Topological Phase Transition driven by Infinitesimal Instability: Majorana Fermions in Non-Hermitian Spintronics

TL;DR

This paper uncovers a novel topological quantum phase transition in non-Hermitian superconducting nanowires driven by infinitesimal instability, revealing Majorana states and a unique bulk-boundary correspondence with potential experimental realization.

Contribution

It introduces a new type of topological phase transition caused by cascade instability in non-Hermitian systems, with a novel bulk-boundary correspondence and experimental proposal.

Findings

Infinitesimal magnetic field suppresses non-Hermitian skin effect.

Topological phase with Majorana boundary states identified.

Non-Hermitian bulk-boundary correspondence established.

Abstract

Quantum phase transitions are intriguing and fundamental cooperative phenomena in physics. Analyzing a superconducting nanowire with spin-dependent non-Hermitian hopping, we discover a topological quantum phase transition driven by infinitesimal cascade instability. The anomalous phase transition is complementary to the universal non-Bloch wave behavior of non-Hermitian systems. We show that an infinite small magnetic field drastically suppresses the non-Hermitian skin effect, inducing a topological phase with Majorana boundary states. Furthermore, by identifying the bulk topological invariant, we establish the non-Hermitian bulk-boundary correspondence that does not have a Hermitian counterpart. We also discuss an experimental realization of the system by using the spin-current injection to a quantum wire.

Equations

$$H=H_N+H_{\Delta },$$

$$H_N=\sum _{k,{\sigma }_z=\pm }\left[-2t\cos k-\frac{i\Gamma }{2}(1+\sin k{\sigma }_z)\right]a_{k,{\sigma }_z}^{†}{a}_{k,{\sigma }_z},$$

$$H_{\Delta }=\sum _k\left[\Delta a_{k,\uparrow }^{†}{a}_{-k,\downarrow }^{†}+H.c.\right],$$

$$H=\sum _{k,a=\pm }{E}_{k,a}{\overline{\alpha }}_{k,a}{\alpha }_{k,a},$$

$$H=\sum _{i,{\sigma }_z=\pm }\left[-t_{{\sigma }_z}{a}_{i+1,{\sigma }_z}^{†}{a}_{i,{\sigma }_z}-t_{(-{\sigma }_z)}{a}_{i,{\sigma }_z}^{†}{a}_{i+1,{\sigma }_z}-i(t_{+}-t_{-})a_{i,{\sigma }_z}^{†}{a}_{i,{\sigma }_z}\right]+\sum _i\left[\Delta a_{i,\uparrow }^{†}{a}_{i,\downarrow }^{†}+H.c.\right],$$

$$a_{i,\uparrow }={\left(\sqrt{\frac{t_{+}}{t_{-}}}\right)}^i{b}_{i,\uparrow },a_{i,\uparrow }^{†}={\left(\sqrt{\frac{t_{+}}{t_{-}}}\right)}^{-i}{b}_{i,\uparrow }^{†},$$

$$a_{i,\downarrow }={\left(\sqrt{\frac{t_{+}}{t_{-}}}\right)}^{-i}{b}_{i,\downarrow },a_{i,\downarrow }^{†}={\left(\sqrt{\frac{t_{+}}{t_{-}}}\right)}^i{b}_{i,\downarrow }^{†},$$

$$-\sqrt{t_{+}{t}_{-}}{b}_{i+1,{\sigma }_z}^{†}{b}_{i,{\sigma }_z}+H.c.,$$

$$E_{\mathbf{k},a}=\sqrt{{\left\{-2\sqrt{t_{+}{t}_{-}}\cos k-i(t_{+}-t_{-})\right\}}^2+{\Delta }^2},$$

$$H_{\mathrm{ex}}=\delta h\sum _i\left[a_{i,\uparrow }^{†}{a}_{i,\downarrow }+H.c.\right].$$

$$H_{\mathrm{ex}}=\delta h\sum _i\left[{\left(\frac{t_{-}}{t_{+}}\right)}^i{b}_{i,\uparrow }^{†}{b}_{i,\downarrow }+{\left(\frac{t_{+}}{t_{-}}\right)}^i{b}_{i,\downarrow }^{†}{b}_{i,\uparrow }\right],$$

$$|\delta h|{\left(\frac{t_{+}}{t_{-}}\right)}^{L/2}$$

$$\iff |\delta h|$$

$$\underset{\delta h\to 0}{\lim }\underset{L\to \infty }{\lim }\ne \underset{L\to \infty }{\lim }\underset{\delta h\to 0}{\lim }.$$

$$H+H_{\mathrm{ex}}=\frac{1}{2}\sum _{i,j}{\Psi }_i^{†}{\mathcal{H}}_{i,j}^{\mathrm{BdG}}{\Psi }_j,$$

$${\overline{\gamma }}_i={\gamma }_i,$$

$$H_{\mathrm{Boundary}}$$

$${\gamma }_1=\alpha +\overline{\alpha },{\gamma }_2=\frac{\alpha -\overline{\alpha }}{i}.$$

$${\mathcal{H}}_k^{\mathrm{BdG}}=\left[-2t\cos k-\frac{i\Gamma }{2}\right]{\hat{\tau }}_z-\frac{i\Gamma }{2}\sin k{\hat{\sigma }}_z-\Delta {\hat{\sigma }}_y{\hat{\tau }}_y,$$

$${\hat{\tau }}_x({\mathcal{H}}_k^{\mathrm{BdG}}{)}^T{\hat{\tau }}_x=-{\mathcal{H}}_{-k}^{\mathrm{BdG}}.$$

$$(-1{)}^{\nu }=\frac{\mathrm{Pf}({\tau }_x{\mathcal{H}}_{k=\pi })}{\mathrm{Pf}({\tau }_x{\mathcal{H}}_{k=0})}\exp \left(-\frac{1}{2}{\int }_0^{\pi }dk\mathrm{Tr}[{\mathcal{H}}_k^{-1}{\partial }_k{\mathcal{H}}_k]\right),$$

Full text

††preprint: APS/123-QED

Recently, non-Hermitian Hamiltonians Hatano and Nelson (1996); Bender and Boettcher (1998); Bender et al. (2002); Bender (2007); Konotop et al. (2016); El-Ganainy et al. (2018); Feng et al. (2017); Miri and Alù (2019) have attracted much interest in various fields such as open systems Konotop et al. (2016); El-Ganainy et al. (2018), optics Malzard et al. (2015); Mochizuki et al. (2016); Xiao et al. (2017); St-Jean et al. (2017); Harari et al. (2018); Bandres et al. (2018); Feng et al. (2017); Miri and Alù (2019); Ozawa et al. (2019), disordered (and correlated) systems Kozii and Fu ; Zyuzin and Zyuzin (2018); Yoshida et al. (2018); Moors et al. (2019); Yoshida et al. (2019); Hamazaki et al. , and quantum critical phenomena Kawabata et al. (2017); Nakagawa et al. (2018); Xiao et al. ; Dóra et al. . Among them, topological properties of such Hamiltonians have been extensively investigated both in gapped Rudner and Levitov (2009); Hu and Hughes (2011); Esaki et al. (2011); Sato et al. (2012); Poli et al. (2015); Lee (2016); Weimann et al. (2017); Leykam et al. (2017); Kim et al. ; Mochizuki et al. (2016); Xiao et al. (2017); St-Jean et al. (2017); Ozawa et al. (2019); Harari et al. (2018); Bandres et al. (2018); Shen et al. (2018); Kawabata et al. (2018a, b); Yao and Wang (2018); Yao et al. (2018); Kunst et al. (2018); Borgnia et al. ; Yokomizo and Murakami ; Lieu (2018); Liu et al. (2019); Gong et al. (2018); Kawabata et al. (2019a, ); Lee and Thomale ; Lee et al. (a) and gapless phases Yoshida et al. (2019); Kozii and Fu ; Zyuzin and Zyuzin (2018); Moors et al. (2019); Malzard et al. (2015); Xu et al. (2017); Cerjan et al. (2018, ); Zhen et al. (2015); González and Molina (2016, 2017); Molina and González (2018); Chernodub (2017); Chernodub and Cortijo ; Zyuzin and Simon ; Zhou et al. (2018); Carlström et al. ; Okugawa and Yokoyama (2019); Budich et al. (2019); Yang and Hu (2019); Zhou et al. (2019); Wang et al. (2019); Luo et al. ; Lee et al. (b); Kawabata et al. (2019b); Yang et al. , and a lot of essential differences from the Hermitian cases have been pointed out. For instance owing to the complex nature of the energy spectrum, there are several distinct definitions of energy gaps, which amplify the possibility of topological phases Gong et al. (2018); Kawabata et al. (2019a, ).

Although the non-Hermitian physics under the periodic boundary condition (PBC) can be investigated by using mathematical tools developed in the Hermitian physics Hasan and Kane (2010); Qi and Zhang (2011); Schnyder et al. (2008); Kitaev et al. (2009); Ryu et al. (2010); Freed and Moore (2013); Gomi ; Morimoto and Furusaki (2013); Shiozaki and Sato (2014); Chiu et al. (2016), it is not easy to treat them under the open boundary condition (OBC) because of the non-Bloch wave behavior Hatano and Nelson (1996). For instance, the phase diagrams under the OBC are different from those under the PBC in several non-Hermitian models Yao and Wang (2018); Yao et al. (2018); Kawabata et al. (2018b), which obscures the conventional bulk-boundary correspondence. Thus far, the non-Hermitian bulk-boundary correspondence has not been established except for several attempts Yao et al. (2018); Yao and Wang (2018); Kunst et al. (2018); Borgnia et al. ; Yokomizo and Murakami .

In this Letter, we construct and analyze a simple non-Hermitian lattice model that describes a one-dimensional $s$-wave superconductor with spin-dependent asymmetric hopping. Although this model is topologically nontrivial under the PBC, the SU(2) imaginary gauge transformation reveals that the system under the OBC does not show any topological boundary modes. Interestingly, however, we find that this mismatching is drastically remedied by an infinitesimal transverse magnetic field in the thermodynamic limit. Performing the numerical diagonalization with a small magnetic field, we find the missing Majorana boundary modes, which are protected by the $\mathbb{Z}_{2}$ topological invariant. This finding establishes the presence of the non-Hermitian bulk-boundary correspondence that has no analog in the Hermitian physics. Finally we also discuss an experimental realization by using the spin current injection to a quantum wire.

Periodic boundary condition.—In order to grasp a rough idea, we first analyze the infinite lattice system under the PBC, where the momentum-space picture is useful. For Hermitian systems, the Majorana fermions are known to appear on boundaries of a spinless $p$-wave superconductor Kitaev (2001), though it is not experimentally relevant thus far. Instead of the direct realization, several schemes have been proposed to effectively create the spinless Cooper pairing Sato and Ando (2017); Sato et al. (2009); Lutchyn et al. (2010); Oreg et al. (2010). For example, in a quantum wire with the Rashba spin-orbit interaction and the Zeeman magnetic field, the spin degree of freedom is frozen due to the spin-momentum locking Lutchyn et al. (2010); Oreg et al. (2010). In this paper, we use the spin-momentum locked dissipation to realize a similar spinless situation.

Let us consider the following Hamiltonian:

[TABLE]

where $(a^{\dagger},a)$ are spin-$1/2$ fermionic (electron) creation and annihilation operators, and real parameters $t,\Gamma$ and $\Delta$ describe the kinetic energy, dissipation (loss for particles and gain for holes), and an $s$-wave gap function 111Since the phase of gap function does not change the physics, the gap function is taken to be real., respectively. The spin- and momentum-dependent dissipation is a non-Hermitian variant of spin-orbit interaction. For $\Gamma>\Delta$, only the left-(right-)going electrons with up-(down-)spin can participate in the Cooper pairing (Fig. 1), and thus we obtain an effective spinless superconductor. Diagonalizing the Hamiltonian, we have

[TABLE]

where $(\overline{\alpha},\alpha)$ are creation and annihilation operators of the Bogoliubov quasi-particles, and $E_{k,\pm}=\sqrt{\left[-2t\cos k-i\Gamma/2(1\pm\sin k)\right]^{2}+\Delta^{2}}$ are their energy dispersion [Fig.2(b)]. Note that $\overline{\alpha}$ is no longer the Hermitian conjugate of $\alpha$ under the non-Hermiticity, while the conventional anti-commutation relation $\{\alpha_{k,a},\overline{\alpha}_{k^{\prime},a^{\prime}}\}=\delta_{kk^{\prime}}\delta_{aa^{\prime}}$ holds, and $\alpha$ annihilates the BCS vacuum $|0\rangle$ (see Supplemental Material (SM) or Ref. Yamamoto et al. ).

In the real-space picture, Eq. (LABEL:momentumspace) can be written as a simple lattice model

[TABLE]

where $i$ is the site index, and $t_{\pm}=t\pm\Gamma/4$. Note that the normal part of the Hamiltonian includes the non-Hermitian asymmetric hopping terms whose asymmetry depends on the $z$-component spin. These hopping terms are regarded as those of a stacked Hatano-Nelson model Hatano and Nelson (1996) with up and down spins.

Open boundary condition.—Thus far, we have introduced the non-Hermitian spin-orbit interaction for the purpose of the realization of Majorana boundary states. This proposal is based on the momentum-space picture, which corresponds to the PBC in real space. In the presence of non-Hermiticity, however, extensive sensitivity of the energy spectrum to boundary conditions obscures the naive bulk-boundary correspondence.

Let us impose the OBC on the Hamiltonian (3). To consider the eigenvalues of Eq. (3), we perform the SU(2) imaginary gauge transformation, which generalizes the imaginary gauge transformation Hatano and Nelson (1996) used in the analysis of the Hatano-Nelson model:

[TABLE]

where $(b^{\dagger},b)$ are creation and annihilation operators in the new basis. Under this transformation, the spin-dependent asymmetric hopping terms are mapped to the spin-independent symmetric ones:

[TABLE]

while the other terms are invariant. Thus, the Hamiltonian (3) is mapped to a conventional $s$-wave superconductor, apart from a constant dissipation term. Although this transformation changes the eigenfunctions drastically, it does not change the eigenvalues because it is a similarity transformation. Thus, the Hamiltonian (3) has the same energy spectrum as that of the mapped $s$-wave superconductor. This implies that the lattice model (3) is topologically trivial under the OBC and has no Majorana boundary modes. The energy spectrum with a constant dissipation does not depend on the boundary condition in the thermodynamic limit as in the case of the Hermitian physics, and it is calculated as

[TABLE]

which does not depend on the band index $a=\pm$ 222Note that the imaginary gauge transformation and the above mapping is not consistent with the PBC.. The obtained spectrum is drastically different from Eq. (2) [Fig.2(a)].

At first sight, the above consideration seems to ruin the scenario of Majorana modes by the non-Hermitian spin-orbit interaction. Interestingly, however, an infinitesimal perturbation resurrects this scenario as shown below.

Phase transition driven by infinitesimal instability.—In the case of the OBC, asymmetry of the hopping induces the accumulation of eigenstates near boundaries. This non-Bloch wave behavior (non-Hermitian skin effect) stems from the exponentially growing form factor in Eq. (4), which breaks the conventional bulk-boundary correspondence. In the case of our model, however, this effect is drastically suppressed by adding the transverse Zeeman term

[TABLE]

This term is not invariant under the SU(2) imaginary gauge transformation,

[TABLE]

and thus our model is no longer equivalent to the trivial $s$-wave superconductor even when $\delta h$ is very small. Roughly speaking, this perturbation cannot be ignored if

[TABLE]

where $L$ is the system size, and $\alpha_{1},\alpha_{2}>0$ are constants. After taking the thermodynamic limit, the infinitesimally small perturbation drastically changes the energy spectrum from the unperturbed one. In other words, the order-of-limits changes the physics:

[TABLE]

Similar high sensitivity of eigenvalues to the pertubation is also discussed in mathematics Reichel and Trefethen (1992). It is important to note that nonlocal perturbations such as the coupling between two ends destroy the OBC nature, while local perturbations such as the transverse magnetic field in our model can preserve it. This difference leads to the emergence of the topological boundary modes. In general, such an infinitesimal instability against a local perturbation occurs when (i) the generalized Bloch Hamiltonian Yokomizo and Murakami is divided into two different eigen sectors of a unitary symmetry operator, (ii) the two sectors are energetically degenerate, but have different values that characterize the non-Hermitian skin effects, and (iii) those two sectors are coupled via the perturbation (see details for SM) 333In our model, the unperturbed Hamiltonian can be block-diagonalized with respect to the $z$-component spin.

The terms in Eq. (8) grow exponentially near boundaries, getting rid of the accumulated states of the skin effect. This implies that the OBC bulk spectrum would be close to the PBC one [Eq.(2)] in the presence of the perturbation. In the following, we perform the numerical diagonalization to confirm this expectation.

Numerical diagonalization.—We rewrite the Hamiltonian (3) with the small perturbation (7) in the Nambu representation:

[TABLE]

where $\Psi^{\dagger}_{i}=(a^{\dagger}_{i,\uparrow},a^{\dagger}_{i,\downarrow},a_{i,\uparrow},a_{i,\downarrow})$ is the Nambu spinor, and $\mathcal{H}^{\mathrm{BdG}}$ is the Bogoliubov-de Gennes (BdG) Hamiltonian matrix (explicit form in SM). Using the Nambu representation, we numerically calculate the energy spectra of the finite lattice system ($L=100$) for various transverse magnetic fields and plot them in Fig. 2 (c). The model parameters are $t_{+}=1,t_{-}=0.7$ and $\Delta=0.2$. $\Gamma$ is set to be larger than $\Delta$ in order to realize the spin-momentum locked Cooper pairing (Fig. 1).

In the absence of the magnetic field, the result is well approximated by Eq. (6). For $\delta h\gtrsim 10^{-8}$, the energy spectrum differs from Eq. (6), which is consistent with the value $(t_{-}/t_{+})^{L/2}\simeq 2\times 10^{-8}$ in Eq. (9). For $\delta h\gtrsim 10^{-4}$, we find two superposition states of Majorana fermions localized on two boundaries of the lattice system (Fig. 3), while the bulk spectrum surrounds the origin of the complex plane and has the similar shape as the spectrum under the PBC. As we expected, the small perturbation changes the spectrum into the topological one with Majorana fermions.

The Majorana fermions satisfy the non-Hermitian Majorana condition (see SM) 444Since the non-Hermitian Majorana fermions also satisfy the important property that the creation and annihilation of particles are equivalent, they have the non-Abelian statistics as in the case of Hermitian one. All known exotic phenomena such as the fractional Josephson effect are expected to be observed.:

[TABLE]

where $i=1,2$ denote the boundaries on which the Majorana fermions localize. The effective theory of the two edges (1 and 2) are given by

[TABLE]

where $\epsilon$ is the complex finite-size coupling, and $(\alpha,\overline{\alpha})$ are the fermion operators constructed from the Majorana fermions:

[TABLE]

In the present numerical calculation, the fermion energy $\epsilon$ takes the imaginary number ($\epsilon=-4\times 10^{-5}i$).

Non-Hermitian topological phase.—In the presence of the non-Hermitian skin effect, it is necessary to define the topological number by using the non-Bloch wave functions Yao et al. (2018); Yao and Wang (2018); Kunst et al. (2018); Yokomizo and Murakami . In our case with small magnetic field, however, the numerical calculation indicates that the non-Bloch wave behavior reduces to the conventional Bloch one in the thermodynamic limit. In fact, it can be proven by combing the method in Ref. Yokomizo and Murakami and a symmetry consideration (see SM). In the following, we identify the topological invariant that protects the Majorana zero mode in terms of the BdG Bloch Hamiltonian in PBC.

Let us consider the BdG Bloch Hamiltonian constructed from Eq. (LABEL:momentumspace)

[TABLE]

where $\hat{\sigma}$s and $\hat{\tau}$s are the Pauli matrices in the spin and particle-hole space, respectively. This Hamiltonian belongs to class D in the the Altland-Zirnbauer Altland and Zirnbauer (1997) classification, and supports the particle-hole symmetry

[TABLE]

Note that the transpose in the charge conjugation is not equivalent to the complex conjugation for the non-Hermitian case cha .

The bulk band is not gapped in a usual sense since the bulk spectrum in Fig.1(b) is totally connected in the complex energy spectrum. Therefore, the conventional class D topological invariant or its non-Hermitian variant is no longer well-defined; our obtained topological phase originates essentially from non-Hermiticity Gong et al. (2018). Hence we use another topological invariant intrinsic to non-Hermitian systems. We propose the following $\mathbb{Z}_{2}$ invariant to characterize the present topological phase:

[TABLE]

where $k=0,\pi$ are the time-reversal-invariant points, and the superscript “BdG” is omitted. The topological invariant is well-defined unless ${\rm det}{\cal H}_{k}=0$ (i.e. $|\Gamma|=|\Delta|$). The competition between $\Delta$ and $\Gamma$ determines the topological phase; the $\mathbb{Z}_{2}$ index is trivial for $|\Gamma|<|\Delta|$ and nontrivial for $|\Gamma|>|\Delta|$ (see SM). Thus, the present strong $\Gamma$ case, where the boundary modes exist, is topologically non-trivial, while the weak $\Gamma$ case, where the boundary modes are absent (see SM), is topologically trivial.

Although non-Hermitian topological phases whose gap-closing point is defined by $\det\mathcal{H}_{k}=0$ had been understood in the absence of boundaries, their topological nature under the OBC was unclear so farGong et al. (2018). Our results firstly establish the bulk-edge correspondence in this context, while whether it can be extended to the other Altland-Zirnbauer classes is a nontrivial open question. The removability of the non-Hermitian skin effect under given symmetry would play an essential role to realize the bulk-edge correspondence.

Spintronic application.—We finally discuss an experimental realization of the Hamiltonian (3). The nontrivial task is to implement the spin-dependent asymmetric hopping terms, or equivalently, the non-Hermitian spin-orbit interaction. Although the full implementation of the $\sin k\ \sigma_{z}$ term seems to be difficult, we may introduce the essentially the same effect near the Fermi level, where the superconducting pairing occurs. In order to introduce the spin-momentum locked effect near the Fermi level, we propose to a pure spin current injection to a quantum wire. Under this non-equilibrium circumstance, modes with spin current $\sigma_{z}\partial E/\partial k$ opposed to the injected spin current would have a shorter life time by scatterings. For sufficiently large imbalance of dissipation $\Gamma$, we obtain the situation in Fig. 1. Another promising platform is ultra-cold atom systems. The possible realization of the asymmetric hopping term has been theoretically proposed in Ref. Gong et al. (2018), which would be generalized to our model with the spin degrees of freedom.

In summary, we have constructed and analyzed a simple non-Hermitian lattice model of an $s$-wave superconductor that realizes a novel topological phase. The topological phase transition is driven by an infinitesimal external magnetic field. We have also discussed an experimental realization in spintronics. The present model provides the first concrete example of the non-Hermitian topological phase that does not have Hermitian counterparts.

Acknowledgements

We acknowledge many fruitful discussions with Ken Shiozaki, Takumi Bessho, and Ryusuke Hamazaki. N.O. is grateful to Kohei Kawabata and Ken Mochizuki for introducing various topics about the non-Hermitian physics. This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science” (KAKENHI Grant No. JP15H05855) from the Japan Society for the Promotion of Science (JSPS). N.O. was supported by KAKENHI Grant No. 18J01610 from the JSPS. M.S. was supported by KAKENHI Grant No. JP17H02922 from the JSPS.