Universal crossovers and critical dynamics of quantum phase transitions: A renormalization group study of the pseudogap Kondo problem

The pseudogap Kondo problem, describing a magnetic impurity embedded in an electronic environment with a power-law density of states, displays continuous quantum phase transitions between free and screened moment phases. In this paper we employ renormalization group techniques to analytically calculate universal crossover functions, associated to these transitions, for various observables. Quantitative agreement with the results of Numerical Renormalization Group (NRG) simulations is obtained for temperature-dependent static and zero-temperature dynamic quantities, at and away from criticality. In the notoriously difficult realm of finite-temperature low-frequency dynamics, usually inaccessible to both NRG and perturbative methods, we show that progress can be made by a suitable renormalization procedure in the framework of the Callan-Symanzik equations. Our general strategy can be extended to other zero-temperature phase transitions, both in quantum impurity models and bulk systems.

Figure 1

(Color online) Generic flow diagrams for (a) the Kondo model (with $r^{\star }<r<1∕2$) and (b) the particle-hole symmetric Anderson model (with $r<1∕2$). In both cases, the universal crossovers (depicted as colored/different grayscale lines) will be computed. See the main text for a description of the nomenclature used in this figure.

Figure 2

(Color online) Low-temperature limit of $T{\chi }_{\mathrm{imp}}$ (continuous upper line) and $T{\chi }_{\mathrm{loc}}$ (dashed lower line) as a function of the Kondo coupling $J$.

Figure 3

Diagrammatic expansion for the impurity susceptibility of the Kondo model, with the tree level contribution (of ${\chi }_{\mathrm{imp},\mathrm{imp}}$ type) plus the order $j$ perturbative correction (of ${\chi }_{\mathrm{u},\mathrm{imp}}$ type). Here full lines denote the Abrikosov fermion propagator, while dashed lines represent the bulk fermions. Each vertex, labeled by a dot, yields the value $J=N_0^{-1}{Z}_j{Z}_f^{-1}{\mu }^{-r}j$.

Figure 4

Impurity susceptibility $T{\chi }_{\mathrm{imp}}\left(T\right)$ at $r=0.1$ given in Eq. (35) (solid), capturing the crossover between SCR at high temperature and LM at low temperature. Also shown are NRG data (dots) for $T{\chi }_{\mathrm{imp}}$, obtained for a pseudogap Anderson model (Ref. 33) with $U∕D=1$, $N_0{V}^2∕D=0.05$. The crossover scale $T^{*}$ has been fitted: $T^{\star }∕D\simeq 1.7\times {10}^{-23}$. [The reason for the very slow crossover from SCR to LM is the large value of the correlation length exponent, $\nu =1∕r+O\left(1\right)\approx 10$.] Note also that deviations from scaling behavior occur when temperature is comparable to the bandwidth $D$, as seen by a kink in the last data point.

Figure 5

Impurity susceptibility $T{\chi }_{\mathrm{imp}}\left(T\right)$, Eq. (35), now for $r=0.3$, together with NRG data for $U∕D=1$, $N_0{V}^2∕D=0.25$, resulting in $T^{\star }=2\times {10}^{-16}$. As discussed in the text, the agreement between the analytical curve and the NRG data is better at low temperature, where the running $j\left(\lambda \right)$ has flown towards weaker coupling.

Figure 6

Diagrams contributing to the local susceptibility ${\chi }_{\mathrm{loc}}$ of the Kondo model up to order $j^2$, with conventions given in Fig. 3. For the second and thir graphs above, the self-energy-like contributions require a mass subtraction.

Figure 7

Rescaled static local susceptibility $T{\chi }_{\mathrm{loc}}\left(T\right)$ for $r=0.3$ according to formula (46). Although not performed here, a comparison to NRG data could be possible.

Figure 8

Rescaled dynamic local susceptibility $\omega {\chi }_{\mathrm{loc}}^{″}(\omega ,T=0)$ at zero temperature for $r=0.1$, with the same parameters as in Fig. 4. NRG data points are the dots, and the solid line follows formula (56). Since the crossover scale $T^{\star }$ was already determined from ${\chi }_{\mathrm{imp}}\left(T\right)$, the only fitting parameter here is the nonuniversal amplitude ${\mathcal{C}}_2$.

Figure 9

Diagrammatic expansion of the T-matrix of the Kondo model up to next-to-leading order $\left(J^3\right)$, with conventions given in Fig. 3.

Figure 10

T-matrix $-{\mathcal{T}}^{″}(\omega ,T=0)$ at zero temperature for the same parameters as in Fig. 4. NRG data point are the dots, and the solid line follows formula (71), where the nonuniversal amplitude $J^2{N}_0{\mu }^{2r}$ has been fitted to match the numerics. Note that the high-frequency peak at $\omega \sim U∕2$ in the numerical result corresponds to charge fluctuations (Hubbard bands) of the Anderson model employed in the NRG simulations, and cannot be accounted for in our calculation based on the Kondo model. The inset shows the same curve on a double-logarithmic scale.

Figure 11

(Color online) T-matrix $-{\mathcal{T}}^{″}(\omega ,T,j)$ for $r=0.3$ at finite temperature from the formula (79) obtained via the analytic RG. Different curves correspond to three values of $j<r$, associated from top to bottom to $T^{\star }∕T=0,1,100$ (the topmost curve corresponds thus to the quantum critical point). Note that no NRG data in the regime $\omega <T$ is available to date.

Figure 12

(Color online) $T{\chi }_{\mathrm{imp}}\left(T\right)$ at $r=0.47$ capturing the crossover between SCR at high temperature and SSC at low temperature. The NRG data (dots) are obtained for $U∕D=1$, $V∕D=0.845$, while the analytical curves are plotted according to the perturbative result, respectively, the full one-loop result (88) for the upper curve, and the strict $O\left(\sqrt{ϵ}\right)$ result (89) for the lower curve (see text). The extracted crossover scale is $T^{\star }∕D\simeq 1\times {10}^{-21}$. Note that the agreement is as always better at low temperature where the calculation is perturbatively controlled.

Figure 13

Diagrammatic contribution to the local susceptibility of the Anderson model up to order $u$, where full lines denote the propagator of the hybridized $d$ level, and squares the $U={\mu }^{-ϵ}{A}_0^{2}u$ vertex.

Figure 14

Imaginary part of the dynamic local susceptibility ${\chi }_{loc}^{″}(\omega ,T=0)$ at zero temperature for $r=0.47$. The analytical result is in Eq. (99).

Figure 15

Self-energy of the Anderson model at order $u^2$.

Figure 16

Rescaled T-matrix $-{\omega }^r{\mathcal{T}}^{″}(\omega ,T=0)$, Eq. (113), at zero temperature for the same parameters as in Fig. 12, showing the crossover on the amplitude of the common power law ${\omega }^{-r}$. The ringing at low frequency in the NRG data comes from numerical errors in the calculated ratio related to the broadening procedure.

Figure 17

(Color online) Impurity susceptibility $T{\chi }_{\mathrm{imp}}\left(T\right)$ for $r=0.4$. The upper curve is obtained from the Kondo expansion result Eq. (35), while the lower curve originates from the Anderson one, Eq. (89). The NRG data (dots) are taken on both sides of the quantum critical point, illustrating flow towards the local moment and the resonant level fixed points, respectively.

Figure 18

T-matrix in diagrammatic form: The double solid line is the full impurity local propagator of the Abrikosov fermions $f_{\sigma }$, the double dashed line represents the full local propagator of the conduction electrons $c_{\sigma }$, and the filled square denotes the full vertex function. The second graph vanishes in absence of a magnetic field.