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arXiv:cond-mat/0207493 · Physical Review Letters

Locally Critical Point in an Anisotropic Kondo Lattice

Daniel R. Grempel, Qimiao Si A

Published Jul 11, 2003• 67 citations
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Abstract

We report the first numerical identification of a locally quantum critical point at which the criticality of the local Kondo physics is embedded in that associated with a magnetic ordering. We are able to numerically access the quantum critical behavior by focusing on a Kondo-lattice model with Ising anisotropy. We also establish that the critical exponent for the q-dependent dynamical spin susceptibility is fractional and compares well with the experimental value for heavy fermions.

DOI
10.1103/physrevlett.91.026401

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We report the first numerical identification of a locally quantum critical point at which the criticality of the local Kondo physics is embedded in that associated with a magnetic ordering. We are able to numerically access the quantum critical behavior by focusing on a Kondo-lattice model with Ising anisotropy. We also establish that the critical exponent for the q-dependent dynamical spin susceptibility is fractional and compares well with the experimental value for heavy fermions.
Motivated by the recent discovery of ferromagnetic quantum criticality in the heavy fermion compound CeRh6Ge4{\mathrm{CeRh}}_{6}{\mathrm{Ge}}_{4}CeRh6​Ge4​, we develop a numerical algorithm of infinite projected entangled pair states for the anisotropic ferromagnetic Kondo-Heisenberg model in two dimensions and study the ferromagnetic quantum phase transitions with varying magnetic and hopping anisotropy. Our calculations reveal a continuous ferromagnetic quantum phase transition in the large anisotropic region and first-order quantum phase transitions for smaller anisotropy. Our results highlight the importance of magnetic anisotropy on ferromagnetic quantum criticality in Kondo lattice systems and provide a possible explanation for the experimental observation in CeRh6Ge4{\mathrm{CeRh}}_{6}{\mathrm{Ge}}_{4}CeRh6​Ge4​ with a quasi-one-dimensional magnetic structure. Our work opens the avenue for future studies of the rich Kondo lattice physics using state-of-the-art tensor network approaches.
The continuous quantum phase transition and the nature of the quantum critical point (QCP) in a modified Kondo lattice model with Ising anisotropic exchange interactions are studied within the density-matrix renormalization group algorithm. We investigate the effect of quantum fluctuations on critical Kondo destruction QCP by probing static and dynamic properties of the magnetic order and the Kondo effect. In particular, we identify that local Kondo physics itself becomes critical at the magnetic phase transition point, providing unbiased evidences for local quantum criticality between two insulators without involving the Fermi surface.
Experiments in heavy-fermion metals and related theoretical work suggest that critical local-moment fluctuations can play an important role near a zero-temperature phase transition. We study such fluctuations at the quantum critical point of a Kondo impurity model in which the density of band states vanishes as /epsilon/(r) at the Fermi energy (epsilon=0). The local spin response is described by a set of critical exponents that vary continuously with r. For 0<r<1, the dynamical susceptibility at the critical point exhibits omega/T scaling with a fractional exponent, implying that the critical point is interacting.
Heavy fermion criticality has been a long-standing problem in condensed matter physics. Here we study a one-dimensional Kondo lattice model through numerical simulation and observe signatures of local criticality. We vary the Kondo coupling J_K J K at fixed doping x. At large positive J_K J K , we confirm the expected conventional Luttinger liquid phase with 2k_F=\frac{1+x}{2} 2 k F = 1 + x 2 (in units of 2\pi 2 π ), an analogue of the heavy Fermi liquid (HFL) in the higher dimension. In the J_K ≤ 0 J K ≤ 0 side, our simulation finds the existence of a fractional Luttinger liquid (LL \star ⋆ ) phase with 2k_F=\frac{x}{2} 2 k F = x 2 , accompanied by a gapless spin mode originating from localized spin moments, which serves as an analogue of the fractional Fermi liquid (FL \star ⋆ ) phase in higher dimensions. The LL \star ⋆ phase becomes unstable and transitions to a spin-gapped Luther-Emery (LE) liquid phase at small positive J_K J K . Then we mainly focus on the “critical regime” between the LE phase and the LL phase. Approaching the critical point from the spin-gapped LE phase, we often find that the spin gap vanishes continuously, while the spin-spin correlation length in real space stays finite and small. For a certain range of doping, in a point (or narrow region) of J_K J K , the dynamical spin structure factor obtained through the time-evolving block decimation (TEBD) simulation shows dispersion-less spin fluctuations in a finite range of momentum space above a small energy scale (around 0.035 J 0.035 J ) that is limited by the TEBD accuracy. All of these results are unexpected for a regular gapless phase (or critical point) described by conformal field theory (CFT). Instead, they are more consistent with exotic ultra-local criticality with an infinite dynamical exponent z=+ z = + . The numerical discovery here may have important implications on our general theoretical understanding of the strange metals in heavy fermion systems. Lastly, we propose to simulate the model in a bilayer optical lattice with a potential difference.
Recent studies of the global phase diagram of quantum-critical heavy-fermion metals prompt consideration of the interplay between the Kondo interactions and quantum fluctuations of the local moments alone. Toward this goal, we study a Bose-Fermi Kondo model (BFKM) with Ising anisotropy in the presence of a local transverse field that generates quantum fluctuations in the local-moment sector. We apply the numerical renormalization-group method to the case of a sub-Ohmic bosonic bath exponent and a constant conduction-electron density of states. Starting in the Kondo phase at zero transverse field, there is a smooth crossover with increasing transverse field from a fully screened to a fully polarized impurity spin. By contrast, if the system starts in its localized phase, then increasing the transverse field causes a continuous, Kondo-destruction transition into the partially polarized Kondo phase. The critical exponents at this quantum phase transition exhibit hyperscaling and take essentially the same values as those of the BFKM in zero transverse field. The many-body spectrum at criticality varies continuously with the bare transverse field, indicating a line of critical points. We discuss implications of these results for the global phase diagram of the Kondo lattice model.
We study quantum phase transitions in the Bose-Fermi Kondo problem, where a local spin is coupled to independent bosonic and fermionic degrees of freedom. Applying a second-order expansion in the anomalous dimension of the Bose field, we analyze the various nontrivial fixed points of this model. We show that anisotropy in the couplings is relevant at the SU(2)\mathrm{SU}(2)SU(2)-invariant non-Fermi-liquid fixed points studied earlier, and thus the quantum phase transition is usually governed by XY\mathrm{XY}XY or Ising-type fixed points. We furthermore derive an exact result that relates the anomalous exponent of the Bose field to that of the susceptibility at any finite coupling fixed point. Implications for the dynamical mean-field approach to locally quantum critical phase transitions are also discussed.