Magnetic impurities at quantum critical points: Large-$N$ expansion and connections to symmetry-protected topological states

In symmetry-protected topological (SPT) phases, the combination of symmetries and a bulk gap stabilizes protected modes at surfaces or at topological defects. Understanding the fate of these modes at a quantum critical point, when the protecting symmetries are on the verge of being broken, is an outstanding problem. This interplay of topology and criticality must incorporate both the bulk dynamics of critical points, often described by nontrivial conformal field theories, and SPT physics. Here, we study the simplest nontrivial setting, that of a $(0+1)$-dimensional topological mode, a quantum spin, coupled to a $(2+1)$-dimensional critical bulk. Using the large-$N$ technique we solve a series of models which, as a consequence of topology, demonstrate intermediate coupling fixed points. We compare our results to previous numerical simulations and find good agreement. We also point out intriguing connections to generalized Kondo problems and Sachdev-Ye-Kitaev (SYK) models. In particular, we show that a Luttinger theorem derived for the complex SYK models, that relates the charge density to particle-hole asymmetry, also holds in our setting. These results should help stimulate further analytical study of the interplay between SPT physics and quantum criticality.

Figure 1

(a) A spin-1 antiferromagnetic Heisenberg model. The Heisenberg couplings on horizontal bonds are $J$, while the couplings on vertical bonds are $J^{\prime }$. When $J^{\prime }/J$ is sufficiently small, the system is in a topological paramagnetic phase protected by the $\text{SO}\left(3\right)$ spin rotation symmetry and translation symmetry. As we increase $J^{\prime }/J$ up to 1, there is a transition to the Néel ordered phase. (b) A dislocation defect with a (1,1) Burgers vector, which supports a spin-$\frac{1}{2}$ topological mode in the paramagnetic phase.

Figure 2

(a) Feynman diagrams at order 1 and $1/N$ for the impurity susceptibility ${\chi }_{\mathrm{imp}}$. Single circles are bulk source terms of the form $\left({\partial }_{\tau }\varphi \right)\varphi$, double circles are bulk source terms of the form ${\varphi }^2$ which couples to $h^2$, and black solid dots are impurity source terms of the form $\gamma \gamma$ [cf. Eq. (30)]. The solid lines are propagators for ${\gamma }_{\alpha }$ with $\alpha \ge 1$ and the dashed lines are ${\gamma }_0$ propagators. $\varphi$ boson propagators are represented by double lines (solid and dashed). (b) The shaded bubble which appears in the first panel. The wavy line represents the propagator of the Lagrange multiplier field $\lambda$ and is of order $1/N$.

Figure 3

$\mathrm{sgn}\left({\tilde{\omega }}_n\right)\mathrm{sgn}\left(J_0\right)\mathrm{Im}{\mathrm{\Phi }}_{\gamma }\left(i{\tilde{\omega }}_n\right)$ as a function of $1/\sqrt{|{\tilde{\omega }}_n|}$ for 100 values of $|{\tilde{\omega }}_n|$. At large $\left|n\right|$, ${\mathrm{\Phi }}_{\gamma }\left(i{\tilde{\omega }}_n\right)$ approaches the zero-temperature form $i\mathrm{sgn}\left(J_0\right)\mathrm{sgn}\left({\tilde{\omega }}_n\right)\sqrt{2\pi }{\left|{\tilde{\omega }}_n\right|}^{-1/2}$, indicated by the red dashed line.

Figure 4

(a) Feynman diagrams relevant in the large-$N$ limit for ${\chi }_{\mathrm{b},\mathrm{b}}$, the bulk-bulk part of the impurity susceptibility to a uniform magnetic field. Single circles are source terms of the form $\left({\partial }_{\tau }\varphi \right)\varphi$ which couples linearly to $h$, and double circles are sources terms of the form ${\varphi }^2$ which couples to $h^2$ [cf. Eq. (50)]. The solid lines are Majorana fermion propagators and the dashed lines are $\varphi$ boson propagators. (b) The shaded bubble appears in the first panel. The wavy line represents the propagator of the Lagrange multiplier field $\lambda$ and is of order $1/N^2$.

Figure 5

(a) Both sides of (81) as functions of ${\theta }_{0+}$. The blue solid line represents the left-hand side. The black dashed lines represent the right-hand side which is a linear function with intercept $b_{\nu }=-4\nu +2$, plotted for a few different values of $\nu$ as indicated in the figure. (b) Solution for $B/A$ as a function of $\nu$ in the case $\mathrm{\Delta }=\frac{1}{4}$.

Figure 6

$\mathrm{sgn}\left(J_0\right)\mathrm{Im}{\mathrm{\Phi }}_{IJ}\left(\mathrm{i}{\tilde{\omega }}_n\right)$ as a function of $1/\sqrt{{\tilde{\omega }}_n}$ for 60 positive values of ${\tilde{\omega }}_n$. Blue squares and black circles represent $IJ=AA$ and $IJ=AB$, respectively. The filling fraction is estimated to be $\nu =0.3354\pm 0.0001$ by a linear fitting of $F_2\left(i{\tilde{\omega }}_n\right)/F_1\left(i{\tilde{\omega }}_n\right)$ with respect to $1/{\tilde{\omega }}_n$ using the 13 smallest positive frequency values. Red dashed lines and purple dotted dashed lines indicate the asymptotic (zero-temperature) behavior for $\nu =\frac{1}{3}$.

Figure 7

(a) ${\mathcal{C}}_{\mathrm{cp}}$ for an antisymmetric tensor impurity as a function of $\nu$ for $\nu \le \frac{1}{2}$. Error bars are indicated. Note that ${\chi }_{\mathrm{imp}}$ for the filling $\nu$ is the same as that for the filling $1-\nu$ due to the particle-hole symmetry. (b) ${\mathcal{C}}_{\mathrm{cp}}/{\mathcal{C}}_{\mathrm{free}}$ as a function of $\nu$.

Figure 8

Bilayer antiferromagnet and an impurity (marked as a red dot) symmetric under the ${\mathbb{Z}}_2$ layer interchange symmetry.