Boundary quantum criticality in models of magnetic impurities coupled to bosonic baths

We investigate quantum impurity problems, where a local magnetic moment is coupled to the spin density of a bosonic environment, leading to bosonic versions of the standard Kondo and Anderson impurity models. In a physical situation, these bosonic environments can correspond either to deconfined spinons in certain classes of $Z_2$ frustrated antiferromagnets, or to particles in a multicomponent Bose gase (in which case the spin degree of freedom is attributed to hyperfine levels). Using renormalization group techniques, we establish that our impurity models, which feature an exchange interaction analogous to Kondo impurities in Fermi liquids, allow flow toward a stable strong-coupling state. Since the low-energy bosons exist around a single point in momentum space, and there is no Fermi surface, an impurity quantum phase transition occurs at intermediate coupling, separating screened and unscreened phases. This behavior is qualitatively different from previously studied spin-isotropic variants of the spin-boson model, which display stable intermediate-coupling fixed points and no screening.

Figure 1

(Color online). RG flow of the spinon Kondo model (16) at the bulk critical point. The axes denote the magnetic (Kondo) coupling $j$ and the potential-scattering term $v$. (a) $ϵ>0$; (b) $ϵ<0$. ($ϵ=d-2$ for noninteracting spinons; note that $ϵ$ may be nonzero in $d=2$ owing to bulk spinon interactions.) QCP represents the boundary quantum critical point, and the bold line is the separatrix, separating a strongly coupled phase from either a local-moment (LM) regime or a potential scattering (PS) at zero Kondo coupling. The curves have been calculated from (18) for $ϵ=\pm 0.1$. Note that the flow is symmetric with respect to $j\leftrightarrow -j$.

Figure 2

(Color online) RG flow of the Kondo model with massless canonical bosons, Eq. (25). (a) $ϵ>0$; (b) $ϵ<0$. ($ϵ=d-2$ for noninteracting bosons; again, $ϵ$ may be nonzero in $d=2$ due to bulk interactions.) The curves have been calculated from (27) for $ϵ=\pm 0.1$.

Figure 3

Graphical representation of all the occurring vertices; $\alpha$ is denoted as a black circle, its counterterm is represented by a big black circle in a box; the small gray dot and the big gray dot with a cross represent $\beta$ and its counterterm; the small (big) gray box (gray box with a cross) represents $\gamma$ (the counterterm of $\gamma$).

Figure 4

Graphical representation of the renormalization condition shown in Eq. (A7) determining the one-loop counterterm ${(Z_{\alpha }-1)}^{\left(1\right)}$ (big black dot in box).

Figure 5

Graphical representation of the renormalization condition shown in Eq. (A7) determining the one-loop counterterm ${(Z_{\beta }-1)}^{\left(1\right)}$ (big gray dot with a cross).

Figure 6

Graphical representation of the renormalization condition shown in Eq. (A7) determining the one-loop counterterm ${(Z_{\gamma }-1)}^{\left(1\right)}$ (big gray box with a cross).

Figure 7

Graphical recursion relation for the two-loop contribution to the renormalization factor ${(Z_{\alpha }-1)}^{\left(2\right)}$ (last term). Note that the counterterms appearing as internal vertices in the diagrams take their one-loop values ${(Z_{\alpha }-1)}^{\left(1\right)}\alpha$, ${(Z_{\beta }-1)}^{\left(1\right)}\beta$, ${(Z_{\gamma }-1)}^{\left(1\right)}\gamma$. (a)–(k) denote topologically distinct diagrams.

Figure 8

Graphical recursion relation for the two-loop contribution to the renormalization factor ${(Z_{\beta }-1)}^{\left(2\right)}$ (last term). Note that the counterterms appearing as internal vertices in the diagrams take their one-loop values ${(Z_{\alpha }-1)}^{\left(1\right)}\alpha$, ${(Z_{\beta }-1)}^{\left(1\right)}\beta$, ${(Z_{\gamma }-1)}^{\left(1\right)}\gamma$.

Figure 9

Recursion relation for the impurity field renormalization factor $(Z_b-1)$ (crossed dot).

Figure 10

(a) Interaction vertex between bulk and impurity in the bosonic Anderson model; the wiggly line denotes the particle $b_0$, the dashed line is $b_{\sigma }$, and the full line stands for $z_{\sigma }$ (incoming arrow) or $z_{\sigma }^{*}$ (outgoing arrow). (b) and (c) denote self-energy corrections to the local levels $b_{\sigma }$ (b) and $b_0$ (c).