Symmetry-protected topological invariant and Majorana impurity states in time-reversal-invariant superconductors
We address the question of whether individual nonmagnetic impurities can induce zero-energy states in time-reversal-invariant topological superconductors, and define a class of symmetries which guarantee the existence of such states for a specific value of the impurity strength. These symmetries allow the definition of a position-space topological invariant, which is related to the standard bulk topological invariant. Our general results are applied to the time-reversal-invariant -wave phase of the doped Kitaev-Heisenberg model, where we demonstrate how a lattice of impurities can drive a topologically trivial system into the nontrivial phase.
pacs:74.62.En, 74.25.Dw, 74.78.Na
Local impurities in superconductors (SCs) give rise to astonishing physics. Abrikosov1960 (); Balatsky2006 (); Alloul2009 (); Hudson2001 (); Sau2012 (); Hu2013 (); Maki1999 (); Khaliullin1997 () Magnetic impurities in -wave SCs lead to pair breaking, and can induce a quantum phase transition to a metallic state with gapless superconductivity near the transition point.Abrikosov1960 () Due to Anderson’s theorem, nonmagnetic impurities have little influence on -wave SCs.Anderson1959 () However, in unconventional SCs, where the sign of the order parameter depends on the direction of momentum, scattering by impurities leads to pair-breaking since the momentum direction of the paired electrons is changed without changing the phase. Balatsky2006 (); Alloul2009 () Thus, impurities give rise to subgap states and can be used to probe high- superconductivity.Balatsky2006 (); Alloul2009 (); Hudson2001 ()
Here, we focus on impurity bound states in time-reversal (TR) invariant odd-parity SCs. These SCs belong to symmetry class DIII of the Altland-Zirnbauer classification Altland1997 () and come in two variants, characterized by a topological invariant . Schnyder2008 (); Kitaev2009 (); Qi2009 (); Hasan2010 (); Qi2011 (); Sato2009 (); Fu2010 (); footnote:invariant3D () The topologically nontrivial SC has protected Majorana boundary modes. It turns out that also predicts the pattern of ground-state degeneracies on a torus, when switching between periodic and antiperiodic boundary conditions. Read2000 () Denoting a pair of states related by time reversal as a Kramers pair, ground states are different depending on whether the number of unpaired Kramers pairs below the Fermi level is even or odd, designated in the following as even or odd “Kramers parity.” Single-band odd-parity SCs have ,Sato2009 () hence their order parameter vanishes at all TR-invariant momenta (TRIM) with up to reciprocal lattice vectors, such that for each TRIM below the Fermi level there is one unpaired Kramers pair. The Kramers parity is thus determined by the number of TRIM enclosed by the Fermi surface, and odd-parity SCs where this number is odd are topologically nontrivial. Sato2009 (); Fu2010 ()
Zero-energy bound states in SCs are intriguing Majorana states. Read2000 (); Kitaev2001 (); Ivanov2001 (); Wilczek2009 () Thus, it may be interesting to artificially create them by tuning an impurity potential, but it is also important to understand how to avoid accidental zero-energy states from nonmagnetic disorder, which may interfere with protocols using protected Majorana zero-energy states,Fulga2013 () occurring for instance in the center of a vortex.Kopnin1991 (); Volovik1999 () In this paper, we derive conditions for the existence of zero-energy impurity states in TR-invariant SCs. To this end, we deduce conditions for the existence of a position-space topological invariant , which for gapped translationally invariant systems is equivalent to and the Kramers parity. We show that upon introduction of a local impurity potential into the system, the conditions for the existence of also guarantee the emergence of zero-energy impurity bound states for a suitably chosen impurity strength. In particular, we find that the existence of symmetries protects zero-energy impurity bound states, such that disorder may introduce states with energies less than the thermal energy even at low temperatures. When an impurity bound state moves through the Fermi level, it changes the Kramers parity and but not , since it is spatially localized and insensitive to boundary conditions. However, a lattice of impurities hosts extended states, and we show that partially moving such an impurity band through zero energy can, for a broad range of potential strengths, turn a topologically trivial SC into a nontrivial one.
Model: We consider a general TR-invariant Bogoliubov–de Gennes Hamiltonian in symmetry class DIII Altland1997 () for an -site lattice in the position-space basis
where , , and annihilates a fermion with spin on site . Hermiticity of the Hamiltonian and Fermi statistics requires , . Hamiltonians in DIII obey both the particle-hole (PH) symmetry , and TR symmetry , . Here and denote the Pauli matrices in PH and spin space, respectively, and is the operator of complex conjugation. Together, these symmetries give rise to the chiral symmetry , .Schnyder2008 () Hence, every eigenvector with energy has a Kramers partner with energy , a PH partner , and a “chiral” partner both with energy .
We describe a local nonmagnetic impurity at site by the Hamiltonian
Results: To get insight into the existence of zero-energy impurity states, we note that in the absence of superconductivity has a zero-energy eigenvalue for a critical impurity strength .footnote:classAII () Without accidental degeneracies, the zero-energy eigenspace is spanned by the mutually orthogonal states and . We now ask whether these states are split by a superconducting coupling in first-order-degenerate perturbation theory, and argue that such a splitting is evidence for an avoided crossing, and thus the absence of a zero-energy state in the full problem. Due to TR and PH symmetry, cannot couple to or , but the coupling to is finite in general and leads to an energy splitting. SupplementalMaterial () However, in the presence of a unitary symmetry , which commutes with and and anticommutes with , the coupling between and vanishes: since with , we find that , and from it follows that . Consequently, vanishes, and there is no energy splitting. This fundamental impact of such a symmetry on the energy of the impurity bound state is illustrated in Fig. 1. There we depict obtained from -matrixBalatsky2006 () calculations for two models: first for the doped Kitaev-Heisenberg (KH) model,Kitaev2006 (); Giniyat0910 () which, as we will demonstrate, has additional symmetries protecting the zero-energy crossings, and second for the case where we added to this model Rashba spin-orbit coupling and modified the order parameter in order to break all these symmetries.
In order to understand the existence of zero-energy states in the full problem, we note that the determinant can be expressed as a product of the eigenvalues of . Thus, if the system without impurity is gapped, a zero of for a critical impurity strength indicates the existence of a zero-energy impurity bound state. As is local in , and since there is a spin and particle-hole degree of freedom at each lattice site, one finds that is a fourth-order polynomial in . For a general Hamiltonian in class DIII, it is difficult to determine under which conditions this polynomial has zeros for a real-valued impurity strength . In the following, we reduce the problem to the analysis of a first-order polynomial by considering the Pfaffian of redundant subblocks of . This will allow us to show non-perturbatively that the presence of a symmetry with and indeed ensures the existence of a zero-energy impurity bound state.
We first use the transformation , which diagonalizes , to bring into a block off-diagonal form
with . Because is antisymmetric, exists and , such that zero-energy eigenvalues of occur whenever . Since appears only in one entry in the upper and lower triangle of the matrix , respectively, is a linear complex function with . If is real, the complex phase of does not depend on and the system is bound to have a single zero-energy crossing of Kramers pairs at . We stress that in general there is no reason for to be real, such that no value of the real control parameter would yield zero-energy states. In the following, we will show that is indeed real provided that a symmetry of the Hamiltonian exists which anticommutes with the chiral operator .
Every possible unitary transformation satisfying has the property SupplementalMaterial ()
with unitary due to the unitarity of and . Provided that is a symmetry of with it follows that
Here, we first used the general properties and of the Pfaffian to write . By utilizing , which is equivalent to the symmetry condition , and the unitarity of , we then arrive at Eq. (5). This equation implies that is a real-valued function, and therefore is real. This demonstrates that in the presence of a symmetry the existence of the zero-energy states is guaranteed for a suitably chosen impurity strength .
To get some intuition about possible symmetries, we first specialize to a situation where can be decomposed into a product of an internal transformation and a lattice transformation , which satisfies as it is a permutation of lattice sites. Then, the condition implies that not all 16 combinations can be used to construct symmetries , but only the eight combinations listed in Table 1. Next, we expand into a spin-independent single-particle part and spin-orbit couplings , , , and decompose into a singlet component and triplet components , , . Then, for every allowed choice of , a subset of the , anticommutes with , and the remaining , commute with ; see Table 1. In the particularly simple case where does not contain a lattice transformation, i.e., , the anticommutation condition implies that the respective , vanish identically, whereas the commutation relation is trivially satisfied.
Now we are in a position to treat the special case of impurity bound states in spin-polarized SCs (belonging to symmetry class D Altland1997 ()) as a first application of our formalism. The specific choice implies that the matrices , , , , which couple up and down spins, have to vanish; see first row in Table 1. Then, the Hamiltonian matrix decomposes into two uncoupled blocks , related by TR symmetry . Each of the blocks is not TR symmetric but still obeys PH symmetry and thus can be an arbitrary member of symmetry class D. From our analysis it follows that hosts a zero-energy impurity bound state for a suitably chosen impurity strength while provides its Kramers partner. This generalizes the result for -wave SCs obtained in Ref. Sau2012, to arbitrary spin-polarized SCs in all spatial dimensions. The symmetries in rows two and three of Table 1 imply a decomposition into two class D blocks as well, with spins polarized in the and directions, respectively.
The symmetry in the fourth row of Table 1 requires the absence of superconductivity. Hence, the coupling between the particle and the hole-sector vanishes, and the Hamiltonian decomposes into two spin-1/2 TR-invariant systems belonging to symmetry class AII. Altland1997 () Thus, we have shown that every gapped system in AII hosts zero-energy impurity bound states for a suitably chosen impurity strength. The last four rows of Table 1 are formally obtained by multiplying the first four rows with the chiral operator . In the context of electronic SCs, there is no obvious example for their use.
More generally, , and the symmetry realizes a combination of a lattice transformation and a rotation in spin and particle-hole space which is required to keep a spin-orbit coupling of angular momentum and spin invariant. An important example are spatial reflections about a mirror plane, accompanied by the appropriate spin rotation.Ueno2013 (); Zhang2013 (); Chiu2013 (); Marimoto2013 () We discuss specific examples for such symmetries in the context of the doped KH model.
The presence of a symmetry is sufficient but not necessary for the existence of zero-energy impurity states. There are conditions not related to symmetries for which has a real zero for some impurity potential.SupplementalMaterial () However, while such conditions can be satisfied in single-particle Hamiltonians, they are expected to be less robust than symmetry conditions when the single-particle Hamiltonian is obtained from a self-consistent mean field approximation to an interacting Hamiltonian which already includes the impurity potential.
Exploiting the constant phase of in the presence of a symmetry , we define a topological invariant , which changes whenever one Kramers pair crosses the Fermi energy. To establish a connection between and the widely used bulk topological invariant for translationally invariant odd-parity single-band SCs, we define for each momentum in analogy to Eq. (3). For a TRIM , and , where is the single-particle energy with respect to the Fermi energy. Hence, is antisymmetric and in agreement with Sato: Sato2009 ()
where , so that counts the number parity of TRIM below the Fermi level and thus the Kramers parity. Consequently, for these systems.footnote:gaplessTopSC () It is straightforward to generalize our definitions to multiband SCs as well. We will make use of this generalization to demonstrate that a lattice of impurity states can drive a SC into a topologically nontrivial phase.
Impurities in the doped KH model: We illustrate our general results by applying them to the TR-invariant -wave phase of the doped KH model on the honeycomb lattice, Kitaev2006 (); Giniyat0910 (); Hyart2012 (); Okamoto2012 (); Scherer2014 () which is paradigmatic for a number of interesting topological phases.Hyart2014 () This phase is a two-dimensional analog of the phase of superfluid He and undergoes a topological phase transition at a critical value of the chemical potential.Hyart2012 () Consider, therefore, the mean-field Hamiltonian
where annihilates a fermion with spin on sublattice , is the chemical potential, is the nearest-neighbor hopping, and are the components of the vector describing spin-triplet pairing; for small , . Here, characterizes the superconducting gap and are the nearest-neighbor vectors.
In Eq. (7) we chose the spin quantization axis such that only equal-spin particles are paired; hence , which is a nonspatial symmetry protecting zero-energy states; cf. Table 1. From the interacting HamiltonianGiniyat0910 () the -wave phase inherits symmetries acting on spin and spatial degrees of freedom. You2012 () Of these symmetries only the three mirror symmetries with respect to the , , or links satisfy Eq. (4), for example , where is the matrix for the mirror permutation of the lattice sites with respect to a link. Hence, also the protect the zero-energy crossings shown in Fig. 1. It is instructive to add Rashba spin-orbit coupling , with , to the Hamiltonian while disregarding the effects that this coupling would have if the order parameter was calculated self-consistently. For this breaks the non-spatial symmetry , but keeps all spatial symmetries intact if , . Anisotropic Rashba coupling with breaks all mirror symmetries except for . By choosing different values for all three one breaks all relevant symmetries and thus avoids the impurity-induced zero-energy crossing. This is illustrated in Fig. 1.
In order to demonstrate that extended impurity states not only change but also , we consider a triangular lattice of impurities with lattice constant , amounting to an impurity density of [see Fig. 2 (b)]. We calculate by evaluating at the four TRIM as well as the Chern number of each spin-resolved impurity band formed by overlapping impurity subgap states, and confirm that . Due to threefold rotational symmetry of You2012 () , where denotes the points and denotes the origin of the Brillouin zone. as well as are the sign of linear functions in , respectively, and thus change independently of each other at critical values and , respectively. Hence, one can change by tuning . In Fig. 2 (a) we show the phase diagram of versus impurity strength and chemical potential . The clean system is in the topologically trivial [nontrivial] phase for . At each value of two transitions occur at and , respectively, and the complicated dependence of and on gives rise to an intricate diagram. Remarkably, it is possible to render the system nontrivial by tuning to values of the order of the hopping .
Conclusion: We described symmetries which guarantee the existence of zero-energy impurity bound states in TR-invariant SCs for a critical value of the impurity strength. The same symmetries allow the definition of the position-space topological invariant which we related to the bulk invariant . The relevance of our findings was demonstrated for the TR-invariant -wave phase of the doped KH model, where symmetries protect the zero-energy crossings and a lattice of impurities can change the bulk topological order of the system. Finally, we have shown that TR invariant topologically non-trivial SCs can be made robust against low-energy impurity states by strongly breaking all additional symmetries. This improves prospects for protocols utilizing topologically protected Majorana zero-energy states.
Acknowledgments: We acknowledge valuable discussions with E. Demler, J. Moore, and B. Zocher, and would like to thank G. Khaliullin for collaboration in an early stage of this work. L.K. acknowledges financial support by ESF and T.H. by the Dutch Science Foundation NWO/FOM. This work was supported in part by the National Science Foundation under Grant No. PHYS-1066293 and the hospitality of the Aspen Center for Physics.
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.1 Accidental zero-energy crossings in class DIII
In this section we show that besides the symmetries which make the phase of independent of the impurity strenght, there are other sufficient conditions which yield the same result, although they are not related to symmetries. We believe that these conditions are less relevant, because they will probably not continue to hold when the order parameter is calculated self-consistently. Moreover, we utilize again the TR invariant -wave phase of the doped KH model to demonstrate that the symmetry unrelated conditions do matter when one investigates theoretically whether certain mean field Hamiltonians can have impurity induced zero-energy states.
Observe first that
for arbitrary and complex matrices with , which follows directly from the definition of the Pfaffian
where defined above is only a function of the product but not of or indepenently. Moreover, in the presence of TR symmetry one has and , so that one can write
where we used and the identity . Starting from this equation, one can make use of Eq. (8) to show that any of the four conditions
suffices to have where . If, for example, the first condition is satisfied one may write
We revisit the example from the main text where a Rashba spin-orbit coupling term was added to the mean field Hamiltonian. Since for finite the zero-energy states are protected only by spatial symmetries one expects any spatially random perturbation to cause avoidance of the impurity induced zero-energy crossing. Hence, it is surprising to find that for , , the presence of spatially random entries in the matrices and does not lead to avoided zero-energy crossings. This finding can be understood by observing that the vector and the isotropic Rashba coupling in position space obey the relation while which is the first of the four conditions stated above. However, when choosing , this condition is violated, and in this case only such nonmagnetic disorder which is compatible with the remaining mirror symmetry preserves the zero-energy crossing.
.2 Coupling of symmetry-related states
In this section we show explicitly that states which are related through one of the symmetry operators can () or cannot () be coupled to each other by a TR invariant BdG Hamiltonian. The state vector under consideration is written as .
where the last equality holds because and .
where we used the TR invariance of the Hamiltonian which implies that and are antisymmetric matrices and that .
This demonstrates that and in general can be coupled by TR invariant BdG Hamiltonians. In the main text we showed, that this coupling must vanish if there is a symmetry of the Hamiltonian that anticommutes with .
.3 Explanation of Eq. (4)
In this section we explain why every unitary operator which anticommutes with the chiral symmetry operator and is compatible with the PH redundancy in Eq. (1) automatically obeys Eq. (4).
Because of the PH redundancy in the Hamiltonian Eq. (1), the general unitary transformation reads in matrix form
i.e. every transformation matrix , which respects the PH redundancy of the Hamiltonian and thus is physically meaningful, has to commute with the PH operator . On the other hand, the most general unitary operator anticommuting with the chiral symmetry operator reads
Hence one infers that every unitary operator which anticommutes with and is compatible with the PH redundancy in Eq. (1) is of the form