# Superconducting Diode Effect in Multiphase Superconductors

Daniel Shaffer  Department of Physics, University of Wisconsin-Madison,
Madison, Wisconsin 53706, USA  Dmitry V. Chichinadze  National High Magnetic
Field Laboratory, Tallahassee, Florida 32310, USA  Alex Levchenko  Department
of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA

###### Abstract

We identify a new mechanism for the intrinsic superconducting diode effect
(SDE) in multiphase superconductors. Using a Ginzburg-Landau and a microscopic
two-band model, we find phase transitions into a mixed phase with finite-
momentum Cooper pairs and SDE with high (including maximal) diode
efficiencies, despite the individual phases exhibiting no SDE and equal
inversion parity. We thus show that parity mixing – invoked in previous
proposals – is not a crucial ingredient for SDE. The new mechanism may be
relevant in a multitude of known multiphase superconductors like UTe2.

_Introduction –_ Barring early observations from boundary effects [1, 2] and
theoretical predictions [3, 4, 5, 6], the superconducting diode effect (SDE)
has only recently become a subject of much interest, spurred by its
experimental discovery in bulk materials and Josephson junctions [7, 8, 9, 10,
11, 12, 13, 14] and subsequent improved theoretical understanding [15, 16, 17,
18, 19, 20, 21, 22]; for reviews see [23, 24]. Defined as the non-reciprocity
of critical currents in bulk superconductors, SDE has many potential
applications in superconducting and quantum electronics. There is consequently
a high demand for novel realizations of SDE, with recent proposals including
chiral SDE [25], unidirectional SDE [26], SDE due to magnetization gradients
[27], SDE due to spontaneous symmetry breaking and coexisting non-SC order
[28], and SDE in altermagnets [29, 30].

All proposals necessarily require the explicit or spontaneous breaking of both
inversion $\mathcal{I}$ and time reversal $\mathcal{T}$ symmetries. The
crystallographic symmetry imposes additional constrains on the Lifshitz
invariants responsible for the finite SDE. The Lifshitz invariants for
different crystallographic point groups are tabulated in Ref. [31]. Breaking
of $\mathcal{I}$ and $\mathcal{T}$ symmetries is generically accompanied by
finite-momentum pairing similar to that in the FFLO state [32, 33, 34] or
$\mathcal{T}$-breaking pair-density waves (PDW) [35, 36, 37, 38, 39, 40, 30].
The finite-momentum pairing is strictly speaking neither necessary nor
sufficient for SDE to occur, though it breaks the same symmetries. This is
especially true for weak-coupling PDWs that take place due to Fermi surface
nesting, with the Fermi surface symmetric under
$\mathbf{p+q}\rightarrow-\mathbf{p+q}$: that same symmetry, if exact,
precludes the SDE.

The FFLO as a mechanism for SDE was considered in Ref. [16]. Similarly, SDE
was initially theorised [5, 15, 16] and reported [7, 9] in finite-momentum
carrying helical SCs [41, 42, 31], i.e. noncentrosymmetric SCs with spin-orbit
coupling (SOC) placed in in-plane magnetic fields. Due to the broken
$\mathcal{I}$, these systems also in general exhibit parity mixing, e.g.
$s$/$p$-wave or singlet/triplet mixing SC [43, 44, 45, 46, 47, 37]. On the
other hand, mixed-parity SC has also been considered in multiphase SCs near
phase boundaries between SCs of opposite parity in systems without an explicit
$\mathcal{I}$-breaking (with or without $\mathcal{T}$-breaking). In that case
$\mathcal{I}$ is instead spontaneously broken in the mixed or coexistence
phase [48, 49, 50, 51]. In particular, a phase transition between SCs of
opposite parity has been reported in CeRh2As2 under an applied magnetic field
[52, 53, 54, 55]; another historical example is 3He [56, 57]. It has recently
been shown [58] that SDE can indeed be realized in such coexisting mixed-
parity SCs in the particular case of anapole SCs (which preserve
$\mathcal{IT}$) [59], theoretically considered in UTe2 [60, 61].

In this Letter, we show that parity mixing is not essential for producing the
SDE, and thus establish a new and more general mechanism for realizing SDE in
multiphase superconductors. This greatly extends the list of materials with
potential for SDE, including $s+d$ [62, 63] and $s+is$ [64, 65] SC proposed in
iron-based high-Tc SCs and moiré SCs like twisted bilayer graphene [66]. The
key theoretical observation behind the proposed mechanism is that even if the
two nearby SC order parameters are uniform (have zero momentum pairing) and
have equal parity, finite-momentum pairing with momentum $q$ can still emerge
in the coexistence phase.

We demonstrate this using a minimal Ginzburg-Landau (GL) free energy with two
order parameters $\Delta_{a}$ and $\Delta_{b}$ with critical temperatures that
can be tuned by some external parameter $p$ (pressure, magnetic or electric
field, doping, etc.). Crucially, a mixing term of the form
$\alpha_{ab}(q)\Delta_{a}^{*}\Delta_{b}+c.c.$ with $\alpha_{ab}\propto q^{2}$
is allowed by symmetry. A first or second order phase transition then takes
place into the nonuniform phase depending on the strength of $\alpha_{ab}$, as
shown in Fig. 1, spontaneously breaking both $\mathcal{I}$ and $\mathcal{T}$
symmetries (unless $p$ is itself a $\mathcal{T}$-breaking field). High
(including perfect) diode efficiencies in the vicinity of the phase
transitions are found. We confirm the plausibility of the proposed mechanism
by calculating the GL coefficients in a two-band microscopic toy model.

Figure 1: (a) Phase diagram of free energy Eq. Superconducting Diode Effect in
Multiphase Superconductors with parameters $T$, $p$ and $\alpha_{ab,2}$. Blue
surface indicates the phase boundary of the non-uniform finite-momentum phase
exhibiting SDE. Uniform (zero-momentum) SC boundaries for both $a$ and $b$
phases are shown in gray. Red and blue points indicate numerically computed
first and second order phase transitions, respectively. (b) and (c) show
representative cuts of the phase diagram at fixed $T$ and $\alpha_{ab,2}$,
respectively, with first and second order phase transitions shown in red and
blue. (d), (e), (f) show the schematic evolution of the condensation energy
$F$ (blue) and the supercurrent $J$ (orange) vs Cooper pair momentum $q$
across the lower-temperature second order, first order, and higher-temperature
second order transitions, respectively (computed at various points
$(p,T,\alpha_{ab,2})$ in the phase diagram indicated on bottom). All plots are
in arbitrary units and centered at zero. Marked maxima (minima) of $J$
correspond to the critical suppercurents $J_{c+}$ ($J_{c-}$).

_Ginzburg-Landau Model–_ We consider the simplest possible Ginzburg-Landau
theory of two coexisting superconducting orders $\Delta_{a}$ and $\Delta_{b}$
(working in momentum space and assuming plane wave $e^{i\mathbf{q\cdot r}}$
solutions):

$\displaystyle\mathcal{F}$
$\displaystyle=\alpha_{a}(q)|\Delta_{a}|^{2}+(\alpha_{ab}(q)\Delta_{a}^{*}\Delta_{b}+c.c.)+\alpha_{b}(q)|\Delta_{b}|^{2}+$
$\displaystyle+\beta_{a}|\Delta_{a}|^{4}+\beta_{b}|\Delta_{b}|^{4}\,,$ (1)

where $\Delta_{a}$ and $\Delta_{b}$ are distinguished by their symmetry
properties (i.e. they belong to different irreducible representations, e.g.
they may be $s$\- and $d$-wave, etc.) and we assume inversion symmetry. For
simplicity we neglect the momentum dependence of fourth order coefficients,
and assume both $\Delta_{a}$ and $\Delta_{b}$ carry the same momentum in the
coexistence phases. The mixed coefficient $\alpha_{ab}(\mathbf{q})$ has to
vanish at $\mathbf{q}=0$ and is either odd or even in $\mathbf{q}$ if
$\Delta_{a}$ and $\Delta_{b}$ have opposite or equal parity under inversion.
The former case includes the anapole SC [59] as a special case.

Here we consider the equal parity case and work to fourth order in $q$, taking
$\alpha_{ab}(\mathbf{q})=\alpha_{ab,2}(q_{x}^{2}-q_{y}^{2})/2=\alpha_{ab2}q^{2}\cos\left(2\theta\right)/2$
where $\theta$ is the momentum polar angle (note that by symmetry, it cannot
be constant in $\theta$). Minimizing with respect to $\theta$ and the relative
phase between $\Delta_{a}$ and $\Delta_{b}$, we obtain

$\displaystyle\mathcal{F}$
$\displaystyle=\alpha_{a}(q)|\Delta_{a}|^{2}-\alpha_{ab,2}q^{2}|\Delta_{a}||\Delta_{b}|+\alpha_{b}(q)|\Delta_{b}|^{2}+$
$\displaystyle+\beta_{a}|\Delta_{a}|^{4}+\beta_{b}|\Delta_{b}|^{4}$ (2)

with $\alpha_{f}(q)=\alpha_{f,0}+\alpha_{f,2}q^{2}+\alpha_{f,4}q^{4}$ for
$f=a,b$, and where $\alpha_{ab,2}$ is now redefined to be real and positive.
We also take $\alpha_{f,0}(T,p)=A(T-T_{cf}(p))$ where $T$ is temperature and
$p$ is some tuning parameter (modelling e.g. pressure, magnetic field, doping,
etc.) that tunes between the $a$ and $b$ phases; for simplicity we take
$T_{ca}=1-p$ and $T_{cb}=p$.

The mixed GL coefficient $\alpha_{ab,2}$ is the new element of the model
responsible for the field-free SDE. To understand this mechanism, we consider
the GL saddle-point equations

$\displaystyle\frac{\delta\mathcal{F}}{\delta|\Delta_{f}|}$
$\displaystyle=2\alpha_{f}(q)|\Delta_{f}|-\alpha_{ab,2}q^{2}|\Delta_{\bar{f}}|+4\beta_{f}|\Delta_{f}|^{3}=0$
$\displaystyle\frac{\delta\mathcal{F}}{\delta q^{2}}$
$\displaystyle=(\alpha_{a,2}|\Delta_{a}|^{2}-\alpha_{ab,2}|\Delta_{a}||\Delta_{b}|+\alpha_{b,2}|\Delta_{b}|^{2})+$
$\displaystyle+2\left(\alpha_{a,4}|\Delta_{a}|^{2}+\alpha_{b,4}|\Delta_{b}|^{2}\right)q^{2}=0\,,$
(3)

where $\bar{f}=a,b\neq f$. The SDE occurs when $q\neq 0$ in the ground state
(i.e. at the lowest energy saddle point). In a pure state ($\Delta_{a}\neq 0$
and $\Delta_{b}=0$, or vice versa), this is only possible if $\alpha_{f,2}<0$
for either $f=a,b$, which happens in the FFLO mechanism [33, 32, 34]. Assuming
no FFLO physics, i.e. $\alpha_{f,2}>0$, it is still possible to have $q=\pm
q_{0}\neq 0$ in the mixed state if $\alpha_{ab,2}>0$ is sufficiently large.

The GL model is simple enough that we can determine some of the conditions for
the finite-momentum pairing to appear analytically. First, we note that
$\mathcal{F}$ always has at most three minima: at $q=0$ or at $q=\pm q_{0}$
(see [67]). We also observe that $\delta\mathcal{F}/\delta q$ is always zero
at $q=0$, so it is always a saddle point. This makes a first order phase
transition possible across which $q=0$ remains a local minimum solution while
the $q=q_{0}$ becomes the global minimum solution, see Fig. 1(e). This is in
contrast to the anapole model, in which $\alpha_{ab}$ is linear in $q$ to
leading order and a kink occurs at $q=0$ as a result, which makes the first
order phase transition unlikely.

Next, optimizing $\mathcal{F}$ over $q$ at fixed $|\Delta_{a}|$ and
$|\Delta_{b}|$ we have

$q_{0}^{2}=-\frac{\alpha_{a,2}r^{2}-\alpha_{ab,2}r+\alpha_{b,2}}{2\left(\alpha_{a,4}r^{2}+\alpha_{b,4}\right)}=-\frac{C(r)}{2\left(\alpha_{a,4}r^{2}+\alpha_{b,4}\right)}$
(4)

where $r=|\Delta_{a}/\Delta_{b}|$. For $q_{0}$ to exist $C(r)$ must be
negative. Note that $C(r)=R$ with fixed $R$ defines a conic section in the
$(|\Delta_{a}|,|\Delta_{b}|)$, with discriminant
$D=\alpha_{ab,2}^{2}-4\alpha_{a,2}\alpha_{b,2}$.111Recall that if $D<0$, $C=R$
defines an ellipse and $C\sim\tilde{x}^{2}+\tilde{y}^{2}\geq 0$. If $D>0$,
$C=R$ defines hyperbolas, and $C\sim\tilde{x}^{2}-\tilde{y}^{2}$ can change
sign. $C=0$ in particular defines two lines,
$|\Delta_{a}|=r_{c}(1\pm\sqrt{1-d})|\Delta_{b}|$, across which $C$ changes
sign, with $r_{c}=\alpha_{ab,2}/(2\alpha_{a,2})$ and
$d=4\alpha_{a,2}\alpha_{b,2}/\alpha_{ab,2}^{2}$. For $C(r)$ to be negative,
the gap ratio $r\equiv|\Delta_{a}|/|\Delta_{b}|$ must be in the range $r\in
r_{c}\left[1-\sqrt{1-d},1+\sqrt{1-d}\right]$.222We also note that there is an
upper bound on $q_{0}$ that can be obtained by solving for Eq. (4) for
$r(q_{0})$: real solutions exist only if
$4(\alpha_{a,2}+2\alpha_{a,4}q_{0}^{2})(\alpha_{b,2}+2\alpha_{b,4}q_{0}^{2})<\alpha_{ab,2}^{2}$
(this can be easily reformulated as an explicit condition for $q_{0}$, but the
expression is a bit lengthy; for large $\alpha_{ab,2}$, the condition is
simply $q_{0}^{2}<\alpha_{ab,2}/(4\sqrt{\alpha_{a,4}\alpha_{b,4}})$). In
particular, it is necessary that $d<1$, which yields a necessary condition on
the GL coefficients for SDE to exist:
$\alpha_{ab,2}^{2}>4\alpha_{a,2}\alpha_{b,2}$.

At a second order phase transition (if it occurs) inside the coexistence
region, the gap ratio is the same as in the uniform $q=0$ state,
$r=r_{0}=\sqrt{\alpha_{a,0}\beta_{b}/(\alpha_{b,0}\beta_{a}})$. One can also
check that the $q=0$ saddle point becomes a maximum at the same time, since
$\delta^{2}\mathcal{F}/\delta q^{2}|_{q=0}=2C|\Delta_{b}|^{2}$, and
consequently the $q=0$ solution becomes a local maximum when $C$ changes sign
from positive to negative, which happens precisely when
$r=r_{0}=r_{c}(1\pm\sqrt{1-d})\equiv r_{c\pm}$. As we find below in our
numerical study, however, a first order phase transitions generically happen
when $\alpha_{ab,2}^{2}\gg 4\alpha_{a,2}\alpha_{b,2}$, in which case the $q=0$
saddle point remains a local minimum at the transition.

Figure 2: Order parameters $\Delta_{a}$ (blue), $\Delta_{b}$ (orange) and $q$
(green) computed across various phase transitions along cuts (i-vi) indicated
in Fig. 1(b), with numerical values in arbitrary units on the left. The
corresponding superconducting diode coefficient $\eta$ is shown in magenta
with numerical values on the right axis.

_Phase Diagram and SDE –_ We next study the phase diagram of the free energy
Eq. Superconducting Diode Effect in Multiphase Superconductors numerically,
treating $T$, $p$ and $\alpha_{ab,2}$ as variable parameters and taking
$\alpha_{f,2}=0.5$, $\alpha_{f,4}=1$ and $\beta_{f}=1$ for both $f=a,b$ (such
that the phase diagram is symmetric with respect to $p\rightarrow 1-p$). The
results are shown in Fig. 1. In Fig. 1(a) the red/blue surface separates the
uniform ($q=0$) and nonuniform ($q\neq 0$) phases. $\Delta_{a}$ and
$\Delta_{b}$ always coexist in the nonuniform phase, and coexist in the
uniform phase when both $T<T_{ca}(p)$ and $T<T_{cb}(p)$ (shown as gray surface
in Fig. 1(a) and gray dashed lines in Fig. 1(c)). See also Fig. 2 where
$\Delta_{a},\Delta_{b}$, and $q$ are shown in blue, orange and green,
respectively, for various fixed temperature cuts of the phase diagram
indicated in Fig. 1(b). Fig. 1(d-f) additionally shows the condensation energy
$F(q)$ (the minimum of $\mathcal{F}$ at fixed $q$) and the corresponding
supercurrent $J(q)=2\partial_{q}F(q)$ at various points in the phase diagram.

The nonuniform phase occurs only for
$\alpha_{ab,2}^{2}>4\alpha_{a,2}\alpha_{b,2}=1$. For $\alpha_{ab,2}$ not too
far above that value the nonuniform phase occurs mostly in the coexistence
region of the uniform phase (with $T<\min[T_{ca}(p),T_{cb}(p)]$). Close to the
degeneracy point and for larger $\alpha_{ab,2}$, however, the nonuniform phase
appears outside of the uniform coexistence region, even including the region
where no SC exists for $\alpha_{ab,2}<1$ (namely when
$T>\max[T_{ca}(p),T_{cb}(p)]$. Fig. 1(b) shows the typical phase boundaries at
fixed $T$ below (solid line) and above (dashed line) the uniform degeneracy
point at $T=0.5$. For $T<0.5$, at lower $\alpha_{ab,2}$ the phase transition
is second order: the minimum at $q=0$ becomes unstable at the transition and
the two minima at $q=\pm q_{0}$ continuously emerge (see Fig. 1(d)). Below the
transition in the uniform phase with $q=0$, the critical currents $J_{c+}$ and
$J_{c-}$ occur at $q=q_{c+}>0$ and $q=q_{c-}=-q_{c+}<0$. In the non-uniform
phase, while $J_{c+}$ still occurs at a slightly shifted $q_{c+}>0$, the lower
critical current corresponds to a new minimum of $J(q)$ at $q_{c-}>0$.
Approaching the phase transition from above, $J_{c-}$ therefore approaches
zero, and the corresponding superconducting diode coefficient
$\eta=(J_{c+}+J_{c-})/(J_{c+}-J_{c-})$ (with $J_{c-}<0$) therefore approaches
its maximum possible value of $1$, technically resulting in a “perfect” diode
effect. This can be seen in Fig. 2(i) and (iii), where $\eta$ is shown in
magenta. A similar “perfect” diode effect was identified at the tricritical
point of the FFLO state [16]. We note, however, and as noted in the
supplementary in [16], that $\eta$ likely does not reach the maximal value in
real systems as the nonuniform state can jump from the $q_{0}$ to $-q_{0}$
state before the transition happens, as the energy barrier for that jump
vanishes at the transition.

At larger $\alpha_{ab,2}$, the transition is similar for $T$ either above or
below the uniform degeneracy point $T=0.5$ and becomes first order (see Fig.
1(e)): the minimum at $q_{0}$ becomes a global minimum before the saddle-point
at $q=0$ switches from a minimum to a maximum. In this case $q$, as well as
$\Delta_{a}$ and $\Delta_{b}$, jump by a finite value at the phase transition,
as shown in Fig. 2(ii) and (v), as well as the transitions in (iv) at the
smallest and largest $p$ values. In this case both $J_{c+}$ and $J_{c-}$ jump
at the transition, so that $\eta$ is in general not equal to $1$ at the
transition, but it is also not equal to zero and in general increases as the
transition is approached from the nonuniform side.

Finally, a different kind of second order phase transition happens for
$T>T_{ca},T_{cb}$, as shown in Fig. 1(e): in this case the uniform solution
does not exist, so no phase transition can occur between the uniform and
nonuniform states. The order parameter $\Delta_{a}$ and $\Delta_{b}$, however,
still vanish at the transition into the normal state (see Fig. 2(iv) and the
phase transitions closest to $p=0.5$ in (iv)). Consequently, while $q$ still
jumps by a finite value the superconducting diode coefficient $\eta$ vanishes
at this transition.

_Microscopic derivation of GL coefficients –_ In order for SDE to exist the GL
coefficients must satisfy the necessary condition
$\alpha_{ab,2}^{2}>4\alpha_{a,2}\alpha_{b,2}$, as explained above. It is
reasonable to ask whether this is a realistic condition from a microscopic
perspective. Here we show that this it is indeed possible in a reasonable
microscopic toy model, namely a two-band (or two-flavor) model with normal
state Hamiltonian
$\mathcal{H}=\sum_{\mathbf{k},f}\left(\xi(\mathbf{k})c^{\dagger}_{\mathbf{k}f}c_{\mathbf{k}f}+tc^{\dagger}_{\mathbf{k}f}c_{\mathbf{k}\bar{f}}\right)$
where $c_{\mathbf{k}f}$ annihilate electrons with momentum $\mathbf{k}$ on
band $f=a,b$, and $\bar{f}=a,b\neq f$. Assuming that the gap functions
$\hat{\Delta}_{f}\sim\langle c_{\mathbf{k}f}c_{-\mathbf{k+q}f}\rangle$, that
there is no interband pairing, and taking for simplicity
$\hat{\Delta}_{a}=\Delta_{a}$ and $\hat{\Delta}_{b}=\Delta_{b}\cos 2\theta$,
we find that in the weak-coupling approximation (with
$\xi(\mathbf{k})=\xi_{0}+\mathbf{v}_{F}\cdot\mathbf{k}$ linearized around the
Fermi momentum)

$\frac{\alpha_{ab,2}^{2}}{4\alpha_{a,2}\alpha_{b,2}}=\frac{1}{2}\left(\frac{14\zeta(3)+\text{Re}[\psi^{(2)}(1/2+it/(2\pi
T))]}{14\zeta(3)-\text{Re}[\psi^{(2)}(1/2+it/(2\pi T))]}\right)^{2}$ (5)

(see [67] for calculation details). The function on the RHS takes values in
the range $[0,1.1665]$, being zero at $t=0$, peaking at $t=\pi T$, and tending
to $0.5$ as $t\rightarrow\infty$. There is therefore a range of $t$ close to
$\pi T$ where the SDE indeed takes place in the phase diagram. It is then
reasonable to assume that the condition
$\alpha_{ab,2}^{2}>4\alpha_{a,2}\alpha_{b,2}$ for SDE can generically be
realized in more sophisticated models relevant to real materials.

_Discussion –_ From a materials perspective, the possibility of SDE in
multiphase SCs with equal parity mixing proposed in this Letter greatly
extends the possibilities for searching for SDE. Though rare, there is now
evidence for many multiphase SCs, including UPt3 [70], UBe13 [71], UTe2 [72,
73, 74, 75, 76, 77, 78, 79], Sr2RuO4 [80, 81], Cd2Re2O7 [82], and PrOs4Sb12
[83]. From our microscopic two-band model we further learned that multiband
SCs with dominant intraband pairing may favor the SDE. This suggests that
multiband and multicomponent SCs including high $T_{c}$ [84, 63] but
especially those that may favor intraband pairing – such as locally [85, 86]
and globally [42, 31] noncentrosymmetric, sublattice-polarized [87, 36, 39],
and (twisted moiré or untwisted) multilayer superconductors [88, 89, 90, 91] –
are particularly promising candidates for realizing SDE. Given the high diode
efficiencies that we find in the vicinity of the phase transitions where the
SDE can be turned on and off (i.e. an implementation of a SC transistor) by
small changes in external parameters without leaving the SC regime, such
multiphase SC-based superconducting diodes may be particularly suitable for
future technological applications.

_Acknowledgements –_ This work was financially supported by the National
Science Foundation, Quantum Leap Challenge Institute for Hybrid Quantum
Architectures and Networks Grant No. OMA-2016136 (D.S. and A.L.). A. L.
gratefully acknowledges H. I. Romnes Faculty Fellowship provided by the
University of Wisconsin-Madison Office of the Vice Chancellor for Research and
Graduate Education with funding from the Wisconsin Alumni Research Foundation.
D.V.C. acknowledges financial support from the National High Magnetic Field
Laboratory through a Dirac Fellowship, which is funded by the National Science
Foundation (Grant No. DMR-1644779) and the State of Florida. This paper was
written in-part at the Aspen Center for Physics, during the program “Quantum
Matter Through the Lens of Moiré Materials”, which is supported by the NSF
Grant No. PHY-2210452.

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Supplementary Material for Superconducting Diode Effect in Multiphase
Superconductors

##  I Ginzburg-Landau Saddle Point Equations for Mixed-Parity Case

For the case of mixed-parity mixing considered, e.g., for the case of anapole
SC in [59], $\alpha_{ab}(q)$ is an odd function of $q$. To leading order in
$q$, we can thus take $|\alpha_{ab}|=\alpha_{ab,1}|q|$ with $\alpha_{ab,1}\geq
0$. The saddle-point equations are then

$\displaystyle\frac{\delta\mathcal{F}}{\delta|\Delta_{a}|}$
$\displaystyle=2\alpha_{a}=|\Delta_{a}|-2\alpha_{ab,1}q|\Delta_{b}|+4\beta_{a}|\Delta_{a}|^{3}=0$
$\displaystyle\frac{\delta\mathcal{F}}{\delta|\Delta_{b}|}$
$\displaystyle=2\alpha_{b}|\Delta_{b}|-2\alpha_{ab,1}q|\Delta_{a}|+4\beta_{b}|\Delta_{b}|^{3}=0$
$\displaystyle\frac{\delta\mathcal{F}}{\delta q}$
$\displaystyle=2(\alpha_{a,2}|\Delta_{a}|^{2}+\alpha_{b,2}|\Delta_{b}|^{2})q-2\alpha_{ab,1}|\Delta_{a}||\Delta_{b}|=0$

The last equation implies that in the ground state

$q_{0}=\frac{\alpha_{ab,1}|\Delta_{a}||\Delta_{b}|}{\alpha_{a,2}|\Delta_{a}|^{2}+\alpha_{b,2}|\Delta_{b}|^{2}}\,;$
(S1)

note that $\Delta_{a/b}$ are themselves functions of $q$ in general, but we
can see immediately that $q_{0}\neq 0$ iff $|\Delta_{a}||\Delta_{b}|\neq 0$,
i.e. the mixed state is always non-uniform. Moreover, at $q=0$ we have

$\left.\frac{\delta^{2}\mathcal{F}}{\delta
q^{2}}\right|_{q=0}=2(\alpha_{a,2}|\Delta_{a}|^{2}+\alpha_{b,2}|\Delta_{b}|^{2})>0$
(S2)

but $\delta\mathcal{F}/\delta
q|_{q=0}=-2\alpha_{ab,1}|\Delta_{a}||\Delta_{b}|\leq 0$. This means that the
$q=0$ state ceases to be a minimum in the mixed state whenever
$\alpha_{ab,1}\neq 0$. In particular, this means that no first order phase
transitions are possible in the mixed-parity case.

##  II Additional Properties of Saddle Points of Ginzburg-Landau Free Energy
Eq. (2)

After some algebra, the equation for the mixed state at fixed $q$ (i.e. the
first equation in Eq. (2)) turns out to be simplest to express in terms of
$x_{a/b}=2\beta_{a/b}(|\Delta_{a/b}|^{2}-|\Delta_{a0/b0}|^{2})$ where
$|\Delta_{a0/b0}|^{2}=-\alpha_{a/b}/(2\beta_{a/b})$. The equations become

$\displaystyle
x_{a}x_{b}=\frac{\alpha_{ab,2}^{2}q^{4}}{4}=\frac{\alpha_{ab}^{2}}{4}$ (S3)
$\displaystyle
16\frac{\beta_{b}}{\beta_{a}}x_{a}^{3}(x_{a}-\alpha_{a})+4\alpha_{b}\alpha_{ab}^{2}x_{a}=\alpha_{ab}^{4}$
(S4)

Note that $x_{a}-\alpha_{a}\geq 0$ and $x_{b}-\alpha_{b}\geq 0$ (another
notable relation at the saddle point is
$|\Delta_{a}/\Delta_{b}|^{2}=x_{b}/x_{a}$). A non-trivial consequence of the
saddle-point equations is that $x_{a},x_{b}\geq 0$ at the saddle points. As a
consequence of that, there is always a single unique solution for $x_{a}$: the
LHS of Eq. (S4) can only change sign once at some $x_{a}$ (and only if
$\alpha_{b}<0$), and it is positive and monotonic for $x_{a}$ above that
value. To find the critical current, we need to plug in the optimal solution
into $\mathcal{F}$ to obtain the condensation energy $F(q)$. The current is
then given by $J(q)=2\partial_{q}F(q)$. We do this numerically to obtain the
plots in Fig. 1 (d-f).

##  III Details of Derivation of GL Coefficients from Microscopic Toy Model

Here we derive the coefficients $\alpha_{f,2}$ and $\alpha_{ab,2}$ for the
two-band microscopic toy model in the main text with Hamiltonian

$\mathcal{H}=\sum_{\mathbf{k},f}\left(\xi(\mathbf{k})c^{\dagger}_{\mathbf{k}f}c_{\mathbf{k}f}+tc^{\dagger}_{\mathbf{k}f}c_{\mathbf{k}\bar{f}}\right)$
(S5)

The Ginzburg-Landau free energy is obtained in the standard way by introducing
the Hubbard-Stratonovich fields $\widehat{\Delta}_{f}$ which enter the
transformed Hamiltonian as
$\widehat{\Delta}_{f}(\mathbf{p;q})c^{\dagger}_{\mathbf{k+q}/2,f}c^{\dagger}_{-\mathbf{k+q}/2,f}$
(and we assume no interband pairing between $a$ and $b$ bands):

$\mathcal{F}[\widehat{\Delta},\widehat{\Delta}^{\dagger}]=-T\sum_{n\mathbf{k}j}\frac{1}{2j}\text{Tr}\left[\left(\widehat{\Delta}^{\dagger}(\mathbf{k;q})G^{(0)}(i\omega_{n},\mathbf{k+q}/2)\widehat{\Delta}(\mathbf{k;q})G^{(0,h)}(i\omega_{n},\mathbf{k-q}/2)\right)^{j}\right]+H_{\Delta^{2}}$
(S6)

where

$\widehat{\Delta}=\left(\begin{array}[]{cc}\widehat{\Delta}_{a}&0\\\
0&\widehat{\Delta}_{b}\end{array}\right)$ (S7)

and

$H_{\Delta^{2}}=\sum_{\begin{subarray}{c}\mathbf{p,k}\\\
\end{subarray}}\widehat{\Delta}^{\dagger}_{f_{3}f_{4}}(\mathbf{k;q})\left[V^{-1}(\mathbf{p,k;q})\right]^{f_{3}f_{4}}_{f_{1}f_{2}}\widehat{\Delta}_{f_{1}f_{2}}(\mathbf{p;q})\,.$
(S8)

Here $V(\mathbf{p,k;q})$ are the pairing interactions and $f_{j}$ are band
indices (summed over implicitly). Assuming the interactions do not depend on
the total Cooper pair momentum $q$ and that the gap functions are self-
consistent, the free energy to second order in $\Delta$ is

$\displaystyle\mathcal{F}_{2}$
$\displaystyle=\frac{T}{2}\sum_{n\mathbf{k}}\text{Tr}\left[\widehat{\Delta}^{\dagger}G^{(0)}(i\omega_{n},\mathbf{k+q/2})\widehat{\Delta}G^{(0,h)}(i\omega_{n},\mathbf{k-q/2})\right]-\frac{T_{c}(0)}{2}\sum_{n\mathbf{k}}\text{Tr}\left[\widehat{\Delta}^{\dagger}G^{(0)}(i\omega_{n}|_{T_{c}},\mathbf{k})\widehat{\Delta}G^{(0,h)}(i\omega_{n}|_{T_{c}},\mathbf{k})\right]$
(S9)

where $G^{(0)}$ is the Green’s function which is $2\times 2$ in the sublattice
indices, $G^{(0,h)}(i\omega,\mathbf{k})=-G^{T}(-i\omega,-\mathbf{k})$. In the
simplified model,

$G^{(0)}(i\omega_{n},\mathbf{k})=\left(\begin{array}[]{cc}G_{1}(i\omega_{n},\mathbf{k})&G_{2}(i\omega_{n},\mathbf{k})\\\
G_{2}(i\omega_{n},\mathbf{k})&G_{1}(i\omega_{n},\mathbf{k})\end{array}\right)=\frac{1}{(i\omega_{n}-\xi(\mathbf{k}))^{2}-t^{2}}\left(\begin{array}[]{cc}i\omega_{n}-\xi(\mathbf{k})&t\\\
t&i\omega_{n}-\xi(\mathbf{k})\end{array}\right)$ (S10)

Taking $\xi(\mathbf{k})=\xi_{0}+\mathbf{v}_{F}\cdot\mathbf{k}$ and
$\widehat{\Delta}_{f}(\mathbf{k;q})=\Delta_{f}d_{f}(\mathbf{k})$, we have

$\mathcal{F}_{2}=\left\langle\Pi_{1}(\theta,\mathbf{q})\left(d_{a}^{2}(\mathbf{k})|\Delta_{a}|^{2}+d_{b}^{2}(\mathbf{k})|\Delta_{b}|^{2}\right)+2\Pi_{2}(\theta,\mathbf{q})d_{a}(\mathbf{k})d_{b}(\mathbf{k})|\Delta_{a}||\Delta_{b}|\right\rangle_{\theta}$
(S11)

where

$\displaystyle\Pi_{1}(\theta,\mathbf{q})$
$\displaystyle=N\left(-\log\frac{T_{c}}{T}+\digamma_{1}\left(\delta\xi(\theta,\mathbf{q})/T,t/T\right)\right)$
$\displaystyle\Pi_{2}(\theta,\mathbf{q})$
$\displaystyle=N\digamma_{2}\left(\delta\xi(\theta,\mathbf{q})/T,t/T\right)$
(S12)

where $N$ is the DOS,
$\delta\xi(\theta,\mathbf{q})=\mathbf{v}_{F}(\theta)\cdot\mathbf{q}/2=(v_{Fx}\cos\theta
q_{x}+v_{Fy}\sin\theta q_{x})/2$, and

$\displaystyle\digamma_{1}\left(x,y\right)$
$\displaystyle=\frac{1}{4}\text{Re}\left[2\psi\left(\frac{1}{2}+\frac{ix}{2\pi}\right)+\psi\left(\frac{1}{2}+\frac{i(x+y)}{2\pi}\right)+\psi\left(\frac{1}{2}+\frac{i(x-y)}{2\pi}\right)-2\psi\left(\frac{1}{2}+\frac{iy}{2\pi}\right)-2\psi\left(\frac{1}{2}\right)\right]$
$\displaystyle\digamma_{2}\left(x,y\right)$
$\displaystyle=\frac{1}{4}\text{Re}\left[2\psi\left(\frac{1}{2}+\frac{ix}{2\pi}\right)-\psi\left(\frac{1}{2}+\frac{i(x+y)}{2\pi}\right)-\psi\left(\frac{1}{2}-\frac{i(x-y)}{2\pi}\right)+2\psi\left(\frac{1}{2}+\frac{iy}{2\pi}\right)-2\psi\left(\frac{1}{2}\right)\right]$
(S13)

and $\langle\dots\rangle_{\theta}=\int_{0}^{2\pi}\dots d\theta/(2\pi)$, with
$\theta$ the angle around the FS (we assume everything can be considered as a
function of this angle and is thus independent of $|\mathbf{k}|$).

Figure S1: Plot of the dimensionless ratio of GL coefficients defined by Eq.
(S16).

If we now take $d_{a}=1$ and $d_{b}=\cos 2\theta$ for sake of argument (i.e.
$s$-wave and $d$-wave pairing) and expand in $q$, we get

$\displaystyle\alpha_{a}(\mathbf{q})$
$\displaystyle=\alpha_{a,0}+\frac{14\zeta(3)-\text{Re}[\psi^{(2)}(1/2+it/(2\pi
T))]}{64\pi^{2}T^{2}}N(v_{Fx}^{2}q_{x}^{2}+v_{Fy}^{2}q_{y}^{2})+\dots$
$\displaystyle\alpha_{b}(\mathbf{q})$
$\displaystyle=\alpha_{b,0}+\frac{14\zeta(3)-\text{Re}[\psi^{(2)}(1/2+it/(2\pi
T))]}{128\pi^{2}T^{2}}N(v_{Fx}^{2}q_{x}^{2}+v_{Fy}^{2}q_{y}^{2})+\dots$
$\displaystyle\alpha_{ab}(\mathbf{q})$
$\displaystyle=\frac{14\zeta(3)+\text{Re}[\psi^{(2)}(1/2+it/(2\pi
T))]}{64\pi^{2}T^{2}}N(v^{2}_{Fx}q_{x}^{2}-v_{Fy}^{2}q_{y}^{2})+\dots$ (S14)

We thus have

$\frac{\alpha_{ab,2}^{2}}{4\alpha_{a,2}\alpha_{b,2}}\propto\left(\frac{14\zeta(3)+\text{Re}[\psi^{(2)}(1/2+it/(2\pi
T))]}{14\zeta(3)-\text{Re}[\psi^{(2)}(1/2+it/(2\pi
T))]}\right)^{2}\in[0,2.333]$ (S15)

(the ratio is zero at $t=0$, maximized for $t=\pi T$, and tends to a constant
value of $1$ as $t/T\to\infty$). Assuming $v_{Fx}=v_{Fy}$, the free energy is
maximized for $\mathbf{q}$ along $x$ or $y$ directions, and

$\frac{\alpha_{ab,2}^{2}}{4\alpha_{a,2}\alpha_{b,2}}=\frac{1}{2}\left(\frac{14\zeta(3)+\text{Re}[\psi^{(2)}(1/2+it/(2\pi
T))]}{14\zeta(3)-\text{Re}[\psi^{(2)}(1/2+it/(2\pi
T))]}\right)^{2}\in[0,1.1665]\,;$ (S16)

see Fig. S1. There is therefore a range of $t$ (close to $\pi T$) for which
the mixed phase does appear, and it is not unreasonable to expect even larger
values of $\frac{\alpha_{ab,2}^{2}}{4\alpha_{a,2}\alpha_{b,2}}$ in more
sophisticated models.