Effective Dirac equations in honeycomb structures Young Researchers - - PowerPoint PPT Presentation

Effective Dirac equations in honeycomb structures

Effective Dirac equations in honeycomb structures

Young Researchers Seminar, CERMICS, Ecole des Ponts ParisTech William Borrelli

CEREMADE, Universit´ e Paris Dauphine

11 April 2018

Effective Dirac equations in honeycomb structures Dirac in 2D

The 2D Dirac operator

The 2D Dirac operator is defined as D = D0 +mσ3 = −i(σ1∂1 + σ2∂2) + mσ3. (1) where σk are the Pauli matrices and m ≥ 0 is the mass of the

• particle. It acts on C2-valued spinors.

Effective Dirac equations in honeycomb structures Dirac in 2D

The 2D Dirac operator

The 2D Dirac operator is defined as D = D0 +mσ3 = −i(σ1∂1 + σ2∂2) + mσ3. (1) where σk are the Pauli matrices and m ≥ 0 is the mass of the

• particle. It acts on C2-valued spinors.

It is self-adjoint on L2(R2, C2) and the spectrum is given by σ(D0) = R, σ(D) = (−∞, −m] ∪ [m, +∞)

Effective Dirac equations in honeycomb structures Dirac in 2D

The 2D Dirac operator

The 2D Dirac operator is defined as D = D0 +mσ3 = −i(σ1∂1 + σ2∂2) + mσ3. (1) where σk are the Pauli matrices and m ≥ 0 is the mass of the

• particle. It acts on C2-valued spinors.

It is self-adjoint on L2(R2, C2) and the spectrum is given by σ(D0) = R, σ(D) = (−∞, −m] ∪ [m, +∞) The domain of the operator and form domain are H1(R2, C2) and H

1 2 (R2, C2), respectively.

Remark The negative spectrum is associated with antiparticles, in relativistic theories.

Effective Dirac equations in honeycomb structures Dirac in 2D

Honeycomb structures

Recently, new two dimensional materials possessing Dirac fermion low-energy excitations have been discovered.

Effective Dirac equations in honeycomb structures Dirac in 2D

Honeycomb structures

Recently, new two dimensional materials possessing Dirac fermion low-energy excitations have been discovered. The most famous example is graphene, which can be modeled as 2D honeycomb lattice of carbon atoms:

Figure: The hexagonal lattice H is a superposition of two copies of a triangular lattice Λ : H = (A + Λ) ∪ (B + Λ)

Effective Dirac equations in honeycomb structures Dirac in 2D

Honeycomb potentials

Let Λ := v1Z ⊕ v2Z be a triangular lattice, and consider its dual Λ∗ := {k ∈ R2|k · v ∈ 2πZ, ∀v ∈ Λ}. The dual lattice H∗ = (K + Λ∗) ∪ (K ′ + Λ∗) is also hexagonal, and its primitive cell B is called the Brillouin zone of the lattice.

Effective Dirac equations in honeycomb structures Dirac in 2D

Honeycomb potentials

Let Λ := v1Z ⊕ v2Z be a triangular lattice, and consider its dual Λ∗ := {k ∈ R2|k · v ∈ 2πZ, ∀v ∈ Λ}. The dual lattice H∗ = (K + Λ∗) ∪ (K ′ + Λ∗) is also hexagonal, and its primitive cell B is called the Brillouin zone of the lattice. Definition A function V ∈ C ∞(R2, R) is called a honeycomb potential if there exists x0 ∈ R2 such that ˜ V (x) := V (x − x0) satisfies: ˜ V is Λ-periodic: ˜ V (x + v) = ˜ V (x), ∀x ∈ R2, ∀v ∈ Λ; ˜ V is even: ˜ V (−x) = ˜ V (x), ∀x ∈ R2; ˜ V is invariant by 2π

3 rotations.

In the sequel V will denote a honeycomb potential.

Effective Dirac equations in honeycomb structures Dirac in 2D

The case of graphene

Graphene can be described by a periodic Schr¨

• dinger operator

−∆ + V , where V ∈ C ∞(R2, R) is a honeycomb potential.

Effective Dirac equations in honeycomb structures Dirac in 2D

The case of graphene

Graphene can be described by a periodic Schr¨

• dinger operator

−∆ + V , where V ∈ C ∞(R2, R) is a honeycomb potential. The spectrum has a band structure, possibly with gaps. It exhibits conical intersections in the low-lying dispersion relations, around the so-called Dirac points:

Effective Dirac equations in honeycomb structures Dirac in 2D

The effective operator

Fefferman and Weinstein ’12 proved that for ”most” honeycomb potentials Dirac points appear on the corners of the Brillouin zone

• f the honeycomb lattice.

Effective Dirac equations in honeycomb structures Dirac in 2D

The effective operator

Fefferman and Weinstein ’12 proved that for ”most” honeycomb potentials Dirac points appear on the corners of the Brillouin zone

• f the honeycomb lattice.

One expects the effective operator around a conical point to be the (massless) Dirac operator, acting on C2-valued spinors: D0 = −i(σ1∂1 + σ2∂2)

Effective Dirac equations in honeycomb structures Dirac in 2D

The effective operator

Fefferman and Weinstein ’12 proved that for ”most” honeycomb potentials Dirac points appear on the corners of the Brillouin zone

• f the honeycomb lattice.

One expects the effective operator around a conical point to be the (massless) Dirac operator, acting on C2-valued spinors: D0 = −i(σ1∂1 + σ2∂2) Remark In this case the vertex of the cone is the Fermi level, and there is no particles/antiparticles interpretation, but rather: positive energies = conduction electrons; negative energies = valence electrons.

Effective Dirac equations in honeycomb structures Dirac in 2D

Linear Dirac dynamics

Consider a wave packet spectrally concentrated around K∗: uε

0(x) = √ε(ψ1,0(εx)Φ1(x) + ψ2,0(εx)Φ2(x)),

ε > 0 (2)

Effective Dirac equations in honeycomb structures Dirac in 2D

Linear Dirac dynamics

Consider a wave packet spectrally concentrated around K∗: uε

0(x) = √ε(ψ1,0(εx)Φ1(x) + ψ2,0(εx)Φ2(x)),

ε > 0 (2) Theorem (Fefferman, Weinstein ’13) Fix ρ > 0,δ > 0,N ∈ N. Then the linear Schr¨

• dinger equation

i∂tu = (−∆ + V )u has a unique solution of the form uε(t, x) = e−iµ∗t  

2

• j=1

√εψj(εt, εx)Φj(x) + ηε(t, x)   (3) with uε(0, x) = uε

0(x), ηε(0, x) = 0. For any |β| ≤ N we have

sup

0≤t≤ρε−2+δ ∂β x ηε(t, x)L2

x(R2)

ε→0

− − − → 0 .

Effective Dirac equations in honeycomb structures Dirac in 2D

Linear Dirac dynamics

The functions Φj are Bloch functions at a Dirac point, i.e. a corner

• f the Brillouin zone.

Effective Dirac equations in honeycomb structures Dirac in 2D

Linear Dirac dynamics

The functions Φj are Bloch functions at a Dirac point, i.e. a corner

• f the Brillouin zone.

The coefficients ψj form a global-in-time solution to the following Dirac equation i∂t ψ1 ψ2

• =
• λ(∂1 + i∂2)

λ(∂1 − i∂2) ψ1 ψ2

• ,

0 = λ ∈ C with initial data ψ1(0, x) ψ2(0, x)

• =

ψ1,0(x) ψ2,0(x)

• ∈
• S(R2)

2. The parameter λ ∈ C depends on the potential V .

Effective Dirac equations in honeycomb structures Dirac in 2D

Linear Dirac dynamics

The functions Φj are Bloch functions at a Dirac point, i.e. a corner

• f the Brillouin zone.

The coefficients ψj form a global-in-time solution to the following Dirac equation i∂t ψ1 ψ2

• =
• λ(∂1 + i∂2)

λ(∂1 − i∂2) ψ1 ψ2

• ,

0 = λ ∈ C with initial data ψ1(0, x) ψ2(0, x)

• =

ψ1,0(x) ψ2,0(x)

• ∈
• S(R2)

2. The parameter λ ∈ C depends on the potential V . Remark It is conceivable that the condition on the initial data can be weakened with additional work.

Effective Dirac equations in honeycomb structures The cubic Dirac equation

From NLS/GP to cubic Dirac

Consider the following nonlinear Schr¨

• dinger/Gross-Pitaevskii

equation: i∂tu = (−∆ + V )u + κ|u|2u where κ ∈ R, and V is a honeycomb potential.

Effective Dirac equations in honeycomb structures The cubic Dirac equation

From NLS/GP to cubic Dirac

Consider the following nonlinear Schr¨

• dinger/Gross-Pitaevskii

equation: i∂tu = (−∆ + V )u + κ|u|2u where κ ∈ R, and V is a honeycomb potential. The effective equation around a Dirac point is (Fefferman-Weinstein ’12, formal derivation):

• ∂tψ1 + λ(∂1 + i∂2)ψ2 = −iκ(β1|ψ1|2 + 2β2|ψ2|2)ψ1

∂tψ2 + λ(∂1 − i∂2)ψ1 = −iκ(β1|ψ2|2 + 2β2|ψ1|2)ψ2 (4) with 0 = λ ∈ C, βj > 0 and ψ = (ψ1, ψ2)T is a C2-spinor.

Effective Dirac equations in honeycomb structures The cubic Dirac equation

Nonlinear Dirac dynamics

Let uε

0(x) = √ε(ψ1,0(εx)Φ1(x) + ψ2,0(εx)Φ2(x)),

ε > 0.

Effective Dirac equations in honeycomb structures The cubic Dirac equation

Nonlinear Dirac dynamics

Let uε

0(x) = √ε(ψ1,0(εx)Φ1(x) + ψ2,0(εx)Φ2(x)),

ε > 0. Theorem (Arbunich,Sparber ’16) Consider the equation i∂tu = (−∆ + V )u + κ|u|2u, and let s > 1, S > 3. There exists T ε ∼ ε−1, s.t. the solution uε ∈ C 0([0, T ε), Hs(R2)) of the equation with uε(0, x) = uε

0(x) is

• f the form

uε(t, x) = e−iµ∗t  

2

• j=1

√εψj(εt, εx)Φj(x) + ηε(t, x)   , provided that ψ = (ψ1, ψ2)T ∈ C 0([0, T ε), HS(R2, C2)) is a solution of (4). In this case the approximation is valid on a time interval O(ε−1).

Effective Dirac equations in honeycomb structures The cubic Dirac equation

Towards Dirac solitons for NLS/GP

We are interested in stationary solutions of the focusing NLS/GP, e−iµ∗tu(x) where µ∗ ∈ σ(−∆ + V ) is the energy of a Dirac point. Then u solves (−∆ + V − µ∗)u = |u|2u. In particular, we may look for solutions of the form uε(x) = 2

j=1

√εψj(εx)Φj(x) + ηε(x)

Effective Dirac equations in honeycomb structures The cubic Dirac equation

Towards Dirac solitons for NLS/GP

We are interested in stationary solutions of the focusing NLS/GP, e−iµ∗tu(x) where µ∗ ∈ σ(−∆ + V ) is the energy of a Dirac point. Then u solves (−∆ + V − µ∗)u = |u|2u. In particular, we may look for solutions of the form uε(x) = 2

j=1

√εψj(εx)Φj(x) + ηε(x)

• µ∗ ∈ σ(−∆ + V ): generally speaking existence is not trivial,

For the same reason, one expects u / ∈ L2(R2), µ∗ correponds to 0 ∈ σ(D) for the effective operator.

Effective Dirac equations in honeycomb structures The cubic Dirac equation

Towards Dirac solitons for NLS/GP

Then one is lead to study the following effective equation for ψ = (ψ1, ψ2)T:

• (∂1 + i∂2)ψ2 = i(β1|ψ2|2 + 2β2|ψ1|2)ψ1

(∂1 − i∂2)ψ1 = i(β1|ψ1|2 + 2β2|ψ2|2)ψ2 (5) Theorem (W.B. ’18) The above equation D ψ = Gβ1,β2(ψ)ψ admits infinitely many solutions ψ ∈ C ∞(R2, C2) of the form ψ(r, ϑ) = iu(r)eiϑ v(r)

• with

u, v : [0, +∞) − → R, (r, ϑ) are polar coordinates. Moreover |u(r)| ∼ 1 r , |v(r)| ∼ 1 r2 , as r → +∞. In particular, ψ ∈ Lp(R2, C2), ∀p > 2, but ψ / ∈ L2(R2, C2).

Effective Dirac equations in honeycomb structures The cubic Dirac equation

Towards Dirac solitons for NLS/GP

Plugging the radial ansatz into the equation we get    ˙ u + u r = (2β2u2 + β1v2)v, u(0) = 0 ˙ v = −(2β2u2 + β1v2)u, v(0) = λ = 0 A solution of the ODE system (v(r), u(r)) → (0, 0), as r → +∞, corresponds to a localized (non-trivial) solution of the PDE.

Effective Dirac equations in honeycomb structures The cubic Dirac equation

Towards Dirac solitons for NLS/GP

Plugging the radial ansatz into the equation we get    ˙ u + u r = (2β2u2 + β1v2)v, u(0) = 0 ˙ v = −(2β2u2 + β1v2)u, v(0) = λ = 0 A solution of the ODE system (v(r), u(r)) → (0, 0), as r → +∞, corresponds to a localized (non-trivial) solution of the PDE. The equation is scale-invariant and odd; For the above reason it suffices to prove the existence of only

• ne solution;

The solutions admit a variational characterization. No gap and conical degeneracy at the Dirac point: the rigorous justification of the effective equation is a challenging problem.

Effective Dirac equations in honeycomb structures The cubic Dirac equation

The massive case

Dirac points are stable w.r.t. small honeycomb perturbations. Adding a suitable perturbation breaking parity opens a gap at a Dirac point (Fefferman-Weinstein ’12). This results in a mass term for the effective operator.

Effective Dirac equations in honeycomb structures The cubic Dirac equation

The massive case

Dirac points are stable w.r.t. small honeycomb perturbations. Adding a suitable perturbation breaking parity opens a gap at a Dirac point (Fefferman-Weinstein ’12). This results in a mass term for the effective operator. Theorem (W.B. ’17/’18) For a fixed ω ∈ (−m, m), the equation (D0 +mσ3 − ω)ψ = Gβ1,β2(ψ)ψ has a (non-trivial) smooth solution of the form ψ(r, ϑ) = iu(r)eiϑ v(r)

• , with exponential decay at infinity.

Effective Dirac equations in honeycomb structures The cubic Dirac equation

Some remarks

Plugging the radial ansatz into the equation we get    ˙ u + u r = (2β2u2 + β1v2)v − (m − ω)v, u(0) = 0 ˙ v = −(2β2u2 + β1v2)u − (m + ω)u, v(0) = λ = 0 A solution of the ODE system (v(r), u(r)) → (0, 0), as r → +∞, corresponds to a localized (non-trivial) solution of the PDE.

Effective Dirac equations in honeycomb structures The cubic Dirac equation

Some remarks

Plugging the radial ansatz into the equation we get    ˙ u + u r = (2β2u2 + β1v2)v − (m − ω)v, u(0) = 0 ˙ v = −(2β2u2 + β1v2)u − (m + ω)u, v(0) = λ = 0 A solution of the ODE system (v(r), u(r)) → (0, 0), as r → +∞, corresponds to a localized (non-trivial) solution of the PDE. In the massive case nonlinear bound states of arbitrary form have exponential decay (Boussa¨ ıd-Comech ’16); Variational characterization not yet available; The equation is odd: actually two non-trivial solutions; The mass term breaks scale-invariance: multiplicity is an open problem.

Effective Dirac equations in honeycomb structures The cubic Dirac equation

Thank you