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

TL;DR

This study investigates the dynamics of heavy-fermion quasiparticles near a quantum critical point in CeCu$_{6-x}$Au$_{x}$, revealing long-lived coherence and unexpected behavior of quasiparticle formation during a quantum phase transition.

Contribution

It provides time-resolved experimental evidence and theoretical insights into quasiparticle collapse and revival near a quantum critical point, challenging existing theories.

Findings

Heavy-fermion state recovers with a coherence time 100 times longer than typical metals.

Quasiparticle weight collapses near QCP while formation temperature remains constant.

Coexistence of quasiparticle collapse and formation challenges current quantum criticality models.

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. Yet, when a system's ground state undergoes a qualitative change at a quantum critical point (QCP), the quasiparticles may disintegrate and give way to an exotic quantum-fluid state of matter. The nature of this breakdown is intensely debated, because the emergent quantum fluid dominates the material properties up to high temperature and might even be related to the occurence of superconductivity in some compounds. Here we trace the dynamics of heavy-fermion quasiparticles in CeCu$_{6-x}$Au$_{x}$ 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 metals. 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.

Equations

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

$$\overline{E}(t)=\frac{{\overline{E}}_0}{{\cosh }^2[2\pi (t/{\tau }_K^{\ast }-1)]},$$

$$\Delta t=\frac{2n_{\mathrm{THz}}d}{c}$$

$$\overline{E}(t)\sim \frac{\mathrm{d}\rho (t)}{\mathrm{d}t}=-{\Gamma }_{c\to hf}\rho (t),$$

$${\Gamma }_{c\to hf}=\frac{2\pi }{\hslash }|⟨f_{hf}|d|i_c⟩{|}^2{\tilde{w}}_K^{\ast }.$$

$${\tilde{w}}_K^{\ast }(t)=w_K^{\ast }\frac{{\rho }_0-\rho (t)}{{\rho }_0},$$

$$w_K^{\ast }=\frac{k_B{T}_K^{\ast }}{\gamma }{N}_c.$$

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

$$n(t)=\frac{1}{2}\left[1-\tanh (2\pi (t-t_0)/{\tau }_K^{\ast })\right].$$

$$\overline{E}(t)=\frac{{\overline{E}}_0}{{\cosh }^2[2\pi (t/{\tau }_K-1)]}.$$

Full text

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

Christoph Wetli

Department of Materials, ETH Zurich, 8093 Zurich, Switzerland

  

Shovon Pal

Department of Materials, ETH Zurich, 8093 Zurich, Switzerland

  

Johann Kroha

Physikalisches Institut and Bethe Center for Theoretical Physics, Universität Bonn, Nussallee 12, D-53115 Bonn, Germany

Center for Correlated Matter, Zhejiang University, Hangzhou, Zhejiang 310058, China

  

Kristin Kliemt

Physikalisches Institut, Goethe-Universität Frankfurt, Max-von-Laue-Str. 1, 60438 Frankfurt, Germany

  

Cornelius Krellner

Physikalisches Institut, Goethe-Universität Frankfurt, Max-von-Laue-Str. 1, 60438 Frankfurt, Germany

  

Oliver Stockert

Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany

  

Hilbert von Löhneysen

Institut für Festkörperphysik and Physikalisches Institut, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany

  

Manfred Fiebig

Department of Materials, ETH Zurich, 8093 Zurich, Switzerland

**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) Loehneysen07 , the quasiparticles may disintegrate and give way to an exotic quantum-fluid state of matter. The nature of this breakdown is intensely debated Si01 ; Coleman01 ; Senthil04 ; Woelfle11 , because the emergent quantum fluid dominates material properties up to high temperatures and might even be related to the occurrence of superconductivity in some compounds Stockert11 . 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 metals Wolf98 ; Kummer12 . 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. **

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) Woelfle18 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 band Loehneysen07 . Towards low temperatures, the conduction electrons form singlets with the $4f$ spins. The energy scale for this singlet formation defines the Kondo-lattice temperature $T_{K}^{*}$, 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) interaction Ruderman54 ; Kasuya56 ; Yosida57 , a paramagnetic Fermi liquid phase is formed. This Kondo state Kondo64 ; Hewson93 is characterized by a sharp resonance in the $4f$ electron spectrum at the Fermi energy. Typically slightly below $T_{K}^{*}$, these resonances become lattice-coherent and form a narrow energy band of

long-lived quasiparticles. Its flat dispersion indicates an approximately $10^{2}$ 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 Millis Hertz76 ; Moriya85 ; Millis93 the heavy quasiparticles remain intact and support critical fluctuations of the magnetization. In many correlated materials, however, this picture is not valid Schroeder00 . In distinguishing these two options, the behaviour of the Kondo scale $T_{K}^{*}$ 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 compounds Loehneysen07 ; Schroeder00 ; Stroka93 ; Ehm07 ; Klein08 . 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 interaction Ruderman54 ; Kasuya56 ; Yosida57 between the $4f$ magnetic moments over the Kondo effect Doniach77 and induces a QPT to an incommensurate, antiferromagnetically ordered state Schroeder94 ; Loehneysen96 at $x=0.1$, see Fig. 1. A variety of experimental investigations by thermodynamic and transport measurements Marabelli90 ; Loehneysen06 ; Loehneysen96 , neutron scattering Stroka93 ; Schroeder00 ; Schroeder94 ; Stockert98 and photoemission spectroscopy Klein08 ; Kummer12 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 behaviour Loehneysen98 ; Schroeder00 with the Hertz-Millis-Moriya scenario Hertz76 ; Moriya85 ; Millis93 points to a breakdown of the heavy-quasiparticle picture Schroeder00 ; Si01 ; Coleman01 . In contrast, its preservation Hertz76 ; Moriya85 ; Millis93 ; Rosch97 may be concluded from signatures of a stable, non-vanishing Kondo temperature Loehneysen06 ; Loehneysen98 ; Klein08 . 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.

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 $T_{K}^{*}$ of the heavy-fermion state within a single experiment. We observe that towards the QCP, the coherent spectral weight $w_{K}^{*}$ collapses, becoming unobservably small below 5 K. The Kondo energy scale, however, remains constant at $T_{K}^{*}\simeq 8$ K. We thus observe a scenario of quasiparticle disintegration where the timescale on which quasiparticles form ($\sim 1/{T_{K}^{*}}$) remains finite, whereas the probability for quasiparticle formation ($\sim w_{K}^{*}$) 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 Figs. 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 $6.2\pm 0.2$ ps and $5.8\pm 0.2$ ps, respectively, while antiferromagnetic Loehneysen07 ; Schroeder94 ; Loehneysen96 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.

Two features about the echo pulse are striking. First, its oscillatory nature indicates quantum coherent dynamics of the terahertz-excited electrons on the order of 10 ps, about two orders of magnitude longer than coherence times typically associated with optically excited electrons in metals Wolf98 . Second, the delayed response does not yield instantaneous exponential decay, as one would expect from single-electron relaxation like exciton recombination (Fig. 3a), but a compact pulse, well separated from the instantaneous reflex by a “dark time”.

The reported Stroka93 ; Klein08 CeCu6 Kondo-lattice temperature $T_{K}^{*}\approx 6$ K corresponds to a coherence time of $\tau_{K}^{*}=h/k_{B}T_{K}^{*}\approx 8$ ps, in remarkable agreement with our observed pulse delay time of about $6$ ps, considering that $T_{K}^{*}$ 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 $\tau_{K}^{*}$ 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 $\sim 10$ 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, $d\rho(t)/dt$, is proportional to the density of photoexcited electrons $\rho(t)$ 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 $t$, $\rho_{0}-\rho(t)$, where $\rho_{0}$ 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, $n(t)=\rho(t)/\rho_{0}$,

[TABLE]

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, $\overline{E}(t)\sim dn/dt$, and is thus given by the solution of equation (1) as,

[TABLE]

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 $w_{K}^{*}$ and the Kondo-lattice temperature $T_{K}^{*}$ in one single experiment. Here, $w_{K}^{*}$ is obtained from the integrated, background-corrected intensity of the delayed terahertz reflex (see Methods). The Kondo-lattice temperature $T_{K}^{*}$ as the energy scale at which the heavy quasiparticles form follows from the inverse pulse delay time as $T_{K}^{*}=h/k_{B}\tau_{K}^{*}$. Determination of $T_{K}^{*}$ is possible even when the coherent quasiparticle weight vanishes because of incoherent Kondo correlations (reported earlier Klein08 ) that are visible in our time traces as wiggles at the same delay $\tau_{K}^{*}$ as the coherent Kondo signature (see discussion in Supplementary Information B).

Figure 4a shows that the Kondo weight in CeCu5.9Au0.1 apears below 150 K and rises logarithmically down to about 30 K. Extension of the Kondo effect Hewson93 to the thermally excited crystal-field states seen in Fig. 4a Stroka93 ; Ehm07 is in line with this high onset temperature. Below 30 K, $w_{K}^{*}$ 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 $w_{K}^{*}\approx 0$ at all temperatures. Apparently, the coherent Kondo signal is fully suppressed by the antiferromagnetic RKKY interaction Nejati17 and not only by critical fluctuations near the antiferromagnetic phase transition at 2.2 K Loehneysen06 ; Loehneysen07 .

It has been suggested Si01 ; Coleman01 that with the quasiparticle breakdown near a QCP the energy scale below which heavy quasiparticles exist vanishes: $T_{K}^{*}\to 0$. In particular, observation of a universal quantum-critical fluctuation spectrum evidenced by the so-called “dynamical scaling” Schroeder00 , indicates the absence of an intrinsic energy scale. This was taken as evidence for a vanishing quasiparticle formation scale, even though other experiments Loehneysen06 ; Loehneysen98 ; Klein08 seem to point to a finite $T_{K}^{*}$ at the QCP.

Our results, shown in Fig. 4, reveal that there is no change of the Kondo scale $T_{K}^{*}$: Within experimental resolution, the time $\tau_{K}^{*}\propto(T_{K}^{*})^{-1}$ 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 becomes 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.