Topological insulators Anderson insulators … Notes/A_Altland.pdf · disorder in topological...
Transcript of Topological insulators Anderson insulators … Notes/A_Altland.pdf · disorder in topological...
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Topological insulators
Anderson insulators
Symmetry classes
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Topological insulators
Anderson insulators
Symmetry classes
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Su-Schrieffer-Haeger chain
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SSH chain
system gappedµ > t
system gapped & topologically chargedµ < t
k
µ > t
k k
µ < tµ = t
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why topological?
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why topological?
introduce Bloch crystal momentumk 2 [0, 2⇡]
Q̂ ! Q(k) 2 C
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why topological?
introduce Bloch crystal momentumk 2 [0, 2⇡]
Q̂ ! Q(k) 2 C
![Page 8: Topological insulators Anderson insulators … Notes/A_Altland.pdf · disorder in topological insulators Alexander Altland, Dmitry Bagrets (Cologne) Alex Kamenev (Minnesota) Lars](https://reader031.fdocuments.us/reader031/viewer/2022013107/5b8b574309d3f20c398ba4de/html5/thumbnails/8.jpg)
why topological?
introduce Bloch crystal momentumk 2 [0, 2⇡]
Q̂ ! Q(k) 2 C
![Page 9: Topological insulators Anderson insulators … Notes/A_Altland.pdf · disorder in topological insulators Alexander Altland, Dmitry Bagrets (Cologne) Alex Kamenev (Minnesota) Lars](https://reader031.fdocuments.us/reader031/viewer/2022013107/5b8b574309d3f20c398ba4de/html5/thumbnails/9.jpg)
why topological?
introduce Bloch crystal momentumk 2 [0, 2⇡]
Q̂ ! Q(k) 2 C
![Page 10: Topological insulators Anderson insulators … Notes/A_Altland.pdf · disorder in topological insulators Alexander Altland, Dmitry Bagrets (Cologne) Alex Kamenev (Minnesota) Lars](https://reader031.fdocuments.us/reader031/viewer/2022013107/5b8b574309d3f20c398ba4de/html5/thumbnails/10.jpg)
Topological insulators
Anderson insulators
Symmetry classes
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Anderson localization
d � 3
d < 3
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one-parameter scaling
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one-parameter scalingAb
raha
ms
et a
l. 79
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one-parameter scalingAb
raha
ms
et a
l. 79
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one-parameter scalingAb
raha
ms
et a
l. 79
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Topological insulators
Anderson insulators
Symmetry classes
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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)
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symmetry classes
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symmetry classes
IQH
SSH
‘Kitaev chain’
‘Majorana quantum wire’
spin QH‘topolgical insulator (HgTe)’
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Topological insulators
Anderson insulators
Symmetry classes
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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)
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case study: IQH
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disordered topological matter — time line
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disordered topological matter — time line
80s
quantum Hall
SSH
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quantum Hall effect (class A)
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quantum Hall effect (class A)
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quantum Hall effect (class A)
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quantum Hall effect (class A)
1998 Nobel prize press release
smooth profiles of , becoming more singular upon lowering temperature
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quantum Hall effect (class A)
1998 Nobel prize press release
smooth profiles of , becoming more singular upon lowering temperature
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quantum Hall effect (class A)
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quantum Hall effect (class A)
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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
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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
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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
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disordered topological matter — time line
80s
quantum Hall
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disordered topological matter — time line
90s
80s
quantum Hall
symmetry classes
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disordered topological matter — time line
90s
00s
80s
quantum Hall
symmetry classes
1d delocalization
2d delocalization
novel quantumHall effects
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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)
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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.)
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disordered topological matter — time line
90s
00s
80s
quantum Hall
symmetry classes
1d delocalization
2d delocalization
novel quantumHall effects
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disordered topological matter — time line
90s
00s
80s
quantum Hall
symmetry classes
1d delocalization
topological matter
2d delocalization
novel quantumHall effects
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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
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disorder vs. topology
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topological Anderson insulator
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topological Anderson insulator
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topological Anderson insulator
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Addition of disorder to a clean band insulator …
invalidate k-theory of band gaps/topological indices
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Addition of disorder to a clean band insulator …
invalidate k-theory of band gaps/topological indices
destroys band gaps and
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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
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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 + (+)
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topological Anderson insulator
cf. Motrunich, Damle and Huse 2001, Groth et al. 2010
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topological Anderson insulator
cf. Motrunich, Damle and Huse 2001, Groth et al. 2010
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topological Anderson insulator
cf. Motrunich, Damle and Huse 2001, Groth et al. 2010
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topological Anderson insulator
early works
cf. Motrunich, Damle and Huse 2001, Groth et al. 2010
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topological Anderson insulator
Brouwer et al., 2012; Groth et al. 2010; …
early works
cf. Motrunich, Damle and Huse 2001, Groth et al. 2010
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topological Anderson insulator
cf. Motrunich, Damle and Huse 2001, Groth et al. 2010
critical theory ?
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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
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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
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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)’
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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)’
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(field) theory
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generalized definition of winding number
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generalized definition of winding number
generalized definition
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generalized definition of winding number
generalized definition
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generalized definition of winding number
generalized definition
want to know of disordered system
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generalized definition of winding number
generalized definition
want to know of disordered system
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field theorymicroscopic system
critical phenomena
Ginzburg Landau theory
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field integral
localization length average index
theta term
matrix fields
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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
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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
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phase diagram
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phase diagram
phase boundaries: lines of half integer bare topological index
flow describes stabilization of self-averaging topological phase/boundary state generation
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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
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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 ]
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physics at the edge
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experiment
Mournik et al., 2012
Churchill et al 2013
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experiment
Mournik et al., 2012
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experiment
class D RMT
cf. A.A. & Bagrets, 12; Beenakker, 12; Lee & Parker, 12
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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
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Kouwenhoven’s setup
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Kouwenhoven’s setup
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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
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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
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Summary
quantum criticality of disordered topological insulators: 2 parameter scaling
stabilization of topology by localization
boundary physics
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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)
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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)
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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 + (-)
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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 + (-)
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generalization to Z2
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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
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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
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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
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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
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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