Tight-Binding Reduction for Continuum IQHE Models Jacob Shapiro - - PowerPoint PPT Presentation
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
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
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.
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).
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.
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.
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 ℓ2(Z2) or ℓ2(Z × N), in the sense
Tight-Binding Reduction for the IQHE August 22, 2019
Ultimate goal
Theorem
σHall(H) = σHall(HTB)
Tight-Binding Reduction for the IQHE August 22, 2019
Ultimate goal
Theorem
σHall(H) = σHall(HTB) H ˆ H HTB ˆ HTB
tight binding reduction R2→R×(0,∞) tight binding reduction Z2→Z×N
Tight-Binding Reduction for the IQHE August 22, 2019
Previous works
[Bellissard ’87], [Helffer-Sj¨
[Carlsson ’90], [Panati-Spohn-Teufel ’03], [de Nittis-Panati ’10], [Freund-Teufel ’16], more...
Tight-Binding Reduction for the IQHE August 22, 2019
Previous works
[Bellissard ’87], [Helffer-Sj¨
[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.
Tight-Binding Reduction for the IQHE August 22, 2019
Strategy: LCAO
Study ground state ϕ of a single magnetic well in R2. lower bd. on inf σ(Π⊥HΠ⊥), full crystal on complement of span of (ϕx)x, the (magnetic) translates of ϕ. Estimate the N.N. hopping term, |ϕ, Hϕe1| from above and below. Schur complement for H = ΠHΠ ΠHΠ⊥ Π⊥HΠ Π⊥HΠ⊥
Tight-Binding Reduction for the IQHE August 22, 2019
The single well
x v
−1
Single well: for b, λ > 0 field strengths, define h ≡ h(b, λ) := (P − 1
2bA)2 + λ2v(X) on L2(R2) where A := 1 2e3 ∧ X
resp., v : R2 → [−1, 0] is a smooth well shape of unique minimum at the origin with diam(supp(v))
a spatially compact perturbation of the Landau Hamiltonian.
Tight-Binding Reduction for the IQHE August 22, 2019
The single well
x v
−1
Single well: for b, λ > 0 field strengths, define h ≡ h(b, λ) := (P − 1
2bA)2 + λ2v(X) on L2(R2) where A := 1 2e3 ∧ X
resp., v : R2 → [−1, 0] is a smooth well shape of unique minimum at the origin with diam(supp(v))
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.
Tight-Binding Reduction for the IQHE August 22, 2019
The single well
x v
−1
Single well: for b, λ > 0 field strengths, define h ≡ h(b, λ) := (P − 1
2bA)2 + λ2v(X) on L2(R2) where A := 1 2e3 ∧ X
resp., v : R2 → [−1, 0] is a smooth well shape of unique minimum at the origin with diam(supp(v))
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.
Tight-Binding Reduction for the IQHE August 22, 2019
A-priori bounds on decay of ϕ at infinity
Estimates on kernel of (P2 + 1)−1 show that |ϕ(x)| ≤ C e−cλx away from the origin (also obtained via Agmon-estimates).
Tight-Binding Reduction for the IQHE August 22, 2019
A-priori bounds on decay of ϕ at infinity
Estimates on kernel of (P2 + 1)−1 show that |ϕ(x)| ≤ C e−cλx away from the origin (also obtained via Agmon-estimates). This may be boosted to |ϕ(x)| ≤ C e−cbx2 using kernel estimates on ((P − bA)2 + 1)−1. (cf. [Erd˝
Tight-Binding Reduction for the IQHE August 22, 2019
A-priori bounds on decay of ϕ at infinity
Estimates on kernel of (P2 + 1)−1 show that |ϕ(x)| ≤ C e−cλx away from the origin (also obtained via Agmon-estimates). This may be boosted to |ϕ(x)| ≤ C e−cbx2 using kernel estimates on ((P − bA)2 + 1)−1. (cf. [Erd˝
Presently we do not have a pointwise lower bound on |ϕ(x)| as x → ∞ for b ∼ λ, which would be useful for tunneling estimates.
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 harmonic approx.: v(x) + 1 ≈ vhar(x) := 1
2ω2x2 for some ω. So
define hhar := (P − bA)2 + λ2vhar(X)
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 harmonic approx.: v(x) + 1 ≈ vhar(x) := 1
2ω2x2 for some ω. So
define hhar := (P − bA)2 + λ2vhar(X) Following [Simon ’83; Matsumoto ’95], with the unitary scaling op. (Uαψ)(x) := α
d 2 ψ(√αx) find
1 λU∗
λhharUλ = (P − b
λA)2 + vhar(X) 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: ej ∼ −λ2 + λ˜ ej (j = 1, . . . , N)
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 harmonic approx.: v(x) + 1 ≈ vhar(x) := 1
2ω2x2 for some ω. So
define hhar := (P − bA)2 + λ2vhar(X) Following [Simon ’83; Matsumoto ’95], with the unitary scaling op. (Uαψ)(x) := α
d 2 ψ(√αx) find
1 λU∗
λhharUλ = (P − b
λA)2 + vhar(X) 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: ej ∼ −λ2 + λ˜ ej (j = 1, . . . , N) Actually all one needs is e ≡ e1 ∼ −λ2 and e2 − e1 ≥ C, indep. of λ, so can take that as hypothesis.
Tight-Binding Reduction for the IQHE August 22, 2019
The full crystal
We define the full crystal Hamiltonian on L2(R2) as H ≡ HG := (P − bA)2 + λ2
x∈G
v(X − x) where G is Z2 (bulk) or Z × N (edge). One could also make λ depend
Tight-Binding Reduction for the IQHE August 22, 2019
The full crystal
We define the full crystal Hamiltonian on L2(R2) as H ≡ HG := (P − bA)2 + λ2
x∈G
v(X − x) where G is Z2 (bulk) or Z × N (edge). One could also make λ depend
e now broadens to a band of width ∼ e−cλ due to tunneling. The next band is order 1 away.
Tight-Binding Reduction for the IQHE August 22, 2019
The full crystal
We define the full crystal Hamiltonian on L2(R2) as H ≡ HG := (P − bA)2 + λ2
x∈G
v(X − x) where G is Z2 (bulk) or Z × N (edge). One could also make λ depend
e now broadens to a band of width ∼ e−cλ due to tunneling. The next band is order 1 away. With Rx the shift operator by x ∈ R2 (so (Rxψ)(y) ≡ ψ(y − x)) we have the magnetic shifts ˆ Rx := exp(− i x · bA)Rx and we define our atomic orbital basis as ϕx := ˆ Rxϕ (x ∈ G) .
Tight-Binding Reduction for the IQHE August 22, 2019
The full crystal
We define the full crystal Hamiltonian on L2(R2) as H ≡ HG := (P − bA)2 + λ2
x∈G
v(X − x) where G is Z2 (bulk) or Z × N (edge). One could also make λ depend
e now broadens to a band of width ∼ e−cλ due to tunneling. The next band is order 1 away. With Rx the shift operator by x ∈ R2 (so (Rxψ)(y) ≡ ψ(y − x)) we have the magnetic shifts ˆ Rx := exp(− i x · bA)Rx and we define our atomic orbital basis as ϕx := ˆ Rxϕ (x ∈ G) . Goal: {ϕx, Hϕx+d}x∈G,d=1 on ℓ2(G).
Tight-Binding Reduction for the IQHE August 22, 2019
The N.N. hopping coefficient
The main object of interest is the hopping coefficient: for d = 1, ρ(d) := |ϕ, (H − e1)ϕd| ≈ λ2|ϕ, v(X) ˆ Rdϕ| + . . . .
Tight-Binding Reduction for the IQHE August 22, 2019
The N.N. hopping coefficient
The main object of interest is the hopping coefficient: for d = 1, ρ(d) := |ϕ, (H − e1)ϕd| ≈ λ2|ϕ, v(X) ˆ Rdϕ| + . . . .
Theorem
ρ(d) ∼ e−λA(d) where A(d) is the minimal Euclidean action over all infinite time paths from min. @ origin to min @ d (i.e. the instanton).
Tight-Binding Reduction for the IQHE August 22, 2019
The N.N. hopping coefficient
The main object of interest is the hopping coefficient: for d = 1, ρ(d) := |ϕ, (H − e1)ϕd| ≈ λ2|ϕ, v(X) ˆ Rdϕ| + . . . .
Theorem
ρ(d) ∼ e−λA(d) where A(d) is the minimal Euclidean action over all infinite time paths from min. @ origin to min @ d (i.e. the instanton). Decay estimates on ϕ(x) give immediate upper bounds ρ(d) ≤ C e−cbd2.
Tight-Binding Reduction for the IQHE August 22, 2019
The N.N. hopping coefficient
The main object of interest is the hopping coefficient: for d = 1, ρ(d) := |ϕ, (H − e1)ϕd| ≈ λ2|ϕ, v(X) ˆ Rdϕ| + . . . .
Theorem
ρ(d) ∼ e−λA(d) where A(d) is the minimal Euclidean action over all infinite time paths from min. @ origin to min @ d (i.e. the instanton). Decay estimates on ϕ(x) give immediate upper bounds ρ(d) ≤ C e−cbd2. Lower bounds are much harder. Either via asymptotic expansion of ϕ [Simon ’83] and explicit calculation for Gaussians or approximate equality with eigenvalue splitting of magnetic double-well (studied problem, .e.g., see [Br¨ uning, Dobrokhotov, Nekrasov ’13] for different kind of double-well, [Helffer-Sj¨
Tight-Binding Reduction for the IQHE August 22, 2019
Building an ONB
Let V := span({ϕx}x∈G) and Π proj. onto V. Gramian G on ℓ2(G) via Gxy := ϕx, ϕyL2(R2), G > 0. So def. M s.t. G −1 ≡ MM∗. |Mxy| ≤ C e−cλx−y too.
Tight-Binding Reduction for the IQHE August 22, 2019
Building an ONB
Let V := span({ϕx}x∈G) and Π proj. onto V. Gramian G on ℓ2(G) via Gxy := ϕx, ϕyL2(R2), G > 0. So def. M s.t. G −1 ≡ MM∗. |Mxy| ≤ C e−cλx−y too. Hence the set of ˜ ϕx :=
y∈G ϕyMyx is an ONB for V that is also
Tight-Binding Reduction for the IQHE August 22, 2019
Building an ONB
Let V := span({ϕx}x∈G) and Π proj. onto V. Gramian G on ℓ2(G) via Gxy := ϕx, ϕyL2(R2), G > 0. So def. M s.t. G −1 ≡ MM∗. |Mxy| ≤ C e−cλx−y too. Hence the set of ˜ ϕx :=
y∈G ϕyMyx is an ONB for V that is also
Define J : V → ℓ2(G) via ˜ ϕx → δx. M := J∗MJ op. on V s.t. Mϕx = ˜ ϕx for all x ∈ G.
Tight-Binding Reduction for the IQHE August 22, 2019
The reduction process
We note (M−1)∗Π(H − e1)ΠM−1 ↔ {ϕx(H − e1)ϕy}x,y∈G
Tight-Binding Reduction for the IQHE August 22, 2019
The reduction process
We note (M−1)∗Π(H − e1)ΠM−1 ↔ {ϕx(H − e1)ϕy}x,y∈G Taking only N.N. elements of this and scaling by ρ ≡ ρ(e1) ∼ e−λ the basic asymptotic unit, we arrived at HTB, the scale-free discrete
Tight-Binding Reduction for the IQHE August 22, 2019
The reduction process
We note (M−1)∗Π(H − e1)ΠM−1 ↔ {ϕx(H − e1)ϕy}x,y∈G Taking only N.N. elements of this and scaling by ρ ≡ ρ(e1) ∼ e−λ the basic asymptotic unit, we arrived at HTB, the scale-free discrete
We need the fact that ρ−1Π(H − e1)Π − J∗HTBJ → 0 which essentially says that all N.N.N. terms and beyond, converge faster than ρ to zero.
Tight-Binding Reduction for the IQHE August 22, 2019
The reduction process
We note (M−1)∗Π(H − e1)ΠM−1 ↔ {ϕx(H − e1)ϕy}x,y∈G Taking only N.N. elements of this and scaling by ρ ≡ ρ(e1) ∼ e−λ the basic asymptotic unit, we arrived at HTB, the scale-free discrete
We need the fact that ρ−1Π(H − e1)Π − J∗HTBJ → 0 which essentially says that all N.N.N. terms and beyond, converge faster than ρ to zero. Energy estimates on Π⊥HΠ⊥ imply that (Π⊥(H − z1)Π⊥)−1 ≤ 1 for z near e. Uses IMS-localization for the magnetic case: For any partition of unity {Θj}j we have H =
Θj(X)HΘj(X) −
(∇Θj)(X)2 .
Tight-Binding Reduction for the IQHE August 22, 2019
Main theorem
Theorem
We have norm-resolvent convergence of ρ−1(H − e1) → HTB as λ → ∞. More precisely, for any z / ∈ σ(HTB), (ρ−1(H − e1) − z)−1 − (J∗HTBJ − z1)−1 ≤ C e−cλ
Tight-Binding Reduction for the IQHE August 22, 2019
Main theorem
Theorem
We have norm-resolvent convergence of ρ−1(H − e1) → HTB as λ → ∞. More precisely, for any z / ∈ σ(HTB), (ρ−1(H − e1) − z)−1 − (J∗HTBJ − z1)−1 ≤ C e−cλ Schur complement gives R(z) = S(z) −S(z)Π(H − z1)Π⊥D(z) ⋆ ⋆⋆
D(z) := (Π⊥(H − z1)Π⊥)−1.
Tight-Binding Reduction for the IQHE August 22, 2019
Main theorem
Theorem
We have norm-resolvent convergence of ρ−1(H − e1) → HTB as λ → ∞. More precisely, for any z / ∈ σ(HTB), (ρ−1(H − e1) − z)−1 − (J∗HTBJ − z1)−1 ≤ C e−cλ Schur complement gives R(z) = S(z) −S(z)Π(H − z1)Π⊥D(z) ⋆ ⋆⋆
D(z) := (Π⊥(H − z1)Π⊥)−1.
Corollary for Fermi projections
For any EF ∈ R \ σ(HTB), χ(−∞,EF )(ρ−1(H − e1)) − χ(−∞,EF )(J∗HTBJ) → 0
Tight-Binding Reduction for the IQHE August 22, 2019
The quantum Hall conductivity
When G = Z2 (bulk), σHall(H) = index PUP + P⊥ where P ≡ χ(−∞,EF )(H) is the Fermi projection and U is some fixed unitary [Bellissard et al ’94]. The previous corollary about convergence
stability of the Fredholm index together imply that the Chern number associated with any isolated sub-band of the lowest band of H agrees with the Chern number of the corresponding sub-band of HTB.
Tight-Binding Reduction for the IQHE August 22, 2019
The quantum Hall conductivity
When G = Z2 (bulk), σHall(H) = index PUP + P⊥ where P ≡ χ(−∞,EF )(H) is the Fermi projection and U is some fixed unitary [Bellissard et al ’94]. The previous corollary about convergence
stability of the Fredholm index together imply that the Chern number associated with any isolated sub-band of the lowest band of H agrees with the Chern number of the corresponding sub-band of HTB. When G = Z × N (edge), σHall( ˆ H) = index Λ1 e− i 2πg( ˆ
H) Λ1 + Λ⊥ 1
where Λ1 is a fixed projection and g is a smooth version of χ(−∞,EF ) [Kellendonk-Schulz-Baldes ’04].
Tight-Binding Reduction for the IQHE August 22, 2019
Epilogue
Accomplished: Proven equality of Hall conductivity of sub-band of scaled lowest band of continuum model with corresponding sub-band
Tight-Binding Reduction for the IQHE August 22, 2019
Epilogue
Accomplished: Proven equality of Hall conductivity of sub-band of scaled lowest band of continuum model with corresponding sub-band
Future work:
Independent study of magnetic double-well eigenvalue splitting.
Tight-Binding Reduction for the IQHE August 22, 2019
Epilogue
Accomplished: Proven equality of Hall conductivity of sub-band of scaled lowest band of continuum model with corresponding sub-band
Future work:
Independent study of magnetic double-well eigenvalue splitting. Handle N bands of the continuum operator, get a discrete matrix model on ℓ2(G) ⊗ CN.
Tight-Binding Reduction for the IQHE August 22, 2019
Epilogue
Accomplished: Proven equality of Hall conductivity of sub-band of scaled lowest band of continuum model with corresponding sub-band
Future work:
Independent study of magnetic double-well eigenvalue splitting. Handle N bands of the continuum operator, get a discrete matrix model on ℓ2(G) ⊗ CN. Handle custom boundary conditions for the edge problem in the continuum and see what they match to after the reduction.
Tight-Binding Reduction for the IQHE August 22, 2019
Epilogue
Accomplished: Proven equality of Hall conductivity of sub-band of scaled lowest band of continuum model with corresponding sub-band
Future work:
Independent study of magnetic double-well eigenvalue splitting. Handle N bands of the continuum operator, get a discrete matrix model on ℓ2(G) ⊗ CN. Handle custom boundary conditions for the edge problem in the continuum and see what they match to after the reduction. Handle disorder in either location of the wells or their coefficients.
Tight-Binding Reduction for the IQHE August 22, 2019
Epilogue
Accomplished: Proven equality of Hall conductivity of sub-band of scaled lowest band of continuum model with corresponding sub-band
Future work:
Independent study of magnetic double-well eigenvalue splitting. Handle N bands of the continuum operator, get a discrete matrix model on ℓ2(G) ⊗ CN. Handle custom boundary conditions for the edge problem in the continuum and see what they match to after the reduction. Handle disorder in either location of the wells or their coefficients. Understand other symmetry classes and dimensions.
Tight-Binding Reduction for the IQHE August 22, 2019