electrical control of the kondo effect at the edge of a...
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Erik Eriksson (University of Gothenburg)
Anders Ström (University of Gothenburg)
Girish Sharma (École Polytechnique)
Henrik Johannesson (University of Gothenburg)
Electrical Control of the Kondo Effectat the Edge of a Quantum Spin Hall System
supported by the Swedish Research Council
Correlations and coherence in quantum systems Évora, Portugal, October 11 2012
Quantum spin Hall effect... some basics
At the edge: A new kind of electron liquid
Adding a magnetic impurity...
... and a Rashba spin-orbit interaction
Electrical control of the Kondo effect!
Summary and outlook
Outline
Quantum spin Hall effect... some basics
Quantum spin Hall effect
no channel for backscattering
B“skipping currents”
quantization
chiral edge states
Quantum Hall effect
ballistic transport along the edge,accounts for the quantization of the Hall conductance
M. Büttiker, PRB 38, 9375 (1988)
Integer
B. I. Halperin, PRB 25, 2185 (1982)
Can a system be stable against local perturbations without breaking time-reversal invariance?
uniformly charged cylinder with electric field
spin-orbit interaction
cf. with the IQHE in a symmetric gauge
Lorentz force
1
E = E(x, y, 0)
(E × k) · σ = Eσz(kyx− kxy)
A =B
2(y,−x, 0)
A · k ∼ eB(kyx− kxy)
G = νe2
h
Mc ≈ 100 meV
D. Grundler, Phys. Rev. Lett. 84, 6074 (2000)
W. Hausler, L. Kecke, and A. H. MacDonald, Phys.
Rev. B 65, 085104 (2002)
�α0 � 2× 10−11
eV m
Λ � 0.5 eV
vF � 1× 106
m/s
Kc + Ks � 1.8
HR
H =
�dx [Hc +Hs]
Hi =vi
2[(∂xϕi)
2+(∂xϑi)
2]−
mi
πacos(
�2πKiϕi), (1)
∆R = γ1 sin(q0a)
with vi and Ki functions of g1τ , g2τ and g4τ
Λ ∼ bandwidth (2)
ϕc ≡ (ϕ+ + ϕ−)/
√2 (3)
ϕs ≡ (ϕ+ − ϕ−)/
√2 (4)
R†τ and L
†τ create excitations at the Fermi points of the
right- and left-moving branches with spin projection τ
(5)
L+
vF = 2a
�t2 + γ
20 and ∆R = γ1 sin(q0a)
Hτ =−ivF
�:R
†τ (x)∂xRτ (x) ::L
†τ (x)∂xLτ (x) :
�
−2∆R cos(Qx)�e−2ik0
F (x+a/2)R
†τ (x)Lτ (x)+H.c.
�,(6)
τ = ±
q0
HR =−i
�
n,µ,ν
(γ0 + γ1 cos (Qna))
�c†n,µσ
yµνcn+1,ν−H.c.
�
γj = αja−1
(j = 0, 1)
ϕc ≡ (ϕ+ + ϕ−)/
√2 (7)
ϕs ≡ (ϕ+ − ϕ−)/
√2 (8)
Hint = g1− :R†τLτL
†−τR−τ : + g2τ :R
†+R+L
†τLτ :
+g4τ
2(:R
†+R+R
†τRτ : +R↔ L) (9)
g2τ ≡g2τ − δτ+g1τ (10)
with vi and K functions of g1τ , g2τ , g3τ (11)
1
E = E(x, y, 0)
(E × k) · σ = Eσz(kyx− kxy)
A =B
2(y,−x, 0)
A · k ∼ eB(kyx− kxy)
G = νe2
h
Mc ≈ 100 meV
D. Grundler, Phys. Rev. Lett. 84, 6074 (2000)
W. Hausler, L. Kecke, and A. H. MacDonald, Phys.
Rev. B 65, 085104 (2002)
�α0 � 2× 10−11
eV m
Λ � 0.5 eV
vF � 1× 106
m/s
Kc + Ks � 1.8
HR
H =
�dx [Hc +Hs]
Hi =vi
2[(∂xϕi)
2+(∂xϑi)
2]−
mi
πacos(
�2πKiϕi), (1)
∆R = γ1 sin(q0a)
with vi and Ki functions of g1τ , g2τ and g4τ
Λ ∼ bandwidth (2)
ϕc ≡ (ϕ+ + ϕ−)/
√2 (3)
ϕs ≡ (ϕ+ − ϕ−)/
√2 (4)
R†τ and L
†τ create excitations at the Fermi points of the
right- and left-moving branches with spin projection τ
(5)
L+
vF = 2a
�t2 + γ
20 and ∆R = γ1 sin(q0a)
Hτ =−ivF
�:R
†τ (x)∂xRτ (x) ::L
†τ (x)∂xLτ (x) :
�
−2∆R cos(Qx)�e−2ik0
F (x+a/2)R
†τ (x)Lτ (x)+H.c.
�,(6)
τ = ±
q0
HR =−i
�
n,µ,ν
(γ0 + γ1 cos (Qna))
�c†n,µσ
yµνcn+1,ν−H.c.
�
γj = αja−1
(j = 0, 1)
ϕc ≡ (ϕ+ + ϕ−)/
√2 (7)
ϕs ≡ (ϕ+ − ϕ−)/
√2 (8)
Hint = g1− :R†τLτL
†−τR−τ : + g2τ :R
†+R+L
†τLτ :
+g4τ
2(:R
†+R+R
†τRτ : +R↔ L) (9)
g2τ ≡g2τ − δτ+g1τ (10)
with vi and K functions of g1τ , g2τ , g3τ (11)
1
E = E(x, y, 0)
(E × k) · σ = Eσz(kyx− kxy)
A =B
2(y,−x, 0)
A · k ∼ eB(kyx− kxy)
G = νe2
h
Mc ≈ 100 meV
D. Grundler, Phys. Rev. Lett. 84, 6074 (2000)
W. Hausler, L. Kecke, and A. H. MacDonald, Phys.
Rev. B 65, 085104 (2002)
�α0 � 2× 10−11
eV m
Λ � 0.5 eV
vF � 1× 106
m/s
Kc + Ks � 1.8
HR
H =
�dx [Hc +Hs]
Hi =vi
2[(∂xϕi)
2+(∂xϑi)
2]−
mi
πacos(
�2πKiϕi), (1)
∆R = γ1 sin(q0a)
with vi and Ki functions of g1τ , g2τ and g4τ
Λ ∼ bandwidth (2)
ϕc ≡ (ϕ+ + ϕ−)/
√2 (3)
ϕs ≡ (ϕ+ − ϕ−)/
√2 (4)
R†τ and L
†τ create excitations at the Fermi points of the
right- and left-moving branches with spin projection τ
(5)
L+
vF = 2a
�t2 + γ
20 and ∆R = γ1 sin(q0a)
Hτ =−ivF
�:R
†τ (x)∂xRτ (x) ::L
†τ (x)∂xLτ (x) :
�
−2∆R cos(Qx)�e−2ik0
F (x+a/2)R
†τ (x)Lτ (x)+H.c.
�,(6)
τ = ±
q0
HR =−i
�
n,µ,ν
(γ0 + γ1 cos (Qna))
�c†n,µσ
yµνcn+1,ν−H.c.
�
γj = αja−1
(j = 0, 1)
ϕc ≡ (ϕ+ + ϕ−)/
√2 (7)
ϕs ≡ (ϕ+ − ϕ−)/
√2 (8)
Hint = g1− :R†τLτL
†−τR−τ : + g2τ :R
†+R+L
†τLτ :
+g4τ
2(:R
†+R+R
†τRτ : +R↔ L) (9)
g2τ ≡g2τ − δτ+g1τ (10)
with vi and K functions of g1τ , g2τ , g3τ (11)
1
E = E(x, y, 0)
(E × k) · σ = Eσz(kyx− kxy)
A =B
2(y,−x, 0)
A · k ∼ eB(kyx− kxy)
G = νe2
h
Mc ≈ 100 meV
D. Grundler, Phys. Rev. Lett. 84, 6074 (2000)
W. Hausler, L. Kecke, and A. H. MacDonald, Phys.
Rev. B 65, 085104 (2002)
�α0 � 2× 10−11
eV m
Λ � 0.5 eV
vF � 1× 106
m/s
Kc + Ks � 1.8
HR
H =
�dx [Hc +Hs]
Hi =vi
2[(∂xϕi)
2+(∂xϑi)
2]−
mi
πacos(
�2πKiϕi), (1)
∆R = γ1 sin(q0a)
with vi and Ki functions of g1τ , g2τ and g4τ
Λ ∼ bandwidth (2)
ϕc ≡ (ϕ+ + ϕ−)/
√2 (3)
ϕs ≡ (ϕ+ − ϕ−)/
√2 (4)
R†τ and L
†τ create excitations at the Fermi points of the
right- and left-moving branches with spin projection τ
(5)
L+
vF = 2a
�t2 + γ
20 and ∆R = γ1 sin(q0a)
Hτ =−ivF
�:R
†τ (x)∂xRτ (x) ::L
†τ (x)∂xLτ (x) :
�
−2∆R cos(Qx)�e−2ik0
F (x+a/2)R
†τ (x)Lτ (x)+H.c.
�,(6)
τ = ±
q0
HR =−i
�
n,µ,ν
(γ0 + γ1 cos (Qna))
�c†n,µσ
yµνcn+1,ν−H.c.
�
γj = αja−1
(j = 0, 1)
ϕc ≡ (ϕ+ + ϕ−)/
√2 (7)
ϕs ≡ (ϕ+ − ϕ−)/
√2 (8)
Hint = g1− :R†τLτL
†−τR−τ : + g2τ :R
†+R+L
†τLτ :
+g4τ
2(:R
†+R+R
†τRτ : +R↔ L) (9)
g2τ ≡g2τ − δτ+g1τ (10)
with vi and K functions of g1τ , g2τ , g3τ (11)
1
B = ∇×A
E = E(x, y, 0)
(E × k) · σ = Eσz(kyx− kxy)
A =B
2(y,−x, 0)
A · k ∼ eB(kyx− kxy)
G = νe2
h
Mc ≈ 100 meV
D. Grundler, Phys. Rev. Lett. 84, 6074 (2000)
W. Hausler, L. Kecke, and A. H. MacDonald, Phys.
Rev. B 65, 085104 (2002)
�α0 � 2× 10−11
eV m
Λ � 0.5 eV
vF � 1× 106
m/s
Kc + Ks � 1.8
HR
H =
�dx [Hc +Hs]
Hi =vi
2[(∂xϕi)
2+(∂xϑi)
2]−
mi
πacos(
�2πKiϕi), (1)
∆R = γ1 sin(q0a)
with vi and Ki functions of g1τ , g2τ and g4τ
Λ ∼ bandwidth (2)
ϕc ≡ (ϕ+ + ϕ−)/
√2 (3)
ϕs ≡ (ϕ+ − ϕ−)/
√2 (4)
R†τ and L
†τ create excitations at the Fermi points of the
right- and left-moving branches with spin projection τ
(5)
L+
vF = 2a
�t2 + γ
20 and ∆R = γ1 sin(q0a)
Hτ =−ivF
�:R
†τ (x)∂xRτ (x) ::L
†τ (x)∂xLτ (x) :
�
−2∆R cos(Qx)�e−2ik0
F (x+a/2)R
†τ (x)Lτ (x)+H.c.
�,(6)
τ = ±
q0
HR =−i
�
n,µ,ν
(γ0 + γ1 cos (Qna))
�c†n,µσ
yµνcn+1,ν−H.c.
�
γj = αja−1
(j = 0, 1)
ϕc ≡ (ϕ+ + ϕ−)/
√2 (7)
ϕs ≡ (ϕ+ − ϕ−)/
√2 (8)
Hint = g1− :R†τLτL
†−τR−τ : + g2τ :R
†+R+L
†τLτ :
+g4τ
2(:R
†+R+R
†τRτ : +R↔ L) (9)
g2τ ≡g2τ − δτ+g1τ (10)
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Consider a Gedanken experiment... B. A. Bernevig and S.-C. Zhang, PRL 96, 106802 (2006)
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First proposed by Kane and Mele for graphene (2005)
Bernevig et al. proposal for HgTe quantum wells (2006)Experimental observation by König et al. (2007)
too weak spin-orbit interaction, doesn’t quite work...
Two copies of a IQH system, bulk insulator with helical edge states
Quantum spin Hall (QSH) system single Kramers pair
from M. König et al., Science 318, 766 (2007)
Bernevig et al. proposal for HgTe quantum wells (2006)Experimental observation by König et al. (2007)
Yes! As long as the perturbationsare time-reversal invariant! Look at the band structure of an HgTe quantum well...
strong spin-orbit interactions in atomic p-orbitals create an inverted band gap (p-band on top of s-band)
Are the helical edge states stable against local perturbations?
supports a single Kramers pair of helical edge states inside the inverted gap
Kramers degeneracy at k=0 protectsthe stability of the edge states
ballistic transport
1
G =2e
2
h
L ∼ 1µm
L ∼ 20µm
g
g ∼ 1/2v
ξ < L
α(±2kF )
±2kF
z⊥ ≡ 2g
√CK
z� ≡ 4K − 2
γ = (α(2kF ) + α(−2kF ))/2
∂xθ
ψ↑ = η↑ exp�i√
π[φ + θ]�/
√2πκ
ψ↓ = η↓ exp�−i√
π[φ− θ]�/
√2πκ
H0 + Hd + Hf + HR
O(α2)
α(x) = �α(x)�+
�
n
α(kn)eikxn
Ψ↑↓(x)=ψ↑↓(x)e±ikF x
Ψ↓(x)=ψ↓(x)e−ikF x
HR = α(x)(z × k) · σ
B = ∇×A
E = E(x, y, 0)
(E × k) · σ = Eσz(kyx− kxy)
A =B
2(y,−x, 0)
A · k ∼ eB(kyx− kxy)
G = νe2
h
Mc ≈ 100 meV
D. Grundler, Phys. Rev. Lett. 84, 6074 (2000)
W. Hausler, L. Kecke, and A. H. MacDonald, Phys.
Rev. B 65, 085104 (2002)
�α0 � 2× 10−11
eV m
Λ � 0.5 eV
vF � 1× 106
m/s
Kc + Ks � 1.8
HR
B. A. Bernevig et al., PRL 95, 066601 (2005)4-terminal measurement, equilibration in contacts
At the edge: A new kind of electron liquid
# of Kramers pair in a nontrivial (trivial) helical liquid
NK = 1 mod 2 = 0 mod 2
2D topological insulator(a.k.a. quantum spin Hall system)
is a ”Z2” topological invariant and can be calculated fromthe band structure of the bulk (”bulk-edge correspondence”)
1
ν = 0, 1
J⊥,z � ω � T
( )
K → K �
E =c
8log(
2�1�22�1 + �2
) + const.
θ
2�1 �2
γ0 ∼ (Jx + J �y)
2T 2(√K−λ/2)2−1, γ�
0 ∼ (Jx − J �y)
2T 2(√K+λ/2)2−1, γE
0 ∼ J2NCT
2K−1, γE0 ∼ J2
NCT
I = G0V − δI
T � TK
K < 1/4
G ∼ (T/TK)2(1/4K−1)
G = 2e2/h
G = 0
L. Fu and C. L. Kane, PRB 76, 045302 (2007)
What if time-reversal symmetry is broken...?
At the edge: A new kind of electron liquid
# of Kramers pair in a nontrivial (trivial) helical liquid
NK = 1 mod 2 = 0 mod 2
2D topological insulator(a.k.a. quantum spin Hall system)
is a ”Z2” topological invariant and can be calculated fromthe band structure of the bulk (”bulk-edge correspondence”)
1
ν = 0, 1
J⊥,z � ω � T
( )
K → K �
E =c
8log(
2�1�22�1 + �2
) + const.
θ
2�1 �2
γ0 ∼ (Jx + J �y)
2T 2(√K−λ/2)2−1, γ�
0 ∼ (Jx − J �y)
2T 2(√K+λ/2)2−1, γE
0 ∼ J2NCT
2K−1, γE0 ∼ J2
NCT
I = G0V − δI
T � TK
K < 1/4
G ∼ (T/TK)2(1/4K−1)
G = 2e2/h
G = 0
L. Fu and C. L. Kane, PRB 76, 045302 (2007)
... for example, by the presence of a magnetic impurity?
case study:
large and positive single-ion anisotropy
1
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
1
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
1
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
anisotropic spin exchange with the edge electrons
1
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
To model the edge electrons, we introduce the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihilates a right-moving
(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-e
interactions, the edge Hamiltonian can then be written as
HK = Ψ†(0)
�J⊥(σ
+S−+ σ−
S+) + Jzσ
zSz�Ψ(0),
1
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
To model the edge electrons, we introduce the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihilates a right-moving
(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-e
interactions, the edge Hamiltonian can then be written as
HK = Ψ†(0)
�J⊥(σ
+S−eff + σ−
S+eff) + Jzσ
zSzeff
�Ψ(0),
1
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
R. Zitko et al., PRB 78, 224404 (2008)
from M. König et al., Science 318, 766 (2007)
Adding a magnetic impurity...
The Kondo interaction is time-reversal invariant! Could it still cause a spontaneous breaking of time reversal invariance and collapse the QSH state?
Adding a magnetic impurity...
Recall the Kondo effect
One-loop RG equations:
strong-coupling physics for
P. W. Anderson, J. Phys. C 3, 2436 (1970)
1
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (1)
J0 ≡ max(J⊥, Jz)D=D0
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
To model the edge electrons, we introduce the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihilates a right-moving
(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-e
interactions, the edge Hamiltonian can then be written as
HK = Ψ†(0)
�J⊥(σ
+S−eff + σ−
S+eff) + Jzσ
zSzeff
�Ψ(0),
1
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (1)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
To model the edge electrons, we introduce the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihilates a right-moving
(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-e
interactions, the edge Hamiltonian can then be written as
HK = Ψ†(0)
�J⊥(σ
+S−eff + σ−
S+eff) + Jzσ
zSzeff
�Ψ(0),
1
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (1)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
T << TK
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
To model the edge electrons, we introduce the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihilates a right-moving
(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-e
interactions, the edge Hamiltonian can then be written as
HK = Ψ†(0)
�J⊥(σ
+S−eff + σ−
S+eff) + Jzσ
zSzeff
�Ψ(0),
1
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (1)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
T << TK
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
To model the edge electrons, we introduce the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihilates a right-moving
(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-e
interactions, the edge Hamiltonian can then be written as
HK = Ψ†(0)
�J⊥(σ
+S−eff + σ−
S+eff) + Jzσ
zSzeff
�Ψ(0),
formation of impurity-electron singlet (”Kondo screening”)
Adding a magnetic impurity...
Pauli principle:punctured 1D lattice
T=0 insulator!
Adding a magnetic impurity...
Does this really happen for the helical liquid?To find out, first add e-e interactions.... important in 1D!
1
k kF − kF
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (1)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
T << TK
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
To model the edge electrons, we introduce the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihilates a right-moving
(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-e
interactions, the edge Hamiltonian can then be written as
HK = Ψ†(0)
�J⊥(σ
+S−eff + σ−
S+eff) + Jzσ
zSzeff
�Ψ(0),
1
k kF − kF
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (1)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
T << TK
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
To model the edge electrons, we introduce the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihilates a right-moving
(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-e
interactions, the edge Hamiltonian can then be written as
HK = Ψ†(0)
�J⊥(σ
+S−eff + σ−
S+eff) + Jzσ
zSzeff
�Ψ(0),
1
k kF − kF
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (1)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
T << TK
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
To model the edge electrons, we introduce the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihilates a right-moving
(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-e
interactions, the edge Hamiltonian can then be written as
HK = Ψ†(0)
�J⊥(σ
+S−eff + σ−
S+eff) + Jzσ
zSzeff
�Ψ(0),
bulk
local (at impurity site)
+
from Kondo
+ +
≠ 0 if U(1) spin symmetry is broken
1
( )
K → K �
E =c
8log(
2�1�22�1 + �2
) + const.
θ
2�1 �2
γ0 ∼ (Jx + J �y)
2T 2(√K−λ/2)2−1, γ�
0 ∼ (Jx − J �y)
2T 2(√K+λ/2)2−1, γE
0 ∼ J2NCT
2K−1, γE0 ∼ J2
NCT
I = G0V − δI
T � TK
K < 1/4
G ∼ (T/TK)2(1/4K−1)
G = 2e2/h
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
1
( )
K → K �
E =c
8log(
2�1�22�1 + �2
) + const.
θ
2�1 �2
γ0 ∼ (Jx + J �y)
2T 2(√K−λ/2)2−1, γ�
0 ∼ (Jx − J �y)
2T 2(√K+λ/2)2−1, γE
0 ∼ J2NCT
2K−1, γE0 ∼ J2
NCT
I = G0V − δI
T � TK
K < 1/4
G ∼ (T/TK)2(1/4K−1)
G = 2e2/h
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
T. L. Schmidt et al., PRL 108, 156402 (2012)
Adding a magnetic impurity...
functions of Kondo couplings
”Luttinger liquid parameter”
1
K
H = (v/2)
�dx
�(∂xϕ)
2+ (∂xϑ)
2�
+A
κcos(
√4πKϕ)+
B
κsin(
√4πKϕ)+
C√K
∂xϑ
+gU
2(πκ)2cos(
√16πKϕ)
+gie
2π2√K
: (∂2xϑ) cos(
√4πKϕ) : (1)
k kF − kF
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (2)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
T << TK
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
To model the edge electrons, we introduce the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihilates a right-moving
(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-e
interactions, the edge Hamiltonian can then be written as
HK = Ψ†(0)
�J⊥(σ
+S−eff + σ−
S+eff) + Jzσ
zSzeff
�Ψ(0),
...perturbative RG and linear response
J. Maciejko et al., PRL 102, 256803 (2009)
Adding the kinetic energy and bosonizing...
Adding a magnetic impurity...
from J. Maciejko et al., PRL 102, 256803 (2009)
”weak” e-e interaction
”strong” e-e interaction
no correction to the edge conductance at T = 0 !1
G = 2e2/h
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
Adding a magnetic impurity...
Due to its topological nature, the QSH state follows the new shape of the edge.Weak coupling QSH states are robust against local breaking of time-reversal symmetry!
J. Maciejko et al., PRL 102, 256803 (2009)
But... one important thing is missing from the analysis!
z
d
zd
Rashba spin-orbit interaction!
2DEG
semiconductor heterostructure
Spatial asymmetry of band edges mimics an E-field in the z-direction
1
HR = α(kxσy − kyσ
x)
HSO = λcrystal(∇V × k) · σ
λcrystal ≈ �2/4m
∗Eg ≈ 106
λvac
λvac = �2/4m
2
0c2 ≈ 3.7× 10−6
A2
HSO = λvac(∇V × k) · σ
Kc ≈ 2.2TK
1
2
K(R)
R
K(R) ∝ (J2/D) cos(kF R)
TK ∝ D exp(−1/ρF J)
K(R) > TK
V
U
Vg
J ∝ V2/U
�d
�F
�d+U
Cimp ∝ (T/TK) ln(TK/T ), Simp = k ln(√
2), ...
Simp = k ln(√
2)
T
K(R)critical
δ = π/2
δ = 0
−ψ(r)
ψ(r)
K(R)→∞
K(R)→ −∞
Hint = J1S1 · σ1 + J2S2 · σ2 + K(R)S1 · S2
J1 �= J2
Yu. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984)
z
d
zd
Rashba spin-orbit interaction!
semiconductor heterostructure
1
HR = α kxσy
K
H = (v/2)
�dx
�(∂xϕ)
2+ (∂xϑ)
2�
+A
κcos(
√4πKϕ)+
B
κsin(
√4πKϕ)+
C√K
∂xϑ
+gU
2(πκ)2cos(
√16πKϕ)
+gie
2π2√K
: (∂2xϑ) cos(
√4πKϕ) : (1)
k kF − kF
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (2)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
T << TK
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
To model the edge electrons, we introduce the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihilates a right-moving
(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-e
interactions, the edge Hamiltonian can then be written as
HK = Ψ†(0)
�J⊥(σ
+S−eff + σ−
S+eff) + Jzσ
zSzeff
�Ψ(0),
1
x
HR = α kxσy
K
H = (v/2)
�dx
�(∂xϕ)
2+ (∂xϑ)
2�
+A
κcos(
√4πKϕ)+
B
κsin(
√4πKϕ)+
C√K
∂xϑ
+gU
2(πκ)2cos(
√16πKϕ)
+gie
2π2√K
: (∂2xϑ) cos(
√4πKϕ) : (1)
k kF − kF
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (2)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
T << TK
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
To model the edge electrons, we introduce the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihilates a right-moving
(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-e
interactions, the edge Hamiltonian can then be written as
HK = Ψ†(0)
�J⊥(σ
+S−eff + σ−
S+eff) + Jzσ
zSzeff
�Ψ(0),
x
doesn’t conserve spin
Adding the Rashba interaction...... breaks the locking of spin to momentum. However, there is still a single Kramers pair on the QSH edge, and this is all that matters!
1
H = vF
�dx Ψ†
(x) [−iσz∂x]Ψ(x) + α
�dx Ψ†
(x) [−iσy∂x]Ψ(x)
x
HR = α kxσy
K
H = (v/2)
�dx
�(∂xϕ)
2+ (∂xϑ)
2�
+A
κcos(
√4πKϕ)+
B
κsin(
√4πKϕ)+
C√K
∂xϑ
+gU
2(πκ)2cos(
√16πKϕ)
+gie
2π2√K
: (∂2xϑ) cos(
√4πKϕ) : (1)
k kF − kF
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (2)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
T << TK
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
To model the edge electrons, we introduce the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihilates a right-moving
(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-e
interactions, the edge Hamiltonian can then be written as
HK = Ψ†(0)
�J⊥(σ
+S−eff + σ−
S+eff) + Jzσ
zSzeff
�Ψ(0),
Electrical control of the Kondo effect in a helical edge liquid
Erik Eriksson,1Anders Strom,
1Girish Sharma,
2and Henrik Johannesson
1
1Department of Physics, University of Gothenburg, SE 412 96 Gothenburg, Sweden2Centre de Physique Theorique, Ecole Polytechnique, 91128 Palaiseau Cedex, France
Magnetic impurities affect the transport properties of the helical edge states of quantum spin
Hall insulators by causing single-electron backscattering. We study such a system in the presence
of a Rashba spin-orbit interaction induced by an external electric field, showing that this can be
used to control the Kondo temperature, as well as the correction to the conductance due to the
impurity. Surprisingly, for a strongly anisotropic electron-impurity spin exchange, Kondo screening
may get obstructed by the presence of a non-collinear spin interaction mediated by the Rashba
coupling. This challenges the expectation that the Kondo effect is stable against time-reversal
invariant perturbations.
PACS numbers: 71.10.Pm, 72.10.Fk, 85.75.-d
Introduction. The discovery that HgTe quantum wells
support a quantum spin Hall (QSH) state?
has set offan avalanche of studies addressing the properties of this
novel phase of matter?. A key issue has been to deter-
mine the conditions for stability of the current-carrying
states at the edge of the sample as this is the feature
that most directly impacts prospects for future applica-
tions in electronics/spintronics. In the simplest picture
of a QSH system the edge states are helical, with counter-
propagating electrons carrying opposite spins. By time-
reversal invariance electron transport then becomes bal-
listic, provided that the electron-electron (e-e) interaction
is sufficiently well-screened so that higher-order scatter-
ing processes do not come into play? ?
.
The picture gets an added twist when including effectsfrom magnetic impurities, contributed by dopant ions or
electrons trapped by potential inhomogeneities. Since an
edge electron can backscatter from an impurity via spin
exchange, time-reversal invariance no longer protects the
helical states from mixing. In addition, correlated two-
electron?
and inelastic single-electron processes? ?
must
now also be accounted for. As a result, at high temper-
atures T electron scattering off the impurity leads to a
ln(T ) correction?
of the conductance at low frequencies
ω, which, however, vanishes? in the dc limit ω → 0. At
low T , for weak e-e interactions, the quantized edge con-
ductance G0 = e2/h is restored as T → 0 with power
laws distinctive of a helical edge liquid. For strong inter-
actions the edge liquid freezes into an insulator at T = 0,
with thermally induced transport via tunneling of frac-
tionalized charge excitations through the impurity?.
A more complete description of edge transport in a
QSH system must include also the presence of a Rashba
spin-orbit interaction. This interaction, which can be
tuned by an external gate voltage, is a built-in feature
of a quantum well?. In fact, HgTe quantum wells ex-
hibit some of the largest known Rashba couplings of
any semiconductor heterostructures?. As a consequence,
spin is no longer conserved, contrary to what is assumed
in the minimal model of a QSH system?. However,
since the Rashba interaction preserves time-reversal in-
variance, Kramers’ theorem guarantees that the edge
states are still connected via a time-reversal transforma-
tion (”Kramers pair”)?. Provided that the Rashba inter-
action is spatially uniform and the e-e interaction is not
too strong, this ensures the robustness of the helical edge
liquid?.
What is the physics with both Kondo and Rashba in-
teractions present? In this Letter we address this ques-
tion with a renormalization group (RG) analysis as well
as a linear-response and rate-equation approach. Specif-
ically, we predict that the Kondo temperature TK −which sets the scale below which the electrons screen the
impurity − can be controlled by varying the strength
of the Rashba interaction. Surprisingly, for a strongly
anisotropic Kondo exchange, a non-collinear spin inter-
action mediated by the Rashba coupling becomes rel-
evant (in the sense of RG) and competes with the
Kondo screening. This challenges the expectation that
the Kondo effect is stable against time-reversal invari-
ant perturbations?. Moreover, we show that the im-
purity contribution to the dc conductance at tempera-
tures T > TK can be switched on and off by adjusting
the Rashba coupling. With the Rashba coupling being
tunable by a gate voltage, this suggests a new inroad to
control charge transport at the edge of a QSH device.
Model. To model the edge electrons, we introduce
the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihi-
lates a right-moving (left-moving) electron with spin-up
(spin-down) along the growth direction of the quantum
well. Neglecting e-e interactions, the edge Hamiltonian
can then be written as
H = vF
�dx Ψ†
(x) [−iσz∂x]Ψ(x) +
+α
�dx Ψ†
(x) [−iσy∂x]Ψ(x) + (1)
+Ψ†(0) [Jxσ
xSx+ Jyσ
ySy+ Jzσ
zSz]Ψ(0),
with vF the Fermi velocity parameterizing the linear ki-
netic energy. The second term encodes the Rashba inter-
action of strength α, with the third term being an anti-
ferromagnetic Kondo interaction between electrons (with
Pauli matrices σi, i = x, y, z) and a spin-1/2 magnetic im-
1
Ψ� = e−iσxθ/2Ψ
H = vF
�dx Ψ†(x) [−iσz∂x]Ψ(x) + α
�dx Ψ†(x) [−iσy∂x]Ψ(x)
x
HR = α kxσy
K
H = (v/2)
�dx
�(∂xϕ)
2 + (∂xϑ)2�
+A
κcos(
√4πKϕ)+
B
κsin(
√4πKϕ)+
C√K
∂xϑ
+gU
2(πκ)2cos(
√16πKϕ)
+gie
2π2√K
: (∂2xϑ) cos(
√4πKϕ) : (1)
k kF − kF
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (2)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
T << TK
Mn2+
kinetic term Rashba
1
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
Ψ� = e−iσxθ/2Ψ
H = vF
�dx Ψ†(x) [−iσz∂x]Ψ(x) + α
�dx Ψ†(x) [−iσy∂x]Ψ(x)
x
HR = α kxσy
K
H = (v/2)
�dx
�(∂xϕ)
2 + (∂xϑ)2�
+A
κcos(
√4πKϕ)+
B
κsin(
√4πKϕ)+
C√K
∂xϑ
+gU
2(πκ)2cos(
√16πKϕ)
+gie
2π2√K
: (∂2xϑ) cos(
√4πKϕ) : (1)
k kF − kF
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (2)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
T << TK
1
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
Ψ� = e−iσxθ/2Ψ
H = vF
�dx Ψ†(x) [−iσz∂x]Ψ(x) + α
�dx Ψ†(x) [−iσy∂x]Ψ(x)
x
HR = α kxσy
K
H = (v/2)
�dx
�(∂xϕ)
2 + (∂xϑ)2�
+A
κcos(
√4πKϕ)+
B
κsin(
√4πKϕ)+
C√K
∂xϑ
+gU
2(πκ)2cos(
√16πKϕ)
+gie
2π2√K
: (∂2xϑ) cos(
√4πKϕ) : (1)
k kF − kF
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (2)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
1
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
Ψ� = e−iσxθ/2Ψ
H = vF
�dx Ψ†(x) [−iσz∂x]Ψ(x) + α
�dx Ψ†(x) [−iσy∂x]Ψ(x)
x
HR = α kxσy
K
H = (v/2)
�dx
�(∂xϕ)
2 + (∂xϑ)2�
+A
κcos(
√4πKϕ)+
B
κsin(
√4πKϕ)+
C√K
∂xϑ
+gU
2(πκ)2cos(
√16πKϕ)
+gie
2π2√K
: (∂2xϑ) cos(
√4πKϕ) : (1)
k kF − kF
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (2)
1
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
Ψ� = e−iσxθ/2Ψ
H = vF
�dx Ψ†(x) [−iσz∂x]Ψ(x) + α
�dx Ψ†(x) [−iσy∂x]Ψ(x)
x
HR = α kxσy
K
H = (v/2)
�dx
�(∂xϕ)
2 + (∂xϑ)2�
+A
κcos(
√4πKϕ)+
B
κsin(
√4πKϕ)+
C√K
∂xϑ
+gU
2(πκ)2cos(
√16πKϕ)
+gie
2π2√K
: (∂2xϑ) cos(
√4πKϕ) : (1)
k kF − kF
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (2)
zz’
The ”Rashba-rotated” spinorstill defines a Kramers pair
1
θ
γ0 ∼ (Jx + J �y)
2T 2(√K−λ/2)2−1, γ�
0 ∼ (Jx − J �y)
2T 2(√K+λ/2)2−1, γE
0 ∼ J2NCT
2K−1, γE0 ∼ J2
NCT
I = G0V − δI
T � TK
K < 1/4
G ∼ (T/TK)2(1/4K−1)
G = 2e2/h
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
C. L. Kane and E. J. Mele, PRL 95, 226801 (2005)
Adding the Rashba interaction...
e-e interaction is invariant under
1
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
Ψ� = e−iσxθ/2Ψ
H = vF
�dx Ψ†(x) [−iσz∂x]Ψ(x) + α
�dx Ψ†(x) [−iσy∂x]Ψ(x)
x
HR = α kxσy
K
H = (v/2)
�dx
�(∂xϕ)
2 + (∂xϑ)2�
+A
κcos(
√4πKϕ)+
B
κsin(
√4πKϕ)+
C√K
∂xϑ
+gU
2(πκ)2cos(
√16πKϕ)
+gie
2π2√K
: (∂2xϑ) cos(
√4πKϕ) : (1)
k kF − kF
Kondo interaction
2
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (2)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
T << TK
Mn2+
(Sz)2
S = 5/2 −→ Seff = 1/2
low T
To model the edge electrons, we introduce the two-spinors ΨT=
�ψ↑,ψ↓
�, where ψ↑ (ψ↓) annihilates a right-moving
(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-e
interactions, the edge Hamiltonian can then be written as
HK = Ψ†(0)
�J⊥(σ
+S−eff + σ−
S+eff) + Jzσ
zSzeff
�Ψ(0),
1
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
Ψ� = e−iσxθ/2Ψ
H = vF
�dx Ψ†(x) [−iσz∂x]Ψ(x) + α
�dx Ψ†(x) [−iσy∂x]Ψ(x)
x
HR = α kxσy
K
H = (v/2)
�dx
�(∂xϕ)
2 + (∂xϑ)2�
+A
κcos(
√4πKϕ)+
B
κsin(
√4πKϕ)+
C√K
∂xϑ
+gU
2(πκ)2cos(
√16πKϕ)
+gie
2π2√K
: (∂2xϑ) cos(
√4πKϕ) : (1)
1
Ψ� = e−iσxθ/2Ψ
H = vF
�dx Ψ†(x) [−iσz∂x]Ψ(x) + α
�dx Ψ†(x) [−iσy∂x]Ψ(x)
x
HR = α kxσy
K
H = (v/2)
�dx
�(∂xϕ)
2 + (∂xϑ)2�
+A
κcos(
√4πKϕ)+
B
κsin(
√4πKϕ)+
C√K
∂xϑ
+gU
2(πκ)2cos(
√16πKϕ)
+gie
2π2√K
: (∂2xϑ) cos(
√4πKϕ) : (1)
k kF − kF
∂J⊥∂D
= −νJ⊥Jz + ...
∂Jz∂D
= −νJ2⊥ + ... (2)
J0 ≡ max(J⊥, Jz)D=D0
TK = D0 exp(−const./J0)
T << TK
Mn2+
1
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
Ψ� = e−iσxθ/2Ψ
H = vF
�dx Ψ†(x) [−iσz∂x]Ψ(x) + α
�dx Ψ†(x) [−iσy∂x]Ψ(x)
x
HR = α kxσy
K
H = (v/2)
�dx
�(∂xϕ)
2 + (∂xϑ)2�
+A
κcos(
√4πKϕ)+
B
κsin(
√4πKϕ)+
C√K
∂xϑ
+gU
2(πκ)2cos(
√16πKϕ)
+gie
2π2√K
: (∂2xϑ) cos(
√4πKϕ) : (1)
XYZ Kondo Non-Collinear term
depend on the Rashba coupling controllable by a gate voltage
1
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
Ψ� = e−iσxθ/2Ψ
H = vF
�dx Ψ†(x) [−iσz∂x]Ψ(x) + α
�dx Ψ†(x) [−iσy∂x]Ψ(x)
x
HR = α kxσy
...bosonization and perturbative RG
E. Eriksson et al., PRB 86, 161103(R) (2012)
Electrical control of the Kondo temperaturevia the ”Rashba angle” ~ gate voltage
easy-axis Kondo
1
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
Ψ� = e−iσxθ/2Ψ
H = vF
�dx Ψ†(x) [−iσz∂x]Ψ(x) + α
�dx Ψ†(x) [−iσy∂x]Ψ(x)
x
1
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
Ψ� = e−iσxθ/2Ψ
Region where dominates the RG flow obstruction of Kondo screening!
1
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
Ψ� = e−iσxθ/2Ψ
challenges the Meir-Wingreen
conjecture that the Kondo effect is
blind to time-reversal invariant
perturbations
Y. Meir and N.S. Wingreen, PRB 50, 4947 (1994)
easy-plane Kondo
2
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = 20 meV, Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
Low-temperature transport, (away from the ”dome”)
1
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
”weak” e-e interaction
1
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
1
G =2e
2
h
L ∼ 1µm
L ∼ 20µm
g
g ∼ 1/2v
ξ < L
α(±2kF )
±2kF
z⊥ ≡ 2g
√CK
z� ≡ 4K − 2
γ = (α(2kF ) + α(−2kF ))/2
∂xθ
ψ↑ = η↑ exp�i√
π[φ + θ]�/
√2πκ
ψ↓ = η↓ exp�−i√
π[φ− θ]�/
√2πκ
H0 + Hd + Hf + HR
O(α2)
α(x) = �α(x)�+
�
n
α(kn)eikxn
Ψ↑↓(x)=ψ↑↓(x)e±ikF x
Ψ↓(x)=ψ↓(x)e−ikF x
HR = α(x)(z × k) · σ
B = ∇×A
E = E(x, y, 0)
(E × k) · σ = Eσz(kyx− kxy)
A =B
2(y,−x, 0)
A · k ∼ eB(kyx− kxy)
G = νe2
h
Mc ≈ 100 meV
D. Grundler, Phys. Rev. Lett. 84, 6074 (2000)
W. Hausler, L. Kecke, and A. H. MacDonald, Phys.
Rev. B 65, 085104 (2002)
�α0 � 2× 10−11
eV m
Λ � 0.5 eV
vF � 1× 106
m/s
Kc + Ks � 1.8
HR
1
T = 0
K>1/4
T � TK
Jx= Jy
= Jz= 10 meV
Jx= Jy
= 5 meV,Jz
= 50 meV
θ
J�y, J
�z, JNC
displaym
ath
α
H�K= Ψ
�† (0)[Jxσ
x Sx + J
�yσy�Sy� + J
�zσz�Sz� + JNC(σ
y�Sz� + σ
z�Sy�)]Ψ
� (0)
S� = e
−iSx θ/2Se
iSx θ/2
Ψ → Ψ�
vα=
�v2F+ α2
cos θ= vF
/vα
sin θ= α/vα
Ψ�T =
�ψ↑�,
ψ↓��
1
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
”weak” e-e interaction
1
1/4 < K < 2/3 K > 2/3
(T/TK)8K−2 (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
1
1/4 < K < 2/3 K > 2/3
(T/TK)8K−2 (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
1
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
1
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
Low-temperature transport, (away from the ”dome”)
1
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
signature of
Rashba!
1
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
”weak” e-e interaction
1
1/4 < K < 2/3 K > 2/3
(T/TK)8K−2 (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
1
1/4 < K < 2/3 K > 2/3
(T/TK)8K−2 (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
”strong” e-e interaction
1
G = 0
1/4 < K < 2/3 K > 2/3
(T/TK)8K−2 (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
1
T = 0
K>1/4
T � TK
Jx= Jy
= Jz= 10 meV
Jx= Jy
= 5 meV,Jz
= 50 meV
θ
J�y, J
�z, JNC
displaym
ath
α
H�K= Ψ
�† (0)[Jxσ
x Sx + J
�yσy�Sy� + J
�zσz�Sz� + JNC(σ
y�Sz� + σ
z�Sy�)]Ψ
� (0)
S� = e
−iSx θ/2Se
iSx θ/2
Ψ → Ψ�
vα=
�v2F+ α2
cos θ= vF
/vα
sin θ= α/vα
Ψ�T =
�ψ↑�,
ψ↓��
1
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
1
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
1
K < 1/4
G ∼ (T/Tbs)2(1/4K−1)
G = 2e2/h
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J �y, J
�z, JNC
displaymath
Low-temperature transport, (away from the ”dome”)
1
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
signature of
Rashba!
1
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
”weak” e-e interaction
1
1/4 < K < 2/3 K > 2/3
(T/TK)8K−2 (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
1
1/4 < K < 2/3 K > 2/3
(T/TK)8K−2 (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
”strong” e-e interaction
1
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
1
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
1
K < 1/4
G ∼ (T/Tbs)2(1/4K−1)
G = 2e2/h
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J �y, J
�z, JNC
displaymath
from instanton processes
J. Maciejko et al., PRL 102, 256803 (2009)
1
T � TK
K < 1/4
G ∼ (T/TK)2(1/4K−1)
G = 2e2/h
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
Low-temperature transport, (away from the ”dome”)
1
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
J�y, J
�z, JNC
displaymath
α
H�K = Ψ�†(0)[Jxσ
xSx + J
�yσ
y�Sy�+ J
�zσ
z�Sz�+ JNC(σ
y�Sz�+ σz�
Sy�)]Ψ�(0)
S� = e−iSxθ/2SeiSxθ/2
Ψ → Ψ�
vα =�
v2F + α2
cos θ = vF /vα
sin θ = α/vα
Ψ�T =�ψ↑�,ψ↓�
�
H� = vα
�dx Ψ�†(x)
�−iσz�
∂x�Ψ�(x)
signature of
Rashba!
blind to
Rashba
1
I = G0V − δI
T � TK
K < 1/4
G ∼ (T/TK)2(1/4K−1)
G = 2e2/h
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
1
T � TK
K < 1/4
G ∼ (T/Tbs)2(1/4K−1)
G = 2e2/h
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
”High-temperature” transport, 1
J⊥,z � ω � T
( )
K → K �
E =c
8log(
2�1�22�1 + �2
) + const.
θ
2�1 �2
γ0 ∼ (Jx + J �y)
2T 2(√K−λ/2)2−1, γ�
0 ∼ (Jx − J �y)
2T 2(√K+λ/2)2−1, γE
0 ∼ J2NCT
2K−1, γE0 ∼ J2
NCT
I = G0V − δI
T � TK
K < 1/4
G ∼ (T/TK)2(1/4K−1)
G = 2e2/h
G = 0
1/4 < K < 2/3 K > 2/3
1
I = G0V − δI
T � TK
K < 1/4
G ∼ (T/TK)2(1/4K−1)
G = 2e2/h
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
conductance correction in dc limit:1
γ0 ∼ (Jx + J �y)
2T 2(√K−λ/2)2−1, γ�
0 ∼ (Jx − J �y)
2T 2(√K+λ/2)2−1, γE
0 ∼ J2NCT
2K−1, γE0 ∼ J2
NCT
I = G0V − δI
T � TK
K < 1/4
G ∼ (T/TK)2(1/4K−1)
G = 2e2/h
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
1
T � TK
K < 1/4
G ∼ (T/Tbs)2(1/4K−1)
G = 2e2/h
G = 0
1/4 < K < 2/3 K > 2/3
∼ (T/TK)8K−2 ∼ (T/TK)2K+2
G =2e2
h− δG
T = 0
K > 1/4
T � TK
Jx = Jy = Jz = 10 meV
Jx = Jy = 5 meV, Jz = 50 meV
θ
”High-temperature” transport, 1
J⊥,z � ω � T
( )
K → K �
E =c
8log(
2�1�22�1 + �2
) + const.
θ
2�1 �2
γ0 ∼ (Jx + J �y)
2T 2(√K−λ/2)2−1, γ�
0 ∼ (Jx − J �y)
2T 2(√K+λ/2)2−1, γE
0 ∼ J2NCT
2K−1, γE0 ∼ J2
NCT
I = G0V − δI
T � TK
K < 1/4
G ∼ (T/TK)2(1/4K−1)
G = 2e2/h
G = 0
1/4 < K < 2/3 K > 2/3
Summary and outlook
Magnetic impurity in a helical edge liquid:Rashba coupling allows electrical control of Kondo temperature and IV-characteristics
blocking of Kondo screening!?
Rashba-induced impurity correction accessible in experiment?
Interesting open problems:Two-loop RG, thermal transport, effects from Dresselhaus spin-orbit interactions, Kondo lattice in a helical liquid, higher-spin impurities... and more!
E. Eriksson et al., PRB 86, 161103(R) (2012)
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