SLAC-PUB-13926 Topological Superconductivity and Superfluidity
Xiao-Liang Qi, Taylor L. Hughes, Srinivas Raghu and Shou-Cheng Zhang Department of Physics, McCullough Building, Stanford University, Stanford, CA 94305-4045
We construct time reversal invariant topological superconductors and superfluids in two and three dimensions which are analogous to the recently discovered quantum spin Hall and three-d Z2 topological insulators respectively. These states have a full pairing gap in the bulk, gapless counter- propagating Majorana states at the boundary, and a pair of Majorana zero modes associated with each vortex. We show that the time reversal symmetry naturally emerges as a supersymmetry, which changes the parity of the fermion number associated with each time-reversal invariant vortex. In the presence of external T-breaking fields, non-local topological correlation is established among these fields, which is an experimentally observable manifestation of the emergent supersymmetry.
PACS numbers: 74.20.Rp, 73.43.-f, 67.30.he, 74.45.+c
The search for topological states of quantum mat- case of the QSH state, a mass term for an odd number ter has become an active and exciting pursuit in con- of pairs of helical Majorana states is forbidden by TR densed matter physics. The quantum Hall (QH) effect [1] symmetry, and therefore, a topologically protected super- provides the first example of a topologically non-trivial conducting or superfluid state can exist in the presence state of matter, where the quantized Hall conductance of time-reversal symmetry. Recently, a Z2 classification is a topological invariant[2]. Recently, the quantum spin of the topological superconductor has been discussed in Hall (QSH) state[3, 4] has been theoretically predicted Refs [12, 13, 14], by noting the similarity between the [5] and experimentally observed in HgTe quantum well Bogoliubov-de Gennes (BdG) superconductor Hamilto- systems[6]. The time reversal invariant (TRI) QSH state nian and the QSH insulator Hamiltonian. The four types is characterized by a bulk gap, a Z2 topological number of topological states of matter discussed here are sum- [7], and gapless helical edge states, where time-reversed marized in Fig. 1. In this work, we give a Z2 classifica- partners counter-propagate[8, 9]. tion of both the 2D and 3D cases which has a profound Chiral superconductors in a time reversal symmetry physical implication. In two dimensions, we show that breaking (TRB) (px+ipy) pairing state in 2d have a sharp a time-reversal invariant topological defect of a Z2 non- topological distinction between the strong and weak pair- trivial superconductor carries a Kramers’ pair of Majo- ing regimes [10]. In the weak pairing regime, the system rana fermions. Let NF be the operator which measures has a full bulk gap and gapless chiral Majorana states at the number of fermions of a general system, then the the edge, which are topologically protected. Moreover, a fermion-number parity operator is given by (−1)NF . This Majorana zero mode is trapped in each vortex core[10], operator is also referred to as the Witten index[15], which which leads to a ground state degeneracy of 2n−1 in the plays a crucial role in supersymmetric theories. We prove presence of 2n vortices. When the vortices wind around the remarkable fact that in the presence of a topological each other a non-Abelian Berry phase is generated in the defect, the TR operator T changes the fermion number 2n−1 dimensional ground state manifold, which implies parity, T −1(−1)NF T = −(−1)NF locally around the de- non-Abelian statistics for the vortices[11]. Chiral super- fect in the Z2 non-trivial state, while it preserves the conductors are analogous to the QH state— they both fermion number parity, T −1(−1)NF T = (−1)NF , in the break time reversal (TR) and have chiral edge states with Z2 trivial state. This fact gives a precise definition of the linear dispersion. However, the edge states of a chiral su- Z2 topological classification of any TRI superconductor perconductor have only half the degrees of freedom com- state and is generally valid in the presence of interactions pared to the QH state, since the negative energy quasi- and disorder. A supersymmetric operation can be de- particle operators on the edge of a chiral superconductor fined as an operation which changes the fermion number describe the same excitations as the positive energy ones. parity; therefore, in this precise sense, we show that the Given the analogy between the chiral superconducting TR symmetry emerges as a supersymmetry in topolog- state and the QH state, and with the recent discovery of ical superconductors. Though supersymmetry has been the TRI QSH state, it is natural to generalize the chiral studied extensively in high energy physics, it has not yet pairing state to the helical pairing state, where fermions been observed in Nature. Our proposal offers the op- with up spins are paired in the (px + ipy) state, while portunity to experimentally observe supersymmetry in fermions with down spins are paired in the (px − ipy) condensed matter systems without any fine tuning of mi- state. Such a TRI state have a full gap in the bulk, croscopic parameters. The physical consequences of such and counter-propagating helical Majorana states at the a supersymmetry is also studied. edge (in contrast, the edge states of the TRI topologi- As the starting point, we consider a TRI p-wave su- cal insulator are helical Dirac fermions). Just as in the perconductor with spin triplet pairing, which has the fol-
SIMES, SLAC National Accelerator Center, 2575 Sand Hill Road, Menlo Park, CA 94309 Work supported in part by US Department of Energy contract DE-AC02-76SF00515. 2
E E opposite chirality: k k Chiral SC QH E E Hedge = vF ky ψ−ky ↑ψky ↑ − ψ−ky ↓ψky ↓ . (3) k k k ≥0 Xy E E The quasi-particle operators ψky ↑, ψky ↓ can be expressed k k in terms of the eigenstates of the BdG Hamiltonian as Helical SC QSH E E
k k 2 † ψky ↑ = d x uky (x)c↑(x)+ vky (x)c↑(x) Z 2 ∗ ∗ † ψky ↓ = d x u−k (x)c↓(x)+ v−k (x)c (x) (4) d y y ↓ FIG. 1: (Top row) Schematic comparison of 2 chiral super- Z conductor and the QH state. In both systems, TR symmetry from which the time-reversal transformation of the quasi- is broken and the edge states carry a definite chirality. (Bot- −1 tom row) Schematic comparison of 2d TRI topological super- particle operators can be determined to be T ψky ↑T = −1 conductor and the QSH insulator. Both systems preserve TR ψ−ky ↓, T ψky ↓T = −ψ−ky ↑. In other words, symmetry and have a helical pair of edge states, where oppo- (ψky ↑, ψ−ky ↓) transforms as a Kramers’ doublet, which site spin states counter-propagate. The dashed lines show forbids a gap in the edge states due to mixing of the that the edge states of the superconductors are Majorana spin-up and spin-down modes when TR is preserved. To E < fermions so that the 0 part of the quasi-particle spectra see this explicitly, notice that the only k -independent are redundant. In terms of the edge state degrees of freedom, y 2 2 4 we have (QSH) = (QH) = (Helical SC) = (Chiral SC) . term that can be added to the edge Hamiltonian (3) is ψ ψ with m ∈ R. However, such a term is im ky −ky ↑ ky ↓ odd under TR, which implies that any back scattering be- P lowing 4 × 4 BdG Hamiltonian: tween the quasi-particles is forbidden by TR symmetry. The discussion above is exactly parallel to the Z2 topo- I αj logical characterization of the quantum spin Hall system. 1 2 † ǫp iσ2σα∆ pj H = d xΨ (x) Ψ(x) (1) In fact, the Hamiltonian (2) has exactly the same form as 2 h.c. −ǫpI Z the four band effective Hamiltonian proposed in Ref.[5] to T describe HgTe quantum wells with the QSH effect. The † † 2 with Ψ(x)= c↑(x),c↓(x),c↑(x),c↓(x) , ǫp = p /2m − edge states of the QSH insulators consist of an odd num- µ the kinetic energy and chemical potential terms and ber of Kramers’ pairs, which remain gapless under any αj † h.c. ≡ (iσ2σα∆ pj ) . The TR transformation is de- small TR-invariant perturbation[8, 9]. A no-go theorem fined as c↑ → c↓, c↓ → −c↑. It can be shown that the states that such a “helical liquid” with an odd number Hamiltonian (1) is time-reversal invariant if ∆αj is a real of Kramers’ pairs at the Fermi energy can not be real- matrix. To show the existence of a topological state, con- ized in any bulk 1d system, but can only appear as an sider the TRI mean-field ansatz ∆α1 = ∆(1, 0, 0), ∆α2 = edge theory of a 2d QSH insulator[9]. Similarly, the edge ∆(0, 1, 0). For such an ansatz the Hamiltonian (1) is state theory (3) can be called a “helical Majorana liquid”, block diagonal with only equal spin pairing: which can only exist on the boundary of a Z2 topolog- ical superconductor. Once such a topological phase is ǫp ∆p+ established, it is robust under any TRI perturbations. 1 2 ˜ † ∆p− −ǫp ˜ The Hamiltonian (1) can be easily generalized to three H = d xΨ Ψ (2) αj 2 ǫp −∆p− dimensions, in which case ∆ becomes a 3 × 3 matrix Z −∆p −ǫp with α = 1, 2, 3 and j = x,y,z. An example of such a + Hamiltonian is given by the well-known 3He BW phase, T for which the order parameter ∆αj is determined by an with Ψ(˜ x) ≡ c (x),c† (x),c (x),c† (x) , and p = ↑ ↑ ↓ ↓ ± orthogonal matrix ∆αj = ∆uαj , u ∈ SO(3)[16]. Here and px ± ipy. From this Hamiltonian we see that the spin below we ignore the dipole-dipole interaction term [17] up (down) electrons form px + ipy (px − ipy) Cooper since it does not affect any essential topological proper- pairs, respectively. In the weak pairing phase with ties. By applying a spin rotation, ∆αj can be diagonal- µ > 0, the (px + ipy) chiral superconductor is known ized to ∆αj = ∆δαj , in which case the Hamiltonian (1) to have chiral Majorana edge states propagating on has the same form as a 3d Dirac Hamiltonian with mo- each boundary, described by the Hamiltonian Hedge = mentum dependent mass ǫ(p) = p2/2m − µ. We know v k ψ ψ ,where ψ = ψ† is the quasipar- ky ≥0 F y −ky ky −ky ky that a band insulator described by the Dirac Hamilto- ticle creation operator [10] and the boundary is taken to nian is a 3d Z topological insulator for µ> 0[18, 19, 20], P 2 be parallel to the y direction. Thus we know that the and has nontrivial surface states. The corresponding su- edge states of the TRI system described by Hamiltonian perconductor Hamiltonian describes a topological super- (2) consist of spin up and spin down quasi-particles with conductor with 2d gapless Majorana surface states. The 3 surface theory can be written as is the annihilation operator of a zero-energy quasipar- ticle. Consequently, the ground state of the system is 1 T at least two-fold degenerate, with two states |G i and Hsurf = vF ψ−k (σzkx + σxky) ψk (5) 0 2 † k |G1i = a |G0i containing 0 and 1 a-fermions. Since X † a a = (1+ iγ↑γ↓) /2, the states G0(1) are eigenstates of which remains gapless under any small TRI perturba- iγ↑γ↓ with eigenvalues −1(+1), respectively. Thus from T tion since the only available mass term m k ψ−kσyψk the oddness of iγ↑γ↓ under TR we know that |G0i and is time-reversal odd. We would like to mention that the |G1i are time-reversed partners. Note that superconduc- 3 P surface Andreev bound states in He-B phase have been tivity breaks the charge U(1) symmetry to Z2, meaning observed experimentally[21]. that the fermion number parity operator (−1)NF is con- To understand the physical consequences of the non- served. Thus, all the eigenstates of the Hamiltonian can NF trivial topology we study the TRI topological defects of be classified by the value of (−1) . If, say, |G0i is a NF † the topological superconductors. We start by consid- state with (−1) = 1, then |G1i = a |G0i must sat- NF ering the equal-spin pairing system with BdG Hamil- isfy (−1) = −1. Since |G0i and |G1i are time-reversal tonian (2) in which spin up and down electrons form partners, we know that in the Hilbert space of the zero- px + ipy and px − ipy Cooper pairs, respectively. A energy states the TR transformation changes the fermion TRI topological defect can be defined as a vortex of number parity: spin-up superfluid coexisting with an anti-vortex of spin- down superfluid at the same position. In the generic T −1(−1)NF T = −(−1)NF . (8) Hamiltonian (1), such a vortex configuration is written αj αj 3j as ∆ = [exp (iσ2θ(r − r0))] , α = 1, 2 and ∆ = 0, Eq. (8) is the central result of this paper. At a first where θ(r − r0) is the angle of r with respect to the vor- glance it seems contradict the fundamental fact that the tex position r0. Since in the vortex core of a weak pair- electron number of the whole system is invariant under ing px + ipy superconductor there is a single Majorana TR. Such a paradox is resolved by noticing that there are zero mode[10, 22], one immediately knows that a pair a always an even number of topological defects in a closed Majorana zero modes exist in the vortex core we study system without boundary. Under the TR transforma- here. In terms of the electron operators, the two Majo- tion, the fermion number parity around each vortex core rana fermion operators can be written as is odd, but the total fermion number parity remains even as expected. Once the anomalous transformation prop- † erty (8) is established for a topological defect in a TRI γ = d2x u (x)c (x)+ u∗(x)c (x) ↑ 0 ↑ 0 ↑ superconductor, it is robust under any TRI perturbation Z 2 ∗ † as long as the bulk quasiparticle gap remains finite and γ↓ = d x u0(x)c↑(x)+ u0(x)c↑(x) (6) other topological defects are far away. Thus Eq. (8) is a Z generic definition of TRI topological superconductors: where we have used the fact that the spin-down zero mode wave function can be obtained from the time- • Definition I. A two-dimensional TRI supercon- reversal transformation of the spin-up one. The Ma- ductor is Z2 nontrivial if and only if fermion num- jorana operators satisfy the anti-commutation relation ber parity around a TRI topological defect is odd {γα,γβ} = 2δαβ. The TR transformation of the Majo- under TR. rana fermions is A transformation changing fermion number by an odd −1 −1 T γ↑T = γ↓, T γ↓T = −γ↑. (7) number is a “supersymmetry”; thus, the TR symmetry emerges as a discrete supersymmetry for each TRI topo- Similar to the case of the edge states studied earlier, the logical defect. The same analysis applies to the edge Majorana zero modes are robust under any small TRI theory (3), which shows that in the 1d helical Majorana perturbation, since the only possible term imγ↑γ↓ which liquid is a theory with TR symmetry as a discrete super- can lift the zero modes to finite energy is TR odd, i.e., symmetry. −1 T iγ↑γ↓T = −iγ↑γ↓. All the conclusions above can be generalized to 3d The properties of such a topological defect appear iden- topological superconductors. In the 3He BW phase the tical to that of a π-flux tube threading into a TRI topo- Goldstone manifold of the order parameter is ∆αj = logical insulator[23, 24], where a Kramers’ pair of com- ∆uαj ∈ SO(3) × U(1)[16, 25]. A time-reversal invari- plex fermions are trapped by the flux tube. However, ant configuration satisfies ∆αj ∈ R, which restricts the there is an essential difference. From the two Majorana order parameter to SO(3). Since Π1(SO(3)) = Z2, the zero modes γ↑,γ↓ a complex fermion operator can be de- TRI topological defects are 1d “vortex” rings. By solv- fined as a = (γ↑ + iγ↓) /2, which satisfies the fermion ing the BdG equations in the presence of such vortex † anticommutation relation a,a = 1. Since γ↑,γ↓ are rings, it can be shown that there are linearly dispersing zero modes, we obtain [a, H] = 0 which implies that a quasiparticles propagating on each vortex ring, similar 4 to the edge states of the 2d topological superconductor. However, for a ring with finite length the quasi-particle spectrum is discrete. Specifically, there may or may not be a pair of Majorana modes at exactly zero energy. The existence of the Majorana zero modes on the vortex rings turns out to be a topological property determined by the linking number between different vortex rings. Due to the length constraints of the present paper, we will write our conclusion and leave the details for a separate work: There are a pair of Majorana fermion zero modes con- fined on a vortex ring if and only if the ring is linked to an odd number of other vortex rings. Such a condition FIG. 3: (a) Illustration of a 2d TRI topological supercon- is shown in Fig. 2. Consequently, the generalization of ductor with four TRI topological defects coupled to a TR- breaking field. The arrows show the sign of the TR-breaking Definition I to 3d is: field sgn(M(rs)) at each topological defect. In the two con- Definition II. figurations shown, only the field around vortex 1 is flipped, • A 3d TRI superconductor is Z2 leading to an opposite fermion number parity in the corre- nontrivial if and only if the fermion number parity sponding ground state |Gi and |G′i (see text). (b) Illustra- around one of the two mutually-linked TRI vortex tion showing the flow of the energy levels when the upper rings is odd under TR. configuration in figure (a) is deformed to the lower one. The flip of the TR-breaking field M(r1) leads to a level crossing at M(r1) = 0, where the fermion number parity in the ground (a) (b) state changes sign.
in each vortex core. Consequently, the perturbed Hamil- E E tonian H[M(r)] to first order can be written as
N
H[M(r)] = iM(rs)asγs↑γs↓ (10) k|| k|| s=1 X in which γs↑(↓) are the Majorana fermion operators, and as ∈ R depend on the details of the perturbation. The important fact is that the mass term induced is linear in FIG. 2: Illustration of a 3d TRI topological superconductor r r with two TRI vortex rings which are (a) linked or (b) un- M( ) at the defect position s, since iγs↑γs↓ is TR odd. Since the superconductor has a full gap, naively one linked. The E − kk dispersion relations show schematically the quasiparticle levels confined on the red vortex ring in both would expect all the correlations of M(r) field to be short cases. “◦” and “×” stand for the quasiparticle levels that are ranged. However, for a system with N topological defects Kramers’ partners of each other. Only case (a) has a pair of the N-point correlation function has a long range order Majorana zero modes located on each vortex ring. when rs,s =1, 2, ..N are chosen to be the coordinates of the topological defects. In other words, the correlation Besides providing a generic definition of the Z2 topo- function logical superconductors, such an emergent supersymme- N try also leads to physical predictions. Consider the lim M(rs) 6=0, (11) 2d topological superconductor coupled to a weak TR- |r −r |→∞, ∀i,j i j *s=1 + breaking field M(r), which is classical but can have ther- Y mal fluctuations. This situation can be realized in an though all the n point correlations in Eq. (9) with n