Quantum phase transitions and thermodynamics of the power-law Kondo model

We revisit the physics of a Kondo impurity coupled to a fermionic host with a diverging power-law density of states near the Fermi level, $\rho \left(\omega \right)\sim {\left|\omega \right|}^r$, with exponent $-1<r<0$. Using the analytical understanding of several fixed points, based partially on powerful mappings between models with bath exponents $r$ and $(-r)$, combined with accurate numerical renormalization group calculations, we determine thermodynamic quantities within the stable phases and also near the various quantum phase transitions. Antiferromagnetic Kondo coupling leads to strong screening with a negative zero-temperature impurity entropy, while ferromagnetic Kondo coupling can induce a stable fractional spin moment. We formulate the quantum field theories for all critical fixed points of the problem, which are fermionic in nature and allow for a perturbative renormalization-group treatment.

Figure 1

Schematic RG flows for the power-law Kondo model with $r<0$, as deduced from NRG calculations. The horizontal axis denotes the (running) Kondo coupling $j$ while the vertical axis denotes the (running) potential scattering $v$. (a) ${\overline{r}}^{*}<r<0$, (b) ${\overline{r}}_{\max }<r<{\overline{r}}^{*}$, (c) $-1<r<{\overline{r}}_{\max }$, with NRG estimates of ${\overline{r}}^{*}=-0.245\pm 0.005$, ${\overline{r}}_{\max }=-0.264\pm 0.001$. The solid dots denote infrared stable fixed points; the open dots are critical fixed points (labeled as in the text). FS denotes the infrared stable intermediate-coupling fixed point with fractional spin.

Figure 2

NRG results for the Kondo temperature, defined by $S_{\mathrm{imp}}\left(T_{\mathrm{K}}\right)=\left(\ln 2\right)/2$, as a function of the Kondo coupling $J$ for a pure power-law bath DOS with different exponents $r=0,-0.1,-0.2,...,-0.9$ decreasing in the direction of the arrow. The dashed lines show the weak-coupling power law of Eq. (13) [dotted line is the standard result, Eq. (12), for the metallic case $r=0$].

Figure 3

Impurity susceptibility $T{\chi }_{\mathrm{imp}}\left(T\right)$ vs $T/D$ across the QPT separating SSC and ALM phases, for antiferromagnetic Kondo coupling $J=0.01$ and $r=-0.6$. The AF-CR critical point (red dotted line) is attained at $V_c\simeq 0.01025...$. The transition is approached progressively from the SSC phase (solid lines, $V<V_c$) or the ALM phase (dashed lines, $V>V_c$) with $V=V_c\pm \delta V$ and $\delta V={10}^{-3},{10}^{-4},{10}^{-5},{10}^{-6},{10}^{-7}$. For comparison, circles are for the direct crossover with $J=0.01$ and $V=0$, while diamonds are for $J=0$ but $V=0.01$. Arrows indicate the fixed point values.

Figure 4

Exact phase boundary between SSC and ALM phases of the Kondo model with pure power-law DOS Eq. (2) and antiferromagnetic Kondo coupling, obtained using NRG.

Figure 5

Exact phase boundaries between TSC and ALM phases of the Kondo model with pure power-law DOS Eq. (2) and ferromagnetic Kondo coupling, obtained using NRG. Shown for bath exponents $r=-0.1$, $-0.25$, and $-0.4$ (black stars, red crosses, and blue plusses, respectively), characteristic of the three distinct regimes. Open circles represent the SCR fixed point [at $(J,V)\simeq (-15.0,0)$ and $(-2.8,0)$ for $r=-0.1$ and $-0.25$], filled circles correspond to the FS fixed point [at $(J,V)\simeq (-0.2,0)$ and $(-1,0)$ for $r=-0.1$ and $-0.25$], and squares are for the ACR fixed point [at $(J,V)\simeq (-3.5,0.3)$ and $(-6,1.35)$ for $r=-0.25$ and $-0.4$].

Figure 6

$T{\chi }_{\mathrm{imp}}\left(T\right)$ vs $T/D$ as in Fig. 3, but now across the QPT separating TSC and ALM phases, for ferromagnetic Kondo coupling $J=-0.01$ and $r=-0.5$. The ACR critical point (red dotted line) is attained at $V_c\simeq 0.001106...$. The transition is approached progressively from the TSC phase (solid lines, $V<V_c$) or the ALM phase (dashed lines, $V>V_c$) with $V=V_c\pm \delta V$ and $\delta V={10}^{-4},{10}^{-5},{10}^{-6},{10}^{-7}$. For comparison, circles are for the direct crossover with $J=-0.01$ and $V=0$, while diamonds are for $J=0$ but $V=0.01$.

Figure 7

Chain representation of the trivial fixed points of the power-law Kondo model. Circles denote fermion sites; the dashed line separates the bath sites (right) from the impurity (left). Full (open) circles denote sites with (without) p-h symmetry; the ellipses indicate strong-coupling singlet or triplet states. For details see text.

Figure 8

Universal scaling curves for impurity entropy $S_{\mathrm{imp}}\left(T\right)$ and impurity susceptibility $T{\chi }_{\mathrm{imp}}\left(T\right)$ in the case of antiferromagnetic Kondo coupling $J>0$ (but $V=0$) for various bath exponents $r=0,-0.1,-0.2,...,-0.9$ (following the arrow). The temperature is plotted as $T/T_{\mathrm{K}}$, with the Kondo temperature $T_{\mathrm{K}}$ defined via $S_{\mathrm{imp}}\left(T_{\mathrm{K}}\right)=\left(\ln 2\right)/2$. In the limit $T\to 0$, we have $S_{\mathrm{imp}}=2r\ln 2$ and $T{\chi }_{\mathrm{imp}}=r/8$ at the SSC fixed point.

Figure 9

Universal $T\to 0$ values of (a) impurity magnetic moment $T{\chi }_{\mathrm{imp}}$ and (b) impurity entropy $S_{\mathrm{imp}}$ as function of the bath exponent $r$ for the fractional-spin fixed point (FS) and the symmetric critical fixed point (SCR), obtained from NRG. Both FS and SCR exist for ${\overline{r}}_{\max }<r<0$ with ${\overline{r}}_{\max }=-0.0264\pm 0.001$. The dashed lines show the perturbative results in Eqs. (19) and (20) for FS; while dotted lines are Eqs. (21) and (22) for SCR. (c) Critical coupling ${\rho }_0{J}_c$, for the TSC–FS transition controlled by SCR and renormalized intermediate coupling at the FS fixed point ${\rho }_0{J}^{*}$, determined from a stationarity condition of the initial NRG flow.

Figure 10

Schematic RG flow for the pseudogap Kondo model with DOS exponent $r^{\prime }>0$ in the regimes (a) $0<r^{\prime }<r^{*}$, (b) $r^{*}<r^{\prime }<r_{\max }$, (c) $r^{\prime }>r_{\max }$, with $r^{*}=0.375\pm 0.002$ and $r_{\max }=\frac{1}{2}$. As in Fig. 1, the axes represent the renormalized Kondo coupling $j$ and potential scattering strength $v$. For details of the fixed points, see text.

Figure 11

Universal $T\to 0$ values of (a) impurity magnetic moment $T{\chi }_{\mathrm{imp}}$ and (b) impurity entropy $S_{\mathrm{imp}}$ as a function of the bath exponent $r$ for the critical fixed points ACR and AF-CR, obtained using NRG (points and solid lines). The dotted lines show the perturbative results for ACR, Eqs. (29) and (30); while the dashed and dot-dashed lines are for AF-CR, Eqs. (24), (25), (27), and (28). The properties of AF-CR display a pronounced change at $r\approx -0.4$, in agreement with the prediction that AF-CR is equivalent to SCR${}^{\prime }$ for $r^{\prime }=-r<r^{*}$, but to ACR${}^{\prime }$ for $r^{\prime }=-r>r^{*}$ (with $r^{*}=0.375$).