Ground state construction of Bilayer Graphene Ian Jauslin joint - PowerPoint PPT Presentation

Ground state construction of Bilayer Graphene Ian Jauslin joint with Alessandro Giuliani arXiv: 1507.06024 http://ian.jauslin.org

Monolayer graphene • 2D crystal of carbon atoms on a honeycomb lattice. 1/ 13

Bilayer graphene • 2 graphene layers in AB stacking. 2/ 13

Bilayer graphene • Rhombic lattice Λ ≡ Z 2 , 4 atoms per site. l 1 l 2 3/ 13

Hamiltonian • Hamiltonian: H = H 0 + UV • Non-interacting Hamiltonian: hoppings γ 0 γ 0 γ 1 γ 3 • Interaction: weak, short-range (screened Coulomb). 4/ 13

Non-interacting Hamiltonian T a †   ˆ   ˆ a k k † ˆ ˆ  ˜  ˜ b  b k  � ˆ   H 0 = k H 0 ( k )     ˆ † ˆ  a k ˜   a ˜    k ∈ ˆ k Λ   ˆ b † ˆ b k k γ 0 Ω ∗ ( k )  0 0  γ 1 γ 1 0 γ 0 Ω( k ) 0 ˆ   H 0 ( k ) :=  γ 0 Ω ∗ ( k ) γ 3 Ω( k ) e 3 ik x  0 0   γ 3 Ω( k ) e − 3 ik x γ 0 Ω( k ) 0 √ Ω( k ) := 1 + 2 e − 3 2 ik x cos( 3 2 k y ) 5/ 13

Non-interacting Hamiltonian • Hopping strengths: γ 0 = 1 , γ 1 = ǫ, γ 3 = 0 . 33 × ǫ • Experimental value ǫ ≈ 0 . 1, here, ǫ ≪ 1. 6/ 13

Interaction � � � � n x − 1 n y − 1 � V = v ( | x − y | ) 2 2 x,y � • : sum over pairs of atoms x,y • v ( | x − y | ) � e − c | x − y | , c > 0 • − 1 2 : half-filling . 7/ 13

Non-interacting Hamiltonian • Eigenvalues of ˆ H 0 ( k ): 8/ 13

Non-interacting Hamiltonian • | k | ≫ ǫ 9/ 13

Non-interacting Hamiltonian • ǫ 2 ≪ | k | ≪ ǫ 10/ 13

Non-interacting Hamiltonian • | k | ≪ ǫ 2 11/ 13

Theorem ∃ U 0 , ǫ 0 > 0, independent, such that, for ǫ < ǫ 0 , | U | < U 0 , • the free energy f := − 1 | Λ | β log T r ( e − β H ) is analytic in U , uniformly in β and | Λ | , • the two-point Schwinger function s 2 ( x − y ) := T r ( e − β H a x a † y ) T r ( e − β H ) is analytic in U , uniformly in β and | Λ | . 12/ 13

Renormalization group flow log 2 ǫ 3 log 2 ǫ 1 h 2 ǫ | log 2 ǫ | ǫ ✐rr❡❧❡✈❛♥t ♠❛r❣✐♥❛❧ ✐rr❡❧❡✈❛♥t ∼ 2 h ∼ ǫ | 2 log 2 ǫ | 2 h − 3 log 2 ǫ ∼ ǫ | h − log 2 ǫ | | W (4) | | U | 13/ 13