topological insulators anderson insulators … notes/a_altland.pdf · disorder in topological...
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Topological insulators
Anderson insulators
Symmetry classes
Topological insulators
Anderson insulators
Symmetry classes
Su-Schrieffer-Haeger chain
SSH chain
system gappedµ > t
system gapped & topologically chargedµ < t
k
µ > t
k k
µ < tµ = t
why topological?
why topological?
introduce Bloch crystal momentumk 2 [0, 2⇡]
Q̂ ! Q(k) 2 C
why topological?
introduce Bloch crystal momentumk 2 [0, 2⇡]
Q̂ ! Q(k) 2 C
why topological?
introduce Bloch crystal momentumk 2 [0, 2⇡]
Q̂ ! Q(k) 2 C
why topological?
introduce Bloch crystal momentumk 2 [0, 2⇡]
Q̂ ! Q(k) 2 C
Topological insulators
Anderson insulators
Symmetry classes
Anderson localization
d � 3
d < 3
one-parameter scaling
one-parameter scalingAb
raha
ms
et a
l. 79
one-parameter scalingAb
raha
ms
et a
l. 79
one-parameter scalingAb
raha
ms
et a
l. 79
Topological insulators
Anderson insulators
Symmetry classes
symmetry classes
unitary symmetries (rotational, translational, …): fragile, not of too much topological significance
anti-unitary symmetries (time reversal, particle hole, chiral symmetries, …): more robust. There exist 10 fundamental symmetry classes, A, AI, AII, AIII, BDI, D, C, CI, CII, DIII (notation: courtesy Cartan)
main principle for the classification of topological insulators: the periodic table (Ludwig et al. 2009)
symmetry classes
symmetry classes
IQH
SSH
‘Kitaev chain’
‘Majorana quantum wire’
spin QH‘topolgical insulator (HgTe)’
Topological insulators
Anderson insulators
Symmetry classes
disorder in topological insulators
Alexander Altland, Dmitry Bagrets (Cologne)Alex Kamenev (Minnesota)Lars Fritz (Utrecht)
disorder vs. topology
bulk quantum criticality
physics at the edge
Utrecht, 12.11.14
PRL 109, 227005 (2012)New J. Phys. 15, 055019 (2013)PRL 112, 206602 (2014)
case study: IQH
disordered topological matter — time line
disordered topological matter — time line
80s
quantum Hall
SSH
quantum Hall effect (class A)
quantum Hall effect (class A)
quantum Hall effect (class A)
quantum Hall effect (class A)
1998 Nobel prize press release
smooth profiles of , becoming more singular upon lowering temperature
quantum Hall effect (class A)
1998 Nobel prize press release
smooth profiles of , becoming more singular upon lowering temperature
quantum Hall effect (class A)
quantum Hall effect (class A)
quantum Hall continued system described by two scaling parameters
: average longitudinal transport coefficient
: average topological index
two parameter criticality
Khm
elni
tski
i, 19
83
quantum Hall continued system described by two scaling parameters
: average longitudinal transport coefficient
: average topological index
two parameter criticality
Khm
elni
tski
i, 19
83
quantum Hall continued system described by two scaling parameters
: average longitudinal transport coefficient
: average topological index
two parameter criticality
Khm
elni
tski
i, 19
83
disordered topological matter — time line
80s
quantum Hall
disordered topological matter — time line
90s
80s
quantum Hall
symmetry classes
disordered topological matter — time line
90s
00s
80s
quantum Hall
symmetry classes
1d delocalization
2d delocalization
novel quantumHall effects
1d delocalization phenomena delocalization in quasi-one dimensional geometries
1998 AIII quantum wire (Brouwer, Mudry, Simons, AA) 1999 D quantum wire (Brouwer, Mudry, Furusaki) 2004 AIII, D, BDI, DIII, CII (Read, Gruzberg, Vishveshwara)
1d delocalization phenomena delocalization in quasi-one dimensional geometries
1998 AIII quantum wire (Brouwer, Mudry, Simons, AA) 1999 D quantum wire (Brouwer, Mudry, Furusaki) 2004 AIII, D, BDI, DIII, CII (Read, Gruzberg, Vishveshwara)
universal interpretation: topological insulators at quantum critical point
Unconventional criticality in 2d
2001 Class C spin quantum Hall effect (Chalker et al.)2001 Class D quantum criticality (Fisher et al., Read et al.)
disordered topological matter — time line
90s
00s
80s
quantum Hall
symmetry classes
1d delocalization
2d delocalization
novel quantumHall effects
disordered topological matter — time line
90s
00s
80s
quantum Hall
symmetry classes
1d delocalization
topological matter
2d delocalization
novel quantumHall effects
disordered topological matter — time line
90s
00s
10s
80s
quantum Hall
symmetry classes
1d delocalization
topological matter
2d delocalization
“topological Anderson insulator”
novel quantumHall effects
disorder vs. topology
topological Anderson insulator
topological Anderson insulator
topological Anderson insulator
Addition of disorder to a clean band insulator …
invalidate k-theory of band gaps/topological indices
Addition of disorder to a clean band insulator …
invalidate k-theory of band gaps/topological indices
destroys band gaps and
Addition of disorder to a clean band insulator …
invalidate k-theory of band gaps/topological indices
destroys band gaps and
turns insulator into nominal metal (-> Anderson ins.)
renders topological indices statistically distributed
Addition of disorder to a clean band insulator …
invalidate k-theory of band gaps/topological indices
destroys band gaps and
turns insulator into nominal metal (-> Anderson ins.)
renders topological indices statistically distributed
clean disorderedk-theory + -band gap + -
index + (+)
topological Anderson insulator
cf. Motrunich, Damle and Huse 2001, Groth et al. 2010
topological Anderson insulator
cf. Motrunich, Damle and Huse 2001, Groth et al. 2010
topological Anderson insulator
cf. Motrunich, Damle and Huse 2001, Groth et al. 2010
topological Anderson insulator
early works
cf. Motrunich, Damle and Huse 2001, Groth et al. 2010
topological Anderson insulator
Brouwer et al., 2012; Groth et al. 2010; …
early works
cf. Motrunich, Damle and Huse 2001, Groth et al. 2010
topological Anderson insulator
cf. Motrunich, Damle and Huse 2001, Groth et al. 2010
critical theory ?
Universal theory of disordered TI — checklist
describe topological invariants without reference to k-space
describe criticality in terms of two observables:
: transport coefficient: configurational average of index
similar architecture in all symmetry classes, d=1,2
describe edge state formation
generically:
critical:
state bare values
Universal theory of disordered TI — checklist
describe topological invariants without reference to k-space
describe criticality in terms of two observables:
: transport coefficient: configurational average of index
similar architecture in all symmetry classes, d=1,2
describe edge state formation
generically:
critical:
state bare values
Universal theory of disordered TI — checklist
describe topological invariants without reference to k-space
describe criticality in terms of two observables:
: transport coefficient: configurational average of index
similar architecture in all symmetry classes, d=1,2
describe edge state formation
generically:
critical:
state bare values
IQH
SSH
‘Kitaev chain’
‘Majorana quantum wire’
spin QH‘topolgical insulator (HgTe)’
Universal theory of disordered TI — checklist
describe topological invariants without reference to k-space
describe criticality in terms of two observables:
: transport coefficient: configurational average of index
similar architecture in all symmetry classes, d=1,2
describe edge state formation
generically:
critical:
state bare values
IQH
SSH
‘Kitaev chain’
‘Majorana quantum wire’
spin QH‘topolgical insulator (HgTe)’
(field) theory
generalized definition of winding number
generalized definition of winding number
generalized definition
generalized definition of winding number
generalized definition
generalized definition of winding number
generalized definition
want to know of disordered system
generalized definition of winding number
generalized definition
want to know of disordered system
field theorymicroscopic system
critical phenomena
Ginzburg Landau theory
field integral
localization length average index
theta term
matrix fields
results (AIII quantum wire)
where
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
c
g
results (AIII quantum wire)
where
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
c
g
phase diagram
phase diagram
phase boundaries: lines of half integer bare topological index
flow describes stabilization of self-averaging topological phase/boundary state generation
phase diagram
phase boundaries: lines of half integer bare topological index
flow describes stabilization of self-averaging topological phase/boundary state generation
· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·
· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·· ·
0 0.5 1.-3
-2
-1
0
1
2
3
wêt
mêt
bI
tI
n=3 2
AI
tAI
n=1
universality
φ
2d A, C, (D)1d AIII, BDI, CII
S[M ] = g̃ Sdi↵
[M ] + �̃Stop
[M ]
σ11σ11
σ12σ12nn n + 1n + 1
σ11
σ11*
generic flow (g,�) L!1�! (0, n)
nStop
[M ] �! nSboundary
[T ]
physics at the edge
experiment
Mournik et al., 2012
Churchill et al 2013
experiment
Mournik et al., 2012
experiment
class D RMT
cf. A.A. & Bagrets, 12; Beenakker, 12; Lee & Parker, 12
field theory results for boundary DoS
level spacing
topologically empty configuration exhibits spectral density similar to that of the topological state.
universality of the DoS profile testifies to relevance beyond confines of the present model
Kouwenhoven’s setup
Kouwenhoven’s setup
Kouwenhoven’s setup
x
z
N T S
L
�
V
µ
B
+ hard interface (cf. Stanescu & Tewari, 2012)
+ system geometry
+ channels structure
realistic modeling of
results
μ = 0.5Δ, B = 0.25Δμ = Δ, B = 2.66Δ
<D
oS
> [
ν]
10
12
14
16
18
20
ε /Δ−1 −0.5 0.5 1
ν
ε
DoS
[ν]
12,5
15
20
22,5
25
ε /Δ−1 −0,5 0,5 1
ν
ε
average DoS ‘typical DoS’
RMT comparison
Summary
quantum criticality of disordered topological insulators: 2 parameter scaling
stabilization of topology by localization
boundary physics
2d Z insulators
classes A, C, D described by unified approach:
S[Q] =g̃
2
Zd
2x str(@µQ@µQ) +
�̃
2
Zd
2x ✏µ⌫ str(Q@µQ@⌫Q)
2d Z insulators
classes A, C, D described by unified approach:
S[Q] =g̃
2
Zd
2x str(@µQ@µQ) +
�̃
2
Zd
2x ✏µ⌫ str(Q@µQ@⌫Q)
2d Z insulators
classes A, C, D described by unified approach:
S[Q] =g̃
2
Zd
2x str(@µQ@µQ) +
�̃
2
Zd
2x ✏µ⌫ str(Q@µQ@⌫Q)
1d 2d2 parameter field theory + +
universal 2 parameter critical flow + (+)gapless boundary modes + +critical theory understood + (-)
2d Z insulators
classes A, C, D described by unified approach:
σ11σ11
σ12σ12nn n + 1n + 1
σ11
σ11*
S[Q] =g̃
2
Zd
2x str(@µQ@µQ) +
�̃
2
Zd
2x ✏µ⌫ str(Q@µQ@⌫Q)
1d 2d2 parameter field theory + +
universal 2 parameter critical flow + (+)gapless boundary modes + +critical theory understood + (-)
generalization to Z2
generalization to Z2
case study: AII, d=2 (Kane & Fu, 2012)
2 parameter criticality
system probed by topological point defects
field theory admits point-like excitations
theta-term —> fugactiy term
generalization to Z2
case study: AII, d=2 (Kane & Fu, 2012)
2 parameter criticality
system probed by topological point defects
field theory admits point-like excitations
theta-term —> fugactiy term
generalization to Z2
case study: AII, d=2 (Kane & Fu, 2012)
2 parameter criticality
system probed by topological point defects
field theory admits point-like excitations
theta-term —> fugactiy term
Summary
quantum criticality of translationally non-invariant topological insulators
2-parameter field theory probed by continuous (Z) or point-like (Z2) topological sources
universal scaling
stabilization of topology by localization
Summary
quantum criticality of translationally non-invariant topological insulators
2-parameter field theory probed by continuous (Z) or point-like (Z2) topological sources
universal scaling
stabilization of topology by localization
Chalker et al. 2d class C
Thouless topology 2d class A
Khmelnitskii/Pruisken criticality 2d class A
Ludwig et al. exact solution 2d class C
Fisher et al., Read et al., Zirnbauer/Serban 2d class D
Zirnbauer 1d super-Fourier analysis
Mirlin, Mudry, Gruzberg et al. disordered topological matter
Beenakker et al., Brouwer et al. scattering theory of topological matter
Read, Gruzberg, Vishveshwara 1d topological quantum criticality
Ludwig et al. disorder vs. bulk-boundary correspondence
Kane/Fu, 2d class AII