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PHYSICAL REVIEW LETTERS
Parity Measurement in Topological Josephson Junctions François Crépin and Björn Trauzettel Institute for Theoretical Physics and Astrophysics, University of Würzburg, 97074 Würzburg, Germany (Received 31 May 2013; published 19 February 2014) We study the properties of a topological Josephson junction made of both edges of a two-dimensional topological insulator. We show that, due to fermion parity pumping across the bulk, the global parity of the junction has a clear signature in the periodicity and critical value of the Josephson current. In particular, we find that the periodicity with the flux changes from 4π in a junction with an even number of quasiparticles to 2π in the odd sector. In the case of long junctions, we exhibit a rigorous mathematical connection between the spectrum of Andreev bound states and the fermion parity anomaly, through bosonization. Additionally, we discuss the rather quantitative effects of Coulomb interactions on the Josephson current. DOI: 10.1103/PhysRevLett.112.077002
PACS numbers: 74.45.+c, 71.10.Pm, 74.78.Na
Introduction.—One-dimensional (1D) topological Josephson junctions are known to exhibit a 4π-periodic Josephson current, a property that arises from a parity constraint on the number of quasiparticles in the junction [1,2]. Depending on the precise system at hand, this effect can be related to the existence of Majorana (bound) states, and hence provide a signature of Majorana fermions in condensed matter [1,2]. Two prominent proposals for the realization of 1D topological superconductors include spinorbit wires in the presence of a magnetic field [3,4], as well as edge states of two-dimensional topological insulators (2D TI) [2], both in proximity to an s-wave superconductor. Recent experiments hinted at 4π Josephson currents in spin-orbit wire junctions [5]. A consequent body of work also describes possible realizations of qubits using Majorana bound states, as well as detailed quantum computation schemes [6,7]. A Majorana qubit consists of four Majorana fermions, which can be combined two by two into a state described by two fermion numbers only. By working in a sector of fixed parity, one can build a qubit using two out of the four available degenerate states. In this Letter,we analyze a topological Josephson junction constructed on both edges of a ring-shaped 2D TI, as sketched in Fig. 1. 2D TIs have been realized in HgTe=CdTe quantum wells [8], as well as recently in InAs=GaSb systems [9,10], with normal and superconducting electrodes [11]. In the normal region, both edges are helical liquids and realize a state of perfect Andreev reflection at the normal-superconductor interfaces. At first sight, this system is a peculiar realization of a spinful 1D Josephson junction [12–14] with the spin degrees of freedom split in two spatially separated regions. Although for large enough widths W of the TI, tunneling between edges and interedge Coulomb interactions are exponentially suppressed, the two halves of the junction are connected by fermion parity pumping, as induced by the flux threaded through the ring [2,15,16]. 0031-9007=14=112(7)=077002(5)
We first analyze the parity constraints on the junction and propose to use properties of the Josephson current— periodicity and critical value—to measure the global parity of the junction. If we were to envision such a setup as a Majorana qubit, with two Majorana fermions at each edge, our method would, in principle, provide a way to read out the parity of such a qubit [17]. We also present a detailed analysis of the long junction case through bosonization, discuss the effect of intraedge interactions, and comment on the similarities and differences of this setup with respect to regular spinful Josephson junctions. Setup and parity measurement.—The system we have in mind is a Josephson junction consisting of a 2D TI upon which an s-wave superconductor is deposited, as pictured in Fig. 1. In contrast with the original proposal of Ref. [2], and subsequent studies [18], we assume that a proximity gap Δ0 opens at both edges. In the normal region, a discrete spectrum of Andreev bound states (ABS)
FIG. 1 (color online). Sketch of a topological Josephson junction. Both edges are contacted by the superconductor deposited on top of a 2D TI. In the simplest case where Sz is conserved, solid red (respectively, blue dashed) lines in the normal region indicate spin-up (respectively, spin-down) electrons. A flux Φ through the hole induces a phase difference ϕ=ð2πÞ ¼ Φ=Φ0 across the Josephson junction with Φ0 ¼ h=2e.
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develops at each edge. Their energies depend on the phase difference ϕ across the junction, which can be induced by a magnetic flux. For a sufficiently large width of the TI, edge states on different edges do not interact and the Josephson current of the junction is given by the sum of two one-dimensional currents mediated by the upper and lower edges. Each current, taken separately, is a 4π-periodic function of the phase difference, owing to the fermion parity anomaly [2] of the topological junction. The explanation of the anomaly relies on the following observation: while the edge Hamiltonian is 2π periodic in the phase difference ϕ, its eigenstates are not. Because of particle-hole symmetry of the Bogoliubov–de Gennes Hamiltonian, each ABS at excitation energy ε comes with a partner at −ε. These two states differ by their fermion parity. When the phase is advanced by 2π, the system switches between them, and therefore changes parity: an extra quasiparticle has been added to the edge. In an ideal environment, that is, in the absence of quasiparticle poisoning, parity is conserved and the system can only return to its ground state by advancing again the phase by 2π, thereby adding (or removing) a quasiparticle and restoring the parity. The origin of the extra quasiparticle lies in the topology of the 2D band insulator, and the existence of a second pair of edge states. In the geometry of Ref. [2], as ϕ is advanced by 2π, a unit of fermion parity is pumped across the sample, from one edge to the other. In the present case of Fig. 1, this so-called Z2 pumping [2,15] effectively transfers a quasiparticle between ABS at the lower and upper edges. As a consequence, even though the fermion parity at each edge changes when ϕ advances by 2π, the global fermion parity of the junction is conserved. We write the Josephson current J of the junction as Jσ;σ0 ½ϕ ¼ I up;σ ½ϕ þ I down;σ 0 ½ϕ;
(1)
where σ ¼ (respectively, σ 0 ¼ ) indicates the fermion number parity (þ being for even and − for odd) of the upper (respectively, lower) edge. The total fermion parity of the junction is Σ ¼ σσ 0 . In Eq. (1) and in the following, we use the letter J for the total current and the letter I for the edge currents. A striking feature of the setup is that the periodicity of the Josephson current depends on the parity of the junction. For an even fermion number (Σ ¼ 1), the current switches between J þ;þ and J−;− at ϕ ¼ π mod 2π, and is thus 4π periodic in ϕ. On the other hand, the odd current (Σ ¼ −1) is 2π periodic. Indeed, it switches at ϕ ¼ π mod 2π between Jþ;− and J−;þ , which are equal functions, provided the upper and lower edges are identical, that is, I up;σ ½ϕ ¼ I down;σ ½ϕ. In addition, the parity of the junction is reflected in the value of the critical current JΣ;c . Indeed, at a given edge, a change of parity reverses the direction of the current, that is, sgnI up;þ ½ϕ ¼ −sgnI up;− ½ϕ, and a similar relation for the
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lower edge. In the even sector, currents at both edges flow in the same direction, while in the odd sector they flow in opposite directions. As a consequence, the value of the critical current in the odd case J−;c is reduced with respect to that in the even case, Jþ;c . The exact ratio J −;c =Jþ;c between even and odd critical currents depends on the length L of the junction. In the short junction regime, L ≪ ξ with ξ ¼ vF =Δ0 the coherence length of the superconducting region, one has, at zero temperature, I up; ½ϕ ¼ I down; ½ϕ ¼ I c sinðϕ=2Þ [2], with I c ¼ Δ0 =2 the edge critical current. Therefore, the even critical current is just J þ;c ¼ 2I c , while the odd critical current vanishes, J−;c ¼ 0. Note that the odd current actually vanishes for all values of ϕ since I up;σ ½ϕ ¼ −I down;−σ ½ϕ. The long junction regime, L ≫ ξ, was studied in Ref. [18]. In that case one finds I up;þ ½ϕ ¼ I down;þ ½ϕ ¼ I 0c ϕ=ð2πÞ and I up;− ½ϕ¼I down;− ½ϕ¼I 0c ðϕ=ð2πÞ−sgnϕÞ, with I 0c ¼ evF =L the edge critical current. As a consequence, Jþ;c ¼ 2I 0c and J−;c ¼ I 0c . In summary, the ratio of critical currents in the odd and even cases interpolates between J−;c =J þ;c ¼ 0 for L ≪ ξ and J −;c =Jþ;c ¼ 1=2 for L ≫ ξ. Long junction regime and bosonization.—The long junction regime is interesting for two reasons. First, it should be of experimental relevance, as argued in Ref. [18]. Second, it allows us to make a clear mathematical connection between the spectrum of ABS and the fermion parity anomaly, through bosonization. At a given edge, helicity imposes perfect Andreev reflection and a nondegenerate spectrum of ABS. In the low-energy limit ε ≪ Δ0 , the ABS spectrum is given by [19] πvF 1 ϕ ε¼ with n ∈ Z; (2) nþ 2 2π L where vF is the Fermi velocity of the linearly dispersing edge states. In the case of a nonhelical, one-dimensional Josephson junction, this spectrum would be twice degenerate due to the spin 1=2 of the electrons. As was first recognized in Ref. [13], the spectrum of Eq. (2) is one of two branches of counterpropagating fermions with a linear spectrum and twisted boundary conditions on a segment of size 2L. Indeed, at low energy, one can write energyindependent boundary conditions for the fermion creation and annihilation operators. At the upper edge, ψ R;↑ ð0Þ ¼ iψ †L;↓ ð0Þ and ψ R;↑ ðLÞ ¼ −ieiϕ ψ †L;↓ ðLÞ which implies ψ α ðx þ 2L; tÞ ¼ −e−iϕ ψ α ðx; tÞ;
(3)
with α ¼ R; ↑ or L; ↓, as well as a relation between right and left movers, ψ R;↑ ðx; tÞ ¼ −iψ †L;↓ ð−x; tÞ;
(4)
for x ∈ ½−L; L. One can then write an effective Hamiltonian for the normal region only with the
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appropriate boundary conditions. Starting from the Hamiltonian of the normal region, at the upper edge, we obtain Z 1 L dxΨ† ðxÞHBdG ΨðxÞ; (5) H up ¼ 2 0 with HBdG ¼ −ivF ∂ x σ z τz the Bogoliubov–de Gennes Hamiltonian, and Ψ ¼ ðψ R;↑ ; ψ L;↓ ; ψ †L;↓ ; −ψ †R;↑ ÞT . σ z (respectively, τz ) is a Pauli matrix acting on the spin-1=2 (respectively, particle-hole) degree of freedom. At low energies, using Eq. (4), the Hamiltonian of Eq. (5) can be unfolded on the segment ½−L; L as Z L v dx½−ψ †R;↑ ∂ x ψ R;↑ þ ψ †L;↓ ∂ x ψ L;↓ ðxÞ; (6) Hup ¼ i F 2 −L with the twisted boundary conditions of Eq. (3). Details of the calculation are given in the Supplemental Material [20]. We then recover the spectrum of Eq. (2). The particle-hole symmetry can be made explicit upon Fourier transform and further use of Eq. (4). Taking the superconducting gap to infinity, Δ0 → ∞, we derive the following normal-ordered form of the Hamiltonian X vF π 1 ϕ Hup ¼ nþ − ∶c†R;n cR;n − cR;n c†R;n ∶; (7) 2L 2 2π n∈Z p1ffiffiffiffi 2L
P
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cR;n ; with the momentum where ψ R;↑ ðx; tÞ ¼ ne kn quantized as kn ¼ ðπ=LÞ½n þ 1=2 − ϕ=ð2πÞ. The spectrum of ABS reduces to that of a single band of fermionic quasiparticles with states labeled by an integer n. States with opposite energies differ in their fermion parity. Note that normal ordering subtracts an infinite ground-state energy coming from quasiparticle states below the Fermi energy, EF ¼ 0. This constant is independent of the phase and therefore does not affect the Josephson current. As the spectrum is unbounded, the Hamiltonian of Eq. (7) is invariant under the transformation ϕ → ϕ þ 2π and n − 1 → n. This transformation leaves the spectrum invariant but changes the parity of the ground state. In this form, the model satisfies all the criteria—a band of fermions with an unbounded, discrete spectrum—for rigorous bosonization [21]. The bosonization transformation reads as
of the Fermi sea. Nˆ R is an operator counting the number of fermions with respect to the Fermi sea. FR is the so-called Klein factor that lowers the fermion number by one. The factor a is a regularization factor. Upon normal ordering ~ e−iϕR ðxÞ , it can be safely taken to be zero. In practice, however, it can be retained and plays the role of a highenergy cutoff on the otherwise unbounded theory. Here, one should take a ≃ ξ ¼ vF =Δ0 . Using Eqs. (8) and (9), the Hamiltonian (7) takes the bosonized form [21] X vF π ˆ ϕ 2 Hup ¼ þ vF qb†R;q bR;q : (10) NR − 2L 2π q>0 Obviously, the transformation ϕ → ϕ þ 2π, changes the topological number N R by 1, a reflection of the fermion parity anomaly. As the Hamiltonian now naturally separates between a phase-dependent topological sector—the number of quasiparticles in the junction—and a nontopological sector—the phase-independent particle-hole excitations— the computation of the current at one edge, taking into account the parity constraints, is straightforward: I up; ½ϕ ¼ −
(8)
Here, ϕ~ R ðxÞ is a chiral bosonic field, periodic on ½−L; L, with the following Fourier decomposition: X rffiffiffiffiffiffi π iqx ðe bR;q þ e−iqx b†R;q Þe−aq=2 ; (9) ϕ~ R ðxÞ ¼ − Lq q>0 where q ¼ ðnπ=LÞ, n ∈ N . The bosonic operators b†R;q and bR;q create and annihilate particle-hole excitations on top
(11)
with Ztup; ½ϕ ¼
ikn x
1 1 ˆ ~ ψ R;↑ ðx; tÞ ≡ FR pffiffiffiffiffiffiffiffi eiðπ=LÞ½N R þ2−ðϕ=2πÞx e−iϕR ðxÞ : 2πa
2e 1 ∂ ln Ztup; ½ϕ; ℏ β ∂ϕ
X
2
e−βðℏπvF =2LÞðN R −ϕ=ð2πÞÞ ;
(12)
N R ∈Z even=odd
the topological part of the partition function. The parity constraint is simply reflected in the parity of the quasiparticle number N R. Interestingly, these partition functions can be expressed analytically in the form of Jacobi’s elliptic theta functions [22] , and we find for the currents, I up; ½ϕ ¼ e
vF ϕ 2e − ∂ ln θ3=2 ½zðϕÞ; q; L 2π ℏβ ϕ
(13)
for jϕj < π. We have defined zðϕÞ ¼ iβðℏπvF =2LÞðϕ=πÞ and q ¼ e−2βðℏπvF =LÞ . We have checked that our Eq. (13) and the formula of Beenakker et al. [18] for the long junction Josephson current, although seemingly different, coincide in the limit Δ0 → ∞ [20] . In particular, the odd current develops a discontinuity at ϕ ¼ 0, as β → ∞. Our bosonization approach is restricted to long junctions; nevertheless, it has the conceptual advantage of putting the emphasis on the fermion parity anomaly through a simple counting of quasiparticles, and leads to a very simple formula for the current. Now, we include the lower edge, which is most easily achieved by exchanging ↑ and ↓ indexes in Eqs. (5) and (6). The effective low-energy Hamiltonian at the lower edge then reads
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PHYSICAL REVIEW LETTERS 21 FEBRUARY 2014 PRL 112, 077002 (2014) X vF π the Supplemental Material to this Letter that, for realistic 1 ϕ nþ þ ∶c†L;n cL;n − cL;n c†L;n ∶; Hdown ¼ junctions, induced fluxes are negligible when compared to 2L 2 2π n∈Z the external flux [20]. (14) Another interesting question raised by this particular pffiffiffiffiffiffi P −ik x setup is its relation to the well-known case of a 1D spinful where ψ L;↑ ðx; tÞ ¼ ð1= 2LÞ n e n cL;n with the momJosephson junction [12–14,24]. One difficulty in that case entum kn quantized as kn ¼ ðπ=LÞ½n þ 1=2 þ ϕ=ð2πÞ. is to incorporate both Andreev reflection and backscatterIn bosonized form, ing at the edge. The bosonization treatment of Ref. [13] 2 was built on the assumption of perfect Andreev reflection, X v π ϕ Hdown ¼ F Nˆ L þ þ vF qb†L;q bL;q : (15) which needed to be corrected to account for normal 2L 2π q>0 reflection. The case of helical liquids provide a natural realization of the assumption of perfect Andreev reflection. Again, the transformation ϕ → ϕ þ 2π, changes the The present setup can be thought of as a way to split topological number N L by 1. The combination of both the ABS of the spinful liquid in two halves, spatially Hamiltonians, (10) and (15), illustrates the transfer of fermion localized in two different regions, the edges, separated by parity from one edge to the other, as the phase is advanced the bulk gap. As we have seen, although the spin degenby 2π. Note that one could also write Hdown in terms of eracy is recovered, the parity pumping across the toporight movers only. This freedom of choice comes from logical insulator entails very different predictions for the Eq. (4), and is due to particle-hole symmetry. It illustrates Josephson current, in particular, 4π periodicity in the even the important fact that only fermion parity is well defined sector. Note that, in a situation where the fermion parity at the edges. The partition function of the lower edge, would not be conserved—for instance if quasiparticle Ztdown; ½ϕ, is given by analogy with Eq. (12). One can poisoning is important [25,26]—one simply recovers the check, by making the change of variables N L → −N L , that Josephson current of the spinful liquid (a sawtooth with I down; ½ϕ ¼ I up; ½ϕ. The partition function of the junction period 2π), by letting the fermion number in Eq. (12) run is simply Ztσ;σ0 ¼ Ztup;σ × Ztdown;σ0 and the Josephson current over both even and odd integers. is indeed the one of Eq. (1). Discussion.—In summary, we have studied a topological Effects of interactions.—Coulomb interactions at the Josephson junction involving both edges of a 2D TI. edges can easily be included in the long junction regime We have put forward a possible nondestructive measureon the basis of the bosonization method. Assuming for now ment of the global parity, which includes the effect of that interedge interactions are made negligible by screenintraedge interactions. Importantly, the topological features ing, the Hamiltonian at either edge conserves its simple of the 2D quantum spin Hall phase are essential for the low-energy form with only renormalized coefficients: fermion parity pumping and thus, the parity meter proposed here. A straightforward extension to, for instance, surface X † vN;α π ˆ ϕ 2 states of three-dimensional topological insulators is not Hα ¼ þ vs;α qBα;q Bα;q ; (16) Nα − possible. As a closing remark, let us briefly elaborate on 2π 2L q>0 interedge interactions. So-called g1;∥ processes [27] do not change the local fermion number and can be easily with α ¼ up, down. We have allowed for the possibility of incorporated in the bosonization treatment, leaving our different interaction strengths at each edge. The new bosonic † † general conclusions unchanged. On the other hand, upon operators B and B are related to the original b and b by a inclusion of g1;⊥ processes, the Hamiltonian no longer Bogoliubov transformation [23]. Quite generally, for shortcommutes with the fermion number, and a simple expresrange interactions, we have vN;α ¼ vF − g2;α =ð2πÞ, where sion of the partition function as in Eq. (12) a priori no g2;α is the original interaction between right and left movers longer exists. Another interesting open question is the at each edge. The Josephson current is simply obtained by influence of disorder on our results [28], in which case the replacing vF by vN;α in Eq. (13). The effects of repulsive two Josephson currents could be separately modified. interactions is therefore to reduce the Josephson current, as was first discovered in the spinful case [13]. One should note We thank Fabrizio Dolcini and Carlo Beenakker for that our original prediction on the relation between parity illuminating discussions. Financial support by the DFG and periodicity will not hold in the case vN;up ≠ vN;down , (German-Japanese research unit “Topotronics” and the since it would imply I down; ½ϕ ≠ I up; ½ϕ. However, the priority program “Topological insulators”) as well as the ratio of critical currents in the odd and even cases is Helmholtz Foundation (VITI) is gratefully acknowledged. unaffected by interactions. Similar conclusions can be reached in the case of equal interactions but unequal lengths of the junctions. At this point, the reader might also wonder about self and mutual induction from Josephson currents that [1] A. Y. Kitaev, Phys. Usp. 44, 131 (2001). would ultimately couple the Hamiltonians (16). We show in [2] L. Fu and C. L. Kane, Phys. Rev. B 79, 161408 (2009). 077002-4
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[3] R. M. Lutchyn, J. D. Sau, and S. D. Sarma, Phys. Rev. Lett. 105, 077001 (2010). [4] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010). [5] L. P. Rokhinson, X. Liu, and J. K. Furdyna, Nat. Phys. 8, 795 (2012). [6] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, Rev. Mod. Phys. 80, 1083 (2008). [7] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Nat. Phys. 7, 412 (2011). [8] M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 (2007). [9] I. Knez, R.-R. Du, and G. Sullivan, Phys. Rev. Lett. 107, 136603 (2011). [10] K. Suzuki, Y. Harada, K. Onomitsu, and K. Muraki, Phys. Rev. B 87, 235311 (2013). [11] I. Knez, R.-R. Du, and G. Sullivan, Phys. Rev. Lett. 109, 186603 (2012). [12] R. Fazio, F. W. J. Hekking, and A. A. Odintsov, Phys. Rev. Lett. 74, 1843 (1995). [13] D. L. Maslov, M. Stone, P. M. Goldbart, and D. Loss, Phys. Rev. B 53, 1548 (1996). [14] I. Affleck, J.-S. Caux, and A. M. Zagoskin, Phys. Rev. B 62, 1433 (2000). [15] L. Fu and C. L. Kane, Phys. Rev. B 74, 195312 (2006). [16] A. Keselman, L. Fu, A. Stern, and E. Berg, Phys. Rev. Lett. 111, 116402 (2013).
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[17] We remark that a related setup was briefly discussed in Ref. [29], in the context of anyons, as a way to realize a so-called braidless operation of the type j00i → j11i. [18] C. W. J. Beenakker, D. I. Pikulin, T. Hyart, H. Schomerus, and J. P. Dahlhaus, Phys. Rev. Lett. 110, 017003 (2013). [19] I. Kulik, Sov. Phys. JETP 30, 944 (1970). [20] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.077002 for where we outline the steps of the bosonization calculation, discuss our Eq. (13) for the current in light of Ref. [18], and compute inductive effects in the superconducting loop. [21] J. von Delft and H. Schoeller, Ann. Phys. (Berlin) 7, 225 (1998). [22] NIST Digital Library of Mathematical Functions, http:// dlmf.nist.gov/20.2. [23] M. Fabrizio and A. O. Gogolin, Phys. Rev. B 51, 17 827 (1995). [24] R. Fazio, F. W. J. Hekking, and A. A. Odintsov, Phys. Rev. B 53, 6653 (1996). [25] J. C. Budich, S. Walter, and B. Trauzettel, Phys. Rev. B 85, 121405 (2012). [26] D. Rainis and D. Loss, Phys. Rev. B 85, 174533 (2012). [27] T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, Oxford, 2004). [28] C. W. J. Beenakker, J. M. Edge, J. P. Dahlhaus, D. I. Pikulin, S. Mi, and M. Wimmer, Phys. Rev. Lett. 111, 037001 (2013). [29] J. C. Y. Teo and C. L. Kane, Phys. Rev. Lett. 104, 046401 (2010).
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Parity Measurement in Topological Josephson Junctions François Crépin and Björn Trauzettel Institute for Theoretical Physics and Astrophysics, University of Würzburg, 97074 Würzburg, Germany (Received 31 May 2013; published 19 February 2014) We study the properties of a topological Josephson junction made of both edges of a two-dimensional topological insulator. We show that, due to fermion parity pumping across the bulk, the global parity of the junction has a clear signature in the periodicity and critical value of the Josephson current. In particular, we find that the periodicity with the flux changes from 4π in a junction with an even number of quasiparticles to 2π in the odd sector. In the case of long junctions, we exhibit a rigorous mathematical connection between the spectrum of Andreev bound states and the fermion parity anomaly, through bosonization. Additionally, we discuss the rather quantitative effects of Coulomb interactions on the Josephson current. DOI: 10.1103/PhysRevLett.112.077002
PACS numbers: 74.45.+c, 71.10.Pm, 74.78.Na
Introduction.—One-dimensional (1D) topological Josephson junctions are known to exhibit a 4π-periodic Josephson current, a property that arises from a parity constraint on the number of quasiparticles in the junction [1,2]. Depending on the precise system at hand, this effect can be related to the existence of Majorana (bound) states, and hence provide a signature of Majorana fermions in condensed matter [1,2]. Two prominent proposals for the realization of 1D topological superconductors include spinorbit wires in the presence of a magnetic field [3,4], as well as edge states of two-dimensional topological insulators (2D TI) [2], both in proximity to an s-wave superconductor. Recent experiments hinted at 4π Josephson currents in spin-orbit wire junctions [5]. A consequent body of work also describes possible realizations of qubits using Majorana bound states, as well as detailed quantum computation schemes [6,7]. A Majorana qubit consists of four Majorana fermions, which can be combined two by two into a state described by two fermion numbers only. By working in a sector of fixed parity, one can build a qubit using two out of the four available degenerate states. In this Letter,we analyze a topological Josephson junction constructed on both edges of a ring-shaped 2D TI, as sketched in Fig. 1. 2D TIs have been realized in HgTe=CdTe quantum wells [8], as well as recently in InAs=GaSb systems [9,10], with normal and superconducting electrodes [11]. In the normal region, both edges are helical liquids and realize a state of perfect Andreev reflection at the normal-superconductor interfaces. At first sight, this system is a peculiar realization of a spinful 1D Josephson junction [12–14] with the spin degrees of freedom split in two spatially separated regions. Although for large enough widths W of the TI, tunneling between edges and interedge Coulomb interactions are exponentially suppressed, the two halves of the junction are connected by fermion parity pumping, as induced by the flux threaded through the ring [2,15,16]. 0031-9007=14=112(7)=077002(5)
We first analyze the parity constraints on the junction and propose to use properties of the Josephson current— periodicity and critical value—to measure the global parity of the junction. If we were to envision such a setup as a Majorana qubit, with two Majorana fermions at each edge, our method would, in principle, provide a way to read out the parity of such a qubit [17]. We also present a detailed analysis of the long junction case through bosonization, discuss the effect of intraedge interactions, and comment on the similarities and differences of this setup with respect to regular spinful Josephson junctions. Setup and parity measurement.—The system we have in mind is a Josephson junction consisting of a 2D TI upon which an s-wave superconductor is deposited, as pictured in Fig. 1. In contrast with the original proposal of Ref. [2], and subsequent studies [18], we assume that a proximity gap Δ0 opens at both edges. In the normal region, a discrete spectrum of Andreev bound states (ABS)
FIG. 1 (color online). Sketch of a topological Josephson junction. Both edges are contacted by the superconductor deposited on top of a 2D TI. In the simplest case where Sz is conserved, solid red (respectively, blue dashed) lines in the normal region indicate spin-up (respectively, spin-down) electrons. A flux Φ through the hole induces a phase difference ϕ=ð2πÞ ¼ Φ=Φ0 across the Josephson junction with Φ0 ¼ h=2e.
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develops at each edge. Their energies depend on the phase difference ϕ across the junction, which can be induced by a magnetic flux. For a sufficiently large width of the TI, edge states on different edges do not interact and the Josephson current of the junction is given by the sum of two one-dimensional currents mediated by the upper and lower edges. Each current, taken separately, is a 4π-periodic function of the phase difference, owing to the fermion parity anomaly [2] of the topological junction. The explanation of the anomaly relies on the following observation: while the edge Hamiltonian is 2π periodic in the phase difference ϕ, its eigenstates are not. Because of particle-hole symmetry of the Bogoliubov–de Gennes Hamiltonian, each ABS at excitation energy ε comes with a partner at −ε. These two states differ by their fermion parity. When the phase is advanced by 2π, the system switches between them, and therefore changes parity: an extra quasiparticle has been added to the edge. In an ideal environment, that is, in the absence of quasiparticle poisoning, parity is conserved and the system can only return to its ground state by advancing again the phase by 2π, thereby adding (or removing) a quasiparticle and restoring the parity. The origin of the extra quasiparticle lies in the topology of the 2D band insulator, and the existence of a second pair of edge states. In the geometry of Ref. [2], as ϕ is advanced by 2π, a unit of fermion parity is pumped across the sample, from one edge to the other. In the present case of Fig. 1, this so-called Z2 pumping [2,15] effectively transfers a quasiparticle between ABS at the lower and upper edges. As a consequence, even though the fermion parity at each edge changes when ϕ advances by 2π, the global fermion parity of the junction is conserved. We write the Josephson current J of the junction as Jσ;σ0 ½ϕ ¼ I up;σ ½ϕ þ I down;σ 0 ½ϕ;
(1)
where σ ¼ (respectively, σ 0 ¼ ) indicates the fermion number parity (þ being for even and − for odd) of the upper (respectively, lower) edge. The total fermion parity of the junction is Σ ¼ σσ 0 . In Eq. (1) and in the following, we use the letter J for the total current and the letter I for the edge currents. A striking feature of the setup is that the periodicity of the Josephson current depends on the parity of the junction. For an even fermion number (Σ ¼ 1), the current switches between J þ;þ and J−;− at ϕ ¼ π mod 2π, and is thus 4π periodic in ϕ. On the other hand, the odd current (Σ ¼ −1) is 2π periodic. Indeed, it switches at ϕ ¼ π mod 2π between Jþ;− and J−;þ , which are equal functions, provided the upper and lower edges are identical, that is, I up;σ ½ϕ ¼ I down;σ ½ϕ. In addition, the parity of the junction is reflected in the value of the critical current JΣ;c . Indeed, at a given edge, a change of parity reverses the direction of the current, that is, sgnI up;þ ½ϕ ¼ −sgnI up;− ½ϕ, and a similar relation for the
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lower edge. In the even sector, currents at both edges flow in the same direction, while in the odd sector they flow in opposite directions. As a consequence, the value of the critical current in the odd case J−;c is reduced with respect to that in the even case, Jþ;c . The exact ratio J −;c =Jþ;c between even and odd critical currents depends on the length L of the junction. In the short junction regime, L ≪ ξ with ξ ¼ vF =Δ0 the coherence length of the superconducting region, one has, at zero temperature, I up; ½ϕ ¼ I down; ½ϕ ¼ I c sinðϕ=2Þ [2], with I c ¼ Δ0 =2 the edge critical current. Therefore, the even critical current is just J þ;c ¼ 2I c , while the odd critical current vanishes, J−;c ¼ 0. Note that the odd current actually vanishes for all values of ϕ since I up;σ ½ϕ ¼ −I down;−σ ½ϕ. The long junction regime, L ≫ ξ, was studied in Ref. [18]. In that case one finds I up;þ ½ϕ ¼ I down;þ ½ϕ ¼ I 0c ϕ=ð2πÞ and I up;− ½ϕ¼I down;− ½ϕ¼I 0c ðϕ=ð2πÞ−sgnϕÞ, with I 0c ¼ evF =L the edge critical current. As a consequence, Jþ;c ¼ 2I 0c and J−;c ¼ I 0c . In summary, the ratio of critical currents in the odd and even cases interpolates between J−;c =J þ;c ¼ 0 for L ≪ ξ and J −;c =Jþ;c ¼ 1=2 for L ≫ ξ. Long junction regime and bosonization.—The long junction regime is interesting for two reasons. First, it should be of experimental relevance, as argued in Ref. [18]. Second, it allows us to make a clear mathematical connection between the spectrum of ABS and the fermion parity anomaly, through bosonization. At a given edge, helicity imposes perfect Andreev reflection and a nondegenerate spectrum of ABS. In the low-energy limit ε ≪ Δ0 , the ABS spectrum is given by [19] πvF 1 ϕ ε¼ with n ∈ Z; (2) nþ 2 2π L where vF is the Fermi velocity of the linearly dispersing edge states. In the case of a nonhelical, one-dimensional Josephson junction, this spectrum would be twice degenerate due to the spin 1=2 of the electrons. As was first recognized in Ref. [13], the spectrum of Eq. (2) is one of two branches of counterpropagating fermions with a linear spectrum and twisted boundary conditions on a segment of size 2L. Indeed, at low energy, one can write energyindependent boundary conditions for the fermion creation and annihilation operators. At the upper edge, ψ R;↑ ð0Þ ¼ iψ †L;↓ ð0Þ and ψ R;↑ ðLÞ ¼ −ieiϕ ψ †L;↓ ðLÞ which implies ψ α ðx þ 2L; tÞ ¼ −e−iϕ ψ α ðx; tÞ;
(3)
with α ¼ R; ↑ or L; ↓, as well as a relation between right and left movers, ψ R;↑ ðx; tÞ ¼ −iψ †L;↓ ð−x; tÞ;
(4)
for x ∈ ½−L; L. One can then write an effective Hamiltonian for the normal region only with the
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appropriate boundary conditions. Starting from the Hamiltonian of the normal region, at the upper edge, we obtain Z 1 L dxΨ† ðxÞHBdG ΨðxÞ; (5) H up ¼ 2 0 with HBdG ¼ −ivF ∂ x σ z τz the Bogoliubov–de Gennes Hamiltonian, and Ψ ¼ ðψ R;↑ ; ψ L;↓ ; ψ †L;↓ ; −ψ †R;↑ ÞT . σ z (respectively, τz ) is a Pauli matrix acting on the spin-1=2 (respectively, particle-hole) degree of freedom. At low energies, using Eq. (4), the Hamiltonian of Eq. (5) can be unfolded on the segment ½−L; L as Z L v dx½−ψ †R;↑ ∂ x ψ R;↑ þ ψ †L;↓ ∂ x ψ L;↓ ðxÞ; (6) Hup ¼ i F 2 −L with the twisted boundary conditions of Eq. (3). Details of the calculation are given in the Supplemental Material [20]. We then recover the spectrum of Eq. (2). The particle-hole symmetry can be made explicit upon Fourier transform and further use of Eq. (4). Taking the superconducting gap to infinity, Δ0 → ∞, we derive the following normal-ordered form of the Hamiltonian X vF π 1 ϕ Hup ¼ nþ − ∶c†R;n cR;n − cR;n c†R;n ∶; (7) 2L 2 2π n∈Z p1ffiffiffiffi 2L
P
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cR;n ; with the momentum where ψ R;↑ ðx; tÞ ¼ ne kn quantized as kn ¼ ðπ=LÞ½n þ 1=2 − ϕ=ð2πÞ. The spectrum of ABS reduces to that of a single band of fermionic quasiparticles with states labeled by an integer n. States with opposite energies differ in their fermion parity. Note that normal ordering subtracts an infinite ground-state energy coming from quasiparticle states below the Fermi energy, EF ¼ 0. This constant is independent of the phase and therefore does not affect the Josephson current. As the spectrum is unbounded, the Hamiltonian of Eq. (7) is invariant under the transformation ϕ → ϕ þ 2π and n − 1 → n. This transformation leaves the spectrum invariant but changes the parity of the ground state. In this form, the model satisfies all the criteria—a band of fermions with an unbounded, discrete spectrum—for rigorous bosonization [21]. The bosonization transformation reads as
of the Fermi sea. Nˆ R is an operator counting the number of fermions with respect to the Fermi sea. FR is the so-called Klein factor that lowers the fermion number by one. The factor a is a regularization factor. Upon normal ordering ~ e−iϕR ðxÞ , it can be safely taken to be zero. In practice, however, it can be retained and plays the role of a highenergy cutoff on the otherwise unbounded theory. Here, one should take a ≃ ξ ¼ vF =Δ0 . Using Eqs. (8) and (9), the Hamiltonian (7) takes the bosonized form [21] X vF π ˆ ϕ 2 Hup ¼ þ vF qb†R;q bR;q : (10) NR − 2L 2π q>0 Obviously, the transformation ϕ → ϕ þ 2π, changes the topological number N R by 1, a reflection of the fermion parity anomaly. As the Hamiltonian now naturally separates between a phase-dependent topological sector—the number of quasiparticles in the junction—and a nontopological sector—the phase-independent particle-hole excitations— the computation of the current at one edge, taking into account the parity constraints, is straightforward: I up; ½ϕ ¼ −
(8)
Here, ϕ~ R ðxÞ is a chiral bosonic field, periodic on ½−L; L, with the following Fourier decomposition: X rffiffiffiffiffiffi π iqx ðe bR;q þ e−iqx b†R;q Þe−aq=2 ; (9) ϕ~ R ðxÞ ¼ − Lq q>0 where q ¼ ðnπ=LÞ, n ∈ N . The bosonic operators b†R;q and bR;q create and annihilate particle-hole excitations on top
(11)
with Ztup; ½ϕ ¼
ikn x
1 1 ˆ ~ ψ R;↑ ðx; tÞ ≡ FR pffiffiffiffiffiffiffiffi eiðπ=LÞ½N R þ2−ðϕ=2πÞx e−iϕR ðxÞ : 2πa
2e 1 ∂ ln Ztup; ½ϕ; ℏ β ∂ϕ
X
2
e−βðℏπvF =2LÞðN R −ϕ=ð2πÞÞ ;
(12)
N R ∈Z even=odd
the topological part of the partition function. The parity constraint is simply reflected in the parity of the quasiparticle number N R. Interestingly, these partition functions can be expressed analytically in the form of Jacobi’s elliptic theta functions [22] , and we find for the currents, I up; ½ϕ ¼ e
vF ϕ 2e − ∂ ln θ3=2 ½zðϕÞ; q; L 2π ℏβ ϕ
(13)
for jϕj < π. We have defined zðϕÞ ¼ iβðℏπvF =2LÞðϕ=πÞ and q ¼ e−2βðℏπvF =LÞ . We have checked that our Eq. (13) and the formula of Beenakker et al. [18] for the long junction Josephson current, although seemingly different, coincide in the limit Δ0 → ∞ [20] . In particular, the odd current develops a discontinuity at ϕ ¼ 0, as β → ∞. Our bosonization approach is restricted to long junctions; nevertheless, it has the conceptual advantage of putting the emphasis on the fermion parity anomaly through a simple counting of quasiparticles, and leads to a very simple formula for the current. Now, we include the lower edge, which is most easily achieved by exchanging ↑ and ↓ indexes in Eqs. (5) and (6). The effective low-energy Hamiltonian at the lower edge then reads
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PHYSICAL REVIEW LETTERS 21 FEBRUARY 2014 PRL 112, 077002 (2014) X vF π the Supplemental Material to this Letter that, for realistic 1 ϕ nþ þ ∶c†L;n cL;n − cL;n c†L;n ∶; Hdown ¼ junctions, induced fluxes are negligible when compared to 2L 2 2π n∈Z the external flux [20]. (14) Another interesting question raised by this particular pffiffiffiffiffiffi P −ik x setup is its relation to the well-known case of a 1D spinful where ψ L;↑ ðx; tÞ ¼ ð1= 2LÞ n e n cL;n with the momJosephson junction [12–14,24]. One difficulty in that case entum kn quantized as kn ¼ ðπ=LÞ½n þ 1=2 þ ϕ=ð2πÞ. is to incorporate both Andreev reflection and backscatterIn bosonized form, ing at the edge. The bosonization treatment of Ref. [13] 2 was built on the assumption of perfect Andreev reflection, X v π ϕ Hdown ¼ F Nˆ L þ þ vF qb†L;q bL;q : (15) which needed to be corrected to account for normal 2L 2π q>0 reflection. The case of helical liquids provide a natural realization of the assumption of perfect Andreev reflection. Again, the transformation ϕ → ϕ þ 2π, changes the The present setup can be thought of as a way to split topological number N L by 1. The combination of both the ABS of the spinful liquid in two halves, spatially Hamiltonians, (10) and (15), illustrates the transfer of fermion localized in two different regions, the edges, separated by parity from one edge to the other, as the phase is advanced the bulk gap. As we have seen, although the spin degenby 2π. Note that one could also write Hdown in terms of eracy is recovered, the parity pumping across the toporight movers only. This freedom of choice comes from logical insulator entails very different predictions for the Eq. (4), and is due to particle-hole symmetry. It illustrates Josephson current, in particular, 4π periodicity in the even the important fact that only fermion parity is well defined sector. Note that, in a situation where the fermion parity at the edges. The partition function of the lower edge, would not be conserved—for instance if quasiparticle Ztdown; ½ϕ, is given by analogy with Eq. (12). One can poisoning is important [25,26]—one simply recovers the check, by making the change of variables N L → −N L , that Josephson current of the spinful liquid (a sawtooth with I down; ½ϕ ¼ I up; ½ϕ. The partition function of the junction period 2π), by letting the fermion number in Eq. (12) run is simply Ztσ;σ0 ¼ Ztup;σ × Ztdown;σ0 and the Josephson current over both even and odd integers. is indeed the one of Eq. (1). Discussion.—In summary, we have studied a topological Effects of interactions.—Coulomb interactions at the Josephson junction involving both edges of a 2D TI. edges can easily be included in the long junction regime We have put forward a possible nondestructive measureon the basis of the bosonization method. Assuming for now ment of the global parity, which includes the effect of that interedge interactions are made negligible by screenintraedge interactions. Importantly, the topological features ing, the Hamiltonian at either edge conserves its simple of the 2D quantum spin Hall phase are essential for the low-energy form with only renormalized coefficients: fermion parity pumping and thus, the parity meter proposed here. A straightforward extension to, for instance, surface X † vN;α π ˆ ϕ 2 states of three-dimensional topological insulators is not Hα ¼ þ vs;α qBα;q Bα;q ; (16) Nα − possible. As a closing remark, let us briefly elaborate on 2π 2L q>0 interedge interactions. So-called g1;∥ processes [27] do not change the local fermion number and can be easily with α ¼ up, down. We have allowed for the possibility of incorporated in the bosonization treatment, leaving our different interaction strengths at each edge. The new bosonic † † general conclusions unchanged. On the other hand, upon operators B and B are related to the original b and b by a inclusion of g1;⊥ processes, the Hamiltonian no longer Bogoliubov transformation [23]. Quite generally, for shortcommutes with the fermion number, and a simple expresrange interactions, we have vN;α ¼ vF − g2;α =ð2πÞ, where sion of the partition function as in Eq. (12) a priori no g2;α is the original interaction between right and left movers longer exists. Another interesting open question is the at each edge. The Josephson current is simply obtained by influence of disorder on our results [28], in which case the replacing vF by vN;α in Eq. (13). The effects of repulsive two Josephson currents could be separately modified. interactions is therefore to reduce the Josephson current, as was first discovered in the spinful case [13]. One should note We thank Fabrizio Dolcini and Carlo Beenakker for that our original prediction on the relation between parity illuminating discussions. Financial support by the DFG and periodicity will not hold in the case vN;up ≠ vN;down , (German-Japanese research unit “Topotronics” and the since it would imply I down; ½ϕ ≠ I up; ½ϕ. However, the priority program “Topological insulators”) as well as the ratio of critical currents in the odd and even cases is Helmholtz Foundation (VITI) is gratefully acknowledged. unaffected by interactions. Similar conclusions can be reached in the case of equal interactions but unequal lengths of the junctions. At this point, the reader might also wonder about self and mutual induction from Josephson currents that [1] A. Y. Kitaev, Phys. Usp. 44, 131 (2001). would ultimately couple the Hamiltonians (16). We show in [2] L. Fu and C. L. Kane, Phys. Rev. B 79, 161408 (2009). 077002-4
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[3] R. M. Lutchyn, J. D. Sau, and S. D. Sarma, Phys. Rev. Lett. 105, 077001 (2010). [4] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010). [5] L. P. Rokhinson, X. Liu, and J. K. Furdyna, Nat. Phys. 8, 795 (2012). [6] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, Rev. Mod. Phys. 80, 1083 (2008). [7] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Nat. Phys. 7, 412 (2011). [8] M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 (2007). [9] I. Knez, R.-R. Du, and G. Sullivan, Phys. Rev. Lett. 107, 136603 (2011). [10] K. Suzuki, Y. Harada, K. Onomitsu, and K. Muraki, Phys. Rev. B 87, 235311 (2013). [11] I. Knez, R.-R. Du, and G. Sullivan, Phys. Rev. Lett. 109, 186603 (2012). [12] R. Fazio, F. W. J. Hekking, and A. A. Odintsov, Phys. Rev. Lett. 74, 1843 (1995). [13] D. L. Maslov, M. Stone, P. M. Goldbart, and D. Loss, Phys. Rev. B 53, 1548 (1996). [14] I. Affleck, J.-S. Caux, and A. M. Zagoskin, Phys. Rev. B 62, 1433 (2000). [15] L. Fu and C. L. Kane, Phys. Rev. B 74, 195312 (2006). [16] A. Keselman, L. Fu, A. Stern, and E. Berg, Phys. Rev. Lett. 111, 116402 (2013).
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[17] We remark that a related setup was briefly discussed in Ref. [29], in the context of anyons, as a way to realize a so-called braidless operation of the type j00i → j11i. [18] C. W. J. Beenakker, D. I. Pikulin, T. Hyart, H. Schomerus, and J. P. Dahlhaus, Phys. Rev. Lett. 110, 017003 (2013). [19] I. Kulik, Sov. Phys. JETP 30, 944 (1970). [20] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.077002 for where we outline the steps of the bosonization calculation, discuss our Eq. (13) for the current in light of Ref. [18], and compute inductive effects in the superconducting loop. [21] J. von Delft and H. Schoeller, Ann. Phys. (Berlin) 7, 225 (1998). [22] NIST Digital Library of Mathematical Functions, http:// dlmf.nist.gov/20.2. [23] M. Fabrizio and A. O. Gogolin, Phys. Rev. B 51, 17 827 (1995). [24] R. Fazio, F. W. J. Hekking, and A. A. Odintsov, Phys. Rev. B 53, 6653 (1996). [25] J. C. Budich, S. Walter, and B. Trauzettel, Phys. Rev. B 85, 121405 (2012). [26] D. Rainis and D. Loss, Phys. Rev. B 85, 174533 (2012). [27] T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, Oxford, 2004). [28] C. W. J. Beenakker, J. M. Edge, J. P. Dahlhaus, D. I. Pikulin, S. Mi, and M. Wimmer, Phys. Rev. Lett. 111, 037001 (2013). [29] J. C. Y. Teo and C. L. Kane, Phys. Rev. Lett. 104, 046401 (2010).
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