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

When a system close to a continuous phase transition is subjected to perturbations, it takes an exceptionally long time to return to equilibrium. This critical slowing down is observed universally in the dynamics of bosonic excitations, such as order-parameter collective modes, but it is not generally expected to occur for fermionic excitations. Here using terahertz time-domain spectroscopy, we find evidence for fermionic critical slowing down in YbRh2Si2 close to a quantum phase transition between an antiferromagnetic phase and a heavy Fermi liquid. In the latter phase, the relevant quasiparticles are a quantum superposition of itinerant and localized electronic states with a strongly enhanced effective mass. As the temperature is lowered on the heavy-Fermi-liquid side of the transition, the heavy-fermion spectral weight builds up until the Kondo temperature TK ≈ 25 K, then decays towards the quantum phase transition and is, thereafter, followed by a logarithmic rise of the quasiparticle excitation rate below 10 K. A two-band heavy-Fermi-liquid theory shows that this is indicative of the fermionic critical slowing down associated with heavy-fermion breakdown near the quantum phase transition. The critical exponent of this breakdown could be used to classify this system among a wider family of fermionic quantum phase transitions that is yet to be fully explored.

At a continuous phase transition, the ordered and the disordered phases become equal in 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 which are bosonic in nature and vanish at the phase transition, like the magnetization, associated with bosonic magnons, in the case of ferromagnetic order [1].By contrast, 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, i.e., heavy, electron states generated by the Kondo effect [2,3], with a low binding energy parameterized by the Kondo energy scale, the lattice Kondo temperature T K .At a magnetic quantum phase transition (QPT) in such materials (see Fig. 1a), these brittle, heavy quasiparticles are assumed to disintegrate due to critical fluctuations [4][5][6] despite their fermionic nature, and fermionic CSD would be a unique signature of this destruction.
Using time-resolved terahertz (THz) spectroscopy, we directly observe such fermionic CSD as a suppression of the heavy-particle hybridization gap and 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 * qp [7].Moreover, we identify a critical exponent in this behaviour and thus set the stage for the classification of fermionic quantum criticality in analogy to the criticality of thermodynamic phase transitions.
Signatures of Kondo quasiparticle destruction were suspected from the dynamical scaling of the magnetic susceptibility [8], specific heat measurements [9], Hall effect measurements of the carrier density [10], and optical conductivity measurements [11].The conjectures entering the interpretation of these measurements have been challenged, however [12,13].Time-resolved THz spectroscopy is a unique tool to probe the heavy quasiparticle dynamics and to resolve these questions.Specifically, HF materials respond to an incident ultrashort THz pulse by the emission of a time-delayed reflex [14].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 = 2π /k B T K , respectively (with the Planck and k B the Boltzmann constant).As a method, this technique was introduced through experiments on CeCu 6−x Au x [14][15][16].
In YbRh 2 Si 2 we now apply this new method to directly measure the fermionic CSD.YbRh 2 Si 2 is a prototypical HF compound.In zero magnetic field it is antiferromagnetic below the Neél temperature T N = 72 mK.It undergoes a QPT to a Kondo HF liquid at a critical magnetic field of B ⊥ ≈ 66 mT perpendicular to the c axis [17,18] (see 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].YbRh 2 Si 2 has a Kondo temperature of T K ≈ 25 K, high enough to enable a wide quantum-critical region and to permit us to search for signs of CSD in the range T * qp < T < T K , in contrast to the case of CeCu 6−x Au x .In our temperature-dependent, time-resolved THz reflection spectroscopy measurements we crossed the QCP by varying the magnetic field or the Ir concentration.analysed as described elsewhere [14].With T K = 25 K we obtain a delay time τ K ≈ 1.9 ps for these in agreement with the data in Figs.1b and 1c.We therefore chose the time window for the analysis from 1.3 to 2.6 ps.Fig. 2 shows the time-integrated intensity of the THz echo pulse for Yb(Rh 1−x Ir x ) 2 Si 2 for x = 0 and B ⊥ = 214 mT and for x = 0.06 and B ⊥ = 0.All plots exhibit a similar maximum value of the spectral Kondo weight near 25 K in good agreement with T K .Aside from a shift towards higher temperature right at the QCP, the maximum shows no systematic field dependence.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 (Figs. 2a and  2b), the signal continues to decrease but remains finite down to the lowest experimentally achieved temperature of 2 K.We note that this temperature and field range (2.0 K ≤ T ≤ 20 K, B ⊥ 66 mT) is within the white area of the phase diagram in Fig. 1a, the so-called quantum-critical fan [9,24], where ther-modynamic and transport properties are dominated by quantumcritical fluctuations [11,17,18].However, our THz time-delay spectroscopy is not directly sensitive to these fluctuations, but exclusively to the HF quasiparticle spectral weight [14,15].Therefore, the behavior in Figs.2a and 2b indicates that the Kondo effect remains partially intact in this temperature range.This is reasonable since we are still a factor of ∼ 25 above T N so that the heavy quasiparticles are not entirely destroyed by the impending antiferromagnetic order..At quantum criticality (Figs.2d and 2h), the temperature dependence changes drastically.The initial signal decrease with temperature is now followed by a logarithmic increase of the THz echo signal that persists down to the lowest observed temperature.Note qualitative similarity between the field-tuned (Fig. 2d) and chemically tuned (Fig. 2h) quantum critical systems, however 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 (Figs. 2e-g).Its onset may also be conjectured at 50 mT on the antiferromagnetic side.(Fig. 2c).
To understand the striking logarithmic increase towards low temperature, we analyze the THz echo signal theoretically.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 avoided crossing [15,25,26], as shown in Figs.3a to 3c.While low-temperature thermodynamic and transport experiments probe the lower, occupied band only, resonant THz spectroscopy involves transitions between both bands, which calls for a two-band theory.Our according, critical HF liquid theory shows (see Methods) that the n = 1, 2 bands with dispersions ε n p (where p is the crystal-electron momentum) have distinct momentum-and temperature-dependent spectral weights z n p .These cross over from z n p ≈ 1 in the strongly dispersive region to z n p = a z 0 1 in the flat region of these bands, see Figs. 3d to 3f 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 ) α is a suppression factor with a critical exponent α.It describes the destruction of quasiparticle spectral weight as the QCP is approached by lowering the temperature T below the onset temperature for quantum criticality, T 0 .For the latter, experiments revealed T 0 ≈ T K in YbRh 2 Si 2 [27].
As mentioned, the THz echo pulse at τ K = 2π /k B T K is solely sensitive to the breakup-and-recovery dynamics of the HFs and not to 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 ∆ε p = ε 2 p − ε 1 p .Here, f (ε n p ) is the Fermi-Dirac distribution function.In Eq. ( 1), it describes the probability that the n = 1 band is occupied and the n = 2  band is empty before the THz absorption process.W ( ω) is the spectrum of the incident THz pulse, which is a Gaussian distribution of width Γ centered around the central frequency Ω THz .
With ∆ε p ≈ Ω 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 square of the electric-dipole transition-matrix element between the two bands.
When the probability for HF formation, a z 0 ∝ (T /T K ) α , tends to zero at the QCP, the heavy bands flatten according to the twoband 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 , see Figs. 3d to 3f.This means that the oscillation frequency of fermionic quasiparticles, ω n p = (ε n p − E F )/ , 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 broadening of the shaded areas in Figs.3a to 3c.The interplay of these two counteracting effects, quasiparticle destruction and phase-space expansion, leads to a non-monotonic temperature dependence of the THz absorption strength P(T ).An expansion of Eq. ( 1)for small z 0 (T ) predicts a logarithmic increase of P(T ) ∝ ln[1/z 0 (T )] towards low temperatures down to the region of T * qp .The full numerical evaluation of Eq. 1 leads to the behavior shown in Fig. 3g.It reproduces the Kondo maximum near T K ≈ 25 K and at quantum criticality indeed to a logarithmic increase across an intermediate temperature window T * qp < T < T K according to observing this behavior in Figs.2d and 2h is thus a unique experimental signature of fermionic quasiparticle CSD in the Yb(Rh 1−x Ir x ) 2 Si 2 system.Upon further decreasing the temperature to T < T * qp , P(T ) approaches zero as the HF weight disappears altogether, see inset of Fig. 3g.The low-temperature scale T * 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 * qp depends on the critical exponent α and is several orders of magnitude lower than the Kondo scale ∼ T K , possibly undetectably small.A low-temperature scale T * has also been observed as a maximum in the magnetic susceptibility [24] whose microscopic origin has, however, remained unclear.Since in Yb(Rh 1−x Ir x ) 2 Si 2 , T * and our theoretically predicted T * qp are in the same temperature range, we conjecture that both may be of the same physical origin -the competition of fermionic CSD and quasiparticle breakdown at the QCP.As a crossover temperature, T * qp remains non-zero, but can be exceedingly small, depending on α, see inset of Fig. 3g.Note that away from criticality, we have α → 0 so that the logarithmic low-temperature behavior does not occur in agreement with Figs.2e to 2g.The blue curves in Figs.2d and 2h represent the evaluation of Eq. ( 1) for the spectrum W ( ω) 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 the CSD at low temperatures (T * qp < T < T K ) from Eq. ( 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 K /T .This directly leads to α = s low /s high .From the experimental data we find α = 0.10 for Yb(Rh 1−x Ir x ) 2 Si 2 at x = 0, B ⊥ = 66 mT in Fig. 2d and α = 0.28 at x = 0.06, B ⊥ = 0 in Fig. 2h.Note that different critical behavior for QPTs by magnetic fields and chemical pressure have been observed before [28].It is a signature of the different types of quantum-critical fluctuations in the two cases.
Our theory also explains why the logarithmic low-temperature increase of THz absorption indicating CSD cannot be observed in the CeCu 6−x Au x system.For this material, the ratio T K /T * is substantially smaller than in the Yb(Rh 1−x Ir x ) 2 Si 2 system so that the effects of the buildup of 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 experiment [14].
Summarizing in Fig. 4 the measured THz absorption data in a T -versus-B phase diagram reveals that on the HF liquid side of the QCP (B ⊥ > 66 mT) P(T ) is enhanced for T < 10 K due to the CSD effect and as explained by our two-band HF liquid theory.By contrast, on the antiferromagnetic side (B ⊥ 50 mT) we observe a reduction of the Kondo-weight-related absorption but no fermionic CSD.This suggests that in this region the RKKY interaction strongly affects the quasiparticle dynamics such that the the two-band HF liquid theory is not valid here [19][20][21].To conclude, we observed a logarithmic low-temperature increase in the resonant quasiparticle excitation probability P(T ) near a magnetic quantum phase transition in heavy-fermion materials.We identified this logarithmic increase as a unique signature of fermionic quasiparticle CSD, that is, a vanishing quasiparticle frequency, near a quantum phase transition with fermionic breakdown.Since, in contrast to 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 to define the heavy quasiparticle weight as an order parameter for quantum phase transitions with fermionic breakdown.It also sets the stage for a classification of fermionic quantum phase transitions in terms of their critical exponent, analogous to thermodynamic phase transitions.

Methods Experimental
Single-crystalline, c-oriented Yb(Rh 1−x Ir x ) 2 Si 2 platelets (x = 0, 0.06) with dimensions of 2 × 3 × 0.07 mm 3 were grown from indium flux as described in the literature [23].The sample surface is freshly polished before the THz measurements.Samples are mounted onto a Teflon holder, where two permanent magnets placed above and below the sample generate a magnetic field of up to 214 mT in the easy magnetic plane perpendicular to the tetragonal c-axis.We use a temperature-controlled Janis SVT-400 helium-reservoir cryostat operable in the range from 1.9 to 325 K.The THz experiments are performed in a 90 • reflection geometry with THz light in a spectral range from 0.1 to 3 THz polarized perpendicular to the crystallographic c-axis.
The energy of the THz pulse is in the range of a few nJ.The residual 10% of the 800-nm beam is then used for free-space electro-optic sampling of the reflected THz light from the sample.Both, the THz and the 800-nm beams, are collinearly focused onto a 0.5-mm-thick (110)-cut ZnTe detection crystal.To increase the accessible time delay between the THz and the 800-nm pulses, Fabry-Pérot resonances from the faces of the detection crystal are suppressed by a 2-mm-thick THz-inactive (100)-cut ZnTe crystal.The THz-inactive crystal is optically bonded to the back of the detection crystal.The THz-induced ellipticity of the 800-nm beam is measured using a quarter-wave plate, a Wollaston prism and a balanced photodiode.

Theoretical
We construct a phenomenological, critical two-band Fermi liquid theory to describe the THz-induced resonant transitions from the heavy conduction to the light valence band.Electrons in the light band have the dispersion ε (0) p measured relative to the Fermi energy E F , have spectral weight unity, and are assumed to be non-interacting.The parameters of the heavy electron states, generated by the Kondo effect, are controlled by the Kondo scale T K .Their energy lies close to the Fermi level E F , shifted by ∆ ≈ ±k B T K above (particle-like HF systems) or below (hole-like HF systems) E F [3,26].They have a strongly reduced spectral weight a(T ) which reaches a(0) = T K /γ 1 at T = 0 and decreases logarithmically for temperatures T > T K [3].Here, γ is the effective hybridization of the rare-earth 4 f orbitals with the conduction electron states.We also take a residual interaction of the quasiparticles within the heavy band into account.It implies an additional reduction of both, the quasiparticle weight and of the heavy band shift, by the local quasiparticle weight factor z 0 (T ) (see main text), a → z 0 a, ∆ → z 0 ∆.Taking now the hybridization V between the light, uncorrelated band and the heavy band into account and calculating the hybridized band structure, the band energies are obtained as as shown in Figs.2a to 2c.Inserting these expressions into Eq.( 1) and adjusting the two parameters T K /D and α, where D is the free conduction bandwidth, leads to the curves shown in Figs.2d, h and 3g, h.Symmetry implies that this result for the THz absorption is the same for particle-like and hole-like HF systems.

-Figure 1 .
Figure 1.Exploring fermionic quantum criticality by time-resolved THz reflectivity.a, Schematic phase diagram of YbRh 2 Si 2 with characteristic energy scales.b, Reflected THz signal from the YbRh 2 Si 2 sample (red) and from a Pt mirror reference (black).The time traces are normalized by the maximum field amplitude at t = 0 ps.The delayed, purely Kondo-related "echo" pulse is visible in the interval between 1.3 and 2.6 ps, where it distinctly differs from the Pt reference signal.c, Signal of the delayed pulse in YbRh 2 Si 2 at B ⊥ = 0 for various temperatures.
a b s o r p t i o n ( a r b .u n i t ) T e m p e r a t u r e ( K ) a b s o r p t i o n ( a r b .u n i t )

Figure 2 .
Figure 2. Temperature dependence of the resonant THz absorption across the QCP in YbRh 2 Si 2 .a-g, Evolution of the THz absorption by resonant Kondo quasiparticle excitations at various external magnetic fields B ⊥ as a function of temperature.The weights are derived from the integrated intensity of the echo pulses emitted in the time window of 1.3 -2.6 ps time window (see Figs. 1b, c).h, Temperature dependence of the resonant THz absorption in quantum critical Yb(Rh 1.94 Ir 0.06 ) 2 Si 2 .Fits of Eq. 1 are plotted as blue lines in d and h.

p 2 FFigure 3 .
Figure 3. Band structure and Kondo weight calculations towards the QCP.a-c, Band structure of the conduction band (steep slope) and the Kondo state (flat band) resulting in a hybridized lower (brown) and upper (orange) branch.d-f, Momentum-and temperature-dependent quasiparticle weights 1 p , z 2 p in the lower (brown) and upper (orange) bands as well as the product z 1 p • z 2 p , all calculated from the two-band critical Fermi liquid theory.g, h, Resonant THz absorption strength at (red and dark yellow) and away from (blue) the QCP as calculated for the system parameters of Yb(Rh 1−x Ir x ) 2 Si 2 , (x = 0, 0.06) and CeCu 6−x Au x , respectively.

2 S h e e t 3 TFigure 4 .
Figure 4. Phase diagram of field-tuned YbRh 2 Si 2 as measured by THz time-delay spectroscopy, using the discrete magnetic-field values shown in Fig. 2. The color code represents the resonant THz absorption, and the dashed line marks the critical field of 66 mT.