Dispersionless wave packets in graphene and Dirac materials V t - PowerPoint PPT Presentation

Dispersionless wave packets in graphene and Dirac materials V´ ıt Jakubsk´ y in collaboration with Matˇ ej Tuˇ sek Nuclear Physics Institute AVˇ CR June 9, 2016

Low-dimensional Dirac equation Relevant in description of surprising variety of physical systems ◮ Andreev aproximation of BdG equations of superconductivity, high-temperature d-wave superconductors, superfluid phases of 3 He ◮ low-dimensional models in quantum field theory ( GN ,...) ◮ Dirac materials - condensed matter systems where low-energy quasi-particles behave like massless Dirac fermions

Dirac materials ◮ 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 - ultracold atoms in optical lattices, CO molecules assembled on copper surface, drilling holes in hexagonal pattern in plexiglass... Manoharan Lab Tarruell, Nature 483, 302 (2012) Torrent,PRL108,174301

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?

Absence of dispersion in the systems with translational invariance Hamiltonian H ( x , y ) 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 � 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

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 take 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 as I n | β n ( k ) | 2 dk = 1. �

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 )) d k e I n � = e − i � k ( y − v n t ) β n ( k ) F n ( x , k ) dk = e − i i � e n t (2 π � ) − 1 / 2 � e n t Ψ n ( x , y − v n t ) e I n ◮ independent on the actual form of the fiber Hamiltonian ◮ also works for higher-dimensional systems with translational symmetry

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 ⊗ m ( x ) σ 3 . v F Structure of bispinors Ψ = ( ψ K , A , ψ K , B , ψ K ′ , B , ψ K ′ , A ) T Topologically nontrivial mass term x →±∞ m ( x ) = m ± , lim m − < 0 , m + > 0 .

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 F − ( x ) ≡ τ 1 ⊗ σ 2 F + ( x ) = (0 , 0 , 1 , i ) T e − γ 0 0 m ( s ) ds . � vF They satisfy H ( x , k ) F ± ( x ) = ± v F kF ± ( x ) . The nondispersive wave packet 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 ( B ( k ) − B ( s )) 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. 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: technology available 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 (long-living quasiparticles on the ”highways” in bilayer graphene )