Quantum electrodynamics in 2+1 dimensions with quenched disorder: Quantum critical states with interactions and disorder

Quantum electrodynamics in 2+1 dimensions (${\mathrm{QED}}_3$) is a strongly coupled conformal field theory (CFT) of a U(1) gauge field coupled to $2N$ two-component massless fermions. The $N=2$ CFT has been proposed as a ground state of the spin-1/2 kagome Heisenberg antiferromagnet. We study ${\mathrm{QED}}_3$ in the presence of weak quenched disorder in its two spatial directions. When the disorder explicitly breaks the fermion flavor symmetry from $\mathrm{SU}\left(2N\right)$ $\to$ U(1) $\times$ $\mathrm{SU}\left(N\right)$ but preserves time-reversal symmetry, we find that the theory flows to a nontrivial fixed line at nonzero disorder with a continuously varying dynamical critical exponent $z>1$. We determine the zero-temperature flavor (spin) conductivity along the critical line. Our calculations are performed in the large-$N$ limit, and the disorder is handled using the replica method.

Figure 1

The kagome lattice. The arrows indicate the convention chosen for the bond directions of the spin chirality operator, ${\mathit{S}}_i\times {\mathit{S}}_j$, where $i$ and $j$ label nearest-neighbor sites. The order of the cross product is taken such that the first spin sits at the lattice site pointing towards the site of the second spin. Later, we will use the same order-ing convention to define nearest-neighbor bond operators ${\mathit{S}}_i·{\mathit{S}}_j$.

Figure 2

Diagrammatic expression for the effective photon propagator in the large-$N$ limit. The dotted lines indicate the bare photon propagator, $D_{\mu \nu }^0\left(p\right)$, while the fermion bubbles are equal to ${\mathrm{\Pi }}_{\mu \nu }\left(q\right)$. As indicated in the text, only the full photon propagator will be used.

Figure 3

Feynman rules associated with the replicated action, $S_n\left[\psi ,\overline{\psi },A\right]$. The diagrams on the first and second rows are diagonal with respect to the replica and flavor indices. In the four-point diagrams, $\ell$ and $m$ are replica indices while $\alpha ,\beta ,\sigma ,\rho$ label the $2N$ fermion flavors.

Figure 4

Example of a diagram which vanishes in the replica limit, $n\to 0$. The internal fermion loop involves a sum over all replica indices, and multiplies the diagram by an overall factor of $n$.

Figure 5

Feynman diagrams which contribute to the fermion self-energy at $\mathcal{O}(g_{\xi },1/2N)$.

Figure 6

Diagrams which contribute when only $\mathrm{SU}\left(2N\right)$-preserving, bilinear disorder is considered ($g_{t,a}=g_{\mathcal{A},a}=g_{v,a}=0$). Both Figs. 6(c) and 6(d) are accompanied by a diagram with the interaction on the other vertex. Partner diagrams to Fig. 6(e) with the fermion loop direction reversed and/or the vertex switched are also present. These diagrams sum to $\left[\mathbb{1}\otimes \mathbb{1}\right]\left[\mathbb{1}\otimes \mathbb{1}\right]\left\{\frac{g_s^2}{\pi \epsilon }+\frac{64g_s{\mathcal{g}}^4}{{\pi }^2\left(2N\right)\epsilon }-\frac{48g_s{\mathcal{g}}^2}{{\pi }^2\left(2N\right)\epsilon }\right\}$.

Figure 7

The only disorder diagram to contribute to $\mathcal{O}(g_{\mathcal{B}}/2N)$ when $\mathcal{B}\left(x\right)$ is the only random field coupled to ${\mathrm{QED}}_3$. Note that it is subleading to the self-energy diagrams we consider elsewhere in the paper (Fig. 5). It contributes a divergence $-i{p}_0{\gamma }^0\left(\frac{g_{\mathcal{B}}}{2\pi \left(2N\right)\epsilon }\right)$.

Figure 8

RG flow in the (a) $(g_{t,z},g_{\mathcal{A},z})$ plane, (b) $(g_{t,z},g_{v,z})$ plane, and (c) $(g_s,g_{v,z})$ plane with all other couplings set to zero. (d) shows the $(g_{t,\perp },g_{\mathcal{A},\perp })$ plane with $g_{t,z}=c$ and all other couplings vanishing. The critical point with all couplings equal to zero (no disorder) is marked in orange with “A” and the critical point with $g_{t,z}=c$ is marked in green with a “B.” In (a), the critical line is drawn in green. Here $c=128/3\pi \left(2N\right)$.

Figure 9

Bond ordering of bond order and vector chirality operators corresponding to the topological currents, $J_{\mathrm{top}}^j$, and the spin currents $J_B^{a,j}\left(r\right)$ in the $x$ and $y$ directions, respectively. Our convention is that in ${\mathit{C}}_{ij}={\mathit{S}}_i\times {\mathit{S}}_j$, the $i\mathrm{th}$ site points towards the $j\mathrm{th}$. The double arrows in (a) identify the bonds which are weighted twice as strongly as others, while the absence of arrows on the horizonal bonds in (b) implies that they do not contribute at all.

Figure 10

Diagrams which contribute to the current-current correlator.

Figure 11

Feynman rules for diagrams without flavor indices. $a,b,c,d$ on the graphs label the spinor indices, and $\ell$ and $m$ label the replica indices. The vertex on the left describes mass-like disorders, such as $M_s\left(r\right)$ and $M_{t,a}\left(r\right)$, and the diagram on the right corresponds to the SU(2) scalar and vector potential disorder, $V_a\left(r\right)$, and ${\mathcal{A}}_{j,a}\left(r\right)$.

Figure 12

Four-point diagrams with photon and mass-like disorder internal lines. Below each diagram, the divergent piece, if present, is given. [The factor of $2\pi \delta \left(q_0\right)$ has been been suppressed for simplicity.]

Figure 13

Four-point diagrams with photon and gauge-like disorder internal lines. Below each diagram, the divergent piece, if present, is given. [The factor of $2\pi \delta \left(q_0\right)$ has been been suppressed for simplicity.]

Figure 14

Four-point diagrams with both mass-like and gauge-like disorder internal lines. Below each diagram, the divergent piece, if present, is given. [The factor of $2\pi \delta \left(q_0\right)$ has been been suppressed for simplicity.] The $\text{tr}\left[{\mathcal{O}}_{fl}\right]$ term in Figs. 14(i) and 14(j) indicates that once the action on the flavor indices has been specified, a trace over this operator should be taken.

Figure 15

Fermion loop subdiagrams which appear in the $\mathcal{O}(g_{\xi }^2,g_{\xi }/2N)$ bilinear counter terms.

Figure 16

Diagrams which enter into the photon self-energy at leading order. (a) will not renormalize the disorder and (b) vanishes.

Figure 17

Diagrams which renormalize the flux disorder at $\mathcal{O}(g_{\xi },g_{\xi }/2N)$. Depending on whether the internal indices are $(\sigma ,\rho )=(0,0)$ or $(i,j)$, the coupling constant is $-g_{\mathcal{B}}$ or $g_{\mathcal{E}}$, respectively.

Figure 18

Subdiagrams which contribute to the flavor conductivity.