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Effective Dirac Equations in Honeycomb Structures

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Effective Dirac Equations in Honeycomb Structures 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; +1) 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; +1) 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; +1) 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 2 C 1(R2; R) is called a honeycomb potential if there 2 exists x0 2 R such that V~(x) := V (x − x0) satisfies: V~ is Λ-periodic: V~(x + v) = V~(x); 8x 2 R2; 8v 2 Λ; V~ is even: V~(−x) = V~(x); 8x 2 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∗ := fk 2 R2jk · v 2 2πZ; 8v 2 Λg. 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∗ := fk 2 R2jk · v 2 2πZ; 8v 2 Λg. 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 2 C 1(R2; R) is called a honeycomb potential if there 2 exists x0 2 R such that V~(x) := V (x − x0) satisfies: V~ is Λ-periodic: V~(x + v) = V~(x); 8x 2 R2; 8v 2 Λ; V~ is even: V~(−x) = V~(x); 8x 2 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 2 C 1(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 2 C 1(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 2 N. Then the linear Schr¨odingerequation i@t u = (−∆ + V )u has a unique solution of the form 0 1 2 p " −iµ∗t X " u (t; x) = e @ " j ("t;"x)Φj (x) + η (t; x)A (3) j=1 " " " with u (0; x) = u0(x); η (0; x) = 0. For any jβj ≤ N we have β " "!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∗: " p u0(x) = "( 1;0("x)Φ1(x) + 2;0("x)Φ2(x)); " > 0 (2) Effective Dirac equations in honeycomb structures Dirac in 2D Linear Dirac dynamics Consider a wave packet spectrally concentrated around K∗: " p u0(x) = "( 1;0("x)Φ1(x) + 2;0("x)Φ2(x)); " > 0 (2) Theorem (Fefferman, Weinstein '13) Fix ρ > 0,δ > 0,N 2 N. Then the linear Schr¨odingerequation i@t u = (−∆ + V )u has a unique solution of the form 0 1 2 p " −iµ∗t X " u (t; x) = e @ " j ("t;"x)Φj (x) + η (t; x)A (3) j=1 " " " with u (0; x) = u0(x); η (0; x) = 0. For any jβj ≤ N we have β " "!0 sup k@ η (t; x)k 2 2 −−−! 0 x Lx (R ) 0≤t≤ρε−2+δ . The coefficients j form a global-in-time solution to the following Dirac equation 1 0 λ(@1 + i@2) 1 i@t = ; 0 6= λ 2 C 2 λ(@1 − i@2) 0 2 (0; x) (x) with initial data 1 = 1;0 2 S(R2)2. 2(0; x) 2;0(x) The parameter λ 2 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 j form a global-in-time solution to the following Dirac equation 1 0 λ(@1 + i@2) 1 i@t = ; 0 6= λ 2 C 2 λ(@1 − i@2) 0 2 (0; x) (x) with initial data 1 = 1;0 2 S(R2)2. 2(0; x) 2;0(x) The parameter λ 2 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.
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