## Introduction

Understanding the elementary scattering processes which govern the ultrafast response of topological states of matter to femtosecond-laser excitation is crucial for applications in ultrafast spintronics and opto-spintronics1,2,3,4. This often requires the experimental determination of the electronic band dispersion of excited states above the Fermi level (EF), simultaneously facilitating for a more complete understanding of a system’s topological order5.

A prominent example are three-dimensional $${{\mathbb{Z}}}_{2}$$ topological insulators (3D TIs)6,7,8, which possess an inverted band structure arising from a high spin–orbit interaction9, and hence a metallic surface hosting topologically protected surface states10,11 that could be used as channels in which to drive and coherently control pure spin currents and spin-polarized electrical currents on ultrafast time scales12,13,14,15,16.

To date, the experimental verification of these topologically nontrivial states has been possible either through the enumeration of an odd number of Fermi surface contours, contributed by the state of interest, around time-reversal invariant momenta17,18, or by the direct observation of their Dirac-like dispersion with a unique helical spin texture17,18,19,20,21,22. Both are hallmarks of a nontrivial topological invariant, ν0 = 1, distinguishing strong topological materials from topologically weak or trivial materials (ν0 = 0)23,24,25.

The fundamental principle of bulk-boundary correspondence is responsible for the enforced existence of a spin-polarized surface state bridging the bulk band gap of a 3D TI26. The surface and bulk electronic structures are therefore fundamentally coupled9,26,27. Already, the influence of this type of bulk-surface coupling on the electronic and transport properties of TIs was investigated for prototypical systems such as Sb2Te3, Bi2Te3, and Bi2Se327,28,29,30,31,32. For Sb2Te3, it was shown how it leads to relatively small changes in the linear band dispersion of the surface state near its connectivity points with the bulk valence bands below EF27. Such an unexpectedly weak bulk-surface coupling was also confirmed in magnetotransport experiments33, and by the observation that the connectivity between bulk and surface bands is mediated by bulk-derived surface resonances exhibiting a reversed spin texture with respect to that of the topological surface state in Bi2Te334 and Bi2Se335. This scenario was further confirmed by measurements of the electron dynamics, which revealed a highly decoupled surface and bulk state dynamics following ultrafast optical excitation35.

For metals such as Au5,36,37,38 and semimetals such as Bi39,40,41,42 and Sb18,43,44,45, however, due to the absence of a global band gap or to the presence of a very small relative band gap, the identification of topological states and their coupling to the bulk proved to be more complex. Au metal is widely accepted as a conventional spin–orbit material with ν0 = 0, consistent with the even number of concentric Fermi contours formed by the spin sub-bands of the Rashba-split surface state enclosing the $$\overline{{{\Gamma }}}$$ point on the (111) surface36,37,38. Recently, however, the Au(111) surface state was identified as $${{\mathbb{Z}}}_{2}$$ topologically nontrivial based on its connectivity to the bulk bands far above EF and on the parity analysis of the band structure5. In a similar context, the connectivity of the Rashba-split surface state of Ir(111) to bulk bands below EF was discussed in terms of topological properties46. In contrast, Bi was predicted to be a topologically trivial system (ν0 = 0) which can undergo a topological phase transition to a $${{\mathbb{Z}}}_{2}$$ TI47 via the application of strain, or into a topological crystalline insulator phase48. Strikingly, however, based on a detailed experimental analysis of the electronic band structure, Bi was also suggested to be topologically nontrivial with ν0 = 141,49.

The semimetal Sb, on the other hand, is regarded as the parent compound underlying the microscopic origin of topological order in the Bi1−xSbx TI class17,18,39. The similarity between the electronic structures of elemental Sb with the prototypical TI Bi0.9Sb0.1 has prompted the suggestion that Sb itself can be described by ν0 = 1, and that its surface supports a Berry’s phase18. Indeed, several previous direct measurements of the (111) surface band dispersion confirm the presence of a surface state with a chiral, momentum-locked spin texture18,43,50,51,52. However, efforts towards a rigorous demonstration of the nontrivial nature of the surface state have been hindered both due to the presence of bulk pockets at EF45, and due to the fact that the connection of the surface state to the bulk continuum between time-reversal invariant momenta occurs above EF. Therefore, direct observation of the relevant surface-bulk connectivity, crucial for the final verification of topological character, remains out of reach for experimental techniques probing only the occupied ground state electronic structure such as conventional angle-resolved photoemission (ARPES). The question on the band connectivity of Sb is also directly related to how bulk-surface coupling influences the dispersion of the surface state, and whether it is consistent with decoupled bulk and surface electron dynamics as previously suggested for strong $${{\mathbb{Z}}}_{2}$$ TIs35.

To investigate these issues, we experimentally follow, by means of time-resolved ARPES (tr-ARPES) in combination with spin resolution, the energy-momentum dispersion of the Sb(111) surface state beyond EF, as well as its connectivity to bulk states. Our main finding is a kink structure in the band dispersion of the surface state which causes a giant mass enhancement due to coupling to the bulk continuum above EF. Our observation of this band connectivity above EF allows to unambiguously verify the topological character of the surface state. The experimental results are supported by quasiparticle self-consistent GW (QSGW) calculations. The strong bulk-surface coupling is also revealed by our measurements of the electron dynamics, which show that the excited states behave as a single thermalizing electron population due to a dominant contribution from interband electron–electron scatterings. These findings are critically important for the implementation of advanced functionalities in future spintronics based on topological semimetals.

## Results and discussion

### Transient band structure and kink above Fermi level

We performed tr-ARPES experiments on Sb(111) single crystals using pump and probe femtosecond (fs) laser pulses of 1.5 and 6 eV photon energy under the experimental geometry shown in Fig. 1a (see Methods for details). The high-quality of the cleaved surface was confirmed by low-energy electron diffraction, as seen in Fig. 1b.

The energy-momentum band dispersion of Sb(111) measured by tr-ARPES along the $$\overline{\,\text{M}\,}$$-$$\overline{{{\Gamma }}}$$-$$\overline{\,\text{K}\,}$$ direction at a time delay Δt = 300 fs following optical excitation by the pump pulse is shown in Fig. 2a. Above EF, the unoccupied part of the band structure is transiently populated with excited electrons and can be observed directly. One can identify the dispersion of the spin-polarized surface-state bands, as well as bulk states and bulk-derived surface resonances both below and above EF (denoted as SS, BS, and SR, respectively). Here, and in general due to the surface sensitive nature of photoemission (even with the slightly enhanced bulk sensitivity at probe photon energies of 6 eV)53, surface-related bands appear as more-intense narrower features, while less-intense broad features correspond to pure bulk states. The assignment of bulk and surface character to the states in Fig. 2 is further supported by our band structure calculations, which will be discussed in the next section, and consistent with previous theoretical works54,55.

Along the $$\overline{{{\Gamma }}}$$-$$\overline{\,\text{K}\,}$$ direction, one can clearly identify the top of the bulk valence band below EF (k,y = 0.12 Å, E − EF = −0.2 eV), and the bottom of the bulk conduction band above EF closer to the $$\overline{{{\Gamma }}}$$ point (k,y = 0.075 Å, E − EF = 0.1 eV). The two V-shaped surface-state bands, which possess an opposite spin texture18,51,52, exhibit a distinct dispersion along this direction. The inner V-shaped surface band crosses EF and merges with bulk conduction band states above EF, while the outer V-shaped surface band bends back before reaching EF and merges with bulk valence band states below EF. The observed band connectivity along $$\overline{{{\Gamma }}}$$-$$\overline{\,\text{K}\,}$$, in fact, resembles that characterizing 3D TIs8. However, it should be emphasized that, for the semimetal Sb, a direct observation of similar bulk-surface connectivity above EF in between the time-reversal invariant momentum points $$\overline{{{\Gamma }}}$$ and $$\overline{{\rm{M}}}$$ is mandatory to verify the topological character of the surface state.

Along the $$\overline{{{\Gamma }}}$$-$$\overline{\,\text{M}\,}$$ direction, as seen in Fig. 2a, both the local bulk conduction band minimum near $$\overline{{{\Gamma }}}$$, and the bulk valence band top are above EF. The outer V-shaped surface band crosses EF and, in a similar way as for the $$\overline{{{\Gamma }}}$$-$$\overline{\,\text{K}\,}$$ direction, it bends back at k,x = −0.2 Å to couple to the bulk valence band exhibiting hole-like behavior. The inner V-shaped surface band, on the other hand, crosses EF and merges with bulk conduction band states above EF (k,x = 0.08 Å, E − EF = 0.1 eV). This behavior clearly excludes the scenario in which (i) there is a turn back in the dispersion of the inner V-shaped surface band without connectivity to bulk conduction band states above EF, followed by (ii) its subsequent re-appearance below EF as one of the surface resonances18,45 only connecting to the valence band continuum at larger k and closer to $$\overline{{\rm{M}}}$$ point. The observed connectivity, in turn, unambiguously demonstrates both the expected partner-switching behavior of a topologically nontrivial system with ν0 = 18, and that the inner V-shaped surface-state band encloses the $$\overline{{{\Gamma }}}$$ point an odd number of times up to the wave vector of the bulk connectivity point above EF, which has never been measured before.

Looking in more detail at the previously unobserved connectivity region between surface state and bulk conduction band above EF, the most striking observation in Fig. 2a is the appearance of a kink structure in the band dispersion of the surface state as it approaches the connectivity point along the $$\overline{{{\Gamma }}}$$-$$\overline{\,\text{M}\,}$$ direction. A zoom-in of this region, presented in Fig. 2b, demonstrates how the kink is very pronounced, evidencing a significant enhancement of the effective mass. A careful extraction of the photoemission peaks along with linear fits of the band dispersion (red symbols and dashed lines in Fig. 2b, respectively) reveals that the kink structure is characterized by a significant drop of about 70% in the group velocity of the surface state, which changes from 4.67 to 1.45 eVÅ.

Using the definition of the transport mass56, one can extract the corresponding increase of effective mass to the drop in group velocity of this region of the band dispersion. In this way we derive that, due to the kink structure, the effective mass of the surface band increases by a factor of 4, from ~0.1 to ~0.4 me, where me denotes the free-electron mass. Such a large surface mass enhancement near the connectivity point with pure bulk states pinpoints strong bulk-surface coupling as the underlying mechanism responsible for the appearance of the kink structure.

The data in Fig. 2a also demonstrate how, due to the interaction with the bulk continuum, the surface bands preferentially follow the dispersion of the border of the bulk band gap. For the outer V-shaped surface band along both the $$\overline{{{\Gamma }}}$$-$$\overline{\,\text{M}\,}$$ and $$\overline{{{\Gamma }}}$$-$$\overline{\,\text{K}\,}$$ directions, it is thus difficult to extract or visualize a kink structure because the band exhibits hole-like behavior and the band top is flat. Similarly, along the $$\overline{{{\Gamma }}}$$-$$\overline{\,\text{K}\,}$$ direction, the absence of a clear kink structure above EF is related to the fact that the inner V-shaped surface state energetically favors a linear dispersion near the border of the kz-projected bulk manifold closest to the $$\overline{{{\Gamma }}}$$ point57.

Our present finding of a strong bulk-surface coupling in the semimetal Sb differs from the situation in prototypical 3D TIs, where the coupling was reported to be weak due to the presence of surface resonances with high degree of spin polarization inhibiting the interaction with pure bulk states27,35. In this respect, it is important to note that along the $$\overline{{{\Gamma }}}$$-$$\overline{\,\text{M}\,}$$ direction in Fig. 2a, we observe bulk-derived surface-resonance states in the immediate vicinity of the kink structure above EF. Our spin-resolved tr-ARPES measurements taken along $$\overline{{{\Gamma }}}$$-$$\overline{\,\text{M}\,}$$ at k,x = –0.11 Å and Δt = 300 fs in Fig. 2c, d, however, in contrast to the case of 3D TIs, reveal that the bulk-derived surface resonance in Sb is neither spin split nor spin-polarized. We do note that along $$\overline{{{\Gamma }}}$$-$$\overline{\,\text{K}\,}$$, the surface resonance is also visible but with much lower intensity, probably due to photoemission matrix-element effects. The inner and outer V-shaped surface bands, on the other hand, exhibit a clear spin polarization which, in agreement with previous findings18,51,52, reach values of ~60% for the in-plane chiral (Fig. 2c) and of ~15% for the out-of-plane (Fig. 2d) spin components. This observed spin texture is consistent with a Berry’s phase18. Note, however, that without a direct observation of the bulk-surface connectivity above EF, the measurement of the spin texture alone cannot be taken as an exclusive proof of the topological character of the Sb surface state. A prominent example in this context is the helical spin texture observed for topologically trivial surface states near the quantum critical point of a phase transition between a trivial and a topological insulator58.

In Fig. 2e, f, we show a schematic representation of two possible scenarios where the hybridization between surface and bulk states can be differently affected by surface-resonance states. For instance, the interaction between parallel spins (Fig. 2e) could lead to spin-dependent avoided crossings or even a breakdown of the bulk-surface connectivity, which for a system with an even number of band inversions in the bulk, can result in a weak topological or topologically trivial phase (ν0 = 0). In contrast, a strong coupling to the bulk continuum is possible (Fig. 2f) when the surface resonances are spin-degenerate, which for a system with an odd number of band inversions in the bulk, can result in a kink structure and a topologically nontrivial phase (ν0 = 1) in agreement with our experimental findings.

### Ab initio calculations and comparison to experiment

To further verify this scenario and especially to understand whether the kink structure is a property of the ground state, we performed QSGW-based calculations (see Methods) of the Sb(111) band structure as shown in Fig. 3. In Fig. 3a, b, the continuum bulk-energy bands are represented by gray-shaded regions, while the colored lines correspond to the discrete bands of a 100-bilayer slab. Here, the color coding represents the localization of the wave function of each state on the topmost bilayer. In Fig. 3c, d, we show the calculated spin polarization of the bands corresponding to the results in Fig. 3a, b, respectively. The red/blue color representation highlights the magnitude and direction of the chiral spin component perpendicular to electron momentum, with the other spin components negligible. Overall, around the $$\overline{{{\Gamma }}}$$ point the calculation shows qualitatively good agreement with the experimental results of Fig. 2 concerning the dispersion of the bulk bands both below and above EF. The calculated bulk band dispersion near the $$\overline{{\rm{M}}}$$ point is also consistent with previous ARPES measurements accessing only occupied states18. Equally important, the calculated surface-state spin sub-bands exhibit the partner-switching behavior of a topological system with ν0 = 1 and strong bulk-surface coupling (see Fig. 2f). In Fig. 3, we can also identify bulk-derived surface resonances with different degree of surface localization and negligible spin polarization in agreement with the experimental results.

A closer view of the connectivity region between surface and bulk conduction band states is shown in the calculation of Fig. 3b. Similar to the tr-ARPES measurements of Fig. 2b, the surface state exhibits a kink structure in the calculated band dispersion near the connectivity point to the bulk continuum. The kink structure, which up to date has not been reproduced in previous calculations of the Sb band structure solely based on conventional density-functional theory (DFT), is characterized by a change in the calculated group velocity of the surface band from ~2.8 to 1.2 eVÅ. These values represent a drop in the group velocity of 57%, in good agreement with the experimental value. This behavior is equivalent to a large increase in the effective mass of the surface band from ~0.1 to ~ 0.3 me as it crosses the connectivity point. The overall change corresponds to a surface mass enhancement by a factor of 3 which is qualitatively consistent with the one derived from the experimental band dispersion of Fig. 2b. The abrupt change in the surface band dispersion near the bulk connectivity point underlines direct hybridization between surface and pure bulk states as the origin of the kink structure. The negligible spin polarization of surface-resonance states in Fig. 3d is also consistent with this picture. The observed mass enhancement, to the best of our knowledge, exceeds by far the one found in any other prototypical TI up to date27. The strong bulk-surface coupling is also evident in Fig. 3a, b from the decrease seen in the localization of the surface bands caused by the interaction with pure bulk states. In Fig. 3a, it can also be seen that when the bulk band gap is very large (small) the surface states are strongly (weakly) localized on the surface. More in detail, our analysis of the orbital character of the surface state around the bulk connectivity point, shown in Fig. 3e–g, reveals an abrupt change in the relative contribution of py and pz orbitals due to hybridization with pure bulk states.

The QSGW approach is known as a powerful method which substantially improves the description of the electronic band structure with respect to conventional DFT calculations47,59. For instance, band structure calculations within the local-density approximation (LDA) suffer from an underestimation of non-inverted band gaps such as the one at $$\overline{{\rm{M}}}$$ point. In this case, the LDA band gap of 0.07 eV increases to 0.27 eV when QSGW is applied. On the contrary, inverted band gaps, such as the one at $$\overline{{{\Gamma }}}$$, tend to be overestimated by LDA47,59. In the present case, the band gap decreases from 0.59 eV in LDA to 0.53 in QSGW enabling the connectivity of the surface state to the bulk continuum much closer to the $$\overline{{{\Gamma }}}$$ point. This behavior is consistent with our parity analysis within both LDA and QSGW which shows the presence of a band inversion at the Γ point of the bulk Brillouin zone, whereas the band gaps at the three L points projecting onto $$\overline{{\rm{M}}}$$ are non-inverted. Thus, the semimetal Sb is characterized by an odd number of band inversions in the bulk, giving rise to a nontrivial $${{\mathbb{Z}}}_{2}$$ invariant. This situation is in contrast to the case of Bi, in which the band gap at Γ is inverted, but the gaps at the three L points are inverted too, resulting in inverted gaps at four time-reversal invariant momenta, and therefore to a topologically trivial value of the $${{\mathbb{Z}}}_{2}$$ invariant47.

### Ultrafast response to optical excitation

To further verify the crucial role of strong bulk-surface coupling for the appearance of the kink structure, we investigated the ultrafast temporal evolution of the transient electron populations within different states following optical excitation by the pump pulse. These type of measurements also provide insight into the elementary scattering processes underlying thermalization of carriers, and whether the coupling influences the relevant scattering channels of ultrafast charge and energy transfer between different states. In Fig. 4a–d, we show several tr-ARPES dispersions at selected time delays following optical excitation. One can see that after ~2.4 ps, high-energy excited states have completely decayed, and that relaxation of excited electrons along $$\overline{{{\Gamma }}}$$-$$\overline{\,\text{M}\,}$$ and $$\overline{{{\Gamma }}}$$-$$\overline{\,\text{K}\,}$$ directions proceeds according to similar dynamics. This behavior can also be recognized in Fig. 4e, f, where we show momentum-resolved intensities extracted at different energies above EF as a function of pump-probe delay for surface and bulk states (Fig. 4e), as well as for surface-resonance states (Fig. 4f).

The faster relaxation of excited electrons at higher energy seems consistent with the expected energy dependence of the relaxation times of a single thermalizing electronic population of surface and bulk states. In particular, one can see that excited electrons within surface and bulk states decay with similar relaxation rates, behaving as dynamically locked electron populations, which evidences the signatures of the coupling between the different states. From Fig. 4 we also derive that the overall dynamics of bulk and surface bands can be described approximately by a global Fermi–Dirac distribution with a single transient electronic temperature reaching a maximum value of Te ≈ 2100 K at the onset of the excitation. This is shown in Fig. 4g, h, where a momentum-integrated spectrum (Fig. 4g) is fit to a Fermi–Dirac distribution from which the temporal evolution of the electronic temperature is derived (Fig. 4h). The error bars in Fig. 4h are largest at early delay times where a Fermi–Dirac distribution is less appropriate, i.e. when the electronic population is far out-of-equilibrium. This is shown explicitly in Fig. 4i, where fits to Fermi–Dirac distributions are shown for selected delay times. Therefore, whilst interband thermalization proceeds on a time scale of about 0.9 ps which is longer than the laser excitation (inset of Fig. 4h), as in some simple metallic systems where nonthermal effects are important60,61,62, the dynamics of surface and bulk carriers do still proceed on similar time scales due to strong bulk-surface coupling. This is also supported by the fact that this initial process occurs on a time scale corresponding to that in which the electronic distribution relaxes from high energies to energies slightly above the energy of the kink structure (Fig. 4c), pinpointing the crucial role of electronic hybridization in the process of interband thermalization. These findings taken together therefore suggest that, owing to strong bulk-surface coupling, the interband electron scattering processes between bulk and surface states are predominantly responsible for the synchronous progression towards a thermalized state.

In Fig. 5, we analyse this behavior in more detail by comparing the tr-ARPES intensities integrated within small energy-momentum windows distributed throughout the dispersion of different states (Fig. 5a). The fact that the temporal evolution of the photoemission intensity at fixed energy is independent of the nature of the excited states pinpoints the importance of bulk-surface coupling in the process of interband thermalization (Fig. 5b). The symmetric tr-ARPES intensity distribution of the surface bands upon rotation of the electron spin by 180 degrees at opposite wave vectors (Fig. 5c), or by 90 degrees at orthogonal wave vectors (Fig. 5d), indicates that the spin texture of the surface state plays a minor role in the electron dynamics. It is understood that spin selection rules do not influence the probability of interband transitions between spin-degenerate bulk states and spin-polarized surface states which will always be allowed under all possible scattering angles. The crucial role of bulk-assisted interband scattering is further supported by the synchronous evolution of the transient electron populations when compared to that of excited bulk states (Fig. 5e). As a result, surface and bulk states behave as a single thermalizing electron population exhibiting a universal energy dependence in the relaxation times which is independent of electron spin (Fig. 5f) and momentum (Fig. 5g). This situation is in stark contrast to the case of prototypical TIs where it has been shown that as a consequence of the complex spin texture of surface resonances hindering the coupling to the bulk continuum27,35, surface and bulk states behave as two independently thermalizing electron populations relaxing on different time scales35. In this regard, we expect the dynamics of excited states in the semimetal Sb, as observed here, to be unusual in the context of systems where spin–orbit interaction is important, especially in topological systems where surface states and resonances are in most cases strongly spin-polarized22,27,34,35,58.

### Elementary scattering mechanisms of ultrafast charge and energy transfer

More in detail, we derive that the energy dependence of the relaxation times can be well described as $${\tau }^{-1}={\tau }_{0}^{-1}+\alpha {(E-{E}_{{\rm{F}}})}^{\beta }$$ (represented as a black dashed line in Fig. 5f, g). Here, τ0 is an effective time constant containing the contributions from electron–phonon scattering as well as electron transport, and the second term is related to the contribution from electron–electron scattering61. The parameter α is the characteristic electron–electron scattering constant which depends on the averaged electron–electron scattering probability, and therefore on the electron–electron interaction strength61,62.

Although somewhat oversimplified, Fig. 5f, g demonstrate that this model, with fitting parameters of τ0 = 2550 ± 80 fs, α = 0.025 ± 0.003 fs−1 eV−2 and β = 1.82 ± 0.05, is in good agreement to the data. Our observation that the energy dependence of τ behaves with β close to the predicted value of 2 highlights the significance of nonthermal contributions to the electron dynamics, specifically the role of electron–electron scatterings61,62. This is also supported by the fact that the characteristic electron–electron scattering constant α that we obtain is not far from that predicted for simple metallic systems when nonthermal effects are taken into account61, which is in principle expected as Sb lies in the metallic regime of the Bi1−xSbx series18.

To further verify the role of nonthermal contributions to the electron dynamics, we also display as a dashed orange line in Fig. 5f, g the relaxation times obtained based on a two-temperature model63 according to which electron–phonon scatterings are the only mechanism responsible for the relaxation of the electronic population. Nonthermal effects, or in other words, the important contribution of electron–electron interactions to the relaxation61,62 are completely neglected in this model. Instead, thermalization of the electronic population is assumed to be much faster than the laser excitation, or instantaneous, which is clearly not the case here (see e.g. Fig. 4i). Despite the limited applicability of this assumption, the result of Fig. 4h is fit with an electron temperature distinct from the lattice temperature, and the energy-dependence of the relaxation times derived solely based on Fermi–Dirac statistics and by completely neglecting the important influence of nonthermal effects to the relaxation (dashed orange line).

Let us now compare the experimental results along with the initial model, which explicitly accounts for electron–electron scatterings, to the results of the two-temperature model. From Fig. 5f, g, it is evident that the relaxation dynamics from high energies down to energies of ≈0.15 eV proceeds according to faster electron–electron scattering processes. Therefore, in this energy region, electron–electron scatterings are the main driving mechanism for the relaxation of the electronic population. In the low-energy region, on the other hand, we observe that the relaxation times exceed the two-temperature model prediction down to EF.

Using the experimentally determined electron–phonon coupling constant λ = 0.2243, one can determine an electron–phonon scattering time of ≈29 fs. This, in conjunction with ωD = 13 meV, where ωD is the Debye frequency64, provides an estimate of $$\tilde{\tau }\ \approx$$ 1780 fs for an electron to lose 0.8 eV (corresponding to a high-energy excited state in Fig. 5) by assuming a linear dependence of the relaxation rate. This time scale is consistent with the two-temperature model prediction near EF of ~1800 fs. We note that an additional contribution from e.g. electron transport would act to further decrease the relaxation times at low energy, and therefore cannot account for this underestimation of the relaxation times in the low-energy region. Instead, nonthermal effects are responsible for the slower low-energy relaxation times. Following the arguments presented in a recent theoretical work62, in the limit 0.05 < γep/βee < 2 the time in which excited electrons lose their energy can be well approximated as $$\tau ^{\ast }\approx {2.5\cdot \gamma _{\mathrm{ep}}^{-0.75}}{\beta}_{\mathrm{ee}}^{-0.25}$$. Here βee is the relaxation rate for 0.5 eV excitations, representing the strength of the electron–electron interactions. The electron–phonon interaction strength, $${\gamma }_{{\rm{ep}}}{\approx} {\tilde{\tau }}^{-1}$$, is determined from the relaxation rates near EF predicted by the two-temperature model62. For our present case, we derive γep/βee ≈ 0.1 yielding βee/(0.5 eV)2 = 0.025 fs−1 eV−2 and τ* = 2475 fs. These estimations are in excellent agreement to the fitting parameters obtained within the initial model for the values of the characteristic electron–electron scattering constant, α, and τ0, respectively (black dashed line in Fig. 5f, g).

### Pathways for ultrafast charge and energy transfer

Figure 6 shows a schematic representation of the most relevant channels of ultrafast charge and energy transfer between different states based on our experimental findings. After laser excitation, electrons and holes start thermalizing via interband transitions between bulk and surface states caused by electron–electron and electron–phonon scattering processes, with predominance of electron–electron scatterings the higher the energy of the excited electron. Note that these transitions can also be driven between bulk valence and conduction band states. Higher-energy excited states within the bulk conduction band continuum relax through electron–electron scatterings into lower-energy states, and the overall transfer of charge and energy during this process requires low-k transfers. Similarly, electron–electron scatterings lead to a continuous exchange of charge and energy between surface and bulk valence band states, with the difference that this process requires large-k transfers. It is understood that transitions between states of opposite spin are strictly forbidden, and that relaxation of an electron via electron–electron scattering is accompanied by the excitation of a second electron under the conservation of spin, energy, and momentum. Therefore, interband transitions between bulk and surface states are allowed under all possible scattering angles.

The relaxation process via electron–electron scatterings proceeds through energy redistribution between high-energy and lower-energy states. Due to energy conservation, the relaxation of highly excited electrons via electron–electron scatterings results in secondary electron excitations at lower energies. In consequence, as the process of interband thermalization progresses, the number of excited electrons increases by about a factor of two in every thermalization step61. This cascade process results in an excess population at low energies, thereby increasing the number of existing electrons emitting phonons, and affecting the relaxation rates also at low energies60,62, where transiently occupied bulk valence band states do also act as scatterers.

As the Fermi level is approached, electron–electron scatterings are still required to continuously re-establish a thermal equilibrium in every relaxation step of the excess population at low energies, and the efficiency of this process depends on the relative electron–electron and electron–phonon interaction strengths. For sufficiently weak electron–electron interactions, as in the case of Sb, nonthermal effects in this energy region are not negligible, resulting from the fact that, on average, each electron–phonon scattering event sets-in before a thermal state can be completely re-established. As many more electron–phonon scattering events are required to relax the excess population, leading to more phonons being emitted, the overall electron–phonon energy relaxation process in the vicinity of EF is bottlenecked. While electron–phonon scatterings have less strict selection rules, they are still constrained by spin selection. Although the total change in energy achievable is small due to the small energy of the phonons, at longer time delays the energy of the electronic system is released into the lattice through electron–phonon scatterings, which govern the slower relaxation of the excited electron population in the immediate vicinity of EF.

Our findings taken altogether provide evidence of the crucial role of bulk-surface coupling as a ground state property which is not only responsible for the observed mass enhancement, but also for the ultrafast response of excited states in a topological semimetal to fs-laser excitation. The underlying interactions offer a versatile platform to tune essential material properties. Control over the coupling strength can be expected when driving the system into a bulk-insulating or two-dimensional regime across a topological phase transition via intentional doping or quantization, thereby paving the way for a systematic tuning of the mass enhancement and the relevant time scales of ultrafast charge and energy transfer. These properties are key ingredients playing a central role in the generation and manipulation of lossless spin-polarized currents for efficient information processing on ultrafast time scales, which is highly relevant for spintronic applications. In future studies, it would be interesting to explore the influence of intense fs-laser fields on the coupling strength and the possibility of generating and transiently modifying hybrid photon–electron states. Our results are also important for other topological systems such as Weyl semimetals, where depending on the coupling strength, laser-driven surface currents originating from topological Fermi arcs and transient bulk currents originating from tilted Dirac cones could be independently manipulated on ultrafast time scales.

## Conclusion

To summarize, we have investigated the previously unmeasured connectivity of the Sb(111) surface state to the bulk continuum above EF by means of tr-ARPES in combination with spin resolution. We have provided a clear observation of this connectivity in the transiently occupied band structure between time-reversal invariant momenta, enabling the unambiguous verification of the topological character of the surface state. Our main finding has been a kink structure in the energy-momentum dispersion of the surface state above EF that is consistent with a giant mass enhancement originating from strong bulk-surface coupling. We have performed self-consistent QSGW band structure calculations and found good agreement with the experimental results. The strong bulk-surface coupling was also confirmed by our measurements of the electron dynamics, which revealed that excited states behave as a single thermalizing electronic population due to distinct contributions from interband transitions via electron–electron and electron–phonon scatterings. These findings have important implications in designing future strategies to exploit optical control of transient spin currents from topological semimetals in ultrafast spintronics.

## Methods

### Photoemission experiments

We carried out tr-ARPES experiments using linearly polarized 1.5 eV pump and 6 eV probe femtosecond (fs) laser pulses incident on the sample under an angle of 45 degrees. The high-quality Sb(111) single crystals were cleaved in situ at room temperature. The pump-probe time delay Δt was varied using an optical delay stage. The time resolution was ~200 fs, and the pump fluence ~100 μJ/cm2. The pulses were generated with a Ti: Sapphire fs oscillator coupled to an ultrafast amplifier laser system (RegA, Coherent) operated at 150 kHz. Photoelectrons were detected with a Scienta R4000 analyzer at the RGBL-2 station in Helmholtz-Zentrum Berlin, and the base pressure of the setup was ~1 × 10−10 mbar. The angular and energy resolutions of the tr-ARPES measurements were 0.1 and 20 meV, respectively. To combine spin resolution with tr-ARPES, we used a Mott-type spin polarimeter operated at 25 kV and capable of detecting both in-plane and out-of-plane components of the spin polarization. Resolutions of spin-resolved tr-ARPES measurements were 0.75 (angular) and 80 meV (energy).

### Quasiparticle self-consistent GW calculations

The ab initio calculations were carried out with the DFT code fleur (www.flapw.de) and the GW code spex65, based on the all-electron full-potential linearized augmented-plane-wave (FLAPW) formalism. For the DFT calculation that serves as a starting point for the quasiparticle self-consistent QSGW calculation47,66, we used the LDA, an angular momentum cutoff of $${l}_{\max }$$= 10 in the muffin-tin spheres and a plane-wave cutoff of 4.0 bohr−1 in the interstitial region. Angular and linear momentum cutoffs of l = 5 and 2.9 bohr−1 were employed to construct the mixed product basis for the dielectric matrix and the screened interaction67. A 8 × 8 × 8 k-point mesh was used to sample the Brillouin zone. To compute the Green and polarization functions, 500 bands were used. Spin–orbit coupling was included self-consistently47,68. For the surface calculation, a tight-binding Hamiltonian based on Wannier functions69 was constructed from the QSGW calculation of the bulk59.