Abstract
The study of the temporal evolution of systems undergoing symmetry breaking phase transitionsâwhether it is in condensed-matter physics, cosmology or finance1,2,3,4,5,6âis difficult because they are hard to repeat, or they occur very rapidly. Here we report a high-time-resolution study of the evolution of both bosonic and fermionic excitations through an electronic charge-ordering symmetry breaking phase transition. Periodically quenching our system with femtosecond optical pulses, we subsequently detect hitherto-unrecorded coherent aperiodic undulations of the order parameter, critical slowing down of the collective mode and evolution of the particleâhole gap as the system evolves through the transition. Modelling on the basis of GinzburgâLandau theory is used to reproduce the observations without free parameters. Of particular interest is the observation of spectrotemporal distortions arising from spontaneous annihilation of topological defects, analogous to those discussed by the KibbleâZurek cosmological model2,3.
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Main
The behaviour of symmetry breaking phase transitions (SBTs) is commonly studied under near-equilibrium (near-ergodic) conditions, where excitations on all timescales contribute to the process, and the behaviour of physical quantities through the SBT is described by power laws with critical exponents. However, when the ordering proceeds non-ergodically, the quasiparticles and collective boson excitations perceive the crystal background as an effective vacuum. To study this behaviour in condensed-matter systems, we chose crystals with electronically driven instabilities caused by a Fermi-surface nesting, leading to a second-order transition to a charge-density-wave (CDW) ordered state7,8,9,10,11,12. The state is characterized by spatial modulations â¼cos(Q x+Ï) of the electronic density and lattice displacements described by a complex order parameter Ψâ¼ÎeiÏ. The elementary bosonic collective excitations of Î and Ï are the amplitude and the phase modes (AM and PhM), and Ψ may be viewed as a âHiggs fieldâ1, opening a gap 2|Î| in the fermionic spectrum. In this paper we focus on TbTe3 (CDW transition at Tc=336âK; refs 13, 14), but experiments on other microscopically diverse systems (DyTe3, 2H-TaSe2 and K0.3MoO3) are also presented, demonstrating universality.
The main idea realized here is to quench the system into the high-symmetry state with intense âdestructionâ (D) laser pulses and then monitor the time evolution through the SBT by measuring the resulting reflectivity oscillations with a pumpâprobe (Pâp) sequence (Fig. 1a). The D pulse excites electrons and holes, thus suppressing the electronic susceptibility at 2kF, whose divergence is the cause for the CDW formation. Any asymmetry in the band structure also leads to an imbalance of the e and h populations, shifting the chemical potential and causing a disturbance δ q of the Fermi surface kF according to and destroying the CDW. After initial rapid quasiparticle (QP) relaxation, we can expect the appearance of topologically non-trivial local configurationsâdomain walls, solitons and so on, which are allowed by the ground-state degeneracy with respect to Ï (ref. 15). (The experimental details are given in the Supplementary Information).
The transient reflectivity ÎR/R of TbTe3 as a function of DâP time delay Ît12 is shown in Fig. 1b. We distinctly observe an exponential QP transient at short times Ît23<1âps, and an oscillatory response owing to AM and coherent phonon oscillations14. A two-dimensional plot highlighting the QP response is shown in Fig. 2a. Immediately after the quench the QP peak amplitude AQP is completely suppressed, indicating the disappearance of the CDW gap. As the QP peak starts to recover, initially the QP lifetime ÏQP is a few picoseconds, but both AQP and ÏQP recover to their equilibrium values within 1â2âps. A single-exponential fit to both ÏQP(t) and AQP(t) is shown in Fig. 2b, giving the QP gap recovery time ÏÏQP =ÏAQP =650±50âfs. This behaviour is consistent with the previously reported relation ÏQPâ¼1/Î(T) (refs 14, 16).
Figure 1c shows ÎR/R with the QP signal subtracted, showing an aperiodic oscillatory response for short Ît12. The fast Fourier transform power spectra of these data as a function of Ît12 are plotted in Fig. 2c. The most obvious non-trivial observation in the Ï âÎt12 plots is that the intensity of the AM fluctuates strongly up to â¼7âps. The fluctuations are irregular at first, showing a distinct slowing down in the critical region Ît12=1.5âps. The AM, whose frequency in the equilibrium broken-symmetry state is 2.18âTHz, shows a dramatic softening for Ît12<2âps. In the course of the system recovery, the AM crosses the 1.75âTHz phonon mode, and a Fano interference effect is clearly observed around t12â1âps, similar to the one observed in the temperature dependence14. Significantly, the spectra seem strongly distorted around Ît12=3.5â4âps, showing asymmetric diagonal âblobsâ. After 6âps the fluctuations die down and the AM intensity eventually reaches full amplitude in approximately 60âps.
To model the evolution of the system through the SBT, we describe Ψ using a weakly dispersive GinzburgâLandau model. Ignoring phase fluctuations (phase perturbations can be ignored on short timescales, because they require significant time for nucleation), the potential energy of the system can be described by a double-well, rather than a âMexican hatâ, potential:
Its time dependence is shown schematically in Fig. 3a. Here A(t,z)=Î(t,z)/Îeq is the time-space-dependent amplitude of Ψ, normalized to the equilibrium value Îeq, and ξ is the coherence length. The function 1âη(t,z) is a parameter describing the perturbation, akin to the temperature deviation (TâTc) from criticality in usual GinzburgâLandau theory. For spatially uniform A(t), η(t)=η(0)exp(ât/ÏAQP ) as shown in Fig. 3b, where ÏAQP is determined from fits to Fig. 2b. The time evolution of U and μ=1âη with η(0)=2 is shown schematically in Fig. 3. Before the D pulse, and for large t or z, η=0, the system is ordered and |A|=1. After the D pulse, 1âη<0 and the double-well potential disappears in favour of a single energy minimum at A=0. As 1âη increases and becomes positive, two minima emerge at ±Amin=±(1âη)1/2, and start to attract the system, breaking the symmetry. From equation (1), the equation of motion can be written as
Here Ï0 is the angular frequency of the bare (2kF) phonon mode responsible for the CDW formation. The second term describes its damping αâ¤ÎνAM/νAM. The exponentially decaying light intensity due to the finite penetration depth of light is accounted for by the excitation function ηâ²(t,z)=η(t)exp(âz/λ), where λ=20ânm is the light absorption depth of TbTe3 at 800ânm. Using the experimental values for ÏQP, , linewidth ÎνAM=0.2âTHz and coherence length ξ=1.2ânm (ref. 9), we can compute A(t,z). In Fig. 3b we first plot the spatially homogeneous solution with ξ=0, with and without the P pulse. The final ground state is ergodically uncorrelated with the initial one, so the formation of domains is expected under inhomogeneous conditions.
The full inhomogeneous solution A(t,z) to equation (2) is plotted in Fig. 4a. We see that after â¼1âps four domains are formed parallel to the surface, with A(t,z) oscillating either around 1 or â1 (orange or blue respectively), accompanied by the emission of propagating A(t,z)-field waves. At â¼3âps we observe the fusion of two domain walls, which is accompanied by the emission of field waves of A(t,z) propagating towards the surface and into the bulk (arrows). They seem to reach the surface around Ît12â 4â5âps, whichâas we shall seeâcauses detectable distortions of the spectra at around 5â6âps. (More A-field wave dynamics is shown in the accompanying movies.)
In Fig. 4b we show the predicted fast Fourier transform power spectra taking full account of spatial inhomogeneity for the D, P and p pulses. (The response function is derived in the Supplementary Information). The main features of our data in Fig. 2c are unmistakably present: oscillations of A(t) at short times and the critical slowing of the AM oscillations close to the critical point tcâ1.5 ps. The calculation also reproduces the softening of the AM for Ît12<2 ps. After 2âps, the ripples in A(t,z) discussed above cause a temporal deformation of the spectral profiles, giving diagonal blobs at 5â¼6âps shown in Fig. 4b. These are remarkably similar to the diagonal spectral distortions observed in the experimental data in Fig. 2c. Exhaustive modelling presented in the Supplementary Information unambiguously shows that the diagonal distortions in νât12 plots are caused by the annihilation of defects.
Further experiments on other microscopically diverse systems (2H-TaSe2, K0.3MoO3 and DyTe3) presented in the Supplementary Information show that the sequence of events after the quench, ultrafast QP gap recovery âΨ-field amplitude fluctuations â critical slowing down through ts and domain creation â coherent defect annihilation, is commonly observed in the tellurides and the selenide. The microscopic properties of the underlying vacuum such as λ and ξ may change the details, so K0.3MoO3 does not show step (4), which we attribute to departure from universality16. Note that the mechanism described here for topological defect creation is conceptually and historically related not only to vortex formation in superconductivity, but also to the KibbleâZurek mechanism for the formation of cosmic strings. The Ψ(t,z) waves such as we observe after annihilation events have a direct analogue in the Higgs spontaneous symmetry breaking mechanism. All these models share a common underlying potential, albeit with different microscopic properties of the underlying vacuum and different symmetries of order parameter 1,2,3,5,17. A notable distinction of our system is that Ï relaxation is slow compared with the relaxation of the potential itself, enabling the collective mode and topological defect dynamics to be uniquely observed.
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Acknowledgements
We wish to acknowledge facilities of the CENN-Nanocenter and thank C. Gadermaier for critical reading of the manuscript. Work at Stanford University was supported by the Department of Energy, Office of Basic Energy Sciences, under contract DE-AC02-76SF00515. S.B. acknowledges support from the ANR project BLAN07-3-192276.
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R.Y., P.K. and T.M. carried out the experimental work and analysis. I.R.F. and J-H.C. provided the samples; S.B. formulated the model, and together with V.V.K. did the theoretical analysis; D.M. devised the experiment, carried out modelling and wrote the paper.
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Yusupov, R., Mertelj, T., Kabanov, V. et al. Coherent dynamics of macroscopic electronic order through a symmetry breaking transition. Nature Phys 6, 681â684 (2010). https://doi.org/10.1038/nphys1738
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DOI: https://doi.org/10.1038/nphys1738
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