Theory of quantum impurities in spin liquids

We describe spin correlations in the vicinity of a generalized impurity in a wide class of fractionalized spin liquid states. We argue that the primary characterization of the impurity is its electric charge under the gauge field describing singlet excitations in the spin liquid. We focus on two gapless $U\left(1\right)$ spin liquids described by $(2+1)$-dimensional conformal field theories (CFT): the staggered flux (sF) spin liquid, and the deconfined critical point between the Néel and valence-bond-solid (VBS) states. In these cases, the electric charge is argued to be an exactly marginal perturbation of the CFT. Consequently, the impurity susceptibility has a $1∕T$ temperature dependence, with an anomalous Curie constant, which is a universal number associated with the CFT. One unexpected feature of the CFT of the sF state is that an applied magnetic field does not induce any staggered spin polarization in the vicinity of the impurity (while such a staggered magnetization is present for the Néel-VBS case). These results differ significantly from earlier theories of vacancies in the sF state, and we explicitly demonstrate how our gauge theory corrects these works. We discuss implications of our results for the cuprate superconductors, organic Mott insulators, and graphene.

Figure 1

Feynman diagrams which contribute to the impurity-dependent renormalization of vertices such as those in Eq. (2.23). The full line is the $z_{\alpha }$ propagator, the wavy line is the $A_{\mu }$ propagator, and the filled square is the source term in ${\mathcal{S}}_{\mathrm{imp}}$ [also in Fig. 2e].

Figure 2

Vertices appearing in the calculation of response to applied field; $H_u$ and $H_s$ denote uniform and staggered magnetic fields, respectively.

Figure 3

Feynman diagrams determining the leading-order contribution to the free energy in presence of uniform [(a)–(g)] and staggered [diagrams (h), (i)] magnetic fields.

Figure 4

Self-energy diagrams determining the thermal screening masses (2.33) for the “photon” $A_{\mu }$ (a) and for $z$ boson (b).

Figure 5

Feynman diagrams to order $1∕N_f$ for the lhs of Eq. (3.9). The full line is the Dirac fermion propagator, the wave line is the photon propagator, the filled square is the impurity source term in ${\mathcal{S}}_{\mathrm{imp}}$, and the filled circle is ${\Psi }^{†}\Psi$ vertex. As noted in the text, (b) has an odd number of photon vertices and, thus, vanishes by Furry’s theorem.[44] Here, and henceforth, we do not show a number of other diagrams, which vanish because of Furry’s theorem.

Figure 6

Feynman diagrams for the impurity susceptibility and Knight shifts of the staggered flux spin liquid. The gray circle vertices depend on the correlator being evaluated: they equal (i) $T^a{\gamma }_{\tau }$ for the ferromagnetic spin density $J_{\tau }^a$, (ii) $T^a$ for the order parameter $N^a$, and (iii) unity for $M$.

Figure 7

(Color online) Numerically calculated scaling function ${\tilde{\Phi }}_u\left(y\right)$ defined by (3.19, 3.20, 3.21, 3.22, 3.23). The inset shows large-$y$ behavior together with the least-squares fit (dashed line) to a power law for $y>40$: the fit yields ${\tilde{\Phi }}_u\left(y\right)\simeq {\mathcal{C}}^{\prime }∕y^{\alpha }$ with ${\mathcal{C}}^{\prime }=24.7\left(1\right)$ and $\alpha =2.070\left(5\right)$.