Unveiling Topological Kondo Effect: Majorana Fermions and Their Manufacture

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 r S 2= 3/4. b, c = bc, +,, 2 abc b c, a= a

How to manufacture a Majorana fermion? = c +c+ electrons have spin! It looks similar to Bogolyubov quasiparticle, but Semiconducting wire with a strong spin-orbit interaction in a transverse magnetic field on top of a superconductor (Sau, Lutchyn, Tewari, Das Sarma, 2010): dx 1 ( ) 2m + x 2 +i + y x + +B + x + + y ++h.c. H = With a proper choice of parameters Majorana zero modes emerge at the boundary.

Majorana-Coulomb box: semiconducting wires with strong spin- orbital interaction on top of a mesoscopic superconducting island. Red dots: E=0 Majorana modes located at the ends of the wires. a, b = ab, hij inter-wire tunneling

Derivation of the Kondo model Beri,Cooper 2012 2 (k)Cj(k)+Htunn+ Eci j,k + H = (k)Cj q /2, j Htunn=ei /2 tij i Cj(xj)+H.c. Ec is the charging energy of the island, is its phase, subscript j labels the wires. By Schrieffer-Wolf transformation one obtains the low energy Kondo model: j,k + +Cj Cj +Ci)x=0, H = (k)Cj (k)Cj(k)+Jij i j(Ci J ~ t2/Ec.

Where else to get Majoranas? Transverse field Ising chains. The model of Y-junction J12 1 3 x1 ( ) 2 x1 ( ) J23 2 x1 ( ) 3 x1 ( ) J13 1 x1 ( ) 3 x1 ( ), H = Is Hp p=1 N J xj ( ) xj +1 ( )+ h z(j) HIs= . j=1 I assume that Jpq << J and J=h: the chains are critical and one can use a continuous description.

Ising model on a star graph: Jordan-Wigner transformation (Crampe, Trombettoni 2012, Tsvelik 2013). , j 1 (j) = pcp(j)exp i +(k)cp(k) p cp k=1 +(j)cp(j) 1/2, =2 pq, p z(j) =cp c+p(j),cq(k) = pq jk, p, q { ,c} =0. p,q label the chains. The difference with the conventional version is the presence of the Klein factors. p are zero energy Majorana fermions.

The fermionic version of the Ising model emerges after Jordan-Wigner transformation. J cp M cp + hcp +(j +1) cp(j +1) cq +(j)+cp(j) . +(j)cp(j) H = + p=1 j=1 +(1) cp(1) +(1) cq(1) p qcp Jpq p q . The boundary spins are +(1) cp(1) p x(1) = pcp

Single Ising chain with a free boundary. The continuous description (Ghosal, Zamolodchikov). (j) =i c+(j) c(j) / 2a. , (j) =c+(j)+c(j), R,L(x = ja) = (j) (j) +1 L dx1 ) R+1 ( ( ) L+im R L L = 2 R iv x m = J h. 2 L +iv x 2 , 0 v = Ja, R(0) = L(0) R , L are real (Majorana) fermionic fields propagating in the bulk. The boundary condition corresponds to the free boundary spin.

For critical chains (m=0), one can introduce chiral fermions: (x) = R (x) (x)+ L (x) ( x) and extend x integration over entire axis: M H =1 p,q piv x ( ) p+ Jpq p q q(0) p(0), += pq. dx 2p=1 ax ( ), by ( ) p, q += (x y), Jpq = q p 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: 2 i Cp ( )= i +Cq Cq +Cp p (a) q (a)= Jpq (2) , a=1 Cj=2 (2):+1 dx 2+( x )2 + x (2) dx iCj dx :Jpq Jpq M 2 Only the current operator interacts with the impurity spin i p q This allows one to identify the critical point as O2(M) WZNW boundary CFT and also to conjecture the Bethe ansatz.

The most interesting results are for the Majorana-Coulomb box model with M leads. At energies << TK ~Ecexp[-2 ( )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)[ jk -1/M]

Coulomb-Majorana box model with M=3 leads. Abelian bosonization: the Gaussian model of 2 fields At the QCP: 8 /3 1(0) d dx x a x a+i a ( ), ) A = a =1,2. , ( ( )e i 2 2(0) S = cos Sz= zcos 2 /3 1(0) 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: Sjk=i j k

Non-trivial 3-point function for M=3 For M=3 spin operators are components of S=1/2: Sa =i abc b c ) abc The long time T Sa 1 ( )Sb 2 ( )Sc 3 ( ) = Tk( 12 13 23)1/3, asymptotic: Response to h cos( t) S3: ) T S F(x) = x 2(t2) ~ hcos[ (t1+t2)]F[ (t1 t2)], 1/6[Y 1/6(x)+J1/6(x)]. 1(t1)S

Ising junctions. What is the spin in this case? It is made of zero modes of different chains: Sp=i 2 pqt q t, 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 n n+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 n n+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) = ln2 (Wiegmann and Tsvelik, 1984). The effective action for energies < Tkondo is (Ioffe et. al. 1994) L*=1 2 +g 1(0) 2(0) 3(0)+1 p iv x ( ) pdx 2 For equal couplings Tkondo ~ Jexp(- G). One local Majorana zero mode remains unquenched. Its relation to the bare Majoranas 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. j=1,2 2 j j + +(0) H =i dx j x j +Gp pSp j (0), +1 2a=1,2,3 j=1,2; =1,2 x j =1 dx a x a+1 ( ) 2 + dx j 2+ x dx a x a i dx , 2a=1,2,3 j=1,2; =1,2 p j =i + 2 pqt q t. The Hamiltonian of SU(2)xSU(2) Dirac fermions can be written as a sum H of free boson field and two triads of Majorana fermions j=1,2 These fermionic bilinears have the same commutation relations. The Y-junction describes just the (spin) sector of the Kondo Hamiltonian, but this sector contains all interactions.

So, what to measure? Energy levels. Ising chains can be made of Josephson junctions (M. Gershenson). Correlation functions from different chains, for instance: z ( ) 2 1 z( ) ~ J12ln2(J/ ) In the 1st order of perturbation theory.

Topological Kondo effect in Coulomb-Majorana box: SO2 (M) Kondo model. ( ) j p+ (k) j(k)+g g ~t2 k, j j + +(k) p(k') p +(k) j(k') H = (k) j Nj,p;k,k' Ec ihjp j p. Spinless fermions in the leads. Majorana zero modes on top of the superconducting box.

Supersymmetry of 2-channel Kondo model. H =Q2, 2 3+Ga a0 ( )Sa. Q = dx 1 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 As quasiparticles: the history starts at 1987 Volovik, Salomaa. 2D Ising model = free Majorana fermions (Onsager 1945, Kaufman 1949, Baruch, Tracy, McCoy, 1976, Schroer, Truong 1978). 1D Quantum Ising model Jordan-Wigner transformation. Both paths meet at defects where propagating particles become zero modes. 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.