aspects of symmetry and topology for free-fermion ground ......aspects of symmetry and topology for...
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Aspects of Symmetry and Topology for Free-Fermion Ground States
with Disorder
Martin R. Zirnbauer
@ Bad Honnef (CRC 183) Summer School
June 22-24, 2017
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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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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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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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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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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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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,
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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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Kitaev Sequence (“real” and “complex” classes)
class symmetries pseudo-syms
Fermi constraint
see below
class symmetries pseudo-syms
Fermi constraint
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Tool: (1,1) periodicity.
Lemma.
Double the dimension ( ) and let
Note: is “imaginary”:
Q: Why do 3 spin generators amount to 4 pseudo-symmetries?
Proof.
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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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Diagonal Map: heuristic
Under the Fermi constraint refines to
(for all ).
Thus our free-fermion ground states are vector bundlessubject to a equivariance condition
with non-trivial involution
Clean limit: translations are symmetries
Conserved momentum decomposition
Gapped system (insulator) vector bundle
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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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Example 1.
(1,1) doubling tensor with and let ,
, .
In BCS form: where .
For more general
DIII (T) D (none)
Topological 1d superconductor with spin-triplet pairing and T-invariance
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Example 2.
(1,1) doubling tensor with band space
Topological 1d class- superconductor (after ph-transformation):
are isolated zeros of Kane-Mele Pfaffian
Kane-Mele invariant non-trivial (QSHI)
Note:
DIII (T) AII (T, Q)
Next SPT phases in line: 3d insulator (CII), 4d superconductor (C), etc.
Apply 1-par. group:
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