Asymptotics for Fermi curves of electric and magnetic periodic - PowerPoint PPT Presentation

Asymptotics for Fermi curves of electric and magnetic periodic fields Gustavo de Oliveira UBC – July 2009

Outline Introduction New results Comments on the proof

Lattice ◮ Γ is a lattice in R 2 : For example Γ = Z 2 .

Periodic potentials ◮ A 1 , A 2 and V are functions from R 2 to R periodic with respect to Γ . ◮ A := ( A 1 , A 2 ) is the magnetic potential. ◮ V is the electric potential.

Periodic potentials ◮ A 1 , A 2 and V are functions from R 2 to R periodic with respect to Γ . ◮ A := ( A 1 , A 2 ) is the magnetic potential. ◮ V is the electric potential.

Periodic potentials ◮ A 1 , A 2 and V are functions from R 2 to R periodic with respect to Γ . ◮ A := ( A 1 , A 2 ) is the magnetic potential. ◮ V is the electric potential.

Periodic potentials ◮ A 1 , A 2 and V are functions from R 2 to R periodic with respect to Γ . ◮ A := ( A 1 , A 2 ) is the magnetic potential. ◮ V is the electric potential.

Hamiltonian ◮ Hamiltonian H = ( i ∇ + A ) 2 + V acting on L 2 ( R 2 ) , where ∇ is the gradient on R 2 . ◮ Spectrum of H is continuous: H has no eigenfunctions in L 2 ( R 2 ) .

Hamiltonian ◮ Hamiltonian H = ( i ∇ + A ) 2 + V acting on L 2 ( R 2 ) , where ∇ is the gradient on R 2 . ◮ Spectrum of H is continuous: H has no eigenfunctions in L 2 ( R 2 ) .

Translational symmetry ◮ But H commutes with translations: for all γ ∈ Γ , HT γ = T γ H where T γ : ϕ ( x ) �→ ϕ ( x + γ ) .

Bloch theory ◮ Hence there are simultaneous eigenvectors for { H and T γ for all γ ∈ Γ } H ϕ n , k = E n ( k ) ϕ n , k , T γ ϕ n , k = e ik · γ ϕ n , k ϕ n , k ( · + γ ) = for all γ ∈ Γ , where k ∈ R 2 and n ∈ { 1 , 2 , 3 , . . . } .

Bloch theory ◮ Hence there are simultaneous eigenvectors for { H and T γ for all γ ∈ Γ } H ϕ n , k = E n ( k ) ϕ n , k , T γ ϕ n , k = e ik · γ ϕ n , k ϕ n , k ( · + γ ) = for all γ ∈ Γ , where k ∈ R 2 and n ∈ { 1 , 2 , 3 , . . . } .

Bloch theory ◮ Hence there are simultaneous eigenvectors for { H and T γ for all γ ∈ Γ } H ϕ n , k = E n ( k ) ϕ n , k , T γ ϕ n , k = e ik · γ ϕ n , k ϕ n , k ( · + γ ) = for all γ ∈ Γ , where k ∈ R 2 and n ∈ { 1 , 2 , 3 , . . . } .

Bloch theory ◮ Equivalently, if we define H k := e − ik · x H e ik · x = ( i ∇ + A − k ) 2 + V , we may consider the k -family of problems ψ n , k ∈ L 2 ( R 2 / Γ) . H k ψ n , k = E n ( k ) ψ n , k for ◮ The spectrum of H k is discrete: E 1 ( k ) ≤ E 2 ( k ) ≤ · · · ≤ E n ( k ) ≤ · · ·

Bloch theory ◮ Equivalently, if we define H k := e − ik · x H e ik · x = ( i ∇ + A − k ) 2 + V , we may consider the k -family of problems ψ n , k ∈ L 2 ( R 2 / Γ) . H k ψ n , k = E n ( k ) ψ n , k for ◮ The spectrum of H k is discrete: E 1 ( k ) ≤ E 2 ( k ) ≤ · · · ≤ E n ( k ) ≤ · · ·

Bloch theory ◮ Equivalently, if we define H k := e − ik · x H e ik · x = ( i ∇ + A − k ) 2 + V , we may consider the k -family of problems ψ n , k ∈ L 2 ( R 2 / Γ) . H k ψ n , k = E n ( k ) ψ n , k for ◮ The spectrum of H k is discrete: E 1 ( k ) ≤ E 2 ( k ) ≤ · · · ≤ E n ( k ) ≤ · · ·

Bloch theory ◮ Equivalently, if we define H k := e − ik · x H e ik · x = ( i ∇ + A − k ) 2 + V , we may consider the k -family of problems ψ n , k ∈ L 2 ( R 2 / Γ) . H k ψ n , k = E n ( k ) ψ n , k for ◮ The spectrum of H k is discrete: E 1 ( k ) ≤ E 2 ( k ) ≤ · · · ≤ E n ( k ) ≤ · · ·

Bloch theory ◮ The function k �→ E n ( k ) is periodic with respect to the dual lattice Γ # := { b ∈ R 2 | b · γ ∈ 2 π Z for all γ ∈ Γ } . ◮ This framework is preserved if we complexify: k ∈ C 2 . A 1 , A 2 , V ∈ C and

Bloch theory ◮ The function k �→ E n ( k ) is periodic with respect to the dual lattice Γ # := { b ∈ R 2 | b · γ ∈ 2 π Z for all γ ∈ Γ } . ◮ This framework is preserved if we complexify: k ∈ C 2 . A 1 , A 2 , V ∈ C and

Fermi curve ◮ The real lifted Fermi curve: F λ, R := { k ∈ R 2 | E n ( k ) = λ � for some n ≥ 1 } = { k ∈ R 2 | ( H k − λ ) ϕ = 0 for some ϕ ∈ D H k \ { 0 }} . ◮ Without loss of generality � � A − A → A , V − λ → V , k → k + A . ◮ The complex lifted Fermi curve: F := { k ∈ C 2 | H k ϕ = 0 � for some ϕ ∈ D H k \ { 0 }} .

Fermi curve ◮ The real lifted Fermi curve: F λ, R := { k ∈ R 2 | E n ( k ) = λ � for some n ≥ 1 } = { k ∈ R 2 | ( H k − λ ) ϕ = 0 for some ϕ ∈ D H k \ { 0 }} . ◮ Without loss of generality � � A − A → A , V − λ → V , k → k + A . ◮ The complex lifted Fermi curve: F := { k ∈ C 2 | H k ϕ = 0 � for some ϕ ∈ D H k \ { 0 }} .

Fermi curve ◮ The real lifted Fermi curve: F λ, R := { k ∈ R 2 | E n ( k ) = λ � for some n ≥ 1 } = { k ∈ R 2 | ( H k − λ ) ϕ = 0 for some ϕ ∈ D H k \ { 0 }} . ◮ Without loss of generality � � A − A → A , V − λ → V , k → k + A . ◮ The complex lifted Fermi curve: F := { k ∈ C 2 | H k ϕ = 0 � for some ϕ ∈ D H k \ { 0 }} .

Fermi curve ◮ The real lifted Fermi curve: F λ, R := { k ∈ R 2 | E n ( k ) = λ � for some n ≥ 1 } = { k ∈ R 2 | ( H k − λ ) ϕ = 0 for some ϕ ∈ D H k \ { 0 }} . ◮ Without loss of generality � � A − A → A , V − λ → V , k → k + A . ◮ The complex lifted Fermi curve: F := { k ∈ C 2 | H k ϕ = 0 � for some ϕ ∈ D H k \ { 0 }} .

Fermi curve: properties The Fermi curve is: 1. Analytic: F = { k ∈ C 2 | F ( k ) = 0 } , � where F ( k ) is an analytic function on C 2 . 2. Periodic with respect to Γ # : F + b = � � for all b ∈ Γ # . F 3. Gauge invariant: � F is invariant under A → A + ∇ Ψ .

Fermi curve: properties The Fermi curve is: 1. Analytic: F = { k ∈ C 2 | F ( k ) = 0 } , � where F ( k ) is an analytic function on C 2 . 2. Periodic with respect to Γ # : F + b = � � for all b ∈ Γ # . F 3. Gauge invariant: � F is invariant under A → A + ∇ Ψ .

Fermi curve: properties The Fermi curve is: 1. Analytic: F = { k ∈ C 2 | F ( k ) = 0 } , � where F ( k ) is an analytic function on C 2 . 2. Periodic with respect to Γ # : F + b = � � for all b ∈ Γ # . F 3. Gauge invariant: � F is invariant under A → A + ∇ Ψ .

The free Hamiltonian ◮ Set A = 0 and V = 0. Then { e ib · x | b ∈ Γ # } is a basis of L 2 ( R 2 / Γ) of eigenfunctions of H k : H k e ib · x = ( i ∇ − k ) 2 e ib · x = ( − b − k ) 2 e ib · x =: N b ( k ) e ib · x = N b , 1 ( k ) N b , 2 ( k ) e ib · x where N b ,ν ( k ) := ( k 1 + b 1 ) + i ( − 1 ) ν ( k 2 + b 2 ) .

The free Hamiltonian ◮ Set A = 0 and V = 0. Then { e ib · x | b ∈ Γ # } is a basis of L 2 ( R 2 / Γ) of eigenfunctions of H k : H k e ib · x = ( i ∇ − k ) 2 e ib · x = ( − b − k ) 2 e ib · x =: N b ( k ) e ib · x = N b , 1 ( k ) N b , 2 ( k ) e ib · x where N b ,ν ( k ) := ( k 1 + b 1 ) + i ( − 1 ) ν ( k 2 + b 2 ) .

The free Hamiltonian ◮ Set A = 0 and V = 0. Then { e ib · x | b ∈ Γ # } is a basis of L 2 ( R 2 / Γ) of eigenfunctions of H k : H k e ib · x = ( i ∇ − k ) 2 e ib · x = ( − b − k ) 2 e ib · x =: N b ( k ) e ib · x = N b , 1 ( k ) N b , 2 ( k ) e ib · x where N b ,ν ( k ) := ( k 1 + b 1 ) + i ( − 1 ) ν ( k 2 + b 2 ) .

The free Hamiltonian ◮ Set A = 0 and V = 0. Then { e ib · x | b ∈ Γ # } is a basis of L 2 ( R 2 / Γ) of eigenfunctions of H k : H k e ib · x = ( i ∇ − k ) 2 e ib · x = ( − b − k ) 2 e ib · x =: N b ( k ) e ib · x = N b , 1 ( k ) N b , 2 ( k ) e ib · x where N b ,ν ( k ) := ( k 1 + b 1 ) + i ( − 1 ) ν ( k 2 + b 2 ) .

The free Hamiltonian ◮ Set A = 0 and V = 0. Then { e ib · x | b ∈ Γ # } is a basis of L 2 ( R 2 / Γ) of eigenfunctions of H k : H k e ib · x = ( i ∇ − k ) 2 e ib · x = ( − b − k ) 2 e ib · x =: N b ( k ) e ib · x = N b , 1 ( k ) N b , 2 ( k ) e ib · x where N b ,ν ( k ) := ( k 1 + b 1 ) + i ( − 1 ) ν ( k 2 + b 2 ) .

The free Hamiltonian ◮ Set A = 0 and V = 0. Then { e ib · x | b ∈ Γ # } is a basis of L 2 ( R 2 / Γ) of eigenfunctions of H k : H k e ib · x = ( i ∇ − k ) 2 e ib · x = ( − b − k ) 2 e ib · x =: N b ( k ) e ib · x = N b , 1 ( k ) N b , 2 ( k ) e ib · x where N b ,ν ( k ) := ( k 1 + b 1 ) + i ( − 1 ) ν ( k 2 + b 2 ) .

The free Fermi curve ◮ Define N b := { k ∈ C 2 | ( k 1 + b 1 ) 2 + ( k 2 + b 2 ) 2 = 0 } , N ν ( b ) := { k ∈ C 2 | ( k 1 + b 1 ) + i ( − 1 ) ν ( k 2 + b 2 ) = 0 } . Hence, for A = 0 and V = 0, F = { k ∈ C 2 | N b ( k ) = 0 � for some b ∈ Γ # } � � = N b = N ν ( b ) . b ∈ Γ # b ∈ Γ # ν ∈{ 1 , 2 }

The free Fermi curve ◮ Define N b := { k ∈ C 2 | ( k 1 + b 1 ) 2 + ( k 2 + b 2 ) 2 = 0 } , N ν ( b ) := { k ∈ C 2 | ( k 1 + b 1 ) + i ( − 1 ) ν ( k 2 + b 2 ) = 0 } . Hence, for A = 0 and V = 0, F = { k ∈ C 2 | N b ( k ) = 0 � for some b ∈ Γ # } � � = N b = N ν ( b ) . b ∈ Γ # b ∈ Γ # ν ∈{ 1 , 2 }

The free Fermi curve ◮ Define N b := { k ∈ C 2 | ( k 1 + b 1 ) 2 + ( k 2 + b 2 ) 2 = 0 } , N ν ( b ) := { k ∈ C 2 | ( k 1 + b 1 ) + i ( − 1 ) ν ( k 2 + b 2 ) = 0 } . Hence, for A = 0 and V = 0, F = { k ∈ C 2 | N b ( k ) = 0 � for some b ∈ Γ # } � � = N b = N ν ( b ) . b ∈ Γ # b ∈ Γ # ν ∈{ 1 , 2 }