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Time-resolved collapse and revival of the Kondo state near a quantum phase transition

Nature Physics (2018) | Download Citation

Abstract

One of the most successful paradigms of many-body physics is the concept of quasiparticles: excitations in strongly interacting matter behaving like weakly interacting particles in free space. Quasiparticles in metals are very robust objects. Nevertheless, when a system’s ground state undergoes a qualitative change at a quantum critical point (QCP)1, the quasiparticles may disintegrate and give way to an exotic quantum-fluid state of matter. The nature of this breakdown is intensely debated2,3,4,5, because the emergent quantum fluid dominates material properties up to high temperatures and might even be related to the occurrence of superconductivity in some compounds6. Here we trace the dynamics of heavy-fermion quasiparticles in CeCu6−xAux and monitor their evolution towards the QCP in time-resolved experiments, supported by many-body calculations. A terahertz pulse disrupts the many-body heavy-fermion state. Under emission of a delayed, phase-coherent terahertz reflex the heavy-fermion state recovers, with a coherence time 100 times longer than typically associated with correlated metals7,8. The quasiparticle weight collapses towards the QCP, yet its formation temperature remains constant—phenomena believed to be mutually exclusive. Coexistence in the same experiment calls for revisions in our view on quantum criticality.

Main

All across condensed-matter physics, coherent excitations of many-body systems can be described in a simple picture of strongly renormalized, but weakly interacting particle-like objects—the so-called quasiparticles. In metals, for example, heavy Landau fermionic quasiparticles of remarkable stability are enforced by the Pauli principle. Despite the Pauli stabilization, however, these heavy-fermion quasiparticles are amenable to disintegration near a quantum phase transition (QPT)9 because of their low binding energy, thus opening an avenue to novel quantum states of matter, governed by quantum fluctuations. At a QPT, near zero temperature, the critical fluctuations are generated by the energy allowed by Heisenberg’s uncertainty principle rather than thermal excitations. As a consequence, non-Fermi-liquid behaviour, frustrated magnetism or unconventional superconductivity may emerge around the associated QCP.

In heavy-fermion compounds, the 4f magnetic moments localized on rare-earth ions in the crystal lattice are exchange-coupled to the electron spins residing in the conduction band1. Towards low temperatures, the conduction electrons form singlets with the 4f spins. The energy scale for this singlet formation defines the Kondo-lattice temperature \(T_{\mathrm{K}}^ \ast\), a crossover temperature that typically is of the order of 10 K. Thus, instead of establishing long-range magnetic order mediated by the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction10,11,12, a paramagnetic Fermi liquid phase is formed. This Kondo state13,14 is characterized by a sharp resonance in the 4f electron spectrum at the Fermi energy. Typically slightly below \(T_{\mathrm{K}}^ \ast\), these resonances become lattice-coherent and form a narrow energy band of long-lived quasiparticles. Its flat dispersion indicates an approximately 102 times heavier effective mass than that of free electrons; hence the name ‘heavy fermion’. In the conventional scenario of an antiferromagnetic QCP due to Hertz, Moriya and Millis15,16,17, the heavy quasiparticles remain intact and support critical fluctuations of the magnetization. In many correlated materials, however, this picture is not valid18. In distinguishing between these two options, the behaviour of the Kondo scale \(T_{\mathrm{K}}^ \ast\) is an important characteristic as it reveals if and how the heavy quasiparticles disintegrate towards the QCP.

CeCu6−xAux is one of the best-known heavy-Fermi-liquid compounds1,18,19,20,21. It is paramagnetic down to very low temperatures but exhibits short-range intersite magnetic correlations and hence appears as an ideal system for studying the change in ground state that a material may undergo near a QCP. Replacing a small fraction of Cu by Au expands the CeCu6 lattice and leads to a decrease of the 4f-conduction electron exchange. This favours the RKKY interaction10,11,12 between the 4f magnetic moments over the Kondo effect22 and induces a QPT to an incommensurate, antiferromagnetically ordered state25,26 at x = 0.1 (see Fig. 1). A variety of experimental investigations by thermodynamic and transport measurements23,24,25, neutron scattering18,19,26,27 and photoemission spectroscopy8,21 revealed valuable aspects of the phases and properties near the QCP of the CeCu6−xAux system, but have also revealed puzzling controversies. In particular, the incompatibility of the critical behaviour18,28 with the Hertz–Millis–Moriya scenario15,16,17 points to a breakdown of the heavy-quasiparticle picture2,3,18. In contrast, its preservation15,16,17,29 may be concluded from signatures of a stable, non-vanishing Kondo temperature21,24,28. Such conflicts are caused by the diversity and, sometimes, indirectness of techniques used to probe QPTs—techniques that do not permit the measurement of fundamental properties such as the Kondo weight and temperature simultaneously.

Fig. 1: Heavy-fermion quasiparticles near a QCP.
Fig. 1

We consider a scenario as in CeCu6−xAux, where the quasiparticles disintegrate near the QCP, giving way to an exotic state of matter dominated by quantum fluctuations. In the quantum-critical region (yellow) the quantum energy of fluctuations is larger than the thermal energy kBT. As a consequence, the system can counter-intuitively enter the quantum-critical region when the temperature is raised.

In this work, we enter the realm of nonequilibrium dynamics to obtain direct and comprehensive information on the evolution of the quasiparticle picture towards the QCP in CeCu6−xAux. We extinguish part of the Kondo state by irradiation with a terahertz pulse and monitor its time-resolved resurgence. The resulting terahertz echo pulse provides a direct measure for the spectral weight and for the Kondo temperature \(T_{\mathrm{K}}^ \ast\) of the heavy-fermion state within a single experiment. We observe that towards the QCP, the coherent spectral weight \(w_{\mathrm{K}}^ \ast\) collapses, becoming unobservably small below 5 K. The Kondo energy scale, however, remains constant at \(T_{\mathrm{K}}^ \ast\) 8 K. We thus observe a scenario of quasiparticle disintegration where the timescale on which quasiparticles form (\(\sim 1{\mathrm{/}}T_{\mathrm{K}}^ \ast\)) remains finite, whereas the probability for quasiparticle formation \(\left( {\sim w_{\mathrm{K}}^ \ast } \right)\) collapses. Up to now, these two phenomena were assumed to exclude each other. Now their simultaneous observation calls for revisions in our view on quantum criticality.

We irradiate a set of CeCu6−xAux samples covering the Fermi-liquid (x = 0), the quantum-critical (x = 0.1) and the antiferromagnetic (x = 1) state with pulses at 0.1–3 THz. We then record the reflected terahertz wave by free-space electrooptical sampling (see Methods). It is crucial that our terahertz frequencies are just enough to excite electrons from the heavy-fermion band to the conduction band. Our pulses thus disrupt the correlated heavy-fermion state, but they cannot ionize the system, excite electrons out of the tightly bound Ce 4f single-particle states, or induce spin–orbit transitions. The time traces in Fig. 2a–c exhibit an instantaneous reflex (t = 0) as response of the light conduction electrons. In CeCu6 and CeCu5.9Au0.1 this is followed by a weaker ‘echo’ pulse delayed by 6.2 ± 0.2 ps and 5.8 ± 0.2 ps, respectively, while antiferromagnetic CeCu5Au shows a negligible echo pulse. This delayed pulse is not to be confused with trivial etalon (Fabry-Perot) reflexes from the terahertz generation crystal, cryostat windows and so on, which are identified at different delay times (see Methods and Supplementary Information A). Unlike these artefacts, our echo does not appear on a Pt reference sample. It furthermore displays a complex temperature dependence, elaborated on in the following, that none of the trivial reflex pulses show.

Fig. 2: Time-resolved terahertz reflectivity of the heavy-fermion system CeCu6−xAux.
Fig. 2

ac, Time dependence of the normalized terahertz electric-field amplitude at three different temperatures (data vertically displaced) for reflection from the heavy-Fermi-liquid compound CeCu6 (a), the quantum-critical CeCu5.9Au0.1 (b) and the antiferromagnetic compound CeCu5Au (c). Each time trace is normalized such that its entire integrated intensity is unity (see Methods). The echo pulse is highlighted in the insets. The green-shaded area shows the envelope of this time-delayed terahertz reflex which contains coherent and certain background contributions as explained in the text. The envelope was derived from a fit of equation (2) to the measured data.

Two features about the echo pulse are striking. First, its oscillatory nature indicates quantum coherent dynamics of the terahertz-excited electrons of the order of 10 ps, about two orders of magnitude longer than coherence times typically associated with optically excited electrons in metals7. Second, the delayed response does not yield instantaneous exponential decay, as one would expect from a single-electron relaxation like exciton recombination (Fig. 3a), but a compact pulse, well separated from the instantaneous reflex by a ‘dark time’.

Fig. 3: Dynamics of heavy-fermion quasiparticle formation.
Fig. 3

a, Sketch of the relaxation dynamics after photoexcitation of carriers into the conduction band. Top: immediate, exponential relaxation of the reemitted electric field resulting from a linear rate equation with constant relaxation rate 1/τlin, describing, for example, exciton recombination in semiconductors. Bottom: solution (equation (2)) of the nonlinear rate equation (equation (1)). Here, the heavy-fermion band is of many-body origin—that is, its spectral weight and, hence, the relaxation rate dynamically follow the heavy-fermion band population. The pronounced echo pulse with delay time \(\tau _{\mathrm{K}}^ \ast\) is a signature of the many-body Kondo physics. b, Band structure of a multi-orbital Anderson lattice model, calculated by nonequilibrium dynamical mean field theory (see Methods). Before the terahertz excitation (t < 0) the nearly flat heavy-fermion band near the Fermi energy EF = 0, accompanied by its crystal-field satellites at 8 and 13 meV (ref. 19), are visible. Excitation with a terahertz pulse (t = 0) destroys the heavy-fermion state. The associated band collapses and all charge carriers become light electrons. After a time \(\tau _{\mathrm{K}}^ \ast\), resurgence of the heavy-fermion state occurs and the excess energy is released as a time-delayed terahertz pulse. The colour scale represents the spectral density in units of the inverse conduction electron bandwidth and ranges linearly from 0 (black) to 3.5 (bright yellow). The pink curves on the left of each panel represent the momentum-integrated, cerium-4f partial density of states, ranging from 0 to 1 in units of the conduction band width.

The reported19,21 CeCu6 Kondo-lattice temperature \(T_{\mathrm{K}}^ \ast \approx 6\) K corresponds to a coherence time of \(\tau _{\mathrm{K}}^ \ast = h{\mathrm{/}}k_{\mathrm{B}}T_{\mathrm{K}}^ \ast \approx 8\) ps, in remarkable agreement with our observed pulse delay time of about 6 ps, considering that \(T_{\mathrm{K}}^ \ast\) merely marks a crossover scale, not a phase transition temperature. This concordance, the logarithmic temperature dependence of the echo-pulse weight discussed below, and the fact that a delayed pulse is observed only in the heavy-fermion phase but strongly suppressed in the antiferromagnetic phase (see Fig. 2), establish unambiguous evidence for the Kondo-related origin of the delayed pulse: By the terahertz excitation, heavy electrons are excited instantaneously (within the terahertz pulse duration) from the heavy-fermion band into the light part of the conduction band, as shown in Fig. 3b. This transition breaks the Kondo singlet and deletes the associated spectral weight of the heavy band. It then takes the coherence time \(\tau _{\mathrm{K}}^ \ast\) to recreate this spectral weight, into which the excited electrons relax with emission of the initially absorbed energy as a delayed terahertz pulse. The oscillatory, time-delayed pulse thus represents the coherent spectral weight of heavy quasiparticles. The phase space for energy- and momentum-conserving scattering of terahertz-excited electrons is drastically reduced compared to optical excitation because of the lower excitation energy. This, together with the large effective heavy-fermion mass, explains the exceptionally long coherence time of >~10 ps in our experiment.

The unusual echo pulse resulting from this dynamics is quantitatively described by a semiclassical rate-equation model. According to Fermi’s golden rule, the density of photoexcited electrons making a transition to the heavy-fermion band per time interval, dρ(t)/dt, is proportional to the density of photoexcited electrons ρ(t) and to the spectral density of heavy-fermion final states. The latter is created by the Kondo singlet formation and is therefore proportional to the density of electrons residing in the heavy-fermion band at time t, ρ0 − ρ(t), where ρ0 is the density of terahertz-excited electrons from the heavy-fermion band. We thus have a convolution of the recreation of the ground-state spectral density and the repopulation of this recreating spectral density. This leads to a nonlinear rate equation for the normalized density of photoexcited electrons, n(t) = ρ(t)/ρ0,

$$\frac{{{\mathrm{d}}n}}{{{\mathrm{d}}t}} = - \frac{{4\pi }}{{\tau _{\mathrm{K}}^ \ast }}(1 - n)n$$
(1)

see Supplementary Information C for a detailed derivation of this simplest possible approach. The electric-field envelope of the emitted terahertz reflex is proportional to this temporal change of occupation, \(\overline E (t)\sim \mathrm{d}n{\mathrm{/}}\mathrm{d}t\), and is thus given by the solution of equation (1) as

$$\overline E (t) = \frac{{\overline E _0}}{{{\mathrm{cosh}}^2\left[ {2\pi \left( {t{\mathrm{/}}\tau _{\mathrm{K}}^ \ast - 1} \right)} \right]}}$$
(2)

where \(\overline E _0\) is the maximum pulse amplitude. This envelope is shown in Fig. 2a and agrees well with our experiment. The time-delayed revival of the signal is caused by the dynamical change of spectral weight and is thus an unambiguous signature of correlated many-body dynamics.

We now scrutinize the behaviour of the heavy quasiparticles as the QCP is approached. Our time-resolved terahertz reflectometry allows one to measure both the (integrated) heavy-quasiparticle spectral ‘Kondo’ weight \(w_{\mathrm{K}}^ \ast\) and the Kondo-lattice temperature \(T_{\mathrm{K}}^ \ast\) in one single experiment. Here, \(w_{\mathrm{K}}^ \ast\) is obtained from the integrated, background-corrected intensity of the delayed terahertz reflex (see Methods). The Kondo-lattice temperature \(T_{\mathrm{K}}^ \ast\) as the energy scale at which the heavy quasiparticles form follows from the inverse pulse delay time as \(T_{\mathrm{K}}^ \ast = h{\mathrm{/}}k_{\mathrm{B}}\tau _{\mathrm{K}}^ \ast\). Determination of \(T_{\mathrm{K}}^ \ast\) is possible even when the coherent quasiparticle weight vanishes because of incoherent Kondo correlations (reported earlier21) that are visible in our time traces as wiggles at the same delay \(\tau _{\mathrm{K}}^ \ast\) as the coherent Kondo signature (see discussion in Supplementary Information B).

Figure 4a shows that the Kondo weight in CeCu5.9Au0.1 appears below 150 K and rises logarithmically down to about 30 K. Extension of the Kondo effect14 to the thermally excited crystal-field states19,20 is in line with this high onset temperature. Below 30 K, \(w_{\mathrm{K}}^ \ast\) drops continuously, becoming unobservably small near 5 K: direct and striking evidence for quasiparticle disintegration near a QCP. In the non-quantum-critical Fermi-liquid compound CeCu6, we also observe the logarithmic increase of the Kondo weight between 150 and 30 K. However, towards 2 K it drops by only 40%, reflecting the proximity, yet finite distance to the QCP. In the antiferromagnetic compound CeCu5Au, we have \(w_{\mathrm{K}}^ \ast \approx 0\) at all temperatures. Apparently, the coherent Kondo signal is fully suppressed by the antiferromagnetic RKKY interaction30 and not only by critical fluctuations near the antiferromagnetic phase transition at 2.2 K (refs. 1,24).

Fig. 4: Evolution of Kondo weight and Kondo temperature towards the QCP in the CeCu6−xAux system.
Fig. 4

a, Temperature dependence of the Kondo weight per sample area. The weight is derived from the integrated intensity of the echo pulse emitted in the 3.5–8.5 ps window (see the insets of Fig. 2) applying the background correction described in the text. In the quantum-critical compound CeCu5.9Au0.1 the Kondo weight rises logarithmically between 150 and 30 K, revealed by the linear slope in the logarithmic plot. Below 30 K the Kondo weight collapses, becoming unobservably small at 5 K. In the Fermi-liquid compound CeCu6 the behaviour is the same as in CeCu5.9Au0.1 down to 30 K, but the decrease towards lower temperature leaves a finite Kondo weight. In antiferromagnetic CeCu5Au, the Kondo weight is zero at all temperatures within the statistical error (that is, the standard deviation, see error bar). b, Evolution of the normalized Kondo temperature towards 0 K for CeCu5.9Au0.1 and CeCu5Au. Within the statistical error, the Kondo temperature remains constant. The Kondo temperature is derived from the envelope of the echo pulse as sketched in the inset.

It has been suggested2,3 that with the quasiparticle breakdown near a QCP the energy scale below which heavy quasiparticles exist vanishes: \(T_{\mathrm{K}}^ \ast \to 0\). In particular, observation of a universal quantum-critical fluctuation spectrum evidenced by the so-called ‘dynamical scaling’18, indicates the absence of an intrinsic energy scale. This was taken as evidence for a vanishing quasiparticle formation scale, even though other experiments21,24,28 seem to point to a finite \(T_{\mathrm{K}}^ \ast\) at the QCP.

Our results, shown in Fig. 4, reveal that there is no change of the Kondo scale \(T_{\mathrm{K}}^ \ast\): Within experimental resolution, the time \(\tau _{\mathrm{K}}^ \ast \propto \left( {T_{\mathrm{K}}^ \ast } \right)^{ - 1}\) it takes the quasiparticles to form remains constant. Nevertheless, the Kondo weight (that is, the probability for the very existence of quasiparticles) collapses at the QCP. Thus, our measurements reconcile the seemingly contradictory phenomena of quasiparticle destruction and their formation energy scale remaining finite. The fact that these two key observations are obtained simultaneously within one single terahertz reflectometry experiment establishes this as a consistent quantum critical scenario. It also appears to be consistent with the observed dynamical scaling in that, although the intrinsic energy scale remains non-zero, its signature at the QCP become unobservably weak due to the coherent spectral weight collapse. Although it remains to be seen if this scenario is realized more universally in other heavy-fermion compounds, it suggests a new way of thinking about quantum critical quasiparticle breakdown.

Methods

Experiment

The CeCu6−xAux samples used in this study were cut from single crystals, and faces perpendicular to the principal axes were polished using SiC. The specimens were mounted in a temperature-controlled Janis SVT-400 helium reservoir cryostat with Tsurupica windows. A Ti:sapphire laser (800 nm, 130 fs, 1 kHz, 2 mJ per pulse) generated single-cycle terahertz pulses of a few nanojoules by optical rectification in a 0.5 mm ZnTe(110) single crystal. Terahertz radiation with a spectral range between 0.1–3 THz was incident on the sample under 45° with the terahertz electric field parallel to its b axis. Because of the low pulse energies and the high reflectivity23 of CeCu6−xAux, heating by our terahertz pulses is negligible.

Electrooptical sampling was performed using the fundamental Ti:sapphire laser pulse as a probe pulse that was time-delayed by the sampling time t with respect to the reflected terahertz wave. The terahertz and probe beams were collinearly focused onto a ZnTe(110) crystal. The terahertz-induced ellipticity of the probe light is measured using a quarter-wave plate, a Wollaston polarizer and a balanced photodiode. Output from the latter was analysed with a lock-in amplifier. In order to increase the accessible time delay between terahertz and probe pulses, Fabry–Pérot resonances from the faces of the 0.5 mm thin ZnTe(110) crystal were suppressed by extending the crystal with a 2-mm-thick terahertz-inactive optically bonded ZnTe(100) crystal.

Prior to the experiment we identified all the terahertz reflexes generated by the optical components so as to avoid confusion with the true terahertz signal generated in the CeCu6−xAux. A measurement of all these terahertz signals is summarized in Supplementary Information A. The earliest of these artefacts appeared at a delay of 10 ps, outside the range discussed in this work.

A set of data points from the electrooptical sampling represents the time trace of the terahertz field pulse between t = −4.0 ps and t = +8.5 ps. Each set was normalized by a factor such that its integrated intensity (the sum of the square of all the normalized data points in the set) equals unity (that is, all traces are scaled to the same overall reflected power). Such normalized data are shown in Fig. 2. They were used to derive the heavy-fermion quasiparticle weight and the Kondo temperature as described in the following. (Refer to Supplementary Information B for an in-detail technical description of this procedure.)

The power of the echo pulse was calculated by integrating the squared electric field of the normalized time traces over the interval from 3.5 to 8.5 ps where the echo pulse appears (see insets of Fig. 2). These raw data exhibit an offset, caused by noise, the residual tail of the undelayed main pulse, and the incoherent Kondo correlations21 mentioned in the main text and discussed in detail in Supplementary Information B. In the range between 150 and 300 K, the Kondo weight of all our CeCu6−xAux samples is known to be zero so that only the temperature-independent offset is present. We subtract the integrated offset signal derived in this temperature range from our raw data to obtain \(w_{\mathrm{K}}^ \ast\) as shown in Fig. 4a. The Kondo temperature \(T_{\mathrm{K}}^ \ast\) is derived from the relation \(T_{\mathrm{K}}^ \ast = h{\mathrm{/}}k_{\mathrm{B}}\tau _{\mathrm{K}}^ \ast\). Here \(\tau _{\mathrm{K}}^ \ast\) is taken as peak of the fitted envelope of the terahertz echo pulse according to equation (2).

Theory

The dynamical band structure of the terahertz-excited heavy-fermion system, including the crystal-field satellite bands, was calculated using a nonequilibrium generalization of the dynamical mean-field theory (DMFT)31 for the multi-orbital Anderson lattice model with six local orbitals, grouped in three Kramers doublets. These orbitals represent the crystal-field-split J = 5/2 ground-state multiplet of the Ce 4f shell. An infinite onsite repulsion within the Ce 4f orbitals was assumed, enforcing the overall single-electron occupancy of the Ce 4f shell. As the impurity solver of the DMFT, an auxiliary-particle representation of the Ce 4f electron fields was employed within the non-crossing approximation20,21,32 (NCA), reaching down to base temperatures (before terahertz excitation) of \(T \approx 0.1 T_{\mathrm{K}}^ \ast\). The DMFT and NCA were generalized to the nonequilibrium case using the Keldysh technique33. The efficient implementation of the NCA algorithm in ref. 32 was adapted for solving the DMFT with the multi-orbital NCA out of equilibrium.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

Additional information

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Acknowledgements

The authors are grateful for financial support by the SNSF via project No. 200021-14708 (M.F., C.W.) and by the DFG via SFB/TR 185 (J.K).

Author information

Affiliations

  1. Department of Materials, ETH Zurich, Zurich, Switzerland

    • C. Wetli
    • , S. Pal
    •  & M. Fiebig
  2. Physikalisches Institut and Bethe Center for Theoretical Physics, Universität Bonn, Bonn, Germany

    • J. Kroha
  3. Center for Correlated Matter, Zhejiang University, Hangzhou, China

    • J. Kroha
  4. Physikalisches Institut, Goethe-Universität Frankfurt, Frankfurt, Germany

    • K. Kliemt
    •  & C. Krellner
  5. Max Planck Institute for Chemical Physics of Solids, Dresden, Germany

    • O. Stockert
  6. Institut für Festkörperphysik and Physikalisches Institut, Karlsruhe Institute of Technology, Karlsruhe, Germany

    • H. v. Löhneysen

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Contributions

All authors contributed to the discussion and interpretation of the experiment and to the completion of the manuscript. C.W. and S.P. performed the experiment and the data analysis. O.S. and H.v.L. provided the CeCu6−xAux samples. K.K. and C.K. provided YbRh2Si2 samples for reference experiments. J.K. performed the theoretical analysis. J.K. and M.F. initiated the experiment and supervised the work.

Competing interests

The authors declare no competing interests.

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Correspondence to J. Kroha or M. Fiebig.

Supplementary information

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https://doi.org/10.1038/s41567-018-0228-3