# Spin fluctuations in the ultranodal superconducting state of Fe(Se,S)

Yifu Cao1, Chandan Setty2, Andreas Kreisel3, Laura Fanfarillo1,4 and P.J.
Hirschfeld1 1Department of Physics, University of Florida, Gainesville,
Florida 32603, USA  2Department of Physics and Astronomy, Rice Center for
Quantum Materials, Rice University, Houston, Texas 77005, USA  3Niels Bohr
Institute, University of Copenhagen, Jagtvej 155 , DK-2200, Copenhagen,
Denmark  4Istituto dei Sistemi Complessi (ISC-CNR), Via dei Taurini 19,
I-00185 Rome, Italy

###### Abstract

The iron-based superconductor FeSe isovalently substituted with S displays an
abundance of remarkable phenomena that have not been fully understood, at the
center of which are apparent zero-energy excitations in the superconducting
state in the tetragonal phase. The phenomenology has been generally consistent
with the proposal of the so-called ultranodal states where Bogoliubov Fermi
surfaces are present. Recently, nuclear magnetic resonance measurements have
seen unusually large upturns in the relaxation rate as temperature decreases
to nearly zero in these systems, calling for theoretical investigations. In
this paper, we calculate the spin susceptibility of an ultranodal
superconductor including correlation effects within the random phase
approximation. Although the non-interacting mean-field calculation rarely
gives an upturn in the low temperature relaxation rate within our model, we
found that correlation strongly enhances scattering between hot spots on the
Bogoliubov Fermi surface, resulting in robust upturns when the interaction is
strong. Our results suggest that in addition to the presence of Bogoliubov
Fermi surfaces, correlation and multiband physics also play important roles in
the system’s low energy excitations.

##  I Introduction

Iron-based superconductors have been drawing lots of research interest for
almost two decades since their discovery, for their relatively high $T_{c}$,
simple structure and the interplay between rich phenomena including
nematicity, magnetization and non-trivial topology[1]. Among all families of
iron-based superconductors, the chalcogenide 11 material FeSe has a distinct
phase diagram where a nematic transition can occur without accompanying
magnetic order[2, 3, 4, 5, 6, 7]. The parent compound FeSe shows a nematic
transition at around $90$K, a superconducting (SC) transition at $9$K and no
magnetic order under ambient pressure. Upon applying hydrostatic pressure the
nematicity is suppressed and antiferromagnetic (AFM) order develops. The AFM
order in FeSe under pressure should resemble that observed in iron pnictides,
and is likely a stripe order with in-plane magnetic moments [1, 8, 9, 10].

On the other hand, the S-substituted FeSe does not show strong evidence for
long-ranged magnetic order, but exhibits peculiar changes in its
superconducting states across the nematic quantum critical point at around
$0.17$ sulfur substitution. For $x>0.17$, the normal state of FeSe1-xSx is
tetragonal, established by various measurements of the electronic
structure[11, 12, 2]; and the transport properties show non-Fermi liquid
behavior near the quantum critical point[13]. The superconducting state shows
curiously large zero energy density of states (DOS), which has so far been
evidenced by specific heat and thermal transport measurements[14], scanning
tunneling microscopy (STM)[15], angular-resolved photoemission spectroscopy
(ARPES)[16] and most recently, by nuclear magnetic resonance (NMR)
studies[17].

Possible origins of such residual DOS in the superconducting states of the
heavily S-substituted FeSe has been discussed in the aforementioned Ref. [16,
17]. Impurity effects or coexistence of spatially separated SC and normal
phase are excluded, because the samples are clean and homogeneous as seen from
quantum oscillation[2] and STM experiments[15]. For measurements done under
external field such as the NMR measurements, another possible explanation for
the observed residual DOS is the Volovik effect[18]. However, the Volovik
effect cannot account for the order of magnitude difference in the relaxation
rate across samples with different substitution levels but the same external
field. It has been suggested[19, 20, 21] that the so-called ultranodal
superconducting state, which by definition hosts Bogoliubov Fermi surfaces
(BFS), is responsible for the large residual DOS in these systems.

Ultranodal states are superconducting states with extended gap nodes that, in
contrast to usual point nodes or line nodes in three dimension, have the same
dimension as the underlying normal state Fermi surface. Such extended nodes
are called Bogoliubov Fermi surfaces[22, 23, 24]. The existence of BFS does
not necessarily require non-trivial topology, as is the case in [25], but they
are topologically protected by a $\mathbb{Z}_{2}$ invariant if the
superconducting state possesses inversion symmetry. In a multiband spin-$1/2$
superconductor, BFS can arise from a interband non-unitary triplet pairing
term or from a magnetic order that breaks time-reversal symmetry, and may[19]
or may not[21, 25] preserve the inversion symmetry. It has also been shown[21]
that the non-unitarity of the interband triplet pairing can be induced by
driving the system close to a magnetic instability, in which case the magnetic
moment of the the non-unitary triplet pair aligns with the fluctuating
magnetic order.

The existence of a BFS explains well the residual DOS in the tetragonal
Fe(Se,S) as seen from specific heat or STS experiments (see however Ref. [26]
for an alternative picture), as well as the possible $C_{4}$ symmetry breaking
in the superconducting phase as seen in the ARPES experiment[16]. However, it
has not been fully understood how the ultranodal scenario can fit the recent
NMR data presented in Ref. [17].

The NMR measurement in Ref. [17], performed on FeSe1-xSx at several
S-substitution level across the nematic QCP with in-plane applied field and
temperature down to $100$ mK, shows not only a finite value of $1/(T_{1}T)$ at
zero temperature for the $x=0.18$ and $x=0.23$ samples, but also an unusual
upturn as temperature decreases to zero. While the former can be understood
fairly straightforwardly as yet another signature of the zero energy residual
DOS in these materials, the latter requires a more sophisticated
understanding. In this paper, we study the models for BFS systems discussed in
[19, 20, 21] to further calculate the spin fluctuations in the ultranodal
states. We compare our calculations of $1/(T_{1}T)$ to the experimental data,
and show that the upturn is likely due to the interplay between strong
magnetic fluctuation and multiband physics in such systems.

Figure 1: (a) Normal (black) and Bogoliubov (colored) Fermi surfaces for the
two-band model (1), with $\mu_{1}=3.5$, $\mu_{2}=3.2$ and
$\Delta_{1}(\mathbf{k})=\Delta_{2}(\mathbf{k})=0.06|\cos
2\theta_{\mathbf{k}}|$, where $\theta_{\mathbf{k}}$ is the angle of
$\mathbf{k}$ on the 2D Fermi surface.
$\Delta_{\uparrow\uparrow}(\mathbf{k})=0.15$,
$\Delta_{\downarrow\downarrow}(\mathbf{k})=0$ at $T=0$. Within the same color
of the BFS the coherent scattering amplitudes are larger than between points
on BFSs with different color (See main text Sec. III). Arrows shows some of
the dominant scattering processes as seen from panel (d-g). (b)The normalized
$1/(T_{1}T)$ as a function of $T$ below $T_{c}$. Cyan and blue curves
calculated from bare and RPA susceptibility near a magnetic instability
respectively. Red dots are experimental data taken from Ref. [17]. $T_{c}$ is
taken to be $0.08$. The critical $U$ is determined from the normal and
Bogoliubov band structure, and $U_{c}(T_{c})=10.8$, $U_{c}(0)=9.8$. (c) Zero
temperature DOS. (d-g) Real and imaginary parts of the spin susceptibility at
$T=0$. The arrows are the same as in panel (a). One can see that there is a
shift of the dominant contribution from the red arrow to green arrow as $U$
increases. The color bar maxima are 0.7, 0.03, 8, 5 for panel (d), (e), (f),
(g) respectively.

##  II Model

We adopt a minimal two-band mean field model with intraband spin-singlet
pairing and interband non-unitary spin-triplet pairing from previous works[19,
20, 21]

$\displaystyle H=$
$\displaystyle\sum_{\mathbf{k},\sigma,i}\epsilon_{i\mathbf{k}\sigma}c^{\dagger}_{i\mathbf{k}\sigma}c_{i\mathbf{k}\sigma}-\sum_{\mathbf{k},i}\Delta_{i}(\mathbf{k})(c^{\dagger}_{i\mathbf{k}\uparrow}c^{\dagger}_{i-\mathbf{k}\downarrow}+h.c.)$
$\displaystyle-\sum_{\mathbf{k},\sigma}\Delta_{\sigma\sigma}(\mathbf{k})(c^{\dagger}_{1\mathbf{k}\sigma}c^{\dagger}_{2-\mathbf{k}\sigma}+h.c.).$
(1)

Here $i=1,2$ is the band index. Seeking qualitative results at low
temperatures, we make the assumption that all gaps correspond to a single
$T_{c}$ and follow a BCS-like temperature dependence, where the key feature is
that the deviation from the $T=0$ value is exponentially or power law small at
low temperature. Also, for simplicity, we consider a tight-binding model with
only nearest neighbor hopping and $\epsilon_{i\mathbf{k}\sigma}=2(\cos
k_{x}+\cos k_{y})-\mu_{i}$. We have set the nearest neighbor hopping parameter
$t=1$ and adopt it as our unit of energy throughout the calculations below.

We wish to calculate the spin susceptibility

$\displaystyle\chi^{uv}(\mathbf{q},t)$
$\displaystyle=-\theta(t)\sum_{\mathbf{q^{\prime}}}\langle\left[S^{u}(\mathbf{q},t),S^{v}(\mathbf{q^{\prime}},0)\right]\rangle,$
(2)

where
$S^{u}(\mathbf{q},t=0)=\sum_{i,\mathbf{k},\alpha,\beta}c^{\dagger}_{i\mathbf{k}\alpha}\sigma^{u}_{\alpha\beta}c_{i\mathbf{k+q}\beta}$
is the total spin operator summed over the two bands, and $u,v=x,y,z$. In
particular, since the $z$ axis of the Pauli matrices denotes the direction of
the magnetic moment of our non-unitary triplet pair, which in our model is the
direction the magnetization would condense in[21]. We can choose $z$ to be in
the crystalline $ab$ plane, i.e. $z\perp c$ corresponding to the direction of
the magnetic field in the experiment of Ref. [17], and focus on the
$zz$-component of the spin susceptibility, $\chi^{zz}$.

To this end, we first find the Nambu Green’s function $G_{\mathbf{k}}(\omega)$
by diagonalizing the Nambu Hamiltonian corresponding to Eq. (1). The Nambu
basis we use is
$\psi_{\mathbf{k}}=[c_{1\mathbf{k}\uparrow},c_{1\mathbf{-k}\downarrow},c_{2\mathbf{-k}\uparrow},c_{2\mathbf{k}\downarrow},c_{1\mathbf{k}\uparrow}^{\dagger},c_{1\mathbf{-k}\downarrow}^{\dagger},c_{2\mathbf{-k}\uparrow}^{\dagger},c_{2\mathbf{k}\downarrow}^{\dagger}]^{T}$.
With the eigenvalues $E_{l\mathbf{k}}$ and the eigenvector matrix
$U_{\mathbf{k}}$ of the Nambu Hamiltonian, the Nambu Green’s function can now
be expressed as

$\displaystyle
G_{\mathbf{k}}(\omega)=U_{\mathbf{k}}^{\dagger}\text{diag}\left(\frac{1}{\omega-
E_{1\mathbf{k}}},\ ...,\ \frac{1}{\omega-
E_{8\mathbf{k}}}\right)U_{\mathbf{k}}.$ (3)

At this point, we would like to also define the $8\times 8$ Nambu spin
matrices
$\Sigma^{u}\equiv\text{diag}(\sigma^{u},\sigma^{u},-(\sigma^{u})^{T},-(\sigma^{u})^{T})$,
composed of $2\times 2$ Pauli matrices on their diagonal blocks. The bare
spin-spin correlation function in the Matsubara representation is
$C^{uv}(\mathbf{q},i\nu)=-\frac{1}{2}\frac{1}{\beta}\sum_{i\omega_{n}}\sum_{\mathbf{k}}\operatorname{Tr}\left(\Sigma^{u}G_{\mathbf{k+q}}(i\omega_{n}+i\nu)\Sigma^{v}G_{\mathbf{k}}(i\omega_{n})\right)$.
Substituting in Eq.(3) and having performed the Matsubara sum and the analytic
continuation to the real axis, we obtain

$\displaystyle\chi^{(0)zz}(\mathbf{q},\omega)=\frac{1}{2}\sum_{l,m}\Sigma^{z}_{ll}\Sigma^{z}_{mm}\chi_{llmm}^{(0)}(\mathbf{q},\omega),$
(4)

where the bare density-density bubble in the quasiparticle band space reads

$\displaystyle\chi_{llmm}^{(0)}(\mathbf{q},\omega)=-\sum_{\mathbf{k},r,s}W_{ll}(r\mathbf{k},s\mathbf{k+q})W_{mm}^{*}(r\mathbf{k},s\mathbf{k+q})$
$\displaystyle\times\frac{f(E_{s\mathbf{k+q}})-f(E_{r\mathbf{k}})}{E_{s\mathbf{k+q}}-E_{r\mathbf{k}}-\omega-i0^{+}}.$
(5)

Here we defined the coherence factors
$W_{ll}(r\mathbf{k},s\mathbf{k^{\prime}})\equiv
U_{rl\mathbf{k}}U_{sl\mathbf{k^{\prime}}}^{*}$, where $r\mathbf{k}$ is a
composite label referring to the Bogoliubov quasiparticle at momentum
$\mathbf{k}$ in the $r$th quasiparticle band.

We can further investigate using a random phase approximation (RPA)
calculation of the effect of a residual interaction in the particle-hole
channel. We consider an interaction of the Hubbard type

$\displaystyle H_{U}=$
$\displaystyle\frac{1}{2}\sum_{\mathbf{r},i}Un_{i\mathbf{r}\uparrow}n_{i\mathbf{r}\downarrow}$
$\displaystyle\equiv$
$\displaystyle\frac{1}{8}\sum_{\mathbf{k,k^{\prime},q},l,m}\Gamma_{llmm}\psi^{\dagger}_{l\mathbf{k}}\psi_{l\mathbf{k-q}}\psi^{\dagger}_{m\mathbf{k^{\prime}}}\psi_{m\mathbf{k^{\prime}+q}}$
(6)

In the last step we have rewritten the interaction using the Nambu basis and
defined a coupling tensor $\Gamma$ that is diagonal in the dimensions labeled
by the 1st, 2nd and 3th, 4th indices and a constant function of momentum
transfer $\mathbf{q}$. As shown in Appendix, it is sufficient to keep only the
partially diagonal part of the full tensor for our purpose of calculating
$\chi^{\mathrm{(RPA)}zz}$, and both tensors in Eq. (5) and (6) can be treated
as $8\times 8$ matrices $\tilde{\chi}^{(0)}_{lm}\equiv\chi^{(0)}_{llmm}$,
$\tilde{\Gamma}_{lm}\equiv\Gamma_{llmm}$. The RPA density-density bubble is
related to the bare bubble through

$\displaystyle\tilde{\chi}^{\mathrm{(RPA)}}(\mathbf{q},\omega)=\tilde{\chi}^{(0)}(\mathbf{q},\omega)\left(\tilde{I}+\tilde{\Gamma}\tilde{\chi}^{(0)}(\mathbf{q},\omega)\right)^{-1}.$
(7)

The sign convention for the above equation is explained in the appendix. The
RPA spin susceptibility is

$\displaystyle\chi^{\mathrm{(RPA)}zz}(\mathbf{q},\omega)=\frac{1}{2}\sum_{l,m}\Sigma^{z}_{ll}\Sigma^{z}_{mm}\tilde{\chi}_{lm}^{\mathrm{(RPA)}}(\mathbf{q},\omega),$
(8)

by analogy to Eq. (4) with
$\chi^{(0)}\rightarrow\tilde{\chi}^{\mathrm{(RPA)}}$. The NMR relaxation rate
probes the spin fluctuation in the system, and is proportional to the
imaginary part of the spin susceptibility,

$\displaystyle\frac{1}{T_{1}}\propto T\lim_{\omega\rightarrow
0}\sum_{\mathbf{q}}\frac{\operatorname{Im}\chi^{zz}(\mathbf{q},\omega)}{\omega}$
(9)

##  III Results

We numerically calculated the spin susceptibility of the model Hamiltonian
BFS. To summarize the result, we found that the bare susceptibility
calculation always give rise to non-zero residual $1/(T_{1}T)$ at zero
temperature when BFS are present, as expected due to the zero energy residual
DOS. However, the bare $1/(T_{1}T)$ rarely increases as temperature decrease
near $T=0$, unless van Hove singularities of the Bogoliubov quasiparticle
bands are tuned to the Fermi level, contributing to a large zero energy peak
in the DOS. On the other hand, if we take into account the correlation effects
using the RPA calculation, certain scattering between hot spots on the BFS can
get strongly enhanced, resulting in an upturn in the $1/(T_{1}T)$ as
temperature decreases. This can happen when the zero energy DOS is not peaked,
or when the BFS are not strongly nested. Below we discuss in details these
results.

In Fig 1 and 2 we show in parallel two examples of having upturns in the
$1/(T_{1}T)$ at low temperature as a result of correlations and multiband
effects. Fig. 1 corresponds to a scenario where the intraband singlet
$\Delta_{i}(\mathbf{k})$ is taken to be nodal s-wave with accidental nodes
along the 45 degree directions, and the interband triplet pairing
$\Delta_{\uparrow\uparrow}(\mathbf{k})$ isotropic. Fig. 2 corresponds

Figure 2: Same as Fig. 1 but with a different set of parameters:
$\mu_{1}=3.7$, $\mu_{2}=3.2$,
$\Delta_{1}(\mathbf{k})=\Delta_{2}(\mathbf{k})=0.05$,
$\Delta_{\uparrow\uparrow}(\mathbf{k})=0.3\cos\theta_{\mathbf{k}}$,
$\Delta_{\downarrow\downarrow}(\mathbf{k})=0$, $T_{c}=0.1$.
$U_{c}(T_{c})=10.8$, $U_{c}(0)=10$. The color bar maxima are 0.7, 0.026, 15,
2.3 for panel (d), (e), (f), (g) respectively.

to the $C_{2}$ symmetric scenario discussed in Ref. [21], where the interband
triplet pairing $\Delta_{\uparrow\uparrow}(\mathbf{k})$ is assumed to be
p-wave and the intraband singlet pairing $\Delta_{i}(\mathbf{k})$ is taken to
be isotropic for simplicity. A BFS then forms only when
$\Delta_{\uparrow\uparrow}$ is sufficiently large.

In both cases we have set $\Delta_{\downarrow\downarrow}(\mathbf{k})=0$. In
Fig. 1, the bare susceptibility already gives rise to an upturn in the
$1/(T_{1}T)$ (panel (b) cyan curve). This is because the van Hove singularity
(band extremum corresponding to where the superconducting gap opens) of the
Bogoliubov band has been tuned at the Fermi level by changing the interband
order parameter, and a peak in the quasiparticle density of states exists at
exactly zero energy (panel (c)). We found that upturns in the bare
$1/(T_{1}T)$ at low temperature seem to be always associated with such peaks
at zero energy in the DOS. Although such peaks in the DOS are not desired as
they are not consistent with the spectroscopic data[15], they are not required
once we include correlations. This can be seen in Fig. 2 (b) and (c), where
the density of states is not peaked at zero energy and the bare $1/(T_{1}T)$
curve does not have an upturn while the RPA curve close to magnetic
instability does. The qualitative difference between the $U=0$ curve and the
$U\lessapprox U_{c}(0)$ curve in Fig. 2 (b) is unusual, since normally one
would expect, from its simplest form for the normal metal as in Eq. (A.13),
the RPA susceptibility to be enhanced further where the bare susceptibility is
already large.

To better understand this unusual behavior, we first divide the BFS into
several segments, as shown by the color scheme in panel (a) of Fig. 1 and 2,
within each segment the scattering is much stronger than across different
segments. This is done by treating the $\mathbf{k}$-points on the BFS as
vertices of an weighted undirected graphs with weights given by the coherence
factors defined in Eq. (5), and employing the Leiden algorithm for community
detection[27, 28]. We see that the parts of the BFS that follow the shape of
the normal Fermi surfaces (red and blue in panel (a)) are well separated from
the rest of the BFSs in terms of scattering processes. Then in panel (d,e) we
plot the bare spin susceptibility as in Eq. (4) at zero temperature, and in
panel (f,g) we show the RPA spin susceptibility (8) at $U\lessapprox U_{c}(0)$
and zero temperature. We identify the important $\mathbf{q}$ vectors as the
red and green arrows connecting segments of BFSs with the same color shown in
Fig. 2(a). From panel (d) we see that the real part of the bare susceptibility
is the largest at the $\mathbf{q}$ vectors connecting the red part of the
BFSs, but is not strongly peaked at any particular $\mathbf{q}$ vector. The
latter observation is an indication of no strong nesting between the the BFSs.
Secondly, by comparing panel (d,e) with panel (f,g), we see that although the
$\mathbf{q}$ vectors that connect the green part of the BFSs are only
subdominant in the bare susceptibility, they become the dominant $\mathbf{q}$
vectors that connect ”hot spots” on the BFSs near the magnetic instability.
This shift of the dominant $\mathbf{q}$ vectors as the interaction $U$
increases within an RPA calculation can only be explained by nontrivial
multiband effects embedded in the coherence factor $W$, which is consistent
with the unusual change in the shape of the normalized $1/(T_{1}T)$ curve as
$U$ increases.

##  IV Conclusion

To summarize, we have calculated the spin susceptibilities for the ultranodal
states in a minimal two-band model, where the interband non-unitary spin-
triplet pairing is responsible for the Bogoliubov Fermi surfaces. We found
that the existence of BFSs in such models naturally gives rise to finite
residual value in the $1/T_{1}T$ at zero temperature, but does not necessarily
produce the large upturns at low temperature, as seen in the experiments [17]
on the Fe(Se,S) system, in a non-interacting calculation. We then studied the
effect of correlation within random phase approximation in the ultranodal
state. By adding a Hubbard interaction in the particle-hole channel while not
changing the pre-assumed pairing gaps, we see that the spin susceptibilities
at $\mathbf{q}$ vectors connecting hot spots on the BFS get strongly enhanced
at low temperature when the interaction is strong, resulting in upturns in
$1/T_{1}T$ irrespective of the presence or absence of upturns in the bare
calculation. The hot spots have strong interband character as indicated from
their position on the BFSs, and do not have particularly large contribution to
the spin susceptibilities at weak interaction. Therefore, we conclude the
experimentally observed upturn in $1/T_{1}T$ can be explained as a combined
effect of the presence of BFS, interband physics and correlation.

Our theory is primarily applicable to the tetragonal phase of FeSe1-xSx with
$x>0.17$ at ambient pressure. For the nematic phase with $x<0.17$ at ambient
pressure and the tetragonal phase with $x<0.17$ under pressure[29], the low
temperature $1/T_{1}T$ seems to have a Korringa behavior, i.e. constant in
temperature, with smaller but finite residual values. Our calculation of the
$1/T_{1}T$ is consistent with these data assuming weak correlation or small
BFSs, but whether the ultranodal scenario can apply to these situations
requires a more careful and comprehensive study in the future.

##  V Acknowledgements

L. F. acknowledges support by the European Union’s Horizon 2020 research and
innovation programme through the Marie Skłodowska-Curie grant SuperCoop (Grant
No 838526). P.J.H. and Y.C. were supported by DOE grant number DE-
FG02-05ER46236. A.K. acknowledges support by the Danish National Committee for
Research Infrastructure (NUFI) through the ESS-Lighthouse Q-MAT.

## Appendix

In this appendix we provide information about our sign convention in Eq. (7),
and how the standard multiorbital RPA formalism in e.g. Ref. [30, 31]
corresponds to our $8\times 8$ matrix formalism in Nambu space. We start by
considering a bare bubble that is Eq. (5) in its general form

$\displaystyle\chi_{ll^{\prime}mm^{\prime}}^{(0)}(\mathbf{q},\omega)=-\sum_{\mathbf{k},r,s}U_{rl\mathbf{k}}U_{sl^{\prime}\mathbf{k+q}}^{*}U_{rm\mathbf{k}}^{*}U_{sm^{\prime}\mathbf{k+q}}\frac{f(E_{s\mathbf{k+q}})-f(E_{r\mathbf{k}})}{E_{s\mathbf{k+q}}-E_{r\mathbf{k}}-\omega-i0^{+}}$
(A.1)

It transforms like a $8\times 8\times 8\times 8$ rank-4 tensor as the Numbu
basis transforms, while contracts like a $64\times 64$ rank-2 tensor (matrix
with $ll^{\prime}$ a composite row index and $mm^{\prime}$ a composite column
index) in the RPA series. Accordingly the full coupling tensor corresponding
to the Hubbard interaction in Eq. (6) can be redefined as

$\displaystyle
H_{U}=\frac{1}{2}\sum_{\mathbf{r},i}Un_{i\mathbf{r}\uparrow}n_{i\mathbf{r}\downarrow}=\frac{1}{16}\sum_{\mathbf{k,k^{\prime},q},l,m,l^{\prime},m^{\prime}}\Gamma_{ll^{\prime}mm^{\prime}}\psi^{\dagger}_{l\mathbf{k}}\psi_{l^{\prime}\mathbf{k-q}}\psi^{\dagger}_{m\mathbf{k^{\prime}}}\psi_{m^{\prime}\mathbf{k^{\prime}+q}}$
(A.2)

where on the RHS the terms quadratic in $c$,$c^{\dagger}$ has been discarded
since they only renormalize the chemical potential. Let’s represent
$\Gamma_{ll^{\prime}mm^{\prime}}$ as a $64\times 64$ matrix which only has two
$8\times 8$ non-zero blocks, ordered as following:

$\displaystyle[\Gamma_{ll^{\prime}mm^{\prime}}]_{64\times
64}=\begin{pmatrix}[\Gamma_{llmm}]_{8\times 8}&0_{8\times 48}&0_{8\times 8}\\\
0_{48\times 8}&0_{48\times 48}&0_{8\times 48}\\\ 0_{8\times 8}&0_{8\times
48}&[\Gamma_{l\bar{l}m\bar{m}}]_{8\times 8}\end{pmatrix}$ (A.3)

The first $8\times 8$ block $[\Gamma_{llmm}]$ is the same as the
$\tilde{\Gamma}$ in Eq. (6), and is the direct interaction. The other $8\times
8$ block $[\Gamma_{l\bar{l}m\bar{m}}]$ is the exchange interaction. Here
$\bar{l}$ is defined as the the Nambu index that corresponds to the time-
reversed $l$th operator in the Nambu basis. For example, $\bar{1}=2$ and
$\bar{8}=7$ (c.f. the choice of our Nambu basis above Eq. (3)). The matrix
elements of $[\Gamma_{llmm}]$ and $[\Gamma_{l\bar{l}m\bar{m}}]$ are tabulated
as in Table 1.

$ll^{\prime}$ $mm^{\prime}$ | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88
---|---|---|---|---|---|---|---|---
11 | 0 | U | 0 | 0 | 0 | 0 | 0 | 0
22 | U | 0 | 0 | 0 | 0 | 0 | 0 | 0
33 | 0 | 0 | 0 | U | 0 | 0 | 0 | 0
44 | 0 | 0 | U | 0 | 0 | 0 | 0 | 0
55 | 0 | 0 | 0 | 0 | 0 | U | 0 | 0
66 | 0 | 0 | 0 | 0 | U | 0 | 0 | 0
77 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | U
88 | 0 | 0 | 0 | 0 | 0 | 0 | U | 0

$ll^{\prime}$ $mm^{\prime}$ | 12 | 21 | 34 | 43 | 56 | 65 | 78 | 87
---|---|---|---|---|---|---|---|---
12 | 0 | -U | 0 | 0 | 0 | 0 | 0 | 0
21 | -U | 0 | 0 | 0 | 0 | 0 | 0 | 0
34 | 0 | 0 | 0 | -U | 0 | 0 | 0 | 0
43 | 0 | 0 | -U | 0 | 0 | 0 | 0 | 0
56 | 0 | 0 | 0 | 0 | 0 | -U | 0 | 0
65 | 0 | 0 | 0 | 0 | -U | 0 | 0 | 0
78 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -U
87 | 0 | 0 | 0 | 0 | 0 | 0 | -U | 0

Table 1: Non-zero matrix elements of $[\Gamma_{ll^{\prime}mm^{\prime}}]$. The
left panel corresponds to the $8\times 8$ block $[\Gamma_{llmm}]$ in Eq.(A.3),
and the right panel corresponds to the $8\times 8$ block
$[\Gamma_{l\bar{l}m\bar{m}}]$.

To calculate the $\chi^{zz}$ component of the spin susceptibility, we can
write the $\Sigma^{z}$ operator (defined in the main text below Eq. (3)) as a
$1\times 64$ vector, which in the same basis as for Eq. (A.3) is

$\displaystyle[\Sigma^{z}_{ll^{\prime}}]_{1\times
64}=(\text{diag}(\Sigma^{z}),0_{1\times 48},0_{1\times 8})$ (A.4)

The RPA spin susceptibility then is

$\displaystyle\chi^{\mathrm{(RPA)}zz}(\mathbf{q},\omega)=\frac{1}{2}[\Sigma^{z}][\chi^{(0)}](\mathbf{q},\omega)\left(I_{64\times
64}+[\Gamma][\chi^{(0)}](\mathbf{q},\omega)\right)^{-1}[\Sigma^{z}]^{T}$ (A.5)

We want to show that Eq. (A.5) is equivalent to Eq. (7) and (8) where the
tilded matrices are essentially the first $8\times 8$ blocks of the $[\cdot]$
matrices. Since the vector $[\Sigma^{z}]$ bears the form of Eq. (A.4), it
suffices to show that

$\displaystyle[\chi^{(0)}](\mathbf{q},\omega)\left(I_{64\times
64}+[\Gamma][\chi^{(0)}](\mathbf{q},\omega)\right)^{-1}=\begin{pmatrix}\tilde{\chi}^{\mathrm{(RPA)}}(\mathbf{q},\omega)&*&*\\\
*&*&*\\\ *&*&*\end{pmatrix}$ (A.6)

where $\tilde{\chi}^{\mathrm{(RPA)}}(\mathbf{q},\omega)$ is the $8\times 8$
matrix defined in Eq. (7), and $*$‘s are placeholders for unimportant blocks.
Since $[\Gamma]$ has the form of Eq. (A.3), without using any knowledge about
$[\chi^{(0)}]$ except that the first $8\times 8$ block of it is
$\tilde{\chi}^{(0)}$, we can see that $I_{64\times
64}+[\Gamma][\chi^{(0)}](\mathbf{q},\omega)$ should have the following form

$\displaystyle I_{64\times
64}+[\Gamma][\chi^{(0)}](\mathbf{q},\omega)=\begin{pmatrix}\tilde{I}+\tilde{\Gamma}\tilde{\chi}^{(0)}(\mathbf{q},\omega)&B_{8\times
48}&C_{8\times 8}\\\ 0_{48\times 8}&I_{48\times 48}&0_{48\times 8}\\\
D_{8\times 8}&E_{8\times 48}&F_{8\times 8}\end{pmatrix}$ (A.7)

Its inverse can be calculated as

$\displaystyle\left(I_{64\times
64}+[\Gamma][\chi^{(0)}](\mathbf{q},\omega)\right)^{-1}=\begin{pmatrix}A^{-1}+A^{-1}C(F-DA^{-1}C)^{-1}DA^{-1}&*&*\\\
0&*&*\\\ *&*&*\end{pmatrix}$ (A.8)

where we have denoted
$A\equiv\tilde{I}+\tilde{\Gamma}\tilde{\chi}^{(0)}(\mathbf{q},\omega)$. Now if
we take a close look at the structure of $[\chi^{(0)}]$, we will see that it
is zero at the off-diagnoal block $\chi^{(0)}_{llm\bar{m}}$, namely

$\displaystyle[\chi^{(0)}](\mathbf{q},\omega)=\begin{pmatrix}\tilde{\chi}^{(0)}(\mathbf{q},\omega)&*&0_{8\times
8}\\\ *&*&*\\\ 0_{8\times 8}&*&[\chi^{(0)}_{l\bar{l}m\bar{m}}]_{8\times
8}\end{pmatrix}$ (A.9)

This is because the pair of Green’s functions
$G_{\bar{m}l\mathbf{k}}(\omega^{\prime})$ and
$G_{lm\mathbf{k+q}}(\omega^{\prime}+\omega)$ cannot be both non-zero at the
same time given that there is only the intraband singlet pairing and interband
triplet pairing processes in the Hamlitonian (1). Now if we combine Eq. (A.8)
and (A.9) and notice that $C=D=0$ once we know (A.9), we see that Eq. (A.6) is
true. Therefore, we have shown that the full RPA treatment of our model can be
reduced to an $8\times 8$ matrix formalism, which is easier to deal with
numerically.

As a further sanity check, we turn off the pairing amplitudes and calculate
the charge and spin susceptibility in the normal state starting from Eq. (7).
In this case $\tilde{\chi}^{(0)}(\mathbf{q},\omega)$ is a diagonal matrix, and
$\tilde{\Gamma}\tilde{\chi}^{(0)}(\mathbf{q},\omega)$ is block diagonal with
four $2\times 2$ blocks on the diagonal. Let’s take the first block as an
example (for a one band system) and denote the $2\times 2$ matrices as hatted
matrices and the matrix elements
$\chi_{1111}^{(0)}=\chi_{2222}^{(0)}\equiv\chi_{1}^{(0)}$. We have

$\displaystyle\hat{I}+\hat{\Gamma}\hat{\chi}^{(0)}(\mathbf{q},\omega)=\begin{pmatrix}1&U\chi_{1}^{(0)}(\mathbf{q},\omega)\\\
U\chi_{1}^{(0)}(\mathbf{q},\omega)&1\end{pmatrix}$ (A.10)

Inverting the above matrix gives

$\displaystyle\hat{\chi}^{\mathrm{(RPA)}}(\mathbf{q},\omega)=\hat{\chi}^{(0)}(\mathbf{q},\omega)\left(\hat{I}+\hat{\Gamma}\hat{\chi}^{(0)}(\mathbf{q},\omega)\right)^{-1}=\frac{\chi^{(0)}_{1}(\mathbf{q},\omega)}{1-U^{2}\chi^{(0)}_{1}(\mathbf{q},\omega)^{2}}\begin{pmatrix}1&-U\chi^{(0)}_{1}(\mathbf{q},\omega)\\\
-U\chi^{(0)}_{1}(\mathbf{q},\omega)&1\end{pmatrix}$ (A.11)

The diagonal elements
$\chi_{1111}^{\mathrm{(RPA)}}(\mathbf{q},\omega)=\chi_{2222}^{\mathrm{(RPA)}}(\mathbf{q},\omega)$
are the polarization bubbles with the same spin on the left and right vertices
(Here $1$ stands for up and $2$ stands for down, c.f. the choice of our Nambu
basis above Eq. (3)). The off-diagonal elements are the bubble with opposite
spins on the vertices. The charge susceptibility of the first band is given by

$\displaystyle\chi^{\mathrm{(RPA)}c}(\mathbf{q},\omega)=\chi_{1111}^{\mathrm{(RPA)}}+\chi_{1122}^{\mathrm{(RPA)}}+\chi_{2211}^{\mathrm{(RPA)}}+\chi_{2222}^{\mathrm{(RPA)}}=\frac{2\chi^{(0)}_{1}(\mathbf{q},\omega)}{1+U\chi^{(0)}_{1}(\mathbf{q},\omega)}$
(A.12)

And the spin susceptibility of the first band is

$\displaystyle\chi^{\mathrm{(RPA)}zz}(\mathbf{q},\omega)=\chi_{1111}^{\mathrm{(RPA)}}-\chi_{1122}^{\mathrm{(RPA)}}-\chi_{2211}^{\mathrm{(RPA)}}+\chi_{2222}^{\mathrm{(RPA)}}=\frac{2\chi^{(0)}_{1}(\mathbf{q},\omega)}{1-U\chi^{(0)}_{1}(\mathbf{q},\omega)}$
(A.13)

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$\displaystyle=$ $\displaystyle C_{E},$ (156)
$\displaystyle\gamma^{5}_{E}\gamma^{\mu\,\dagger}_{E}\gamma^{5}_{E}$
$\displaystyle=$ $\displaystyle-\gamma^{\mu}_{E},$ (157) $\displaystyle
C^{-1}_{E}\gamma^{\mu}_{E}C_{E}=-\gamma_{E}^{\mu*}$ $\displaystyle=$
$\displaystyle-\gamma_{E}^{\mu T},$ (158)
$\displaystyle\left(\gamma^{5}_{E}\right)^{2}=-C_{E}^{2}$ $\displaystyle=$
$\displaystyle 1$ (159)

In particular, by using Eq. (157) and Eq. (III.1)—or Eq. (169)—one can prove
the $\gamma^{5}$-hermiticity of the Wilson-Dirac operator $D$, or equivalently
the $\gamma^{5}$-hermiticity of the fermion propagator $S(x,y)$, as follows,

$\displaystyle\gamma^{5}_{E}D_{xy}^{R,\dagger}\gamma^{5}_{E}$ $\displaystyle=$
$\displaystyle\gamma^{5}_{E}(S_{R}(x,y)^{-1})^{\dagger}\gamma^{5}_{E}$ (160)
$\displaystyle=$
$\displaystyle\left(4+am_{0}^{R}\right)\delta_{xy}-\frac{1}{2}\sum_{\mu}\left((1+\gamma_{E\,\mu})U^{(R),\dagger}_{\mu}(x)\delta_{x+\mu,y}+(1-\gamma_{E\,\mu})U_{\mu}^{(R)}(y)\delta_{x,y+\mu}\right)$
$\displaystyle=$ $\displaystyle S_{R}(y,x)^{-1}=D_{y,x}^{R}.$

For the combinations of gamma matrices
$\left(\Gamma^{1},\Gamma^{2}\right)=\left(C\gamma^{5},\mathbb{1}_{4}\right)$,
that appear in Sect. IV, the following useful relation, which enters the
derivation of Eq. (108), holds:

$\left(\gamma^{0}_{E}\Gamma^{1\ast}\gamma^{0}_{E}\right)^{\alpha\beta}\left(\gamma^{0}_{E}\Gamma^{2\dagger}\gamma^{0}_{E}\right)^{\gamma\delta}=\left(\Gamma^{1}\right)^{\alpha\beta}\left(\Gamma^{2}\right)^{\gamma\delta}\,,$
(161)

which descends from the fact that that $\gamma_{E}^{0}$, $\gamma_{E}^{2}$,
$\gamma_{E}^{5}$ are real and Hermitian.

###  A.3 A basis of generators for $(f)$ and $(as)$ representations of $Sp(4)$

In the HMC/RHMC algorithms it is necessary to have an explicit expression for
the generators for a given representation $R$ of $Sp(4)$ in order to compute
the MD forces associated with the HMC/RHMC Hamiltonian. We make an explicit
choice of basis, and report it here, for completeness. For the fundamental
representation $T^{A}_{(f)}$, with $A=1,2,\cdots,10$, our choice is the
following:

$T^{1}_{(f)}=\frac{1}{2}\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&0&0\\\
0&0&-1&0\\\
0&0&0&0\end{array}\right),~{}T^{2}_{(f)}=\frac{1}{2}\left(\begin{array}[]{cccc}0&0&0&0\\\
0&1&0&0\\\ 0&0&0&0\\\
0&0&0&-1\end{array}\right),~{}T^{3}_{(f)}=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{cccc}0&i&0&0\\\
-i&0&0&0\\\ 0&0&0&i\\\ 0&0&-i&0\end{array}\right),$
$T^{4}_{(f)}=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{cccc}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&-1\\\
0&0&-1&0\end{array}\right),~{}T^{5}_{(f)}=\frac{1}{2}\left(\begin{array}[]{cccc}0&0&1&0\\\
0&0&0&0\\\ 1&0&0&0\\\
0&0&0&0\end{array}\right),~{}T^{6}_{(f)}=\frac{1}{2}\left(\begin{array}[]{cccc}0&0&i&0\\\
0&0&0&0\\\ -i&0&0&0\\\ 0&0&0&0\end{array}\right),$
$T^{7}_{(f)}=\frac{1}{2}\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&1\\\
0&0&0&0\\\
0&1&0&0\end{array}\right),~{}T^{8}_{(f)}=\frac{1}{2}\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&i\\\ 0&0&0&0\\\
0&-i&0&0\end{array}\right),~{}T^{9}_{(f)}=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{cccc}0&0&0&i\\\
0&0&i&0\\\ 0&-i&0&0\\\ -i&0&0&0\end{array}\right),$ (162)
$T^{10}_{(f)}=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{cccc}0&0&0&1\\\
0&0&1&0\\\ 0&1&0&0\\\ 1&0&0&0\end{array}\right).$

As discussed in Section III.1, the generators for the antisymmetric
representation of $Sp(4)$ appear in the infinitesimal transformation of the
antisymmetric link variable in Eq. (75). We adopt the conventional basis of
matrices $e_{(as)}$ given by

$e^{(12)}_{(as)}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cccc}0&-1&0&0\\\
1&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right),~{}e^{(23)}_{(as)}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&-1&0\\\ 0&1&0&0\\\
0&0&0&0\end{array}\right),~{}e^{(14)}_{(as)}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cccc}0&0&0&-1\\\
0&0&0&0\\\ 0&0&0&0\\\ 1&0&0&0\end{array}\right),$ (163)
$e^{(24)}_{(as)}=\frac{1}{2}\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&-1\\\
-1&0&0&0\\\
0&1&0&0\end{array}\right),~{}e^{(34)}_{(as)}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&-1\\\ 0&0&1&0\end{array}\right),$

With this convention we find the following expressions of the generators for
the antisymmetric representation of $Sp(4)$:

$T^{1}_{(as)}=\frac{1}{2}\left(\begin{array}[]{ccccc}1&0&0&0&0\\\
0&-1&0&0&0\\\ 0&0&1&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&-1\end{array}\right),~{}T^{2}_{(as)}=\frac{1}{2}\left(\begin{array}[]{ccccc}1&0&0&0&0\\\
0&1&0&0&0\\\ 0&0&-1&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&-1\end{array}\right),$
$T^{3}_{(as)}=\frac{-i}{2}\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&-1&0\\\ 0&0&0&-1&0\\\ 0&1&1&0&0\\\
0&0&0&0&0\end{array}\right),~{}T^{4}_{(as)}=\frac{1}{2}\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&-1&0\\\ 0&0&0&1&0\\\ 0&-1&1&0&0\\\ 0&0&0&0&0\end{array}\right),$
$T^{5}_{(as)}=\frac{1}{2}\left(\begin{array}[]{ccccc}0&-1&0&0&0\\\
-1&0&0&0&0\\\ 0&0&0&0&1\\\ 0&0&0&0&0\\\
0&0&1&0&0\end{array}\right),~{}T^{6}_{(as)}=\frac{-i}{2}\left(\begin{array}[]{ccccc}0&1&0&0&0\\\
-1&0&0&0&0\\\ 0&0&0&0&-1\\\ 0&0&0&0&0\\\ 0&0&1&0&0\end{array}\right),$ (164)
$T^{7}_{(as)}=\frac{1}{2}\left(\begin{array}[]{ccccc}0&0&1&0&0\\\
0&0&0&0&-1\\\ 1&0&0&0&0\\\ 0&0&0&0&0\\\
0&-1&0&0&0\end{array}\right),~{}T^{8}_{(as)}=\frac{-i}{2}\left(\begin{array}[]{ccccc}0&0&-1&0&0\\\
0&0&0&0&1\\\ 1&0&0&0&0\\\ 0&0&0&0&0\\\ 0&-1&0&0&0\end{array}\right),$
$T^{9}_{(as)}=\frac{-i}{2}\left(\begin{array}[]{ccccc}0&0&0&1&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ -1&0&0&0&-1\\\
0&0&0&1&0\end{array}\right),~{}T^{10}_{(as)}=\frac{1}{2}\left(\begin{array}[]{ccccc}0&0&0&-1&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ -1&0&0&0&1\\\ 0&0&0&1&0\end{array}\right).$

##  Appendix B More about chimera baryons on the lattice

In the discussion in the main text, we wrote the correlation function $C_{\rm
CB}(t-t_{0})$ involving the chimera baryon operator appearing in
$\mathcal{O}_{{\rm CB},\,4}$ and $\mathcal{O}_{{\rm CB},\,5}$ in Eq. (67). It
is worth checking that $C_{\rm CB}(t-t_{0})$ built from a different choice of
element of $\mathcal{O}_{\rm CB}\sim 4$ of the global $Sp(4)$ symmetry gives
rise to the same results. To this purpose, let us consider the combination of
the interpolating operators $\frac{1}{2}\left(\mathcal{O}_{{\rm
CB},1}+i\mathcal{O}_{{\rm CB},2}\right)$:

$\mathcal{O}^{k\,\gamma}_{\rm
CB}(x)=(\overline{Q^{1\,a}}(x)\gamma^{5}Q^{2\,b}(x))\Omega_{bc}\delta^{\gamma\delta}\Psi^{k\,ca}_{\delta}(x)\,,$
(165)

and its Dirac conjugate

$\overline{\mathcal{O}^{k}_{\rm
CB}}^{\,\gamma}(x)=\delta^{\gamma\delta}\overline{\Psi^{k\,ca}}_{\delta}(x)\Omega^{cb}(\overline{Q^{2\,b}}(x)\gamma^{5}Q^{1\,a}(x)).$
(166)

Then, the corresponding $2$-point correlation function is

$\displaystyle\langle\mathcal{O}^{k\,\gamma}_{{\rm
CB}}(x)\overline{\mathcal{O}^{k}_{{\rm CB}}}^{\,\gamma^{\prime}}(y)\rangle$
$\displaystyle=$
$\displaystyle-\Omega_{bc}\Omega^{c^{\prime}b^{\prime}}\delta^{\gamma\delta}\delta^{\gamma^{\prime}\delta^{\prime}}S^{k\,ca}_{\Psi\,\,\,\,\,c^{\prime}a^{\prime}\,\delta\delta^{\prime}}(x,y)S^{2\,b}_{Q\,\,\,b^{\prime}\,\beta\beta^{\prime}}(x,y)\gamma^{5\,\beta^{\prime}\alpha^{\prime}}S_{Q\,\,\,\,a\,\alpha^{\prime}\alpha}^{1\,a^{\prime}}(y,x)\gamma^{5\,\alpha\beta}\,.$
(167)

To see the equivalence between Eqs. (167) and (109) with the choice of
$(\Gamma^{1},\Gamma^{2})=(C\gamma^{5},\mathbb{1})$, we will use the following
properties. First of all, for a symplectic unitary matrix $\mathcal{U}\in
Sp(4)$:

$\Omega^{-1}\mathcal{U}\Omega=\mathcal{U}^{*}.$ (168)

We next consider the inverse of the fermion propagator in the Wilson-Dirac
formalism

$\displaystyle S_{Q}(x,y)^{-1}$ $\displaystyle=$ $\displaystyle\langle
Q(x)\overline{Q}(y)\rangle^{-1}$ (169) $\displaystyle=$
$\displaystyle\left(4+am_{0}^{f}\right)\delta_{xy}-\frac{1}{2}\sum_{\mu}\left((1-\gamma_{\mu})U_{\mu}^{(f)}(x)\delta_{x+\mu,y}+(1+\gamma_{\mu})U_{\mu}^{(f),\dagger}(y)\delta_{x,y+\mu}\right)\,.$

By applying the transpose and the charge conjugation operator to $S_{Q}^{-1}$,
we have

$\displaystyle C^{T}(S_{Q}(x,y)^{-1})^{T}C$ $\displaystyle=$
$\displaystyle\left(4+am_{0}^{f}\right)\delta_{xy}-\frac{1}{2}\sum_{\mu}\left((1+\gamma_{\mu})U^{(f),T}_{\mu}(x)\delta_{x+\mu,y}+(1-\gamma_{\mu})U_{\mu}^{(f),\ast}(y)\delta_{x,y+\mu}\right)\,.$
(170)

Using Eq. (168), we arrive at

$\displaystyle\Omega^{-1}C^{T}(S_{Q}(x,y)^{-1})^{T}C\Omega$ $\displaystyle=$
$\displaystyle\left(4+am_{0}^{f}\right)\delta_{xy}-\frac{1}{2}\sum_{\mu}\left((1+\gamma_{\mu})U^{(f),\dagger}_{\mu}(x)\delta_{x+\mu,y}+(1-\gamma_{\mu})U_{\mu}^{(f)}(y)\delta_{x,y+\mu}\right)$
(171) $\displaystyle=$ $\displaystyle S_{Q}(y,x)^{-1},$

which in turn implies that

$\Omega^{-1}C^{T}S_{Q}^{T}(x,y)C\Omega=S_{Q}(y,x).$ (172)

Using this result, with $\Gamma^{1}=C\gamma^{5}$ and $\Gamma^{2}=\mathbb{1}$,
we can rewrite Eq. (109) as

$\displaystyle\langle{\mathcal{O}^{k}_{{\rm
CB}}}^{\gamma}(x)\overline{\mathcal{O}^{k}_{{\rm
CB}}}^{\,\gamma^{\prime}}(y)\rangle$ $\displaystyle=$
$\displaystyle\Omega_{da}\Omega^{d^{\prime}a^{\prime}}\delta^{\gamma\delta}\delta^{\gamma^{\prime}\delta^{\prime}}S^{k\,cd}_{\Psi\,\,\,\,\,c^{\prime}d^{\prime}\,\delta\delta^{\prime}}(x,y)\,\times$
(173)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times{\rm
Tr\,}_{s}\left[S_{Q\,\,\,a^{\prime}}^{2\,a}(x,y)\gamma^{5}\left(\Omega^{-1}C^{T}\left(S_{Q}^{1}(x,y)\right)^{T}C\Omega\right)^{c^{\prime}}_{\,\,\,c}\gamma^{5}\right]$
$\displaystyle=$
$\displaystyle\Omega_{da}\Omega^{d^{\prime}a^{\prime}}\delta^{\gamma\delta}\delta^{\gamma^{\prime}\delta^{\prime}}S^{k\,cd}_{\Psi\,\,\,\,\,c^{\prime}d^{\prime}\,\delta\delta^{\prime}}(x,y)\,~{}{\rm
Tr\,}_{s}\left[S_{Q\,\,\,a^{\prime}}^{2\,a}(x,y)\gamma^{5}S^{1\,c^{\prime}}_{Q\,\,\,c}(y,x)\gamma^{5}\right].$

Comparing Eqs. (167) and (173), we conclude that the chimera propagators built
out of $\mathcal{O}_{{\rm CB},1(2)}$ and $\mathcal{O}_{{\rm CB},4(5)}$ are
identical to one another.

##  Appendix C Tables of numerical results

Table 3: Ensembles generated for the numerical study of finite volume effects reported in Section V.3. $N_{t}$ and $N_{s}$ are the temporal and spatial extents of the lattice, while $N_{\rm conf}$ and $\delta_{\rm traj}$ denote the number of configurations and the length of the Monte Carlo trajectory between adjacent configurations. In the last column, we show the average plaquette value $\langle{\cal P}\rangle$.  Ensemble |  $N_{t}\times N_{s}^{3}$ |  $N_{conf}$ |  $\delta_{\rm traj}$ | $\langle\cal P\rangle$
---|---|---|---|---
E1 | $36\times 8^{3}$ | 160 | 24 |  0.585758(87)
E2 | $48\times 12^{3}$ | 130 | 24 | 0.585447(51)
E3 | $48\times 16^{3}$ | 140 | 20 | 0.585233(34)
E4 | $48\times 18^{3}$ | 180 | 12 | 0.585234(22)
E5 | $48\times 20^{3}$ | 130 | 12 | 0.585137(20)
E6 | $48\times 24^{3}$ | 165 | 8 | 0.585148(13)
E7 | $54\times 28^{3}$ | 180 | 12 | 0.585144(11)

In this appendix, we tabulate some numerical information relevant to the
discussions in Sections V.3-V.5. The parameters of the lattice theory are
$\beta=6.5$, $am_{0}^{as}=-1.01$ and $am_{0}^{f}=-0.71$. The baryonic and
mesonic observables are measured using point and stochastic wall sources,
respectively. The numerical results are presented in lattice units.

In Table 3, we list the details characterising the ensembles used for our
investigations of finite volume effects. The ensembles denoted by E6 and E7
are also used for numerical studies of the chimera baryon and the combined
spectrum, respectively. We save configurations separated by $\delta_{\rm
traj}$ trajectories, after discarding a sufficient large number of initial
trajectories to allow for the thermalisation, so that those are independent to
each other. We determine $\delta_{\rm traj}$ by monitoring the average
plaquette values $\langle{\cal P}\rangle$, and chose it to be comparable to
one autocorrelation length.

Table 4: Numerical results for the masses and decay constants of pseudoscalar mesons (PS), and the masses of vector mesons (V), used to investigate finite volume effects in Section V.3. The constituent fermions are in the fundamental representation. The pseudoscalar mass at infinite volume, $am_{\rm PS}^{f,\,{\rm inf}}$, is the one extracted from the ensemble with the largest volume, $54\times 28^{3}$.  Ensemble | $am_{\rm PS}^{f}$ | $am_{\rm V}^{f}$ | $af_{\rm PS}^{f}$ | $m_{\rm PS}^{f,\,{\rm inf}}\,L$
---|---|---|---|---
E1 |  $0.7488(64)$ |  $0.7982(72)$ |  $0.0349(20)$ |  $2.8783(76)$
E2 | $0.5171(48)$ | $0.5685(51)$ | $0.0419(17)$ | $4.317(11)$
E3 | $0.3849(45)$ | $0.4238(58)$ | $0.0427(14)$ | $5.757(15)$
E4 | $0.3778(22)$ | $0.4290(24)$ | $0.0461(11)$ | $6.476(17)$
E5 | $0.3702(16)$ | $0.4142(22)$ | $0.05151(88)$ | $7.196(19)$
E6 | $0.3640(19)$ | $0.4067(20)$ | $0.04992(87)$ | $8.635(23)$
E7 | $0.35979(95)$ | $0.4009(11)$ | $0.05058(61)$ | $10.074(27)$
Table 5: Numerical results for the masses and decay constants of pseudoscalar mesons (PS), and the masses of vector mesons (V), used to investigate finite volume effects in Section V.3. The constituent fermions are in the antisymmetric representation. We also list, in the last column, the mass of chimera baryons with positive parity.  Ensemble | $am_{\rm PS}^{as}$ | $am_{\rm V}^{as}$ | $af_{\rm PS}^{as}$ | $am^{+}_{\rm CB}$
---|---|---|---|---
E1 |  $0.4277(53)$ |  $0.4411(60)$ |  $0.0843(32)$ |  $1.012(16)$
E2 | $0.5499(35)$ | $0.5814(47)$ | $0.0781(20)$ | $0.927(15)$
E3 | $0.5858(21)$ | $0.6241(33)$ | $0.0767(15)$ | $0.768(13)$
E4 | $0.5956(14)$ | $0.6395(21)$ | $0.0794(12)$ | $0.7974(72)$
E5 | $0.6017(10)$ | $0.6491(15)$ | $0.08349(96)$ | $0.7803(60)$
E6 | $0.6023(12)$ | $0.6481(14)$ | $0.0805(12)$ | $0.7654(50)$
E7 | $0.60205(92)$ | $0.6450(15)$ | $0.08313(88)$ | $0.7636(28)$
Table 6: Numerical results for the masses of mesons in additional spin-0 and spin-1 channels, sourced by the interpolating operators in Eq. (93). The representation of the constituent fermions are denoted by superscripts $f$ and $as$. The measurements are performed on the ensemble with the largest-volume, E7. Ensemble | $am_{\rm T}^{f}$ | $am_{\rm AV}^{f}$ | $am_{\rm AT}^{f}$ | $am^{f}_{\rm S}$ | $am_{\rm T}^{as}$ | $am_{\rm AV}^{as}$ | $am_{\rm AT}^{as}$ | $am^{f}_{\rm S}$
---|---|---|---|---|---|---|---|---
E7 |  $0.3995(13)$ |  $0.544(10)$ |  $0.543(11)$ |  $0.508(12)$ |  $0.6461(14)$ |  $0.859(12)$ |  $0.889(14)$ |  $0.843(11)$

In Tables 4 and 5, we present the results of the measurements of the masses of
the pseudoscalar (PS) and vector (V) mesons composed of fermionic constituents
in the fundamental and antisymmetric representations, and the decay constant
of the pseudoscalar meson. We also show the mass of the chimera baryon (CB)
with positive parity, and $m_{\rm PS}^{f,\,{\rm inf}}\,L$—$am_{\rm
PS}^{f,\,{\rm inf}}$ is extracted from the measurement on the ensemble with
the largest available lattice.

In Table 6, we present the numerical results for the masses of the other
mesons in the spin-0 and spin-1 channels, besides to the ones we have already
presented in Tables 4 and 5. These are sourced by the tensor (T), axial-vector
(AV), axial-tensor (AT) and scalar (S) interpolating operators defined with
the gamma structures in Eq. (93). These measurements have been carried out by
using ensemble E7, the one that has the largest volume.

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