Spectral gap-labelling conjecture for magnetic Schrdinger operators - PowerPoint PPT Presentation

Spectral gap-labelling conjecture for magnetic Schrödinger operators and recent progress Recent progress in mathematics of topological insulators September 3-6, 2018. ETH Zürich Mathai Varghese

Collaborators and references joint work with: • Moulay Tahar Benameur (University of Montpellier, France); [BM15] M-T. Benameur and V. M., Gap-labelling conjecture with non-zero magnetic field. Advances in Mathematics , 325 , (2018) 116–164. [BM18] M-T. Benameur and V. M., Proof of the magnetic gap-labelling conjecture for principal solenoidal tori. 9 pages, [1806.06302] .

Outline of talk Magnetic Schrödinger operators 1 Motivations for the magnetic gap-labelling conjecture 2 Magnetic gap-labelling conjecture 3 What is known about the conjecture. 4 Solonoidal tori and principal solenoidal tori. 5 Recent progress in higher dimensions 6

Magnetic Schrodinger operators - Consider Euclidean space R d equipped with its usual Riemannian metric � d j = 1 dx 2 j . - the magnetic field B = 1 j , k Θ jk dx j ∧ dx k = 1 2 dx t Θ dx , where � 2 Θ is a constant ( d × d ) skew-symmetric matrix, dx is the column vector with entries dx j and dx t is the corresponding row vector, and matrix multiplication is used. B is closed. -Let us now pick a 1-form η such that d η = B . This is always possible since B is a closed 2-form and R d is contractible. We may regard η as defining a connection ∇ = d + i η on the trivial line bundle L over R d , whose curvature is iB . Physically we can think of η as the electromagnetic vector potential for a uniform magnetic field B normal to R d .

Magnetic Schrodinger operators Using the Riemannian metric the Hamiltonian of an electron in this magnetic field is given by H = 1 2 ∇ † ∇ + V = 1 2 ( d + i η ) † ( d + i η ) + V , acting on L 2 ( R d ) where † denotes the adjoint and V is a smooth real-valued bounded function. H is (formally) self-adjoint & bounded below. The restriction of H to a bounded domain Ω (with piecewise smooth boundary ∂ Ω ) in R d is denoted by H Ω . Imposing self-adjoint (Dirichlet/Neumann...) boundary conditions, then since H Ω becomes a self-adjoint elliptic operator, it has an (unbounded) purely discrete real spectrum, but which is bounded below. Moreover the eigenvalues all have finite multiplicity.

Magnetic Schrodinger operators Define the (spectral) counting function N ( H Ω , λ ) = # { µ ∈ spec ( H Ω ) : µ ≤ λ } = Tr ( χ ( −∞ ,λ ] ( H Ω )) This is a step function, and the values N ( H Ω , E ) is a gap-label whenever E lies in a spectral gap, i.e. for all E ∈ R \ spec ( H Ω ) . Properties of the (spectral) counting function N ( H Ω , λ ) N ( H Ω , λ ) is non-decreasing; 1 N ( H Ω , λ ) = 0 for all λ < inf spec ( H Ω ) ; 2 N ( H Ω , λ ) ∼ λ d / 2 as λ → ∞ (Weyl law); 3 N ( H Ω , λ ) is constant on spectral gaps. 4

Integrated Density of States

Magnetic Schrodinger operators What we would like to have is a counting function for the operator H on R d . This is trickier to define, as H contains continuous spectrum in general. Define the integrated density of states (IDS) 1 N ( H , λ ) = lim vol (Ω) N ( H Ω , λ ) Ω ↑ R d (which exists because R d is amenable, and such a sequence of open sets { Ω } is a Folner sequence .) Properties of the IDS N ( H , λ ) N ( H , λ ) is non-decreasing; 1 N ( H , λ ) = 0 for all λ < inf spec ( H ) ; 2 N ( H , λ ) ∼ λ d / 2 as λ → ∞ (Weyl law); 3 N ( H , λ ) is constant on spectral gaps. 4

Integrated Density of States NB Gap labels are no longer integers and more interesting!

Some C ∗ -algebras Difficult to work with this definition of IDS - seek an alternatives. Let U ( γ ) denote the unitary operator on L 2 ( R d ) given by translation by γ ∈ Z d . Consider the set consisting of translates of the resolvent operator: U ( γ )( H − zI ) − 1 U ( γ ) − 1 : γ ∈ Z d � � Σ 0 ( z ) = & assume that it is precompact for some z ∈ C with ℑ ( z ) � = 0. This is the case for any V ∈ L ∞ ( R d , R ) ∩ C ∞ ( R d ) . In the case when V is smooth and periodic, then this set is a point.

Some C ∗ -algebras s be the compact set that is the strong closure. Let Σ = Σ 0 ( z ) It turns out to be independent of z with ℑ ( z ) � = 0, modulo homeomorphism. It is called the disorder set associated to H , or the hull . Z d acts on Σ by homeomorphisms. The most interesting case is when Σ is a Cantor set . and Z d acts on Σ minimally (i.e. having dense orbit) Let µ be a Z d -invariant probability Borel measure on Σ .

Twisted crossed product algebra Let σ : Z d × Z d − → U ( 1 ) be a 2-cocycle on Z d , so it satisfies, γ 1 , γ 2 , γ 3 ∈ Z d σ ( γ 1 , γ 2 ) σ ( γ 1 + γ 2 , γ 3 ) = σ ( γ 1 , γ 2 + γ 3 ) σ ( γ 2 , γ 3 ) , Then the twisted crossed product C ∗ -algebra A = C (Σ) ⋊ σ Z d is constructed as follows. Let a , b ∈ A 0 = C c (Σ × Z d ) . Product: γ ′ ∈ Z d a ( ω, γ ′ ) b ( γ ′− 1 ω, γ − γ ′ ) σ ( γ − γ ′ , γ ′ ) ; ab ( ω, γ ) = � The adjoint: a ∗ ( ω, γ ) = a ( γ − 1 ω, − γ ) σ ( − γ, γ ) ; The regular representation for a ∈ A 0 and ψ ∈ L 2 ( Z d ) : γ ′ ∈ Z d a ( γ ′− 1 ω, γ − γ ′ ) ψ ( γ ′ ) σ ( γ − γ ′ , γ ′ ) ; π ω ( a ) ψ ( γ ) = � The norm: || a || = sup ω ∈ Σ || π ω ( a ) || ; ||·|| The twisted crossed product C ∗ -algebra: A = A 0

The trace functional Let µ : C (Σ) − → C be an invariant measure on Σ . Then it induces a trace τ µ on the twisted crossed product C ∗ -algebra A = C (Σ) ⋊ σ Z d as follows: define for a ∈ A , � τ µ ( a ) = a ( ω, 0 ) d µ ( ω ) . Σ Then for a , b ∈ A , one has τ µ ( ab ) = τ µ ( ba ) , and if a ≥ 0, τ µ ( a ) ≥ 0 . The trace actually extends to the bigger von Neumann algebra, L ∞ (Σ) ⋊ σ Z d , using the same formula.

Magnetic gap-labelling conjecture Now the spectral projections of H are bounded measurable functions of H , therefore χ ( −∞ ,λ ] ( H ) ∈ L ∞ (Σ) ⋊ σ Z d ⊗ K , the von Neumann algebra. Then one has the useful Theorem (Shubin) IDS has the following expression, at a point of continuity: N ( H , λ ) = τ µ ( χ ( −∞ ,λ ] ( H )) , λ ∈ R When λ is in a spectral gap of H , i.e. λ ∈ R \ spec ( H ) , then the projection χ ( −∞ ,λ ] ( H ) is the smaller algebra C (Σ) ⋊ σ Z d ⊗ K , Lemma E �∈ spec ( H ) ⇒ χ ( −∞ , E ] ( H )) ∈ C (Σ) ⋊ σ Z d ⊗ K . Proof . Suppose that, spec ( H ) ⊂ [ − A , ∞ ) and that the open interval ( a , b ) is a spectral gap of H , i.e. ( a , b ) ∩ spec ( H ) = ∅ .

Magnetic gap-labelling conjecture Suppose that E ∈ ( a , b ) i.e. E / ∈ spec ( H ) . Then there is a holomorphic function φ on a neighbourhood of spec ( H ) ∩ [ − A , a ] such that d λ � χ ( −∞ , E ] ( H ) = φ ( H ) = λ − H C where C is a closed contour enclosing the interval [ − A , a ] to the left of E , and is the Riesz projection . Since C (Σ) ⋊ σ Z d ⊗ K is closed under the continuous functional calculus, it follows that χ ( −∞ , E ] ( H ) ∈ C (Σ) ⋊ σ Z d ⊗ K .

Magnetic gap-labelling conjecture It follows that the spectral gap-labels of H are contained in the countable subgroup of R , τ µ ( K 0 ( C (Σ) ⋊ σ Z d )) . Then the magnetic gap-labelling conjecture is about finding an expression for τ µ ( K 0 ( C (Σ) ⋊ σ Z d )) =????

K-theory of C ∗ -algebras Two projections P , Q in the C ∗ -algebra A ⊗ K , where K is the algebra of compact operators on a separable Hilbert space, are said to be (Murray-von Neumann) equivalent P ∼ Q whenever there is an element a ∈ A such that P = a ∗ a and Q = aa ∗ . Recall that the K-theory K 0 ( A ) is defined as stable equivalence classes of pairs of projections ( P , Q ) in A ⊗ K , where ( P , Q ) and ( P ′ , Q ′ ) are stably equivalent whenever P ⊕ Q ′ ⊕ R ∼ P ′ ⊕ Q ⊕ R , for some projection R in A ⊗ K . K 0 ( A ) is a countable group. Clearly a trace on the algebra A , τ µ : A → C , induces a morphism τ µ : K 0 ( A ) − → R .

Motivation: Magic formula Let Λ[ dx ] = Λ[ dx 1 , . . . , dx d ] denote the exterior algebra with generators dx 1 , . . . , dx d . It has basis the monomials dx I = dx i 1 , . . . , dx i p , I = { i 1 , . . . , i p } , i 1 < · · · < i p , 1 ≤ p ≤ d . Given a skew-symmetric matrix Θ , we can associate a quadratic element 1 2 dx t Θ dx in Λ[ dx ] . Recall that the Pfaffian of the skew-symmetric matrix Θ , Pf (Θ) can be defined as � m 1 � 1 2 dx t Θ dx = Pf (Θ) dx 1 ∧ dx 2 ∧ . . . ∧ dx d m ! where d = 2 m .

Motivation: Magic formula By section 1 in [Mathai-Quillen86], 2 dx t Θ dx = 1 � e Pf (Θ I ) dx I I where I runs over subsets of { 1 , . . . , d } with an even number of elements, and Θ I denotes the submatrix of Θ = (Θ ij ) with i , j ∈ I , which is clearly also skew-symmetric. This was a key formula in the paper above, to construct the Chern-Weil representative of the Thom class of an oriented vector bundle.