Two-channel pseudogap Kondo and Anderson models: Quantum phase transitions and non-Fermi liquids

We discuss the two-channel Kondo problem with a pseudogap density of states $\rho \left(\omega \right)\propto {\left|\omega \right|}^r$ 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 $r=0$ case, is replaced by a stable particle-hole-symmetric intermediate-coupling fixed point for $0<r<r_{\max }\approx 0.23$. For $r>r_{\max }$, 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 $\epsilon$ expansion techniques to calculate critical properties near the critical dimensions $r=0$ 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.

Figure 1

Schematic RG flow diagrams for the two-channel Kondo model with a pseudogap DOS $\rho \left(\omega \right)\propto {\left|\omega \right|}^r$. The horizontal axis denotes the renormalized Kondo coupling $j$, and the vertical axis is the renormalized potential scattering $v$, representing particle-hole asymmetry. Dashed flow lines symbolize a flow out of the plane shown here. The thick lines correspond to continuous boundary phase transitions; the full (open) circles are stable (unstable) fixed points. (a) $r=0$, i.e., the familiar metallic case. For any finite $j$ the flow is toward the line of NFL fixed points, describing two-channel non-Fermi-liquid behavior. (b) $0<r<r_{\max }$: p-h asymmetry is irrelevant in the NFL phase, such that the line of fixed points is replaced by a single p-h-symmetric NFL fixed point. The local-moment fixed point LM is stable and separated from NFL by a p-h-symmetric critical (SCR) fixed point. For $r\to 0$ ($r\to r_{\max })$, SCR approaches LM (NFL). Depending on microscopic details, a second NFL${}^{\prime }$ phase may be reached at large couplings and asymmetries, separated by a critical ACR fixed point; see text. (c) $r\ge r_{\max }$: The NFL phase disappears, and the only phase transition is between LM and LM${}^{\prime }$, the latter representing a free-channel (i.e., flavor) moment. This transition is controlled by ACR. Note that the character of this transition changes at $r=1$, where ACR merges with FImp; see text.

Figure 2

As Fig. 1, but for the two-channel Anderson model. The horizontal axis denotes the energy difference $\varepsilon$ between spin and flavor impurity levels, and the vertical axis is the renormalized hybridization $g$. The diagrams represent cuts, taken at $v=0$, through the full RG flow. The flow diagrams are mirror symmetric by virtue of the $Z_2$ transformation (4). (a) $r=0$. The line of NFL fixed points, describing two-channel non-Fermi-liquid behavior crosses the $v=0$ plane at $\varepsilon =0$. (b) $0<r<r_{\max }$: The fixed points LM, LM${}^{\prime }$ are stable, corresponding to unscreened spin and flavor moments, respectively. Within the $v=0$ plane, NFL is replaced by ACR, whereas two isolated p-h-symmetric NFL and NFL${}^{\prime }$ fixed points exist outside this plane. The transitions to NFL and NFL${}^{\prime }$ are controlled by the p-h-symmetric SCR and SCR${}^{\prime }$ fixed points (located outside the $v=0$ plane). (c) $r_{\max }\le r<1$: $g$ is still relevant at $\varepsilon =0$. However, NFL is now replaced by a single unstable fixed point (ACR) located in the $v=0$ plane. At finite $g$, the transition between the two stable fixed points LM and LM${}^{\prime }$ is controlled by ACR. (d) $r\ge 1$: $g$ is irrelevant, and the only transition is a level crossing (with perturbative corrections) occurring at $g=\varepsilon =0$, i.e., at the free-impurity fixed point (FImp).

Figure 3

NRG results for the impurity susceptibility $T{\chi }_{\mathrm{imp}}$ at the intermediate-coupling fixed points ACR ($red\square$), SCR ($green▵$), and NFL ($blue\diamond$). Also shown are the results from the renormalized perturbation theory of the Kondo model, which allows us to access SCR near $r=0$ [Sec. 4, Eq. (14), dashed], and those from the Anderson model, appropriate for ACR for $r\lesssim 1$ [Sec. 5, Eq. (49), dash-dotted]. The SCR data are partially taken from Ref. 19. The vertical dashed lines indicate the critical dimensions $r=r_{\max }$ and $r=1$.

Figure 4

As Fig. 3, but for the impurity entropy $S_{\mathrm{imp}}$. The perturbative expansions are in Eq. (15) (dashed) and Eq. (53) (dash-dotted). The dotted horizontal lines correspond to $S_{\mathrm{imp}}=0.5\ln 2$, $\ln 2$, and $\ln 4$. The inset shows a zoom onto the region $r\lesssim r_{\max }$.

Figure 5

Phase diagram of the two-channel Anderson model as a function of the energy difference ${\varepsilon }_0$ and the bath exponent $r$, keeping $g_0^2=4$ fixed. The discontinuous change at $r=r_{\max }$ is apparent.

Figure 6

Inverse correlation-length exponent $1/\nu$ obtained from NRG calculations for the ACR ($red\square$) and SCR ($green▵$) critical points, together with the analytical RG results from the expansions in $r$ [Sec. 4, Eq. (12), dashed] and in $(1-r)$ [Sec. 5, Eq. (38), dash-dotted]. Note that $\nu$ of ACR becomes very large for small $r$—this corresponds to the extremely slow flow from ACR to NFL for $0<r<r_{\max }$.

Figure 7

NRG results for (a) the impurity susceptibility $T{\chi }_{\mathrm{imp}}$ and (b) the impurity entropy $S_{\mathrm{imp}}$ of the two-channel Anderson model at $r=0.22$. Parameter values are $g_0^2=2$ and ${\varepsilon }_0=0$ (dashed), 0.5 (solid), 1 (long-dashed), and 2 (dash-dotted). Both ${\varepsilon }_0=0.5$ and 1 are located in the NFL phase while ${\varepsilon }_0=2$ corresponds to the LM phase. The ${\varepsilon }_0=0.5$ curves show the flow from ACR to NFL upon lowering $T$, where in particular the impurity entropy increases along the RG flow. (The flow is too slow to reach its fixed point within the accessible temperature range.)

Figure 8

Feynman diagrams entering the self-energies up to quadratic order in $g_0$. Full, dashed, and wiggly lines denote $f_{\sigma }$, $c_{\sigma }$, and $b_{\overline{\alpha }}$ propagators, respectively; the full dots are $g_0$ interaction vertices.

Figure 9

Feynman diagrams entering ${\langle {\chi }_{\mathrm{loc}}\rangle }_{{\lambda }_0}$ up to quadratic order in $g_0$. Open circles are sources, and blank boxes denotes the counterterms $\delta {\lambda }_f$. Notation otherwise as in Fig. 8

Figure 10

Feynman diagrams entering the corrections to the unperturbed part of ${\langle \widehat{Q}\rangle }_{{\lambda }_0}$ up to quadratic order in $g_0$. Notation as in the previous figures. Black boxes denote the counterterms $\delta {\lambda }_b$.

Figure 11

Further Feynman diagrams entering corrections to the impurity susceptibility to quadratic order in $g_0$. Notation as in the previous figures.