Effective Dirac equations in honeycomb structures

Effective Dirac equations in honeycomb structures Young Researchers Seminar, CERMICS, Ecole des Ponts ParisTech

William Borrelli

CEREMADE, Universit´eParis Dauphine

11 April 2018 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 1 2 2 H 2 (R , C ), respectively. Remark The negative spectrum is associated with antiparticles, in relativistic theories.

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. The domain of the operator and form domain are H1(R2, C2) and 1 2 2 H 2 (R , C ), respectively. Remark The negative spectrum is associated with antiparticles, in relativistic theories.

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 1 2 2 H 2 (R , C ), respectively. Remark The negative spectrum is associated with antiparticles, in relativistic theories. 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 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) Definition A function V ∈ C ∞(R2, R) is called a honeycomb potential if there 2 exists x0 ∈ R 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; ˜ 2π V is invariant by 3 rotations. In the sequel V will denote a honeycomb potential.

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 0 + 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 0 + 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 2 exists x0 ∈ R 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; ˜ 2π V is invariant by 3 rotations. In the sequel V will denote 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 case of graphene

Graphene can be described by a periodic Schr¨odingeroperator −∆ + 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¨odingeroperator −∆ + 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: 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 The effective operator

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

Fefferman and Weinstein ’12 proved that for ”most” honeycomb potentials Dirac points appear on the corners of the Brillouin zone of 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 of 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. Theorem (Fefferman, Weinstein ’13) Fix ρ > 0,δ > 0,N ∈ N. Then the linear Schr¨odingerequation i∂t u = (−∆ + V )u has a unique solution of the form   2 √ ε −iµ∗t X ε u (t, x) = e  εψj (εt, εx)Φj (x) + η (t, x) (3) j=1

ε ε ε with u (0, x) = u0(x), η (0, x) = 0. For any |β| ≤ N we have

β ε ε→0 sup k∂ η (t, x)k 2 2 −−−→ 0 x Lx (R ) 0≤t≤ρε−2+δ .

Effective Dirac equations in honeycomb structures Dirac in 2D Linear Dirac dynamics

Consider a wave packet spectrally concentrated around K∗: ε √ u0(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∗: ε √ u0(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¨odingerequation i∂t u = (−∆ + V )u has a unique solution of the form   2 √ ε −iµ∗t X ε u (t, x) = e  εψj (εt, εx)Φj (x) + η (t, x) (3) j=1

ε ε ε with u (0, x) = u0(x), η (0, x) = 0. For any |β| ≤ N we have

β ε ε→0 sup k∂ η (t, x)k 2 2 −−−→ 0 x Lx (R ) 0≤t≤ρε−2+δ . The coefficients ψj form a global-in-time solution to the following Dirac equation       ψ1 0 λ(∂1 + i∂2) ψ1 i∂t = , 0 6= λ ∈ C ψ2 λ(∂1 − i∂2) 0 ψ2

ψ (0, x) ψ (x) with initial data 1 = 1,0 ∈ S(R2)2. ψ2(0, x) ψ2,0(x) 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 Dirac in 2D Linear Dirac dynamics

The functions Φj are Bloch functions at a Dirac point, i.e. a corner of the Brillouin zone. Remark It is conceivable that the condition on the initial data can be weakened with additional work.

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 of the Brillouin zone. The coefficients ψj form a global-in-time solution to the following Dirac equation       ψ1 0 λ(∂1 + i∂2) ψ1 i∂t = , 0 6= λ ∈ C ψ2 λ(∂1 − i∂2) 0 ψ2

ψ (0, x) ψ (x) with initial data 1 = 1,0 ∈ S(R2)2. ψ2(0, x) ψ2,0(x) 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 of the Brillouin zone. The coefficients ψj form a global-in-time solution to the following Dirac equation       ψ1 0 λ(∂1 + i∂2) ψ1 i∂t = , 0 6= λ ∈ C ψ2 λ(∂1 − i∂2) 0 ψ2

ψ (0, x) ψ (x) with initial data 1 = 1,0 ∈ S(R2)2. ψ2(0, x) ψ2,0(x) 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. The effective equation around a Dirac point is (Fefferman-Weinstein ’12, formal derivation):

( 2 2 ∂t ψ1 + λ(∂1 + i∂2)ψ2 = −iκ(β1|ψ1| + 2β2|ψ2| )ψ1 2 2 (4) ∂t ψ2 + λ(∂1 − i∂2)ψ1 = −iκ(β1|ψ2| + 2β2|ψ1| )ψ2

T 2 with 0 6= λ ∈ C, βj > 0 and ψ = (ψ1, ψ2) is a C -spinor.

Effective Dirac equations in honeycomb structures The cubic Dirac equation From NLS/GP to cubic Dirac

Consider the following nonlinear Schr¨odinger/Gross-Pitaevskii equation: 2 i∂t u = (−∆ + V )u + κ|u| u 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¨odinger/Gross-Pitaevskii equation: 2 i∂t u = (−∆ + V )u + κ|u| u where κ ∈ R, and V is a honeycomb potential. The effective equation around a Dirac point is (Fefferman-Weinstein ’12, formal derivation):

( 2 2 ∂t ψ1 + λ(∂1 + i∂2)ψ2 = −iκ(β1|ψ1| + 2β2|ψ2| )ψ1 2 2 (4) ∂t ψ2 + λ(∂1 − i∂2)ψ1 = −iκ(β1|ψ2| + 2β2|ψ1| )ψ2

T 2 with 0 6= λ ∈ C, βj > 0 and ψ = (ψ1, ψ2) is a C -spinor. Theorem (Arbunich,Sparber ’16) 2 Consider the equation i∂t u = (−∆ + V )u + κ|u| u, and let s > 1, S > 3. There exists T ε ∼ ε−1, s.t. the solution ε 0 ε s 2 ε ε u ∈ C ([0, T ), H (R )) of the equation with u (0, x) = u0(x) is of the form   2 √ ε −iµ∗t X ε u (t, x) = e  εψj (εt, εx)Φj (x) + η (t, x) , j=1

T 0 ε S 2 2 provided that ψ = (ψ1, ψ2) ∈ C ([0, T ), H (R , C )) 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 Nonlinear Dirac dynamics

ε √ Let u0(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

ε √ Let u0(x) = ε(ψ1,0(εx)Φ1(x) + ψ2,0(εx)Φ2(x)), ε > 0. Theorem (Arbunich,Sparber ’16) 2 Consider the equation i∂t u = (−∆ + V )u + κ|u| u, and let s > 1, S > 3. There exists T ε ∼ ε−1, s.t. the solution ε 0 ε s 2 ε ε u ∈ C ([0, T ), H (R )) of the equation with u (0, x) = u0(x) is of the form   2 √ ε −iµ∗t X ε u (t, x) = e  εψj (εt, εx)Φj (x) + η (t, x) , j=1

T 0 ε S 2 2 provided that ψ = (ψ1, ψ2) ∈ C ([0, T ), H (R , C )) is a solution of (4). In this case the approximation is valid on a time interval O(ε−1). 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

We are interested in stationary solutions of the focusing NLS/GP,

e−iµ∗t u(x)

where µ∗ ∈ σ(−∆ + V ) is the energy of a Dirac point. Then u solves 2 (−∆ + V − µ∗)u = |u| u. In particular, we may look for solutions of the form ε P2 √ ε  u (x) = j=1 εψj (εx)Φj (x) + η (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∗t u(x)

where µ∗ ∈ σ(−∆ + V ) is the energy of a Dirac point. Then u solves 2 (−∆ + V − µ∗)u = |u| u. In particular, we may look for solutions of the form ε P2 √ ε  u (x) = j=1 εψj (εx)Φj (x) + η (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 T ψ = (ψ1, ψ2) : ( 2 2 (∂1 + i∂2)ψ2 = i(β1|ψ2| + 2β2|ψ1| )ψ1 2 2 (5) (∂1 − i∂2)ψ1 = i(β1|ψ1| + 2β2|ψ2| )ψ2

Theorem (W.B. ’18)

The above equation D ψ = Gβ1,β2 (ψ)ψ admits infinitely many iu(r)eiϑ solutions ψ ∈ C ∞(R2, C2) of the form ψ(r, ϑ) = with v(r) u, v : [0, +∞) −→ R,(r, ϑ) are polar coordinates. Moreover

1 1 |u(r)| ∼ , |v(r)| ∼ , as r → +∞. r r 2 In particular, ψ ∈ Lp(R2, C2), ∀p > 2, but ψ∈ / L2(R2, C2). The equation is scale-invariant and odd; For the above reason it suffices to prove the existence of only one 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 Towards Dirac solitons for NLS/GP

Plugging the radial ansatz into the equation we get

 u 2 2  u˙ + = (2β2u + β1v )v, u(0) = 0 r 2 2  v˙ = −(2β2u + β1v )u, v(0) = λ 6= 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 2 2  u˙ + = (2β2u + β1v )v, u(0) = 0 r 2 2  v˙ = −(2β2u + β1v )u, v(0) = λ 6= 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 one 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. 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 iu(r)eiϑ ψ(r, ϑ) = , with exponential decay at infinity. v(r)

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 iu(r)eiϑ ψ(r, ϑ) = , with exponential decay at infinity. v(r) 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 Some remarks

Plugging the radial ansatz into the equation we get

 u 2 2  u˙ + = (2β2u + β1v )v − (m − ω)v, u(0) = 0 r 2 2  v˙ = −(2β2u + β1v )u − (m + ω)u, v(0) = λ 6= 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 2 2  u˙ + = (2β2u + β1v )v − (m − ω)v, u(0) = 0 r 2 2  v˙ = −(2β2u + β1v )u − (m + ω)u, v(0) = λ 6= 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

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