Introduction

A central driving force of modern condensed matter physics is the realization of novel phases of matter out-of equilibrium like transient superconductivity1,2,3,4,5 or excitonic insulators in transition metal dichalcogenides (TMDCs)6,7,8,9,10,11,12. These are typically created from a highly complex interplay of band structure, (electronic) interactions, and excitations by a light-field. Nowadays, experimental techniques allow to actively “engineer” properties of many-body quantum systems in such out-of-equilibrium systems in a highly controlled way13,14,15. This opens up the possibility to create behavior that is not even possible in equilibrium setups. An important pathway is to induce excitations whose interplay with the electronic interactions can lead to intriguing transient behavior. Such excitations can be realized in experiments, e.g., by ultrashort laser pulses in so-called pump-probe setups16,17,18,19, or by continuous periodic driving of the systems, e.g., by shaking ultracold atom systems on optical lattices20,21,22,23,24,25,26. A much studied theoretical idealization is (infinite) periodic driving, which can be addressed by Floquet theory. In this framework and within certain limits, the properties of the system can be described by a time-independent effective Hamiltonian27,28. Control over the parameters of the driving translates to control over the effective Hamiltonian and this, in turn, allows to manipulate order parameters29, induce topological order30 etc.

It is possible to derive time-independent approximate effective Hamiltonians in the low-31 and high-frequency27,32 regimes, and development of methods in this vein is ongoing33. In addition, the periodic driving generically leads to energy absorption evolving the system towards an infinite-temperature state34,35. This restricts the relevance of such effective Hamiltonians to regimes in which the energy absorption is suppressed—like the high-frequency regime. Additional transient dynamics can be induced by the switching-on procedure28,36,37,38,39. One major topic of interest is the role of interactions in strongly driven systems. It has been shown that in Hubbard systems at resonance the interaction can be renormalized, and double occupancies can be enabled27,32,40; one can even tune the parameters of the driving so that the fermions behave like free particles41. In experiments on topological insulators, the formation of Floquet side-bands has been reported42,43 in time-resolved angle-resolved photoemission (ARPES) spectra. For strongly correlated systems, however, their experimental observation and theoretical understanding are still open questions44,45,46, and it is interesting to search for additional interaction-induced effects in nonequilibrium spectra.

In our work, we investigate periodically driven strongly interacting fermions after a sudden switching-on of the drive and without assuming the high-frequency approximation. We shed light on the effect of interactions on the transient dynamics, and we find the formation of a cosine-like band inside the gap-region of the spectral function, even though in the ground state single particle excitations have no support in this region. This is reminiscent of the creation or excitation of quasiparticles47 and their dispersion in the transient state before thermalization. To this end, we calculate non-equilibrium spectral functions with unbiased matrix product state (MPS)48,49 approaches, and systematically investigate correlation effects by comparing to non-interacting and mean-field scenarios. This is in contrast to other approaches, which e.g. rely on the Floquet-Magnus expansion27,41 in terms of the inverse driving frequency. We focus on a simple paradigmatic model for strongly correlated physics, namely a chain of spinless fermions with nearest-neighbor interactions. At half filling and zero temperature the model is known to undergo a Berezhinskii-Kosterlitz-Thouless (BKT) type transition50 from a Luttinger liquid (LL)51 to an interacting charge density wave (CDW) insulator52 when increasing the interaction strength. Driving the system with frequencies much larger than the gap ("Magnus case”), a renormalization of the gap size for this system is predicted29.

As periodic driving heats up the system, we compare our nonequilibrium results to spectral functions at finite temperatures. There, it is known from the structure factors of quantum magnets that at finite temperatures features in the gap region can be realized (e.g., the so-called “Villain mode”53,54,55,56, scattering of magnons57, or bound states58). Here, we will discuss the behavior of the periodically driven fermionic system in light of these findings.

Results

Model

We consider a periodically driven chain of interacting spinless fermions described by the Hamiltonian

$$\hat{H}(t) = -{t}_{h}\mathop{\sum }\limits_{l=1}^{L-1}\left({{{{{\mathrm{e}}}}}}^{iA(t)}{c}_{l}^{{{{\dagger}}} }{c}_{l+1}^{}+{{{{\mathrm{H.c.}}}}}\,\right)\\ +V\mathop{\sum }\limits_{l=1}^{L-1}\left({n}_{l}-\frac{1}{2}\right)\left({n}_{l+1}-\frac{1}{2}\right)\\ +\mathop{\sum }\limits_{l=1}^{L}{\mu }_{l}{n}_{l},$$
(1)

where V is the strength of the density-density interaction and μl is an additional, l-dependent on-site potential (e.g., a pinning field at the edge), which in the numerical calculations allows us to select one of the symmetry-broken CDW states, see the Methods section. \(A(t)=\theta (t){A}_{0}\sin \left(\Omega t\right)\) is a time-dependent vector potential, which is switched on at time t = 0. This approach is known as Peierls substitution59 and models a classical light field, which couples to the electrons in the system (here: spinless fermions). At equilibrium, this model can be mapped to a spin-1/2 XXZ quantum magnet via Jordan-Wigner transform. We will mostly consider open boundary conditions (OBC) and for comparison periodic boundary conditions (PBC). For A(t) ≡ 0, Bethe ansatz (BA)60 gives the BKT transition at V/th = 2. In the following, we will drive a system starting in the temperature T = 0 CDW ground state at V/th = 5, for which the energy gap according to BA60 is Δ/th ≈ 1.576.

We define the non-equilibrium generalization of the spectral function via the Fourier transform of the retarded Green’s function

$${A}_{k}^{\,{{{{\mathrm{ret}}}}}}(t,\omega ) =-{{{{{{\mathrm{Im}}}}}}}\,\frac{1}{\sqrt{2\pi }}\int\nolimits_{-\infty }^{\infty }{{{{\mathrm{d}}}}}\tau \ {e}^{(i\omega -\eta )\tau }{G}_{kk}^{\,{{{{\mathrm{ret}}}}}\,}(t,\tau ),\\ {G}_{\alpha \beta }^{\,{{{{\mathrm{ret}}}}}\,}({t}^{{\prime} },\tau ) := -i\theta \left(\tau \right)\left\langle \left\{{c}_{\alpha }({t}^{{\prime} }+\tau ),{c}_{\beta }^{{{{\dagger}}} }({t}^{{\prime} })\right\}\right\rangle \,$$
(2)

with a damping factor η ≈ 0.1 as further explained in the Methods section. As discussed there, t is the waiting time after turning on the periodic drive (note that waiting time t = 0 is not the same as equilibrium, since these are the results immediately after turning on the driving, and the time evolution over τ is performed with the driven, time-dependent Hamiltonian). This quantity relates to measurements in time-dependent angle-resolved photoemission spectroscopy (trARPES), although a more detailed modelling is required for a direct comparison to experiments17,61,62, as well as for its interpretation in the deep nonequilibrium regime63. Integration of \({A}_{k}^{{{{{{{{\rm{ret}}}}}}}}}(t,\omega )\) over the crystal momenta k directly yields the time-dependent density of states (tDOS). These quantities allow us to study qualitative changes of the spectral function with time, such as the formation of additional branches, or a change of the band structure due to the excitation.

We complement our study by considering the time evolution of the CDW order parameter

$${{{{{{{{\mathcal{O}}}}}}}}}_{{{{{\mathrm{CDW}}}}}}(t):= \frac{2}{L}\,\,\mathop{\sum}\limits_{i\,\,{{{{\mathrm{even}}}}}\,}\left\langle{n}_{i+1}(t)-{n}_{i}(t)\right\rangle\,;$$
(3)

further details are explained in the Methods section. We also keep track of the time evolution of the energy, E(t) = 〈H(t)〉, which serves as a measure for the heating of the system.

Driving a strongly correlated charge-density-wave insulator

For frequencies Ω well above the band gap (“Magnus case”) the properties of the periodically driven model are expected to be well-described by an effective Hamiltonian according to the Floquet-Magnus/high-frequency expansion29,32. It is given by the original Hamiltonian but with a renormalized hopping parameter64 \({t}_{h}^{\,{{{{\mathrm{eff}}}}}\,}={J}_{0}({A}_{0}){t}_{h}\approx 0.7652{t}_{h}\) for A0 = 1. This corresponds to a model with \(V/{t}_{h}^{\,{{{{\mathrm{eff}}}}}\,}\simeq 1.3V/{t}_{h}\). In the following, we consider these aspects by studying the system at V/th = 5, which is deep in the CDW insulating phase.

Figures 1 and 2 show results for the equilibrium and for the non-equilibrium spectral functions at waiting times \(t=0,\,10{t}_{h}^{-1},\) and \(20{t}_{h}^{-1}\). The driving frequency of Ω/th = 10 > Δ/th ≈ 1.576 is larger than the spectral gap but not substantially larger than the interaction strength so that we are not deeply in the Magnus regime. Figure 1e shows the equilibrium spectral function for the effective Hamiltonian with the renormalized hopping matrix element. Let us consider the equilibrium results first: In Fig. 1a we can identify the equilibrium continuum of excitations65 as well as the spectral gap located around ω = 0. Despite finite size effects its minimal size around k = π/2 is in agreement with the BA prediction. The spectral function of the effective Hamiltonian in Fig. 1e looks similar to Fig. 1a but the width of the continuum is smaller and the gap is larger. Turning to the non-equilibrium results, we see that the spectral function at waiting time t = 0 in Fig. 1b looks quite similar to the equilibrium result for the effective Hamiltonian in Fig. 1e, but it possesses additional features. In particular, an in-gap band comes into appearance and the continuum changes slightly its size and form. Around frequencies ω/th = ± 10 weak signals appear, which seem to echo the in-gap feature. These are reminiscent of Floquet sidebands, which are observed in time-resolved ARPES experiments42. It is noteworthy, however, that one seems to obtain these “echoes” only for the in-gap signal, but not for the main spectral features. This needs further investigations, which go beyond the scope of the present paper. At later waiting times \(t=10{t}_{h}^{-1}\) in Fig. 1c and \(t=20{t}_{h}^{-1}\) in Fig. 1d, the shape of the continuum does not further change, but the in-gap signal becomes more pronounced. This is further confirmed by Fig. 2, which shows a momentum cut through the central region of the spectral function at k ≈ π/2 for waiting times t = 0, \(10{t}_{h}^{-1}\) and \(20{t}_{h}^{-1}\). In order to better focus on the relative distribution of spectral weight we normalized all spectral functions to their maximum value at that k-slice. In all cases, we identify two main lobes and smaller peaks. Let us follow the behavior of the main lobes and of the largest peaks at ω = ± 10 and ω = 0: At waiting time t = 0, the lobes show a small difference between the nonequilibrium result and the result of the effective Hamiltonian, which becomes even smaller at later waiting times. It is noteworthy that the similarity to the effective description is already obtained at waiting time t = 0, although the effective Hamiltonian relied on the infinite-driving assumption. The additional signals at ω/th = ± 10 oscillate in time, but are suppressed with increasing time. Nevertheless, the peak at ω = 0 becomes more pronounced with time and so the cosine-like in-gap feature of Fig. 1 appears stable on the time scales treated by us. We checked that it is also present for other system sizes and periodic boundary conditions (PBC) so that boundary effects can be ruled out as an explanation (cf. Supplementary Note 1). By comparing results for L = 32 and L = 64 we find that the peak gets sharper for larger system size, while keeping the relative weight. At early waiting times negative weight appears in the spectral function. This is not an artefact and traces back to the non-equilibrium nature of the state. It was reported recently37,66 that upon averaging of the Wigner coordinate \({t}_{{{{{\mathrm{ave}}}}}}=(t+{t}^{{\prime} })/2\) over a driving period, the non-equilibrium density of states for fermions can be shown to be positive. However, at later waiting times away from the turning-on of the field at time t = 0, our spectral function (obtained using “horizontal time coordinates”37) is also almost completely positive without this procedure.

Fig. 1: (Non-)Equilibrium spectral functions of the periodically driven charge-density wave (CDW) state.
figure 1

The system is prepared initially in the ground state at interaction strength V/th = 5 and is driven with a periodic drive of frequency Ω/th = 10. a Equilibrium spectral function \({A}_{k}^{\,{{\mbox{ret}}}\,}(\omega )\) of the system. b–d Non-equilibrium spectral functions \({A}_{k}^{\,{{\mbox{ret}}}\,}(t,\omega )\) upon driving at the instances indicated. Times are measured in units of the inverse hopping amplitude \({t}_{h}^{-1}\). e Equilibrium spectral function \({A}_{k}^{\,{{\mbox{ret}}}\,}(\omega )\) of the system with renormalized hopping parameter \({t}_{h}^{\,{{\mbox{eff}}}\,}\), for comparison. The spectral features in e are also present in the spectral function of the driven system (b–d). In addition, there is clearly additional spectral weight appearing out-of-equilibrium, whose main feature is well approximated by \(f(k)\approx -2.5\cos (ka)\) (black line in d)). All data is obtained with time evolution based on matrix product states for a chain with L = 64 lattice sites and open boundary conditions. The color scheme encodes the values of \({A}_{k}^{\,{{\mbox{ret}}}\,}(t,\omega )\), where blue represents negative and red positive values.

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Fig. 2: Cross section plots through the data in Fig. 1 at electronic crystal momentum \(k\approx \frac{\pi }{2}\).
figure 2

The spectral functions in the panels are obtained at times (a) t = 0, (b) t = 10 and (c) t = 20, respectively. Dotted grey lines: Rescaled equilibrium spectral function \({\tilde{A}}_{k}^{\,{{\mbox{ret}}}\,}(\omega )\) of the initial charge-density wave state without driving and for a hopping amplitude th = 1. Dashed black lines: analogous data for a renormalized hopping amplitude \({t}_{h}^{\,{{\mbox{eff}}}\,}\approx 0.7652{t}_{h}\). Solid lines: Rescaled non-equilibrium spectral functions \({\tilde{A}}_{k}^{\,{{\mbox{ret}}}\,}(t,\omega )\) (red color = positive, blue color = negative). All data shown is rescaled such that the maximum value of the data at this k-slice is equal to one for better visual comparison of the spectral weight distribution, \({\tilde{A}}_{k}^{\,{{\mbox{ret}}}}(t,\omega )={A}_{k}^{{{\mbox{ret}}}\,}(t,\omega )\ /\ \mathop{\max }\nolimits_{\omega }\,{A}_{k}^{\,{{\mbox{ret}}}\,}(t,\omega )\).

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To better understand our findings, we study in Fig. 3 the system at the same value of V/th = 5 but with a driving frequency closer to resonance Ω/th ≈ 42 ⋅ 10−1 for comparison. The additional feature in the gap region in this case is even more strongly pronounced and it goes hand in hand with a significant reduction of the original spectral features of the CDW insulator. The question arises how this disappearance of the holon continuum is connected to a destruction of the CDW state. To study this, we calculate the CDW order parameter \({{{{{{{{\mathcal{O}}}}}}}}}_{{{{{{{{\rm{CDW}}}}}}}}}(t)\), which is displayed in Fig. 4a for driving in the Magnus regime and closer to resonance. In the latter case, \({{{{{{{{\mathcal{O}}}}}}}}}_{{{{{{{{\rm{CDW}}}}}}}}}(t)\) completely vanishes on a time scale \(t \approx 10{t}_{h}^{-1}\), which is in agreement with the time scale on which the holon continuum disappears in the spectral function. The behavior for Ω/th = 10 is more complicated, but also here a partial reduction of the CDW order is realized, which continues over times longer than the ones treated by us. Note that the parameters of the effective Hamiltonian are deeper in the CDW phase such that in equilibrium one would expect a larger CDW order parameter. In contrast, here we observe a partial melting of the order, which is due to the absorption of energy. Driven systems in the long-time limit will realize an infinite-temperature state34,35. In our case, as can be seen in Fig. 4b, the energy continues to increase as a function of time indicating that, on the transient time scale treated by us, the infinite temperature state is not yet fully reached. Clearly, energy absorption is increased closer to resonance.

Fig. 3: Spectral functions of the periodically driven charge-density wave state with drive frequency Ω ≈ 4.2th.
figure 3

The system is prepared initially in the ground state at interaction strength V/th = 5. a Equilibrium spectral function, b–d Nonequilibrium spectral functions \({A}_{k}^{\,{{\mbox{ret}}}\,}(t,\omega )\) of the driven system. The color scheme encodes the value of \({A}_{k}^{\,{{\mbox{ret}}}\,}(t,\omega )\), where blue represents negative and red positive values. The dotted lines in c and d show the function \(f(k) \sim -\frac{2.5}{4}\cos (ka)\) as a guide to the eye (note the difference to the guiding line in Fig. 1). The data in this figure was obtained from time evolution with matrix product states (MPS) for a chain length of L = 64 sites and with open boundary conditions.

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Fig. 4: Dynamics of the total energy change ΔE(t) and the charge-density wave order parameter \({{{{{{{\mathcal{O}}}}}}}}(t)\) after the sudden switch-on of the periodic driving.
figure 4

a Shows the order parameter, which was calculated with a weak pinning potential (as outlined in the Technical aspects in the Methods section) that leads to an explicit breaking of the translational symmetry. b Shows the difference in total energy to the initial state in units of the hopping amplitude th. The data in this figure was obtained from time evolution with matrix product states (MPS) for a chain length of L = 32 sites and both periodic (PBC) and open boundary conditions (OBC) are used.

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Driven CDW without interactions

We would like to distinguish our finding further from a known effect: In earlier works on electron-mediated CDW melting67,68, the appearance of in-gap spectral weight was reported already in a pumped non-interacting fermion model as a genuine non-equilibrium effect. Hence, the question arises, if the in-gap band can be obtained also in a continuously driven CDW system without interactions. To study this, we adopt the “A-B model” by Shen et al.67 at half filling,

$$\hat{H} = -{t}_{h}\mathop{\sum}\limits_{j}\left({{{{{\mathrm{e}}}}}}^{iA(t)}{c}_{j}^{{{{\dagger}}} }{c}_{j+1}^{}+{{{{\mathrm{H.c.}}}}}\,\right)\\ +\frac{U}{2}\left(\mathop{\sum}\limits_{i\in A}{c}_{i}^{{{{\dagger}}} }{c}_{i}-\mathop{\sum}\limits_{i\in B}{c}_{i}^{{{{\dagger}}} }{c}_{i}\right),$$
(4)

and apply the same semi-infinite driving protocol used for the t-V chain. The CDW order in the model is due to the presence of a staggered on-site potential and leads to a spectral gap of size Δ ≈ U. We use the same Trotterized time-evolution as in the original work67 and choose a step size of \(1{0}^{-6}{t}_{h}^{-1}\). The results of the simulations for a driving frequency of Ω = 10th and a gap of U = 5th are shown in Fig. 5. Panels Fig. 5a–d show spectral functions as in Fig. 1a–d. The dynamics of the order parameter in Fig. 5e is more oscillatory than in Fig. 4a. Nevertheless, the mean value of the oscillations decreases such that one finds a reduction of the CDW order, similar to the data with Ω/th = 10 in Fig. 4. The momentum cuts through the spectral function in Fig. 5f–h, however, show that this is not connected with the formation of a peak in the spectral gap. One should note that the spectral function does not become stationary in the model but in the in-gap region the only effect appears to be a small shift [compare Fig. 5g and h].

Fig. 5: Simulation results for the non-interacting A-B model (4) under semi-infinite sinusoidal driving.
figure 5

In this figure, an on-site potential U = 5th and semi-infinite sinusoidal driving with amplitude A0 = 1 and frequency Ω = 10th is used. The chain length is L = 128 sites. The panels a–d show results for the non-equilibrium spectral functions \({A}_{k}^{\,{{\mbox{ret}}}\,}(t,\omega )\) at different times. Panel e shows the dynamics of the charge-density wave order parameter as a function of time and panels f–h show cross sections through the spectral functions of b–d. The color scheme encodes the values of \({A}_{k}^{\,{{\mbox{ret}}}\,}(t,\omega )\), where blue represents negative and red positive values.

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Mean-field dynamics

To go beyond the purely non-interacting limit we treat the dynamics of the CDW phase in the t-V model within a Hartree-Fock mean-field (MF) approach, whose results are shown in Fig. 6. This allows us to investigate the role of interaction-induced doublon excitations for the in-gap feature. A more detailed discussion can be found in the Methods section. The equilibrium MF band structure is similar to the one in the A-B model. In the driven model, however, we obtain, in addition to the Floquet replicas of the equilibrium bands, a signal in the band gap around ω/th ≈ 0, in contrast to the findings for the noninteracting system of the previous section. This shows that an interaction-induced CDW state is essential for this feature to appear.

Fig. 6: Simulation results for the t-V model under semi-infinite sinusoidal driving within Hartree-Fock mean-field theory.
figure 6

The data in this figure was obtained for an interaction strength V = 8th and a vector potential with amplitude A0 = 1 and frequency Ω = 30th. Panels a–d show results for the non-equilibrium spectral functions \({A}_{k}^{\,{{\mbox{ret}}}\,}(t,\omega )\) at different times. Panel e shows the dynamics of the charge-density wave order parameter as a function of time and panels f–h show cross sections through the spectral functions of b–d. The color scheme encodes the values of \({A}_{k}^{\,{{\mbox{ret}}}\,}(t,\omega )\), where blue represents negative and red positive values. The length of the chain is L = 64 sites and periodic boundary conditions have been used. The heatmaps in a–d are drawn for the reduced Brillouin zone.

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Comparison to finite temperatures and quench dynamics

Additional insights can be gained by comparing the results of the periodic driving to a quench to the high-frequency effective Hamiltonian as well as to finite-temperature results at equilibrium. Figure 7 displays exact diagonalization results (ED) for \({A}_{k}^{\,{{{{\mathrm{ret}}}}}\,}(\omega )\) at finite and infinite inverse temperatures (see Methods section) β, and \({A}_{k}^{\,{{{{\mathrm{ret}}}}}\,}(t,\omega )\) at times t = 0 and t = 10 after the quench, respectively. As can be seen in Fig. 7a and b, at low temperatures the in-gap band is not visible in the ED results, but at intermediate temperatures [β = 1, Fig. 7c] a band-like feature in the gap region becomes visible, which at β = 0 (Fig. 7d) is replaced by a flat signal around ω = 0 and additional features at ω ~ ± V/2. We focus on the feature around ω = 0 at intermediate temperatures, since it is most similar to the results in the periodically driven system. We checked for different values of V/th and find that the bandwidth and position of the in-gap band at β = 1 seems not to depend on V, similar to the driven system (cf. Supplementary Fig. 4). In the quench dynamics, already immediately after the quench a weak in-gap band is visible (Fig. 7e), which is stronger at time t = 10 (Fig. 7f), whose intensity, however, depends non-monotonically on time. This is in contrast to the periodically driven system, where the in-gap band seems to be a stable feature on the time scales investigated by us. These findings further support the picture that the in-gap band can be understood already in the lowest order of the high-frequency expansion. It also shows that in our case the quench dynamics contains the main spectral features of the periodically driven system.

Fig. 7: Spectral functions at nonzero temperature T and after a quench.
figure 7

a–d Exact diagonalization results for the equilibrium spectral function \({A}_{k}^{\,{{\mbox{ret}}}\,}(\omega )\) of a chain with interaction strength V/th = 5 and inverse temperature β = 1/T, as indicated in the panels. The dashed line in c is a guide to the eye indicating the in-gap band displayed in Fig. 1d. The data was obtained for a chain with L = 12 lattice sites and open boundary conditions. e, f Nonequilibrium spectral function \({A}_{k}^{\,{{\mbox{ret}}}\,}(t,\omega )\) after a sudden quench (no periodic driving) to the zeroth order high-frequency effective Hamiltonian. The data is obtained from time evolution with matrix product states in a system with L = 32 lattice sites for open boundary conditions. The color scheme encodes the values of the spectral functions, where in e and f blue color represents negative values.

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A similar band around ω ~ 0 is known to appear in the dynamical structure factor of quantum antiferromagnets, where thermal excitation leads to the formation and scattering of moving domain walls, the so-called “Villain mode”, which has been observed in neutron scattering experiments53,54,55,56. Our findings indicate that a similar mode is obtained also in the present system at finite temperatures, and in nonequilibrium after a sudden quench or when periodically driving: In the CDW state treated here, such a domain wall is realized by a doublon, i.e. two adjacent electrons, which can freely itinerate through the system unless it recombines with the holon. The formation and delocalization of the doublon lead to a reduction of the CDW order parameter. A contribution to the spectral function stems from the scattering between populated eigenstates of the excited system with at least one doublon. Removing (adding) an electron without changing the doublon number connects two such states, generically at different momenta k and k + q. For quantum antiferromagnets, similarly, the scattering between states formed of a delocalized single domain wall—within the subspace of one domain wall—leads to a sinusoidal signal around ω = 0 in the longitudinal dynamical structure factor, which then is called the Villain mode (or resonance)53,56,69. This is also found for the transversal dynamical structure factor of the XXZ antiferromagnet69. The breaking of a doublon, in contrast, leads to a signal at ω ~ V, as seen in our finite-temperature results at higher temperatures.

The MPS results show an asymmetric line-shape (see Supplementary Figs. 1–3), which is also reported for the Villain mode in antiferromagnets69. Although the fundamental mechanism behind the Villain mode, i.e. free doublon propagation, can also occur in mean-field theory, it clearly does not capture those asymmetries. Nevertheless, our data indicates Villain-like resonances as a likely explanation of the observed in-gap feature. To further understand the properties of the in-gap band, a similar calculation to the one pursued by Jones et al.69 for the spinless-fermion model is needed, which goes beyond the scope of this paper.

Conclusions

We report the formation of a cosine-shaped signal in the time-dependent (retarded) spectral function upon periodic driving of an initial CDW insulator. For driving frequencies Ω larger than the spectral gap, it occurs in addition to the equilibrium spectral features, which are modified according to the prediction by the high-frequency effective Hamiltonian29. When approaching resonance, the spectral function changes significantly as compared to the equilibrium case and a flat band around ω = 0 becomes the dominant feature.

We find these results based on quasi-exact time evolution using matrix product states. The additional signal is not observed when periodically driving a non-interacting CDW state, but a mean-field treatment of the Hamiltonian (1) reveals a similar in-gap signal around ω = 0. However, its properties still differ significantly from the MPS results, in particular with respect to the line shape.

Comparing to finite-temperature results and to the dynamics of a quench to the high-frequency effective Hamiltonian reveals that also in these cases the in-gap band is present, and its properties are very similar to the one seen in the periodically driven system. This indicates that exciting the system by adding a finite amount of energy populates certain states not present in the ground state spectral function. This is reminiscent of the behavior of the dynamical structure factor of quantum magnets, where different possible scenarios for temperature-induced additional branches are known, e.g., i) Villain mode of moving domain walls in spin-1/2 quantum antiferromagnets53,54,55,56; ii) scattering of elementary excitations (e.g. magnons in spin-1 chains57); iii) population of bound states58. The bandwidth of the observed in-gap band appears to be independent of V, similar to the Villain resonance53, so that our fermionic system likely realizes a Villain-like band at finite (intermediate) temperatures and out-of-equilibrium, in both the quench dynamics and under periodic driving. At high temperatures, a flat band is obtained at ω ≈ 0, which is also observed when driving close to resonance, where the results are very similar to the β = 0 results, indicating heating to infinite temperature.

Note that a Villain-like mode has also been reported for the fermionic Hubbard chain at finite temperatures, however, there it is seen in the structure factor of the density-density correlations70. It will be interesting to further investigate this feature in this simple correlated system, e.g., using perturbative approaches69 or Bethe ansatz, or semiclassical approaches, such as fermionic truncated Wigner approximations71,72,73. It is an open question to see whether the interplay of the melting of CDW states and electron correlations can lead to similar additional resonances in the spectral function also in periodically driven interacting two-dimensional systems. This is relevant for predictions for ongoing pump-probe investigations, e.g., in two-dimensional transition metal dichalcogenides (TMDCs)74. It will be fruitful to explore the interplay of such in-gap features, Floquet side-bands and dephasing mechanisms in strongly correlated materials, and also in related model realizations by quantum simulators with ultracold gases on optical lattices20,21,75.

Methods

Green’s functions

All Green’s functions (GFs) are derived from the contour-ordered single-particle Green’s function76

$${G}_{\alpha \beta }(t,{t}^{{\prime} })=-i\left\langle {\hat{T}}_{C}{c}_{\alpha }(t){c}_{\beta }^{{{{\dagger}}} }({t}^{{\prime} })\right\rangle \,,$$
(5)

which can be written in a matrix representation with respect to the forward and backward branches of the real-time axis. In this representation the greater, \({G}_{\alpha \beta }^{\ > \ }(t,{t}^{{\prime} })\), and lesser Green’s function \({G}_{\alpha \beta }^{ < }(t,{t}^{{\prime} })\) each have one time argument lying on the forward and one on the backward branch of the real-time contour. The retarded GF is a linear combination of the two with an additional theta function,

$${G}_{\alpha \beta }^{\,{{{{\mathrm{ret}}}}}\,}(t,{t}^{{\prime} }) = \, \theta (t-{t}^{{\prime} })\left({G}_{\alpha \beta }^{\ > \ }(t,{t}^{{\prime} })-{G}_{\alpha \beta }^{ < }(t,{t}^{{\prime} })\right)\\ := -i\theta \left(t-{t}^{{\prime} }\right)\left(\left\langle {c}_{\alpha }(t){c}_{\beta }^{{{{\dagger}}} }({t}^{{\prime} })\right\rangle +\left\langle {c}_{\beta }^{{{{\dagger}}} }({t}^{{\prime} }){c}_{\alpha }(t)\right\rangle \right)\,.$$
(6)

At equilibrium, one of the two time variables can be suppressed due to time-translational invariance. Out-of-equilibrium, however, one needs to consider both time variables, and the Fourier transform to frequency space is not unique any more. In order to minimize the numerical costs, we choose to use “horizontal” time coordinates, in which we evolve the wavefunction up to a time \({t}^{{\prime} }\) and then perform the Fourier transform with respect to the relative time \(\tau =t-{t}^{{\prime} }\), after further evolving the system in time, with \(t > {t}^{{\prime} }\). Kalthoff et al.37 obtained similar results for this choice of the waiting time and the Wigner choice \((t+{t}^{{\prime} })/2\) in the case of a free fermion model. The states are labelled by momentum indices k (depending on boundary conditions, see below). In the numerics we calculate the auxiliary quantities

$$\begin{array}{l}{C}_{kl}^{ < }({t}^{{\prime} },\tau )=\left\langle \right.{c}_{k}^{{{{\dagger}}} }({t}^{{\prime} }+\tau ){c}_{l}({t}^{{\prime} })\rangle \\ {C}_{kl}^{\ > \ }({t}^{{\prime} },\tau )=\left\langle \right.{c}_{l}({t}^{{\prime} }){c}_{k}^{{{{\dagger}}} }({t}^{{\prime} }+\tau )\rangle \,,\end{array}$$
(7)

such that

$${G}_{kk}^{\,{{{{\mathrm{ret}}}}}\,,\lessgtr }({t}^{{\prime} },\tau )=-i\theta (\tau ){C}_{kk}^{\lessgtr }{({t}^{{\prime} },\tau )}^{* },$$
(8)

and we finally obtain the nonequilibrium spectral functions

$${A}_{k}^{\,{{{{\mathrm{ret}}}}}\,,\lessgtr }(t,\omega ) = -{{{{{{\mathrm{Im}}}}}}}\,\frac{1}{\sqrt{2\pi }}\int\nolimits_{-\infty }^{\infty }\,{{{{\mathrm{d}}}}}\,\tau \ {{{{{\mathrm{e}}}}}}^{(i\omega -\eta )\tau }{G}_{kk}^{{{{{\mathrm{ret}}}}}\,,\lessgtr }(t,\tau )\,,\\ {A}_{k}^{\,{{{{\mathrm{ret}}}}}\,}(t,\omega ) = \,{A}_{k}^{\,{{{{\mathrm{ret}}}}}, < }(t,\omega )+{A}_{k}^{{{{{\mathrm{ret}}}}}\,,\ > \ }(t,\omega )\,.$$
(9)

In equilibrium the retarded GF contains information about the density of states while the lesser and greater GFs contain information about occupations of the states. The latter is reflected in the fact that due to the absence of the θ(τ) the whole “history” of the lesser and greater GFs needs to be traced back until time −∞. In this work we only compute the retarded Green’s function. Integration of \({A}_{k}^{{{{{{{{\rm{ret}}}}}}}}}(t,\omega )\) over k directly yields the time-dependent local density of states. In contrast to the equilibrium situation the quantity \({A}_{k}^{{{{{{{{\rm{ret}}}}}}}}}(t,\omega )\) is not necessarily positive, so that care needs to be taken with this interpretation63.

A more subtle issue concerns the gauge-invariance of the calculated spectral function. In accordance with other studies in the literature44,77, we perform calculations in a fixed gauge. Gauge-invariant formulations have been proposed17,61,62, in particular for overlapping pump and probe pulses, or if the system possesses multiple bands. These are, however, difficult to implement in practice and fixed-gauge calculations already yield qualitatively insightful results.

Therefore, we focus on the most important qualitative features evolving with time. Central predictions of Floquet theory, like the effective Hamiltonian picture, are captured with a good accuracy, so that we believe that our results not only provide a qualitative picture, but also quantitative predictions with a good accuracy.

Technical aspects

Spatial Fourier transform

For periodic boundary conditions (PBC) we use \(k\in \frac{2\pi }{L}\cdot \{0,\ldots ,L\}\) while for open boundary conditions (OBC) \(k\in \frac{\pi }{L+1}\cdot \{1,\ldots ,L\}\) corresponding to a sine transform78,79 with

$${c}_{k}=\sqrt{\frac{2}{L+1}}\mathop{\sum}\limits_{k}\sin (k{r}_{i}){c}_{i}.$$
(10)

The OBC momenta differ slightly ( ~ 1/(L + 1)) from simple fractions of π, e.g. π/2, so we always write ≈ and the closest k-value.

Temporal Fourier transform

Due to the finite maximal τ that we are able to reach and due to the theta function, the temporal Fourier transform produces a non-zero background ~ 10−5 signal everywhere in the spectral function. Since we expect the exact spectral function to have value zero if no signal is present, we decided to subtract this background. We add a damping factor η ≈ 0.1 to regularize the finite time-propagation with respect to τ. In addition, to improve the ω-resolution we padded the τ-data with zeros to obtain at least 4096 frequency points.

Symmetry-broken CDW state

For MPS calculations with OBCs we prepare the system at half filling with μl = 0. In the CDW phase, this leads to an exact superposition of the two possible symmetry-broken ground states, so that for a finite system \({{{{{{{{\mathcal{O}}}}}}}}}_{{{{{\mathrm{CDW}}}}}}(t)=0\) for all times t. In order to be able to keep track of the dynamics of the CDW order parameter, we performed additional simulations where we applied a ‘pinning field’ at the edge μ1,L ≠ 0, which selects one of the ground states and allows us to study \({{{{{{{{\mathcal{O}}}}}}}}}_{{{{{\mathrm{CDW}}}}}}(t)\). We checked that the spectral function does not differ if calculated with or without the pinning field. In order to minimize possible boundary effects in the order parameter, we perform the sum in Eq. (3) only over the four unit cells in the center of the system.

MPS calculations

For comprehensive reviews we refer to the literature48,49. Here we only focus on the main aspects concerning the computation of the time-dependent spectral functions. In order to compute \({G}_{kk}^{\,{{{{\mathrm{ret}}}}}\,,\lessgtr }(t,\tau )\) using matrix product states we start from the system’s ground state \(\left\vert \,{{{{\mathrm{GS}}}}}\,\right\rangle\) and consider the following quantum states

$$\big\vert \psi (t)\big\rangle = {U}_{{{{{\mathrm{TDVP}}}}}}(t,0)\big\vert \,{{{{\mathrm{GS}}}}}\,\big\rangle ,\\ \big\vert {\phi }_{l}^{ < }(t)\big\rangle = {c}_{l}\big\vert \psi (t)\big\rangle ,\\ \big\vert {\phi }_{l}^{\ > \ }(t)\big\rangle = {c}_{l}^{{{{\dagger}}} }\big\vert \psi (t)\big\rangle ,\\ \big\vert {\phi }_{l}^{\lessgtr }(t+\tau )\big\rangle = {U}_{{{{{\mathrm{TDVP}}}}}}(t+\tau ,t)\big\vert {\phi }_{l}^{\lessgtr }(t)\big\rangle ,$$
(11)

where UTDVP(t, t0) denotes time-evolution using the MPS realization of the time-dependent variational principle (TDVP). Here, we always apply a two-site TDVP algorithm49. The operators ck carry momentum space labels. However, we always work in position space and exploit that we can write momentum space annihilation and creation operators as a sum of local operators ck = ∑jPk,jcj, where we have introduced the transformation matrix P, allowing us to compute \({C}_{kl}^{\lessgtr }(t,\tau )\) through a series of local MPO-MPS applications. P depends on the boundary conditions used. Using the states (11) we calculate the quantities

$${C}_{kl}^{ < }(t,\tau )=\left\langle \psi (t+\tau )| {c}_{k}^{{{{\dagger}}} }| {\phi }_{l}^{ < }(t+\tau )\right\rangle ,$$
(12)
$${C}_{kl}^{\ > \ }(t,\tau )=\left\langle {\phi }_{l}^{\ > \ }(t+\tau )| {c}_{k}^{{{{\dagger}}} }| \psi (t+\tau )\right\rangle ,$$
(13)

which are related to \({G}_{kk}^{\,{{{{\mathrm{ret}}}}}\,,\lessgtr }(t,\tau )\), see Eq. (8). For PBC we implement a “snake geometry” (Paeckel, S. & Köhler, T. Private communication)80 for the labelling of the sites in the chain. It turned out that the DMRG ground state search always choses one of the degenerate ground states which allowed us to calculate the order parameter directly. Further details on the calculation of the non-equilibrium spectral function with MPS methods can be found elsewhere79.

Hartree-Fock time evolution

We start from a Hartree-Fock decoupling of the interaction term and assume a two-site unit cell with sublattices A and B. Let us denote

$$\begin{array}{l}{\rho }_{A}:= {\langle {c}_{i}^{{{{\dagger}}} }{c}_{i}\rangle }_{i\in A},\quad {\rho }_{0}:= {\langle {c}_{i}^{{{{\dagger}}} }{c}_{i+1}\rangle }_{i\in A},\\ {\rho }_{B}:= {\langle {c}_{i}^{{{{\dagger}}} }{c}_{i}\rangle }_{i\in B},\quad {\rho }_{1}:= {\langle {c}_{i}^{{{{\dagger}}} }{c}_{i+1}\rangle }_{i\in B}.\end{array}$$
(14)

Using the Fourier basis (Q = π)

$$\begin{array}{l}{c}_{i\in A}^{{{{\dagger}}} }=\frac{1}{\sqrt{V}}\mathop{\sum}\limits_{k\in \,{{{{\mathrm{rBZ}}}}}}{{{{{\mathrm{e}}}}}}^{-ik{r}_{i}}\left({c}_{k}^{{{{\dagger}}} }+{c}_{k+Q}^{{{{\dagger}}} }\right)\\ {c}_{i\in B}^{{{{\dagger}}} }=\frac{1}{\sqrt{V}}\mathop{\sum}\limits_{k\in \,{{{{\mathrm{rBZ}}}}}}{{{{{\mathrm{e}}}}}}^{-ik{r}_{i}}\left({c}_{k}^{{{{\dagger}}} }-{c}_{k+Q}^{{{{\dagger}}} }\right)\end{array}$$
(15)

we obtain, using the definitions \({\epsilon }_{k}=-2{t}_{h}\cos (k)\), \({\chi }_{k}=V\left({\rho }_{0}{{{{{\mathrm{e}}}}}}^{-ik}+{\rho }_{1}^{* }{{{{{\mathrm{e}}}}}}^{ik}\right)\), the following representation of the Hamiltonian

$$(H) = \left(\begin{array}{cc}{c}_{k}^{{{{\dagger}}} }&{c}_{k+Q}^{{{{\dagger}}} }\end{array}\right)\\ \left(\begin{array}{cc}{\epsilon }_{k}-{{{{{{\mathrm{Re}}}}}}}\,({\chi }_{k})+V({\rho }_{A}+{\rho }_{B})-\mu &i{{{{{{\mathrm{Im}}}}}}}\,({\chi }_{k})+V({\rho }_{B}-{\rho }_{A})\\ -i{{{{{{\mathrm{Im}}}}}}}\,({\chi }_{k})+V({\rho }_{B}-{\rho }_{A})&-{\epsilon }_{k}+{{{{{{\mathrm{Re}}}}}}}\,({\chi }_{k})+V({\rho }_{A}+{\rho }_{B})-\mu \end{array}\right)\\ \left(\begin{array}{c}{c}_{k}\\ {c}_{k+Q}\end{array}\right)\,.$$
(16)

In the following we consider half filling μ = V(ρA + ρB). The saddle point values of ρA, ρB, etc. are determined with a simulated annealing approach. Diagonalization of the Hamiltonian yields the eigenenergies

$${E}_{k} = \pm {\left\{-\left[-{\left({\epsilon }_{k}-{{{{{{\mathrm{Re}}}}}}}\,({\chi }_{k})\right)}^{2}-{{{{{{\mathrm{Im}}}}}}}\,{({\chi }_{k})}^{2}-{V}^{2}{\left({\rho }_{B}-{\rho }_{A}\right)}^{2}\right]\right\}}^{1/2}\\ = \pm {\left\{\left\vert \right.{\epsilon }_{k}-{\chi }_{k}{\left\vert \right.}^{2}+{V}^{2}{({\rho }_{B}-{\rho }_{A})}^{2}\right\}}^{1/2}.$$
(17)

Hence, the spectral gap is given by 2V(ρB − ρA).

For the dynamics we first solve the time-diagonal problem and obtain the full one-particle reduced density matrix \({\rho }_{ij}(t)=\langle {c}_{i}^{{{{\dagger}}} }(t){c}_{j}(t)\rangle\). In a second iteration we solve the equation of motion for the relative time τ using the time-diagonal data from the first iteration. This corresponds to solving the Kadanoff-Baym equations with a Hartree-Fock self-energy76.

Computation of the finite-temperature spectral function

We use the Lehmann representation for the equilibrium retarded spectral function at finite temperatures,

$${A}_{k}^{{{{{{{{\rm{ret}}}}}}}}}(\omega ) = \frac{1}{Z}\mathop{\sum}\limits_{m,n}{\left\vert \langle m| {c}_{k}| n\rangle \right\vert }^{2}\left({{{{{\mathrm{e}}}}}}^{-\beta {E}_{n}}+{{{{{\mathrm{e}}}}}}^{-\beta {E}_{m}}\right)\cdot \\ \delta \left(({E}_{n}-{E}_{m})-\omega \right)\,,$$
(18)

where \(Z={\sum }_{n}{e}^{-\beta {E}_{n}}\) is the partition function. The eigenstates \(\left\vert m\right\rangle\) and \(\left\vert n\right\rangle\) as well as the eigenvalues En of the Hamiltonian (1) evaluated at time t = 0 were obtained using QuSpin81,82. We grouped the frequencies into bins of width 10−2.