Phase transitions in the pseudogap Anderson and Kondo models:  Critical dimensions, renormalization group, and local-moment criticality

The pseudogap Kondo problem, describing quantum impurities coupled to fermionic quasiparticles with a pseudogap density of states $\rho \left(\omega \right)\propto {\mid \omega \mid }^r$ shows a rich zero-temperature phase diagram, with different screened and free moment phases and associated transitions. We analyze both the particle-hole symmetric and asymmetric cases using renormalization group techniques. In the vicinity of $r=0$, which plays the role of a lower-critical dimension, an expansion in the Kondo coupling is appropriate. In contrast, $r=1$ is the upper-critical dimension in the absence of particle-hole symmetry, and here insight can be gained using an expansion in the hybridization strength of the Anderson model. As a by-product, we show that the particle-hole symmetric strong-coupling fixed point for $r<1$ is described by a resonant level model, and corresponds to an intermediate-coupling fixed point in the renormalization group language. Interestingly, the value $r=1∕2$ plays the role of a second lower-critical dimension in the particle-hole symmetric case, and there we can make progress by an expansion performed around a resonant level model. The different expansions allow a complete description of all critical fixed points of the models and can be used to compute a variety of properties near criticality, describing universal local-moment fluctuations at these impurity quantum phase transitions.

Figure 1

Schematic RG flow diagrams for the particle-hole symmetric single impurity Anderson model with a pseudogap DOS, $\rho \left(\omega \right)\propto {\mid \omega \mid }^r$. The horizontal axis denotes the renormalized on-site level energy $\epsilon$ (related to the on-site repulsion $u$ by $u=-2\epsilon$), the vertical axis is the renormalized hybridization $v$. The thick lines correspond to continuous boundary phase transitions; the full (open) circles are stable (unstable) fixed points, for details see text. All fixed points at nonzero $\epsilon$ have a mirror image at $-\epsilon$, related by the particle-hole transformation (3). (a) $r=0$, i.e., the familiar metallic case. For any finite $v$ the flow is towards the strong-coupling fixed point (SC), describing Kondo screening. (b) $0<r<\frac{1}{2}$: The local-moment (LM) fixed point is stable, and the transition to symmetric strong coupling (SSC) is controlled by the SCR fixed point. For $r\to 0$, SCR approaches LM, and the critical behavior at SCR is accessible via an expansion in the Kondo coupling $j$. In contrast, for $r\to \frac{1}{2}$, SCR approaches SSC, and the critical behavior can be accessed by expanding in the deviation from SCR, i.e., in $\epsilon =-u∕2$. (c) $\frac{1}{2}⩽r<1$: $v$ is still relevant at $u=0$. However, SSC is now unstable with respect to finite $u$. At finite $v$, the transition between the two stable fixed points LM and ${\mathrm{LM}}^{\prime }$ is controlled by SSC (which is now a critical fixed point). (d) $r⩾1$: $v$ is irrelevant, and the only transition is a level crossing (with perturbative corrections) occurring at $v=u=0$, i.e., at the free-impurity fixed point (FImp).

Figure 2

Schematic RG flow diagram for the maximally particle-hole asymmetric pseudogap Anderson impurity model. The horizontal axis denotes the on-site impurity energy $\epsilon$ the vertical axis is the fermionic coupling $v$, the bare on-site repulsion is fixed at $u_0=\infty$. The symbols are as in Fig. 1. (a) $r*<r<1$: $v$ is relevant, and the transition is controlled by an interacting fixed point (ACR). As $r\to r*\approx 0.375$, $p\text{−}h$ symmetry at the critical fixed point is dynamically restored, and ACR merges into the SCR fixed point of Fig. 1—this cannot be described using the RG of Sec. 5. In the metallic $r=0$ situation, studied by Haldane (Ref. 13), the flow from any point with $v\ne 0$ is towards the screened singlet fixed point with $\epsilon =\infty$. (b) $r⩾1$: $v$ is irrelevant, and the transition is a level crossing with perturbative corrections, occuring at $v=\epsilon =0$, i.e., the valence-fluctuation fixed point (VFl).

Figure 3

Inverse correlation length exponent $1∕\nu$ obtained from NRG, at both the symmetric (squares) and asymmetric (triangles) critical points, together with the analytical RG results from the expansions in $r$ [Sec. 3, Eq. (20), solid], in $(\frac{1}{2}-r)$ [Sec. 4, Eq. (43), dashed] and in $(1-r)$ [Sec. 5, Eq. (72), dash-dot]. The numerical data have been partially extracted from Ref. 7 using hyperscaling relations; for the symmetric model data are from Ref. 5.

Figure 4

Feynman diagrams for the symmetric Anderson impurity model, where the expansion is done in $U_0$ around the resonant level model fixed point. Full lines are dressed $f_{\sigma }$ propagators with the self-energy arising from the conduction electrons already taken into account, i.e., with the propagator given in Eq. (26). (a) Bare interaction vertex $U_0$. (b) Correlation function involving a composite $\left(\overline{f}f\right)$ operator (wiggly line), together with the first perturbative correction. (c) $U_0$ correction to the local susceptibility ${\chi }_{\mathrm{imp},\mathrm{imp}}={\chi }_{\mathrm{loc}}$, open circles are sources. (d) $U_0$ correction to ${\chi }_{u,u}$, which contributes to the Curie term of the impurity susceptibility ${\chi }_{\mathrm{imp}}$. Dashed lines are conduction electron lines, and the full dots are the $V_0$ vertices. (e) $U_0^2$ contribution to the impurity free energy.

Figure 5

NRG data (Ref. 7) for the local susceptibility exponent ${\eta }_{\chi }$, defined through ${\chi }_{\mathrm{loc}}\propto T^{-1+{\eta }_{\chi }}$, at both the symmetric (squares) and asymmetric (triangles) critical points, together with the renormalized perturbation theory results from the expansions in $r$ [Sec. 3 Eq. (23), solid], in $(\frac{1}{2}-r)$ [Sec. 4 Eq. (47), dashed], and in $(1-r)$ [Sec. 5 Eq. (75), dash dotted].

Figure 6

Numerical data for the impurity susceptibility $T{\chi }_{\mathrm{imp}}$ at both the symmetric (squares) and asymmetric (triangles) critical points, together with the renormalized perturbation theory results from the expansions in $r$ [Sec. 3, Eq. (21), solid], in $(\frac{1}{2}-r)$ [Sec. 4, Eq. (51), dashed], and in $(1-r)$ [Sec. 5, Eq. (85), dash-dot]. The thin line is the value of the SSC fixed point $T{\chi }_{\mathrm{imp}}=r∕8$. The numerical data are partially taken from Ref. 6; we have recalculated the data points near $r=1$, as the logarithmically slow flow at $r=1$ complicates the data analysis.

Figure 7

As Fig. 6, but for the impurity entropy $S_{\mathrm{imp}}$; the NRG values at the SSC fixed point are hardly distinguishable from $\ln 2$. The perturbative expansions are in Eq. (22) (solid), Eq. (55) (dashed), and Eq. (89) (dash-dot); the thin line is the value of the SSC fixed point $T{\chi }_{\mathrm{imp}}=2r\ln 2$.

Figure 8

Feynman diagrams for the infinite-$U$ Anderson model. Full/wiggly/dashed lines denote $f_{\sigma }∕b_s∕c_{\sigma }$ propagators, the cross is the counterterm (60). (a) Bare interaction vertex $V_0$. (b) $V_0^2$ contribution to the partition function, which also appears in the denominator of Eq. (59). (c), (d) Diagrams entering the local susceptibility ${\chi }_{\mathrm{imp},\mathrm{imp}}$ to order $V_0^2$. (e) $V_0^2$ contribution to ${\chi }_{u,\mathrm{imp}}$. (f) $V_0^2$ contributions to ${\chi }_{u,u}$.

Figure 9

NRG results for a slightly $p\text{−}h$ asymmetric Anderson model at $r=0.45$ close to criticality (ACR). Parameter values are a hybridization strength $\pi V_0^2\rho \left(0\right)=1$, ${\epsilon }_0=-0.5$, and $U_0=0.9949$, 0.99493 (dashed), 0.9949213, 0.9949215 (solid). The thin lines are for a $p\text{−}h$ symmetric model very close to criticality (SCR), with ${\epsilon }_0=-0.487$, $U_0=0.974$. The NRG runs were done using a discretization parameter of $\Lambda =9$, keeping $N_s=650$ levels. The two-stage flow described in the text can be clearly seen, i.e., there is a small energy scale $T*$ where the system flows from ACR to ASC or LM (solid: $T*\approx {10}^{-28}$, dashed: $T*\approx {10}^{-22}$), and a larger scale $T_{\mathrm{ACR}}\approx {10}^{-12}$ where the system flows from SCR to ACR. (a) Impurity susceptibility $T{\chi }_{\mathrm{imp}}$. (b) Impurity entropy $S_{\mathrm{imp}}$. The behavior of the entropy illustrates the point made in Sec. 5F3: $S_{\mathrm{imp}}\left(T\right)$ decreases as function of $T$, i.e., increases along the RG flow, for $T*<T<T_{\mathrm{ACR}}$ due to $S_{\mathrm{imp},\mathrm{ACR}}>S_{\mathrm{imp},\mathrm{SCR}}$. This “uphill flow” does not violate thermodynamic stability criteria, as the total entropy (impurity plus bath) of the system still decreases under RG.

Figure 10

Schematic $T=0$ phase diagram of the pseudogap Kondo model in a local magnetic field $B$. At $B=0$, there is a phase transition at $J_K=J_c$ between LM and a strong-coupling phase SSC or ASC. Upon application of a field, an unscreened spin becomes strongly polarized (POL), whereas a screened spin in a strong-coupling phase is only weakly polarized. Increasing the field at $J_K>J_c$ drives a phase transition—such a transition is not present in the metallic case $r=0$. The phase diagram for a pseudogap Anderson model is similar. As discussed in the text, the zero-field critical fixed point is always unstable with respect to finite $B$ in the RG sense; thus the two transitions are in general in different universality classes (which is already clear from symmetry considerations).

Figure 11

Feynman diagrams for the spin susceptibility of the noninteracting resonant level model of Sec. 4B. Dashed/full lines denote $c_{\sigma }∕\mathit{\text{dressed}}$ $f_{\sigma }$ propagators, respectively; the full dots are $V_0$ vertices, and the open circles are sources. (a) ${\chi }_{\mathrm{imp},\mathrm{imp}}$. (b) ${\chi }_{u,\mathrm{imp}}$. (c) ${\chi }_{u,u}$. Note that the displayed diagrams give the exact results for the susceptibilities, as all self-energies are contained in the dressed $f$ propagator.

Figure 12

Diagrams occurring in the perturbative RG for the noninteracting resonant level model of Sec. 4B. Here, full lines are bare $f_{\sigma }$ propagators. (a) Bare interaction vertex $V_0$. (b) $f_{\sigma }$ self-energy. (c) Diagram for the $v^2$ entropy correction. (d) Diagrams entering the local susceptibility to order $v^2$. Within the renormalized perturbation theory scheme all corrections beyond second order in $v$ vanish.

Figure 13

Diagrams for the RG of the symmetric Anderson impurity model, with notation as in Fig. 4. (a) $U_0^2$ self-energy contribution, which vanishes. (b) One-loop vertex renormalizations, which vanish due to $p\text{−}h$ symmetry. (c) Two-loop vertex renormalizations—these are the only contributions to the beta function (41).

Figure 14

Diagrams for the RG of the infinite-$U$ Anderson model. Notations are as in Fig. 8. (a), (b) Self-energies for $f_{\sigma }$ and $b_s$ to two-loop order. The crosses denotes the counterterms which cancel the real parts of the self-energy insertions. (c) Two-loop vertex renormalization; there is no contribution to one-loop order. (d) Diagrams with mass insertions (squares), needed to determine the correlation length exponent.