Tight-Binding Reduction for Continuum IQHE Models Jacob Shapiro (ongoing project with Michael I. Weinstein) Venice 2019 - Quantissima in the Serenissima III August 22, 2019 J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
Motivation for studying continuum models Study similarities as well as discrepancies between continuum and discrete space models for topological matter. J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
Motivation for studying continuum models Study similarities as well as discrepancies between continuum and discrete space models for topological matter. Justify the ubiquitous usage of discrete models for studying topological matter (if one takes the continuum setting as the more fundamental one, coming from first principles). J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
Motivation for studying continuum models Study similarities as well as discrepancies between continuum and discrete space models for topological matter. Justify the ubiquitous usage of discrete models for studying topological matter (if one takes the continuum setting as the more fundamental one, coming from first principles). Allow for disorder effects (which are essential), as well as edge effects, so no argument in the process may rely on Bloch decomposition or Wannier functions. J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
Motivation for studying continuum models Study similarities as well as discrepancies between continuum and discrete space models for topological matter. Justify the ubiquitous usage of discrete models for studying topological matter (if one takes the continuum setting as the more fundamental one, coming from first principles). Allow for disorder effects (which are essential), as well as edge effects, so no argument in the process may rely on Bloch decomposition or Wannier functions. Ideally be able to deal with both large and small magnetic field strength in the semiclassical limit. J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
Today Reduction of the (scaled) lowest band of a continuum 2D IQHE model to a scale-free N.N. tight-binding model on ℓ 2 ( Z 2 ) or ℓ 2 ( Z × N ), in the sense of norm-resolvent convergence. J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
Ultimate goal Theorem σ Hall ( H ) = σ Hall ( H TB ) J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
Ultimate goal Theorem σ Hall ( H ) = σ Hall ( H TB ) ˆ H H R 2 → R × (0 , ∞ ) � tight binding reduction tight binding reduction Z 2 → Z × N ˆ H TB H TB J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
Previous works [Bellissard ’87], [Helffer-Sj¨ ostrand ’89, ...], [Nenciu ’90, ...], [Carlsson ’90], [Panati-Spohn-Teufel ’03], [de Nittis-Panati ’10], [Freund-Teufel ’16], more... J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
Previous works [Bellissard ’87], [Helffer-Sj¨ ostrand ’89, ...], [Nenciu ’90, ...], [Carlsson ’90], [Panati-Spohn-Teufel ’03], [de Nittis-Panati ’10], [Freund-Teufel ’16], more... [Fefferman-Lee-Thorp-Weinstein ’17]: The the graphene tight-binding reduction is made for translation invariance models respecting the Dirac point. We follow the philosophy of this approach closely. J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
Strategy: LCAO Study ground state ϕ of a single magnetic well in R 2 . lower bd. on inf σ (Π ⊥ H Π ⊥ ), Estimate the N.N. hopping term, |� ϕ, H ϕ e 1 �| from above full crystal on complement of and below. span of ( ϕ x ) x , the (magnetic) translates of ϕ . Schur complement for � Π H Π Π H Π ⊥ � H = . Π ⊥ H Π Π ⊥ H Π ⊥ J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
The single well v � x � − 1 Single well: for b , λ > 0 field strengths, define 2 bA ) 2 + λ 2 v ( X ) on L 2 ( R 2 ) where A := 1 h ≡ h ( b , λ ) := ( P − 1 2 e 3 ∧ X mag. vector pot. (symm. gauge), P , X are the mom. and pos. op. resp., v : R 2 → [ − 1 , 0] is a smooth well shape of unique minimum at the origin with diam(supp( v )) suff. small compared with lattice const. and fixed in advance. So h is a spatially compact perturbation of the Landau Hamiltonian. J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
The single well v � x � − 1 Single well: for b , λ > 0 field strengths, define 2 bA ) 2 + λ 2 v ( X ) on L 2 ( R 2 ) where A := 1 h ≡ h ( b , λ ) := ( P − 1 2 e 3 ∧ X mag. vector pot. (symm. gauge), P , X are the mom. and pos. op. resp., v : R 2 → [ − 1 , 0] is a smooth well shape of unique minimum at the origin with diam(supp( v )) suff. small compared with lattice const. and fixed in advance. So h is a spatially compact perturbation of the Landau Hamiltonian. Tight-binding regime: λ → ∞ (or � → 0 semiclassical regime), however, also have to decide how b scales w.r.t. λ , if at all. J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
The single well v � x � − 1 Single well: for b , λ > 0 field strengths, define 2 bA ) 2 + λ 2 v ( X ) on L 2 ( R 2 ) where A := 1 h ≡ h ( b , λ ) := ( P − 1 2 e 3 ∧ X mag. vector pot. (symm. gauge), P , X are the mom. and pos. op. resp., v : R 2 → [ − 1 , 0] is a smooth well shape of unique minimum at the origin with diam(supp( v )) suff. small compared with lattice const. and fixed in advance. So h is a spatially compact perturbation of the Landau Hamiltonian. Tight-binding regime: λ → ∞ (or � → 0 semiclassical regime), however, also have to decide how b scales w.r.t. λ , if at all. Denote by ( ϕ, e ) the ground state eigenpair of h . J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
A-priori bounds on decay of ϕ at infinity Estimates on kernel of ( P 2 + 1 ) − 1 show that | ϕ ( x ) | ≤ C e − c λ � x � away from the origin (also obtained via Agmon-estimates). J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
A-priori bounds on decay of ϕ at infinity Estimates on kernel of ( P 2 + 1 ) − 1 show that | ϕ ( x ) | ≤ C e − c λ � x � away from the origin (also obtained via Agmon-estimates). This may be boosted to | ϕ ( x ) | ≤ C e − cb � x � 2 using kernel estimates on (( P − bA ) 2 + 1 ) − 1 . (cf. [Erd˝ os ’96; Nakamura ’96]) J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
A-priori bounds on decay of ϕ at infinity Estimates on kernel of ( P 2 + 1 ) − 1 show that | ϕ ( x ) | ≤ C e − c λ � x � away from the origin (also obtained via Agmon-estimates). This may be boosted to | ϕ ( x ) | ≤ C e − cb � x � 2 using kernel estimates on (( P − bA ) 2 + 1 ) − 1 . (cf. [Erd˝ os ’96; Nakamura ’96]) Presently we do not have a pointwise lower bound on | ϕ ( x ) | as � x � → ∞ for b ∼ λ , which would be useful for tunneling estimates. J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
The harmonic approximation v has a unique min., so as λ → ∞ we may gain info about h via the 2 ω 2 x 2 for some ω . So harmonic approx.: v ( x ) + 1 ≈ v har ( x ) := 1 define h har := ( P − bA ) 2 + λ 2 v har ( X ) J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
The harmonic approximation v has a unique min., so as λ → ∞ we may gain info about h via the 2 ω 2 x 2 for some ω . So harmonic approx.: v ( x ) + 1 ≈ v har ( x ) := 1 define h har := ( P − bA ) 2 + λ 2 v har ( X ) Following [Simon ’83; Matsumoto ’95], with the unitary scaling op. 2 ψ ( √ α x ) find d ( U α ψ )( x ) := α 1 λ h har U λ = ( P − b λ A ) 2 + v har ( X ) λ U ∗ so if we take b λ → 1 we get a scale-free harmonic osc. in a const. magnetic field, and hence, for any N ∈ N , explicit asymptotics of the lowest N eigenvectors / eigenvalues of h for λ large: e j ∼ − λ 2 + λ ˜ e j ( j = 1 , . . . , N ) J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
The harmonic approximation v has a unique min., so as λ → ∞ we may gain info about h via the 2 ω 2 x 2 for some ω . So harmonic approx.: v ( x ) + 1 ≈ v har ( x ) := 1 define h har := ( P − bA ) 2 + λ 2 v har ( X ) Following [Simon ’83; Matsumoto ’95], with the unitary scaling op. 2 ψ ( √ α x ) find d ( U α ψ )( x ) := α 1 λ h har U λ = ( P − b λ A ) 2 + v har ( X ) λ U ∗ so if we take b λ → 1 we get a scale-free harmonic osc. in a const. magnetic field, and hence, for any N ∈ N , explicit asymptotics of the lowest N eigenvectors / eigenvalues of h for λ large: e j ∼ − λ 2 + λ ˜ e j ( j = 1 , . . . , N ) Actually all one needs is e ≡ e 1 ∼ − λ 2 and e 2 − e 1 ≥ C , indep. of λ , so can take that as hypothesis. J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
The full crystal We define the full crystal Hamiltonian on L 2 ( R 2 ) as H ≡ H G := ( P − bA ) 2 + λ 2 � v ( X − x ) x ∈ G where G is Z 2 (bulk) or Z × N (edge). One could also make λ depend on space (e.g. i.i.d. { λ x } x ∈ G ⊆ [ λ − , λ + ]). J. Shapiro (Columbia U.) Tight-Binding Reduction for the IQHE August 22, 2019
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