Topological states of matter with cold
gases and Rydberg atoms
Hans Peter Büchler
Institut für theoretische Physik III,
Universität Stuttgart, Germany
Collaboration
Norman Yao, Mikhail Lukin,
Sebastian Huber
Research group:
David Peter, Nicolai Lang,
Sebastian Weber
Adam Bühler, Przemek Bienias
Krzysztof Jachymski, Jan Kumlin
Strongly interacting
Rydberg slow light
polaritons
SFB TRR21:
Tailored quantum matter
Outline
Topological band structures
with dipolar interactions
- realized with polar molecules and
Rydberg atoms
- Chern number C=2
Majorana Edge modes
- exact solvable system with
fixed particle number
- double wire system
Topological band structure
singel particle band structure
independent on statistics of particles
- topological quantum numbers
- edge states
Requirements for topological states/phase
Fermions
Bosons
integer filling+
weak interactions
topological insulators
flat bands +
strong interactions
fractional topological
insulators
flat bands +
strong interactions
bosonic fractional
topological insulators
motivation: difficulties
Topological band structure
Homogeneous magnetic fields
- integer quantum Hall effect
- Hofstadter butterfly
Time-reversal symmetry
breaking
- lattice shaking
- artificial gauge fields
- synthetic dimension
(Esslinger, Bloch, Spielman, Fallani,…
- Haldane model
Here:
product state
of lowest band
Topological band structure
characterized by a Chern
number
1
C=
2⇡
Z
F (✓x , ✓y ) = Im h@✓y |@✓x i
✓y
2⇡
d✓x
0
h@✓x |@✓y i
Z
2⇡
d✓y F (✓x , ✓y )
0
✓x
Dipolar interactions
Dipole-dipole interaction
di
- anisotropic interaction
nij
dipole operator
Vdd =
di · dj
3 (di · nij ) (di · nij )
|ri rj |3
- coupling between orbital degree of freedom
and internal degree of freedom
dj
Observation in cold gases
- Einstein de Haas effect
(Y. Kawaguchi et al, PRL 2006)
“spin-orbit”
coupling
- demagnetization cooling
(M. Fattori et al, Nature Phys 2006)
- pattern formation in spinor condensates
(D. Stamper-Kurn, M. Ueda, RMP 2013).
Dipolar interactions
di
Exploit this spin orbit coupling
for the generation
of topological band structures
nij
dj
title
Requirements:
Systems:
- particles in
a 2D lattice
- polar molecules
- Rydberg atoms
- internal “Spin” structure
- NV centers
- atoms with large magnetic
dipole moments
- strong dipole-dipole interaction
1
Rydberg atoms in an optical lattice
Setup
- one atom per lattice site
with quenched tunneling
- static external electric field
and magnetic field
- select three internal states
| ii |+ii
|0ii
: two excited states
: ground state
Mapping onto two hard-core bosons:
- bosonic creation operators for excitations
|+ii = b†i,+ |0i
| ii = b†i, |0i
Rydberg atoms in an optical lattice
Setup
- one atom per lattice site
with quenched tunneling
- static external electric field
and magnetic field
| ii
|+ii
- select three internal states
| ii |+ii
|0ii
: ground state
: two excited states
Mapping onto two hard-core bosons:
- bosonic creation operators for excitations
|+ii = b†i,+ |0i
| ii = b†i, |0i
|0ii
Dipolar exchange interactions
Hamiltonian
- dipolar interaction restricted to
the three internal levels
Hij =
1
|ri
rj |3

d0i d0j
d0 ⌘ dz
1 +
+
di dj + di d+
j
2
dipole
interaction
d+ ⌘ dx + idy
3 + +
di dj e
2
i2
d ⌘ dx
ij
+ di dj ei2
spin-orbit
coupling
exchange
interaction
di
nij
ij
dj
hopping of excitation
idy
ij
Dipolar exchange interactions
Hamiltonian
- dipolar interaction restricted to
the three internal levels
Hij =
1
|ri
rj |3

d0i d0j
d0 ⌘ dz
1 +
+
di dj + di d+
j
2
dipole
interaction
d+ ⌘ dx + idy
3 + +
di dj e
2
i2
d ⌘ dx
ij
+ di dj ei2
spin-orbit
coupling
exchange
interaction
di
nij
ij
dj
hopping with spin flip
idy
ij
Single excitation Hamiltonian
Hamiltonian
i
- single particle hoping with
spin orbit interaction
H=
X
i6=j
a
|ri
3
rj
|3
†
i
✓
t
e
+
i2
Time-reversal symmetry breaking
- different hopping for excitations
t 6= t+
- energy shift on levels
H=
X
i
†
i
✓
µ
0
0
µ
◆
i
ij
we
i2
t
ij
◆
i
=
✓
b+,i
b ,i
◆
Single excitation Hamiltonian
X 1
ik·ri +i2
✏ˆk =
e
|ri |3
Hamiltonian
i6=0
- single particle hoping with
spin orbit interaction
H=
X
†
k
k
=
X
k
†
k
⇥
n0k
✓
+
t ✏k + µ
wˆ
✏⇤k
+ nk
⇤
wˆ
✏
t ✏k + µ
X 1
ik·ri
✏k =
e
|ri |3
k
i6=0
- characterized by a Chern number
C=
dk
nk ·
v0
✓
@nk @nk
^
@kx
@ky
◆
k
Chern
number
k as a winding number
Topological character of band
Z
i
◆
2Z
Topological band structure
band structure
With time reversal symmetry
- square lattice
- two bands with quadratic
band touching points
Linear dispersion:
- consequence of slow
dipolar decay of hopping
12
quadratic band
touching point
Dirac points in
rectangular lattice
see also Syzranov et al Nat. Comm. 2014
Topological band structure
Without time reversal symmetry
band structure
- square lattice
topological ph
- gap openings
Topological bands
- Chern number
C=
Z
dk
nk ·
v0
✓
@nk @nk
^
@kx
@ky
◆
2Z
12
Topological band structure
Without time reversal symmetry
- optimal experimental parameters
- Chern number C=2
band
Edge states
Finite system in y-direction
- bulk edge correspondence
(Hatsugai PRL 1993)
- exponential localization in
presence of long-range hopping
C= 2 implies two edge states
edge states
Stability under disorder
Disorder
Chern number in the disordered system
- missing molecules in the lattice
- stabilized by long-range hopping
edge states: disorder
18
26
Flat topological bands
Honeycomb lattice
- much flatter bands accessible
- very rich topological structure
C 2 {0, ±1, ±2, ±3, ±4}
honeycomb lattice: flat bands
- even richer for Kagame lattice
honeycomb lattice: topological phase diagram
0 µs
0.1 µs
0.2 µs
Energy (MHz)
20
10
0
Finite systems
Propagation of edge excitation
-10
Minimal system for observation
of edge states
-20
K
localized
at edge
3 µs - exponentially
0.4K
µs 0.5
µs
0.0 µs
M
0.2 µs
M 0.4 µsKK
- uni-directional motion
0.0
0.3
0.60.6
0.0
0.3
Density (1 / volume
BZ)
Density (1 / of
volume
of BZ)
0.6 µs K
0.8 µs
Band
structure
DOS
along
path
DOS
Band
structure
M
Bulk
densityBand structure
Density
of states
S
0.
1.0 µs
excitation
ofOverlap
states
edge Overlap
Overlapwith
with with
excitation
(A)
with(B)
excitation
(B) Location
Location
20
0
0 µs -10
0.2 µs
6 µs
0.4 µs
-20
0.8 µs
1.0 Kµs
0
10
Propagation of edge excitation
0
-10
-20
M0.0
0 µs K
0
0.6 µs
0
f revolutions
10
Energy (MHz)
Energy (MHz)
20
5
4
3
0.
0 0
µs
-10 0.4 µs
0
0
-10
0.8 µs
Finite systems
-20for observation
0-20 Minimal system
µs of
1.0edge
µs states
0
K
00 M
K
Direct observation of excitation
0
of edge excit
atPropagation
edge
- round structure with a single edge
0.0Bulk 0.3
0.6
Edge Bulk
0.0 0.0 0.3
0.6 0.3
0.6
Density
(1 / volume
Density (1 / volume
of(1BZ)
Density
/ volume
of BZ) of BZ)
Edge
Spectroscopy
DOS
Unwrapped
DOS
Unwrapped
band
structure band structure
Band
-DOS
round structure
withstructure
a single edge
0.0 µs
0.6 µs
Overlap
with
excitation
(C)
Location
Overlap
with
excitation
(C)
Location
Overlap with excitation (B)
Location
20
Energy (MHz)
0 201.0
0 100.8
0 00.6
0-100.4
0it-200.2
0
0.0
Density of
10
1.2 µs
0
-10
1.8 µs
5
4
3
2
1
0
0
Linear fit
Linear fit
-20
50 0 100
1.0 µs
Number of revolutions
Energy (
0.6 µs
00
50
50
100
0100
0
Density of states in round sample
states with
0.000
0.0025 0.050
Bulk
0.000
Bulk
Edge
Bulk
Edge
0.00.0025
0.3 0.050
0.6
edge
Density (1 / volume of BZ)
50
2.4 µs
100
3.0 µs
Edge
Propagation of edge excit
Outlook on topological bands
Dipolar interaction provides
natural spin orbit coupling
- topological band structures
with Chern number C=2
Robust to disorder
- topological nature is very robust
to missing particles in the lattice
Towards bosonic fractional
Chern insulators
- strongly interacting system
- is flatness high enough for
topological phases
- candidate expected at 2/3 filling
edge states: disorder
Outline
Topological band structures
with dipolar interactions
- realized with polar molecules and
Rydberg atoms
- Chern number C=2
Majorana Edge modes
- exact solvable system with
fixed particle number
- double wire system
Kitaev’s Majorana chain
Kitaev’s Majorana chain
- fermions on a 1D lattice
with supefluid pairing term
H=
L
X1 h
wa†i ai+1
ai ai+1 + h.c.
i=1
µ
L
X
a†i ai
i
Trivial phase:
Dominant chemical potential
i=1
- exact solution by introducing
Majorana fermions
c2i 1 + ic2i
ai =
2
Topological phase:
Dominant hopping & pairing
Topological state
- robust ground state
degeneracy
- non-local order
parameter
- localized edge states
⇒ Zero-energy edge mode
ai =
c2i
1
+ ic2i
2
Why should we care
Topological invariant edge states
- ground state degeneracy is robust
to local perturbations
robust quantum
memory?
Non-abelian anyons
- localized edge modes obey
non-abelian braiding statistics
topological quantum
computation
Topological Phase:
(Alicea
et al Nat. Phys. 2011)
Anyonic
Statistics:
Topological invariant protects
- Novel
state of matter
ground state
degeneracy
Localized edge modes obey
non-abelian anyonic statistics
&
↓
- Toy model of a
topological phase
Inherently robust way of
storing & manipulating quantum information
Beyond mean-field
H=
L
X1 h
wa†i ai+1
ai ai+1 + h.c.
i=1
i
- mean-field coupling
- violates particle conservation
exist a particle conserving theory
with Majorana modes in one-dimension?
Beyond mean-field
Previous work in this context:
→ M. Cheng and H.-H. Tu (2011). Physical Review B, 84(9), 094503.
Majorana edge states in interacting two-chain ladders of fermions.
→ J. D. Sau et al. (2011). Physical Review B, 84(14), 144509.
Number conserving theory for topologically protected degeneracy
in one-dimensional fermions.
→ L. Fidkowski et al. (2011). Physical Review B, 84(19), 195436.
Majorana zero modes in one-dimensional quantum wires
without long-ranged superconducting order.
Bosonization
Bosonization
→ J. Ruhman et al. (2014). arXiv:1412.3444
Topological States in a One-Dimensional Fermi Gas
with Attractive Interactions.
→ C. V. Kraus et al. (2013). Physical Review Letters, 111(17), 173004.
Majorana Edge States in Atomic Wires Coupled by Pair Hopping.
Numerical
(DMRG)
Numerical
→ G. Ortiz et al. (2014). arXiv:1407.3793
Many-body characterization of topological superconductivity:
The Richardson-Gaudin-Kitaev chain.
Long-Range
Long-range
Here: Sort-range
theory
⇒ Here: Short-range
interactinginteracting
Theory
exact
Exact ground
stateground state
Majorana
edge modes
Majorana Modes
on edges
N. Lang and H. P. Büchler, Phys. Rev. B 92, 041118 (2015).
F.Iemini,et al., Phys. Rev. Lett. 115, 156402 (2015).
Microscopic model
Hamiltonian
- double wire system
H = Ha + Hb + Hab
- intra-chain contribution
Ha =
X
Aai (1 + Aai )
i
- inter-chain contribution
Hab =
X
Bi (1 + Bi )
i
Symmetries
- total number of particles
N
- time reversal symmetry
T
P
- sub-chain parity
Microscopic model
Inter-chain Hamiltonian
Ha =
X
Aai (1 + Aai )
i
Aai = a†i ai+1 + a†i+1 ai
Intrachain Interactions:
- positive Hamiltonian
- zero-energy state
is ground state
Single-particle hopping &
NN density-density inter
X
|ni =
{ni }
| . . . , ni , ni+1 , . . .i
equal weight superposition of all
possible distribution of n fermions
Inter-chain Hamiltonian (expanded)
Expanded form:
Hia = ai a†i+1 + ai+1 a†i + nai 1
nai+1 + nai+1 (1
nai )
Microscopic model
Intra-chain Hamiltonian
Hab =
X
Bi (1 + Bi )
i
Bi = a†i a†i+1 bi bi+1 + b†i b†i+1 ai ai+1
pair-hopping between chains
Interchain Interaction:
- positive Hamiltonian & Pair-density-densityX
Pair-hopping
interactions
- zero-energy state
is ground state
- fixed total number
of particles
| i=
n
|ni|N
ni
equal weight superposition of all possible
distribution of N fermions between the two wires
Intra-chain Hamiltonian (expanded)
i
Hab
= a†i a†i+1 bi bi+1 +b†i b†i+1 ai ai+1 +nai nai+1 1 nbi
1 nbi+1 + nbi nbi+1 (1 nai ) 1 nai+1
Ground state degeneracy
GS = Equal-weight superposition
Do the math ...
total particle number & subc
Two-open chains
Degeneracy for open boundar
Even total number
of particles
Two
GS for each particle
- two-fold ground state
degeneracy
|
|
even i
=
X n2even
X
|ni|N
Do the math ...
even
odd
even
odd
Degeneracy for open boundary conditions
GS = Equal-weight superposition
Two GS for with
eachfixed
particle number s
ni
Note:
total particle number & subchain parity
even not work
odd for periodic
Does
even
Do the math ...
odd
with
GS = Equal-weight
superposition with fixed
ni
Note:
total particle number
& subchain
parity
superposition
with fixed
Does not work for periodic boundary
total particle number & subchain parity
i =math ... |ni|N
Do
oddthe
GS = Equal-weight
n2odd
Degeneracy for open boundary conditions:
with
Odd total
number
of particles
Two GS for each particle
number
sector:
Two- closed chains
even with
odd
even
odd
Degeneracy for open boundary
conditions:
even
odd
odd
even
- only one zero energy state
for GS for each particle number sector:
Two
Degeneracy
open boundary
conditions:
total even
number offor
particles
Note:
even
oddfor periodic boundary even
Two GS for each particleDoes
number
sector:
not work
conditions odd
in all sectors!
even
even
Note:
odd
Note:
odd
even
odd
even
odd
odd
even
odd
even
Does not work for periodic boundary conditions in all sectors!
Do the math ...
Do the math ...
GS
= Equal-weight
w
GS = Equal-weight
superposition
withsuperposition
fixed
total&particle
number
total particle number
subchain
parity
with & subc
Wire networks
Degeneracy for open boundary conditions: Degeneracy for open boundary
Do the
math ...number sector:
Two GS for each particle n
Two GS for each
particle
Networks of wires
GS = Equal-weight superposition with fixed
with
even
even even
odd
odd odd
total
particle
number
&
subchain
parity
Degeneracy for open boundary conditions:
even
odd even even odd
Two GSodd
for each particle number
sector:
even Degeneracy
even
oddconditions:
for open boundary
withconditions
Note:
Degeneracy for open
boundary
- exact ground states for arbitraryNote:
networks
Does
not
work
fornumber
periodic
b
Two GS
for
each
particle
Two for
GS periodic
for each
particle
number
sector:
even
Does not work
boundary
conditions
in
all sectors!
odd
odd se
- degeneracy consistent with
majorana modes at edges
2E/2
1
number of edges
even
even
odd
odd
Note:
Degeneracy
boundary
conditions:
even
even for openodd
odd
even
odd
Does
notfor
work
for particle
periodic boundary
conditions in all
Two GS
each
number sector:
Note:
even Note:
odd
even
Does not work for
periodic
boundary
in all sect
Does not
work forconditions
periodic boundary
c
even
odd
odd
Note:
Does not work for periodic boundary conditions in all
Ground state properties
Density-density correlations
i
- independent on ground state
0
hni nj i = ⇢2
i 6= j
i
Superfluid correlations
ha†i a†i+1 aj aj+1 i
= ⇢(1
j
⇢)
- long-range superfluid p-wave pairing
i+l
Greens Function:
- exponential decay
j+l
Vanishes exponentially in the bulk & revival at t
Green’s function
ha†i aj i
j
existence of
edge modes
- revival at the edge
We conclude:
Ground state properties
Stability of ground state degeneracy
of edge states
- stable under all local perturbations
- splitting decays exponentially
Stability of ground state degeneracy
for open wires
- Protected by either time-reversal
symmetry or subchain parity
a†i bi + b†i ai
: stable under time
reversal hopping
ia†i bi
: finite overlap between
two ground states
ib†i ai
Excitation spectrum
Low-energy excitations
- Goldstone mode due to broken
U(1) symmetry
- exact wave function for single phase
kink excitation
|k, i =
X
j
h
eikj ( 1)
na
j
+ ( 1)
- quadratic excitation spectrum
✏k = 4 sin2 k/2 ⇠ k 2
nbj
System is in a critical state
i
- vanishing compressibility
| i
- Goldstone mode with
quadratic dispersion
Non-abelian Braiding statistics
Idea:
Setup for braiding of
two edge states
Braid edge-modes on subchains by adiabat
- wire network with two edges
- restriction to the low energy
diabatic
deformation of Hamiltonian
sector
"Bath"
- very weak coupling terms:
adiabatic switching between them
Majorana
Low-energy
8 relevant
states sector:
Negative total parity
states
Weak couplings
- characterized by
→ 8parity
ground
subchain
Subchain-pari
Non-abelian Braiding statistics
Idea:
Braid edge-modes on subchains by adiabatic deformatio
"Bath"
Negative t
→ 8 groun
Adiabatic switching of coupling
- transformation of the ground state
according to the non-abelian statistic
of Majorana modes
Low-energy
Majorana
Weak couplings
Subchain-parities:
Conclusion
Topological band structure
with dipolar exchange
interactions
- spin-orbit coupling natural present
in dipolar system
- existence of topological band structures
Majorana Edge modes
- exact solvable system with
fixed particle number
- analytical demonstration of
Majorana edge modes