Topological Kondo effect Alexei Tsvelik,

Capri, April 2014,

Topological Kondo effect occurs when bulk quasiparticles scatter on composite spin nonlocally encoded by Majorana fermions located at different space points.

Simplest example: spin S=1/2 can be made of 3 Majorana zero energy modes:

Sa i

2 abcbc , b ,c bc, a a

,,r S 2 3/4.

How to manufacture a Majorana fermion?

Semiconducting wire with a strong spin-orbit interaction in a transverse magnetic field on top of a superconductor (Sau, Lutchyn, Tewari, Das Sarma, 2010):

H dx 1

2m x

2 i yx B x y h.c.

With a proper choice of parameters Majorana zero modes emerge at the boundary.

c c It looks similar to Bogolyubov quasiparticle, but electrons have spin!

Majorana-Coulomb box: semiconducting wires with strong spin-orbital interaction on top of a mesoscopic superconducting island.

Red dots: E=0Majorana

modes g located at the ends of the wires.

a ,b ab ,

hij – inter-wire tunneling

Derivation of the Kondo model

H (k)C j

j,k

(k)C j (k) H tunn E c i

q

2

/2,

H tunn e i / 2 tijij

C j (x j ) H.c.

Ec is the charging energy of the island, f is its phase, subscript j labels the wires.

H (k)C j

j,k

(k)C j (k) Jiji j (CiC j C j

Ci)x0 ,

J ~ t 2 /E c .

By Schrieffer-Wolf transformation one obtains the low energy Kondo model:

Beri,Cooper 2012

Where else to get Majoranas? Transverse field Ising chains. The model of Y-junction

I assume that Jpq << J and

J=h: the chains are critical and one can use a continuous description.

H H pIs

p1

3

J121x 1 2x 1 J232

x 1 3x 1 J131x 1 3x 1 ,

H Is Jx j x j 1 hz ( j) j 1

N

.

Ising model on a star graph: Jordan-Wigner transformation (Crampe, Trombettoni 2012, Tsvelik 2013).

p ( j) pc p ( j)exp i c p

(k)c p (k)k1

j 1

,

pz ( j) c p

( j)c p ( j) 1/2,

p ,q 2 pq, cp ( j),cq (k) pq jk, {,c}0.

p,q label the chains.

The difference with the conventional version is the presence of the Klein factors.

gp are zero energy Majorana fermions.

The fermionic version of the Ising model emerges after Jordan-Wigner transformation.

The boundary spins are

H J c p( j 1) c p ( j 1) c p

( j)c p ( j) hc p( j)c p ( j)

j 1

p1

M

Jpqpq

pq c p(1) c p (1) cq

(1) cq (1) .

px (1) p c p

(1) c p (1) .

Single Ising chain with a free boundary. The continuous description (Ghosal, Zamolodchikov).

( j) c( j)c( j), ( j) i c( j) c( j) ,R ,L (x ja) ( j)( j) / 2a.

L 0

L

dx1

2R ivx R

1

2L ivx L imR L

1

2,

v Ja, m J h.

R (0) L (0)

The boundary condition corresponds to the free boundary spin.

cR ,cL are real (Majorana) fermionic fields propagating in the bulk.

For critical chains (m=0), one can introduce chiral fermions: (c x) = cR (x) (q x)+cL (x) (-q x) and extend x integration over entire axis:

H 1

2 p1

M

dx

p ivx p Jpqpqp,q

q (0)p (0),

a x ,b y (x y), p ,q

pq .

Jpq = cq cp are O(M)1 Kac-Moody currents.

Thus the junction of M critical Ising chains is described by the new type Kondo model. For M=3 it equivalent to 2-channel Kondo model in the Majorana fermion formulation of Ioffe et al. (1994).

Both the model with the Majorana-Coulomb box and the model of Ising star graph are exactly solvable.

Star graph model has a singlet ground state for M even,

and Quantum critical point for M odd (Tsvelik, 2013,2014).

The Majorana-Coulomb box model is Quantum Critical (Altland et.al, 2013).

Solution: Bethe ansatz, bosonization, boundary Conformal Field Theory.

Back to the Majorana-Coulomb box. Non-Abelian bosonization.

In Kondo models the bulk can be represented by 1D chiral fermions with linear spectrum.

We have the following remarkable identities:

i CpCq Cq

Cp i p(a )q

(a ) Jpq(2)

a1

2

,

dx iC jx C j

2M

dx : Jpq(2) Jpq

(2) :12

dx 2 (x)2

Only the current operator interacts with the impurity “spin”

ipq

This allows one to identify the critical point as O2(M) WZNW boundary CFT and also to conjecture the Bethe ansatz.

At energies << TK ~Ecexp[-2 /( -2)p M J] the model is quantum critical. The universality class is O2(M) Wess-Zumino-Novikov-Witten theory with a boundary c=M-1.

There is a finite entropy in the ground state

S(0) = ln dM, dM = (M)1/2 (M odd), (M/2)1/2 (M – even).

The most unusual feature is non-trivial M-point correlation functions (the next page).

Universal conductance Gjk=(e2/h)[djk -1/M]

The most interesting results are for the Majorana-Coulomb box model with M leads.

Coulomb-Majorana box model with M=3 leads.

Abelian bosonization: the Gaussian model of 2 fields f1,2.

At the QCP:

A ddxxa xa ia , a 1,2.

S z z cos 8 /31(0) , S cos 2 /31(0) ei 2 2 (0)

ea

are cocycles. Thus at low energies the “spin” components become primary fields of the boundary CFT with dimension 1/3.

M=3 problem is equivalent to 4-channel S=1/2 Kondo model. Here the thermodynamics and the response functions are the most singular.

For M>3 the singularities are weaker.

There are non-trivial M-point functions of “spin” components. The “spin” operators are defined as tunneling ones:

S jk i jk

Non-trivial 3-point function for M=3

For M=3 “spin” operators are components of S=1/2: Sa =ieabcgbgc

The long time

asymptotic:

) T Sa 1 Sb 2 S c 3

abc

Tk (121323)1/ 3 ,

Response to h cos(wt) S3:

) T S1(t1)S

2(t2) ~ hcos[(t1 t2)]F[(t1 t2)],

F(x) x 1/ 6[Y 1/ 6(x) J1/ 6(x)].

Ising junctions.

What is the “spin” in this case? It is made of zero modes of different chains:

S p i

2 pqtqt ,

But the zero mode operators are nonlocal in Ising spins and one cannot probe it with any local “magnetic” field.

The nonlocality preserves the critical point,

but makes it difficult

to observe. Thing-in-itself (almost).

Although for critical Ising chains the “spin” is not directly observable, one can use the relevant operator

h gngn+1 as a formal device to generate a crossover

between M and M-2.

To understand the difference between even and odd M, add the relevant operator h gngn+1which drives the system from M to M-2. The minimal even M is 2 (Fermi liquid), the minimal odd is 1 (single boundary Majorana).

Observables: energy levels and thermodynamics are as for the Kondo model: C/T ~ln(Tkondo /T) , but the correlation functions are different.

The star junction of M critical Ising chains.The most interesting case is M=3.

There is nonzero ground state entropy S(0) = ½ ln2 (Wiegmann and Tsvelik, 1984).

The effective action for energies < Tkondo is (Ioffe et. al. 1994)

L* 1

2 g1(0)2(0)3(0)

1

2p ivx pdx

For equal couplings Tkondo ~ Jexp(- /p G).

One local Majorana zero mode eremains unquenched. Its relation to the bare Majoranas g is complicated.

Conclusions for the Ising model

Star-junction of three critical Ising chains is an active element where the boundary states undergo a renormalization.

Existence of these boundary degrees of freedom comes from topology – in the given case just from the fact that the chains have ends.

The M=3 case realizes 2-channel Kondo model. For all odd M there is a non-Fermi liquid Quantum Critical point. In the process the boundary “spin” is quenched, but not completely.

Conclusions for the Majorana-Coulomb box model.

The model is always quantum critical.

There are multiple possibilities for observing the critical properties: one can measure the conductance, a response to external gate potentials etc.

For M=3 there is an interesting “spin” response related to existence of nontrivial 3-point correlation function of the “spins”.

Main conclusion

Models of topological Kondo effect provide natural settings for Quantum Critical Points.

Despite the screening of the local degrees of freedom by the gapless bulk excitations the non-Abelian nature of the “spins” reveals itself in the ground state in non-trivial multi-point correlation functions.

Two-channel Kondo model of electrons in the Majorana representation. The equivalency is derived by non-Abelian bosonization.

H ij 1,2;1,2

dx j x j G p

j 1,2

j (0)

p S p j (0),

ij 1,2;1,2

dx j x j

1

2dx 2 x 2 12 a1,2,3

dxaxa 1

2 a1,2,3

dxaxa ,

j 1,2

j

p j i

2 pqtq t .

The Y-junction describes just the c(spin) sector of the Kondo Hamiltonian,but this sector contains all interactions.

These fermionic bilinears have the same commutation relations.

The Hamiltonian of SU(2)xSU(2)

Dirac fermions ycan be written

as a sum H of free boson field fand two

triads of Majorana fermions ,xc

So, what to measure? Energy levels. Ising chains can be made of Josephson junctions (M. Gershenson). Correlation functions from different chains, for instance:

1z 2z( ) ~ J12 ln

2(J /)

In the 1st order of perturbation theory.

Topological Kondo effect in Coulomb-Majorana box: SO2

(M) Kondo model.

H (k) j

k, j

(k) j (k)g

N j ,p;k,k'

j(k) p (k') p

(k) j (k ') jp

ih jp jp .

Spinless fermions in the leads. Majorana zero modes on top of the superconducting box.

g ~t 2

Ec

Supersymmetry of 2-channel Kondo model.

H Q2,

Q dx1 23 Gaa 0 Sa .

Another idea from particle theory. Is it of any use?

Happy Anniversary!

An early image of Paul Wiegmann from Louvre (Fra Angelico):

Majorana fermions in condensed matter physics.

As collective excitations –

Long history

2D Ising model = free Majorana fermions (Onsager 1945,

Kaufman 1949, Baruch, Tracy, McCoy, 1976, Schroer, Truong 1978).

1D Quantum Ising model – Jordan-Wigner transformation.

80-ties: 2-channel Kondo model (Wiegmann, Tsvelik), S=1 magnets (Takhtadjan, Babujan), SU2 (2) Wess-Zumino model (Fateev, Zamolodchikov).

90-ties: spin ladders (Nersesyan, Tsvelik).

In all these models Majoranas emerge as collective excitations.

As quasiparticles:the history starts at 1987 Volovik, Salomaa.

Both paths meet at defects where propagating particles become zero modes.