 3 / 15

SUSY

Supersymmetric formulation

• Let Λ = [0, LN]3, {ω(x)} Gaussian i.i.d.. Let {ψ±

x,σ} = Grassmann vars:

{ψε

x,σ , ψε′ y,σ′} = 0 ,

• dψε

x,σ = 0 ,

• dψε

x,σψε x,σ = 1 .

Then, Gω(x, y; µ + iη) =

• [

x,σ dψ+ x,σdψ− x,σ]e−(ψ+,G−1

ω ψ−)ψ−

x,σψ+ y,σ

• [

x,σ dψ+ x,σdψ− x,σ]e−(ψ+,G−1

ω ψ−)

True for any invertible matrix. Choose G−1

ω

= (Hω − µ) − iη.

• EωGω? Difficulty: the denominator. Trick:
• [
• x,σ

dψ+

x,σdψ− x,σ]e−(ψ+,iG−1

ω ψ−) = det iG−1

ω

= [

• x,σ

dφ+

x,σdφ− x,σ]e−(φ+,iG−1

ω φ−)−1

where φ+

x,σ ∈ C and φ+ = φ−. Integral makes sense for η > 0!

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SUSY

Supersymmetric formulation

• Therefore:

EωiGω(x, y; µ + iη) = Eω

• [
• x∈Λ

dψ−

x dψ+ x ][

• x∈Λ

dφ−

x dφ+ x ]e−(ψ+,iG−1

ω ψ−)e−(φ+,iG−1 ω φ−)ψ−

x ψ+ y

=: Eω

• DΦ e−(Φ+,iG−1

ω Φ−)ψ−

x ψ+ y

where Φ±

x = (φ± x , ψ± x ) = Gaussian superfield. Using that:

e−(Φ+,iG−1

ω Φ−) = e−(Φ+,iG−1Φ−)e−iγ x,σ ωxΦ+ x,σΦ− x,σ

and taking the average with respect of {ω(x)}, we get, for λ = γ2/2: EωiGω(x, y; µ + iη) =

• DΦ e−(Φ+,iG−1Φ−)e−λ

x(Φ+ x ·Φ− x )2ψ−

x ψ+ y

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SUSY

Remarks

EωiGω(x, y; µ + iη) =

• DΦ e−(Φ+,iG−1Φ−)e−λ

x(Φ+ x ·Φ− x )2ψ−

x ψ+ 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. Shcherbina2 ’08+: Application to random matrix models.

• Today. RG construction of the SUSY Gibbs state, in the hierarchical

approximation.

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Hierarchical model

Hierarchical model for 3d semimetals

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Hierarchical model

Hierarchical model

• Hierarchical block spin decomposition: [Gawedzki-Kupiainen ’82]

Hierarchical SUSY field: Φx :=

N

• h=0

L−hA⌊L−hx⌋ζ(h)

⌊L−h−1x⌋

where

y∈

Ay = 0 , ζ−(h)

⌊L−h−1x⌋ζ+(h) ⌊L−h−1y⌋ = −iδ⌊L−h−1x⌋,⌊L−h−1y⌋ .

That is: Φx ≡ Φ(≥0)

x

= 1 LΦ(≥1)

⌊x/L⌋

• average “background spin” on x

+ Axζ(0)

⌊x/L⌋

• zero-sum fluctuation
• It mimics the block spin decomp. of the true Gaussian free field [GK85]

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Hierarchical model

Covariance of the hierarchical model

• Hierarchical SUSY field:

Φx :=

N

• h=0

L−hA⌊L−hx⌋ζ(h)

⌊L−h−1x⌋

Hierarchical covariance: CN(x, y) :=

N

• h=0

L−2hA⌊L−hx⌋A⌊L−hy⌋ζ−(h)

⌊L−h−1x⌋ζ+(h) ⌊L−h−1y⌋

= −i d(x, y)2

N

• h=k

A⌊L−hx⌋A⌊L−hy⌋ L2(h−k) where d(x, y) is the hierarchical distance of x and y: d(x, y) := Lk , k := min{j ∈ N | ⌊x/Lj⌋ = ⌊y/Lj⌋}

• CN mimics the decay of the Green’s function of 3d semimetals.

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Hierarchical model

Main result

• Let:

V (Φ) := λ

• x∈Λ

(Φ+

x · Φ− x )2 + iµ

• x∈Λ

(Φ+

x · Φ− x )

with µ playing the role of chemical potential. Define: P(Φ)N := 1 ZN

• [

N

• h=0

dµ(ζ(h))] e−V (h)(Φ)P(Φ) . Theorem (Antinucci, Fresta, P. 2019) For λ > 0 small enough, there exists µ ≡ µ(λ) = O(λ) such that: φ+

x,σφ− y,σ′N

= −ψ+

x,σψ− y,σ′N

= −iδσ,σ′ d(x, y)2

N

• h=k

A⌊L−hx⌋A⌊L−hy⌋ L2(h−k) + EN(x, y) |EN(x, y)| ≤ Kλ

1 2 −ε

d(x, y)

5 2 −ε ,

• unif. in N.

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Hierarchical model

Remarks

• Theorem based on RG analysis of the model. Inspired by the analysis of

[GK82] for the hierarchical φ4

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, 1d hier. Anderson with summable hopping. van Soosten-Warzel ’17: RG analysis for [B90]. Proof of localization.

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Sketch of the proof

Sketch of the proof

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Sketch of the proof

Multiscale analysis

• Iterative scheme. Main ingredient: flow of effective interactions.
• Let:

U(0)(Φ) := e−λ

x(Φ+ x ·Φ− x )2−iµ x(Φ+ x ·Φ− x )

We define the effective interaction on scale h = 1 as: U(1)(Φ(≥1)) := TRGU(0)(Φ(≥0)) :=

• dµ(ζ(0)) U(0)(L−1Φ(≥1) + Aζ(0)) .
• Nice thing about hier. models: TRG equivalent to a local map.

Let Λ(1) := Z3 ∩ L−1Λ. Then: U(1)(Φ(≥1)) =

x∈Λ(1) U (1)(Φ(≥1) x

), U (1)(Φ(≥1)

x

) =

• dµ(ζ(0)

x )

• y∈x

U (0)(L−1Φ(≥1)

x

+ Ayζ(0)

x )

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Sketch of the proof

Multiscale analysis

• Iteration to higher scales. For all x ∈ Λ(h) = Z(3) ∩ L−hΛ:

U (h+1)(Φ(≥h+1)

x

) =

• dµ(ζ(h)

x )

• y∈x

U (h)(L−1Φ(≥h+1)

x

+ Ayζ(h)

x )

Using that

x∈ Ax = 0:

U (h+1)(Φ) =

• dµ(ζ) [U (h)(Φ/L + ζ)U (h)(Φ/L − ζ)]

L3 2

• Simple power counting:

Φ4 → Φ4/L: the quartic interaction is RG-irrelevant Φ2 → LΦ2: the chemical potential is RG-relevant.

• Show:

U (h) ≃ e−λh(Φ+·Φ−)2−iµh(Φ+·Φ−), λh ∼

λ Lh ,

µh ∼ λh.

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Sketch of the proof

Small and large fields

• φ±

σ = φ1,σ ± iφ2,σ. Complex extension: φ1,σ, φ2,σ ∈ C.

• S(h) = small fields, L(h) = large fields. Analyticity in S(h) ∪ L(h).

S(h) := {φ | φ ≤ λ−1/4

h

} , L(h) :=

• φ | φ > λ−1/4

h

, Im φ ≤ λ−1/4

h

ℓ

• λ−1/4

h λ−1/4

h

ℓ

Reφ Imφ Marcello Porta SUSY semimetals August 19, 2019 12 / 15

Sketch of the proof

Sketch of the proof of the proposition

• For h ≥ 0:

U (h+1)(Φ) =

• dµ(ζ) [U (h)(Φ/L + ζ)U (h)(Φ/L − ζ)]

L3 2

≡ e−

λh L (Φ+·Φ−)2−iLµh(Φ+·Φ−)R(h)(Φ)

with R(h)(Φ) of the form: R(h)(Φ) =

• n=0,1,2

E(h)

n (φ)(ψ+ · ψ−)n

for some functions E(h)

n (φ) analytic in LS(h) ∪ LL(h).

• The ζφ integration is performed via a stationary phase expansion:
• dζφ e−i(ζ+

φ ·ζ− φ )f(ζφ) =

m−1

• j=0

(i∆ζφ)j 4jj! f(0) + Em(f) for f ∈ S(R4) and for any m ∈ N.

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Sketch of the proof

Small fields: renormalization

L1/4λ−1/4

h

Lλ−1/4

h

L

λ−1/4

h

ℓ

L1/4 λ−1/4

h

ℓ Reφ

Imφ λ−1/4

h λ−1/4

h

ℓ

Figure: white: S(h) ∪ L(h); light grey: S(h+1) ∪ L(h+1); grey: LS(h) ∪ LL(h)

• R(h)(Φ) =

n E(h) n (φ)(ψ+ · ψ−)n.

E(h)

n (φ) analytic in φ ∈ LS(h).

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Sketch of the proof

Small fields: renormalization

L1/4λ−1/4

h

Lλ−1/4

h

L

λ−1/4

h

ℓ

L1/4 λ−1/4

h

ℓ Reφ

Imφ λ−1/4

h λ−1/4

h

ℓ

Figure: white: S(h) ∪ L(h); light grey: S(h+1) ∪ L(h+1); grey: LS(h) ∪ LL(h)

• R(h)(Φ) =

n E(h) n (φ)(ψ+ · ψ−)n.

E(h)

n (φ) analytic in φ ∈ LS(h).

• Renormalization: extract the Φ2 terms from R(h)(Φ), redefine µh:

µh+1 = Lµh + βh , |βh| ≤ CLλh . Remainder irrelevant, improved λh-dep for φ ∈ S(h+1) ⊂ LS(h) (Cauchy).

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Conclusions

Conclusions and perspectives

• We proved algebraic decay for the 2-point correlation function of a

hierarchical SUSY model for 3d semimetals.

• In the full lattice model, no nice exact decomposition in blocks.

Distant blocks are (weakly) interacting. For φ4

4, connection between hierarchical and full lattice model provided

by a cluster expansion (high T exp.) [GK85], [BBS14].

• Open problems:

d = 2: Graphene (Φ4 is marginal). Universality of σ(η)? For staggered {ω(x)}, univ. class of random Ising [Dotsenko2, ’83] Big open problem: extended Fermi surface in d = 3. Nontrivial fixed point?

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Sketch of the proof

Thank you!

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