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Bott Periodicity and the“Periodic Table” of
Topological Insulatorsand Superconductors
Martin R. Zirnbauer
@ Mathem. Kolloquium, Uni Paderborn
July 24, 2017
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Kitaev’s Periodic Table of topological insulators/
superconductors
Schnyder, Ryu, Furusaki, Ludwig (2008); Kitaev (2009); Teo & Kane (2010); Stone, Chiu, Roy (2011); Freedman, Hastings, Nayak, Qi, Walker, Wang (2011); Abramovici & Kalugin (2012); Freed & Moore (2013)
from Hasan & Kane, Rev. Mod. Phys. (2011):
Quantum Hall Effect
He-3 (B phase)
QSHI: HgTe
Majorana
Bi Se2 3
QSHI = Quantum Spin Hall Insulator
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Integer Quantum Hall Effect
D.J. Thouless (1982-5; Nobel Prize Physics 2016):
Ground state = complex line bundle A over
Hall conductance
= = (integral of) first Chern class of A
2D disordered electrons; high B, low T
bulk-boundary correspondence
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Quantum Spin Hall Insulator(Kane & Mele, 2005)
Bulk-boundary correspondence:
non-trivial bulk topological invariant
perfectly conducting surface mode
Molenkamp group
(Würzburg, 2007)
This twisting is detected by the Kane-Mele index.
Strong spin-orbit scattering (preserves time-reversal invariance) causes band inversion the bundle of Fermi projections is twisted.
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Majorana chain (gapped 1d superconductor)
No symmetries, “spinless fermions”, single band; momentum
Weak pairing: bulk-boundary correspondence gapless edge state
Bogoliubov transformation:
Fermi constraint:
Weak pairing: (topol. invt.)
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Bott Periodicity
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Bott Periodicity (I)
compact Riemannian manifold
triple
space of minimal geodesics of class
minimum over all non-minimal geodesics of class of no. of negative eigenvalues of Hessian of length function
Thm (Bott, 1959). If is a symmetric space, then so is ,and
Example.
(Freudenthal suspension theorem).
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What’s a symmetric space?
Def.: A (locally) symmetric space is a Riemannian manifold
with covariantly constant curvature:
Globally symmetric spaces classified by E. Cartan (1926)
10 large families: A, AI, AII, AIII, BD, BDI, C, CI, CII, DIII
Classification:
Ex. 1: the round two-sphere
Riemann tensor:
Ex. 2: the set
of all subspaces
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n-sequences: complex and real
Bott Periodicity (II)
Raoul Bott(1923-2005)
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Bott Periodicity (III): Morse Theory
Manifold , function (smooth and proper).
homotopy-equivalent
Thm 1. If has no critical values in , then .
Thm 2. Let be a non-degenerate critical point of of index . If is the only critical point in , then ( -cell ).
Example: 2-torus
Fact. Functions of the needed kind (Morse functions) do exist.
height function
critical values
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Bott Periodicity (IV): Idea of proof
(space of paths in from to of homology class ),
Morse function length of path.
if geodesic distance from to ,
if
Hence if
where
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Bott Periodicity (V): Clifford algebra
Let Grassmann m’fld of complex planes inDefine
Lemma.
generators of Clifford algebra on :
Hermitian vector space with compatible symmetric bilinear form
complex Bott sequence,
real Bott sequence.
Note:
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More precisely,
Sketch of proof for
Let and
Then for it follows that
The action on is transitive.
The stabilizer of is Hence
and
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This is the Bott Map …
Bott Periodicity (VI): Bott Map
Make the identifications and paths from to
Givenassign to a minimal geodesicby
Comment. Same for instead of
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n-sequences: complex and real
Bott Periodicity (II’)
Bott Map
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Free-Fermion Ground States
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Universal model (notation/setting)
Quasi-particle vacuum free-fermion ground state Hartree-Fock-Bogoliubov mean-field ground state
Single-particle annihilation (creation) operators
space of field operators (Nambu space)
Hermitian conjugation
CAR bilinear formStructure on :
Fact. Free-fermion ground states
where and the are
quasi-particle annihilation ops:
Fermi constraint:
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Eugene P. Wigner
Q: What’s a symmetry in quantum mechanics?
A: An operator on Hilbert rays that preserves
all transition probabilities:
Symmetries in quantum mechanics
Remark 2: Symmetries commute with the Hamiltonian ( ).
Thus “chiral symmetry” ( ) is not a symmetry.
Remark 1: The symmetries form a group,
Wigner’s Theorem:
A symmetry in quantum mechanics can always
be represented on Hilbert space by an operator
which is either unitary or anti-unitary.
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J. Math. Phys. 3 (1962) 1199
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Tenfold Way
In this paper, it is proved that the symmetry classes of disordered fermions arein one-to-one correspondence with the 10 large families of symmetric spaces.
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Symmetry operations of the Tenfold Way
Anti-unitary symmetries:
1. Time reversal
2. Particle-hole conjugation
Unitary symmetries include
symmetry (charge operator ),
spin rotations generated by
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Universal Model (including symmetries)
Clifford algebra (Kitaev) of pseudo-symmetries :
Example. Time-reversal symmetry (spin ½):
( Hermitian conj.)
Indeed, and
Definition. A free-fermion ground state of symmetry class is a polarization where the complex vector spaceis subject to
- Fermi constraint:
- pseudo-symmetries:
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Kitaev Sequence (“real” and “complex” classes)
class symmetries pseudo-syms
Fermi constraint
see below
class symmetries pseudo-syms
Fermi constraint
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Diagonal Map
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from Hasan & Kane, Rev. Mod. Phys. (2011):
Quantum Hall Effect
He-3 (B phase)
QSHI: HgTe
Majorana
Bi Se2 3
Question: does there exist a ``Diagonal Map’’?
Bott-Kitaev Periodic Table
?
?
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Clean & Disordered Systems
Clean limit: translations are symmetries
Disorder: Non-Commutative Geometry approach, developed for IQHE by Bellissard et al. (following A. Connes)
Prodan, Schulz-Baldes (2013), Thiang (2014), Kellendonk;Boersema, Loring (2015), Carey et al. (2016)
Conserved momentum decomposition
Gapped system (insulator) vector bundle
algebra of bounded operators
Pairing between cyclic cohomology and
Hall conductance = non-commutative Chern number
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Diagonal Map: heuristic
Under the decomposition (conserved momentum)
the Fermi constraint refines to
(for all ).
Thus our free-fermion ground states are vector bundlessubject to a equivariance condition
with non-trivial involution
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Diagonal Map
Starting point: and
Define:
Preparatory step: jack up by (1,1) periodicity
New starting point: and
Note:
1. Fermi constraint:
2. Pseudo-syms:
3. Degeneration:
Outcome: V.B. in class on
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Diagonal Map = Bott Map2
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Summary of colloquium
Motivation: topological insulators and superconductors
Bott periodicity theorem from Morse theory
Free-fermion ground states with symmetries (imposed in a
prescribed order) realize the complex/real Bott sequences.
For Kitaev’s Periodic Table replace Bott Map by Diagonal Map
Reference: R. Kennedy and M.R. Zirnbauer, arXiv:1509.2537,
Commun. Math. Phys. 342 (2016) 909—963
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Thank you!(The End)