Kondo Breakdown Quantum Critical Point Research Articles

Quantum phase transition in a two-dimensional Kondo-Heisenberg model: A dynamical Schwinger-boson large- N approach

Strange metal behavior arises in heavy fermion metals close to antiferromagnetic transitions. An increasing amount of experiments indicates a link of such behavior to a Kondo breakdown quantum critical point. To shed light on this intriguing problem, we study the 2D Kondo-Heisenberg model using a dynamical large-N multichannel Schwinger boson approach. We identify and characterize the quantum phase transition from an antiferromagnetically ordered ground state to a Kondo-dominated paramagnetic state, and attribute a jump in certain phase shift to Kondo breakdown. In addition, we calculate transport and thermodynamic quantities and discuss them in the context of the experimental observations in quantum critical heavy fermion systems.

Heavy fermions: superconductivity and its relationship to quantum criticality

Anti-ferromagnetic (AF) heavy-fermion (HF) metals frequently exhibit a quantum critical point (QCP), i.e. a continuous disappearance of AF order at zero temperature as a function of a non-thermal control parameter, like pressure. Commonly, unconventional (d-wave) superconductivity (SC) is observed in the vicinity of such a QCP. In this article, the interplay between quantum criticality and SC in HF metals is discussed. The latter can be well described in the framework of the Kondo lattice (KL) model. After introducing HFs and giving an overview on HF superconductivity, the characteristic energy scales in KL systems are addressed. Subsequently, we concentrate on the isostructural compounds CeCu2Si2 and YbRh2Si2. CeCu2Si2 is a prototypical HF superconductor which shows a QCP of itinerant, i.e. three-dimensional (3D) spin-density-wave (SDW) type. Overdamped, nearly quantum-critical SDW fluctuations are concluded from inelastic-neutron-scattering results to be the driving force for SC (Tc ≈ 0.6 K) in this compound. While the QCP in several other Ce-based HF superconductors is presumably of the SDW variety too, it is likely to be of the local, i.e. Kondo destroying type in the pressure-induced superconductor CeRhIn5. YbRh2Si2 has been identified as a prototype HF metal exhibiting such a Kondo breakdown QCP, often named a (T = 0) 4f-orbital selective Mott transition. This compound is not a superconductor at T ˃ 10 mK. Systematically searching for and studying SC near Kondo destroying QCPs may offer a link to unconventional SC in other families of correlated materials, such as the doped Mott insulators in the high-Tc cuprates and some of the organic charge-transfer salts.

Routes to heavy-fermion superconductivity

Superconductivity in lanthanide- and actinide-based heavy-fermion (HF) metals can have different microscopic origins. Among others, Cooper pair formation based on fluctuations of the valence, of the quadrupole moment or of the spin of the localized 4f/5f shell have been proposed. Spin-fluctuation mediated superconductivity in CeCu2Si2 was demonstrated by inelastic neutron scattering to exist in the vicinity of a spin-density-wave (SDW) quantum critical point (QCP). The isostructural HF compound YbRh2Si2 which is prototypical for a Kondo-breakdown QCP has so far not shown any sign of superconductivity down to T ≈ 10 mK. In contrast, results of de-Haas-van-Alphen experiments by Shishido et al. (J. Phys. Soc. Jpn. 74, 1103 (2005)) suggest superconductivity in CeRhIn5 close to an antiferromagnetic QCP beyond the SDW type, at which the Kondo effect breaks down. For the related compound CeCoIn5 however, a field-induced QCP of SDW type is extrapolated to exist inside the superconducting phase.

Role of vertex corrections in theT-linear resistivity at the Kondo-breakdown quantum critical point

The Kondo-breakdown scenario has been claimed to allow the $T$-linear resistivity in the vicinity of the Kondo-breakdown quantum critical point, two cornerstones of which are the dynamical exponent $z=3$ quantum criticality for hybridization fluctuations in three dimensions and irrelevance of vertex corrections for transport due to the presence of localized electrons. We revisit the issue of vertex corrections in electrical transport coefficients. Assuming that two kinds of bosonic degrees of freedom, e.g., hybridization excitations and gauge fluctuations, are in equilibrium, we derive coupled quantum Boltzmann equations for two kinds of fermions, e.g., conduction electrons and spinons. We reveal that vertex corrections play a certain role, changing the $T$-linear behavior into ${T}^{5/3}$ in three dimensions. However, the ${T}^{5/3}$ regime turns out to be narrow, and the $T$-linear resistivity is still expected in most temperature ranges at the Kondo-breakdown quantum critical point in spite of the presence of vertex corrections. We justify our evaluation, showing that the Hall coefficient is not renormalized to remain as the Fermi-liquid value at the Kondo-breakdown quantum critical point.

The Luttinger–Ward functional approach in the Eliashberg framework: a systematicderivation of scaling for thermodynamics near the quantum critical point

Scaling expressions for the free energy are derived, using the Luttinger–Ward (LW) functionalapproach in the Eliashberg framework, for two different models of the quantum critical point(QCP). First, we consider the spin-density-wave model for which the effective theory is theHertz–Moriya–Millis theory, describing the interaction between itinerant electrons and collectivespin fluctuations. The dynamics of the latter are described using a dynamical exponentz depending on the nature of the transition. Second, we consider the Kondo breakdownmodel for QCPs, one possible scenario for heavy-fermion quantum transitions,for which the effective theory is given by a gauge theory in terms of conductionelectrons, spinons for localized spins, holons for hybridization fluctuations, andgauge bosons for collective spin excitations. For both models, we construct thethermodynamic potential, in the whole phase diagram, including all kinds of self-energycorrections in a self-consistent way, at the one-loop level. We show how the Eliashbergframework emerges at this level and use the resulting Eliashberg equations tosimplify the LW expression for the free energy. It is found that collective bosonexcitations play a central role. The scaling expression for the singular part of the freeenergy near the Kondo breakdown QCP is characterized by two length scales:one is the correlation length for hybridization fluctuations, and the other is thatfor gauge fluctuations, analogous to the penetration depth for superconductors.

Thermopower as a fingerprint of the Kondo breakdown quantum critical point

We propose that the thermoelectric power distinguishes two competing scenarios for quantum phase transitions in heavy fermions: the spin-density-wave (SDW) theory and breakdown of the Kondo effect. In the Kondo breakdown scenario, the Seebeck coefficient turns out to collapse from the temperature scale ${E}^{*}$ associated with quantum fluctuations of the Fermi surface reconfiguration. This feature differs radically from the physics of the SDW theory, where no reconstruction of the Fermi surface occurs, and can be considered the hallmark of the Kondo breakdown theory. We test these ideas on experimental results for YbRh${}_{2}$Si${}_{2}$.

Break Up of Heavy Fermions at an Antiferromagnetic Instability

We present results of high-resolution, low-temperature measurements of the Hall coefficient, thermopower, and specific heat on stoichiometric YbRh 2 Si 2 . They support earlier conclusions of an electronic (Kondo-breakdown) quantum critical point concurring with a field induced antiferromagnetic one. We also discuss the detachment of the two instabilities under chemical pressure. Volume compression/expansion (via substituting Rh by Co/Ir) results in a stabilization/weakening of magnetic order. Moderate Ir substitution leads to a non-Fermi-liquid phase, in which the magnetic moments are neither ordered nor screened by the Kondo effect. The so-derived zero-temperature global phase diagram promises future studies to explore the nature of the Kondo breakdown quantum critical point without any interfering magnetism.

Thermopower as a signature of quantum criticality in heavy fermions

We present a series of arguments showing that the Seebeck coefficient can be used as a decisive experiment to characterize the nature of the quantum critical point (QCP) in heavy fermion compounds. Being reactive almost exclusively to the presence of delocalized entropic carriers, the Seebeck coefficient shows a drastic collapse at the Kondo breakdown QCP, as the reconstruction of the Fermi surface takes place. In contrast, around a Spin Density Wave (SDW) QCP, the Seebeck coefficient is broadly symmetric. We discuss the possibility of a change of sign at the QCP, the characteristic variation of $| S / T | $ with temperature and external parameter, as well as the capacity of the Seebeck coefficient to distinguish between localized and itinerant anti-ferromagnetism (AF). Suggestions of experiments are given in the case of four non conventional compounds : YbRh$_2$Si$_2$, Ce(Mn)In$_5$, CeCu$_{6-x}$Au$_x$ and URu$_2$Si$_2$.

Signature of Kondo breakdown quantum criticality in optical conductivity

We study the finite-frequency inter-band transition peak in the optical conductivity of a heavy fermion system close to a Kondo breakdown quantum critical point, where the lattice Kondo temperature vanishes. As the system approaches the phase transition from the heavy Fermi liquid side, we find a new cross-over regime where the peak position is related to, but is not directly proportional to, the lattice Kondo scale. In particular, the position of the peak moves to lower energies, but remains finite at the critical point. On the other hand, the peak value changes non-monotonically and eventually the peak disappears at the quantum critical point, indicating the decoupling of the narrow band of f-electrons from the conduction band. We argue that these are unique signatures of a Kondo breakdown transition, and therefore can be useful to distinguish it experimentally from a spin density wave instability.

Grüneisen Ratio at the Kondo-Breakdown Quantum Critical Point

We show that the scenario of a multiscale Kondo-breakdown quantum critical point gives rise to a divergent Grüneisen ratio with an anomalous exponent 0.7. In particular, we fit the experimental data of YbRh2(Si0.95Ge0.05)2 for a specific heat, thermal expansion, and Grüneisen ratio based on our simple analytic expressions. A reasonable agreement between the experiment and theory is found for the temperature range between 0.4 and 10 K. We discuss how the Grüneisen ratio is a key measurement to discriminate between the Kondo-breakdown and spin-density wave theories.

Multiscale fluctuations near a Kondo breakdown quantum critical point

We study the Kondo-Heisenberg model using a fermionic representation for the localized spins. The mean-field phase diagram exhibits a zero-temperature quantum critical point separating a spin liquid phase where the $f$-conduction hybridization vanishes and a Kondo phase where it does not. Two solutions can be stabilized in the Kondo phase, namely, a uniform hybridization when the band masses of the conduction electrons and the $f$ spinons have the same sign and a modulated one when they have opposite sign. For the uniform case, we show that above a very small Fermi-liquid temperature scale, the critical fluctuations associated with the vanishing hybridization have dynamical exponent $z=3$, giving rise to a specific-heat coefficient that diverges logarithmically in temperature, as well as a conduction-electron inverse lifetime that has a $T\text{ }\text{log}\text{ }T$ behavior. Because the $f$ spinons do not carry current, but act as an effective bath for the relaxation of the current carried by the conduction electrons, the latter result also gives rise to a $T\text{ }\text{log}\text{ }T$ behavior in the resistivity. This behavior is consistent with observations in a number of heavy fermion metals.

Selective Mott transition and heavy fermions

Starting with an extended version of the Anderson lattice where the $f$ electrons are allowed a weak dispersion, we examine the possibility of a Mott localization of the $f$ electrons for a finite value of the hybridization $V$. We study the fluctuations at the quantum critical point (QCP) where the $f$ electrons localize. We find that they are in the same universality class as for the Kondo breakdown QCP, with the following notable features. The quantum critical regime sees the appearance of an additional energy scale separating two universality classes. In the low energy regime, the fluctuations are dominated by massless gauge modes, while in the intermediate energy regime, the fluctuations of the modulus of the order parameter are the most relevant ones. In the latter regime, electric transport simplifies drastically, leading to a quasilinear resistivity in three dimensional and anomalous exponents lower than $T$ in two dimensional. This rather unique feature of the quantum critical regime enables us to make experimentally testable predictions.

Kondo Breakdown as a Selective Mott Transition in the Anderson Lattice

We show within the slave-boson technique that the Anderson lattice model exhibits a Kondo breakdown quantum critical point where the hybridization goes to zero at zero temperature. At this fixed point, the f electrons experience as well a selective Mott transition separating a local-moment phase from a Kondo-screened phase. The presence of a multiscale quantum critical point in the Anderson lattice in the absence of magnetism is discussed in the context of heavy fermion compounds. This study is the first evidence for a selective Mott transition in the Anderson lattice.