Benefits of weak disorder in one dimensional topological superconductors
Abstract
Majorana bound states are zero-energy modes localized at the ends of a one-dimensional (1D) topological superconductor. Introducing disorder usually increases the Majorana localization length, until eventually inducing a topological phase transition to a trivial phase. In this work we show that in some cases weak disorder causes the Majorana localization length to decrease, making the topological phase more robust. Increasing the disorder further eventually leads to a change of trend and to a phase transition to a trivial phase. Interestingly the transition occurs at , where is the disorder mean-free path and is the localization length in the clean limit. Our results are particularly relevant to a 1D topological superconductors formed in planar Josephson junctions.
pacs:
Introduction.—
Understanding the effect of unavoidable disorder on topological superconductivity is of great interest. Of particular interest is its effect on the localization length of the zero-energy Majorana bound states (MBSs), and the critical strength for transition to a trivial state.
Effects of disorder on spinless single-channel p-wave superconductor Kitaev (2001); Read and Green (2000) - the canonical model for topological superconductivity (TSC) Qi and Zhang (2011); Alicea (2012); Beenakker (2013); Lutchyn et al. (2018); Aguado (2017) - were previously studied Motrunich et al. (2001); Brouwer et al. (2011a); Lobos et al. (2012); Pientka et al. (2013); Huse et al. (2013); Adagideli et al. (2014). Disorder was found to increase the Majorana localization length, , according to , with being the localization length (or coherence length) in the clean limit, and being the impurity-induced mean free path Brouwer et al. (2011a). At the critical value the localization length diverges leading to a phase transition to a trivial phase. Accordingly, the critical mean-free time, , is determined by the excitation gap of the clean system, ^{1}^{1}1This result is valid when the Fermi energy is large compared with , which is the limit of interest here. For the opposite limit see Ref. Pientka et al. (2013)..
For a multi-channel 1D system Potter and Lee (2010); Rieder et al. (2013); Rieder and Brouwer (2014); Lu et al. (2016); Burset et al. (2017), at weak-enough disorder the behavior is similar to the single-channel case with monotonically-increasing . For stronger disorder, multiple transitions between trivial and topological occur at with the number of channels Rieder et al. (2013); Rieder and Brouwer (2014).
In this paper we study the effect of disorder on a novel realization of a 1D topological superconductor: a planar Josephson junction (JJ), implemented in a Rashba two-dimensional electron gas (2DEG), and subject to in-plane magnetic field Hell et al. (2017a); Pientka et al. (2017); Hell et al. (2017b); Hart et al. (2017) (see Fig. 1). We find that in this system weak potential disorder causes to decrease [see. Fig. 1(b)]. For strong disorder, the trend eventually reverses and the localization length increases back until finally diverging at the transition to the trivial phase. Importantly, this transition occurs at a critical disorder strength, which is typically much larger than the gap of the clean system.
Studying a general low-energy model for a multi-channel TSC, we show that disorder can cause to increase or decrease, depending on the relative phases of the pairing potentials in different channels, and the structure of the inter-channel impurity scattering (see also Fig. 2). Scattering between modes of equal-phase pairing potential increases the “effective” pairing gap, while scattering between modes of opposite-phase potentials decreases the effective gap. Due to the p-wave nature of the pairing within each channel, intra-channel backscattering always decreases the effective gap, and de-localizes the MBS.
We find that the enhancement of localization by weak disorder in the planar JJ is related to the structure of the low-energy excitations confined to the junction. The excitations carry a longitudinal momentum . The spectrum is gapped, and the smallest gap is at large , close the Fermi momenta of the 2DEG Pientka et al. (2017). At these ’s spin-orbit coupling (SOC) dominates over the Zeeman field, causing the spins of opposite-momenta modes in each channel to be oppositely polarized, thereby suppressing the detrimental intra-channel backscattering Brouwer et al. (2011b). Consequently, disorder effectively increases the gap of the large-momentum channels. In contrast, at small Zeeman field dominates over SOC, allowing for intra-channel backscattering, which decreases the effective gap. The smallest of the gaps determines . Weak disorder then increases the large momentum gap and enhances localization. As disorder is increased, the trend changes when the gaps at small and large momentum become equal (see also Fig. 3).
We begin with a numerical analysis of the dependence of on disorder in a planar JJ. We then consider a low-energy model of a multi-channel TSC. Finally, we construct a simplified model of the planar JJ which qualitatively reproduces the numerical results.
Numerical analysis of the planar Josephson junction.—
The planar JJ consists of two conventional superconductors in proximity to a Rashba-spin-orbit-coupled 2DEG Hart et al. (2017). The superconductors are separated by a distance , and are of length in the direction, see Fig. 1. As shown theoretically Hell et al. (2017a); Pientka et al. (2017), by applying an in-plane magnetic field and controlling the phase bias, the junction can realize a 1D TSC. Experimental evidence for a TSC has been recently reported Ren et al. ; Fornieri et al. (2018).
In the presence of impurity-potential disorder, the system’s Hamiltonian is
(1) |
where is the effective electron mass in the 2DEG, is the chemical potential, with () its value in the junction (below the superconductors), is the Rashba spin-orbit coupling coefficient, is the Zeeman splitting due to the in-plane magnetic field, with () being its values in the junction (below the superconductors), and is the electrons’ pairing potential, being the phase difference between the two superconductors. Here, is a random disorder potential having zero average and short-range correlations, , where is the mean free time for disorder scattering in the bare 2DEG. In writing Eq. (1) we have used the Nambu basis, , where creates an electron in the 2DEG with spin at position . Accordingly, the sets of Pauli matrices, and , operate on the spin and particle-hole degrees of freedom, respectively.
(a) (b) |
To analyze the disordered system numerically, we use a lattice model and construct a corresponding tight-binding Hamiltonian. The topological invariant and the localization length can be obtained from the scattering matrix between two fictitious leads at and (which extend throughout the direction). The scattering matrix is calculated numerically using a recursive Green-function method Lee and Fisher (1981); SM .
Let be the reflection matrix for electrons and holes incident on the left at energy . The topological invariant satisfies Akhmerov et al. (2011); Fulga et al. (2011), , which in the limit takes the values in the trivial phase and in the topological phase.
We obtain from finite-size scaling of the zero-energy transmission probability matrix, . Except for the phase transition, the eigenvalues of decay exponentially with Beenakker (1997); Evers and Mirlin (2008). The smallest exponent determines the localization length of mid-gap zero energy states. In the topological phase, this defines the Majorana localization length . We average over many disorder realizations.
Figure 1(a) presents the phase diagram of the clean system [], previously obtained in Ref. Pientka et al. (2017). We note the chemical potential need not be fine-tuned for the system to be topological; in particular, it can be substantially larger than . In the topological phase, the junction hosts zero-energy Majorana bound states (MBS) at the junction’s ends near and .
Figure 1(b) presents versus disorder strength, represented by the inverse mean free time of the underlying 2DEG, , for different values of and [see markers in Fig. 1(a)]. In all cases shown, first decreases as a function of , reaching a minimum which can be an order of magnitude smaller than its value in the clean system. This makes the Majorana bound states more protected against perturbations that can potentially couple them. When increasing disorder strength further, eventually increases, diverging at the phase transition to the trivial phase, shown by the vertical dashed lines. Notice the phase transition occurs at a critical disorder strength, , much larger than .
Low-energy model.—
To understand the above results, we consider a more general model of a 1D multi-channel TSC, comprising of linearly-dispersing electronic modes, and given by , with
(2) |
Here each of the conducting channels contains a right-moving mode () and a left-moving mode (), is the Fermi momentum of the -th mode, is the mode velocity, is a pairing potential in the -th channel, and are scattering terms arising from disorder. Notice due to hermiticity, and due to the anticommutativity of . In the clean limit, the system is topological for odd , and trivial for even . The Majorana localization length (for odd ) is determined by the maximal
This model can be related to the planar JJ, at low energies, by first solving the Hamiltonian of Eq. (1) inside the junction () in the absence of coupling to the SCs, i.e., when the reflection from the SCs is purely normal [see e.g. Fig. 3(a)], then linearizing the spectrum near the Fermi points to obtain and , and finally considering the induced superconductivity in the form of the pairing potentials, SM . This is justified when the Fermi level is far enough from the bottom of the band, compared with , . Omitting inter-channel pairings is justified whenever the energy mismatch, is large compared with the inter-channel pairing.
In the above model, Eq. (2), we assume that , , and . This will indeed be the case in the planar JJ due to a reflection symmetry, , present in the clean limit SM . The elements of the disorder matrix are normally distributed, with zero mean and short-range correlations, , where the upper bar denotes disorder averaging, and is related to the disorder-induced transition rate from mode to , through . While is generally complex, in our case it may be chosen real and positive, thanks to a time-reversal-like symmetry, , which exists in the clean limit Pientka et al. (2017); SM . We make this choice here.
(a) | (b) |
To obtain a correction to in the form of a disorder self energy, we examine the Nambu-Gor’kov Green function, where . In the absence of disorder, the momentum-space Green function reads
(3) |
For weak disorder, we can obtain the self-energy within the Born approximation SM ,
(4) |
where . Comparing with the unperturbed Green function, we see that disorder changes the effective pairing potentials according to
(5) |
Notice that the contribution of mode to depends on the inter-channel scattering rate, , the scattering phase, , and on the relative phase between and . Importantly, disorder can either decrease or increase (and therefore increase or decrease ). The process underlying Eq. (5) is depicted in Fig. 2.
Disordered s-wave vs. disordered p-wave superconductor.—
We explore two special cases of the multi-channel superconductor: (i) a single-channel p-wave SC and (ii) a single-(spinful)-channel s-wave SC. These cases clarify the non-monotonic behavior of for the disordered planar JJ, observed in Fig. 1(b).
(a) | (b) |
(c) | (d) |
The low-energy Hamiltonian of a single-channel p-wave SC is obtained by setting in Eq. (2), with , , and . Equation (5) then yields
(6) |
The localization length can then be obtained by , yielding the known result Brouwer et al. (2011a), where is the mean free path, and .
For a single-channel -wave superconductor there are no zero-energy end modes and is the length to which a single electron at zero energy penetrates the superconductor before being reflected. The index corresponds to the two spin directions. The spin-singlet nature of the pairing dictates and the spin-independence of the disorder forbids intra-channel scattering and dictates . Furthermore, the two velocities are the same,
Setting this in Eq. (5), we have
(7) |
where , and correspondingly , where . Unlike the case of the single-channel p-wave SC, the localization length in the s-wave case decreases monotonically. We emphasize that the relative sign difference in Eq. (6), compared to Eq. (7), stems from (i) lack of scattering between opposite spins and (ii) the s-wave spin-singlet nature of the pairing. While these results for and were obtained using a weak-disorder perturbative analysis, they are actually exact for the linearized model of Eq. (2), as shown in the Supplemental Material SM .
The results for and let us understand the non-monotonic behavior of in the planar JJ [see Fig. 1(b)]. The low-energy spectrum of sub-gap excitations confined between the two superconductors may be seen as coming out of superconducting pairing of several low-energy modes. Figure 3(a) presents an example of the spectrum of the Hamiltonian, Eq. (1), confined within the junction under the assumption of full normal reflection. The red arrows, representing the spin expectation values of the modes, indicate that the outer channels (larger Fermi momentum) are largely spin polarized due to the spin-orbit coupling, with opposite-momentum modes having approximately opposite spins. For the inner channel, the spin varies along the –direction, resulting in a smaller expectation value.
Since pairing is induced by an s-wave SC, the large-momentum channels, being spin-polarized, behave as a s-wave SC, with a localization length, , that decreases with disorder. In contrast, the small-momentum channel is not spin-polarized, and allows for intra-channel backscattering. Consequently this channel behaves as a disordered p-wave SC, with localization length, , that increases with disorder. The overall localization length of the system, , is then the larger between and .
The behavior of versus disorder therefore depends on the relative size of and . Assuming, for simplicity , and , we find that when [Fig. 3(c)], the localization length decreases for weak disorder. With stronger disorder the two gaps approach one another. Consequently, scattering between the large-momentum and the low-momentum channels causes “level repulsion” between and , as depicted in Fig. 3(c,d), and increases with disorder (blue solid line). Notice that the critical disorder strength, , can be much larger than the gap of the clean system, |. In contrast, if [Fig. 3(d)], disorder causes to increase monotonically, diverging at the critical disorder , which now equals the gap of the clean system.
In the planar JJ, the gap of the large-momentum channels is indeed the smaller one [see Fig. 3(b)]. For a not-too-narrow junction, the low-momentum gap is approximately , while the large-momentum gap is approximately . This difference may be viewed as originating from the fact that high-momentum electrons propagate almost parallel to the SCs, and are therefore only weakly coupled to the SCs. For this system, then, disorder may increase the effective gap from the scale of to the scale of .
We test our understanding by studying two geometries of superconducting proximity [Fig. 4(a)], where a strip of 2DEG is coupled to a single SC from the side (blue line) or from above the strip (red line). While in the former the large-momentum (s-wave) gap is the smallest, giving rise to behavior similar to the planar JJ, this is not the case in the latter geometry, resulting in a monotonically-increasing .
As another test, we add to Eq. (1) a magnetic disorder term . Here is a random field with zero average and correlations ^{2}^{2}2We limit to the junction since we are not interested here in its effect on the SCs.. Figure 4(b) presents for different values of the ratio between magnetic and potential disorder, , where . Since magnetic disorder can scatter between the opposite-spin states, the large-momentum channels do not behave anymore as an s-wave SC, and instead are more similar to a multi-channel p-wave SC Rieder et al. (2013); Rieder and Brouwer (2014); Potter and Lee (2010). Indeed, with increasing , the disorder-induced decrease in diminishes.
(a) | (b) |
Acknowledgments.—
We have benefited from the insightful comments of B Halperin and Y Oreg. We also thank M Buchhold for useful discussions. We acknowledge support from the Walter Burke Institute for Theoretical Physics at Caltech (AH), the Israel Science Foundation (AS), the European Research Council under the European Community Seventh Framework Program (FP7/2007- 2013)/ERC Project MUNATOP (AS), Microsoft Station Q (AS), and the DFG (CRC/Transregio 183, EI 519/7-1) (AS).
References
- Kitaev (2001) A. Kitaev, Phys. Usp. 44, 131 (2001).
- Read and Green (2000) N. Read and D. Green, Phys. Rev. B 61, 10267 (2000).
- Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
- Alicea (2012) J. Alicea, Rep. Prog. Phys. 75, 076501 (2012).
- Beenakker (2013) C. W. J. Beenakker, Annu. Rev. Condens. Matt. Phys. 4, 113 (2013).
- Lutchyn et al. (2018) R. Lutchyn, E. Bakkers, L. Kouwenhoven, P. Krogstrup, C. Marcus, and Y. Oreg, Nat. Rev. Mat. 3, 52 (2018).
- Aguado (2017) R. Aguado, La Rivista Del Nuovo Cimento 40, 523 (2017).
- Motrunich et al. (2001) O. Motrunich, K. Damle, and D. A. Huse, Phys. Rev. B 63, 224204 (2001).
- Brouwer et al. (2011a) P. W. Brouwer, M. Duckheim, A. Romito, and F. von Oppen, Phys. Rev. Lett. 107, 196804 (2011a).
- Lobos et al. (2012) A. M. Lobos, R. M. Lutchyn, and S. Das Sarma, Phys. Rev. Lett. 109, 146403 (2012).
- Pientka et al. (2013) F. Pientka, A. Romito, M. Duckheim, Y. Oreg, and F. von Oppen, New J. Phys. 15, 025001 (2013).
- Huse et al. (2013) D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal, and S. L. Sondhi, Phys. Rev. B 88, 014206 (2013).
- Adagideli et al. (2014) I. Adagideli, M. Wimmer, and A. Teker, Phys. Rev. B 89, 144506 (2014).
- (14) This result is valid when the Fermi energy is large compared with , which is the limit of interest here. For the opposite limit see Ref. Pientka et al. (2013).
- Potter and Lee (2010) A. C. Potter and P. A. Lee, Phys. Rev. Lett. 105, 227003 (2010).
- Rieder et al. (2013) M.-T. Rieder, P. W. Brouwer, and I. Adagideli, Phys. Rev. B 88, 060509 (2013).
- Rieder and Brouwer (2014) M.-T. Rieder and P. W. Brouwer, Phys. Rev. B 90, 205404 (2014).
- Lu et al. (2016) B. Lu, P. Burset, Y. Tanuma, A. A. Golubov, Y. Asano, and Y. Tanaka, Phys. Rev. B 94, 014504 (2016).
- Burset et al. (2017) P. Burset, B. Lu, S. Tamura, and Y. Tanaka, Phys. Rev. B 95, 224502 (2017).
- Hell et al. (2017a) M. Hell, M. Leijnse, and K. Flensberg, Phys. Rev. Lett. 118, 107701 (2017a).
- Pientka et al. (2017) F. Pientka, A. Keselman, E. Berg, A. Yacoby, A. Stern, and B. I. Halperin, Phys. Rev. X 7, 021032 (2017).
- Hell et al. (2017b) M. Hell, K. Flensberg, and M. Leijnse, Phys. Rev. B 96, 035444 (2017b).
- Hart et al. (2017) S. Hart, H. Ren, M. Kosowsky, G. Ben-Shach, P. Leubner, C. Brüne, H. Buhmann, L. W. Molenkamp, B. I. Halperin, and A. Yacoby, Nat. Phys. 13, 87 (2017).
- Brouwer et al. (2011b) P. W. Brouwer, M. Duckheim, A. Romito, and F. von Oppen, Phys. Rev. B 84, 144526 (2011b).
- (25) H. Ren, F. Pientka, S. Hart, A. Pierce, M. Kosowsky, L. Lunczer, R. Schlereth, B. Scharf, E. M. Hankiewicz, L. W. Molenkamp, et al., arXiv:1809.03076 .
- Fornieri et al. (2018) A. Fornieri, A. M. Whiticar, F. Setiawan, E. P. Marín, A. C. Drachmann, A. Keselman, S. Gronin, C. Thomas, T. Wang, R. Kallaher, et al., arXiv:1809.03037 (2018).
- Lee and Fisher (1981) P. A. Lee and D. S. Fisher, Phys. Rev. Lett. 47, 882 (1981).
- (28) See Supplemental Material for details on: (i) numerical simulations, (ii) analysis of the low-energy model, and (iii) calculation of the localization length in the case of a single-channel -wave and -wave SCs, which include Refs. Fisher and Lee (1981); Iida et al. (1990); Potter and Lee (2011); Bardeen (1961); Haim et al. (2016); Halperin (1967); Dorokhov (1982); Mello et al. (1988).
- Akhmerov et al. (2011) A. R. Akhmerov, J. P. Dahlhaus, F. Hassler, M. Wimmer, and C. W. J. Beenakker, Phys. Rev. Lett. 106, 057001 (2011).
- Fulga et al. (2011) I. C. Fulga, F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker, Phys. Rev. B 83, 155429 (2011).
- Beenakker (1997) C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).
- Evers and Mirlin (2008) F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008).
- (33) We limit to the junction since we are not interested here in its effect on the SCs.
- Fisher and Lee (1981) D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981).
- Iida et al. (1990) S. Iida, H. A. Weidenmüller, and J. Zuk, Ann. Phys. 200, 219 (1990).
- Potter and Lee (2011) A. C. Potter and P. A. Lee, Phys. Rev. B 83, 094525 (2011).
- Bardeen (1961) J. Bardeen, Phys. Rev. Lett. 6, 57 (1961).
- Haim et al. (2016) A. Haim, K. Wölms, E. Berg, Y. Oreg, and K. Flensberg, Phys. Rev. B 94, 115124 (2016).
- Halperin (1967) B. I. Halperin, Properties of a Particle in a One-Dimensional Random Potential, edited by I. Prigogine, Vol. 13 (Wiley Online Library, 1967) pp. 123–177.
- Dorokhov (1982) O. Dorokhov, JETP Lett 36, 318 (1982).
- Mello et al. (1988) P. Mello, P. Pereyra, and N. Kumar, Ann. Phys. 181, 290 (1988).
Supplemental Material
I Details of numerical simulations
In this section we present details of the numerical simulations whose results are summarized in Fig. 1 of the main text. We begin by presenting the lattice model used for simulating the system. We then explain the procedure for obtaining the reflection matrix and extracting the Majorana localization length.
i.1 The lattice model
For the purpose of numerically simulating the planar Josephson junction [Eq. (1) of the main text], we replace it with a model of a square lattice of lattice constant , whose Hamiltonian is given by
(8) |
where creates an electron on site , , , , , , , , and . In the present work, we use .
i.2 The reflection matrix
We begin by rewriting the Hamiltonian in the following form
(9) |
where is a vector of creation and annihilation operators, and where and are matrices.
We place two normal-metal leads, at and . The reflection matrix for electrons and holes incident from the right is given by Fisher and Lee (1981); Iida et al. (1990)
(10) |
where , with being the density of states in the right lead, and is the Green function matrix at the right-most sites of the system, obtained through the recursive relation Lee and Fisher (1981)
(11) |
Here, is a matrix for every (indices running over spin, particle-hole and ), and , with being the density of states in the left lead.
i.3 Topological invariant and localization length
Given the reflection matrix, the topological invariant is given by Akhmerov et al. (2011); Fulga et al. (2011) , which takes the value () in the trivial (topological) phase. As an example, in Fig. 5(a) we present as a function of system’s length, , for four different disorder realizations, with increasing value of disorder strength. The rest of the system parameters are as in Fig. 1 of the main text, with and . When calculating the topological invariant for a clean system [Fig. 1(a) of the main text] we have instead used the Pfaffian invariant introduced in Ref. Kitaev (2001).
To obtain the localization length, the transmission probability matrix is obtained through , where is the transmission matrix, and we used the fact that the scattering matrix is unitary. The Majorana localization length is determined by the decay of the largest eigenvalue of This eigenvalue is shown in Fig. 5(b) as a function of for four different value of disorder strength. We then extract the localization length by computing
(12) |
where by we denote the largest eigenvalue of the transmission probability, for a system of length Notice that for an exponentially decaying transmission, , this indeed yields the decay length, assuming the lattice spacing is taking to be small enough . In the simulations presented in this work we averaged over a 100 realizations for every data point, and the maximal system’s length was .
Finally, in Fig. 5(c) we present the profile of the zero-energy local density of states, , for the four disorder realizations corresponding to Figs. 5(a) and 5(b). The local density of states, , was calculated according to the method described in Ref. Potter and Lee (2011).
(a) | (b) | (c) |
i.4 results for different parameters
In Fig. 6, we present results for a junction with parameters different from those shown in Fig. 1 of the main text. Figure 6(a) presents the phase diagram in the clean limit, and Fig. 6(b) presents the Majorana localization length as a function of disorder strength, for chemical potential . The rest of the parameters are the same as in Fig. 1 of the main text. The same qualitative behavior is observed as in Fig. 1 of the main text.
In Fig. 6(c) we examine the effect of disorder on the system’s phase diagram, for , and for a narrower junction, (compared with in the main text). The rest of the system parameters are the same as in Fig. 1 of the main text. The topological invariant, , is shown for a disorder strength of . The black dashed line represents the phase boundaries in the case of the clean system with the same parameters. Interestingly, for some magnetic fields, , and phase biases, , disorder drives the system from the trivial phase to the topological phase. A similar effect was previously observed in Refs. Adagideli et al. (2014); Pientka et al. (2013).
(a) | (b) | (c) |
Ii The analysis of the linearized multi-channel model
In the main text we have studied a linearized low-energy model describing a disordered multi-channel superconductor, Eq. (2), and performed a perturbative analysis of the disorder, which resulted in new effective pairing potentials, Eq. (5). In this section we explain how this model can arise from a microscopic model, such as the planar JJ, Eq. (1) of the main text, and provide details regarding the calculation of the self energy which yielded the expression for the effective pairing potentials.
ii.1 Origin of the model
We start from the 2d model of Eq. (1) of the main text, and separate the system to two parts: the normal part which is the strip defined by , and the superconducting part, . Following the Bardeen tunneling-Hamiltonian approach Bardeen (1961), we then write the overall Hamiltonian as a combination of the three terms, describing the normal part, the SC part, and the coupling between them,
(13) |
where () is the Hamiltonian obtained by imposing hard-wall boundary conditions for (). This treatment is valid when the normal reflection at the N-S interfaces () is strong, such that the normal part as weakly coupled to the SC. This is the case, in particular, for the high-momentum modes as they impinge upon the N-S interface at large angles. Regardless of the above considerations, our numerical analysis shows that the qualitative conclusions drawn from the low-energy model of Eq. (2) of the main text hold much more generally.
We write the normal part, , as a combination as two terms,
(14) |
where describes the system in the clean limit, and is the part coming from disorder. Our treatment of the system is composed of two steps: (i) first we solve for , and (ii) the disorder term and the induced superconductivity are then projected onto the basis diagonalizing .
The clean part of the Hamiltonian, , is generally solved by a set of eigenstates,
(15) |
with corresponding eigen-energies, , and Here, is the momentum in the direction, while labels the transverse channels.
ii.1.1 Reflection Symmetry
The clean part of the Hamiltonian obeys the following symmetry,
(16) |
as can be checked by setting in Eq. (1) of the main text. The eigenstates can therefore be chosen to obey
(17) |
ii.1.2 Conducting channels
Depending on the chemical potential, some of the bands labeled by will cross zero energy, , for some momentum . Due to the above reflection symmetry, these momenta will come in opposite-momentum pairs (except for potentially a single Fermi point at , which can occur when the chemical potential is at the bottom of one of the bands). The number of bands crossing zero energy, , defines the number of conducting channels in the model. Correspondingly, we label the Fermi momenta by , where , and where . Below we will be interested only in the modes having momentum near .
ii.1.3 Projection and Linearization
We first project the disorder part of the Hamiltonian onto the new basis. To this end, we first make the transformation
(18) |
where by definition, creates an electron in the state described by . Setting in Eq. (14), one then has
(19) |
Since we are concerned only with the low-energy modes, we can project out all the bands not crossing the Fermi energy. Out of the sum over , this leaves us only with a sum over Furthermore, we can limit the integral over to momenta close to the Fermi points, . This is done by defining the fields living close to the Fermi momenta, where Finally, if the bottom of all the bands is far enough from the Fermi energy (which we shall assume to be the case), then we can approximate the dispersions of the modes near the Fermi points by , and take . Note also that due to the symmetry, Eq. (17), one has Applying the above procedure to Eq. (19), one has
(20) |
where we have defined
(21) |
Finally, we account for the coupling to the superconducting region. At least in principle, one can integrate out the degrees of freedom of the SC region Alicea (2012); Haim et al. (2016). This will result in induced pairing potential operating on the modes living in the normal region,
(22) |
Importantly, only pairing between modes of opposite momenta will open a gap at the Fermi energy. Assuming the Fermi momenta, , are not degenerate (this will generally be the case when breaking symmetry), we can therefore omit all the pairing terms except for . Combining with Eq. (20), the Hamiltonian describing the overall system at low energies is given by
(23) |
which is the Hamiltonian introduced in Eq. (2) of the main text.
ii.1.4 Properties of the disorder term
ii.2 Derivation of the self energy
ii.2.1 Gauging out the diagonal scattering terms
Starting from the low-energy Hamiltonian, Eq. (2) of the main text, we define the fields
(27) |
Inserting this definition into Eq. (2) of the main text, we arrive at an identical Hamiltonian, except that now intra-mode scattering is absent,
(28) |
where,
(29) |
and we have used the fact that To leading order in the disorder strength, the correlations of the new disorder term are unaltered,
(30) |
ii.2.2 Born approximation
We begin by rewriting the Hamiltonian in a BdG form
(31) |
where
(32) |
where