kondo physics in 2d topological insulators with rashba

Erik Eriksson (University of Gothenburg)Anders Ström (currently at TU Braunschweig)

Girish Sharma (currently at Clemson University)Henrik Johannesson (University of Gothenburg)

Kondo Physics in 2D Topological Insulators with Rashba Interactions

Cargèse International School 2013 Topological Phases in Condensed Matter and Cold Atom Systems

June 25, 2013

Outline

2D topological insulators... some basics


At the edge: A new kind of electron liquid

Outline



Adding a magnetic impurity...

Outline




... and a Rashba spin-orbit interaction

Outline





Electrical control of the Kondo effect!

Outline





Electrical control of the Kondo effect!

Summary and outlook

Outline

”topologically ordered”FQHE, Z2 spin liquids,...

”Thouless type”IQHE,...

”symmetry protected”topological insulators,...

....

”symmetry protected”

topological insulatorsprotected by time-reversal invariance

2D topological insulatorsF. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988)C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005)B. A. Bernevig and S. C. Zhang, Phys. Rev. Lett. 96, 106802 (2006)B. A. Bernevig, T. A. Hughes, and S. C. Zhang, Science 314, 1757 (2006)

experiment: M. König et al., Science 318, 766 (2007) C. Brüne et al., Nature Physics 8, 486 (2012)



”reference state”: 2D bulk insulatorshowing quantumspin Hall (QSH) effect

| >

| >

electric field

spin current

1

⇤+ | S | +⌅

spin-polarized helical edgestates for constant �1,2

choose spin quantizationaxis along ⇤+ | S | +⌅

�1,2

⇤sxy = Cs

e

4⇥, Cs =

�±2 if QSH insulator0 if ordinary insulator

⇤sxy = �

e

2⇥, � =

�1 if 2D topological insulator0 if ordinary insulator

� =Cs

2mod 2 =

�1 ”2D topological insulator”0 ordinary insulator

� = 1

� = 0

⇤sxy = ± e

2⇥

t > t0

⇥ | �q(t) ⌅ � e�iHqt | �o ⌅

H = J ⇥(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇥ Hq = J ⇥(J1�1 · �2 + J2�1 · �3) + J ⇥(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⌅

F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988)C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005)B. A. Bernevig and S. C. Zhang, Phys. Rev. Lett. 96, 106802 (2006)B. A. Bernevig, T. A. Hughes, and S. C. Zhang, Science 314, 1757 (2006)



1

⇤+ | S | +⌅



�1,2

⇤sxy = Cs

e

4⇥, Cs =


⇤sxy = �

e

2⇥, � =


� =�

2mod 2 =


� = 1

� = 0

⇤sxy = ± e

4⇥

t > t0

⇥ | �q(t) ⌅ � e�iHqt | �o ⌅

H = J ⇥(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇥ Hq = J ⇥(J1�1 · �2 + J2�1 · �3) + J ⇥(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⌅

2D topological insulators

| >

| >

electric field

spin current



1

⇤+ | S | +⌅



�1,2

⇤sxy = Cs

e

4⇥, Cs =


⇤sxy = �

e

2⇥, � =


� =�

2mod 2 =


� = 1

� = 0

⇤sxy = ± e

4⇥

t > t0

⇥ | �q(t) ⌅ � e�iHqt | �o ⌅

H = J ⇥(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇥ Hq = J ⇥(J1�1 · �2 + J2�1 · �3) + J ⇥(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⌅



Perturb adiabatic

ally with a

time-rev

ersal invar

iant spin-

nonconserving in

teraction | >

| >

electric field

spin current



electric field

1

⇤+ | S | +⌅



�1,2

⇤sxy = Cs

e

4⇥, Cs =


⇤sxy = �

e

2⇥, � =


� =�

2mod 2 =


� = 1

� = 0

⇤sxy = ± e

4⇥

t > t0

⇥ | �q(t) ⌅ � e�iHqt | �o ⌅

H = J ⇥(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇥ Hq = J ⇥(J1�1 · �2 + J2�1 · �3) + J ⇥(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⌅



Perturb adiabatic

ally with a

time-rev

ersal invar

iant spin-

nonconserving in

teraction



1

⇤+ | S | +⌅



�1,2

⇤sxy = Cs

e

4⇥, Cs =


⇤sxy = �

e

2⇥, � =


� =�

2mod 2 =


� = 1

� = 0

⇤sxy = ± e

4⇥

t > t0

⇥ | �q(t) ⌅ � e�iHqt | �o ⌅

H = J ⇥(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇥ Hq = J ⇥(J1�1 · �2 + J2�1 · �3) + J ⇥(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⌅



Perturb adiabatic

ally with a

time-rev

ersal invar

iant spin-

nonconserving in

teraction

1

⇤+ |S |+⌅ =⇥⇧3(⇥�

2⇥1 +⇥�1⇥2)X

+i⇥⇧3(�⇥�

2⇥1 +⇥�1⇥2)Y

+ (|⇥2 |2 [1+ |� |2] + |⇥1⇥ |2

3)Z

⇤� |S |�⌅ = �⇤+ |S |+⌅

|+⌅ = ⇥1 | E1+⌅+⇥2 | H1+⌅

|�⌅ = T |+⌅ = �⇥�1 | E1�⌅ �⇥�

2 | H1�⌅

|E1±⌅ = � |�6,±1

2⌅+ ⇥ |�8,±

1

2⌅

displaymath

|H1±⌅ = � |�8,±3

2⌅

Jx = Jy = 3Jz = 30 meV, a0 = 0.5 nm, vF = 5⇥ 105 m/s, and D = 300 meV.

⇤+ | S | +⌅



�1,2

⇧sxy = Cs

e

4⌅, Cs =


⇧sxy = ⇤

e

2⌅, ⇤ =


⇤ =


2D topological insulator ’’adiabatically connected’’to an ideal quantum spinHall (QSH) insulator



Z2 topological invariantencodes Berry curvature structure of the bulk bands


1

⇤+ |S |+⌅ =⇥⇧3(⇥�

2⇥1 +⇥�1⇥2)X

+i⇥⇧3(�⇥�

2⇥1 +⇥�1⇥2)Y

+ (|⇥2 |2 [1+ |� |2] + |⇥1⇥ |2

3)Z

⇤� |S |�⌅ = �⇤+ |S |+⌅

|+⌅ = ⇥1 | E1+⌅+⇥2 | H1+⌅

|�⌅ = T |+⌅ = �⇥�1 | E1�⌅ �⇥�

2 | H1�⌅

|E1±⌅ = � |�6,±1

2⌅+ ⇥ |�8,±

1

2⌅

displaymath

|H1±⌅ = � |�8,±3

2⌅


⇤+ | S | +⌅



�1,2

⇧sxy = Cs

e

4⌅, Cs =


⇧sxy = ⇤

e

2⌅, ⇤ =


⇤ =


1

⌃sxy = ⇥

e

2⌅, ⇥ =

�1 if QSH insulator0 if ordinary insulator

⌃sxy = ⇥

e

2⌅, ⇥ =


⇥ = 1

⇥ = 0

⌃sxy =

e

2⌅

t > t0

⇤ | �q(t) ⇧ ⇥ e�iHqt | �o ⇧

H = J ⌅(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇤ Hq = J ⌅(J1�1 · �2 + J2�1 · �3) + J ⌅(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⇧

⇤K

SA(r) = �Tr(⇧[0,r] log ⇧[0,r]) (1)

⇧[0,r] = Tr[r,⇧]⇧ = Tr[r,⇧] |�⇧⌅� | (2)

H = H⇥ + �1O1 + ...

SA(r) = � limn⇤1

⌥

⌥nTr⇧nr = � lim

n⇤1

⌥

⌥n

ZRn

Zn

1

⌃sxy = ⇥

e

2⌅, ⇥ =


⌃sxy = ⇥

e

2⌅, ⇥ =


⇥ = 1

⇥ = 0

⌃sxy =

e

2⌅

t > t0

⇤ | �q(t) ⇧ ⇥ e�iHqt | �o ⇧

H = J ⌅(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇤ Hq = J ⌅(J1�1 · �2 + J2�1 · �3) + J ⌅(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⇧

⇤K

SA(r) = �Tr(⇧[0,r] log ⇧[0,r]) (1)

⇧[0,r] = Tr[r,⇧]⇧ = Tr[r,⇧] |�⇧⌅� | (2)

H = H⇥ + �1O1 + ...


⌥


n⇤1

⌥

⌥n

ZRn

Zn


experiment: M. König et al., Science 318, 766 (2007 C. Brüne et al., Nature Physics 8, 486 (2012)


1

⌃sxy = ⇥

e

2⌅, ⇥ =


⌃sxy = ⇥

e

2⌅, ⇥ =


⇥ = 1

⇥ = 0

⌃sxy =

e

2⌅

t > t0

⇤ | �q(t) ⇧ ⇥ e�iHqt | �o ⇧

H = J ⌅(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇤ Hq = J ⌅(J1�1 · �2 + J2�1 · �3) + J ⌅(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⇧

⇤K

SA(r) = �Tr(⇧[0,r] log ⇧[0,r]) (1)

⇧[0,r] = Tr[r,⇧]⇧ = Tr[r,⇧] |�⇧⌅� | (2)

H = H⇥ + �1O1 + ...


⌥


n⇤1

⌥

⌥n

ZRn

Zn

1

⌃sxy = ⇥

e

2⌅, ⇥ =


⌃sxy = ⇥

e

2⌅, ⇥ =


⇥ = 1

⇥ = 0

⌃sxy =

e

2⌅

t > t0

⇤ | �q(t) ⇧ ⇥ e�iHqt | �o ⇧

H = J ⌅(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇤ Hq = J ⌅(J1�1 · �2 + J2�1 · �3) + J ⌅(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⇧

⇤K

SA(r) = �Tr(⇧[0,r] log ⇧[0,r]) (1)

⇧[0,r] = Tr[r,⇧]⇧ = Tr[r,⇧] |�⇧⌅� | (2)

H = H⇥ + �1O1 + ...


⌥


n⇤1

⌥

⌥n

ZRn

Zn



helical edge states


| >

| >

| >

1

⌃sxy = ⇥

e

2⌅, ⇥ =


⌃sxy = ⇥

e

2⌅, ⇥ =


⇥ = 1

⇥ = 0

⌃sxy =

e

2⌅

t > t0

⇤ | �q(t) ⇧ ⇥ e�iHqt | �o ⇧

H = J ⌅(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇤ Hq = J ⌅(J1�1 · �2 + J2�1 · �3) + J ⌅(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⇧

⇤K

SA(r) = �Tr(⇧[0,r] log ⇧[0,r]) (1)

⇧[0,r] = Tr[r,⇧]⇧ = Tr[r,⇧] |�⇧⌅� | (2)

H = H⇥ + �1O1 + ...


⌥


n⇤1

⌥

⌥n

ZRn

Zn

1

⌃sxy = ⇥

e

2⌅, ⇥ =


⌃sxy = ⇥

e

2⌅, ⇥ =


⇥ = 1

⇥ = 0

⌃sxy =

e

2⌅

t > t0

⇤ | �q(t) ⇧ ⇥ e�iHqt | �o ⇧

H = J ⌅(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇤ Hq = J ⌅(J1�1 · �2 + J2�1 · �3) + J ⌅(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⇧

⇤K

SA(r) = �Tr(⇧[0,r] log ⇧[0,r]) (1)

⇧[0,r] = Tr[r,⇧]⇧ = Tr[r,⇧] |�⇧⌅� | (2)

H = H⇥ + �1O1 + ...


⌥


n⇤1

⌥

⌥n

ZRn

Zn

helical edge states

are pseudospins that keep track on the total angularmomentum sectors (good quantum numbers in thepresence of spin-nonconserving interactions)

| >



| > | >

1

⌃sxy = ⇥

e

2⌅, ⇥ =


⌃sxy = ⇥

e

2⌅, ⇥ =


⇥ = 1

⇥ = 0

⌃sxy =

e

2⌅

t > t0

⇤ | �q(t) ⇧ ⇥ e�iHqt | �o ⇧

H = J ⌅(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇤ Hq = J ⌅(J1�1 · �2 + J2�1 · �3) + J ⌅(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⇧

⇤K

SA(r) = �Tr(⇧[0,r] log ⇧[0,r]) (1)

⇧[0,r] = Tr[r,⇧]⇧ = Tr[r,⇧] |�⇧⌅� | (2)

H = H⇥ + �1O1 + ...


⌥


n⇤1

⌥

⌥n

ZRn

Zn

1

⌃sxy = ⇥

e

2⌅, ⇥ =


⌃sxy = ⇥

e

2⌅, ⇥ =


⇥ = 1

⇥ = 0

⌃sxy =

e

2⌅

t > t0

⇤ | �q(t) ⇧ ⇥ e�iHqt | �o ⇧

H = J ⌅(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇤ Hq = J ⌅(J1�1 · �2 + J2�1 · �3) + J ⌅(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⇧

⇤K

SA(r) = �Tr(⇧[0,r] log ⇧[0,r]) (1)

⇧[0,r] = Tr[r,⇧]⇧ = Tr[r,⇧] |�⇧⌅� | (2)

H = H⇥ + �1O1 + ...


⌥


n⇤1

⌥

⌥n

ZRn

Zn


experiment: M. König et al., Science 318, 766 (2007)

helical edge states

HgTe/CdTeheterostructure

But the spin polarization of helicaledge states has been measured experimentally in HgTe quantum wells!

C. Brüne et al., Nature Physics 8, 486 (2012)

?


| >

| >


| >

1

⌅+ | S | +⇧

choose spin quantizationaxis along ⌅+ | S | +⇧

�1,2

⇧sxy = �

e

2⇤, � =


⇧sxy = �

e

2⇤, � =


� = 1

� = 0

⇧sxy =

e

2⇤

t > t0

⇤ | �q(t) ⇧ ⇥ e�iHqt | �o ⇧

H = J ⇥(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇤ Hq = J ⇥(J1�1 · �2 + J2�1 · �3) + J ⇥(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⇧

⇥K

SA(r) = �Tr(⌅[0,r] log ⌅[0,r]) (1)

1

⇤+ | S | +⌅



�1,2

⌅sxy = �

e

2⇤, � =


⌅sxy = �

e

2⇤, � =


� = 1

� = 0

⌅sxy =

e

2⇤

t > t0

⇥ | �q(t) ⌅ � e�iHqt | �o ⌅

H = J ⇥(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇥ Hq = J ⇥(J1�1 · �2 + J2�1 · �3) + J ⇥(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

| �o ⌅

⇥K

OK in a small energy range!

P. Michetti and P. Recher,Phys. Rev. B 83, 125420 (2011)


”BHZ model”1

|+⌃ = �1 | E1+⌃+�2 | H1+⌃

|�⌃ = T |+⌃ = ��⇥1 | E1�⌃ ��⇥

2 | H1�⌃


⇧+ | S | +⌃


choose spin quantizationaxis along ⇧+ | S | +⌃

�1,2

⇤sxy = Cs

e

4⇥, Cs =


⇤sxy = �

e

2⇥, � =


� =Cs

2mod 2 =


� = 1

� = 0

⇤sxy = ± e

2⇥

t > t0

⌅ | �q(t) ⌃ ⇤ e�iHqt | �o ⌃

1

|+⌃ = �1 | E1+⌃+�2 | H1+⌃

|�⌃ = T |+⌃ = ��⇥1 | E1�⌃ ��⇥

2 | H1�⌃


⇧+ | S | +⌃


choose spin quantizationaxis along ⇧+ | S | +⌃

�1,2

⇤sxy = Cs

e

4⇥, Cs =


⇤sxy = �

e

2⇥, � =


� =Cs

2mod 2 =


� = 1

� = 0

⇤sxy = ± e

2⇥

t > t0

⌅ | �q(t) ⌃ ⇤ e�iHqt | �o ⌃

| >

1

|+⌅ = ⇥1 | E1+⌅+⇥2 | H1+⌅

|�⌅ = T |+⌅ = �⇥�1 | E1�⌅ �⇥�

2 | H1�⌅

|E1±⌅ = � |�6,±1

2⌅+ ⇥ |�8,±

1

2⌅

displaymath

|H1±⌅ = � |�8,±3

2⌅


⇤+ | S | +⌅



�1,2

⇧sxy = Cs

e

4⌅, Cs =


⇧sxy = ⇤

e

2⌅, ⇤ =


⇤ =Cs

2mod 2 =


⇤ = 1

⇤ = 0

⇧sxy = ± e

2⌅

1

|+⌅ = ⇥1 | E1+⌅+⇥2 | H1+⌅

|�⌅ = T |+⌅ = �⇥�1 | E1�⌅ �⇥�

2 | H1�⌅

|E1±⌅ = � |�6,±1

2⌅+ ⇥ |�8,±

1

2⌅

displaymath

|H1±⌅ = � |�8,±3

2⌅


⇤+ | S | +⌅



�1,2

⇧sxy = Cs

e

4⌅, Cs =


⇧sxy = ⇤

e

2⌅, ⇤ =


⇤ =Cs

2mod 2 =


⇤ = 1

⇤ = 0

⇧sxy = ± e

2⌅

1

⇤+ |S |+⌅ =⇥⇧3(⇥�

2⇥1 +⇥�1⇥2)X

+i⇥⇧3(�⇥�

2⇥1 +⇥�1⇥2)Y

+ (|⇥2 |2 [1+ |� |2] + |⇥1⇥ |2

3)Z

⇤� |S |�⌅ = �⇤+ |S |+⌅

|+⌅ = ⇥1 | E1+⌅+⇥2 | H1+⌅

|�⌅ = T |+⌅ = �⇥�1 | E1�⌅ �⇥�

2 | H1�⌅

|E1±⌅ = � |�6,±1

2⌅+ ⇥ |�8,±

1

2⌅

displaymath

|H1±⌅ = � |�8,±3

2⌅


⇤+ | S | +⌅



�1,2

⇧sxy = Cs

e

4⌅, Cs =


⇧sxy = ⇤

e

2⌅, ⇤ =


⇤ =Cs

2mod 2 =


What about the ”symmetry protection”by time-reversal invariance?

The helical edge states are stable against time-reversal invariant perturbations!

strong spin-orbit interactions in atomic p-orbitals create an inverted band gap (p-band on top of s-band)

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 =2e2

h

L ⌃ 1µm

L ⌃ 20µm

g

g ⌃ 1/2v

� < L

�(±2kF )

±2kF

z⇧ ⇧ 2g�

CK

z⌃ ⇧ 4K � 2

⇥ = (�(2kF ) + �(�2kF ))/2

⌘x⌅

�⇤ = ⇤⇤ exp�i� [↵+ ⌅]

⇥/�

2 ⇧

�⌅ = ⇤⌅ exp��i� [↵� ⌅]

⇥/�

2 ⇧

H0 + Hd + Hf + HR

O(�2)

�(x) = ⌦�(x)↵+⇤

n

�(kn)eikxn

⇥⇤⌅(x)=�⇤⌅(x)e±ikF x

⇥⌅(x)=�⌅(x)e�ikF x

HR = �(x)(z ⇤ k) · �

B = �⇤A

E = E(x, y, 0)

(E ⇤ k) · � = E⌦z(kyx� kxy)

A =B

2(y,�x, 0)

A · k ⌃ eB(kyx� kxy)

G = ⌥e2

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)

��0 � 2⇤ 10�11 eV m� � 0.5 eV

vF � 1⇤ 106 m/sKc + Ks � 1.8

HR

B. A. Bernevig et al., PRL 95, 066601 (2005)


What if time-reversal symmetry gets broken...?

The helical edge states are stable against time-reversal invariant perturbations!


... for example, by putting in a magnetic impurity at the edge?

from M. König et al., Science 318, 766 (2007)

case study:

1

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

1

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

1

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

1

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T

J. K. Furdyna, JAP 64, R29 (1988) large and positive single-ion anisotropy



case study:

large and positive single-ion anisotropy

1

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

1

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

1

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T

anisotropic spin exchange with the edge electrons

1

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T

To model the edge electrons, we introduce the two-spinors �T =�⇥⇥,⇥⇤

�, where ⇥⇥ (⇥⇤) annihilates a right-moving

(left-moving) electron with spin-up (spin-down) along the growth direction of the quantum well. Neglecting e-einteractions, the edge Hamiltonian can then be written as

HK = �†(0)⇥J⌅(�

+S� + ��S+) + Jz�zSz

⇤�(0),

1

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T

To model the edge electrons, we introduce the two-spinors �T =�⇥⇥,⇥⇤

�, where ⇥⇥ (⇥⇤) annihilates a right-moving


HK = �†(0)⇥J⌅(�

+S�e� + ��S+

e�) + Jz�zSz

e�

⇤�(0),

1

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

R. Zitko et al., PRB 78, 224404 (2008)




The Kondo interaction is time-reversal invariant! Could it still cause a spontaneous breaking of time reversal invariance and localize the edge states?


Recall the Kondo effect

One-loop RG equations:

strong-coupling physics for

P. W. Anderson, J. Phys. C 3, 2436 (1970)

1

⌅J⌅⌅D

= ��J⌅Jz + ...

⌅Jz⌅D

= ��J2⌅ + ... (1)

J0 ⌘ max(J⌅, Jz)D=D0

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T

To model the edge electrons, we introduce the two-spinors �T =�⇤⇥,⇤⇤

�, where ⇤⇥ (⇤⇤) annihilates a right-moving


HK = �†(0)⇥J⌅(⇥

+S�e� + ⇥�S+

e�) + Jz⇥zSz

e�

⇤�(0),

1

⌅J⌅⌅D

= ��J⌅Jz + ...

⌅Jz⌅D

= ��J2⌅ + ... (1)


TK = D0 exp(�const./J0)

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⇥

+S�e� + ⇥�S+

e�) + Jz⇥zSz

e�

⇤�(0),

1

⌅J⌅⌅D

= ��J⌅Jz + ...

⌅Jz⌅D

= ��J2⌅ + ... (1)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⇥

+S�e� + ⇥�S+

e�) + Jz⇥zSz

e�

⇤�(0),

1

⌅J⌅⌅D

= ��J⌅Jz + ...

⌅Jz⌅D

= ��J2⌅ + ... (1)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⇥

+S�e� + ⇥�S+

e�) + Jz⇥zSz

e�

⇤�(0),

formation of impurity-electron singlet (”Kondo screening”)


Pauli principle:punctured 1D lattice


Pauli principle:punctured 1D lattice

T=0 insulator!


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

= ��J⌅Jz + ...

⌅Jz⌅D

= ��J2⌅ + ... (1)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⇥

+S�e� + ⇥�S+

e�) + Jz⇥zSz

e�

⇤�(0),

1

k kF � kF

⌅J⌅⌅D

= ��J⌅Jz + ...

⌅Jz⌅D

= ��J2⌅ + ... (1)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⇥

+S�e� + ⇥�S+

e�) + Jz⇥zSz

e�

⇤�(0),

1

k kF � kF

⌅J⌅⌅D

= ��J⌅Jz + ...

⌅Jz⌅D

= ��J2⌅ + ... (1)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⇥

+S�e� + ⇥�S+

e�) + Jz⇥zSz

e�

⇤�(0),

”bulk”

local (at impurity site)

+

from Kondo

+ +


Adding the kinetic energy and bosonizing...

Kondo

local Umklapp

local 1-particle

inelastic term

1

K

H = (v/2)

Zdx

�(�x⌥)

2 + (�x⌃)2�

+A

�cos(

p4⇤K⌥)+

B

�sin(

p4⇤K⌥)+

CpK

�x⌃

+gU

2(⇤�)2cos(

p16⇤K⌥)

+gie

2⇤2pK

: (�2x⌃) cos(

p4⇤K⌥) : (1)

k kF � kF

�J⌅�D

= �⇥J⌅Jz + ...

�Jz�D

= �⇥J2⌅ + ... (2)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T

To model the edge electrons, we introduce the two-spinors �T =�⇧⇥,⇧⇤

�, where ⇧⇥ (⇧⇤) annihilates a right-moving


HK = �†(0)⇥J⌅(⌅

+S�e� + ⌅�S+

e�) + Jz⌅zSz

e�

⇤�(0),


Bosonization...”Luttinger liquid parameter”

1

K

H = (v/2)

Zdx

�(�x⌥)

2 + (�x⌃)2�

+A

�cos(

p4⇤K⌥)+

B

�sin(

p4⇤K⌥)+

CpK

�x⌃

+gU

2(⇤�)2cos(

p16⇤K⌥)

+gie

2⇤2pK

: (�2x⌃) cos(

p4⇤K⌥) : (1)

k kF � kF

�J⌅�D

= �⇥J⌅Jz + ...

�Jz�D

= �⇥J2⌅ + ... (2)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⌅

+S�e� + ⌅�S+

e�) + Jz⌅zSz

e�

⇤�(0),


functions of Kondo couplings

”Luttinger liquid parameter”Bosonization...

1

K

H = (v/2)

Zdx

�(�x⌥)

2 + (�x⌃)2�

+A

�cos(

p4⇤K⌥)+

B

�sin(

p4⇤K⌥)+

CpK

�x⌃

+gU

2(⇤�)2cos(

p16⇤K⌥)

+gie

2⇤2pK

: (�2x⌃) cos(

p4⇤K⌥) : (1)

k kF � kF

�J⌅�D

= �⇥J⌅Jz + ...

�Jz�D

= �⇥J2⌅ + ... (2)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⌅

+S�e� + ⌅�S+

e�) + Jz⌅zSz

e�

⇤�(0),


functions of Kondo couplings

”Luttinger liquid parameter”Bosonization...

1

K

H = (v/2)

Zdx

�(�x⌥)

2 + (�x⌃)2�

+A

�cos(

p4⇤K⌥)+

B

�sin(

p4⇤K⌥)+

CpK

�x⌃

+gU

2(⇤�)2cos(

p16⇤K⌥)

+gie

2⇤2pK

: (�2x⌃) cos(

p4⇤K⌥) : (1)

k kF � kF

�J⌅�D

= �⇥J⌅Jz + ...

�Jz�D

= �⇥J2⌅ + ... (2)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⌅

+S�e� + ⌅�S+

e�) + Jz⌅zSz

e�

⇤�(0),

...perturbative RG and linear responseJ. Maciejko et al., PRL 102, 256803 (2009)


from J. Maciejko et al., PRL 102, 256803 (2009)

”weak” e-e interaction

”strong” e-e interaction

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� ⇥G

T = 0

K > 1/4

T ⇤ TK

Jx = Jy = Jz = 10 meV

Jx = Jy = 5 meV, Jz = 50 meV

⇤

J ⇥y, J

⇥z, JNC

displaymath

�

H ⇥K = �⇥†(0)[Jx⌅

xSx + J ⇥y⌅

y0Sy0

+ J ⇥z⌅

z0Sz0

+ JNC(⌅y0Sz0

+ ⌅z0Sy0

)]�⇥(0)


from J. Maciejko et al., PRL 102, 256803 (2009)

”weak” 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� ⇥G

T = 0

K > 1/4

T ⇤ TK



⇤

J ⇥y, J

⇥z, JNC

displaymath

�

H ⇥K = �⇥†(0)[Jx⌅

xSx + J ⇥y⌅

y0Sy0

+ J ⇥z⌅

z0Sz0

+ JNC(⌅y0Sz0

+ ⌅z0Sy0

)]�⇥(0)


Due to its topological nature, the edge states follow the new shape of the edge.Weak coupling helical edge states are robust against spontaneous breaking of time-reversal symmetry!

J. Maciejko et al., PRL 102, 256803 (2009)

But... one important thing is missing from the analysis!

zd

Rashba spin-orbit interaction!

2DEG

semiconductor heterostructure

Spatial asymmetry of band edges mimics an E-field in the z-direction

1

HR = �(kx⌥y � ky⌥x)

HSO = ⌅crystal(⌦V ⇤ k) · �

⌅crystal ⌅ �2/4m⇥Eg ⌅ 106⌅vac

⌅vac = �2/4m20c

2 ⌅ 3.7⇤ 10�6A2

HSO = ⌅vac(⌦V ⇤ k) · �

Kc ⌅ 2.2TK

1

2

K(R)

R

K(R) ⌃ (J2/D) cos(kF R)

TK ⌃ D exp(�1/⌃F J)

K(R) > TK

V

U

Vg

J ⌃ V 2/U

⇤d

⇤F

⇤d+U

Cimp ⌃ (T/TK) ln(TK/T ), Simp = k ln(

2), ...

Simp = k ln(

2)

T

K(R)critical

⇥ = ⇧/2

⇥ = 0

��(r)

�(r)

K(R)⇧⌥

K(R)⇧ �⌥

Hint = J1S1 · �1 + J2S2 · �2 + 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

2DEG


1

HR = �(kx⌥y � ky⌥x)

HSO = ⌅crystal(⌦V ⇤ k) · �

⌅crystal ⌅ �2/4m⇥Eg ⌅ 106⌅vac

⌅vac = �2/4m20c

2 ⌅ 3.7⇤ 10�6A2

HSO = ⌅vac(⌦V ⇤ k) · �

Kc ⌅ 2.2TK

1

2

K(R)

R

K(R) ⌃ (J2/D) cos(kF R)

TK ⌃ D exp(�1/⌃F J)

K(R) > TK

V

U

Vg

J ⌃ V 2/U

⇤d

⇤F

⇤d+U

Cimp ⌃ (T/TK) ln(TK/T ), Simp = k ln(

2), ...

Simp = k ln(

2)

T

K(R)critical

⇥ = ⇧/2

⇥ = 0

��(r)

�(r)

K(R)⇧⌥

K(R)⇧ �⌥

Hint = J1S1 · �1 + J2S2 · �2 + K(R)S1 · S2

J1 �= J2

Yu. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984)Spatial asymmetry of

band edges mimics an E-field in the z-direction,tunable by gate voltage

z

d

zd



1

x

HR = � kx⇧y

K

H = (v/2)

Zdx

�( x�)

2 + ( x⌥)2�

+A

⇥cos(

p4⌅K�)+

B

⇥sin(

p4⌅K�)+

CpK x⌥

+gU

2(⌅⇥)2cos(

p16⌅K�)

+gie

2⌅2pK

: ( 2x⌥) cos(

p4⌅K�) : (1)

k kF � kF

J⌅ D

= �⇤J⌅Jz + ...

Jz D

= �⇤J2⌅ + ... (2)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T

To model the edge electrons, we introduce the two-spinors �T =�⌃⇥,⌃⇤

�, where ⌃⇥ (⌃⇤) annihilates a right-moving


HK = �†(0)⇥J⌅(⇧

+S�e� + ⇧�S+

e�) + Jz⇧zSz

e�

⇤�(0),

z

d

x

1

HR = � kx⇧y

K

H = (v/2)

Zdx

�( x�)

2 + ( x⌥)2�

+A

⇥cos(

p4⌅K�)+

B

⇥sin(

p4⌅K�)+

CpK x⌥

+gU

2(⌅⇥)2cos(

p16⌅K�)

+gie

2⌅2pK

: ( 2x⌥) cos(

p4⌅K�) : (1)

k kF � kF

J⌅ D

= �⇤J⌅Jz + ...

Jz D

= �⇤J2⌅ + ... (2)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⇧

+S�e� + ⇧�S+

e�) + Jz⇧zSz

e�

⇤�(0),

zd


1

HR = � kx⇧y

K

H = (v/2)

Zdx

�( x�)

2 + ( x⌥)2�

+A

⇥cos(

p4⌅K�)+

B

⇥sin(

p4⌅K�)+

CpK x⌥

+gU

2(⌅⇥)2cos(

p16⌅K�)

+gie

2⌅2pK

: ( 2x⌥) cos(

p4⌅K�) : (1)

k kF � kF

J⌅ D

= �⇤J⌅Jz + ...

Jz D

= �⇤J2⌅ + ... (2)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⇧

+S�e� + ⇧�S+

e�) + Jz⇧zSz

e�

⇤�(0),

1

x

HR = � kx⇧y

K

H = (v/2)

Zdx

�( x�)

2 + ( x⌥)2�

+A

⇥cos(

p4⌅K�)+

B

⇥sin(

p4⌅K�)+

CpK x⌥

+gU

2(⌅⇥)2cos(

p16⌅K�)

+gie

2⌅2pK

: ( 2x⌥) cos(

p4⌅K�) : (1)

k kF � kF

J⌅ D

= �⇤J⌅Jz + ...

Jz D

= �⇤J2⌅ + ... (2)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⇧

+S�e� + ⇧�S+

e�) + Jz⇧zSz

e�

⇤�(0),

+ .....

HgTe quantum well:weaker-amplitude kx3 correctionsD.G. Rothe et al., New J. Phys. 12, 065012 (2010)


z

d

x

zd


1

HR = � kx⇧y

K

H = (v/2)

Zdx

�( x�)

2 + ( x⌥)2�

+A

⇥cos(

p4⌅K�)+

B

⇥sin(

p4⌅K�)+

CpK x⌥

+gU

2(⌅⇥)2cos(

p16⌅K�)

+gie

2⌅2pK

: ( 2x⌥) cos(

p4⌅K�) : (1)

k kF � kF

J⌅ D

= �⇤J⌅Jz + ...

Jz D

= �⇤J2⌅ + ... (2)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⇧

+S�e� + ⇧�S+

e�) + Jz⇧zSz

e�

⇤�(0),

1

x

HR = � kx⇧y

K

H = (v/2)

Zdx

�( x�)

2 + ( x⌥)2�

+A

⇥cos(

p4⌅K�)+

B

⇥sin(

p4⌅K�)+

CpK x⌥

+gU

2(⌅⇥)2cos(

p16⌅K�)

+gie

2⌅2pK

: ( 2x⌥) cos(

p4⌅K�) : (1)

k kF � kF

J⌅ D

= �⇤J⌅Jz + ...

Jz D

= �⇤J2⌅ + ... (2)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⇧

+S�e� + ⇧�S+

e�) + Jz⇧zSz

e�

⇤�(0),


z

d

x

Adding the Rashba interaction...... breaks the locking of spin to momentum. However, there is still a single Kramers pair on the edge, and this is all that matters!

Adding the Rashba interaction...... breaks the locking of spin to momentum. However, there is still a single Kramers pair on the edge, and this is all that matters! In fact, we can recover helicity by rotating the spin quantization axis: 1

H = vF

Zdx �†(x) [�i⇧z x]�(x) + �

Zdx �†(x) [�i⇧y x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

Zdx

�( x�)

2 + ( x⌥)2�

+A

⇥cos(

p4⌅K�)+

B

⇥sin(

p4⌅K�)+

CpK x⌥

+gU

2(⌅⇥)2cos(

p16⌅K�)

+gie

2⌅2pK

: ( 2x⌥) cos(

p4⌅K�) : (1)

k kF � kF

J⌅ D

= �⇤J⌅Jz + ...

Jz D

= �⇤J2⌅ + ... (2)



T << TK

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

low T




HK = �†(0)⇥J⌅(⇧

+S�e� + ⇧�S+

e�) + Jz⇧zSz

e�

⇤�(0),

Electrical control of the Kondo e�ect in a helical edge liquid

Erik Eriksson,1 Anders Strom,1 Girish Sharma,2 and Henrik Johannesson1

1Department of Physics, University of Gothenburg, SE 412 96 Gothenburg, Sweden2Centre de Physique Theorique, Ecole Polytechnique, 91128 Palaiseau Cedex, France

Magnetic impurities a�ect the transport properties of the helical edge states of quantum spinHall insulators by causing single-electron backscattering. We study such a system in the presenceof a Rashba spin-orbit interaction induced by an external electric field, showing that this can beused to control the Kondo temperature, as well as the correction to the conductance due to theimpurity. Surprisingly, for a strongly anisotropic electron-impurity spin exchange, Kondo screeningmay get obstructed by the presence of a non-collinear spin interaction mediated by the Rashbacoupling. This challenges the expectation that the Kondo e�ect is stable against time-reversalinvariant perturbations.

PACS numbers: 71.10.Pm, 72.10.Fk, 85.75.-d

Introduction. The discovery that HgTe quantum wellssupport a quantum spin Hall (QSH) state? has set o⇥an avalanche of studies addressing the properties of thisnovel phase of matter? . A key issue has been to deter-mine the conditions for stability of the current-carryingstates at the edge of the sample as this is the featurethat most directly impacts prospects for future applica-tions in electronics/spintronics. In the simplest pictureof 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) interactionis su⇤ciently well-screened so that higher-order scatter-ing processes do not come into play? ? .

The picture gets an added twist when including e⇥ectsfrom magnetic impurities, contributed by dopant ions orelectrons trapped by potential inhomogeneities. Since anedge electron can backscatter from an impurity via spinexchange, time-reversal invariance no longer protects thehelical states from mixing. In addition, correlated two-electron? and inelastic single-electron processes? ? mustnow also be accounted for. As a result, at high temper-atures T electron scattering o⇥ the impurity leads to aln(T ) correction? of the conductance at low frequencies⌅, which, however, vanishes? in the dc limit ⌅ ⇥ 0. Atlow T , for weak e-e interactions, the quantized edge con-ductance G0 = e2/h is restored as T ⇥ 0 with powerlaws 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 aQSH system must include also the presence of a Rashbaspin-orbit interaction. This interaction, which can betuned by an external gate voltage, is a built-in featureof a quantum well? . In fact, HgTe quantum wells ex-hibit some of the largest known Rashba couplings ofany semiconductor heterostructures? . As a consequence,spin is no longer conserved, contrary to what is assumedin 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 nottoo strong, this ensures the robustness of the helical edgeliquid? .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 wellas 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 theimpurity � can be controlled by varying the strengthof the Rashba interaction. Surprisingly, for a stronglyanisotropic Kondo exchange, a non-collinear spin inter-action mediated by the Rashba coupling becomes rel-evant (in the sense of RG) and competes with theKondo screening. This challenges the expectation thatthe Kondo e⇥ect 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 o⇥ by adjustingthe Rashba coupling. With the Rashba coupling beingtunable by a gate voltage, this suggests a new inroad tocontrol charge transport at the edge of a QSH device.Model. To model the edge electrons, we introduce

the two-spinors �T =�⇤�,⇤⇥

⇥, where ⇤� (⇤⇥) annihi-

lates a right-moving (left-moving) electron with spin-up(spin-down) along the growth direction of the quantumwell. Neglecting e-e interactions, the edge Hamiltoniancan then be written as

H = vF

⇤dx �†(x) [�i⇥z⇧x]�(x) +

+�

⇤dx �†(x) [�i⇥y⇧x]�(x) + (1)

+�†(0) [Jx⇥xSx + Jy⇥

ySy + Jz⇥zSz]�(0),

with vF the Fermi velocity parameterizing the linear ki-netic energy. The second term encodes the Rashba inter-action of strength �, with the third term being an anti-ferromagnetic Kondo interaction between electrons (withPauli matrices ⇥i, i = x, y, z) and a spin-1/2 magnetic im-

1

�⇥ = e�i⇥x�/2�

H = vF

⇤dx �†(x) [�i⇧z�x]�(x) + �

⇤dx �†(x) [�i⇧y�x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�(�x⌥)

2 + (�x⌃)2⇥

+A

⇥cos(

⇤4⌅K⌥)+

B

⇥sin(

⇤4⌅K⌥)+

C⇤K

�x⌃

+gU

2(⌅⇥)2cos(

⇤16⌅K⌥)

+gie

2⌅2⇤K

: (�2x⌃) cos(

⇤4⌅K⌥) : (1)

k kF � kF

�J⇤�D

= �⇤J⇤Jz + ...

�Jz�D

= �⇤J2⇤ + ... (2)

J0 ⇥ max(J⇤, Jz)D=D0


T << TK

Mn2+

kinetic term Rashba

1

H ⇥ = v�

⇤dx �⇥†(x)

⌅�i⇧z0

�x⇧�⇥(x)

�⇥ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇧z�x]�(x) + �

⇤dx �†(x) [�i⇧y�x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�(�x⌥)

2 + (�x⌃)2⇥

+A

⇥cos(

⇤4⌅K⌥)+

B

⇥sin(

⇤4⌅K⌥)+

C⇤K

�x⌃

+gU

2(⌅⇥)2cos(

⇤16⌅K⌥)

+gie

2⌅2⇤K

: (�2x⌃) cos(

⇤4⌅K⌥) : (1)

k kF � kF

�J⇤�D

= �⇤J⇤Jz + ...

�Jz�D

= �⇤J2⇤ + ... (2)



T << TK

1

�⌅T =�⌃⇥⌅,⌃⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇧z0

x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇧z x]�(x) + �

⇤dx �†(x) [�i⇧y x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�( x�)

2 + ( x⌥)2⇥

+A

⇥cos(

⌅4⌅K�)+

B

⇥sin(

⌅4⌅K�)+

C⌅K x⌥

+gU

2(⌅⇥)2cos(

⌅16⌅K�)

+gie

2⌅2⌅K

: ( 2x⌥) cos(

⌅4⌅K�) : (1)

k kF � kF

J⇧ D

= �⇤J⇧Jz + ...

Jz D

= �⇤J2⇧ + ... (2)

J0 ⇥ max(J⇧, Jz)D=D0


1

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌥⇥⌅,⌥⇤0

⇥

H⇥ = v�

⇤dx �⌅†(x)

⌅�i⌃z0

⌦x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⌃z⌦x]�(x) + �

⇤dx �†(x) [�i⌃y⌦x]�(x)

x

HR = � kx⌃y

K

H = (v/2)

⇤dx

�(⌦x )

2 + (⌦x�)2⇥

+A

⇤cos(

⇤4⇧K )+

B

⇤sin(

⇤4⇧K )+

C⇤K⌦x�

+gU

2(⇧⇤)2cos(

⇤16⇧K )

+gie

2⇧2⇤K

: (⌦2x�) cos(⇤4⇧K ) : (1)

k kF � kF

⌦J⇧⌦D

= �⌅J⇧Jz + ...

⌦Jz⌦D

= �⌅J2⇧ + ... (2)

1

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌥⇥⌅,⌥⇤0

⇥

H⇥ = v�

⇤dx �⌅†(x)

⌅�i⌃z0

⌦x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⌃z⌦x]�(x) + �

⇤dx �†(x) [�i⌃y⌦x]�(x)

x

HR = � kx⌃y

K

H = (v/2)

⇤dx

�(⌦x )

2 + (⌦x�)2⇥

+A

⇤cos(

⇤4⇧K )+

B

⇤sin(

⇤4⇧K )+

C⇤K⌦x�

+gU

2(⇧⇤)2cos(

⇤16⇧K )

+gie

2⇧2⇤K

: (⌦2x�) cos(⇤4⇧K ) : (1)

k kF � kF

⌦J⇧⌦D

= �⌅J⇧Jz + ...

⌦Jz⌦D

= �⌅J2⇧ + ... (2)

zz’1

⇤

�0 ⇥ (Jx + J ⇥y)

2T 2(⇤K��/2)2�1, �⇥

0 ⇥ (Jx � J ⇥y)

2T 2(⇤K+�/2)2�1, �E

0 ⇥ J2NCT

2K�1, �E0 ⇥ J2

NCT

I = G0V � ⇥I

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� ⇥G

T = 0

K > 1/4

T ⇤ TK

Adding the Rashba interaction...

e-e interaction is invariant under

1

� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌃⇥⌅,⌃⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇧z0

x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇧z x]�(x) + �

⇤dx �†(x) [�i⇧y x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�( x�)

2 + ( x⌥)2⇥

+A

⇤cos(

⌅4⌅K�)+

B

⇤sin(

⌅4⌅K�)+

C⌅K x⌥

+gU

2(⌅⇤)2cos(

⌅16⌅K�)

+gie

2⌅2⌅K

: ( 2x⌥) cos(

⌅4⌅K�) : (1)

k kF � kF

Kondo interaction

2

⌅J⌅⌅D

= ��J⌅Jz + ...

⌅Jz⌅D

= ��J2⌅ + ... (2)

J0 ⇥ max(J⌅, Jz)D=D0


T << TK

Mn2+

(Sz)2

S = 5/2 �⇤ Se� = 1/2

low T


⇥, where ⇤⇥ (⇤⇤) annihilates a right-moving


HK = �†(0)⇤J⌅(⇥

+S�e� + ⇥�S+

e�) + Jz⇥zSz

e�

⌅�(0),

1

H ⌅K = �⌅†(0)[Jx⇧

xSx + J ⌅y⇧

y0Sy0

+ J ⌅z⇧

z0Sz0

+ JNC(⇧y0Sz0

+ ⇧z0Sy0

)]�⌅(0)

S⌅ = e�iSx⇥/2SeiSx⇥/2

� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌃⇥⌅,⌃⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇧z0

x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇧z x]�(x) + �

⇤dx �†(x) [�i⇧y x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�( x�)

2 + ( x⌥)2⇥

+A

⇤cos(

⌅4⌅K�)+

B

⇤sin(

⌅4⌅K�)+

C⌅K x⌥

+gU

2(⌅⇤)2cos(

⌅16⌅K�)

+gie

2⌅2⌅K

: ( 2x⌥) cos(

⌅4⌅K�) : (1)

1

�⇥ = e�i⇥x�/2�

H = vF

⇤dx �†(x) [�i⇧z�x]�(x) + �

⇤dx �†(x) [�i⇧y�x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�(�x⌥)

2 + (�x⌃)2⇥

+A

⇥cos(

⇤4⌅K⌥)+

B

⇥sin(

⇤4⌅K⌥)+

C⇤K

�x⌃

+gU

2(⌅⇥)2cos(

⇤16⌅K⌥)

+gie

2⌅2⇤K

: (�2x⌃) cos(

⇤4⌅K⌥) : (1)

k kF � kF

�J⇤�D

= �⇤J⇤Jz + ...

�Jz�D

= �⇤J2⇤ + ... (2)



T << TK

Mn2+

1

H ⌅K = �⌅†(0)[Jx⇧

xSx + J ⌅y⇧

y0Sy0

+ J ⌅z⇧

z0Sz0

+ JNC(⇧y0Sz0

+ ⇧z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌃⇥⌅,⌃⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇧z0

x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇧z x]�(x) + �

⇤dx �†(x) [�i⇧y x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�( x�)

2 + ( x⌥)2⇥

+A

⇤cos(

⌅4⌅K�)+

B

⇤sin(

⌅4⌅K�)+

C⌅K x⌥

+gU

2(⌅⇤)2cos(

⌅16⌅K�)

+gie

2⌅2⌅K

: ( 2x⌥) cos(

⌅4⌅K�) : (1)



1

� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌃⇥⌅,⌃⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇧z0

x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇧z x]�(x) + �

⇤dx �†(x) [�i⇧y x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�( x�)

2 + ( x⌥)2⇥

+A

⇤cos(

⌅4⌅K�)+

B

⇤sin(

⌅4⌅K�)+

C⌅K x⌥

+gU

2(⌅⇤)2cos(

⌅16⌅K�)

+gie

2⌅2⌅K

: ( 2x⌥) cos(

⌅4⌅K�) : (1)

k kF � kF

Kondo interaction

2

⌅J⌅⌅D

= ��J⌅Jz + ...

⌅Jz⌅D

= ��J2⌅ + ... (2)



T << TK

Mn2+

(Sz)2

S = 5/2 �⇤ Se� = 1/2

low T




HK = �†(0)⇤J⌅(⇥

+S�e� + ⇥�S+

e�) + Jz⇥zSz

e�

⌅�(0),

1

H ⌅K = �⌅†(0)[Jx⇧

xSx + J ⌅y⇧

y0Sy0

+ J ⌅z⇧

z0Sz0

+ JNC(⇧y0Sz0

+ ⇧z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌃⇥⌅,⌃⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇧z0

x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇧z x]�(x) + �

⇤dx �†(x) [�i⇧y x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�( x�)

2 + ( x⌥)2⇥

+A

⇤cos(

⌅4⌅K�)+

B

⇤sin(

⌅4⌅K�)+

C⌅K x⌥

+gU

2(⌅⇤)2cos(

⌅16⌅K�)

+gie

2⌅2⌅K

: ( 2x⌥) cos(

⌅4⌅K�) : (1)

1

�⇥ = e�i⇥x�/2�

H = vF

⇤dx �†(x) [�i⇧z�x]�(x) + �

⇤dx �†(x) [�i⇧y�x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�(�x⌥)

2 + (�x⌃)2⇥

+A

⇥cos(

⇤4⌅K⌥)+

B

⇥sin(

⇤4⌅K⌥)+

C⇤K

�x⌃

+gU

2(⌅⇥)2cos(

⇤16⌅K⌥)

+gie

2⌅2⇤K

: (�2x⌃) cos(

⇤4⌅K⌥) : (1)

k kF � kF

�J⇤�D

= �⇤J⇤Jz + ...

�Jz�D

= �⇤J2⇤ + ... (2)



T << TK

Mn2+

1

H ⌅K = �⌅†(0)[Jx⇧

xSx + J ⌅y⇧

y0Sy0

+ J ⌅z⇧

z0Sz0

+ JNC(⇧y0Sz0

+ ⇧z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌃⇥⌅,⌃⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇧z0

x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇧z x]�(x) + �

⇤dx �†(x) [�i⇧y x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�( x�)

2 + ( x⌥)2⇥

+A

⇤cos(

⌅4⌅K�)+

B

⇤sin(

⌅4⌅K�)+

C⌅K x⌥

+gU

2(⌅⇤)2cos(

⌅16⌅K�)

+gie

2⌅2⌅K

: ( 2x⌥) cos(

⌅4⌅K�) : (1)

XYZ Kondo Non-Collinear term

depend on the Rashba coupling controllable by a gate voltage

1

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇤z⇧x]�(x) + �

⇤dx �†(x) [�i⇤y⇧x]�(x)

x

HR = � kx⇤y



1

� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌃⇥⌅,⌃⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇧z0

x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇧z x]�(x) + �

⇤dx �†(x) [�i⇧y x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�( x�)

2 + ( x⌥)2⇥

+A

⇤cos(

⌅4⌅K�)+

B

⇤sin(

⌅4⌅K�)+

C⌅K x⌥

+gU

2(⌅⇤)2cos(

⌅16⌅K�)

+gie

2⌅2⌅K

: ( 2x⌥) cos(

⌅4⌅K�) : (1)

k kF � kF

Kondo interaction

2

⌅J⌅⌅D

= ��J⌅Jz + ...

⌅Jz⌅D

= ��J2⌅ + ... (2)



T << TK

Mn2+

(Sz)2

S = 5/2 �⇤ Se� = 1/2

low T




HK = �†(0)⇤J⌅(⇥

+S�e� + ⇥�S+

e�) + Jz⇥zSz

e�

⌅�(0),

1

H ⌅K = �⌅†(0)[Jx⇧

xSx + J ⌅y⇧

y0Sy0

+ J ⌅z⇧

z0Sz0

+ JNC(⇧y0Sz0

+ ⇧z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌃⇥⌅,⌃⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇧z0

x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇧z x]�(x) + �

⇤dx �†(x) [�i⇧y x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�( x�)

2 + ( x⌥)2⇥

+A

⇤cos(

⌅4⌅K�)+

B

⇤sin(

⌅4⌅K�)+

C⌅K x⌥

+gU

2(⌅⇤)2cos(

⌅16⌅K�)

+gie

2⌅2⌅K

: ( 2x⌥) cos(

⌅4⌅K�) : (1)

1

�⇥ = e�i⇥x�/2�

H = vF

⇤dx �†(x) [�i⇧z�x]�(x) + �

⇤dx �†(x) [�i⇧y�x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�(�x⌥)

2 + (�x⌃)2⇥

+A

⇥cos(

⇤4⌅K⌥)+

B

⇥sin(

⇤4⌅K⌥)+

C⇤K

�x⌃

+gU

2(⌅⇥)2cos(

⇤16⌅K⌥)

+gie

2⌅2⇤K

: (�2x⌃) cos(

⇤4⌅K⌥) : (1)

k kF � kF

�J⇤�D

= �⇤J⇤Jz + ...

�Jz�D

= �⇤J2⇤ + ... (2)



T << TK

Mn2+

1

H ⌅K = �⌅†(0)[Jx⇧

xSx + J ⌅y⇧

y0Sy0

+ J ⌅z⇧

z0Sz0

+ JNC(⇧y0Sz0

+ ⇧z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌃⇥⌅,⌃⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇧z0

x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇧z x]�(x) + �

⇤dx �†(x) [�i⇧y x]�(x)

x

HR = � kx⇧y

K

H = (v/2)

⇤dx

�( x�)

2 + ( x⌥)2⇥

+A

⇤cos(

⌅4⌅K�)+

B

⇤sin(

⌅4⌅K�)+

C⌅K x⌥

+gU

2(⌅⇤)2cos(

⌅16⌅K�)

+gie

2⌅2⌅K

: ( 2x⌥) cos(

⌅4⌅K�) : (1)

XYZ Kondo Non-Collinear term

depend on the Rashba coupling controllable by a gate voltage

1

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇤z⇧x]�(x) + �

⇤dx �†(x) [�i⇤y⇧x]�(x)

x

HR = � kx⇤y

...bosonization and perturbative RGE. Eriksson, A. Ström, G. Sharma, H.J., PRB 86, 161103(R) (2012)

Electrical control of the Kondo temperaturevia the ”Rashba angle” ~ gate voltage

easy-plane Kondo

easy-axis Kondo

1

⇥

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇤z⇧x]�(x) + �

⇤dx �†(x) [�i⇤y⇧x]�(x)

x

1



⇥

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

2

⇥ (T/TK)8K�2 ⇥ (T/TK)2K+2

G =2e2

h� ⇥G

T = 0

K > 1/4

T ⇤ TK



⇤

J ⇥y, J

⇥z, JNC

displaymath

�

H ⇥K = �⇥†(0)[Jx⌅

xSx + J ⇥y⌅

y0Sy0

+ J ⇥z⌅

z0Sz0

+ JNC(⌅y0Sz0

+ ⌅z0Sy0

)]�⇥(0)

S⇥ = e�iSx⇥/2SeiSx⇥/2

� ⌅ �⇥

v� =�

v2F + �2

cos ⇤ = vF /v�


easy-plane Kondo

1

⇥

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇤z⇧x]�(x) + �

⇤dx �†(x) [�i⇤y⇧x]�(x)

x

2

⇥ (T/TK)8K�2 ⇥ (T/TK)2K+2

G =2e2

h� ⇥G

T = 0

K > 1/4

T ⇤ TK



⇤

J ⇥y, J

⇥z, JNC

displaymath

�

H ⇥K = �⇥†(0)[Jx⌅

xSx + J ⇥y⌅

y0Sy0

+ J ⇥z⌅

z0Sz0

+ JNC(⌅y0Sz0

+ ⌅z0Sy0

)]�⇥(0)


� ⌅ �⇥

v� =�

v2F + �2

cos ⇤ = vF /v�

experimentally probed range in a HgTe quantum well by tuning the bias of a top gate from -2V to 2VJ. Hinz et al., Semicond. Sci. Technol. 21, 501 (2006)

1

Mn2+

(Sz)2

S = 5/2 �! Se� = 1/2

J.K. Furdyna, J. Appl. Phys. 64, R29 (1988)


easy-plane Kondo

easy-axis Kondo

1

⇥

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇤z⇧x]�(x) + �

⇤dx �†(x) [�i⇤y⇧x]�(x)

x

1



⇥

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

2

⇥ (T/TK)8K�2 ⇥ (T/TK)2K+2

G =2e2

h� ⇥G

T = 0

K > 1/4

T ⇤ TK



⇤

J ⇥y, J

⇥z, JNC

displaymath

�

H ⇥K = �⇥†(0)[Jx⌅

xSx + J ⇥y⌅

y0Sy0

+ J ⇥z⌅

z0Sz0

+ JNC(⌅y0Sz0

+ ⌅z0Sy0

)]�⇥(0)


� ⌅ �⇥

v� =�

v2F + �2

cos ⇤ = vF /v�

Note: Kondo temperatures modified by spin-orbit interactions or spin-dependenthopping have been proposed also for ordinary (non-helical) conduction electrons:

M. Pletyukhov and D. Schuricht, PRB 84, 041309(R) (2011)X.-Y. Feng and F.-C. Zhang, J. Phys.: Cond. Matt. 23, 105602 (2011)R. Zitko and J. Bonca, PRB 84, 193411 (2011)M. Zarea, S. E. Ulloa, and N. Sandler, PRL 108, 046601 (2012)L. Isaev, L. Agterberg, and I. Vekhter, PRB 85, 081107 (2012)


easy-plane Kondo

easy-axis Kondo

1

⇥

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇤z⇧x]�(x) + �

⇤dx �†(x) [�i⇤y⇧x]�(x)

x

1



⇥

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

2

⇥ (T/TK)8K�2 ⇥ (T/TK)2K+2

G =2e2

h� ⇥G

T = 0

K > 1/4

T ⇤ TK



⇤

J ⇥y, J

⇥z, JNC

displaymath

�

H ⇥K = �⇥†(0)[Jx⌅

xSx + J ⇥y⌅

y0Sy0

+ J ⇥z⌅

z0Sz0

+ JNC(⌅y0Sz0

+ ⌅z0Sy0

)]�⇥(0)


� ⌅ �⇥

v� =�

v2F + �2

cos ⇤ = vF /v�


easy-axis Kondo

1

⇥

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇤z⇧x]�(x) + �

⇤dx �†(x) [�i⇤y⇧x]�(x)

x

1



⇥

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

Region where dominates the RG flow obstruction of Kondo screening!

1



⇥

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

easy-plane Kondo

2

⇥ (T/TK)8K�2 ⇥ (T/TK)2K+2

G =2e2

h� ⇥G

T = 0

K > 1/4

T ⇤ TK



⇤

J ⇥y, J

⇥z, JNC

displaymath

�

H ⇥K = �⇥†(0)[Jx⌅

xSx + J ⇥y⌅

y0Sy0

+ J ⇥z⌅

z0Sz0

+ JNC(⌅y0Sz0

+ ⌅z0Sy0

)]�⇥(0)


� ⌅ �⇥

v� =�

v2F + �2

cos ⇤ = vF /v�


easy-axis Kondo

1

⇥

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

H = vF

⇤dx �†(x) [�i⇤z⇧x]�(x) + �

⇤dx �†(x) [�i⇤y⇧x]�(x)

x

1



⇥

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

Region where dominates the RG flow obstruction of Kondo screening!

1



⇥

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇥ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⇤ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

�⌅ = e�i⇤x⇥/2�

easy-plane Kondo

2

⇥ (T/TK)8K�2 ⇥ (T/TK)2K+2

G =2e2

h� ⇥G

T = 0

K > 1/4

T ⇤ TK



⇤

J ⇥y, J

⇥z, JNC

displaymath

�

H ⇥K = �⇥†(0)[Jx⌅

xSx + J ⇥y⌅

y0Sy0

+ J ⇥z⌅

z0Sz0

+ JNC(⌅y0Sz0

+ ⌅z0Sy0

)]�⇥(0)


� ⌅ �⇥

v� =�

v2F + �2

cos ⇤ = vF /v�

challenges ”conventional wisdom” that

Kondo screening cannot be killed off

by time-reversal invariant perturbations!

drawn fromY. Meir and N.S. Wingreen, PRB 50, 4947 (1994)

no signa

ture of

Rashba

1

G =2e2

h� ⇥G

T = 0

K > 1/4

T ⇥ TK



⇤

J ⇥y, J

⇥z, JNC

displaymath

�

H ⇥K = �⇥†(0)[Jx⌅

xSx + J ⇥y⌅

y0Sy0

+ J ⇥z⌅

z0Sz0

+ JNC(⌅y0Sz0

+ ⌅z0Sy0

)]�⇥(0)


� ⇤ �⇥

v� =�

v2F + �2

cos ⇤ = vF /v�

sin ⇤ = �/v�

”weak” e-e interaction 1

1/4 < K < 2/3 K > 2/3

(T/TK)8K�2 (T/TK)2K+2

G =2e2

h� ⇥G

T = 0

K > 1/4

T ⇥ TK



⇤

J ⇥y, J

⇥z, JNC

displaymath

�

H ⇥K = �⇥†(0)[Jx⌅

xSx + J ⇥y⌅

y0Sy0

+ J ⇥z⌅

z0Sz0

+ JNC(⌅y0Sz0

+ ⌅z0Sy0

)]�⇥(0)


� ⇤ �⇥

v� =�

v2F + �2

1

1/4 < K < 2/3 K > 2/3

(T/TK)8K�2 (T/TK)2K+2

G =2e2

h� ⇥G

T = 0

K > 1/4

T ⇥ TK



⇤

J ⇥y, J

⇥z, JNC

displaymath

�

H ⇥K = �⇥†(0)[Jx⌅

xSx + J ⇥y⌅

y0Sy0

+ J ⇥z⌅

z0Sz0

+ JNC(⌅y0Sz0

+ ⌅z0Sy0

)]�⇥(0)


� ⇤ �⇥

v� =�

v2F + �2


1

G = 0

1/4 < K < 2/3 K > 2/3

⇥ (T/TK)8K�2 ⇥ (T/TK)2K+2

G =2e2

h� ⇥G

T = 0

K > 1/4

T ⇤ TK



⇤

J ⇥y, J

⇥z, JNC

displaymath

�

H ⇥K = �⇥†(0)[Jx⌅

xSx + J ⇥y⌅

y0Sy0

+ J ⇥z⌅

z0Sz0

+ JNC(⌅y0Sz0

+ ⌅z0Sy0

)]�⇥(0)

S⇥ = e�iSx�/2SeiSx�/2

� ⌅ �⇥

1

G = 0

1/4 < K < 2/3 K > 2/3

⇥ (T/TK)8K�2 ⇥ (T/TK)2K+2

G =2e2

h� ⇥G

T = 0

K > 1/4

T ⇤ TK



⇤

J ⇥y, J

⇥z, JNC

displaymath

�

H ⇥K = �⇥†(0)[Jx⌅

xSx + J ⇥y⌅

y0Sy0

+ J ⇥z⌅

z0Sz0

+ JNC(⌅y0Sz0

+ ⌅z0Sy0

)]�⇥(0)

S⇥ = e�iSx�/2SeiSx�/2

� ⌅ �⇥

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� �G

T = 0

K > 1/4

T ⇤ TK



⇥

J ⇥y, J

⇥z, JNC

displaymath

from instanton processes J. Maciejko et al., PRL 102, 256803 (2009)J. Maciejko, PRB B 85, 245108 (2012)

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� �G

T = 0

K > 1/4

T ⇤ TK



⇥

Low-temperature transport, (away from the ”dome”)

1

T ⇥ TK



⇥

J ⌅y, J

⌅z, JNC

displaymath

�

H ⌅K = �⌅†(0)[Jx⇤

xSx + J ⌅y⇤

y0Sy0

+ J ⌅z⇤

z0Sz0

+ JNC(⇤y0Sz0

+ ⇤z0Sy0

)]�⌅(0)


� ⇤ �⌅

v� =⌃

v2F + �2

cos ⇥ = vF /v�

sin ⇥ = �/v�

�⌅T =�⌅⇥⌅,⌅⇤0

⇥

H⌅ = v�

⇤dx �⌅†(x)

⌅�i⇤z0

⇧x⇧�⌅(x)

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� �G

T = 0

K > 1/4

T ⇤ TK



⇥

”High-temperature” transport,

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� �G

T = 0

K > 1/4

T ⇤ TK



⇥

1

Jx = Jy = 3Jz = 30 meV, a0 = 0.5 nm,

⇤+ | S | +⌅



�1,2

⇤sxy = Cs

e

4⇥, Cs =


⇤sxy = �

e

2⇥, � =


� =Cs

2mod 2 =


� = 1

� = 0

⇤sxy = ± e

2⇥

t > t0

⇥ | �q(t) ⌅ � e�iHqt | �o ⌅

H = J ⇥(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇥ Hq = J ⇥(J1�1 · �2 + J2�1 · �3) + J ⇥(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

1

Jx = Jy = 3Jz = 30 meV, a0 = 0.5 nm, vF = 5� 105 m/s, and D = 300 meV.

⌅+ | S | +⇧


choose spin quantizationaxis along ⌅+ | S | +⇧

�1,2

⇤sxy = Cs

e

4⇥, Cs =


⇤sxy = �

e

2⇥, � =


� =Cs

2mod 2 =


� = 1

� = 0

⇤sxy = ± e

2⇥

t > t0

⇤ | �q(t) ⇧ ⇥ e�iHqt | �o ⇧

H = J ⇥(J1�1 · �2 + J2�1 · �3) + J1

N�1⇥

i=2

�i · �i+1 + J2

N�2⇥

i=2

�i · �i+2

⇤ Hq = J ⇥(J1�1 · �2 + J2�1 · �3) + J ⇥(J1�N�1 · �N + J2�N�2 · �N ) + J1

N�2⇥

i=2

�i · �i+1 + J2

N�3⇥

i=2

�i · �i+2

1

⇤I = I � I0 (1)

⇤+ |S |+⌅ =⇥⇧3(⇥�

2⇥1 +⇥�1⇥2)X

+i⇥⇧3(�⇥�

2⇥1 +⇥�1⇥2)Y

+ (|⇥2 |2 [1+ |� |2] + |⇥1⇥ |2

3)Z

⇤� |S |�⌅ = �⇤+ |S |+⌅

|+⌅ = ⇥1 | E1+⌅+⇥2 | H1+⌅

|�⌅ = T |+⌅ = �⇥�1 | E1�⌅ �⇥�

2 | H1�⌅

|E1±⌅ = � |�6,±1

2⌅+ ⇥ |�8,±

1

2⌅

displaymath

|H1±⌅ = � |�8,±3

2⌅


⇤+ | S | +⌅



�1,2

⌃sxy = Cs

e

4⇧, Cs =


⌃sxy = ⌅

e

2⇧, ⌅ =


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� �G

T = 0

K > 1/4

T ⇤ TK



⇥


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� �G

T = 0

K > 1/4

T ⇤ TK



⇥

The Rashba-driven correction for a fixed voltage increaseswith the e-e interaction. Estimated values of K for the HgTe quantum wells used in the Würzburg experiments range from K = 0.55 to K = 0.98.

1

⇤I = I � I0 (1)

⇤+ |S |+⌅ =⇥⇧3(⇥�

2⇥1 +⇥�1⇥2)X

+i⇥⇧3(�⇥�

2⇥1 +⇥�1⇥2)Y

+ (|⇥2 |2 [1+ |� |2] + |⇥1⇥ |2

3)Z

⇤� |S |�⌅ = �⇤+ |S |+⌅

|+⌅ = ⇥1 | E1+⌅+⇥2 | H1+⌅

|�⌅ = T |+⌅ = �⇥�1 | E1�⌅ �⇥�

2 | H1�⌅

|E1±⌅ = � |�6,±1

2⌅+ ⇥ |�8,±

1

2⌅

displaymath

|H1±⌅ = � |�8,±3

2⌅


⇤+ | S | +⌅



�1,2

⌃sxy = Cs

e

4⇧, Cs =


⌃sxy = ⌅

e

2⇧, ⌅ =


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� �G

T = 0

K > 1/4

T ⇤ TK



⇥


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� �G

T = 0

K > 1/4

T ⇤ TK



⇥

For more results on transport, seeE. Eriksson et al., PRB 86, 161103(R) (2012); PRB 87, 079902(E) (2013)

1

⇤I = I � I0 (1)

⇤+ |S |+⌅ =⇥⇧3(⇥�

2⇥1 +⇥�1⇥2)X

+i⇥⇧3(�⇥�

2⇥1 +⇥�1⇥2)Y

+ (|⇥2 |2 [1+ |� |2] + |⇥1⇥ |2

3)Z

⇤� |S |�⌅ = �⇤+ |S |+⌅

|+⌅ = ⇥1 | E1+⌅+⇥2 | H1+⌅

|�⌅ = T |+⌅ = �⇥�1 | E1�⌅ �⇥�

2 | H1�⌅

|E1±⌅ = � |�6,±1

2⌅+ ⇥ |�8,±

1

2⌅

displaymath

|H1±⌅ = � |�8,±3

2⌅


⇤+ | S | +⌅



�1,2

⌃sxy = Cs

e

4⇧, Cs =


⌃sxy = ⌅

e

2⇧, ⌅ =


Summary

Magnetic impurity in a helical edge liquid:Rashba coupling allows electrical ”control” of Kondo temperature and IV-characteristics

blocking of Kondo screening (one-loop RG) Rashba-induced

impurity correction, accessible in experiment?

More results: Current fluctuations, thermal transport, effects from Dresselhaus interactions,...

E. Eriksson et al., PRB 86, 161103(R) (2012)


E. Eriksson, PRB 87, 235414 (2013)

PRB 87, 079902(E) (2013)

Addendum...

Conventional picture...

Kondo resonance at the Fermi level from divergence in the electron self-energy

multiple electron-impurity scattering



non-conservation of electron spin in the presence of spin-orbit interactions




non-conservation of electron spin in the presence of spin-orbit interactions!

But,... experiments show that the Kondo effect is insensitive to spin-orbit scattering...

G. Bergmann, PRL 57, 1460 (1986)




non-conservation of electron spin in the presence of spin-orbit interactions!

But,... experiments show that the Kondo effect is insensitive to spin-orbit scattering...

G. Bergmann, PRL 57, 1460 (1986)

...protected by time-reversal invarianceY. Meir and N.S. Wingreen, PRB 50, 4947 (1994)

-

= 0

+


kondo physics in 2d topological insulators with rashba

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