On dispersion of wave packets in Dirac materials V t Jakubsk y in - PowerPoint PPT Presentation

On dispersion of wave packets in Dirac materials V´ ıt Jakubsk´ y in collaboration with Matˇ ej Tuˇ sek arXiv:1604.00157 Nuclear Physics Institute of the CAS, Czech Republic QMATH13, Atlanta October 8th, 2016

Dirac materials ◮ systems (mostly in cond.mat.), where low-energy spectrum has linear dependence on the momentum - dynamics is well approximated by 2D or 1D Dirac equation! 2D stationary equation (for interactions changing smoothly on the interatomic distance and preserving spin) � V ( x , y ) + M ( x , y ) � Π x − i Π y Ψ = E Ψ Π x + i Π y V ( x , y ) − M ( x , y ) where Π x = − i ∂ x + A x ( x , y ) , Π y = − i ∂ y + A y ( x , y ) ◮ interesting toy for mathematical physicists! Relevant in description of surprising variety of physical systems ◮ Andreev approximation of BdG equations of superconductivity, high-temperature d-wave superconductors, superfluid phases of 3 He ◮ low-dimensional models in quantum field theory ( GN ,...) ◮ condensed matter systems where low-energy quasi-particles behave like massless Dirac fermions

Graphene and its cousins ◮ graphene, silicene, germanene, stanene, h-BN, dichalcogenides Trivedi, J. Comp. Theor. NanoSci. 11, 1 (2014) dichalcogenides low-energy approximation of TBM of hexagonal lattice with nearest neighbor interaction, Hasegawa, PRB74, 033413

Artificial graphene ◮ artificial graphene - ultracold atoms in optical lattices, CO molecules assembled on copper surface, drilling holes in hexagonal pattern in plexiglass... Manoharan, Nature 483, 306 (2012) Tarruell, Nature 483, 302 (2012) Torrent,PRL108,174301 Dirac materials - rapidly expanding ZOO of physical systems!

1D Dirac Hamiltonian - qualitative spectral analysis Spectral properties of the Hamiltonian h = ( − i σ 1 ∂ x + W ( x ) σ 2 + M σ 3 ) with x →±∞ W ′ ( x ) = 0 , x →±∞ W ( x ) = W ± , lim lim | W − | ≤ | W + | . Sufficient conditions for existence of bound states in the spectrum ( VJ,D.Krejˇ rik Ann.Phys.349,268 (2014) ), e.g.: ciˇ ”When � ∞ ( W 2 − W 2 − ) < 0 , −∞ then the Hamiltonian has at least one bound state with the energy � � � � W 2 W 2 − + M 2 , − + M 2 E ∈ − . ” Question: What kind of observable phenomena can be attributed to the bound states? Let’s make wave packets!

Absence of dispersion in the systems with translational invariance Mathematical abstraction of the setting (forget about Dirac operator for now): ◮ translational invariance Let’s have a (generic) Hamiltonian H ( x , y ) that commutes with the generator of translations ˆ k y = − i � ∂ y , [ H ( x , y ) , ˆ k y ] = 0 . After the partial Fourier transform F y → k , the action of the Hamiltonian can be written as (direct integral decomposition) � i H ( x , y ) ψ ( x , y ) = (2 π � ) − 1 / 2 � ky H ( x , k ) ψ ( x , k ) dk , e R where H ( x , k ) = F y → k H ( x , y ) F − 1 y → k , and � e − i ψ ( x , k ) = F y → k ψ ( x , y ) = (2 π � ) − 1 / 2 � ky ψ ( x , y ) dy . R

◮ discrete energies for fiber operators Assume H ( x , k ) has a non-empty set of discrete eigenvalues E n ( k ) for each k ∈ J n ⊂ R . The associated normalized bound states F n ( x , k ) satisfy ( H ( x , k ) − E n ( k )) F n ( x , k ) = 0 , k ∈ J n . We make a “linear combination” composed of F n ( x , k ) with fixed n � i Ψ n ( x , y ) = (2 π � ) − 1 / 2 � ky β n ( k ) F n ( x , k ) dk e I n where β n ( k ) = 0 for all k / ∈ I n ⊂ J n . Ψ n is normalized as long I n | β n ( k ) | 2 dk = 1. � as

◮ Suppose that E n ( k ) is linear on I n , E n ( k ) = e n + v n k , k ∈ I n . Then Ψ n evolves with a uniform speed without any dispersion, e − i � H ( x , y ) t Ψ n ( x , y ) = c n ( t )Ψ n ( x , y − v n t ) , | c n ( t ) | = 1 . Indeed, we have � e − i � ky e − i i � H ( x , y ) t Ψ n ( x , y ) = (2 π � ) − 1 / 2 � H ( x , k ) t ( β n ( k ) F n ( x , k )) e I n � = e − i i � e n t (2 π � ) − 1 / 2 � k ( y − v n t ) β n ( k ) F n ( x , k ) dk e I n = e − i � e n t Ψ n ( x , y − v n t ) . ◮ independent on the actual form of H ( x , k ) ◮ can be generalized to higher-dimensional systems with the translational symmetry ◮ simple observation relevant for Dirac materials!

Realization of dispersionless wave packets Linear dispersion relation - hard to get with Schr¨ odinger operator, but available in Dirac systems We fix the Hamiltonian in the following form � � − i � σ 1 ∂ x − i � σ 2 ∂ y + γ 0 H ( x , y ) = v F τ 3 ⊗ m ( x ) σ 3 , v F whose fiber operator reads � � − i � σ 1 ∂ x + k σ 2 + γ 0 H ( x , k ) = v F τ 3 ⊗ v F m ( x ) σ 3 . Structure of bispinors: Ψ = ( ψ K , A , ψ K , B , ψ K ′ , B , ψ K ′ , A ) T Topologically nontrivial mass term: lim x →±∞ m ( x ) = m ± , m + m − < 0 Mass term arises when sublattice symmetry is broken Drummond et al, PRB 85, 075423 (2012)

Realization of dispersionless wave packets Linear dispersion relation - hard to get with Schr¨ odinger operator, but available in Dirac systems! We fix the Hamiltonian in the following form � � − i � σ 1 ∂ x − i � σ 2 ∂ y + γ 0 H ( x , y ) = v F τ 3 ⊗ m ( x ) σ 3 , v F whose fiber operator reads � � − i � σ 1 ∂ x + k σ 2 + γ 0 H ( x , k ) = v F τ 3 ⊗ v F m ( x ) σ 3 . Structure of bispinors: Ψ = ( ψ K , A , ψ K , B , ψ K ′ , B , ψ K ′ , A ) T Topologically nontrivial mass term: lim x →±∞ m ( x ) = m ± , m + m − < 0 Mass term arises when sublattice symmetry is broken Drummond et al, PRB 85, 075423 (2012)

Then H ( x , k ) has two nodeless bound states localized at the domain wall where the mass changes sign Semenoff, PRL 101,87204 (2008) . � x − γ 0 F + ( x ) ≡ F 0 ( x , k ) = (1 , i , 0 , 0) T e 0 m ( s ) ds , � vF � x − γ 0 F − ( x ) ≡ τ 1 ⊗ σ 2 F + ( x ) = (0 , 0 , 1 , i ) T e 0 m ( s ) ds . � vF They satisfy H ( x , k ) F ± ( x ) = ± v F kF ± ( x ) . As F ± ( x ) do not depend on k , the nondispersive wave packet can be written as Ψ ± ( x , y ) = F ± ( x ) G ± ( y ) , where G ± ( y ) are arbitrary square integrable functions ◮ There are two counterpropagating dispersionless wave packets, one for each Dirac point (valleytronics)

Slowly dispersing wave packets Assume the dispersion relation E = E ( k ) is not linear. We define B ( k ) = E n ( k ) − ( e + vk ) , k ∈ I n , where e and v are free parameters so far. We are interested in the transition probability A ( t ) = |� Ψ n ( x , y − vt ) , e − i � H ( x , y ) t Ψ n ( x , y ) �| 2 � ( B ( k ) − B ( s )) t dkds | β n ( k ) | 2 | β n ( s ) | 2 cos � � = � I n × I n Let us find the lower bound of A ( t ) ( B ( k ) − B ( s )) t � � A ( t ) ≥ inf cos � ( k , s ) ∈ I n × I n ≥ 1 − t 2 ( B ( k ) − B ( s )) 2 ≥ 1 − 2 t 2 | B ( k ) | 2 . sup � 2 sup 2 � 2 ( k , s ) ∈ I n × I n k ∈ I n � In E ′ n ( k ) dk = E n ( b ) − E n ( a ) We set average speed v = , and e such b − a b − a that sup k ∈ I n ( E n ( k ) − vk − e ) = − inf k ∈ I n ( E n ( k ) − vk − e ).

Example The fiber Hamiltonian is ˜ H K ( x , k ) = − i σ 1 ∂ x − ωα tanh( α x ) σ 2 + k σ 3 . The solutions of stationary equation are H K ( x , k )˜ ˜ ± E n ( k )˜ F ± F ± n ( x , k ) = n ( x , k ) , � � f n ( x ) � � ˜ � 1 0 H K ( x , 0) � ˜ F ± n ( x , k ) = 1 + , ǫ ± ( k , n ) E n (0) 2 0 0 � n ( − n + 2 ω ) α 2 + k 2 E n ( k ) = E n (0) where we denoted ǫ ± ( k , n ) = ± √ E n (0) 2 + k 2 + k and � 1 � sech − n + ω ( α x ) 2 F 1 f n ( x ) = − n , 1 − n + 2 ω, 1 − n + ω, . 1 + e 2 α x The zero modes are (˜ H ( x , k ) − k )˜ F + ( x ) = 0, ˜ F + ( x ) = (sech ω ( α x ) , 0) T .

� � 1 β 1 ( k ) = C b exp − , β 1 ( k ) = 0 for k � = ( c − b , c + b ). b 2 − ( k − c ) 2 � � ˜ e iky β 1 ( k )˜ Ψ + = ˜ ˜ F + e iky β 1 ( k ) d k , Ψ 1 = 1 ( x , k ) d k , F + ( x ) I 1 I 1

Discussion and Outlook ◮ insight into experimental data (e.g. existence of slowly dispersing wave packets in bilayer graphene ”highways”) Martin et al, PRL100,036804 (2008) ◮ realization of quantum states following classical trajectories seeked already by Sch¨ odinger (free particle Berry, Am. J. Phys. 47, 264 (1979) , Trojan states for Rydberg atoms Bialnicki-Birula et al, PRL 73,1777 (1994) ) ◮ experimental preparation of the disperionless wave packets requires precise control of quantum states: achieved by laser pulses for Rydberg atoms ( Weinacht, Nature 397 (1999), 233; Verlet, Phys. Rev. Lett. (2002) 89, 263004 ) generalizations ◮ improvements of estimates for slowly dispersing wp (lower bound for transition amplitude, weighted group velocity of the packet) ◮ extension to other geometries ◮ (geometrically) imperfect systems, crossroads