Kondo volume collapse, Kondo breakdown, and Fermi surface transitions in heavy-fermion metals

The unconventional critical behavior near magnetic quantum phase transitions in various heavy-fermion metals, apparently inconsistent with the standard spin-density-wave scenario, has triggered proposals on the breakdown of the Kondo effect at the critical point. Here, we investigate, within one specific scenario, the fate of such a zero-temperature transition upon coupling of the electronic to lattice degrees of freedom. We study a Kondo–Heisenberg model with volume-dependent Kondo coupling—this model displays both Kondo volume collapse and Kondo-breakdown transitions, as well as Lifshitz transitions associated with a change in the Fermi-surface topology. Within a large-$N$ treatment, we find that the Lifshitz transition tends to merge with the Kondo volume collapse and hence becomes first order, whereas the Kondo breakdown transition remains of second order except for very soft lattices. Interesting physics emerges at the zero-temperature endpoint of the Kondo volume collapse: For electrons in two space dimensions, this endpoint is located at the Lifshitz line for a large range of parameters, thus two continuous quantum phase transitions coincide without fine tuning. We analyze the effective Landau theory for such a situation and discuss critical exponents. Finally, we relate our findings to current heavy-fermion experiments.

Figure 1

Fermi surface evolution from ${\text{FL}}^{\ast }$ to FL, where shaded areas correspond to occupied states. Left: ${\text{FL}}^{\ast }$, with one spinon (dark) and one conduction electron (light) sheet. Note that the spinon band is holelike. Middle: ${\text{FL}}_2$, with two sheets, where the outer one represents heavy quasiparticles with primarily $f$ character. Right: ${\text{FL}}_1$, with one heavy-electron sheet. ${\text{FL}}_2$ and ${\text{FL}}_1$ are separated by a Lifshitz transition where the outer Fermi sheet disappears at critical value of the ratio $J_H/J_K$. The conduction band has a filling of $n_c=0.8$. (The corresponding band structures are also shown in Fig. 4 below.)

Figure 2

Large-$N$ ground state phase diagrams of Kondo-Heisenberg model (6) on a $2d$ square lattice, as function of pressure $p$ and intermoment exchange $J_H$. The panels (a)–(d) are for bulk moduli $B\text{=0}\text{.005}$, 0.0055, 0.007, and 0.015; the other parameters are $t\text{=1}$, $n_c=0.8$, $J_K=1.5$, and $\gamma =0.05$. Thin (thick) lines are second (first) order transitions. The Lifshitz transition line separates ${\text{FL}}_1$ and ${\text{FL}}_2$, the Kondo breakdown transition separates FL from ${\text{FL}}^{\ast }$, for details see text. (The calculations have been at a low $T=0.005$, the dashed lines are extrapolations of the phase boundaries obtained from runs at lower $T$.)

Figure 3

As in Fig. 2, but for electrons on a $3d$ cubic lattice, for $B=0.005$.

Figure 4

Quasiparticle dispersions for different parameters along a constant-$J_H$ cut of the phase diagram in Fig. 2, for parameters $t=1$, $n_c=0.8$, $J_H=0.1$, $J_K=1.5$, $\gamma =0.05$, $B=0.01$, and $T=0$. Upper panel: Bands along the $-\left(\pi ,\pi \right)\to \left(\pi ,\pi \right)$ direction for pressures in the (a) ${\text{FL}}_1$, (b) ${\text{FL}}_2$ (close to the Lifshitz transition), and (c) ${\text{FL}}^{\ast }$ phases ($p=0$, 0.0085, 0.1, respectively). Lower panel: Comparison of the low-energy part of the dispersions. Case (a) has one heavy electronlike band intersecting the Fermi level. Increasing pressure lowers the energy at the zone boundary and eventually causes the emergence of a holelike sheet [case (b)]. At higher pressures, the two bands evolve into conduction electron and spinon bands, which each intersect the Fermi energy once—this is the ${\text{FL}}^{\ast }$ phase [case (c)].

Figure 5

(Color online) Samples of the mean-field free energy $F\left(ϵ\right)$ [Eq. (14)], for various values of the parameter $v$. The left figure corresponds to $v>0$ and $d=2$, the right figure to $v<0$ and $d=3$. If $m$ drops below $m_{c2}$, a second minimum evolves at some $ϵ\ne 0$. If even $m<m_{c1}$, this minimum becomes the absolute minimum (see also Tables ).

Figure 6

Two representative mean-field phase diagrams in two dimensions, derived from the Landau theory. Thick or thin lines represent first- or second-order transitions. (b) If the parameters $m_{c1}$ and $m_{c2}$ coincide, the quantum critical endpoint of the lattice transition is located on the Lifshitz transition line. (a) In the other case, $m_{c1}\ne m_{c2}$, the first-order lattice transition line turns away from the Lifshitz line and ends at a critical endpoint.

Figure 7

Additional cases for mean-field phase diagrams that occur in $d=3$. Both situations the first-order lattice transition line bifurcates at a point which (a) does [(b) does not] coincide with the point where the Lifshitz transition changes from first to second order.