# Enhanced Superconducting Diode Effect due to coexisting Phases

Sayan Banerjee  Institute for Theoretical Physics, University of Innsbruck,
Innsbruck A-6020, Austria  Mathias S. Scheurer  Institute for Theoretical
Physics, University of Innsbruck, Innsbruck A-6020, Austria

###### Abstract

The superconducting diode effect refers to an asymmetry in the critical
supercurrent $J_{c}(\hat{n})$ along opposite directions, $J_{c}(\hat{n})\neq
J_{c}(-\hat{n})$. While the basic symmetry requirements for this effect are
known, it is, for junction-free systems, difficult to capture within current
theoretical models the large current asymmetries
$J_{c}(\hat{n})/J_{c}(-\hat{n})$ recently observed in experiment. We here
propose and develop a theory for an enhancement mechanism of the diode effect
arising from spontaneous symmetry breaking. We show—both within a
phenomenological and a microscopic theory—that there is a coupling of the
supercurrent and the underlying symmetry-breaking order parameter. This
coupling can enhance the current asymmetry significantly. Our work might not
only provide a possible explanation for recent experiments on trilayer
graphene but also pave the way for future realizations of the superconducting
diode effect with large current asymmetries.

Diodes, which are characterized by a highly asymmetric relation between
resistance $R$ and current $\boldsymbol{J}$, $R(\boldsymbol{J})\neq
R(-\boldsymbol{J})$, are an integral part of modern-day electronics. A
superconductor where the critical current $J_{c}$ is different along opposite
directions, $J_{c}(\hat{n})>J_{c}(-\hat{n})$, can realize a superconducting
analogue of a diode in the sense that $R(-J\hat{n})\neq R(J\hat{n})=0$, if
$J_{c}(-\hat{n})<J<J_{c}(\hat{n})$. While asymmetries in superconducting
current-voltage relations in low-symmetry superconductors had been observed
before, see, e.g., Broussard and Geballe (1988); Jiang  _et al._ (1994); Papon
_et al._ (2008), the potential technological applications of and fundamental
scientific questions associated with this superconducting diode effect (SDE)
have attracted significant experimental Ando  _et al._ (2020); Lyu  _et al._
(2021); Du  _et al._ (2023); Sundaresh  _et al._ (2023); Kealhofer  _et al._
(2023); Hou  _et al._ (2023); Chen  _et al._ (2023); Gupta  _et al._ (2022);
Baumgartner  _et al._ (2022); Banerjee  _et al._ (2023); Pal  _et al._ (2022);
Kim  _et al._ (2023); Gupta  _et al._ (2022); Turini  _et al._ (2022);
Bauriedl  _et al._ (2022); Wu  _et al._ (2022); Diez-Merida  _et al._ (2021);
Golod and Krasnov (2022); Chiles  _et al._ (2022); Zhang  _et al._ (2023a);
Shin  _et al._ (2021); Jeon  _et al._ (2022); Kealhofer  _et al._ (2023); Hou
_et al._ (2023); Lin  _et al._ (2022); Anwar  _et al._ (2022); Narita  _et
al._ (2022); Gutfreund  _et al._ (2023) and theoretical Daido and Yanase
(2022); Daido  _et al._ (2022); Yuan and Fu (2022); He  _et al._ (2022a); Ilić
and Bergeret (2022); Scammell  _et al._ (2022); Zinkl  _et al._ (2022); Zhang
_et al._ (2022a); He  _et al._ (2022b); Zhai  _et al._ (2022); Jiang  _et al._
(2022); Kokkeler  _et al._ (2022); Chazono  _et al._ (2022); Vodolazov and
Peeters (2005); de Picoli  _et al._ (2023); Kochan  _et al._ (2023); Ikeda
_et al._ (2022); Tanaka  _et al._ (2022); Wang  _et al._ (2022); Haenel and
Can (2022); Legg  _et al._ (2023); Cuozzo  _et al._ (2023); Souto  _et al._
(2022); Cheng and Sun (2023); Steiner  _et al._ (2022); Costa  _et al._
(2023); Wei  _et al._ (2022); Legg  _et al._ (2022); Karabassov  _et al._
(2022); Hu  _et al._ (2022) attention in recent years; this led to a variety
of different realizations both in tunnel-junction setups Chen  _et al._
(2023); Gupta  _et al._ (2022); Baumgartner  _et al._ (2022); Banerjee  _et
al._ (2023); Pal  _et al._ (2022); Kim  _et al._ (2023); Gupta  _et al._
(2022); Turini  _et al._ (2022); Bauriedl  _et al._ (2022); Wu  _et al._
(2022); Diez-Merida  _et al._ (2021); Golod and Krasnov (2022); Chiles  _et
al._ (2022); Zhang  _et al._ (2023a); Shin  _et al._ (2021); Jeon  _et al._
(2022) and in single junction-free superconducting phases Ando  _et al._
(2020); Lyu  _et al._ (2021); Du  _et al._ (2023); Sundaresh  _et al._ (2023);
Kealhofer  _et al._ (2023); Hou  _et al._ (2023); Lin  _et al._ (2022); Anwar
_et al._ (2022); Narita  _et al._ (2022); Gutfreund  _et al._ (2023).
Importantly, critical current asymmetries in superconductors require broken
time-reversal symmetry (TRS), which can not only be achieved by applying a
magnetic field Chen  _et al._ (2023); Gupta  _et al._ (2022); Baumgartner  _et
al._ (2022); Banerjee  _et al._ (2023); Pal  _et al._ (2022); Kim  _et al._
(2023); Gupta  _et al._ (2022); Turini  _et al._ (2022); Bauriedl  _et al._
(2022); Ando  _et al._ (2020); Lyu  _et al._ (2021); Du  _et al._ (2023);
Sundaresh  _et al._ (2023); Kealhofer  _et al._ (2023); Hou  _et al._ (2023),
a current Chiles  _et al._ (2022); Zhang  _et al._ (2023a), or magnetic
proximity Shin  _et al._ (2021); Narita  _et al._ (2022); Gutfreund  _et al._
(2023); Jeon  _et al._ (2022), but also result from spontaneous TRS-breaking
in a junction Wu  _et al._ (2022); Diez-Merida  _et al._ (2021); Golod and
Krasnov (2022) or homogeneous superconductor Lin  _et al._ (2022); Anwar  _et
al._ (2022); aside from the fundamental interest in the competition of
magnetism and superconductivity in a single electron liquid, this might also
be useful for integrated designs where external fields are not practical.
Another crucial aspect for applications is the degree of current asymmetry,
conveniently measured by the dimensionless diode efficiency
$\eta(\hat{n})=|J_{c}(\hat{n})-J_{c}(-\hat{n})|/(J_{c}(\hat{n})+J_{c}(-\hat{n}))$.
A large efficiency, as close to $1$ as possible, is desirable for the system
to operate as a superconducting diode without having to fine-tune the current
magnitude.

In this work, we propose and demonstrate theoretically an enhancement
mechanism for the SDE efficiency $\eta$, specifically for systems with
spontaneously broken TRS. While it is clear by symmetry that the condensation
of a time-reversal-odd order parameter can affect the critical supercurrent
and induce a SDE, we here demonstrate that the supercurrent also couples back
to the underlying symmetry-breaking order parameter. If superconductivity and
the TRS-breaking order have similar energy scales, this “back action” can be
quite large and, as we demonstrate, can enhance $\eta$ significantly. Although
this mechanism is more generally valid and should apply to different models
and forms of the time-reversal-odd order parameter, we use a model inspired by
graphene moiré systems; this is motivated by Ref. Lin  _et al._ , 2022 where a
junction-free sample of twisted trilayer graphene was studied and a zero-field
SDE with a particularly large $\eta$ was observed. Our work, thus, both
provides a possible route to explaining the large $\eta$ of Ref. Lin  _et al._
, 2022 and paves the way for the design of zero-field superconducting diodes
with large current asymmetries.

Model for SDE.—As motivated above, we study an electronic Hamiltonian of the
form

$\displaystyle\begin{split}\mathcal{H}_{c}=&\sum_{\boldsymbol{k},\eta}\xi_{\boldsymbol{\boldsymbol{k}},\eta}c^{\dagger}_{\boldsymbol{k},\eta}c^{\phantom{\dagger}}_{\boldsymbol{k},\eta}+\Phi_{V}\sum_{\boldsymbol{k},\eta}\eta
c^{\dagger}_{\boldsymbol{k},\eta}c^{\phantom{\dagger}}_{\boldsymbol{k},\eta}\\\
&+\sum_{\boldsymbol{k},\boldsymbol{q}}\left[\Delta_{\boldsymbol{q}}c^{\dagger}_{\boldsymbol{k}+\boldsymbol{q}/2,+}c^{\dagger}_{-\boldsymbol{k}+\boldsymbol{q}/2,-}+\text{H.c.}\right],\end{split}$
(1)

where $c_{\boldsymbol{k},\eta}^{\dagger}$ creates an electron in valley
$\eta=\pm$ and at momentum $\boldsymbol{k}$. The three terms in
$\mathcal{H}_{c}$ capture the non-interacting bandstructure of the nearly flat
bands at the chemical potential, which obey
$\xi_{\boldsymbol{\boldsymbol{k}},\eta}=\xi_{-\boldsymbol{\boldsymbol{k}},-\eta}\equiv\xi_{\boldsymbol{k}\cdot\eta}$
due to TRS, the coupling of the TRS-breaking normal state order (in our case,
valley imbalance $\Phi_{V}$) and of the superconducting order parameter to the
electrons, respectively. For notational simplicity, we suppress the spin index
in Eq. (1) and in the following, but emphasize that our results apply both for
spin polarized bands Morissette  _et al._ (2022); Zondiner  _et al._ (2020) as
well as in the spinful case, with singlet or triplet pairing.

To stabilize superconductivity, we take an attractive ($g_{c}>0$) interaction
$\mathcal{H}_{g_{c}}=-\frac{g_{c}}{2}\sum_{\boldsymbol{q},\eta,\eta^{\prime}}C^{\dagger}_{\boldsymbol{q};\eta,\eta^{\prime}}C^{\phantom{\dagger}}_{\boldsymbol{q};\eta,\eta^{\prime}}$
with
$C_{\boldsymbol{q};\eta,\eta^{\prime}}=\sum_{\boldsymbol{k}}c_{\boldsymbol{k}+\boldsymbol{q}/2,\eta}c_{-\boldsymbol{k}+\boldsymbol{q}/2,\eta^{\prime}}$
and perform a decoupling in the intervalley pairing channel,
$\braket{C_{\boldsymbol{q};\eta,\eta^{\prime}}}=\eta\delta_{\eta,\eta^{\prime}}\Delta_{\boldsymbol{q}}/g_{c}$;
we focus on intervalley pairing not only because it is favored as a
consequence of TRS for a finite range of $\Phi_{V}$ Scammell  _et al._ (2022)
but also because intravalley pairing is more susceptible to impurities and,
hence, likely less relevant experimentally. The resulting mean-field
Hamiltonian for superconductivity for given valley polarization $\Phi_{V}$
then reads as
$\mathcal{H}_{\text{S}}=\mathcal{H}_{c}+\sum_{\boldsymbol{q}}|\Delta_{\boldsymbol{q}}|^{2}/g_{c}$.
While we will compute the supercurrent systematically below, we start for
illustration purposes with a semi-phenomenological approach based on a
Ginzburg-Landau expansion. Integrating out the electrons in
$\mathcal{H}_{\text{S}}$ and expanding up to second order in
$\Delta_{\boldsymbol{q}}$, the change
$\delta\mathcal{F}_{\text{S}}:=\mathcal{F}[\Delta_{\boldsymbol{q}},\Phi_{V}]-\mathcal{F}[0,\Phi_{V}]$
of the free-energy with superconductivity reads as

$\delta\mathcal{F}_{\text{S}}\sim\sum_{\boldsymbol{q}}a^{\text{S}}_{\boldsymbol{q}}\,|\Delta_{\boldsymbol{q}}|^{2}+b^{\text{S}}\sum_{\boldsymbol{q}_{i}}\Delta^{*}_{\boldsymbol{q}_{1}}\Delta^{*}_{\boldsymbol{q}_{2}}\Delta^{\phantom{*}}_{\boldsymbol{q}_{3}}\Delta^{\phantom{*}}_{\boldsymbol{q}_{4}}\delta_{\boldsymbol{q}_{1}+\boldsymbol{q}_{2},\boldsymbol{q}_{3}+\boldsymbol{q}_{4}},$

where we neglected the momentum dependence of the quartic term. It holds
$a^{\text{S}}_{\boldsymbol{q}}=1/g_{c}-\Gamma_{\boldsymbol{q}}$ with Scammell
_et al._ (2022)

$\Gamma_{\boldsymbol{q}}=\frac{1}{2N}\sum_{\boldsymbol{k}}\frac{\tanh{\frac{E_{\boldsymbol{k},\boldsymbol{q},+}}{2T}}+\tanh{\frac{E_{\boldsymbol{k},\boldsymbol{q},-}}{2T}}}{E_{\boldsymbol{k},\boldsymbol{q},+}+E_{\boldsymbol{k},\boldsymbol{q},-}},$
(2)

where $N$ is the number of unit cells and
$E_{\boldsymbol{k},\boldsymbol{q},\eta}=\xi_{\boldsymbol{k}+\eta\boldsymbol{q}/2}+\eta\,\Phi_{V}$
encodes the normal-state dispersion, associated with the first line of Eq.
(1).

The equilibrium superconducting state at given $\Phi_{V}$ is found by
minimizing $\delta\mathcal{F}_{\text{S}}$. Restricting the analysis to
single-$\boldsymbol{q}$ states,
$\Delta_{\boldsymbol{q}}\propto\delta_{\boldsymbol{q},\boldsymbol{q}_{0}}$,
the value of $\boldsymbol{q}_{0}$ is determined by the minimum of
$a^{\text{S}}_{\boldsymbol{q}}$ (the maximum of $\Gamma_{\boldsymbol{q}}$). As
expected, the current Daido  _et al._ (2022); Yuan and Fu (2022); Scammell
_et al._ (2022)

$\boldsymbol{J}(\boldsymbol{q})=2e\Delta_{\boldsymbol{q}}\boldsymbol{v}_{\boldsymbol{q}},\quad\boldsymbol{v}_{\boldsymbol{q}}=\boldsymbol{\nabla}_{\boldsymbol{q}}a^{\text{S}}_{\boldsymbol{q}},$
(3)

vanishes in equilibrium, $\boldsymbol{J}(\boldsymbol{q}_{0})=0$; a
supercurrent-carrying state, therefore, corresponds to pairing with
$\boldsymbol{q}\neq\boldsymbol{q}_{0}$. We define the critical current
$J_{c}(\hat{n})$ along $\hat{n}$ as the maximal magnitude of
$\boldsymbol{J}(\boldsymbol{q})$ oriented along $\hat{n}$.

For $\Phi_{V}=0$, it holds
$E_{\boldsymbol{k},\boldsymbol{q},\eta}=E_{\boldsymbol{k},-\boldsymbol{q},-\eta}$
and thus $\Gamma_{\boldsymbol{q}}=\Gamma_{-\boldsymbol{q}}$ in Eq. (2). This,
in turn, immediately implies
$\boldsymbol{J}(\boldsymbol{q})=-\boldsymbol{J}(-\boldsymbol{q})$ and
$J_{c}(\hat{n})=J_{c}(-\hat{n})$, i.e., there is no SDE. This was expected as
a consequence of TRS or two-fold rotational symmetry, $C_{2z}$, for that
matter—both have to be broken to get $J_{c}(\hat{n})\neq J_{c}(-\hat{n})$.
This is achieved by $\Phi_{V}\neq 0$ Scammell  _et al._ (2022), leading to a
finite SDE efficiency,

$\eta=\max_{\hat{n}}\frac{|J_{c}(\hat{n})-J_{c}(-\hat{n})|}{J_{c}(\hat{n})+J_{c}(-\hat{n})}\in[0,1].$
(4)

Although generically non-zero by symmetry, it does not guarantee that $\eta$
can reach high values corresponding to large current asymmetries,
$J_{c}(\hat{n})\gg J_{c}(-\hat{n})$—as mentioned above, a property desirable
for applications of the SDE and relevant fundamentally to understand recent
experiments Lin  _et al._ (2022).

Phenomenological coupling.—The main result of this work is that the diode-
effect efficiency $\eta$ can be significantly enhanced by taking into account
that the strength of the valley polarization is affected by the supercurrent
or, put differently, $\Phi_{V}$ entering
$E_{\boldsymbol{k},\boldsymbol{q},\eta}$ in Eq. (2) also depends on the center
of mass momentum $\boldsymbol{q}$ of the Cooper pairs,
$\Phi_{V}\rightarrow\Phi_{V}(\boldsymbol{q})$. Postponing a systematic
microscopic computation that treats $\Delta_{\boldsymbol{q}}$ and
$\Phi_{V}(\boldsymbol{q})$ on equal footing, we start with a simpler
phenomenological analysis. In this simplified description, we determine
$\Phi_{V}(\boldsymbol{q})$ by minimizing the free energy

$\mathcal{F}_{\text{V}}\sim
a_{\Phi}\Phi_{V}^{2}+b_{\Phi}\Phi_{V}^{4}+\Phi_{V}\sum_{\boldsymbol{q}}|\Delta_{\boldsymbol{q}}|^{2}\left[F(\boldsymbol{q})+\alpha^{\prime\prime}\Phi_{V}\right]$
(5)

with respect to $\Phi_{V}$, where terms involving fifth and higher powers of
the order parameters $\Phi_{V}$, $\Delta_{\boldsymbol{q}}$ are neglected. The
first two terms in $\mathcal{F}_{\text{V}}$ determine the value of $\Phi_{V}$
in the absence of superconductivity. Meanwhile, the remaining terms in Eq. (5)
capture the key coupling between the momentum of the Cooper pairs and valley
polarization. The last term, $\propto\alpha^{\prime\prime}$, simply describes
the fact that increasing (decreasing) the superconducting order parameter
magnitude suppresses (enhances) valley polarization. The other term with
$F(\boldsymbol{q})=-F(-\boldsymbol{q})$ describes the fact that
superconductivity at finite $\boldsymbol{q}$ breaks TRS and $C_{2z}$ and can,
hence, couple to the first (and in general an odd) power of $\Phi_{V}$.
Motivated by twisted graphene systems, which exhibit $C_{3z}$ rotational
symmetry, we write

$F(\boldsymbol{q})=\alpha\,\hat{\boldsymbol{e}}_{\epsilon}\cdot\boldsymbol{X}(\boldsymbol{q})+\alpha^{\prime}\,\chi(\boldsymbol{q}),$
(6)

where $\boldsymbol{X}(\boldsymbol{q})$ and $\chi(\boldsymbol{q})$ are real-
valued, periodic on the Brillouin zone, odd in $\boldsymbol{q}$, and transform
as a vector and scalar under $C_{3z}$, respectively; in the limit of small
$\boldsymbol{q}$, it holds $\boldsymbol{X}\sim(q_{x},q_{y})^{T}$ and
$\chi(\boldsymbol{q})\sim q_{x}(q_{x}^{2}-3q_{y}^{2})$, using a convention
where one of the primitive translation vectors, $\boldsymbol{a}_{j}$, of the
underlying triangular Bravais lattice points along the $x$ direction. For the
following calculations, we extend these functions to the Brillouin zone using
the leading lattice harmonics (see Appendix A). Importantly, the first term in
Eq. (6), requires finite strain or nematic order which break $C_{3z}$ symmetry
explicitly. These phenomena, which seem to be ubiquitous in graphene moiré
systems Jiang  _et al._ (2019); Kerelsky  _et al._ (2019); Cao  _et al._
(2021); Rubio-Verdú  _et al._ (2022); Samajdar  _et al._ (2021); Sobral  _et
al._ (2023); Zhang  _et al._ (2022b), define a preferred in-plane direction,
captured by the unit vector $\hat{\boldsymbol{e}}_{\epsilon}$. For instance,
in the case of strain,
$\hat{\boldsymbol{e}}_{\epsilon}\propto(\epsilon_{xx}-\epsilon_{yy},\epsilon_{xy}+\epsilon_{yz})^{T}$,
where $\epsilon_{ij}$ are the strain-tensor elements. We note that the
coupling would also be present when $\Phi_{V}$ is replaced by another order
parameter that itself already breaks both TRS and $C_{3z}$ simultaneously
Zhang  _et al._ (2022b); Lin  _et al._ (2023). The second term in Eq. (6),
albeit subleading for small $\boldsymbol{q}$, is finite regardless of whether
$C_{3z}$ is broken.

To study the SDE, we self-consistently minimize
$\mathcal{F}_{V}[\Phi_{V},\Delta_{\boldsymbol{q}}]$ and
$\delta\mathcal{F}_{\text{S}}[\Phi_{V},\Delta_{\boldsymbol{q}}]$ with respect
to $\Phi_{V}$ and $\Delta_{\boldsymbol{q}}$, respectively, for given
$\boldsymbol{q}$, and compute the current according to Eq. (3). For
concreteness, we use a nearest-neighbor dispersion with finite flux $\phi$ on
the triangular lattice,
$\xi_{\boldsymbol{k}}=-\sum_{j=1}^{3}t_{j}\cos(\boldsymbol{a}_{j}\cdot\boldsymbol{k}-\phi/3)$
where $\boldsymbol{a}_{j}$ are three $C_{3z}$-related primitive vectors. We
choose $t_{1}=t$, $t_{2}=t_{3}=t(1+\beta)$ to parameterize the impact of
strain ($\propto\beta$) on the dispersion and, for notational simplicity,
measure all energies in units of $t$ in the following.

Figure 1: Phenomenological theory with leading nematic coupling, $\alpha\neq
0$, $\alpha^{\prime}=\alpha^{\prime\prime}=0$ in Eq. (5).
$\Phi_{v}(\boldsymbol{q})$ and $\Gamma(\boldsymbol{q})$ for (a) $\alpha=-0.5$
and (b) $\alpha=-3.0$ are shown in upper and lower panels, respectively. (c)
Angular ($\hat{n}$) dependence of $J_{c}$ for different $\alpha$, taking
$|\Phi(\boldsymbol{q}=0)|=0.32$. (d) SDE efficiency $\eta$ for different
$\alpha$ as a function of valley polarization at $\boldsymbol{q}=0$. We use
$g_{c}^{-1}=0.8\Gamma_{\boldsymbol{q}=0}[\Phi_{V}(\boldsymbol{q}=0)]$,
$\mu=-0.68$, $\phi=-0.7\pi$, $T=0.2$,
$\hat{\boldsymbol{e}}_{\epsilon}=(\cos{\pi/3},\sin{\pi/3})$.

Back action and SDE.—We start with the first term in Eq. (6), i.e., set
$\alpha^{\prime}=\alpha^{\prime\prime}=0$ in $\mathcal{F}_{V}$. As can be seen
in Fig. 1(a), finite $\alpha$ now induces a $\boldsymbol{q}$ dependence in
$\Phi_{V}$, which breaks $C_{3z}$ symmetry as a result of the preferred
direction associated with $\hat{\boldsymbol{e}}_{\epsilon}$. As we set
$\beta=0$ for now, $\Gamma_{\boldsymbol{q}}$ looks virtually $C_{3z}$
symmetric, while larger $\alpha$ will lead to a noticeable asymmetry in
$\Gamma_{\boldsymbol{q}}$ as well, see Fig. 1(b). This results from a “back
action mechanism”, where a finite supercurrent, which is associated with
detuning the center of mass momentum
$\boldsymbol{q}_{0}\rightarrow\boldsymbol{q}+\delta\boldsymbol{q}_{0}$ of the
Cooper pairs away from their equilibrium value $\boldsymbol{q}_{0}$, changes
the strength of $\Phi_{V}$; as increasing (decreasing) $\Phi_{V}$ will weaken
(strengthen) superconductivity, this, in turn, influences
$\Gamma_{\boldsymbol{q}}$. This can be thought of as the supercurrent analogue
of the coupling between the dissipative current and valley polarization
observed in graphene moiré systems Serlin  _et al._ (2020); Sharpe  _et al._
(2019); Ying  _et al._ (2021). We note in passing that a sufficiently strong
supercurrent can also flip the sign $\Phi_{V}$ in our theory.

This back action also affects the $\boldsymbol{q}$ dependence of the
supercurrent, which allows to enhance the SDE as can be seen in Fig. 1(c). We
further present in Fig. 1(d) the diode efficiency $\eta$ as a function of
$\Phi_{V}(\boldsymbol{q}=0)$ for different strengths of the back action.
Without back action ($\alpha=0$, red dashed line), $\eta$ does not exceed
$30\%$, corresponding to $J_{c}(\hat{n})\simeq 1.86J_{c}(-\hat{n})$. In
contrast, increasing $\alpha$ and, thus, boosting the back action mechanism
can significantly enhance the SDE, even reaching efficiencies as high as 75%
(solid black line), i.e., $J_{c}(\hat{n})\simeq 7J_{c}(-\hat{n})$.

This enhancement mechanism of the SDE is also possible in the presence of
$C_{3z}$, where $\alpha=0$, if we take into account finite $\alpha^{\prime}$
in Eq. (6), see Fig. 2. We can clearly see the induced $\boldsymbol{q}$ (and
thus supercurrent) dependence of $\Phi_{V}$ in Fig. 2(a). However, the back
action onto $\Gamma_{\boldsymbol{q}}$ is less clearly seen in Fig. 2(b) at a
single $\alpha^{\prime}$, since it is $C_{3z}$ symmetric with and without it.
Most importantly, though, Fig. 2(c) reveals that also the subleading back
action ($\alpha=0$) enhances the current asymmetry.

Figure 2: Phenomenological theory with $C_{3z}$-symmetric coupling,
$\alpha^{\prime}\neq 0$, $\alpha=0$, $\alpha^{\prime\prime}=1.0$ in Eq. (5).
$\Phi_{v}(\boldsymbol{q})$ and $\Gamma(\boldsymbol{q})$ in (a) and (b) are
shown for $\alpha^{\prime}=0.8$ and (c) displays the angular ($\hat{n}$)
dependence of $J_{c}$ for different $\alpha^{\prime}$. We use
$g_{c}^{-1}=0.7\Gamma_{\boldsymbol{q}=0}[\Phi_{V}=0.32]$, $\mu=-0.68$,
$\phi=-0.7\pi$, $T=0.2$.

Self-consistent theory.—Having established a phenomenological understanding of
the enhancement of the SDE from the back action mechanism, we next turn our
attention to a systematic self-consistent formalism. We switch to an action
formalism and redefine $c_{\boldsymbol{k},\eta}$ in Eq. (1) as Grassmann
variables depending on imaginary time $\tau$. We start from the effective
action

$\mathcal{S}=\int\mathrm{d}\tau\Biggl{[}\sum_{\boldsymbol{k},\eta}c^{\dagger}_{\boldsymbol{k},\eta}\partial_{\tau}c^{\phantom{\dagger}}_{\boldsymbol{k},\eta}+\mathcal{H}_{c}+\mathcal{S}_{\Delta}+\mathcal{S}_{\Phi}\Biggr{]},$
(7)

that captures the desired phenomenology in a minimal setting. In Eq. (7),
$\mathcal{S}_{\Delta}=\sum_{\boldsymbol{q}}(\frac{1}{g_{c}}|\Delta_{\boldsymbol{q}}|^{2}+u_{\Delta}|\Delta_{\boldsymbol{q}}|^{4})$
and $\mathcal{S}_{\Phi}=\frac{1}{g_{v}}\Phi_{V}^{2}+v_{\Phi_{V}}\Phi_{V}^{4}$
are the bare actions of the Hubbard-Stratonovich fields associated with
superconductivity and valley polarization, respectively. We include terms up
to quartic order (which, e.g., arise after having integrated out electronic
degrees of freedom at higher energies) to stabilize coexistence of these two
orders, as is seen in experiment Lin  _et al._ (2022); Zhang  _et al._
(2023b); Scammell  _et al._ (2022).

The saddle-point equations for $\Phi_{V}$ and $\Delta_{\boldsymbol{q}}$ read
as

$\frac{\Phi_{V}}{g_{v}}+2v_{\Phi}\Phi_{V}^{3}=\frac{1}{4}\sum_{\boldsymbol{k},p=\pm}p\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},p}}{2T}},$
(8a)
$\frac{1}{g_{c}}+2u_{\Delta}|\Delta_{\mathbf{q}}|^{2}=\sum_{\boldsymbol{k}}\frac{(\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},+}}{2T}}+\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},-}}{2T}})}{2(\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},+}+\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},-})},$
(8b)

or $\Delta_{\boldsymbol{q}}=0$, where
$\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},p}=\sqrt{\zeta_{\boldsymbol{k},\boldsymbol{q},+}^{2}+|\Delta_{\boldsymbol{q}}|^{2}}+p\,\zeta_{\boldsymbol{k},\boldsymbol{q},-}$
with
$\zeta_{\boldsymbol{k},\boldsymbol{q},\pm}=(E_{\boldsymbol{k},\boldsymbol{q},+}\pm
E_{\boldsymbol{k},\boldsymbol{q},-})/2$. As required, Eq. (8b) becomes
equivalent to $a^{\text{S}}_{\boldsymbol{q}}=0$ when expanded to leading order
in $\Delta_{\boldsymbol{q}}$.

Figure 3: Coupling of supercurrent to valley polarization within microscopic
theory. $\Phi_{V}(\boldsymbol{q})$ and $\Delta(\boldsymbol{q})$ along one-
dimensional momentum cut with angle $\pi/6$ relative to the $q_{x}$ axis (a)
without form factors ($f_{\boldsymbol{k}}=1$) and (b) with form factors,
$f_{\boldsymbol{k}}=\sum_{j=1}^{3}\sin{\boldsymbol{a}_{j}\cdot\boldsymbol{k}}$.
The red (blue) hexagon denotes the maximum (minimum) of
$\Delta({\boldsymbol{q}})$ and $\Phi_{V}(\boldsymbol{q})$, respectively. We
use $\beta=0.6$, $\mu=-1.36$, $g_{c}=5.6$, $g_{v}=11$, $u_{\Delta}=0.25$,
$v_{\Phi}=0.125$, $T=0.4$.

We solve Eq. (8b) self-consistently for both $\Delta_{\boldsymbol{q}}$ and
$\Phi_{V}$ for given $\boldsymbol{q}$. This captures the mutual influence of
these two orders, as $\Delta_{\boldsymbol{q}}$ and $\Phi_{V}$ enter each
other’s saddle-point equation, Eqs. (8a) and (8b), respectively, via the
dispersion $\mathcal{E}$. Broken $C_{3z}$ symmetry is entirely encoded in the
parameter $\beta$ entering the bare normal-state dispersion
$\xi_{\boldsymbol{k}}$. Generically, the two coupled saddle-point equations
are expected to capture any symmetry-allowed coupling term between
$\Delta_{\boldsymbol{q}}$ and $\Phi_{V}$ in an effective free-energy
expansion, including those in Eq. (5) but also terms involving higher powers
of the order parameters. It turns out, though, that for the simple coupling in
$\mathcal{H}_{c}$, where valley polarization only enters as an imbalance of
the chemical potential in the two valleys, we find $\alpha=0$ in Eq. (6) for
any $\beta$. This is readily shown by expanding Eq. (8a) in terms of
$\delta\Phi_{V}(\boldsymbol{q})=\Phi_{V}(\boldsymbol{q})-\Phi_{V}(0)$ at fixed
$\Delta(\boldsymbol{q})$, see Appendix C. This is why the back action is
dominated by the term
$\alpha^{\prime\prime}|\Delta_{\boldsymbol{q}}|^{2}\Phi_{V}^{2}$ in Eq. (6)
for small $\boldsymbol{q}$ as can be seen in Fig. 3(a), where
$\delta\Phi_{V}(\boldsymbol{q})\propto\delta\Delta(\boldsymbol{q})\sim\boldsymbol{q}^{2}$
for small $\boldsymbol{q}$. We have checked that generalizing the coupling to
$\Phi_{V}\sum_{\boldsymbol{k},\eta}\eta
f_{\boldsymbol{k}\cdot\eta}c^{\dagger}_{\boldsymbol{k},\eta}c^{\phantom{\dagger}}_{\boldsymbol{k},\eta}$,
with $C_{3z}$-invariant form factors $f_{\boldsymbol{k}}$, leads to finite
$\alpha$; this is clearly visible in Fig. 3(b), where
$\Phi_{V}(\boldsymbol{q})$ is found to have a finite slope at the maximum (red
hexagon) of $\Delta(\boldsymbol{q})$.

Figure 4: SDE within self-consistent theory, Eq. (7). (a) shows
$\Phi_{V}(\boldsymbol{q})$ and $\Delta(\boldsymbol{q})$ in the upper and lower
panel, for $|\Phi_{V}(\boldsymbol{q}=0)|=0.48$ and $\beta=0.6$. Resulting
angular dependence of $J_{c}$ is shown without strain in (b) and with strain
($\beta=0.6$) in (c). (d) Efficiency $\eta$ as a function of
$\Phi_{V}(\boldsymbol{q}=0)$ in presence of strain with back action (black
solid line) and without back action (red dotted line). The parameters for
$g_{c}$, $T$, $\mu$, $v_{\Phi}$, $u_{\Delta}$ are the same as in Fig. 3; only
$g_{v}$ is varied to change $\Phi_{V}(\boldsymbol{q}=0)$.

SDE in self-consistent theory.—To demonstrate that the enhancement mechanism
for the SDE based on the back action is a robust phenomenon, we here discuss
it for the original from of the order parameter $\Phi_{V}$ without form
factors, $f_{\boldsymbol{k}}=1$, where the leading coupling ($\propto\alpha$)
in the phenomenological theory in Eq. (5) is absent. In Fig. 4(a), we plot
$\Phi_{V}(\boldsymbol{q})$ and $\Delta(\boldsymbol{q})$, obtained by solving
Eq. (8b). We can clearly see the supercurrent-induced change of valley
polarization, which is manifestly not $C_{3z}$ invariant, due to $\beta\neq
0$, and exhibits the strongest asymmetry under
$\boldsymbol{q}\rightarrow-\boldsymbol{q}$ for $\boldsymbol{q}$ along $q_{x}$.
Due to the additional coupling terms beyond those in Eq. (5),
$\Phi_{V}(\boldsymbol{q})$ looks different at large $\boldsymbol{q}$ from that
shown in Fig. 2(a). However, the for our purpose more important
$\Delta(\boldsymbol{q})$ looks very similar to $\Gamma_{\boldsymbol{q}}$ in
Fig. 2(b). Evaluating the thermal expectation value of the current operator
within the full theory (see Appendix D) as a function of $\boldsymbol{q}$, we
compute the directional dependence of the critical current, shown in Fig. 4(b)
and (c) for $\beta=0$ and $\beta\neq 0$, respectively. Exactly as in the
phenonemological theory, we find that broken $C_{3z}$ symmetry can enhance the
SDE and that the current asymmetry first increases with small $\Phi_{V}$ (as
significant TRS breaking is needed to generate a large current asymmetry) but
then eventually decreases with large $\Phi_{V}$ (as it destabilizes
superconductivity and also weakens the back action). This is also visible in
Fig. 4(d), where we plot the SDE efficiency $\eta$ of the full self-consistent
theory (black line) as a function of valley polarization at $\boldsymbol{q}=0$
and compare it with the situation without back action (red dashed line); for
the latter, we just fix $\Phi_{V}$ at the indicated value
$\Phi_{V}(\boldsymbol{q}=0)$, independent of $\boldsymbol{q}$ and solve for
superconductivity via Eq. (2). We observe that the back action not only
enhances, as before, the maximum value of $\eta$ that can be achieved, but
also increases the efficiency significantly at smaller values of valley
polarization. The reason for this amplification is the aforementioned
dominance of the $\alpha^{\prime\prime}$-like couplings in the self-consistent
theory, which enhance $\Phi_{V}(\boldsymbol{q})$ when suppressing
$\Delta_{\boldsymbol{q}}$ with finite $\boldsymbol{q}$, see also Fig. 3(a). So
even if we have very small $\Phi_{V}(\boldsymbol{q}=0)$, valley polarization
can reach sizable values at non-zero $\boldsymbol{q}$ due to the coupling to
the supercurrent, which in turn enhances the diode effect. This is why $\eta$
reaches finite values in the limit $\Phi_{V}(\boldsymbol{q}=0)\rightarrow
0^{+}$ with back action, while it has to vanish linearly with
$\Phi_{V}(\boldsymbol{q}=0)$ without it in Fig. 4(d). It shows that the
proposed back action mechanism also enhances the typical $\eta$ which can
reach large values without the need of fine-tuning
$\Phi_{V}(\boldsymbol{q}=0)$ to its optimal strength.

Conclusion.—We have shown, both using a simple free-energy expansion and a
self-consistent theory, that the supercurrent can couple to and, hence, affect
the valley polarization, $\Phi_{V}$, which in turn can enhance the SDE
efficiency significantly, see Fig. 1(d) and Fig. 4(d). Motivated by Ref. Lin
_et al._ , 2022, this was formulated for a theory applicable to graphene-based
systems; however, we expect that the basic mechanism based on coupling of the
supercurrent to a TRS-breaking order parameter is more generally valid for
zero-field superconducting diodes.

###### Acknowledgements.

S.B. and M.S.S. acknowledge funding by the European Union (ERC-2021-STG,
Project 101040651—SuperCorr). Views and opinions expressed are however those
of the authors only and do not necessarily reflect those of the European Union
or the European Research Council Executive Agency. Neither the European Union
nor the granting authority can be held responsible for them. M.S. thanks
J.I.A. Li and H. Scammell for insightful discussions and comments on the
manuscript. S.B. is grateful for discussions with J.A. Sobral, A. Rastogi, B.
Putzer and P. Wilhelm.

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##  Appendix A Details of models

We here state the explicit form of the basis functions used to parameterize
the phenomenological coupling in Eq. (6) of the main text. Applying the
convention, where one of the primitive lattice vectors of the underlying
triangular lattice is given by $\boldsymbol{a}_{1}=(1,0)^{T}$, these functions
read as

$\displaystyle\boldsymbol{X}(\boldsymbol{q})$
$\displaystyle=\left(\frac{2}{3}(2\cos{\frac{q_{x}}{2}}+\cos{\frac{\sqrt{3}q_{y}}{2}})\sin{\frac{q_{x}}{2}}\right.,\left.{}\frac{2}{\sqrt{3}}\cos{\frac{q_{x}}{2}}\sin{\frac{\sqrt{3}q_{y}}{2}}\right),$
(9) $\displaystyle\chi(\boldsymbol{q})$
$\displaystyle=16\left(-\cos{\frac{q_{x}}{2}}+\cos{\frac{\sqrt{3}q_{y}}{2}}\right)\sin{\frac{q_{x}}{2}}.$
(10)

These are the leading (in the sense of having the fewest number of nodes/nodal
lines) functions which obey the required symmetry constraints while being
periodic on the Brillouin zone. They are illustrated in Fig. 5.

Figure 5: Momentum dependence of the two components of the function (a)
$\boldsymbol{X}(\boldsymbol{q})$ and of (b) $\chi(\boldsymbol{q})$ as defined
by Eq. (10). The black dashed lines indicate the lines across which the
function changes sign.

##  Appendix B Derivation of the saddle-point equations

To derive saddle point equations, we generalize Eq. (7) to include non-trivial
form factors, $f_{\boldsymbol{k}}\neq\text{const.}$, for the coupling of
valley polarization to the electrons. The associated action in Matsubara-
frequency notation becomes

$\mathcal{S}[\Delta_{\boldsymbol{q}},\Phi_{V}]=\frac{1}{g_{v}}\Phi_{V}^{2}+\frac{1}{g_{c}}|\Delta_{\boldsymbol{q}}|^{2}+v_{\Phi_{V}}\Phi_{V}^{4}+u_{\Delta}|\Delta_{\boldsymbol{q}}|^{4}+\sum_{k}\Psi_{\boldsymbol{k,q}}^{\dagger}\mathcal{G}^{-1}_{\boldsymbol{k,q}}\Psi_{\boldsymbol{k,q}},$
(11)

where the sum is done over $k=(\boldsymbol{k},\omega_{n})$, comprising
momentum $\boldsymbol{k}$ and Matsubara frequencies $\omega_{n}$.
$\mathcal{G}_{k,\boldsymbol{q}}$ and $\Psi_{k,\boldsymbol{q}}$ are given by

$\mathcal{G}^{-1}_{k,\boldsymbol{q}}=\begin{bmatrix}-i\omega_{n}+\xi_{\boldsymbol{k}+\frac{\boldsymbol{q}}{2},+}+f_{\boldsymbol{k}+\frac{\boldsymbol{q}}{2},+}\Phi_{V}&\Delta_{\boldsymbol{q}}\\\
\Delta_{\boldsymbol{q}}^{\dagger}&-i\omega_{n}-\xi_{-\boldsymbol{k}+\frac{\boldsymbol{q}}{2},-}+f_{-\boldsymbol{k}+\frac{\boldsymbol{q}}{2},-}\Phi_{V}\end{bmatrix},\quad\Psi_{k,\boldsymbol{q}}=\binom{c_{\mathbf{k+\frac{q}{2},+}}}{c_{\mathbf{-k+\frac{q}{2},-}}^{\dagger}}.$
(12)

Integrating out the fermions, we obtain the effective action as,

$\mathcal{S}_{\text{eff}}=\frac{\beta}{g_{v}}\Phi_{V}^{2}+\frac{\beta}{g_{c}}|\Delta_{\boldsymbol{q}}|^{2}+\beta
v_{\Phi_{V}}\Phi_{V}^{4}+\beta
u_{\Delta}|\Delta_{\boldsymbol{q}}|^{4}-\sum_{k}\text{tr}\ln{\mathcal{G}^{-1}_{\boldsymbol{k,q}}}.$
(13)

Now, we calculate the saddle point equation for $\Delta_{\boldsymbol{q}}$ we
set $\delta S_{\text{eff}}/\delta\Delta_{\boldsymbol{q}}\overset{!}{=}0$

$\displaystyle\beta\frac{\Delta_{\boldsymbol{q}}^{\dagger}}{g_{c}}+2\beta
u_{\Delta}|\Delta_{\boldsymbol{q}}|^{2}\Delta_{\boldsymbol{q}}^{\dagger}-\sum_{k}\text{tr}\left[\mathcal{G}_{\boldsymbol{k,q}}\frac{\delta}{\delta\Delta_{\boldsymbol{q}}}\begin{pmatrix}-i\omega_{n}+E_{\boldsymbol{k,q},+}&\Delta_{\boldsymbol{q}}\\\
\Delta_{\boldsymbol{q}}^{\dagger}&-i\omega_{n}-E_{\boldsymbol{k,q},-}\end{pmatrix}\right]$
$\displaystyle=0$ (14)
$\displaystyle\beta\frac{\Delta_{\boldsymbol{q}}^{\dagger}}{g_{c}}+2\beta
u_{\Delta}|\Delta_{\boldsymbol{q}}|^{2}\Delta_{\boldsymbol{q}}^{\dagger}-\sum_{k}\text{tr}\left[\mathcal{G}_{\boldsymbol{k,q}}\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\right]$ $\displaystyle=0$ (15)

Now,

$\mathcal{G}_{\boldsymbol{k,q}}=\frac{-1}{(-i\omega_{n}+E_{\boldsymbol{k,q},+})(-i\omega_{n}-E_{\boldsymbol{k,q},-})-\Delta_{\boldsymbol{q}}^{2}}\begin{bmatrix}i\omega_{n}+E_{\boldsymbol{k,q},-}&\Delta_{\boldsymbol{q}}\\\
\Delta_{\boldsymbol{q}}^{\dagger}&i\omega_{n}-E_{\boldsymbol{k,q},+}\end{bmatrix}$
(16)

where we have redefined our energies as,

$E_{\boldsymbol{k,q},+}=\xi_{\boldsymbol{k}+\frac{\boldsymbol{q}}{2},+}+f_{\boldsymbol{k}+\frac{\boldsymbol{q}}{2},+}\Phi_{V},\quad
E_{\boldsymbol{k,q},-}=\xi_{-\boldsymbol{k}+\frac{\boldsymbol{q}}{2},-}-f_{-\boldsymbol{k}+\frac{\boldsymbol{q}}{2},-}\Phi_{V},$
(17)

which is the generalization of the energies defined below Eq. (2) of the main
text to include form factors. Therefore, the trace term reduces to

$\sum_{k}\text{tr}\left[\mathcal{G}_{\boldsymbol{k,q}}\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\right]=\sum_{\boldsymbol{k},n}\frac{\Delta_{\boldsymbol{q}}^{\dagger}}{\Delta_{\boldsymbol{q}}^{2}+\zeta_{\boldsymbol{k,q},+}^{2}+(\omega_{n}+i\zeta_{\boldsymbol{k,q},-})^{2}},\quad\text{with}\quad\zeta_{\boldsymbol{k},\boldsymbol{q},\pm}=(E_{\boldsymbol{k},\boldsymbol{q},+}\pm
E_{\boldsymbol{k},\boldsymbol{q},-})/2.$ (18)

Performing the sum over Matsubara frequencies, we get

$\text{tr}\left[\mathcal{G}_{\boldsymbol{k,q}}\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\right]=\sum_{\boldsymbol{k}}\frac{\Delta_{\boldsymbol{q}}^{\dagger}(\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},+}}{2T}}+\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},-}}{2T}})}{2T(\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},+}+\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},-})}$
(19)

where
$\mathcal{E}_{\boldsymbol{k,q},p}=\sqrt{\zeta_{\boldsymbol{k},\boldsymbol{q},+}^{2}+|\Delta_{\boldsymbol{q}}|^{2}}+p\,\zeta_{\boldsymbol{k},\boldsymbol{q},-}$.
Consequentially, this leads us to our saddle point equation for
$\Delta_{\boldsymbol{q}}$,

$\frac{\Delta_{\boldsymbol{q}}^{\dagger}}{g_{c}}+2u_{\Delta}|\Delta_{\mathbf{q}}|^{2}\Delta_{\boldsymbol{q}}^{\dagger}=\sum_{\boldsymbol{k}}\frac{\Delta_{\boldsymbol{q}}^{\dagger}(\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},+}}{2T}}+\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},-}}{2T}})}{2(\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},+}+\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},-})}$
(20)

as stated in Eq. (8b) of the main text. Moving onto the saddle point equation
for valley polarization, we start similarly by setting $\delta
S_{\text{eff}}/\delta\Phi_{V}\overset{!}{=}0$

$\displaystyle\frac{2\beta\Phi_{V}}{g_{v}}+4v_{\Phi}\beta\Phi_{V}^{3}-\sum_{k}\text{tr}\left[\mathcal{G}_{\boldsymbol{k,q}}\frac{\delta}{\delta\Phi_{V}}\begin{pmatrix}-i\omega_{n}+E_{\boldsymbol{k,q},+}&\Delta_{\boldsymbol{q}}\\\
\Delta_{\boldsymbol{q}}^{\dagger}&-i\omega_{n}-E_{\boldsymbol{k,q},-}\end{pmatrix}\right]$
$\displaystyle=0$ (21)
$\displaystyle\frac{2\beta\Phi_{V}}{g_{v}}+4v_{\Phi}\Phi_{V}^{3}-\sum_{k}\text{tr}\left[\mathcal{G}_{\boldsymbol{k,q}}\begin{pmatrix}\mathcal{F}_{\boldsymbol{k,q},+}&0\\\
0&\mathcal{F}_{\boldsymbol{k,q},-}\end{pmatrix}\right]$ $\displaystyle=0$ (22)

where
$\mathcal{F}_{\boldsymbol{k},\boldsymbol{q},+}=f_{\boldsymbol{k}+\frac{\boldsymbol{q}}{2},+}$
and
$\mathcal{F}_{\boldsymbol{k},\boldsymbol{q},-}=f_{-\boldsymbol{k}+\frac{\boldsymbol{q}}{2},-}$.
Evaluating the trace, we get

$\sum_{k}\text{tr}\left[\mathcal{G}_{\boldsymbol{k,q}}\begin{pmatrix}\mathcal{F}_{\boldsymbol{k,q},+}&0\\\
0&\mathcal{F}_{\boldsymbol{k,q},-}\end{pmatrix}\right]=\sum_{\boldsymbol{k},n}\frac{(i\omega_{n}+E_{\boldsymbol{k,q},-})\mathcal{F}_{\boldsymbol{k,q},+}+(i\omega_{n}-E_{\boldsymbol{k,q},+})\mathcal{F}_{\boldsymbol{k,q},-}}{\Delta_{\boldsymbol{q}}^{2}+\zeta_{\boldsymbol{k,q},+}^{2}+(\omega_{n}+i\zeta_{\boldsymbol{k,q},-})^{2}}$
(23)

If we now evaluate the Matsubara sums, we arrive at

$\sum_{k}\text{tr}\left[\mathcal{G}_{\boldsymbol{k,q}}\begin{pmatrix}\mathcal{F}_{\boldsymbol{k,q},+}&0\\\
0&\mathcal{F}_{\boldsymbol{k,q},-}\end{pmatrix}\right]=\frac{1}{2}\sum_{\boldsymbol{k}}\left(F_{\boldsymbol{k,q},+}\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},+}}{2T}}-F_{\boldsymbol{k,q},-}\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},-}}{2T}}\right)$
(24)

where the coefficients are given by

$F_{\boldsymbol{k,q},+}=\mathcal{F}_{\boldsymbol{k,q}+}l_{\boldsymbol{k,q}}+\mathcal{F}_{\boldsymbol{k,q}-}m_{\boldsymbol{k,q}},\quad
F_{\boldsymbol{k,q},-}=\mathcal{F}_{\boldsymbol{k,q}-}l_{\boldsymbol{k,q}}+\mathcal{F}_{\boldsymbol{k,q}+}m_{\boldsymbol{k,q}}$
(25)
$l_{\boldsymbol{k,q}}=\frac{1}{2}\left(1+\frac{\zeta_{\boldsymbol{k,q},+}}{\sqrt{\zeta_{\boldsymbol{k,q},+}^{2}+\Delta_{\boldsymbol{q}}^{2}}}\right),\quad
m_{\boldsymbol{k,q}}=\frac{1}{2}\left(1-\frac{\zeta_{\boldsymbol{k,q},+}}{\sqrt{\zeta_{\boldsymbol{k,q},+}^{2}+\Delta_{\boldsymbol{q}}^{2}}}\right)$
(26)

Therefore, we obtain our saddle point equation for valley polarization as,

$\frac{\Phi_{V}}{g_{v}}+2v_{\Phi}\Phi_{V}^{3}=\frac{1}{4}\sum_{\boldsymbol{k}}\left(F_{\boldsymbol{k,q},+}\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},+}}{2T}}-F_{\boldsymbol{k,q},-}\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},-}}{2T}}\right)$
(27)

which reduces to Eq. (8a) when the form factors are trivial
[$f_{\boldsymbol{k},\pm}=1$ in Eq. (12)].

##  Appendix C Coupling of supercurrent and valley polarization within full
self-consistent theory

In this part of the appendix, we elucidate the impact of finite
$\boldsymbol{q}$ on superconductivity and valley polarization within the
coupled non-linear self-consistency equations in Eq. (8b) of the main text. We
choose a gauge where $\Delta(\boldsymbol{q})\in\mathbb{R}$, for notational
simplicity, and begin with Eq. (8a) which we expand to linear order in
$\boldsymbol{q}$,
$\delta\Phi_{V}(\boldsymbol{q}):=\Phi_{V}(\boldsymbol{q})-\Phi_{V}(\boldsymbol{q}=0)$,
and
$\delta\Delta(\boldsymbol{q}):=\Delta(\boldsymbol{q})-\Delta(\boldsymbol{q}=0)$.
This leads to

$C_{\Phi\Phi}\delta\Phi_{V}(\boldsymbol{q})+C_{\Phi\Delta}\delta\Delta(\boldsymbol{q})=\boldsymbol{n}_{\Phi}\cdot\boldsymbol{q}+\mathcal{O}(\boldsymbol{q}^{2},\delta\Phi_{V}^{2},\delta\Delta^{2}),$
(28)

where

$\displaystyle C_{\Phi\Phi}$
$\displaystyle=\frac{1}{g_{v}}+6v_{\Phi}\Phi_{V}^{2}(\boldsymbol{q}=0)-\frac{1}{8T}\sum_{\boldsymbol{k},p}\frac{1}{\cosh^{2}(\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q}=0,p}}{2T})},$
(29) $\displaystyle C_{\Phi\Delta}$
$\displaystyle=-\frac{\Delta(\boldsymbol{q}=0)}{8T}\sum_{\boldsymbol{k},p}\frac{1}{\sqrt{\xi_{\boldsymbol{k}}^{2}+\Delta^{2}(\boldsymbol{q}=0)}}\frac{p}{\cosh^{2}(\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q}=0,p}}{2T})},$
(30) $\displaystyle\boldsymbol{n}_{\Phi}$
$\displaystyle=\frac{1}{16T}\sum_{\boldsymbol{k},p}\frac{\boldsymbol{\nabla}_{\boldsymbol{k}}\xi_{\boldsymbol{k}}}{\cosh^{2}(\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q}=0,p}}{2T})}.$
(31)

In our case,
$\mathcal{E}_{\boldsymbol{k},\boldsymbol{q}=0,p}=\sqrt{\xi_{\boldsymbol{k}}^{2}+\Delta^{2}(\boldsymbol{q}=0)}+p\Phi_{V}(\boldsymbol{q}=0)$
only depends on $\boldsymbol{k}$ through $\xi_{\boldsymbol{k}}$, leading to
$\boldsymbol{n}_{\Phi}=0$. Note this would generically be different if the
order parameter of superconductivity or valley polarization
[$f_{\boldsymbol{k}}\neq\text{const.}$ in Eq. (12)] depended on momentum. We
thus conclude that
$\delta\Phi_{V}(\boldsymbol{q})\propto\delta\Delta(\boldsymbol{q})+\mathcal{O}(\boldsymbol{q}^{2})$.
This is consistent with Fig. 3(a) and $\alpha=0$ in Eq. (6) of the main text.

We can learn more by expanding Eq. (8b) in the same way,

$C_{\Delta\Delta}\delta\Delta(\boldsymbol{q})+C_{\Delta\Phi}\delta\Phi(\boldsymbol{q})=\boldsymbol{n}_{\Delta}\cdot\boldsymbol{q}+\mathcal{O}(\boldsymbol{q}^{2},\delta\Phi_{V}^{2},\delta\Delta^{2}),$
(32)

where the coefficients are given by

$\displaystyle C_{\Delta\Delta}$
$\displaystyle=4u_{\Delta}\Delta(\boldsymbol{q}=0)+\frac{1}{2}\sum_{\boldsymbol{k},p}\tanh\left(\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q}=0,p}}{2T}\right)\frac{\Delta(\boldsymbol{q}=0)}{(\xi_{\boldsymbol{k}}^{2}+\Delta^{2}(\boldsymbol{q}=0))^{\frac{3}{2}}}-\frac{1}{4T}\sum_{\boldsymbol{k},p}\frac{\Delta(\boldsymbol{q}=0)}{\xi_{\boldsymbol{k}}^{2}+\Delta^{2}(\boldsymbol{q}=0)}\frac{1}{\cosh^{2}(\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q}=0,p}}{2T})},$
(33) $\displaystyle C_{\Delta\Phi}$
$\displaystyle=-\frac{1}{2T}\sum_{\boldsymbol{k},p}\frac{1}{\sqrt{\xi_{\boldsymbol{k}}^{2}+\Delta^{2}(\boldsymbol{q}=0)}}\frac{p}{\cosh^{2}(\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q}=0,p}}{2T})},$
(34) $\displaystyle\boldsymbol{n}_{\Delta}$
$\displaystyle=\frac{1}{8T}\sum_{\boldsymbol{k},p}\frac{1}{\sqrt{\xi_{\boldsymbol{k}}^{2}+\Delta^{2}(\boldsymbol{q}=0)}}\frac{p\boldsymbol{\nabla}_{\boldsymbol{k}}\xi_{\boldsymbol{k}}}{\cosh^{2}(\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q}=0,p}}{2T})}.$
(35)

The same comments as those above for $\boldsymbol{n}_{\Phi}$ also apply for
$\boldsymbol{n}_{\Delta}$, which thus vanishes, $\boldsymbol{n}_{\Delta}=0$.
Without fine-tuning, we expect $C_{\Phi\Phi}C_{\Delta\Delta}\neq
C_{\Phi\Delta}C_{\Delta\Phi}$ and Eqs. (28) and (32) therefore imply
$\delta\Delta(\boldsymbol{q}),\delta\Phi_{V}(\boldsymbol{q})=\mathcal{O}(\boldsymbol{q}^{2})$
exactly as we find numerically, see Fig. 3(a). This is the reason why the
maximum of $\Delta(\boldsymbol{q})$ [minimum of $\Phi_{V}(\boldsymbol{q})$]
can be pinned to $\boldsymbol{q}=0$ even when $C_{3z}$ is broken. As
$\boldsymbol{n}_{\Phi},\boldsymbol{n}_{\Delta}$ become non-zero when form
factors are introduced, the maximum of $\Delta(\boldsymbol{q})$ [minimum of
$\Phi_{V}(\boldsymbol{q})$] move to finite values when $C_{3z}$ is broken as
is observed in Fig. 3(b).

##  Appendix D Microscopic calculation of current

We can write a BdG Hamiltonian for our system as,

$H_{\text{BdG}}=\begin{bmatrix}E_{\mathbf{k+\frac{q}{2}},+}&\Delta_{\mathbf{q}}\\\
\Delta_{\mathbf{q}}^{*}&-E_{\mathbf{-k+q/2},-}\end{bmatrix},\quad\Psi=\binom{c_{\mathbf{k+\frac{q}{2},+}}}{c_{\mathbf{-k+\frac{q}{2},-}}^{\dagger}},\quad\Psi^{\dagger}=\begin{pmatrix}c_{\mathbf{k+\frac{q}{2},+}}^{\dagger}&c_{\mathbf{-k+\frac{q}{2},-}}\end{pmatrix}$
(36)

and

$\binom{c_{\mathbf{k+\frac{q}{2},+}}}{c_{\mathbf{-k+\frac{q}{2},-}}^{\dagger}}=\begin{pmatrix}u_{\mathbf{k,q}}&v_{\mathbf{k,q}}\\\
-v_{\mathbf{k,q}}^{*}&u_{\mathbf{k,q}}^{*}\end{pmatrix}\binom{\gamma_{\mathbf{k,q},+}}{\gamma^{\dagger}_{\mathbf{-k,q},-}}.$
(37)

The BdG Hamiltonian in diagonal basis can be now written as,

$H_{\text{BdG}}=UH_{\text{diag}}U^{\dagger},\quad
U=\begin{pmatrix}u_{\mathbf{k,q}}&v_{\mathbf{k,q}}\\\
-v_{\mathbf{k,q}}^{*}&u_{\mathbf{k,q}}^{*}\end{pmatrix},\quad
U^{\dagger}=\begin{pmatrix}u_{\mathbf{k,q}}^{*}&-v_{\mathbf{k,q}}\\\
v_{\mathbf{k,q}}^{*}&u_{\mathbf{k,q}}\end{pmatrix}$ (38)

The expression for the current (see, e.g., Daido  _et al._ (2022)) reads as

$\displaystyle J$
$\displaystyle=\sum_{\mathbf{k}}\text{tr}\left[\partial_{\mathbf{q}}H_{\text{N}}n_{F}(H_{\text{BdG}})\right]$
(39)
$\displaystyle=\sum_{\mathbf{k}}\text{tr}\left[\begin{pmatrix}\partial_{\mathbf{q}}E_{\mathbf{k+\frac{q}{2}},+}&0\\\
0&-\partial_{\mathbf{q}}E_{\mathbf{-k+q/2},-}\end{pmatrix}n_{F}(H_{\text{BdG}})\right],$
(40)

where $n_{F}(\epsilon)=(1+e^{\beta\epsilon})^{-1}$ is the Fermi distribution
function and

$H_{\text{N}}=\begin{pmatrix}E_{\mathbf{k+\frac{q}{2}},+}&0\\\
0&-E_{\mathbf{-k+q/2},-}\end{pmatrix}.$ (41)

Therefore we have,

$\displaystyle n(H_{\text{BdG}})$
$\displaystyle=Un_{F}(H_{\text{diag}})U^{\dagger}$ (42)
$\displaystyle=U\begin{pmatrix}\frac{1}{1+e^{\beta E_{1,\mathbf{k,q}}}}&0\\\
0&\frac{1}{1+e^{-\beta E_{2,\mathbf{k,q}}}}\end{pmatrix}U^{\dagger}$ (43)

Plugging this into the expression for current we get:

$\displaystyle J$
$\displaystyle=\sum_{\mathbf{k}}\text{tr}\left[U^{\dagger}\begin{pmatrix}\partial_{\mathbf{q}}E_{\mathbf{k+\frac{q}{2}},+}&0\\\
0&-\partial_{\mathbf{q}}E_{\mathbf{-k+q/2},-}\end{pmatrix}Un_{F}(H_{\text{diag}})\right]$
(44)
$\displaystyle=\sum_{\mathbf{k}}\text{tr}\left[\begin{pmatrix}u_{\mathbf{k,q}}^{*}&-v_{\mathbf{k,q}}\\\
v_{\mathbf{k,q}}^{*}&u_{\mathbf{k,q}}\end{pmatrix}\begin{pmatrix}u_{\mathbf{k,q}}\partial_{\mathbf{q}}E_{\mathbf{k+\frac{q}{2}},+}&v_{\mathbf{k,q}}\partial_{\mathbf{q}}E_{\mathbf{k+\frac{q}{2}},+}\\\
v_{\mathbf{k,q}}^{*}\partial_{\mathbf{q}}E_{\mathbf{-k+\frac{q}{2}},-}&-u^{*}_{\mathbf{k,q}}\partial_{\mathbf{q}}E_{\mathbf{-k+q/2},-}\end{pmatrix}n_{F}(H_{\text{diag}})\right]$
(45)
$\displaystyle=\sum_{\mathbf{k}}\text{tr}\left[\begin{pmatrix}|u_{\mathbf{k,q}}|^{2}\partial_{\mathbf{q}}E_{\mathbf{k+\frac{q}{2}},+}-|v_{\mathbf{k,q}}|^{2}\partial_{\mathbf{q}}E_{\mathbf{-k+\frac{q}{2}},-}&u_{\mathbf{k,q}}^{*}v_{\mathbf{k,q}}\partial_{\mathbf{q}}E_{\mathbf{k+\frac{q}{2}},+}+u^{*}_{\mathbf{k,q}}v_{\mathbf{k,q}}\partial_{\mathbf{q}}E_{\mathbf{-k+q/2},-}\\\
v_{\mathbf{k,q}}^{*}u_{\mathbf{k,q}}\partial_{\mathbf{q}}E_{\mathbf{k+\frac{q}{2}},+}+u_{\mathbf{k,q}}v_{\mathbf{k,q}}^{*}\partial_{\mathbf{q}}E_{\mathbf{-k+\frac{q}{2}},-}&|v_{\mathbf{k,q}}|^{2}\partial_{\mathbf{q}}E_{\mathbf{k+\frac{q}{2}},+}-|u_{\mathbf{k,q}}|^{2}\partial_{\mathbf{q}}E_{\mathbf{-k+\frac{q}{2}},-}\end{pmatrix}n_{F}(H_{\text{diag}})\right]$
(46)
$J=\sum_{\mathbf{k}}(|u_{\mathbf{k,q}}|^{2}\partial_{\mathbf{q}}E_{\mathbf{k+\frac{q}{2}},+}-|v_{\mathbf{k,q}}|^{2}\partial_{\mathbf{q}}E_{\mathbf{-k+\frac{q}{2}},-})n_{F}(E_{1,\mathbf{k,q}})\\\
+\sum_{\mathbf{k}}(|v_{\mathbf{k,q}}|^{2}\partial_{\mathbf{q}}E_{\mathbf{k+\frac{q}{2}},+}-|u_{\mathbf{k,q}}|^{2}\partial_{\mathbf{q}}E_{\mathbf{-k+\frac{q}{2}},-})n_{F}(-E_{2,\mathbf{k,q}})$
(47)


and for each $s\in S$, there is some
$\sigma^{\prime}.\alpha(s)\langle\psi\rangle\in\sigma(b)$, such that
$f(\sigma^{\prime})R_{al(s)}f(\sigma^{\prime}.\alpha(s)\langle\psi\rangle)$
and
$\mathcal{M},f(\sigma^{\prime}.\alpha(s)\langle\psi\rangle)\models{\mathbf{finb}}(nx(s),\varepsilon)$,
or

$T\xrightarrow{g}S$:

and for each $s\in S$, there is some
$\sigma^{\prime}.\alpha(s)\langle\psi\rangle\in\sigma(b)$, such that
$f(\sigma^{\prime})R_{al(s)}f(\sigma^{\prime}.\alpha(s)\langle\psi\rangle)$
and
$\mathcal{M},f(\sigma^{\prime}.\alpha(s)\langle\psi\rangle)\models{\mathbf{finb}}(nx(s),\varepsilon)$
and for some $s\in S$, there is some
$\sigma^{\prime}.\alpha(s)\langle\psi\rangle\in\sigma(b)$, such that
$f(\sigma^{\prime})R_{al(s)}f(\sigma^{\prime}.\alpha(s)\langle\psi\rangle)$
and
$\mathcal{M},f(\sigma^{\prime}.\alpha(s)\langle\psi\rangle)\models{\mathbf{finb}}^{X}(nx(s),\varpi)$.

In both cases, by the inductive hypothesis there is a finite
$\xrightarrow{X}$-path in $b$, from $\sigma^{\prime}~{}\psi_{1}^{\prime}$ to
$\sigma~{}\psi_{2}^{\prime}$, where $X$ appears. Therefore, all five
conditions for $cyc_{X}(T,\sigma^{\prime})$ are satisfied and
$f(\sigma^{\prime})\in C_{Z_{T}^{X}}$, completeing the proof of the first
statement, and the induction.

We now prove that $cyc_{X}((\psi_{1},\psi_{2}),\sigma)$ implies
$\mathcal{M},f(\sigma)\models{\mathbf{finb}}^{X}((\psi_{1},\psi_{2}),\varepsilon)$.
By Condition 5 from $cyc_{X}((\psi_{1},\psi_{2}),\sigma)$, there is a finite
$\xrightarrow{X}$-path from $\psi_{1}$ to $\psi_{2}$ that only visits prefixes
that extend $\sigma$, including $\sigma$. Let $k$ be the length of the largest
$\sigma^{\prime}$, such that the path goes through $\sigma.\sigma^{\prime}$.
We proceed by induction on $k$. Let $T=(\psi_{1},\psi_{2}).$

If $k=0$,:

then the finite path remains in $\sigma$, and therefore
$(\psi_{1},\psi_{2})\xrightarrow{g(\sigma),X}\emptyset$. As
$\mathcal{M},f(\sigma)\models g(\sigma)$, we conclude that
$\mathcal{M},f(\sigma)\models_{\rho}{\mathbf{finb}}^{X}((\psi_{1},\psi_{2}),\varepsilon)$.

If $k>0$,:

we demonstrate that
$\mathcal{M},f(\sigma)\models{\mathbf{finb}}^{X}((\psi_{1},\psi_{2}),\varepsilon,g(\sigma))[{\mathbf{finb}}^{X}((\psi_{1},\psi_{2}),\varepsilon)/Z^{X}_{T}]$.
Let $p$ be the finite $\xrightarrow{X}$-path from $\psi_{1}$ to $\psi_{2}$.
Since $k>0$, $p$ will leave $\sigma$ and return, perhaps multiple times. As
$p$ leaves and returns to $\sigma$ $r$ times, in $\sigma$ it will visit
exactly the formulas
$\sigma~{}\chi_{1},\sigma~{}\chi_{2},\ldots,\sigma~{}\chi_{2r}\in b$, in that
order. We note that for every $1\leq i\leq r$,
$(out,bot)\in\ell_{g(\sigma)}(\chi_{2i-1})$ and
$(in,bot)\in\ell_{g(\sigma)}(\chi_{2i})$. Let $S=\\{s_{i}\mid 1\leq i\leq
r\\}$, where $s_{i}=(\chi_{2i-1},\chi_{2i})$ for every $i$; we see that
$(\psi_{1},\psi_{2})\xrightarrow{g(\sigma)}S$. Furthermore, if $p$ goes
through $\sigma~{}X$, then $(\psi_{1},\psi_{2})\xrightarrow{g(\sigma),X}S$,
and otherwise $p$ goes through some $\sigma^{\prime}~{}X$ between some
$\sigma~{}\chi_{2j-1}$ and $\sigma~{}\chi_{2j}$. For every $i\leq r$, there
are some $\sigma.\alpha(s_{i})\langle\psi\rangle~{}\chi^{\prime}_{2i-1}\in b$
and $\sigma.\alpha(s_{i})\langle\psi\rangle~{}\chi^{\prime}_{2i}\in b$, such
that
$\sigma~{}\chi_{2i-1}\xrightarrow{X}\sigma.\alpha(s_{i})\langle\psi\rangle~{}\chi^{\prime}_{2i}$
and
$\sigma.\alpha(s_{i})\langle\psi\rangle~{}\chi^{\prime}_{2i}\xrightarrow{X}\sigma~{}chi_{2i}$.
Then, the part of $p$ from
$\sigma.\alpha(s_{i})\langle\psi\rangle~{}\chi^{\prime}_{2i-1}$ to
$\sigma.\alpha(s_{i})\langle\psi\rangle~{}\chi^{\prime}_{2i}$ is a finite
$\xrightarrow{X}$-path that only goes through extensions of
$\sigma.\alpha(s_{i})\langle\psi\rangle$ that extend the prefix by a length at
most $k-1$. Therefore,
$cyc_{X}(nx(s_{i}),\sigma.\alpha(s_{i})\langle\psi\rangle)$, and by the
inductive hypothesis,
$\mathcal{M},f(\sigma.\alpha(s_{i})\langle\psi\rangle)\models{\mathbf{finb}}(nx(s_{i}),\varepsilon)$,
and if it is not the case that
$(\psi_{1},\psi_{2})\xrightarrow{g(\sigma),X}S$, and $i=j$,
$\mathcal{M},f(\sigma.\alpha(s_{i})\langle\psi\rangle)\models{\mathbf{finb}}^{X}(nx(s_{i}),\varepsilon)$.
Using Lemma 30 and the fact that
$f(\sigma)R_{\alpha(s_{i})}f(\sigma.\alpha(s_{i})\langle\psi\rangle)$, we
conclude that

$\mathcal{M},f(\sigma)\models{\mathbf{finb}}^{X}((\psi_{1},\psi_{2}),\varepsilon,g(\sigma))[{\mathbf{finb}}^{X}((\psi_{1},\psi_{2}),\varepsilon)/Z^{X}_{T}].$

Again, using Lemma 30, this yields that

$\displaystyle\mathcal{M},f(\sigma)\models{\mathbf{finb}}^{X}((\psi_{1},\psi_{2}),\varepsilon,g(\sigma))[{\mathbf{finb}}^{X}((\psi_{1},\psi_{2}),\varepsilon)/Z^{X}_{T}].$

We now express that there is an infinite path in $b$, where $X$ appears
infinitely often, starting from $\sigma~{}\psi$.

$\displaystyle{\mathbf{infb}}^{X}(\psi,\varpi,\varpi_{X})$
$\displaystyle:=\begin{cases}Z_{\psi}^{X}&\text{ if $\psi$ appears in
}\varpi_{X}\\\ Z_{\psi}&\text{ if }\psi\in\varpi\text{ and
}\not\in\varpi_{X}\\\ \nu Z_{\psi}.\mu
Z_{\psi}^{X}.\displaystyle\bigvee_{(\psi,in,top)\atop\in
g}g\land{\mathbf{infb}}^{X}(\psi,\varpi,\varpi_{X},g)&\text{
otherwise}\end{cases}$
$\displaystyle{\mathbf{infb}}^{X}(\psi,\varpi,\varpi_{X},g)$ $\displaystyle:=$
$\displaystyle\bigvee_{\psi\xrightarrow{g,X}S,\psi^{\prime}}$
$\displaystyle\bigwedge_{s\in
S}\langle\alpha(s)\rangle{\mathbf{finb}}(nx(s),\varepsilon)\land{\mathbf{infb}}^{X}(\psi^{\prime},\varpi\psi,\varepsilon)$
$\displaystyle\lor$
$\displaystyle\bigvee_{\psi\xrightarrow{g}S,\psi^{\prime}}$
$\displaystyle\left(\begin{array}[]{l}\left(\begin{array}[]{l}\displaystyle\bigvee_{s\in
S}\langle\alpha(s)\rangle{\mathbf{finb}}^{X}(nx(s),\varepsilon)\\\
\qquad\land\\\ \displaystyle\bigwedge_{s\in
S}\langle\alpha(s)\rangle{\mathbf{finb}}(nx(s),\varepsilon)\land{\mathbf{infb}}^{X}(\psi^{\prime},\varpi\psi,\varepsilon)\end{array}\right)\\\
\qquad\qquad\qquad\lor\\\ \bigwedge_{s\in
S}\langle\alpha(s)\rangle{\mathbf{finb}}(nx(s),\varepsilon)\land{\mathbf{infb}}^{X}(\psi^{\prime},\varpi\psi,\varpi_{X}\psi)\end{array}\right)$
$\displaystyle{\mathbf{inf\\_path}}_{X}$
$\displaystyle:={\mathbf{finb}}^{X}(\emptyset,\epsilon)\lor\bigvee_{\psi\in
g}\left(g\land{\mathbf{infb}}^{X}(\psi,\epsilon,\epsilon)\right)$
$\displaystyle{\mathbf{inf\\_path}}$ $\displaystyle:=\bigvee_{X\in
mV}{\mathbf{inf\\_path}}_{X}$

######  Lemma 32.

Let $X\in mV$, $f:b\xrightarrow{r}\mathcal{M},s$, and $\sigma\in\sigma(b)$.
$\mathcal{M},f(\sigma)\models{\mathbf{infb}}^{X}(\psi,\varepsilon,\varepsilon)$
if and only if

  1. (1)

$\sigma~{}\psi\in b$, and

  2. (2)

there is an infinite $\xrightarrow{X}$-path in $b$, from $\sigma~{}\psi$,
where $X$ appears infinitely often, that only visits prefixes that extend
$\sigma$.

######  Proof 6.28.

We first assume that $\sigma~{}\psi\in b$, and that there is an infinite
$\xrightarrow{X}$-path in $b$, from $\sigma~{}\psi$, where $X$ appears
infinitely often, that only visits prefixes that extend $\sigma$, and prove
that
$\mathcal{M},f(\sigma)\models{\mathbf{infb}}^{X}(\psi,\varepsilon,\varepsilon)$.
For every $\psi^{\prime}\in\overline{\mathsf{sub}}(\varphi)$, let
$IP_{\psi^{\prime}}$ be the set of prefixes $\sigma^{\prime}\in\sigma(b)$,
such that $\sigma^{\prime}~{}\psi^{\prime}\in b$, and that there is an
infinite $\xrightarrow{X}$-path in $b$, from
$\sigma^{\prime}~{}\psi^{\prime}$, where $X$ appears infinitely often, that
only visits prefixes that extend $\sigma^{\prime}$. To prove the implication,
we prove the stronger statement that
$\mathcal{M},f(\sigma)\models_{\rho}{\mathbf{infb}}^{X}(\psi,\varpi,\varpi_{X})$,
where for every $\psi^{\prime}\in\overline{\mathsf{sub}}(\varphi)$,
$\rho(Z_{\psi^{\prime}})=\rho(Z_{\psi^{\prime}}^{X})=\\{f(\sigma^{\prime})\mid\sigma^{\prime}\in
IP_{\psi^{\prime}}\\}$. We proceed by induction on the structure of the
formula.

If $\psi$ appears in $\varpi$ (or in $\varpi_{X}$), then
${\left\llbracket{\mathbf{infb}}^{X}(\psi,\varpi,\varpi_{X}),\rho\right\rrbracket}={\left\llbracket
Z_{\psi}\right\rrbracket}=\\{f(\sigma^{\prime})\mid\sigma^{\prime}\in
IP_{\psi}\\}\ni f(\sigma)$, and we are done. Otherwise, using the semantics of
Table 1, since

$f(\sigma)\in\\{f(\sigma^{\prime})\mid\sigma^{\prime}\in
IP_{\psi^{\prime}}\\}=\rho(Z_{\psi}),$

it suffices to prove that

$\\{f(\sigma^{\prime})\mid\sigma^{\prime}\in
IP_{\psi^{\prime}}\\}\subseteq{\left\llbracket\mu
Z_{\psi}^{X}.\bigvee_{(\psi,in,top)\in
g}{\mathbf{infb}}^{X}(\psi,\varpi,\varpi_{X},g),\rho\right\rrbracket}.$

Let $\sigma^{\prime}\in IP_{\psi^{\prime}}$. There must be an infinite
$\xrightarrow{X}$-path in $b$ from $\sigma^{\prime}~{}\psi$ that visits $X$
infinitely often, and avoids strict prefixes of $\sigma$. That path may visit
certain prefixes more than once. Let $\sigma^{\prime}.\sigma^{\prime\prime}$
be minimal, such that the path goes through some
$\sigma^{\prime}.\sigma^{\prime\prime}~{}\psi^{\prime}\in b$ after visiting
$X$ at least once after $\sigma^{\prime}~{}\psi$. Let $k$ be the length of
$\sigma^{\prime\prime}$. We prove that
$\mathcal{M},f(\sigma^{\prime})\models_{\rho}\mu
Z_{\psi}^{X}.\bigvee_{(\psi,in,top)\in
g}{\mathbf{infb}}^{X}(\psi,\varpi,\varpi_{X},g)$ and we do so by induction on
$k$. The case of $k=0$ is straightforward, using Lemma 31, and similar
reasoning as in the proof of Lemma 31, as $\mu Z_{\psi}^{X}$ is not free in
${\mathbf{infb}}^{X}(\psi^{\prime},\varpi\psi,\varepsilon)$. The case of $k>0$
is also straightforward, using Lemma 31, and similar reasoning as in the proof
of Lemma 31, but also the inductive hypothesis for $k-1$ for the case of
${\mathbf{infb}}^{X}(\psi^{\prime},\varpi\psi,\varepsilon)$. Thus, the
induction on $k$ is complete, and we proved that
$\mathcal{M},f(\sigma^{\prime})\models_{\rho}\mu
Z_{\psi}^{X}.\bigvee_{(\psi,in,top)\in
g}{\mathbf{infb}}^{X}(\psi,\varpi,\varpi_{X},g)$, and therefore the original
implication.

We now assume that
$\mathcal{M},f(\sigma)\models{\mathbf{infb}}^{X}(\psi,\varepsilon,\varepsilon)$
and prove that

  1. (1)

$\sigma~{}\psi\in b$, and

  2. (2)

there is an infinite $\xrightarrow{X}$-path in $b$, from $\sigma~{}\psi$,
where $X$ appears infinitely often, that only visits prefixes that extend
$\sigma$.

It is straightforward to see that for every prefix $\sigma\in\sigma(b)$,
$\mathcal{M},f(\sigma)\models{\mathbf{infb}}^{X}(\psi,\varpi,\varpi_{X})$
implies that $\sigma~{}\psi\in b$.

Let $\rightarrow$ be an lfp-finite dependency relation. We demonstrate how to
recursively construct an infinite $\xrightarrow{X}$-path in $b$, starting from
$\sigma~{}\psi$, where $X$ appears infinitely often. Let
$p_{0}=\sigma~{}\psi=\sigma_{0}~{}\psi_{0}$. For a finite
$\xrightarrow{X}$-path $p_{i}$ in $b$ that ends in $\sigma_{i}~{}\psi_{i}$,
such that
$\mathcal{M},f(\sigma_{i})\models{\mathbf{infb}}^{X}(\psi_{i},\varpi^{i},\varpi^{i}_{X})$,
we show how to extend the path to $p_{i+1}$ that ends in
$\sigma_{i+1}~{}\psi_{i+1}$, such that
$\mathcal{M},f(\sigma_{i+1})\models{\mathbf{infb}}^{X}(\psi_{i+1},\varpi^{i+1},\varpi^{i+1}_{X})$,
and

$(\sigma_{i},{\mathbf{infb}}^{X}(\psi_{i},\varpi^{i},\varpi^{i}_{X}))\rightarrow^{+}(\sigma_{i+1},{\mathbf{infb}}^{X}(\psi_{i+1},\varpi^{i+1},\varpi^{i+1}_{X}))~{}.$

By observing the definition of
$(\sigma_{i},{\mathbf{infb}}^{X}(\psi_{i},\varpi^{i},\varpi^{i}_{X}))$, and
using Lemma 31, it is not hard to see that

$(\sigma_{i},{\mathbf{infb}}^{X}(\psi_{i},\varpi^{i},\varpi^{i}_{X}))\rightarrow^{+}(\sigma_{i+1},{\mathbf{infb}}^{X}(\psi_{i+1},\varpi^{i+1},\varpi^{i+1}_{X}))$

for some
$(\sigma_{i+1},{\mathbf{infb}}^{X}(\psi_{i+1},\varpi^{i+1},\varpi^{i+1}_{X}))$,
where

  1. (1)

$\sigma_{i}R_{\alpha}\sigma_{i+1}$ for some $\alpha\in\text{\sc Ag}$;

  2. (2)

there is a finite $\xrightarrow{X}$-path $\varkappa_{i}$ in $b$ from
$\sigma_{i}~{}\psi_{i}$ to $\sigma_{i+1}~{}\psi_{i+1}$; and

  3. (3)

$\varkappa_{i}$ goes through some $\sigma^{\prime}~{}X$ if and only if
$\varpi_{X}^{i+1}=\varepsilon$.

We define $p_{i+1}=p_{i}\varkappa_{i}$. The infinite path $p$ is the path that
has every $p_{i}$ as a prefix. If $X$ does not appear infinitely often in $p$,
then there is some $i>0$, such that every $\varkappa_{j}$, where $j>i$, does
not go through any $\sigma^{\prime}~{}X$. The dependency relation
$\rightarrow$ goes through finitey many formulas, and therefore it goes
through some infinitely many times. Let
${\mathbf{infb}}^{X}(\psi^{\prime},\varpi,\varpi_{X})$ be the largest
subformula of ${\mathbf{infb}}^{X}(\psi,\varepsilon,\varepsilon)$ that appears
infinitely often as a
${\mathbf{infb}}^{X}(\psi_{j},\varpi^{j},\varpi^{j}_{X})$. Then, we observe
that there is an infinite $\xrightarrow{Z_{\psi^{\prime}}^{X}}$-path from some
$(f(\sigma_{j}),{\mathbf{infb}}^{X}(\psi^{\prime},\varpi,\varpi_{X}))$ that
visits $Z_{\psi^{\prime}}^{X}$ infinitely often. Therefore, the dependency
relation is not lfp-finite, contradicting our assumption that
$\mathcal{M},f(\sigma)\models{\mathbf{infb}}^{X}(\psi,\varepsilon,\varepsilon)$.
We conclude that $X$ appears infinitely often in $p$, which completes our
proof.

######  Theorem 33.

The formula ${\mathbf{rules}}\land
Inv(\neg{\mathbf{inf\\_path}})\land\bigvee_{\varphi\in g\in G^{t}}g$ is
satisfiable if and only if $\varphi$ is ${\mathbf{L}}$-satisfiable.

######  Proof 6.29.

If ${\mathbf{rules}}\land
Inv(\neg{\mathbf{inf\\_path}})\land\bigvee_{\varphi\in g\in G^{t}}g$ is
satisfiable, then there is a model $\mathcal{M},s\models{\mathbf{rules}}\land
Inv(\neg{\mathbf{inf\\_path}})\land\bigvee_{\varphi\in g\in G^{t}}g$, and
therefore $\mathcal{M},s\models{\mathbf{rules}}$. By Lemma 28, $\mathcal{M},s$
represent a maximal branch $b$ that is not locally closed. From
$\mathcal{M},s\models\bigvee_{\varphi\in g\in G^{t}}g$, $b$ is a branch of a
tableau for $\varphi$, and from $\mathcal{M},s\models
Inv(\neg{\mathbf{inf\\_path}})$ and Lemma 32, $b$ is open.

On the other hand, if $\varphi$ is ${\mathbf{L}}$-satisfiable, then its
tableau has an open branch, and similarly, by Lemmata 29, 28, and 32,
${\mathbf{rules}}\land Inv(\neg{\mathbf{inf\\_path}})\land\bigvee_{\varphi\in
g\in G^{t}}g$ is satisfiable Theorem 20 now yields that $\varphi$ is
${\mathbf{L}}$ satisfiable and we are done.

######  Corollary 34.

${\mathbf{L}}$-satisfiability is in ${\mathsf{2EXP}}$, for every logic
${\mathbf{L}}$, where no agent has constraint $5$.

######  Proof 6.30.

Given a formula $\varphi$, the height of the syntax tree of each
${\mathbf{finb}}(T,\varepsilon),{\mathbf{finb}}^{X}(T,\varepsilon)$, and
${\mathbf{infb}}^{X}(\psi,\varepsilon)$, due to the dependency on the
permutations $\varpi$, is $O(|\varphi|^{2})$. The syntax tree furthermore
branches with at most $2^{|\varphi|}$ disjunctions or conjunctions at each
node. Therefore, the formula ${\mathbf{rules}}\land
Inv(\neg{\mathbf{inf\\_path}})\land\bigvee_{\varphi\in g\in G^{t}}g$ has size
at most $2^{O(|\varphi|)}$. Each step of the construction only requires up to
double-exponential time with respect to $|\varphi|$. Theorem 3 then yields the
resulting double-exponential upper bound.

##### Negative introspection, again

We now explain how to adjust the construction of ${\mathbf{rules}}$ and
${\mathbf{inf\\_path}}$, to also handle the case of agents with negative
introspection. What differs in this case, is that the tableau rules for these
agents relate prefixes that are not necessarily consecutive. These prefixes
are, however, close, in that their closest common ancestor is at worst their
parent prefix. Therefore, the tableau rules are still, in that sense, local.
We adjust the contents of the graphs that we use as variables, to take into
account these extended formula interactions through the tableau rules. The
version of the $\mu$-calculus to which we translate is on the set of agents

$\text{\sc
Ag}_{5}^{\varphi}=\\{\gamma,\alpha\langle\psi_{1}\rangle\beta,\alpha\langle\psi_{1}\rangle\alpha\langle\psi_{2}\rangle\beta\mid\psi_{1},\psi_{2}\in\overline{\mathsf{sub}}(\varphi),~{}{\mathbf{\gamma}}\text{
does not have }5,{\mathbf{\alpha}}\text{ does },\alpha\neq\beta\\}.$

Let, for each $\alpha\in\text{\sc Ag}$, where ${\mathbf{L}}(\alpha)$ has
constraint $5$,

$\overline{\mathsf{sub}}_{5}(\varphi)=\\{\varepsilon~{}\psi,~{}\alpha\langle\psi_{1}\rangle~{}\psi,~{}\alpha\langle\psi_{1}\rangle.\alpha\langle\psi_{2}\rangle.\psi\mid\psi,\psi_{1},\psi_{2}\in\overline{\mathsf{sub}}(\varphi)\text{
and }{\mathbf{L}}(\alpha)\text{ has constraint }5\\}.$

The set ${\mathbf{G}}$ of labelled directed graphs is extended so that for
each $g\in G$, $\Phi_{g}\subseteq\overline{\mathsf{sub}}_{5}(\varphi)$.

We now say that $g=(\Phi,(\xrightarrow{X})_{X\in mV},\ell)\in{\mathbf{G}}$ is
_compatible with the tableau rules_ and write $com(g)$ when for every
$\delta=\varsigma~{}\psi\in\Phi$,
$\delta^{\prime},\delta_{1},\delta_{2}\in\overline{\mathsf{sub}}_{5}(\varphi)$,
$\beta\neq\alpha$, and $X\in mV$, where $\psi\neq Y$ for any $X<Y$:

  1. (1)

for every tableau rule of the form
$\frac{\sigma.~{}\delta}{\sigma.\varsigma^{\prime}~{}\delta^{\prime}}$,
$\delta^{\prime}\in\Phi$ and $\delta\xrightarrow{X}\delta^{\prime}$;

  2. (2)

for every tableau rule of the form
$\frac{\sigma~{}\delta}{\genfrac{}{}{0.0pt}{1}{\sigma~{}\delta_{1}}{\sigma~{}\delta_{2}}}$,
$\delta_{1}\in\Phi$, $\delta\xrightarrow{X}\delta_{1}$, $\delta_{2}\in\Phi$,
and $\delta\xrightarrow{X}\delta_{2}$;

  3. (3)

for every tableau rule of the form
$\frac{\sigma~{}\delta}{\sigma~{}\delta_{1}~{}\mid~{}\sigma~{}\delta_{2}}$,
$\delta_{1}\in\Phi$ and $\delta\xrightarrow{X}\delta_{1}$, or
$\delta_{2}\in\Phi$ and $\delta\xrightarrow{X}\delta_{2}$;

  4. (4)

for every tableau rule of the form
$\frac{\sigma~{}\delta}{\sigma.\beta\langle\psi_{1}\rangle~{}\psi_{2}}$,
$(out,bot)\in\ell(\delta)$;

  5. (5)

for every tableau rule of the form
$\frac{\sigma~{}\delta}{\sigma^{\prime}~{}\delta^{\prime}}$, where $\sigma$ is
a strict extension of $\sigma^{\prime}$, $(out,top)\in\ell(\delta)$; and

  6. (6)

$\varsigma~{}\neg\psi\notin\Phi$.

${\mathbf{G}}^{t}$ is then, the restriction of ${\mathbf{G}}$ on graphs that
are compatible with the tableau rules.

Let $a=a^{\prime}\beta\in\text{\sc Ag}_{5}^{\varphi}$,
$\chi\in\overline{\mathsf{sub}}(\varphi)$, and
$g=(\Phi_{g},(\xrightarrow{X}_{g})_{X\in
mV},\ell_{g}),~{}h=(\Phi_{h},(\xrightarrow{X}_{h})_{X\in mV},\ell_{h})\in
G^{t}$. We say that $g$ is an $a\langle\chi\rangle$-child of $h$, and write
$c_{a\langle\chi\rangle}(h,g)$ when:

  1. (1)

$a^{\prime}~{}\langle\beta\rangle\chi\in\Phi_{h}$ and $\chi\in\Phi_{g}$ and
$(in,top)\in\ell_{g}(\chi)$;

  2. (2)

if $a^{\prime}~{}[\beta]\psi\in\Phi_{h}$ and
$\frac{\sigma.a^{\prime}~{}[\beta]\psi}{\sigma.a\langle\psi_{1}\rangle~{}\psi_{2}}$
is a tableau rule, then $\psi_{2}\in\Phi_{g}$ and
$(in,top)\in\ell_{g}(\psi_{2})$; and

  3. (3)

if $[\beta]\psi\in\Phi_{g}$ and
$\frac{\sigma.\beta\langle\psi_{1}\rangle~{}[\beta]\psi}{\sigma~{}\psi_{2}}$
is a tableau rule, then $a^{\prime}~{}\psi_{2}\in\Phi_{h}$ and
$(in,bot)\in\ell_{h}(a^{\prime}~{}\psi_{2})$.

We can similarly define that $g$ is an $a$-child of $h$, and write
$c_{a}(h,g)$, by omitting the first condition. To simplify the notation
further on, we may write $\langle a\rangle\psi$ instead of
$a^{\prime}~{}\langle\beta\rangle\psi$.

The size of ${\mathbf{G}}$ is still at most exponential with respect to
$\varphi$ and it takes at most polynomial time to verify $com(g)$ and
$c(g,h)$; furthermore, $|\text{\sc Ag}_{5}^{\varphi}|\leq 2|\text{\sc
Ag}|\cdot|\overline{\mathsf{sub}}(\varphi)|^{2}$.

We use each $g\in G^{t}$ and $\ulcorner\psi\urcorner$, where
$\psi\in\overline{\mathsf{sub}}(\varphi)$, as a propositional variable. Let

$\displaystyle{\mathbf{rules}}:=Inv\left(\bigvee_{g\in
G^{t}}g\land\bigwedge_{g,h\in G^{t}}\neg(g\land
h)\land\bigwedge_{\begin{subarray}{c}g\in G^{t}\\\ \langle
a\rangle\psi\in\Phi_{g}\end{subarray}}g\rightarrow\langle
a\rangle\bigvee_{c_{a\langle\psi\rangle}(g,h)}h\land\bigwedge_{g\in
G^{t}}g\rightarrow[a]\bigvee_{c_{a}(g,h)}h\right)$

Let $b$ be a tableau branch. We define
$\sigma_{5}(b)=\sigma(b)\cap\\{a\langle\psi\rangle\mid a\in\text{\sc
Ag}_{5}^{\varphi},~{}\psi\in\overline{\mathsf{sub}}(\varphi)\\}^{*}$. For the
remaining construction, to express when there exists a path that visits some
$a~{}X$ infinitely often, it does not matter what objects the vertices of the
graphs are. We can then continue with the construction, similarly as for the
case of logics with no agents with negative introspection, using
$\sigma_{5}(b)$ instead of $\sigma(b)$; $\text{\sc Ag}_{5}^{\varphi}$ instead
of Ag, and $a$ instead of $\alpha$; $\overline{\mathsf{sub}}_{5}(\varphi)$
instead of $\overline{\mathsf{sub}}(\varphi)$ and $\delta$ instead of $\psi$;
and $a~{}X$ instead of $X$. That construction yields the following result.

######  Corollary 35.

${\mathbf{L}}$-satisfiability is in ${\mathsf{2EXP}}$.

##  7\. Conclusions

In this work we studied the complexity of the satisfiability problem for
multi-modal logics with recursion. These logics mix the frame conditions from
epistemic modal logic and the recursion operators of the $\mu$-calculus.

In the first part we gave simple translations among several of these logics
that connect their satisfiability problems. This approach allowed us to offer
complexity bounds for satisfiability and to prove finite-model results. We
also presented a sound and complete tableau that has termination guarantees,
conditional on a logic’s finite-model property. At the end of this work we had
several hardness and completeness results for individual logics and their
combinations, with a few isolated cases missing (results summarized in Table
2).

In the second part of this work we aimed to fill these gaps in complexity
results, specifically for the case of the multi-agent $\mu$-calculus over
frames containing the ${\mathbf{B}}$ and ${\mathbf{K5}}$ restrictions and the
case of single-agent $\mu$-calculus over transitive frames. To tackle the
multi-agent case, we gave a translation from tableaux to $\mu$-calculus
formulas, which allowed us to prove the missing complexity results and to use
the known algorithm for testing satisfiability over the $\mu$-calculus for
those cases. The reason why our translation-based approach did not work for
some of the logics considered in Table 2 is that, in those cases, it produced
formulas whose size was not polynomial in that of the input formulas. In those
cases, the resulting upper and lower bounds did not match. Of course, it is
possible that translations different from those we proposed might work.
However, we showed that a straightforward, one-step translation would not work
even in the single-agent setting. To address this limitation of the
translation-based approach, we optimized the tableaux exploration algorithm we
already had from the first part of this work by employing the transitivity
frame restriction. This optimization yielded the tight $\mathsf{PSPACE}$
bound.

Table 5 summarizes the results.

$\\#$ agents | Restrictions on syntax/frames | Upper Bound | Lower Bound
---|---|---|---
| frames with B or 5 |  $\mathsf{2EXP}$ 35 |  $\mathsf{EXP}$-hard 6
$\geq 2$ | not B , 5 |  $\mathsf{EXP}$ 16 |  $\mathsf{EXP}$-hard 6
| not 5, not $\mu$. X |  $\mathsf{EXP}$ 17 |  $\mathsf{EXP}$-hard 5
| with 5 (or B4) |  $\mathsf{NP}$ 1 |  $\mathsf{NP}$-hard 1
1 | with 4 |  $\mathsf{PSPACE}$ 27 |  $\mathsf{PSPACE}$-hard 1
| Any other restrictions |  $\mathsf{EXP}$ 3 |  $\mathsf{EXP}$-hard 3
Table 5. The updated summary of the complexity of satisfiability checking for
various modal logics with recursion. Our additional contributions are
highlighted.

##### Conjectures and Future Work

We currently do not have a tight complexity bound for the case of the multi-
agent $\mu$-calculus over symmetric or euclidean frames. What is interesting
is that we also do not have a counterexample to prove that the translations of
Section 3.1.3 that we already have, as well as other ones we attempted, do not
preserve satisfiability.

The complexity of the model checking problem for the $\mu$-calculus is an
important open problem, known to have a quasi-polynomial time solution, but
not known whether it is in $\mathsf{P}$ [LPSW22, Leh18, JL17, CJK+17, FJS+17].
The problem does not depend on the frame restrictions of the particular logic,
though one may wonder whether additional frame restrictions would help solve
the problem more efficiently. Currently we are not aware of a way to use our
translations to obtain such an improvement.

As, to the best of our knowledge, most of the logics described in this chapter
have not been explicitly defined before, with notable exceptions such as
[DL10, DL13, AF09], they also lack any axiomatizations and completeness
theorems. We do expect the classical methods from [Koz83, Lad77, HM92] and
others to work in these cases as well. However, it would be interesting and
desirable to flesh out the details and see if there are any unexpected
situations that arise.

Given the importance of common knowledge for epistemic logic and the fact that
it has been known that common knowledge can be thought of as a (greatest)
fixed-point already from [Har77, Bar88], we consider the logics that we
presented to be natural extensions of modal logic. Besides the examples given
in Section 2, we are interested in exploring what other natural concepts can
be defined with this enlarged language.

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