Role of Majorana Fermions in high-harmonic generation from Kitaev chain

Emission of radiation at prominently higher integer multiple frequencies of the incident laser frequency due to strong nonlinear interaction of intense laser fields with matter is known as high-harmonic generation (HHG) ^{1; 2; 3}. The emitted radiation encodes the information about laser-driven sub-cycle electron dynamics which forms the basis of HHG spectroscopy. In recent years, HHG in solids has become a method of choice to probe various aspects of condensed matter such as the observation of Bloch oscillations ^{4; 5}, examining the dynamics of the defects in solids ^{6; 7}, band structure tomography ^{8; 9; 10}, observation of the valley pseudospin ^{11; 12; 13}, imaging valence electrons ^{14; 15}, realisation of petahertz currents in solids ^{16; 17}, probing Berry phases ^{18; 19}, and phase transitions driven by both light and topology ^{20; 21; 22; 23; 24; 25; 26; 27; 28}.

In the field of strongly correlated matter, topological superconductivity and associated phase transitions is presently one of the major topics ^{29; 30} as it is associated with the emergence of the Majorana fermion – particles that are their own antiparticles ³¹. The notion of the Majorana fermion is also of great importance in nuclear and particle physics, apart from solid-state physics ³². In a condensed matter context, Majorana fermions are emergent quasiparticle excitations that have been proposed as the physical basis to realize qubits that are robust against decoherence and fault-tolerant topological quantum computation  ^{33; 34; 35; 36}. In this work, we discuss the role of the Majorana fermions by probing the topological–trivial superconducting phase transition using HHG spectroscopy.

Experimental observation of Majorana fermions is an ongoing quest, and several experiments on semiconductor nanowires, magnetic chains and superconductors, using transport and electrical measurements with scanning tunneling spectroscopy and Coulomb-blockade spectroscopy, have been reported ^{37; 38; 39; 40; 41; 42; 43; 44}. Topological superconductors are considered as a promising platform for Majorana fermions since the original proposal of Kitaev. ⁴⁵ Thus for this study, we choose the paradigmatic one dimensional (1D) Kitaev chain of spinless fermions as a suitable model system. The many-body Hamiltonian for this 1D superconducting chain is

Here, $\mu$ is the onsite energy, $t_{0}$ is a nearest neighbour hopping term, and $\Delta$ is an order parameter (chosen to be purely real). The term with $\Delta$ is a superconducting pairing term in mean-field approximation, which creates or annihilates pairs of particles at neighbouring lattice sites. $c_{j}^{\dagger}$ ($c_{j}$) is fermionic creation (annihilation) operator at site $j$ and $L$ represents the number of lattice sites.

In the appropriate region of parameter space, the system described by Eq. (1) can host a topological superconductor that leads to the emergence of Majorana zero-energy edge modes (MZMs) ^{32; 42; 43}. The boundary of the phase transition from the topological to the trivial superconducting side depends on the critical values of $\mu$, $t_{0}$ and $\Delta$ ^{29; 32}. For example, for $\Delta=t_{0}$, the system is in the trivial phase when $\left|\mu/t_{0}\right|>2$ in the limit $L\rightarrow\infty$. This is qualitatively similar to the limit $\left|\mu/t_{0}\right|\rightarrow\infty$. For $\left|\mu/t_{0}\right|<2$, the system behaves as a topological superconductor with boundary MZMs in presence of edges. At $\left|\mu/t_{0}\right|=2$, the gap closes which causes the boundary zero modes to hybridize and disappear ³⁴.

To set the stage for the HHG study, we briefly recall this topological–trivial phase structure and its association with the MZMs. Following Ref. ⁴⁵, one rewrites Eq. (1) in terms of Majorana operators $c^{\dagger}_{j}=\frac{1}{2}(\gamma_{j,1}+i\gamma_{j,2})$ and $c_{j}=\frac{1}{2}(\gamma_{j,1}-i\gamma_{j,2})$ to arrive at $\mathcal{H}=\frac{i}{2}\Big{[}-\mu\sum_{j=1}^{L}\gamma_{j,1}\gamma_{j,2}+(-t_{0}+\Delta)\sum_{j=1}^{L}\gamma_{j,1}\gamma_{j+1,2}+(t_{0}+\Delta)\sum_{j=1}^{L}\gamma_{j,2}\gamma_{j+1,1}\Big{]}$ up to a constant. The system is in the trivial phase when two Majorana fermions are bound on the same site as sketched in Fig 1(a). This is clearly seen when $\mu$ is the dominant scale in $\mathcal{H}$ as expressed in terms of the Majorana operators. On the other hand, two unpaired Majorana fermions emerge at the boundary when Majorana fermions are bound in pairs on the neighbouring sites as sketched in Fig 1(b) that corresponds with the system being in the topological superconducting phase. This is easiest to see for $t_{0}=\Delta$ and $\mu=0$. More generally in the topological phase, it is understood that the two unpaired Majorana fermions (or MZMs) are exponentially localised at each physical boundary ⁴⁵. This notion of emergent unpaired Majorana fermions at the boundary or edge, expectedly, gets lost when closed (periodic) boundary condition (PBC) is employed as sketched in Fig 1(c). The topological–trivial transition can be seen by tracking the lowest many-body excitation energy as a function of $\left|\mu/t_{0}\right|$ for the open chain as shown in Fig 1(d). On the topological side, there is an effectively zero energy (fermionic) excitation which is composed from the unpaired Majorana modes localised at the opposite edges as in Fig. 1(b), whereas on the trivial side, such zero energy modes are lost.

High-harmonic spectra for the 1D Kitaev chain with open boundary conditions (OBC) and PBC are presented in Fig. 2. The left and right panels in the figure correspond to topological and trivial phases, respectively, on either side of the transition at $\mu/t_{0}=2.0$. To observe the sensitivity of HHG spectroscopy on the two phases, we have chosen the values of $\mu/t_{0}$ close to transition point. A gradual decrease in the harmonic intensity up to 15${}^{\textrm{th}}$ harmonics is visible when the system is in the topological phase at $\mu/t_{0}=1.8$ [see Fig. 2(a)].

However, the spectrum for the trivial phase at $\mu/t_{0}=2.2$ is significantly different from the topological phase, as evident from the right panel of the figure [see Fig. 2(b)]. In this case, the extent of the harmonics is relatively broad, ranging from 3${}^{\textrm{rd}}$ to 15${}^{\textrm{th}}$ order. The intensity of the harmonics in the plateau region is comparable, and the spectra extend up to the 17${}^{\textrm{th}}$ harmonic. In both the phases, the harmonic spectra exhibit odd harmonics only as expected from the inversion symmetry present in the 1D system. Note that energy eigen-spectra are almost same at $\mu/t_{0}=1.8$ and 2.2 as evident from Fig. S1. The band gap at $\mu/t_{0}=1.8$ is $\sim$0.109 eV (if MZM states at zero energy are excluded), and at $\mu/t_{0}=2.2$ is $\sim$0.11 eV. In Fig. S7, we also present a series of harmonic spectra for different values of $\mu/t_{0}$ (= 1.6, 1.7, 1.8, 1.9, 2.1, 2.2, 2.3, 2.4) across the transition through which their differences on the two sides of the transition becomes evident.

The situation becomes very different when PBC is imposed on the 1D Kitaev chain such that there are no boundaries. In this case, the harmonic spectra corresponding to $\mu/t_{0}=1.8$ and 2.2 are very similar as reflected from Figs. 2(c) and 2(d). The small differences between both the spectra arise due to slight changes in the eigen-spectra at $\mu/t_{0}=1.8$ and 2.2. The analysis of Fig. 2 concludes that the harmonic spectra are very similar when PBC is employed, whereas markedly different for both the phases when OBC is employed. This indicates that the MZMs are playing a distinct role in the harmonic generation as seen by the contrast in the HHG spectra in both the phases with different boundary conditions. In the following, we will investigate how the MZMs are associated with the different harmonic spectra in the two phases.

To elucidate the role of the MZMs in the harmonic generation, population dynamics of the doubly degenerate MZMs in Bogoliubov-de Gennes (BdG) spectrum on the topological side is presented in Fig. 3. We recall that the degeneracy of the two zero modes is lifted in the trivial phase, whereby they lose their Majorana character. When compared to these corresponding non-degenerate and non-zero modes in the trivial phase, the population dynamics during the laser pulse on the topological side at $\mu/t_{0}=1.8$ shows that the degenerate Majorana states show drastically different behaviours [see Figs. 3(a) and  3(b)]. This noticeable difference in population dynamics in both the phases owing to the topologically enforced zero mode character of the Majorana excitations or lack thereof can be correlated with the different harmonic spectra observed in both the phases as shown in Fig. 2.

In presence of PBC, when there are no MZMs present, the temporal evolution of the corresponding modes are quite similar at $\mu/t_{0}=1.8$ and 2.2 [see Figs. 3(c) and  3(d)]. This similarity in their population evolution indicates that both the modes are contributing in the same way to the harmonic generation at $\mu/t_{0}=1.8$ and 2.2; this correlates with almost the same harmonic spectra for both the phases with PBC as shown in Figs. 2(c) and  2(d) in contrast to the above discussion on the previous case with OBC. Moreover, their behaviour is similar to the OBC case on the trivial side [Fig. 3(b)] when these modes have become part of the bulk and not localized at the boundary anymore. We recall again that the notion of the unpaired Majorana fermions exists only on the topological side with OBC which is in line with the preceding discussions. The present findings thus establish that the harmonic spectra and the population dynamics of the MZMs are different for both the phases for the 1D Kitaev chain with OBC. This concludes that high-harmonic spectroscopy is sensitive to the presence of the MZMs. The notion of unpaired Majorana fermion does not exist for the 1D Kitaev chain with PBC. As a result, the harmonic spectra and the population dynamics are similar for different values of $\mu/t_{0}$.

In order to highlight further the role of MZMs in the differences of the harmonic profiles in presence of OBC, we also show the harmonic spectra obtained by artificially suppressing the presence of MZMs in the eigen-spectra. The harmonic spectra at $\mu/t_{0}=1.8$ and 2.2 with OBC are almost same as evident from Figs. 4(a) and  4(b). The minor difference in the spectra are originating from the minor difference in the eigen-spectra. Moreover, the harmonic spectra solely originating from one occupied MZM at $\mu/t_{0}=1.8$ and corresponding non-zero mode at $\mu/t_{0}=2.2$ is shown in Figs. 4(c) and  4(d). As reflected from the figure, the spectra are drastically different from each other. Furthermore, the population dynamics of the highest occupied and lowest unoccupied eigen-states (representative of the bulk modes) corresponding to the spectra in Figs. 4(a) and  4(b) are similar as expected (see Fig. S8). On the other hand, the population dynamics corresponding to spectra in Figs. 4(c) and  4(d) are drastically different (see Fig. S8). Results presented in Fig. 4 unequivocally establish that the similarities and differences in the spectra under different situations are due to the presence or absence of the MZM modes. Also, the system continues to exhibit different harmonic profiles in the two phases (for OBC) for different laser intensities (see Figs. S3 and S4) and wavelengths (see Figs. S5 and S6).

We now briefly comment on the domain of validity for the observations made earlier. Keeping our choice of the laser parameters fixed, we find that a variation of $\sim 11\%$ in the $t_{0}=\Delta$ scale does not change our conclusions. Keeping the $t_{0}$ scale fixed, a variation of $~{}\sim\pm 8\%$ in $\Delta/t_{0}$ keeps conclusions unchanged as well. Beyond these variations, the laser parameters have to be tuned accordingly to find the suitable values. Thus, analogous simulations can be helpful in determining the appropriate laser characteristics when applying HHG to different physical systems. Finally, it is expected that the HHG obtained from the time evolution of the many-body Hamiltonian in Eq. (1) and from that of the BdG-recasted “one-particle” Hamiltonian should be the same in principle. We have confirmed this expectation by explicit numerical simulations (results not shown here), which is an independent check on the accuracy of our numerical time evolution procedures.

In summary, our work demonstrates that HHG spectroscopy is sensitive to the topological–trivial phase transition in 1D superconducting Kitaev chain. The extent of the harmonic plateau and energy cutoff in the trivial phase are distinguishable from the harmonic spectra in the topological phase. This distinguishability is further consolidated by various harmonic spectra for different intensities and wavelengths of driving laser as presented in supplementary material. The population dynamics of the two Majorana states and their contributions to the total harmonic spectra are very different for both the phases, which we established to be correlated with the differences in the HHG spectra of the two phases. This distinct signature may potentially point to a new probe of Majorana modes in solids. A physical explanation for the numerically observed differences in the population dynamics of MZMs and bulk modes that were shown to underlie the HHG spectra is desirable for better understanding. Finally, the feasibility of coupling existing realisations of topological superconductivity with potential MZMs to intense laser fields in the laboratory is an open issue ⁴⁶, and we hope our work stimulates attempts in this direction.