Introduction Building up our toolbox The Model Analysis Floquet Topological Insulator: Understanding“Floquet topological insulator in semiconductor quantum wells”by Lindner et al. Condensed Matter Journal Club Caltech February 12 2014 Floquet TIs Journal club
Introduction Building up our toolbox Motivation The Model Analysis Motivation Helical edge states require a band inversion (saw this in previous papers) *References: (RRL) 7, no. 1-2 (2013): 101-108 institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Motivation The Model Analysis Motivation Helical edge states require a band inversion (saw this in previous papers) This requires careful band structure engineering *References: (RRL) 7, no. 1-2 (2013): 101-108 institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Motivation The Model Analysis Motivation Helical edge states require a band inversion (saw this in previous papers) This requires careful band structure engineering The topological property of the material is permanent; cannot widen well or change strain (?) *References: (RRL) 7, no. 1-2 (2013): 101-108 institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Motivation The Model Analysis Motivation Helical edge states require a band inversion (saw this in previous papers) This requires careful band structure engineering The topological property of the material is permanent; cannot widen well or change strain (?) ... What about time periodic perturbations? *References: (RRL) 7, no. 1-2 (2013): 101-108 institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Motivation The Model Analysis Time periodic perturbations They can induce non-equilibrium protected edge states! institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Motivation The Model Analysis Time periodic perturbations They can induce non-equilibrium protected edge states! Controlling low-frequency electromagnetic modes is a mature technology. institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Motivation The Model Analysis Time periodic perturbations They can induce non-equilibrium protected edge states! Controlling low-frequency electromagnetic modes is a mature technology. Could allow fast switching of edge state transport and control the spectral properties (velocity) of the edge states. institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Motivation The Model Analysis Time periodic perturbations They can induce non-equilibrium protected edge states! Controlling low-frequency electromagnetic modes is a mature technology. Could allow fast switching of edge state transport and control the spectral properties (velocity) of the edge states. We will treat these perturbations classically, and to solve for the dynamics, we will need Floquet theory. institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Floquet Theory The Model Back to Chern numbers Analysis Spatial Periodicity in Hamiltonians: A familiar case Consider a system that has some discrete translation symmetry by some lattice vectors R . Then, an eigenstate, | ψ � , of the Hamiltonian of that system, ˆ H , is of the following form: R | u � | ψ k � = e − i k . ˆ (1) where | u � = � u ( r ) | r � d r is an arbitrary periodic wavefunction and k is an arbitrary wavevector called the Bloch wavevector. In addition, | u � is an eigenstate of the effective Bloch Hamiltonian institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Floquet Theory The Model Back to Chern numbers Analysis Effective Bloch Hamiltonian Let us substitute a Bloch wavefunction | ψ k � = e − i k . ˆ R | u � into the Schrodinger’s equation � 1 � 2 �� R | u � He − i k . ˆ e − i k . ˆ � � = Ee − i k . ˆ ˆ ˆ R R P − � k = + V R (2) 2 m and hence 1 � 2 � � � ˆ ˆ ˆ P − � k H bloch = + V R (3) 2 m is the effective Bloch Hamiltonian for | u � institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Floquet Theory The Model Back to Chern numbers Analysis Some Properties Let K be a reciprocal lattice vector. Hence, | ψ k + K � is the same eigenvector as | ψ k � and k is defined modulo K . institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Floquet Theory The Model Back to Chern numbers Analysis Some Properties Let K be a reciprocal lattice vector. Hence, | ψ k + K � is the same eigenvector as | ψ k � and k is defined modulo K . Mean velocity is given by v n ( k ) = 1 � ∇ k E n ( k ) (4) where n is a band index (obtained by solving ˆ H bloch ( k ) | u n � = E n ( k ) | u n � ) institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Floquet Theory The Model Back to Chern numbers Analysis Some Properties Let K be a reciprocal lattice vector. Hence, | ψ k + K � is the same eigenvector as | ψ k � and k is defined modulo K . Mean velocity is given by v n ( k ) = 1 � ∇ k E n ( k ) (4) where n is a band index (obtained by solving ˆ H bloch ( k ) | u n � = E n ( k ) | u n � ) Weak periodic potentials break degeneracies. institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Floquet Theory The Model Back to Chern numbers Analysis Some Properties Let K be a reciprocal lattice vector. Hence, | ψ k + K � is the same eigenvector as | ψ k � and k is defined modulo K . Mean velocity is given by v n ( k ) = 1 � ∇ k E n ( k ) (4) where n is a band index (obtained by solving ˆ H bloch ( k ) | u n � = E n ( k ) | u n � ) Weak periodic potentials break degeneracies. Plus one more very important property ... institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Floquet Theory The Model Back to Chern numbers Analysis Floquet theorem Consider the differential equation − i d dx ψ ( x ) = H ( x ) ψ ( x ) = ˆ P | ψ � (5) then according to the Floquet theorem we have that the unitary operator is given by U ( x ) = B ( x ) e − iH B x (6) where H B is the effective Bloch Hamiltonian and B ( x ) is a periodic function in R institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Floquet Theory The Model Back to Chern numbers Analysis From spatial periodicity to time periodicity Consider a time periodic Hamiltonian H ( t ). Its evolution is governed by: i ∂ t | ψ � = H ( t ) | ψ � (7) ω , that is the analogue as ˆ I will consider i ∂ t to be an operator, ˆ P (and so t would be the analog of ˆ R ) ω − ǫ (in analogy with ˆ P → ˆ Then ˆ ω → ˆ P − � k ), where ǫ is called a quasi energy. Hence, Schrodinger’s equation becomes (for the floquet states e i ǫ t | φ � ):( i ∂ t − ǫ ) | φ � = H ( t ) | φ � which can be rewritten as ( H ( t ) − i ∂ t ) | φ � = ǫ | φ � (8) institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Floquet Theory The Model Back to Chern numbers Analysis From spatial periodicity to time periodicity Consider a time periodic Hamiltonian H ( t ). Its evolution is governed by: i ∂ t | ψ � = H ( t ) | ψ � (7) ω , that is the analogue as ˆ I will consider i ∂ t to be an operator, ˆ P (and so t would be the analog of ˆ R ) ω − ǫ (in analogy with ˆ P → ˆ Then ˆ ω → ˆ P − � k ), where ǫ is called a quasi energy. Hence, Schrodinger’s equation becomes (for the floquet states e i ǫ t | φ � ):( i ∂ t − ǫ ) | φ � = H ( t ) | φ � which can be rewritten as ( H ( t ) − i ∂ t ) | φ � = ǫ | φ � (8) In analogy to the effective Bloch Hamiltonian, we have the effective Floquet Hamiltonian/operator: H F ( t ) ≡ H ( t ) − i ∂ t (9) institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Floquet Theory The Model Back to Chern numbers Analysis Some Properties H F ( t ) ≡ H ( t ) − i ∂ t ; H F ( t ) | φ � = ǫ | φ � (10) In analogy to Bloch wavevectors being defined modulo the reciprocal lattice vectors K , the quasi-energies are defined modulo the frequency ω = 2 π/ T (where T is the periodicity of the Hamiltonian) institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Floquet Theory The Model Back to Chern numbers Analysis Some Properties H F ( t ) ≡ H ( t ) − i ∂ t ; H F ( t ) | φ � = ǫ | φ � (10) In analogy to Bloch wavevectors being defined modulo the reciprocal lattice vectors K , the quasi-energies are defined modulo the frequency ω = 2 π/ T (where T is the periodicity of the Hamiltonian) The unitary evolution operator for | ψ ( t ) � = e − i ǫ t is given by S k ( t ) = P b ( t ) exp [ − iH F ( k ) t ] ; P k ( t ) = P ( t + T ) (11) institution-logo-filenam Floquet TIs Journal club
Introduction Building up our toolbox Floquet Theory The Model Back to Chern numbers Analysis Definitions Berry curvature: � � ∗ � � � � F n = ∇ � k ( � r ) × ∇ � k ( � r ) d � (12) k k u n ,� k u n ,� r unit cell where u n ,� k is a Bloch wave at band n . institution-logo-filenam Floquet TIs Journal club
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