Riding a wild horse: Majorana fermions interacting with solitons of fast bosonic fields
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Riding a wild horse: Majorana fermions interacting with solitons of fast bosonic fields.
A. M. Tsvelik
Department of Condensed Matter Physics and Materials Science,Brookhaven National Laboratory, Upton, NY 11973-5000, USA
I consider a class of one-dimensional models where Majorana fermions interact with bosonicfields. Contrary to a more familiar situation where bosonic degrees of freedom are phonons andas such form a slow subsystem, I consider fast bosons. Such situation exists when the bosonicmodes appear as collective excitations of interacting electrons as, for instance, in superconductorsor carbon nanotubes. It is shown that an entire new class of excitations emerge, namely boundstates of solitons and Majorana fermions. The latter bound states are not topological and theirexistence and number depend on the interactions and the soliton’s velocity. Intriguingly the numberof bound states increases with the soliton’s velocity.
I. INTRODUCTION
Models describing Majorana fermions have received alot of attention, mostly in the context of quantum com-putation. In the process of the discussion several modelswith interesting common features have emerged. In allthese models Majorana fermions interact with solitons ofbosonic fields representing collective degrees of freedomof electronic systems (usually phase fields of supercon-ducting order parameters). Although models of interact-ing solitons and fermions have been considered before andthere is an extensive literature on the subject, there aresome new features which merit a discussion. The mostimportant new feature shared by the models in questionis that Majorana fermions are slow in comparison to thebosonic modes. This is opposite to a more familiar situa-tion of the Peierls-Froelich model of polyacetylene wherea slow optical phonons interact with fast fermions.As I will demonstrate, in this situation new branches offermion-soliton bound states emerge some of which existonly in a finite region of momentum space.
II. MODELSA. p-wave Josephson junction
This model was formulated by Grosfeld and Stern [1]who considered a long insulating one-dimensional Joseph-son junction between two p-wave superconductors. As itis well known, p-wave superconductors have zero energyMajorana modes as boundary states. When the bound-ary is extended, as in the case of a long junction, thesemodes propagate along the edge with velocity v ∆ ∼ ∆,the superconducting gap in the bulk. A conventionallong Josephson junction is described by the sine-Gordonmodel; the p-wave one acquires an additional term inthe Hamiltonian corresponding to Majorana fermions.The resulting model of the junction has the following Lagrangian density: L = (1)˜ c β h ˜ c − ( ∂ τ Φ) + ( ∂ x Φ) + 4 λ J sin (Φ / i +i2 (cid:16) r∂ τ r − vr∂ x r + l∂ τ l + vl∂ x l (cid:17) + i mrl cos(Φ / , where r, l are right- and left-moving Majorana fermionfields propagating along different sides of the junctionand Φ-field represents the phase difference between thetwo superconductors. The parameters of the model are˜ c = c r dd + 2 λ L , β / π = 2 e ~ c s d ( d + 2 λ ) h z , (2)where d is the thickness of the barrier, λ L is the Londonpenetration depth, h z is the hight of the junction and λ J and m are determined by characteristics of the junction.The magnetic field changes the argument of the sinus toΦ / → Φ / − eBdx/c ~ . (3)Model (1) is a generalization of the Super sine-Gordon(SSG) model; for the latter case m = v/λ J , ˜ c = v. (4)so that the model is Lorentz invariant. The SSG model isintegrable [2] and its excitations and S-matrix are known[3]. In particular, it is known that solitons of SSG modelobey non-Abelian statistics.Since all candidates for p-wave superconductivity havesmall critical temperatures corresponding to small v , theratio v/ ˜ c is also likely to be very small (however, it maybe increased somewhat by putting the junction on top ofa dielectric with large ǫ ). B. Combination of spin-orbit interaction andsuperconductivity
Although this model is not qualitatively different fromthe previous one, there are certain features which makeits material realization easier. Namely, the chiral Majo-rana modes emerge here not from a p -wave supercon-ductor, which remains a rather exotic object, but byother means. Namely, Majorana fermions may emergein a semiconductor with a strong spin-orbit interac-tion subject to external magnetic field brought into con-tact with an s -wave superconductor (see, for example[4],[5]). Following [4] I write down the Hamiltonian oftwo-dimensional film of such material: H = Z Ψ + ( x, y ) H Ψ( x, y )d x d y, (5)Ψ + = ( ψ + ↑ , ψ + ↓ , ψ ↓ , − ψ ↑ ) , H = [ p / m − µ ] τ z + u ( p y σ z − p x σ y ) τ z + B ( y ) σ x + ∆( y ) τ x , where τ a act in particle-hole and σ a in spin space re-spectively. As was shown in [4], when function V ( y ) = B ( y ) + ∆( y ) changes sign Majorana zero modes emerge.The corresponding operators of right- and left movingmodes are made of combinations r = 12 ( ψ ↑ − i ψ ↓ + ψ + ↑ + i ψ + ↓ ) , (6) l = 12 ( ψ ↑ + i ψ ↓ + ψ + ↑ − i ψ + ↓ ) , (7)These modes are spatially separated: the mode r emergesat the edge with d V / d y > l -mode at the edged V / d y <
0. These edges are boundaries between thetopological insulator and the superconducting state. Be-ing projected onto these modes the Hamiltonian densitybecomes H = u Z d x ( − r∂ x r + l∂ x l ) . (8)Since the spin-orbit interaction is typically much greaterin magnitude than the p -wave order parameter, one canincrease the ratio u/ ˜ c . In [4] the authors cite u = 7 . × cm/sec for InAr. With λ L ∼ A and d ∼
10A onecan get u/ ˜ c ∼ − .Now following [6] consider a situation when a narrowsuperconducting region is sandwiched between two topo-logical insulators. Then instead of one bosonic mode asin (1) we will have more. Namely, if the superconductingregion between two topological insulators is sufficientlynarrow, we have the following action [6]: − J cos( φ a − φ b ) + (9)i rl h t cos (cid:16) φ a − φ b (cid:17) + t cos (cid:16) φ a + φ b − φ m (cid:17)i where φ a,b are superconducting phases on the left andright from the superconducting strip and φ m is a phaseon the strip. Therefore there are two independent bosonicmodes. C. Half filled carbon nanotube
In [7] the author and Nersesyan derived an effectivefield theory for armchair carbon nanotubes using thebosonization approach with a partial refermionization.At half filling the model is similar to (1), but the num-ber of Majorana fermions is not one, but 6 with differentmass parameters m a such that m − = m − = m c , m = m = m = m t , m c + m t + m = 0 . (10)The role of the bosonic field Φ is played by the totalcharge field Φ c . The smallness of parameter β and alarge value of ˜ c/v originate from the unscreened Coulombinteraction. The Lagrangian density is L = 12 v ( ∂ t Φ c ) − v K ( ∂ x Φ c ) +i2 X a = − ¯ χ a (cid:16) γ ∂ t + vγ ∂ x (cid:17) χ a − V , (11) V = (12)i cos[ √ π Φ c ] h m − X a = − ¯ χ a χ a + m X a =1 ¯ χ a χ a + m ¯ χ χ i , where γ , γ are Dirac gamma matrices, K << c is the total charge field and χ a are Majorana fermions made of chiral components of Φ f ( a = − , −
1) and Φ s , Φ sf fields ( a = 0 , , ,
3) [7]. The V -term represents the leading interaction generated by theUmklapp processes; the interactions between the Majo-rana fermions are small in comparison. The symmetryof (11) is U(1) × U(1) × SU(2) × Z . The Majorana modeswith a = 1 , , K << c is essentially a classical field and itsdynamics is determined by the equation − v − ∂ t Φ c ( t, x ) + vK − ∂ x Φ c ( t, x ) + (13) √ π sin[ √ π Φ c ( t, x )] X a m a h χ a ( t, x ) ¯ χ a ( t, x ) i = 0 , At K << − v − ∂ t Φ( t, x ) + K − ∂ x Φ( t, x ) + M sin[Φ( t, x )] = 0 ,M ≈ X a m a v ln(Λ / | m a | ) , (14)where Φ = √ π Φ c and M is calculated with the loga-rithmic accuracy. This description is valid for excitationsmoving with velocities < v . So we see that the solitonsof Φ c are solutions of the sine-Gordon equation. As faras the fermionic excitations are concerned, they are de-termined by the same equations as for model (1). In thenotations of (1) we have1 /λ J = KM. (15)As is shown the subsequent Section, the fermions havebound states with the solitons such that the number offinite energy bound states in each channel is N ,a = m a λ J /v = m a K h P b m b ln(Λ / | m b | ) i / . (16)It is clear that for the Lorentz invariant case K = 1 thereare no finite energy bound states. More than that, theyappear only if the Coulomb interaction is quite strong. III. SEMICLASSICAL ANALYSIS
In all models described above the bosonic action is ofthe sine-Gordon type. The sine-Gordon subsystem hastwo types of excitations: kinks and breathers. Kinksstrongly interact with the Majorana fermions since thelatter ones create bound states with kinks. There aretwo types of bound states: one type is the Majoranazero modes which modify the kink’s quantum numbersand the other are massive ones. Below I do the analy-sis for model (1), generalizations for models (9,11) arestraightforward. In particular, in model (11) ˜ c = v/K .It is convenient to introduce new fermionic fields: χ = ( r + l ) / √ , χ = ( r − l ) / √ x the fermion operators can berepresented by the mode expansion (19)[8]. The eigen-functions satisfy Eψ = i( v∂ x − W ) ψ , Eψ = i( v∂ x + W ) ψ ,E ψ = ( − v ∂ x + W − v∂ x W ) ψ (18)where W = m cos(Φ / (cid:16) χ ( x, t ) χ ( x, t ) (cid:17) = γ (cid:16) ψ ( x − x )0 (cid:17) + (19) X E n > nh e − i E n t ˆ γ n ( x ) + ei E n t ˆ γ + n ( x ) i(cid:16) ψ ,n ( x − x )0 (cid:17) +i E n h e − i E n t ˆ γ n ( x ) − ei E n t ˆ γ + n ( x ) i × (cid:16) v∂ x + W ) ψ ,n ( x − x ) (cid:17)o A general single-soliton solution isΦ = 4 tan − h exp (cid:16) x − x − utλ J p − ( u/ ˜ c ) (cid:17)i . (20) In the static case u = 0 this gives rise to the followingpotential: W = m tanh( x/λ J ) . (21)Substituting it in (18) and using the results from [9], Iobtain the following eigenvalues for the bound states: E n = (cid:16) mN (cid:17) n (2 N − n ) ,n = 0 , , ... < N = mλ J /v. (22)and the eigenfunctions are ψ = (cosh ξ ) ( − N + n ) × (23) F (cid:16) − n, N − n + 1 , N − n + 1; 1e ξ + 1 (cid:17) ,ξ = x/λ J . Notice that in the Lorentz invariant case (4) N = 1 andthe only bound state is the topological one n = 0.Now let us consider the case of moving soliton u = 0.Let us introduce new coordinates: x ′ = γ ( x − ut ) , t ′ = γ ( t − xu/v ) ,γ = [1 − ( u/v ) ] − / . (24)This Lorentz transformation leaves the fermionic actioninvariant and puts us in the reference frame of the movingsoliton. The mass term in (18) becomes W (cid:16) x ′ λ ′ (cid:17) , λ ′ = λ J h − ( u/ ˜ c ) − ( u/v ) i / . (25)Notice that in the Lorentz invariant case ˜ c = v the scaledoes not change. However, if v < ˜ c , the soliton sizeincreases and, as a consequence, the number of boundstates N also increases: N = N h − ( u/ ˜ c ) − ( u/v ) i / . (26)This is a somewhat unexpected result. The energy in thisreference frame is given by (22) with N replaced by N .In the laboratory reference frame the energy and mo-mentum of the fermionic part of the bound state are (Iset ˜ c → ∞ ): E n ( u ) = 2 mN s n h N p − ( u/v ) − n i , (27) P n ( u ) = uE n ( u ) /v , n = 0 , ...N / p − ( u/v ) − , To obtain the total energy and momentum of the kinkand the bound state one has to add the energy and mo-mentum of the kink: E k ( u ) = M k p − ( u/ ˜ c ) , P k ( u ) = M k u ˜ c p − ( u/ ˜ c ) . (28) p ã FIG. 1: The spectrum e = N [ E ( p ) − M k ] / m (27) for N = 5and n = 6. The spectrum exists only for momenta larger thancritical as explained in the text. The fermion velocity is set v = 1. Consider n = 0 mode first. For zero velocity this modealways exists, but for finite velocities its existence is re-stricted by the condition u < v . Assuming that ˜ c/v isso large that the kink’s momentum is much smaller thanthe fermionic part and its dispersion is slow, we can set E k = M k , P k = 0. Then from (28) we extract its disper-sion: E ( p ) = q M k + (˜ cp ) , (29) | p | < M k ˜ c [(˜ c/v ) − − / ≈ vM k / ˜ c . Thus the zero energy bound state always exists, thoughin a limited region of the Brillouin zone.Now consider the finite energy bound states. They arenot topological and their existence is conditional. Some-what unexpectedly (27) shows that the conditions fortheir existence improve when the kink’s velocity increasestowards v . There are solutions with N [1 − ( u/v ) ] − / >n > N existing only for finite velocities (momenta). Itis illustrated on Fig. 1.To derive the dispersion of the bound states we haveto take into account the fact that their energies and mo-menta are sums of (27) and (28). If we assume that2 m/N >> M k ( v/ ˜ c ) , (30)then the inertia of the kinks is very small and their contri-bution to the total momentum can be neglected in com-parison with the momentum of the bound state (27). Asa result one gets the picture of the dispersion depictedon Fig. 2. p e FIG. 2: The spectrum e = N [ E n ( p ) − M k ] / m (27) for N =5 and n = 1 , ,
3. The fermion velocity is set v = 1. IV. QUANTUM NUMBERS ANDCORRELATION FUNCTIONS
As we see from (19) the operators of fermion-kink zeromodes γ ( x ) compose a Clifford algebra. For modelswith several species of Majorana fermions, such as (11), γ a create a spinor representation of the correspondinggroup (for (11) the group is O(6) ∼ SU(4) ) and thebound states of solitons and Majorana fermions trans-form according to this spinor representation. For thecase of half filled carbon nanotube these excitations carrythe same quantum numbers as the original fermions andtherefore are quasiparticles. The situation with n = 0bound states is quite different. They transform accordingto the vector representation of the corresponding groupand therefore can be created only by pairs of fermionicoperators.In order to get a better grip of the picture, let usconsider the model of carbon nanotube. The symmetrygroup of model (11) is U(1) × U(1) × O(3) × Z . As an ex-ample of a local field having nonzero matrix elements be-tween the vacuum and the aforementioned bound stateswe haveei √ πϕ c cos[ √ πϕ f ] ∼ R +1 ↑ R +1 ↓ + R +2 ↑ R +2 ↓ , (31)where ϕ c,f are right-moving components of the corre-sponding bosonic fields, c labels the total charge, f la-bels the relative one and 1 , f -sector (asymmetric charge) and theother soliton remains unbounded. From this example onecan see that the bound states can be observed only asparts of continua. In the above example the continuumconsists of a ”naked” soliton and a soliton-fermion boundstate. Existence of ”naked” solitons, i.e. ones which donot carry any fermionic modes is guaranteed by the factthat ˜ c > v and so their is plenty of room in momentumspace for solitons which velocity exceeds the one of thefermions and those, as we know, do not create boundstates. V. CONCLUSIONS
This paper demonstrates that in field theories with-out Lorentz invariance (quite a common thing in con-densed matter physics) one has to expect appearance ofnew types of bound states, some of them existing only in a limited region of momentum space.I am grateful to Alexander Nersesyan and RobertKonik for interesting discussions. AMT was supportedby US DOE under contract number DE-AC02 -98 CH10886. [1] E. Grosfeld and A. Stern, Proc. Natl. Acad. Sci. USA,2011 Jul 19; (29); 11810-4.[2] A. Tsvelik, Sov. J. Nucl. Phys. ( Yad. Fis. ) 47, 172 (1988).[3] C. Ahn, Nucl. Phys. B , 57 (1991).[4] Y. Oreg, G. Refael and F. von Oppen, Phys. Rev. Lett. , 177002 (2010).[5] R. M. Lutchyn, T. D. Stanescu, and S. Das Sarma, Phys.Rev. Lett. , 127001 (2011).[6] L. Jiang, D. Pekker, J. Alicea, G. Refael, Y. Oreg, and F. von Oppen, arXiv: 1107.4102.[7] A. A. Nersesyan and A. M. Tsvelik, Phys. Rev. B ,235419 (2003).[8] E. Witten, Nucl. Phys. B142