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 = D 0 + m σ 3 = − i ( σ 1 ∂ 1 + σ 2 ∂ 2 ) + m σ 3 . (1) where σ k are the Pauli matrices and m ≥ 0 is the mass of the particle. It acts on C 2 -valued spinors.

Effective Dirac equations in honeycomb structures Dirac in 2D The 2D Dirac operator The 2D Dirac operator is defined as D = D 0 + m σ 3 = − i ( σ 1 ∂ 1 + σ 2 ∂ 2 ) + m σ 3 . (1) where σ k are the Pauli matrices and m ≥ 0 is the mass of the particle. It acts on C 2 -valued spinors. It is self-adjoint on L 2 ( R 2 , C 2 ) and the spectrum is given by σ ( D 0 ) = 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 = D 0 + m σ 3 = − i ( σ 1 ∂ 1 + σ 2 ∂ 2 ) + m σ 3 . (1) where σ k are the Pauli matrices and m ≥ 0 is the mass of the particle. It acts on C 2 -valued spinors. It is self-adjoint on L 2 ( R 2 , C 2 ) and the spectrum is given by σ ( D 0 ) = R , σ ( D ) = ( −∞ , − m ] ∪ [ m , + ∞ ) The domain of the operator and form domain are H 1 ( R 2 , C 2 ) and 1 2 ( R 2 , C 2 ), respectively. H 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 := v 1 Z ⊕ v 2 Z be a triangular lattice, and consider its dual Λ ∗ := { k ∈ R 2 | k · v ∈ 2 π Z , ∀ 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 := v 1 Z ⊕ v 2 Z be a triangular lattice, and consider its dual Λ ∗ := { k ∈ R 2 | k · v ∈ 2 π Z , ∀ 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 ∞ ( R 2 , R ) is called a honeycomb potential if there exists x 0 ∈ R 2 such that ˜ V ( x ) := V ( x − x 0 ) satisfies: V is Λ-periodic: ˜ ˜ V ( x + v ) = ˜ V ( x ) , ∀ x ∈ R 2 , ∀ v ∈ Λ; V is even: ˜ ˜ V ( − x ) = ˜ V ( x ) , ∀ x ∈ R 2 ; ˜ V is invariant by 2 π 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¨ odinger operator − ∆ + V , where V ∈ C ∞ ( R 2 , 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¨ odinger operator − ∆ + V , where V ∈ C ∞ ( R 2 , 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 of 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 of the honeycomb lattice. One expects the effective operator around a conical point to be the (massless) Dirac operator, acting on C 2 -valued spinors: D 0 = − i ( σ 1 ∂ 1 + σ 2 ∂ 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 C 2 -valued spinors: D 0 = − i ( σ 1 ∂ 1 + σ 2 ∂ 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 ∗ : 0 ( x ) = √ ε ( ψ 1 , 0 ( ε x )Φ 1 ( x ) + ψ 2 , 0 ( ε x )Φ 2 ( x )) , u ε ε > 0 (2)

Effective Dirac equations in honeycomb structures Dirac in 2D Linear Dirac dynamics Consider a wave packet spectrally concentrated around K ∗ : 0 ( x ) = √ ε ( ψ 1 , 0 ( ε x )Φ 1 ( x ) + ψ 2 , 0 ( ε x )Φ 2 ( x )) , u ε ε > 0 (2) Theorem (Fefferman, Weinstein ’13) Fix ρ > 0, δ > 0, N ∈ N . Then the linear Schr¨ odinger equation i ∂ t u = ( − ∆ + V ) u has a unique solution of the form   2 √ εψ j ( ε t , ε x )Φ j ( x ) + η ε ( t , x ) u ε ( t , x ) = e − i µ ∗ t � (3)   j =1 with u ε (0 , x ) = u ε 0 ( x ) , η ε (0 , x ) = 0. For any | β | ≤ N we have ε → 0 0 ≤ t ≤ ρε − 2+ δ � ∂ β x η ε ( t , x ) � L 2 sup − − − → 0 x ( R 2 ) .

Effective Dirac equations in honeycomb structures Dirac in 2D Linear Dirac dynamics The functions Φ j are Bloch functions at a Dirac point, i.e. a corner of the Brillouin zone.

Effective Dirac equations in honeycomb structures Dirac in 2D Linear Dirac dynamics The functions Φ j are Bloch functions at a Dirac point, i.e. a corner of the Brillouin zone. The coefficients ψ j form a global-in-time solution to the following Dirac equation � ψ 1 � � � � ψ 1 � 0 λ ( ∂ 1 + i ∂ 2 ) i ∂ t = , 0 � = λ ∈ C ψ 2 λ ( ∂ 1 − i ∂ 2 ) 0 ψ 2 � ψ 1 (0 , x ) � � ψ 1 , 0 ( x ) � � 2 . � S ( R 2 ) with initial data = ∈ ψ 2 (0 , x ) ψ 2 , 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 Φ j are Bloch functions at a Dirac point, i.e. a corner of the Brillouin zone. The coefficients ψ j form a global-in-time solution to the following Dirac equation � ψ 1 � � � � ψ 1 � 0 λ ( ∂ 1 + i ∂ 2 ) i ∂ t = , 0 � = λ ∈ C ψ 2 λ ( ∂ 1 − i ∂ 2 ) 0 ψ 2 � ψ 1 (0 , x ) � � ψ 1 , 0 ( x ) � � 2 . � S ( R 2 ) with initial data = ∈ ψ 2 (0 , x ) ψ 2 , 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 The cubic Dirac equation From NLS/GP to cubic Dirac Consider the following nonlinear Schr¨ odinger/Gross-Pitaevskii equation : i ∂ t u = ( − ∆ + V ) u + κ | u | 2 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 : i ∂ t u = ( − ∆ + V ) u + κ | u | 2 u where κ ∈ R , and V is a honeycomb potential. The effective equation around a Dirac point is (Fefferman-Weinstein ’12, formal derivation): ∂ t ψ 1 + λ ( ∂ 1 + i ∂ 2 ) ψ 2 = − i κ ( β 1 | ψ 1 | 2 + 2 β 2 | ψ 2 | 2 ) ψ 1 � (4) ∂ t ψ 2 + λ ( ∂ 1 − i ∂ 2 ) ψ 1 = − i κ ( β 1 | ψ 2 | 2 + 2 β 2 | ψ 1 | 2 ) ψ 2 with 0 � = λ ∈ C , β j > 0 and ψ = ( ψ 1 , ψ 2 ) T is a C 2 -spinor.

Effective Dirac equations in honeycomb structures The cubic Dirac equation Nonlinear Dirac dynamics 0 ( x ) = √ ε ( ψ 1 , 0 ( ε x )Φ 1 ( x ) + ψ 2 , 0 ( ε x )Φ 2 ( x )) , Let u ε ε > 0 .

Effective Dirac equations in honeycomb structures The cubic Dirac equation Nonlinear Dirac dynamics 0 ( x ) = √ ε ( ψ 1 , 0 ( ε x )Φ 1 ( x ) + ψ 2 , 0 ( ε x )Φ 2 ( x )) , Let u ε ε > 0 . Theorem (Arbunich,Sparber ’16) Consider the equation i ∂ t u = ( − ∆ + V ) u + κ | u | 2 u , and let s > 1 , S > 3. There exists T ε ∼ ε − 1 , s.t. the solution u ε ∈ C 0 ([0 , T ε ) , H s ( R 2 )) of the equation with u ε (0 , x ) = u ε 0 ( x ) is of the form   2 √ εψ j ( ε t , ε x )Φ j ( x ) + η ε ( t , x ) � u ε ( t , x ) = e − i µ ∗ t  ,  j =1 provided that ψ = ( ψ 1 , ψ 2 ) T ∈ C 0 ([0 , T ε ) , H S ( R 2 , C 2 )) is a solution of (4). In this case the approximation is valid on a time interval O ( ε − 1 ).