Quantum criticality in
quasi-one dimensional
topological nanowires
Dmitry Bagrets
A. Altland, DB, A. Kamenev, L. Fritz, H. Schmiedt, arXiv:
1402.1738
“New frontiers for Majorana fermions” , Frascati, Italy, 5-6th May
Overview
•  Historical excursus to (disordered) topological matter
•  Paradigmatic models of 1d AIII and BDI nanowires
•  Topological Z-index with disorder
•  Field theory and quantum criticality
•  Conclusions & outlook
IQHE (Class A in 2d)
•  Two parameter flow
Khmelnitskii’84 & Pruisken‘85
Fixed points (g*, n+1/2)
with g*~1
(strong coupling limit)
?
σxx - average longitudinal conductivity
σxy - average topological index
quantum critical point
80s
Altland-Zirnbauer classification
•  Symmetries of mesoscopic system
Time reversal: U T† Hˆ TU T = Hˆ ,
U T*U T = ±1
ˆ TU = − Hˆ , U *U = ±1
Particle-hole: UC† H
C
C C
ˆ = − Hˆ
Chiral (parity): PHP
Altland & Zirnbauer‘ 97
90s
1d Delocalization
90s
•  Disorder-induced localization
D ≤ 2: all electron states are localized (Anderson’58)
•  Delocalization in quasi-1d geometries (DMPK)
98: AIII quantum wire (Brouwer, Mudry, Simons, Altalnd)
99: D quantum wire (Brouwer, Mudry, Furusaki)
04: AIII, BDI, CII, D, DIII (Gruzberg, Read, Vishveshwara)
•  Universal interpretation
Topological insulators at quantum critical point
Classification table of topological
insulators & superconductors
Kitaev‘ 2009
Ludwig, Ryu, Schnyder, Furusaki‘
2009
00s
Classification table of topological
insulators & superconductors
Kitaev‘ 2009
Ludwig, Ryu, Schnyder, Furusaki‘
2009
00s
IQHE
SQHE
Classification table of topological
insulators & superconductors
Kitaev‘ 2009
Ludwig, Ryu, Schnyder, Furusaki‘
2009
this talk
00s
IQHE
SQHE
Topological spin-orbit nanowire
•  1D spin-orbit-coupled wire in proximity to s-wave
superconductor
Class BDI or D
i ( Δ ± h) ⎞
⎛ −iu∂ x
H ± = ⎜
⎟
−
i
(
Δ
±
h
)
i
u
∂
x
⎝
⎠
Lutchyn, Sau, Das Sarma‘ 2010
Oreg, Refael, von Oppen‘ 2010
Topological spin-orbit nanowire
Mourik, Zuo, Frolov, Plissard, Bakkers, Kouwenhoven‘ 2012
p-wave nanowire (Kitaev’s chain)
Majoranas
j+1
j
j-1
N
•  Model Hamiltonian:
L
l-1
(
l
l+1
)
H = ∑l =1 ⎡ψ l ( µ + tˆ ' )σ 3phψ l + ψ l ( −tσ 3ph + iΔσ 2ph )ψ l +1 + h.c ⎤
⎣
⎦
p-wave nanowire (Kitaev’s chain)
Majoranas
tˆ '
j+1
j
j-1
N
•  Model Hamiltonian:
L
l-1
(
l
l+1
)
H = ∑l =1 ⎡ψ l ( µ + tˆ ' )σ 3phψ l + ψ l ( −tσ 3ph + iΔσ 2ph )ψ l +1 + h.c ⎤
⎣
⎦
p-wave nanowire (Kitaev’s chain)
Majoranas
tˆ, Δ
j+1
j
j-1
N
•  Model Hamiltonian:
L
l-1
(
l
l+1
)
H = ∑l =1 ⎡ψ l ( µ + tˆ ' )σ 3phψ l + ψ l ( −tσ 3ph + iΔσ 2ph )ψ l +1 + h.c ⎤
⎣
⎦
p-wave nanowire (Kitaev’s chain)
Majoranas
tˆ, Δ
j+1
j
j-1
N
•  Model Hamiltonian:
L
l-1
l
(
l+1
)
H = ∑l =1 ⎡ψ l ( µ + tˆ ' )σ 3phψ l + ψ l ( −tσ 3ph + iΔσ 2ph )ψ l +1 + h.c ⎤
⎣
⎦
•  Class BDI:
Hˆ , σ 1ph = 0, σ 1ph Hˆ T σ 1ph = − Hˆ
{
}
p-wave nanowire (Kitaev’s chain)
Majoranas
tˆ, Δ
j+1
j
j-1
N
•  Model Hamiltonian:
L
l-1
l
(
l+1
)
H = ∑l =1 ⎡ψ l ( µ + tˆ ' )σ 3phψ l + ψ l ( −tσ 3ph + iΔσ 2ph )ψ l +1 + h.c ⎤
⎣
⎦
•  Class BDI:
Hˆ , σ 1ph = 0, σ 1ph Hˆ T σ 1ph = − Hˆ
{
}
p-wave nanowire (Kitaev’s chain)
Majoranas
tˆ, Δ
j+1
j
j-1
N
•  Model Hamiltonian:
L
l-1
l
(
l+1
)
H = ∑l =1 ⎡ψ l ( µ + tˆ ' )σ 3phψ l + ψ l ( −tσ 3ph + iΔσ 2ph )ψ l +1 + h.c ⎤
⎣
⎦
•  Class BDI:
Hˆ , σ 1ph = 0, σ 1ph Hˆ T σ 1ph = − Hˆ
{
}
p-wave nanowire (Kitaev’s chain)
tˆ, Δ
j+1
j
j-1
N
•  Model Hamiltonian:
L
l-1
l
(
l+1
)
H = ∑l =1 ⎡ψ l ( µ + tˆ ' )σ 3phψ l + ψ l ( −tσ 3ph + iΔσ 2ph )ψ l +1 + h.c ⎤
⎣
⎦
•  Class BDI:
Hˆ , σ 1ph = 0, σ 1ph Hˆ T σ 1ph = − Hˆ
{
}
p-wave nanowire (Kitaev’s chain)
lattice constant
•  Long-wave limit:
1
m= 2,
ta
u = Δa
⎛ µ + p 2 2m
⎞
−iup
⎟ ψ p
H = ∑ψ p ⎜
2
⎜
⎟
iu
p
−
µ
+
p
2
m
(
)
p
⎝
⎠
•  Class BDI:
Hˆ , σ 1ph = 0, σ 1ph Hˆ T σ 1ph = − Hˆ
{
}
µ
µ3
µ2
µ1
Topological Z-index
n=0
+π
n=1
n=2
n=3
dk
n = −i ∫
tr(Qk−1∂ k Qk )
2π
−π
Chiral basis:
0
tI
1
tdeik
n=
∫ µ + teik
2π i —
⎛
H k = ⎜ T
⎝ Qk
Qk ⎞
⎟
⎠
Im(eik)
Re(eik)
bI
Disordered p-wave nanowire
Majoranas
j+1
j
j-1
N
•  Inter-chain random hopping:
L
H V = ∑ l =1ψ l Vˆ3,lσ 3phψ l ,
l-1
l
l+1
Vˆ3,ijlVˆ3,i l' j ' = ( w2 N ) (δ ii 'δ jj ' + δ ij 'δ ji ' )
RMT approach
Disordered p-wave nanowire
Majoranas
j+1
j
j-1
N
•  Inter-chain random hopping:
L
H V = ∑ l =1ψ l Vˆ3,lσ 3phψ l ,
l-1
l
l+1
Vˆ3,ijlVˆ3,i l' j ' = ( w2 N ) (δ ii 'δ jj ' + δ ij 'δ ji ' )
RMT approach
µ
µ3
µ2
µ1
Phase diagram of tI-wire
n=0
?
n=1
n=2
Disorder
n=3
0
tI
bI
w
tAI
AI
Phase-transition points
quantum critical lines
Winding number in k-space obviously does not work (:
How to define Z-index of
a random chain?
Scattering matrix
•  Landauer-Büttiker approach to e/h transport
!r t'"
ˆ
S =#
$
t
r
'
%
&
•  DMPK decomposition
Lyapunov exponents
"U
# " th(λ / 2) 1/ ch(λ / 2) # " U '
#
ˆ
S =$
% $ 1/ ch(λ / 2) − th(λ / 2) % $
%
V
V
'
'
('
('
(
λ=diag(λ1, ... λ4N),
λ‘s can have either sign!
Bound states
Ψ= rTrSΨ
det (1 − rˆT rˆS ) = 0
Majorana = 1/2 ( Andreev bound state at E=EF)
Scattering approach
Fulga, Hassler, Akhmerov, Beenakker‘ 2011
symmetry class
D
topological phase
Z2
DIII
Z2
x
S = −S T
spin-rotation sym.
chiral symmetry
reflection matrix
topological number
AIII
Z
Z
S = ST
x
x
S = S*
p-h symmetry
time-reversal sym.
BDI
x
S 2 = −1
x
r=r
yes, x
*
Z
S = Σ y S *Σ y
S = ΣyST Σy
x
S2 = 1
r = r * = −r T r = r * = r T
sgn Det r sgn Pf r
CII
ν (r )
r = r+
ν (r )
v = number of negative (w.r.t trivial insulator) eigenvalues
r = r + = Σ y rT Σ y
1
2
ν (r )
Parity current
{P, Hˆ } = 0
•  Quest for the field theory?
- one needs „sources“
ξ=ξ1– ξ0
iP(ξ 1 −ξ 0 )
iPξ 0
iPξ 1
ˆ
ˆ
ˆ
H 01 → e H 01e = H 01e
Axial flux ξ=ξ1– ξ0 is attached to a link
0
1
2
Parity current
{P, Hˆ } = 0
•  SUSY partition function
Z (θ ,ϕ ) =
Axial gauge field
fermions (ξ->θ)
det ⎡⎣i 0+ − H F (θ ) ⎤⎦
det ⎡⎣i 0+ − H B ( −iϕ ) ⎤⎦
bosons (ξ->-i!)
dis
Dyson symmetry classes: Nazarov’94; Rejaei’ 96
Frahm & Brouwer’ 96
BdG classes (CI,DIII): Lamacraft, Simons, Zirnbauer‘2004
Parity current
Conserved current, a ‘‘counter‘‘ position is irrelevant!
{P, Hˆ } = 0
•  SUSY partition function
Z (θ ,ϕ ) =
Axial gauge field
fermions (ξ->θ)
det ⎡⎣i 0+ − H F (θ ) ⎤⎦
det ⎡⎣i 0+ − H B ( −iϕ ) ⎤⎦
•  Axial current of P-symmetry
J P (ϕ ) = ∂θ Z (θ , ϕ ) θ =− iϕ
bosons (ξ->-i!)
dis
Parity current
•  P-current is a ‘CGF’ of Lyapunov exponents
i
J P (ϕ ) =
2
2N
⎛ ϕ − λn ⎞
th ⎜
∑
⎟
2
⎝
⎠
n =1
dis
Parity current
•  P-current is a ‘CGF’ of Lyapunov exponents
i
J P (ϕ ) =
2
2N
⎛ ϕ − λn ⎞
th ⎜
∑
⎟
2
⎝
⎠
n =1
dis
•  (Thermal) conductance
g = −4i∂ϕ J (0) = ∑ ch −2 ( λn 2 )= tr ( tˆ+tˆ )
n
R. Landauer’57, M. Büttiker‘88
Parity current
•  P-current is a ‘CGF’ of Lyapunov exponents
i
J P (ϕ ) =
2
2N
⎛ ϕ − λn ⎞
th ⎜
∑
⎟
2
⎝
⎠
n =1
dis
•  Topological index
1
1
χ = iJ P (0) = ∑ th ( λn 2 )= tr ( rˆP )
2 n
2
Fulga, Hassler, Akhmerov, Beenakker‘ 2011
Parity current
•  P-current is a ‘CGF’ of Lyapunov exponents
i
J P (ϕ ) =
2
2N
⎛ ϕ − λn ⎞
th ⎜
∑
⎟
2
⎝
⎠
n =1
dis
•  Density of λ‘s
1
π
Re J p (ϕ − iπ ) =
∑ δ (ϕ − λ )
n
n
Cf. Nazarov‘ 94
dis
Field theory (NLσM)
•  SUSY partition function
twisted boundary conditions
T ( L)
Z L (θ ,ϕ ) =
∫
1
DT exp ( − S [T ]) ,
T ( L) = diag ( eϕ , eiθ )
BF
Supermatrix field T lives in a symmetric space
•  Class AIII wire:
T ∈ GL(1 |1) - group manifold
•  BDI p-wave wire: T ∈ GL(2 | 2) / OSp(2|2) -
coset
Field theory (NLσM)
•  SUSY partition function
twisted boundary conditions
T ( L)
Z L (θ ,ϕ ) =
∫
DT exp ( − S [T ]) ,
1
T ( L) = diag ( eϕ , eiθ )
BF
Supermatrix field T lives in a symmetric space
•  Class AIII wire:
T ∈ GL(1 |1) - group manifold
•  BDI p-wave wire: T ∈ GL(2 | 2) / OSp(2|2) •  Action
coset
$ ξ%
%
−1
−1
S [T ] = ∫ dx ' str ( ∂ xT ∂ xT ) + χ%str (T ∂ xT )( π1(TF)=Z
)4
*
0
L
bare localization length
SCBA topological index
Q. M. on the GL(1|1), class AIII
Grassmann field
•  Parametrization
•  Action
$ e y1
T = UTzU , Tz = &
(
−1
metric on the group
ξ%L ⎡
%
$0
,
U
=
exp
'
&σ
eiy0 ) bf
(
ρ%
0 ')
gauge field
L
⎤
2
2
2 ⎛ y1 − iy0 ⎞
S [ z ] = ∫ ⎢dy0 + dy1 + 4sinh ⎜
d
σ
d
ρ
− i χ%∫ ⎡⎣dy0 + idy1 ⎤⎦
⎟
⎥
4 0 ⎣
⎝ 2 ⎠
⎦
0
•  Berezinian
J ( y ) = sdet
depends only on radial variables
1/ 2
det g B 1
−2 " y1 − iy0 #
= sinh $
(g) =
%
det g F 2
& 2 '
“Schrödinger” equation
•  Imaginary time Schrödinger eq.
ξ%∂ x Ψ ( y, x ) =
„vector“ poitential
1
(∂α − iAα ) J ( y )(∂α − iAα ) Ψ ( y, x ) , A = χ%(1, i )
J ( y)
length becomes imaginary time !
•  Spectral decomposition
Ψ ( y, L ) = 1 +
initial conditions
dl1
∑ ∫ ( 2π ) P Ψ ( y ) e
l0 = Z +1/ 2
2
l
l
− ε l L / ξ%
, Pl = 4π ( l0 + il1 )
−1
•  Eigenfunctions & spectrum
2
2
$ y1 + iy0 % ilα yα
Ψ l ( y ) = sinh )
* e , ε l = ( l0 − χ%) + ( l1 − i χ%)
( 2 +
Solution can be found via Sutherland transformation
Flow diagram
L ξ%= 14 , 12 ,1,2,...,32
Universal description of the Z-topological chiral wires:
two parameter scaling
& L
'
1
χ ( L) = n −
erf * %(,l ± ( n − χ%))- +
∑
4 l∈Z +1/ 2,±
. ξ
/
- „running“ topological index
ξ%
−( l − χ%) L / ξ%
g ( L) =
e
∑
Class AIII
π L l∈Z +1/ 2
- (thermal) conductance
2
Phase diagram and criticality
Critical conductance g ( L ) ~
Localization length ξ
-2
%
%
%
( χ ) = ξ χ − n − 1/ 2
ξ%L
- critical
exponent ν=2
Lyapunov exponents (AIII)
L ξ%= 1, 4,16,32
drift
•  Density of λ‘s:
−π 2n 2ξ%/ L
2 %
%
n e
2ξ
π
%L + 2 χ% sh nξ
ρ (λ ) =
+ ∑ (−)
cos
π
n
λ
ξ
2
L
π
n
L
n
( (
))
Lyapunov exponents (AIII)
L ξ%= 1, 4,16,32
delocalization
•  Density of λ‘s:
−π 2n 2ξ%/ L
2 %
%
n e
2ξ
π
%L + 2 χ% sh nξ
ρ (λ ) =
+ ∑ (−)
cos
π
n
λ
ξ
2
L
π
n
L
n
( (
))
NLσM vs. T-matrix
χ=3/2
•  SCBA topological index
−1
2
Σ + = w (i 0 − H 0 − Σ+ )
Half-integer index defines
critical lines in (µ,w)-plane
χ%= − 2i tr G+ Pˆ ∂ k H
(
)
NLσM vs. T-matrix
χ=3/2
•  SCBA topological index
−1
2
Σ + = w (i 0 − H 0 − Σ+ )
Half-integer index defines
critical lines in (µ,w)-plane
χ%= − 2i tr G+ Pˆ ∂ k H
(
)
M.-T. Rieder, P.Brouwer, I. Adagideli‘ 2013 wn
1/ 2
= t ( N ( 2n + 1) )
NLσM vs. T-matrix
χ=3/2
•  T-matrix approach
#ψ l +1 $ # − H1,−l1H 0,l
%ψ &=%
1
' l ( '
− H1,−l1H1,+l $ # ψ l $
&%
&
ψ
0
l
−
1
'
(
(
L
T = ∏ Tl ⇒ exp( Lλ j )
l =0
Z-index = # (negative Lyapunov exponents)
NLσM vs. T-matrix
χ=3/2
•  T-matrix approach
#ψ l +1 $ # − H1,−l1H 0,l
%ψ &=%
1
' l ( '
− H1,−l1H1,+l $ # ψ l $
&%
&
ψ
0
l
−
1
'
(
(
L
T = ∏ Tl ⇒ exp( Lλ j )
l =0
Z-index = # (negative Lyapunov exponents)
Conclusions
•  Classes AIII, BDI & CII described by a unified way
- physical response is probed by a chiral gauge flux
- structurally identical SUSY actions with top. θ-term
- similar flow of the universal pair (g,χ) & phase portraits
•  Extension to Z2 topological wires D and DIII ?
- σ-model manifold is disconnected ( O(N) group )
- one need to introduce kinks in the path integral
- what is the relation to the DMPK approach (?)
Acknowledgment
Alexander Altland
Univ. zu Köln
Lars Fritz
Utrecht University
Alex Kamenev
Univ. of Minnesota
Hanno Schmiedt
Univ. zu Köln
Bipartite disordered chain (AIII)
t1
N
•  Class AIII Hamiltonian:
l-1
l
t2
j+1
j
j-1
l+1
L
H = ∑l =1ψ l ⎡⎣(t1 + t2 ) + (t1 − t2 ) P ⎦⎤ ψ l +1 + h.c
Pψ l , j = ( −)lψ l , j ⇒ Hˆ , P = 0
{
}
Parity operator
P = σ 3AB
Bipartite disordered chain (AIII)
t1
N
•  Bond disorder
L
l-1
H V = ∑l =1ψ l ,iVˆl ψ l +1, j ,
ij
Vˆl Vˆ
ij
*
( )
i' j'
l'
l
t2
l+1
= ( w2 N ) δii 'δ jj 'δ ll '
j+1
j
j-1
Bipartite disordered chain (AIII)
j+1
j
j-1
N
•  Bond disorder
L
l-1
H V = ∑l =1ψ l ,iVˆl ψ l +1, j ,
ij
Vˆl Vˆ
ij
*
( )
i' j'
l'
l
l+1
= ( w2 N ) δii 'δ jj 'δ ll '