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arXiv:cond-mat/0204118 · Physical review. B, Condensed matter

Quantum phase transitions in the Bose-Fermi Kondo model

Gergely Zaránd, Eugene Demler

Published Jul 19, 2002• 102 citations
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Abstract

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 $\mathrm{SU}(2)$-invariant non-Fermi-liquid fixed points studied earlier, and thus the quantum phase transition is usually governed by $\mathrm{XY}$ 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.

DOI
10.1103/physrevb.66.024427

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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.
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In quantum critical heavy fermion systems, local moments are coupled to both collective spin fluctuations and conduction electrons. As such, the Bose-Fermi Kondo model, describing the coupling of a local moment to both a bosonic and a fermionic bath, has been of extensive interest. For the model in the presence of SU(2) spin rotational symmetry, questions have been raised about its phase diagram. Here we develop a version of continuous-time Quantum Monte Carlo (CT-QMC) method suitable for addressing this issue; this procedure can reach sufficiently low temperatures while preserving the SU(2) symmetry. Using this method for the Bose-Fermi Anderson model, we clarify the renormalization-group fixed points and the phase diagram for the case with a constant fermionic-bath density of states and a power-law bosonic-bath spectral function ρb(ω)∝ωs\rho_{b}(\omega) \propto \omega^{s}ρb​(ω)∝ωs (0<s<10<s<10<s<1). We find two types of Kondo destruction QCP, depending on the power-law exponent sss in the bosonic bath spectrum. For s∗<s<1s^{*}<s<1s∗<s<1, both types of QCPs exist and, in the parameter regime accessible by an analytical ϵ\epsilonϵ-expansion renormalization-group calculation (here ϵ=1−s\epsilon=1-sϵ=1−s), the CT-QMC result is fully consistent with prior predictions by the latter method. For s<s∗s<s^{*}s<s∗, there is only one type of QCP. At both type of Kondo destruction QCPs, we find that the exponent of the local spin susceptibility η\etaη obeys the relation η=ϵ\eta=\epsilonη=ϵ, which has important implications for Kondo destruction QCP in the Kondo lattice problem.
We study the Kondo-Heisenberg model on the honeycomb lattice at half filling. Due to the vanishing of the density of states at the fermi level, the Kondo insulator disappears at a finite Kondo coupling even in the absence of the Heisenberg exchange. We adopt a large-NNN formulation of this model and use the renormalization group machinery to study systematically the second order phase transition of the Kondo insulator (KI) to the algebraic spin liquid (ASL). We note that neither phase breaks any physical symmetry, so that the transition is not described by the standard Ginzburg-Landau-Wilson critical point. We find a stable Lorentz-invariant fixed point that controls this second order phase transition. We calculate the exponent $\ensuremath{\nu}$ of the diverging length scale near the transition. The quasiparticle weight of the conduction electron vanishes at this KI-ASL fixed point, indicating non-Fermi-liquid behavior. The algebraic decay exponent of the staggered spin correlation is calculated at the fixed point and in the ASL phase. We find a jump in this exponent at the transition point.
We discuss the two-channel Kondo problem with a pseudogap density of states $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{|\ensuremath{\omega}|}^{r}$ of the bath fermions. Combining both analytical and numerical renormalization group techniques, we characterize the impurity phases and quantum phase transitions of the relevant Kondo and Anderson models. The line of stable points, corresponding to the overscreened non-Fermi-liquid behavior of the metallic r=0r=0r=0 case, is replaced by a stable particle-hole-symmetric intermediate-coupling fixed point for $0<r<{r}_{\mathrm{max}}\ensuremath{\approx}0.23$. For r>rmaxr>{r}_{\mathrm{max}}r>rmax​, this non-Fermi-liquid phase disappears, and instead a critical fixed point with an emergent spin-channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively. We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ $\ensuremath{\epsilon}$ expansion techniques to calculate critical properties near the critical dimensions r=0r=0r=0 and 1, the latter being potentially relevant for two-channel Kondo impurities in neutral graphene. We find the analytical results to be in excellent agreement with those obtained from applying Wilson's numerical renormalization group technique.