Superconducting Transition Edge Sensors & Topological Defects Formation
From the topological invariants to the edge states of ...
Transcript of From the topological invariants to the edge states of ...
From the topological invariants to the edge states of topological insulators
Victor Gurarie
Euler Mathematical Institute, July 2013
1
work with A. Essin
Monday, July 15, 13
In this talk 2
• Bulk-boundary correspondence for the topological invariants of topological insulators with Green’s functions
• Integer quantum Hall effect
• Fractional quantum Hall effect
• 1D topological insulators with and without interactions, spin-1 spin chain
VG, PRB 2011A. Essin and VG, PRB 2012S. Manmana, A. Essin, R. Noack, VG, PRB 2012VG and A. Essin, JETP Lett., 2013
Monday, July 15, 13
Topological insulators 4
1. Topological insulators insulate in the bulk,conduct at the boundary.
2. Topological insulators have topological invariants computed in the boundary-less geometry.
It is well known that
3. The boundary of the topological insulator is a topological (semi)metal, with its own topological invariant.
It is somewhat less known that
Monday, July 15, 13
Bulk-boundary correspondence 5
The bulk invariant and the edge invariant are equal.
Nd = Nd�2(⇤)�Nd�2(�⇤)
Bulk invariant in d dimensions Edge invariant
G. Volovik, 1980s; VG, A. Essin, PRB 2011
The edge is not an insulator; the edge invariant is of the type used to characterize the
topological (semi)metals.
Monday, July 15, 13
Edge topological invariant in d-dimensions 6
1. Take a d-dim insulator with the bulk invariant Nd
Monday, July 15, 13
Edge topological invariant in d-dimensions 6
1. Take a d-dim insulator with the bulk invariant Nd
2. d-1 dimensional edge with d-1 momenta
Example: 2D edge of a 3D insulator (d=3)
Monday, July 15, 13
Edge topological invariant in d-dimensions 6
1. Take a d-dim insulator with the bulk invariant Nd
2. d-1 dimensional edge with d-1 momenta3. Edge is not an insulator
Example: 2D edge of a 3D insulator (d=3)
Fermi surface
px
py
Monday, July 15, 13
Edge topological invariant in d-dimensions 6
1. Take a d-dim insulator with the bulk invariant Nd
2. d-1 dimensional edge with d-1 momenta3. Edge is not an insulator
4. Fix pd-1=Λ some large value
Example: 2D edge of a 3D insulator (d=3)
Fermi surface
px
py⇤
Monday, July 15, 13
Edge topological invariant in d-dimensions 6
1. Take a d-dim insulator with the bulk invariant Nd
2. d-1 dimensional edge with d-1 momenta3. Edge is not an insulator
4. Fix pd-1=Λ some large value
5. Now the edge is an d-2 dim insulator
Example: 2D edge of a 3D insulator (d=3)
Fermi surface
px
py⇤
H⇤(px) = H(px
, py
)|py=⇤
Effective 1D insulator
Monday, July 15, 13
Edge topological invariant in d-dimensions 6
1. Take a d-dim insulator with the bulk invariant Nd
2. d-1 dimensional edge with d-1 momenta3. Edge is not an insulator
4. Fix pd-1=Λ some large value
5. Now the edge is an d-2 dim insulator
6. Calculate its invariant Nd-2(Λ)
Example: 2D edge of a 3D insulator (d=3)
Fermi surface
px
py⇤
H⇤(px) = H(px
, py
)|py=⇤
Effective 1D insulator
Monday, July 15, 13
Edge topological invariant in d-dimensions 6
1. Take a d-dim insulator with the bulk invariant Nd
2. d-1 dimensional edge with d-1 momenta3. Edge is not an insulator
4. Fix pd-1=Λ some large value
5. Now the edge is an d-2 dim insulator
6. Calculate its invariant Nd-2(Λ)
7. Claim: Nd =Nd-2(Λ)-Nd-2(-Λ)
Example: 2D edge of a 3D insulator (d=3)
Fermi surface
px
py
�⇤
⇤H⇤(px) = H(p
x
, py
)|py=⇤
Effective 1D insulator
The edge is a topological metalMonday, July 15, 13
Edge topological invariant 7
Example: 3D edge of a 4D insulator
Fermi surface
1. Take a d-dim insulator with the bulk invariant Nd
2. d-1 dimensional edge with d-1 momenta3. Edge is not an insulator
4. Fix pd-1=Λ some large value
5. Now the edge is an d-2 dim insulator
6. Calculate its invariant Nd-2(Λ)
7. Claim: Nd =Nd-2(Λ)-Nd-2(-Λ)
Monday, July 15, 13
Example: an edge of a 4D TRI insulator 8
This edge is taken as
because it is1. linear in momenta2. time-reversal invariantH(p) = �yH
⇤(�p)�y
But does it have the right edge invariant?
H = vX
i=x,y,z
�i
pi
� µ
Monday, July 15, 13
Example: an edge of a 4D TRI insulator 8
This edge is taken as
because it is1. linear in momenta2. time-reversal invariantH(p) = �yH
⇤(�p)�y
But does it have the right edge invariant?
H = v�x
px
+ v�y
py
± v��z
� µ
Fix pz=+Λ or pz=-ΛEffectively 2D.
N2(�)�N2(��) = 1 Well known relation.Yes, it is an edge.
H = vX
i=x,y,z
�i
pi
� µ
LFSG, 1994Monday, July 15, 13
Topological invariant for even dimensions type 10
Matsubara Green’s function
Z
Gab(!,p) = [i! �H]�1ab
Monday, July 15, 13
Topological invariant for even dimensions type 10
Matsubara Green’s function
known numerical coefficient
Nd = Cd ��0...�d tr
Zd⇥ddpG�1⇤�0G . . .G�1⇤�dG
Summation over each � = ⇥, p1, . . . , pd is implied
topological invariant
Z
Works only if d is even
Gab(!,p) = [i! �H]�1ab
Monday, July 15, 13
Topological invariant for even dimensions type 10
If d=2 this coincides with the TKNN invariant (“Chern number”). Niu, Thouless, Wu (1985)
Matsubara Green’s function
known numerical coefficient
Nd = Cd ��0...�d tr
Zd⇥ddpG�1⇤�0G . . .G�1⇤�dG
Summation over each � = ⇥, p1, . . . , pd is implied
topological invariant
Z
Works only if d is even
Gab(!,p) = [i! �H]�1ab
⇡d+1 (GL(N ,C)) = Z ensures this is a topological invariant
Monday, July 15, 13
Integer QHE 11
p
B ✏
p�⇤
⇤
Edge spectrum
Edge invariant
!
p
tr
Idsµ
2⇡iG�1@µG = tr
Z 1
�1
d!
2⇡iG�1@!G
����p=⇤
p=�⇤
⌘ N0(⇤)�N0(�⇤)
�⇤ ⇤
Monday, July 15, 13
Topological invariant in zero dimensions 12
N0 = tr
Zd!
2⇡iG�1@!G =
Z 1
�1
d!
2⇡i@! ln detG
G = [i! �H]�1
N0 = �X
n
Z 1
�1
d!
2⇡i@! ln (i! � ✏n)
energy levels H n = ✏n n
topological invariant = excess of the number of positive energy states over the number of negative energy states
N0 =1
2
X
n
sign ✏n
Nd = Cd ��0...�d tr
Zd⇥ddpG�1⇤�0G . . .G�1⇤�dG
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Integer QHE 13
N0(⇤)�N0(�⇤) = 1p
✏
p�⇤
⇤
There has to be a level such thatThis is the chiral edge state!
✏m(⇤) > 0, ✏m(�⇤) < 0
B
N0(⇤)�N0(�⇤) =1
2
X
n
[sign ✏n(⇤)� sign ✏n(�⇤)]
Monday, July 15, 13
Integer quantum Hall effect 14
Magnetic fieldthrough this 2D rectangle
�xy
=e2
h
B
Suppose
Monday, July 15, 13
Integer quantum Hall effect 14
Magnetic fieldthrough this 2D rectangle
�xy
=e2
h
B
Suppose
h
e2�xy
= N2 =1
24⇡2
X
↵��
✏↵��
Zd!d2p tr
⇥G�1@
↵
GG�1@�
GG�1@�
G⇤
TKNN invariant constructed out of the Green’s function G(!, p
x
, py
) = [i! �H(px
, py
)]�1
h
e2�xy
= N2 = 1In this example
Monday, July 15, 13
Derivation of the bulk-boundary correspondence 15
B
pG(x0
, x
00) ! G(x, k)
tr
Zd!G
�1@
!
G
����p=⇤
p=�⇤
= tr1
2i
X
a,b=x,k
Zd!dxdk G
�1@
a
GG
�1@
b
GG
�1@
!
G ✏
ab
������
p=⇤
p=�⇤
G
�1(x0, x
00) ! G
�1(x, k) +1
2i
X
a,b=x,k
G
�1@
a
GG
�1@
b
G
�1✏
ab
+ . . .
x
= N2|x=L
x=�L
BoundaryBulk
Monday, July 15, 13
Integer quantum Hall effect with disorder 16
Disorder breaks translational invariance, so no momentum.
Monday, July 15, 13
Integer quantum Hall effect with disorder 16
Old idea of Thouless, Wu, Niu: impose phases across the system
✓x
✓y
Gij
(!, ✓x
, ✓y
. . . )
Summation over each ↵ = !, ✓1, . . . , ✓d is implied
Nd = Cd ✏↵0...↵d tr
Zd!dd✓G�1@↵0G . . .G�1@↵dG
Disorder breaks translational invariance, so no momentum.
Monday, July 15, 13
Integer quantum Hall effect with disorder 16
Old idea of Thouless, Wu, Niu: impose phases across the system
✓x
✓y
Gij
(!, ✓x
, ✓y
. . . )
Summation over each ↵ = !, ✓1, . . . , ✓d is implied
Nd = Cd ✏↵0...↵d tr
Zd!dd✓G�1@↵0G . . .G�1@↵dG
N0(⇤)�N0(�⇤) = 1
✏
�⇤⇤ ✓
d = 2 : This edge level must be delocalized!
Disorder breaks translational invariance, so no momentum.
Monday, July 15, 13
Bulk-boundary valid with interactions 18
Nd = Cd ��0...�d tr
Zd⇥ddpG�1⇤�0G . . .G�1⇤�dG
Summation over each � = ⇥, p1, . . . , pd is implied
Nd = Nd�2(⇤)�Nd�2(�⇤)
Bulk invariant in d dimensions Edge invariant
⇡d+1 (GL(N ,C)) = Z
Bulk-boundary is valid regardless of the origin of G
ensures this is a topological invariant
Monday, July 15, 13
0D invariant with interactions 19
N0 =
Z 1
�1
d!
2⇡i@! ln detG
detG =Y
n
1
i! � ✏n no interactions
detG =
Qm (i! � rn)Qn (i! � ✏n)
with interactions
poles (states)
zeros
VG, PRB 2011
Monday, July 15, 13
0D invariant with interactions 19
N0 =
Z 1
�1
d!
2⇡i@! ln detG
detG =Y
n
1
i! � ✏n no interactions
detG =
Qm (i! � rn)Qn (i! � ✏n)
with interactions
poles (states)
zeros
VG, PRB 2011
N0(⇤)�N0(�⇤) =1
2
X
n
(sign ✏n(⇤)� sign ✏n(�⇤))�1
2
X
n
(sign rn(⇤)� sign rn(�⇤))
relevant formula
Monday, July 15, 13
FQHE at simple fractions 20
⌫ =1
2m+ 1G(x, t) ⇠ 1
(x� vt)2m+1
electron Green’s function
G(!, p) =(i! + vp)2m
i! � vp
Wen, 1995
✏(p) = vp, r(p) = �vp
B
pppole
zero
Monday, July 15, 13
FQHE at simple fractions 20
⌫ =1
2m+ 1G(x, t) ⇠ 1
(x� vt)2m+1
electron Green’s function
G(!, p) =(i! + vp)2m
i! � vp
Wen, 1995
✏(p) = vp, r(p) = �vp
B
p
N0(⇤)�N0(�⇤) =1
2
X
n
(sign ✏n(⇤)� sign ✏n(�⇤))�1
2
X
n
(sign rn(⇤)� sign rn(�⇤))
N0(⇤)�N0(�⇤) = 2m+ 1 = N2
N2 = ✏↵0↵1↵2 tr
Zd!dp
x
dpy
24⇡2G�1@
↵0GG�1@↵1GG�1@
↵2G = 2m+ 1
�xy
=1
2m+ 16= N2 = 2m+ 1
ppole
zero
Monday, July 15, 13
Issues with the winding 21
G =X
n
⇢ni! � ✏n
Can’t have arbitrary winding
Large windings probably are a low energy statement
Z 1
�1
d!
2⇡i@! lnG =
1
2sign
X
n
⇢n✏n
Monday, July 15, 13
Significance of N2 for FQHE 22
• We don’t know G in the bulk of quantum Hall system analytically
• We do know the low energy edge structure analytically, and we used it to calculate the invariant.
• Thus we calculated a quantity in the bulk from knowing the theory structure at the edge
• Numerically we typically calculate the bulk properties, but not the edge properties.
• We could determine N2 in the bulk and compare with the value calculated analytically at the edge
Monday, July 15, 13
Generic Abelian FQH states 23
S =1
4⇡
X
ab
Zdxdt [K
ab
@
t
�
a
@
x
�
b
� V
ab
@
a
�@
b
�]
K-matrix (universal) V-matrix (non-universal)
a = eiP
Kab�b electron operators
Gab(x, t) ⇠Y
c
�ab
(x� vct)↵a
c can be found in terms of K, Vvc, ↵a
c
N0(⇤)�N0(�⇤) =1
2⇡
X
a
Im lnGaa(!, p)|!=1!=�1
�����
p=⇤
p=�⇤
= trK
N2 = trKMonday, July 15, 13
Non-Abelian QH states 24
Read-Rezayi (parafermion) stateslevelN
parameterM , always odd
⌫ =N
MN + 2Filling fraction
Edge Green’s function calculated with the conformal field theory
G(x, t) ⇠ 1
(x� vnt)2� 2
N (x� vct)M+ 2
N
what matters is the sum of these two
N2 = M + 2
N2 = 3For a Pfaffian at ν=5/2, M=1, and For an anti-Pfaffian at ν=5/2, N2 = 1
Monday, July 15, 13
FQHE: conclusions 25
• Chern number-like topological invariant labels FQH states
• The correspondence between the invariant and the states is not one to one: changing the invariant changes the state, but two different states can have the same invariant
• Nevertheless, it should, at least, serve as an additional way to distinguish FQH states numerically
Monday, July 15, 13
1D: Peierls-Hubbard model 27
ax,�
! (�1)xa†x,�
a†x,�
! (�1)xax,�
Chiral transformations
This model is chirally invariant at half filling
t1, t2 real
H =X
x, �=",#
ht1 a
†2x+1,�a2x,� + t2 a
†2x+2,�a2x+1,�
i+h. c.+U
X
x
a†x"a
†x#ax#ax"
H = ⌃H⌃⌃
Monday, July 15, 13
1D: Peierls-Hubbard model 27
ax,�
! (�1)xa†x,�
a†x,�
! (�1)xax,�
Chiral transformations
This model is chirally invariant at half filling
t1, t2 real
U > 0 Mott insulator -> AF spin chain
H ⇡X
x
t21U
S2x · S2x+1 +t22U
S2x+1 · S2x+2
�
H =X
x, �=",#
ht1 a
†2x+1,�a2x,� + t2 a
†2x+2,�a2x+1,�
i+h. c.+U
X
x
a†x"a
†x#ax#ax"
H = ⌃H⌃⌃
Monday, July 15, 13
1D: Peierls-Hubbard model 27
ax,�
! (�1)xa†x,�
a†x,�
! (�1)xax,�
Chiral transformations
This model is chirally invariant at half filling
t1, t2 real
U > 0 Mott insulator -> AF spin chain
H ⇡X
x
t21U
S2x · S2x+1 +t22U
S2x+1 · S2x+2
�
H =X
x, �=",#
ht1 a
†2x+1,�a2x,� + t2 a
†2x+2,�a2x+1,�
i+h. c.+U
X
x
a†x"a
†x#ax#ax"
U
free fermions
Mott insulator withgapless spin excitations
Mott insulator withgapless charge
excitations
U=0 nontopological gapped phase
U=0 topological gapped phase
�
t=
t1 � t2t1 + t2
What about the rest of the
phase diagram?
H = ⌃H⌃⌃
Monday, July 15, 13
Calculating the 1D invariant with DMRG 28
g =
✓0 vv⇤ 0
◆g = �⌃g⌃
N1 = tr
Zdp
4⇡i⌃ g�1@pg = tr
Zdp
2⇡iv�1@pv
winding of v around the complex plane origin
� = arg v
�(pj) = �0 ! N1 =X
j
sign
d�(p)
dp
����p=pj
!
Discussions with A. Kitaev
⌃ =
✓1 00 �1
◆g = G(!, p)|!=0
Monday, July 15, 13
Numerical phase diagram 29
δ/t
U/t
-1 1-0.3 0.3
U/t=10
U/t=0
U/t=2
U/t=10
U/t=0
U/t=2
��
��/2
0
�/2
�
�� ��/2 0 �/2 �
k
L = 22, � = 0.3, OBC; � = i
Winding number of Green’s function frequency � = i
U = 0U = 2
U = 10U = 10, L = 250
��
��/2
0
�/2
�
�� ��/2 0 �/2 �
k
L = 22, � = -0.3, OBC; � = i
Winding number of Green’s function frequency � = i
U = 0U = 2
U = 10
A. Essin, S. Manmana, R. Noack, VGPRB 2012
Monday, July 15, 13
Numerical phase diagram 29
δ/t
U/t
-1 1-0.3 0.3
U/t=10
U/t=0
U/t=2
U/t=10
U/t=0
U/t=2
��
��/2
0
�/2
�
�� ��/2 0 �/2 �
k
L = 22, � = 0.3, OBC; � = i
Winding number of Green’s function frequency � = i
U = 0U = 2
U = 10U = 10, L = 250
��
��/2
0
�/2
�
�� ��/2 0 �/2 �
k
L = 22, � = -0.3, OBC; � = i
Winding number of Green’s function frequency � = i
U = 0U = 2
U = 10
A. Essin, S. Manmana, R. Noack, VGPRB 2012
U
free fermions
Mott insulator withgapless spin excitations
Mott insulator withgapless charge
excitations
Nontopological gapped phase with
N1=0
Topological gapped phase with
N1=2�
t=
t1 � t2t1 + t2
N1 = tr
Zdp
4⇡i⌃ g�1@pg
⌃ =
✓1 00 �1
◆
Edge states only if δ<0
N1 =
⇢0, �t > 02, �t < 0
Monday, July 15, 13
Spin-1 chain 30
H =X
x even,�=",#
ht1a
†x+1,�ax,� + t2a
†x+2,�ax+1,�
i+ h.c.+U
X
x
a†x"a
†x#ax#ax"
�JX
x
S2x · S2x+1 Large U Mott insulator
H ⇡X
x
⇢t21U
� J
�S2x · S2x+1 +
t22U
S2x+1 · S2x+2
�
Sufficiently large J -> Haldane’s spin-1 Heisenberg chain
{ { { {S = 1S = 1
S = 1S = 1
Monday, July 15, 13
Application 6: Haldane spin chain 31
H =X
x even,�=",#
ht1a
†x+1,�ax,� + t2a
†x+2,�ax+1,�
i+ h.c.+U
X
x
a†x"a
†x#ax#ax"
�JX
x
S2x · S2x+1
{ { { {S = 1S = 1
S = 1S = 1
DMRG confirms that N1=2(large positive J)Spin-1/2 edge states are due to the nonzero value of the topological invariant.
A. Essin, S. Manmana, R. Noack, VGPRB 2012
−π
−π/2
0
π/2
π
−π −π/2 0 π/2 π
k
V(k,ω = i), U/t = 10, δt = 0J = 2 (AFM)J = -2 (FM)
Monday, July 15, 13
Conclusions 32
• Topological invariants constructed out of single-particle Green’s functions tell us something about the edge states, even when there are interactions, in 1D and in 2D.
• Yet they are not as powerful as when there are no interactions. In particular, there is no one to one correspondence between the invariant and the phase. The invariant is not equal to Hall conductance in the FQH phase.
• They are still useful to distinguish different topological states, if used with care.
• Generalizations: many particle Green’s function, Thouless’s many body invariant?
Monday, July 15, 13