Periodic quantum graphs in magnetic fields Differential Operators on - PowerPoint PPT Presentation

Periodic quantum graphs in magnetic fields Differential Operators on Graphs and Waveguides – Graz 2019 Simon Becker Cambridge

Consider a lattice Z 2 or the honeycomb lattice.

Consider a lattice Z 2 or the honeycomb lattice. We study quantum graphs with magnetic field and randomness:

Consider a lattice Z 2 or the honeycomb lattice. We study quantum graphs with magnetic field and randomness: B := B dx 1 ∧ dx 2

Consider a lattice Z 2 or the honeycomb lattice. We study quantum graphs with magnetic field and randomness: A = 1 B := B dx 1 ∧ dx 2 = d A , 2 B ( − x 2 dx 1 + x 1 dx 2 ) .

Consider a lattice Z 2 or the honeycomb lattice. We study quantum graphs with magnetic field and randomness: A = 1 B := B dx 1 ∧ dx 2 = d A , 2 B ( − x 2 dx 1 + x 1 dx 2 ) . ( D B ψ ) e := − i ψ ′ ( H B λ,ω ψ ) e := ( D B D B ψ ) e + V ψ e + V ω ψ e , e − A e ψ e � ( D B ψ ) e ( v ) = 0 . v ∈ ∂ e 1 ∩ ∂ e 2 ⇒ ψ e 1 ( v ) = ψ e 2 ( v ) , ∂ e ∋ v

Consider a lattice Z 2 or the honeycomb lattice. We study quantum graphs with magnetic field and randomness: A = 1 B := B dx 1 ∧ dx 2 = d A , 2 B ( − x 2 dx 1 + x 1 dx 2 ) . ( D B ψ ) e := − i ψ ′ ( H B λ,ω ψ ) e := ( D B D B ψ ) e + V ψ e + V ω ψ e , e − A e ψ e � ( D B ψ ) e ( v ) = 0 . v ∈ ∂ e 1 ∩ ∂ e 2 ⇒ ψ e 1 ( v ) = ψ e 2 ( v ) , ∂ e ∋ v The Peierls substitution P : ψ e �→ e iA e t ψ e transforms the magnetic field into the boundary conditions:

Consider a lattice Z 2 or the honeycomb lattice. We study quantum graphs with magnetic field and randomness: A = 1 B := B dx 1 ∧ dx 2 = d A , 2 B ( − x 2 dx 1 + x 1 dx 2 ) . ( D B ψ ) e := − i ψ ′ ( H B λ,ω ψ ) e := ( D B D B ψ ) e + V ψ e + V ω ψ e , e − A e ψ e � ( D B ψ ) e ( v ) = 0 . v ∈ ∂ e 1 ∩ ∂ e 2 ⇒ ψ e 1 ( v ) = ψ e 2 ( v ) , ∂ e ∋ v The Peierls substitution P : ψ e �→ e iA e t ψ e transforms the magnetic field into the boundary conditions: Λ B := P − 1 H B P , (Λ B ψ ) e = − ψ ′′ e + V ψ e ⇒ e i δ + ± A e 1 ψ e 1 ( v ) = e i δ + ± A e 2 ψ e 2 ( v ) , ∂ ± e 1 = ∂ ± e 2 =: v = � e i δ + ± A e ψ ′ e ( v ) = 0 , ∂ ± e ∋ v

Consider a lattice Z 2 or the honeycomb lattice. We study quantum graphs with magnetic field and randomness: A = 1 B := B dx 1 ∧ dx 2 = d A , 2 B ( − x 2 dx 1 + x 1 dx 2 ) . ( D B ψ ) e := − i ψ ′ ( H B λ,ω ψ ) e := ( D B D B ψ ) e + V ψ e + V ω ψ e , e − A e ψ e � ( D B ψ ) e ( v ) = 0 . v ∈ ∂ e 1 ∩ ∂ e 2 ⇒ ψ e 1 ( v ) = ψ e 2 ( v ) , ∂ e ∋ v The Peierls substitution P : ψ e �→ e iA e t ψ e transforms the magnetic field into the boundary conditions: Λ B := P − 1 H B P , (Λ B ψ ) e = − ψ ′′ e + V ψ e ⇒ e i δ + ± A e 1 ψ e 1 ( v ) = e i δ + ± A e 2 ψ e 2 ( v ) , ∂ ± e 1 = ∂ ± e 2 =: v = � e i δ + ± A e ψ ′ e ( v ) = 0 , ∂ ± e ∋ v Br¨ uning–Geyler–Pankrashkin ’07

Roughly speaking, Krein’s formula reduces the study of an operator on the graph to the study of an operator on Z 2 � u )( γ ) = 1 4 ( τ 0 + τ ∗ 0 + τ 1 + τ ∗ ( Q h 1 ) u ( γ ) � � 0 1 + τ 0 + τ 1 ( Q h � u )( γ ) = 1 u ( γ ) 1 + τ ∗ 0 + τ ∗ 3 0 1

Roughly speaking, Krein’s formula reduces the study of an operator on the graph to the study of an operator on Z 2 � u )( γ ) = 1 4 ( τ 0 + τ ∗ 0 + τ 1 + τ ∗ ( Q h 1 ) u ( γ ) � � 0 1 + τ 0 + τ 1 ( Q h � u )( γ ) = 1 u ( γ ) 1 + τ ∗ 0 + τ ∗ 3 0 1 with translations given by τ 0 ( u )( γ ) := u ( γ 1 − 1 , γ 2 ) τ 1 ( u )( γ ) := e ih γ 1 u ( γ 1 , γ 2 − 1) ,

Roughly speaking, Krein’s formula reduces the study of an operator on the graph to the study of an operator on Z 2 � u )( γ ) = 1 4 ( τ 0 + τ ∗ 0 + τ 1 + τ ∗ ( Q h 1 ) u ( γ ) � � 0 1 + τ 0 + τ 1 ( Q h � u )( γ ) = 1 u ( γ ) 1 + τ ∗ 0 + τ ∗ 3 0 1 with translations given by τ 0 ( u )( γ ) := u ( γ 1 − 1 , γ 2 ) τ 1 ( u )( γ ) := e ih γ 1 u ( γ 1 , γ 2 − 1) , here h is the flux through a fundamental cell.

Roughly speaking, Krein’s formula reduces the study of an operator on the graph to the study of an operator on Z 2 � u )( γ ) = 1 4 ( τ 0 + τ ∗ 0 + τ 1 + τ ∗ ( Q h 1 ) u ( γ ) � � 0 1 + τ 0 + τ 1 ( Q h � u )( γ ) = 1 u ( γ ) 1 + τ ∗ 0 + τ ∗ 3 0 1 with translations given by τ 0 ( u )( γ ) := u ( γ 1 − 1 , γ 2 ) τ 1 ( u )( γ ) := e ih γ 1 u ( γ 1 , γ 2 − 1) , here h is the flux through a fundamental cell. The above operators are equivalent to operators on Z with c ( θ ) = 1 + e − 2 π i θ and v ( θ ) = 2 cos(2 πθ ) ( H � u )( n ) = u ( n + 1) + u ( n − 1) + v ( k + n h 2 π ) u ( n ) ( H � u )( n ) = c ( k + n h 2 π ) u ( n + 1) + c ( k + ( n − 1) h 2 π ) u ( n − 1) + v ( k + n h 2 π ) u ( n ) .

Roughly speaking, Krein’s formula reduces the study of an operator on the graph to the study of an operator on Z 2 � u )( γ ) = 1 4 ( τ 0 + τ ∗ 0 + τ 1 + τ ∗ ( Q h 1 ) u ( γ ) � � 0 1 + τ 0 + τ 1 ( Q h � u )( γ ) = 1 u ( γ ) 1 + τ ∗ 0 + τ ∗ 3 0 1 with translations given by τ 0 ( u )( γ ) := u ( γ 1 − 1 , γ 2 ) τ 1 ( u )( γ ) := e ih γ 1 u ( γ 1 , γ 2 − 1) , here h is the flux through a fundamental cell. The above operators are equivalent to operators on Z with c ( θ ) = 1 + e − 2 π i θ and v ( θ ) = 2 cos(2 πθ ) ( H � u )( n ) = u ( n + 1) + u ( n − 1) + v ( k + n h 2 π ) u ( n ) ( H � u )( n ) = c ( k + n h 2 π ) u ( n + 1) + c ( k + ( n − 1) h 2 π ) u ( n − 1) + v ( k + n h 2 π ) u ( n ) .

Theorem (Helffer-Sj¨ ostrand+ B.-Han-Jitomirskaya) The spectrum of Q h � or Q h h � for 2 π ∈ Q is band spectrum and a.c.. h If 2 π ∈ R \ Q the spectrum is a Cantor set (closed, nowhere dense, no isolated points) of Lebesgue measure zero and s.c..

Theorem (Helffer-Sj¨ ostrand+ B.-Han-Jitomirskaya) The spectrum of Q h � or Q h h � for 2 π ∈ Q is band spectrum and a.c.. h If 2 π ∈ R \ Q the spectrum is a Cantor set (closed, nowhere dense, no isolated points) of Lebesgue measure zero and s.c.. The proof- roughly -relies on three main ideas: ◮ Exclude point spectrum from regularity properties of the density of states.

Theorem (Helffer-Sj¨ ostrand+ B.-Han-Jitomirskaya) The spectrum of Q h � or Q h h � for 2 π ∈ Q is band spectrum and a.c.. h If 2 π ∈ R \ Q the spectrum is a Cantor set (closed, nowhere dense, no isolated points) of Lebesgue measure zero and s.c.. The proof- roughly -relies on three main ideas: ◮ Exclude point spectrum from regularity properties of the density of states. ◮ Get estimates on the Lebesgue measure of the spectrum (as a h set) for rational flux 2 π ∈ Q .

Theorem (Helffer-Sj¨ ostrand+ B.-Han-Jitomirskaya) The spectrum of Q h � or Q h h � for 2 π ∈ Q is band spectrum and a.c.. h If 2 π ∈ R \ Q the spectrum is a Cantor set (closed, nowhere dense, no isolated points) of Lebesgue measure zero and s.c.. The proof- roughly -relies on three main ideas: ◮ Exclude point spectrum from regularity properties of the density of states. ◮ Get estimates on the Lebesgue measure of the spectrum (as a h set) for rational flux 2 π ∈ Q . ◮ Prove that the spectrum is H¨ older continuous and approximate irrationals by rationals.

Theorem (Helffer-Sj¨ ostrand+ B.-Han-Jitomirskaya) The spectrum of Q h � or Q h h � for 2 π ∈ Q is band spectrum and a.c.. h If 2 π ∈ R \ Q the spectrum is a Cantor set (closed, nowhere dense, no isolated points) of Lebesgue measure zero and s.c.. The proof- roughly -relies on three main ideas: ◮ Exclude point spectrum from regularity properties of the density of states. ◮ Get estimates on the Lebesgue measure of the spectrum (as a h set) for rational flux 2 π ∈ Q . ◮ Prove that the spectrum is H¨ older continuous and approximate irrationals by rationals.

This is a plot of the spectrum of H B for the hexagonal graph:

This is a plot of the spectrum of H B for the hexagonal graph:

We know what kind of spectrum there is, but now we want to know: Where is the spectrum?-Are there spectral gaps?

We know what kind of spectrum there is, but now we want to know: Where is the spectrum?-Are there spectral gaps? - Therefore, we study the density of states

We know what kind of spectrum there is, but now we want to know: Where is the spectrum?-Are there spectral gaps? - Therefore, we study the density of states tr 1 B ( R ) f ( H B λ,ω ) tr f ( H B � λ,ω ) := lim vol ( B ( R )) R →∞

We know what kind of spectrum there is, but now we want to know: Where is the spectrum?-Are there spectral gaps? - Therefore, we study the density of states � tr 1 B ( R ) f ( H B λ,ω ) tr f ( H B � λ,ω ) := lim = f ( E ) d ρ λ ( E ) . vol ( B ( R )) R →∞ R

We know what kind of spectrum there is, but now we want to know: Where is the spectrum?-Are there spectral gaps? - Therefore, we study the density of states � tr 1 B ( R ) f ( H B λ,ω ) tr f ( H B � λ,ω ) := lim = f ( E ) d ρ λ ( E ) . vol ( B ( R )) R →∞ R The limit does exist and is a.s. non random-just like the spectrum of H B λ,ω .