Time-Reversal Invariant Parafermions in Interacting Rashba Nanowires
TTime-Reversal Invariant Parafermions in Interacting Rashba Nanowires
Jelena Klinovaja and Daniel Loss Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA, and Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland (Dated: October 29, 2018)We propose a scheme to generate pairs of time-reversal invariant parafermions. Our setup consistsof two quantum wires with Rashba spin orbit interactions coupled to an s -wave superconductor, inthe presence of electron-electron interactions. The zero-energy bound states localized at the wireends arise from the interplay between two types of proximity induced superconductivity: the usualintrawire superconductivity and the interwire superconductivity due to crossed Andreev reflections.If the latter dominates, which is the case for strong electron-electron interactions, the system sup-ports Kramers pair of parafermions. Moreover, the scheme can be extended to a two-dimensionalsea of time-reversal invariant parafermions. PACS numbers: 71.10.Pm; 74.45.+c; 05.30.Pr; 73.21.Hb
I. INTRODUCTION
Topological properties of condensed matter systemshave attracted wide attention in recent years. In par-ticular, localized bound states emerging at the inter-face between different topological regions have beenstudied intensely both theoretically and experimentally.Majorana fermions (MFs), zero-energy bound stateswith non-Abelian braid statistics, were predicted inseveral systems such as fractional quantum Hall ef-fect (FQHE) systems, topological insulators, opti-cal lattices, p -wave superconductors, nanowires withRashba spin orbit interaction (SOI), self-tuningRKKY systems, and graphene-like systems. Though MFs possess non-Abelian statistics, it is ofIsing type which is not sufficient for universal quantumcomputation, in contrast to Fibonacci anyons. The ba-sic building blocks for the latter anyons are parafermions(PFs), also referred to as fractional MFs, which allowfor more universal quantum operations than MFs.
Similarly to MFs, PFs are bound states that arise atthe interface between two distinct topological phases. Incontrast to MFs, however, PFs owe their peculiar proper-ties to strong electron-electron interactions. As a result,most proposals to host PFs invoke edge states of FQHEsystems, and to stabilize them at zero energy one re-lies on particle-hole symmetry generated by proximity toa superconductor.
However, while strong mag-netic fields are required for the FQHE, they are detri-mental for superconductivity, making the experimentalrealization of such proposals challenging.
This hasmotivated us to search for alternatives to generate PFswith superconductivity but without magnetic fields. In-deed, we will show that by taking advantage of time-reversal invariance it is possible to construct Kramerspairs of PFs, which can be considered as generaliza-tion of Kramers pairs of MFs studied before.
Weare also motivated to work with one-dimensional systemswhere recent experiments have demonstrated proximity-induced superconductivity of crossed Andreev type, strong electron-electron interaction, and high tun- x y z FIG. 1. Sketch of two Rashba QWs (yellow strips) coupled toan s -wave superconductor (blue strip). The SOI field pointsin positive (negative) direction, say, along the z azis for theupper (lower) QW, τ = 1 ( τ = ¯1). The intrawire proximityinduced superconductivity of strength ∆ τ corresponds to aCooper pair (pair of green dots) tunneling as a whole intothe τ -wire. The interwire proximity induced superconductiv-ity of strength ∆ c corresponds to crossed Andreev reflectioninto both QWs, which dominates for strong electron-electroninteraction assumed here. ability of the chemical potential. Moreover, the classof materials suitable for our scheme is larger than forschemes with magnetic field since we do not require large g -factors.The setup we consider (see Fig. 1) consists of two one-dimensional channels, or quantum wires (QWs) with theRashba SOI. The QWs are close to an s -wave supercon-ductor resulting in proximity induced superconductivity.In general, there are two types of pairing terms. Thefirst one is intrawire pairing corresponding to tunnel-ing of Cooper pairs as a whole to either of the QWs.The second type is the interwire pairing correspond-ing to ‘crossed Andreev reflection’ where the Cooperpair gets split into two different channels. Such pro-cesses dominate in the regime of strong electron-electroninteractions. In this case, the system is in the topo-logical phase with bound states localized at the systemends. If the chemical potential is tuned close to the SOIenergy, the system supports two MFs at each end thatare time-reversal partners of each other.
More strik-ingly, if the chemical potential is lowered, e.g. to onenineth of the SOI energy, and electron-electron interac-tions are strong, the zero-energy ground state containsthree PF Kramers pairs. However, similar to Ref. 34,the degeneracy of our bound states is not protected by a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l a fundamental system property and is susceptible to aspecific kind of disorder.The paper is organized as follows. In Sec. II we in-troduce the model system; in Sec. III we consider thenon-inetracting case and find Kramers pairs of Majoranafermions, first for wires with SOI with opposite signs andthen for wires with equal signs. In Sec. IV we considerthe case with interactions, and using a bosonization ap-proach we derive the parafermion bound states. Finally,we give some conclusions in Sec. V. II. MODEL
We consider a system consisting of two Rashba QWsbrought into the proximity to an s -wave superconductor,see Fig. 1. The upper (lower) QW is labeled by the index τ = 1 ( τ = ¯1) and is aligned in the x direction. Thekinetic part of the Hamiltonian is given by H = (cid:88) τ,σ (cid:90) dx Ψ † τσ ( x ) (cid:20) − (cid:126) ∂ x m − µ τ (cid:21) Ψ τσ ( x ) , (1)where Ψ τσ ( x ) † [Ψ τσ ( x )] is the creation (annihilation) op-erator of an electron of mass m at position x of the τ -wire with spin σ/ ± / z -axis, and µ τ isthe chemical potential. The Rashba SOI field α Rτ , thatcharacterizes the strength and the direction of the spinpolarization caused by SOI, points in the z direction ineach of the two QW, so the Rashba SOI term is writtenas H so = − i (cid:88) τ,σ,σ (cid:48) α Rτ (cid:90) dx τ Ψ † τσ ( σ ) σσ (cid:48) ∂ x Ψ τσ (cid:48) . (2)Here, the Pauli matrices σ , , act on the spin of theelectron. We note that the spin projection on the z di-rection is a good quantum number ( σ ), and the disper-sion relation for the spin component σ at the τ -wire isgiven by E τσ = (cid:126) ( k − τ σk so,τ ) / m , where the chem-ical potential µ is tuned to the crossing point betweentwo spin-polarized bands at k = 0, i.e. µ = E so , seeFig. 2. Here, E so,τ = (cid:126) k so,τ / m is the SOI energy, and k so,τ = mα Rτ / (cid:126) is the SOI wavevector.In addition, the intrawire superconductivity ofstrength ∆ τ is proximity induced in each of the QWsby the tunneling of Cooper pairs as a whole from the su-perconductor to the τ -wire. The corresponding pairingterm is given by H s = (cid:88) τ,σ,σ (cid:48) (cid:90) dx ∆ τ τσ ( iσ ) σσ (cid:48) Ψ τσ (cid:48) + H.c. ] . (3)If the distance between two QWs is shorter than the su-perconductor coherence length then crossed Andreev re-flection is possible where the electrons from the sameCooper pair tunnel into two different QWs, resulting in the interwire proximity induced superconductivity. The corresponding pairing term is given by H c = (cid:88) τ,σ,σ (cid:48) (cid:90) dx ∆ c τσ ( iσ ) σσ (cid:48) Ψ ¯ τσ (cid:48) + H.c. ] , (4)where ∆ c is the strength of the induced inter-wire superconductivity. Such a process is useful inCooper pair splitters where crossed Andreev reflectiondominates, so ∆ c > ∆ τ .Finally, we note that H c becomes equivalent to FFLOpairing if one gauges away the SOI in the wires. Itis known that in one-dimensional wires the RashbaSOI can be gauged away by a spin-dependent gaugetransformation. In our case, we gauge away the RashbaSOI simultaneously in both wires by the following trans-formation Ψ (cid:48) τσ = e iτσk so,τ x Ψ τσ , (5)which is also wire-dependent ( τ ) as a consequence of op-posite Rashba SOI. As a result, the crossed Andreev term H c becomes in this new gauge H (cid:48) c = 12 (cid:88) τ,σ,σ (cid:48) (cid:90) dx (cid:104) ∆ c e − iτσ ( k so, − k so, ¯1 ) x Ψ (cid:48) τσ ( iσ ) σσ (cid:48) Ψ (cid:48) ¯ τσ (cid:48) + H.c. (cid:105) , (6)whereas H s remains unchanged. Thus, the crossed An-dreev superconductivity has a non-uniform pairing term,∆ c e − iτσ ( k so, − k so, ¯1 ) x , which manifestly breaks the trans-lation invariance if k so, (cid:54) = k so, ¯1 . This term is re-lated to the Fulde-Ferrel-Larkin-Ovchinnikov (FFLO)state, where the Cooper pair has finite total mo-mentum. Therefore, all results derived in the main partfor two wires with opposite Rashba SOI are also valid fora system consisting of two wires without SOI but coupledto an FFLO-type superconductor instead of an ordinary s -wave superconductor.The spatial dependence makes it explicit that there canbe ground states in the system with broken symmetries(such as a charge density wave state), and thus states ofdifferent symmetries separated by domain walls that hostbound states. We note that this situation is analogous toRef. 34, which finds parafermions in a one-dimensionalRashba wire coupled to a superconductor and in the pres-ence of magnetic fields. There, it has been pointed out that the resulting gapped state is not within the list ofpossible gapped one-dimensional phases classified in Ref.74. As a consequence, disorder or deviations from themean-field description of superconductivity can lift, inprinciple, the bound state degeneracy. III. KRAMERS PAIRS OF MAJORANAFERMIONSA. SOIs of opposite sign
In this subsection we focus on the case where theRashba SOIs are of opposite sign in the two QWs, α R α R ¯1 <
0. In addition, the chemical potential is tunedto the SOI energy in both QWs, µ τ = E so,τ . To simplifyanalytical calculations, we assume in what follows that α R = − α R ¯1 = α R . We note that the choice of ex-actly opposite SOIs, such that the Fermi velocities υ F are the same in the two QWs, is convenient but not nec-essary. All that is needed is to tune the individual Fermiwavevectors k F τ (via chemical potentials) to the individ-ual k so,τ values (or fractions thereof) in each wire.The proximity-induced superconductivity leads to gapsin the spectrum. Thus, the question arises if thereare zero-energy bound states localized at the ends ofthe wires. To find an answer, we proceed by lineariz-ing the spectrum around the Fermi points k = 0 and k = ± k F ≡ ± k so (see Fig. 2),Ψ = R e ik F x + L , (7)Ψ = L e − ik F x + R , (8)Ψ ¯11 = L ¯11 e − ik F x + R ¯11 , (9)Ψ ¯1¯1 = R ¯1¯1 e ik F x + L ¯1¯1 , (10) FIG. 2. The spectrum of two QWs with positive (negative)Rashba SOI for τ = 1 ( τ = ¯1). The solid (dashed) lines corre-spond to electrons (holes). The chemical potential µ is tunedto the crossing point between spin up (blue) and spin down(red). The superconductivity couples states with oppositemomenta and opposite spins belonging to the same τ -wire(∆ τ ) and belonging to different wires (∆ c ). The spectrumis gapless at k = 0 for ∆ c = ∆ ∆ ¯1 , marking the topologi-cal phase transition that separates the topological phase withtwo localized midgap bound states at each wire end from thetrivial phase without them. where R τσ ( x ) [ L τσ ( x )] are slowly varying right (left)mover fields of the electron with the spin σ/ τ -wire. Thus, H + H so reduces to H kin = i (cid:126) υ F (cid:88) τ,σ (cid:90) dx [ L † τσ ∂ x L τσ − R † τσ ∂ x R τσ ] , (11)and the superconductivity part to H s = (cid:88) τ (cid:90) dx ∆ τ R † τ L † τ ¯1 − L † τ ¯1 R † τ + L † τ R † τ ¯1 − R † τ ¯1 L † τ + H.c. ) , (12) H c = ∆ c (cid:90) dx ( L † L † ¯1¯1 − L † ¯1¯1 L † + R † ¯11 R † − R † R † ¯11 + H.c. ) . (13)Here, υ F = (cid:126) k F /m is the Fermi velocity. We note thatthe interwire superconductivity ∆ c couples only stateswith momenta close to zero, see Fig. 2.Combining together H kin , H s , and H c , we ar-rive at the following Hamiltonian density H , H =(1 / (cid:82) dx ˆΨ † ( x ) H ˆΨ( x ), H = (cid:126) υ F ˆ kρ + ∆ c ( τ η σ + τ η σ ρ ) /
2+ ∆ (1 + τ ) η σ ρ / ¯1 (1 − τ ) η σ ρ / , (14)where the basis is chosen to be ˆΨ=( R , L , R , L , R † , L † , R † , L † , R ¯1¯1 , L ¯11 , R ¯1¯1 , L ¯1¯1 , R † ¯1¯1 , L † ¯11 , R † ¯1¯1 , L † ¯1¯1 ). The Pauli matrices τ , , ( σ , , ) act in the QW(spin) space. The Pauli matrices η , , ( ρ , , ) act inthe electron-hole (right-left mover) subspace. The time-reversal operator U T = σ ρ satisfies U † T H ∗ ( − k ) U T = H ( k ). The particle-hole symmetry operator U P = η satisfies U † P H ∗ ( − k ) U P = −H ( k ). As a result, the sys-tem under consideration belongs to topological symmetryclass DIII. The spectrum of the system is given by E τ, ± = ( (cid:126) υ F k ) + ∆ τ , (15) E , ± , ± = 12 (cid:16) (cid:126) υ F k ) + ∆ + ∆ + 2∆ c (16) ± (cid:113) (∆ − ∆ ) + 4∆ c [4( (cid:126) υ F k ) + (∆ + ∆ ¯1 ) ] (cid:17) , where each level is twofold degenerate due to the time-reversal invariance of the system. The system is gaplessat k = 0 if ∆ c = ∆ ∆ ¯1 and at k = ± (cid:112) ∆ c − ∆ / (cid:126) υ F if∆ = ∆ ¯1 < ∆ c . In the latter case, the gap closes twicesince the levels are twofold degenerate. Although thisdoes not change the number of bound states, the supportsof the corresponding wavefunctions are different.Generally, if ∆ c > ∆ ∆ ¯1 and ∆ (cid:54) = ∆ ¯1 , there are twozero-energy bound states localized at the left end andtwo at the right end of the system. These two states areKramers partners protected by the time-reversal symme-try. Below we provide the wavefunction Φ MF1 ( x ) of oneof these left-localized states written in the basis (Ψ ,Ψ ,Ψ † , Ψ † , Ψ ¯11 , Ψ ¯1 , ¯1 , Ψ † ¯11 , Ψ † ¯1¯1 ). Applying the time-reversal symmetry operator T , we find the wavefunctionof its Kramers partner Φ MF¯1 ( x ) = T Φ MF1 ( x ). The gen-eral form of the Majorana fermion wavefunction is thengiven byΦ MF1 ( x ) = f ( x ) g ( x ) f ∗ ( x ) g ∗ ( x ) f ¯1 ( x ) g ¯1 ( x ) f ∗ ¯1 ( x ) g ∗ ¯1 ( x ) , Φ MF¯1 ( x ) = g ∗ ( x ) − f ∗ ( x ) g ( x ) − f ( x ) g ∗ ¯1 ( x ) − f ∗ ¯1 ( x ) g ¯1 ( x ) − f ¯1 ( x ) , , (17)which follows from the requirement that the Majoranaoperators [belonging to zero-energy eigenstates of Eq.(33)] be self-adjoint: ˆΨ MF1 ( x ) = ˆΨ † MF1 ( x ). From nowon, without loss of generality, we assume that ∆ > ∆ ¯1 .Next, we solve the eigenvalue equation for the Hamil-tonian density given in Eq. (33) for zero eigenenergy ex-plicitly (following Ref. 62). If ∆ + ∆ ¯1 > c , the com-ponents of the corresponding wavefunctions are found tobe given by f ( x ) = − ig ∗ ( x ) = ( e − x/ξ − e − x/ξ ¯2 ) (18) × ∆ c (cid:16) ∆ + ∆ ¯1 + (cid:112) (∆ + ∆ ¯1 ) − c (cid:17) ,f ¯1 ( x ) = − ig ∗ ¯1 ( x ) = − c e − x/ξ (19) − e − x/ξ ¯1 + ik F x (cid:112) (∆ + ∆ ¯1 ) − c × (cid:16) ∆ + ∆ ¯1 + (cid:112) (∆ + ∆ ¯1 ) − c (cid:17) + 12 e − x/ξ ¯2 (cid:16) ∆ + ∆ ¯1 + (cid:112) (∆ + ∆ ¯1 ) − c (cid:17) , where the localization lengths are given by ξ ± = (cid:126) υ F / ∆ ± , (20) ξ ± = 2 (cid:126) υ F / (cid:16) ∆ − ∆ ¯1 ± (cid:112) (∆ + ∆ ¯1 ) − c (cid:17) . If ∆ + ∆ ¯1 < c , the wavefunction components aregiven by f ( x ) = ig ∗ ( x ) = − e − x/ξ sin( k x ) (21) × c (cid:16) ∆ + ∆ ¯1 + i (cid:112) c − (∆ + ∆ ¯1 ) (cid:17) ,f ¯1 ( x ) = ig ∗ ¯1 ( x ) (22)= e − x/ξ ¯1 + ik F x (cid:112) c − (∆ + ∆ ¯1 ) × (cid:16) ∆ + ∆ ¯1 + i (cid:112) c − (∆ + ∆ ¯1 ) (cid:17) − e − x/ξ (cid:16) ∆ + ∆ ¯1 + i (cid:112) c − (∆ + ∆ ¯1 ) (cid:17) × (cid:104) cos( k x ) (cid:112) c − (∆ + ∆ ¯1 ) + sin( k x )(∆ + ∆ ¯1 ) (cid:105) , where the localization length ξ ± and the wavevector k are given by ξ = 2 (cid:126) υ F / (∆ − ∆ ¯1 ) , (23) k = ± (cid:112) c − (∆ + ∆ ¯1 ) / (cid:126) υ F . The case of ∆ + ∆ ¯1 = 2∆ c should be treated sepa-rately leading to f ( x ) = − ig ∗ ( x ) = − ixe − x/ξ (∆ + ∆ ¯1 ) / (cid:126) υ F , (24) f ¯1 ( x ) = − ig ∗ ¯1 ( x ) = − i (cid:16) e − x/ξ ¯1 + ik F x − e − x/ξ [2 + x (∆ + ∆ ¯1 ) / (cid:126) υ F ] (cid:17) . (25)As a result, if ∆ c > ∆ ∆ ¯1 and ∆ (cid:54) = ∆ ¯1 , we findtwo zero-energy bound states at each system end, andwe denote the corresponding Majorana operators (say, atthe left end) as Ψ MF τ = Ψ † MF τ . These MFs are Kramerspartners of each other, so that their wavefunctions arerelated by Φ MF¯1 ( x ) = T Φ MF1 ( x ). Here, the time-reversaloperator T is given by T = iσ K , where K Φ( x ) = Φ ∗ ( x ). B. SOIs of equal sign
In this subsection, we consider the case where the twoQWs have the same sign of Rashba SOI, α R α R ¯1 > α R > α R ¯1 >
0. Otherwise, as mentionedabove, the SOI can be gauged away completely withoutgenerating the position-dependent crossed Andreev pair-ing. Again, MFs emerge as a result of a competition be-tween two pairing terms, and, importantly, the crossedAndreev pairing is possible only at k = 0 but not at finitemomenta, where states with opposite spins do not haveopposite momenta, see Fig. 3. FIG. 3. The spectrum of two QWs with positive RashbaSOI in both QWs. The solid (dashed) lines correspond toelectrons (holes). The chemical potential µ τ is tuned to thecrossing point between spin up (blue) and spin down (red).The superconductivity couples states with opposite momentaand opposite spins belonging to the same τ -wire (∆ τ ) andbelonging to different wires (∆ c ). The spectrum is gapless at k = 0 for ∆ c = ∆ ∆ ¯1 , marking the topological phase tran-sition that separates the topological phase with two localizedmidgap bound states at each wire end from the trivial phasewithout them. In this subsection we use the same notation for Hamil-tonian as in the previous one. We believe that this shouldnot lead to any misinterpretation but could help to makeconnections between two setups. In addition, taking intoaccount that calculations are very similar in the two case,we try to keep the discussion short and omit details.Again, we linearize the spectrum around the Fermipoints k = 0 and k F τ = ± k so,τ ,Ψ = R e ik F x + L , (26)Ψ = L e − ik F x + R , (27)Ψ ¯11 = L ¯11 + R ¯11 e ik F ¯1 x , (28)Ψ ¯1¯1 = R ¯1¯1 + L ¯1¯1 e − ik F ¯1 x . (29)where R τσ ( x ) [ L τσ ( x )] are slowly varying right (left)mover fields of the electron with the spin σ/ τ -wire. Here, we again assume that the chemicalpotentials are tuned to the SO energy, µ τ = E so,τ .The kinetic part of the Hamiltonian H + H so reducesto H kin = (cid:88) τ,σ (cid:90) dx i (cid:126) υ F τ [ L † τσ ∂ x L τσ − R † τσ ∂ x R τσ ] , (30) and the superconductivity part to H s = (cid:88) τ (cid:90) dx ∆ τ R † τ L † τ ¯1 − L † τ ¯1 R † τ + L † τ R † τ ¯1 − R † τ ¯1 L † τ + H.c. ) , (31) H c = ∆ c (cid:90) dx ( L † ¯11 R † − R † L † ¯11 + L † R † ¯1¯1 − R † ¯1¯1 L † + H.c. ) . (32)Here, υ F τ = (cid:126) k F τ /m is the Fermi velocity. Again, the in-terwire superconductivity ∆ c acts only at momenta closeto zero, see Fig. 3.The Hamiltonian density H in terms of Pauli matricesis given by H = (cid:126) υ F ˆ k (1 + τ ) ρ / (cid:126) υ F ¯1 ˆ k (1 − τ ) ρ /
2+ ∆ c τ η ( σ ρ − σ ρ ) /
2+ ∆ (1 + τ ) η σ ρ / ¯1 (1 − τ ) η σ ρ / , (33)where the basis is chosen to be ˆΨ=( R , L , R , L , R † , L † , R † , L † , R ¯1¯1 , L ¯11 , R ¯1¯1 , L ¯1¯1 , R † ¯1¯1 , L † ¯11 , R † ¯1¯1 , L † ¯1¯1 ). The energy spectrum is given by E τ, ± = ( (cid:126) υ F τ k ) + ∆ τ , (34) E , ± , ± = 12 (cid:16) ∆ + ∆ + 2∆ c + (cid:126) ( υ F + υ F ) k (35) ± (cid:113) (∆ − ∆ ) + 4∆ c (∆ + ∆ ¯1 ) + (cid:126) ( υ F − υ F ) k + 4∆ c (cid:126) ( υ F − υ F ) k + 2 (cid:126) ( υ F − υ F )(∆ − ∆ ) k (cid:17) , where each level is twofold degenerate. We note againthat the spectrum is gapless at k = 0 provided that∆ c = ∆ ∆ ¯1 . If ∆ c > ∆ ∆ ¯1 , we find two zero-energybound states at each system end. The correspondingMF wavefunctions are too involved to be displayed in ageneral case. However, in the special simplified case with∆ = ∆ ¯1 and υ F = υ F , the MFs are defined by Eq.(17) with f ( x ) = ig ∗ ( x ) = ( e − ik F x e − x/ξ − e − x/ξ ) , (36) f ¯1 ( x ) = ig ∗ ¯1 ( x ) = ( e − x/ξ − e − ik F ¯1 x e − x/ξ ) (37)The localization length are given by ξ = (cid:126) υ F / ∆ and ξ = (cid:126) υ F / (∆ c − ∆ ). IV. KRAMERS PAIRS OF PARAFERMIONS
Electron-electron interaction effects become importantif the chemical potential is tuned to be, for example, atone third of the SOI energy, µ / ,τ = E so,τ /
9, such thatthe Fermi wavevectors become ± k so,τ (1 ± / g B are taken into account togenerate momentum-conserving terms. Below, wefocus on the second case of Rashba SOI of the same signin both QWs.In particular, the interwire superconductivity Hamil-tonian density in Nambu space is given by H eec = g c (cid:104) L † ¯11 R † ( L † ¯11 R )( L R † ) − R † L † ¯11 × ( R † L )( R ¯11 L † ¯11 ) + L † R † ¯1¯1 ( L † R )( L ¯1¯1 R † ¯1¯1 ) − R † ¯1¯1 L † ( R † ¯1¯1 L ¯1¯1 )( R L † ) + H.c. (cid:105) , (38)where the coupling strength is given by g c ∝ ∆ c g B .The structure of H eec can be understood as follows. Ifa Cooper pair splits and each partner tunnels into adifferent QW ( i.e. L † ¯11 R † ¯1¯1 ), both electrons go to thesame momentum k F , as a result, the finite momentum ofsuch a Cooper pair should be compensated by two back-scattering events taking place inside each of the QWs( i.e. L † ¯11 R and L ¯1¯1 R † ¯1¯1 ).Next, we note that H eec and H s [defined by Eq. (31)] a) b) FIG. 4. The momentum-conserving scattering events cor-responding to a) H eec and b) H ees,τ for the chemical poten-tial µ / ,τ = E so,τ / ± k so (1 ± / do not commute, so these two terms cannot be orderedsimultaneously in the bosonized represenation (see be-low). Thus, only of these term can be dominant andresult in the energy gap. In what follows, we assumethat our setup is in the regime where H eec dominatesover H s . This corresponds to two possible cases: thescaling dimension K c of H eec is the lowest one or the barecoupling constant g c is of order one. The scaling dimen-sion K c = [ K − α + K − δ + 9( K β + K γ )] / φ α,β,γ,δ and θ α,β,γ,δ : χ rτσ = [ rφ α + θ α + τ ( rφ β + θ β ) + σ ( rφ γ + θ γ + τ ( rφ δ + θ δ )] /
2. Here, the bosonic field χ tauσ ( χ ¯1 τσ ) corresponds to the fermion operator R τσ ( L τσ ).The scaling dimension of H s is given in the same basisby K s = [ K − α + K − β + K γ + K δ ] /
4. Comparing K s and K c , we see that in the regime of strong electon-electroninteraction when the Luttinger parameters are substan-tially smaller than one, the crossed Andreev pairing isdominant, K s < K c .The intrawire pairing term H ees = (cid:80) τ H ees,τ that com-mutes with H eec is given by H ees,τ = g τ (cid:104) R † τ L † τ ¯1 ( R † τ L τ )( R τ ¯1 L † τ ¯1 ) − L † τ ¯1 R † τ ( L τ R † τ )( L † τ ¯1 R τ ¯1 ) + H.c. (cid:105) , (39)where g τ ∝ ∆ τ g B .Next, we perform a bosonization of the fermions inNambu space. For this we represent electron (hole) op-erators as R τσ = e iφ τσ and L τσ = e iφ ¯1 τσ ( R † τσ = e i ˜ φ τσ and L † τσ = e i ˜ φ ¯1 τσ ) in terms of chiral fields φ rτσ and ˜ φ rτσ ,where r refers to the right/left movers, and τ ( σ ) labels the QW (spin). We then get, H eec =2 g c (cid:104) cos(2 φ ¯1¯11 − φ − φ + ˜ φ ¯11¯1 ) − cos(2 φ − φ ¯1¯11 − φ ¯11¯1 + ˜ φ )cos(2 φ ¯111 − φ − φ + ˜ φ ¯1¯1¯1 ) − cos(2 φ − φ ¯111 − φ ¯1¯1¯1 + ˜ φ ¯111 ) (cid:105) , (40) H ees,τ =2 g τ [cos(2 φ τ − φ ¯1 τ ¯1 − φ ¯1 τ + ˜ φ τ ¯1 ) − cos(2 φ ¯1 τ ¯1 − φ τ − φ τ ¯1 + ˜ φ ¯1 τ )] . (41)Next, we separate the total Hamiltonian into two un-coupled commuting parts, H + ¯ H , where H ( ¯ H ) operatesin the space spanned by ( φ rτ , ˜ φ rτ ¯1 ) [( φ rτ ¯1 , ˜ φ rτ )]. Thus, H and ¯ H operate in time-reversal conjugated spaces,which we can treat as two independent subsystems.Thus, we will focus only on H , knowing that the solutionfor ¯ H can be obtained by direct analogy or via the re-quirement of time-reversal symmetry. To simplify calcu-lations, we introduce new notations η rτσ = 2 φ rτσ − φ ¯ rτσ and ˜ η rτσ = 2 ˜ φ rτσ − ˜ φ ¯ rτσ . This results in H ee = 2 g cos( η − ˜ η ¯11¯1 ) + 2 g ¯1 cos( η − ˜ η ¯1¯1¯1 )+ 2 g c cos( η ¯1¯11 − ˜ η ) + 2 g c cos( η ¯111 − ˜ η ) . (42)Searching for bound states, we impose vanishing bound-ary conditions at x = 0 , (cid:96) , which couples right and leftmovers, η τσ ( x = 0 , (cid:96) ) = η ¯1 τσ ( x = 0 , (cid:96) ) + π . Next, weunfold the QWs by formally extending themfrom − (cid:96) to (cid:96) by defining new chiral fields such that theboundary conditions are satisfied automatically, ξ τ ( x ) = (cid:40) η τ ( x ) , x > η ¯1 τ ( − x ) + π, x < , (43)and analogously we define ˜ ξ τ with ˜ η ’s. Next, we trans-form the chiral fields to conjugate fields φ, θ , via ξ τ =( φ + θ + 3 τ φ + 3 τ θ ) / ξ r = ( − φ + θ − τ φ +3 τ θ ) /
2. Finally, we arrive at H ee = (cid:40) (cid:80) τ g τ cos( φ + 3 τ φ ) , x > g c cos( φ ) cos(3 θ ) , x < . (44)Working in the limit of strong electron-electron interac-tions, we assume that g τ and g c are large enough, sothat the interaction terms are dominant, resulting in thepinning of the fields to constant values such that the to-tal energy is minimized. Thus, we conclude thatthe field φ = πM is pinned uniformly to minimize thekinetic energy. In addition, the two non-commuting con-jugated fields θ and φ are pinned in two neighbouringregions separated by an infinitesimal interval, θ = π (1 + M + 2 m ) / , x < , (45) φ = π (1 + M + 2 n ) / , x > , (46)where M , n , and m are integer-valued operators. Wenote that the only non-zero commutator is [ m, n ] =3 i/ π , which follows directly from [ φ ( x ) , θ ( x (cid:48) )] = − ( iπ/ x − x (cid:48) ), which in turn follows from the stan-dard commutation relation for the chiral fields ξ τ and ˜ ξ τ defined in Eq. (43). Next, we define two operators thatcommute with the Hamiltonian, so that they correspondto zero energy states, α = e i π ( m − n ) , α ¯1 = e i π ( m + n ) . (47)These operators act at the QW ends and are easilyseen to satisfy α = α = 1, and α α ¯1 = α ¯1 α e − iπ/ .Thus, they form parafermions. We further note that theground state of H is threefold degenerate. Indeed, from( α † α ¯1 ) = 1 we see that α † α ¯1 has three distinct eigenval-ues e iπq/ , where q = 0 , ± | q (cid:105) . With an appropriatephase choice, we find α | q (cid:105) = | q + 1 (cid:105) , so the groundstate is threefold degenerate in the considered subspace.Analogously, we obtain the Kramers partners from ¯ H ,¯ α τ = α τ ( m, n → ¯ m, ¯ n ), where, again, ¯ m , and ¯ n areinteger-valued operators, and ¯ q = 0 , ± | q (cid:105) ⊗ | ¯ q (cid:105) , consist ofthree Kramers pairs of parafermions. However, as shownin Ref. 34, the degeneracy could be lifted by disorder.As a result, the parafermion phase found here does notbelong to the topological phases classified in 74.We note that due to our basis choice the constructedstates | q (cid:105) , | ¯ q (cid:105) are not particle-hole symmetric. However,one can easily find new particle-hole invariant states bycombining two Kramers partners with appropriate phase. FIG. 5. Two-dimensional system of parafermions consistingof an array of coupled QWs with proximity induced inter-wire and intrawire superconductivity, see Fig. 1 in the mainpart. The transition between the interwire-pairing-dominantphase ( µ / ) and the intrawire-pairing-dominant phase ( µ o ) iscontrolled by electrical gates (green slabs). Parafermions areformed initially at the boundaries between these two phases.The tunneling t between two neighbouring QWs not sepa-rated by a superconductor results in deconfinement and ina sea of time-reversal invariant parafermions. So far we have considered QWs of finite length whichare entirely in the non-trivial phase supporting PFs lo-calized at the wire ends. However, by local tuning ofthe chemical potential µ , we can move parafermions in-side the QWs, see Fig. 5. As shown above, if µ = µ / ,the interwire superconductivity dominate. However, if µ = µ o is significantly detuned from µ / , the inter-wire superconductivity H eec is suppressed. Thus, theintrawire superconductivity H s dominates, driving this part of the system into the trivial phase. As before, PFsare localized at the boundary between two phases. Allthis allows us to generate PF networks that can also ex-tend to two-dimensional setups. Introducing couplingbetween parafermions one generates a sea of PFs, whichcan potentially result in the Fibonacci phase as arguedin Ref. 36. At the same time, the extension to a two-dimensional system can help to stabilize this phase andmake it less susceptible to disorder.The presence of the PFs in the gap can be tested insetups similar to the ones developed for MFs.
In par-ticular, one can detect PFs by the zero bias peak in theconductance. The periodicity of the Josephson currentas function of the superconducting phase provides moreinformation. As shown before, the period for Z n PFs is2 πn . For time-reversal invariant PFs, similar to time-reversal invariant MFs, several periods can be observedwith 2 πn being the largest one, i.e., π for the PFs con-sidered in this work. V. CONCLUSIONS
We showed that it is possible to construct Kramerspairs of PFs in a time-reversal invariant system. As anexample of such a setup we considered Rashba QWs cou-pled to a superconductor. Given the rapid experimen-tal progress with similar ultraclean systems designed forMFs, the proposed setup seems to be within ex-perimental reach. In addition, we mention that a similarscheme works also for edge states of fractional topologi-cal insulators (or fractional quantum spin Hall effect sys-tem), where different topological regions can be inducedby superconductivity and transverse hopping. We alsoenvisage the extension of our system to a 2D network thatmight result in a Fibonacci phase. The construction ofquantum gates for time-reversal invariant parafermions isan interesting problem by itself, and could be addressedin further work. We also leave for further work a study ofthe splitting potentially caused by disorder effects. How-ever, we envisage that if disorder effects lift the degen-eracy of the bound states, the resulting energy splittingof states can serve as a useful tool to experimentally ac-cess the level of the initial ground state degeneracy, suchthat we can distinguish directly one Kramers pair of MFsfrom three Kramers pairs of PFs.
ACKNOWLEDGMENTS
We thank the UCSB KITP for hospitality. This re-search is supported by the Harvard Quantum Optics Cen-ter, the Swiss NSF, and the NCCR QSIT.
Appendix A: Alternative way to bosonize
In this appendix we show that the bosonization of theeffective Hamiltonian can also be performed by introduc-ing bosonic operators for electrons only, φ rτσ . Thus, in-troducing bosonic operators for both electrons and holes(‘Nambu space representation’), as done in Sec. IV, is notnecessary. However, the Nambu space representation ismore convenient for time-reversal invariant systems.In a first step, Eqs. (40) - (41) become H eec =2 g c (cid:104) cos(2 φ ¯1¯11 + 2 φ − φ − φ ¯11¯1 )cos(2 φ ¯111 + 2 φ − φ − φ ¯1¯1¯1 ) (cid:105) , H ees,τ =2 g τ cos(2 φ τ + 2 φ ¯1 τ ¯1 − φ ¯1 τ − φ τ ¯1 ) , (A1)leading to H ee = 2 g cos( η + η ¯11¯1 ) + 2 g ¯1 cos( η + η ¯1¯1¯1 )+ 2 g c cos( η ¯1¯11 + η ) + 2 g c cos( η ¯111 + η ) (A2) where we introduced the new chiral fields η rτσ = 2 φ rτσ − φ ¯ rτσ . Again, we double the system in order to satisfy thevanishing boundary conditions at the two system endsautomatically, ξ τ ( x ) = (cid:40) η τ ( x ) , x > η ¯1 τ ( − x ) + π, x < , (A3) ξ ¯1 τ ( x ) = (cid:40) η ¯1 τ ¯1 ( x ) , x > η τ ¯1 ( − x ) + π, x < . (A4)Next, we transform the chiral fields to conjugate fields φ and θ , via ξ sτ = ( φ + sθ + 3 τ φ + 3 τ sθ ) /
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