SYNTHETIC HELICAL LIQUID
IN A QUANTUM WIRE
George I. Japaridze
Ilia state University and
Andronikashvili Institute of Physics
SYNTHETIC HELICAL LIQUID IN A QUANTUM WIRE
 What is a helical liquid (HL)?
Unconventional State of matter with spin-momentum
locking

p

p

S

S
SYNTHETIC HELICAL LIQUID IN A QUANTUM WIRE
 What is a helical liquid (HL)?
Unconventional State of matter with spin-momentum
locking
Helical modes: where do they occur?
topological insulators Pankratov, Pakhomov & Volkov,
SSC 61, 93 (1987) Hasan and
(edges):
Kane, RMP 82, 3045 (2010)
quasi-­­1d Super Cond
superconductors:
Pot t er and Lee, PRL 105, 227003
(2010)
Semiconducing
Streda and Seba, PRL 90, 256601 (2003)
nanowires: -magnetic Braunecker, Japaridze, Klinovaja, DL,
PRB 82, 045127 (2010)
field
carbon nanotubes:
Klinovaja, Schmidt, Braunecker, DL, PRL
106 156809 (2011), and PRB 84, 085452
(2011)
Insulators with Strong Spin-Orbit
Interaction:Spin-Hall Effect: Edge states
L. Fu, C. L. Kane and E. J. Mele PRL 98, 106803 (2007).
J. E. Moore and L. Balents, PRB 75, 121306(R) (2007).
S.-C. Zhang, Nature Physics 1, 6 (2008).
Majorana Fermions in 1d
Alicea, PRB 81, 125318 (2010)
Oreg, Refael, and von Oppen, PRL 105, 177002 (2010)
Lutchyn, Sau, and Das Sarma, PRL 105, 077001 (2010)
Kitaev, Phys.-­Usp. 44, 313 (2001)
Alicea, Oreg, Refael, v.Oppen, Fisher, Nature Phys.7,412 (2011)
Gangadharaiah, Braunecker, Simon, DL, PRL 107, 036801 (2011)
Semiconducting Nanowires
Various materials:
ZnO, InAs, InP, GaAs, AlAs, Ge, Si, SiGe, GaN, GaP, CdS,
…
Operate both in the
conduction band (CB) and valence band (VB)
ELECTRONS
Charge: similar
Spin: very different
HOLES
Particularly characteristic
for semiconductors
Holes turn out to be advantageous in many aspects!
Ge/Si Core/Shell Nanowires
Xiang et al., Nature (2006); Hu et al., Nat. Nano (2007); Hu et al., preprint (2011)
Ex
Nanowire
grown along [110]
Large Ge/Si valence band offset of ~ 0.5 eV, narrow interfaces
→ replace with hard wall at core radius Rc ≡ R
Lauhon et al., Nature (2002), Lu et al., PNAS USA (2005)
v
E
Lorenz transformation
H
•c
v=0
E
• The 1D lattice Hamiltonian:
Hl a t = − t
,
(c†n ,σ cn +1,σ + h.c.)
n ,σ
+
+αR
(c†
, f
,
[
†
−
c
c
n ,↑ n +1,↓
n ,↓ cn +1,↑ ) + h.c.]
n
U ρn , ↑ ρn ,↓ + V ρn ρn +1 + JS n · S n +1 ...
n
V. Gritsev, G. Japaridze, M. Pletyukhov, and D. Baeriswyl , Phys. Rev. Lett. 94, 137207 (2005).
S. Gangadharaiah, J. Sun, and O. A. Starykh, Phys. Rev. B 78, 054436 (2008).
Part I: Magnetic field induced
quasi-helical liquid state
in a disordered 1D electron system
with strong spin-orbit interaction
Anders Ström, Bernd Braunecker and G.J.
PRB 87, 075151 (2013).
Helical mode: spin coupled to momentum
Rashba spin-orbit field along z-axis
H R  px z
04. pptx
e.g. InAs nanowire λSO ~ 100 nm, Fasth et al., PRL 98, 266801 (2007)
Helical Hole States
Kloeffel, Trif, DL, arXiv:1107.4870
E-field
along x:
Rc = 5.0 nm
Rs = 6.5 nm
 strong
Rashba SOI (~ 1-10 meV)
Ex = 6 V/µm
No RSOI!
Helical Hole States
Kloeffel, Trif, DL, arXiv:1107.4870
E-field along x: Ex = 6 V/µm 
strong Rashba SOI (~ 1-10 meV)
Bx opens a gap 0.8 T:~
0.25 meV
0.3 T: ~ 0.10 meV
Rc = 5.0 nm
Rs = 6.5 nm
Helical States from Rotating Zeeman Field
Braunecker, Japaridze, Klinovaja, Loss, PRB 82, 045127 (2010)
2kF

local spin basis transformation:
spin-dependent shift in k-space
2kF
2kF

Effect of Interaction : strong
renormalization of Zeeman gap!
see also, Stoudenmire, Alicea, Starykh, Fisher, PRB 84,
014503 (2011)
Band structures
Spins look antiparallell for opposite
velocities. Helical?
Corrupted by the
magnetic field HS!
Let’s calculate the spin
overlaps!
Band structures
Spin Overlaps
Limits:
Band structures
Spin Overlaps
Detail for aa overlap. 0.3 meV (ca 1 T)
yields overlap squared of about 0.01.
Disorder: Anderson Localization in
1D !!
(absent in the Helical Liquid State!!)
Overlaps:
Limits:
Disorder
RG
Scaling equations!
For full Luttinger liquid with spin:
For spinless or helical Luttinger liquid:
Above
gap
Below
gap
Disorder
Localisation length
Can have loc. from ab disorder, if gap is smaller than ca 6ehjhjhjee5 eV,
almost 1 K.
Below the gap, we only need to consider disordered aa backscattering.
01.pptx
Part II: SYNTHETIC HELICAL
LIQUID IN A QUANTUM WIRE
Mariana Malard, Henrik Johannesson and GJ
PRB 89, 201403(R) (2014).
Band picture
insulating
phase
branch
2
 (k )
_
_
+
+
F
 k F  q0
 k F  q0
 k F  q0
Q  2(k F  q0 )
k
 k F  q0
Band picture
conducting
helicalbranch
electron
2
liquid
 (k )

p
_

S
_
+

p

S
+
F
 k F  q0
 k F  q0
 k F  q0
Q  2(k F  q0 )
 k F  q0
k
Our proposal
HL in a quantum wire using electrical fields only and
standard nanoscale semiconductor technology!
modulation of the
spin-orbit Rashba
interaction
top
nanogates
quantum
wire
Semiconduc
tor
heterostructu
re
+ spin-orbit
Dresselhaus + e-e
interactions
The MODEL
hopping and
chemical
potential
+
uniform Rashba and
Dresselhaus
interactions
+
modulated
Rashba
interaction
+
e-e
interaction
Effective theory
sine-Gordon
potential
relevant for
K<½
gap
helical Dirac
hamiltonian
branch-mixing
potential
For future success in
building of GermanGeorgian Science Bridge
Thank you for your attention!