Critical slowing down near a magnetic quantum phase transition with fermionic breakdown (2024)

At a continuous phase transition, the ordered and the disordered phases have the same energy. As a consequence, the fluctuations between these two states become infinitely slow. This so-called critical slowing down (CSD) is universally observed in the dynamics of classical fields that are bosonic in nature but vanishes at the phase transition, like the magnetization associated with bosonic magnons, in the case of ferromagnetic order¹. In contrast, the CSD of fermionic excitations or quasiparticles is generally not expected to occur since fermions, as elementary particles, are thought to be indestructible. However, certain quantum materials known as heavy-fermion (HF) compounds host composite fermionic quasiparticles. These are quantum superpositions of itinerant and localized (that is, heavy) electron states generated by the Kondo effect^2,3 and have low binding energy parameterized by the Kondo energy scale, the lattice Kondo temperature T_K. At a quantum phase transition (QPT) in such materials (Fig. 1a), these brittle, heavy quasiparticles are assumed to disintegrate^4,5,6 despite their fermionic nature. Defining their spectral weight, that is, the probability of their existence, as the order parameter of such a fermionic QPT, one may expect the CSD of the fermionic quasiparticle oscillations—a unique signature of critical HF quasiparticle destruction, as opposed to bosonic order-parameter fluctuations.

Using time-resolved terahertz (THz) spectroscopy, we directly observe such fermionic CSD as a suppression of the heavy-particle hybridization gap and a flattening of the associated band. This ‘softening’ expands the region in momentum space where resonant THz absorption is allowed. We observe this as an increase instead of a Kondo-weight-loss-generated decrease of the Kondo-related THz signal towards the quantum-critical point (QCP), before the heavy quasiparticle band vanishes altogether below a breakdown temperature \({T}_{{{{\rm{qp}}}}}^{* }\) (ref. ⁷). Moreover, we identify a critical exponent in this behaviour, which may, thus, lead to the classification of fermionic quantum criticality in analogy to the criticality of thermodynamic phase transitions.

Signatures of Kondo quasiparticle destruction were suspected to have been seen in the dynamical scaling of the magnetic susceptibility⁸, specific heat measurements⁹, Hall effect measurements of the carrier density¹⁰ and optical conductivity measurements¹¹. The conjectures required for interpreting these measurements have been challenged, however^12,13. Time-resolved THz spectroscopy is a unique tool for probing heavy quasiparticle dynamics and for resolving these questions. Specifically, HF materials respond to an incident ultrashort THz pulse by emitting a time-delayed reflex¹⁴. This ‘echo’ is a response from the reconstructing Kondo ground state after its destruction by the incident pulse. Hence, time acts as a filter separating the Kondo-sensitive delayed pulse from the Kondo-insensitive main pulse so that the former is a background-free response of the HF state. Specifically, the amplitude and the delay time of the echo pulse are proportional to the HF spectral weight and the Kondo coherence time τ_K = 2πℏ/k_BT_K, respectively (with ℏ the reduced Planck constant and k_B the Boltzmann constant). This technique was introduced in experiments on CeCu_6−xAu_x (refs. ^14,15,16).

We now apply this new method to measure the fermionic CSD of YbRh₂Si₂ directly. YbRh₂Si₂ is a prototypical HF compound. In zero magnetic field, it is antiferromagnetic below the Néel temperature T_N = 72 mK. It undergoes a QPT to a Kondo HF liquid at a critical magnetic field of \({B}_{\perp }^{{{{\rm{cr}}}}}\approx 66\) mT perpendicular to the caxis^17,18 (Fig. 1a) induced by the Ruderman–Kittel–Kasuya–Yosida (RKKY) magnetic interaction between the Yb moments^19,20,21,22. Alternatively, a 6% substitution of Rh by Ir creates a QCP at zero field^23,24. YbRh₂Si₂ has a Kondo temperature of T_K ≈ 25 K, which is high enough to enable a wide quantum-critical region and to permit us to search for signs of CSD in the range \({T}_{{{{\rm{qp}}}}}^{* } < T < {T}_{{{{\rm{K}}}}}\), in contrast to CeCu_6−xAu_x. In our temperature-dependent, time-resolved THz reflection spectroscopy measurements, we cross the QCP by varying the magnetic field or the Ir concentration. The 1.5-cycle THz pulses of approximately 2 ps duration are incident onto the c-cut Yb(Rh_1−xIr_x)₂Si₂ samples (x = 0, 0.06). The echo pulses are analysed as described elsewhere¹⁴. With T_K = 25 K, we obtain a delay time τ_K ≈ 1.9 ps for these in agreement with the data in Fig. 1b,c. We, therefore, choose the time window for the analysis as being from 1.3 to 2.6 ps (see Supplementary Fig. 2 for the verification of the robustness of our results with respect to variations of this time window).

Figure 2 shows the time-integrated intensity of the THz echo pulse for Yb(Rh_1−xIr_x)₂Si₂ for x = 0 at various values of B_⊥ across the QPT, as well as for x = 0.06 at B_⊥ = 0. All plots exhibit the Kondo-like logarithmic increase of the spectral weight down from high temperatures, reaching a maximum in the region around the crossover temperature of T_K = 25 K. The kink near 100 K visible in all plots is caused by the population of the first crystal-electric-field excitation^25,26. Below the peak temperature, the signal initially decreases with temperature for all magnetic fields. This can be attributed to the reduced thermal broadening of THz-induced interband transitions and is reproduced by the theory introduced below.

On the antiferromagnetic side of the QCP (Fig. 2a,b), the signal continues to decrease but remains finite down to the lowest experimentally achieved temperature of 2 K. Note that this temperature range (2.0 K ≤ T ≤ 20 K) and field range (B_⊥ ≲ 66 mT) are both within the white area of the phase diagram in Fig. 1a, the so-called quantum-critical fan^9,24, where the thermodynamic and transport properties are dominated by quantum-critical fluctuations^11,17,18.

Our THz time-delay spectroscopy is not directly sensitive to these fluctuations but exclusively to the HF quasiparticle spectral weight^14,15. Therefore, the behaviour in Fig. 2a,b indicates that the Kondo effect remains partially intact in this temperature range. This is reasonable since we are still a factor of approximately 25 above T_N so that the heavy quasiparticles are not entirely destroyed by the impending antiferromagnetic order.

Near quantum criticality (Fig. 2d–f,l), the temperature dependence changes drastically. The initial signal decrease with temperature is now followed by a logarithmic increase of the THz echo signal, which persists down to the lowest observed temperature. Note the qualitative similarity between the field-tuned (Fig. 2e) and chemically tuned (Fig. 2l) quantum-critical systems, albeit with different logarithmic slopes. On the HF-liquid side of the QCP, the signal increase towards the lowest temperatures is still present and gradually fades away with distance from the QCP (Fig. 2g–j). Its onset may also be conjectured at 50 mT on the antiferromagnetic side (see the inset in Fig. 2c).

To understand the striking logarithmic increase towards low temperature, we analyse the THz echo signal theoretically. We summarize this analysis here and elaborate on its technical aspects in Methods. In a HF system, the strongly correlated, flat band produced by the Kondo effect hybridizes with the light conduction band to generate a structure with a lower (n = 1) and an upper (n = 2) band with an avoided crossing^15,27,28, as shown in Fig. 3a–c. Low-temperature thermodynamic and transport experiments probe the lower, occupied band only, whereas resonant THz spectroscopy covers transitions between both bands, requiring a two-band theory. Our critical-HF-liquid theory shows (Methods) that the n = 1, 2 bands with dispersions \({\varepsilon }_{n\mathbf{p}}\) (where \(\mathbf{p}\) is the crystal-electron momentum) have distinct momentum- and temperature-dependent spectral weights \({z}_{n\mathbf{p}}\). This is a crossover from \({z}_{n\mathbf{p}}\approx 1\) in the strongly dispersive region to \({z}_{n\mathbf{p}}=a\,{z}_{0}\ll 1\) in the flat region of these bands (Fig. 3d–f and insets). Here, a ≪ 1 is the spectral weight of the local, single-ion Kondo resonance, which builds up logarithmically from above T_K and then saturates towards a constant value for T < T_K. Further, \({z}_{0}={(T/{T}_{0})}^{\alpha }\) is a suppression factor with a critical exponent α. It describes the destruction of the quasiparticle spectral weight as the QCP is approached on lowering the temperature T below the onset temperature for quantum criticality, T₀. For the latter, experiments revealed that T₀ ≈ T_K in YbRh₂Si₂ (ref. ²⁹).

As mentioned, the THz echo pulse at τ_K = 2πℏ/k_BT_K is solely sensitive to the breakup-and-recovery dynamics of the HFs and not other THz absorption channels^14,15,16. Therefore, its intensity exclusively depends on the quasiparticle weight and the phase space available for THz-induced excitations. Specifically, the echo-pulse intensity is proportional to the probability:

for the resonant excitation of electrons from the lower to the upper band at an energy difference \({\Delta}{\varepsilon }_{\mathbf{p}}={\varepsilon }_{2\mathbf{p}}-{\varepsilon }_{1\mathbf{p}}\). Here, \(f({\varepsilon }_{n\mathbf{p}})\) is the Fermi–Dirac distribution function. In equation (1), it describes the probability that the n = 1 band is occupied and that the n = 2 band is empty before the THz absorption process. W(ℏω) is the spectrum of the incident THz pulse, which is a Gaussian distribution of width Γ centred around the central frequency Ω_THz. With \({{\Delta }}{\varepsilon }_{\mathbf{p}}\approx \hslash {{{\varOmega }}}_{{{{\rm{THz}}}}}\), the THz-induced interband transition becomes resonantly allowed. The integral runs over all electron momenta, and the factor A is a temperature-independent constant, proportional to the intensity of the incident THz pulse and to the modulus squared of the electric–dipole transition-matrix element between the two bands.

When the probability for HF formation, \(a\,{z}_{0}\propto {(T/{T}_{{{{\rm{K}}}}})}^{\alpha }\), tends to zero at the QCP, the heavy bands flatten according to the two-band HF-liquid theory. Also, the hybridization gap vanishes, and with that the quasiparticle energy in the heavy regions of both bands (n = 1, 2) approaches the Fermi energy E_F (Fig. 3d–f). This means that the oscillation frequency of fermionic quasiparticles, \({\omega }_{n\mathbf{p}}=({\varepsilon }_{n\mathbf{p}}-{E}_{{{{\rm{F}}}}})/\hslash\), vanishes, which is indicative of a fermionic CSD. In turn, it implies an expansion of the region in the momentum space where resonant THz transitions are allowed, seen as a broadening of the shaded areas in Fig. 3a–c. The interplay of these two counteracting effects, namely quasiparticle destruction and phase-space expansion, leads to a non-monotonic temperature dependence of the THz absorption strength P(T). In equation (1), P(T) depends on the bare quasiparticle weight z₀(T) via \({\varepsilon }_{n\mathbf{p}}\), \({z}_{n\mathbf{p}}\), n = 1, 2, and \({{\Delta }}{\varepsilon }_{\mathbf{p}}\); see equations (4) and (5). A careful expansion of all these dependencies for small z₀(T) and performing the integration in equation (1) predicts a logarithmic increase of \(P(T)\propto \ln [1/{z}_{0}(T)]\) towards low temperatures down to the region of \({T}_{{{{\rm{qp}}}}}^{* }\). The full numerical evaluation of equation (1) leads to the behaviour shown in Fig. 3g. This reproduces the Kondo maximum near T_K ≈ 25 K, and at quantum criticality, it indeed shows a logarithmic increase within an intermediate temperature window \({T}_{{{{\rm{qp}}}}}^{* } < T < {T}_{{{{\rm{K}}}}}\) according to

Observing this behaviour in Fig. 2d–g,l is, thus, a unique experimental signature of fermionic quasiparticle CSD in the Yb(Rh_1−xIr_x)₂Si₂ system.

Upon further decreasing the temperature to \(T < {T}_{{{{\rm{qp}}}}}^{* }\), P(T) approaches zero as the HF weight disappears altogether (inset of Fig. 3g). The low-temperature scale \({T}_{{{{\rm{qp}}}}}^{* }\) is, thus, defined as the position of the signal maximum between the logarithmic increase and its ultimate collapse towards T → 0. As seen in Fig. 3g (inset), \({T}_{{{{\rm{qp}}}}}^{* }\) depends on the critical exponent α and is several orders of magnitude lower than the Kondo scale of approximately T_K and therefore possibly undetectably small. A low-temperature scale T* has also been observed as a maximum in the magnetic susceptibility²⁴ whose microscopic origin, however, remains unclear. Since in Yb(Rh_1−xIr_x)₂Si₂, T* and our theoretically predicted \({T}_{{{{\rm{qp}}}}}^{* }\) are in the same temperature range, we conjecture that both may have the same physical origin, namely the competition between fermionic CSD and quasiparticle breakdown at the QCP. As a crossover temperature, \({T}_{{{{\rm{qp}}}}}^{* }\) remains non-zero, but can be exceedingly small, depending on α (inset of Fig. 3g). Note that away from criticality (blue curve, Fig. 3g), quasiparticles persist (\({z}_{0}(T)={{{\rm{const.}}}}\)), so that the logarithmic low-temperature behaviour does not occur, in agreement with Fig. 2h–j. The blue curves in Fig. 2f,l represent the evaluation of equation (1) for the spectrum W(ℏω) of the THz pulses used in our experiment. Considering that α is the only adjustable parameter apart from the overall signal amplitude and that we use the same value T_K ≈ 25 K for both curves, the agreement between theory and data is excellent.

We can now extract the critical exponent α by comparing the logarithmic slope s_low associated with the CSD at low temperatures (\({T}_{{{{\rm{qp}}}}}^{* } < T < {T}_{{{{\rm{K}}}}}\)) from equation (2) with the slope s_high of the standard logarithmic behaviour of the Kondo weight at high temperature (T > T_K) according to \(P(T)=A\,\ln ({T}_{{{{\rm{K}}}}}/T)\). Specifically, s_high is extracted from the temperature window between the signal maximum and the crystal-electric-field kink near 100 K. This directly leads to α = s_low/s_high. We find from the experimental data that α = 0.14 ± 0.02 for Yb(Rh_1−xIr_x)₂Si₂ in the critical region x = 0 and 63 mT ≤ B_⊥ ≤70 mT in Fig. 2d–f and that α = 0.29 at x = 0.06 and B_⊥ = 0 in Fig. 2l. Such different critical behaviours for magnetic-field and chemical-pressure tuning reflects that, for different tuning parameters, the QCPs are of a different nature, which has also been observed in response functions³⁰ quantifying the critical behaviour of bosonic fields.

Our theory also explains why the logarithmic low-temperature increase of THz absorption indicating CSD cannot be observed in the CeCu_6−xAu_x system. For this material, the ratio \({T}_{{{{\rm{K}}}}}/{T}_{{{{\rm{qp}}}}}^{* }\) is substantially smaller than in the Yb(Rh_1−xIr_x)₂Si₂ system, so that the effects of the buildup of the Kondo weight and of the CSD overlap to such an extent that the latter is obscured, as seen in Fig. 3h and in agreement with experiment¹⁴.

Our findings are summarized in the T versus B_⊥ phase diagram of Fig. 4. On the HF-liquid side of the QCP (\(B > {B}_{\perp c}^{{{{\rm{cr}}}}}=66\) mT), P(T) is logarithmically enhanced due to the CSD effect, as explained by our two-band HF-liquid theory, signalling the critical behaviour of the HF quasiparticles up to T ≲ 10 K. In contrast, on the antiferromagnetic side (B_⊥ ≲ 50 mT), we observe a reduction of the Kondo-weight-related absorption but no fermionic CSD. This reduction is different from thermodynamic and response measurements, which are dominated by quantum fluctuations. Our measurements are sensitive to the quasiparticle dynamics only. This suggests that in this region, the RKKY interaction^19,20,21 strongly affects the quasiparticle dynamics such that the two-band HF-liquid theory is not valid here. Figure 4 shows how our measurements connect to the quantum-critical fan³¹ observed at the lowest temperatures (T < 2 K).

To conclude, we observed a logarithmic low-temperature increase in the resonant quasiparticle excitation probability P(T) near a magnetic QPT in HF materials. We identified this logarithmic increase as a unique signature of fermionic quasiparticle CSD, that is, a vanishing quasiparticle frequency near a QPT with fermionic breakdown. Since, in contrast to the thermodynamic and transport properties, our time-resolved THz spectroscopy is exclusively sensitive to the HF quasiparticle dynamics as opposed to thermal fluctuations, we could further extract the fermionic critical exponent α of the vanishing quasiparticle weight. The critical behaviour of α suggests that we can define the heavy quasiparticle weight as an order parameter for QPTs with fermionic breakdown. This work may lead to the classification of fermionic QPTs in terms of their critical exponent, analogous to thermodynamic phase transitions.