A Alternative way to bosonize

Time-Reversal Invariant Parafermions in Interacting Rashba Nanowires

Abstract

We propose a scheme to generate pairs of time-reversal invariant parafermions. Our setup consists of two quantum wires with Rashba spin orbit interactions coupled to an -wave superconductor, in the presence of electron-electron interactions. The zero-energy bound states localized at the wire ends arise from the interplay between two types of proximity induced superconductivity: the usual intrawire 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 supports Kramers pair of parafermions. Moreover, the scheme can be extended to a two-dimensional sea of time-reversal invariant parafermions.

pacs:
71.10.Pm; 74.45.+c; 05.30.Pr; 73.21.Hb

I Introduction

Topological properties of condensed matter systems have attracted wide attention in recent years. In particular, localized bound states emerging at the interface between different topological regions have been studied intensely both theoretically and experimentally. Majorana fermions (MFs), zero-energy bound states with non-Abelian braid statistics, were predicted in several systems such as fractional quantum Hall effect (FQHE) systems,(1) topological insulators, (2); (3); (4) optical lattices, (5); (6) -wave superconductors, (7) nanowires with Rashba spin orbit interaction (SOI), (8); (9); (17); (11); (13); (12); (16); (14); (15); (10) self-tuning RKKY systems, (18); (19); (20) and graphene-like systems.(21); (22); (23); (26); (25); (24)

Though MFs possess non-Abelian statistics, it is of Ising type which is not sufficient for universal quantum computation, in contrast to Fibonacci anyons.(27) The basic building blocks for the latter anyons are parafermions (PFs), also referred to as fractional MFs, which allow for more universal quantum operations than MFs.(28); (29); (30); (31); (32); (34); (33); (36); (35); (37); (38) Similarly to MFs, PFs are bound states that arise at the interface between two distinct topological phases. In contrast to MFs, however, PFs owe their peculiar properties to strong electron-electron interactions. As a result, most proposals to host PFs invoke edge states of FQHE systems, and to stabilize them at zero energy one relies on particle-hole symmetry generated by proximity to a superconductor. (31); (32); (34); (33); (36); (37) However, while strong magnetic fields are required for the FQHE, they are detrimental for superconductivity, making the experimental realization of such proposals challenging. (36); (39) This has motivated us to search for alternatives to generate PFs with superconductivity but without magnetic fields. Indeed, we will show that by taking advantage of time-reversal invariance it is possible to construct Kramers pairs of PFs, which can be considered as generalization of Kramers pairs of MFs studied before. (40); (42); (43); (44); (45); (46); (41); (47); (48); (49) We are also motivated to work with one-dimensional systems where recent experiments have demonstrated proximity-induced superconductivity of crossed Andreev type, (50); (51); (52) strong electron-electron interaction, (53); (54); (55); (56) and high tunability of the chemical potential. (11); (13); (12); (16); (14); (15) Moreover, the class of materials suitable for our scheme is larger than for schemes with magnetic field since we do not require large -factors.

Figure 1: Sketch of two Rashba QWs (yellow strips) coupled to an -wave superconductor (blue strip). The SOI field points in positive (negative) direction, say, along the azis for the upper (lower) QW, (). The intrawire proximity induced superconductivity of strength corresponds to a Cooper pair (pair of green dots) tunneling as a whole into the -wire. The interwire proximity induced superconductivity of strength corresponds to crossed Andreev reflection into both QWs, which dominates for strong electron-electron interaction assumed here.

The setup we consider (see Fig. 1) consists of two one-dimensional channels, or quantum wires (QWs) with the Rashba SOI. The QWs are close to an -wave superconductor resulting in proximity induced superconductivity. In general, there are two types of pairing terms. The first one is intrawire pairing corresponding to tunneling of Cooper pairs as a whole to either of the QWs. The second type is the interwire pairing corresponding to ‘crossed Andreev reflection’ (50) where the Cooper pair gets split into two different channels. Such processes dominate in the regime of strong electron-electron interactions. (57); (58); (59) In this case, the system is in the topological phase with bound states localized at the system ends. If the chemical potential is tuned close to the SOI energy, the system supports two MFs at each end that are time-reversal partners of each other. (47); (45) More strikingly, if the chemical potential is lowered, e.g. to one nineth of the SOI energy, and electron-electron interactions are strong, the zero-energy ground state contains three PF Kramers pairs. However, similar to Ref. (34), the degeneracy of our bound states is not protected by a fundamental system property(74) and is susceptible to a specific kind of disorder.

The paper is organized as follows. In Sec. II we introduce the model system; in Sec. III we consider the non-inetracting case and find Kramers pairs of Majorana fermions, first for wires with SOI with opposite signs and then for wires with equal signs. In Sec. IV we consider the case with interactions, and using a bosonization approach we derive the parafermion bound states. Finally, we give some conclusions in Sec. V.

Ii Model

We consider a system consisting of two Rashba QWs brought into the proximity to an -wave superconductor, see Fig. 1. The upper (lower) QW is labeled by the index () and is aligned in the direction. The kinetic part of the Hamiltonian is given by

(1)

where [] is the creation (annihilation) operator of an electron of mass at position of the -wire with spin along the -axis, and is the chemical potential. The Rashba SOI field , that characterizes the strength and the direction of the spin polarization caused by SOI, points in the direction in each of the two QW, so the Rashba SOI term is written as

(2)

Here, the Pauli matrices act on the spin of the electron. We note that the spin projection on the direction is a good quantum number (), and the dispersion relation for the spin component at the -wire is given by , where the chemical potential is tuned to the crossing point between two spin-polarized bands at , i.e. , see Fig. 2. Here, is the SOI energy, and is the SOI wavevector.

In addition, the intrawire superconductivity of strength is proximity induced in each of the QWs by the tunneling of Cooper pairs as a whole from the superconductor to the -wire. The corresponding pairing term is given by

(3)

If the distance between two QWs is shorter than the superconductor coherence length then crossed Andreev reflection is possible (50) where the electrons from the same Cooper pair tunnel into two different QWs, resulting in the interwire proximity induced superconductivity. (60); (57); (58); (59) The corresponding pairing term is given by

(4)

where is the strength of the induced interwire superconductivity. Such a process is useful in Cooper pair splitters where crossed Andreev reflection dominates,(61); (51); (52) so .

Finally, we note that becomes equivalent to FFLO pairing if one gauges away the SOI in the wires. It is known that in one-dimensional wires the Rashba SOI can be gauged away by a spin-dependent gauge transformation. (39) In our case, we gauge away the Rashba SOI simultaneously in both wires by the following transformation

(5)

which is also wire-dependent () as a consequence of opposite Rashba SOI. As a result, the crossed Andreev term becomes in this new gauge

(6)

whereas remains unchanged. Thus, the crossed Andreev superconductivity has a non-uniform pairing term, , which manifestly breaks the translation invariance if . This term is related to the Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state,(75); (76); (77) where the Cooper pair has finite total momentum. Therefore, all results derived in the main part for two wires with opposite Rashba SOI are also valid for a system consisting of two wires without SOI but coupled to an FFLO-type superconductor instead of an ordinary -wave superconductor.

The spatial dependence makes it explicit that there can be ground states in the system with broken symmetries (such as a charge density wave state), and thus states of different symmetries separated by domain walls that host bound states. We note that this situation is analogous to Ref. (34), which finds parafermions in a one-dimensional Rashba wire coupled to a superconductor and in the presence of magnetic fields. There, it has been pointed out (34) that the resulting gapped state is not within the list of possible gapped one-dimensional phases classified in Ref. (74). As a consequence, disorder or deviations from the mean-field description of superconductivity can lift, in principle, the bound state degeneracy. (34)

Figure 2: The spectrum of two QWs with positive (negative) Rashba SOI for (). The solid (dashed) lines correspond to electrons (holes). The chemical potential is tuned to the crossing point between spin up (blue) and spin down (red). The superconductivity couples states with opposite momenta and opposite spins belonging to the same -wire () and belonging to different wires (). The spectrum is gapless at for , marking the topological phase transition that separates the topological phase with two localized midgap bound states at each wire end from the trivial phase without them.

Iii Kramers pairs of Majorana fermions

iii.1 SOIs of opposite sign

In this subsection we focus on the case where the Rashba SOIs are of opposite sign in the two QWs, . In addition, the chemical potential is tuned to the SOI energy in both QWs, . To simplify analytical calculations, we assume in what follows that . We note that the choice of exactly opposite SOIs, such that the Fermi velocities are the same in the two QWs, is convenient but not necessary. All that is needed is to tune the individual Fermi wavevectors (via chemical potentials) to the individual values (or fractions thereof) in each wire.

The proximity-induced superconductivity leads to gaps in the spectrum. Thus, the question arises if there are zero-energy bound states localized at the ends of the wires. To find an answer, we proceed by linearizing the spectrum around the Fermi points and (see Fig. 2),

(7)
(8)
(9)
(10)

where [] are slowly varying right (left) mover fields of the electron with the spin at the -wire.(62); (10); (63) Thus, reduces to

(11)

and the superconductivity part to

(12)
(13)

Here, is the Fermi velocity. We note that the interwire superconductivity couples only states with momenta close to zero, see Fig. 2.

Combining together , , and , we arrive at the following Hamiltonian density , ,

(14)

where the basis is chosen to be =(, , , , , , , , , , , , , , , . The Pauli matrices () act in the QW (spin) space. The Pauli matrices () act in the electron-hole (right-left mover) subspace. The time-reversal operator satisfies . The particle-hole symmetry operator satisfies . As a result, the system under consideration belongs to topological symmetry class . (78)

The spectrum of the system is given by

(15)
(16)

where each level is twofold degenerate due to the time-reversal invariance of the system. The system is gapless at if and at if . In the latter case, the gap closes twice since the levels are twofold degenerate. Although this does not change the number of bound states, the supports of the corresponding wavefunctions are different.

Generally, if and , there are two zero-energy bound states localized at the left end and two at the right end of the system. These two states are Kramers partners protected by the time-reversal symmetry. Below we provide the wavefunction of one of these left-localized states written in the basis (, ,, , , , , ). Applying the time-reversal symmetry operator , we find the wavefunction of its Kramers partner . The general form of the Majorana fermion wavefunction is then given by

(17)

which follows from the requirement that the Majorana operators [belonging to zero-energy eigenstates of Eq. (33)] be self-adjoint: . From now on, without loss of generality, we assume that .

Next, we solve the eigenvalue equation for the Hamiltonian density given in Eq. (33) for zero eigenenergy explicitly (following Ref. (62)). If , the components of the corresponding wavefunctions are found to be given by

(18)
(19)

where the localization lengths are given by

(20)

If , the wavefunction components are given by

(21)
(22)

where the localization length and the wavevector are given by

(23)

The case of should be treated separately leading to

(24)
(25)
Figure 3: The spectrum of two QWs with positive Rashba SOI in both QWs. The solid (dashed) lines correspond to electrons (holes). The chemical potential is tuned to the crossing point between spin up (blue) and spin down (red). The superconductivity couples states with opposite momenta and opposite spins belonging to the same -wire () and belonging to different wires (). The spectrum is gapless at for , marking the topological phase transition that separates the topological phase with two localized midgap bound states at each wire end from the trivial phase without them.

As a result, if and , we find two zero-energy bound states at each system end, and we denote the corresponding Majorana operators (say, at the left end) as . These MFs are Kramers partners of each other, so that their wavefunctions are related by . Here, the time-reversal operator is given by , where .

iii.2 SOIs of equal sign

In this subsection, we consider the case where the two QWs have the same sign of Rashba SOI, . However, in this case, in contrast to the previous section, we assume that . Otherwise, as mentioned above, the SOI can be gauged away completely without generating the position-dependent crossed Andreev pairing. Again, MFs emerge as a result of a competition between two pairing terms, and, importantly, the crossed Andreev pairing is possible only at but not at finite momenta, where states with opposite spins do not have opposite momenta, see Fig. 3.

In this subsection we use the same notation for Hamiltonian as in the previous one. We believe that this should not lead to any misinterpretation but could help to make connections between two setups. In addition, taking into account 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 Fermi points and ,

(26)
(27)
(28)
(29)

where [] are slowly varying right (left) mover fields of the electron with the spin at the -wire.(62); (10); (63) Here, we again assume that the chemical potentials are tuned to the SO energy, .

The kinetic part of the Hamiltonian reduces to

(30)

and the superconductivity part to

(31)
(32)

Here, is the Fermi velocity. Again, the interwire superconductivity acts only at momenta close to zero, see Fig. 3.

The Hamiltonian density in terms of Pauli matrices is given by

(33)

where the basis is chosen to be =(, , , , , , , , , , , , , , , . The energy spectrum is given by

(34)
(35)

where each level is twofold degenerate. We note again that the spectrum is gapless at provided that . If , we find two zero-energy bound states at each system end. The corresponding MF wavefunctions are too involved to be displayed in a general case. However, in the special simplified case with and , the MFs are defined by Eq. (17) with

(36)
(37)

The localization length are given by and .

Iv Kramers pairs of parafermions

Electron-electron interaction effects become important if the chemical potential is tuned to be, for example, at one third of the SOI energy, , such that the Fermi wavevectors become . In this case, the interwire pairing is possible only if backscattering terms of finite strength are taken into account to generate momentum-conserving terms. (34); (64); (65); (66); (67); (68) Below, we focus on the second case of Rashba SOI of the same sign in both QWs.

Figure 4: The momentum-conserving scattering events corresponding to a) and b) for the chemical potential with associated Fermi wavevectors . See the caption of Fig. 2 for notations.

In particular, the interwire superconductivity Hamiltonian density in Nambu space is given by

(38)

where the coupling strength is given by . The structure of can be understood as follows. If a Cooper pair splits and each partner tunnels into a different QW (i.e. ), both electrons go to the same momentum , as a result, the finite momentum of such a Cooper pair should be compensated by two back-scattering events taking place inside each of the QWs (i.e. and ).

Next, we note that and [defined by Eq. (III.2)] do not commute, so these two terms cannot be ordered simultaneously in the bosonized represenation (see below). Thus, only of these term can be dominant and result in the energy gap. In what follows, we assume that our setup is in the regime where dominates over . This corresponds to two possible cases: the scaling dimension of is the lowest one or the bare coupling constant is of order one. The scaling dimension can be found in a usual way in the basis of conjugated bosonic fields and : . Here, the bosonic field () corresponds to the fermion operator (). The scaling dimension of is given in the same basis by . Comparing and , we see that in the regime of strong electon-electron interaction when the Luttinger parameters are substantially smaller than one, the crossed Andreev pairing is dominant, .

The intrawire pairing term that commutes with is given by

(39)

where .

Next, we perform a bosonization of the fermions (64) in Nambu space. For this we represent electron (hole) operators as and ( and ) in terms of chiral fields and , where refers to the right/left movers, and () labels the QW (spin). We then get,

(40)
(41)

Next, we separate the total Hamiltonian into two uncoupled commuting parts, , where () operates in the space spanned by []. Thus, and operate in time-reversal conjugated spaces, which we can treat as two independent subsystems. Thus, we will focus only on , knowing that the solution for can be obtained by direct analogy or via the requirement of time-reversal symmetry. To simplify calculations, we introduce new notations and . This results in