# Nonreciprocal Meissner Response in Parity-Mixed Superconductors

Hikaru Watanabe  RIKEN Center for Emergent Matter Science (CEMS), Wako
351-0198, Japan  Department of Physics, Graduate School of Science, Kyoto
University, Kyoto 606-8502, Japan  Akito Daido  Department of Physics,
Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan  Youichi
Yanase  Department of Physics, Graduate School of Science, Kyoto University,
Kyoto 606-8502, Japan  Institute for Molecular Science, Okazaki 444-8585,
Japan

###### Abstract

The parity breaking gives rise to rich superconducting properties through the
admixture of even and odd-parity Cooper pairs. A new light has been shed on
parity-breaking superconductors by recent observations of nonreciprocal
responses such as the nonlinear optical responses and the superconducting
diode effect. In this Letter, we demonstrate that the nonreciprocal responses
are characterized by a unidirectional correction to the superfluid density,
which we call nonreciprocal superfluid density. This correction leads to the
nonreciprocal Meissner effect, namely, asymmetric screening of magnetic fields
due to the nonreciprocal magnetic penetration depth. Performing a microscopic
analysis of an exotic superconductor UTe2 and examining the temperature
dependence and renormalization effect, we show that the nonreciprocal Meissner
effect is useful to probe parity-mixing properties and gap structures in
superconductors.

Introduction— The nonlinear and nonreciprocal responses are recently
attracting interest in various fields of condensed matter physics. For
instance, second harmonic generation and photocurrent creation have been
applied to a probe of the symmetry breaking in matters and the topological
nature of electrons Fiebig  _et al._ (2005); Orenstein  _et al._ (2021); Ideue
and Iwasa (2021). Recent studies explored the nonreciprocal responses in
superconductors. The nonreciprocal electric conductivity Tokura and Nagaosa
(2018), a rectified conductivity originating from the parity violation, is
strongly enhanced by the superconducting fluctuation Wakatsuki  _et al._
(2017) and by the vortex dynamics Hoshino  _et al._ (2018); Zhang  _et al._
(2020). Furthermore, the recent efforts have clarified the nonreciprocal
superconducting phenomena such as the nonreciprocal critical current Ando  _et
al._ (2020); Yuan and Fu (2021); Daido  _et al._ (2021); He  _et al._ (2021);
Ilić and Sebastian Bergeret (2021) and Josephson current Baumgartner  _et al._
(2021); Wu  _et al._ (2021); He  _et al._ (2021). The nonreciprocal critical
current realizes the superconducting diode effect, indicating that the
electrical resistivity is zero in a direction while finite in the opposite
direction. The nonreciprocal optical responses have also been observed in
superconducting systems whose parity violation stems from the spontaneous
order intertwined with superconductivity Zhao  _et al._ (2017); Lim  _et al._
(2020); de la Torre  _et al._ (2020) or the injected supercurrent Yang  _et
al._ (2019); Nakamura  _et al._ (2020); Vaswani  _et al._ (2020). Building on
the superconducting properties, the nonreciprocal phenomena imply richer
functionalities.

There is a symmetry requirement of nonreciprocal responses which is unique to
superconductors. In addition to the space-inversion symmetry breaking, its
combination with the gauge symmetry has to be broken. In particular, the odd-
parity superconductivity is insufficient to cause the nonreciprocal response
because the above symmetry holds. Superconductors must have no definite parity
under the $\mathcal{P}$ operation to host nonreciprocal responses, implying
the parity-mixed superconducting state. Conversely, the nonreciprocal response
may be an indicator of the parity mixing in superconductors as proposed in the
prior study of the nonreciprocal conductivity Wakatsuki and Nagaosa (2018).
This potential indicator of the parity-mixed superconducting state may allow
us to identify the relation between the parity violation and superconducting
symmetries which has been intensively investigated with the noncentrosymmetric
superconductors such as CePt3Si Bauer and (eds.).

Considering the high interest in the research community, it is desirable to
further explore nonreciprocal properties of superconductors. While previous
theoretical studies have focused on DC or low-frequency charge transport
Wakatsuki  _et al._ (2017); Hoshino  _et al._ (2018); Wakatsuki and Nagaosa
(2018); Daido  _et al._ (2021); Yuan and Fu (2021); He  _et al._ (2021),
nonreciprocal nature may appear in the other responses as well. Recently the
authors have identified anomalous contributions to the nonlinear optical
conductivity in superconductors, which diverges in the low-frequency limit Wat
as is the case for the linear optical conductivity Tinkham and Ferrell (1959).
This in turn implies an inherent directionality in the Meissner response,
i.e., the nonreciprocal Meissner effect. Since the Meissner effect plays the
central role in superconductivity, the nonreciprocal Meissner effect may also
contribute to a deeper understanding of the parity-breaking superconducting
states.

Figure 1: Nonreciprocal Meissner effect. The nonreciprocal superfluid density
$\bm{f}$ (green arrow) (a) strengthens or (b) diminishes the supercurrent
$\bm{J}_{\text{SC}}$ (orange arrows) shielding the external magnetic field
(blue arrow).

This Letter consists of two parts. First, we show that the anomalous nonlinear
conductivity arising from the parity-mixing leads to the unidirectional
correction to the rigidity of the superconducting state, which we call
nonreciprocal superfluid density. Various nonreciprocal responses of
superconductors are characterized by the nonreciprocal superfluid density.
Second, to demonstrate exotic phenomena arising from the nonreciprocal
superfluid density, we elaborate the nonreciprocal property of the Meissner
effect (Fig. 1). With the microscopic analysis implementing the model of a
putative parity-mixed superconductor UTe2, we show that the nonreciprocal
Meissner effect is sensitive to the parity violation and hence applicable to
detection of the parity-mixed superconducting state.

Nonreciprocal property of the superfluid density — The nonreciprocal electric
conductivity of superconductors has recently been formulated by using the
density matrix and Green function methods Wat . The leading nonreciprocal
correction to the electric current is given by the second-order component

$J_{\alpha}^{(2)}(\omega)=\int\frac{d\Omega}{2\pi}\sigma_{\alpha;\beta\gamma}(\omega,\Omega)E_{\beta}(\Omega)E_{\gamma}(\omega-\Omega).$
(1)

In the low-frequency limit, we obtain the nonlinear optical conductivity

$\displaystyle\sigma_{\alpha;\beta\gamma}(\omega,\Omega)$
$\displaystyle=\frac{f_{\alpha\beta\gamma}}{2\Omega(\omega-\Omega)}$
$\displaystyle-\frac{i}{4}\lim_{\bm{A}\rightarrow\bm{0}}\left(\frac{1}{\omega-\Omega}\partial_{A_{\beta}}\sigma_{\alpha\gamma}^{({\bm{A}})}+\frac{1}{\Omega}\partial_{A_{\gamma}}\sigma_{\alpha\beta}^{({\bm{A}})}\right).$
(2)

The diverging terms proportional to $\Omega^{-2}$ or $\Omega^{-1}$ are unique
to the superconducting state while they are forbidden in the normal state
Michishita and Peters (2021). Thus, we call the terms anomalous contributions.
We suppressed $O(\Omega^{0})$ terms comprising non-divergent nonlinear optical
conductivity since it is negligible in the low-frequency regime.

The anomalous nonreciprocal conductivity in Eq. (2) is determined by the
nonreciprocal superfluid density $f_{\alpha\beta\gamma}$ (NRSF) and the
conductivity derivative
$\partial_{A_{\gamma}}\sigma_{\alpha\beta}^{({\bm{A}})}$. The NRSF is given by

$f_{\alpha\beta\gamma}=\lim_{\bm{A}\rightarrow\bm{0}}\partial_{A_{\alpha}}\partial_{A_{\beta}}\partial_{A_{\gamma}}F_{A},$
(3)

where $F_{A}$ is the free-energy obtained from the Hamiltonian containing the
vector potential ${\bm{A}}$. Since the superfluid density is given by
$\rho^{\text{s}}_{\alpha\beta}=\lim_{\bm{A}\rightarrow\bm{0}}\partial_{A_{\alpha}}\partial_{A_{\beta}}F_{A}$,
the NRSF is regarded as a unidirectional correction to $\rho^{\text{s}}$. When
the superfluid density is isotropic
$\rho^{\text{s}}_{\alpha\beta}=\rho^{\text{s}}\delta_{\alpha\beta}$, the NRSF
is recast as the vector $\bm{f}$ which has the same symmetry as the electric
current and the toroidal moment Spaldin  _et al._ (2008); Hlinka (2014). Thus,
the NRSF is allowed in the absence of both $\mathcal{P}$ and time-reversal
($\mathcal{T}$) symmetries.

In the second term of Eq. (2), $\sigma_{\alpha\beta}^{({\bm{A}})}$ denotes the
regular part of the linear static conductivity calculated with the Bogoliubov-
de Gennes Hamiltonian including ${\bm{A}}$. Its derivative
$\partial_{A_{\gamma}}\sigma_{\alpha\beta}^{({\bm{A}})}$, which we call
conductivity derivative, is decomposed into the symmetric and antisymmetric
parts in terms of the permutation of indices $(\alpha,\beta)$ for
$\sigma_{\alpha\beta}^{({\bm{A}})}$. The symmetric part of
$\sigma_{\alpha\beta}^{({\bm{A}})}$ corresponds to the Drude contribution,
while the anti-symmetric part is the Berry curvature term. Thus, we call the
corresponding components in
$\partial_{A_{\gamma}}\sigma_{\alpha\beta}^{({\bm{A}})}$ the Drude and Berry
curvature derivatives, which vanish in $\mathcal{T}$ and $\mathcal{PT}$
symmetric parity-mixed superconductors, respectively. From Eq. (2), we see
that the nonreciprocal optical responses such as the photocurrent creation
($\omega=0,~{}\Omega\neq 0$) and the second harmonic generation
($\omega=2\Omega$) show a prominent divergent behavior in the low-frequency
regime, which is unique to superconductors Wat .

Recalling the minimal coupling between the electrons and electromagnetic
field, the vector potential twists the phase of superfluid and plays the same
role as the supercurrent. Thus, when the electromagnetic perturbation is weak,
the NRSF determines the nonreciprocal component of the supercurrent induced by
a given phase twist. In the Josephson junction, the phase twist is similarly
accumulated through the junction bridging the superconducting leads, and the
NRSF also participates in the nonreciprocal Josephson current Baumgartner  _et
al._ (2021); He  _et al._ (2021).

According to the Ginzburg-Landau free energy analysis, both the NRSF and the
nonreciprocal critical current are attributed to the cubic gradient component
of the quadratic term as well as the linear gradient component of the quartic
term Daido  _et al._ (2021); sup . Therefore, the NRSF provides a systematic
understanding of various nonreciprocal responses in superconductors, including
the optical response, Josephson effect, and critical current. This is similar
to the case of the conventional superfluid density, which determines the
anomalous linear optical conductivity, Meissner effect, and zero-resistance
phenomenon Tinkham and Ferrell (1959); Tinkham (2004).

Nonreciprocal Meissner effect — We now transform the conductivity into the
susceptibility

$J_{\alpha}(\omega)=K^{(1)}_{\alpha\beta}A_{\beta}(\omega)+\int\frac{d\Omega}{2\pi}K^{(2)}_{\alpha;\beta\gamma}(\omega,\Omega)A_{\beta}(\Omega)A_{\gamma}(\omega-\Omega).$
(4)

The anomalous nonreciprocal conductivity contributes to the response function
$K^{(2)}$ as

$\displaystyle
2K^{(2)}_{\alpha;\beta\gamma}(\omega,\Omega)=-2\Omega\left(\omega-\Omega\right)\sigma_{\alpha;\beta\gamma}(\omega,\Omega),$
(5)
$\displaystyle=-f_{\alpha\beta\gamma}+\frac{i}{2}\lim_{\bm{A}\rightarrow\bm{0}}\left(\Omega\,\partial_{A_{\beta}}\sigma_{\alpha\gamma}^{({\bm{A}})}+\left(\omega-\Omega\right)\partial_{A_{\gamma}}\sigma_{\alpha\beta}^{({\bm{A}})}\right).$
(6)

The anomalous conductivity determines low-frequency behaviors of the nonlinear
coupling between the vector potential and electric current. This indicates
that the NRSF causes the nonreciprocal property of the Meissner response, that
is, nonreciprocal Meissner effect. Note that the nonreciprocal Meissner effect
is unique to the parity-breaking superconductors and distinguished from the
nonlinear Meissner response Yip and Sauls (1992); Xu  _et al._ (1995), which
is reciprocal in terms of magnetic fields. Although the conductivity
derivative may participate in the AC nonreciprocal Meissner response, we
hereafter focus on the static response determined by the NRSF. Classification
of the nonreciprocal Meissner kernel $K^{(2)}$ based on the $\mathcal{T}$ and
$\mathcal{PT}$ symmetries is summarized in Table 1.

Table 1: Classification of the nonreciprocal Meissner kernel $K^{(2)}$ based on $\mathcal{T}$ and $\mathcal{PT}$ symmetries. The $O(\Omega^{n})$ contributions allowed by the symmetry are summarized.  $O(\Omega^{n})$ | $\mathcal{T}$ | $\mathcal{PT}$
---|---|---
$n=0$ | N/A | NRSF
$n=1$ | Berry curvature deriv. | Drude deriv.
$(n\geq 2)$ | (regularized nonlinear conductivity)

We phenomenologically introduce a nonlinear correction to the London theory by

$J_{\alpha}(\bm{r})=-\rho_{\alpha\beta}^{s}A_{\beta}(\bm{r})-f_{\alpha\beta\gamma}A_{\beta}(\bm{r})A_{\gamma}(\bm{r}),$
(7)

where the first and second terms are normal and nonreciprocal supercurrents,
respectively. Here, we consider a superconductor occupying the spatial region
$z\leq 0$ and the NRSF vector $\bm{f}\parallel\hat{x}$ (Fig. 1). When the
magnetic field is applied to the $y$-direction and the nonreciprocal effect is
assumed to be small, the field-dependent magnetic penetration depth is
estimated as
$\lambda(B)=\lambda_{\text{L}}\left(1+\lambda_{\text{L}}Bf/3\rho^{\text{s}}\right)$
with the London penetration depth
$\lambda_{\text{L}}^{-1}=\sqrt{\mu_{0}\rho^{\text{s}}}$. Thus, it shows the
unidirectional magnetic-field dependence. Intuitively, the magnetic flux is
retracted from or drawn into the superconductor when the supercurrent
shielding the magnetic flux is parallel or antiparallel to the NRSF vector
$\bm{f}$. Thus, a careful magnetic penetration depth measurement Hashimoto
_et al._ (2009) can evaluate the NRSF.

The $\mathcal{P}$ and $\mathcal{T}$ symmetries have to be broken in
superconductors that host the NRSF. To our best knowledge, three setups are
available; (i) systems in which $\mathcal{P}$ and $\mathcal{T}$ symmetries are
broken by other spontaneous orders or by the crystal structure, (ii)
superconductors under the supercurrent flow, and (iii) exotic superconductors
whose order parameter spontaneously breaks the symmetries. Case (i) is
realized in various situations, such as the noncentrosymmetric superconductors
under external magnetic fields Ideue and Iwasa (2021) and the superconductors
undergoing the magnetically parity-breaking order Sumita  _et al._ (2017).
Interestingly, Case (ii) was recently supported by an experiment where the
superconducting NbN thin film was probed under the electric current by the
second harmonic generation Nakamura  _et al._ (2020). Case (iii) is further
classified into two classes. First, the multiple transitions of even-parity
and odd-parity superconductivity make both $\mathcal{P}$ and $\mathcal{T}$
parities ill-defined Wang and Fu (2017). Second, these symmetries are broken
by chiral superconductivity and noncentrosymmetric crystal structure Brydon
_et al._ (2019); Ghosh  _et al._ (2020). Later we will investigate the former
class in (iii) by referring to the recent proposal for a heavy fermion
superconductor UTe2 Ishizuka and Yanase (2021).

We discuss the magnitude of the nonreciprocal Meissner response by the ratio

$\eta_{\text{NR}}=\frac{\lambda_{\text{L}}B_{\text{c2}}f}{\rho^{\text{s}}},$
(8)

where $B_{\text{c2}}$ is the upper critical magnetic field. First, our
analysis based on the Ginzburg-Landau theory shows
$\eta_{\text{NR}}\propto|T-T_{\text{c}}|^{1/2}$ sup , and thus, the
nonreciprocal response may be negligible in the vicinity of the transition
temperature $T\lesssim T_{\text{c}}$. Next, we estimate the renormalization
effect on the ratio $\eta_{\text{NR}}$. Interestingly, the correlation-induced
renormalization effect denoted by $z$ positively influences the nonreciprocal
Meissner response. Since $\eta_{\text{NR}}\propto z^{-5/2}$, the nonreciprocal
Meissner effect is significantly enhanced by a strong renormalization. Thus,
strongly-correlated electron systems are potential candidates offering a
sizable nonreciprocal Meissner effect. While we will work on a heavy fermion
system UTe2 in the following, another strongly correlated electron systems
such as cuprate superconductors and twisted bilayer graphene are also of
interest. The cuprates are usually centrosymmetric in the bulk, whereas the
parity violation can be evoked by spontaneous order [Case (i)] and by the
supercurrent injection [Case (ii)]. As for the former case, the loop-current
order has been proposed for the pseudogap phase in cuprate superconductors
Varma (1997); Fauqué  _et al._ (2006); Li  _et al._ (2008); Pershoguba  _et
al._ (2013, 2014); Zhao  _et al._ (2017); Lim  _et al._ (2020). The NRSF may
be a long-sought probe for examining such intertwining order in cuprates.

Microscopic calculations of UTe2 model— UTe2 is recently attracting a lot of
attention as a candidate material for spin-triplet superconductivity Ran  _et
al._ (2019); Aoki  _et al._ (2019a). It is argued that the ferromagnetic
fluctuation plays a key role in the spin-triplet superconductivity as in
ferromagnetic superconductors Ran  _et al._ (2019); Aoki  _et al._ (2019a, b);
Tokunaga  _et al._ (2019); Sundar  _et al._ (2019), whereas the
antiferromagnetic fluctuation has also been observed recently Thomas  _et al._
(2020); Duan  _et al._ (2020); Knafo  _et al._ (2021); Duan  _et al._ (2021).
Since the antiferromagnetic fluctuation usually stabilizes even-parity spin-
singlet superconductivity, multiple magnetic fluctuations are expected to lead
to multiple pairing instabilities Ishizuka and Yanase (2021). Interestingly,
UTe2 shows the multiple superconducting transitions Braithwaite  _et al._
(2019). Thus, the coexisting even- and odd-parity pairing state has been
proposed for the low-temperature phase Ishizuka and Yanase (2021).

Based on a microscopic model, we investigate the NRSF in the putative parity-
mixed phase of UTe2. The model Hamiltonian for the normal state reads

$H_{{\bm{k}}}^{\text{N}}=\left(\varepsilon_{0}-\mu\right)+V\rho_{x}+V^{\prime}\rho_{y}+\bm{g}\cdot\bm{\sigma}\rho_{z},$
(9)

which is described by Pauli matrices representing spin ($\sigma_{\mu}$) and
sublattice ($\rho_{\mu}$) degrees of freedom. The details are given in
Supplemental Materials sup . The model Hamiltonian reproduces the heavy band
mainly consisting of U $5f$-orbitals near the Fermi level, which was obtained
in the DFT+U calculations Ishizuka  _et al._ (2019); Shishidou  _et al._
(2021). We introduce a pair potential for parity-mixed superconductivity

$\hat{\Delta}_{\bm{k}}=\left(\psi_{\bm{k}}+\bm{d}_{\bm{k}}\cdot\bm{\sigma}\right)i\sigma_{y}\rho_{0},$
(10)

where we consider intra-sublattice Cooper pairing. According to theoretical
calculations implementing the Eliashberg theory and the DFT+U calculation, the
even-parity pairing is characterized by the $A_{g}$ irreducible representation
($\psi_{\bm{k}}=\Delta_{\text{e}}\cos{k_{x}}$), while the odd-parity pairing
is either of the $A_{u}$\- or $B_{3u}$-type denoted by
$\bm{d}_{\bm{k}}=\Delta_{\text{o}}\sin{k_{y}}\hat{y}$ or
$\Delta_{\text{o}}\sin{k_{y}}\hat{z}$, respectively Ishizuka and Yanase
(2021). It is energetically favorable for the relative phase between the pair
potentials to be $\pm\pi/2$, when the spin-orbit coupling due to
noncentrosymmetric crystal structures is absent or weak. This choice leads to
the $s+ip$-wave superconductivity preserving the $\mathcal{PT}$ symmetry Wang
and Fu (2017); Kanasugi and Yanase (2021). This case contrasts with the fact
that the spin-orbit coupling in the noncentrosymmetric superconductor leads to
the zero phase difference indicating the $\mathcal{T}$ symmetric state such as
$s+p$-wave superconductivity Bauer and (eds.). Since UTe2 crystallizes in a
centrosymmetric structure, we take the $\mathcal{PT}$ symmetric mean field
$\Delta_{\text{e}}^{(0)}=r\Delta_{\text{e+o}}^{(0)},~{}\Delta_{\text{o}}^{(0)}=i(1-r)\Delta_{\text{e+o}}^{(0)}$
with the parity-mixing ratio $r$. Here we denote pair potentials at zero
temperature by those with the superscript ‘$(0)$’.

Figure 2: Temperature dependence of the NRSF $f_{xxx}$ for the $A_{g}+iB_{3u}$
superconducting state. (a) Plot with several ratios of even- to odd-parity
pair potentials, $r=0.25$, $0.5$, and $0.75$. (b) Decomposition the total NRSF
into the Fermi-sea and Fermi-surface terms in the case of $r=0.5$. The dashed
line guides the transition temperature
$T_{\text{c}}^{\text{(e)}}=T_{\text{c}}^{\text{(o)}}=0.5\Delta_{\text{e+o}}^{(0)}/1.76$.
The inset shows the low-temperature regime with the horizontal axis $T\times
10^{6}$.

Following the symmetry analysis, we obtain the NRSF $f_{xyz}$ for the
$A_{g}+iA_{u}$ state and $f_{xxx},f_{xyy},f_{xzz}$ for the $A_{g}+iB_{3u}$
state. Here, we investigate the temperature and parity-mixing ratio dependence
of the NRSF $f_{xxx}$ in the $A_{g}+iB_{3u}$ state in details, while we obtain
a similar result for the $A_{g}+iA_{u}$ state sup . The temperature dependence
of the pairing potential is assumed to follow the phenomenological formula

$\Delta_{\text{e,o}}(T)=\Delta_{\text{e,o}}^{(0)}\tanh{\left(1.74\sqrt{\dfrac{|\Delta_{\text{e,o}}^{(0)}|}{1.76T}-1}\right)}.$
(11)

Figure 2 shows the NRSF calculated with several parity-mixing ratios $r=0.25$,
$0.5$, $0.75$. We do not have any NRSF in the pure spin-singlet or spin-
triplet state ($r=1,0$) where the $\mathcal{P}$ and U(1)$\times$$\mathcal{P}$
symmetry respectively forbid the NRSF. In Fig. 2, it is clearly shown that the
NRSF arises in the parity-mixed superconducting state. Each plot shows the
maximum value at an intermediate temperature. This is because the NRSF is
almost determined by the Fermi-surface contribution [Fig. 2(b)] given by

$\sum_{a}\left(J_{aa}^{x}\right)^{3}\frac{\partial^{2}f(\varepsilon)}{\partial\varepsilon^{2}}_{|\varepsilon=\varepsilon_{a}},$
(12)

where $\bm{J}_{aa}$ is the paramagnetic current density of the Bogoliubov
quasiparticle labeled by the quantum number $a$. On the other hand, the Fermi-
surface term gets suppressed at low temperature, and then the Fermi-sea term
mainly contributes to the NRSF [inset of Fig. 2(b)]. Decomposition of the NRSF
into the Fermi-surface and Fermi-sea terms is formulated in Supplemental
Materials sup . Since the Fermi-surface contribution is much larger than the
Fermi-sea contribution, the NRSF shows a non-monotonic temperature dependence.
The sizable Fermi-surface contribution is attributed to the almost nodal
superconducting gap. To support this argument, we show that the Fermi-surface
term is negligible in a superconducting state with a nearly isotropic gap sup
. Therefore, the significant NRSF and its non-monotonic temperature dependence
are characteristic behaviors of nodal superconductors, and they are useful in
identifying the nodal texture in the superconducting gap.

The present study clarified that the Fermi-surface effect is much more
significant than the Fermi-sea effect. The behavior is in contrast to the
normal superfluid density, which is usually determined by the Fermi-sea effect
and detrimentally influenced by the Fermi-surface effect. Since the
quasiparticle excitations moderately occur in the intermediate temperature
regime, the NRSF $f_{\alpha\beta\gamma}$ as well as the ratio
$\eta_{\text{NR}}$ in Eq. (8) are enhanced there.

To estimate the ratio $\eta_{\text{NR}}$ of our model, we first take
$\rho_{xx}^{\text{s}}\sim 10^{19}$ A$\cdot$V${}^{-1}\cdot$m${}^{-1}\cdot$s-1
sup , $|f_{xxx}|\sim 10^{22}$ A${}^{2}\cdot$V${}^{-2}\cdot$s-2, and
$B_{\text{c2}}\sim 1\text{\,T}$. The superfluid density leads to the
penetration depth $\lambda_{\text{L}}\sim
0.3\,{\mathrm{\,\text{\textmu}m\,}}$. Then, we obtain $\eta_{\text{NR}}\sim
3\times 10^{-4}$. The ratio may increase due to the renormalization effect by
electron correlations. The adopted model Hamiltonian is based on the DFT+U
calculation and does not sufficiently take into account the electron
correlation effect of UTe2. We consider the renormalization factor $z=0.1$,
which enhances the London penetration depth as much as the observed values
$\lambda_{\text{L}}\sim 1\,{\mathrm{\,\text{\textmu}m\,}}$ Sundar  _et al._
(2019); Metz  _et al._ (2019); Bae  _et al._ (2021). Accordingly, the ratio
$\eta_{\text{NR}}$ increases by $\sim 300$ times larger than the above
estimation. As a result, the enhanced ratio is estimated to be
$\eta_{\text{NR}}\sim 10^{-1}$ which may be within the experimental
sensitivity.

Discussion and Summary— This Letter reveals that the NRSF plays an essential
role in various nonreciprocal responses of superconductors such as optical
responses, supercurrent flow, and Meissner effect. Thus, the NRSF is a
potential indicator of nonreciprocal responses and is helpful to probe the
parity mixing in a superconducting state, which causes parity breaking
required for the nonreciprocal superconducting phenomena. It is noteworthy
that the NRSF can be estimated through various experimental techniques
implemented in nonlinear optics and magnetic penetration depth measurements.
We also note that the NRSF is well-defined in the whole temperature region and
complementary to the fluctuation-assisted normal nonlinear conductivity, which
captures the nonreciprocal response in the vicinity of the transition
temperature Wakatsuki and Nagaosa (2018); Wakatsuki  _et al._ (2017); Hoshino
_et al._ (2018).

We have mainly discussed the NRSF and the resulting nonreciprocal Meissner
response in exotic superconductors, where the competing pairing interaction in
the spin-singlet and spin-triplet channels leads to multiple superconducting
transitions. However, our formulation applies to the NRSF in a broad class of
superconductors with broken $\mathcal{P}$ and $\mathcal{T}$ symmetries. For
instance, the noncentrosymmetric superconductors under the external magnetic
field and some classes of magnetic superconductors satisfy the symmetry
condition. As artificially-engineered noncentrosymmetric superconductors have
realized sizable nonreciprocal transport phenomena in recent experimental
works Ando  _et al._ (2020); Baumgartner  _et al._ (2021), it is also expected
that nonreciprocal properties of optical phenomena and Meissner response due
to the NRSF are observed.

The dynamical nonreciprocal response is also of interest, while we mainly
focused on the static nonreciprocal Meissner response in this study. The low-
frequency behavior of the nonreciprocal Meissner Kernel $K^{(2)}$ is closely
related to the space-time symmetry of the superconductivity as shown in Table
1; $K^{(2)}$ has a static component in the $\mathcal{PT}$ symmetric parity-
mixed superconductor, whereas it contains a $\Omega$ linear term as a leading
order term in the $\mathcal{T}$ symmetric superconductor. Thus, a careful AC
experiment on the magnetic penetration depth may distinguish the symmetry of
parity-mixed superconductors.

To summarize, we proposed that the NRSF provides a systematic understanding of
nonreciprocal responses in superconductors. Accordingly, we clarified the
nonreciprocal magnetic penetration phenomenon, which we call the nonreciprocal
Meissner effect. According to the study of UTe2, the nonreciprocal Meissner
effect mainly arises from the Fermi-surface effect and is sensitive to the
superconducting gap structure and electron correlation effect. Therefore, the
phenomenon is expected to be a key to identify the symmetry of exotic
superconductors such as UTe2.

Supplemental Materials include Refs. Ikeda  _et al._ (2006); Hutanu  _et al._
(2019); Fischer  _et al._ (2011); Maruyama  _et al._ (2012); Basov and Timusk
(2005); Ahn and Nagaosa (2021); Watanabe and Oshikawa (2020); Železný  _et
al._ (2014); Ishihara  _et al._ (2021)

## Acknowledgement

The authors thank Jun Ishizuka, Shota Kanasugi, and Yoshihiro Michishita for
fruitful discussions. This work was supported by JSPS KAKENHI (Grants No.
JP18H05227, No. JP18H01178, and No. 20H05159) and SPIRITS 2020 of Kyoto
University. H.W. is a JSPS research fellow and supported by JSPS KAKENHI
(Grant No. 18J23115 and No. 21J00453). A.D. is supported by JSPS KAKENHI
(Grant No. 21K13880).

Supplemental Materials for
“Nonreciprocal Meissner Response in Parity-Mixed Superconductors”
Hikaru Watanabe, Akito Daido, and Youichi Yanase

##  S1 Ginzburg-Landau analysis of nonreciprocal superfluid density

In this section, we estimate the NRSF based on the Ginzburg-Landau (GL)
analysis. As mentioned in the main text, the NRSF appears in three-fold cases;
(I) superconductors where $\mathcal{P}$ and $\mathcal{T}$ symmetries are
broken in the normal state, (II) superconductors under the supercurrent, and
(III) the cases of superconductivity which itself breaks the symmetry. Most
cases are captured by the single component GL free energy, while the
multicomponent GL theory should be analyzed when the multiple superconducting
transitions break the symmetry as in Case (III). We, therefore, present two GL
analyses dealing with a single component or multicomponent order parameter.

###  S1.1 Single component GL theory

We consider the GL free energy parametrized by the single component order
parameter $\Delta$. The GL free energy is given by Daido  _et al._ (2021)

$F[q,\Delta]=\alpha(q)\Delta^{2}+\frac{1}{2}\beta(q)\Delta^{4},$ (S1)

where the quadratic and quartic terms read

$\displaystyle\alpha(q)=-a_{0}+\frac{1}{2}a_{2}(q-q_{0})^{2}+\frac{1}{6}a_{3}(q-q_{0})^{3},$
(S2) $\displaystyle\beta(q)=b_{0}+b_{1}(q-q_{0}).$ (S3)

The gradient terms are obtained by the expansion of the GL free energy around
$q_{0}$. For instance, a finite $q_{0}$ arises from the cooperation of an
antisymmetric spin-orbit coupling and an external magnetic field [Case (I)] or
from an injected supercurrent [Case (II)]. We also impose the conditions,
$\alpha(q)<0$ and $\beta(q)>0$, to obtain a non-zero proper solution for the
order parameter.

With the saddle-point approximation for the order parameter,
$\partial_{\Delta}F[q,\Delta]=0$, the order parameter is given by

$\overline{\Delta}^{2}=-\frac{\alpha(q)}{\beta(q)}.$ (S4)

Plugging the obtained order parameter into the GL free energy, we obtain

$\mathcal{F}(q)=F[q,\overline{\Delta}]=-\frac{1}{2}\frac{\alpha^{2}(q)}{\beta(q)}.$
(S5)

The NRSF $f_{\text{NRSF}}$ is obtained from the third derivative with respect
to $q$ as

$f_{\text{NRSF}}=\lim_{q\rightarrow
0}\partial_{q}^{3}\mathcal{F}(q)=a_{0}\left(\frac{1}{b_{0}}a_{3}-\frac{3a_{2}}{b_{0}^{2}}b_{1}\right)+O(b_{1}^{3}).$
(S6)

As in the cases of the nonreciprocal critical current and nonreciprocal
Josephson effect Yuan and Fu (2021); Daido  _et al._ (2021); He  _et al._
(2021), higher-order gradient terms denoted by $a_{3},b_{1}$ play essential
roles in the nonreciprocal property of the superfluid density. When we
consider the temperature dependence of the mass term
$a_{0}\propto|T_{\text{c}}-T|$, it follows that
$f_{\text{NRSF}}\propto|T_{\text{c}}-T|$. Taking into account the temperature
dependence of the upper critical field and magnetic penetration depth, we
obtain the critical behavior, $\eta_{\text{NR}}\propto|T-T_{\rm c}|^{1/2}$.

###  S1.2 Multicomponent GL theory

Here we consider the GL free energy consisting of two component order
parameters $\Delta_{1},\,\Delta_{2}$ whose $\mathcal{P}$ parities are
opposite. We assume the relative phase between the two order parameters to be
$\pm\pi/2$ and redefine the order parameters by
$(\Delta_{1},\Delta_{2})\rightarrow(\Delta_{1},i\Delta_{2})$ to suppress the
imaginary unit. The GL free energy is given by

$F[q_{1},q_{2},\Delta_{1},\Delta_{2}]=\alpha_{1}(q_{1})\Delta_{1}^{2}+\alpha_{2}(q_{2})\Delta_{2}^{2}+\frac{1}{2}\beta_{1}(q_{1})\Delta_{1}^{4}+\frac{1}{2}\beta_{2}(q_{2})\Delta_{2}^{4}+F_{12}.$
(S7)

The coupling coefficients of the quadratic and quartic terms are defined as

$\displaystyle\alpha_{i}(q_{i})=-a_{0}^{(i)}+\frac{1}{2}a_{2}^{(i)}q_{i}^{2},$
(S8) $\displaystyle\beta_{i}(q_{i})=b_{0}^{(i)}.$ (S9)

Since the normal state is assumed to be centrosymmetric, we have no components
arising from the parity violation in Eqs. (S8) and (S9). The cross-coupling
term between $\Delta_{1}$ and $\Delta_{2}$ reads

$F_{12}=\frac{c_{3}}{12}(q_{1}^{3}+q_{2}^{3})\Delta_{1}\Delta_{2}+\frac{1}{2}d_{0}\Delta_{1}^{2}\Delta_{2}^{2}.$
(S10)

The coefficient $c_{3}$ is allowed in the $\mathcal{P}$-broken system and thus
characteristic to multi-component superconducting order parameters which
cooperatively break the $\mathcal{P}$ symmetry. Note that we assume the
parity-mixed superconducting state which is nonpolar but noncentrosymmetric
for simplicity. Here, we consider two cases; (1) $\Delta_{1}$ and $\Delta_{2}$
have the same transition temperature and (2) the transition temperature for
$\Delta_{1}$ is much lower than that for $\Delta_{2}$. In the latter case, we
can neglect the temperature dependence of $\Delta_{2}$. In general, the
transition temperatures for $\Delta_{1}$ and $\Delta_{2}$ are not equivalent
without fine tuning.

First, we consider the case (1) where the mass terms $a_{0}^{(i)}$ vanish at
the same temperature. Regarding $c_{3}$ as a small parameter, we apply the
saddle-point approximation to the order parameters. The order parameters
obtained with neglecting $c_{3}$ are given by

$\begin{pmatrix}{\overline{\Delta}_{1}}^{2}\\\
{\overline{\Delta}_{2}}^{2}\end{pmatrix}=\frac{-2}{4b_{0}^{(1)}b_{0}^{(2)}-d_{0}^{2}}\begin{pmatrix}2b_{0}^{(2)}\alpha_{1}(q_{1})-d_{0}\alpha_{2}(q_{2})\\\
-d_{0}\alpha_{1}(q_{1})+2b_{0}^{(1)}\alpha_{2}(q_{2})\end{pmatrix}.$ (S11)

Plugging $\overline{\Delta}_{1}$ and $\overline{\Delta}_{2}$ into the first
term of Eq. (S10), we obtain $f_{\text{NRSF}}$ up to first order in $c_{3}$,

$\displaystyle f_{\text{NRSF}}$ $\displaystyle=\lim_{q_{1},q_{2}\rightarrow
0}\sum_{0\leq
n,m}^{n+m=3}\frac{(n+m)!}{n!~{}m!}~{}\partial_{q_{1}}^{n}\partial_{q_{2}}^{m}F[q_{1},q_{2},\overline{\Delta_{1}},\overline{\Delta_{2}}],$
(S12)
$\displaystyle=\frac{2c_{3}}{4b_{0}^{(1)}b_{0}^{(2)}-d_{0}^{2}}\sqrt{\left(a_{0}^{(2)}d_{0}-2a_{0}^{(1)}b_{0}^{(2)}\right)\left(a_{0}^{(1)}d_{0}-2a_{0}^{(2)}b_{0}^{(1)}\right)}.$
(S13)

Taking into account the temperature dependence of the mass terms
$a_{0}^{(i)}\propto|T_{\text{c}}-T|$, we obtain
$f_{\text{NRSF}}\propto|T_{\text{c}}-T|$. As a result, the critical behavior
of the NRSF is the same as that in the single component GL theory.

Next, we consider the case (2). Neglecting the temperature dependence of an
order parameter $\Delta_{2}$, we discuss the second superconducting transition
due to the appearance of $\Delta_{1}$. Thus, the following calculation is
based on the GL free energy for $\Delta_{1}$

$F^{\prime}[q,\Delta_{1}]=\alpha^{\prime}(q)\Delta_{1}^{2}+\frac{1}{2}\beta^{\prime}(q)\Delta_{1}^{4}+\frac{1}{6}\kappa_{3}q^{3}\Delta_{1},$
(S14)

where we give the coefficients by

$\displaystyle\alpha^{\prime}(q)=-A_{0}+\frac{1}{2}A_{2}q^{2},$ (S15)
$\displaystyle\beta^{\prime}(q)=B_{0}.$ (S16)

The $O(\Delta_{1})$ term denoted by $\kappa_{3}$ originates from the coupling
with the order parameter $\Delta_{2}$, although the odd-order terms
$\Delta^{2n+1}$ of the GL free energy are usually forbidden due to the global
gauge symmetry. If the coupling constant $\kappa_{3}$ is sufficiently small,
the saddle-point approximation can be performed as in Eq. (S4). Accordingly,
the GL free energy is obtained as

$\mathcal{F}^{\prime}(q)=F^{\prime}[q,\overline{\Delta}_{1}],$ (S17)

and the NRSF is evaluated up to $O(\kappa_{3})$ by

$f_{\text{NRSF}}=\lim_{q\rightarrow
0}\partial_{q}^{3}\mathcal{F}^{\prime}(q)=\kappa_{3}\sqrt{\frac{A_{0}}{B_{0}}}.$
(S18)

The temperature dependence of the NRSF is $|T_{\text{c}}-T|^{1/2}$ due to
$A_{0}\propto|T_{\text{c}}-T|$. Note that the normal superfluid density is
less dependent on $T_{\text{c}}-T$ since the order parameter $\Delta_{2}$
makes a dominant contribution to the superfluidity in the critical region of
the second superconducting transition.

To summarize, the critical exponents of the NRSF in multi-component
superconductors are different between the equally pairing case ($r=0.5$) and
the unequally pairing case ($r\neq 0.5$). The critical behaviors of the NRSF
($f_{\text{NRSF}}\propto|T_{\text{c}}-T|^{\gamma}$, $\gamma=1,0.5$) are
confirmed by the numerical analysis of the model Hamiltonian adopted in the
main text. In both cases, we obtain the critical exponent for
$\eta_{\text{NR}}\propto|T-T_{\text{c}}|^{1/2}$.

##  S2 Model study for Uranium Ditelluride

We introduce the model Hamiltonian proposed in Ref. Shishidou  _et al._ (2021)
and investigate the superconductivity of UTe2. UTe2 has the orthorhombic
crystal structure (space group $Immm$, No. 71) and U sites show the locally-
noncentrosymmetric property. The model Hamiltonian for the normal state is
given by

$H_{{\bm{k}}}^{\text{N}}=\left(\varepsilon_{0}-\mu\right)+V\rho_{x}+V^{\prime}\rho_{y}+\bm{g}\cdot\bm{\sigma}\rho_{z},$
(S19)

where the spin and sublattice degrees of freedom are respectively represented
by the Pauli matrices $\bm{\sigma}$ and $\bm{\rho}$. Each component is given
by

$\displaystyle\varepsilon_{0}$
$\displaystyle=t_{1}\cos{k_{x}}+t_{2}\cos{k_{y}},$ (S20) $\displaystyle V$
$\displaystyle=\left(v_{1}+v_{2}\cos{\frac{k_{x}}{2}}\cos{\frac{k_{y}}{2}}\right)\cos{\left(2k_{z}\delta-\frac{k_{z}}{2}\right)},$
(S21) $\displaystyle V^{\prime}$
$\displaystyle=\left(v_{1}+v_{2}\cos{\frac{k_{x}}{2}}\cos{\frac{k_{y}}{2}}\right)\sin{\left(2k_{z}\delta-\frac{k_{z}}{2}\right)},$
(S22) $\displaystyle\bm{g}$
$\displaystyle=\left(\alpha_{1}\sin{k_{y}},\alpha_{2}\sin{k_{x}},\alpha_{3}\sin{\frac{k_{x}}{2}}\sin{\frac{k_{y}}{2}}\sin{\frac{k_{z}}{2}}\right),$
(S23)

with $\delta=0.135$ Ikeda  _et al._ (2006); Hutanu  _et al._ (2019). The
locally noncentrosymmetric property is built into the sublattice-dependent
antisymmetric spin-orbit coupling $\bm{g}\cdot\bm{\sigma}\rho_{z}$ Fischer
_et al._ (2011); Maruyama  _et al._ (2012). The model parameters are
determined to reproduce the Fermi-surface consisting of U $5f$ orbitals
Shishidou  _et al._ (2021); Ishizuka  _et al._ (2019). The parameters in the
unit of electron volt (eV) are given by

$(\mu,t_{1},t_{2},v_{1},v_{2},\alpha_{1},\alpha_{2},\alpha_{3})=(-0.129,-0.0892,0.0678,-0.062,0.0742,0.006,0.008,0.01).$
(S24)

Figure S1(a) shows the Fermi surface. The obtained hyperbolic Fermi surface
has been observed in DFT+U calculations Ishizuka  _et al._ (2019); Shishidou
_et al._ (2021).

With the normal Hamiltonian, we construct the Bogoliubov-de Gennes (BdG)
Hamiltonian

$\mathcal{H}_{\text{BdG}}=\frac{1}{2}\sum_{\bm{k}}\bm{\Psi}_{\bm{k}}^{\dagger}H_{\text{BdG}}({\bm{k}})\bm{\Psi}_{\bm{k}},$
(S25)

where the ${\bm{k}}$-resolved Nambu spinor is
$\bm{\Psi}_{\bm{k}}=\left(\bm{c}_{\bm{k}}^{\dagger},\bm{c}_{-{\bm{k}}}^{T}\right)$.
The electron creation operator $\bm{c}_{\bm{k}}^{\dagger}$ is labeled by the
spin and sublattice degrees of freedom. For each ${\bm{k}}$, the BdG
Hamiltonian is given by

$H_{\text{BdG}}({\bm{k}})=\begin{pmatrix}H_{{\bm{k}}}^{\text{N}}&\hat{\Delta}_{{\bm{k}}}\\\
\hat{\Delta}_{{\bm{k}}}^{\dagger}&-\left(H_{-{\bm{k}}}^{\text{N}}\right)^{T}\end{pmatrix}.$
(S26)

The superconducting pair potential is given by the mixture of the odd and
even-parity pair potentials

$\hat{\Delta}_{\bm{k}}=\left(\psi_{\bm{k}}+\bm{d}_{\bm{k}}\cdot\bm{\sigma}\right)i\sigma_{y}\rho_{0},$
(S27)

with $\psi_{\bm{k}}=\Delta_{\text{e}}\cos{k_{x}}$ and
$\bm{d}_{\bm{k}}=\Delta_{\text{o}}\sin{k_{y}}\hat{y}$ ($A_{u}$ state) or
$\bm{d}_{\bm{k}}=\Delta_{\text{o}}\sin{k_{y}}\hat{z}$ ($B_{3u}$ state). Here
we do not take into account the inter-sublattice pairing for simplicity.

We adopt the total magnitude of the pairing potential
$\Delta_{\text{e+o}}^{(0)}=|\Delta_{\text{e}}^{(0)}|+|\Delta_{\text{o}}^{(0)}|=0.0005$
at the zero temperature, where the pair potential amplitudes are parametrized
by the ratio $r$ as
$|\Delta_{\text{e}}^{(0)}|/|\Delta_{\text{o}}^{(0)}|=r/(1-r)$. The temperature
dependence is assumed to be phenomenological form

$\Delta_{\text{e,o}}(T)=\Delta_{\text{e,o}}^{(0)}\tanh{\left(1.74\sqrt{\dfrac{|\Delta_{\text{e,o}}^{(0)}|}{1.76T}-1}\right)}.$
(S28)

The transition temperatures
$T_{\text{c}}^{\text{(e,o)}}=|\Delta_{\text{e,o}}^{(0)}|/1.76$ roughly
correspond to the observed superconducting transition temperature Ran  _et
al._ (2019); Braithwaite  _et al._ (2019). We show the momentum dependence of
the even and odd-parity superconducting gap amplitudes in Figs. S1(b) and (c),
which are parametrized by $|\cos{k_{x}}|$ and $|\sin{k_{y}}|$, respectively.

Figure S1: (a) Fermi surface of the effective single orbital model for UTe2.
(b) and (c) Momentum dependence of the gap amplitude for (b) the even-parity
pair potential and (c) the odd-parity pair potential.

In the following subsections, we show the numerical results of all the
components of normal and nonreciprocal superfluid densities, although we show
a component of the NRSF $f_{xxx}$ for the $A_{g}+iB_{3u}$ state in the main
text.

###  S2.1 Normal superfluid density

The linear optical conductivity has been calculated in the framework of the
BdG formulation Tinkham (2004). The total optical conductivity is given by Wat

$\sigma_{\alpha\beta}(\omega)=\sigma_{\alpha\beta}^{\text{reg}}(\omega)+\sigma_{\alpha\beta}^{\text{M}}(\omega).$
(S29)

The first component is the normal optical conductivity

$2\sigma^{\text{reg}}_{\alpha\beta}(\omega)=\sum_{a}\frac{1}{i\omega-\eta}J_{aa}^{\alpha}J_{aa}^{\beta}\partial_{\varepsilon}f_{a}+i\sum_{a\neq
b}\frac{J^{\alpha}_{ab}J^{\beta}_{ba}f_{ab}}{(\omega+i\eta-\varepsilon_{ba})\varepsilon_{ab}},$
(S30)

which consists of the intraband and interband terms. The energy difference
$\varepsilon_{ab}=\varepsilon_{a}-\varepsilon_{b}$ and the Fermi-Dirac
distribution function $f_{a}$ are defined with the energy eigenvalue
$\varepsilon_{a}$ of the BdG Hamiltonian. We introduced the adiabaticity
parameter $\eta$ and the current operators $J^{\alpha}$ and $J^{\alpha\beta}$
which are called paramagnetic and diamagnetic current operators, respectively.
The basis for the single particle energy eigenstates $\\{\ket{a}\\}$ is
spanned by the crystal momentum ${\bm{k}}$ and the band indices. In the Nambu
representation, the paramagnetic and diamagnetic currents are given by

$\mathcal{J}^{\alpha}=\frac{1}{2}\sum_{{\bm{k}},p,q}\Psi_{{\bm{k}}p}^{\dagger}J_{pq}^{\alpha}\Psi_{{\bm{k}}q},~{}\mathcal{J}^{\alpha\beta}=\frac{1}{2}\sum_{{\bm{k}},p,q}\Psi_{{\bm{k}}p}^{\dagger}J_{pq}^{\alpha\beta}\Psi_{{\bm{k}}q},$
(S31)

where the ${\bm{k}}$-resolved matrices are
$J^{\alpha}=\partial_{\alpha}H^{\text{N}}_{\text{BdG}}({\bm{k}})\tau_{z}$ and
$J^{\alpha\beta}=\partial_{\alpha}\partial_{\beta}H^{\text{N}}_{\text{BdG}}({\bm{k}})$
spanned by the band indices $(p,q)$. The Pauli matrices $\bm{\tau}$ represent
the Nambu degree of freedom and $H^{\text{N}}_{\text{BdG}}$ denotes the BdG
Hamiltonian without the pair potential. Although Eq. (S30) is diverging in the
low-frequency limit when the superconducting gap has a nodal structure, it is
regularized by a proper scattering effect such as the electron correlation and
impurity scattering. Thus, the contribution is denoted as the ‘regular’ term
Tinkham and Ferrell (1959); Basov and Timusk (2005). On the other hand, the
second term in Eq. (S29) represents a divergent contribution written by

$2\sigma_{\alpha\beta}^{\text{M}}(\omega)=-\frac{2\rho^{\text{s}}_{\alpha\beta}}{i\omega-\eta}.$
(S32)

We introduced the (normal) superfluid density $\rho^{\text{s}}_{\alpha\beta}$

$\rho^{\text{s}}_{\alpha\beta}=\lim_{\bm{A}\rightarrow\bm{0}}\partial_{A_{\alpha}}\partial_{A_{\beta}}F_{A}=\frac{1}{2}\lim_{\bm{A}\rightarrow\bm{0}}\sum_{a}\partial_{A_{\beta}}\left(\partial_{A_{\alpha}}\varepsilon_{a}f_{a}\right),$
(S33)

where $F_{A}$ is the free energy calculated with the BdG Hamiltonian including
the vector potential ${\bm{A}}$. The contribution (S32) gives an infinite
static conductivity since
$\left(w+i\eta\right)^{-1}=\text{P}\,\omega^{-1}-i\pi\delta(\omega)$ where P
represents the principal integral for $\omega$. This reveals that the
superfluid density ensures the zero-resistivity phenomenon in superconductors.
Thus, the $\omega^{-1}$ divergence in
$\Im{[\,\sigma_{\alpha\beta}^{\text{M}}(\omega)]}$ is robust to a scattering
effect unless the superfluid density vanishes.

On the basis of the London gauge ($\nabla\cdot{\bm{A}}=0$), the superfluid
density gives rise to the static coupling between the electric current and
vector potential, that is, Meissner response. The response formula is given by

$J_{\alpha}(\bm{r})=-\rho^{\text{s}}A_{\alpha}(\bm{r}),$ (S34)

for an isotropic superconductor
$\rho^{\text{s}}_{\alpha\beta}=\rho^{\text{s}}\delta_{\alpha\beta}$.
Accordingly, the London penetration depth is obtained as

$\lambda_{\text{L}}=\frac{1}{\sqrt{\mu_{0}\rho^{\text{s}}}},$ (S35)

with the vacuum permeability $\mu_{0}$.

Here we calculate the superfluid density in the UTe2 model. The numerical
calculation is performed with the $N^{3}$-discretized Brillouin zone. It is
convenient for numerical calculations to rewrite the superfluid density by the
paramagnetic and diamagnetic current operators. The expression is given by

$\displaystyle\sum_{a}\partial_{A_{\alpha}}\left(\partial_{A_{\beta}}\varepsilon_{a}f_{a}\right)$
$\displaystyle=\sum_{a}\partial_{A_{\alpha}}\left(-J^{\beta}_{aa}f_{a}\right),$
(S36) $\displaystyle=\sum_{a}\left[J^{\alpha\beta}_{aa}+\sum_{b\neq
a}\frac{1}{\varepsilon_{ab}}\left(J^{\alpha}_{ab}J^{\beta}_{ba}+{\rm
c.c.}\right)\right]f_{a}+\sum_{a}J^{\alpha}_{aa}J^{\beta}_{aa}\partial_{\varepsilon}f_{a},$
(S37)

where we used the Hermann-Feynman relation between the paramagnetic and
diamagnetic current operators. In the final line, the first two components are
Fermi-sea contributions, whereas the last term is the Fermi-surface
contribution. When we adopt a simple model Hamiltonian such as that preserving
the $\mathcal{P}$ symmetry, the interband matrix element of the paramagnetic
current operator is forbidden by the emergent particle-hole symmetry Ahn and
Nagaosa (2021). Thus, in superconductors with gapped Bogoliubov quasiparticle
spectrum and such an emergent symmetry, the superfluid density at low
temperatures is determined by the diamagnetic contribution, which is closely
related to the total electron density Tinkham and Ferrell (1959); Watanabe and
Oshikawa (2020).

Figure S2 shows the numerical results of the $A_{g}+iB_{3u}$ state with the
parameter $r=0.5$. We obtained qualitatively the same results for the
$A_{g}+iA_{u}$ state, and another ratio for the pairing potentials $r$ gives
similar results. Owing to the orthorhombic symmetry of UTe2, the superfluid
density is anisotropic, and thus, we show $\rho_{xx}^{s}$, $\rho_{yy}^{s}$,
and $\rho_{zz}^{s}$. We also show the Fermi-sea and Fermi-surface
contributions in Fig. S2(b). Although the two contributions are completely
canceled in the normal state ($T\geq T_{\text{c}}$), a finite superfluid
density appears as the Fermi-surface term is suppressed in the superconducting
state ($T<T_{\text{c}}$). The superfluid density usually takes the maximum
value determined at zero temperature, and it is determined by the Fermi-sea
term. Thus, the Fermi-surface contribution is detrimental to the Meissner
response. The power-law temperature dependence of the superfluid density
indicates anisotropic superconducting gap structure, as seen in an experiment
for UTe2 Ishihara  _et al._ (2021). From Fig. S2 the London penetration depth
is evaluated as $\lambda_{\text{L}}\sim 0.3\,{\mathrm{\,\text{\textmu}m\,}}$
by adopting the superfluid density $\sim 10^{19}$
A$\cdot$V${}^{-1}\cdot$m${}^{-1}\cdot$s-1.

Figure S2: Superfluid density (A$\cdot$V${}^{-1}\cdot$m${}^{-1}\cdot$s-1) in
the $A_{g}+iB_{3u}$ pairing state. (a) Temperature dependence of
$\rho^{\text{s}}_{xx}$ (blue), $\rho^{\text{s}}_{yy}$ (green), and
$\rho^{\text{s}}_{zz}$ (red). (b) Decomposition of $\rho^{\text{s}}_{xx}$ into
the Fermi-sea (green) and Fermi-surface (red) contributions. The blue-colored
plot is the total superfluid density. We adopt $r=0.5$ and the lattice
constants reported in Ref. Ikeda  _et al._ (2006). We choose $N=300$ for
numerical calculations.

###  S2.2 Nonreciprocal superfluid density

The numerical calculation of the NRSF,
$f_{\alpha\beta\gamma}=\lim_{\bm{A}\rightarrow\bm{0}}\partial_{A_{\alpha}}\partial_{A_{\beta}}\partial_{A_{\gamma}}F_{A}$,
can be performed as in Sec. S2.1. When we rewrite the NRSF by the velocity
operators, the expression is

$f_{\alpha\beta\gamma}=f_{\alpha\beta\gamma}^{\text{(surf)}}+f_{\alpha\beta\gamma}^{\text{(sea)}},$
(S38)

which consists of the Fermi-surface contribution

$\displaystyle-2f_{\alpha\beta\gamma}^{\text{(surf)}}$
$\displaystyle=\sum_{a}\left(J^{\mu\nu}_{aa}J^{\lambda}_{aa}+J^{\mu\lambda}_{aa}J^{\nu}_{aa}+J^{\nu\lambda}_{aa}J^{\mu}_{aa}\right)\partial_{\varepsilon}f_{a}+\sum_{a}J^{\mu}_{aa}J^{\nu}_{aa}J^{\lambda}_{aa}\partial^{2}_{\varepsilon}f_{a}$
$\displaystyle+\sum_{a\neq
b}\frac{1}{\varepsilon_{ab}}\left[J^{\mu}_{aa}\left(J^{\nu}_{ab}J^{\lambda}_{ba}+{\rm
c.c.}\right)+J^{\nu}_{aa}\left(J^{\lambda}_{ab}J^{\mu}_{ba}+{\rm
c.c.}\right)+J^{\lambda}_{aa}\left(J^{\mu}_{ab}J^{\nu}_{ba}+{\rm
c.c.}\right)\right]\partial_{\varepsilon}f_{a},$ (S39)

and the Fermi-sea contribution

$\displaystyle-2f_{\alpha\beta\gamma}^{\text{(sea)}}$
$\displaystyle=\sum_{a}J^{\alpha\beta\gamma}_{aa}f_{a}+\sum_{a\neq
b}\frac{1}{\varepsilon_{ab}}\left(J^{\alpha}_{ab}J^{\beta\gamma}_{ba}+J^{\beta}_{ab}J^{\gamma\alpha}_{ba}+J^{\gamma}_{ab}J^{\alpha\beta}_{ba}+{\rm
c.c.}.\right)f_{a}-\sum_{a\neq
b}\frac{\Delta^{\gamma}_{ab}}{\varepsilon_{ab}^{2}}\left(J^{\beta}_{ab}J^{\alpha}_{ba}+J^{\beta}_{ba}J^{\alpha}_{ab}\right)f_{a}$
$\displaystyle+\sum_{a,b,c}^{a\neq
b,c}\frac{1}{\varepsilon_{ab}\varepsilon_{ac}}f_{a}\left[J^{\alpha}_{ab}J^{\beta}_{bc}J^{\gamma}_{ca}+J^{\beta}_{ab}J^{\alpha}_{bc}J^{\gamma}_{ca}+{\rm
c.c.}\right]+\sum_{a,b,c}^{b\neq
a,c}\frac{1}{\varepsilon_{ab}\varepsilon_{bc}}f_{a}\left[J^{\beta}_{ab}J^{\gamma}_{bc}J^{\alpha}_{ca}+J^{\alpha}_{ab}J^{\gamma}_{bc}J^{\beta}_{ca}+{\rm
c.c.}\right].$ (S40)

We defined the third-order generalized velocity operator, whose momentum
representation is
$J^{\alpha\beta\gamma}=\partial_{\alpha}\partial_{\beta}\partial_{\gamma}H^{\text{N}}_{\text{BdG}}({\bm{k}})\tau_{z}$.
The velocity difference matrix is defined by
$\Delta^{\alpha}_{ab}=J^{\alpha}_{aa}-J^{\alpha}_{bb}=-\partial_{\alpha}\varepsilon_{ab}$.

First, we investigate the NRSF for the $A_{g}+iB_{3u}$ pairing state.
According to the symmetry analysis, the allowed NRSF components are $f_{xxx}$,
$f_{xyy}$, and $f_{xzz}$. We plot the temperature dependence of each component
in Fig. S3. The NRSF $f_{xyy}$ and $f_{xzz}$ are almost determined by the
Fermi-surface effect as we show in the main text for $f_{xxx}$. By decomposing
the Fermi-surface contribution, we identified that the NRSF is almost
determined by one of the Fermi-surface terms

$-\frac{1}{2}\sum_{a}J^{\mu}_{aa}J^{\nu}_{aa}J^{\lambda}_{aa}\partial^{2}_{\varepsilon}f_{a}.$
(S41)

This term plays an essential role in (nearly) nodal superconducting states,
although it is strongly suppressed in fully gapped superconductors (see Sec.
S3).

Figure S3: Temperature dependence of the NRSF tensors $f_{\alpha\beta\gamma}$
in the $A_{g}+iB_{3u}$ equally-pairing ($r=0.5$) state. The unit of the NRSF
is A$\cdot$V${}^{-2}\cdot$s-2, and we implement the $N^{3}=200^{3}$
discretized momentum summation. The vertical dashed line guides the transition
temperature.

Following the parallel calculations, we obtain the NRSF for the $A_{g}+iA_{u}$
pairing state. The allowed NRSF component is $f_{xyz}$ which indicates that
the nonreciprocal Meissner response occurs under the magnetic field
perpendicular to the $[111]$-direction. The dependences on the temperature $T$
and the pairing ratio $r$ are found to be qualitatively similar to the
$A_{g}+iB_{3u}$ state. For example, we show the temperature dependence of the
NRSF $f_{xyz}$ for the equally-pairing case ($r=0.5$) in Fig. S4. The
prominent temperature dependence originates from the Fermi-surface term in Eq.
(S41).

Figure S4: Temperature dependence of the NRSF tensor $f_{xyz}$ in the
$A_{g}+iA_{u}$ equally-pairing ($r=0.5$) state. The unit and parameters for
the numerical calculation are the same as Fig. S3.

##  S3 Nonreciprocal superfluid density with isotropic s-wave pairing

We studied the model of UTe2 in the main text and Sec. S2, where we found a
sizable Fermi-surface contribution to the NRSF arising from the nodal-like gap
structure. In this section, we present another model study to demonstrate that
the origin and temperature dependence of the NRSF drastically change in the
presence of the isotropic superconducting gap. We discuss two examples. One is
the staggered Rashba Hamiltonian. The other is the same Hamiltonian as in the
main text and Sec. S2, although the even-parity pair potential is assumed to
be an isotropic $s$-wave one.

###  S3.1 Staggered Rashba model

The model represents a two-dimensional locally-noncentrosymmetric system. We
consider the square lattice and the Rashba-type sublattice-dependent spin-
orbit coupling Železný  _et al._ (2014). The Hamiltonian for the normal state
is given by

$H_{{\bm{k}}}^{\text{N}}=\left(\varepsilon_{0}-\mu\right)+V\rho_{x}+\bm{g}\cdot\bm{\sigma}\rho_{z},$
(S42)

where we introduced the Pauli matrices $\bm{\sigma}$ and $\bm{\rho}$ for the
spin and sublattice degrees of freedom. Each component reads

$\displaystyle\varepsilon_{0}=-4t_{1}\cos{k_{x}}\cos{k_{y}},$ (S43)
$\displaystyle V=-2t_{2}\left(\cos{k_{x}}+\cos{k_{y}}\right),$ (S44)
$\displaystyle\bm{g}=4\alpha\left(\cos{k_{x}}\sin{k_{y}},-\sin{k_{x}}\cos{k_{y}},0\right).$
(S45)

The model parameters are chosen as

$\mu=-4,~{}\alpha=0.3,~{}t_{1}=0.6,~{}t_{2}=1.0.$ (S46)

The mixed-parity superconducting pair potential is introduced by the molecular
field

$\hat{\Delta}_{\bm{k}}=\left(\psi_{\bm{k}}+i\bm{d}_{\bm{k}}\cdot\bm{\sigma}\right)i\sigma_{y}\rho_{0}.$
(S47)

The spin-singlet ($\psi_{\bm{k}}$) and spin-triplet ($\bm{d}_{\bm{k}}$)
components respectively correspond to the even and odd-parity pairings, and we
assume

$\psi_{\bm{k}}=\Delta_{\text{e}},~{}\bm{d}_{\bm{k}}=\Delta_{\text{o}}\sin{k_{x}}\cos{k_{y}}\hat{z}.$
(S48)

The symmetry of the parity-mixed superconductivity is denoted by the
$A_{g}+iE_{u}$ representation of a tetragonal point group $D_{4h}$.
Accordingly, the allowed NRSF components are given as in the case of the
$A_{g}+iB_{3u}$ pairing of UTe2. In the following numerical calculations, we
set the total amplitude of the pair potentials to
$\Delta_{\text{e+o}}=|\Delta_{\text{e}}|+|\Delta_{\text{o}}|=0.1$. We adopt a
series of the ratio $r$ defined by
$|\Delta_{\text{e}}|=r\,\Delta_{\text{e+o}}$.

The model illustrates the crossover from a gapful superconductor ($r=1$) to a
nodal superconductor ($r=0$). Thus, we can see how quasiparticle excitation
contributes to the NRSF by changing the ratio $r$. The temperature dependence
of the pair potentials is phenomenologically assumed by Eq. (S28). Figure S5
shows the NRSF with the ratio $r=0.1,~{}0.5,~{}0.9$. For a moderate weight of
$p$-wave pairing ($r=0.1,~{}0.5$), the Fermi-surface contribution plays a
major role. On the other hand, in the $s$-wave dominant regime ($r=0.9$), the
Fermi-sea contribution is dominant, while the Fermi-surface contribution is
negligible due to the absence of thermal excitations. Such behavior is also
found in the low-temperature regime for $r=0.1,~{}0.5$ [see insets of Figs.
S5(a) and S5(b)]. The NRSF is remarkably enhanced by the Fermi-surface
contribution, which can be more than $10^{3}$ times larger than the Fermi-sea
contribution when thermal excitations moderately occur. Therefore, the NRSF in
anisotropic superconductors can be much larger than that in the isotropic
superconductors.

Figure S5: Temperature dependence of the NRSF $f_{xxx}$
(A$\cdot$V${}^{-2}\cdot$s-2) in the staggered Rashba model. The total NRSF
(blue) is decomposed into the Fermi-sea (green) and Fermi-surface (red)
contributions. (a) $p$-wave (odd-parity) dominant regime ($r=0.1$). (b)
Equally-pairing regime ($r=0.5$). (c) $s$-wave (even-parity) dominant regime
($r=0.9$). The horizontal axis of the insets is $T\times 10^{3}$. The dashed
lines guide the transition temperature of even-parity
($T_{\text{c}}^{\text{(e)}}$) and odd-parity ($T_{\text{c}}^{\text{(o)}}$)
superconductivity. We use the $N^{2}=1500^{2}$ discretized Brillouin zone in
the numerical calculation.

###  S3.2 Uranium ditelluride model with isotropic $s$-wave pairing

Instead of the extended $s$-wave pairing
$\psi_{\bm{k}}=\Delta_{\text{e}}\cos{k_{x}}$ adopted in the main text and Sec.
S2, we take the isotropic form $\psi^{\prime}_{\bm{k}}=\Delta_{\text{e}}$.
Except for the $s$-wave pair potential, we adopt the same Hamiltonian as in
Sec. S2.

Figure S6 shows the temperature dependence of the NRSF $f_{xxx}$ when the
several pairing ratios $r=0.1,0.5,0.9$ are taken. Similarly to Fig. S5, the
NRSF is strongly enhanced by the Fermi-surface term and reaches the maximum
value in the intermediate temperature regime for the the cases with $r=0.1$
and $0.5$ [Figs. S6(a) and S6(b)]. On the other hand, the Fermi-surface term
plays a minor role in the $s$-wave dominant state with $r=0.9$ [Fig. S6(c)],
because the nearly isotropic superconducting gap suppresses the quasiparticle
excitation. Instead, the Fermi sea determines the NRSF but makes a much
smaller contribution than the Fermi-surface effect does in the presence of a
sizable $p$-wave component.

In conclusion, the structure of the superconducting gap significantly
influences the nonreciprocal property of the superfluid density due to a
sizable contribution from the quasiparticle excitations. Thus, the temperature
dependence of the NRSF will be informative in determining the superconducting
gap structure. For instance, if we carefully control the temperature $T$ and
pairing ratio $r$, the NRSF and the associated nonreciprocal Meissner response
show a drastic change.

Figure S6: Temperature dependence of the NRSF $f_{xxx}$ in the UTe2 model with
the isotropic $s$-wave pairing. The convention is the same as in Fig. S5. (a)
$p$-wave (odd-parity) dominant regime ($r=0.1$). (b) Equally-pairing regime
($r=0.5$). (c) $s$-wave (even-parity) dominant regime ($r=0.9$). We use the
$N^{3}=150^{3}$ discretized Brillouin zone in the numerical calculation.

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