Exceptional degeneracies in non-Hermitian Rashba semiconductors

The effect of dissipation, often seen as detrimental, has recently attracted a paramount attention in physics due its potential to induce novel phenomena with technological applications [1, 2, 3, 4, 5, 6, 7]. Dissipation naturally occurs in open systems and is effectively described by non-Hermitian (NH) Hamiltonians [8, 9, 10]. The most salient property of these NH models is the emergence of a complex spectrum with degeneracies known as exceptional points (EPs) [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], where eigenstates and eigenvalues coalesce, in stark contrast to Hermitian systems. While EPs were initially seen as a mathematical curiosity, it has been recently shown that they represent truly topological objects enabling topological phases with no counterpart in Hermitian setups [3, 4, 7].

The concept of EPs and their topological properties have recently been generalized to higher dimensions, giving rise to exceptional degeneracies in the form of lines, rings, and surfaces as generic and stable bulk phenomena. These exceptional degeneracies have already proven crucial to enable unique topological effects [3, 4], such as enhanced sensing [22, 23], unidirectional lasing [24, 25], and bulk Fermi arcs [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], which do not have a Hermitian analog. Despite the numerous theoretical and experimental studies, however, the majority of them has investigated exceptional degeneracies mostly in optical and photonic systems [1, 2, 6, 5].

Material junctions have been shown to offer another powerful and experimentally relevant platform for the realization of exceptional degeneracies [39, 40, 41, 42, 43, 44, 45, 46, 47]. Material junctions constitute electronic open systems with a clear NH description that is well-established in quantum transport [48]. In this regard, open semiconductor-superconductor junctions have been shown to host several classes of exceptional degeneracies [41, 42, 47], which characterize distinct NH topological phases without analog in the Hermitian regime. Notwithstanding the importance of this study, it only focused on the impact of non-Hermiticity on the superconducting properties, such as on its particle-hole symmetry and energy gap, leaving largely unexplored the role of non-Hermiticity on the semiconductor. Of particular importance in such semiconductors is their intrinsic Rashba spin-orbit coupling (SOC) [49, 50, 51], which arises due to the lack of structural inversion symmetry and induces a spin-momentum locking [52, 53]. This property of the Rashba SOC has been shown to enable a great control of the electron’s spin, a crucial ingredient for several spintronics and topological phenomena [54], already proven useful in recent experiments [54, 55, 56, 57, 58, 53]. However, despite the advances, the interplay between Rashba SOC and non-Hermiticity still remains unknown, specially the potential of this combination for inducing exceptional degeneracies.

In this work we consider a realistic NH Rashba semiconductor and discover the formation of stable and highly tunable bulk exceptional degeneracies. In particular, we engineer a NH Rashba system by coupling a two-dimensional (2D) semiconductor with Rashba SOC to a semi-infinite ferromagnet lead, an easily achievable heterostructure using e.g., InAs or InSb semiconductors [54, 55, 56, 57, 58, 53], see also [59]. We discover that EPs appear along rings in 2D momentum space and mark the ends of lines formed by the coalescence of eigenvalues at finite real energy. The emergence of eigenvalues at the same real energy resembles the formation of bulk Fermi arcs, which, although initially conceived at zero real energy, have recently been generalized to finite real energies [60]. We also show that the exceptional degeneracies found here can be controlled by an in-plane Zeeman field but then higher values of non-Hermiticity are required. Furthermore, we find that the spin projections coalesce at the exceptional degeneracies and can even develop larger values than in the Hermitian phase due to non-Hermiticity. Finally, we find that the exceptional degeneracies induce large spectral features, which can be detected, e.g., using angle-resolved photoemission spectroscopy (ARPES).

We consider an open system by coupling a 2D semiconductor with Rashba SOC to a semi-infinite ferromagnet lead, as schematically shown in Fig. 1. This open system is modelled by the following effective NH Hamiltonian

where $H_{\rm R}$ describes the closed system, which is Hermitian, and $\Sigma^{r}(\omega=0)$ is the zero-frequency retarded self-energy due to the coupling to the semi-infinite ferromagnet lead. More specifically, the closed system corresponds to a 2D Rashba semiconductor described by

where $\xi_{k}=\hbar^{2}(k_{x}^{2}+k_{y}^{2})/2m-\mu$ is the kinetic energy, $k_{x(y)}$ the momentum along $x(y)$, $\mu$ is the chemical potential, $\alpha$ is the Rashba SOC strength, and $\sigma_{j}$ the $j$-th Pauli matrix in spin space, and without loss of generality we assume $\hbar=m=1$. The Hamiltonian $H_{\rm R}$ in Eq. (2) describes well the Rashba SOC in 2D semiconductors, such as in InAs or InSb, which are also within experimental reach [54, 55, 56, 57, 58, 53, 59]. As an external control knob, we also consider that the closed system is subjected to an applied magnetic field along $x$ which produces a Zeeman field $B$, denoted by the brown arrow in Fig. 1. The effect of this Zeeman field is modelled by adding $B\sigma_{x}$ to $H_{\rm R}$ in Eq. (2) which induces a renormalization to the SOC term $\alpha k_{y}\sigma_{x}$.

The zero-frequency self-energy $\Sigma^{r}(\omega=0)$ in Eq. (1), whose independence of frequency $\omega$ is well justified in the wide-band limit [48], is analytically obtained and given by [45, 46]

where $\Gamma=(\Gamma_{\uparrow}+\Gamma_{\downarrow})/2$ and $\gamma=(\Gamma_{\uparrow}-\Gamma_{\downarrow})/2$, with $\Gamma_{\sigma}=\pi|t^{\prime}|^{2}\rho_{\rm L}^{\sigma}$, being $t^{\prime}$ the hopping amplitude into the lead from the 2D Rashba semiconductor and $\rho_{\rm L}^{\sigma}$ the surface density of states of the lead for spin $\sigma=\uparrow,\downarrow$. It is thus evident that $\Gamma_{\sigma}$ characterizes the coupling amplitude between the lead and the 2D Rashba semiconductor. For completeness, the derivation of the self-energy given by Eq. (3) is presented in Appendix A.

The self-energy in Eq. (3) is imaginary and thus NH, a unique effect emerging due to the coupling to the semi-infinite ferromagnet lead. Thus, the imaginary self-energy renders the total effective Hamiltonian $H_{\rm eff}$ to be NH, introducing dramatic changes in the properties of the closed system $H_{\rm R}$, which is the focus of this work here. In particular, we are interested in investigating the interplay between non-Hermiticity and Rashba SOC in 2D semiconductors and how it leads to the formation of bulk exceptional degeneracies.

To identify the emergence of bulk exceptional degeneracies, we obtain the eigenvalues and eigenvectors of the effective Hamiltonian $H_{\rm eff}$ given by Eq. (1). At zero Zeeman field $B=0$, they are given by

where $|k|^{2}=k_{x}^{2}+k_{y}^{2}$ and $\pm$ labels the two distinct bands which have a mixture of $\uparrow$ and $\downarrow$ spins. At finite Zeeman fields $B$, the eigenvalues and eigenvectors can be obtained by replacing $\alpha k_{y}\rightarrow B+\alpha k_{y}$ in Eqs. (4) and (5). An immediate observation in the energies and wavefunctions is their dependence on the couplings $\Gamma_{\uparrow,\downarrow}$ via $\gamma$ and $\Gamma$, already revealing a clear impact of the NH self energy given by Eq. (3). This can be visualized in Fig. 2, where we plot the eigenvalues and eigenvectors as a function of momenta $k_{x}$ and $k_{y}$ at zero Zeeman field. At $\Gamma_{\uparrow,\downarrow}=0$, the system described by Eq. (1) is Hermitian and its two eigenvalues in Eq. (4) are real: they correspond to two parabolas shifted by $\pm k_{\rm soc}=\pm m\alpha/\hbar^{2}$ that intersect at $k_{x,y}=0$, see gray curves in Fig. 2(a). Here, their respective eigenvectors are orthogonal as expected for Hermitian systems, see Eq. (5). While this Rashba system is gapless at zero momenta, finite values of $k_{y}$ opens a gap at $k_{x}=0$ even at zero Zeeman field. A finite in-plane Zeeman field opens a gap at $k_{x,y}=0$, also known as helical gap, where states are counter propagating and have distinct spins [61, 59, 62, 63].

At any $\Gamma_{\uparrow,\downarrow}\neq 0$, the two eigenvalues $E_{\pm}$ acquire finite imaginary parts that strongly depend on momenta, see Eq. (4). The formation of eigenvalues with imaginary terms signals the emergence of NH physics as a pure effect due to the ferromagnet lead [45, 46, 47]. The inverse of these imaginary parts define the quasiparticle lifetime in the 2D Rashba semiconductor, thus offering a clear physical meaning of non-Hermiticity [48]. From the dependence of the eigenvalues on $\gamma$ in Eq. (4), we note that their imaginary parts exhibit a non trivial behaviour. In fact, at $\gamma=0$, which is satisfied when $\Gamma_{\uparrow}=\Gamma_{\downarrow}$, the two eigenvalues acquire the same imaginary part equal to $-i\Gamma$. This situation remains for $\gamma\neq 0$ only when $|\gamma|<\alpha|k|$. At these conditions, therefore, quasiparticles in the Rashba semiconductor have the same and constant lifetime.

The behaviour of the eigenvalues becomes more interesting when $\gamma\neq 0$ and $|\gamma|>\alpha|k|$, which then allows the two eigenvalues to acquire distinct imaginary parts. This is visualized in Fig. 2(a) where we plot the real and imaginary parts of the eigenvalues at zero Zeeman field and at $k_{y}=0$, see solid blue and dashed red curves. Surprisingly, we observe that both the real and imaginary parts simultaneously merge at finite energy into a single value at special positive and negative momenta. The regime with $\gamma\neq 0$ and $|\gamma|>\alpha|k|$ not only affects the eigenvalues but also the eigenvectors, which can be noticed by inspecting their inner product or overlap $\psi_{+-}=\bra{\psi_{+}}\ket{\psi_{-}}$, $\psi_{\pm}$ is given by Eq. (5). This overlap is depicted by the green curve in Fig. 2(a), where we see that it reaches $1$ at the special momenta, a situation that can only occur if the eigenvectors are parallel. Although unusual, the behaviour of eigenvectors and eigenvalues seen in Fig. 2(a) is common in NH Hermitian systems. In particular, the spectral degeneracies occurring at the special momenta discussed here signal the emergence of EPs, whose formation can be understood by noting that they occur when the square root in Eq. (4) vanishes. At zero Zeeman field, the condition for the formation of EPs is given by

while at finite Zeeman field we have to change $\alpha k_{y}\rightarrow B+\alpha k_{y}$. At this EP condition, the eigenvalues and eigenvectors become,

where the values of momenta satisfy the condition given by Eq. (6). Hence, the two eigenvalues (eigenvectors) coalesce at EPs: instead of having two eigenvalues (eigenvectors), only one eigenvalue (eigenvector) remains at EPs, see Fig. 2(a). For $k_{y}=0$, the EP occur at positive and negative momenta given by $\pm k^{\rm EP}_{x}=\pm(\gamma/\alpha)^{2}$ at $B=0$, marking the ends of the cyan region in Fig. 2(a). Between these two EP points, the eigenvalues have the same real part determined by the quadratic dispersion $\xi_{k}$ and different imaginary parts determined by $-i\Gamma\pm i\sqrt{\gamma^{2}-\alpha^{2}|k|^{2}}$, see cyan region in Fig. 2(a). In relation to the eigenvectors, the fact that both merge into a single eigenvector implies that they are parallel, a situation that has no analog in the Hermitian regime but expected at EPs of NH systems [3]. This effect can be also seen in the inner product (or overlap) between the two eigenvectors $\bra{\psi_{+}}\ket{\psi_{-}}$ in Eq. (5), where it reaches 1 at the EPs, see green curve in Fig. 2(a). We remark that for EPs to emerge it is crucial to have eigenvalues with different imaginary parts, which is only achieved when $\Gamma_{\uparrow}\neq\Gamma_{\downarrow}$, thus pointing out the necessity of a ferromagnet lead for the NH features discussed in Fig. 2(a). In passing, we note that having bulk energy lines due to eigenvalues with the same real part resembles the formation of bulk Fermi arcs [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], although here they occur at finite real energy in contrast to the common expectation at zero real energy. In this regard, very recently, the definition of bulk Fermi arcs has been generalized to any two eigenvalues with the same real energy [60], suggesting that the bulk energy lines found here might be an example of bulk Fermi arcs. However, the detail properties of this NH bulk effect require a throughout investigation which will be addressed elsewhere.

Furthermore, another property of the EPs determined by the condition in Eq. (6) is that they occur along a ring defined by $\alpha^{2}(k_{x}^{2}+k_{y}^{2})=\gamma^{2}$ at $B=0$ or by $\alpha^{2}k_{x}^{2}+(\alpha k_{y}+B)^{2}=\gamma^{2}$ at $B\neq 0$. To support this idea, in Fig. 2(b,d) we plot the difference between real and imaginary parts of the eigenvalues, namely, ${\rm Re}\Delta E={\rm Re}(E_{+}-E_{-})$ and ${\rm Im}\Delta E={\rm Im}(E_{+}-E_{-})$, as a function of $k_{x}$ and $k_{y}$. In this case, the blue regions indicate ${\rm Re}\Delta E=0$ and ${\rm Im}\Delta E=0$, with their borders marking the occurrence of rings. To highlight these rings, in Fig. 2(b,d) we also plot the condition given by Eq. (6) in dashed cyan color. The nature of these rings can be also seen in Fig. 2(c), where we plot the eigenvector overlap, which acquires $1$ exactly along them which implies that the eigenvectors here become parallel as at the EPs discussed previously. This thus demonstrates that the rings seen in Fig. 2 truly represent bulk exceptional degeneracies of NH 2D Rashba semiconductors and can be referred to as exceptional rings.

To induce the formation of the exceptional degeneracies, or exceptional rings, it is sufficient the interplay between non-Hermiticity and SOC, as clearly seen in Eq. (6). While this conclusion is already evident in Fig. 2, to further support it, in Fig. 3(a) we present ${\rm Re}\Delta E$ as a function of the SOC strength $\alpha$ and coupling $\Gamma_{\uparrow}$ at finite momenta and zero Zeeman field. We obtain that the region with ${\rm Re}\Delta E=0$ increases following a triangular-shaped profile depicted in blue, which is delimited by $\pm\sqrt{\gamma^{2}/|k|^{2}}$ indicated by cyan dashed lines. At fixed momenta, the SOC drives the formation of EPs, requiring lower SOC when non-Hermiticity is small. By fixing only one momentum coordinate, it is also possible to induce EPs, as seen in Fig. 3(d). Furthermore, another possibility to control the appearance of EPs is by an in-plane Zeeman field along $x$ as considered in Fig. 1. Thus, in Fig. 3(b) we show ${\rm Re}\Delta E$ as a function of $\alpha$ and $\Gamma_{\uparrow}$ at finite $B$, while in Fig. 3(c) we show ${\rm Re}\Delta E$ as a function of $B$ and $\Gamma_{\uparrow}$. In this case, we identify two relevant features. First, at finite SOC and finite momenta, the non-Hermiticity needed to induce EPs needs to overcome the effect of the Zeeman field $B$, thus requiring larger non-Hermiticity than in the absence of $B$ [Fig. 3(b)]. Second, at all fixed parameters, the Zeeman field drives the emergence of EPs [Fig. 3(b)]; the EPs here are marked by the cyan curves which correspond to $-\alpha k_{y}\pm\sqrt{\gamma^{2}+\alpha^{2}k_{x}^{2}}$. In sum, the bulk exceptional degeneracies found in 2D Rashba semiconductors exhibit a high degree of tunability by SOC, momenta, and Zeeman field, which could be relevant for their realization and subsequent observation.

Having established the emergence of exceptional degeneracies in the bulk of 2D Rashba semiconductors, now we turn our attention to how the spins here behave under non-Hermiticity. This is motivated by the fact that it is the spin an important quantity for several phenomena in semiconductors, useful for spintronics and topological phenomena. In particular, in this part we focus on the spin expectation values, which here will be referred to as spin projections and are obtained by

where $\Psi^{\dagger}_{\eta}$ is given by Eq. (5) with $\eta=\pm$ and $\sigma_{j}$ the $j$-th spin Pauli matrix. Thus, $S^{\eta}_{j}$ represents the spin projection along $j$ axis associated to $\eta=\pm$. By plugging Eq. (5) into Eq. (8), we obtain

for the spin projections along $x$ and $y$, respectively, while $S^{\pm}_{z}=0$ along $z$. Note that here $|k|^{2}=k_{x}^{2}+k_{y}^{2}$ and $\gamma=(\Gamma_{\uparrow}-\Gamma_{\downarrow})/2$ characterizes the amount of non-Hermiticity due to the ferromagnet lead, see Eqs. (1) and (3). As before, the effect of the Zeeman field along $x$ considered in Fig. 1 can be included by replacing $\alpha k_{y}\rightarrow B+\alpha k_{y}$. The expressions given by Eqs. (9) are relatively simple and permit us to identify the impact of non-Hermiticity on the spin projections by naked eye. In the Hermitian regime, when $\gamma=0$, the spin projections reduce to $S^{\pm}_{x}=\pm k_{y}/|k|$ and $S^{\pm}_{y}=\mp k_{x}/|k|$, as expected [52, 53]. Note that $S^{+}_{y(x)}$ and $S^{-}_{y(x)}$ change their sign when $k_{x(y)}$ varies from negative to positive values passing through $k_{x(y)}=0$, see gray and pink curves in Fig. 4(a) showing the behaviour of $S^{+}_{y}$. The sign of $S^{\pm}_{y(x)}$ remains, however, upon variations of $k_{y(x)}$, as depicted in gray and pink curves in Fig. 4(c).

For finite non-Hermiticity, characterized by $\gamma\neq 0$, the behaviour of the spin projections $S^{\pm}_{x(y)}$ is highly unusual. A finite $\gamma$ generates a linear in momentum term proportional to $k_{x(y)}\gamma$ for $S_{x(y)}$ and renormalizes the Hermitian component with $\sqrt{\alpha^{2}|k|^{2}-\gamma^{2}}$, see first and second terms in Eqs. (9). Both terms reveal a unique effect of non-Hermiticity. The first part of $S^{\pm}_{x(y)}$, proportional to $k_{x(y)}\gamma$, is always real and appears along the same direction of the spin projection, in contrast to the Hermitian contribution where $S^{\pm}_{x(y)}$ is only finite along $y(x)$. The second part of $S^{\pm}_{x(y)}$ is real for $|\alpha||k|>|\gamma|$, which then adds up to the first part, but becomes imaginary for $|\alpha||k|<|\gamma|$. Thus, the appearance of an imaginary part in the spin projections for $|\alpha||k|<|\gamma|$ can be interpreted as a signal of their lifetime, which becomes highly anisotropic in momentum space. At $\alpha^{2}|k|^{2}-\gamma^{2}=0$, the second term in Eqs. (9) vanish and the spin projections $S^{\pm}_{j}$ coalesce, namely, they merge into a single value that is given by

Interestingly, the condition $\alpha^{2}|k|^{2}-\gamma^{2}=0$, which leads to this spin projection coalescence, is the same condition that determines the formation of exceptional degeneracies discussed in previous section, see Eq. (6). Thus, the coalescence effect of $S^{\pm}_{j}$ can be seen as unique NH effect without analog in Hermitian systems. We also note that along the lines connecting these exceptional degeneracies, which correspond to energy lines that resemble bulk Fermi arcs, the spin projections acquire a finite imaginary part with a natural physical interpretation as discussed in previous paragraph.

In order to gain visual understanding of the spin projection coalescence, in Fig. 4 we plot the real and imaginary parts of $S^{\pm}_{y}$ as a function of momenta (a-d) and in the $B-\gamma$ plane (e,f). At $k_{y}=0$, the real part of the spin projections $S^{\pm}_{y}$ vanishes along a line of $k_{x}$ and the ends of such line mark the EP momenta obtained from Eq. (6) and given by $|k_{x}^{\rm EP}|=|\gamma|/|\alpha|$, see solid and dashed blue curves in Fig. 4(a); see also Eqs. (9) and (10). The imaginary part of $S^{\pm}_{y}$ undergoes a coalescence effect as well at the EP momenta $\pm k_{x}^{\rm EP}$ but acquires large values between them and vanish at $k_{x}=0$ [Fig. 4(b)]. For $k_{y}>0$, the coalescence effect persists, with smaller imaginary parts, but the real part does not vanish anymore and, instead, develops a maximum at $k_{x}=0$ favouring a large positive spin projection along $y$, see Eqs. (9) and (10). For $k_{y}<0$, the spin projection $S^{\pm}_{y}$ has instead a minimum at $k_{x}=0$, favouring a large negative spin projection along $y$. The coalescence of spin projections is also observed in Fig. 4(c,d), where we plot the real and imaginary parts of $S^{\pm}_{y}$ as a function of $k_{y}$ at fixed values of $k_{x}$. At $k_{x}=0$, no EP transition is observed in $S^{\pm}_{y}$ because the square root term that gives rise to EPs is multiplied by zero and hence vanishes, see blue curves in Fig. 4(c,d) and also Eqs. (9). However, for finite $k_{x}$, the spin projections develop a clear EP transition, revealing that their coalescence is a highly tunable NH bulk effect.

The spin projection coalescence discussed above requires finite momenta, Rashba SOC, and non-Hermiticity, a combination of ingredients inherent to NH Rashba semiconductors. Furthermore, it is also possible to tune and control the spin projections by an in-plane Zeeman field $B$, e.g., along $x$ as sketched in Fig. 1, see also discussions below Eqs. (2) and (9). To support this idea, in Fig. 4(e,f) we present the Re and the Im parts of the difference between spin projection along $y$, ${\rm Re}\Delta S_{y}={\rm Re}(S^{-}_{y}-S^{+}_{y)}$ and ${\rm Im}\Delta S_{y}={\rm Im}(S^{-}_{y}-S^{+}_{y)}$, at fixed $k_{x,y}$ as a function of $B$ and $\gamma$. Here, the blue regions indicate ${\rm Re}\Delta S_{y}=0$ and ${\rm Im}\Delta S_{y}=0$ and their borders show the exceptional degeneracies, indicated in cyan dashed curves. At fixed non-Hermiticity ($\gamma$), the Zeeman field $B$ induces the coalescence of spin projections $S_{y}^{\pm}$ at EPs, which requires small (large) $B$ for weak (strong) non-Hermiticity. Therefore, Zeeman fields offer another possibility for tuning and controlling the spin-projection coalescence at exceptional degeneracies in 2D Rashba semiconductors.

In this part we explore the spectral function as a potential way for detecting EPs in NH Rashba semiconductors, which can be measured by tools such as ARPES [64, 65, 66, 67, 68, 69]. The spectral function can be obtained as $A(k,\omega)=-{\rm ImTr}{(G^{r}-G^{a})}$, where $G^{r}=(\omega-H_{\rm eff})^{-1}$ and $G^{a}=(G^{r})^{\dagger}$ are the retarded and advanced Green’s functions, respectively [70, 71]. The expressions for $G^{r(a)}$ are not complicated, which allows us to write down the spectral function as

where $E_{\pm}(k)$ are given by Eq. (4). Although this expression already reveals the behaviour of the spectral function in our system, it is useful to write it as

with $D(\omega,E_{i})=(\omega-{\rm Re}E_{i})^{2}+({\rm Im}E_{i})^{2}$. Now, we clearly see that the spectral function in our system is a sum of two Lorentzians centered at $\omega={\rm Re}E_{\pm}$ with their height and width characterized by ${\rm Im}E_{\pm}$. It is thus evident in Eq. (12) that, at the EPs and at momenta between them where eigenvalues merge, instead of two Lorentzian resonances we only have one.

The behaviour of the spectral function can be further visualized in Fig. 5(a), where we plot $A$ as a function of $\omega$ and $k_{x}$ at $k_{y}=0$, $B=0.5$; the real and imaginary parts of the eigenvalues are shown in dashed blue and dashed red curves. We also plot line cuts in Fig. 5(b) for distinct $k$-values spanning the momenta at which EP occur (yellow curves), and momenta between EPs (red curves). The immediate observation is that $A$ develops high intensity regions for a line of momenta bounded by the EP momenta, revealing both the position of EPs and the formation of the finite real energy Fermi arcs. At the EPs, marked by yellow lines in Fig. 5(b), the spectral function undergoes a transition along $k$ from having two resonances to having a single resonance centered at ${\rm Re}E_{1,2}\equiv\xi_{k}$, see also Eq. (12). For momenta between the EPs, the real parts stick together ${\rm Re}E_{1,2}\equiv\xi_{k}$ and a single resonance remains all over these momenta, see red curves. Interestingly, the resonances between the EPs acquire very large values which could be also useful for identifying the bulk Fermi arc at real energies found here. Before ending this part, we would like to mention that the length of the high intensity region seen in Fig. 5(a) along $k$, which is the length of the arc, not only permits to estimate the EP momenta but it also allows to identify the amount of non-Hermiticity $\gamma$ according to Eq. (6). Alternatively, this feature could provide a way to assess the strength of SOC $\alpha$, provided $\gamma$ is known as dictated by Eq. (6). In sum, the spectral function reveals unique features of EPs and offers a powerful route for their detection.

We have demonstrated that the interplay between non-Hermiticity and Rashba spin-orbit coupling in semiconductors gives rise to the emergence of stable and highly tunable bulk exceptional degeneracies. We have found that these degeneracies form rings in two-dimensional momentum space and signal the ends of lines forming due to the coalescence of eigenvalues at finite real energy. Interestingly, the lines at finite real energies have the appearance of bulk Fermi arcs but now at finite energies, suggesting new possibilities for non-Hermitian bulk phenomena [60], whose detail properties, however, deserve a proper investigation and will be addressed in a future study. We have also shown that the exceptional degeneracies and bulk Fermi arcs can be controlled by an in-plane Zeeman field, albeit larger non-Hermiticity values are then needed. Furthermore, we have discovered that the spin projections coalesce at the exceptional degeneracies and can easily achieve higher values than in the Hermitian regime. We also demonstrate that the exceptional degeneracies induce large spectral features along momenta connecting them, unique features that can be directly detected by ARPES. Taken together, the results presented here put semiconductors with Rashba SOC as an interesting arena for the realization of highly tunable non-Hermitian bulk phenomena.

We also note that there is reasonable evidence suggesting that the system studied here as well as the main findings are within experimental reach. In fact, very similar systems as studied here have already been fabricated, including semiconductors with SOC made of InAs [72, 73, 74, 75, 76, 77] or InSb [78, 79, 80, 81, 82] and ferromagnets producing sizeable Zeeman fields [83, 84, 85, 86, 87, 88, 89]. The coupling between ferromagnet and semiconductor, here characterized by $\Gamma_{\sigma}$, can be controlled by an appropriate manipulation of both the spin-dependent density of states in the lead and the tunneling between lead and semiconductor. While the Zeeman field of the ferromagnet enables a distinct density of states for different spins, thus producing different couplings $\Gamma_{\sigma}$, the overall strength of such couplings can be tuned by inserting a normal potential barrier of finite thickness between the semiconductor and ferromagnet lead, e.g., by using a few nm thick InGaAs layer [73]. We have estimated that, using $\mu_{\rm L}=1$ meV, $t_{z}=2$ meV, a ferromagnet with a Zeeman field of the order of $3$meV would be necessary to achieve the conditions of distinct couplings of $\Gamma_{\uparrow}=0.4$ meV and $\Gamma_{\downarrow}=0$, thus giving $\gamma=0.2$ meV. Under these conditions and considering $k_{y}=0$ and $\alpha=20$ meVnm, which is in the range of the SOC in InAs and InSb, we obtain that the EP conditions are satisfied for $k_{x}\approx 0.7$ nm^-1 or $k_{x}^{-1}=150$ nm, which is clearly in the range of reasonable length scales in these systems, such as the length scale related to SOC [55]. We can thus conclude that, despite the possible challenges, there already exist experiments suggesting that the NH semiconductor and the exceptional points studied here represent a feasible idea.