Download Bosonic Symmetry Protected Topological States: Theory, Numerics

Bosonic Symmetry Protected Topological States:
Theory, Numerics, And Experimental Platform
Zhen Bi University of California, Santa Barbara
Acknowledgements
•
Collaborators from UCSB:
Cenke Xu, Leon Balents, Andrea Young,
Kevin Slagle, Yi-Zhuang You.
•
Collaborators from Penn. State U:
Ruixing Zhang, Chao-Xing Liu.
•
Collaborators from China:
Zi-Yang Meng (IOP, CAS)
Yuan-Yao He, Han-Qing Wu, Zhong-Yi Lu (Renming U.)
"Oversimplified" Introduction to SPT States
• Bosonic Symmetry Protected Topological (SPT) States
• Generalization of TI/TSC to spin/boson systems
• Bulk: gapped and non-degenerate; Boundary: gapless
• Always require strong interactions
1d Haldane phase of spin-1 chain
• Example:
1
Haldane 1983
H=
Affleck, Kennedy, Lieb, Tasaki 1987
•
Si · S j +
〈i j〉
3
Si · S j
2
Field theory: O(3) NLSM + Θ term ( π2 S2 = ℤ )
S=
1
ⅈΘ
2
n · ∂t n × ∂x n
ⅆx ⅆτ (∂μ n) +
g
4π
Θ=2π
build with Neél order parameter n~(-)i S i
Haldane 1988, Ng 1994, Coleman 1976
"Oversimplified" Introduction to SPT States
dimensional bosonic SPT states are much more
• Higher
complicated, they can be classified mathematically.
Chen, Gu, Liu, Wen 2011; Kapustin 2014; Wen 2014; Kitaev …
• What about lattice model/Hamiltonian?
model
• Levin-Gu
ⅈπ
HLG = -
Xi , Xi = -ⅈ Xi
i
• CZX model
exp
〈 jk〉∈
4
Z j Zk
�
�
�
• The boundary is gapless assuming the Z2 symmetry.
Levin, Gu 2012
Chen, Liu, Wen 2012
"Oversimplified" Introduction to SPT States
• More generic properties:
boundary of many 2d bosonic SPT states can be
• The
thought of as 1+1d O(4) WZW CFT with anisotropies:
L=
Z
1
dxd⌧ (@µ~n)2 +
g
Z
1
0
i2⇡
du
✏abcd na @x nb @⌧ nc @u nd
⌦3
• SO(4)~SU(2) x SU(2) .
The boundary of bosonic integer quantum Hall
• Example:
state, corresponds to breaking the SU(2) symmetry
L
R
L
completely, but break the SU(2)R symmetry to U(1) charge
Senthil, Levin 2012
conservation symmetry.
To find a realistic condensed matter system to realize/
• Goal:
mimic bosonic SPT state in 2d.
Realize 2d Bosonic SPT States in Bilayer Graphene
• Proposal:
Bilayer graphene under (strong) magnetic field can be driven
into a "bosonic" SPT state with U(1)×U(1) symmetry by
Coulomb interaction.
Bi, Zhang, You, Young, Balents, Liu, Xu (2016)
• Meaning:
gapless boson modes with U(1)×U(1) symmetry,
• Boundary:
fermion modes gapped out by interaction.
Realize 2d Bosonic SPT States in Bilayer Graphene
• Proposal:
Bilayer graphene under (strong) magnetic field can be driven
into a "bosonic" SPT state with U(1)×U(1) symmetry by
Coulomb interaction.
Bi, Zhang, You, Young, Balents, Liu, Xu (2016)
• Meaning:
quantum phase transition between BSPT and trivial
• Bulk:
state only closes boson gap, fermions remain gapped.
Boundary Analysis
fermion modes gapped out under interaction,
• Boundary:
remaining gapless boson modes with U(1)×U(1) symmetry.
layer graphene under perpendicular magnetic field
• Single
without interactions.
Abanin, Lee, Levitov 2006, Fertig, Brey 2006
Young et.al. 2014
edge mode: a pair of counter-propagating
• Helical
fermion modes (c=1 CFT)
Boundary Analysis
• Bilayer graphene
• Noninteracting: QSH×2, two helical edge modes (c=2)
interaction is relevant → gaps out all the fermion
• Coulomb
modes → only a pair of gapless counter-propagating boson
modes (c=1 CFT)
• Bosonization
Z
X
2
H0 =
H0 =
dx
Z
¯l,L iv@x
H̃ =
¯l,R iv@x
l,R
l=1
2
v X 1
dx
(@x ✓l )2 + K(@x l )2
2⇡
K
l,L/R
l=1
Coulomb Hv ⇠ cos(2(
Z
l,L
dx
v
2⇡
✓
1
2 )) ⇠
†
1,L
1
(@x ✓+ )2 + K̃(@x
K̃
†
2,R
1,R
+)
2
◆
2,L
⇠ ei(✓l ±
l)
Boundary Analysis
• BoundaryZ theory✓
1
(@x ✓+ )2 + K̃(@x
K̃
Boundary collective modes:
H̃ =
•
dx
• SC
• XY SDW
v
2⇡
+)
2
◆
n1 + in2 ⇠ ei✓+ ⇠ ✏↵ 1,↵ 2,
X
i2 +
n3 + in4 ⇠ e
⇠
( 1)l
† +
l
l
l
can derive the boundary effective theory. It’s an O(4) WZW
• We
model at level-1 (with anisotropy)
L=
Z
1
dxd⌧ (@µ~n)2 +
g
Z
1
0
i2⇡
du
✏abcd na @x nb @⌧ nc @u nd
⌦3
Boundary Analysis
fermion modes gapped out under interaction,
• Boundary:
remaining gapless boson modes with U(1)×U(1) symmetry.
picture for why the boundary must be gapless:
• Naïve
spin defect carries charge, charge defect carries spin
current is transported by the bosonic edge modes
• Edge
(charge 2e Cooper pairs) → shot noise measurement
• Tunneling from a normal metal → single particle gap
purely bosonic gapless boundary cannot occur with
• Such
only one layer of QSH insulator
Wu, Bernevig, Zhang (2005)
Xu, Moore (2005)
Boundary Analysis
wave function can be derived from boundary CFT correlation
• Bulk
according to the bulk-boundary correspondence.
Moore, Read (1991)
hei✓+ (z,z̄) e
hei✓+ (z,z̄) e
hei2
+ (z,z̄)
[wi , zj ] = h
e
Y
e
i
= F (|zi
i✓+ (0)
i2
+ (0)
i2
+ (0)
i✓+ (wi )
i = |z|
K̃/2
i = |z|
2/K̃
i = z̄/|z|
Y
j
zj |, |wi
ei2
+ (zj )
Obg i
wj |, |zi
wj |, K̃)
Y
i,j
(zi
wj )e
1
4
P
i (|wi |
2
+|zi |2 )
last factor encodes the essential physics that the spin and
• The
charge view each other as flux. Consistent with the flux attachment
picture of Senthil & Levin 2012.
Senthil, Levin (2012)
Bulk Analysis
quantum phase transition between BSPT and trivial state
• Bulk:
only closes boson gap, fermions remain gapped.
theory can be build from boundary with a Chalker• Bulk
Coddington / coupled-wire type of model
to>te, trivial
to<te, Chern insulator
Chern insulator & trivial insulator. We can build the
• Example:
bulk with coupled chiral fermions. The quantum critical point
between Chern insulator and trivial insulator is precisely a
2+1d Dirac fermion.
Bulk Analysis
quantum phase transition between BSPT and trivial state
• Bulk:
only closes boson gap, fermions remain gapped.
theory can be build from boundary with a Chalker• Bulk
Coddington / coupled-wire type of model
to>te, trivial
to<te, BSPT
theory only has gapless bosons (at low energy)
• Boundary
→ expect (and supported by numerics) that bulk transition
is also "bosonic" → mimic a bosonic SPT-trivial transition.
Sign Problem Free Lattice Model
• The spirit: spherical chicken
Leonard Hofstadter from the Big Bang Theory:
There's this farmer, and he has these chickens,
but they won't lay any eggs. So, he calls a
physicist to help. The physicist then does some
calculations, and he says, um, I have a solution,
but it only works with spherical chickens in a
vacuum!
• Topological state, is a chicken that can be thought of as a sphere,
so seemingly different chickens can behave exactly the same.
Sign Problem Free Lattice Model
designed a lattice model with all the key physics and with
• We
no sign problem
H = Hband + Hint
H
band
c†iℓ
= -t
c jℓ +
〈i j〉,ℓ
Hint
=J
X
i
ⅈ
λi j c†iℓ
σ c jℓ
〈〈i j〉〉,ℓ
1
(Si1 · Si2 + (ni1
4
�
z
λ
�
1)(ni2
�
�
1))
• Simple limits of this model:
• Free limit: bilayer QSH, σ = ±2 (depending on λ)
• Strong J-interacting limit: trivial Mott, σ = 0
sH
sH
c†1↑ c†2↓ - c†1↓ c†2↑ 0〉
Ψ〉 =
rung singlet product state
i
Slagle, You, Xu (2014)
Sign Problem Free Lattice Model
H = Hband + Hint
= -t
Hband
〈i j〉,ℓ
•
X
i
〈〈i j〉〉,ℓ
1
(Si1 · Si2 + (ni1
4
1)(ni2
1))
������� ����
σ sH = 0
���
σ sH = -2
Jc
��
Hint
=J
ⅈ λi j c†iℓ σ z c jℓ
c†iℓ c jℓ +
����������� �� �
designed a lattice model with all the key physics and with
• We
no sign problem
���
σ sH = +2
�
λ<�
λ>�
Simple limits of this model:
Free limit: bilayer QSH, σsH = ±2 (depending on λ)
Strong J-interacting limit: trivial Mott, σsH = 0
•
•
c†1↑ c†2↓ - c†1↓ c†2↑ 0〉
Ψ〉 =
rung singlet product state
i
Slagle, You, Xu (2014)
Sign Problem Free Lattice Model
• Determinant QMC (Edge)
fermion correlation
2
SDW/SC correlation
• When the fermion Green’s function already decays
FIG. 4. (Color online) The log-log plot of equal-time twoFIG. 2. (Color online) The log-log plot of single-particle
particle
O(4)
vector correlation
at the boundary for
exponentially
boundary,
bosonic
modesfunction
still have
Green’s function
at the boundaryat
as the
a function
of interlayer
(a)
= 0(b)
and (b) Jz = t. Both panels show the power-law
antiferromagnetic
interaction
J/t when (a) Jzuntil
= 0Jzand
power
law
correlation,
the
system
hits the bulk
decay behaviors
before the bulk topological phase transition
Jz = t. In both cases, results show the exponential
decay
transition
into transition
the trivial
Mott
Jc . phase.
before the bulk
topological phase
Jc . at
Wu et al. 2016
spin (SDW-XY) correlation function and superconduct-
Sign Problem Free Lattice Model
• Determinant QMC (Bulk)
• Fermion gap always finite.
modes become
• Bosonic
gapless at the SPT-trivial
critical point.
Fundamentally different
from free fermion QSH
transition.
Because the fermionic degrees
of freedom never show up at
either the boundary or the bulk
quantum transition, the whole
system can be viewed as a
bosonic SPT state.
•
•
He et al. 2015
Sign Problem Free Lattice Model
There’s a family of 2d Sign Problem Free lattice model for Bosonic
•SPTs
with Sp(N)xSp(N) symmetry. And the boundary realizes Sp(N)
1
xSp(N)-1 CFT.
copies of QSH insulators. Free boundary theory:
•Take 2NXidentical
Z
⇣
⌘
2N
Hbdy =
dx
†
l,L i@x l,L
†
l,R i@x
l,R
l=1
⇠ U (2N )1 ⇥ U (2N )
1
U (2N ) = Sp(N ) + SU (2)
•CFT decomposition
Interactions gap out SU(2) sector and all the fermions, while leaves
•the
Sp(N) sector intact (bosonic).
1
Hint ⇠
1
N
a
a
JSU
J
(2)L SU (2)R
An effective time reversal symmetry guarantees the model is sign
•problem
free.
You, Bi, Mao, Xu (2015)
Barkeshli, Wen (2010)
A Theory for the Bulk Transition
• A conjectured field theory for the bulk transition:
is the tuning parameter of the transition. Across the
• mtransition,
the Hall conductivity of external field A (B)
Senthil, Fisher 2005
changes by +2 (-2) → consistent with the BSPT physics
It has the desired SO(4) ~ SU(2)xSU(2) symmetry, because
of the self-dual structure Xu, You 2015; Karch, Tong 2016; Hsin, Seiberg, 2016
•
our numerical results, other numerics studies also
• Besides
suggest that N = 2 QED (at m = 0) is indeed a CFT.
f
3
Karthik, Narayanan 2016
Summary
• Predictions:
boundary is a conductor with
• The
single particle gap
between transport and
• Contrast
tunneling
competition between magnetic and electric field in the
• The
bulk may lead to a purely bosonic quantum phase transition,
with gapped electron but gapless bosonic collective modes;
• Other possible systems:
• Topological mirror insulator
Zhang, Xu, Liu (2014)