Session T16: Kondo Lattice Theory

A new paradigm for heavy electron materials

Recent experiments on the emergence of heavy electrons that display universal behavior below a characteristic temperature T$^{\ast }$ have shown that the well-known Doniach phase diagram does not apply to most materials. Here we introduce the concept of hybridization effectiveness as the organizing principle and show that it makes possible a consistent and quantitative description of the low temperature emergent behaviors of a number of heavy electron materials. We propose a new phase diagram and predict a delocalization line in the pressure/temperature phase diagram that is to be examined in future experiment.   

Superconductivity in Heavy Fermion Materials

Superconductivity in heavy fermion materials is a complex phenomenon since it often emerges from a strongly entangled state, the Kondo screened heavy Fermi liquid. In this talk, we discuss possible pairing interactions that arises in the heavy Fermi liquid, and the resulting symmetries and momentum dependencies of the superconducting order parameter. In particular, we show that the interplay between the large Fermi surface of the Fermi liquid state and the momentum dependence of the pairing interaction naturally leads to an unconventional symmetry of the superconducting order parameter in general, and a $d_{x^2-y^2}$-symmetry in particular. Moreover, we identify the signatures of unconventional pairing in the differential conductance, $dI/dV$, measured in scanning tunneling spectroscopy as well as in the quasi-particle interference pattern. Finally, we discuss the effects of impurities on the spatial dependence of both the superconducting order parameter and the hybridization.   

Single-particle and optical self-energies of a Fermi liquid revisited

We discuss the conditions under which the imaginary part of the single-particle self-energy at the Fermi surface $\Sigma (\omega, T)$ and the optical scattering rate $1/\tau ( \Omega, T)$ have particular simple scaling forms $\mathrm{Im} \Sigma (\omega, T) \propto \omega^2 + \pi^2 T^2 $ and $1/\tau (\Omega, T) \propto \Omega^2 + 4\pi^2 T^2$. We show that these relations follow from particular analytic properties of the effective fermion-fermion interaction and are only satisfied when the single-particle and optical self-energies are analytic functions of the frequency. When they are not, the scaling forms are more complex even if the system remains a Fermi liquid. We also address recently observed violation of the $\Omega^2 + 4\pi^2 T^2$ form of $1/\tau$ in URu$_2$Si$_2$ and discuss possible mechanisms of this violation.   

Singular Valence Fluctuations at a Kondo Destroyed Quantum Critical Point

Recent experiments on the heavy fermion superconductor $\backslash $beta-YbAlB4 have indicated that this compound satisfies quantum critical scaling [1]. Motivated by the observation of mixed valency in this material [2], we study the Kondo destruction physics in the mixed-valence regime [3] of a particle-hole asymmetric Anderson impurity model with a pseudogapped density of states. In the vicinity of the quantum critical point we determine the finite temperature spin and charge susceptibilities by utilizing a continuous time quantum Monte Carlo method [4] and the numerical renormalization group. We show that this mixed-valence quantum critical point displays a Kondo breakdown effect. Furthermore, we find that both dynamic spin and charge susceptibilities obey frequency over temperature scaling, and that the static charge susceptibility diverges with a universal exponent. Possible implications of our results for $\backslash $beta-YbAlB4 are discussed. [1] Matsumoto et al, Science \textbf{331}, 316 (2011). [2] Okawaet al, Physical Review Letters \textbf{104}, 247201 (2010). [3] J. H. Pixley, S. Kirchner, Kevin Ingersent and Q. Si, arXiv:1108.5227v1 (2011). [4] M. Glossop, S. Kirchner, J. H. Pixley and Q. Si, Phys. Rev. Lett. \textbf{107}, 076404 (2011).   

Kondo Physics in a Rare Earth Ion with well localized {\it 4f } electrons

We observe a rise in the low temperature resistivity of dilute La$_{1-x}$Nd$_x$B$_6$ single crystals. The specific heat data for the same samples show that there is an entropy of $R$ln4 per mole Nd involved in the excess specific heat above that of LaB$_6$. All these features are consistent with a Kondo scale of $T_K$ = 1 K. However, Nd has a well localized {\it 4f }$^{ 3}$ {\it J} = 9/2 Hund's Rule configuration which is not expected to be Kondo coupled to the conduction electrons in LaB$_6$. We conjecture that the unexpected Kondo effect arises via participation of {\it 4f} quadrupolar degrees of freedom of the Nd crystal field ground state quartet. Raman experiments as well as detailed theoretical studies do indicate that one expects strong quadrupolar influence in the properties of NdB$_6$.   

Partial disorder in the periodic Anderson model on a triangular lattice

In Kondo lattice systems, keen competition between the Kondo coupling and RKKY interaction leads to a quantum critical point between a magnetically-ordered state and a Fermi liquid state. This is a source of novel phenomena, such as a non-Fermi liquid behavior and a superconductivity. In the present study, we explore a new quantum phase related to the competition by introducing a new parameter, geometrical frustration of the lattice structure. Especially, we focus on the possibility of a partial disordered (PD) state, in which a magnetically ordered and nonmagnetic sites coexist. We consider a fundamental model for the rare-earth systems, a periodic Anderson model on a frustrated triangular lattice, and investigate the ground-state phase diagram by the Hartree-Fock approximation with assuming three-site unit cell. As a result, we find a PD state at half filling between a noncollinear antiferromagnetic metal and a Kondo insulator [1]. The PD state is stabilized by relaxing the frustration by forming collinear antiferromagnetic order on an unfrustrated honeycomb subnetwork, while leaving remaining sites non-magnetic. We will also discuss the effects of spin anisotropy and carrier doping to the PD state.\\[4pt] [1] S. Hayami, M. Udagawa, and Y. Motome, J. Phys. Soc. Jpn. 80, 073704 (2011).   

Quantum critical Kondo destruction of the Bose-Fermi Kondo model in the presence of a local transverse field

Recent studies of the global phase diagram of quantum critical heavy fermion metals [1] have motivated us to consider the interplay between the quantum fluctuations within the local-moment system and those associated with the Kondo interaction. Towards this goal, we study a Bose-Fermi Kondo model with Ising anisotropy in the presence of a local transverse field. We apply the numerical renormalization group method for the case with a sub-ohmic bosonic bath exponent and a constant conduction electron density of states [2]. Upon increasing the strength of the transverse field in the Kondo screened phase we find a crossover from a fully screened to a fully polarized impurity spin with no transition in between. Increasing the strength of the transverse coupling in the localized phase, on the other hand, we identify signatures for a continuous transition into a partially polarized Kondo phase. We discuss the implications of our results for the global phase diagram of the Kondo lattice system. \\[4pt] [1] Q. Si and F. Steglich, Science 329, 1161 (2010). \\[0pt] [2] M. T. Glossop and K. Ingersent, Phys. Rev. B 75, 104410 (2007).   

A spin-selective Kondo-insulator: Cooperation between Ferromagnetism and Kondo-effect

Taking ferromagnetic heavy fermion compounds as motivation we analyze the mechanism stabilizing the ferromagnetic state in the antiferromagnetically coupled Kondo lattice model. We find that even for this ferromagnetic state Kondo screening plays an essential role in stabilizing the ferromagnetic state at zero temperature leading to very interesting properties: while the majority-spin electrons are metallic, the minority-spin electrons form an insulating state. We clarify that this state is due to partial Kondo screening, so that parts of the local moments are bound to the electrons, resulting in a dynamically-induced commensurability which is essential for producing the gap in the minority-spin electrons. We believe that the mechanism proposed here, the dynamically generated commensurability, is generic for the ferromagnetic phase in the antiferromagnetically coupled Kondo lattice model, thus providing new insights into the zero temperature physics for the Kondo lattice model.   

Phase diagram of half-filled $J_1-J_2$-Heisenberg Kondo lattice model on honeycomb lattice

Recent studies in quantum critical heavy fermion metals have opened up a rich phase diagram. The zero-temperature global phase diagram involves a combination of phases, featuring Kondo screening/destruction and antiferromagnetic order/disorder as the quantum fluctuations of the local moments are tuned relative to their Kondo coupling with the spins of the conduction electrons. Recently, we have studied a one-dimensional Heisenberg-Kondo model to reveal a competition between a Kondo-screened paramagnetic phase with a large Fermi surface and a Kondo-destroyed paramagnetic spin-Peierls phase with a small Fermi surface [1]. To take advantage of the intuitions gained through that work, we study here a half-filled, $J_1-J_2$ Heisenberg-Kondo lattice model on a honeycomb lattice. We have obtained a rich phase diagram as a function of magnetic frustration and Kondo coupling. Apart from antiferromagnetic and Kondo insulators, we also find Kondo-destroyed phases that are semi-metallic with Fermi points. There are also particular critical points which are gapless in both charge and spin sectors. \\[4pt] [1] P. Goswami and Q. Si, Phys. Rev. Lett. {\bf 107}, 126404 (2011).   

Cluster dynamical mean-field study of charge ordering in the Kondo lattice model at quarter filling

The Kondo lattice model is one of fundamental models for heavy-fermion systems, where exchange interactions between itinerant electrons and localized spins play an important role. Among many different phases described by this model, an interesting possibility is a charge-ordered state, in particular when considering the fact that the model does not include any bare repulsive interaction between electrons. The possibility was first pointed out by a perturbation expansion in the strong Kondo coupling limit at quarter filling~[1], and recently examined by the dynamical mean-field theory in infinite dimensions~[2]. However, it remains unclear whether the charge-ordered state still survives in two or three dimensions and what type of magnetic order is accommodated in the charge-ordered state. To clarify these issues, we investigate the quarter-filled Kondo lattice model on a square lattice by using cluster dynamical mean-field theory, which can include both spatial and dynamical correlations. We found that charge-ordered state appears in the intermediate coupling region. We will discuss electronic and magnetic properties in the charge-ordered state in detail.\\[4pt] [1] H. E. Hirsch, Phys. Rev. B {\bf 30}, 5383 (1984).\\[0pt] [2] J. Otsuki {\it et al}., J. Phys. Soc. Jpn. {\bf 78}, 034719 (2009).   

Phase diagram of the non-centrosymmetric Kondo lattice model

Kondo lattice model is prototypical for studying materials with localized f-electrons such as heavy-fermion compounds that exhibit a competition between Kondo screening and magnetism. This competition was argued to be crucial for superconductivity in these systems. We study the effects of spin-orbit interaction (SOI) in the conduction band (due to the lack of inversion symmetry), on the interplay between Kondo, superconducting and magnetic phases by considering the $S=1/2$ 2D Kondo lattice with short-range Heisenberg coupling between localized moments and utilizing a pseudo-fermion hybridization mean-field theory. In particular, we demonstrate that the heavy-fermion state, and hence superconductivity, is suppressed with increasing SOI. Our results are of relevance for ${\rm Ce}$- and ${\rm U}$-based heavy-fermion superconductors without inversion symmetry.   

Simulation of the triangular Kondo-lattice model using the gradient kernel polynomial method

We introduce a method to study systems, such as the Kondo-lattice model, where a classical field is coupled to fermionic degrees of freedom. Such systems are computationally challenging because each change to the classical field requires recalculation of the density of states for the bilinear fermionic Hamiltonian. The kernel polynomial method (KPM) is a useful tool to approximate the density of states at a cost linear in the system size. We extend KPM to approximate the gradient of the density of states at the same cost, allowing fast updates of the entire classical field. Simulations of the triangular Kondo-lattice model indicate phases with exotic non-coplanar spin ordering and spontaneous quantum Hall effect.   

Monte Carlo simulations of magnetic clustering at a quantum critical point

We present the results of Monte Carlo simulations on a percolating magnetic system with relevance to quantum critical point materials. It has previously been shown that, for heavily doped quantum critical point compounds such as Ce(Ru$_{0.24}$Fe$_{0.76})_{2}$Ge$_{2}$, the formation and dynamics of magnetic clusters strongly influences the physical response of the system at low temperature. Our simulation is based on the idea that finite-size effects force small magnetic clusters to order at comparatively high temperatures and, once formed, are impervious to Kondo shielding. Disorder acts to introduce a distribution of Kondo temperatures which, in turn, governs the formation of clusters as the temperature is lowered. We implement a percolation model based on such a distribution-- first introduced by Bernal et al.-- and with a restriction whereby Kondo shielding is allowed to remove moments from the infinite cluster \textit{only}. We investigate how this influences thermodynamic quantities as well as how well the simulations align with our analytic theory that is based on the same restriction.   

Monte Carlo study of a spin-ice type Kondo lattice model on a pyrochlore lattice

Recent experiments on geometrically frustrated metallic oxides, such as pyrochlore oxides R$_2$Mo$_2$O$_7$ and R$_2$Ir$_2$O$_7$, have drawn increasing interest in the effect of geometrical frustration in itinerant electron systems. In these systems, the coupling between electrons and frustrated magnetism leads to various fascinating phenomena. However, much less theoretical studies have been done by treating the interplay of localized spins and itinerant electrons in an unbiased manner. To clarify electronic and magnetic behaviors in such spin-charge coupled systems on frustrated lattice structures, we investigate a spin-ice type Kondo lattice model on a pyrochlore lattice by a real-space Monte Carlo simulation. By employing a sophisticated algorithm, we conduct calculations up to 2048 sites. We show that the system exhibits keen competition between various magnetic phases depending on the spin-charge coupling and electron density. As a consequence, the system shows rich phase diagram with complex magnetic orderings and phase separations between them. Furthermore, in applied magnetic field, we identify a magnetization plateau for one of the novel magnetic phases. In the presentation, the phase diagram and the mechanism of the magnetic orderings will be discussed in details.   

O(3)-symmetric variational wave functions for the Kondo lattice model

We construct and investigate a class of correlated wave functions for the Kondo lattice model. The construction is based on the faithful fermionic representation of the Kondo lattice model that was introduced recently by the author in PRB 83, 235103 (2011). In the limit of small exchange coupling the wave functions allow for both local singlet correlations between the localized moments and the conduction electrons, as well as an essentially arbitrary spin correlation function for the localized moments. We compare and contrast our results with those of previous studies.