aainstitutetext: Departamento de Física, Universidade Federal do Maranhão,
Campus Universitario do Bacanga, São Luís (MA), 65080-805,
Brazil.bbinstitutetext: Instituto de Física, Universidade Federal do Rio de
Janeiro, Rio de Janeiro, RJ, 21941-909, Brazil

# Geometric Josephson junction

Fabiano F. Santos b Henrique Boschi-Filho<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work we present a gravitational dual of a Josephson junction
constructed from the AdS/BCFT correspondence. On the gravity side, we consider
a planar AdS-Schwarzschild black hole. Our junction is connected by the
boundary $Q$ with tension $\Sigma$ of the boundary CFT. Our computations on
the gravity side reproduce the standard relation between the current across
the junction and the phase difference of the condensate controlled by the
tension $\Sigma$. We also study the maximum current’s dependence on the
junction’s tension and size and reproduce familiar results.

††dedication: This work is dedicated to the memory of Roberto Nicolsky.

##  1 Introduction

More than one century ago, Einstein provided his ground-breaking General
Theory of Relativity, whose main character is the gravitational interaction
due to the curvature of the spacetime. Since then, many remarkable results
have emerged from this theory, such as the FRW cosmology. Besides, quantum
theory, usually formulated in flat spacetime, can be extended to curved
spacetimes. This is relevant, for instance, for the early universe and black
holes (see e.g. Clifton:2011jh ; Ydri:2017oja ; Mann:2021mnc ; Haco:2018ske ).
Recently, this domain of applications was enlarged with the proposal of the
Anti-de-Sitter/Conformal Field Theory (AdS/CFT) correspondence
Maldacena:1997re ; Witten:1998qj  that relates string theory on asymptotically
anti-de Sitter spacetimes to conformal field theories on its boundary. This
duality provides a wide range of possible relations between gravitation and
condensed matter, where gravitational non-perturbative methods can be applied
in the description of macroscopic properties of strongly coupled matter
Santos:2023flb ; Hartnoll:2009sz ; Hartnoll:2008hs ; Hartnoll:2007ai ;
Hartnoll:2007ih ; Hartnoll:2007ip ; Hartnoll:2008vx ; Horowitz:2011dz ;
Fujita:2012fp ; Hovhannisyan:2022gkp ; Liu:2022bdu ; Melnikov:2012tb . In this
framework, condensed matter physicists have a novel way to study the
properties of strongly correlated transport and dynamics in theories without
quasiparticles where charged black holes become dual states of matter at
finite charge density and temperature.

Over the last two and half decades, it becomes clear that this holographic
correspondence between a gravitational theory and a quantum field theory can
be extended to understand the aspects of nuclear physics Boschi-Filho:2006jiv
; Mateos:2007ay ; BehnamPourhassan:2022rkg  and condensed matter systems
Hartnoll:2008hs ; Hartnoll:2009sz . Phenomena like the Hall Hartnoll:2008vx ;
Melnikov:2012tb  and the Nernst Hartnoll:2007ai ; Hartnoll:2007ih ;
Hartnoll:2007ip ; Hartnoll:2008hs  effects have dual gravitational
descriptions. Beyond, the dual gravitational description of superconductivity
was performed in Hartnoll:2008vx . In this scenario, a critical temperature
below a charged condensate forms through a second-order phase transition, and
the conductivity (DC) becomes infinite.

Another physical problem that can be considered in the framework of the
holographic physics of superconductors is the Josephson junction
Horowitz:2011dz ; Hovhannisyan:2022gkp . As known in the last decades, the
Josephson junction arrays, are very well characterized, leading to exciting
and robust connection between experiments, computer simulations, and applied
mathematics. The nature of Josephson’s junction and his known physics allows
easy simulations which can be compared to and suggest experiments
Josephson:1962zz ; NicolskyPRB1990 ; Newrock2000 .

The Josephson junctions are supported by two superconductor’s weak links
coupled by a region that may be either a normal conductor or a superconductor
Newrock2000  like superconductor-insulator-superconductor (SIS),
superconductor-normal-superconductor (SNS) junctions. Furthermore, Josephson
junctions can also be fabricated by making a constriction in a film (a
microbridge) or by coupling through a weaker superconductor (SS’S). From the
point of view of the holographic correspondence Horowitz:2011dz ;
Hovhannisyan:2022gkp , the normal conductor between superconductors to study
Josephson junction in a holographic way has been proposed by Hartnoll:2008vx .
Some elements are essential to constructing a holographic superconductor, like
the gravitational scenario, the introduction of temperature through the black
hole, and gauge fields. These ingredients are responsible for extending the
range of condensed matter phenomena that can be described gravitationally, and
with them, new insight into Josephson junctions was made to superconductors
Horowitz:2011dz . The last is the simplest model to study holographic
superconductors. Although the chemical potential is not constant, the
stationary solutions are due to the nonzero electric field on the boundary
near the edges of the gap, which becomes the charge density time-dependent.

This work proposes geometrically building an Anti-de-Sitter/Boundary Conformal
Field Theory (AdS/BCFT) Josephson junctions. Our model considers that one of
these superconductors is on the AdS side and the other in BCFT. This last is
represented by boundary $\partial\Omega$ (Fig. 1). The building AdS/BCFT
correspondence proposed by Takayanagi Takayanagi:2011zk  is an extension of
the AdS/CFT correspondence Maldacena:1997re , defining in the CFT a boundary
in $D$-dimensional manifold $\mathcal{M}$ for a $D+1$-dimensional
asymptotically AdS space so that $\Omega=\mathcal{M}\cup\partial\Omega$, where
$\partial\Omega$ is a $d$-dimensional manifold that satisfies
$\partial\Omega\leavevmode\nobreak\ \cap\leavevmode\nobreak\
\mathcal{M}=\mathcal{P}$ (Fig. 1). This new extension has brought attention to
investigating the transport coefficients like the Hall effect, fluid/gravity
correspondence, and the Hawking-Page phase transition Takayanagi:2011zk ;
Fujita:2011fp ; Melnikov:2012tb ; Cavalcanti:2018pta ; Magan:2014dwa ;
Santos:2021orr ; Sokoliuk:2022llp ; Santos:2023mee .

$\partial\Omega$$\mathcal{M}$$\Omega$$\mathcal{P}$ Figure 1: AdS/CFT
correspondence in the presence of boundary hypersurface $\partial\Omega$.

Besides, recent investigations with two holographic CFTs in two dimensions
Afrasiar:2023nir  have been proposed to obtain Page curves for the Hawking
radiation in Jackiw-Teitelboim (JT) braneworld Jackiw:1984je ;
Teitelboim:1983ux ; Almheiri:2019qdq ; Goto:2020wnk . This setting is
performed by two CFT2 coupled along a shared interface where the bulk dual
geometry has two AdS3 spacetimes, truncated by a shared Karch-Randall brane.
These geometry setups generalize the entanglement entropy for bipartite states
through the island prescription in the compelling lower-dimensional picture.
As shown by Afrasiar:2023nir , perfect agreement in the limit of large brane
tension with doubly holographic computations in the bulk geometry.
Gravitational duals on AdSD+1 with a subcritical Karch-Randall (KR) brane can
be used to connect an AdSD gravity located on the brane coupled with a CFT
with a non-gravitational bath that shares the same CFT Randall:1999vf ;
Karch:2000ct ; Brito:2018pwe ; Santos:2023eqp ; Almheiri:2019psf ;
Penington:2019npb ; Almheiri:2019hni ; Almheiri:2019psy . In our holographic
Josephson junction, two AdS4 planar Schwarzschild black hole in bulk theory
will be connected by a dual AdS3 profile, which is described through a CFT3 in
a thermal state residing in an eternal black hole background.

With the above, we construct our Josephson junction with two geometries, which
can give Josephson junction-like constriction Josephson:1962zz ;
Hovhannisyan:2022gkp  where this hypersurface partitions spacetime are
composed by two regions as presented by Fig. 2.

$\displaystyle\partial{\Omega}_{-}$$\displaystyle\partial{\Omega}_{+}$$\displaystyle{\Omega}_{-}$$\displaystyle{\Omega}_{+}$Constriction
Junction Figure 2: The figure presents the bulk manifold into the two domains
$\Omega_{\pm}$. In the AdS/BCFT correspondence scheme $\mathcal{M}_{\pm}$ are
asymptotically AdS spaces so that
$\Omega_{\pm}=\mathcal{M}_{\pm}\cup\partial\Omega_{\pm}$ where
$\partial\Omega_{\pm}$ are manifolds that satisfy
$\partial\Omega_{\pm}=\mathcal{M}_{\pm}=\mathcal{P}_{\pm}$.

In our setup, the geometric profile of the junctions is drawn as Fig. 2 where
the junctions between the AdS and BCFT side, i.e. the boundary
$\partial\Omega=\partial\Omega_{+}\cup\leavevmode\nobreak\
\partial\Omega_{-}$. The action that describes the geometry profile of these
junctions is supported by

$\displaystyle S_{\rm Profile}=S^{\Omega}+S^{\Omega}_{\rm
mat}+S^{\partial\Omega}+S^{\partial\Omega}_{\rm mat}+S^{\mathcal{P}}_{\rm ct}$
(1)

where $S^{\Omega}_{\rm mat}$ describes a perfect fluid and

$\displaystyle S^{\Omega}=\frac{1}{2}\kappa_{\rm
G}\left(\int_{\Omega_{+}}+\int_{\Omega_{-}}\right){d^{D+1}x\sqrt{-g}\leavevmode\nobreak\
(R-2\Lambda)},$ (2) $\displaystyle S^{\partial\Omega}=\kappa_{\rm
G}\left(\int_{\partial\Omega_{+}}+\int_{\partial\Omega_{-}}\right){d^{D}x\sqrt{-h}\leavevmode\nobreak\
(K-\Sigma)},$ (3) $\displaystyle S^{\partial\Omega}_{\rm mat}=\kappa_{\rm
G}\left(\int_{\partial\Omega_{+}}+\int_{\partial\Omega_{-}}\right){d^{D}x\sqrt{-h}\leavevmode\nobreak\
\mathcal{L}_{\rm mat}},$ (4) $\displaystyle S^{\mathcal{P}}_{\rm
ct}=\left(\int_{\mathcal{P}_{+}}+\int_{\mathcal{P}_{-}}\right){d^{D}x\sqrt{-h}\leavevmode\nobreak\
\mathcal{L}_{\rm ct}}.$ (5)

Here $\kappa_{\rm G}^{-1}=8\pi\,G$ is the gravitational coupling constant,
$g_{\mu\nu}$ is the bulk metric, $\Omega=\Omega_{+}\cup\leavevmode\nobreak\
\leavevmode\nobreak\ \Omega_{-}$. With respect to the boundary
$\partial\Omega$, we define its tension $\Sigma$, the extrinsic curvature
$K_{\mu\nu}=h^{\phantom{\mu}\beta}_{\mu}\nabla_{\beta}n_{\nu}$, its trace
$K=h^{\mu\nu}K_{\mu\nu}$, the induced metric $h_{\mu\nu}$, and an outward
pointing unit normal vector $n^{\mu}$. $\mathcal{L}_{\rm mat}$ is the matter
Lagrangian on $\partial\Omega$. We need to impose the boundary to provide an
asymptotic AdS spacetime. $S^{\mathcal{P}}_{\rm ct}$ is the action that
contains possible counter-terms and contract terms, localized on
$\mathcal{P}$.

The superconducting phase of these junctions has a complex order parameter,
which supports the phase difference between the condensate of these two
superconductors. This ingredient is essential to constructing the
gravitational dual of a Josephson junction Horowitz:2011dz ; Hartnoll:2008vx .
With this, we propose that such phase difference can be regarding
$\Psi=\tilde{\Psi}\,e^{i\theta}$ where this field is complex. Indeed, $\Psi$
is the charged scalar field, $\tilde{\Psi}$ is the fundamental scalar field,
and $\theta$ is the Stückelberg field Liu:2022bdu . The Lagrangian that
represents all these fields can be written as:

$\displaystyle{\cal L}_{\rm
FF}=-\dfrac{\kappa}{4q^{2}}F^{\mu\nu}F_{\mu\nu}-(\partial_{\mu}\Psi-
qA_{\mu})^{2}-\frac{1}{2}m^{2}\Psi^{2},$ (6)

where $F=dA$ and $m$ is the mass of the scalar field $\Psi$. We will restrict
our analysis to the probe approximation scenario. In our setup, we performing
a rescale like $\Psi=\tilde{\Psi}/q$, $A=\tilde{A}/q$ and consider
$q\to\infty$, keeping $\tilde{\Psi}$ and $\tilde{A}$ fixed. Such
considerations are important in neglecting the backreaction of these fields on
the metric.

Holographic superconductors explored in the AdS/CFT correspondence are
performed by imposing the Dirichlet boundary condition at the boundary of AdS.
Therefore, we require the Dirichlet boundary condition on $\mathcal{M}$.
Nevertheless, in our scenario for the holographic Josephson junction according
to Takayanagi:2011zk ; Fujita:2011fp , for AdS/BCFT duality, a Neumann
Boundary Condition (NBC) on $\partial\Omega$ is required, given that this
boundary should be dynamical. There is no natural definite metric on
$\partial\Omega$ specified from the CFT side Nozaki:2012qd . With this, the
duality finds its natural roots in the holographic derivation of entanglement
entropy Ryu:2006bv ; DosSantos:2022exb  as well as in the Randall-Sundrum
model Randall:1999vf . The extension of the CFT’s boundary inside the bulk of
the AdS space is considered a modification of a thin Randall-Sundrum brane,
which intersects the AdS boundary. Such brane must be a dynamic object
Takayanagi:2011zk ; Fujita:2011fp , where is imposed Neumann Boundary
Condition (NBC). In this framework, the discontinuity in the bulk extrinsic
curvature across the defect is compensated by the tension from the brane
Santos:2021orr ; Santos:2023flb .

In our assignments, the Neumann Boundary Condition (NBC) provides boundary
$\mathcal{P}_{\pm}$ specified by the condition $y_{\pm}=const$, where
$y_{\pm}$ is one of the coordinates on $\mathcal{M}_{\pm}$ (Fig. 2), this
AdS/BCFT case corresponds to the problem on a complete Minkowski space
Santos:2021orr ; Santos:2023flb ; Sokoliuk:2022llp . We derive the transport
coefficients through the Neumann Boundary Condition (NBC) and analyze the
profiles of the Josephson junction. The strong physics motivation to study
this model is to extend the range of condensed matter phenomena that can be
described gravitationally.

This work is organized as follows: In Sec. 2, we present the holographic
framework necessary to construct the AdS/BCFT Josephson junction. In Sec. 3,
we consider probe approximation to provide the AdS planar Schwarzschild black
hole solution and, with this analysis, the boundary conditions to gauge
fields. In Sec. 4, we derive the boundary conditions to the boundary Q and
relate them with the Josephson current, where the boundary tension controls
the profile of the Junction. In Sec. 5, we present conclusions and discussions
about our proposal.

##  2 Holographic framework

In this section, we construct a holographic model to study our Josephson
junction. This will be described through the following total action

$\displaystyle S=S_{\rm Profile}+S^{\Omega}_{FF}$ (7)

where $S_{\rm Profile}$ is defined by Eq. (1) and

$\displaystyle S^{\Omega}_{FF}=\frac{1}{2}\kappa_{\rm
G}\left(\int_{\Omega_{+}}+\int_{\Omega_{-}}\right){d^{D+1}\sqrt{-g}\left(-\dfrac{1}{4q^{2}}F^{\mu\nu}F_{\mu\nu}-(\partial_{\mu}\Psi-
qA_{\mu})^{2}-\frac{1}{2}m^{2}\Psi^{2}\right)}.$

From the action (7) and considering $\Psi=\tilde{\Psi}/q$, $A=\tilde{A}/q$
with $q\to\infty$ we derive the equations of motion as $\delta\,S^{\Omega}$,
and assume $S^{\Omega}_{\rm mat}=$constant where $\delta\,S^{\Omega}_{\rm
mat}=0$ the gravitational equations of motion can be write as

$(G_{\mu\nu}+\Lambda
g_{\mu\nu})_{\Omega}=0,\quad\Omega=\Omega_{+}\cup\leavevmode\nobreak\
\Omega_{-}$ (9)

Now, we can choose the Neumann boundary conditions alternatively. Let us do
this for the induced metric on $\partial\Omega$, that is let us impose
Santos:2021orr ; Santos:2023flb :

$\displaystyle K_{\mu\nu}-h_{\mu\nu}(K-\Sigma)=\kappa\,{\cal
S}^{\partial\Omega}_{\mu\nu}\,,$ (10)

where

$\displaystyle{\cal
S}^{\partial\Omega}_{\mu\nu}=-\frac{2}{\sqrt{-h}}\frac{\delta
S^{\partial\Omega}_{\rm mat}}{\delta h^{\mu\nu}}\,.$ (11)

Considering the matter stress-energy tensor on $\partial\Omega$ as a constant
(this is ${\cal S}^{\partial\Omega}_{\alpha\beta}=0$), we can write

$\displaystyle[K_{\mu\nu}-h_{\mu\nu}(K-\Sigma)]_{\partial\Omega}=0;\quad\partial\Omega=\partial\Omega_{+}\cup\leavevmode\nobreak\
\partial\Omega_{-}.$ (12)

This is the dynamical equation for the induced metric $h_{\mu\nu}$, that is,
for the profile of the hypersurface $\partial\Omega$. The solution of the
AdS/BCFT then requires a solution of the equations of motion, derived from
action (7), which additionally satisfies the Dirichlet boundary conditions on
$\Omega$ and Neumann boundary conditions (12) on $\partial\Omega$.

##  3 AdS planar Schwarzschild black hole

In this section, we present a fixed metric background AdS planar Schwarzschild
black hole

$ds^{2}=-f(u)dt^{2}+\frac{du^{2}}{f(u)}+u^{2}(dx^{2}+dy^{2}),$ (13)

Through the equation (9), we have black hole solution as

$\displaystyle f(u)=\frac{1}{L^{2}}\left(u^{2}-\frac{u^{3}_{h}}{u}\right),$
(14)

Where $u_{h}$ is the black hole horizon. The solution (14) resemble the case
of Horowitz:2011dz . A crucial difference appears in the absence of the
Horndeski parameters $\alpha$ and $\gamma$ in $f(u)$. The temperature of this
black hole can be written as

$\displaystyle T=\frac{3u_{h}}{4\pi\,L^{2}}.$ (15)

Now, we will create the junction along the $y$ direction. Our fields are
functions of $u$ and $y$. The phase difference regards the Josephson junction,
and then we include a phase in our scalar field as $\Psi=\phi\,e^{i\theta}$.
We consider solutions of the form

$\displaystyle\tilde{A}=A_{t}dt+A_{u}du+A_{y}dy,$ (16)

where $|\Psi|$, $A_{t}$, $A_{u}$ and $A_{y}$ are all real functions of $u$ and
$y$. Although of $A$, we wish to work with the gauge-invariant fields
$M=A-d\theta$. Now, with these gauge transformations, we derive the equations
of motion in the function of gauge-invariant as

$\displaystyle\partial^{2}_{u}|\Psi|+\frac{1}{u^{2}f}\partial^{2}_{y}|\Psi|+\left(\frac{f^{{}^{\prime}}}{f}+\frac{2}{u}\right)\partial_{u}|\Psi|+\frac{1}{f}\left(\frac{M^{2}_{t}}{f}-fM^{2}_{u}-\frac{M^{2}_{y}}{u^{2}}-m^{2}\right)|\Psi|=0,$
(17)
$\displaystyle\partial^{2}_{u}M_{t}+\frac{1}{u^{2}f}\partial^{2}_{y}M_{t}+\frac{2}{u}\partial_{u}M_{t}-\frac{2|\Psi|^{2}}{f}M_{t}=0,$
(18)
$\displaystyle\partial^{2}_{u}M_{u}-\partial_{u}\partial_{x}M_{y}-2u^{2}|\Psi|^{2}M_{u}=0,$
(19)
$\displaystyle\partial^{2}_{u}M_{y}-\partial_{u}\partial_{y}\,M_{u}+\frac{f^{{}^{\prime}}}{f}(\partial_{u}M_{y}-\partial_{y}M_{u})-\frac{2|\Psi|^{2}}{f}M_{y}=0,$
(20)
$\displaystyle\partial_{u}M_{u}+\frac{1}{u^{2}f}\partial_{y}M_{y}+\frac{2}{|\Psi|}\left(M_{u}\partial_{u}|\Psi|+\frac{M_{y}}{u^{2}f}\partial_{y}|\Psi|\right)+\left(\frac{f^{{}^{\prime}}}{f}+\frac{2}{u}\right)M_{u}=0.$
(21)

The above equations (17-20) are second-order dynamical equations, while the
last one (21) is a first-order constraint equation coming from the
conservation of the source in Maxwell’s equations. So, the boundary conditions
at $u\to\infty$ the scalar have an asymptotic behavior like:

$\displaystyle|\Psi|=\frac{\Psi^{(1)}(y)}{u}+\frac{\Psi^{(2)}(y)}{u^{2}}+\mathcal{O}\left(\frac{1}{u^{3}}\right),$
(22)

In our expansion, we consider for simplicity $L=1$ and $m=-2$. Besides, as
discussed by Horowitz:2011dz , both $\Psi^{(1)}$ and $\Psi^{(2)}$ are
renormalizable, when we choice the boundary conditions. Choosing $\Psi^{(1)}$,
we have that via gauge/gravity duality that $\Psi^{(2)}$ provide the
expectation value of a dimension two operator in the boundary field theory
($<\mathcal{O}>=\Psi^{(2)}$), which is interpreted as the operator of the
superconducting condensate. With this, we can present the asymptotic form of
the Maxwell fields as

$\displaystyle
M_{t}=\mu-\frac{\rho(y)}{u}+\mathcal{O}\left(\frac{1}{u^{2}}\right),$ (23)
$\displaystyle M_{u}=\mathcal{O}\left(\frac{1}{u^{3}}\right),$ (24)
$\displaystyle M_{y}=\nu(y)+\frac{B}{u}.$ (25)

The quantities $\mu$, $\rho(y)$, $B$, and $\nu(y)$ are interpreted in the
boundary field theory as the chemical potential, charge density, magnetic
field, superfluid velocity, and current, respectively. Here, we found these
equations of motion near $u=\infty$. Such a condition was imposed by the
constraint (21) that enforces current conservation.

##  4 Profile description of the Josephson junction

For the metric (13), we can find the normal vectors about surface $Q$ that are
of the form

$\displaystyle(n^{t},n^{x},n^{y},n^{u})=\left(0,0,\frac{1}{ug(u)},-\frac{uf(u)y^{{}^{\prime}}(u)}{g(u)}\right),$
(26)

Where $g^{2}(z)=1+u^{2}f(u)y^{{}^{\prime}2}(u)$ and
$y^{{}^{\prime}}=\frac{dy}{du}$ and the induced metric is given by

$\displaystyle
ds^{2}_{ind}=-f(u)dt^{2}+\frac{g^{2}(u)du^{2}}{f(u)}+u^{2}dx^{2},$ (27)

Now let us solve equation (12) for the induced metric in the boundary $Q$, we
have

$\displaystyle
y^{{}^{\prime}}_{\pm}(u)=\pm\frac{\Sigma}{\sqrt{4-\Sigma^{2}u^{2}f}}.$ (28)

The behavior of the above equation is presented in Fig. 3. Our Josephson
junctions are fabricated by two profiles that give the constriction junction.

$\theta$$y$$\partial\Omega_{-}$$\partial\Omega_{+}$ Figure 3: The figure
presents the junction of two profiles with the two domains
$\partial\Omega_{\pm}$ for the values $L=1$ with $\Sigma=0.22$ (solid),
$\Sigma=0.24$ (dashed), $\Sigma=0.26$ (dot dashed), and $\Sigma=0.28$ (thick).

The phase difference between the condensate of two superconductors is given
through the gauge invariance

$\displaystyle\gamma=\Delta\theta-\int{A_{y}},$ (29)

where the integral crosses the gap. Besides, as we know Horowitz:2011dz , the
edges of the gap will not be completely sharp (for numerical reasons). Here in
our analysis, it is more convenient to use

$\displaystyle\gamma=-\int^{+\infty}_{-\infty}{dy[\nu(y)-\nu(\pm\infty)-\mathcal{O}{\rm(complex\,\,\,terms)}]},$
(30)

The equations (17)-(21) are invariant under the following scaling symmetry
like

$\displaystyle
u\to\,au,\quad(t,x,y)\to(t,x,y)/a,\quad\,M_{u}\to\,M/a,\quad(M_{t},M_{y})\to\,a(M_{t},M_{y}.)$
(31)

For the case of homogenous superconductors, this scale invariant temperature
is $T/\mu$, so changing $\mu$ is equivalent changes the temperature $T$. In
this case, $T$ must still be constant. On the other hand, in our case, $y(u)$
plays the hole of $\mu(x)$ as present in Horowitz:2011dz  where the weak link
between the constriction junction is supported by the boundary tension
$\Sigma$. Furthermore, the Josephson’s current

$\displaystyle J=J_{max}\sin(\gamma),$ (32)

which exists with no applied voltage. Some types of Josephson junctions depend
on the nature of the link. But here, the nature of the link is related to the
tension $\Sigma$. Using the Neumann boundary conditions
$n^{\mu}M_{\mu\nu}|_{Q}=0$ provide Santos:2023flb , we have that

$\displaystyle\nu(y)=\frac{B}{u},$ (33)

where $y=\int{duy^{{}^{\prime}}(u)}$. Recalling the equation (35) with
$\nu(\pm\infty)\to 0$ with boundary conditions at $u\to\infty$ the function
$f(u)$ have an asymptotic behavior like $f(u)\sim\,u^{2}/L^{2}$, we have

$\displaystyle\gamma=-\chi\int^{+\infty}_{-\infty}{du\frac{y^{{}^{\prime}}(u)}{u}}=-\Sigma\,B\int^{+\infty}_{-\infty}{\frac{du}{u\sqrt{4-(\Sigma\,u)^{2}\left(\frac{u^{2}}{L^{2}}\right)}}},$
(34)

Solving this equation, we have

$\displaystyle\gamma=\frac{\Sigma\,B}{4}\ln\left(\frac{\Sigma}{L}\right),$
(35)

and Josephson’s current can be written as

$\displaystyle
J=J_{max}\sin\left[\frac{\Sigma\,B}{4}\ln\left(\frac{\Sigma}{L}\right)\right],$
(36)

Figure 4: The figure shows the behavior of the equation (36) varying with
$\Sigma$ for the values of $B=0.1$ (solid pink), $B=0.2$ (dot dashed blue),
$B=0.3$ (dashed red) and $B=0.4$ (thick green), respectively.

The behavior of the Josephson current (Fig. 4) indicates that the AdS planar
black hole allows us to implement a local magnetic field. In the field of
magnetic sensors, there is a compromise between spatial resolution and field
sensitivity, where the resolution is improved due to the decrease in sensor
size. However, the sensitivity deteriorates Hovhannisyan:2022gkp . We can
observe that we analytically calculated the Josephson current (36) as a
function of the magnetic field and is the width ($\Sigma$) of our junction.
Therefore, when $\Sigma$ increases, it is clear that $J$ increases
considerably. This current varying with the magnetic field (Fig. 5) has a
similar behavior to that demonstrated by Hovhannisyan:2022gkp  from which we
can conclude that our analytical results have a similar behavior with the
experimental data for Josephson Junctions.

Figure 5: The figure shows the behavior of the equation (36) varying with $B$
for the values of $\Sigma=0.1$ (solid pink), $\Sigma=0.2$ (dot dashed blue),
$\Sigma=0.3$ (dashed red) and $\Sigma=0.4$ (thick green), respectively.

Now, extract the expectation value of the operator $\mathcal{O}$ is given by

$\displaystyle\frac{<\mathcal{O}>_{y=0,J=0}}{T^{2}_{c}}=\frac{A_{1}}{e^{\frac{\Sigma}{2\zeta}}}.$
(37)

With this, we can that the dependence of $J$ with $\Sigma$ (Fig. 4) where this
curve is in good agreement with the experimental behavior Hovhannisyan:2022gkp
, which for Josephson junctions, predicts an exponential behavior:

$\displaystyle\frac{J_{max}}{T^{2}_{c}}=\frac{A_{0}}{e^{\frac{\Sigma}{\zeta}}}.$
(38)

Figure 6: The figure shows the behavior of the equations (37) and (38) for the
values $A_{0}=A_{1}=1$ with $\zeta=0.1$ (solid pink), $\zeta=0.2$ (dot dashed
blue), $\zeta=0.3$ (dashed red) and $\Sigma=0.4$ (thick green), respectively.

As expected by the predictions of Hovhannisyan:2022gkp , the planar Josephson
junction can present an example of quantum holographic of electronic wave
functions, which corresponds to the macroscopic superconducting condensate
phase (Fig. 6).

##  5 Conclusions and discussions

In four dimensions with AdS planar Schwarzschild black hole, we construct the
simplest model of a holographic superconductor to build a holographic
Josephson junction connecting two geometries through the profile $y(u)$. In
our case, the Josephson junction with constant chemical potential ($\mu$) was
constructed by modifying the boundary condition for the scalar field inspired
by the works Hartnoll:2008vx ; Horowitz:2011dz ; Hovhannisyan:2022gkp . As we
show, on the BCFT side, the effects through the profile description of the
Josephson junction represent an example of quantum holographic electronic wave
functions, which correspond to a macroscopic superconducting condensate.

The spatial dependence of the profile $y(u)$ doesn’t imply the vanishing $J$,
and we find that the superfluid current is proportional to the sine of the
phase difference controlled by the tension $\Sigma$ of the boundary $Q$.
Furthermore, the constriction junction is predicted by a graph with an
exponential decay with the growing width $\Sigma$ of the gap in $J_{max}$. We
can conclude that the result of this paper, one can see that the model of the
holographic DC Josephson junction is valid for the (3+1)-dimensional case
through the AdS5/BCFT4 correspondence.

The Josephson effect says that when a voltage difference $V$ is applied across
a junction, the solution depends on time, and the phase difference satisfies

$\displaystyle\frac{d\gamma}{dt}=qV$ (39)

which provides an alternating current. However, time dependence makes it more
difficult to construct appropriate gravitational solutions. On the other hand,
the Josephson effect is expected to be reproduced on the gravity side, which
is obtained through a gauge transformation in the bulk. Adding a voltage
difference $V$ corresponds to adding a constant $V$ to $A_{t}$, but this is
equivalent in gauge terms to changing the phase of our charged scalar $\Psi$
by $qVt$.

Then, in this work, we described tunneling supercurrents where we concluded
tunnel currents do not go to zero when the voltage across the barrier equals
zero. This way, we can observe that the phase transition within our model is
due to the critical magnetic field. Thus, we described the normal
superconducting phase boundaries in the region of penetration into a magnetic
field.

In future work, we intend to extend these results by considering modified
theories of gravity rather than Einstein’s General Relativity, for instance,
introducing a Horndeski scalar field representing a Josephson junction with a
topological defect.

###### Acknowledgements.

This work was supported by Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES) under finance code 001. Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq) partially funds H.B.-F. under grants $\\#$s
311079/2019-9 and 310346/2023-1.

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  * (1)
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# MPI-based Evaluation of Coordinator Election Algorithms

Filip De Turck, Ghent University - imec, Belgium

###### Abstract

In this paper, we detail how two types of distributed coordinator election
algorithms can be compared in terms of performance based on an evaluation on
the High Performance Computing (HPC) infrastructure. An experimental approach
based on an MPI (Message Passing Interface) implementation is presented, with
the goal to characterize the relevant evaluation metrics based on statistical
processing of the results. The presented approach can be used to learn master
students of a course on distributed software the basics of algorithms for
coordinator election, and how to conduct an experimental performance
evaluation study. Finally, use cases where distributed coordinator election
algorithms are useful are presented.

##  1 Problem statement

The coordinator election algorithms are stated as follows. We consider a set
of N processes ${p_{1},...,p_{N}}$ that (i) have no shared variables, and (ii)
can communicate by message passing. From this set, we want to select one
process to play a special role (e.g. coordinator), such that (i) every process
$p_{i}$ should have the same coordinator, (ii) if the elected process fails a
new election round is started. In order to be able to make this choice, we
assume that each process has a unique ID, and the problem is to elect the
process with the largest ID. This ID could e.g. be related to the inverse of
the load on the process, in order to elect a relatively unloaded process. This
problem statement implies that IDs are unique and that IDs are totally
ordered.

##  2 Evaluation metrics

There are two important metrics to characterize the performance of algorithms
for coordinator election:

  * •

Bandwidth: the number of messages used to elect a new coordinator.

  * •

Turnaround time: time needed for the election (measured from the moment the
election process is started until all processes have agreed on a common
coordinator). This turnaround time is typically expressed as a function of
$\delta$, the one-way communication delay between process nodes).

##  3 Two considered algorithms

###  3.1 Chang-Roberts Algorithm

The Chang-Roberts election algorithm assumes that all processes are organized
in a logical ring, where each process can communicate with its successor in
the ring. Each process has a unique ID, and the purpose of the algorithm is to
elect the process with the largest ID as coordinator. The algorithm assumes
reliable communication channels between live processes (no crashes while the
election is ongoing). No assumptions are made on channel or processing delays,
so the algorithm supports asynchronous systems.

###  3.2 Franklin Algorithm

This election algorithm is a variant of the Chang-Roberts algorithm. It is
based on bidirectional communication in a ring, whereas the Chang-Roberts
algorithm is based on unidirectional communication in a ring. Nodes are either
active or passive. Initially, all nodes are active.
The algorithm works in multiple rounds. In each round, each active process
takes the following actions: (i) it sends the token with process ID to both
neighbours, (ii) it examines the two tokens received from its neighbours: if
one or two tokens contain ID $>$ process ID, the process turns passive, i.e.
only forwards tokens to neighbours (i.e. acts as a router), and (iii) when
there is only one active process left (i.e. process receives its own tokens),
this process becomes the coordinator.
In a ring with N active processes, at least $N/2$ turn passive after one
round. After at most ${log}_{2}N$ rounds, there is only one active process
left. Then one additional round is needed to become leader. In total,
$1+{log}_{2}N$ rounds are needed to elect an unique coordinator.

##  4 Approach

Consider the following two algorithms for distributed coordinator election:

  1. 1.

implement the Chang-Roberts algorithm based on MPI for execution on the HPC
infrastructure. Make sure to keep track of the amount of messages that were
sent by the algorithm. Run the solution 10 times, each time using 16
processes. Give the minimum, maximum and average (averaged over 10 runs)
amount of messages that your solution needs to elect a coordinator. In
addition, also give the minimum, maximum and average (averaged over 10 runs)
execution times.

  2. 2.

implement the Franklin algorithm based on MPI for execution on the HPC
infrastructure. Make sure to keep track of the amount of rounds that were
needed to elect a coordinator by the algorithm. Run the solution 10 times,
each time using 16 processes. Give the minimum, maximum and average (averaged
over 10 runs) number of rounds that your solution needs to elect a
coordinator. In addition, also give the minimum, maximum and average (averaged
over 10 runs) execution times.

Assume $N$ equals 16, implement the two algorithms above in the C++
programming language and by making use of the MPI (Message Passing Interface)
library. Determine the used bandwidth and obtained turnaround times for the
two algorithms.

##  5 HPC infrastructure

All code is compiled and executed using the UGent HPC (High Performance
Computing) infrastructure. Once connected to the HPC infrastructure, students
are logged on to one of the interface nodes. These nodes can be used to
compile software and submit jobs to the different clusters, but running
software on these interface nodes is generally considered bad practice. For
this experimental evaluation, however, it is fine to run lightweight examples
on the interface node. Hence, you may simply execute the commands in the
assignments on the interface nodes. The project should be compiled and run on
a interface nodes. In order to compile all source files, please use the
provided compile.sh script (that relies on CMake internally to generate a
Makefile). You can also change the number of parallel processes MPI should
start, which is specified by -np.

##  6 Discussion

The comparison study of the two algorithms for coordinator election
contributes to acquiring knowledge and getting acquainted with:

  1. 1.

the internal details of both algorithms (both ring-based, one algorithm is an
extension of the other),

  2. 2.

the typical communication delays and processing delays obtained on the High
Performance Cluster,

  3. 3.

the experimental evaluation setup on the High Performance Cluster and
statistical processing of the obtained numerical results, and

  4. 4.

typical numerical values for the two important evaluation metrics: bandwidth
and turnaround time.

Coordination election algorithms are very useful and required in several
application domains. Important use cases where coordinator election algorithms
can be applied to optimize the system behaviour are:

  * •

hierarchical network management systems [1, 2], where the coordinator can
collect additional data and initiate the decision making process,

  * •

softwarized network management [3, 4], where the coordinators can contribute
to the dynamic resource allocation decisions,

  * •

adaptive video delivery [5, 6, 7], where the coordinators estimate the most
appropriate video quality levels and inform the other involved nodes,

  * •

efficient virtual desktop cloud computing [8], where coordinators in a
hierarchical setting can optimize the resource allocation process,

  * •

smart city applications [9, 10] and mobile augmented reality applications
[11], where the coordinator can collect and aggregate additional data for
informing the other nodes,

  * •

replica placement in ring based content delivery networks [12], where
coordinator election algorithms are important to generate a hierarchical and
scalable operation,

  * •

optimization the delivery of adaptive video streaming services by in network
optimizations [13, 14].

##  7 GitHub repository

https://github.ugent.be/fdeturck/PDS/tree/main/HWA5

## References

  * [1] J. Famaey, S. Latré, J. Strassner, F. De Turck, A hierarchical approach to autonomic network management, 2010 IEEE/IFIP Network Operations and Management Symposium Workshops, IEEE NOMS 2010, Osaka, Japan, pp 225-232.
  * [2] H Moens, B Hanssens, B Dhoedt, F De Turck, Hierarchical network-aware placement of service oriented applications in clouds, IEEE Network Operations and Management Symposium, IEEE NOMS 2014, Cracow, Poland, pp. 1-8.
  * [3] H. Moens, F. De Turck, VNF-P: A model for efficient placement of virtualized network functions, IEEE International Conference on Network and Service Management (CNSM), 2014, Rio De Janeiro, Brazil, pp. 418-423.
  * [4] H. Moens, F. De Turck, Customizable function chains: Managing service chain variability in hybrid NFV networks, IEEE Transactions on Network and Service Management 13 (4), 2016, pp. 711-724.
  * [5] S. Laga, T. Van Cleemput, F. Van Raemdonck, F. Vanhoutte, N. Bouten, M. Claeys, F. De Turck, Optimizing scalable video delivery through OpenFlow layer-based routing, 2014 IEEE Network Operations and Management Symposium (NOMS), IEEE NOMS 2014, pp. 1-4.
  * [6] J. van der Hooft, T. Wauters, F. De Turck, C. Timmerer, H. Hellwagner, Towards 6dof http adaptive streaming through point cloud compression, Proceedings of the 27th ACM International Conference on Multimedia, ACM Multimedia, 2019, pp. 2405-2413.
  * [7] M. Torres Vega, C. Perra, F. De Turck, A. Liotta, A review of predictive quality of experience management in video streaming services, IEEE Transactions on Broadcasting, Volume 64, Issue 2, IEEE, pp. 432-445.
  * [8] L. Deboosere, B. Vankeirsbilck, P. Simoens, F. De Turck, B. Dhoedt, P Demeester, Efficient resource management for virtual desktop cloud computing, Springer Journal of Supercomputing 62 (2), 2012, pp. 741-767.
  * [9] J. Santos, T. Wauters, B. Volckaert, F. De Turck, Resource provisioning for IoT application services in Smart Cities, IEEE International Conference on Network and Service Management (CNSM), 2017, Tokyo, Japan, pp. 1-9.
  * [10] J. Santos, P. Leroux, T. Wauters, B. Volckaert, F. De Turck, Anomaly detection for smart city applications over 5g low power wide area networks, NOMS 2018-2018 IEEE/IFIP Network Operations and Management Symposium, IEEE NOMS 2018, pp. 1-9.
  * [11] T. Verbelen, T. Stevens, P. Simoens, F. De Turck, B. Dhoedt, Dynamic deployment and quality adaptation for mobile augmented reality applications, Journal of Systems and Software, Volume 84, Issue 11, 2011, Elsevier, pp. 1871-1882.
  * [12] T. Wauters, J. Coppens, F. De Turck, B. Dhoedt, P. Demeester, Replica placement in ring based content delivery networks, Elsevier Journal on Computer Communications, Volume 29, Issue 16, 2006, pp. 3313-3326.
  * [13] N. Bouten, S. Latré, J. Famaey, W. Van Leekwijck, F. De Turck, In-network quality optimization for adaptive video streaming services, IEEE Transactions on Multimedia, 2014, Volume 16, Issue 8, pp. 2281-2293.
  * [14] N. Bouten, R. de O Schmidt, J. Famaey, S. Latré, A. Pras, F. De Turck, QoE-driven in-network optimization for Adaptive Video Streaming based on packet sampling measurements, Elsevier Computer Networks, Volume 81, 2015, pp. 96-115.


# Triplet pairing mechanisms from Hund’s-Kondo models:
applications to UTe2 and CeRh2As2

Tamaghna Hazra1<EMAIL_ADDRESS>Piers Coleman1,2 1Center
for Materials Theory, Rutgers University, Piscataway, New Jersey, 08854, USA
2Department of Physics, Royal Holloway, University of London, Egham, Surrey
TW20 0EX, UK

(August 27, 2024)

###### Abstract

Observing that several U and Ce based candidate triplet superconductors share
a common structural motif, with pairs of magnetic atoms separated by an
inversion center, we hypothesize a triplet pairing mechanism based on an
interplay of Hund’s and Kondo interactions that is unique to this structure.
In the presence of Hund’s interactions, valence fluctuations generate a
triplet superexchange between electrons and local moments. The offset from the
center of symmetry allows spin-triplet pairs formed by the resulting Kondo
effect to delocalize onto the Fermi surface, precipitating superconductivity.
We demonstrate this mechanism within a minimal two-channel Kondo lattice model
and present support for this pairing mechanism from existing experiments.

Hund’s coupling is a direct manifestation of the Coulomb interaction in multi-
electron atoms. One well-known consequence is the development of large moments
in partially filled $d$ and $f$-shell materials. Much less discussed are the
effects of Hund’s interactions on valence fluctuations and the Kondo effect in
such high-spin systems.

A possible role of Hund’s coupling in pairing mechanisms for triplet
superconductivity has been discussed by Anderson [1], Norman [2], Hotta and
Ueda [3] and more recently in the context of triplet resonating valence bonds
(tRVBs) in various settings [4, 5]. Here we discuss the implication of these
ideas for triplet-paired heavy fermion superconductors in a two-channel Kondo
lattice model [6, 7, 8], with two key observations. First, we show how Hund’s
coupling modifies the structure of the Kondo interaction, inducing triplet
correlations between scattered electrons and local moments. Second, we
highlight a structural motif common to candidate triplet-paired heavy fermion
materials (Table 1 and Figure 1) - an even number of magnetic ions separated
by a center of symmetry. This sublattice structure enables the triplet paired
correlations induced in the Kondo effect to develop coherence on the Fermi
surface, inducing a Cooper instability.

| UTe2 | UGe2 | UCoGe | URhGe | UBe13 | UPt3 | CeRh2As2
---|---|---|---|---|---|---|---
Ref | [9, 10] | [11] | [11] | [11] | [12, 13] | [14, 15] | [16]
nM | 2 | 2 | 4 | 4 | 2 | 2 | 2
Tc |  $2$K |  $0.8$K |  $0.8$K |  $0.25$K |  $0.95$K |  $0.5$K |  $0.26$K
HP |  $3.7$T |  $1.5$T |  $1.5$T |  $0.5$T |  $1.8$T |  $0.9$T |  $0.5$T
Hc2 |  $>60$T |  $3$T |  $18$T |  $13$T |  $14$T |  $2.8$T |  $14$T
TM | - |  $52$K |  $2.7$K |  $9.5$K | - | - | -
Table 1: Candidate heavy fermion triplet superconductors have either
n${}_{M}=2$ or 4 magnetic U/Ce atoms in the primitive unit cell separated by
an inversion center. Maximum superconducting Tc, Pauli-limiting critical field
HP, highest measured upper critical field Hc2, Curie temperature TM for
ferromagnetic superconductors.

Our observations are built on the tRVB concept [4, 5], whereby valence
fluctuations into entangled, high-spin configurations (Figure 2) can
delocalize into a coherent triplet RVB superconducting state. Symmetry plays a
central role in this process, for triplet-paired configurations in the excited
state of a magnetic atom are even under inversion about the atom, while
triplet pairing on a Fermi surface is necessarily odd parity. Anderson [1]
recognized that if local moments are situated at distinct sublattices away
from the inversion centers, the onsite triplet pairs could acquire an
inversion-odd sublattice form factor, allowing them to coherently couple to
triplet Cooper pairs on the Fermi surface [2, 17, 3]. Remarkably, forty years
later, the structural motif identified by Anderson appears to characterize the
entire family of candidate triplet-paired uranium-based superconductors [18]

Figure 1:  Two-sublattice structure of UPt3, UBe13 and UTe2, shown in the
primitive unit cell. The inversion center is shown in red, and the two
distinct sublattices of U in blue and yellow.

(see Table 1 and Figure 1). Recently, CeRh2As2 has also emerged as a new
member [16] of this class of compounds 111As pointed out in the Supplementary
of [16], an odd-parity spin-triplet order parameter with the d-vector along
$z$ leads to the same pseudo-spin triplet pairing order parameter on the Fermi
surface as an odd-parity spin-singlet pairing order parameter..

Figure 2: Delocalization of Hund’s coupled triplet pairs:  (a) Hund’s coupling
($J_{H}$) and crystal field ($\Delta_{CF}$) entangle two spins in an $f^{2}$
local moment into an $S_{z}=0$ triplet state. (b) Triplet valence bond
representation of $S_{z}=0$ triplet state. (c) Hybridization causes a transfer
of triplet pairs (red) into the conduction (orange) sea.

To illustrate the interplay between Hund’s coupling and valence fluctuations,
consider a magnetic ion in which the f-electrons exist in three valence
states, which for simplicity we take to be 4f0, 4f1 and 4f2. Suppose that
Hund’s and spin-orbit coupling in the 4f2 configuration stabilize a state in
which two f-orbitals of symmetry $\Gamma_{1}$ and $\Gamma_{2}$ are entangled
into an $S=1$, $S_{z}=0$ state (Fig. 2a), forming a triplet valence bond
between the two orbitals (Fig. 2b). When the ion hybridizes with the
conduction sea, valence fluctuations will now allow the entangled triplet pair
to escape into the conduction sea (Fig. 2c).

To understand how Hund’s interactions affect super-exchange, suppose the 4f1
$\Gamma_{1}$ Kramer’s configuration is the most stable, forming a local
moment. Virtual valence fluctuations, $4{\rm f}^{1}\rightleftharpoons 4{\rm
f}^{0}+e^{-}$ and $e^{-}+4{\rm f}^{1}\rightleftharpoons 4{\rm f}^{2}$ now
generate super-exchange interactions with the conduction sea (Fig. 3).

Figure 3:  Low energy Hilbert space of the local moment, showing associated
Hubbard operators:  $\Gamma_{1}$ is the irreducible representation of the
spatial wavefunction of the ground-state $f^{1}$ doublet. The 4f2 excited
stated is a entangled spin-triplet state with $S_{z}=0$, formed from electrons
in the $\Gamma_{1}$ and $\Gamma_{2}$ orbitals.

Integrating out the virtual valence fluctuations via a Schrieffer-Wolff
transformation generates a second-order perturbation in the energy of
electrons scattering in the $\Gamma_{1,2}$ channel according to

$H_{K}=-\frac{|V_{\Gamma_{1}}|^{2}}{E_{0}}P^{\Gamma_{1}}_{S}-\frac{|V_{\Gamma_{2}}|^{2}}{E_{2}}P^{\Gamma_{2}}_{T}$
(1)

where $V_{\Gamma_{1,2}}$ are the hybridization matrix elements in the two
channels, while $E_{0}$ and $E_{2}$ are the corresponding excitation energies.
The operators
$P^{\Gamma_{1}}_{S}=\frac{1}{2}-\vec{\sigma}^{\Gamma_{1}}\cdot\vec{S}$ and
$P^{\Gamma_{2}}_{T}=\sigma_{z}P^{\Gamma_{2}}_{S}\sigma_{z}$ project the
incoming quasiparticles into the singlet and triplet states of the excited 4f0
and 4f2 states. Here $\vec{\sigma}^{\Gamma}$ is the spin of the conduction
electron in channel $\Gamma$. By noting that a $S_{z}=0$ triplet
$\mid\Downarrow\uparrow+\Uparrow\downarrow\rangle=\sigma_{z}\mid\Downarrow\uparrow-\Uparrow\downarrow\rangle$
is obtained from a singlet $\mid\Downarrow\uparrow-\Uparrow\downarrow\rangle$
by rotating the conduction electron spin through 180∘ about the $z-$axis, we
have written $P^{\Gamma_{2}}_{T}=\sigma_{z}P^{\Gamma_{2}}_{S}\sigma_{z}$ as a
unitary transform of the singlet operator $P^{\Gamma_{2}}_{S}$.

Omitting potential scattering terms, it follows that Hund’s interactions cause
the Kondo interaction to develop a triplet-superexchange with XXZ anisotropy,

$\displaystyle H_{K}$ $\displaystyle=$ $\displaystyle J_{1}{\bf
S}\cdot\boldsymbol{\sigma}^{\Gamma_{1}}+J_{2}{\bf
S}\cdot(\sigma_{z}\boldsymbol{\sigma}^{\Gamma_{2}}\sigma_{z})$ (2)

where $J_{1}=|V_{\Gamma_{1}}|^{2}/E_{0}$ and
$J_{2}=|V_{\Gamma_{2}}|^{2}/E_{2}$. The Hund’s-Kondo term can alteratively be
written
$H_{K2}=J_{2}[S_{z}\sigma^{\Gamma_{2}}_{z}-S_{x}\sigma^{\Gamma_{2}}_{x}-S_{y}\sigma^{\Gamma_{2}}_{y}]$.
Acting in isolation (i.e. if $J_{1}=0$) the Hund’s Kondo coupling $J_{2}$
flows to strong-coupling like its antiferromagnetic counterpart [20], but
forms a “screened” triplet state
($\mid\Uparrow\downarrow+\Downarrow\uparrow\rangle$). We are interested in the
interplay of the two terms in the lattice.

We now incorporate the above effects into a two channel Kondo lattice model
$H=H_{c}+H_{K1}+H_{K2}$, where

$\displaystyle H_{c}$ $\displaystyle=$
$\displaystyle-\sum_{\mathbf{k}}c^{\dagger}_{{\bf
k}}\bigl{[}(t_{0}\gamma_{\mathbf{k}}+\mu)+t_{1}\gamma_{\mathbf{k}}\alpha^{x}\bigr{]}c_{\mathbf{k}},$
(3) $\displaystyle H_{K1}$ $\displaystyle=$ $\displaystyle
J_{1}\sum_{j\alpha}\psi^{\dagger}_{1\alpha}(j)\boldsymbol{\sigma}\psi_{1\alpha}(j)\cdot\mathbf{S}_{j\alpha},$
(4) $\displaystyle H_{K2}$ $\displaystyle=$ $\displaystyle
J_{2}\sum_{j\alpha}\psi^{\dagger}_{2\alpha}(j)\sigma_{z}\boldsymbol{\sigma}\sigma_{z}\psi_{2\alpha}(j)\cdot\mathbf{S}_{j\alpha}.$
(5)

Here $H_{c}$ describes electron hopping on a two-sublattice body-centered
crystal, reminiscent of UTe2 [21], where $c^{\dagger}_{{\bf k}}\equiv
c^{\dagger}_{{\bf k}\alpha\sigma}$ creates an electron of wavevector ${\bf k}$
on sublattice $\alpha=\pm 1$ with spin component $\sigma^{z}=\sigma$, $t_{0}$
and $t_{1}$ are the intra- and inter-sublattice hopping integrals,
respectively, $({\alpha}^{x},{\alpha}^{y},{\alpha}^{z})$ are the sublattice
Pauli matrices and
$\gamma_{\mathbf{k}}=8\cos\frac{k_{x}}{2}\cos\frac{k_{y}}{2}\cos\frac{k_{z}}{2}$
is the nearest neighbor form-factor that is invariant under the D2h point
group. $H_{K1}$ and $H_{K2}$ are the Kondo interaction in channels
$\Gamma=(\Gamma_{1},\Gamma_{2})$, where

$\psi_{\Gamma\alpha}^{\dagger}(j)=\frac{1}{\sqrt{N_{s}}}\sum_{{\bf
k}}c^{\dagger}_{\mathbf{k}\alpha}\Phi_{\Gamma{\bf k}}e^{-i{\bf k}\cdot{\bf
R}_{j}}$ (6)

create electrons in Wannier states of symmetry $\Gamma$ coupled to spins
$\mathbf{S}_{j\alpha}$ at site $j,\alpha$ ($N_{s}$ is the number of unit
cells). We choose $\Phi_{1{\bf k}}=1$ and $\Phi_{2{\bf
k}}=i\sigma_{x}(\sqrt{1-\zeta^{2}}+i\zeta\alpha^{z}p^{z}_{{\bf k}})$, for the
two Kondo channels, consistent with time-reversal and inversion symmetry. Here
$p^{z}_{{\bf
k}}=\cos(\frac{k_{x}}{2})\cos(\frac{k_{y}}{2})\sin(\frac{k_{z}}{2})$ is an
odd-parity crystal harmonic transforming under the $B_{1u}$ representation.
The coefficient $\zeta$ is finite when the two magnetic atoms are displaced
from their common center of symmetry, activating a coupling between tRVBs
(Fig. 2b) and triplet Cooper pairs.

The key physics of this model involves a co-operative action of the Kondo
effect in the two channels [22]. At high temperatures in the lattice, the
Kondo coupling in both channels renormalizes to strong coupling according the
scaling equation
$\partial(J_{\Gamma}\rho)/\partial\ln\Lambda=-(J_{\Gamma}\rho)^{2}$ [23],
where $\rho$ is the conduction electron density of states, and $\Lambda$ the
energy cut-off. Suppose channel one is the strongest channel with the largest
Kondo temperature $T_{K1}\sim De^{-1/J_{1}\rho}$, where $D$ is the bandwidth.
Then the logarithmic renormalization in channel two is interrupted at a scale
$\Lambda=T_{K1}$, with a renormalized Kondo coupling constant given by

$\frac{1}{J_{2}^{*}}=\frac{1}{J_{2}^{0}}-\rho\
{\ln\left(\frac{D}{T_{K1}}\right)}$ (7)

At temperatures $T{{\lower 4.0pt\hbox{$<$}}\atop\widetilde{\ \ \ }}T_{K1}$ the
local moments fractionalize into deconfined heavy fermions
$\mathbf{S}_{j}\rightarrow
f^{\dagger}_{j}\left(\frac{\mathbf{\sigma}}{2}\right)f_{j}$: in an impurity
model, a Kondo resonance forms in the strongest channel and nothing further
happens.

However, in the Kondo lattice, the heavy fermions hybridize with the
conduction electrons to form a large Fermi surface [24, 25]. Moreover, in the
presence of a finite $\zeta$ the residual interaction $J_{2}$ created by
valence fluctuations into Hund’s-coupled spin-triplet states, couples triplet
Cooper pairs on the Fermi surface, which reactivates its scaling, causing it
to resume its upward logarithmic renormalization, ultimately diverging at
$T_{c}$ to form a Hund’s driven triplet superconductor.

To examine this process, we note that action of $H_{K2}$ on the deconfined
fermions is given by

$H_{K2}=-J_{2}^{*}\sum_{j}(\psi_{2j}^{\dagger}\sigma_{z}(-i\sigma_{y})f^{\dagger}_{j})(f_{j}(i\sigma_{y})\sigma_{z}\psi_{2j})$
(8)

where the sublattice indices have been suppressed. Here we have used the
particle-hole symmetry of the fractionalized spin operator to replace
$f_{j}\rightarrow-i\sigma_{y}f^{\dagger}_{j}$ in the usual hybridized form of
the Kondo interaction. Now the hybridized quasiparticle operators formed in
channel one have the form

$a_{{\bf k}\sigma}=u_{{\bf k}}f_{{\bf k}\sigma}+v_{{\bf k}}\Phi_{1{\bf
k}}c_{{\bf k}\sigma},$ (9)

where $u_{{\bf k}}\sim 1$, while $v_{{\bf k}}^{2}\sim\frac{m}{m^{*}}\ll 1$ is
set by the inverse mass renormalization of the heavy fermions 222For
simplicity, we have ignored the inter-sublattice hopping for now. In general,
$f_{{\bf k}\alpha},c_{{\bf k}\alpha}$ in (8) must be replaced by the
corresponding band eigenstates. If we now decompose $H_{K2}$ as a pair
scattering potential acting on the quasiparticle pairs at the Fermi level, we
obtain

$H^{*}=-J_{2}^{*}|u_{{\bf k}}v_{{\bf k}}|_{\rm FS}^{2}\sum_{{\bf k},{\bf
k}^{\prime}}\Lambda_{{\bf k}}^{\dagger}\Lambda_{{\bf k}^{\prime}}$ (10)

where $|u_{{\bf k}}v_{{\bf k}}|_{\rm FS}^{2}$ denotes the Fermi surface
average of the coherence factors and $\Lambda_{{\bf k}}=a_{-{\bf
k}}(i\sigma_{y})(\mathbf{d}^{A}_{{\bf k}}\cdot\mathbf{\sigma})a_{{\bf k}}$ is
the projection of the triplet pair operators in (8) onto the Fermi surface,
where $\mathbf{d}^{A}_{{\bf k}}=(\mathbf{d}_{{\bf k}}-\mathbf{d}_{-{\bf
k}})/2$ is the odd-parity component of the form factor $\mathbf{d}_{{\bf
k}}\cdot\mathbf{\sigma}=\alpha_{z}\sigma_{z}\Phi_{2{\bf k}}\Phi_{1{\bf k}}$,
projected into the band eigenstates. This antisymmetric term is absent if the
magnetic ions lie at a center of symmetry, both $\Phi_{1{\bf k}}$ and
$\Phi_{2{\bf k}}$ have the same parity under $k\to-k$, so
$\mathbf{d}^{A}_{{\bf k}}=0$ and the triplet pseudopotential vanishes.
However, in the presence of an offset center of symmetry, ($\zeta>0$) it
becomes finite. In our model calculation, $\mathbf{d}^{A}_{{\bf k}}\sim i\zeta
p^{z}_{{\bf k}}\hat{\mathbf{y}}$ (See Appendix B).

From these arguments, we see that the triplet coupling constant induced by the
Hund’s Kondo effect is now $g_{t}\sim J_{2}^{*}\rho^{*}(uv)^{2}_{FS}\approx
J_{2}^{*}\rho^{*}v^{2}$ because $u_{{\bf k}}\sim 1$. Now at first sight, the
small size of $v^{2}\sim m/m^{*}\ll 1$ is cause for concern, but it is
compensated by the large density of states of the heavy Fermi liquid
$\rho^{*}\sim 1/T_{K1}$. In fact the product $v^{2}\rho^{*}=dN/d\mu$ is
recognized as the charge susceptibility of the heavy fermi liquid, and since
the f-component of the heavy fermi liquid is incompressible, this quantity is
equal to the unrenormalized density of states of the conduction fluid,
$dN/d\mu\sim\rho$, so that $g_{t}\sim\zeta J_{2}^{*}\rho$ is essentially
unaffected by the large mass renormalization of the heavy electrons. A Cooper
instability will develop when

$\frac{1}{J_{2}^{*}}-\rho d^{2}\ln\left(\frac{T_{K1}}{T_{c}}\right)=0,$ (11)

where we have denoted the average d-vector magnitude by
$d^{2}=\langle|{\mathbf{d}}^{A}_{{\bf k}}|^{2}\rangle_{FS}$. Combining with
(7), a superconducting instability will take place when

$\frac{1}{J_{2}^{0}}-\rho\ {\ln\left(\frac{D}{T_{K1}}\right)}-\rho
d^{2}\ln\left(\frac{T_{K1}}{T_{c}}\right)=0$ (12)

Remarkably, this expression contains two logs - a Kondo log that renormalizes
$J_{2}$ from the band-width down to $T_{K1}$, followed by a further Cooper
renormalization of the coupling constant below $T_{K1}$. Thus the second
channel Kondo effect plays a vital co-operative role in enhancing the pairing
process. Solving (12), we find that

$T_{c}=(T_{K2})^{\frac{1}{d^{2}}}(T_{K1})^{1-\frac{1}{d^{2}}}$ (13)

is a weighted geometric mean of the Kondo temperatures
$T_{K\Gamma}=De^{-1/J_{\Gamma}\rho}$ in the two channels, emphasizing the co-
operative nature of the Hund’s-Kondo effect.

To quantify this effect in greater detail, we employ an SU(2) gauge theory[27,
28] description of the Kondo effect. The local moment on each sublattice may
be represented in terms of Abrikosov pseudo-fermions,
$\mathbf{S}=\frac{1}{2}(f^{\dagger}\boldsymbol{\sigma}f)$. In the Nambu basis
$F=\big{(}\begin{array}[]{cc}f,&(i\sigma_{y})^{\dagger}f^{\dagger}\end{array}\big{)}^{T}$,
this corresponds to
$\mathbf{S}=\frac{1}{4}F^{\dagger}(\boldsymbol{\sigma}\otimes\tau_{0})F$, a
representation that is invariant under SU(2) particle-hole transformations of
the spinons $F\rightarrow e^{i\mathbf{n\cdot}\boldsymbol{\tau}}F$, where
$\tau_{i}$ are Pauli matrices in Nambu space. In this representation, the
Kondo couplings become four-fermion interactions which are then decoupled
using a hybridization mean-field $V_{\Gamma\alpha}$ and a ‘pairing’ field
$\Delta_{\Gamma\alpha}$ in each channel and sublattice to get $H=H_{c}+H_{K}$,
where

$\displaystyle H_{K}$ $\displaystyle=$
$\displaystyle\sum_{\Gamma\mathbf{k}\alpha}\left(V_{\Gamma\alpha}\psi_{\Gamma\mathbf{k}\alpha}^{\dagger}f_{\mathbf{k}\alpha}+\Delta_{\Gamma\alpha}\psi_{\Gamma\mathbf{k}\alpha}^{\dagger}(i\sigma_{y})^{\dagger}f_{-\mathbf{k}\alpha}^{\dagger}+\mathrm{H.c.}\right)$
(14) $\displaystyle+$ $\displaystyle
N_{s}\sum_{\Gamma\alpha}\frac{\left|V_{\Gamma\alpha}\right|^{2}+\left|\Delta_{\Gamma\alpha}\right|^{2}}{J_{\Gamma}}.$

With a suitable SU(2) gauge transformation on the spinons $F\rightarrow
e^{i\mathbf{n\cdot}\boldsymbol{\tau}}F$, one can choose a gauge in which
$\Delta_{1\alpha}=0$, and the hybridization $V_{1\alpha}$ is real. This
hybridization in the first (singlet) channel then locks the U(1) gauge of the
spinons to the electrons, so that they have the same charge.

We shall focus on the situation in which $J_{2}$ is smaller than $J_{1}$ and
seek superconducting solutions in which the normal state $V_{1\alpha}=V$ is
uniform and $\Delta_{2\alpha}$ are non-zero, leading to the gap equation

$\displaystyle\frac{\Delta_{2\alpha}}{J_{2}}=-\frac{1}{N_{s}}\sum_{\mathbf{k}}\left\langle
f_{-\mathbf{k}\alpha}(i\sigma_{y})\sigma_{y}\biggl{(}\sqrt{1-\zeta^{2}}+i{\zeta}\alpha
p^{z}_{{\bf k}}\biggr{)}c_{\mathbf{k}\alpha}\right\rangle$ (15)

with the constraint $n_{f\alpha}=\frac{1}{N_{s}}\sum_{\mathbf{k}}\left\langle
f_{\mathbf{k}\alpha}^{\dagger}f_{\mathbf{k}\alpha}\right\rangle=1$. We see
that there are two types of superconducting solution:

  1. 1.

a uniform solution where $\Delta_{2\alpha}=\Delta_{2}$ and the triplet pairing
term $\langle f_{-{\bf k}\alpha}c_{{\bf k}\alpha}\rangle$ is an even function
of ${\bf k}$. The even character of the gap function means that this is an
inter-band gap function without a Cooper instability, that will only develop
for $J_{2}$ above a critical value.

  2. 2.

a staggered gap where $\Delta_{2\alpha}\propto\alpha\Delta_{2}$ is odd on the
two sublattices and $\langle f_{-{\bf k}\alpha}c_{{\bf k}\alpha}\rangle$ is an
odd function of momentum. This solution has a Cooper instability.

We find a sublattice-odd mean-field ground state
$V_{1A}=V_{1B},\Delta_{2A}=-\Delta_{2B},V_{2\alpha}=0$ corresponding to an
odd-parity triplet superconductor. Approaching the state from high temperature
(Figure (4)) we find a sequence of transitions, first the Kondo hybridization
in the first channel sets in at $T_{K}$333It is well known that fluctuations
not captured by a mean-field description make this a crossover, rather than a
sharp phase transition. leading to an expansion of the Fermi surface, and then
at a lower temperature $T_{c}$ the pairing order parameter in the triplet
channel turns on.

Figure 4: Temperature dependence of order parameters. Calculations were made
with parameters
$J_{K1}=2.67t_{0},J_{K2}=0.067t_{0},\mu=2t_{0},t_{1}=0.67t_{0},\zeta=0.995$.

The product of the order parameters in the two channels corresponds to a
composite pairing operator[22]

$\displaystyle\Psi=\langle
V_{1\alpha}\Delta_{2\alpha}\rangle\propto\sum_{\mathbf{k},j,\alpha}\langle\psi_{2,-\mathbf{k}\alpha}\sigma_{z}(\boldsymbol{\sigma}i\sigma_{y})\psi_{1\mathbf{k}\alpha}\cdot\boldsymbol{S}_{j\alpha}\rangle,$
(17)

which transforms as the $S_{y}=0$ component of a spin-triplet due to the non-
trivial form-factor between the two channels
$\langle\psi_{2,-\mathbf{k}\alpha}\sigma_{z}(i\sigma_{y})\psi_{1\mathbf{k}\alpha}\rangle\sim\sigma_{y}$.

The triplet pair potential we discuss here transforms according to the
$B_{3u}$ representation of $D_{2h}$ and has line nodes corresponding to the
intersection of $p_{\bf k}^{z}=0$ and the Fermi surfaces. It has its d-vector
aligned along the $y$-axis. Since the pair spin is pinned to the $xz$ plane,
upper critical fields in the $y$ direction are expected to significantly
overshoot the Pauli limit [30]. This is in addition to the inherent magnetic
anisotropies in the crystal-field split, spin-orbit coupled magnetic ion,
which can lead to violations of the Pauli limit by trivial means (viz. a large
g-factor).

Discussion: The core idea that is demonstrated by a simple concrete example in
this paper, is that strong Hund’s coupling at the atomic level induces triplet
correlations between electron pairs trapped inside local moments. Kondo
hybridization transfers these correlations to the dispersive electrons which
then condense into a triplet superconductor. How would this simple idea be
generalized to account for the complexity of the various heavy fermion
superconductors in Table 1?

The presence of a sublattice degree of freedom, common to the f-shell atoms of
all of these materials, allows the possibility of an odd-parity local triplet
pair, which after Kondo hybridization, overlaps with triplet pairs on the
Fermi surface. The form of the pair potential depends on three atomic
ingredients, specific to each of the materials. First is the spin-structure of
the pairs trapped on the local moments - here we have chosen the
$|f^{2}\rangle$ state to be a $S_{z}=0$ spin-triplet. Second is the different
orbital content of the f-electrons forming the local moment - here we have
chosen one orbital in the pair to transform like $\sigma_{x}$ relative to the
other. Both of these are a direct consequence of Hund’s rules, projected into
the crystal-field eigenstates. Both types of quenched angular momentum - spin
and orbital - effectively contribute to rotating the Kondo coupling into the
anisotropic spin exchange that engenders triplet pairing - what we call
triplet Kondo coupling. The d-vector of the resulting triplet superconductor
is then determined by the symmetry of the atomic ground states i.e. the
irreducible representation of $|f^{n+2}\rangle\langle f^{n}|$ which can in
principle be experimentally accessed by resonant inelastic X-ray scattering
[31, 32] and may be used to constrain such models.

The third ingredient is the position of the sublattices in the unit cell, and
this determines the odd-parity form-factor of the non-trivial hybridization.
Here, our choice of $p_{{\bf k}}^{z}$ is motivated by the vertical rung
structure of UTe2, and the k-dependence of the pairing form factor for
specific materials is determined by the irreducible representation of the
sublattice-odd form factor. This third ingredient determines the nodal
structure of the superconducting gap once all possible symmetry allowed form-
factors are included.

In UTe2, the experimental evidence for point-like gap nodes can be reconciled
by noting that an additional secondary order parameter with d-vector
$\mathbf{d}_{\bf k}=p_{\bf k}^{y}\hat{z}$ in the same irreducible
representation will be induced which will gap out the line nodes except where
$p_{\bf k}^{y}=0$. This results in the d-vector of the pair potential having a
two-dimensional texture in momentum space, and points to one of the many ways
one could engineer non-trivial topology in this system. In our simple
demonstration, the gaps on the two Fermi surfaces are identical and the
superconductor is topologically trivial because any time-reversal invariant
$\mathbb{Z}_{2}$ topological index is doubled. In UBe13, a possible extension
of this model would replace $p_{{\bf k}}^{z}$ by an f-wave form factor that
transforms like $k_{x}k_{y}k_{z}$. However, the observation that
superconductivity emerges directly from a local moment regime suggests that
the coupling strengths in the two channels are similar $J_{1}\approx J_{2}$,
so that $T_{c}\approx T_{K}$. In the high temperature A phase of UPt3, there
is strong evidence [14, 33] for an f-wave superconducting gap and $S_{z}=0$
spin-triplet pairing, readily captured by a variation of our model in which
the local pairs have $S_{z}=0$ spin-structure and the orbital content differs
by the f-wave form factor $\Gamma_{2}\sim\Gamma_{1}\times(x^{2}-y^{2})z$.

Several of the heavy fermion superconductors in Table 1 have multiple
superconducting phases [34, 35, 16], which are hard to reconcile with simple
spin-fluctuation mediated pairing scenarios. How would these be captured using
a Hund’s-Kondo pairing mechanism? In general, for valence fluctuations between
$f^{n-1},f^{n},f^{n+1}$ configurations, each of the crystal-field split
$f^{n\pm 1}$ levels provides a distinct symmetry channel for the conduction
electrons to resonate into the local moment, leading to a distinct triplet
Kondo coupling, with the $f^{n\pm 1}$ ground states determining the channels
with the strongest Kondo coupling. Pressure and magnetic field move the atomic
levels and may change the relative strengths of the Kondo hybridization in
different channels, thereby favoring competing pairing tendencies.

There is good reason to suspect that the Kondo effect is intimately linked
with the superconductivity in these materials. Tunneling experiments [36] in
UTe2 find a Fano lineshape in the differential conductance, that is
characteristic of cotunneling into a local moment. The electrical resistivity
of UTe2[18, 10] and CeRh2As2[16] both show clear broad maxima in the
temperature dependence, putatively signaling the onset of Kondo hybridization.
The superconducting $T_{c}$ is well below the Kondo temperature - the local
moments are not spectators to the development of triplet pairing correlations.

In contrast, among the many theoretical proposals [37, 38, 39, 40, 41, 42, 43,
44, 45, 46] for the pairing symmetry of UTe2 and CeRh2As2, with the exception
of [45], the Kondo character is typically considered inconsequential for the
low temperature physics, including superconductivity. Instead, the common
theme is to consider a Kramer’s doublet at each site, forming narrow
dispersive bands, whose Fermi surfaces are unstable to pairing by intersite
magnetic exchange interactions. The symmetry of the superconducting order
parameter is then determined either by the anisotropy of ferromagnetic
fluctuations [21] or by the anisotropy of the band spin-orbit coupling in
presence of isotropic ferromagnetic exchange [39].

We emphasize however, that the symmetric-spin pairing correlations ascribed to
inter-site ferromagnetic interactions, are already present at the atomic
level, driven directly by the largest energy scale in the system - atomic
Coulomb repulsion. The tendency for spin-triplet pairing is thus divorced from
the inter-site exchange, ferromagnetic or antiferromagnetic. In this picture,
the role of intersite hybridization or interactions is to provide a channel
for triplet pairing correlations inside local pre-formed pairs to delocalize
and condense. In the Hund’s-Kondo mechanism we present, the pairing symmetry
is determined by the symmetry of the strongly correlated _atomic_ states, via
the non-trivial form factor of the triplet Kondo coupling. This follows
earlier proposals of two-channel Kondo models as a pairing mechanism for
d-wave singlet pairing in the Ce- and Np-based heavy fermion superconductors
[8], supported by valence fluctuations [47] into a spin-singlet f2 ground
state.

The conceptual appeal of a Hund’s-Kondo pairing mechanism lies in the fact
that it captures in a unified framework both the large pairing energy from
strongly-coupled pre-formed triplet pairs, driven by Hund’s coupling and the
substantial phase stiffness that results from Kondo hybridization with a
dispersive weakly-interacting conduction band. It raises the prospect of a
strong-coupling framework that unifies the diverse set of heavy fermion
superconductors in Table 1, bound by the common structural motif of local
moments with a sublattice degree of freedom, that allows local triplet pairs
to overlap with odd-parity triplet pairs on the Fermi surface.

_Acknowledgments:_ This work was supported by the Department of Energy, Basic
Energy Sciences grant DE-FG02-99ER45790 (TH) and the U.S. National Science
Foundation grant DMR-1830707 (PC). We gratefully acknowledge discussions with
Steven Kivelson, Vidya Madhavan, Daniel Agterberg, Silke Buhler-Paschen,
Priscila Rosa, Yashar Komijani and Victor Drouin-Touchette.

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##  Appendix A Derivation of singlet-triplet Kondo model from mixed-valent
description

Here we derive the two channel Kondo model discussed in the main text starting
from a minimal periodic Anderson model that captures the charge fluctuations
between three low-lying f-states mediated by the conduction electrons. The
single-particle hybridization between the conduction and f electrons at a
single impurity site is projected into the truncated Hilbert space of the
f-electron vacuum, a single f1 Kramers doublet and a Hund’s coupled high-spin
f2 singlet as shown in Figure 3. Given that
$|f^{2}\rangle=f_{\Gamma_{1}s}^{\dagger}\left(\sigma_{z}i\sigma_{y}\right)_{ss^{\prime}}^{\dagger}f_{\Gamma_{2}s^{\prime}}^{\dagger}|0\rangle$,
the matrix element $\langle
f^{2}|f_{\Gamma_{2}\sigma}^{\dagger}|f_{\Gamma_{1}\sigma^{\prime}}^{1}\rangle\sim\left(\sigma_{z}i\sigma_{y}\right)_{\sigma^{\prime}\sigma}$.
Thus the projected hybridization Hamiltonian is given by

$\displaystyle H_{cf}$ $\displaystyle=\tilde{V_{1}}\sum_{{\bf r}{\bf
k}}\left(X_{\sigma 0}^{\Gamma_{1}}({\bf r})\psi_{1{\bf k}\sigma}e^{i{\bf
k}\cdot{\bf r}}+\mathrm{h.c.}\right)$ $\displaystyle+\tilde{V_{2}}\sum_{{\bf
r}{\bf k}}\left(X_{2\sigma^{\prime}}^{\Gamma_{2}}({\bf
r})\left(\sigma_{z}i\sigma_{y}\right)_{\sigma^{\prime}\sigma}\psi_{2{\bf
k}\sigma}e^{i{\bf k}\cdot{\bf{\bf r}}}+\mathrm{h.c.}\right)$ (18)

where $X_{\sigma 0}^{\Gamma_{1}}({\bf
r})=|f_{\Gamma_{1}\sigma}^{1}\rangle_{{\bf r}}\langle 0|$ and
$X_{2\sigma^{\prime}}^{\Gamma_{2}}({\bf r})=|f^{2}\rangle_{{\bf r}}\langle
f_{\Gamma_{1}\sigma^{\prime}}^{1}|_{{\bf r}}$ are the Hubbard operators at
site ${\bf r}$, $\psi_{1(2){\bf k}}=\sum_{\bf r}\psi_{{\bf
r}}^{\Gamma_{1(2)}}e^{i{\bf k}\cdot{\bf r}}/\sqrt{N_{s}}$, $N_{s}$ is the
number of sites and $\psi_{{\bf r}}^{\Gamma_{i}}$ is the conduction electron
annihilation operator at site ${\bf r}$ weighted by a local form-factor that
transforms according to the irreducible representation $\Gamma_{i}$ of the
site symmetry group. The important point for our pairing mechanism is that the
transitions between $f^{0}\leftrightarrow f^{1}$ and $f^{1}\leftrightarrow
f^{2}$ generically have different symmetries and therefore hybridize with
conduction electrons in different representations, leading to a two-channel
periodic Anderson model.

We now consider this very general model on a lattice with a two-sublattice
basis. For concreteness, we assume that the conduction band Wannier functions
transform according to $\Gamma_{1}$ so that $\psi_{1{\bf k}\alpha}=c_{{\bf
k}\alpha}$, where $\alpha$ is the sublattice index. Furthermore, we assume
that the product of irreps $\Gamma_{2}\times\Gamma_{1}$ is isomorphic to
$\sigma_{x}$ so that a symmetry-allowed form-factor in the second channel
$\psi_{2{\bf k}\alpha}=\Phi_{2{\bf k}}c_{{\bf k}\alpha}$ is $\Phi_{2{\bf
k}}=i\sigma_{x}(\sqrt{1-\zeta^{2}}+i\zeta\alpha_{z}p_{{\bf k}}^{z})$. Here
$\alpha_{i}(\sigma_{i})$ are Pauli matrices in sublattice (spin) space and
$p_{\bf k}^{z}$ is the nearest neighbor crystal harmonic that transforms like
$z$. Note that the inversion interchanges the sublattices so that
$\alpha_{z}p_{\bf k}^{z}$ is even parity. The construction is motivated by the
structure of UTe2 (see Figure 1) but is readily generalized. This leads to the
following Anderson model,

$\displaystyle H_{cf}$ $\displaystyle=\tilde{V_{1}}\sum_{j{\bf
k}\alpha}\left(X_{10j\alpha}^{\Gamma_{1}}c_{{\bf k}\alpha}e^{i{\bf k}\cdot{\bf
r}_{j\alpha}}+\mathrm{h.c.}\right)$
$\displaystyle+\tilde{V_{2}}\sum_{jk\alpha}\left(X_{21j\alpha}^{\Gamma_{2}}\sigma_{z}i\sigma_{y}\Phi_{2{\bf
k}}c_{k\alpha}e^{ik\cdot r_{j\alpha}}+\mathrm{h.c.}\right)$ (19)

where $j$ is a unit cell index. Note that the triplet hybridization in the
second term is symmetry-equivalent to the antisymmetric spin-orbit coupling
(ASOC) discussed in the context of non-centrosymmetric superconductors [48]
and more recently, locally-noncentrosymmetric superconductors like UTe2 and
CeRh2As2[49, 21].

Assuming that the atomic energy levels are captured by the Hamiltonian $H_{0}$
with the energy levels shown in Figure 3, the charge fluctuations can now be
integrated out by a Schrieffer-Wolff transformation which decouples the low-
energy sector $e^{iS}(H_{0}+H_{cf})e^{-iS}$ with

$\displaystyle S=$ $\displaystyle\sum_{j{\bf
k}\alpha}\bigg{(}-\frac{\tilde{V_{1}}}{E_{0}}X_{01j\alpha}^{\Gamma_{1}}c_{{\bf
k}\alpha}e^{i{\bf k}\cdot{\bf r}_{j\alpha}}$
$\displaystyle+\frac{\tilde{V_{2}}}{E_{2}}X_{12j\alpha}^{\Gamma_{2}}\sigma_{z}i\sigma_{y}\Phi_{2{\bf
k}}c_{{\bf k}\alpha}e^{i{\bf k}\cdot{\bf r}_{j\alpha}}+\mathrm{h.c.}\bigg{)}$
(20)

to give the two-channel Kondo model introduced in (3) with
$J_{K1}\sim\left|\tilde{V}_{1}\right|^{2}/E_{0},\>J_{K2}\sim\left|\tilde{V}_{2}\right|^{2}/E_{2},$
and the spin
$\mathbf{S}=|f^{1s}\rangle\boldsymbol{\sigma}_{ss^{\prime}}\langle
f^{1s^{\prime}}|/2$. Note that it is the charge fluctuation with the
Hund’s-coupled high-spin f2 state that results in a non-trivial coupling of
the conduction electron and local moment spin, that we dub triplet Kondo
coupling.

##  Appendix B Triplet Kondo interaction as an interband pair scattering term

Here we show explicitly that the Hund’s-driven triplet Kondo coupling (6) in
our model is an attractive pair scattering interaction for triplet pairs on
the Fermi surface, as indicated in (8). First we transform to the eigenbasis
of the conduction Hamiltonian $H_{c}$ with $\tilde{c}_{{\bf k}\eta}=U_{{\bf
k}\eta\alpha}c_{{\bf k}\alpha},\tilde{f}_{{\bf k}\eta}=U_{{\bf
k}\eta\alpha}f_{{\bf k}\alpha}$ where $\eta$ labels the two conduction bands
and $U$ is the unitary transform that diagonalizes $H_{c}$. Next we
diagonalize the singlet Kondo hybridization to get the heavy fermion band
eigenstates

$\displaystyle a_{{\bf k}\eta}=u_{{\bf k}\eta}\tilde{f}_{{\bf k}\eta}+v_{{\bf
k}\eta}\Phi_{1{\bf k}\eta\eta}\tilde{c}_{{\bf k}\eta}$ (21)

where $\Phi_{1{\bf k}\eta\eta}$ is the form-factor of the first channel
projected into the band basis. We now consider the pair operators in (6) after
inserting a resolution of the identity $\alpha_{z}^{2}=1$ to allow the
fermions pairs to acquire a sublattice-odd form factor

$\displaystyle\Lambda_{{\bf k}}=f_{-{\bf
k}\alpha}(i\sigma_{y})\sigma_{z}\alpha_{z}\Phi_{2{\bf k}\alpha\beta}c_{{\bf
k}\beta}$ (22)

To project them into the band basis, we invert the unitary transforms above to
get

$\displaystyle P_{\eta}f_{{\bf k}\alpha}P_{\eta}$
$\displaystyle=U^{\dagger}_{{\bf k}\alpha\eta}u_{{\bf k}\eta}a_{{\bf k}\eta}$
$\displaystyle P_{\eta}c_{{\bf k}\alpha}P_{\eta}$
$\displaystyle=U^{\dagger}_{{\bf k}\alpha\eta}\Phi^{\dagger}_{1{\bf
k}\eta\eta}v_{{\bf k}\eta}a_{{\bf k}\eta}$ (23)

and insert these into (22) to get

$\displaystyle P_{\eta}\Lambda_{{\bf k}}P_{\eta}$
$\displaystyle=P_{\eta}f_{-{\bf
k}\alpha}(i\sigma_{y})\sigma_{z}(\alpha_{z}\Phi_{2{\bf
k}})_{\alpha\beta}c_{{\bf k}\beta}P_{\eta}$ $\displaystyle=a_{-{\bf
k}\eta}\mathbf{d}_{{\bf k}\eta}\cdot\boldsymbol{\sigma}(i\sigma_{y})a_{{\bf
k}\eta}$ (24)

with the d-vector of the intra-band triplet pair given by

$\displaystyle\mathbf{d}_{{\bf
k}\eta}\cdot\boldsymbol{\sigma}=\sigma_{z}\left(U^{\dagger}_{-{\bf
k}\alpha\eta}(\alpha_{z}\Phi_{2{\bf k}})_{\alpha\beta}U^{\dagger}_{{\bf
k}\beta\eta}\right)\left(u_{-{\bf k}\eta}\Phi_{1{\bf
k}\eta\eta}^{\dagger}v_{{\bf k}\eta}\right)$

Being a triplet-pair, the d-vector inherits the full antisymmetry of the pair
potential $d_{{\bf k}\eta}=-d_{-{\bf k}\eta}^{T}$. Since there is no spin-
orbit coupling in either the conducting hopping or the singlet hybridization
in our model, $u_{-{\bf k}}=u_{\bf k},U_{-{\bf k}}=U_{{\bf k}}$ and the
antisymmetrized d-vector for our model is then

$\displaystyle d_{{\bf k}\eta}\cdot\boldsymbol{\sigma}-d_{-{\bf
k}\eta}^{T}\cdot\boldsymbol{\sigma}$
$\displaystyle=\sigma_{z}\left((U^{\dagger}_{{\bf
k}\alpha\eta})^{2}(2\zeta\sigma_{x}p_{\bf k}^{z})\right)\left(u_{{\bf
k}\eta}v_{{\bf k}\eta}\right)$ (25) $\displaystyle=2i\zeta\sigma_{y}p_{\bf
k}^{z}C_{{\bf k}\eta}$ (26)

where $C_{{\bf k}\eta}=\sum_{\alpha}u_{{\bf k}\eta}v_{{\bf
k}\eta}(U^{\dagger}_{{\bf k}\alpha\eta})^{2}$. Note that without the
$\alpha_{z}$ form-factor in (22), this interband pair potential would vanish
for the even-parity order parameter $\Delta_{2\alpha}=\Delta_{2}$.