Classification of "Real" Bloch-bundles: topological insulators of type AI Giuseppe De Nittis ( FAU, Universität Erlangen-Nürnberg ) ——————————————————————————– EPSRC Symposium: Many-Body Quantum Systems University of Warwick, U.K. 17-21 March, 2014 ——————————————————————————– Joint work with: K. Gomi Reference: arXiv:1402.1284
Outline Topological Insulators and symmetries 1 What is a Topological Insulator? What it means to classify Topological Insulators? The rôle of symmetries Classification of “Real” Bloch-bundles 2 The Borel equivariant cohomology The classification table The case d = 4
Band Insulators For an enlightened explanation about the physical point of view the main reference is !! Graf’s talk of last Tuesday 18th !! I will focus only on the mathematical (topological) aspects. The Bloch-Floquet theory exploits the translational symmetry of a crystal structure to describe electronic states in terms of their crystal momentum k , defined in a periodic Brillouin zone B . A little bit more in general one can assume that: “ the electronic properties of a crystal are described by a family of Hamiltonians labelled by points of a manifold B ” B ∋ k �− → H ( k ) .
In a band insulator an energy gap separates the filled valence bands from the empty conduction bands. The Fermi level E F characterizes the gap. The energy bands E ( k ) are the eigenvalues of H ( k ) H ( k ) ψ ( k ) = E ( k ) ψ ( k ) k ∈ B .
Outline Topological Insulators and symmetries 1 What is a Topological Insulator? What it means to classify Topological Insulators? The rôle of symmetries Classification of “Real” Bloch-bundles 2 The Borel equivariant cohomology The classification table The case d = 4
A rigorous classification scheme requires (in my opinion !! ) three ingredients: «A» The interpretation of the “vague” notion of topological insulator in terms of a mathematical structure (category) for which the notion of classification makes sense (objects, isomorphisms, equivalence classes, ...). «B» A classification theorem. «C» An (hopefully !! ) algorithmic method to compute the classification and a set of proper labels to discern between different (non-isomorphic) objects.
«A» Bloch Bundle and Vector Bundle Theory For all k ∈ B the operator H ( k ) is a self-adjoint N × N matrix with real eigenvalues E 1 ( k ) � E 2 ( k ) � ... � E N − 1 ( k ) � E N ( k ) and related eigenvectors ψ j ( k ) , j = 1 ,..., N . Definition (Gap condition) There exists a E F ∈ R and an integer 1 < M < N such that: � E M ( k ) < E F ∀ k ∈ B . E M + 1 ( k ) > E F The Fermi projection onto the filled states is the matrix-valued map B ∋ k �→ P F ( k ) defined by M ∑ P F ( k ) := | ψ j ( k ) �� ψ j ( k ) | . j = 1
For each k ∈ B H k := Ran P F ( k ) ⊂ H is a subspace of C N of dimension M . The collection � E F := H k k ∈ B is a topological space (said total space) and the map π : E F − → B defined by π ( k , v ) = k is continuous (and open). π : E F → B is a complex vector bundle called Bloch bundle.
«B» Classification Theorem Gapped band insulator (of type A) at Fermi energy E F � Rank M complex vector bundle over B � (homotopy classification theorem ) Vec M C ( B ) ≃ [ B , Gr M ( C N )] ( N ≫ 1 ) The space Gr M ( C N ) := U ( N ) / � U ( M ) × U ( N − M ) � . is the Grasmannian of M -planes in C N . ✑ Remark: The computation of [ B , Gr M ( C N )] is, generally, an extremely difficult task (non algorithmic problem !! ). Explicit computations are available only for simple spaces B .
The Case of Free Fermions: B ≡ S d For a system of free fermions (after a Fourier transform) S d := k ∈ R d + 1 | � k � = 1 ≃ R d ∪{ ∞ } . � � Number of different phases of a band insulator of type A � � Gr M ( C N ) � := [ S d , Gr M ( C N )] π d π d ( X ) is the d -th homotopy group of the space X . ✻ Problem: How to compute the homotopy of Gr M ( C N ) ? Theorem (Bott, 1959) Gr M ( C N ) � � � � π d = π d − 1 U ( M ) if 2 N � 2 M + d + 1 .
Homotopy groups of U ( M ) � � π d U ( M ) d = 0 d = 1 d = 2 d = 3 d = 4 d = 5 M=1 0 Z 0 0 0 0 M=2 0 Z 0 Z Z 2 Z 2 M=3 0 0 0 Z Z Z M=4 0 Z 0 Z 0 Z M=5 0 0 0 Z Z Z The stable regime is defined by d < 2 M (in blu the values for the unstable case). In the stable regime one has the Bott periodicity 0 if d even or d = 0 � � π d U ( M ) = Z if d odd ( d � 2 M ) if d = 2 M . Z M !
Topological Insulators in class A ( B = S d ) The number of topological phases depends on the dimension d and on the number of filled states M (this is missed in K -theory !! ) d = 1 d = 2 d = 3 d = 4 d = 5 ... 0 ( M = 1) 0 ( M = 1) 0 Z 0 Z 2 ( M = 2) ... Z ( M � 2) 0 ( M � 3) d = 1 Band insulators show only the trivial phase (ordinary insulators). d = 2 For every integer there exists a topological phase and band insulators in different phases cannot be deformed into each other without “altering the nature” of the system ( e.g. quantum Hall insulators). d = 3 As in the case d = 1. d = 4 A difference between the non-stable case M = 1 and the stable case M � 2 appears. The value of M is dictated by physics !!
Ordinary insulator: Band insulator in a trivial phase � Trivial vector bundle � Exists a global frame of continuous Bloch functions Allowed (adiabatic) deformations: Transformations which doesn’t alter the nature of the system � Stability of the topological phase � Vector bundle isomorphism
The Case of Bloch Electrons: B ≡ T d «C» Electrons interacting with the crystalline structure of a metal (Bloch-Floquet) B = T d := S 1 × ... × S 1 ( d -times ) . The computation of [ T d , Gr M ( C N )] is non trivial. The theory of characteristic class becomes relevant (since algorithmic !! ) . Theorem (Peterson, 1959) If dim ( X ) � 4 then Vec 1 C ( X ) ≃ H 2 ( X , Z ) Vec M C ( X ) ≃ H 2 ( X , Z ) ⊕ H 4 ( X , Z ) ( M � 2 ) and the isomorphism Vec M → ( c 1 , c 2 ) ∈ H 2 ( X , Z ) ⊕ H 4 ( X , Z ) C ( X ) ∋ [ E ] �− is given by the first two Chern classes (c 2 = 0 if M = 1 ).
d = 1 d = 2 d = 3 d = 4 0 ( M = 1) B = S d 0 Z 0 Z ( M � 2) Z 6 ( M = 1) B = T d Z 3 0 Z Z 7 ( M � 2) Table taken from [SRFL] d = 3 The cases B = S 3 and B = T 3 are different. In the periodic case one has Z 3 distinct quantum phases. These are three-dimensional versions of a 2D quantum Hall insulators.
Outline Topological Insulators and symmetries 1 What is a Topological Insulator? What it means to classify Topological Insulators? The rôle of symmetries Classification of “Real” Bloch-bundles 2 The Borel equivariant cohomology The classification table The case d = 4
Table taken from [SRFL] Let H acts on a space H and C is a (anti-linear) complex conjugation on H . Definition (Time Reversal Symmetry) The Hamiltonian H has a Time Reversal Symmetry (TRS) if there exists a unitary operator U such that: U H U ∗ = C H C . � CUC = + U ∗ AI if (even) H is in class CUC = − U ∗ AII if (odd) .
Involutions over the Brillouin Zone Let B ∋ k �→ P F ( k ) be the fibered Fermi projection of a band insulator H . If H has a TRS, U acts by “reshuffling the fibers” U P F ( k ) U ∗ = C P F ( τ ( k )) C ∀ k ∈ B . Here τ : B → B is an involution: Definition (Involution) Let X be a topological space and τ : X → X a homeomorphism. We said that τ is an involution if τ 2 = Id X . The pair ( X , τ ) is called an involutive space. ✑ Remark: Each space X admits the trivial involution τ triv := Id X .
Continuous case B = S d τ d ✲ S d S d S d := ( S d , τ d ) ˜ (+ k 0 , + k 1 ,..., + k d ) τ d ✲ (+ k 0 , − k 1 ,..., − k d )
Periodic case B = T d τ d := τ 1 × ... × τ 1 T d = S 1 × ... × S 1 ✲ T d = S 1 × ... × S 1 T d := ( T d , τ d ) ˜
«A» Vector Bundles over Involutive Spaces U P F ( k ) U ∗ = C P F ( τ ( k )) C ∀ k ∈ B induces an additional structure on the Bloch-bundle E → B . Definition (Atiyah, 1966) Let ( X , τ ) be an involutive space and E → X an complex vector bundle. Let Θ : E → E an homeomorphism such that Θ : E | x − → E | τ ( x ) is anti-linear . The pair ( E , τ ) is a “Real”-bundle over ( X , τ ) if Θ 2 : E | x + 1 − → E | x ∀ x ∈ X ; The pair ( E , τ ) is a “Quaternionic”-bundle over ( X , τ ) if Θ 2 : E | x − 1 − → E | x ∀ x ∈ X .
AZC TRS Category VB Vec M A 0 complex C ( X ) Vec M AI + “Real” R ( X , τ ) Vec M − AII “Quaternionic” Q ( X , τ ) The names are justified by the following isomorphisms: Vec M Vec M R ( X , Id X ) ≃ R ( X ) Vec M Vec M Q ( X , Id X ) ≃ H ( X )
Outline Topological Insulators and symmetries 1 What is a Topological Insulator? What it means to classify Topological Insulators? The rôle of symmetries Classification of “Real” Bloch-bundles 2 The Borel equivariant cohomology The classification table The case d = 4
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