# Unconventional superconductivity protected from disorder on the kagome
lattice

Sofie Castro Holbæk  Niels Bohr Institute, University of Copenhagen, DK-2200
Copenhagen, Denmark  Morten H. Christensen  Niels Bohr Institute, University
of Copenhagen, DK-2200 Copenhagen, Denmark  Andreas Kreisel  Niels Bohr
Institute, University of Copenhagen, DK-2200 Copenhagen, Denmark  Brian M.
Andersen  Niels Bohr Institute, University of Copenhagen, DK-2200 Copenhagen,
Denmark

###### Abstract

Motivated by the recent discovery of superconductivity in the kagome $A$V3Sb5
($A$: K, Rb, Cs) metals, we perform a theoretical study of the symmetry-
allowed superconducting orders on the two-dimensional kagome lattice with
focus on their response to disorder. We uncover a qualitative difference
between the robustness of intraband spin-singlet (even-parity) and spin-
triplet (odd-parity) unconventional superconductivity to atomic-scale
nonmagnetic disorder. Due to the particular sublattice character of the
electronic states on the kagome lattice, disorder in spin-singlet
superconducting phases is only weakly pair-breaking despite the fact that the
gap structure features sign changes. By contrast, spin-triplet condensates
remain fragile to disorder on the kagome lattice. We demonstrate these effects
in terms of the absence of impurity bound states and an associated weak
disorder-induced $T_{c}$-suppression for spin-singlet order. We also discuss
the consequences for quasi-particle interference and their inherent tendency
for momentum-space anisotropy due to sublattice effects on the kagome lattice.
For unconventional kagome superconductors, our results imply that any allowed
spin-singlet order, including for example $d+id$-wave superconductivity,
exhibits a disorder-response qualitatively similar to standard conventional
$s$-wave superconductors.

##  I Introduction

The irreducible representations of a given point group dictate the allowed
homogeneous superconducting order parameters of materials [1]. However,
additional complexity of e.g. sublattice degrees of freedom, multiple active
orbitals at the Fermi level, associated Hund’s interactions, or strong spin-
orbit coupling adds significant richness to the problem [2, 3, 4, 5, 6, 7, 8,
9, 10, 11]. Whatever superconducting gap structure eventually gets singled out
in a given material depends on the particular pairing mechanism operative in
that system. Determining experimentally which order parameter is present in
specific materials can be a tremendous challenge due to the low temperature
and often minute energy scales of the superconducting problem. In this
respect, disorder can play an important role because it can act as a phase-
sensitive probe. Disorder effects can be studied in terms of e.g. atomically-
resolved impurity bound states detectable by local probes as well as the
overall disorder-averaged superconducting response as seen by e.g.
thermodynamic probes or transport measurements. If nonmagnetic disorder is
able to generate in-gap bound states or if it severely affects
superconductivity, it is typically a strong indicator of an unconventional
superconducting condensate [12].

The discovery of superconductivity in vanadium-based kagome metals $A$V3Sb5
($A$: K, Rb, Cs) has reinvigorated the discussion of conventional versus
unconventional pairing in novel quantum materials [13]. The kagome lattice is
particularly intriguing since its basic electronic structure features both
flat bands, van Hove singularities, and Dirac points, as seen in Fig. 1. In
addition, the Fermi surface distribution of sublattice weights of the
eigenstates of the kagome tight-binding bands, also illustrated in Fig. 1, can
play important roles for determining e.g. the leading instabilities arising
from electronic interactions [14, 15]. This may indeed be of relevance in the
$A$V3Sb5 compounds where superconductivity appears in proximity to a charge-
density wave (CDW) phase, which has been studied intensely both theoretically
[16, 17, 18, 19, 20, 21, 22] and experimentally [23, 24, 25, 26, 27]. For the
superconducting phase, some theoretical studies have explored Cooper pairing
arising from purely electronic fluctuations [28, 14, 29, 16, 30, 19, 31, 32,
30, 33, 17, 34]. Other works have pointed to the important role of phonons for
the generation of superconductivity, either on their own or in conjunction
with electronic correlations [35, 36, 37, 38, 39, 40].

Experimentally, the $A$V3Sb5 kagome metals enter their superconducting phase
at $T_{c}\sim 1-3$K [13, 41, 42]. The critical temperature $T_{c}$, however,
may be significantly enhanced by uniaxial strain or hydrostatic pressure,
e.g., for CsV3Sb5, $T_{c}\sim 8$K at $\sim 2$ GPa [43, 44, 45, 46]. The
detailed nature and origin of electronic pairing in $A$V3Sb5 remains
controversial at present, with evidence for both standard nodeless non-sign-
changing gaps and nodal unconventional superconducting order. For example,
several STM measurements have reported ’V’-shaped STM conductance spectra [26,
47, 48], and thermal conductivity data has been interpreted in favor of nodal
superconductivity [49]. Similarly, muon spin spectroscopy experiments on
RbV3Sb5 and KV3Sb5 samples report nodal gaps at ambient pressure [50], and a
pressure-tuned transition to nodeless order with additional evidence for
spontaneous time-reversal symmetry breaking (TRSB) setting in at $T_{c}$ for
high pressures $\sim 2$ GPa [51, 50]. On the other hand, a Knight shift
suppression and the existence of a Hebel-Slichter coherence peak in the spin-
lattice relaxation observed by nuclear magnetic resonance measurements point
to $s$-wave spin-singlet superconductivity [52]. Penetration depth
measurements and specific heat data on CsV3Sb5 have also been analysed in
terms of an anisotropic non-sign-changing gap with a finite small minimum gap
[41, 53, 54, 55, 45]. Recent laser ARPES measurements find isotropic
(momentum-independent) spectroscopic gaps [56]. Nodeless superconductivity
also appears consistent with multiband features and a ’U’-shaped conductance
behavior near zero bias as seen by some scanning tunneling microscopy (STM)
measurements [48]. In addition, the lack of sign-changes in the gap function
appears consistent with the absence of in-gap states near nonmagnetic
impurities [48] and a weak dependence of $T_{c}$ on residual resistivity
ratios (sample quality) [57].

A main point of the current work is that one needs to be careful applying
standard arguments about disorder and pair-breaking when the impurities are
located on sites that are not centers of the point group operations. This is
indeed the case for the kagome lattice, where the site-symmetry of the
individual lattice sites does not possess the full point group symmetry. In a
nutshell, even though a gap function averages to zero over the Fermi surface,
$\sum_{\bf k}\Delta({\bf{k}})=0$, atomic-scale disorder necessarily selects a
specific sublattice site and thereby only scatters to certain allowed regions
of the Fermi surface. For spin-singlet order on the kagome lattice this
mechanism of sublattice-selective scattering in fact renders nonmagnetic
disorder nearly blind to sign changes in the gap structure. In this sense, the
kagome lattice protects spin-singlet superconductivity from nonmagnetic
disorder.

Figure 1: (a) Illustration of the kagome lattice with lattice vectors
$\mathbf{t}_{1}$ and $\mathbf{t}_{2}$, and sublattice vectors
$\mathbf{a}_{1}$, $\mathbf{a}_{2}$ and $\mathbf{a}_{3}$. The grey area denotes
the unit cell with the three different sublattice sites denoted A (blue), B
(orange) and C (green). (b) Tight-binding energy bands plotted along the high-
symmetry path shown in the inset. Here, $n$ is a band label, introduced for
later convenience. (c) Distribution of the sublattice weights across the
Brillouin zone for the second band ($n=2$). The Fermi surface for $\mu=0$,
corresponding to the upper van Hove singularity in (b), is shown in black.

Here, we perform a theoretical study of the basic properties of the
superconducting phases on the kagome lattice and their response to pointlike
defects, and determine the robustness of different symmetry-allowed
superconducting order parameters to disorder. As mentioned above, we
demonstrate that the sublattice weight of the band eigenstates play a crucial
role for determining the response of superconductivity to disorder.
Specifically, while intraband spin-triplet pairing remains fragile to
disorder, spin-singlet superconductivity is robust to atomic-scale disorder
due to the sublattice dependence of the eigenstates. We stress that this is a
generic property of the kagome lattice, and not restricted to a specific
electron concentration and associated Fermi surface. More generally, we expect
that other lattices with substantial sublattice weight variation present at
the Fermi surface will exhibit similar behavior. We analyze the impact of
atomic scale disorder in terms of allowed impurity bound states and the
resulting local density of states, and discuss implications for quasi-particle
interference. Additionally, we discuss the robustness of $T_{c}$ within
Abrikosov-Gor’kov theory [58] applied to both spin-triplet and spin-singlet
superconducting states, and compare to similar calculations obtained on the
square lattice.

The paper is organized as follows: In Sec. II we review the electronic
structure of the kagome lattice and in Sec. III we construct the symmetry-
allowed superconducting orders assuming a single $s$-orbital degree of freedom
on each sublattice site. From these candidates we next explore their response
and robustness to atomic scale disorder. Section IV.1 focuses on the existence
of impurity bound states and the response to local probes, whereas the
disorder-averaged impact on $T_{c}$ within an Abrikosov-Gor’kov approach is
described in Sec. IV.2. Finally, Sec. V contains a discussion of the relevance
of the current findings to the $A$V3Sb5 materials, and our conclusions.

##  II Phenomenology of the kagome lattice

The kagome lattice is a network of corner-sharing triangles with three sites
in the unit cell, as shown in Fig. 1. The normal state Hamiltonian reads

$\mathcal{H}_{0}=\sum_{\begin{subarray}{c}\mathbf{k}\sigma\\\
\alpha\beta\end{subarray}}\left(h_{\alpha\beta}(\mathbf{k})-\mu\delta_{\alpha\beta}\right)c^{\dagger}_{\mathbf{k}\sigma\alpha}c^{\phantom{\dagger}}_{\mathbf{k}\sigma\beta}\,,$
(1)

where $\sigma$ denotes spin and $\alpha,\beta=A,B,C$ are sublattice indices.
The kinetic part is given by

$\displaystyle h(\mathbf{k})=-2t\begin{pmatrix}0&\cos k_{3}&\cos k_{1}\\\ \cos
k_{3}&0&\cos k_{2}\\\ \cos k_{1}&\cos k_{2}&0\end{pmatrix}\,,$ (2)

where we have restricted to nearest-neighbor (NN) hoppings. In what follows we
will set the hopping amplitude $t=1$. Here,
$k_{i}=\mathbf{k}\cdot\mathbf{a}_{i}$ where

$\mathbf{a}_{1}=\frac{1}{2}\begin{pmatrix}1&0\end{pmatrix}\,,\quad\mathbf{a}_{2}=\frac{1}{2}\begin{pmatrix}\frac{1}{2}&\frac{\sqrt{3}}{2}\end{pmatrix}\,,\quad\mathbf{a}_{3}=\frac{1}{2}\begin{pmatrix}-\frac{1}{2}&\frac{\sqrt{3}}{2}\end{pmatrix}\,,$
(3)

are the sublattice vectors, where we have put the distance between neighboring
unit cells to one.

The Hamiltonian is diagonalized by a unitary transformation,
$u_{n\alpha}^{\ast}(\mathbf{k})h_{\alpha\beta}(\mathbf{k})u_{\beta
m}(\mathbf{k})=\xi_{n}(\mathbf{k})\delta_{nm}$ yielding the band energies
$\xi_{n}(\mathbf{k})$ and the eigenstates $u_{\alpha n}(\mathbf{k})$ of band
$n$. The resulting band structure is shown in Fig. 1 and features a Dirac
point at the K point and two van Hove singularities at the M point. In
addition, for the NN tight-binding model there is a flat band which acquires a
weak dispersion upon including further neighbor hoppings. Furthermore, the
kagome lattice is endowed with a property which has become known as sublattice
interference [14] for which specific hopping trajectories interfere
destructively resulting in electronic wavefunctions that localize on specific
sites inside the unit cell. Specifically, electronic states at the upper van
Hove singularity, at $\mu=0$, are localized on only one of the three sites in
the unit cell, as shown in Fig. 1. In contrast, states at the lower van Hove
singularity, at $\mu=-2$, localize on two of the three sublattice sites.

##  III Superconducting pairing states

Superconductivity is included at the mean-field level through the Bogoliubov-
de Gennes (BdG) framework. In Nambu basis, the Hamiltonian reads

$\mathcal{H}_{\rm
BdG}=\sum_{\mathbf{k}}\Psi^{\dagger}_{\mathbf{k}}\widehat{H}_{\rm
BdG}(\mathbf{k})\Psi_{\mathbf{k}}\,,$ (4)

where

$\widehat{H}_{\rm
BdG}(\mathbf{k})=\begin{pmatrix}h(\mathbf{k})-\mu\mathds{1}&-\Delta(\mathbf{k})\\\
-\Delta(\mathbf{k})^{\dagger}&-h(-\mathbf{k})^{T}+\mu\mathds{1}\end{pmatrix}\,,$
(5)

and
$\Psi_{\mathbf{k}}^{\dagger}=\begin{pmatrix}c^{\dagger}_{\mathbf{k}\uparrow}&c^{\phantom{\dagger}}_{-\mathbf{k}\downarrow}\end{pmatrix}$
with
$c^{\dagger}_{\mathbf{k}\sigma}=\begin{pmatrix}c^{\dagger}_{\mathbf{k}\sigma
A}&c^{\dagger}_{\mathbf{k}\sigma B}&c^{\dagger}_{\mathbf{k}\sigma
C}\end{pmatrix}$ and the hat denotes $6\times 6$ matrices in Nambu and
sublattice space. In this section we discuss which form the $3\times 3$
matrices $\Delta(\mathbf{k})$ can take on the kagome lattice. Readers
interested mainly in the effects of disorder can skip directly to Sec. IV. In
what follows, we will not be concerned with the microscopic origin of
superconductivity and instead focus on which states are allowed to exist by
symmetry. A momentum independent microscopic pairing interaction induces only
on-site terms while any momentum dependence of the pairing interaction
generically yields both on-site terms and further neighbor couplings. As shown
in Fig. 1, the two-dimensional (2D) kagome lattice has three distinct sites in
the unit cell and the point group is $D_{6}$. Barring any accidental near-
degeneracies, the superconducting order parameter will therefore transform as
an irreducible representation (irrep) of $D_{6}$.

Irrep | Real space | Momentum space
---|---|---
Singlet | Triplet | Singlet | Triplet
On-site |  |  |  |
$A_{1}$ |  |  |  |
$E_{2}^{(1)}$ |  |  |  |
$E_{2}^{(2)}$ |  |  |  |
Nearest-neighbor |  |  |  |
$A_{1}$ |  |  |  |
$A_{2}$ |  |  |  | Interband
$B_{1}$ |  |  |  |
$B_{2}$ |  |  | Interband |
$E_{1}^{(1)}$ |  |  | Interband |
$E_{1}^{(2)}$ |  |  | Interband |
$E_{2}^{(1)}$ |  |  |  | Interband
$E_{2}^{(2)}$ |  |  |  | Interband

Table 1: Summary of the different pairing states on the kagome lattice. The
first column gives the irrep, while the second and third columns provide
schematic illustrations of the form of the pairing state in real space. The
fourth and fifth columns display the projection of the form factors on the
$n=2$ band, as obtained from Eq. (16). The irreps on the upper three rows
originate from on-site terms while the rest are from nearest-neighbor terms.
The color and its intensity denotes the relative sign and strength of the
pairing condensate in both the real and momentum space cases. In the triplet
case, the sign is denoted by the direction of the arrows, to highlight their
antisymmetry under exchange of the real-space indices.

There are four one-dimensional (1D) irreps, $A_{1}$, $A_{2}$, $B_{1}$, and
$B_{2}$, and two 2D ones, $E_{1}$ and $E_{2}$. In what follows, we will
decompose the possible pairing states in terms of these irreps. For
simplicity, we will confine our attention to pairing states for which the two
electrons of the Cooper pair are either on the same site or on neighboring
sites. For on-site (OS) Cooper pairs, the pairing state can be decomposed as

$f^{S}_{\rm OS}=A_{1}\oplus E_{2}\,.$ (6)

The real- and momentum-space structures of these states are shown in Table 1.
Accordingly, the two matrices corresponding to the two on-site components of
$E_{2}$ are

$\displaystyle f^{S}_{{\rm OS},E_{2}^{(1)}}$
$\displaystyle=\frac{1}{\sqrt{6}}\begin{pmatrix}+1&0&0\\\ 0&-2&0\\\
0&0&+1\end{pmatrix}\,,$ (7) $\displaystyle f^{S}_{{\rm OS},E_{2}^{(2)}}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}+1&0&0\\\ 0&0&0\\\
0&0&-1\end{pmatrix}\,.$ (8)

Since there is only a single orbital per site in our model, on-site states are
restricted to being of spin-singlet character. This is no longer the case for
pairing states involving NN, and we find both singlet and triplet candidates:

$\displaystyle f^{S}_{\rm NN}$ $\displaystyle=A_{1}\oplus B_{2}\oplus
E_{1}\oplus E_{2},$ (9) $\displaystyle f^{T}_{\rm NN}$
$\displaystyle=A_{2}\oplus B_{1}\oplus E_{1}\oplus E_{2}\,,$ (10)

which are illustrated in Table 1. For example, the two components of the
singlet NN $E_{2}$ form factor written in momentum space are

$\displaystyle f^{S}_{{\rm NN},E_{2}^{(1)}}$
$\displaystyle=\frac{1}{\sqrt{12}}\begin{pmatrix}0&-\cos k_{3}&2\cos k_{1}\\\
-\cos k_{3}&0&-\cos k_{2}\\\ 2\cos k_{1}&-\cos k_{2}&0\end{pmatrix}\,,$ (11)
$\displaystyle f^{S}_{{\rm NN},E_{2}^{(2)}}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&\cos k_{3}&0\\\ \cos k_{3}&0&-\cos
k_{2}\\\ 0&-\cos k_{2}&0\end{pmatrix}\,.$ (12)

We illustrate the triplet order parameters using arrows to highlight the fact
that these, by their nature, are antisymmetric under exchange of the real-
space indices. We obtain the triplet $E_{1}$ components

$\displaystyle f^{T}_{{\rm NN},E_{1}^{(1)}}$
$\displaystyle=\frac{i}{\sqrt{12}}\begin{pmatrix}0&\sin k_{3}&-2\sin k_{1}\\\
\sin k_{3}&0&-\sin k_{2}\\\ -2\sin k_{1}&-\sin k_{2}&0\end{pmatrix}\,,$ (13)
$\displaystyle f^{T}_{{\rm NN},E_{1}^{(2)}}$
$\displaystyle=\frac{i}{2}\begin{pmatrix}0&\sin k_{3}&0\\\ \sin k_{3}&0&\sin
k_{2}\\\ 0&\sin k_{2}&0\end{pmatrix}\,.$ (14)

The form factors for all the cases shown in Table 1 are provided in Appendix
A. We note that the appearance of the $B_{2}$ and $E_{1}$ singlet states and
the $A_{2}$ and $E_{2}$ triplet states is facilitated only by the internal
sublattice structure of the kagome lattice. Consequently, these lead to
_inter_ -band pairing in momentum space [59]. For the filling factors we
consider, none of these lead to a gap opening at the Fermi level, and the
electronic structure of the Bogoliubov quasiparticles in this energy range is
unchanged from the normal state. Consequently, we do not discuss these states
further in what follows. The $B_{2}$ triplet and $A_{2}$ singlet gap
structures familiar from previous studies of superconductivity on the kagome
lattice [30, 32] are obtained at second- and third-nearest neighbors,
respectively.

The gap structure is generally a sum of multiple terms. For a generic irrep,
$\Gamma$, the singlet or triplet order parameter is

$\Delta_{\Gamma}=\Delta_{0}f_{{\rm OS},\Gamma}+\Delta_{1}f_{{\rm
NN},\Gamma}+\cdots\,,$ (15)

where $\Delta_{i}$ denote constants, possibly zero, which depend on the
specific irrep and pairing interaction considered and $\cdots$ denotes terms
beyond NN which we do not consider in this work. Here, we choose
$\Delta_{0}=0.2$ and $\Delta_{1}=\Delta_{0}/2$ for the $A_{1}$ and $E_{2}$
cases and, in the remaining cases, $\Delta_{0}=0$, $\Delta_{1}=0.2$. These
choices imply that the superconducting coherence peaks are at comparable
energies for the different cases. For the 2D irreps, $E_{1}$ and $E_{2}$, the
system will choose a linear combination of the two components,
$aE_{1,2}^{(1)}+bE_{1,2}^{(2)}$, where $a$ and $b$ are complex numbers. Below,
we will focus on the cases where $a=1$, $b=i$, which breaks time-reversal
symmetry, and $a=1$, $b=0$ (or $a=0$, $b=1$) which breaks lattice rotational
symmetry. $\Delta_{\Gamma}$ of Eq. (15) corresponds to the $3\times 3$
matrices appearing in the BdG Hamiltonian, Eq. (5). These can be transformed
to band space through a unitary transformation

$\Delta_{nm}(\mathbf{k})=u_{n\alpha}^{\ast}(\mathbf{k})\Delta_{\alpha\beta}(\mathbf{k})u_{\beta
m}^{\ast}(-\mathbf{k})\,,$ (16)

where $u_{n\alpha}(\mathbf{k})$ are the eigenstates of band $n$. We note that
$\Delta_{nm}(\mathbf{k})$ is not necessarily a diagonal matrix; the off-
diagonal components correspond to interband pairing. The results for the
middle band ($n=m=2$) are shown in the fourth and fifth column of Table 1. We
note that the projection onto the $n=m=1$ band leads to similar results, as
expected from the symmetry of the lattice. The ‘Interband’ cases in Table 1
denote the ones where the order parameter consists of electrons from two
different bands, i.e. where the off-diagonal elements of
$\Delta_{nm}(\mathbf{k})$ are finite while the diagonal components vanish.
These states do not generally lead to a gap opening at the Fermi level as
mentioned above.

Figure 2: DOS near the Fermi energy for systems with either singlet (left
column) or triplet (right column) superconducting order. The singlet cases are
plotted for $\mu=0$ while the triplet cases are plotted for $\mu=0.08$. (a)
Singlet $A_{1}$, resulting in the usual fully gapped $s$-wave spectrum. (b)
Singlet $E_{2}$ which is 2D and corresponds to $d$-wave at lowest order in
momentum. In black we illustrate the $d+id$ TRSB combination. This combination
results in a fully gapped spectrum. In orange and blue we show the results for
the two individual components, $d_{x^{2}-y^{2}}$ (orange) and $d_{xy}$ (blue).
Both exhibit nodes on the Fermi surface and lead to V-shaped gaps. They also
break rotational symmetry and the LDOS differs between the different
sublattice sites. Here we show the case corresponding to the $A$ sublattice.
(c) Triplet $B_{1}$, which is commonly denoted $f$-wave. This has nodes on the
Fermi surface (see Table 1) resulting in a V-shaped gap. (d) Triplet $E_{1}$
which is also 2D and corresponds to $p$-wave at lowest order in momentum. We
show the TRSB $p+ip$ variant in black, while the $p_{x}$ and $p_{y}$
components are shown in orange and blue, respectively. Both $p_{x}$ and
$p_{y}$ break lattice rotational symmetry. Here, we stress that we only plot
the DOS on the $A$ sublattice which is why the $p_{x}$ component has a
U-shaped gap; the $A$ sublattice has vanishing spectral weight in the regions
of the BZ where $p_{x}$ has nodes, compare Fig. 1(c) and Table 1.

In Fig. 2 we show the density of states (DOS) for different choices of the
superconducting order parameter corresponding to the singlets $A_{1}$
($s$-wave) and $E_{2}$ ($d+id$) and the triplets $E_{1}$ ($p+ip$) and $B_{1}$
($f$). The $s$-wave, $d+id$ and $p+ip$ cases all exhibit a full gap whereas
the $f$-wave state is nodal. We denote the location of the superconducting
coherence peak by $\Delta_{c}$. For $d+id$ and $p+ip$ pairing, the full gap is
facilitated by the breaking of time-reversal symmetry, which allows the system
to avoid possible gap nodes on the Fermi surface. For the $d+id$ and the
$p+ip$ cases we also include the DOS for the individual components. In this
case, rotational symmetry is broken by the superconducting order, and the DOS
on the $A$, $B$, and $C$ sublattices can be different. The cases shown in Fig.
2 correspond to the local density of states (LDOS) at an $A$ sublattice site.

##  IV Effects of disorder on kagome superconductivity

In this section we turn to a discussion of the effects of disorder on the
symmetry-distinct pairing states introduced above. The discussion is divided
into two parts. First, we address the response of the system to a single
impurity and the possibility of localized in-gap bound states and second, we
discuss the disorder-averaged suppression of $T_{c}$ versus the impurity
scattering rate.

###  IV.1 Impurity bound states and sublattice interference

In this section, we discuss the effects of a single impurity on
superconductivity on the kagome lattice. We focus on isolated lattice site-
centered defects, and refer to the discussion section for a more general
discussion of the effects of other kinds of disorder. In the following, the
existence or absence of bound states and the resulting LDOS is studied within
the $T$-matrix framework. In this methodology, bound state solutions are
analyzed in terms of the existence of poles in the $T$-matrix. These poles can
occur at real or complex energies $\omega$ inside the superconducting gap,
referring to bound or resonant impurity states, respectively. Generally, sign-
changing gap structures support in-gap bound or resonant states from
nonmagnetic disorder as opposed to standard $s$-wave non-sign-changing
superconductivity which forbids such solutions [60, 12]

In order to investigate the response of the pairing candidates to single
impurities on the kagome lattice, we introduce either a nonmagnetic impurity
or a (classical) magnetic impurity. We arbitrarily locate the impurity at a
specific atomic site of the lattice. For concreteness, in the following we
choose sublattice site $A$ located at $\mathbf{r}=\mathbf{0}$ to host the
impurity. The impurity Hamiltonian matrices in Nambu formalism are then given
by

$\displaystyle\widehat{H}_{\text{imp}}=V\tau^{z}\otimes\begin{pmatrix}1&0&0\\\
0&0&0\\\ 0&0&0\end{pmatrix},$ (17)

for the potential scatterer, and

$\displaystyle\widehat{H}_{\text{imp}}=S_{z}\tau^{0}\otimes\begin{pmatrix}1&0&0\\\
0&0&0\\\ 0&0&0\end{pmatrix},$ (18)

for the case of a magnetic impurity. Here, $\tau^{z}$ and $\tau^{0}$ denote
Pauli matrices in Nambu space. The full Green function described by the
Hamiltonian including the impurity potential can be obtained within the
$T$-matrix approach:

$\displaystyle\widehat{G}(\mathbf{r},\mathbf{r}^{\prime},\omega)=\widehat{G}^{(0)}(\mathbf{r}-\mathbf{r}^{\prime},\omega)+\widehat{G}^{(0)}(\mathbf{r},\omega)\widehat{T}(\omega)\widehat{G}^{(0)}(-\mathbf{r}^{\prime},\omega),$
(19)

with the $T$-matrix defined by
$\widehat{T}(\omega)\equiv\widehat{D}^{-1}(\omega)\widehat{H}_{\rm imp}$ where

$\displaystyle\widehat{D}(\omega)\equiv\left[\mathds{1}-\widehat{H}_{\rm
imp}\widehat{G}^{(0)}(\mathbf{0},\omega)\right].$ (20)

Here, the free real-space Green function is calculated from the retarded Green
function

$\widehat{G}^{(0)}(\mathbf{k},\omega)=\left[(\omega+i\eta)\mathds{1}-\widehat{H}_{\rm
BdG}(\mathbf{k})\right]^{-1}\,,$ (21)

by the Fourier transform
$\widehat{G}^{(0)}(\mathbf{r},\omega)=\frac{1}{N}\sum_{\mathbf{k}}\widehat{G}^{(0)}(\mathbf{k},\omega)e^{i{\mathbf{k}}\cdot{\mathbf{r}}}$
where $N$ is number of points in momentum space. The parameter $\eta$ in Eq.
(21) is an infinitesimal positive smearing resulting from the analytic
continuation to the real frequency axis. The spin-summed electronic LDOS
$\rho_{\alpha}({\mathbf{r}},\omega)$ at sublattice position $\alpha$ is
obtained by

$\displaystyle\rho_{\alpha}(\mathbf{r},\omega)=-\frac{1}{\pi}\text{Im}\left[G_{\alpha\alpha}(\mathbf{r},\mathbf{r},\omega)+G_{\bar{\alpha}\bar{\alpha}}(\mathbf{r},\mathbf{r},-\omega)\right],$
(22)

where $\bar{\alpha}$ refers to the same sublattice site as $\alpha$ but in the
complementary Nambu block. Throughout, we will use system sizes of at least
$N_{k_{x}}\times N_{k_{y}}=1500\times 1500$ to calculate the LDOS and values
of the smearing $\eta\leq 0.0025$. As mentioned above, the existence and
location of possible impurity bound states is typically analyzed by studying
the poles of the $T$-matrix. In the current situation, exemplified with a
single impurity on a sublattice $A$ site, we search for solutions of
$\mbox{det}[\widehat{D}(\omega)]=0$, which, for a nonmagnetic scattering
potential, yields

$\displaystyle
V^{2}\left[G^{(0)}_{AA}(\omega)G^{(0)}_{\bar{A}\bar{A}}(\omega)-G^{(0)}_{A\bar{A}}(\omega)G^{(0)}_{\bar{A}A}(\omega)\right]+V\left[G^{(0)}_{AA}(\omega)-G^{(0)}_{\bar{A}\bar{A}}(\omega)\right]-1=0\,,$
(23)

and

$\displaystyle
S_{z}^{2}\left[G^{(0)}_{AA}(\omega)G^{(0)}_{\bar{A}\bar{A}}(\omega)-G^{(0)}_{A\bar{A}}(\omega)G^{(0)}_{\bar{A}A}(\omega)\right]-S_{z}\left[G^{(0)}_{AA}(\omega)+G^{(0)}_{\bar{A}\bar{A}}(\omega)\right]+1=0\,,$
(24)

for a magnetic scattering potential. Here, the notation refers to momentum-
summed Green functions, i.e. $G^{(0)}_{\alpha\beta}(\omega)\equiv
G^{(0)}_{\alpha\beta}(\mathbf{0},\omega)=\frac{1}{N}\sum_{\mathbf{k}}G^{(0)}_{\alpha\beta}({\mathbf{k}},\omega)$.
Further investigation of these equations requires determination of the
functional form of these local free Green function entries.

As a short intermezzo, we briefly summarize the physics of impurity bound
states in standard one-band superconductors. In that case, Eqs. (23) and (24)
are unchanged except for the absence of the sublattice degree of freedom. For
a conventional $s$-wave superconductor with an isotropic full gap, Eqs. (23)
and (24) allow for bound state solutions only for the case of magnetic
impurities as discovered by Yu, Shiba, and Rusinov [61, 62, 63]. In contrast,
no in-gap solutions exist for nonmagnetic potentials in agreement with
Anderson’s theorem [60]. This can be easily verified by utilizing analytical
expressions for the momentum-summed diagonal component

$\displaystyle
G^{(0)}_{11}(\omega)=-\frac{\pi\rho(0)\omega}{\sqrt{|\Delta|^{2}-\omega^{2}}}\,,$
(25)

and the off-diagonal (anomalous) component

$\displaystyle
G^{(0)}_{1\bar{1}}(\omega)=\frac{\pi\rho(0)\Delta}{\sqrt{|\Delta|^{2}-\omega^{2}}},$
(26)

of the Green function at energies inside the gap, $\omega<|\Delta|$. Utilizing
Eqs. (23) and (24) for determining the existence of bound states, one sees
that the properties of the anomalous Green function are crucial. Typically,
for sign-preserving pairing such as standard $s$-wave superconductivity
$\sum_{\mathbf{k}}\Delta({\mathbf{k}})\neq 0$, in agreement with Eq. (26),
disallowing bound state solutions. On the other hand, for unconventional
pairing where $\sum_{\mathbf{k}}\Delta({\mathbf{k}})=0$, the anomalous
momentum-summed Nambu components vanish, and bound state solutions may exist
for both magnetic and nonmagnetic impurities. In the case of nodal gaps, e.g.
$d_{x^{2}-y^{2}}$-wave pairing as in the cuprates, these states acquire a
lifetime and the energies of the associated resonant or virtual states move
off the real axis [64, 12].

Figure 3: Energy dependence of the real (a) and imaginary (b) parts of
$\mbox{det}[\widehat{D}(\omega)]$ as given by Eq. (23) for a range of
nonmagnetic scattering potentials from $V=-10$ to $V=10$, denoted by the
colors, for a singlet $d_{x^{2}-y^{2}}+id_{xy}$ gap at $\mu=0.0$ with the DOS
in the clean case shown by the red line. The absence of impurity bound states
inside the gapped region is evident from panel (a). We note that the finite
$\text{Im}(\mbox{det}[\widehat{D}(\omega)])$ inside the hard gap in (b) is due
to the finite smearing $\eta=0.002$. Vanishing values of $\eta$ lead to a
vanishing imaginary part inside the hard gap. Figure 4:  Momentum dependence
of the real (a)–(d) and imaginary (e)–(h) parts of the anomalous free Green
function components at $\omega=0$ for $d_{x^{2}-y^{2}}+id_{xy}$
superconductivity at $\mu=0.0$. Panels (a) and (e) display the Green function
entries for the second band ($n=2$), reflecting the $d_{x^{2}-y^{2}}+id_{xy}$
gap structure at the Fermi surface. The lower six panels refer to the
anomalous entries in sublattice space. Importantly, the individual components
of the Green function in sublattice space, when integrated over the Brillouin
zone, do not vanish, implying that Eq. (23) may have no real solutions.

Turning back to the kagome lattice and the relevant pairing states discussed
in the previous section, we find an interesting twist not present in the
standard cases summarized above: despite the fact that
$\sum_{\mathbf{k}}\Delta_{nm}({\mathbf{k}})=0$ for the spin-singlet order
parameters, there are no low-energy in-gap bound state solutions generated by
nonmagnetic impurities, i.e. no bona fide bound state solutions to Eq. (23).
By contrast, magnetic impurities allow bound state solutions for all
superconducting states on the kagome lattice, and will not be further
discussed here.

For the kagome lattice it is not straightforward to obtain analytical
expressions for $\sum_{\mathbf{k}}G^{(0)}_{\alpha\beta}(\mathbf{k},\omega)$
and directly analyze Eq. (23). However, the absence of low-energy in-gap bound
states for spin-singlet order can be illustrated in several ways. In the
following, we focus initially on the fully-gapped $d_{x^{2}-y^{2}}+id_{xy}$
spin-singlet phase with the homogeneous DOS shown in Fig. 2(b). Figure 3 shows
the real (a) and imaginary (b) parts of the determinant of the $T$-matrix
denominator $\widehat{D}(\omega)$ for a range of different scattering
potentials. As seen, inside the fully-gapped region the left-hand side of Eq.
(23) becomes real [Fig. 3(b)], up to a small imaginary part proportional to
the smearing $\eta$, and exhibits no bound state solutions [Fig. 3(a)]: the
low-energy gapped region is protected from any solutions.

What is the origin of the absence of bound states? Figures 4(a) and (e) show
the gap structure of $d_{x^{2}-y^{2}}+id_{xy}$ on the Fermi surface. Clearly
both the real and imaginary parts sum to zero. However, for the scattering
problem with pointlike defects located at specific sublattice sites, the
relevant quantities are the anomalous Green function components in sublattice
space. Figures 4(b)–4(d) and 4(f)–4(h) display the entries of the off-diagonal
sublattice-resolved components of
$G_{\alpha\bar{\alpha}}^{(0)}(\mathbf{k},\omega)$ at $\omega=0$ in momentum
space. As seen, the sublattice weight of the Fermi surface is directly
imprinted on the different Green function sublattice components. This implies
that the momentum-summed anomalous components,
$\sum_{\mathbf{k}}G_{\alpha\bar{\alpha}}^{(0)}(\mathbf{k},\omega)$, are finite
despite the fact that the momentum structure of $d_{x^{2}-y^{2}}+id_{xy}$
itself averages to zero over the Fermi surface. In essence, the sublattice
weights in conjunction with the even-parity nature of the spin-singlet pairing
structure restore the “standard $s$-wave scenario” for atomic-scale impurities
on the kagome lattice. This is the main finding of the present work.

Figure 5:  LDOS along a cut through the impurity site, shown in the inset, for
the case with $d_{x^{2}-y^{2}}+id_{xy}$ superconductivity at $\mu=0.0$ and an
impurity potential $V=-4.0$. The different curves correspond to different
sites along the cut with the black curve denoting the impurity site, while
blue and green curves denote $A$ and $C$ sites, respectively, as shown in the
inset. The impurity is located at an $A$ site.

In order to further understand the emergence of the above property, we can
approximately calculate the relevant components of the anomalous Green
function as follows. First, we consider the Green function in band space which
(in the absence of interband pairing) becomes block diagonal with the
anomalous part

$G_{n\bar{n}}({\mathbf{k}},\omega)=-\frac{\Delta_{n}({\mathbf{k}})}{\omega^{2}-\left(\xi_{n}(\mathbf{k})-\mu\right)^{2}-|\Delta_{n}(\mathbf{k})|^{2}}\,.$
(27)

At low energies, only the band $n^{*}$ that crosses the Fermi level
contributes, and we can ignore anomalous parts of the Green function of other
bands. Next, we transform the Green function back to sublattice space (where
we want to calculate the local Green function), and use the matrix elements of
the unitary transformation $u_{\alpha n}(\mathbf{k})$ to obtain approximately
$G_{\alpha\bar{\alpha}}({\mathbf{k}},\omega)\approx u_{\alpha
n^{*}}(\mathbf{k})G_{n^{*}\bar{n}^{*}}(\mathbf{k},\omega)u_{n^{*}\alpha}(-\mathbf{k})$.
Combining this with Eq. (27), one sees that the relevant quantity is the
product of the sublattice weight and the order parameter
$|u_{n^{*}\alpha}(\mathbf{k})|^{2}\Delta_{n^{*}}(\mathbf{k})$, which combined
does not average to zero on the Fermi surface even though
$\Delta_{n^{*}}(\mathbf{k})$ does.

Finally, to complete the discussion of the $d_{x^{2}-y^{2}}+id_{xy}$ spin-
singlet superconductivity, we show in Fig. 5 the LDOS curves in the vicinity
of an impurity. In agreement with Fig. 3, the inner gap is void of any bound
state peaks. Interestingly, the gapped region is lined by resonant states,
also in agreement with the $T$-matrix analysis from Fig. 3.

Figure 6: Energy dependence of the real (a) and imaginary (b) parts of
$\mbox{det}[\widehat{D}(\omega)]$ as given by Eq. (23) for a range of
nonmagnetic scattering potentials from $V=-10$ to $V=10$, denoted by the
colors, for a triplet $p_{x}+ip_{y}$ gap at $\mu=0.08$ with the DOS for the
clean case shown by the red line. In this case, the gapped region exhibits
impurity bound states, in contrast to the case shown in Fig. 3. As in Fig. 3,
it is the finite value of $\eta$ that causes a finite imaginary part inside
the hard gap in (b).

From the above discussion of the mechanism of the protection of spin-singlet
superconductivity from nonmagnetic atomic-scale disorder, it is evident that
spin-triplet pairing should not enjoy such privilege. This can be verified
explicitly by turning to, e.g., the $E_{1}$ ($p_{x}+ip_{y}$) spin-triplet
irrep. Figure 6 is equivalent to Fig. 3, i.e., it shows the real and imaginary
parts of $\mbox{det}[\widehat{D}(\omega)]$. Note that the $E_{1}$ intraband
irreps all have nodes at the M points, see Table 1. Hence, to achieve a fully
gapped spectrum, we therefore shift the chemical potential to $\mu=0.08$ in
this case to move the Fermi surface away from the nodes. In this setting, we
observe that there are bound state solutions at arbitrarily low energies
inside the gap. In other words, the odd parity of the triplet order cannot
exploit the sublattice weights to “restore $s$-wave behavior”. Figure 7, which
can be compared directly to Fig. 4, shows the sublattice-resolved off-diagonal
Green function components in momentum space for a spin-triplet $p_{x}+ip_{y}$
superconductor. In this case, the sign-changes of the gap function are
preserved, also for the relevant combined object
$|u_{n^{*}\alpha}(\mathbf{k})|^{2}\Delta_{n^{*}}(\mathbf{k})$. Therefore, the
momentum-summed components vanish, allowing bound state solutions. Finally, in
Fig. 8 we show the resulting LDOS at the same near-impurity sites as in Fig.
5, all clearly featuring in-gap bound state peaks extended in real space.

Figure 7: Momentum dependence of the real (a-d) and imaginary (e-h) parts of
the anomalous free Green function components at $\omega=0$ for $p_{x}+ip_{y}$
superconductivity. Panels (a,e) display the Green function entries in band
space, simply reflecting the $p_{x}+ip_{y}$ gap structure at the Fermi
surface. The lower six panels refer to the anomalous entries in sublattice
space. All plots here are for the $p_{x}+ip_{y}$ phase at $\mu=0.08$. In
contrast to the $d_{x^{2}-y^{2}}+id_{xy}$ case, the individual components of
the sublattice Green function, when integrated over the Brillouin zone, all
vanish. As a consequence Eq. (23) always has real solutions. Figure 8:  LDOS
along a cut through the impurity site, shown in the inset, for the case with
$p_{x}+ip_{y}$ superconductivity at $\mu=0.08$ and an impurity potential
$V=-4.0$. The different curves correspond to different sites along the cut
with the black curve denoting the impurity site, while blue and green curves
denote $A$ and $C$ sites, respectively, as shown in the inset. The impurity is
located on an $A$ site. In contrast to the $d_{x^{2}-y^{2}}+id_{xy}$ case
shown in Fig. 5, an impurity bound state appears inside the gap for the
$p_{x}+ip_{y}$ order parameter, and extends along the
$\mathbf{a}_{1}$-direction as shown.

Finally we turn to some of the other allowed superconducting order parameters
on the kagome lattice and discuss their ability to host bound state solutions
from nonmagnetic impurities. First of all, while we demonstrated the relevance
(irrelevance) of sublattice weights for fully-gapped spin-singlet (spin-
triplet) states of the form $d_{x^{2}-y^{2}}+id_{xy}$ ($p_{x}+ip_{y}$), we
stress that our conclusions remain valid for the individual components,
$d_{x^{2}-y^{2}}$, $d_{xy}$ ($p_{x}$, $p_{y}$), or a real combination of the
two. In that case, the LDOS spectrum depends on the sublattice site. In Fig. 9
we show cases where the LDOS is nodal and our discussion refers to the absence
(presence) of in-gap resonant states for spin-singlet (spin-triplet) order.
This is seen from Fig. 9 where we show the LDOS similar to Figs. 5 and 8 for
(a) $B_{1}$ ($f$-wave), (b) $E^{(1)}_{1}$ ($p_{x}$), (c) $E^{(1)}_{2}$
($d_{x^{2}-y^{2}}$) and (d) $E^{(2)}_{2}$ ($d_{xy}$) superconducting order. As
discussed above, the crucial property of the gap function is its sign under a
parity operation. Therefore, the odd-parity spin-triplet states exhibit in-gap
resonant states, whereas the even-parity spin-singlet cases are protected. The
$A_{2}$ spin-singlet even-parity order parameter always allows for in-gap
bound states on the kagome lattice, due to the fact that it is odd under
$C_{2}$ rotations around the in-plane axes.

Figure 9: LDOS along a horizontal cut (same as Figs. 5 and 8) near the
impurity site (black) for the case of a (a) $B_{1}$ ($f$-wave), (b)
$E^{(1)}_{1}$ ($p_{x}$), (c) $E^{(1)}_{2}$ ($d_{x^{2}-y^{2}}$) and (d)
$E^{(2)}_{2}$ ($d_{xy}$) superconducting order parameter with an impurity
strength of $V=-4$ in all cases. The spin-singlet cases are obtained with a
chemical potential $\mu=0$, while the triplet cases are calculated with
$\mu=0.08$, i.e. both near the upper van Hove filling. The LDOS for spin-
triplet order parameters is visibly affected by the impurity and features low-
energy resonant states whereas the singlet $d$-wave order parameters are
protected from in-gap resonant states.

The interplay between sublattice weights and localized impurities may also
have important consequences for other probes and measurable quantities. Future
studies might explore how competing order may locally dress the impurity
potentials. This topic has been studied extensively on other lattices where
nonmagnetic impurity bound/resonant states were crucial for generating the
induced order [65, 66, 67, 68, 69, 70, 69, 71, 72, 73]. Here, we focus on
consequences for quasi-particle interference (QPI) measurements from pointlike
nonmagnetic defects [74, 75, 76]. Figure 10 shows the LDOS in real-space in
the vicinity of a nonmagnetic impurity at the $A$ sublattice site in the case
of $d_{x^{2}-y^{2}}+id_{xy}$ superconductivity. As seen, the resulting LDOS
modulations are highly asymmetric, preserving only $D_{2}$ symmetry around the
impurity site. This is a trivial consequence of the fact that the site-
symmetry of the lattice sites is lower than the point group symmetry.
Naturally, this anisotropy will also carry over to the QPI images, which will
only become approximately symmetric when averaged over large fields of view
containing pointlike disorder equally distributed on the three sublattice
sites.

Figure 10: LDOS $\rho(\mathbf{r},0)$ plotted for a fixed energy $\omega=0$
with $d_{x^{2}-y^{2}}+id_{xy}$ superconductivity and an impurity of strength
$V=-4$ located at an A sublattice site. The breaking of several point group
symmetries by the impurity position is reflected in the associated LDOS
modulations in the vicinity of the impurity.

###  IV.2 Disorder-averaged $T_{c}$-suppression

The absence of in-gap bound states from nonmagnetic disorder in the case of
even-parity unconventional superconducting order on the kagome lattice
suggests that atomic-scale disorder may only be weakly pair-breaking. As a
consequence, its impact on the superconducting critical temperature $T_{c}$
may be smaller than naively expected. In order to explore this question, we
have calculated the disorder-averaged suppression of the order parameter and
$T_{c}$ within the standard Abrikosov-Gor’kov (AG) framework [58]. While this
method disallows actual spatial inhomogeneity and neglects local properties
when feedback or unusual bandstructure effects are important [77, 78, 79], it
provides a simple method to probe and compare the robustness of various
superconducting condensates to disorder.

Within AG-theory, translational invariance is restored upon averaging over
random impurity distributions, and the full Green function is

$\displaystyle\widehat{G}(\mathbf{k},i\omega_{n})^{-1}=\widehat{G}^{(0)}(\mathbf{k},i\omega_{n})^{-1}-\widehat{\Sigma}(\mathbf{k},i\omega_{n}),$
(28)

where $\widehat{G}^{(0)}(\mathbf{k},i\omega_{n})$ is the Green function of the
superconducting system and $\widehat{\Sigma}(\mathbf{k},i\omega_{n})$ denotes
the electronic self-energy. In the Born approximation, the self-energy is

$\displaystyle\widehat{\Sigma}(i\omega_{n})=\frac{nV^{2}}{N}\sum_{\mathbf{k}}\widehat{h}_{\rm
imp}\widehat{G}^{(0)}(\mathbf{k},i\omega_{n})\widehat{h}_{\rm imp},$ (29)

where $N$ is the number of points in the momentum sum, $n$ is the impurity
concentration, and $h_{\rm imp}$ is the impurity potential as given by Eq.
(17) without the impurity strength prefactor, i.e. $\widehat{H}_{\rm
imp}=V\widehat{h}_{\rm imp}$. The superconducting order parameter is given by

$\displaystyle\Delta_{\alpha\beta}(\mathbf{k})=\frac{TV_{\rm
SC}}{N}\left(f_{\mathbf{k}}^{\beta\alpha}\right)^{\ast}\sum_{\begin{subarray}{c}\mathbf{k}^{\prime}\omega_{n}\\\
\gamma\delta\end{subarray}}f^{\gamma\delta}_{\mathbf{k}^{\prime}}G_{\gamma\bar{\delta}}(\mathbf{k}^{\prime},i\omega_{n})\,.$
(30)

The above expression corresponds to an assumed pairing interaction of the form
$V^{\alpha\beta\gamma\delta}_{\mathbf{k},\mathbf{k}^{\prime}}=-V_{\rm
SC}\left(f_{\mathbf{k}}^{\beta\alpha}\right)^{\ast}f^{\gamma\delta}_{\mathbf{k^{\prime}}}$
where $f^{\alpha\beta}_{\mathbf{k}}$ is a form factor transforming as the
irrep of our interest, as given by, e.g., the matrices in Sec. III. In this
manner, a superconducting order parameter transforming as a specific irrep can
be stabilized. Thus, we have two inter-dependent equations, Eqs. (28) and
(30), which are solved selfconsistently to obtain the order parameter
$\Delta(\mathbf{k})$ as a function of temperature, $T$.

Figure 11: (a) The temperature dependence of the $d_{x^{2}-y^{2}}+id_{xy}$
order parameter $\Delta(T)$ on the kagome lattice for a range of $nV^{2}$. The
AG results (circles) have been fitted with the solutions to the mean-field
equation $x=\tanh(x/T)$ (full lines) to obtain critical temperatures $T_{c}$.
(b) The critical temperatures $T_{c}/T_{c0}$ as a function of $nV^{2}$, where
$T_{c0}=T_{c}(nV^{2}=0)$, for an $s$-wave (blue), $d_{x^{2}-y^{2}}$ (red),
$d+id$ (purple), $p_{x}$ (dark green) and $p+ip$ (light green) order
parameter. Note that the $p_{x}$ and $p+ip$ results for the kagome lattice
nearly coincide. The results are for $\mu=0$ in the kagome lattice and
$\mu=-1$ for the square lattice. In the case of the rotational symmetry
breaking order parameters the critical temperatures were calculated with the
impurities located on sites A, B and C, respectively, and an average of the
three values is plotted. The critical temperatures have been matched in the
$nV^{2}=0$ case by self-consistently solving the BdG gap equation. The solid
line in panel (b) shows the $T_{c}$-suppression from the universal AG curve
[58, 77].

In Fig. 11(a) we show the temperature dependence of the superconducting gap,

$\Delta\equiv\frac{TV_{\rm
SC}}{N}\sum_{\begin{subarray}{c}\mathbf{k}\,\omega_{n}\\\
\gamma\delta\end{subarray}}f^{\gamma\delta}_{\mathbf{k}}G_{\gamma\bar{\delta}}(\mathbf{k},i\omega_{n})\,,$
(31)

in the $d_{x^{2}-y^{2}}+id_{xy}$ case for different impurity concentrations.
Note that this deviates from Eq. (30) by the form factor,
$(f^{\beta\alpha}_{\mathbf{k}})^{\ast}$, which only serves to ensure the
correct sublattice and momentum dependence of the order parameter. To
determine the critical temperature $T_{c}$, we fit the solutions to the mean-
field equation $x=\tanh(x/T)$, where $x$ is the mean-field order parameter, to
our results. This yields the solid lines in Fig. 11(a). In Fig. 11(b), we plot
$T_{c}$, determined from fitting to the AG results in Fig. 11(a), as a
function of impurity concentration $nV^{2}$ for different superconducting
order parameters (colored lines) and square and kagome lattices, corresponding
to dashed and full lines, respectively. The $d_{x^{2}-y^{2}}+id_{xy}$ case
shown in Fig. 11(a) leads to the full purple curve in Fig. 11(b). The same
suppression rate is obtained for $d_{x^{2}-y^{2}}$ superconductivity on the
kagome lattice, as shown by the full red curve in Fig. 11(b). In this case, we
have averaged over impurities on the $A$, $B$, and $C$ sublattices and the
horizontal axis is rescaled taking into account that the impurity potential
only acts on $1/3$ of the sites in the kagome lattice. The latter applies to
all kagome results in Fig. 11(b). This is to be contrasted with
$d_{x^{2}-y^{2}}$ superconductivity on the square lattice, which is shown by
the red dashed line in Fig. 11(b), which follows the standard AG
$T_{c}$-suppression curve [58, 77]. The AG-curve in Fig. 11(b) corresponds to
a case with $\rho(0)=0.14$, the DOS at $\mu=-1$ for the square lattice, and is
also representative for the kagome DOS near $\mu=0$ (averaged over the gap
energy scale). Remarkably, the $T_{c}$-suppression of even-parity
unconventional superconductivity on the kagome lattice is qualitatively
distinct from the $T_{c}$ suppression of unconventional superconductivity on
the square lattice. We ascribe the origin of the reduced pair-breaking on the
kagome lattice to the interplay between sublattice weights and localized
impurities, as discussed above. In Fig. 11(b) we additionally compare the
$T_{c}$-suppression to standard $s$-wave order and triplet order on the square
and kagome lattices. As seen, standard $s$-wave order exhibits the expected
slow dependence on impurity concentration in agreement with Anderson’s
theorem, whereas all triplet orders remain fragile to nonmagnetic disorder,
also on the kagome lattice since it is not sublattice-protected.

##  V Discussion and conclusions

In this work, after a general discussion of the allowed homogeneous
superconducting order parameters on the kagome lattice, we have focused on
their properties in the presence of atomic-scale defects. By picking specific
sublattice sites, such isolated impurities limit scattering between Bloch
eigenstates to selected regions of the Fermi surface. This leads to an
important difference between even- and odd-parity superconducting order
parameters in terms of the ability of nonmagnetic impurities to generate bound
states and break Cooper pairs. This fundamental distinction between the
response to nonmagnetic disorder between spin-singlet and spin-triplet order
is unlike, e.g., the square lattice [80, 12, 81] or earlier results relevant
for graphene on the honeycomb lattice[82, 83] where both such superconducting
condensates support impurity bound or resonant states, and become fragile to
nonmagnetic disorder. For even-parity superconductivity on the kagome lattice,
pointlike disorder studies are not phase-sensitive probes of
superconductivity.

We stress that the absence of in-gap bound states in even-parity
superconductors from nonmagnetic disorder is a generic property of the kagome
lattice, and not tied to fine-tuned parameters of, e.g., the Fermi surface
topology or the amplitude of the impurity potential. We have focused on the
upper van Hove filling due to its relevance for the $A$V3Sb5 compounds and, at
that filling, the protected gap region is very pronounced. However, the Fermi
surface states exhibit significant momentum-dependent variation of the
sublattice weight also at other fillings, thus leading to qualitatively
similar effects.

As mentioned above, the amplitude of the pointlike impurity potential is not
important. Likewise, whether the impurity resides on sublattice site $A$, $B$
or $C$, is also not important. These conclusions are based on single-impurity
properties and thus valid in the dilute disorder limit. When disorder cross-
talk becomes important or if the impurities correspond to spatially extended
objects, the disorder-response is altered. For example, if the disorder
potential extends equally to all three sublattice sites, impurity scattering
probes the whole Fermi surface and bound states may get generated, also for an
even-parity spin-singlet sign-changing superconducting order. This is
illustrated in Fig. 12 where we show that there are indeed in-gap bound state
solutions for an extended triangular impurity. Since interstitials or
impurities off the kagome plane are expected to produce extended disorder
potentials, such perturbations can generate in-gap bound states. However, in
such cases there is the additional complication of the strengths of the
potentials generated by more extended defects and whether they are strong
enough to produce low-energy bound states. In particular, if the disorder
potentials arise from out-of-plane defects or interstitials, the resulting
scattering potentials in the kagome planes may simply be too weak to generate
in-gap bound states.

Figure 12: LDOS near an impurity potential extended equally to the $A$, $B$
and $C$ sublattice sites, as shown in the inset, for a
$d_{x^{2}-y^{2}}+id_{xy}$ superconductor at $\mu=0.0$ and an impurity
potential $V=-2.0$. In the present case impurity bound states emerge inside
the gap and the in-gap bound state position is tuned by $V$.

An outstanding question relates to the relevance of the mechanism described in
this work of how sublattice weights can “restore $s$-wave behavior” for the
$A$V3Sb5 superconductors. These materials exhibit a significantly more complex
Fermi surface than the NN tight-binding band utilized in this work [13]. Thus,
for the current mechanism to be active in $A$V3Sb5, superconductivity in these
compounds should primarily be present on the vanadium $d$-orbital bands
featuring significant sublattice weight variation in momentum space.
Furthermore, these materials exhibit CDW order in addition to
superconductivity. However, the CDW order leads to only a weak electronic
reconstruction [84, 85] and it should therefore not be important for the
current discussion. Lastly, as we have stressed, the main source of disorder
has to be pointlike defects located on specific sublattice sites. This seems
to be relevant, for example, to the STM experiments reported in Ref. 48
finding no impurity-bound states near vanadium vacancy defects. Likewise, the
irradiation studies of the penetration depth in Ref. 55 introduce mainly
pointlike defects. Within our scenario, the in-plane vanadium defects are not
phase sensitive to the superconducting gap structure, whereas in-plane Sb or
out-of-plane Sb and Cs defects might be. The latter depends on their strength
on the vanadium sites in the kagome planes remaining after screening.

The understanding of slow $T_{c}$-suppression rates has also been recently
discussed in the context of e.g. PdTe2 and CuxBi2Se3 [86, 87, 88, 89, 90].
Related theoretical studies have formulated a “generalized Anderson theorem”,
clarifying under what circumstances the nature of the disorder is benign for
the particular superconducting state under consideration [91, 87, 88]. In the
present case, the weak $T_{c}$-suppression rate cannot be straightforwardly
explained from the generalized Anderson theorem since the gap structure and
the disorder potential do not fulfil the basic requirements of the theorem.
This is mainly due to the existence of interband components of the
superconducting gap structure. Within the formulation of Ref. 88, an
additional distinction of the current work is the presence of momentum
dependence of the superconducting gap structure in sublattice space. Thus, the
current “robustness” of superconductivity is not quantitatively in line with
Anderson’s theorem, in agreement with the weaker robustness seen in Fig. 11.
Finally, we note an interesting study of robust superfluidity in the polar
phase of 3He in the presence of oriented columnar defects [92]. In that case
the resulting anisotropic scattering of the columnar disorder is blind to the
putative nodal line of the gap, not unlike the present case where the
pointlike defects are also highly anisotropic scatterers in momentum space.

In this manuscript we have mainly studied unconventional superconductivity on
the kagome lattice. Naturally, if it turns out that the $A$V3Sb5 materials are
conventional superconductors with standard non-sign-changing $s$-wave
superconductivity, there is nothing for the sublattice weights “to restore”
and the robustness to nonmagnetic disorder follows from Anderson’s theorem
[60]. The main point of the present paper is to point out that even
unconventional sign-changing spin-singlet superconducting orders on the kagome
lattice exhibit an intrinsic robustness to nonmagnetic disorder.

###### Acknowledgements.

We acknowledge fruitful discussions with F. Ferrari, M. Scheurer, and T.
Shibauchi. This project (M.H.C.) has received funding from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement No 101024210. A.K. acknowledges support by
the Danish National Committee for Research Infrastructure (NUFI) through the
ESS-Lighthouse Q-MAT.

##  Appendix A Pairing matrices

Here we include the explicit form of all the form factor matrices as defined
in Eqs. (6), (9), and (10) for both singlet and triplet order parameters up to
nearest neighbor. For the $A_{1}$ form factor we find

$\displaystyle f^{S}_{{\rm OS},A_{1}}$
$\displaystyle=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}\,,$ (32) $\displaystyle f^{S}_{{\rm NN},A_{1}}$
$\displaystyle=\frac{1}{\sqrt{6}}\begin{pmatrix}0&\cos k_{3}&\cos k_{1}\\\
\cos k_{3}&0&\cos k_{2}\\\ \cos k_{1}&\cos k_{2}&0\end{pmatrix}\,,$ (33)

while for the singlet $B_{2}$ we have:

$\displaystyle f^{S}_{{\rm NN},B_{2}}=\frac{i}{\sqrt{6}}\begin{pmatrix}0&-\sin
k_{3}&\sin k_{1}\\\ \sin k_{3}&0&\sin k_{2}\\\ -\sin k_{1}&-\sin
k_{2}&0\end{pmatrix}\,.$ (34)

Due to the sublattice degree of freedom, there is both a singlet and a triplet
order parameter with $E_{1}$ character. The singlet form factors are

$\displaystyle f^{S}_{{\rm NN},E_{1}^{(1)}}$
$\displaystyle=\frac{i}{2}\begin{pmatrix}0&\sin k_{3}&0\\\ -\sin k_{3}&0&\sin
k_{2}\\\ 0&-\sin k_{2}&0\end{pmatrix}\,,$ (35) $\displaystyle f^{S}_{{\rm
NN},E_{1}^{(2)}}$ $\displaystyle=\frac{i}{\sqrt{12}}\begin{pmatrix}0&\sin
k_{3}&2\sin k_{1}\\\ -\sin k_{3}&0&-\sin k_{2}\\\ -2\sin k_{1}&\sin
k_{2}&0\end{pmatrix}\,.$ (36)

The singlet $E_{2}$ form factor has both OS and NN contributions:

$\displaystyle f^{S}_{{\rm OS},E_{2}^{(1)}}$
$\displaystyle=\frac{1}{\sqrt{6}}\begin{pmatrix}+1&0&0\\\ 0&-2&0\\\
0&0&+1\end{pmatrix}\,,$ (37) $\displaystyle f^{S}_{{\rm OS},E_{2}^{(2)}}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}+1&0&0\\\ 0&0&0\\\
0&0&-1\end{pmatrix}\,,$ (38) $\displaystyle f^{S}_{{\rm NN},E_{2}^{(1)}}$
$\displaystyle=\frac{1}{\sqrt{12}}\begin{pmatrix}0&-\cos k_{3}&2\cos k_{1}\\\
-\cos k_{3}&0&-\cos k_{2}\\\ 2\cos k_{1}&-\cos k_{2}&0\end{pmatrix}\,,$ (39)
$\displaystyle f^{S}_{{\rm NN},E_{2}^{(2)}}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&\cos k_{3}&0\\\ \cos k_{3}&0&-\cos
k_{2}\\\ 0&-\cos k_{2}&0\end{pmatrix}$ (40)

The triplet orders occur at nearest-neighbor and beyond. For the $A_{2}$ form
factor, we find

$\displaystyle f^{T}_{{\rm NN},A_{2}}=\frac{1}{\sqrt{6}}\begin{pmatrix}0&-\cos
k_{3}&\cos k_{1}\\\ \cos k_{3}&0&-\cos k_{2}\\\ -\cos k_{1}&\cos
k_{2}&0\end{pmatrix}\,,$ (41)

and for $B_{1}$:

$\displaystyle f^{T}_{{\rm NN},B_{1}}=\frac{i}{\sqrt{6}}\begin{pmatrix}0&-\sin
k_{3}&-\sin k_{1}\\\ -\sin k_{3}&0&\sin k_{2}\\\ -\sin k_{1}&\sin
k_{2}&0\end{pmatrix}\,.$ (42)

The $E_{1}$ triplet form factor takes the form

$\displaystyle f^{T}_{{\rm NN},E_{1}^{(1)}}$
$\displaystyle=\frac{i}{\sqrt{12}}\begin{pmatrix}0&\sin k_{3}&-2\sin k_{1}\\\
\sin k_{3}&0&-\sin k_{2}\\\ -2\sin k_{1}&-\sin k_{2}&0\end{pmatrix}\,,$ (43)
$\displaystyle f^{T}_{{\rm NN},E_{1}^{(2)}}$
$\displaystyle=\frac{i}{2}\begin{pmatrix}0&\sin k_{3}&0\\\ \sin k_{3}&0&\sin
k_{2}\\\ 0&\sin k_{2}&0\end{pmatrix}\,,$ (44)

while we find the $E_{2}$ triplet form factor to be

$\displaystyle f^{T}_{{\rm NN},E_{2}^{(1)}}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&-\cos k_{3}&0\\\ \cos k_{3}&0&\cos
k_{2}\\\ 0&-\cos k_{2}&0\end{pmatrix}\,,$ (45) $\displaystyle f^{T}_{{\rm
NN},E_{2}^{(2)}}$ $\displaystyle=\frac{1}{\sqrt{12}}\begin{pmatrix}0&\cos
k_{3}&2\cos k_{1}\\\ -\cos k_{3}&0&\cos k_{2}\\\ -2\cos k_{1}&-\cos
k_{2}&0\end{pmatrix}\,.$ (46)

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