The remarkable feature associated with spontaneous symmetry breaking (SSB) is emergent scale-invariant dynamics in approaching a thermal critical point¹. There have been strong incentives to understand how this self-organization may proceed out of thermal equilibrium^1,2. Studying the nonequilibrium phase transition introduced via a swift change of the system parameters, also called a quench, is one of the most active areas in nonequilibrium physics, impacting diverse fields from condensed matter^3,4, quantum gases^2,5,6, to cosmology^4,7. It is widely believed that, after an interaction quench, isolated systems generically approach a thermal state²; however, before thermalization can occur a transient nonthermal stationary state may emerge with properties unlike their equilibrium counterparts⁸. Such investigations have been carried out using ultracold atoms^2,5. Meanwhile, recently ultrafast pump-probe studies made surprising discoveries of light-induced superconductivity⁹ and insulator-metal transitions in hidden charge-density wave (CDW) states^10,11,12, hinting undisclosed routes towards new broken-symmetry ground states through light excitation far from equilibrium.

In this article, we demonstrate a prototypical case of nonequilibrium SSB into a hidden ground state through a laser-assisted nonthermal quench of the system parameters that define competing orders. The system is CeTe₃¹³ in which the naturally occurring SSB ground state is the stripe-phase c-CDW order^14,15,16; see Fig. 1a. Upon femtosecond (fs) near-infrared pulse excitation the pre-existing order is transiently suppressed; however, the system develops a new preference of SSB from the stripe order to a bi-directional state beyond a well-defined critical excitation. Furthermore, this ultrafast phase transition displays key characteristics of nonequilibrium SSB processes^4,5,17: the spontaneous emergence of soft-mode instabilities of the new CDW order, followed by a slow onset of the phase ordering stage to develop the long-range correlations. Finally, after the system relaxation back to the thermal ground state the remnants of transient orders survive as long-lived topological defects^4,6,7 for more than 2 ns. The dynamics observed here open an intriguing perspective of controlling phase transitions in quantum materials far from equilibrium.

The CeTe₃ studied here belongs to the rare-earth tritelluride (RTe₃) family^13,14,16 where the incommensurate CDW develops inside the double Te square lattice sheets, which are isolated by the buckled insulating CeTe layer (Fig. 1a). The 2D-layered RTe₃ compounds are ideal systems for studying SSB because in the square net of Te layers two competing stripe density waves (along the a or c-axis) can appear^14,15,16,18. The formation of different types of CDWs is subject to the nesting in the electronic structure^16,18 but the strong momentum-dependent electron-phonon coupling (EPC) is believed to be essential to form single-wavevector CDW at dimensionality higher than one^19,20,21,22. The shape of the 2D metallic Fermi surface (FS) depends on the relative coupling strengths between neighboring 5p_x and 5p_z orbitals (t_⊥ and t_// in Fig. 1a), which play a key role in these interactions^16,18,23,24. Nonetheless, the tendency for the RTe₃ system to form c-CDW over a-CDW as the preferred broken-symmetry ground state is facilitated by a subtle bi-layer coupling that weakly breaks the C4 symmetry in the Te lattice¹⁸. Especially in CeTe₃, a large c-CDW gap removes a significant amount of the potential a-CDW spectral weight^16,25, which completely excludes subsequent formation of a-CDW in its equilibrium phase diagram (Fig. 1a)¹⁵. Only in the heavier members of RTe₃ (see Fig. 1a) where the smaller lattice constant weakens the c-CDW can the a-CDW develop at a second lower critical temperature^14,15,16.

Our CeTe₃ sample is a single crystal prepared by tape exfoliation to a thickness of ≈25 nm²⁶ with a transverse size ≈30 μm; see “Methods” section. The sample film thickness is matched to the penetration depth of the pump laser pulse^27,28, establishing a nearly homogeneous excitation profile facilitated by the thin-film interference effect (Supplementary Fig. 1)²⁹. The uniformity and thickness of the exfoliated samples are checked using a TEM, where we also conduct the temperature-dependent studies; see Supplementary Fig. 2 for the results. The sample is gently placed on a thin TEM grid in the vacuum specimen chamber where a relatively large pump laser beam (500 μm) illuminates the sample uniformly. Due to the good thermal isolation in this sample setting, the experiments show no visible dissipation of the pump energy absorbed into the materials over the observation window (2 ns). To optimize the probe beam brightness, we deliver intense electron pulses (~10⁶ e/pulse) at 100 keV generated from a silver photo-cathode just below the virtual cathode limit^30,31. The phase space of the space-charge-dominated pulse is actively manipulated through placing a radio-frequency cavity in the optical column acting as the longitudinal lens^30,32 (Fig. 1b) to reach ≈100 fs pulse duration. This is accomplished under a condition that its transverse phase space, controlled through a series of magnetic lenses, produces a highly collimated beam³³ with a beam coherence length up to 40 nm³⁰. A high-quality pattern produced by this fs coherent scattering setup is shown in Fig. 1c, where the incommensurate CDW state evidenced by the satellite peaks at Q_c ≈ 2/7c*^13,14 stands sharply apart from the lattice Bragg peaks (G) in Miller indices. Due to the large intensity difference between the two (~100:1), typically the lattice peaks are intentionally saturated to provide sufficient dynamical range for investigating the nonequilibrium CDW dynamics.

Dynamics of CDW phase transition in RTe₃ have been investigated using the ultrafast pump-probe techniques^{24,25,28,34,35,36,37,38,39,40}. The pump fs laser pulses couple to the CDW materials with a broad excitation spectrum across the gap, leading to rapid carrier heating. Hence, the natural setting of these experiments has been to investigate the melting of the pre-existing c-CDW in this system. The time- and angle-resolved photoemission spectroscopy (trARPES) found that within ≈250 fs upon applying laser pulses, the spectroscopic gap at the Brillouin zone momentum c* − Q_c is smeared^{24,25,37,38,39}. The gap dynamics is coupled to the amplitude suppression of the density wave at the momentum wavevector Q_c as resolved with the fs electron^36,41 and x-ray scattering^40,42. Time-domain spectroscopies gain accesses to the lower frequency collective modes, which are coherently excited upon applying a weaker laser pulse^{20,28,40,43,44}. The BCS-type behavior^14,23,45 has been identified by monitoring the collective mode softening in temperature (T) and the laser-fluence (F) dependent investigations near T_c,1^20,43,44. Focused on the dynamics of the recovery, a study joining ultrafast electron diffraction (UED), optical reflectivity, and trARPES to investigate LaTe₃ reported a much faster recovery of gap size than in recuperating the diffraction correlation length. The authors attributed the phenomenon to the presence of topological defects induced in the processes³⁸. Interestingly, a more recent MeV-UED investigation following the work found the emergence of a light-induced a-CDW state and interpreted the phenomenon as a transient order developed by nucleating around the topological defects⁴¹. Whereas the various pump-probe investigations have illustrated the ultrafast dynamics of melting and the possibility of defect-induced state in RTe₃, the symmetry-breaking aspect that underpins the formation of the hidden order and the associated nonequilibrium phase ordering dynamics have not been clearly elaborated.

Here, with a high momentum resolution fs scattering probe, our work has addressed the nonequilibrium structural formation of a hidden a-CDW in CeTe₃. The slow onset of the new phase and the coupled soft modes directly involved in the ordering processes lead us to conclude that a nonequilibrium SSB is responsible. The new understanding also bears on a recent clear demonstration of hidden co-instabilities occurring along both symmetry-breaking axes approaching T_c,1 in a similar DyTe₃ system, probed by the inelastic x-ray scattering (IXS)²³. This result suggests that the underlying energy landscape supports two stripe-phase CDW ground states – even though a-CDW remains unsubstantiated throughout the phase transition. The reversal of the symmetry-breaking preference by light excitations evidenced here hence opens a window of opportunities for controlled investigation of the competitively ordered systems.

A phenomenological Landau–Ginzburg (L–G) theory is adopted to capture the essences of nonequilibrium phase transitions. We describe the two CDW states in terms of the order parameter η_l = |η_l|e^iϕl, where l = a or c, and |(η_l)| and ϕ_l represent the amplitude and phase fields. The L–G free energy modeled on the amplitude field is written as^18,46

with α_l and A_l > 0 for supporting the two symmetry-breaking states at T^(l) < T_c. The inclusion of the repulsive bi-quadratic coupling term where Ã > 0 makes the eventual symmetry breaking a competition between the two. In this regard, an anisotropy in α_l/A_l makes c-CDW instead of a-CDW the global ground state. The competing strengths of the order parameters can be evaluated at the stationary conditions, \(\frac{{\partial f}}{{\partial \left| {\eta _a} \right|}} \,=\, \frac{{\partial f}}{{\partial \left| {\eta _c} \right|}} \,=\, 0\), which define the coordinates of the free energy minimum where \(\left| {\eta _a} \right| \,=\, \sqrt {\frac{{\alpha_a\left( {T_c \,-\, T} \right) \,-\, \widetilde A\left| {\eta _c} \right|^2}}{{A_a}}}\) . From this, it is established that a stationary state with a non-zero |η_a| occurs only if |η_c| is suppressed beyond a critical threshold \(\left| {\eta _{c,th}} \right| \,=\, \sqrt {\frac{{\alpha_a\left( {T_c \,-\, T} \right)}}{{\tilde A}}}\), which can occur under the nonequilibrium condition.

In Fig. 2a this nonequilibrium route of phase transition is illustrated. The key step, modeled after the scenario of F = 1.85 mJ/cm², is the sudden unfolding of the L–G landscape from the uniaxial to the bi-directional one, driven by an interaction quench that shifts the pre-existing order parameter |η_a| from 1 to ≈\(\sqrt {0.17}\) (see stage II). Here, governing the a-CDW formation following this change is the projected free energy, f(|η_a|), as depicted in the inset panel. The f(|η_a|) unfolds from an uphill into a double well under the presupposition that the subsystem effective temperature, T^(a), remains near that of the ambient, in contrast to the excited c-CDW state. This non-ergodicity presupposition is justified by the high quench speed and the involvement of slow modes mediating the phase transitions. The nonequilibrium scenario makes the bi-directional ordering possible as opposed to the equilibrium route (also shown on the right of Fig. 2a). For details of parameterizing the L–G equation to describe the thermal and nonthermal phase transitions, see “Methods” section.

One chief goal in the studies of nonequilibrium SSB dynamics is to see how the rapidly quenched system adopts the long-range correlations from the deeply unstable order parameter fields^3,4,5,7,17, often referred to as coarsening^1,3. While at the late stage of coarsening the system is expected to follow universal laws^1,3, at the shortest time the evolution is dominated by the spinodal (unfolding) instabilities^17,47. In CDW phase transitions, the instabilities are mediated by soft modes, which occur in both the amplitude and phase fields^23,48,49,50, and the order parameter dynamics are governed by a 2D Mexican hat free energy (see Fig. 2b) in which the phase fluctuation acts as a Goldstone mode with essentially a zero energy cost^48,49,50,51. It has been theorized that to excite a collective mode in the CDW fields at the momentum wavevector k ≡ q − Q_a, two lattice soft phonons at q = k ± Q_a must be joined coherently⁴⁹.

The static single-wavevector lattice distortion wave (LDW), u_η(Q_l r), is a distinct physical manifestation of the CDW state which can be probed directly with the scattering techniques. The conversion between the order parameters (with |η_c| = 1, t < 0) modeled by the L–G theory and the LDW can be established with coefficient \(A_{\eta u} \,=\, \frac{{\left| {u_\eta } \right|}}{{\left| \eta \right|}}\). The nonequilibrium LDW with field variations is described by \({\mathbf{u}}_\eta \left( {{\mathbf{r}},t} \right) = u_{0,\eta }{\hat{\mathbf{e}}}_\eta \left( {1 \,+\, \delta u_\eta \left( {{\mathbf{r}},t} \right)} \right) \sin\left[ {{\mathbf{Q}} \cdot {\mathbf{r}} \,+\, \delta \phi \left( {{\mathbf{r}},t} \right)} \right]\), where u_0,η and \({\hat{\mathbf{e}}}_\eta\) are the amplitude and polarization vector. These field variations, which can be expanded in series of fluctuation waves of wavevector k, manifest in the diffuse scattering centered at Q₁; see Supplementary Note 1. The sum rules allow a simple expression \(S_{Q_l}\left( {{\mathbf{k}},t} \right) \approx | {u_{0,\eta _l}(t)} |^2f\left( {{\mathbf{k}},t} \right)\) in which f(k; t) is a normalized function containing the spatial and ensemble averaging of a system much greater than the CDW correlation length ξ; see Supplementary Note 2. In this expression, the structure factor S_Ql (k;t) simply is the Fourier counterpart of the autocorrelation function, \(S_{\eta _l}\left( {{\mathbf{r}},t} \right) \approx {\mathbf{u}}_{\eta _l}\left( {{\mathbf{r}}^\prime,t} \right){\mathbf{u}}_{\eta _l}\left( {{\mathbf{r}}^{\prime\prime},t} \right)\), with r = r′ − r″ and the bracket denoting the averaging^1,3,17,48. The coarsening process, during which fluctuation waves condensed into the LDW, is evidenced in the dynamical structure factor that sharpens over time, as illustrated in Fig. 2b. It is easy to see the amplitude of LDW is given by integrating structure factor over the momenta, \(m_{Q_l} = {\int} {S_{Q_l}\left( {{\mathbf{k}},t} \right)d{\mathbf{k}} \approx | {u_{0,\eta _l}(t)} |^2}\). Meanwhile, to probe the fluctuation effects, it is key to resolving \(S_{Q_l}\left( {{\mathbf{k}},t} \right)\), typically a Lorentzian with \(f\left( {\mathbf{k}} \right)\sim \frac{1}{{1 + \xi ^2k^2}}\) (assuming the correlation decays exponentially over the length ξ), in the momentum space.