Bimodal conductance distribution of Kitaev edge modes in topological superconductors

A two-dimensional superconductor with spin-triplet p-wave pairing supports chiral or helical Majorana edge modes with a quantized (length $L$-independent) thermal conductance. Sufficiently strong anisotropy removes both chirality and helicity, doubling the conductance in the clean system and imposing a super-Ohmic $1/\sqrt{L}$ decay in the presence of disorder. We explain the absence of localization in the framework of the Kitaev Hamiltonian, contrasting the edge modes of the two-dimensional system with the one-dimensional Kitaev chain. While the disordered Kitaev chain has a log-normal conductance distribution peaked at an exponentially small value, the Kitaev edge has a bimodal distribution with a second peak near the conductance quantum. Shot noise provides an alternative, purely electrical method of detection of these charge-neutral edge modes.


Introduction
Gapless edge states are a striking manifestation of topological protection in two-dimensional systems. First studied in connection with the quantum Hall effect in a strong magnetic field [1,2], they are now known to exist also in the presence of time-reversal symmetry (for topological insulators (TIs)) or particle-hole symmetry (for topological superconductors) [3,4]. The edge current can carry charge or heat, it can be uni-directional (chiral) or bi-directional (helical), but in each manifestation there is no backscattering-so that the corresponding electrical or thermal conductance is quantized, independent of system size.
Isotropic two-dimensional superconductors with spin-triplet p-wave pairing belong to the class of topological superconductors, with charge-neutral gapless edge states. Depending on the absence or presence of time-reversal symmetry, the edge modes can be chiral (class D) or helical (class DIII). It is known that the quantization of the thermal conductance breaks down if the two-dimensional superconductor is strongly anisotropic; the edge states remain, but backscattering by disorder is no longer forbidden by a topological invariant [5][6][7][8]. One might surmise that the edge states will localize on length scales L larger than the mean free path ℓ, with an exponentially decaying conductance ∝ −L exp ( /ℓ), but that is not what happens. Instead, in [6] an anomalously slow (super-Ohmic) scaling ∝ L ℓ/ was found, unlike that of any known one-dimensional system. A statistical symmetry (translational invariance of the disorder distribution) was identified as the origin of the topological protection [6].
Here we study this remarkable delocalization of edge states in the framework of the Kitaev Hamiltonian [9] of randomly coupled Majorana fermions. We contrast the two realizations of the model illustrated in figure 1: at the edge of a two-dimensional superconductor (Kitaev edge, figure 1(a)) and as a one-dimensional chain of nanoparticles (Kitaev chain, figure 1(b)). While the Kitaev chain allows for delocalization, this requires a fine-tuning to the critical point of the topological phase diagram [10,11]. Generically, the conductance of the Kitaev chain has a lognormal distribution peaked at an exponentially small value [12][13][14], because disorder drives the system away from the gapless critical point into the gapped phase. In contrast, for the Kitaev edge we find a bimodal conductance distribution, with a second peak near the quantized  (Kitaev edge, panel (a)) and in a chain of magnetic nanoparticles (Kitaev chain, panel (b), adapted from [15,16]). Statistical translational invariance at the Kitaev edge means that all couplings κ n between Majorana fermion operators γ n and γ + n 1 have the same statistical distribution. In the Kitaev chain it means that the couplings κ n 2 between nanoparticles have the same distribution, as well as the couplings κ − n 2 1 of pairs of Majorana fermions within a nanoparticle-while the sets κ n 2 and κ − n 2 1 are unrelated. This difference is the reason that statistical translational invariance protects the Kitaev edge from localization, but not the Kitaev chain. As a consequence, the thermal conductance has a lognormal distribution in the Kitaev chain (dashed curve in panel (c)), but a bimodal distribution in the Kitaev edge (solid curve, with a second peak of weight ∝ L 1/ at the conductance quantum G 0 ).
conductance of the clean system. (Compare dashed and solid curves in figure 1(c).) The L ℓ/ weight of this second peak produces the super-Ohmic conductance scaling of [6].
We explain the difference in conductance distributions in terms of the different way in which translational invariance of the disorder distribution is realized in the two systems: in the Kitaev edge all nearest-neighbor coupling strengths of Majorana fermions are statistically equivalent, while in the Kitaev chain even and odd-numbered couplings are inequivalent. Finally, we show how the charge-neutral edge modes of the topological superconductor can be detected in an electrical-rather than thermal-measurement, by considering the shot noise of time-dependent current fluctuations.
The outline of this paper is as follows. In section 2 we introduce model Hamiltonians for pwave superconductors with chiral or helical edge states and calculate the topological phase diagram in the presence of both anisotropy and disorder. The topological phase transitions are identified by considering the bulk conductance and the associated topological invariants. Edge conductance in the topologically non-trivial phases is studied in section 3. In section 4 we contrast the conductance distributions of the Kitaev edge and the Kitaev chain. Electrical, rather than thermal, detection of the edge modes is discussed in section 5. We conclude in section 6.

Topological phase diagrams of chiral and helical p-wave superconductors
The topological phase diagram of clean chiral p-wave superconductors (or superfluids) was studied in [5,7,8]. Here we show how the topologically distinct phases evolve when we include disorder, for both chiral and helical pair potentials, which as we will see have a qualitatively different phase diagram. Numerical calculations on a disordered tight-binding model are compared with analytical calculations of the phase boundaries in self-consistent Born approximation (SCBA).

Model Hamiltonians
Superconductors with broken spin-rotation symmetry are in symmetry class D or DIII in the Altland-Zirnbauer classification [17], depending on whether time-reversal symmetry is broken or not. In both symmetry classes the Bogoliubov-De Gennes Hamiltonian  k ( ) has electronhole symmetry, where the Pauli matrix τ i acts on the electron-hole degree of freedom. In class DIII there is additionally the time-reversal symmetry x y is the anisotropic amplitude of the chiral p-wave pair potential (in a gauge where it is real), x y is the anisotropic hopping amplitude, and μ is the chemical potential. The parameter α ∈ [0, 1] measures the degree of anisotropy, with α → 1 the isotropic limit. We consider equal-spin pairing, so the spin degree of freedom does not appear in  D .
In class DIII the additional constraint (2) is satisfied by taking two time-reversed copies of the Hamiltonian (3), . We study thermal conduction by attaching disorder-free leads at two ends of the lattice, connected to reservoirs at temperature T 0 and δ + T T 0 (see figure 2). The scattering matrix, evaluated at the Fermi level (E = 0) determines the thermal conductance In the μ-α plane the four gapless lines are boundaries separating five topologically distinct insulating phases, see figure 3.
The number of chiral edge modes is given by the Chern number ν [3,4,19], being the winding number of eigenstates in the Brillouin zone. We compute this two-dimensional topological invariant from the scattering matrix rather than from an integral over the Brillouin zone, in a formulation that can be applied directly to disordered systems [20,21], Here ϕ r ( ) is the reflection block of the scattering matrix (5) in the geometry of figure 2(a), with leads attached to x = 0 and = x L x and twisted periodic boundary conditions 3 on the scattering state ψ x y ( , ) in the y-direction: ψ ψ the system is topologically non-trivial, with ν μ = sign and a chiral Majorana edge mode. (The sign of ν gives the direction of propagation.) The absence of backscattering leads to a quantized thermal edge conductance = G G 0 . This characterizes the strong TI. When ) the Chern number ν = 0, so there are no chiral edge modes. These regions in the phase diagram are distinguished by an alternative 'weak' topological invariant ν y [22,23]. Again, to prepare ourselves for disorder effects, we use a scattering matrix formulation rather than a Brillouin zone formulation [5,7]. The twodimensional weak topological invariant is the strong topological invariant in one dimension lower, which in class D is given by the determinant of the reflection matrix [11], The dimensional reduction is implemented by evaluating ϕ r ( ) at ϕ = 0, so for periodic boundary conditions in the y-direction. When the Chern number ν = 0 the weak invariant ν y may be equal to +1 (trivial insulator) or −1 (weak topological insulator, WTI).
For μ α | | > + t 2 (1 ) we are in the topologically trivial phase, with ν = 1 y and no edge modes at all. In contrast, when μ α | | < − t 2 (1 ) the system is a WTI, with ν = − 1 y and nonchiral Majorana modes on the edges in the y-direction.

Disorder effects.
Having described the phase diagram of the system in the clean limit, we now turn to the effects of disorder. Sufficiently strong disorder can convert a class-D superconductor that is insulating in the bulk into a thermal metal [24][25][26]. To search for this topological phase transition we take the two-terminal geometry of figure 2(a) with periodic boundary conditions in the y-direction, in order to focus on the metallic or insulating nature of the bulk. (We will consider edge conduction in section 3.) Numerical results for the disorder-averaged thermal conductance G are shown in figure 4, as a function of chemical potential μ and disorder strength δ. One can see that both the TI and WTI phases are robust to disorder, up to about δ ≈ t. For stronger disorder there is a TIto-thermal metal phase transition, followed by a transition to a topologically trivial Anderson insulator. The phase boundaries between TI, WTI, and thermal metal are in quite good agreement with those calculated in SCBA (dashed lines in figure 4, see the appendix for details of the calculation). The transition to an Anderson insulator at strong disorder is out of reach of that approximation.
The distinct topological nature of the TI, WTI, and trivial phase is confirmed by a calculation of the topological invariants ν ν , y , see figure 5. In the bulk insulating phases these are quantized numbers: . At the topological phase transitions, when the bulk gap closes, both ν and ν y are free to vary between these integer values, resulting in the smooth transitions shown in figure 5.
2.3. Class DIII phase diagram 2.3.1. Clean limit. We now turn to the phase diagram of the class DIII Hamiltonian (4), first without disorder. It is convenient to rotate the Hamiltonian to a block off-diagonal form, Δσ αΔσ x y x y The gap closings identify the boundaries of insulating phases, as shown in figure 6. While in class D the gap closes along a line in the phase diagram, in class DIII there are extended gapless regions of a metallic phase separating the insulating phases. (This is a generic feature of helical p-wave superconductors [27]. ) We can distinguish five distinct insulating regions of phase space. For weak anisotropy, α Δ > K / , we find two μ ± symmetric insulating phases bounded by When the anisotropy reaches the critical value c c Additionally, for any amount of anisotropy there are insulating phases at large chemical potentials, with boundaries given by   To test the topological properties of these phases, we compute the associated DIII topological invariants in a scattering formulation (so that we can directly apply it to disordered systems in the next subsection). The strong topological invariant [21],  Figure 6. Phase diagram of the class-DIII Hamiltonian (4) without disorder, for Δ = K / 0.35. Topologically distinct insulating phases are separated by gapless metallic regions (blue). The topologically trivial insulator, without edge states, exists for any amount of anisotropy, while the TI and WTI phases with edge states require, respectively, α Δ > K / and α y y is determined by the Pfaffians of the reflection matrix with periodic (ϕ = 0) and anti-periodic (ϕ π = ) boundary conditions in the y-direction. In view of the time-reversal symmetry condition (2), the matrix σ ϕ r i ( ) y is anti-symmetric for ϕ π = 0, , so the Pfaffian exists. The insulating regions delimited by equation (11) are topologically non-trivial ( = − Q 1), with helical Majorana edge states and quantized thermal conductance = G G 2 0 . All other insulating phases have Q = 1. The ones appearing at large chemical potentials, bounded by equation (14), are topologically trivial, without edge states. However, the phase which develops at the critical anisotropy α c , bounded by equation (13), has Q = 1 but still supports gapless modes on edges oriented in the y-direction. The weak topological invariant = − Q 1 y of this phase is obtained by dimensional reduction to the one-dimensional class-DIII topological invariant [28],  Figure 7 shows the effect of disorder on the topological phases, probed by calculating the thermal conductance in the geometry of figure 2(a) with periodic boundary conditions in the y-direction. Comparison with the class-D phase diagram of figure 4 shows as a qualitative difference that the thermal metal phase extends down to zero disorder. This behavior is also captured by the SCBA (dashed curves), see the appendix for details of the calculation. The scattering matrix formulas (15) and (16) for the class-DIII strong and weak topological invariants are applied to a disordered system in figure 8. These are both  2 invariants, meaning that they take on the values ±1 when the bulk is insulating: in the TI, and Q = 1, = − Q 1 y in the WTI.

Edge conduction
So far we studied thermal conduction in the geometry of figure 2(a) with periodic boundary conditions in the transverse direction, in order to eliminate edge contributions and focus on bulk properties. To study edge conduction in the TI and WTI phases we now take the geometry of figure 2(b), with leads attached to = y L 0, y and hard-wall boundary conditions at = x L 0, x . We again first consider the clean case and then add the effects of disorder, for both symmetry classes D and DIII.

Clean case
The TI and WTI phases both have gapless edge states, the difference being that the TI edge states appear on all edges while the WTI edge states exist only at two of the four edges (see figure 9). In the geometry of figure 2(b) we can probe the edge conductance in both phases. Without disorder the conductance is system-size independent, because there is no backscattering, and the difference between the TI and WTI phases is simply a factor of two.
This conductance doubling at the TI-to-WTI transition is shown in figure 10. In class D it happens because the chiral edge state of the TI phase can propagate in both directions in the WTI phase. In class DIII we start out with helical edge states in the TI phase, with direction of propagation tied to the spin degree of freedom. In the WTI phase this helicity is lost, so now both spin-up and spin-down can propagate in both directions and the conductance is doubled.

Disorder effects
The addition of disorder has no effect on the conductance in the TI phase, since backscattering of chiral or helical edge states is forbidden. The edge states in the WTI are neither chiral nor helical, so disorder does cause backscattering and reduces the edge conductance. However, as discovered in [6], the recovery of translational invariance upon ensemble averaging prevents localization of the WTI edge states. Instead of an exponential decay of the conductance with the length ≡ L L y of the edge, there is only an algebraic L 1/ decay. In figure 11 we show this super-Ohmic conductance scaling for the WTI phase of the class-D and class-DIII Hamiltonians (3) and (4).

Kitaev chain versus Kitaev edge
The absence of localization at the edge of the anisotropic p-wave superconductor is puzzling if one tries to understand it starting from the limit α → 0 of strong anisotropy. Then the system can be thought of as an array of weakly coupled nanowires with overlapping Majorana zeromodes at the end points, a so-called Kitaev ladder [29][30][31]. The effective edge Hamiltonian is the Kitaev Hamiltonian [9] in class D, or two time-reversed copies of it in class DIII. The disordered one-dimensional Kitaev model, called the Kitaev chain, is known to be an insulator [12][13][14] . This one-dimensional system in symmetry class D has a  2 topological invariant [9], K determined by the Pfaffians of a pair of real antisymmetric matrices ± A , having non-zero matrix n n n n n , 1 1, Translational invariance of the disorder ensemble means two completely different things for the Kitaev edge and for the Kitaev chain. For the Kitaev edge it means that the coupling strengths κ n between adjacent Majoranaʼs all have the same distribution. The disorder average Q K of the topological invariant then vanishes, which is why the Kitaev edge is called a critical WTI [6]. In contrast, as illustrated in figure 1, for the Kitaev chain translational invariance means that the κ n with n even or those with n odd have the same distribution, but the distributions of κ n 2 and κ − n 2 1 are unrelated. The topological invariant is then non-zero on average, so the Kitaev chain is not critical [12,14].
The implication for the transmission probability T follows if we remove the coupling between γ N 2 and γ 1 , so that we can introduce transmission and reflection amplitudes α = t 1/cosh , α = r tanh . The Lyapunov exponent α determines both T and Q K , K 2 and has a Gaussian distribution α P ( ) [10,14]. The variance α = L Var /ℓ is determined by the mean free path ℓ for backscattering along the edge, of length ≫ L ℓ. The mean α ξ = L/ defines the localization length ξ. A vanishing Q K implies that the median of α P ( ) is zero, and since itʼs Gaussian also α ξ = ⇒ = ∞ 0 . For the Kitaev chain ≠ Q 0 K and hence ξ is finite, so the transmission probability has a log-normal distribution peaked at = ξ − T e L 2 / , with an exponentially decaying average transmission [12,14]. In contrast, for the Kitaev edge The Gaussian distribution of the Lyapunov exponent then produces a bimodal distribution of the transmission probability for ≫ L ℓ, x L x contribute to the thermal conductance in the WTI phase, but one edge can be removed by reducing the width of the contacts to the interval ⩽ ⩽ x L 0 /2 x . Results are shown in figure 12. With the mean free path ℓ as a single fit parameter, the transition from a uni-modal distribution to a bimodal distribution upon increasing L/ℓ is well-described by equation (22).

Electrical detection of Kitaev edge modes
So far we considered thermal conduction as the probe of edge state transport. Electrical detection would be more convenient experimentally, and this is possible by adapting the nanowire setup of [11]. All contacts are now at the same temperature T 0 , the superconductor is grounded as well as one of the metal contacts (number 2), and the other contact (number 1) is biased at voltage V 1 . The electrical current into the grounded contact 2 fluctuates in time with noise power P 12 . This is dominated by shot noise, at low temperatures when thermal noise can be neglected ( ≪ k T eV B 0 1 ). The noise power is given in terms of the transmission matrix by [32] τ τ

Tr Tr
Tr Tr , The subscripts e h , indicate transmission from electron to electron (t ee ) or from electron to hole (t he ), and we used electron-hole symmetry in the second equation to rewrite the whole expression in terms of the full transmission matrix t.
As derived in [11], when the transmission is via an unpaired Majorana mode, the second trace in equation (23) vanishes identically so the electrical shot noise is directly related to the thermal conductance:  both symmetry classes D and DIII the two quantities P 12 and G have the same L 1/ scaling in the WTI phase 4 , compare figures 11 and 13.

Conclusion
The Kitaev model [9] is paradigmatic for topological superconductivity and Majorana zeromodes, and for that reason has been studied extensively [33,34]. Here we have shown that the realization of this model at the edge of a two-dimensional system (what we have called the 'Kitaev edge') is fundamentally different from its strictly one-dimensional counterpart, the Kitaev chain. The difference, summarized in figure 1, manifests itself in the different distribution of the thermal conductance, peaked at exponentially small value in the Kitaev chain [12][13][14] while the Kitaev edge has a second peak at the conductance quantum.
As a possible physical realization of Kitaev edge modes we have studied in some detail a model of an anisotropic two-dimensional chiral p-wave superconductor [5][6][7][8], as well as its time-reversally symmetric (helical) counterpart. Both can produce WTI with Kitaev edge modes, but while they appear at any amount of anisotropy for chiral p-wave pairing, the helical p-wave pairing requires a threshold anisotropy (compare figures 3 and 6). We have demonstrated the robustness of the WTI phase to disorder by numerical simulations, in good . Incidentally, we note that this also implies that the 'thermal Fano factor' = − − F T T T (1 ) 1 has the same L-independent value 1/3 as the electrical Fano factor of a metallic diffusive conductor-even though the conductance in that system has a Gaussian rather than bimodal distribution. agreement with analytical calculations of the phase boundaries in self-consistent Born approximation (figures 4 and 7). Experimentally the transition into the WTI phase can be detected, on length scales below the mean free path, via the doubling of the thermal conductance (figure 10), and on larger length scales via the super-Ohmic scaling ( figure 11). Because of the complexity of thermal transport measurements at low temperatures, we have proposed an alternative fully electrical method of detection, using the electrical shot noise power (figure 13). The SCBA self-energy is a k-independent 4×4 matrix with spin and electron-hole degrees of freedom The terms δμ and δK renormalize the chemical potential and coupling respectively. Both terms account for a disorder induced shift of the phase boundaries between the trivial insulator, TI, WTI, and thermal metal. The term γ produces a finite density of states, induced by the disorder. Such a finite density of states may indicate a thermal metal or a trivial Anderson insulator, but it cannot distinguish between the two.