Instabilities of Weyl loop semimetals

The simplest realistic description of non-interacting fermions whose dispersion admits a nodal line Fermi surface in three dimensions is given by [40, 42]

$$\begin{eqnarray}&&{S}_{0}=\int {\rm{d}}{K}\,{{\rm{\Psi }}}^{\dagger }(K)[{{\rm{ik}}}_{0}{\sigma }_{0}+E({\bf{K}})]{\rm{\Psi }}(K),\end{eqnarray} \tag{ 1 }$$

where ${\rm{d}}{K}\equiv \tfrac{{{\rm{d}}{k}}_{0}{{\rm{d}}{K}}_{x}{{\rm{d}}{K}}_{y}{{\rm{d}}{K}}_{z}}{{(2\pi )}^{4}}$, k₀ is the Euclidean (Matsubara) frequency, ${\bf{K}}$ denotes three dimensional momentum, σ₀ is the 2 × 2 identity matrix, and ${\rm{\Psi }}{(K)=({c}_{1}(K),{c}_{2}(K))}^{\top }$ is a spinor representing fermions $({c}_{\mathrm{1,2}}(K))$ from $1,2$ orbitals.

The dispersion is

$$\begin{eqnarray}&&E({\bf{K}})={\tilde{v}}_{r}(| \vec{K}| )\,(| \vec{K}| -\kappa )\,{\sigma }_{1}+{v}_{z}{K}_{z}{\sigma }_{2},\end{eqnarray} \tag{ 2 }$$

where ${\tilde{v}}_{r}(| \vec{K}| )=(| \vec{K}| +\kappa )/(2m)$. Here we have distinguished between the three dimensional momentum from its projection, $\vec{K}$, on the plane of the Weyl-loop. We have chosen the loop to lie on the x –y plane, and it is defined by $| \vec{K}| =\sqrt{{K}_{x}^{2}+{K}_{y}^{2}}=\kappa$. Here σ₁ and σ₂ are the first two Pauli matrices which encode the orbital degrees of freedom, and m and v_z are bandstructure parameters. We note that at finite doping, i.e. away from perfect compensation, the non-interacting theory is modified by replacing ${{ik}}_{0}\mapsto {{ik}}_{0}-\mu$. Our theory will focus on $\mu =0$.

Diagonalizing $E({\bf{K}})$ yields two bands that disperse as

$$\begin{eqnarray}&&{\xi }_{\pm }({\bf{K}})=\pm \sqrt{{\tilde{v}}_{r}^{2}(\vec{K}){(| \vec{K}| -\kappa )}^{2}+{v}_{z}^{2}{K}_{z}^{2}}.\end{eqnarray} \tag{ 3 }$$

Since the chemical potential $\mu =0$, the ground state is defined by the configuration where $| \vec{K}| =\kappa$ and K_z = 0, which precisely corresponds to the loop. Thus at low energy $| \vec{K}| \approx \kappa$ and ${K}_{z}\approx 0$, and the dispersion, equation (2), can be approximated to

$$\begin{eqnarray}&&\varepsilon ({k}_{r},{k}_{z})={v}_{r}\,{k}_{r}\,{\sigma }_{1}+{v}_{z}{k}_{z}{\sigma }_{2},\end{eqnarray} \tag{ 4 }$$

where ${v}_{r}\equiv {\tilde{v}}_{r}(\kappa )=\kappa /m$, and ${k}_{r}\equiv | \vec{K}| -\kappa$ and k_z are deviations of momentum in the radial and z directions, respectively. The band dispersion simplifies to ${\xi }_{\pm }({k}_{r},{k}_{z})=\pm \sqrt{{v}_{r}^{2}{k}_{r}^{2}+{v}_{z}^{2}{k}_{z}^{2}}$.

We scale $({k}_{0},{k}_{z})\mapsto {v}_{r}({k}_{0},{k}_{z}/{v}_{z})$, and identify the long wavelength fluctuations of the fermions (low energy modes) through the relation,

$$\begin{eqnarray}&&{\rm{\Psi }}(\tau ,{\bf{r}})\sim {{\rm{e}}}^{{\rm{i}}\vec{\kappa }\cdot {\bf{r}}}\,\psi (\tau ,{\bf{r}})+{\rm{fast}}\,{\rm{modes}},\end{eqnarray} \tag{ 5 }$$

where $\vec{\kappa }=\kappa (\cos \theta ,\sin \theta ,0)$ with θ specifying the position on the loop. We further sharpen the definition of the low energy modes by requiring that the momentum carried by these modes to be such that $\sqrt{{k}_{r}^{2}+{k}_{z}^{2}}\lt {\rm{\Lambda }}\ll \kappa$, where Λ is a UV cutoff on the ${k}_{r}{\rm{-}}{k}_{z}$ plane, measured from the loop (see figure 1). Integrating out the modes which modulate over length scales ≲Λ⁻¹ we obtain

$$\begin{eqnarray}{S}_{0} & \simeq & \displaystyle \frac{{v}_{r}^{3}\kappa }{{v}_{z}}{\displaystyle \int }_{0}^{2\pi }\displaystyle \frac{{\rm{d}}\theta }{2\pi }{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{{\rm{d}}{k}}_{0}{{\rm{d}}{k}}_{r}{{\rm{d}}{k}}_{z}}{{(2\pi )}^{3}}\,{f}_{{\rm{\Lambda }}}({k}_{r},{k}_{z})\\ & & \times {\psi }^{\dagger }({k}_{0},\vec{k},\theta )[{{\rm{i}}{k}}_{0}{\sigma }_{0}+{k}_{r}{\sigma }_{1}+{k}_{z}{\sigma }_{2}]\psi ({k}_{0},\vec{k},\theta ),\end{eqnarray} \tag{ 6 }$$

where ${f}_{{\rm{\Lambda }}}({k}_{r},{k}_{z})$ is a cutoff function which suppresses modes with ${k}_{r}^{2}+{k}_{z}^{2}\gtrsim {{\rm{\Lambda }}}^{2}$ ³ . We choose ${f}_{{\rm{\Lambda }}}({k}_{r},{k}_{z})$ to be rotationally symmetric in ${k}_{r}{\rm{-}}{k}_{z}$ plane. The two dimensional vector $\vec{k}$ is the deviation of momentum from the loop, and is defined on the ${k}_{r}{\rm{-}}{k}_{z}$ plane by $\vec{k}\equiv ({k}_{r},{k}_{z})$. We emphasize that $\vec{k}$ is not linearly related to the deviations from the loop in ${\bf{K}}$ space. In particular, under inversion ${\bf{K}}\mapsto -{\bf{K}}$, but $\vec{k}=({k}_{r},{k}_{z})\mapsto ({k}_{r},-{k}_{z})$.

Zoom In Zoom Out Reset image size

2.1.1. Symmetries

The dynamics of the non-interacting fermions described by equation (6) enjoys a set of continuous and discrete symmetries. In this section we describe these symmetries, and the respective symmetry transformations.

Since the fermion dynamics is independent of the position of the fermionic momentum on the loop, the loop coordinate θ acts as a label for the ψ fields. The cyclic nature of θ leads to three continuous symmetries of S₀. The first is a SO(2) rotational invariance of the action under ${{ \mathcal R }}_{\theta }\,:\theta \mapsto \theta +{\theta }_{0}$. In order to isolate the second symmetry let us write the spatial part of the propagator in terms of polar coordinates $(k,\varphi )$ as $k(\cos \varphi \,{\sigma }_{1}+\sin \varphi \,{\sigma }_{2})$, where $({k}_{r},{k}_{z})=(k\cos \varphi ,k\sin \varphi )$. Under the transformation ${{ \mathcal R }}_{\varphi }\,:\psi \mapsto {{\rm{e}}}^{{\rm{i}}{\xi }_{\theta }{ \mathcal J }}\psi$, with ${ \mathcal J }={\sigma }_{3}/2-{\rm{i}}{\partial }_{\varphi }$, the lagrangian

$$\begin{eqnarray}&&{L}_{0}[{k}_{0},k,\theta ,\varphi ;\psi ]={\psi }^{\dagger }({k}_{0},k,\theta ,\varphi )[{{\rm{i}}{k}}_{0}{\sigma }_{0}+k(\cos \varphi \,{\sigma }_{1}+\sin \varphi \,{\sigma }_{2})]\psi ({k}_{0},k,\theta ,\varphi )\end{eqnarray} \tag{ 7 }$$

transforms to ${L}_{0}[{k}_{0},k,\theta ,\varphi +{\xi }_{\theta };\psi ]$, implying ${{ \mathcal R }}_{\varphi }$ is a symmetry of S₀. ${{ \mathcal R }}_{\varphi }$ corresponds to a rotation in the plane perpendicular to the loop at each point θ. The third symmetry is the invariance of the action under a θ-dependent U(1) transformation $\psi \mapsto {{\rm{e}}}^{{\rm{i}}{\omega }_{\theta }}\psi$. Since the latter two symmetry transformations are locally defined on the loop, they lead to distinct emergent $U{(1)}_{\infty }$ symmetries which we will distinguish as pseudospin-$U{(1)}_{\infty }$ and charge-$U{(1)}_{\infty }$, respectively. While the former corresponds to the conservation of $\hat{\theta }$ component of total angular momentum, the latter originates from particle number conservation at each θ. We note that the charge-$U{(1)}_{\infty }$ symmetry is present in any non-interacting theory where the single particle dispersion is minimized on a degenerate manifold. Since short-range interactions mix momenta at different parts of the loop, these $U{(1)}_{\infty }$ symmetries are broken by generic scatterings among the fermions. Nevertheless, it is possible for subgroups of the $U{(1)}_{\infty }$ symmetries to emerge at fixed points of the interacting theory [43].

The action is also invariant under three sets of discrete transformations. The first is a mirror-plane symmetry which originates from the symmetry between the dynamics above and below the k_z = 0 plane. It is effected by the transformation ${{ \mathcal P }}_{z}^{-1}\,{L}_{0}[{k}_{0},k,\theta ,\varphi ;\psi ]\,{{ \mathcal P }}_{z}={L}_{0}[{k}_{0},k,\theta ,-\varphi ;\psi ]$, where the 'operator' ${{ \mathcal P }}_{z}$ flips the sign of k_z such that ${{ \mathcal P }}_{z}^{-1}\,\{{k}_{0},k,\theta ,\varphi \}\,{{ \mathcal P }}_{z}=\{{k}_{0},k,\theta ,-\varphi \}$ and ${{ \mathcal P }}_{z}^{-1}\psi ({k}_{0},k,\theta ,\varphi ){{ \mathcal P }}_{z}={\sigma }_{1}\psi ({k}_{0},k,\theta ,-\varphi )$. The second is a pair of 'anti-unitary' symmetries, the first of which is defined through the transformation ${{ \mathcal P }}_{{\bf{K}}}^{-1}\,{L}_{0}[{k}_{0},k,\theta ,\varphi ;\psi ]\,{{ \mathcal P }}_{{\bf{K}}}={L}_{0}[{k}_{0},k,\theta +\pi ,-\varphi ;\psi ]$. Here ${{ \mathcal P }}_{{\bf{K}}}$ inverts the three-momentum ${\bf{K}}$, and acts on the fermion fields as ${{ \mathcal P }}_{{\bf{K}}}^{-1}\{\psi ({k}_{0},k,\theta ,\varphi ),\,$ ${\psi }^{\dagger }({k}_{0},k,\theta ,\varphi )\}$ ${{ \mathcal P }}_{{\bf{K}}}=\{{\psi }^{* }({k}_{0},k,\theta +\pi ,-\varphi ),-{\psi }^{\top }$ $({k}_{0},k,\theta +\pi ,-\varphi )\}$ with ${\psi }^{* }\equiv {({\psi }^{\dagger })}^{\top }$. The second element of the pair is obtained by combining ${{ \mathcal P }}_{z}$ with ${{ \mathcal P }}_{{\bf{K}}}$. We note that while these are symmetries of the action, they act on the Hamiltonian in an unusual way. In particular at the level of a first quantized Hamiltonian they change the sign of the ${\sigma }_{1}$ term, and hence effectively connect the Hamiltonian with $| \vec{K}| -\kappa ={k}_{r}\lt 0$ to the Hamiltonian with $| \vec{K}| -\kappa \gt 0$. The last of the sets of discrete symmetries is another pair of antiunitary transformations whose first element is defined by ${{ \mathcal P }}_{0}^{-1}\,{L}_{0}[{k}_{0},k,\theta ,\varphi ;\psi ]\,{{ \mathcal P }}_{0}={L}_{0}[-{k}_{0},k,\theta ,\varphi ;\psi ]$, where ${{ \mathcal P }}_{0}$ inverts the Euclidean frequency k₀ and transforms the fields as ${{ \mathcal P }}_{0}^{-1}\{\psi ({k}_{0},k,\theta ,\varphi ),\,$ ${\psi }^{\dagger }({k}_{0},k,\theta ,\varphi )\}$ ${{ \mathcal P }}_{0}=\{{\sigma }_{2}{\psi }^{* }$ $(-{k}_{0},k,\theta ,\varphi ),{\psi }^{\top }(-{k}_{0},k,\theta ,\varphi ){\sigma }_{2}\}$. The second element of the pair is obtained by combining ${{ \mathcal P }}_{z}$ with ${{ \mathcal P }}_{0}$. We summarize all the symmetries of S₀ in table 1.

Table 1.  The symmetries of the Gaussian fixed point action equation (6). ${\theta }_{0}$ is a constant, and ${\omega }_{\theta }$ and ${\xi }_{\theta }$ are functions of θ. The first three are continuous symmetries, while the last five are discrete. The first three antiunitaries square to −1 while the last squares to +1.

Symmetry	Operation
θ-rotation	${{ \mathcal R }}_{\theta }\,:\theta \mapsto \theta +{\theta }_{0}$
Pseudospin-$U{(1)}_{\infty }$	${{ \mathcal R }}_{\varphi }\,:\psi \mapsto \exp ({\rm{i}}{\xi }_{\theta }{ \mathcal J })\,\psi$
Charge-$U{(1)}_{\infty }$	$\psi \mapsto \exp ({\rm{i}}{\omega }_{\theta })\,\psi$
Mirror plane (${{ \mathcal P }}_{z}$)	${k}_{z}\mapsto -{k}_{z}$ and $\psi \mapsto {\sigma }_{1}\,\psi$
Antiunitary (${{ \mathcal P }}_{{\bf{K}}}$)	${\bf{K}}\mapsto -{\bf{K}}$ and $\{\psi ,{\psi }^{\dagger }\}\mapsto \{{({\psi }^{\dagger })}^{\top },-{\psi }^{\top }\}$
Antiunitary(${{ \mathcal P }}_{z}{{ \mathcal P }}_{{\bf{K}}}$)	${\bf{K}}\mapsto -{\bf{K}}$, $\{\psi ,{\psi }^{\dagger }\}\mapsto \{{\sigma }_{1}{({\psi }^{\dagger })}^{\top },-{\psi }^{\top }{\sigma }_{1}\}$
Antiunitary (${{ \mathcal P }}_{0}$)	${k}_{0}\mapsto -{k}_{0}$, $\{\psi ,{\psi }^{\dagger }\}\mapsto \{{\sigma }_{2}{({\psi }^{\dagger })}^{\top },{\psi }^{\top }{\sigma }_{2}\}$
Antiunitary (${{ \mathcal P }}_{z}{{ \mathcal P }}_{0}$)	${k}_{0}\mapsto -{k}_{0}$ $\{\psi ,{\psi }^{\dagger }\}\mapsto \{{\sigma }_{3}{({\psi }^{\dagger })}^{\top },{\psi }^{\top }{\sigma }_{3}\}$

2.1.2. Generalization

As we will show later in the paper, in order to obtain a controlled truncation of the beta functions of various operators it is convenient to generalize the linear band-touching model to higher order band-touchings. We use the polar coordinates introduced in section 2.1.1 to generalize the dispersion of fermions as ${\varepsilon }_{\eta }(k,\varphi )={k}^{\eta }\,(\cos \varphi \,{\sigma }_{1}+\sin \varphi \,{\sigma }_{2})$ for any real number $\eta \gt 0$. The band dispersion is modified accordingly as ${\xi }_{\eta ;\pm }(k,\varphi )=\pm {k}^{\eta }$. Thus, the dynamics of the non-interacting fermions with η-th order band-touching is given by

$$\begin{eqnarray}{S}_{\eta ;0} & = & \displaystyle \frac{{v}_{r}^{3}\kappa }{{v}_{z}}{\displaystyle \int }_{-\pi }^{\pi }\displaystyle \frac{{\rm{d}}\theta {\rm{d}}\varphi }{{(2\pi )}^{2}}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{{\rm{d}}{k}}_{0}}{2\pi }{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{\rm{d}}{k}\,k}{2\pi }\,{f}_{{\rm{\Lambda }}}(k){\psi }^{\dagger }({k}_{0},k,\theta ,\varphi )[{{\rm{i}}{k}}_{0}{\sigma }_{0}\\ & & +{k}^{\eta }\,(\cos (\varphi ){\sigma }_{1}+\sin (\varphi ){\sigma }_{2})]\psi ({k}_{0},k,\theta ,\varphi ).\end{eqnarray} \tag{ 8 }$$

Since the value of η does not affect the symmetry transformations, ${S}_{\eta ;0}$ and ${S}_{1;0}$ share the same set of symmetries. The Gaussian fixed point described by ${S}_{\eta ;0}$ is invariant under the following choice of scaling,

$$\begin{eqnarray}&&[\theta ],[\varphi ]=0,\quad [k]=1,\quad [{k}_{0}]=\eta ,\quad [\psi ]=-(\eta +1),\end{eqnarray} \tag{ 9 }$$

where the quantity X scales as $X^{\prime} ={{\rm{e}}}^{[X]{\rm{d}}{\ell }}X$ with ${\rm{d}}{\ell }$ being a logarithmic momentum scale. We note that within our scheme the radius of the loop (κ) is dimensionless, which implies coarse-graining towards the loop in momentum space [43, 44].

In this section we introduce the vertices describing instantaneous short-range interactions that are consistent with the discrete symmetries of the non-interacting theory. After scaling k₀ and K_z, as was done for S₀, to obtain

$$\begin{eqnarray}{\tilde{S}}_{{int}} & = & {\left(\displaystyle \frac{{v}_{r}^{2}}{{v}_{z}}\right)}^{3}\displaystyle \sum _{\mu ,\nu =0}^{3}{\displaystyle \int }_{{\rm{\Lambda }}}\left(\displaystyle \prod _{n=1}^{4}{{\rm{d}}{K}}_{n}\right){(2\pi )}^{4}\,{\delta }^{(4)}({K}_{1}-{K}_{2}+{K}_{3}-{K}_{4}){{ \mathcal U }}_{\mu ,\nu }(\{{{\bf{K}}}_{n}\})\\ & & \times \,[{\psi }^{\dagger }({K}_{1}){\sigma }_{\mu }\psi ({K}_{2})][{\psi }^{\dagger }({K}_{3}){\sigma }_{\nu }\psi ({K}_{4})],\end{eqnarray} \tag{ 10 }$$

where ${\int }_{{\rm{\Lambda }}}{\rm{d}}{K}=\int {\rm{d}}{K}\,{f}_{{\rm{\Lambda }}}(| \vec{K}| -\kappa ,{K}_{z})$. We assume ${{ \mathcal U }}_{\mu ,\nu }(\{{{\bf{K}}}_{n}\})={{ \mathcal U }}_{\nu ,\mu }(\{{{\bf{K}}}_{n}\})$ to be real valued functions of momentum. In general, there are 16 vertices corresponding to four choices each for μ and ν. In this paper we focus on only those vertices that are invariant under the discrete symmetries of S₀, viz. ${{ \mathcal P }}_{z}$, ${{ \mathcal P }}_{{\bf{K}}}$, and ${{ \mathcal P }}_{0}$. The terms that have odd parity under these symmetries are marked with corresponding labels in table 2. From the table we see that only those terms for which $\mu =\nu$ have the same parity as S₀. Thus, the minimal set of interactions that respect the discrete symmetries of S₀ is given by

$$\begin{eqnarray}{S}_{{int}} & = & \displaystyle \frac{{v}_{r}^{6}}{{v}_{z}^{3}}{\displaystyle \int }_{{\rm{\Lambda }}}\left(\displaystyle \prod _{n=1}^{4}{{\rm{d}}{K}}_{n}\right)\,{(2\pi )}^{4}{\delta }^{(4)}({K}_{1}-{K}_{2}+{K}_{3}-{K}_{4})\\ & & \times \displaystyle \sum _{\mu }{\tilde{g}}_{\mu }(\{{\widehat{K}}_{n}\})({\psi }^{\dagger }({K}_{1})\,{\sigma }_{\mu }\,\psi ({K}_{2}))({\psi }^{\dagger }({K}_{3})\,{\sigma }_{\mu }\,\psi ({K}_{4})),\end{eqnarray} \tag{ 11 }$$

where ${\widehat{K}}_{n}={\vec{K}}_{n}/| {\vec{K}}_{n}|$, and we have Taylor expanded ${{ \mathcal U }}_{\mu ,\mu }({{\bf{K}}}_{1},{{\bf{K}}}_{2},{{\bf{K}}}_{3},{{\bf{K}}}_{4})$ about the loop to obtain the coupling functions ${\tilde{g}}_{\mu }({\widehat{K}}_{1},{\widehat{K}}_{2},{\widehat{K}}_{3},{\widehat{K}}_{4})$. Since, $\widehat{K}$ and θ are physically equivalent, ${\tilde{g}}_{\mu }(\{{\widehat{K}}_{n}\})$ are functions of the loop coordinates only.

Table 2.  The vertices in equation (10) that are disallowed by the discrete symmetries of S₀ (see table 1). The $(\mu ,\nu )$-th cell in the table represents the corresponding vertex. The cells are labeled by the symmetry transformations under which the parity of the vertex is odd.

ν⧹μ	0	1	2	3
0		${{ \mathcal P }}_{0}$	${{ \mathcal P }}_{z},{{ \mathcal P }}_{{\bf{K}}},{{ \mathcal P }}_{0}$	${{ \mathcal P }}_{z}$, ${{ \mathcal P }}_{0}$
1	${{ \mathcal P }}_{0}$		${{ \mathcal P }}_{z},{{ \mathcal P }}_{{\bf{K}}}$	${{ \mathcal P }}_{z}$
2	${{ \mathcal P }}_{z},{{ \mathcal P }}_{{\bf{K}}},{{ \mathcal P }}_{0}$	${{ \mathcal P }}_{z},{{ \mathcal P }}_{{\bf{K}}}$		${{ \mathcal P }}_{{\bf{K}}}$
3	${{ \mathcal P }}_{z},{{ \mathcal P }}_{0}$	${{ \mathcal P }}_{z}$	${{ \mathcal P }}_{{\bf{K}}}$

We add equations (8) to (11), and scale $\psi \mapsto \sqrt{({v}_{z}/{v}_{r}^{3})}\,\psi$, to obtain the action for the interacting theory for any η. It is useful to introduce $({\gamma }_{0},{\gamma }_{1},{\gamma }_{2},{\gamma }_{3})\equiv ({\sigma }_{3},{\sigma }_{2},-{\sigma }_{1},{\sigma }_{0})$, and define the conjugate field

$$\begin{eqnarray}&&\bar{\psi }(K)={\psi }^{\dagger }(K)\,{\gamma }_{0},\end{eqnarray} \tag{ 12 }$$

such that the generalized low energy effective theory is

$$\begin{eqnarray}S(\eta ) & = & {\rm{i}}{\displaystyle \int }_{{\rm{\Lambda }}}{\rm{d}}{K}\bar{\psi }(K)\left[{k}_{0}{\gamma }_{0}+{({(| \vec{K}| -\kappa )}^{2}+{K}_{z}^{2})}^{\tfrac{(\eta -1)}{2}}\,((| \vec{K}| -\kappa )\,{\gamma }_{1}+{K}_{z}{\gamma }_{2})\right]\psi (K)\\ & & +{\displaystyle \int }_{{\rm{\Lambda }}}\left(\displaystyle \prod _{n=1}^{4}{{\rm{d}}{K}}_{n}\right)\,{(2\pi )}^{4}\,{\delta }^{(4)}({K}_{1}-{K}_{2}+{K}_{3}-{K}_{4})\left[\displaystyle \frac{{g}_{1}(\{{\widehat{K}}_{n}\})}{\kappa }(\bar{\psi }({K}_{1}){\gamma }_{0}\psi ({K}_{2}))(\bar{\psi }({K}_{3}){\gamma }_{0}\psi ({K}_{4}))\right.\\ & & -\displaystyle \frac{{g}_{2+}(\{{\widehat{K}}_{n}\})}{\kappa }\{(\bar{\psi }({K}_{1}){\gamma }_{1}\psi ({K}_{2}))(\bar{\psi }({K}_{3}){\gamma }_{1}\psi ({K}_{4}))+({\gamma }_{1}\to {\gamma }_{2})\}\\ & & -\displaystyle \frac{{g}_{2-}(\{{\widehat{K}}_{n}\})}{\kappa }\{(\bar{\psi }({K}_{1}){\gamma }_{1}\psi ({K}_{2}))(\bar{\psi }({K}_{3}){\gamma }_{1}\psi ({K}_{4}))-({\gamma }_{1}\to {\gamma }_{2})\}\\ & & \left.+\displaystyle \frac{{g}_{3}(\{{\widehat{K}}_{n}\})}{\kappa }(\bar{\psi }({K}_{1})\psi ({K}_{2}))(\bar{\psi }({K}_{3})\psi ({K}_{4}))\right],\end{eqnarray} \tag{ 13 }$$

where $({g}_{1},{g}_{2+},\,{g}_{2-},\,{g}_{3})\equiv \tfrac{\kappa }{{v}_{z}}\,({\tilde{g}}_{0},\tfrac{1}{2}({\tilde{g}}_{1}+{\tilde{g}}_{2}),\tfrac{1}{2}({\tilde{g}}_{1}-{\tilde{g}}_{2}),{\tilde{g}}_{3})$. We note that in contrast to the Gaussian part, the interaction terms generally do not admit a straightforward decomposition in terms of angular patches because the short-range scatterings mix angular coordinates. Nevertheless, the diagonal structure of the Gaussian part in terms of the patch index θ is useful for the evaluation of quantum corrections.

From equation (13) we deduce the bare propagator

$$\begin{eqnarray}&&{\langle {\psi }_{a}(K^{\prime} ){\bar{\psi }}_{b}(K)\rangle }_{0}={(2\pi )}^{4}\,{\delta }^{(4)}(K-K^{\prime} )\,{{ \mathcal G }}_{a,b}(K),\end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray}&&{ \mathcal G }(K)\equiv { \mathcal G }({k}_{0},k,\varphi )=-{\rm{i}}\,\displaystyle \frac{{k}_{0}{\gamma }_{0}+{k}^{\eta }(\cos \varphi \,{\gamma }_{1}+\sin \varphi \,{\gamma }_{2})}{{k}_{0}^{2}+{k}^{2\eta }},\end{eqnarray} \tag{ 15 }$$

By applying the scaling relations in equation (9) to the interaction vertices, we obtain the scaling dimension for the coupling functions,

$$\begin{eqnarray}&&[{g}_{i}]=\eta -2.\end{eqnarray} \tag{ 16 }$$

Therefore, for $\eta \lt 2$ ($\eta \gt 2$) the interactions are irrelevant (relevant) at the Gaussian fixed point. In the presence of irrelevant interactions the Gaussian fixed point is stable and has a finite basin of attraction, whose volume in coupling space is controlled by the parameter $\epsilon =2-\eta$. Thus, it is expected that for $\epsilon \gt 0$ weak short range interactions cannot lead to new phases, and the nodal loop semi-metal is stable. The Weyl-loop semi-metal corresponds to $\epsilon =1$, where the short-range interactions have bare scaling dimension −1, and are strongly irrelevant. Although irrelevant in RG sense, microscopically strong interactions can still drive the system towards a non-trivial phase by pushing the couplings out of the basin of attraction of the Gaussian fixed point. Such finite-coupling instabilities of the $\eta =1$ ($\epsilon =1$) system is the focus of this paper. However, at $\eta =1$ the bare scaling dimension of the couplings are ${ \mathcal O }(1)$, which obstructs a controlled access to the potential finite-coupling fixed points and instabilities. In order to achieve perturbative control we turn to the limit where $0\leqslant \epsilon \ll 1$. In particular, $\epsilon =0$ corresponds to a Weyl-loop semi-metal with quadratic band-touching. Here the short-range interactions are either marginally relevant or irrelevant, as is the case for Fermi surfaces with unit codimension. Motivated by the smooth interpolation between the quadratic and linear band-touching models at tree level, we analyze the finite coupling instabilities close to $\eta =2$, using as the control parameter. In the spirit of all expansions, it is hoped that the small analysis would be able to access qualitative elements of the $\eta =1$ theory ⁴ . We note that our approach is complementary to reference [45], where the codimension of a one-dimensional Fermi surface was used as a tuning parameter to controllably access a finite coupling pairing instability. Similar strategies based on tuning the dispersion of the dynamical modes [46, 47], and the codimension of the Fermi surface [48–51] have been applied to the study of strongly coupled field theories in the presence of a Fermi surface. Although here we adopt the former strategy, it is possible to utilize the latter approach with the deviation from spatial dimension d = 2 as the small parameter.

The expressions of the beta-functions are simplified by changing coordinates in the coupling space as

$$\begin{eqnarray}&&{f}_{1\pm ;{\rm{J}}}={V}_{2+;{\rm{J}}}\pm \displaystyle \frac{1}{2}({V}_{1;{\rm{J}}}-{V}_{3;{\rm{J}}}),\quad {f}_{2\pm ;{\rm{J}}}={V}_{2-;{\rm{J}}}\pm \displaystyle \frac{1}{2}({V}_{1;{\rm{J}}}+{V}_{3;{\rm{J}}}).\end{eqnarray} \tag{ 22 }$$

The flows of ${f}_{n\pm ;{\rm{J}}}$ for $\eta =2-\epsilon \lt 2$ are governed by

$$\begin{eqnarray}&&{\partial }_{{\ell }}{f}_{1+;{\rm{J}}}=-{f}_{1+;{\rm{J}}}\left[\epsilon +\displaystyle \frac{1}{2\pi }{f}_{1+;{\rm{J}}}\right],\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{\partial }_{{\ell }}{f}_{1-;{\rm{J}}}=-{f}_{1-;{\rm{J}}}\left[\epsilon -\displaystyle \frac{1}{2\pi }{f}_{1-;{\rm{J}}}\right],\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{\partial }_{{\ell }}{f}_{2+;{\rm{J}}}=-{f}_{2+;{\rm{J}}}\left[\epsilon +\displaystyle \frac{1}{\pi }{f}_{2+;{\rm{J}}}\right],\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{\partial }_{{\ell }}{f}_{2-;{\rm{J}}}=-\epsilon {f}_{2-;{\rm{J}}}.\end{eqnarray} \tag{ 26 }$$

The beta-functions imply that ${f}_{n\pm ;{\rm{J}}}$ do not mix at one-loop order. We plot the projections of the four dimensional RG flow on the $({f}_{1+;{\rm{J}}},{f}_{1-;{\rm{J}}})$ and $({f}_{2+;{\rm{J}}},{f}_{2-;{\rm{J}}})$ planes in figure 5. The fixed points are derived from the conditions for simultaneous vanishing of the four beta functions, which result in 3 quadratic and 1 linear equations that have 2³ solutions. The Gaussian fixed point is the only stable fixed point of the RG flow. It has a finite basin of attraction, whose volume is controlled by . The non-Gaussian fixed points have at least one relevant direction which take(s) the flow towards the Gaussian fixed point or strong-coupling, depending on which side of the separatrices the couplings lie. In table 3 we list all non-Gaussian fixed points according to the number of relevant directions they possess.

Zoom In Zoom Out Reset image size

Table 3.  Non-Gaussian fixed points in units of $\pi \epsilon$ for the BCS couplings, with at least one relevant direction. The number of relevant couplings required to be tuned to achieve the criticality determines its order, viz. stable (0), critical (1), bicritical (2), and tricritical (3), where the numbers within the parentheses are the number of relevant couplings. Thus, fixed points ${{\rm{I}}}_{(\mathrm{BCS})}$ - ${\mathrm{III}}_{(\mathrm{BCS})}$ are critical, ${\mathrm{IV}}_{(\mathrm{BCS})}$ - ${\mathrm{VI}}_{(\mathrm{BCS})}$ are bicritical, and ${\mathrm{VII}}_{(\mathrm{BCS})}$ is tricritical.

#	${f}_{1+;{\rm{J}}}^{* }$	${f}_{1-;{\rm{J}}}^{* }$	${f}_{2+;{\rm{J}}}^{* }$	${f}_{2-;{\rm{J}}}^{* }$	Tune
${{\rm{I}}}_{(\mathrm{BCS})}$	−2	0	0	0	${f}_{1+}$
${\mathrm{II}}_{(\mathrm{BCS})}$	0	2	0	0	${f}_{1-}$
${\mathrm{III}}_{(\mathrm{BCS})}$	0	0	−1	0	${f}_{2+}$
${\mathrm{IV}}_{(\mathrm{BCS})}$	−2	2	0	0	${f}_{1\pm }$
${{\rm{V}}}_{(\mathrm{BCS})}$	−2	0	−1	0	${f}_{1+},{f}_{2+}$
${\mathrm{VI}}_{(\mathrm{BCS})}$	0	2	−1	0	${f}_{1-},{f}_{2+}$
${\mathrm{VII}}_{(\mathrm{BCS})}$	−2	2	−1	0	${f}_{1\pm },{f}_{2+}$

In order to interpret the fixed points in terms of the original couplings of the model we invert the relations in equation (22) to obtain

$$\begin{eqnarray}{V}_{1;{\rm{J}}} & = & \displaystyle \frac{1}{2}(({f}_{2+;{\rm{J}}}-{f}_{2-;{\rm{J}}})+({f}_{1+;{\rm{J}}}-{f}_{1-;{\rm{J}}})),\\ {V}_{2+;{\rm{J}}} & = & \displaystyle \frac{1}{2}({f}_{1+;{\rm{J}}}+{f}_{1-;{\rm{J}}}),\\ {V}_{2-;{\rm{J}}} & = & \displaystyle \frac{1}{2}({f}_{2+;{\rm{J}}}+{f}_{2-;{\rm{J}}}),\\ {V}_{3;{\rm{J}}} & = & \displaystyle \frac{1}{2}(({f}_{2+;{\rm{J}}}-{f}_{2-;{\rm{J}}})-({f}_{1+;{\rm{J}}}-{f}_{1-;{\rm{J}}})).\end{eqnarray} \tag{ 27 }$$

From table 3 we note that at ${\mathrm{III}}_{(\mathrm{BCS})}$ ${V}_{2+;{\rm{J}}}$ vanishes, which implies the emergence of the BCS-U(1) symmetry at the critical fixed point. The BCS-U(1) symmetry is also present at the bi- and tri-critical fixed points ${\mathrm{IV}}_{(\mathrm{BCS})}$ and ${\mathrm{VII}}_{(\mathrm{BCS})}$, respectively. For the rest of this section we focus on the subspace where ${V}_{2+;{\rm{J}}}=0$, i.e. the subspace invariant under BCS-U(1). To motivate this approximation, note that it is natural to take the bare UV scale interaction to be pure density-density, without any pseudospin structure. An interacting theory with only density-density interactions will have this BCS-U(1) symmetry. Interactions with non-trivial pseudospin structure will then be generated under the RG, but only those interactions that lie within the 'maximally symmetric subspace.' In particular, ${V}_{2+}$, which breaks the BCS-U(1) symmetry, will not be generated. Additionally of course, restricting to the maximally symmetric subspace has the advantage of providing us with a 'toy model' that is more amenable to analysis.

We therefore restrict ourselves to the subspace with ${V}_{2+}=0$. Since ${V}_{2+;{\rm{J}}}$ was a relevant perturbation at ${\mathrm{IV}}_{(\mathrm{BCS})}$ and ${\mathrm{VII}}_{(\mathrm{BCS})}$, these fixed points become critical and bicritical, respectively, with respect to BCS-U(1) invariant perturbations. Thus, we obtain one Gaussian, two critical, and one bi-critical fixed points. The non-Gaussian fixed points are listed in table 4 in terms of ${V}_{n;{\rm{J}}}$.

Table 4.  The fixed points (in units of $\pi \epsilon$) in table 3 that lie in the ${V}_{2+;{\rm{J}}}=0$ subspace. The first two are critical fixed points, while the last one is bicritical.

#	${V}_{1;{\rm{J}}}^{* }$	${V}_{2-;{\rm{J}}}^{* }$	${V}_{3;{\rm{J}}}^{* }$	Tune
${\mathrm{III}}_{(\mathrm{BCS})}$	$-1/2$	$-1/2$	$-1/2$	${f}_{2+}$
${\mathrm{IV}}_{(\mathrm{BCS})}$	−2	0	2	$({V}_{1}-{V}_{3})$
${\mathrm{VII}}_{(\mathrm{BCS})}$	$-5/2$	$-1/2$	3/2	$({V}_{1}-{V}_{3})$, ${f}_{2+}$

Since both ${\mathrm{III}}_{(\mathrm{BCS})}$ and ${\mathrm{IV}}_{(\mathrm{BCS})}$ are critical fixed points, they potentially separate the non-interacting Gaussian fixed point (Weyl-loop semi-metal phase) from superconducting phases. We first discuss the stability of ${\mathrm{III}}_{(\mathrm{BCS})}$, and then apply the same analysis to ${\mathrm{IV}}_{(\mathrm{BCS})}$. Since the RG flow of ${f}_{n\pm ;{\rm{J}}}$ mutually decouple and they are irrelevant when $| {f}_{n\pm ;{\rm{J}}}| \ll \epsilon$, the critical fixed point ${\mathrm{III}}_{(\mathrm{BCS})}$, which is realized at ${f}_{1\pm ;{\rm{J}}},{f}_{2-;{\rm{J}}}=0$, can only be destabilized by perturbations with non-zero component along ${f}_{2+;{\rm{J}}}$. As depicted in figure 5(b), the sign of the deviation $\delta {f}_{2+;{\rm{J}}}={f}_{2+;{\rm{J}}}-{f}_{2+;{\rm{J}}}^{* }$ determines whether the perturbation takes the flow towards the Gaussian fixed point or towards strong coupling where ${f}_{2+;{\rm{J}}}$ is large and negative.

In contrast to ${\mathrm{III}}_{(\mathrm{BCS})}$, ${\mathrm{IV}}_{(\mathrm{BCS})}$ is located on the $({f}_{1+;{\rm{J}}},{f}_{1-;{\rm{J}}})$ plane with ${f}_{2\pm ;{\rm{J}}}=0$. Since in the ${V}_{2+;{\rm{J}}}=0$ subspace ${f}_{1\pm ;{\rm{J}}}$ are equivalent, the RG flow in the neighborhood of ${\mathrm{IV}}_{(\mathrm{BCS})}$ is governed by

$$\begin{eqnarray}&&{\partial }_{{\ell }}\delta {f}_{1-;{\rm{J}}}=\epsilon \delta {f}_{1-;{\rm{J}}},\quad {\partial }_{{\ell }}\delta {f}_{2\pm ;{\rm{J}}}=-\epsilon \delta {f}_{2\pm ;{\rm{J}}}.\end{eqnarray} \tag{ 28 }$$

Therefore, perturbations with $\delta {f}_{1-;{\rm{J}}}\ne 0$ are relevant and, depending on its sign, take the flow either towards the Gaussian fixed point, or towards strong coupling where ${f}_{1-;{\rm{J}}}$ is large and positive.

In this subsection we identify the effective interactions along the stable RG flow trajectory that takes the theory towards strong coupling, as we tune away from the critical fixed points. As in the preceding subsection, we discuss the flow away from ${\mathrm{III}}_{(\mathrm{BCS})}$ first, followed by ${\mathrm{IV}}_{(\mathrm{BCS})}$.

Let us label the flow away from the critical point towards strong coupling on the ${f}_{2+;{\rm{J}}}$ axis as a strong coupling trajectory (SCT). Due to the stability of the Gaussian fixed point in the $({f}_{1+;{\rm{J}}},{f}_{1-;{\rm{J}}},{f}_{2-;{\rm{J}}})$ subspace, the SCT is stable against perturbations perpendicular to it. With the aid of equation (27) we note that on the SCT ${V}_{1;{\rm{J}}}={V}_{2-;{\rm{J}}}={V}_{3;{\rm{J}}}=\tfrac{1}{2}{f}_{2+;{\rm{J}}}\lt -\tfrac{\pi \epsilon }{2}$. Thus, the BCS vertices on the SCT are given by

$$\begin{eqnarray}&&{L}_{\mathrm{III}}^{(\mathrm{BCS})}(\{{K}_{n}\})=\displaystyle \frac{1}{\kappa }{V}_{1}({\hat{K}}_{1},{\hat{K}}_{2})[{(\bar{\psi }{\gamma }_{0}\psi )}^{2}+{(\bar{\psi }\psi )}^{2}-{(\bar{\psi }{\gamma }_{1}\psi )}^{2}+{(\bar{\psi }{\gamma }_{2}\psi )}^{2}],\end{eqnarray} \tag{ 29 }$$

where we have suppressed the momentum dependence of the fields. On the plane of the Weyl-loop the vertices simplify to

$$\begin{eqnarray}&&{L}_{\mathrm{III}}^{(\mathrm{BCS})}({\vec{K}}_{1},{\vec{K}}_{2})=\displaystyle \frac{1}{2\kappa }{V}_{1}({\hat{K}}_{1},{\hat{K}}_{2})\{{\psi }^{\dagger }({\vec{K}}_{1}){({\psi }^{\dagger }(-{\vec{K}}_{1}))}^{\top }\}\{{\psi }^{\top }(-{\vec{K}}_{2})\psi ({\vec{K}}_{2})\}.\end{eqnarray} \tag{ 30 }$$

Since ${V}_{1}\lt 0$ on the SCT, this indicates that a pairing instability is driven by the condensation of ${\psi }^{\top }(-\vec{K})\psi (\vec{K})$.

The SCT originating at ${\mathrm{IV}}_{(\mathrm{BCS})}$ is defined by $({V}_{1;{\rm{J}}},{V}_{2\pm ;{\rm{J}}},{V}_{3;{\rm{J}}})=(-{f}_{1-;{\rm{J}}},0,{f}_{1-;{\rm{J}}})$. Therefore, along the SCT the BCS vertices with momenta on the plane of the Weyl-loop simplifies to

$$\begin{eqnarray}{L}_{\mathrm{IV}}^{(\mathrm{BCS})}({\vec{K}}_{1},{\vec{K}}_{2}) & = & {V}_{1}({\hat{K}}_{1},{\hat{K}}_{2})[\{{\psi }^{\dagger }({\vec{K}}_{1})\,{\sigma }_{1}\,{({\psi }^{\dagger }(-{\vec{K}}_{1}))}^{\top }\}\{{\psi }^{\top }(-{\vec{K}}_{2}){\sigma }_{1}\psi ({\vec{K}}_{2})\}\\ & & +\{{\psi }^{\dagger }({\vec{K}}_{1})\,{\sigma }_{2}\,({\psi }^{\dagger }{(-{\vec{K}}_{1}))}^{\top }\}\{{\psi }^{\top }(-{\vec{K}}_{2})\,{\sigma }_{2}\,\psi ({\vec{K}}_{2})\}].\end{eqnarray} \tag{ 31 }$$

Since ${V}_{1}\lt 0$ along the SCT, both vertices in equation (31) can drive a pairing instability. In the following subsection we verify the identity of the superconducting states indicated above through explicit computation of the anomalous scaling dimension of various pairing susceptibilities along the two SCTs.

In this subsection we determine the nature of the superconducting states that arise as instabilities of the critical points in the ${V}_{2+;{\rm{J}}}=0$ subspace. In particular, we compute the change of scaling dimension of the pairing susceptibilities along the SCT as the system flows towards strong-coupling.

We consider insertions of fermion pairs,

$$\begin{eqnarray}&&{S}_{2}^{(\mu )}=\int {\rm{d}}{K}\,{{\rm{\Delta }}}_{\mu }^{* }(\widehat{K})\,{{\rm{\Psi }}}^{\top }(-K)\,{\gamma }_{\mu }\,{\rm{\Psi }}(K)+{\rm{h.c.}},\end{eqnarray} \tag{ 32 }$$

where $\mu =0,1,2,3$. The pairing amplitude ${{\rm{\Delta }}}_{\mu }(\widehat{K}){\gamma }_{\mu }=-{{\rm{\Delta }}}_{\mu }(-\widehat{K}){\gamma }_{\mu }^{\top }$, with ${{\rm{\Delta }}}_{\mu }(\widehat{K})$ being a complex valued function. The 'singlet' pairing corresponds to ${\gamma }_{\mu }^{\top }=-{\gamma }_{\mu }$ and ${{\rm{\Delta }}}_{\mu }(\hat{K})={{\rm{\Delta }}}_{\mu }(-\hat{K})$, while for ${\gamma }_{\mu }^{\top }={\gamma }_{\mu }$ and ${{\rm{\Delta }}}_{\mu }(\hat{K})=-{{\rm{\Delta }}}_{\mu }(-\hat{K})$ the pairing occurs in a 'triplet' channel. At one-loop order ${{\rm{\Delta }}}_{\mu }(\widehat{K})$ is renormalized by figure 6. As derived in appendix B the RG flow of each angular momentum harmonic of ${{\rm{\Delta }}}_{\mu }(\widehat{K})$ is governed by

$$\begin{eqnarray}&&{\partial }_{{\ell }}{{\rm{\Delta }}}_{\mu ;{\rm{J}}}={{\rm{\Delta }}}_{\mu ;{\rm{J}}}[2+{\delta }_{\mu }(\{{V}_{i;{\rm{J}}}\})],\end{eqnarray} \tag{ 33 }$$

where the anomalous dimension of ${{\rm{\Delta }}}_{\mu ;{\rm{J}}}$, ${\delta }_{\mu }(\{{V}_{i;{\rm{J}}}\})$, is defined in equation (B6).

Zoom In Zoom Out Reset image size

Along the SCT originating from ${\mathrm{III}}_{(\mathrm{BCS})}$ the susceptibility for γ₃ is most strongly enhanced, and the RG flow of ${{\rm{\Delta }}}_{3}(\widehat{K})$ is governed by

$$\begin{eqnarray}&&{\partial }_{{\ell }}{{\rm{\Delta }}}_{3;{\rm{J}}}={{\rm{\Delta }}}_{3;{\rm{J}}}\left[2-\displaystyle \frac{{V}_{1;{\rm{J}}}}{\pi }\right],\end{eqnarray} \tag{ 34 }$$

where we have used the fact that along the SCT ${V}_{1;{\rm{J}}}={V}_{3;{\rm{J}}}={V}_{2-;{\rm{J}}}=\tfrac{1}{2}{f}_{2+;{\rm{J}}}$. Since ${V}_{1;{\rm{J}}}\lt 0$, ${{\rm{\Delta }}}_{3}(\widehat{K})$ obtains a positive anomalous dimension. The scaling dimensions of ${{\rm{\Delta }}}_{\mu }(\widehat{K})$ for $\mu \ne 3$ do not change because ${\delta }_{\mu \ne 3}=0$.

From the symmetry properties of the gap function, we obtain ${{\rm{\Delta }}}_{3}(-\widehat{K})=-{{\rm{\Delta }}}_{3}(\widehat{K})$, which is equivalent to ${{\rm{\Delta }}}_{3}(\theta +\pi )=-{{\rm{\Delta }}}_{3}(\theta )$ in terms of the loop coordinate θ. Decomposing ${{\rm{\Delta }}}_{3}(\theta )$ in terms of angular momentum harmonics,

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{3}(\theta )=\displaystyle \sum _{J=-\infty }^{\infty }{{\rm{e}}}^{-{\rm{i}}\theta J}\,{{\rm{\Delta }}}_{3;{\rm{J}}},\end{eqnarray} \tag{ 35 }$$

we note that only the odd J harmonics are non-zero. While the flow equations in different odd angular momentum channels are identical, the degeneracy between different angular momentum channels will be broken by the initial conditions. The largest 'bare interaction' is likely to arise in the lowest allowed angular momentum channel (i.e. $J=\pm 1$) which will then be the leading instability. Therefore, the leading instability is expected to be 'p-wave' consistent with general arguments [42]. Retaining only the $J=\pm 1$ harmonic we express

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{3}(\theta )={\tilde{{\rm{\Delta }}}}_{+}\cos \theta +{\rm{i}}{\tilde{{\rm{\Delta }}}}_{-}\sin \theta ,\end{eqnarray} \tag{ 36 }$$

where the constants ${\tilde{{\rm{\Delta }}}}_{\pm }=2({{\rm{\Delta }}}_{3;-1}\pm {{\rm{\Delta }}}_{3;1})$. Allowing for weak radial momentum dependence on the plane of the loop we generalize ${{\rm{\Delta }}}_{3}(\theta )$ to

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{3}(\vec{K})={\tilde{{\rm{\Delta }}}}_{+}\,\displaystyle \frac{{K}_{x}}{\kappa }+{\rm{i}}{\tilde{{\rm{\Delta }}}}_{-}\,\displaystyle \frac{{K}_{y}}{\kappa }.\end{eqnarray} \tag{ 37 }$$

In terms of the generalized expression for a superconducting order parameter

$$\begin{eqnarray}&&\overleftrightarrow{{\rm{\Delta }}}(\vec{K})={\rm{i}}({d}_{0}(\vec{K})+{\bf{d}}(\vec{K})\cdot {\boldsymbol{\sigma }}){\sigma }_{2},\end{eqnarray} \tag{ 38 }$$

the current state corresponds to ${d}_{0}(\vec{K}),{d}_{1}(\vec{K}),{d}_{3}(\vec{K})=0$, and ${d}_{2}(\vec{K})=-{\rm{i}}{{\rm{\Delta }}}_{3}(\vec{K})$.

Note that we have two degenerate channels $J=\pm 1$ which are related by time reversal symmetry. We now discuss the competition of these two channels below T_c. For the superconducting state where $J=\pm 1$ components are 'in-phase' (${{\rm{\Delta }}}_{3;1}={{\rm{\Delta }}}_{3;-1}$) and 'out-of-phase' (${{\rm{\Delta }}}_{3;1}=-{{\rm{\Delta }}}_{3;-1}$), $({\tilde{{\rm{\Delta }}}}_{+},{\tilde{{\rm{\Delta }}}}_{-})=(4{{\rm{\Delta }}}_{3;-1},0)$ and $({\tilde{{\rm{\Delta }}}}_{+},{\tilde{{\rm{\Delta }}}}_{-})=(0,4{{\rm{\Delta }}}_{3;-1})$, respectively. Therefore, in this state the gap function vanishes at two points on the Fermi surface — this is nodal superconductivity which spontaneously breaks rotation symmetry. In contrast if only one out of the J_± channels develops a non-zero order parameter this corresponds to

$$\begin{eqnarray}{{\rm{\Delta }}}_{3}(\vec{K})=\left\{\begin{array}{ll}\displaystyle \frac{2{{\rm{\Delta }}}_{3;-1}}{\kappa }\,({K}_{x}+{{\rm{i}}{K}}_{y}) & {\rm{for}}\,J=-1\\ -\displaystyle \frac{2{{\rm{\Delta }}}_{3;1}}{\kappa }\,({K}_{x}-{{\rm{i}}{K}}_{y}) & {\rm{for}}\,J=+1.\end{array}\right.\end{eqnarray} \tag{ 39 }$$

This type of ordering spontaneously breaks time reversal symmetry and corresponds to chiral superconductivity. Note that these gap functions do not vanish on the Fermi surface, and thus are expected to have a larger condensation energy. We show this explicitly by minimizing the Ginzburg-Landau free energy, similar to [53–57].

Since ${V}_{1}\lt 0$ along the SCT, we introduce an auxiliary field, $\phi (k)$, to decompose the first term in equation (30). Integrating out the fermions generates an effective Ginzburg–Landau action for $\phi (k)$. In the symmetry broken state below the critical temperature we ignore spatiotemporal fluctuations of $\phi (k)$, and focus on the 'potential' part of the effective action. We ignore contributions from scattering between Cooper pairs at different parts of the loop, and express the total free energy as a sum over free energy per unit length of the loop,

$$\begin{eqnarray}&&{ \mathcal F }={\int }_{-\pi }^{\pi }\displaystyle \frac{{\rm{d}}\theta }{2\pi }\,{ \mathcal F }^{\prime} (\theta )={\int }_{-\pi }^{\pi }\displaystyle \frac{{\rm{d}}\theta }{2\pi }\,(a| \phi (\theta ){| }^{2}+b| \phi (\theta ){| }^{4}),\end{eqnarray} \tag{ 40 }$$

where a and b are effective parameters. Substituting $\phi (\theta )={{\rm{\Delta }}}_{3}(\theta )$ (defined in equation (36)) leads to

$$\begin{eqnarray}&&{ \mathcal F }=\displaystyle \frac{a}{2}(| {\tilde{{\rm{\Delta }}}}_{+}{| }^{2}+| {\tilde{{\rm{\Delta }}}}_{-}{| }^{2})+\displaystyle \frac{3b}{8}(| {\tilde{{\rm{\Delta }}}}_{+}{| }^{4}+| {\tilde{{\rm{\Delta }}}}_{-}{| }^{4})+\displaystyle \frac{b}{4}| {\tilde{{\rm{\Delta }}}}_{+}{| }^{2}| {\tilde{{\rm{\Delta }}}}_{-}{| }^{2}\left(1-\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{\tilde{{\rm{\Delta }}}}_{+}^{* }{\tilde{{\rm{\Delta }}}}_{-}}{| {\tilde{{\rm{\Delta }}}}_{+}| | {\tilde{{\rm{\Delta }}}}_{-}| }-{\rm{c.c.}}\right)}^{2}\right).\end{eqnarray} \tag{ 41 }$$

We note that the third term represents a repulsion between the two components of ${{\rm{\Delta }}}_{3}(\theta )$. For a condensate to form the condensation energy needs to overcome the energy barrier due to the repulsion. Since in the superconducting phase $a\lt 0$ and $b\gt 0$, the configuration that minimizes ${ \mathcal F }$ is determined by the relative magnitude of $| a|$ and b. By expressing $({\tilde{{\rm{\Delta }}}}_{+},{\tilde{{\rm{\Delta }}}}_{-})=\tilde{{{\rm{\Delta }}}_{+}}(1,\sqrt{x}{{\rm{e}}}^{{\rm{i}}{A}})$, in units of $b| {\tilde{{\rm{\Delta }}}}_{+}{| }^{4}$ (or equivalently $b| {\tilde{{\rm{\Delta }}}}_{-}{| }^{4}$) we obtain

$$\begin{eqnarray}&&\displaystyle \frac{2{ \mathcal F }}{b| {\tilde{{\rm{\Delta }}}}_{+}{| }^{4}}=-\displaystyle \frac{| a| }{b| {\tilde{{\rm{\Delta }}}}_{+}{| }^{2}}(1+x)+\displaystyle \frac{3}{4}{(1-x)}^{2}+2x\left(1+\displaystyle \frac{1}{2}{\sin }^{2}A\right).\end{eqnarray} \tag{ 42 }$$

Minimization of the scaled free energy with respect to the relative magnitude x and the relative phase A leads to three possible states corresponding to x = 0, $(x\gt 0,A=\pi /2\,{\rm{or}}\,3\pi /2)$, and $(x\gt 0,A=0\,{\rm{or}}\,\pi )$ as long as $\tfrac{| a| }{b| {\tilde{{\rm{\Delta }}}}_{+}{| }^{2}}\gt 1/2$. The conditional inequality is selfconsistently satisfied by $| {\tilde{{\rm{\Delta }}}}_{+}|$ at each minimum. The first two minima correspond to nodal states, while the last one is a pair of nodeless states which are equivalent to those in equation (39) for x = 1. The lower bound on the dimensionless ratio $\tfrac{| a| }{b| {{\rm{\Delta }}}_{+}{| }^{2}}$ originates from the competition between the repulsion and condensation energy. By comparing the free energy at these minima we conclude that the nodeless state is realized at the global minimum of the free energy. Thus the leading instability associated with the flow to strong coupling emerging from the ${\mathrm{III}}_{(\mathrm{BCS})}$ vertex is to a fully gapped chiral state with odd angular momentum and with ${\rm{\Delta }}(\vec{k})$ proportional to the unit matrix in pseudospin space (i.e. ${d}_{1}={d}_{3}=0$). Further, we note that this is the only state that involves solely intra-band pairing, and is smoothly connected to paired states in both the conduction and valence band. Thus this state is expected to be most robust to chemical potential disorder [42]. Indeed, it may even be enhanced by disorder through the mechanism discussed in [58–60].

Applying the above analysis to the SCT originating from ${\mathrm{IV}}_{(\mathrm{BCS})}$, we find that pairing susceptibilities for γ₁ and γ₂ vertices are enhanced identically, while γ₀ and γ₃ are unaffected. From the symmetry of ${{\rm{\Delta }}}_{\mu }{\gamma }_{\mu }$ we identify the γ₁ (γ₂) pairing vertex as a singlet (triplet). The triplet pairing is distinguished from the one associated with ${\mathrm{III}}_{(\mathrm{BCS})}$ with the aid of equation (38), and it corresponds to ${d}_{\mu \ne 3}(\widehat{K})=0$ and ${d}_{3}(\widehat{K})={{\rm{\Delta }}}_{3}(\widehat{K})$. The singlet corresponds to ${\bf{d}}(\widehat{K})=0$ and ${d}_{0}(\widehat{K})=-{\rm{i}}{{\rm{\Delta }}}_{1}(\widehat{K})$. While the quantum scaling dimensions of the singlet and triplet pairings are identical due to the BCS-U(1) symmetry, the pairings occur in distinct angular momentum channels: the singlet (triplet) pairing occurs in even (odd) angular momentum channel. Since there is no reason why the bare couplings (which set the initial conditions for the RG flow) should be equal in distinct angular momentum channels, the apparent degeneracy will thus be broken by the initial conditions, and the leading instability will occur in the channel ${d}_{\mu \ne 3}(\widehat{K})=0$ and ${d}_{3}(\widehat{K})={{\rm{\Delta }}}_{3}(\widehat{K})$ if the most attractive bare coupling is an odd angular momentum channel, and in the channel ${\bf{d}}(\widehat{K})=0$ and ${d}_{0}(\widehat{K})=-{\rm{i}}{{\rm{\Delta }}}_{1}(\widehat{K})$ if the most attractive bare coupling is in an even angular momentum channel. In the case where the leading instability is in a channel with non-zero J, the $\pm J$ channels will again be degenerate, and one may have either fully gapped chiral superconductors or gapless non-chiral superconductors. An analysis of the most likely symmetry broken state resulting from the instability driven by the γ₂ pairing vertex indicates a fully gapped p-wave state as obtained above. However, it is distinguished from the same through the nontrivial matrix structure of the order parameter in the pseudospin space since ${d}_{3}\ne 0$. The most likely candidate for the symmetry broken state for the singlet pairing is a uniform s-wave superconductor. We note that these states involve interband pairing [42] and thus will likely be rapidly disrupted by chemical potential disorder, unlike the state arising from the flow out of ${\mathrm{III}}_{(\mathrm{BCS})}$.

In the forward scattering channel, even at zero energy, the net incoming momentum is generically non-zero as the momenta of typical incoming states are not anti-parallel in ${\bf{K}}$-space. In order to transfer the finite momentum of the incoming states to the outgoing states, the virtual excitations must carry a net momentum. Therefore, scattering processes that favor virtual excitations with zero net momentum are suppressed for forward scattering channels at low energy, as is the case for the BCS diagram. The internal loop in the other three diagrams in figure 4 carry a net momentum, and renormalizes U_j at leading order in Λ/κ. The FS couplings flow according to

$$\begin{eqnarray}&&{\partial }_{{\ell }}{U}_{1;{\rm{J}}}=-\epsilon {U}_{1;{\rm{J}}},\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray}&&{\partial }_{{\ell }}{U}_{2+;{\rm{J}}}=-{U}_{2+;{\rm{J}}}\left[\epsilon +\displaystyle \frac{2}{\pi }{U}_{2+;{\rm{J}}}\right]-\displaystyle \frac{2}{\pi }{U}_{2-;{\rm{J}}}^{2},\end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}&&{\partial }_{{\ell }}{U}_{2-;{\rm{J}}}=-{U}_{2-;{\rm{J}}}\left[\epsilon +\displaystyle \frac{4}{\pi }{U}_{2+;{\rm{J}}}\right],\end{eqnarray} \tag{ 46 }$$

$$\begin{eqnarray}&&{\partial }_{{\ell }}{U}_{3;{\rm{J}}}=-{U}_{3;{\rm{J}}}\left[\epsilon +\displaystyle \frac{4}{\pi }{U}_{3;{\rm{J}}}\right].\end{eqnarray} \tag{ 47 }$$

It is interesting to note that when all four couplings are repulsive, they are irrelevant. Moreover, the U₁ vertex which mediates scatterings between total densities in momentum space, ${\sum }_{i=\mathrm{1,2}}{c}_{i}^{\dagger }({\bf{K}}){c}_{i}({\bf{K}})$, remains unrenormalized.

There are 2³ solutions to ${\partial }_{{\ell }}{U}_{i;{\rm{J}}}=0$, which correspond to distinct combinations of the fixed points of ${U}_{i;{\rm{J}}}$. The non-Gaussian fixed points are listed in table 5. There are 3 critical, 3 bicritical, and 1 tricritical fixed points in the four dimensional coupling space. Among the critical fixed points, the FS-U(1) symmetry emerges at ${{\rm{I}}}_{(\mathrm{FS})}$, due to the vanishing of ${U}_{2-;{\rm{J}}}$. Since it is protected by an emergent symmetry, in the rest of the section we focus on the ${U}_{2-;{\rm{J}}}=0$ subspace. Two more interacting fixed points are present in the subspace, both of which lose a relevant direction due to the projection to the subspace. Thus, in the ${U}_{2-;{\rm{J}}}=0$ subspace there are 2 critical (${{\rm{I}}}_{(\mathrm{FS})}$, ${\mathrm{IV}}_{(\mathrm{FS})}$) and 1 bicritical (${\mathrm{VII}}_{(\mathrm{FS})}$) fixed points.

Table 5.  The non-Gaussian fixed points for the couplings in the forward scattering channel, in units of $\pi \epsilon$. In the four dimensional coupling space ${{\rm{I}}}_{(\mathrm{FS})}$ - ${\mathrm{III}}_{(\mathrm{FS})}$ are critical, ${\mathrm{IV}}_{(\mathrm{FS})}$ - ${\mathrm{VI}}_{(\mathrm{FS})}$ are bicritical, and ${\mathrm{VII}}_{(\mathrm{FS})}$ is tricritical.

#	${U}_{1;{\rm{J}}}^{* }$	${U}_{2+;{\rm{J}}}^{* }$	${U}_{2-;{\rm{J}}}^{* }$	${U}_{3;{\rm{J}}}^{* }$	Tune
${{\rm{I}}}_{(\mathrm{FS})}$	0	0	0	$-1/4$	${U}_{3;{\rm{J}}}$
${\mathrm{II}}_{(\mathrm{FS})}$	0	$-1/4$	$-1/4$	0	$({U}_{2+;{\rm{J}}}+{U}_{2-;{\rm{J}}})$
${\mathrm{III}}_{(\mathrm{FS})}$	0	$-1/4$	1/4	0	$({U}_{2+;{\rm{J}}}-{U}_{2-;{\rm{J}}})$
${\mathrm{IV}}_{(\mathrm{FS})}$	0	$-1/2$	0	0	${U}_{2+;{\rm{J}}},{U}_{2-;{\rm{J}}}$
${{\rm{V}}}_{(\mathrm{FS})}$	0	$-1/4$	$-1/4$	$-1/4$	${U}_{3;{\rm{J}}},({U}_{2+;{\rm{J}}}+{U}_{2-;{\rm{J}}})$
${\mathrm{VI}}_{(\mathrm{FS})}$	0	$-1/4$	1/4	$-1/4$	${U}_{3;{\rm{J}}},({U}_{2+;{\rm{J}}}-{U}_{2-;{\rm{J}}})$
${\mathrm{VII}}_{(\mathrm{FS})}$	0	$-1/2$	0	$-1/4$	${U}_{3;{\rm{J}}},{U}_{2+;{\rm{J}}},{U}_{2-;{\rm{J}}}$

The critical points are expected to separate the Weyl-loop semi-metal phase from symmetry broken phases, which are realized by tuning a single parameter. In the ${U}_{2-;{\rm{J}}}=0$ subspace the two critical points ${{\rm{I}}}_{(\mathrm{FS})}$ and ${\mathrm{IV}}_{(\mathrm{FS})}$ are achieved by tuning ${U}_{3;{\rm{J}}}$ and ${U}_{2+;{\rm{J}}}$, respectively. On tuning these coupling beyond their critical values the system is set to flow towards two distinct strong coupling fixed points. In this section we determine the stable RG flow trajectories that lead to those fixed points, which will help us identify the possible symmetry broken states that can be realized at finite (or strong) coupling. We first discuss the SCT originating from ${{\rm{I}}}_{(\mathrm{FS})}$, followed by ${\mathrm{IV}}_{(\mathrm{FS})}$.

Since all couplings but ${U}_{3;{\rm{J}}}$ vanish at ${{\rm{I}}}_{(\mathrm{FS})}$, it is easy to see that the SCT must lie along ${U}_{3;{\rm{J}}}\lt -\pi \epsilon /4$. This trajectory is stable against small perturbations since the Gaussian fixed points of ${U}_{i\ne 3;{\rm{J}}}$ are stable. The effective interaction along the SCT,

$$\begin{eqnarray}&&{L}_{{\rm{I}}}^{(\mathrm{FS})}=\displaystyle \frac{{U}_{3}({\widehat{K}}_{1},{\widehat{K}}_{2})}{\kappa }\,({\psi }^{\dagger }({{\bf{K}}}_{1}){\sigma }_{3}\psi ({{\bf{K}}}_{1}))({\psi }^{\dagger }({{\bf{K}}}_{2}){\sigma }_{3}\psi ({{\bf{K}}}_{2})),\end{eqnarray} \tag{ 48 }$$

indicates that particle-hole pairs ${\psi }^{\dagger }({\bf{K}}){\sigma }_{3}\psi ({\bf{K}})$ are progressively favored as $| {U}_{3}({\widehat{K}}_{1},{\widehat{K}}_{2})|$ increases. Condensation of ${\psi }^{\dagger }({\bf{K}}){\sigma }_{3}\psi ({\bf{K}})$ produces a mass term for the fermions, which gaps out the fermionic excitations. An identical analysis for the SCT originating at ${\mathrm{IV}}_{(\mathrm{FS})}$ reveals a stable SCT along ${U}_{2+;{\rm{J}}}\lt -\pi \epsilon /2$, and effective interaction on the SCT,

$$\begin{eqnarray}&&{L}_{\mathrm{IV}}^{(\mathrm{FS})}=\displaystyle \frac{{U}_{2+}({\widehat{K}}_{1},{\widehat{K}}_{2})}{\kappa }\,[({\psi }^{\dagger }({{\bf{K}}}_{1}){\sigma }_{1}\psi ({{\bf{K}}}_{1}))({\psi }^{\dagger }({{\bf{K}}}_{2}){\sigma }_{1}\psi ({{\bf{K}}}_{2}))+({\psi }^{\dagger }({{\bf{K}}}_{1}){\sigma }_{2}\psi ({{\bf{K}}}_{1}))({\psi }^{\dagger }({{\bf{K}}}_{2}){\sigma }_{2}\psi ({{\bf{K}}}_{2}))].\end{eqnarray} \tag{ 49 }$$

In this subsection we compute the anomalous scaling dimension of susceptibility along the two SCTs identified above. We also identify the symmetry broken strong coupling fixed points to which the SCTs flow.

Let us consider an insertion of particle-hole pairs carrying a net momentum $\vec{P}$ on the plane of the loop,

$$\begin{eqnarray}&&{S}_{2,\mu }^{({DW})}(\vec{P})={\int }_{{\rm{\Lambda }}}{\rm{d}}K{{\rm{\Phi }}}_{\mu }(\vec{P};\widehat{K})\bar{\psi }(P+K){\gamma }_{\mu }\psi (K)+{\rm{h}}.{\rm{c}}.\end{eqnarray} \tag{ 50 }$$

Here the four-dimensional vector $P\equiv (0,\vec{P},0)$. In contrast to the pairing susceptibility, the density wave susceptibility obtains quantum correction from the two diagrams in figure 7. At low energy quantum corrections to the susceptibility at any finite $\vec{P}$ are suppressed, compared to $\vec{P}=0$. This is because of a lack of phase space for both the virtual particle and hole to be near the loop when $\vec{P}\ne 0$. Thus we consider the susceptibility for density wave states with $\vec{P}=0$.

Zoom In Zoom Out Reset image size

The source ${{\rm{\Phi }}}_{\mu }(0;\widehat{K})$ scales as

$$\begin{eqnarray}&&{\partial }_{{\ell }}{{\rm{\Phi }}}_{0;{\rm{J}}}=\eta {{\rm{\Phi }}}_{\mu ;{\rm{J}}},\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}&&{\partial }_{{\ell }}{{\rm{\Phi }}}_{1;{\rm{J}}}={{\rm{\Phi }}}_{1;{\rm{J}}}\left[\eta -\displaystyle \frac{1}{\pi }({U}_{2+;{\rm{J}}}+{U}_{2-;{\rm{J}}})\right],\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}&&{\partial }_{{\ell }}{{\rm{\Phi }}}_{2;{\rm{J}}}={{\rm{\Phi }}}_{2;{\rm{J}}}\left[\eta -\displaystyle \frac{1}{\pi }({U}_{2+;{\rm{J}}}-{U}_{2-;{\rm{J}}})\right],\end{eqnarray} \tag{ 53 }$$

$$\begin{eqnarray}&&{\partial }_{{\ell }}{{\rm{\Phi }}}_{3;{\rm{J}}}={{\rm{\Phi }}}_{3;{\rm{J}}}\left[\eta -\displaystyle \frac{2}{\pi }{U}_{3;{\rm{J}}}\right],\end{eqnarray} \tag{ 54 }$$

where $\eta =2-\epsilon$ is the bare scaling dimension of ${{\rm{\Phi }}}_{\mu ;{\rm{J}}}$. Thus at the critical point ${{\rm{I}}}_{(\mathrm{FS})}$ and the ensuing SCT, only ${{\rm{\Phi }}}_{3;{\rm{J}}}$ is enhanced, while the scaling dimension of ${{\rm{\Phi }}}_{\mu \ne 3;{\rm{J}}}$ remain unchanged. For ${\mathrm{IV}}_{(\mathrm{FS})}$ and the associated SCT ${{\rm{\Phi }}}_{1;{\rm{J}}}$ and ${{\rm{\Phi }}}_{2;{\rm{J}}}$ are equally enhanced. This degeneracy is protected by the FS-U(1) symmetry. Again the flow equations do not distinguish between angular momentum channel, and the leading instability will be determined by which angular momentum channel has the largest bare couplings (and J = 0 is allowed). There is however a constraint, namely that the overall Hamiltonian must be Hermitian. This then enforces that the order parameter must be real i.e. either the instability will be in the J = 0 channel, or if the instability is in a channel with non-zero angular momentum then a real superposition of $\pm J$ states must arise (i.e. $\propto \,\sin J\theta$ or $\cos J\theta$).

The flow out of ${{\rm{I}}}_{(\mathrm{FS})}$ is associated with the condensation of ${\psi }^{\dagger }{\sigma }_{3}\psi$. If this occurs in a channel with non-zero J then it leads to a low energy Hamiltonian $H\sim (| \vec{K}| -\kappa ){\sigma }_{1}+{K}_{z}{\sigma }_{2}+{{\rm{\Delta }}}_{3}\sin (J\theta -{\theta }_{3}){\sigma }_{3}$, where Δ₃ and ${\theta }_{3}$ are real parameters. Such an instability opens a gap almost everywhere on the Weyl loop, with nodes surviving at $\theta ={\theta }_{3}+n\pi /J$ (integer n) i.e. this is a gap opening instability that simultaneously breaks the θ-rotational symmetry. It also breaks several discrete symmetries, in particular the antiunitary symmetries ${{ \mathcal P }}_{z}{{ \mathcal P }}_{{\bf{K}}}$ (${{ \mathcal P }}_{{\bf{K}}}$) for even (odd) J, ${{ \mathcal P }}_{z}{{ \mathcal P }}_{0}$, and the mirror symmetry ${{ \mathcal P }}_{z}$. However the symmetries ${{ \mathcal P }}_{{\bf{K}}}$ (${{ \mathcal P }}_{z}{{ \mathcal P }}_{{\bf{K}}}$) for even (odd) J, and ${{ \mathcal P }}_{0}$ are preserved. Meanwhile, if this occurs in a channel with J = 0 then the gap function is independent of θ, and the θ-rotation symmetry is preserved, while the discrete symmetries identified above are still broken. Condensation in the J = 0 channel uniformly gaps out the Weyl loop, with an effective Hamiltonian of the form $H\sim (| \vec{K}| -\kappa ){\sigma }_{1}+{K}_{z}{\sigma }_{2}+{{\rm{\Delta }}}_{3}{\sigma }_{3}$ (real ${{\rm{\Delta }}}_{3}$) and a dispersion $E\sim \pm \sqrt{{(| \vec{K}| -\kappa )}^{2}+{K}_{z}^{2}+{{\rm{\Delta }}}_{3}^{2}}$. Since this is a gap opening instability that preserves an antiunitary symmetry ${{ \mathcal P }}_{{\bf{K}}}$, which squares to −1, one can ask whether the resulting insulating state is topological or trivial. To address this issue, note that the Weyl loop can be obtained by starting with (spinless) graphene in the y – z plane with it's two (opposite sense) Dirac points located at $\pm \kappa \hat{y}$ and rotating it through 180 degrees about the $\hat{z}$ axis. Gapping out the two Dirac points of (spinless) graphene with a mass term of the same sign on each Dirac point yields a trivial insulator, and rotating a trivial two dimensional insulator through 180 degrees should yield a trivial three dimensional insulator. Nonetheless, we note that on the plane of the Weyl-loop ${{ \mathcal P }}_{{\bf{K}}}$ has the interesting property of mapping the region outside the loop to its interior, which is an unusual implementation of an anti-unitary symmetry that does not appear to fit naturally into the existing classifications.

The flow out of critical point ${\mathrm{IV}}_{(\mathrm{FS})}$ is associated with the condensation of either ${\psi }^{\dagger }{\sigma }_{1}\psi$ or ${\psi }^{\dagger }{\sigma }_{2}\psi$. If this occurs in the J = 0 channel it leads to an effective Hamiltonian of the form $H\sim (| \vec{K}| -\kappa )$ ${\sigma }_{1}+{K}_{z}{\sigma }_{2}+{{\rm{\Delta }}}_{1}$ ${\sigma }_{1}+{{\rm{\Delta }}}_{2}{\sigma }_{2}$ where ${{\rm{\Delta }}}_{\mathrm{1,2}}$ are real parameters. Such a perturbation shifts the radius of the Weyl loop to $\kappa -{{\rm{\Delta }}}_{1}$, and shifts it into the plane with ${K}_{z}=-{{\rm{\Delta }}}_{2}$. A non-zero ${{\rm{\Delta }}}_{1}$ does not break any symmetries and can be absorbed into a redefinition of the Weyl loop radius κ. A non-zero ${{\rm{\Delta }}}_{2}$ breaks the mirror symmetry, and also the discrete antiunitary symmetries ${{ \mathcal P }}_{{\bf{K}}}$ and ${{ \mathcal P }}_{z}{{ \mathcal P }}_{0}$, but preserves ${{ \mathcal P }}_{z}{{ \mathcal P }}_{{\bf{K}}}$ and ${{ \mathcal P }}_{0}$ - it simply shifts the Weyl loop out of the k_z = 0 plane. More interesting is the situation where the instability develops in a channel with $J\ne 0$ such that the effective Hamiltonian takes the form $H\sim (| \vec{K}| -\kappa )$ ${\sigma }_{1}+{K}_{z}{\sigma }_{2}+{{\rm{\Delta }}}_{1}$ $\sin (J\theta +{\theta }_{1})$ ${\sigma }_{1}+{{\rm{\Delta }}}_{2}\cos (J\theta +{\theta }_{2}){\sigma }_{2}$ where ${\theta }_{\mathrm{1,2}}$ are constants and ${{\rm{\Delta }}}_{\mathrm{1,2}}$ are real. Non-zero ${{\rm{\Delta }}}_{1}$ will lead to a θ-dependent distortion of the nodal ring in the $\vec{K}$-plane, whereas non-zero ${{\rm{\Delta }}}_{2}$ will lead to a θ dependent distortion perpendicular to the x-y plane. These order parameters break the θ-rotation symmetry, and correspond to Pomeranchuk instabilities. The competition between ${{\rm{\Delta }}}_{1}$ and ${{\rm{\Delta }}}_{2}$ (in particular whether both ${{\rm{\Delta }}}_{1}$ and ${{\rm{\Delta }}}_{2}$ are non-zero, or only one is) will be determined by a Landau Ginzburg calculation similar to those that have already been performed. Note that all of these are gapless phases which continue to have a loop of Dirac nodes.

Contraction of two vertices in the BCS channel leads to the quantum correction,

$$\begin{eqnarray}{\delta {S}_{{\rm{int}}}^{(\mu ,\nu )}| }_{{BCS}} & = & -\displaystyle \frac{4}{2{\kappa }^{2}}\displaystyle \int \displaystyle \prod _{n=1}^{4}{{\rm{d}}{K}}_{n}\,{{\rm{d}}{K}}_{n}^{\prime }\,{\delta }^{(4)}({K}_{1}-{K}_{2}+{K}_{3}-{K}_{4})\,{\delta }^{(4)}({K}_{1}^{\prime }-{K}_{2}^{\prime }+{K}_{3}^{\prime }-{K}_{4}^{\prime })\\ & & \times {\gamma }_{\mu }^{{a}_{1},{b}_{1}}{\gamma }_{\mu }^{{a}_{2},{b}_{2}}{\gamma }_{\nu }^{{a}_{1}^{\prime },{b}_{1}^{\prime }}{\gamma }_{\nu }^{{a}_{2}^{\prime },{b}_{2}^{\prime }}\,{u}_{\mu }({\widehat{K}}_{1},{\widehat{K}}_{2},{\widehat{K}}_{3},{\widehat{K}}_{4})\,{u}_{\nu }({\widehat{K}}_{1}^{\prime },{\widehat{K}}_{2}^{\prime },{\widehat{K}}_{3}^{\prime },{\widehat{K}}_{4}^{\prime })\\ & & \times \langle {\psi }_{{b}_{2}^{\prime }}({K}_{4}^{\prime }){\bar{\psi }}_{{a}_{1}}({K}_{1})\rangle \langle {\psi }_{{b}_{1}^{\prime }}({K}_{2}^{\prime }){\bar{\psi }}_{{a}_{2}}({K}_{3})\rangle {\bar{\psi }}_{{a}_{2}^{\prime }}({K}_{3}^{\prime }){\psi }_{{b}_{1}}({K}_{2}){\bar{\psi }}_{{a}_{1}^{\prime }}({K}_{1}^{\prime }){\psi }_{{b}_{2}}({K}_{4}).\end{eqnarray} \tag{ A1 }$$

Here for notational and computational convenience we have used u_μ to identify the coupling functions for the ${(\bar{\psi }{\gamma }_{\mu }\psi )}^{2}$ vertex. In particular

$$\begin{eqnarray}&&{u}_{0}={g}_{1},\quad {u}_{1}=-({g}_{2+}\,+\,{g}_{2-}),\quad {u}_{2}=-({g}_{2+}\,-\,{g}_{2-}),\quad {u}_{3}={g}_{3},\end{eqnarray} \tag{ A2 }$$

where we have suppressed the dependence on loop-coordinates on both sides. Utilizing the definition of the propagator in equation (15), and integrating over ${K}_{2}^{\prime }$, ${K}_{4}^{\prime }$, and K₃ leads to

$$\begin{eqnarray}{\delta {S}_{{\rm{int}}}^{(\mu ,\nu )}| }_{{BCS}} & = & -\displaystyle \frac{2}{{\kappa }^{2}}\displaystyle \int {{\rm{d}}{K}}_{2}{{\rm{d}}{K}}_{4}{{\rm{d}}{K}}_{1}^{\prime }{{\rm{d}}{K}}_{3}^{\prime }\,{(2\pi )}^{4}{\delta }^{(4)}({K}_{3}^{\prime }-{K}_{2}+{K}_{1}^{\prime }-{K}_{4})\\ & & \times {{\rm{\Upsilon }}}_{\mu \nu }^{{a}_{2}^{\prime },{b}_{1};{a}_{1}^{\prime },{b}_{2}}({K}_{3}^{\prime },{K}_{2},{K}_{1}^{\prime },{K}_{4})\,\,{\bar{\psi }}_{{a}_{2}^{\prime }}({K}_{3}^{\prime }){\psi }_{{b}_{1}}({K}_{2}){\bar{\psi }}_{{a}_{1}^{\prime }}({K}_{1}^{\prime }){\psi }_{{b}_{2}}({K}_{4}),\end{eqnarray} \tag{ A3 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Upsilon }}}_{\mu \nu }({K}_{1},{K}_{2},{K}_{3},{K}_{4}) & = & \displaystyle \int {\rm{d}}{Q}\,[{\gamma }_{\nu }\,G(Q)\,{\gamma }_{\mu }]\circ [{\gamma }_{\nu }\,G({K}_{4}+{K}_{2}-Q)\,{\gamma }_{\mu }]\\ & & \times {u}_{\mu }(\widehat{Q},{\widehat{K}}_{4},({K}_{2}\widehat{+{K}_{4}}-Q),{\widehat{K}}_{2})\,{u}_{\nu }({\widehat{K}}_{1},({K}_{2}\widehat{+{K}_{4}}-Q),{\widehat{K}}_{3},\widehat{Q}),\end{array}\end{eqnarray} \tag{ A4 }$$

with $A\,\circ \,B\equiv {A}_{{a}_{1},{b}_{1}}{B}_{{a}_{2},{b}_{2}}$, and $({K}_{2}\widehat{\,+\,{K}_{4}}-Q)$ being the unit vector along $({\vec{K}}_{2}+{\vec{K}}_{4}-\vec{Q})$. Here the internal momentum ${\bf{Q}}$ is restricted to lie within the shell being eliminated (c f figure 2).

The BCS channel is defined by $({\widehat{K}}_{1},{\widehat{K}}_{2})=(-{\widehat{K}}_{3},-{\widehat{K}}_{4})$. Because we are interested in the IR, we set the external frequency to 0, and external momenta to lie on the loop. Combined with the angular constraint, this implies ${\vec{K}}_{2}+{\vec{K}}_{4}=0={\vec{K}}_{1}+{\vec{K}}_{3}$. Therefore,

$$\begin{eqnarray}{{\rm{\Upsilon }}}_{\mu \nu }({\widehat{K}}_{1},{\widehat{K}}_{2},-{\widehat{K}}_{1},-{\widehat{K}}_{2}) & = & \displaystyle \int {\rm{d}}{Q}\,[{\gamma }_{\nu }\,G(Q)\,{\gamma }_{\mu }]\circ [{\gamma }_{\nu }\,G(-Q)\,{\gamma }_{\mu }]\\ & & \times {u}_{\mu }(\widehat{Q},-{\widehat{K}}_{2},-\widehat{Q},{\widehat{K}}_{2})\,{u}_{\nu }({\widehat{K}}_{1},-\widehat{Q},-{\widehat{K}}_{1},\widehat{Q}).\end{eqnarray} \tag{ A5 }$$

Let us define

$$\begin{eqnarray}&&{{\rm{\Lambda }}}^{-\epsilon }{V}_{\mu }^{\prime }({\theta }_{1}-{\theta }_{2})={u}_{\mu }({\widehat{K}}_{1},{\widehat{K}}_{2},-{\widehat{K}}_{1},-{\widehat{K}}_{2}),\end{eqnarray} \tag{ A6 }$$

where ${\theta }_{i}$ is the angle the loop momentum $\kappa {\widehat{K}}_{i}$ makes with respect to the x-axis. Thus, we obtain an equivalent expression to equation (A5) in terms of the angles,

$$\begin{eqnarray}{{\rm{\Upsilon }}}_{\mu \nu }({\theta }_{1},{\theta }_{2},{\theta }_{1}+\pi ,{\theta }_{2}+\pi ) & = & \kappa {{\rm{\Lambda }}}^{-2\epsilon }{\displaystyle \int }_{-\pi }^{\pi }\displaystyle \frac{{\rm{d}}\theta {\rm{d}}\varphi }{{(2\pi )}^{2}}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{{\rm{dq}}}_{0}}{2\pi }{\displaystyle \int }_{(1-{\rm{d}}{\ell }){\rm{\Lambda }}}^{{\rm{\Lambda }}}\displaystyle \frac{{\rm{dq}}\,q}{2\pi }\,{V}_{\mu }^{\prime }(\theta -{\theta }_{2}+\pi ){V}_{\nu }^{\prime }({\theta }_{1}-\theta -\pi )\\ & & \times [{\gamma }_{\nu }\,G({q}_{0},q,\varphi )\,{\gamma }_{\mu }]\circ [{\gamma }_{\nu }\,G(-{q}_{0},q,-\varphi )\,{\gamma }_{\mu }].\end{eqnarray} \tag{ A7 }$$

We note that $G({q}_{0},q,\varphi )$ does not have a specific parity under spacetime inversion. As a result, the integrand of equation (A7), up to terms that are even in φ equals,

$$\begin{eqnarray}&&\displaystyle \frac{{q}_{0}^{2}[{\gamma }_{\nu }{\gamma }_{0}{\gamma }_{\mu }]\,\circ \,[{\gamma }_{\nu }{\gamma }_{0}{\gamma }_{\mu }]-{q}^{2(2-\epsilon )}({\cos }^{2}\varphi \,[{\gamma }_{\nu }{\gamma }_{1}{\gamma }_{\mu }]\circ [{\gamma }_{\nu }{\gamma }_{1}{\gamma }_{\mu }]-{\sin }^{2}\varphi \,[{\gamma }_{\nu }{\gamma }_{2}{\gamma }_{\mu }]\circ [{\gamma }_{\nu }{\gamma }_{2}{\gamma }_{\mu }])}{{({q}_{0}^{2}+{q}^{2(2-\epsilon )})}^{2}},\end{eqnarray} \tag{ A8 }$$

The opposite sign for the γ₁ and γ₂ terms will lead to unequal quantum corrections to the γ₁ and γ₂ vertices, as we will see below. This is a manifestation of the absence of ${{ \mathcal R }}_{\varphi }$ symmetry for the interaction vertices, in general.

Noting that θ decouples from rest of the internal variables, it is simplest to integrate in the order φ, q, and q₀. We cannot explicitly integrate over θ, but we can simplify the ${V}_{\mu }^{\prime }$ dependence of the quantum correction by expressing the coupling functions in terms of angular momentum harmonics. The inverse Fourier transform of ${V}_{\mu }^{\prime }(\theta )$ is given by

$$\begin{eqnarray}&&{V}_{\mu }^{\prime }(\theta )=\displaystyle \sum _{J=-\infty }^{\infty }\,{{\rm{e}}}^{-{\rm{i}}\theta J}\,{\tilde{V}}_{\mu ;{\rm{J}}},\end{eqnarray} \tag{ A9 }$$

which leads to

$$\begin{eqnarray}&&{\int }_{-\pi }^{\pi }\displaystyle \frac{{\rm{d}}\theta }{2\pi }\,{V}_{\mu }^{\prime }(\theta -{\theta }_{2}+\pi ){V}_{\nu }^{\prime }({\theta }_{1}-\theta -\pi )=\displaystyle \sum _{J}{{\rm{e}}}^{-{\rm{i}}({\theta }_{1}-{\theta }_{2})J}\,{\tilde{V}}_{\mu ;{\rm{J}}}{\tilde{V}}_{\nu ;{\rm{J}}.}\end{eqnarray} \tag{ A10 }$$

Therefore, equation (A7) evaluates to,

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Upsilon }}}_{\mu \nu }({\theta }_{1},{\theta }_{2},{\theta }_{1}+\pi ,{\theta }_{2}+\pi )=-d{\ell }\,\displaystyle \frac{\kappa {{\rm{\Lambda }}}^{-\epsilon }}{16\pi }\displaystyle \sum _{J}{{\rm{e}}}^{-{\rm{i}}({\theta }_{1}-{\theta }_{2})J}\,{\tilde{V}}_{\mu ;{\rm{J}}}{\tilde{V}}_{\nu ;{\rm{J}}}\\ \qquad \times [({\gamma }_{\nu }{\gamma }_{1}{\gamma }_{\mu })\,\,\circ \,\,({\gamma }_{\nu }{\gamma }_{1}{\gamma }_{\mu })-({\gamma }_{\nu }{\gamma }_{2}{\gamma }_{\mu })\circ ({\gamma }_{\nu }{\gamma }_{2}{\gamma }_{\mu })-2({\gamma }_{\nu }{\gamma }_{0}{\gamma }_{\mu })\,\circ \,({\gamma }_{\nu }{\gamma }_{0}{\gamma }_{\mu })].\end{array}\end{eqnarray} \tag{ A11 }$$

This leads to quantum corrections to the BCS channel,

$$\begin{eqnarray}{\delta {S}_{{\rm{int}}}^{(\mu ,\nu )}| }_{\mathrm{BCS}} & = & \displaystyle \frac{{{\rm{\Lambda }}}^{-\epsilon }{\rm{d}}{\ell }}{8\pi \kappa }\displaystyle \sum _{J}\displaystyle \int \displaystyle \prod _{n=1}^{4}{{\rm{d}}{K}}_{n}\,{(2\pi )}^{4}{\delta }^{(4)}({K}_{1}-{K}_{2}+{K}_{3}-{K}_{4})\,\delta ({{\bf{K}}}_{1}+{{\bf{K}}}_{3})\,\delta ({{\bf{K}}}_{2}+{{\bf{K}}}_{4})\\ & & \times {{\rm{e}}}^{-{\rm{i}}({\theta }_{1}-{\theta }_{2})J}\,{\tilde{V}}_{\mu ;{\rm{J}}}{\tilde{V}}_{\nu ;{\rm{J}}}[\{\bar{\psi }({K}_{1})\,{\gamma }_{\mu }{\gamma }_{1}{\gamma }_{\nu }\,\psi ({K}_{2})\}\{\bar{\psi }({K}_{3})\,{\gamma }_{\mu }{\gamma }_{1}{\gamma }_{\nu }\,\psi ({K}_{4})\}\\ & & -\{\bar{\psi }({K}_{1})\,{\gamma }_{\mu }{\gamma }_{2}{\gamma }_{\nu }\,\psi ({K}_{2})\}\{\bar{\psi }({K}_{3})\,{\gamma }_{\mu }{\gamma }_{2}{\gamma }_{\nu }\,\psi ({K}_{4})\}\\ & & -2\{\bar{\psi }({K}_{1})\,{\gamma }_{\mu }{\gamma }_{0}{\gamma }_{\nu }\,\psi ({K}_{2})\}\{\bar{\psi }({K}_{3})\,{\gamma }_{\mu }{\gamma }_{0}{\gamma }_{\nu }\,\psi ({K}_{4})\}].\end{eqnarray} \tag{ A12 }$$

The net quantum correction is obtained by summing over μ and ν,

$$\begin{eqnarray}&&\delta {S}_{{\rm{int}}}^{(\mathrm{BCS})}={\left.\displaystyle \sum _{\mu ,\nu =0}^{3}\delta {S}_{{\rm{int}}}^{(\mu ,\nu )}\right|}_{\mathrm{BCS}}\\ &&\quad =-\displaystyle \frac{{{\rm{\Lambda }}}^{-\epsilon }{\rm{d}}{\ell }}{4\pi \kappa }\displaystyle \sum _{J}\displaystyle \int \displaystyle \prod _{n=1}^{4}{{\rm{d}}{K}}_{n}\,{(2\pi )}^{4}{\delta }^{(4)}({K}_{1}-{K}_{2}+{K}_{3}-{K}_{4})\,\delta ({{\bf{K}}}_{1}+{{\bf{K}}}_{3})\,\delta ({{\bf{K}}}_{2}+{{\bf{K}}}_{4})\,{{\rm{e}}}^{-{\rm{i}}({\theta }_{1}-{\theta }_{2})J}\\ &&\quad \times [\{2({\tilde{V}}_{3;{\rm{J}}}{\tilde{V}}_{0;{\rm{J}}}-{\tilde{V}}_{1;{\rm{J}}}{\tilde{V}}_{2;{\rm{J}}})+({\tilde{V}}_{0;{\rm{J}}}{\tilde{V}}_{2;{\rm{J}}}+{\tilde{V}}_{3;{\rm{J}}}{\tilde{V}}_{2;{\rm{J}}}-{\tilde{V}}_{3;{\rm{J}}}{\tilde{V}}_{1;{\rm{J}}}-{\tilde{V}}_{0;{\rm{J}}}{\tilde{V}}_{1;{\rm{J}}})\}\,{(\bar{\psi }\psi )}^{2}\\ &&\quad +\{({({\tilde{V}}_{3;{\rm{J}}})}^{2}+{({\tilde{V}}_{0;{\rm{J}}})}^{2}+{({\tilde{V}}_{1;{\rm{J}}})}^{2}+{({\tilde{V}}_{2;{\rm{J}}})}^{2})+({\tilde{V}}_{0;{\rm{J}}}{\tilde{V}}_{2;{\rm{J}}}+{\tilde{V}}_{3;{\rm{J}}}{\tilde{V}}_{2;{\rm{J}}}-{\tilde{V}}_{0;{\rm{J}}}{\tilde{V}}_{1;{\rm{J}}}-{\tilde{V}}_{3;{\rm{J}}}{\tilde{V}}_{1;{\rm{J}}})\}\,{(\bar{\psi }{\gamma }_{0}\psi )}^{2}\\ &&\quad +\left\{2({\tilde{V}}_{0;{\rm{J}}}{\tilde{V}}_{1;{\rm{J}}}-{\tilde{V}}_{3;{\rm{J}}}{\tilde{V}}_{2;{\rm{J}}})+({\tilde{V}}_{1;{\rm{J}}}{\tilde{V}}_{2;{\rm{J}}}-{\tilde{V}}_{3;{\rm{J}}}{\tilde{V}}_{0;{\rm{J}}})-\displaystyle \frac{1}{2}({({\tilde{V}}_{3;{\rm{J}}})}^{2}+{({\tilde{V}}_{0;{\rm{J}}})}^{2}+{({\tilde{V}}_{1;{\rm{J}}})}^{2}+{({\tilde{V}}_{2;{\rm{J}}})}^{2})\right\}{(\bar{\psi }{\gamma }_{1}\psi )}^{2}\\ &&\quad \left.+\left\{2({\tilde{V}}_{0;{\rm{J}}}{\tilde{V}}_{2;{\rm{J}}}-{\tilde{V}}_{3;{\rm{J}}}{\tilde{V}}_{1;{\rm{J}}})+({\tilde{V}}_{3;{\rm{J}}}{\tilde{V}}_{0;{\rm{J}}}-{\tilde{V}}_{1;{\rm{J}}}{\tilde{V}}_{2;{\rm{J}}})+\displaystyle \frac{1}{2}({({\tilde{V}}_{3;{\rm{J}}})}^{2}+{({\tilde{V}}_{0;{\rm{J}}})}^{2}+{({\tilde{V}}_{1;{\rm{J}}})}^{2}+{({\tilde{V}}_{2;{\rm{J}}})}^{2})\right\}{(\bar{\psi }{\gamma }_{2}\psi )}^{2}\right],\end{eqnarray} \tag{ A13 }$$

where the dependence of ψ on K_n is made implicit for notational convenience.

The forward scattering channels are renormalized by figures 4(b), (c), and (d). In order to compute their contributions it is convenient to distinguish between the FS and ES channels at an intermediate step, and unify them at the end through the relationship

$$\begin{eqnarray}&&\left(\begin{array}{c}{{U}_{0}^{\prime }}^{(\mathrm{ES})}\\ {{U}_{1}^{\prime }}^{(\mathrm{ES})}\\ {{U}_{2}^{\prime }}^{(\mathrm{ES})}\\ {{U}_{3}^{\prime }}^{(\mathrm{ES})}\end{array}\right)=\displaystyle \frac{1}{2}\left(\begin{array}{cccc}-1 & -1 & 1 & 1\\ -1 & 1 & -1 & 1\\ -1 & 1 & 1 & -1\\ -1 & -1 & -1 & -1\end{array}\right)\left(\begin{array}{c}{{U}_{0}^{\prime }}^{(\mathrm{FS})}\\ {{U}_{1}^{\prime }}^{(\mathrm{FS})}\\ {{U}_{2}^{\prime }}^{(\mathrm{FS})}\\ {{U}_{3}^{\prime }}^{(\mathrm{FS})}\end{array}\right),\end{eqnarray} \tag{ A14 }$$

where the primed and their unprimed counterparts defined in the main section are related in the same way as equations (A2) and (A6). The relationship between the ES and FS representation of the coupling functions in the forward scattering channel is defined on the loop through the equivalence of ${L}_{{\rm{int}}}^{(\mathrm{ES})}\equiv {\sum }_{\mu }{{U}_{\mu }^{\prime }}^{(\mathrm{ES})}$ $(\bar{\psi }{\gamma }_{\mu }\psi {)}_{(\mathrm{ES})}^{2}={\sum }_{\mu }{{U}_{\mu }^{\prime }}^{(\mathrm{FS})}(\bar{\psi }{\gamma }_{\mu }\psi {)}_{(\mathrm{FS})}^{2}\equiv {L}_{{\rm{int}}}^{(\mathrm{FS})},$ where the subscript in the ${\psi }^{4}$ term denotes the arrangement of the fermion momenta in accordance with the definition of the FS and ES channels.

At leading order in Λ/κ the external legs of the ${ZS}^{\prime}$ diagram are arranged as in the ES channel, while those of the ZS and P diagrams are arranged as in the FS channel. Repeating the computation presented above for the BCS diagram to the present set of diagrams leads to the quantum corrections

$$\begin{eqnarray}\delta {S}_{{\rm{int}}}^{({ZS}^{\prime} )} & = & \displaystyle \frac{{{\rm{\Lambda }}}^{-\epsilon }{\rm{d}}{\ell }}{4\pi \kappa }\displaystyle \sum _{J}\displaystyle \int \displaystyle \prod _{n=1}^{4}{{\rm{d}}{K}}_{n}\,{(2\pi )}^{4}{\delta }^{(4)}({K}_{1}-{K}_{2}+{K}_{3}-{K}_{4})\,\delta ({{\bf{K}}}_{1}-{{\bf{K}}}_{4})\,\delta ({{\bf{K}}}_{3}-{{\bf{K}}}_{2})\,{{\rm{e}}}^{-{\rm{i}}({\theta }_{1}-{\theta }_{2})J}\\ & & \times [\{2({\tilde{U}}_{0;{\rm{J}}}^{(\mathrm{ES})}{\tilde{U}}_{3;{\rm{J}}}^{(\mathrm{ES})}+{\tilde{U}}_{1;{\rm{J}}}^{(\mathrm{ES})}{\tilde{U}}_{2;{\rm{J}}}^{(\mathrm{ES})})+{\tilde{U}}_{1;{\rm{J}}}^{(\mathrm{ES})}{\tilde{U}}_{3;{\rm{J}}}^{(\mathrm{ES})}+{\tilde{U}}_{0;{\rm{J}}}^{(\mathrm{ES})}{\tilde{U}}_{2;{\rm{J}}}^{(\mathrm{ES})}+{\tilde{U}}_{2;{\rm{J}}}^{(\mathrm{ES})}{\tilde{U}}_{3;{\rm{J}}}^{(\mathrm{ES})}+{\tilde{U}}_{0;{\rm{J}}}^{(\mathrm{ES})}{\tilde{U}}_{1;{\rm{J}}}^{(\mathrm{ES})}\}\,{(\bar{\psi }\psi )}^{2}\\ & & +\left\{\displaystyle \sum _{\mu }{({\tilde{U}}_{\mu ;{\rm{J}}}^{(\mathrm{ES})})}^{2}+({\tilde{U}}_{0;{\rm{J}}}^{(\mathrm{ES})}+{\tilde{U}}_{3;{\rm{J}}}^{(\mathrm{ES})})({\tilde{U}}_{1;{\rm{J}}}^{(\mathrm{ES})}+{\tilde{U}}_{2;{\rm{J}}}^{(\mathrm{ES})})\right\}\,{(\bar{\psi }{\gamma }_{0}\psi )}^{2}\\ & & +\left\{\displaystyle \frac{1}{2}\displaystyle \sum _{\mu }{({\tilde{U}}_{\mu ;{\rm{J}}}^{(\mathrm{ES})})}^{2}+2({\tilde{U}}_{0;{\rm{J}}}^{(\mathrm{ES})}{\tilde{U}}_{1;{\rm{J}}}^{(\mathrm{ES})}+{\tilde{U}}_{2;{\rm{J}}}^{(\mathrm{ES})}{\tilde{U}}_{3;{\rm{J}}}^{(\mathrm{ES})})+({\tilde{U}}_{0;{\rm{J}}}^{(\mathrm{ES})}{\tilde{U}}_{3;{\rm{J}}}^{(\mathrm{ES})}+{\tilde{U}}_{1;{\rm{J}}}^{(\mathrm{ES})}{\tilde{U}}_{2;{\rm{J}}}^{(\mathrm{ES})})\right\}{(\bar{\psi }{\gamma }_{1}\psi )}^{2}\\ & & \left.+\left\{\displaystyle \frac{1}{2}\displaystyle \sum _{\mu }{({\tilde{U}}_{\mu ;{\rm{J}}}^{(\mathrm{ES})})}^{2}+2({\tilde{U}}_{0;{\rm{J}}}^{(\mathrm{ES})}{\tilde{U}}_{2;{\rm{J}}}^{(\mathrm{ES})}+{\tilde{U}}_{1;{\rm{J}}}^{(\mathrm{ES})}{\tilde{U}}_{3;{\rm{J}}}^{(\mathrm{ES})})+({\tilde{U}}_{0;{\rm{J}}}^{(\mathrm{ES})}{\tilde{U}}_{3;{\rm{J}}}^{(\mathrm{ES})}+{\tilde{U}}_{1;{\rm{J}}}^{(\mathrm{ES})}{\tilde{U}}_{2;{\rm{J}}}^{(\mathrm{ES})})\right\}{(\bar{\psi }{\gamma }_{2}\psi )}^{2}\right],\end{eqnarray} \tag{ A15 }$$

$$\begin{eqnarray}\delta {S}_{{\rm{int}}}^{({ZS})} & = & -\displaystyle \frac{{{\rm{\Lambda }}}^{-\epsilon }{\rm{d}}{\ell }}{2\pi \kappa }\displaystyle \sum _{J}\displaystyle \int \displaystyle \prod _{n=1}^{4}{{\rm{d}}{K}}_{n}\,{(2\pi )}^{4}{\delta }^{(4)}({K}_{1}-{K}_{2}+{K}_{3}-{K}_{4})\,\delta ({{\bf{K}}}_{1}-{{\bf{K}}}_{2})\,\delta ({{\bf{K}}}_{3}-{{\bf{K}}}_{4})\,{{\rm{e}}}^{-{\rm{i}}({\theta }_{1}-{\theta }_{2})J}\\ & & \times [2{({\tilde{U}}_{3;{\rm{J}}}^{(\mathrm{FS})})}^{2}\,{(\bar{\psi }\psi )}^{2}-{({\tilde{U}}_{1;{\rm{J}}}^{(\mathrm{FS})})}^{2}\,{(\bar{\psi }{\gamma }_{1}\psi )}^{2}-{({\tilde{U}}_{2;{\rm{J}}}^{(\mathrm{FS})})}^{2}\,{(\bar{\psi }{\gamma }_{2}\psi )}^{2}]\end{eqnarray} \tag{ A16 }$$

$$\begin{eqnarray}\delta {S}_{{\rm{int}}}^{(P)} & = & \displaystyle \frac{{{\rm{\Lambda }}}^{-\epsilon }{\rm{d}}{\ell }}{2\pi \kappa }\displaystyle \sum _{J}\displaystyle \int \displaystyle \prod _{n=1}^{4}{{\rm{d}}{K}}_{n}\,{(2\pi )}^{4}{\delta }^{(4)}({K}_{1}-{K}_{2}+{K}_{3}-{K}_{4})\,\delta ({{\bf{K}}}_{1}-{{\bf{K}}}_{2})\,\delta ({{\bf{K}}}_{3}-{{\bf{K}}}_{4})\,{{\rm{e}}}^{-{\rm{i}}({\theta }_{1}-{\theta }_{2})J}\\ & & \times \left[2{\tilde{U}}_{3;{\rm{J}}}^{(\mathrm{FS})}(\displaystyle \sum _{\mu }{\tilde{U}}_{\mu ;{\rm{J}}}^{(\mathrm{ES})})\,{(\bar{\psi }\psi )}^{2}-{\tilde{U}}_{1;{\rm{J}}}^{(\mathrm{FS})}({\tilde{U}}_{3;{\rm{J}}}^{(\mathrm{ES})}-{\tilde{U}}_{0;{\rm{J}}}^{(\mathrm{ES})}+{\tilde{U}}_{1;{\rm{J}}}^{(\mathrm{ES})}-{\tilde{U}}_{2;{\rm{J}}}^{(\mathrm{ES})})\,{(\bar{\psi }{\gamma }_{1}\psi )}^{2}\right.\\ & & -{\tilde{U}}_{2;{\rm{J}}}^{(\mathrm{FS})}({\tilde{U}}_{3;{\rm{J}}}^{(\mathrm{ES})}-{\tilde{U}}_{0;{\rm{J}}}^{(\mathrm{ES})}-{\tilde{U}}_{1;{\rm{J}}}^{(\mathrm{ES})}+{\tilde{U}}_{2;{\rm{J}}}^{(\mathrm{ES})})\,{(\bar{\psi }{\gamma }_{2}\psi )}^{2}]\end{eqnarray} \tag{ A17 }$$

Applying the transformation in equation (A14) leads to the net quantum correction to the forward scattering vertices in the FS representation (we drop explicit reference to FS),

$$\begin{eqnarray}\delta {S}_{{\rm{int}}}^{(\mathrm{FS})} & = & -\displaystyle \frac{2{{\rm{\Lambda }}}^{-\epsilon }{\rm{d}}{\ell }}{\pi \kappa }\displaystyle \sum _{J}\displaystyle \int \displaystyle \prod _{n=1}^{4}{{\rm{d}}{K}}_{n}\,{(2\pi )}^{4}{\delta }^{(4)}({K}_{1}-{K}_{2}+{K}_{3}-{K}_{4})\,\delta ({{\bf{K}}}_{1}-{{\bf{K}}}_{2})\,\delta ({{\bf{K}}}_{3}-{{\bf{K}}}_{4})\,{{\rm{e}}}^{-{\rm{i}}({\theta }_{1}-{\theta }_{2})J}\\ & & \times [2{({\tilde{U}}_{3;{\rm{J}}})}^{2}\,{(\bar{\psi }\psi )}^{2}-{({\tilde{U}}_{1;{\rm{J}}})}^{2}\,{(\bar{\psi }{\gamma }_{1}\psi )}^{2}-{({\tilde{U}}_{2;{\rm{J}}})}^{2}\,{(\bar{\psi }{\gamma }_{2}\psi )}^{2}].\end{eqnarray} \tag{ A18 }$$

Let the quadratic-insertion be

$$\begin{eqnarray}&&{S}_{2,\mu }^{(\mathrm{BCS})}={\int }_{{\rm{\Lambda }}}{\rm{d}}{K}\,{{\rm{\Delta }}}_{\mu }^{* }(\widehat{K}){\psi }^{\top }(-K){\gamma }_{\mu }\psi (K)+{\rm{h}}.{\rm{c}}.\end{eqnarray} \tag{ B1 }$$

The label Λ reminds us of the cutoff for the effective action. The quantum corrections to equation (B1) are generated by contracting it with the quartic vertices in the action, which result in processes represented by figure 6,

$$\begin{eqnarray}&&\delta {S}_{2,\mu }^{(\mathrm{BCS})}=-\displaystyle \sum _{\nu }{\int }_{(1-d{\ell }){\rm{\Lambda }}}{\rm{d}}K{\psi }^{\top }(-K)\,{\gamma }_{\nu }^{\top }{{\rm{\Gamma }}}_{(\mathrm{BCS})}^{\mu \nu }(\hat{K}){\gamma }_{\nu }\,\psi (K)+{\rm{h}}.{\rm{c}}.,\end{eqnarray} \tag{ B2 }$$

where

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{(\mathrm{BCS})}^{\mu \nu }(\hat{K})=\displaystyle \frac{{{\rm{\Lambda }}}^{-\epsilon }}{\kappa }{\int }^{\prime }{\rm{d}}{{QV}}_{\nu }^{\prime }(\hat{Q},\hat{K}){{\rm{\Delta }}}_{\mu }^{* }(\hat{Q})\,{G}^{\top }(-Q){\gamma }_{\mu }G(Q).\end{eqnarray} \tag{ B3 }$$

Here the primed integration sign implies that the integral is restricted within the high-energy region which corresponds to $k\in [(1-{\rm{d}}{\ell }){\rm{\Lambda }},{\rm{\Lambda }}]$. After introducing the angular momentum harmonics, the mode elimination leads to

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{(\mathrm{BCS})}^{\mu \nu }({\theta }_{k})={\rm{d}}{\ell }\displaystyle \frac{1}{8\pi }\left[{\gamma }_{0}{\gamma }_{\mu }{\gamma }_{0}+\displaystyle \frac{1}{2}\displaystyle \sum _{{\rm{i}}=1,2}{\gamma }_{{\rm{i}}}{\gamma }_{\mu }{\gamma }_{{\rm{i}}}\right]{\displaystyle \sum _{J}}^{\prime }\,{{\rm{e}}}^{-{\rm{i}}{l}{\theta }_{k}}\,{\tilde{V}}_{\nu ;{\rm{J}}}{{\rm{\Delta }}}_{\mu ;{\rm{J}}}^{* },\end{eqnarray} \tag{ B4 }$$

where the prime over the sum represent the restriction of J to even (odd) integers when ${\gamma }_{\mu }^{\top }={\gamma }_{\mu }$ (${\gamma }_{\mu }^{\top }=-{\gamma }_{\mu }$). Substituting ${{\rm{\Gamma }}}_{(\mathrm{BCS})}^{\mu \nu }$ into the expression of quantum correction, and summing over ν results in

$$\begin{eqnarray}&&\delta {S}_{2,\mu }^{(\mathrm{BCS})}={\rm{d}}{\ell }{\displaystyle \sum _{J}}^{\prime }{\int }_{(1-{\rm{d}}{\ell }){\rm{\Lambda }}}{\rm{d}}K{\delta }_{\mu }(\{{V}_{i;{\rm{J}}}\})\,{{\rm{\Delta }}}_{\mu ;{\rm{J}}}^{* }\,{\psi }^{\top }(-K)\,{\gamma }_{\mu }\,\psi (K)+{\rm{h}}.{\rm{c}}.,\end{eqnarray} \tag{ B5 }$$

where

$$\begin{eqnarray}8\pi \,{\delta }_{0}(\{{V}_{i;{\rm{J}}}\}) & = & 0,\\ 8\pi \,{\delta }_{1}(\{{V}_{i;{\rm{J}}}\}) & = & -({V}_{1;{\rm{J}}}-{V}_{3;{\rm{J}}})+2{V}_{2+;{\rm{J}}}=2{f}_{1-;{\rm{J}}},\\ 8\pi \,{\delta }_{2}(\{{V}_{i;{\rm{J}}}\}) & = & -({V}_{1;{\rm{J}}}-{V}_{3;{\rm{J}}})-2{V}_{2+;{\rm{J}}}=-2{f}_{1+;{\rm{J}}},\\ 8\pi \,{\delta }_{3}(\{{V}_{i;{\rm{J}}}\}) & = & -2({V}_{1;{\rm{J}}}+{V}_{3;{\rm{J}}})-4{V}_{2-;{\rm{J}}}=-4{f}_{2+;{\rm{J}}}.\end{eqnarray} \tag{ B6 }$$

Adding the quantum correction to $\delta {S}_{2,\mu }^{(\mathrm{BCS})}$ and rescaling all dimensionful quantities to restore the cutoff to Λ gives the beta function of ${{\rm{\Delta }}}_{\mu ,l}$,

$$\begin{eqnarray}&&{\partial }_{{\ell }}{{\rm{\Delta }}}_{\mu ;{\rm{J}}}={{\rm{\Delta }}}_{\mu ;{\rm{J}}}[\eta +{\delta }_{\mu }(\{{V}_{i;{\rm{J}}}\})].\end{eqnarray} \tag{ B7 }$$

Figures 7(a) and (b) leads to the quantum corrections,

$$\begin{eqnarray}{\delta {S}_{2,\mu }^{(\mathrm{DW})}(\vec{P})| }_{a} & = & -2\displaystyle \sum _{\nu }{\displaystyle \int }_{{\rm{\Lambda }}(1-d{\ell })}{\rm{d}}K\bar{\psi }(P+K)\,{\gamma }_{\nu }{{\rm{\Gamma }}}_{(\mathrm{DW});{\rm{a}}}^{\mu \nu }(\widehat{P},\widehat{K}){\gamma }_{\nu }\,\psi (K)+{\rm{h}}.{\rm{c}}.,\\ {\delta {S}_{2,\mu }^{(\mathrm{DW})}(\vec{P})| }_{b} & = & 2\displaystyle \sum _{\nu }{\displaystyle \int }_{{\rm{\Lambda }}(1-d{\ell })}{\rm{d}}K{{\rm{\Gamma }}}_{(\mathrm{DW});{\rm{b}}}^{\mu \nu }(\widehat{P},\widehat{K})\,\bar{\psi }(P+K)\,{\gamma }_{\nu }\,\psi (K)+{\rm{h}}.{\rm{c}}.,\end{eqnarray} \tag{ B8 }$$

respectively, where (u_μ are defined in equation (A2))

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{(\mathrm{DW});{\rm{a}}}^{\mu \nu }(\widehat{P},\widehat{K}) & = & \displaystyle \frac{1}{\kappa }{\displaystyle \int }^{\prime }{\rm{d}}Q{{\rm{\Phi }}}_{\mu }(\vec{P};\widehat{Q})\,{u}_{\nu }(\widehat{Q},\widehat{K},\widehat{(K+P)},\widehat{(Q+P)})\,G(Q+P){\gamma }_{\mu }G(Q),\\ {{\rm{\Gamma }}}_{(\mathrm{DW});{\rm{b}}}^{\mu \nu }(\widehat{P},\widehat{K}) & = & \displaystyle \frac{1}{\kappa }{\displaystyle \int }^{\prime }{\rm{d}}Q{{\rm{\Phi }}}_{\mu }(\vec{P};\widehat{Q})\,{u}_{\nu }(\widehat{Q},\widehat{(Q+P)},\widehat{(K+P)},\widehat{K})\,\mathrm{tr}\,\{{\gamma }_{\nu }G(Q+P){\gamma }_{\mu }G(Q)\}.\end{array}\end{eqnarray} \tag{ B9 }$$

The difference in overall sign between the two quantum corrections in equation (B8) arises from the fermion-loop in figure 7(b). We set $\vec{P}=0$, and apply the steps in appendices A2 and B1 to obtain the beta functions for ${{\rm{\Phi }}}_{\mu ,J}$ quoted in the main section.