ν=0 Quantum Hall state in Bilayer graphene: collective modes

Transcription

1 ν= Quantum Hall state in Bilayer graphene: collective modes Bilayer graphene: Band structure Quantum Hall effect ν= state: Phase diagram Time-dependent Hartree-Fock approximation Neutral collective excitations J. R. M. de Nova, I. Z. & E. Demler

2 Band structure I

3 Band structure I-bis 3

4 Band structure II 4

5 Band structure III 5

6 Hamiltonian: No interactions Interlayer voltage ε V Zeeman term ε Z QH in bilayer graphene I ˆ ˆ ˆ B V zz Z z H d x ( x) H T ( x) H T B a B B ( ab ) 1 KK AB ij i j s eb 1 B 6.81 B[ T] Hz m l B y ix 5.7 ab, lb nm eb B [ T] ˆ ( ) ˆ KA x ( x) KA ˆ ( ) ˆ ( ) KB KB ˆ ( ) x x x, ˆ ( x) ˆ ( ) KB KA x ˆ ( ) ˆ KA x ( ) KB x 6

7 Orbital spin-valley decomposition:, ( x) ( x) n, k n, k QH in bilayer graphene II Orbital wave functions (Landau gauge) χ are spinors in valley-spin. Shown spinor is in sub-lattice QN n=landau level index n, 1 iky e 1 sgn n n ( x kl ) B nk, ( x) L n ( x klb) y n sgn n n ( n 1) B ˆ ( ) ˆ KA x ( x) KA ˆ ( ) ˆ ( ) KB KB ˆ ( ) x x x, ˆ ( x) ˆ ( ) KB KA x ˆ ( ) ˆ KA x ( ) KB x ν= state: ZLL, n= and 1 iky nk, ( x) L n ( x klb) y n e 7

8 ZLL projection ˆ ( x) ( x) cˆ n,1 k, n, k, n, k, QH in bilayer graphene III () 1 Hˆ d xˆ ( x) ˆ ˆ ˆ ˆ ˆ VTz Z z ( x) d d :[ ( ) ( )] ( )[ ( ) ( )]: x x x x V x x x x 14 d g :[ ˆ ( ) ˆ( )] : d d ˆ ( ) (, ) ˆ i Ti V DS ( ) m x x x x x x x x x i T g g g KK i i x y z, g HF homogeneous and Hund s rule ansatz half-filling GS cˆ ˆ ˆ ˆ kac1 kackbc1 kb k HF equations for spin-valley spinors (a b filled, c d empty levels) F F P T u [tr( PT )] T T PT n n, n V z Z z i i i i i i F F P, n n a a b b n,( a, b) ( a, b) n,( c, d ) ( c, d ) u 4 g F (3 / ) F F (5 / 4) F i B i 1 F e c l B 8

9 occupied, ab, c,d Ferromagnetic (F): VALLEY SPIN a z z empty n s, n s b z z n s, n s c u u z z d z z z V Z Full layer-polarized (FLP): n s, n s a z z b z z n s, n s c z z d z z Phase diagram I Canted-Antiferromagnetic (CAF): n s, n s a z a b z n s, n s s c [ sin cos, sin sin, cos ] a, b s s s s s cos u z s z a d z b Z / u V Z u u u u Partially layer-polarized (PLP): n s, n s n [sin cos,sin sin, cos ] cos / ( u z n s, n s a z b z c v z v v v v z u d ) v b z v V u u * * (, z ) ( Z /, V Z / ) Actual graphene: u u, u, u ~.1 z z B 9

10 Phase diagram II 1

11 Hartree-Fock (Thouless approach): ˆ 1 H ( H ) cˆ cˆ V cˆ cˆ cˆ cˆ sp lk l k lk, jm l j m k l, k l, k, j, m N 1 e w ˆ Slater det. ansatz occupied c cˆ cˆ Small variations empty Hˆ cˆ cˆ Hˆ HF equations 1 ( Hˆ EHF ) W XW w N A W, X * * w A N HF stability quadratic form N, thus X Hermitian N cˆ cˆ ( Hˆ E ) cˆ cˆ ( ) V V A, HF,,, cˆ cˆ cˆ cˆ ( Hˆ E ) cˆ cˆ cˆ, HF ĉ Hˆ V V,, 11

12 TD Hartree-Fock (Thouless approach) : ˆ 1 H ( H ) cˆˆ c V cˆ cˆ cˆ cˆ sp lk l k lk, jm l j m k l, k l, k, j, m N 1 ˆ c Slater det. ansatz occupied E HF i t w() t cˆ cˆ ( t) f ( t) e e Small variations empty ˆ d ( t) H i ( t) Variational principle dt w ( t) u e v e int * int n, n, n N A un un Bosonic Bogoliubov-de Gennes * n A N v n v n 1

13 Conclusions from BdG analogy: There is a BdG Scalar product Stable system: positive energy modes carry positive normalization and vice versa Modes come in pairs ω, -ω* If A=, there are no dynamical instabilities (DI) CAF and PLP phases are candidates to DI, but before that they are energetically unstable F and FLP have no DI No DI if the stability matrix is positive Continuous degeneracy of non-symmetric modes are gapless The following mode analysis ensures that Hund s rule states are local minima 13

14 Collective neutral excitations I Magneto-exciton (MO) wave functions 1 M e c c N ˆ iq kxlb ( ) ˆ ˆ nn k ky 1 M e c N B q ky n, q, n, q, ˆ iq kxlb ( ) ˆ ˆ n n k k c y ab, Occupied HF levels B q ky n, q, n, q, k momentum of the magnetoexciton (conserved) n,1 Landau orbital k y q Landau gauge momentum, ' K, K c,d Empty HF levels ˆ ( k, t) M ( k, t) e Mˆ ( k, t) u ( k) Mˆ ( k) e v ( k) Mˆ ( k) e i( k) t * i( k) t 1 u e u p p i kxb l ( k) p p, ky NB p, p 1 v e v p p i kxb l ( k) p p, k, y NB p, p 14

15 Magneto-exciton (MO) symmetries Lie algebras generators 1 Sˆ ˆ ˆ i cn, p,, ( i ) cn, p,, np,,, ˆ 1 L cˆ ( T ) cˆ i n, p,, i, n, p,, np,,, ˆ 1 O cˆ ( ) cˆ i n, p,, i nn n, p,, p,, n, n MO transformation properties (here shown spin, similar valley and orbital) ˆ ˆ ˆ [ S, M ( k)] ( G ) M ( k) i n n i, n n 1 ( G ) ( ) ( ) i, i i Collective neutral excitations II MO spin triplet: Mˆ ( k) Mˆ ( k) nn11 n n 1 Mˆ ˆ ˆ nn1( k) M n n ( ) M n n ( ) k k Mˆ ( k) Mˆ ( k) nn11 n n MO spin singlet: 1 Mˆ ˆ ˆ nn( k) M n n ( ) M n n ( ) k k H conmutes with: S, L, Sˆ, Lˆ z z MO diagonal in OO z (k = ) Isotropic system: => no direction dependence k : k 15

16 F phase: Ground state: L, L z =, S, S z = 4N B / MO: S = 1, S = 1 z LL z (k) ω OOz Collective neutral excitations III Spin-flip: L = 1, L z = ω 1 k = = F-CAF (Goldstone) Full-flip: L = 1 L z = ±1 ω 11 k = = F-FLP (gapless) Continuous degeneracy of states: ( ) cos( ) exp itan Mˆ z z ( k ) () ( / ) FM NB OO, LL 11 FM FLP Indices: Valley L, L z Spin S, S z Orbital pseudo-spin O, O z (k= prolongation of the MO) M k ω OOz ω k = < ω 11 k = = M k = <ω 1 k = =ω 1 1 ω M k = = marks the phase Boundaries 16

17 Collective neutral excitations IV N =,1,,3 => OO z =,11,1 1,1 17

18 Collective neutral excitations V CAF phase: Ground state: L z = MO: L z Prolongation of F spin-flip MO: L ω z =,± OOz (k) L z =,± k = = CAF-F (Goldstone) ω Hybridized prolongation of F full-flip MO: L ω z =±1 OOz (k) ω 11 k = = CAF-PLP (gapless) Can be complex inside the PLP stable region Indices: Valley L, L z Spin S, S z Orbital pseudo-spin O, O z (k= prolongation of the MO) M k ω OOz ω k = < ω 11 k = = M k = <ω 1 k = =ω 1 1 ω M k = = marks the phase boundaries 18

19 Collective neutral excitations VI N =,1,,3 => OO z =,11,1 1,1 19

20 Collective neutral excitations VII PLP phase: Ground state: S, S z =, L, L n = 4N B / MO: L = 1, L n = 1 SS z (k) ω OOz Prolongation of FLP valley-flip MO: ω S OOz (k) k = = FLP-PLP (Goldstone) ω Hybridized prolongation of FLP full-flip MO: ω 1±1 OOz (k) ω 11 k = = PLP-CAF (gapless) Can be complex inside the CAF stable region Indices: Valley L, L z Spin S, S z Orbital pseud-spin O, O z (k= prolongation of the MO) M k ω OOz ω k = < ω 11 k = = M k = <ω 1 k = =ω 1 1 ω M o k = = marks the phase boundaries