Common non-Fermi liquid phases in quantum impurity physics

We study correlated quantum impurity models that undergo a local quantum phase transition (QPT) from a strong coupling, Fermi liquid phase to a non-Fermi liquid phase with a globally doubly degenerate ground state. Our aim is to establish what can be shown exactly about such “local moment” (LM) phases, of which the permanent (zero-field) local magnetization is a hallmark, and an order parameter for the QPT. A description of the zero-field LM phase is shown to require two distinct self-energies, which reflect the broken symmetry nature of the phase and together determine the single self-energy of standard field theory. Distinct Friedel sum rules for each phase are obtained, via a Luttinger theorem embodied in the vanishing of appropriate Luttinger integrals. By contrast, the standard Luttinger integral is nonzero in the LM phase but found to have universal magnitude. A range of spin susceptibilites are also considered, including that corresponding to the local order parameter, whose exact form is shown to be RPA-like, and to diverge as the QPT is approached. Particular attention is given to the pseudogap Anderson model, including the basic physical picture of the transition, the low-energy behavior of single-particle dynamics, the quantum critical point itself, and the rather subtle effect of an applied local field. A two-level impurity model that undergoes a QPT (“singlet-triplet”) to an underscreened LM phase is also considered, for which we derive on general grounds some key results for the zero-bias conductance in both phases.

Figure 1

A zero-field QPT ($T=0$) occurs at a critical interaction $U_c$. It separates an FL (or strong coupling) phase for $U<U_c$, which is adiabatically connected to the noninteracting limit, from an NFL, local moment phase for $U>U_c$. Solid lines denote low-energy scales characteristic of each phase, which vanish as the QPT is approached and set the scale for crossover to quantum critical behavior at finite $T$ or field.

Figure 2

(Top) Schematic of $T=0$ local magnetization $m$ in a LM phase: $m\left(h\right)=-m(-h)$ vs. $h$ for fixed $U>U_c$. As $h\to 0\pm$, $m\left(h\right)\to \pm |\tilde{\mu }|$, with $|\tilde{\mu }|$ the magnitude of the permanent local moment. (Bottom) Qualitative behavior of $m$ as a function of $U$ and $h$, with the critical $U_c$ indicated.

Figure 3

Evolution of the zero field, permanent local moment $|\tilde{\mu }|$ [$=m_A\left(0\right)=m(0+)$, Eqs. (15) and (16)], and the excess magnetization $m_{\mathrm{imp},A}\left(0\right)$ [$=m_{\mathrm{imp}}(0+$)], on crossing the transition at $U=U_c$ to the LM phase. $|\tilde{\mu }|$ vanishes continuously as $U\to U_c+$, while $m_{\mathrm{imp},A}\left(0\right)$ generically changes discontinuously to full saturation on crossing the transition. ($|\tilde{\mu }|$ shown here has been obtained from NRG for $r=0.4$, see text.)

Figure 4

Phase diagram for the PAIM with $r=0.4$, determined numerically via the FDM-NRG [49, 51, 52]. Shown in the $(U,x)$ plane, with $x=\epsilon +\frac{U}{2}$ (and ${\left[{\Gamma }_0\right]}^{1/(1-r)}\equiv 1$ as the unit of energy). SC and LM phases are separated by a (solid) line of quantum critical points (see Sec. 7). Characteristic $n_{\mathrm{imp}}\left(0\right)$'s for the SC and LM phases (Secs. 5a and 5b) are indicated; the dotted line merely shows the p-h symmetric line ($x=0$) for the symmetric SC phase, on which $n_{\mathrm{imp}}\left(0\right)=1$.

Figure 5

For $r=0.45$, FDM-NRG determined scaling spectra shown as $|\tilde{\omega }{|}^r{\Psi }_{\alpha }^I\left(\tilde{\omega }\right)\equiv \pi {\Gamma }_0{\left|\omega \right|}^{r}D\left(\omega \right)$ vs $\tilde{\omega }=\omega /{\omega }_{*}$ on a logarithmic scale. (Top) Asymmetric SC phase (with $x=\epsilon +\frac{U}{2}<0$), for both $|\tilde{\omega }|<0$ (solid line) and $|\tilde{\omega }|>0$ (dashed line) and symmetric SC phase ($x=0$, dotted line). (Bottom) Corresponding LM phase results for both asymmetric and symmetric cases. For the asymmetric cases, results shown are independent of $\left|x\right|$. For both SC and LM phases, coefficients of proportionality in ${\omega }_{*}$ have been (freely) chosen such that spectral maxima occur at $|\tilde{\omega }|=1$. The coefficients $C_s\left(r\right)$ and $C_a\left(r\right)$ relevant to the QCP as discussed in Sec. 7a, are indicated by arrows.

Figure 6

$r$ dependence of coefficients $C\equiv C\left(r\right)$ [Eq. (89)] describing the QCP spectrum, $\pi {\Gamma }_{0}D\left(\omega \right)=C{\left|\omega \right|}^{-r}$. Calculated from FDM-NRG, showing both $C\equiv C_s\left(r\right)$ characteristic of the symmetric QCP for $r\le \frac{1}{2}$ (points, with solid line as guide to eye) and $C\equiv C_a\left(r\right)$ (points with dashed line) characteristic of the asymmetric QCP for $r>r^{*}\simeq 0.375$. As $r\to 0$, the asymptotic behavior is found to be $C_s\left(r\right)=3{\pi }^2{r}^2/16$ (see inset, where dotted line shows $3{\pi }^2{r}^2/16$), while $C_s\left(r\right)=\frac{1}{2}$ as $r\to \frac{1}{2}$, as explained in text.

Figure 7

(Top) Schematic in the $(U,h)$-plane for the asymmetric PAIM ($x=\epsilon +\frac{U}{2}\ne 0$), and a typical $r\in (0,1)$. Dashed line shows points for which ${\epsilon }_{\sigma }^{*}\left(h\right)=0$ with ${\epsilon }_{-\sigma }^{*}\left(h\right)\ne 0$ (terminating at $h=\left|\epsilon \right|$ in the noninteracting limit). The zero-field QCP at $U=U_c\left(r\right)$ is indicated, and is the only local QCP in the $(U,h)$ plane (see text). (Bottom) Renormalized levels ${\epsilon }_{\sigma }^{*}\left(h\right)$ vs $U$, for $h=0.1$ (squares, with solid line as guide to eye) and $h=0$ (circles and dashed line); shown for $r=0.35$ and fixed $x=\epsilon +\frac{U}{2}=-1$, and calculated from FDM-NRG. For $h=0.1$, ${\epsilon }_{\downarrow }^{*}\left(h\right)$ changes sign at $U\simeq 4.1$ [$<U_c\left(r\right)\simeq 7.8$], while ${\epsilon }_{\uparrow }^{*}\left(h\right)<0$ remains. ${\left[{\Gamma }_0\right]}^{1/(1-r)}\equiv 1$ is taken as the energy unit.