Non-Renormalization of the `Chiral Anomaly' in Interacting Lattice Weyl

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof

Non-renormalization of the ‘chiral anomaly’ in interacting lattice Weyl semimetals

Alessandro Giuliani Univ. Roma Tre & Centro Linceo Interdisciplinare B. Segre Conferenza del Centro Linceo

Based on a joint work with V. Mastropietro and M. Porta

MCQM Seminar, Politecnico di Milano, December 14, 2020 Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Outline

1 Overview

2 The chiral anomaly in QED4

3 Lattice Weyl semimetals & Main results

4 Sketch of the proof Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Outline

1 Overview

2 The chiral anomaly in QED4

3 Lattice Weyl semimetals & Main results

4 Sketch of the proof Under special circumstances, these equations also describe quasi-particles in a crystal. Well-known examples: graphene and surface states of 3D topological insulators (TI). Recent: 3D Weyl semimetals with point-like Fermi surface.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Overview

The Dirac equation describes relativistic electrons µ iγ ∂µψ − mψ = 0 and, if m = 0, is equivalent to two Weyl equations µ µ σ ∂µψ+ = 0, σ¯ ∂µψ− = 0 1±γ5 µ 1 for ψ± = 2 ψ, whereσ ¯ = ( , −σ1, −σ2, −σ3). Under special circumstances, these equations also describe quasi-particles in a crystal. Well-known examples: graphene and surface states of 3D topological insulators (TI). Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Overview

The Dirac equation describes relativistic electrons µ iγ ∂µψ − mψ = 0 and, if m = 0, is equivalent to two Weyl equations µ µ σ ∂µψ+ = 0, σ¯ ∂µψ− = 0 1±γ5 µ 1 for ψ± = 2 ψ, whereσ ¯ = ( , −σ1, −σ2, −σ3). Under special circumstances, these equations also describe quasi-particles in a crystal. Well-known examples: graphene and surface states of 3D topological insulators (TI). Recent: 3D Weyl semimetals with point-like Fermi surface. Do Weyl semimetals support the analogue of chiral anomaly? Nielsen-Ninomiya 1983: YES, the analogue being a ﬂow of quasi-particles from one Weyl node to the other: 1 ∂ hN (t) − N (t)i = E · B. t + − 2π2

1 The 2π2 is the analogue of the Adler-Bell-Jackiw anomaly.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Overview

Low-energy excitations of Weyl semimetals behave like Dirac fermions in d = 3 + 1 ⇒ at low T they mimick infrared QED4. Nielsen-Ninomiya 1983: YES, the analogue being a ﬂow of quasi-particles from one Weyl node to the other: 1 ∂ hN (t) − N (t)i = E · B. t + − 2π2

1 The 2π2 is the analogue of the Adler-Bell-Jackiw anomaly.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Overview

Low-energy excitations of Weyl semimetals behave like Dirac fermions in d = 3 + 1 ⇒ at low T they mimick infrared QED4.

Do Weyl semimetals support the analogue of chiral anomaly? 1 The 2π2 is the analogue of the Adler-Bell-Jackiw anomaly.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Overview

Low-energy excitations of Weyl semimetals behave like Dirac fermions in d = 3 + 1 ⇒ at low T they mimick infrared QED4.

Do Weyl semimetals support the analogue of chiral anomaly? Nielsen-Ninomiya 1983: YES, the analogue being a ﬂow of quasi-particles from one Weyl node to the other: 1 ∂ hN (t) − N (t)i = E · B. t + − 2π2 Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Overview

Low-energy excitations of Weyl semimetals behave like Dirac fermions in d = 3 + 1 ⇒ at low T they mimick infrared QED4.

1 The 2π2 is the analogue of the Adler-Bell-Jackiw anomaly. Its perturbative proof is based on cancellations of loop integrals, which require exact relativistic invariance.

Should the same universal coeﬃcient be observed in interacting Weyl semimetals? Remarkably, YES!

We consider a class of lattice Weyl semimetals with short range interactions, and prove universality of the ABJ anomaly.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Overview

In QED, ABJ anomaly not dressed by radiative corrections: Adler-Bardeen anomaly non-renormalization theorem. Should the same universal coeﬃcient be observed in interacting Weyl semimetals? Remarkably, YES!

We consider a class of lattice Weyl semimetals with short range interactions, and prove universality of the ABJ anomaly.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Overview

In QED, ABJ anomaly not dressed by radiative corrections: Adler-Bardeen anomaly non-renormalization theorem.

Its perturbative proof is based on cancellations of loop integrals, which require exact relativistic invariance. Remarkably, YES!

We consider a class of lattice Weyl semimetals with short range interactions, and prove universality of the ABJ anomaly.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Overview

In QED, ABJ anomaly not dressed by radiative corrections: Adler-Bardeen anomaly non-renormalization theorem.

Its perturbative proof is based on cancellations of loop integrals, which require exact relativistic invariance.

Should the same universal coeﬃcient be observed in interacting Weyl semimetals? We consider a class of lattice Weyl semimetals with short range interactions, and prove universality of the ABJ anomaly.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Overview

In QED, ABJ anomaly not dressed by radiative corrections: Adler-Bardeen anomaly non-renormalization theorem.

Its perturbative proof is based on cancellations of loop integrals, which require exact relativistic invariance.

Should the same universal coeﬃcient be observed in interacting Weyl semimetals? Remarkably, YES! Mechanism: very diﬀerent from Adler-Bardeen proof. Our ingredients: rigorous RG, regularity of current correlations, lattice Ward Identities. Important fact: short range interactions are irrelevant in the IR.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Overview

In QED, ABJ anomaly not dressed by radiative corrections: Adler-Bardeen anomaly non-renormalization theorem.

Its perturbative proof is based on cancellations of loop integrals, which require exact relativistic invariance.

Should the same universal coeﬃcient be observed in interacting Weyl semimetals? Remarkably, YES!

We consider a class of lattice Weyl semimetals with short range interactions, and prove universality of the ABJ anomaly. Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Overview

In QED, ABJ anomaly not dressed by radiative corrections: Adler-Bardeen anomaly non-renormalization theorem.

Its perturbative proof is based on cancellations of loop integrals, which require exact relativistic invariance.

Should the same universal coeﬃcient be observed in interacting Weyl semimetals? Remarkably, YES!

We consider a class of lattice Weyl semimetals with short range interactions, and prove universality of the ABJ anomaly.

Mechanism: very diﬀerent from Adler-Bardeen proof. Our ingredients: rigorous RG, regularity of current correlations, lattice Ward Identities. Important fact: short range interactions are irrelevant in the IR. Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Outline

1 Overview

2 The chiral anomaly in QED4

3 Lattice Weyl semimetals & Main results

4 Sketch of the proof L covariant under local U(1) gauge transformation: −iα(x) † † +iα(x) ψx → e ψx , ψx → ψxe , Aµ,x → Aµ,x + ∂µα(x).

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Massless Dirac fermions, gauge symmetries

Consider massless 4D Dirac fermions in a background ﬁeld: ¯ L(ψ, A) = ψγµ(i∂µ − Aµ)ψ ¯ † where ψ = ψ γ0 and γµ are Euclidean Gamma matrices: 0 1 0 iσj {γµ, γν} = 2δµ,ν, e.g. : γ0 = , γj = . 1 0 −iσj 0 L covariant under local U(1) gauge transformation: −iα(x) † † +iα(x) ψx → e ψx , ψx → ψxe , Aµ,x → Aµ,x + ∂µα(x). Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Massless Dirac fermions, gauge symmetries

Consider massless 4D Dirac fermions in a background ﬁeld: ¯ L(ψ, A) = ψγµ(i∂µ − Aµ)ψ ¯ † where ψ = ψ γ0 and γµ are Euclidean Gamma matrices: 0 1 0 iσj {γµ, γν} = 2δµ,ν, e.g. : γ0 = , γj = . 1 0 −iσj 0 L covariant under local U(1) gauge transformation: −iα(x) † † +iα(x) ψx → e ψx , ψx → ψxe , Aµ,x → Aµ,x + ∂µα(x). L invariant under global axial U(1) gauge transformation: 5 1 0 ψ → eiγ5α ψ , γ = γ γ γ γ = . x x 5 0 1 2 3 0 −1 Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Massless Dirac fermions, gauge symmetries

ψ 1 + γ 0 1 − γ ψ + = 5 ψ, = 5 ψ, i.e., ψ = + . 0 2 ψ− 2 ψ− Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Massless Dirac fermions, gauge symmetries

Consider massless 4D Dirac fermions in a background ﬁeld: ¯ L(ψ, A) = ψγµ(i∂µ − Aµ)ψ ¯ † where ψ = ψ γ0 and γµ are Euclidean Gamma matrices: 0 1 0 iσj {γµ, γν} = 2δµ,ν, e.g. : γ0 = , γj = . 1 0 −iσj 0 L covariant under local U(1) gauge transformation: −iα(x) † † +iα(x) ψx → e ψx , ψx → ψxe , Aµ,x → Aµ,x + ∂µα(x). Axial symm. can be promoted to local U(1), by adding to L 5 ¯ an auxiliary term −Aµψγµγ5ψ, and letting

−iγ5α5(x) † † +iγ5α5(x) 5 5 5 ψx → e ψx , ψx → ψxe , Aµ,x → Aµ,x+∂µα (x) These conservation laws might be broken in quantum theory, due to UV regularization. Z 5 ¯ 5 5 eW (A,A ) ∝ Dψe−i(ψ,6∂ψ)+(Aµ, jµ)+(Aµ, jµ).

Formally, W (A, A5) = W (A + ∂α, A5 + ∂α5), from which 5 h∂µ jµiA = 0, h∂µ jµiA = 0, R Dψe−i(ψ,¯ 6∂ψ)+(Aµ, jµ)O(ψ) where hO(ψ)iA = ¯ . R Dψe−i(ψ,6∂ψ)+(Aµ, jµ)

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Conserved currents: classical and quantum

5 Classically, Noether’s theorem ⇒ ∂µ jµ = 0 and ∂µ jµ = 0 ¯ 5 ¯ 5 with jµ = ψγµψ and jµ = ψγ γµψ, states conservation of the total and axial charges: † X † 5 † X † j0 = ψ ψ = ψωψω, j0 = ψ γ5ψ = ωψωψω. ω=± ω=± Formally, W (A, A5) = W (A + ∂α, A5 + ∂α5), from which 5 h∂µ jµiA = 0, h∂µ jµiA = 0, R Dψe−i(ψ,¯ 6∂ψ)+(Aµ, jµ)O(ψ) where hO(ψ)iA = ¯ . R Dψe−i(ψ,6∂ψ)+(Aµ, jµ)

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Conserved currents: classical and quantum

5 Classically, Noether’s theorem ⇒ ∂µ jµ = 0 and ∂µ jµ = 0 ¯ 5 ¯ 5 with jµ = ψγµψ and jµ = ψγ γµψ, states conservation of the total and axial charges: † X † 5 † X † j0 = ψ ψ = ψωψω, j0 = ψ γ5ψ = ωψωψω. ω=± ω=± These conservation laws might be broken in quantum theory, due to UV regularization. Z 5 ¯ 5 5 eW (A,A ) ∝ Dψe−i(ψ,6∂ψ)+(Aµ, jµ)+(Aµ, jµ). Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Conserved currents: classical and quantum

5 Classically, Noether’s theorem ⇒ ∂µ jµ = 0 and ∂µ jµ = 0 ¯ 5 ¯ 5 with jµ = ψγµψ and jµ = ψγ γµψ, states conservation of the total and axial charges: † X † 5 † X † j0 = ψ ψ = ψωψω, j0 = ψ γ5ψ = ωψωψω. ω=± ω=± These conservation laws might be broken in quantum theory, due to UV regularization. Z 5 ¯ 5 5 eW (A,A ) ∝ Dψe−i(ψ,6∂ψ)+(Aµ, jµ)+(Aµ, jµ).

Formally, W (A, A5) = W (A + ∂α, A5 + ∂α5), from which 5 h∂µ jµiA = 0, h∂µ jµiA = 0, R Dψe−i(ψ,¯ 6∂ψ)+(Aµ, jµ)O(ψ) where hO(ψ)iA = ¯ . R Dψe−i(ψ,6∂ψ)+(Aµ, jµ) That is, all loop diagrams cancel:

However, the loop diagrams with n ≤ 3 are UV divergent! We need an UV regularization (to be eventually removed) in order to give the diagrams and to the cancellations a meaning.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Loop cancellation and UV divergences

] Note: h∂µ µiA = 0 is the same as [letting p = p1 + ··· + pn] X 1 p hˆ] i = p Aˆ ··· Aˆ hj ] ; j ; ··· ; j i = 0. µ µ,p A n! µ µ1,p1 µn,pn µ,p µ1,p1 mn,pn 0 n≥1 However, the loop diagrams with n ≤ 3 are UV divergent! We need an UV regularization (to be eventually removed) in order to give the diagrams and to the cancellations a meaning.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Loop cancellation and UV divergences

] Note: h∂µ µiA = 0 is the same as [letting p = p1 + ··· + pn] X 1 p hˆ] i = p Aˆ ··· Aˆ hj ] ; j ; ··· ; j i = 0. µ µ,p A n! µ µ1,p1 µn,pn µ,p µ1,p1 mn,pn 0 n≥1 That is, all loop diagrams cancel: Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Loop cancellation and UV divergences

] Note: h∂µ µiA = 0 is the same as [letting p = p1 + ··· + pn] X 1 p hˆ] i = p Aˆ ··· Aˆ hj ] ; j ; ··· ; j i = 0. µ µ,p A n! µ µ1,p1 µn,pn µ,p µ1,p1 mn,pn 0 n≥1 That is, all loop diagrams cancel:

However, the loop diagrams with n ≤ 3 are UV divergent! We need an UV regularization (to be eventually removed) in order to give the diagrams and to the cancellations a meaning. Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof The axial anomaly

Fact: there is no way to add an UV regularization preserving both the vectorial and axial current conservations. If we choose to preserve the vectorial U(1) gauge symmetry, then i h∂ j 5i = − ε ∂ A ∂ A . µ µ A 2π2 αβνσ α ν β σ 1 2π2 is the ABJ anomaly, determined by the triangle graph: Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Radiative corrections, Adler-Bardeen theorem

What if we add interactions, i.e., coupling with dynamical e.m. ﬁeld? Is the triangle graph dressed by radiative corrections?

Adler-Bardeen theorem:NO! All possible dressings of the triangle cancel exactly. Required: speciﬁc UV regularization, exact relativistic invariance of the fermionic propagator. [Deep consequences in QED and Standard Model: exact decay rate of π0 → γγ, constraint on the number of lepton/quark families.] Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Outline

1 Overview

2 The chiral anomaly in QED4

3 Lattice Weyl semimetals & Main results

4 Sketch of the proof α(k) β(k) Bloch Hamiltonian: Hˆ0(k) = , β(k) −α(k)

where α(k) = t3(2 + ζ − cos k1 − cos k2 − cos k3) and β(k) = t1 sin k1 − it2 sin k2, with t1, t2, t3 > 0 hopping parameters, and −1 < ζ < 1 tunes location of Weyl nodes.

Energy bands touch with ± conical singularity at k = pF , ± with pF = (0, 0, ± arccos ζ): Weyl nodes merge as ζ → 1−

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof The non-interacting model

Spinless electrons on a simple cubic lattice, with two orbitals of opposite parity at each site. Energy bands touch with ± conical singularity at k = pF , ± with pF = (0, 0, ± arccos ζ): Weyl nodes merge as ζ → 1−

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof The non-interacting model

Spinless electrons on a simple cubic lattice, with two orbitals of opposite parity at each site. α(k) β(k) Bloch Hamiltonian: Hˆ0(k) = , β(k) −α(k)

where α(k) = t3(2 + ζ − cos k1 − cos k2 − cos k3) and β(k) = t1 sin k1 − it2 sin k2, with t1, t2, t3 > 0 hopping parameters, and −1 < ζ < 1 tunes location of Weyl nodes. Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof The non-interacting model

Spinless electrons on a simple cubic lattice, with two orbitals of opposite parity at each site. α(k) β(k) Bloch Hamiltonian: Hˆ0(k) = , β(k) −α(k)

where α(k) = t3(2 + ζ − cos k1 − cos k2 − cos k3) and β(k) = t1 sin k1 − it2 sin k2, with t1, t2, t3 > 0 hopping parameters, and −1 < ζ < 1 tunes location of Weyl nodes.

Energy bands touch with ± conical singularity at k = pF , ± with pF = (0, 0, ± arccos ζ): Weyl nodes merge as ζ → 1− The reﬂection symmetries

ˆ0 ˆ0 ˆ0 ˆ0 H (k) = H (k1, k2, −k3) and H (k) = −σ1H (−k1, k2, k3) σ1

guarantee that the Weyl nodes are located alonge ˆ3. The WSM phase of Hˆ0(k) appears as a transitional state between a NI (ζ < −1) and a TI (ζ > 1).

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Symmetries

The Hamiltonian breaks time-reversal symmetry:

Hˆ0(k) 6= Hˆ0(−k)

and is inversion-symmetric:

ˆ0 ˆ0 H (k) = σ3H (−k)σ3. The WSM phase of Hˆ0(k) appears as a transitional state between a NI (ζ < −1) and a TI (ζ > 1).

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Symmetries

The Hamiltonian breaks time-reversal symmetry:

Hˆ0(k) 6= Hˆ0(−k)

and is inversion-symmetric:

ˆ0 ˆ0 H (k) = σ3H (−k)σ3. The reﬂection symmetries

ˆ0 ˆ0 ˆ0 ˆ0 H (k) = H (k1, k2, −k3) and H (k) = −σ1H (−k1, k2, k3) σ1

guarantee that the Weyl nodes are located alonge ˆ3. Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Symmetries

The Hamiltonian breaks time-reversal symmetry:

Hˆ0(k) 6= Hˆ0(−k)

and is inversion-symmetric:

ˆ0 ˆ0 H (k) = σ3H (−k)σ3. The reﬂection symmetries

ˆ0 ˆ0 ˆ0 ˆ0 H (k) = H (k1, k2, −k3) and H (k) = −σ1H (−k1, k2, k3) σ1

guarantee that the Weyl nodes are located alonge ˆ3. The WSM phase of Hˆ0(k) appears as a transitional state between a NI (ζ < −1) and a TI (ζ > 1). Z −βH0 dk + ˆ0 − Tr(e ·) If H0 = aˆk H (k)ˆak and h·i0 = lim , (2π)3 β,L→∞ Tr e−βH0 [−π,π]3 then the two-point function reads:

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Eﬀective Weyl Hamiltonian and two-point function

ω In the vicinity of the Weyl nodes pF = (0, 0, ω arccos ζ), ˆ0 0 0 0 0 0 0 H (k) ' v1 k1σ1 + v2 k2σ2 + ωv3 σ3k3, 0 0 0 p 2 where (v1 , v2 , v3 ) = (t1, t2, t3 1 − ζ ) are the free Fermi 0 ω velocities, ω is the chirality index, and k = k − pF . Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Eﬀective Weyl Hamiltonian and two-point function

ω In the vicinity of the Weyl nodes pF = (0, 0, ω arccos ζ), ˆ0 0 0 0 0 0 0 H (k) ' v1 k1σ1 + v2 k2σ2 + ωv3 σ3k3, 0 0 0 p 2 where (v1 , v2 , v3 ) = (t1, t2, t3 1 − ζ ) are the free Fermi 0 ω velocities, ω is the chirality index, and k = k − pF . Z −βH0 dk + ˆ0 − Tr(e ·) If H0 = aˆk H (k)ˆak and h·i0 = lim , (2π)3 β,L→∞ Tr e−βH0 [−π,π]3 then the two-point function reads: Z dk d 3k ha−a+i = 0 (−ik + Hˆ (k))−1e−ik·(x−y) x y 0 (2π)4 0 0 Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Eﬀective Weyl Hamiltonian and two-point function

Z 3 0 3 − + X −ipω·(x−y) dk0 d k X ω 0 0 −1 −ik0·(x−y) ha a i ' e F ( σ v k ) e x y 0 (2π)4 µ µ µ ω=± ω µ=0 k'pF ω 1 0 0 where (σµ ) = (−i , σ1, σ2, ωσ3), and v0 ≡ 1, k0 ≡ k0. −βH −βH We let h·i = limβ,L→∞ Tr(e ·)/Tr e . We are interested 5 P 5 in the response of N = x nx to an external e.m. ﬁeld, where Z 5 X n5 = bare (ia+a− − ia+ a−). Note: x 4 x x+δe3 x+δe3 x δ=± Z Z 0 5 5 dk + − 5 + X dk + − N = Zbare (2π)3 sin k3aˆk aˆk ' Zbare sin pF ,3 ω (2π)3 aˆω,k0 aˆω,k0 , ω=± k0'0 ± ± 5 wherea ˆ 0 =a ˆ 0 ω . N is lattice analogue of the chiral charge. ω,k k +pF 5 Zbare ﬁxed by imposing: dressed chiral charge ≡ electric charge.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Lattice interacting model and lattice chiral charge

We consider an interacting version of the model:

H = H0 + λV0 + νN3

where V0 is a short-range density-density interaction, N3 = R dk ˆ+ ˆ− ω = (2π)3 ψk σ3ψk , and ν is used to fix the location of pF . We are interested 5 P 5 in the response of N = x nx to an external e.m. field, where Z 5 X n5 = bare (ia+a− − ia+ a−). Note: x 4 x x+δe3 x+δe3 x δ=± Z Z 0 5 5 dk + − 5 + X dk + − N = Zbare (2π)3 sin k3aˆk aˆk ' Zbare sin pF ,3 ω (2π)3 aˆω,k0 aˆω,k0 , ω=± k0'0 ± ± 5 wherea ˆ 0 =a ˆ 0 ω . N is lattice analogue of the chiral charge. ω,k k +pF 5 Zbare fixed by imposing: dressed chiral charge ≡ electric charge.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Lattice interacting model and lattice chiral charge

We consider an interacting version of the model:

H = H0 + λV0 + νN3

where V0 is a short-range density-density interaction, N3 = R dk ˆ+ ˆ− ω = (2π)3 ψk σ3ψk , and ν is used to ﬁx the location of pF .

−βH −βH We let h·i = limβ,L→∞ Tr(e ·)/Tr e . Note:

Z Z 0 5 5 dk + − 5 + X dk + − N = Zbare (2π)3 sin k3aˆk aˆk ' Zbare sin pF ,3 ω (2π)3 aˆω,k0 aˆω,k0 , ω=± k0'0 ± ± 5 wherea ˆ 0 =a ˆ 0 ω . N is lattice analogue of the chiral charge. ω,k k +pF 5 Zbare ﬁxed by imposing: dressed chiral charge ≡ electric charge.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Lattice interacting model and lattice chiral charge

We consider an interacting version of the model:

H = H0 + λV0 + νN3

where V0 is a short-range density-density interaction, N3 = R dk ˆ+ ˆ− ω = (2π)3 ψk σ3ψk , and ν is used to ﬁx the location of pF .

−βH −βH We let h·i = limβ,L→∞ Tr(e ·)/Tr e . We are interested 5 P 5 in the response of N = x nx to an external e.m. ﬁeld, where Z 5 X n5 = bare (ia+a− − ia+ a−). x 4 x x+δe3 x+δe3 x δ=± Z Z 0 5 5 dk + − 5 + X dk + − N = Zbare (2π)3 sin k3aˆk aˆk ' Zbare sin pF ,3 ω (2π)3 aˆω,k0 aˆω,k0 , ω=± k0'0 ± ± 5 wherea ˆ 0 =a ˆ 0 ω . N is lattice analogue of the chiral charge. ω,k k +pF 5 Zbare ﬁxed by imposing: dressed chiral charge ≡ electric charge.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Lattice interacting model and lattice chiral charge

We consider an interacting version of the model:

H = H0 + λV0 + νN3

where V0 is a short-range density-density interaction, N3 = R dk ˆ+ ˆ− ω = (2π)3 ψk σ3ψk , and ν is used to ﬁx the location of pF .

−βH −βH We let h·i = limβ,L→∞ Tr(e ·)/Tr e . We are interested 5 P 5 in the response of N = x nx to an external e.m. ﬁeld, where Z 5 X n5 = bare (ia+a− − ia+ a−). Note: x 4 x x+δe3 x+δe3 x δ=± 5 Zbare ﬁxed by imposing: dressed chiral charge ≡ electric charge.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Lattice interacting model and lattice chiral charge

We consider an interacting version of the model:

H = H0 + λV0 + νN3

where V0 is a short-range density-density interaction, N3 = R dk ˆ+ ˆ− ω = (2π)3 ψk σ3ψk , and ν is used to ﬁx the location of pF .

−βH −βH We let h·i = limβ,L→∞ Tr(e ·)/Tr e . We are interested 5 P 5 in the response of N = x nx to an external e.m. ﬁeld, where Z 5 X n5 = bare (ia+a− − ia+ a−). Note: x 4 x x+δe3 x+δe3 x δ=± Z Z 0 5 5 dk + − 5 + X dk + − N = Zbare (2π)3 sin k3aˆk aˆk ' Zbare sin pF ,3 ω (2π)3 aˆω,k0 aˆω,k0 , ω=± k0'0 ± ± 5 wherea ˆ 0 =a ˆ 0 ω . N is lattice analogue of the chiral charge. ω,k k +pF Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Lattice interacting model and lattice chiral charge

We consider an interacting version of the model:

H = H0 + λV0 + νN3

where V0 is a short-range density-density interaction, N3 = R dk ˆ+ ˆ− ω = (2π)3 ψk σ3ψk , and ν is used to ﬁx the location of pF .

−βH −βH We let h·i = limβ,L→∞ Tr(e ·)/Tr e . We are interested 5 P 5 in the response of N = x nx to an external e.m. ﬁeld, where Z 5 X n5 = bare (ia+a− − ia+ a−). Note: x 4 x x+δe3 x+δe3 x δ=± Z Z 0 5 5 dk + − 5 + X dk + − N = Zbare (2π)3 sin k3aˆk aˆk ' Zbare sin pF ,3 ω (2π)3 aˆω,k0 aˆω,k0 , ω=± k0'0 ± ± 5 wherea ˆ 0 =a ˆ 0 ω . N is lattice analogue of the chiral charge. ω,k k +pF 5 Zbare ﬁxed by imposing: dressed chiral charge ≡ electric charge. The A-dependent hopping term is gauge covariant under

± ±iα(x) ± ai,x → e ai,x , A(x) → A(x) + ∂α(x).

We let H0(A) be the gauge covariant hopping term. Note: the interaction is gauge invariant. Also the chiral density is coupled to the e.m. ﬁeld:

5 5 Z X + − i R 1 A (x+sδe )ds n (A) = bare (ia a e 0 3 3 + H.c.) x 4 x x+e3 δ=±

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Coupling to an external e.m. ﬁeld

Gauge invariant coupling to an external e.m. ﬁeld: any + − hopping tx,y ai,x aj,y is modiﬁed into (Peierl’s substitution):

R + − + − i x→y A(`)·d` + − tx,y ai,x aj,y −→ tx,y (A)ai,x aj,y = tx,y e ai,x aj,y . We let H0(A) be the gauge covariant hopping term. Note: the interaction is gauge invariant. Also the chiral density is coupled to the e.m. ﬁeld:

5 5 Z X + − i R 1 A (x+sδe )ds n (A) = bare (ia a e 0 3 3 + H.c.) x 4 x x+e3 δ=±

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Coupling to an external e.m. ﬁeld

Gauge invariant coupling to an external e.m. ﬁeld: any + − hopping tx,y ai,x aj,y is modiﬁed into (Peierl’s substitution):

R + − + − i x→y A(`)·d` + − tx,y ai,x aj,y −→ tx,y (A)ai,x aj,y = tx,y e ai,x aj,y . The A-dependent hopping term is gauge covariant under

± ±iα(x) ± ai,x → e ai,x , A(x) → A(x) + ∂α(x). Also the chiral density is coupled to the e.m. ﬁeld:

5 5 Z X + − i R 1 A (x+sδe )ds n (A) = bare (ia a e 0 3 3 + H.c.) x 4 x x+e3 δ=±

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Coupling to an external e.m. ﬁeld

Gauge invariant coupling to an external e.m. ﬁeld: any + − hopping tx,y ai,x aj,y is modiﬁed into (Peierl’s substitution):

R + − + − i x→y A(`)·d` + − tx,y ai,x aj,y −→ tx,y (A)ai,x aj,y = tx,y e ai,x aj,y . The A-dependent hopping term is gauge covariant under

± ±iα(x) ± ai,x → e ai,x , A(x) → A(x) + ∂α(x).

We let H0(A) be the gauge covariant hopping term. Note: the interaction is gauge invariant. Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Coupling to an external e.m. ﬁeld

Gauge invariant coupling to an external e.m. ﬁeld: any + − hopping tx,y ai,x aj,y is modiﬁed into (Peierl’s substitution):

R + − + − i x→y A(`)·d` + − tx,y ai,x aj,y −→ tx,y (A)ai,x aj,y = tx,y e ai,x aj,y . The A-dependent hopping term is gauge covariant under

± ±iα(x) ± ai,x → e ai,x , A(x) → A(x) + ∂α(x).

We let H0(A) be the gauge covariant hopping term. Note: the interaction is gauge invariant. Also the chiral density is coupled to the e.m. ﬁeld:

5 5 Z X + − i R 1 A (x+sδe )ds n (A) = bare (ia a e 0 3 3 + H.c.) x 4 x x+e3 δ=± h·iβ,L;t ≡ Tr · ρ(t): time-dependent state of the system, with −βH ρ(t) solution of i∂t ρ(t) = [H(A(t)), ρ(t)] s.t. ρ(−∞) ∝ e . By formally expanding in A, for A small with zero average: Z −3 5 i dp ˆ ˆ 5 ˆ 3 lim L hN (A(t))iβ,L;t = Ap(t), Π0(η, p)A−p(t) +O(A ). β,L→∞ 2 (2π)3

ˆ 5 By definition, Π0(η, p) is the quadratic response coefficient of the chiral density to the external e.m. field A.

It is the analogue of the dressed triangle graph in QED4.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Quadratic response coeﬃcient of the chiral density

ηt External e.m. potential: Ax (t) = e Ax for t ≤ 0, with η > 0 small (adiabatic switching), Ax slowly varying in space. By formally expanding in A, for A small with zero average: Z −3 5 i dp ˆ ˆ 5 ˆ 3 lim L hN (A(t))iβ,L;t = Ap(t), Π0(η, p)A−p(t) +O(A ). β,L→∞ 2 (2π)3

ˆ 5 By definition, Π0(η, p) is the quadratic response coefficient of the chiral density to the external e.m. field A.

It is the analogue of the dressed triangle graph in QED4.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Quadratic response coeﬃcient of the chiral density

ηt External e.m. potential: Ax (t) = e Ax for t ≤ 0, with η > 0 small (adiabatic switching), Ax slowly varying in space.

h·iβ,L;t ≡ Tr · ρ(t): time-dependent state of the system, with −βH ρ(t) solution of i∂t ρ(t) = [H(A(t)), ρ(t)] s.t. ρ(−∞) ∝ e . ˆ 5 By definition, Π0(η, p) is the quadratic response coefficient of the chiral density to the external e.m. field A.

It is the analogue of the dressed triangle graph in QED4.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Quadratic response coeﬃcient of the chiral density

ηt External e.m. potential: Ax (t) = e Ax for t ≤ 0, with η > 0 small (adiabatic switching), Ax slowly varying in space.

h·iβ,L;t ≡ Tr · ρ(t): time-dependent state of the system, with −βH ρ(t) solution of i∂t ρ(t) = [H(A(t)), ρ(t)] s.t. ρ(−∞) ∝ e . By formally expanding in A, for A small with zero average: Z −3 5 i dp ˆ ˆ 5 ˆ 3 lim L hN (A(t))iβ,L;t = Ap(t), Π0(η, p)A−p(t) +O(A ). β,L→∞ 2 (2π)3 It is the analogue of the dressed triangle graph in QED4.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Quadratic response coeﬃcient of the chiral density

ηt External e.m. potential: Ax (t) = e Ax for t ≤ 0, with η > 0 small (adiabatic switching), Ax slowly varying in space.

ˆ 5 By definition, Π0(η, p) is the quadratic response coefficient of the chiral density to the external e.m. field A. Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Quadratic response coefficient of the chiral density

ηt External e.m. potential: Ax (t) = e Ax for t ≤ 0, with η > 0 small (adiabatic switching), Ax slowly varying in space.

ˆ 5 By definition, Π0(η, p) is the quadratic response coefficient of the chiral density to the external e.m. field A.

It is the analogue of the dressed triangle graph in QED4. Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Main result

Theorem (G., Mastropietro, Porta 2020) For |λ| small enough and any ζ ∈ (−1, 1), there exists ν = ν(λ) = O(λ) s.t. interacting two-point function is ± 5 5 singular at pF , Zbare = Zbare(λ) = 1 + O(λ) such that the chiral dressed charge equals the electric charge. 5 Fixing ν and Zbare this way, for small enough η, p:

3 1 X Πˆ 5 (η, p) = − ε p + Rˆ5(η, p), 0,i,j 2π2 ijl l l=1

where εijl is the Levi-Civita symbol and, for any θ ∈ (0, 1):

|Rˆ5(η, p)| = O([max{η, |p|}]1+θ) 5 ˆ 5 ν, Zbare and Π0 are analytic in λ. 3 2 ˆ 5 1 P Plugging Π0,i,j (η, p) ' − 2π2 l=1 εijl pl into 5 −3 5 hn (A(t))it ≡ lim L hN (A(t))iβ,L;t β,L→∞ i Z dp ' Aˆ (t), Πˆ 5(η, p)Aˆ (t) 2 (2π)3 p 0 −p we ﬁnd that: 1 Z ∂ hn5(A(t))i ' dx E (t) · B (t), t t 2π2 x x which is the same as Nielsen-Ninomiya but in the interacting case! This equation has measurable implications (negative longitudinal magnetoresistance) for real Weyl SM.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Remarks

1 The result holds uniformly in the distance between the Weyl nodes. 3 2 ˆ 5 1 P Plugging Π0,i,j (η, p) ' − 2π2 l=1 εijl pl into 5 −3 5 hn (A(t))it ≡ lim L hN (A(t))iβ,L;t β,L→∞ i Z dp ' Aˆ (t), Πˆ 5(η, p)Aˆ (t) 2 (2π)3 p 0 −p we ﬁnd that: 1 Z ∂ hn5(A(t))i ' dx E (t) · B (t), t t 2π2 x x which is the same as Nielsen-Ninomiya but in the interacting case! This equation has measurable implications (negative longitudinal magnetoresistance) for real Weyl SM.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Remarks

1 The result holds uniformly in the distance between the 5 ˆ 5 Weyl nodes. ν, Zbare and Π0 are analytic in λ. 1 Z ∂ hn5(A(t))i ' dx E (t) · B (t), t t 2π2 x x which is the same as Nielsen-Ninomiya but in the interacting case! This equation has measurable implications (negative longitudinal magnetoresistance) for real Weyl SM.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Remarks

1 The result holds uniformly in the distance between the 5 ˆ 5 Weyl nodes. ν, Zbare and Π0 are analytic in λ. 3 2 ˆ 5 1 P Plugging Π0,i,j (η, p) ' − 2π2 l=1 εijl pl into 5 −3 5 hn (A(t))it ≡ lim L hN (A(t))iβ,L;t β,L→∞ i Z dp ' Aˆ (t), Πˆ 5(η, p)Aˆ (t) 2 (2π)3 p 0 −p we ﬁnd that: This equation has measurable implications (negative longitudinal magnetoresistance) for real Weyl SM.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Remarks

1 The result holds uniformly in the distance between the 5 ˆ 5 Weyl nodes. ν, Zbare and Π0 are analytic in λ. 3 2 ˆ 5 1 P Plugging Π0,i,j (η, p) ' − 2π2 l=1 εijl pl into 5 −3 5 hn (A(t))it ≡ lim L hN (A(t))iβ,L;t β,L→∞ i Z dp ' Aˆ (t), Πˆ 5(η, p)Aˆ (t) 2 (2π)3 p 0 −p we ﬁnd that: 1 Z ∂ hn5(A(t))i ' dx E (t) · B (t), t t 2π2 x x which is the same as Nielsen-Ninomiya but in the interacting case! Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Remarks

1 The result holds uniformly in the distance between the 5 ˆ 5 Weyl nodes. ν, Zbare and Π0 are analytic in λ. 3 2 ˆ 5 1 P Plugging Π0,i,j (η, p) ' − 2π2 l=1 εijl pl into 5 −3 5 hn (A(t))it ≡ lim L hN (A(t))iβ,L;t β,L→∞ i Z dp ' Aˆ (t), Πˆ 5(η, p)Aˆ (t) 2 (2π)3 p 0 −p we ﬁnd that: 1 Z ∂ hn5(A(t))i ' dx E (t) · B (t), t t 2π2 x x which is the same as Nielsen-Ninomiya but in the interacting case! This equation has measurable implications (negative longitudinal magnetoresistance) for real Weyl SM. Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Outline

1 Overview

2 The chiral anomaly in QED4

3 Lattice Weyl semimetals & Main results

4 Sketch of the proof Large distance decay of two-point and vertex functions

ˆ − + ˆ − + S2(k) := haˆk aˆk i and Γ0(k, p) := hnˆp;a ˆk+paˆk i same as for non-interacting model, up to ﬁnite multiplicative renormalization and ﬁnite dressing of the Fermi velocities.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Asymptotic freedom of the infrared theory

Ground state correlations constructed via a rigorous RG, analogous to the one used by G-Mastropietro for the half-ﬁlled Hubbard model on the hexagonal lattice. IR theory of WSM, even if massless, very well behaved: quartic interaction is irrelevant ⇒ scales to zero at large distances n m 3 (scaling dimension of ψ A kernels: D = 4 − 2 n − m). Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Asymptotic freedom of the infrared theory

Ground state correlations constructed via a rigorous RG, analogous to the one used by G-Mastropietro for the half-ﬁlled Hubbard model on the hexagonal lattice. IR theory of WSM, even if massless, very well behaved: quartic interaction is irrelevant ⇒ scales to zero at large distances n m 3 (scaling dimension of ψ A kernels: D = 4 − 2 n − m). Large distance decay of two-point and vertex functions

ˆ 0 ω ˆ 0 ω ˆ 0 ω Γ0(k + pF , p) ' Z0S2(k + p + pF )S2(k + pF ).

where Z0 = Z0(λ) = 1 + O(λ).

A Ward Identity guarantees that Z0 = Z Physical meaning: dressed electric charge ≡ bare charge.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Dressed two-point and vertex function

Interacting two-point function. For k0 small:

3 1 X −1 Sˆ (k0 + pω ) ' σωv k0 , 2 F Z µ µ µ µ=0

0 with Z = Z(λ) = 1 + O(λ) and vµ = vµ(1 + O(λ)), v0 ≡ 1. A Ward Identity guarantees that Z0 = Z Physical meaning: dressed electric charge ≡ bare charge.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Dressed two-point and vertex function

Interacting two-point function. For k0 small:

3 1 X −1 Sˆ (k0 + pω ) ' σωv k0 , 2 F Z µ µ µ µ=0

0 with Z = Z(λ) = 1 + O(λ) and vµ = vµ(1 + O(λ)), v0 ≡ 1. Interacting vertex function. For k0, p small:

ˆ 0 ω ˆ 0 ω ˆ 0 ω Γ0(k + pF , p) ' Z0S2(k + p + pF )S2(k + pF ).

where Z0 = Z0(λ) = 1 + O(λ). Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Dressed two-point and vertex function

Interacting two-point function. For k0 small:

3 1 X −1 Sˆ (k0 + pω ) ' σωv k0 , 2 F Z µ µ µ µ=0

0 with Z = Z(λ) = 1 + O(λ) and vµ = vµ(1 + O(λ)), v0 ≡ 1. Interacting vertex function. For k0, p small:

ˆ 0 ω ˆ 0 ω ˆ 0 ω Γ0(k + pF , p) ' Z0S2(k + p + pF )S2(k + pF ).

where Z0 = Z0(λ) = 1 + O(λ).

A Ward Identity guarantees that Z0 = Z Physical meaning: dressed electric charge ≡ bare charge. For k0, p small:

ˆ 0 ω ˜ 5 ˆ 0 ω ˆ 0 ω Γ0(k + pF , p) ' ZZbareωS2(k + p + pF )S2(k + pF ). with Z˜ = Z˜(λ) = 1 + O(λ). NOTE: no chiral gauge symmetry 5 ˜ 5 ⇒ no a priori relation between relating Z0 := ZZbare and Z. 5 Criterium for ﬁxing Zbare (‘same’ used by Adler in QED4): 5 ˜ 5 Z0 = ZZbare ≡ Z which means: dressed chiral charge ≡ electric charge.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Dressed chiral vertex

Interacting chiral vertex function:

ˆ5 5 − + Γ0(k, p) := hnˆp;a ˆk+paˆk i. NOTE: no chiral gauge symmetry 5 ˜ 5 ⇒ no a priori relation between relating Z0 := ZZbare and Z. 5 Criterium for ﬁxing Zbare (‘same’ used by Adler in QED4): 5 ˜ 5 Z0 = ZZbare ≡ Z which means: dressed chiral charge ≡ electric charge.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Dressed chiral vertex

Interacting chiral vertex function:

ˆ5 5 − + Γ0(k, p) := hnˆp;a ˆk+paˆk i. For k0, p small:

ˆ 0 ω ˜ 5 ˆ 0 ω ˆ 0 ω Γ0(k + pF , p) ' ZZbareωS2(k + p + pF )S2(k + pF ). with Z˜ = Z˜(λ) = 1 + O(λ). 5 Criterium for ﬁxing Zbare (‘same’ used by Adler in QED4): 5 ˜ 5 Z0 = ZZbare ≡ Z which means: dressed chiral charge ≡ electric charge.

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Dressed chiral vertex

Interacting chiral vertex function:

ˆ5 5 − + Γ0(k, p) := hnˆp;a ˆk+paˆk i. For k0, p small:

ˆ 0 ω ˜ 5 ˆ 0 ω ˆ 0 ω Γ0(k + pF , p) ' ZZbareωS2(k + p + pF )S2(k + pF ). with Z˜ = Z˜(λ) = 1 + O(λ). NOTE: no chiral gauge symmetry 5 ˜ 5 ⇒ no a priori relation between relating Z0 := ZZbare and Z. Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Dressed chiral vertex

Interacting chiral vertex function:

ˆ5 5 − + Γ0(k, p) := hnˆp;a ˆk+paˆk i. For k0, p small:

5 5 Bare constants Zµ,bare ﬁxed s.t. Zµ = Zµ. ˆ5 ˆ 5 Quadratic response of Jµ,p to external e.m. ﬁeld: Πµ,ν,σ(p1, p2) 5 Asymptotic freedom in IR + Ward Identities + (Zµ = Zµ) ⇒

ˆ 5 vµvνvσ 5 Πµ,ν,σ(p1, p2) = Iµ,ν,σ(p1, p2) + Hµ,ν,σ(p1, p2) (∗) v1v2v3

where: Iµ,ν,σ(p1, p2) is the relativistic chiral triangle graph with momentum cutoﬀ, pi,µ = vµpi,µ and the subdominant term 5 1+θ Hµ,ν,σ is C in a neighborhood of (0, 0) (while Iµ,ν,σ has discontinuous derivatives at the origin).

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof IR decomposition of quadratic response

Vertex functions for (non-)chiral density can be generalized to ˆ ˆ5 (non-)chiral lattice currents Jµ,p and Jµ,p, µ = 0, 1, 2, 3. ˆ5 ˆ 5 Quadratic response of Jµ,p to external e.m. ﬁeld: Πµ,ν,σ(p1, p2) 5 Asymptotic freedom in IR + Ward Identities + (Zµ = Zµ) ⇒

ˆ 5 vµvνvσ 5 Πµ,ν,σ(p1, p2) = Iµ,ν,σ(p1, p2) + Hµ,ν,σ(p1, p2) (∗) v1v2v3

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof IR decomposition of quadratic response

Vertex functions for (non-)chiral density can be generalized to ˆ ˆ5 (non-)chiral lattice currents Jµ,p and Jµ,p, µ = 0, 1, 2, 3.

Ward Identities ⇒ Zµ = Zvµ

5 5 Bare constants Zµ,bare ﬁxed s.t. Zµ = Zµ. Iµ,ν,σ(p1, p2) is the relativistic chiral triangle graph with momentum cutoﬀ, pi,µ = vµpi,µ and the subdominant term 5 1+θ Hµ,ν,σ is C in a neighborhood of (0, 0) (while Iµ,ν,σ has discontinuous derivatives at the origin).

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof IR decomposition of quadratic response

Vertex functions for (non-)chiral density can be generalized to ˆ ˆ5 (non-)chiral lattice currents Jµ,p and Jµ,p, µ = 0, 1, 2, 3.

Ward Identities ⇒ Zµ = Zvµ

5 5 Bare constants Zµ,bare ﬁxed s.t. Zµ = Zµ. ˆ5 ˆ 5 Quadratic response of Jµ,p to external e.m. ﬁeld: Πµ,ν,σ(p1, p2) 5 Asymptotic freedom in IR + Ward Identities + (Zµ = Zµ) ⇒

ˆ 5 vµvνvσ 5 Πµ,ν,σ(p1, p2) = Iµ,ν,σ(p1, p2) + Hµ,ν,σ(p1, p2) (∗) v1v2v3 where: Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof IR decomposition of quadratic response

Vertex functions for (non-)chiral density can be generalized to ˆ ˆ5 (non-)chiral lattice currents Jµ,p and Jµ,p, µ = 0, 1, 2, 3.

Ward Identities ⇒ Zµ = Zvµ

5 5 Bare constants Zµ,bare ﬁxed s.t. Zµ = Zµ. ˆ5 ˆ 5 Quadratic response of Jµ,p to external e.m. ﬁeld: Πµ,ν,σ(p1, p2) 5 Asymptotic freedom in IR + Ward Identities + (Zµ = Zµ) ⇒

ˆ 5 vµvνvσ 5 Πµ,ν,σ(p1, p2) = Iµ,ν,σ(p1, p2) + Hµ,ν,σ(p1, p2) (∗) v1v2v3

3 X 1 (p + p )I (p , p ) = p p ε + h.o., 1,µ 2,µ µ,ν,σ 1 2 6π2 1,α 2,β αβνσ µ=0 3 X 1 p I (p , p ) = p p ε + h.o. 1,ν µ,ν,σ 1 2 6π2 1,α 2,β αβµσ ν=0

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof The relativistic chiral triangle graph

ˆ 5 vµvνvσ 5 Πµ,ν,σ(p1, p2) = Iµ,ν,σ(p1, p2) + Hµ,ν,σ(p1, p2) (∗) v1v2v3

5 1+θ where pi = (pi,0, v1pi,1, v2pi,2, v3pi,3), Hµ,ν,σ ∈ C and Z dk nχ(k) χ(k + p ) χ(k + p ) o I (p , p ) = Tr γ γ 1 γ 2 γ + µ,ν,σ 1 2 (2π)4 k/ µ 5 k/ + p ν k/ + p σ /1 /2

+ [(ν, p1) ↔ (σ, p2)]. Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof The relativistic chiral triangle graph

ˆ 5 vµvνvσ 5 Πµ,ν,σ(p1, p2) = Iµ,ν,σ(p1, p2) + Hµ,ν,σ(p1, p2) (∗) v1v2v3

5 1+θ where pi = (pi,0, v1pi,1, v2pi,2, v3pi,3), Hµ,ν,σ ∈ C and Z dk nχ(k) χ(k + p ) χ(k + p ) o I (p , p ) = Tr γ γ 1 γ 2 γ + µ,ν,σ 1 2 (2π)4 k/ µ 5 k/ + p ν k/ + p σ /1 /2

+ [(ν, p1) ↔ (σ, p2)]. A computation shows that 3 X 1 (p + p )I (p , p ) = p p ε + h.o., 1,µ 2,µ µ,ν,σ 1 2 6π2 1,α 2,β αβνσ µ=0 3 X 1 p I (p , p ) = p p ε + h.o. 1,ν µ,ν,σ 1 2 6π2 1,α 2,β αβµσ ν=0 5 From this: Hµ,ν,σ(0, 0) = 0 and 5 5 ∂Hµ,ν,σ 1 ∂Hµ,ν,σ 1 (0, 0) = − 2 ενβµσ, (0, 0) = − 2 εσαµν (∗∗) ∂p2,β 6π ∂p1,α 6π ˆ 5 We now contract pµ = −p1,µ − p2,µ with Πµ,ν,σ(p1, p2): X X vµvνvσ X p Πˆ 5 (p) = p I (p , p )+ p H5 (p , p ). µ µ,ν,σ µ v v v µ,ν,σ 1 2 µ µ,ν,σ 1 2 µ µ 1 2 3 µ First term: we computed it explicitly. Second term: use (**). X 1 p Πˆ 5 (p , p ) = − p p ε + OP2+θ). µ µ,ν,σ 1 2 2π2 1,α 2,β αβνσ µ

with P = max{|p1|, |p2|}. ˆ 5 We now contract pµ = −p1,µ − p2,µ with Πµ,ν,σ(p1, p2): X X vµvνvσ X p Πˆ 5 (p) = p I (p , p )+ p H5 (p , p ). µ µ,ν,σ µ v v v µ,ν,σ 1 2 µ µ,ν,σ 1 2 µ µ 1 2 3 µ First term: we computed it explicitly. Second term: use (**). X 1 p Πˆ 5 (p , p ) = − p p ε + OP2+θ). µ µ,ν,σ 1 2 2π2 1,α 2,β αβνσ µ

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Lattice Ward identities P ˆ 5 We now use ν p1,νΠµ,ν,σ(p1, p2) = 0, which implies, together 1+θ 5 with (∗) and the C diﬀerentiability of Hµ,ν,σ: 5 p1,αp2,βεαβµσ 5 X ∂Hµ,ν,σ 2+θ 2 +p1,ν Hµ,ν,σ(0, 0)+ pj,α (0, 0) = O(P ), 6π ∂pj,α j=1,2 5 with P = max{|p1|, |p2|}. From this: Hµ,ν,σ(0, 0) = 0 and 5 5 ∂Hµ,ν,σ 1 ∂Hµ,ν,σ 1 (0, 0) = − 2 ενβµσ, (0, 0) = − 2 εσαµν (∗∗) ∂p2,β 6π ∂p1,α 6π ˆ 5 We now contract pµ = −p1,µ − p2,µ with Πµ,ν,σ(p1, p2): X X vµvνvσ X p Πˆ 5 (p) = p I (p , p )+ p H5 (p , p ). µ µ,ν,σ µ v v v µ,ν,σ 1 2 µ µ,ν,σ 1 2 µ µ 1 2 3 µ First term: we computed it explicitly. Second term: use (**). Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Lattice Ward identities P ˆ 5 We now use ν p1,νΠµ,ν,σ(p1, p2) = 0, which implies, together 1+θ 5 with (∗) and the C diﬀerentiability of Hµ,ν,σ: 5 p1,αp2,βεαβµσ 5 X ∂Hµ,ν,σ 2+θ 2 +p1,ν Hµ,ν,σ(0, 0)+ pj,α (0, 0) = O(P ), 6π ∂pj,α j=1,2 5 with P = max{|p1|, |p2|}. From this: Hµ,ν,σ(0, 0) = 0 and 5 5 ∂Hµ,ν,σ 1 ∂Hµ,ν,σ 1 (0, 0) = − 2 ενβµσ, (0, 0) = − 2 εσαµν (∗∗) ∂p2,β 6π ∂p1,α 6π ˆ 5 We now contract pµ = −p1,µ − p2,µ with Πµ,ν,σ(p1, p2): X X vµvνvσ X p Πˆ 5 (p) = p I (p , p )+ p H5 (p , p ). µ µ,ν,σ µ v v v µ,ν,σ 1 2 µ µ,ν,σ 1 2 µ µ 1 2 3 µ First term: we computed it explicitly. Second term: use (**). X 1 p Πˆ 5 (p , p ) = − p p ε + OP2+θ). µ µ,ν,σ 1 2 2π2 1,α 2,β αβνσ µ – Proof based on constructive RG methods combined with lattice Ward Identities + bounds on regularity of correlations and of the Schwinger terms.

– Open problems: 1 Eﬀects of disorder? 2 Coupling to a dynamical e.m. ﬁeld: rigorous construction of infrared QED4 [without photon mass counterterms]? Dynamical restoration of Lorentz invariance in the IR? Non-renormalization of the chiral anomaly?

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Conclusions

– The chiral anomaly of QED4 has a cond-mat counterpart in Weyl semimetals. We proved the nonperturbative analogue of the Adler-Bardeen thm for interacting lattice Weyl fermions: non-renormalization of the anomaly. – Open problems: 1 Eﬀects of disorder? 2 Coupling to a dynamical e.m. ﬁeld: rigorous construction of infrared QED4 [without photon mass counterterms]? Dynamical restoration of Lorentz invariance in the IR? Non-renormalization of the chiral anomaly?

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Conclusions

– Proof based on constructive RG methods combined with lattice Ward Identities + bounds on regularity of correlations and of the Schwinger terms. 2 Coupling to a dynamical e.m. ﬁeld: rigorous construction of infrared QED4 [without photon mass counterterms]? Dynamical restoration of Lorentz invariance in the IR? Non-renormalization of the chiral anomaly?

Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Conclusions

– Proof based on constructive RG methods combined with lattice Ward Identities + bounds on regularity of correlations and of the Schwinger terms.

– Open problems: 1 Eﬀects of disorder? Overview The chiral anomaly in QED4 Lattice Weyl semimetals & Main results Sketch of the proof Conclusions

– Proof based on constructive RG methods combined with lattice Ward Identities + bounds on regularity of correlations and of the Schwinger terms.

Thank you!