SMSupplementary References

Topological semimetals without quasiparticles

Haoyu Hu1, Lei Chen1, Chandan Setty1, Mikel Garcia-Diez2,3, Sarah E. Grefe4,

Andrey Prokofiev5, Stefan Kirchner6,7, Maia G. Vergniory2,8, Silke Paschen5,1,

Jennifer Cano9,10, and Qimiao Si1,∗

1Department of Physics and Astronomy, Rice Center for Quantum Materials, Rice
University, Houston, Texas, 77005, USA

2Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018
Donostia-San Sebastian, Spain

3Department of Physics, University of the Basque Country (UPV-EHU), Bilbao,
Spain

4Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA

5Institute of Solid State Physics, TU Wien, Wiedner Hauptstr. 8-10, 1040
Vienna, Austria

6Department of Electrophysics, National Yang Ming Chiao Tung University,
Hsinchu 30010, Taiwan

7Center for Emergent Functional Matter Science, National Yang Ming Chiao Tung
University, Hsinchu 30010, Taiwan

8Max Planck Institute for Chemical Physics of Solids, Noethnitzer Str. 40,
01187 Dresden, Germany

9Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY
11794, USA

10Center for Computational Quantum Physics, Flatiron Institute, New York, NY
10010, USA

The interplay between interactions and topology in quantum materials is of
extensive current interest. Strong correlations are known to be important for
insulating topological states, as exemplified by the fractional quantum Hall
effect. For the metallic case, whether and how they can drive topological
states that have no free-electron counterparts is an open and pressing
question. We introduce a general framework for lattice symmetries to constrain
single-particle excitations even when they are not quasiparticles, and
substantiate it in a periodic Anderson model with two channels of conduction
electrons. We demonstrate that symmetry constrains correlation-induced
emergent excitations to produce non-Fermi liquid topological phases. The loss
of quasiparticles in these phases is manifested in a non-Fermi liquid form of
spectral and transport properties, whereas its topological nature is
characterized by surface states and valley and spin Hall conductivities. We
also identify candidate materials to realize the proposed phases. Our work
opens a door to a variety of non-Fermi liquid topological phases in a broad
range of strongly correlated materials.

States of matter and their transitions have traditionally been classified in
terms of Landau’s order-parameter framework, which is based on global symmetry
and its spontaneous breaking (?). For free-electron systems, typically with
large spin-orbit couplings, the topological paradigm distinguishes electronic
states in terms of certain topological invariants associated with their
quantum-mechanical wavefunctions (?). The electronic topology is responsible
for various novel phenomena, such as unusual Hall effects and development of
nontrivial surface states. In recent years, much progress has been made in
realizing metallic topological phases, including those that harbor Weyl and
Dirac fermions (?, ?). Crystalline symmetries play an important role in
stabilizing such topological semimetals (?, ?, ?, ?, ?, ?).

In metallic topological states, electron correlations are typically treated
perturbatively (?, ?). The pursuit of strongly correlated topological
semimetals is largely in its infancy (?, ?). Nonetheless, recent studies –
both theoretical (?, ?) and experimental (?, ?) – have demonstrated that
correlations drive composite fermions with highly renormalized topological
nodal excitations that are pinned to the Fermi energy. An important ingredient
in realizing the correlated topological semimetals is that strong correlations
and space-group symmetry cooperate (?). As exemplified by the fractional
quantum Hall effect (?), strong correlations can lead to insulating
topological states that do not have noninteracting counterparts. In the
metallic case, realizing such topological states represents an outstanding
challenge.

Here we advance the notion that space-group symmetry constrains emergent (low-
energy) electronic excitations, which are induced by strong correlations and
particularly electron fractionalization, to produce non-Fermi liquid
topological phases. Such phases lose Landau quasiparticles and have no free-
electron counterpart. Figure 1A illustrates the idea.

We start by providing a general theoretical foundation for the lattice
symmetries of interacting systems to constrain dispersive electronic
excitations, even when they are no longer quasiparticles. The Green’s
functions describe the electronic (single-particle) excitations of an
interacting electron system. The Green’s function matrix $[G]$ can be
expressed in terms of their eigenvectors, $v_{i}(\omega,\bm{k})$, and
eigenfunctions (?); the imaginary part of the eigenfunctions corresponds to
the spectral functions, with their peaks describing dispersive electronic
excitations regardless of whether they are quasiparticles. Consider the action
of an element, $h$, of the space group $H$ on an electron annihilation
operator $\psi$ in the space of internal quantum numbers ($a$), frequency
($\omega$) and wave vector ($\bm{k}$):

$\displaystyle h\in
H:\,\,\psi_{a,i\omega,\bm{k}}\rightarrow\sum_{b}[U_{h}^{\dagger}(\bm{k})]_{ab}\psi_{b,i\omega,h\bm{k}}\,,$
(1)

where $U_{h}(\bm{k})$ is a unitary matrix. It follows from symmetry that
$[\mathcal{U}_{h}v_{i}]_{(a,i\omega,\bm{k})}$ is also an eigenvector of $[G]$,
where
$[\mathcal{U}_{h}]_{(a,i\omega,\bm{k}),(b,i\omega^{\prime},\bm{k}^{\prime})}=\delta_{\omega,\omega^{\prime}}\delta_{\bm{k}^{\prime},h\bm{k}}U_{h}(\bm{k})$.
[The details are described in the Supplementary Materials (SM), Sec. A.]

We therefore reach the important conclusion that the set of all eigenvectors
$\\{v_{i}\\}_{i=1,2,...}$ form a representation space of the space group $H$.
This sets the stage to examine the dispersive modes through the subset of
eigenvectors that form certain irreducible representations of the space group
associated with a given wavevector $\bm{k}$, similarly as for the Bloch
functions in the case of noninteracting electrons (?) (SM, Sec. A). As we tune
the wavevector to some high symmetry points, where additional symmetry leads
to a two or higher dimensional irreducible representation, the entire
(frequency-dependent) spectral functions of the corresponding dispersive modes
must be degenerate. The same conclusion is reached if, alternatively, as we
continuously tune the wavevector, the peaks of two dispersive modes from two
different irreducible representations cross; this node cannot be removed by
symmetry preserving disruptions. As such, our procedure provides a robust
route to correlated topological semimetals, including ones that cannot be
adiabatically connected to free-electron systems.

We now turn to a proof-of-principle demonstration of this general framework in
a well-defined model. Our focus is on a periodic Anderson model, in which
correlated electrons act as local moments that are coupled to two degenerate
channels of conduction electrons. As illustrated in Fig. 1B, the local degrees
of freedom at any given site correspond to a doublet in the ground state
manifold, with (pseudo-)spins $\sigma=\uparrow,\downarrow$, which hybridizes
with two channels of conduction electrons, labeled by $\alpha=1,2$, via an
excited doublet. Two channels of Kondo couplings lead to frustration in the
development of Kondo entanglement, providing a means of fractionalizing the
electronic excitations (?, ?, ?, ?). The model, thus, provides an ideal
setting to explore lattice symmetry constraints on emergent excitations for
non-Fermi liquid topology. Further details of the model and solution method
are provided in the SM (Secs. B,C,D).

To address the symmetry constraints on emergent excitations, we choose to
analyze the model in a three-dimensional (3D) kagome lattice, where the space
group allows for protected nodes and, in addition, whose inversion symmetry
breaking can be readily tuned (SM, Secs. B,C). Here, $c_{i,\sigma,\alpha}$
($d_{i,\sigma,\alpha})$ denotes a conduction (local) electron, describing the
physical $spd$\- ($f$-) electron, on site $i$, which marks both the unit cell
position and sub-lattice index. We solve the model within dynamical mean-field
theory (DMFT) (?), while ensuring that our results capture the asymptotically-
exact $\sqrt{|\omega|}$ low-energy behavior in the multi-channel Kondo model
(?, ?). The Green’s function of the conduction electrons are calculated via
$G_{c}(\omega,\bm{k})=[\omega-t_{\bm{k}}-\Sigma_{c}(\omega)]^{-1}$. Here,
$t_{\bm{k}}$ is the hopping matrix, and $\Sigma_{c}(\omega)$ is the local
self-energy that encodes the correlation effects on the single-electron
excitations. We solve the dynamical equations in real frequency and reach
extremely low temperatures (compared to $T_{K}$, the Kondo temperature, which
is the dynamically generated low-energy scale) (SM, Sec. D) in order to reveal
the asymptotic low-energy behavior.

The imaginary part of the conduction-electron self-energy, ${\rm
Im}\Sigma_{c}(\omega)$, is found to depend on frequency in a square-root
fashion in the low-energy limit (SM, Fig. S2A). The $\sqrt{|\omega|}$ form
ensures that the spectral width of any dispersive mode will become
progressively larger than its energy on approach of the Fermi energy, in
contrast to what happens when the electronic excitations are quasiparticles.
It describes a non-Fermi liquid, as characterized by an electrical resistivity
that depends on temperature ($T$) in a $\sqrt{T}$ (SM, Fig. S2B). These
asymptotic results are found only when our calculations are done at
sufficiently low temperatures (SM, Sec. E). The asymptotic form of the self-
energy, however, is not adequate: On its own, it would not lead to new
dispersive modes near the Fermi energy. Importantly, we find that the self-
energy is described by a pole in the complex energy plane, with relevant
energy scales of the order $T_{K}$. This pole, in turn, gives rise to narrow
bands that are pinned to the Fermi energy (Secs. E,F and Fig. S6). The results
for the narrow bands are shown in Fig. 2A.

In the above sense, the dispersive modes near the Fermi energy are emergent
electronic excitations. Crucially, they go beyond Landau quasiparticles: For
reasons described earlier and as seen in Fig. 2A, any peak of the spectral
function vs. energy for a given wavevector $\bm{k}$ has a width that is always
larger than the peak energy as measured from the Fermi energy; this persists
on approach of the Fermi energy, and is opposite to what happens with Landau
quasiparticles. The disperse modes capture the development in the spectral
weight of the physical $f$-electrons (represented in the model by the
$d$-operators) in the immediate vicinity of the Fermi energy (SM, Fig. S4A):
they appear within a range of the Kondo energy.

What is remarkable is that these dispersive modes form Weyl nodes which are
bound near to the Fermi energy, as seen in Fig. 2A (along the high symmetry
line, $-H^{\prime}$-$K$-$H$, of the Brillouin zone, as marked by the red line
that is specified in Figs. 2B,C). They comprise two close-by branches, which
are illustrated more clearly in Fig. 2B and Fig. 2C, respectively. To
understand the Weyl nodes, we analyze the structure of the interacting Green’s
function in terms of the Green’s function eigenvectors and eigenfunctions. As
discussed earlier (and in the SM, Sec. A), the eigenvectors are labeled by the
wave vector $\bm{k}$ and are subject to the same crystalline symmetry
constraints as Bloch functions. Indeed, the two dispersive modes shown in Fig.
2A have different symmetry eigenvalues and form a symmetry-protected node (see
the SM, Sec. H); a similar mechanism produces additional non-Fermi-liquid
Dirac nodes along the high symmetry line $-A$-$\Gamma$-$A$. In other words, in
this model, a robust mechanism underlies the emergence of the Weyl nodes and
the associated nodal excitations that go beyond quasiparticles.

We characterize the topological nature of these non-Fermi liquid phases in
several ways. We start by analyzing the surface states. Their spectral
functions are plotted in Fig. 3A; their exponential decay from the surface to
the interior of the system is demonstrated in the SM (Sec. I and Fig. S8A). In
addition, in Fig. 3B, we show the in-plane spin-Hall conductivity as a
function of $k_{z}$ (see the SM, Sec. J), which implicates the topological
nature of the Weyl nodes. The overall spin-Hall conductivity has a $\sqrt{T}$
dependence, as shown in Fig. 3C. At the same time, the nonlinear Hall effect
is found to display a singular $\frac{1}{\sqrt{T}}$ dependence below the Kondo
temperature scale (see the SM, Sec. K). We also expect the valley Hall
conductivity to have a $\sqrt{T}$ dependence, as explicitly calculated for a
model that shows Dirac nodal states without quasiparticles (see the SM, Fig.
S7E). The surface states and the various Hall effects also provide the means
to identify the topological semimetal phases experimentally.

Having advanced genuinely interacting Weyl semimetal phases, we now turn to
their materials realization. So far we have analyzed the two-channel periodic
Anderson model on the 3D kagome lattice, by assuming the existence of the
pertinent local degrees of freedom as illustrated in Fig. 1B. From the model
studies, we gain the key insight that the form of the electron self-energy is
insensitive to the underlying lattice, provided the required local degrees of
freedom are in place; for example, its form on the 2D kagome or 3D cubic
lattice is essentially the same (see the SM, Fig. S5) as what we have
described for the 3D kagome lattice case. We thus look for space group
symmetries that constrain nodes in the dispersion while also allowing for the
local degrees of freedom of Fig. 1B.

This prescription has led us to three materials, the centrosymmetric PrBi (?)
and PrFe4P12 (?) as well as the noncentrosymmetric UPt3Au2 (?), with the space
groups $\\#225$, $\\#204$, and $\\#216$, respectively, and the point group
symmetries $O_{h}$, $T_{h}$, and $T_{d}$ at the Pr/U site. The Pr3+ and U4+
ions have the electronic configuration $4f^{2}$ which, for these point group
symmetries, can lead to a $\Gamma_{3}$ non-Kramers doublet ground state. A
non-Kramers doublet can be treated as pseudo-spin degree of freedom. The first
excited levels are then Kramers doublets, whose spin degrees of freedom
provide the pseudo-channel index (?). PrBi has a low charge carrier
concentration (?), PrFe4P12 can be readily tuned to such a state (?), and
UPt3Au2 (?) exhibits a very weakly temperature-dependent resistivity, all
suggestive of semimetallic ground states (see the SM, Sec. L, for further
details). Our analysis of space group symmetry constraints suggests the
emergence of symmetry protected Dirac nodes in PrBi and PrFe4P12 and symmetry
protected Weyl nodes in UPt3Au2 (see the SM, Sec. M). We have verified our
symmetry argument in the representative case of PrBi: Its electronic structure
of the non-$f$ conduction electrons (SM, Sec. N) is shown in Fig. S11;
symmetry-protected Dirac points away from the Fermi energy are indeed found
along the $\Gamma$-$X$ line (which also implies that the Kondo-generated
emergent excitations in PrBi will have Dirac nodes near the Fermi energy).
Thus, to realize the proposed correlated gapless topological phases, we put
forward the three as candidate materials for systematic low-temperature
studies in their respective phase diagrams.

We close with several important remarks. Firstly, we have already discussed
the robustness of our theoretical results for the two-channel Kondo/Anderson
lattice models, in that we recover the asymptotically exact low-energy
behavior of the self-energy while also uncovering the emergence of a pole in
the self-energy. Furthermore, the low-temperature non-Fermi liquid behavior we
have presented in the models is consistent with experimental results observed
in some (generic-lattice, in particular cubic) heavy-fermion metals that host
non-Kramers doublets in the ground-state manifold and, hence, two-channel
Kondo lattices (?). Secondly, the loss of quasiparticles in our case is
associated with emergent nodal states created by the Kondo effect near the
Fermi energy. This is in contrast to the perturbative effect of interactions
on pre-existing nodal states of conduction electrons (?, ?, ?). Thirdly, there
can be additional broken-symmetry states at low temperatures, following the
paradigm in strongly correlated systems that non-Fermi liquids nucleate novel
phases (?). For generic lattices, the multi-channel Kondo correlations may
lead to broken-symmetry phases such as superconductivity (?), multipolar order
(?) or even time-reversal-symmetry-broken “hastatic” order (?). It will be
instructive to explore any new physics that the nontrivial topology brings in
these states. Fourthly, the role of electron fractionalization can be further
explored in the proposed non-Fermi liquid topological semimetals. The two-
channel Kondo effect fractionalizes electrons into Majorana fermions (?),
which underlie the loss of Landau quasiparticles. Finally, we have emphasized
the constraints of lattice symmetries on dispersive modes associated with
peaks of the electron spectral functions. Because it keeps track of both
energy and wave vector, our approach will likely allow for constraining the
nature of topological states with other types of features in correlations,
such as zeros of Green’s functions in topological Mott states (?, ?).

To summarize, our work has advanced the first topological semimetal that has
emergent electronic topological excitations beyond quasiparticles and that has
no free-electron counterpart. We have advanced a general framework, viz. to
place symmetry constraints on the dispersive modes of single-particle Green’s
functions, which allows us to establish the development of non-Fermi liquid
topological semimetals in a robust way. In specific lattices, we have shown
how their space group symmetries constrain the two-channel Kondo correlations
to produce Weyl (and Dirac) nodal states that are pinned near the Fermi energy
but are not quasiparticles. Guided by the considerations of both symmetry and
strong correlations, we have proposed several materials to realize these non-
Fermi liquid topological semimetals. The proposed approach has the potential
to help realize gapless topological phases without free-electron counterparts
in a variety of other strongly correlated materials and structures.

## References

  * 1. L. Landau, Nature 138, 840 (1936).
  * 2. D. Xiao, M.-C. Chang, Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
  * 3. N. P. Armitage, E. J. Mele, A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018).
  * 4. N. Nagaosa, T. Morimoto, Y. Tokura, Nature Reviews Materials 5, 621 (2020).
  * 5. B. Bradlyn, et al., Nature 547, 298 (2017).
  * 6. J. Cano, et al., Phys. Rev. B 97, 035139 (2018).
  * 7. H. C. Po, A. Vishwanath, H. Watanabe, Nature Communications 8, 50 (2017).
  * 8. H. Watanabe, H. C. Po, M. P. Zaletel, A. Vishwanath, Phys. Rev. Lett. 117, 096404 (2016).
  * 9. B. Bradlyn, et al., Science 353 (2016).
  * 10. J. Cano, B. Bradlyn, Annu. Rev. Condens. Matter Phys. 12, 225 (2021).
  * 11. W. Witczak-Krempa, G. Chen, Y. B. Kim, L. Balents, Annu. Rev. Condens. Matter Phys. 5, 57 (2014).
  * 12. S. Paschen, Q. Si, Nat. Rev. Phys. 3, 9 (2021).
  * 13. H.-H. Lai, S. E. Grefe, S. Paschen, Q. Si, Proceedings of the National Academy of Sciences 115, 93 (2018).
  * 14. S. E. Grefe, H.-H. Lai, S. Paschen, Q. Si, Phys. Rev. B 101, 075138 (2020).
  * 15. S. Dzsaber, et al., Phys. Rev. Lett. 118, 246601 (2017).
  * 16. S. Dzsaber, et al., Proceedings of the National Academy of Sciences 118 (2021).
  * 17. L. Chen, et al., arXiv preprint arXiv:2107.10837; Nat. Phys., in press (2021).
  * 18. H. L. Stormer, D. C. Tsui, A. C. Gossard, Rev. Mod. Phys. 71, S298 (1999).
  * 19. A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Methods of quantum field theory in statistical physics (Courier Corporation, 2012).
  * 20. S. Kirchner, Advanced Quantum Technologies 3, 1900128 (2020).
  * 21. I. Affleck, A. W. W. Ludwig, Phys. Rev. B 48, 7297 (1993).
  * 22. V. J. Emery, S. Kivelson, Phys. Rev. B 46, 10812 (1992).
  * 23. J. M. Maldacena, A. W. Ludwig, Nuclear Physics B 506, 565 (1997).
  * 24. G. Kotliar, D. Vollhardt, Physics today 57, 53 (2004).
  * 25. D. L. Cox, A. E. Ruckenstein, Phys. Rev. Lett. 71, 1613 (1993).
  * 26. X. He, et al., Phys. Rev. B 101, 075106 (2020).
  * 27. H. Hidaka, et al., Phys. Rev. B 71, 073102 (2005).
  * 28. C. Quitmann, et al., Phys. Rev. B 38, 6432 (1988).
  * 29. D. L. Cox, A. Zawadowski, Advances in Physics 47, 599 (1998).
  * 30. A. Sakai, S. Nakatsuji, J. Phys. Soc. Japan 80, 063701 (2011).
  * 31. E.-G. Moon, C. Xu, Y. B. Kim, L. Balents, Phys. Rev. Lett. 111, 206401 (2013).
  * 32. M. Tsujimoto, Y. Matsumoto, T. Tomita, A. Sakai, S. Nakatsuji, Phys. Rev. Lett. 113, 267001 (2014).
  * 33. P. Chandra, P. Coleman, R. Flint, Nature 493, 621 (2013).
  * 34. T. Morimoto, N. Nagaosa, Sci. Rep. 6, 19853 (2016).
  * 35. P. Mai, B. Feldman, P. W. Phillips, arXiv preprint arXiv:2207.01638 (2022).
  * 36. V. Gurarie, Phys. Rev. B 83, 085426 (2011).
  * 37. Z. Wang, X.-L. Qi, S.-C. Zhang, Phys. Rev. B 85, 165126 (2012).
  * 38. P. Nozieres, A. Blandin, Journal de Physique 41, 193 (1980).
  * 39. M. Pustilnik, L. Borda, L. I. Glazman, J. von Delft, Phys. Rev. B 69, 115316 (2004).
  * 40. A. J. Keller, et al., Nature 526, 237 (2015).
  * 41. Z. Iftikhar, et al., Nature 526, 233 (2015).
  * 42. S.-S. Yeh, et al., Nature Communications 11, 4749 (2020).
  * 43. A. Cai, Z. Yu, H. Hu, S. Kirchner, Q. Si, Phys. Rev. Lett. 124, 027205 (2020).
  * 44. A. Georges, G. Kotliar, Q. Si, International Journal of Modern Physics B 06, 705 (1992).
  * 45. J. Kroha, P. Wölfle, Advances in Solid State Physics 39 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1999), pp. 271–280.
  * 46. M. H. Hettler, J. Kroha, S. Hershfield, Phys. Rev. B 58, 5649 (1998).
  * 47. F. B. Anders, Phys. Rev. B 71, 121101 (2005).
  * 48. M. Jarrell, H. Pang, D. L. Cox, K. H. Luk, Phys. Rev. Lett. 77, 1612 (1996).
  * 49. Y. Michishita, R. Peters, Phys. Rev. B 103, 195133 (2021).
  * 50. Z. Z. Du, C. M. Wang, H.-P. Sun, H.-Z. Lu, X. C. Xie, Nature Communications 12, 5038 (2021).
  * 51. A. Vashist, R. K. Gopal, D. Srivastava, M. Karppinen, Y. Singh, Phys. Rev. B 99, 245131 (2019).
  * 52. Z. Wu, et al., Phys. Rev. B 99, 035158 (2019).
  * 53. S. H. Curnoe, H. Harima, K. Takegahara, K. Ueda, Phys. Rev. B 70, 245112 (2004).
  * 54. H. Sugawara, et al., Phys. Rev. B 66, 134411 (2002).
  * 55. G. Kresse, J. Hafner, Phys. Rev. B 48, 13115 (1993).
  * 56. G. Kresse, J. Furthmüller, Computational Materials Science 6, 15 (1996).
  * 57. A. D. Becke, E. R. Johnson, The Journal of chemical physics 124, 221101 (2006).
  * 58. F. Tran, P. Blaha, Phys. Rev. Lett. 102, 226401 (2009).
  * 59. G. Kresse, D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * 60. D. Hobbs, G. Kresse, J. Hafner, Phys. Rev. B 62, 11556 (2000).

Acknowledgments: Work at Rice has been supported by the the National Science
Foundation under Grant No. DMR-2220603 (L.C. and Q.S.), the Air Force Office
of Scientific Research under Grant No. FA9550-21-1-0356 (C.S.) and the Robert
A. Welch Foundation Grant No. C-1411 (H.H.). The majority of the computational
calculations have been performed on the Shared University Grid at Rice funded
by NSF under Grant EIA-0216467, a partnership between Rice University, Sun
Microsystems, and Sigma Solutions, Inc., the Big-Data Private-Cloud Research
Cyberinfrastructure MRI-award funded by NSF under Grant No. CNS-1338099, and
the Extreme Science and Engineering Discovery Environment (XSEDE) by NSF under
Grant No. DMR170109. M.G.-D. and M.G.V. acknowledge support from the Spanish
Ministry of Science and Innovation Grant No. PID2019-109905GB-C21 and the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) GA 3314/1-1
– FOR 5249 (QUAST). Work at Los Alamos was carried out under the auspices of
the U.S. Department of Energy (DOE) National Nuclear Security Administration
under Contract No. 89233218CNA000001, and was supported by LANL LDRD Program.
A.P. and S.P. acknowledge funding from the Austrian Science Fund (projects No.
I4047 and 29279), the European Microkelvin Platform (H2020 project No. 824109)
and the FWF via the Research unit QUAST-FOR5249. S.K. was in part supported by
the Ministry of Science and Technology, Taiwan (grant No. MOST
111-2634-F-A49-007) and the Center for Emergent Functional Matter Science of
National Yang Ming Chiao Tung University from “The Featured Areas Research
Center Program” within the framework of the “Higher Education Sprout Project
by the Ministry of Education (MOE)” in Taiwan, by the Yusha Fellowship Program
of the MOE Taiwan, and by the National Key R&D Program of the MOST of China,
Grant No. 2016YFA0300202 and the National Science Foundation of China, Grant
No. 11774307. J.C. acknowledges the support of the National Science Foundation
under Grant No. DMR-1942447 and the support of the Flatiron Institute, a
division of the Simons Foundation. Four of us (S.G., S.P., J.C. and Q.S.)
acknowledge the hospitality of the Aspen Center for Physics, which is
supported by NSF grant No. PHY-1607611.

Authors contributions: All authors contributed to the research of the work and
the writing of the paper.

Competing interests: The authors have no competing interests.

Data availability: The data that support the findings of this study are
available from the corresponding author upon reasonable request.

∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>
Figure 1:  Illustration of the proposed route for electronic topological
phases without quasiparticles.  (A) An approach for topological phases without
free-electron counterparts: Lattice symmetries constrain emergent excitations
that are induced by strong correlations and particularly fractionalization.
(B) Strongly correlated local degrees of freedom involved in the two-channel
Anderson lattice model.

Figure 2:  Emergence of Weyl nodal excitations beyond quasiparticles. (A) The
$d$ electron spectral functions ($\rho_{d}$) at various $\bm{k}$ points
(marked by the red bar on the right) along a high symmetry line in the 3D
kagome lattice. Solid and dashed lines denote two dispersive modes from two
eigenvalues of the Green’s function. The blue and orange dots label the
positions of the spectral peaks, which meet at a Weyl point. Each $\rho_{d}$
curve has been shifted vertically to avoid overlapping. (B),(C) illustrate the
spectral functions of the two Weyl-point-bearing branches. Here, the dashed
line denotes the energy spectral peaks. The red bar marks the cut of wave
vectors in the Brillouin zone along which the spectral functions are shown in
panel A. The calculations are done at the temperature $T=10^{-3}T_{K}$.

Figure 3:  Characterizing the topological nature of the correlated phase. (A)
The spectrum of the edge mode at $k_{x}=4\pi/3$ in the 3D kagome lattice at
$T/T_{K}=10^{-3}$ with an open boundary along the $y$ direction; the two red
dots mark the Weyl nodes; the dashed line denotes the energy spectral peak.
(B) In-plane spin-Hall conductivity $\sigma_{xy}^{s}$ as a function of $k_{z}$
in the 3D kagome model at $T/T_{K}=10^{-3}$. The vertical solid/dashed lines
mark the $k_{z}$ coordinates of the planes in which the Weyl/Dirac nodes
reside. A rapid change (in the form of a smeared jump) occurs when $k_{z}$
crosses the Weyl/Dirac-node-hosting planes. (C) The total spin Hall
conductivity of the 3D kagome model showing a $\sqrt{T}$ dependence. Here, the
extrapolated zero-temperature value ($\sim 0.25$) is subtracted in order to
highlight the nature of the $T$-dependent component.

Supplementary Materials

Topological semimetals without quasiparticles

Haoyu Hu, Lei Chen, Chandan Setty, Mikel Garcia-Diez, Sarah E. Grefe, Andrey
Prokofiev,
Stefan Kirchner, Maia G. Vergniory, Silke Paschen, Jennifer Cano, and Qimiao
Si∗

∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>

Materials and Methods
Figs. S1–S11
References (36–60, see above)

## Materials and Methods

### A. Symmetry of Green’s function: Space group symmetry constraints on
interacting systems

We study the symmetry properties of the Green’s functions:

$\displaystyle[G(\tau,\bm{r},\tau^{\prime},\bm{r^{\prime}})]_{ab}=-\langle
T_{\tau}\psi_{a,\bm{r},\tau}\psi^{\dagger}_{b,\bm{r},\tau^{\prime}}\rangle\,,$

where $\psi_{a,\bm{r},\tau}^{\dagger}$ creates an electron with index $a$,
which characterizes its spin, orbital and sublattice, at position $\bm{r}$ and
imaginary time $\tau$. We will first prove that, in a quantum phase with
symmetry of the space group $H$, the eigenvectors of the Green’s function form
a representation of both $H$ and the corresponding little group $H_{\bm{k}}$.
Based on these symmetry properties, we can analyze the symmetry-protected
crossing of single-particle excitations.

In general, it is more convenient to work with Matsubara frequency and wave
vector by introducing Fourier transformation of electron operators:

$\displaystyle\psi_{a,i\omega,\bm{k}}$ $\displaystyle=$
$\displaystyle\frac{1}{\beta
N}\sum_{\bm{r}}\int_{0}^{\beta}\psi_{a,\bm{r},\tau}e^{i\omega\tau-i\bm{k}\cdot\bm{r}}d\tau\,,$

where $N$ is the total number of unit cells and $\beta$ is the inverse
temperature. The Fourier transformation leads to following Green’s function,
$[G]_{(a,i\omega,\bm{k}),(b,i\omega^{\prime},\bm{k^{\prime}})}=-\langle\psi_{a,i\omega,\bm{k}}\psi_{b,i\omega^{\prime},\bm{k^{\prime}}}^{\dagger}\rangle$,
which can be treated as a matrix with row index $(a,i\omega,\bm{k})$ and
column index $(b,i\omega^{\prime},\bm{k^{\prime}})$. As a matrix, we can
introduce the eigenvectors and eigenvalues:

$\displaystyle\frac{1}{N\beta}\sum_{b,i\omega^{\prime},\bm{k^{\prime}}}[G]_{(a,i\omega,\bm{k}),(b,i\omega^{\prime},\bm{k^{\prime}})}[v_{i}]_{(b,i\omega^{\prime},\bm{k^{\prime}})}=g_{i}[v_{i}]_{(a,i\omega,\bm{k})}\,,$

where $g_{i}$ and $[v_{i}]_{(a,i\omega,\bm{k})}$ denote the $i$-th eigenvalue
and $i$-th eigenvector.

We then prove the set of all eigenvectors $\\{v_{i}\\}_{i=1,2,...}$, form a
representation space of the space group $H$ whose element is defined as

$\displaystyle h\in H$ $\displaystyle:$
$\displaystyle\psi_{a,i\omega,\bm{k}}\rightarrow\sum_{b}[U_{h}^{\dagger}(\bm{k})]_{ab}\psi_{b,i\omega,h\bm{k}}\,,$

where $U_{h}(\bm{k})$ is a unitary matrix. The symmetry of the system requires
the Green’s function to satisfy

$\displaystyle[\mathcal{U}_{h}^{{\dagger}}\,G\,\mathcal{U}_{h}]_{(a,i\omega,\bm{k}),(b,i\omega^{\prime},\bm{k^{\prime}})}=[G]_{(a,i\omega,\bm{k}),(b,i\omega^{\prime},\bm{k^{\prime}})}\,,$

where
$[\mathcal{U}_{h}]_{(a,i\omega,\bm{k}),(b,i\omega^{\prime},\bm{k}^{\prime})}=\delta_{\omega,\omega^{\prime}}\delta_{\bm{k}^{\prime},h\bm{k}}U_{h}(\bm{k})$.
Consequently, $[\mathcal{U}_{h}v_{i}]_{(a,i\omega,\bm{k})}$ is also an
eigenvector of the Green’s function matrix $[G]$ and we can then introduce the
matrix representation $\mathcal{D}[h]$ of $h\in H$ via

$\displaystyle[\mathcal{U}_{h}\,v_{i}]_{a,i\omega,\bm{k}}=\sum_{j}\mathcal{D}[h]_{ij}\,[v_{j}]_{a,i\omega,\bm{k}}\,.$

Therefore, the set of all eigenvectors $\\{v_{i}\\}_{i=1,2,...}$ form a
representation of space group $H$ with matrix representation $\mathcal{D}[h]$.

In practice, the systems we deal with hold space and time translation
symmetry, which allows us to decompose $G$ into block diagonal forms

$\displaystyle[G]_{(a,i\omega,\bm{k}),(b,i\omega^{\prime},\bm{k^{\prime}})}=\delta_{\omega,\omega^{\prime}}\delta_{\bm{k},\bm{k^{\prime}}}[G(i\omega,\bm{k})]_{ab}\,.$

We also perform an analytical continuation by replacing $i\omega$ with
$\omega+i0^{+}$ and work with the retarded Green’s function. Due to the block
diagonalized structure, it’s enough to consider the eigenvalues and
eigenvectors of each block characterized by $(\omega,\bm{k})$ (?, ?, ?):

$\displaystyle\sum_{b}[G(\omega,\bm{k})]_{ab}[v_{i}(\omega,\bm{k})]_{b}=g_{i}(\omega,\bm{k})[v_{i}(\omega,\bm{k})]_{a}\,,$

where $g_{i}(\omega,\bm{k})$ and $[v_{i}(\omega,\bm{k})]_{b}$ are the $i$-th
eigenvalue and $i$-th eigenvector of each block. We define the $i$-th mode in
terms of the combination of $g_{i}(\omega,\bm{k})$ and $v_{i}(\omega,\bm{k})$.
The corresponding spectral function of the $i$-th mode is

$\displaystyle\rho_{i}(\omega,\bm{k})=-\frac{1}{\pi}\text{Im}[g_{i}(\omega,\bm{k})]\,.$

The full spectral function is then the sum of spectral functions of each mode

$\displaystyle\rho(\omega,\bm{k})=-\frac{1}{\pi}\text{Tr}\text{Im}[G(\omega,\bm{k})]=\sum_{i}\rho_{i}(\omega,\bm{k}).$

To analyze the symmetry properties of the dispersive modes, it’s useful to
consider the little group $H_{\bm{k}}\leq H$. For a given $\bm{k}$ in the
first Brillouin zone, the little group $H_{\bm{k}}$ is defined as
$H_{\bm{k}}=\\{h\in H|h\bm{k}=\bm{k}\\}$ (?). Since the symmetry operation
$h\in H_{\bm{k}}$ won’t change the wave vector $\bm{k}$, we can focus on the
set of eigenvectors $\\{v_{i}(\omega,\bm{k})\\}_{i=1,2,...}$ with the wave
vector index $\bm{k}$. As we mentioned earlier, the eigenvectors form a
representation space of the space group $H$. Thus, the eigenvectors of the
$\bm{k}$ block, $\\{v_{i}(\omega,\bm{k})\\}_{i=1,2,...}$, form a
representation space of the little group $H_{\bm{k}}$, which is also a
subgroup of $H$. The matrix representation $D_{\omega,\bm{k}}[h]$ for $h\in
H_{\bm{k}}$ is defined via

$\displaystyle[U_{h}(\bm{k})v_{i}(\omega,\bm{k})]_{a}=\sum_{j}\bigg{[}D_{\omega,\bm{k}}[h]\bigg{]}_{ij}[v_{j}(\omega,\bm{k})]_{a}\,.$
(S1)

We can further decompose the representation into a direct sum of irreducible
representations and analyze their symmetry properties. In this way, the
lattice symmetry operates in interacting systems on the eigenfunctions of the
Green’s function in parallel to how it acts in noninteracting systems on the
Bloch functions (?, ?, ?, ?, ?, ?).

Based on the representation decomposition, we can inspect the dispersive modes
of each irreducible representation. As we continuously tune $\bm{k}$, if the
peaks of two dispersive modes from two different irreducible representations
cross, then such crossing cannot be gapped out by symmetry preserved
perturbations. Another type of symmetry-protected crossing could appear when
we tune the $\bm{k}$ to certain high symmetry points, where additional
symmetry leads to a two or higher dimensional irreducible representation. In
this case, the spectral functions of the corresponding dispersive modes must
degenerate. A detailed explanation of the symmetry-protected Weyl/Dirac node
in our microscopic model is provided in this SM (Sec. H).

### B. Periodic Anderson model with two channels of conduction electrons

We consider a two-channel periodic Anderson model. While such a model can be
considered in a generic setting for correlation physics, we focus on two
representative lattices in which crystalline symmetries promote Dirac and Weyl
nodes. At every site, the local degrees of freedom (Fig. 1B) hybridize with
two channels of conduction electrons, labeled by $\alpha=1,2$; the
hybridization involves the change of local electron number by $1$, which turns
the local configuration to $|\alpha=1,2\rangle$, and likewise from
$|\alpha\rangle$ to $|\sigma\rangle$. In practice, one way of realizing the
two-channel periodic Kondo/Anderson model is to use an orbital degree of
freedom (non-Kramers doublet such as $\Gamma_{3}$) to represent the pseudo-
spin $\sigma$ leaving the physical spins of the conduction electrons and a
higher Kramers doublet to represent the channel $\alpha$ (?, ?). The single-
impurity version of this model, the two-channel Anderson/Kondo model, is well
established to have non-Fermi liquid behavior (?, ?, ?, ?, ?, ?, ?, ?); the
electron correlations fractionalize the $d$-electron, leading to an emergent
Majorana fermion (?, ?).

We consider both 3D and 2D kagome lattices, where the 3D kagome lattice
corresponds to the stacking of the kagome planes with two added ingredients –
the inversion symmetry is broken and a non-zero spin orbit coupling reduces
the $SU(2)$ spin symmetry down to the $U(1)$ spin symmetry (see Sec. C). The
model can be expressed in terms of the following Hamiltonian:

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\sum_{ij,\sigma,\alpha}(t_{ij,\sigma\sigma^{\prime}}-\mu\delta_{ij}\delta_{\sigma\sigma^{\prime}})c_{i,\sigma,\alpha}^{\dagger}c_{j,\sigma^{\prime},\alpha}$
(S2)
$\displaystyle+P\bigg{[}\epsilon_{f}\sum_{i,\sigma,\alpha}d_{i,\sigma,\alpha}^{\dagger}d_{i,\sigma,\alpha}+V\sum_{i,\sigma,\alpha}(c_{i,\sigma,\alpha}^{\dagger}d_{i,\sigma,\alpha}+\text{h.c.})\bigg{]}P\,.$

Here, $c_{i,\sigma,\alpha}$ ($d_{i,\sigma,\alpha})$ denotes a conduction
(local) electron of spin $\sigma$ and channel $\alpha$ on site $i$, which
marks both the unit cell position and sub-lattice index. The projection
operator $P$ allows for the strongly correlated $d$-electrons to have two
empty states $|1\rangle,|2\rangle$ and two singly occupied states
$|\uparrow\rangle,|\downarrow\rangle$ at every site. In addition,
$t_{ij,\sigma\sigma^{\prime}}$ is the hopping matrix of the conduction
electrons, with $\mu$ acting as the chemical potential; $\epsilon_{f}$
specifies the spacing of the $d$-electron levels (Fig. 1B), and $V$ is the
hybridization strength between the two types of electrons. We solve the model
within dynamical mean-field theory (DMFT) (?) as described in this SM, Sec. D;
importantly, we utilize a dynamical large-$N$ method that captures the
asymptotic low-energy behavior of the two-channel Kondo model (?, ?) (see Sec.
D).

### C. Noninteracting band structure

The noninteracting Hamiltonian of the 3D kagome lattice is

$\displaystyle{\cal H}_{0}$ $\displaystyle=$
$\displaystyle\sum\begin{bmatrix}c_{\bm{k},\alpha,\uparrow}^{\dagger}&c_{\bm{k},\alpha,\downarrow}^{\dagger}\end{bmatrix}\begin{bmatrix}h(k)-\mu\mathbb{I}&\\\
&&h^{*}(-k)-\mu\mathbb{I}\end{bmatrix}\begin{bmatrix}c_{\bm{k},\alpha,\uparrow}\\\
c_{\bm{k},\alpha,\downarrow}\end{bmatrix}$ $\displaystyle
c_{\bm{k},\alpha,\sigma}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}c_{A,\bm{k},\alpha,\sigma}&c_{B,\bm{k},\alpha,\sigma}&c_{C,\bm{k},\alpha,\sigma}\end{bmatrix}^{T}\,.$
(S3)

Here, $A,B,C$ denote three sublattices located at
$(\frac{1}{4},\frac{\sqrt{3}}{4},0)a,(\frac{1}{2},0,0)a,(\frac{1}{4},-\frac{\sqrt{3}}{4},0)a$
respectively (thus, each unit cell has three sites). The hopping matrix in the
spin up sector is:

$\displaystyle h(k)$ $\displaystyle=$ $\displaystyle[t-t_{z,2}\cos(k_{z})]$
$\displaystyle\begin{bmatrix}0&[(1+\alpha)+(1-\alpha)e^{-ik_{2}}]&[(1+\alpha)e^{ik_{1}}+(1-\alpha)e^{-ik_{2}}]\\\
[(1+\alpha)+(1-\alpha)e^{ik_{2}}]&0&[(1+\alpha)e^{ik_{1}}+(1-\alpha)]\\\
[(1+\alpha)e^{-ik_{1}}+(1-\alpha)e^{ik_{2}}]&[(1+\alpha)e^{-ik_{1}}+(1-\alpha)]&0\end{bmatrix}$
$\displaystyle-t_{z}\cos(k_{z})\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix}-t_{2}\begin{bmatrix}0&[e^{i(-k_{1}-k_{2})}+e^{ik_{1}}]&(1+e^{i(k_{1}-k_{2})})\\\
[e^{i(k_{1}+k_{2})}+e^{-ik_{1}}]&0&[e^{i(k_{1}+k_{2})}+e^{-ik_{2}}]\\\
[1+e^{-i(k_{1}-k_{2})}]&[e^{-i(k_{1}+k_{2})}+e^{ik_{2}}]&0\end{bmatrix}$
$\displaystyle+[i\lambda+i\lambda_{z}\cos(k_{z})]\begin{bmatrix}0&[1+e^{-ik_{2}}]&[-e^{ik_{1}}-e^{-ik_{2}}]\\\
-[1+e^{ik_{2}}]&0&[1+e^{ik_{1}}]\\\
[e^{-ik_{1}}+e^{ik_{2}}]&-[1+e^{-ik_{1}}]&0\end{bmatrix}\,,$

where

$\displaystyle k_{1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(k_{x}\pm\frac{k_{y}}{\sqrt{3}})\,.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,.$

In addition, $t,t_{2}$ are the nearest-neighbor and next-nearest-neighbor
intra-layer hopping parameters, and $t_{z}$ and $t_{z,2}$ are the nearest-
neighbor and next-nearest-neighbor inter-layer hopping elements. $\alpha$
denotes the anisotropy of $t_{1}$ and $t_{z,2}$ hopping that breaks mirror
symmetry $M_{y}$ and inversion symmetry. $\lambda$ and $\lambda_{z}$ are the
intra-layer and inter-layer spin orbit couplings, respectively, which reduce
the $SU(2)$ spin symmetry down to $U(1)$. The corresponding 2D kagome model
without spin-orbit coupling and inversion symmetry breaking can be realized by
setting $t_{z}=t_{z,2}=\lambda=\lambda_{z}=0$, $\alpha=0$ and dropping all the
terms that contain $k_{z}$.

In practice, we set
$t_{1}=0.29,t_{2}=0.14,t_{z}=-0.17,t_{z,2}=0.26,\lambda=0.43,\lambda_{z}=0.06,\alpha=1.4,\mu=-1.09$
for the 3D kagome lattice. As shown in Fig. S1B, Fig. S1E and Fig. S1F, we
observe two Weyl nodes at $(1/3,1/3,z)$, $(2/3,2/3,-z)$, two anti-Weyl nodes
at $(1/3,1/3,-z)$, $(2/3,2/3,z)$, and two Dirac nodes at $(0,0,\pm
z^{\prime})$ where $z\approx 0.19,z^{\prime}\approx 0.32$. Weyl and Dirac
nodes are all protected by $C_{3z}$ symmetry. In Fig. S1E and Fig. S1F, we
label the relevant bands by their eigenvalues of $C_{3z}$ symmetry, where
$\eta=e^{i2\pi/6}$. The $C_{3z}$ symmetry excludes the hybridization between
the two bands with different eigenvalues, and thus protects the nodes. The
bands along the high symmetry line ($-A$-$\Gamma$-$A$) are two-fold degenerate
due to the mirror symmetry $M_{z}$ and time-reversal symmetry. The eigenvalues
inside (outside) the brackets are for the spin-up (spin-down) sector. Because
the hybridization between spin-up and spin-down electrons is forbidden by the
$U(1)$ spin symmetry, there can be no gap opened by hybridizing two bands with
the same $C_{3z}$ eigenvalue but opposite spin indices.

For the 2D kagome lattice, we set $t_{1}=1,t_{2}=0.3,\mu=-1.0$. (We have
chosen generic parameters so that the band structure is relatively simple.)
The noninteracting dispersion and Dirac node are shown in Fig. S1D. At the
high symmetry point $K/K^{\prime}$, three bands form
$\Gamma_{1}\oplus\Gamma_{3}$ representations of the little group $C_{3v}$,
where $\Gamma_{3}$ is a two-dimensional irreducible representation and thus
protects the Dirac node. Here, due to the $SU(2)$ symmetry, we consider the
spinless representations of the group.

### D. Solution methods: Dynamical mean-field theory and the dynamical
large-$N$ approach

The two-channel Kondo effect is a robust mechanism for non-Fermi liquid
physics (?). Here, we describe the dynamical method that captures the
asymptotic low-energy form of the non-Fermi liquid behavior (?, ?), while
incorporating the lattice symmetries that can constrain the emergent
electronic excitations.

To solve the lattice model, we first introduce the auxiliary-particle
representation of the projected local electron annihilation operator:
$Pd_{i,\sigma,\alpha}P=f_{i,\sigma}b_{i,\alpha}^{\dagger}$. Here,
$f_{i,\sigma}$ is a pseudo-fermion annihilation operator and
$b_{i,\alpha}^{\dagger}$ is a pseudo-boson creation operator. The Hilbert
space of the $d$ electron can be expressed as:
$|\sigma\rangle=f_{\sigma}^{\dagger}|\text{vac}\rangle$,
$|\alpha\rangle=b_{\alpha}^{\dagger}|\text{vac}\rangle$ with the vacuum state
$|\text{vac}\rangle$. There is also a local constraint on the pseudo-particle
occupation numbers,
$Q_{i}=\sum_{\sigma}f_{i,\sigma}^{\dagger}f_{i,\sigma}+\sum_{\alpha}b_{i,\alpha}^{\dagger}b_{i,\alpha}=1$.

In DMFT, the local correlators of the two-channel periodic Anderson model is
determined by the self-consistent local model (?):

$\displaystyle S_{loc}$ $\displaystyle=$
$\displaystyle\int\sum_{\sigma,\alpha}c_{\sigma,\alpha}^{\dagger}(\tau)G_{0}(\tau,\tau^{\prime})^{-1}c_{\sigma,\alpha}(\tau^{\prime})d\tau
d\tau^{\prime}$ (S4) $\displaystyle+$
$\displaystyle\int\Bigg{[}\sum_{\sigma}f_{\sigma}^{\dagger}(\tau)(\partial_{\tau}-\epsilon_{f})f_{\sigma}(\tau)+\sum_{\alpha}b_{\alpha}^{\dagger}(\tau)\partial_{\tau}b(\tau)\bigg{]}d\tau$
$\displaystyle+$ $\displaystyle\int
V\sum_{\sigma,\alpha}(c_{\alpha,\sigma}^{\dagger}(\tau)f_{\sigma}(\tau)b_{\alpha}^{\dagger}(\tau)+h.c.)d\tau$
$\displaystyle+$ $\displaystyle\int
i\lambda(\tau)\bigg{(}\sum_{\sigma}f_{\sigma}^{\dagger}(\tau)f_{\sigma}(\tau)+\sum_{\alpha}b_{\alpha}^{\dagger}(\tau)b_{\alpha}(\tau)-Q\bigg{)}d\tau\,.$

Here, $\lambda$ is a Lagrangian multiplier enforcing the pseudo-particle
constraint. Whereas $G_{0}$ is the bath function of the conduction electrons,
which is determined from the DMFT equations:

$\displaystyle G_{c,loc}(\omega)=\frac{1}{L^{d}}\sum_{\bm{k}}\frac{1}{\omega-
t_{\bm{k}}-\Sigma_{c}(\omega)}$ $\displaystyle
G_{0}^{-1}(\omega)-G_{c,loc}^{-1}(\omega)=\Sigma_{c}(\omega)\,.$ (S5)

$G_{c,loc}(\omega)$ is the local Green’s function of the $c$ electrons,
$\Sigma_{c}(\omega)$ is the associated self-energy which is taken to be local,
i.e., $\bm{k}$-independent within the DMFT. We note that the effective local
model in the 3D lattice model has an effective $SU(2)$ spin symmetry. The bath
functions for the spin up and spin down conduction electrons are the same due
to the time reversal symmetry and no hybridization between spin up and spin
down electrons is allowed given the $U(1)$ spin symmetry.

The local model is solved via the saddle point equations (?, ?, ?, ?) of a
dynamical large-$N$ limit [with a generalization to an $SU(N)$ symmetry for
the local electrons’ spins and an $SU(N)\times SU(M)$ symmetry for the
conduction electrons, while taking the large-$N,M$ limit with a fixed
$N/M=1$]. Here, the self-energies of the pseudo-fermion and pseudo-boson are

$\displaystyle\Sigma_{f}(\omega)=MV^{2}\int\rho_{0}(\omega-\epsilon)f(\epsilon-\omega)G_{b}(\epsilon)d\epsilon$
$\displaystyle\Sigma_{b}(\omega)=NV^{2}\int\rho_{0}(\epsilon-\omega)f(\epsilon-\omega)G_{f}(\epsilon)d\epsilon$

where $\rho_{0}$ is the density of states associated with the bath function
$G_{0}$ and $f(x)$ is the Fermi-Dirac function. In conjunction with the Dyson
equations for the pseudo-particle propagators $G_{f}$,$G_{b}$,

$\displaystyle
G_{f}^{-1}(\omega)=\omega-\epsilon_{f}-\lambda-\Sigma_{f}(\omega)$
$\displaystyle G_{b}^{-1}(\omega)=\omega-\lambda-\Sigma_{b}(\omega)\,,$

we have a complete set of equations. The local Green’s function is calculated
via
$G_{c,loc}(\omega)=G_{0}(\omega)+V^{2}G_{0}(\omega)G_{d}(\omega)G_{0}(\omega)$,
with $G_{d}(\omega)$ being the correlator of the $d$ electron. We note that
$V$ is at the order of $1/\sqrt{N}$ in the large-$N$ approach. These equations
are known to capture the asymptotic low-energy behavior – including the non-
Fermi liquid exponents - of the multi-channel Kondo/Anderson models (?).
Importantly, the $1/N$ vertex corrections do not change the asymptotic low-
energy behavior (?, ?).

In practice, we take $V=1.0,\epsilon_{f}=-0.5$ for the 2D kagome model and
$V=1.5,\epsilon_{f}=-1.0$ for the 3D kagome model. We work on the retarded
Green’s functions and solve the DMFT self-consistent equation by varying the
bath function $G_{c,0}$ and stop the procedure when the difference between two
iterations is less then $1\%$.

We solve the dynamical large-$N$ equations in real frequency. It turns out to
be important to reach sufficiently low temperatures (down to about
$10^{-3}-10^{-5}$ of the Kondo temperature $T_{K}$) in order to reveal the
asymptotic low-energy behavior. We find $T_{K}/t\approx 0.03$ for the model on
the 3D kagome lattice and $T_{K}/t\approx 0.01$ for the model on the 2D kagome
lattice.

In Fig. S3A and Fig. S3B, we show the local spectral functions of the pseudo-
bosons and pseudo-fermions of the 3D kagome lattice. We observe an $\omega/T$
scaling with

$\displaystyle\rho_{f}(\omega,T)$
$\displaystyle=(\frac{T}{T_{K}})^{1/2}g_{f}(\omega/T)$
$\displaystyle\rho_{b}(\omega,T)$
$\displaystyle=(\frac{T}{T_{K}})^{1/2}g_{b}(\omega/T)\,,$ (S6)

where $g_{f}(x),g_{b}(x)$ are universal functions. The results for the 2D
kagome lattice (not shown) are very similar. In addition, to illustrate the
power-law dependence, Fig. S3C and Fig. S3D depict $\rho_{b}$ vs. frequency at
a low temperature for the 3D and 2D kagome lattices, respectively.

### E. Breakdown of quasiparticles

The imaginary part of the conduction-electron self-energy, ${\rm
Im}\Sigma_{c}(\omega)$, is found to depend on frequency in a square-root
fashion in the low-energy limit, as shown in Fig. S2A. This is in contrast to
the Fermi liquid case, where ${\rm Im}\Sigma_{c}(\omega)$ goes quadratically
in $\omega$. The square-root frequency dependence of the self-energy describes
a non-Fermi liquid, with a loss of quasiparticles. The non-Fermi liquid
behavior is characterized by an electrical resistivity that depends on
temperature ($T$) in a $\sqrt{T}$ way, which is illustrated in Fig. S2B. In
practice, we find that the asymptotic behavior, in both the frequency and
temperature dependences, appears only at sufficiently low temperatures; it was
not found in previous DMFT solutions of the multichannel Kondo lattice model
on the infinite dimensional hypercubic lattice (?) solved at much higher
temperatures (about $0.1$ $T_{K}$ and above).

The self-energy can be described by a form that contains both the asymptotic
low-energy behavior and a pole in the complex energy plane.

$\displaystyle\Sigma_{c}(\omega)\sim
A\frac{\gamma}{\omega-\omega_{0}+i\gamma}+g(\omega)\,\,\Theta(\Lambda-|\omega|)\,,$
(S7)

where $\Theta(x)$ is the Heaviside step function, and $g(\omega)\sim
a+c\sqrt{|\omega|}$ describes the low-energy asymptotic behavior that appears
below the energy cutoff $\Lambda$. (For further details, see Sec. F.) To our
knowledge, the existence of a pole in the self-energy has not been recognized
in any two-channel Kondo model; its uncovering is made possible by our working
on the real-frequency axis and solving the model at very low temperatures.
Physically, the pole captures the development of the $f$-electron spectral
weight within an energy range of $T_{K}$ near the Fermi energy. Because the
$f$-electron state can decay into the fractionalized (Majorana) excitations,
the damping rate $\gamma$ remains nonzero even at the zero-temperature limit.

The structure of the self-energy implies two branches of dispersive modes near
the Fermi energy $E_{F}$, with velocities highly reduced by the factor $\sim
T_{K}/D$, where $D$ is the bare bandwidth (see Sec. F).

### F. Self-energy: asymptotic low energy behavior, a pole and the resulting
emergent dispersive modes

In this section, we describe the nature of the self-energy $\Sigma_{c}$ in
some detail. Due to the symmetry requirement, the local self-energy
$\Sigma_{c}$ is diagonal in spin, sublattice and channel spaces, which
simplifies the analysis. As shown in Fig. S4A, $\text{Im}[\Sigma_{c}](\omega)$
has a peak near the Fermi energy. We find that this peak is well represented
by a Lorentzian function; we define the peak center and width as $\omega_{0}$
and $\gamma$, respectively. Accordingly, we obtain $\Sigma_{c}(\omega)\sim
A\frac{\gamma}{\omega-\omega_{0}+i\gamma}$. In addition, we observe a
$\sqrt{|\omega|}$ behavior in the low energy limit as shown in Fig. S2A. Such
a dependence fully captures the asymptotic low-energy behavior for the two-
channel Kondo model (?, ?). Combining both contributions, we have the form of
the self-energy given in the previous section (Eq. S7).

As shown in Fig. S4A, we fit the imaginary part of the self-energy with this
form. The Heaviside step function has been replaced by a soft cutoff via a
hyperbolic tangent function in the fitting. In Fig. S4B, we also plot the
evolution of $\gamma$, which characterize the width of the peak, as a function
of temperature. At low temperature, we find that $\gamma$ saturates to a
finite value at a fraction of $T_{K}$. This saturation further supports the
robustness of the pole in the complex plane.

Physically, the Kondo effect always leads to the development of $f$-electron
spectral weight within an energy range of $T_{K}$ near the Fermi energy. In
the single-channel Kondo/Anderson lattice, such spectral weight is carried by
Landau quasiparticles; a pole in $\Sigma_{c}(\omega)$ develops whose decay
rate must satisfy the Fermi liquid form and depends on $\omega$ in an
$\omega^{2}$ form (where $\omega$ is measured from the Fermi energy). By
contrast, in the two-channel Kondo/Anderson lattice model, the $f$-electron
state can decay into the fractionalized (Majorana) excitations. Consequently,
the damping rate $\gamma$ remains nonzero even at the zero-temperature and
zero-frequency limit.

The existence of a pole in the self-energy leads to a band reconstruction in
the interacting model, as shown in Fig. S6. Combining the self-energy and
symmetry group, we are able to analyze the structure of the Green’s function
near the Dirac and Weyl nodes.

To further explicate the consequence of the pole in the self-energy, we stress
that emergent excitations develop at low energies (within the Kondo scale near
the Fermi energy) out of this pole. They take the form of hybridized bands,
except that these are not quasiparticles. Specifically, in the vicinity of the
crossing wave vector $\tilde{k}$, we have

$\tilde{P}G^{-1}_{c}(\omega,\tilde{k}+k)\tilde{P}\approx
U^{\dagger}\begin{bmatrix}\omega-
E_{0}-v_{z}k_{z}-\Sigma_{c}(\omega)&-v_{x}k_{x}+iv_{y}k_{y}\\\
-v_{x}k_{x}-iv_{y}k_{y}&\omega-E_{0}+v_{z}k_{z}-\Sigma_{c}(\omega)\\\
\end{bmatrix}U$

where $\tilde{P}$ is the projection matrix that only keeps the relevant bands,
$U$ is a unitary matrix that brings the dispersion near a Weyl/Dirac node into
the canonical form and $E_{0}$ is the energy of the noninteracting Weyl/Dirac
nodes.

Importantly, a pole of $\Sigma_{c}(\omega)$ in the complex-frequency plane
leads to a doubling of the bands, creating dispersive modes within the Kondo
energy $T_{K}$ near the Fermi energy that are not quasiparticles. We
illustrate this explicitly for the case of Dirac nodes in 2D, where we can set
$v_{z}=0$ and keep implicit the two-fold degeneracy in spin space. The
corresponding spectral functions would have low-energy peaks at

$\displaystyle\omega\approx\omega_{0}+{\gamma}\frac{A}{E_{0}}(1\pm\sqrt{v_{x}^{2}k_{x}^{2}+v_{y}^{2}k_{y}^{2}+v_{z}^{2}k_{z}^{2}}/E_{0})$
(S8)

We re-iterate that this equation describes the emergent dispersive modes
located near Fermi energy. The results are showin in Fig. 7A,B,C. The
corresponding results for the 3D kagome lattice have already been shown in the
main text, Fig. 2A,B,C.

Recognizing that both $E_{0}$ and ${A}$ are non-universal and generically of
the order of the bare conduction electron bandwidth $D$, and $\gamma$ is of
the order of $T_{K}$, the velocity of the emergent Weyl nodal states is
renormalized down from its noninteracting counterpart by a factor of
$\sim(T_{K}/D)$. Each peak corresponds to one branch of the Dirac/Weyl node in
the non-Fermi liquid dispersive modes.

In the broad energy range, the pole in the self-energy essentially leads to a
band reconstruction in the interacting model, as shown in Fig. S6A,B. The
corresponding results for the 3D kagome lattice are shown in Fig. S6C,D.

### G. Topological Dirac semimetal without quasiparticles in the 2D model

The Dirac spectrum appearing near the Fermi energy, for the case of the 2D
kagome lattice, is shown in Fig. S7 A,B,C. The emergent dispersive modes near
the Fermi energy form a symmetry-protected Dirac node at the $K$ point. The
non-Fermi liquid properties in the 2D kagome lattice are also characterized by
the $\sqrt{T}$ dependence in both the resistivity and valley Hall
conductivity, as shown in Fig. S7 D,E.

### H. Symmetry of Green’s function in the microscopic models

Based on the eigenvectors and eigenvalues of the single-particle Green’s
function, we now prove that the Weyl (Dirac) nodes in the 3D kagome model and
the Dirac nodes in the 2D kagome model are both protected by crystalline
symmetry, but via different mechanisms. Since single-particle excitations are
composite fermions, it is adequate to focus on the Green’s function of the $d$
electrons, $G_{d}.$

The Weyl nodes in 3D kagome model come from the band inversion mechanism.
Along the high symmetry line $K-H$, we can assign an eigenvalue of $C_{3z}$ to
each dispersive mode (eigenvectors of Green’s function). In Fig. 2A, two modes
with left-moving and right-moving peaks have symmetry eigenvalues $\eta^{3}$
and $\eta^{5}$, respectively. The different eigenvalues would protect the
crossing of the two modes under symmetry-preserved perturbations. A similar
mechanism protects the Weyl nodes on $-H$-$K^{\prime}$-$H^{\prime}$ and Dirac
nodes on $-A$-$\Gamma$-$A$.

In the 2D kagome lattice, the Dirac nodes are symmetry enforced.
$\\{v_{d,i}(\omega,\bm{k}=K/K^{\prime})\\}_{i}$ forms the
$\Gamma_{1}\oplus\Gamma_{3}$ representation of the little group
$G_{K/K^{\prime}}=C_{3v}$. Due to the $SU(2)$ symmetry, we can ignore the spin
index and use a spinless representation here. The two-dimensional irreducible
representation $\Gamma_{3}$ enforces a two-fold-degenerate (four-fold-
degenerate including the spin degree of freedom) spectrum at the
$K/K^{\prime}$ point. Consequently, the two dispersive modes have to coincide
with each other at the $K/K^{\prime}$ point as shown in Fig. S7A.

### I. Edge modes

To evaluate the edge mode in the 3D kagome model, we take a slab geometry with
50 unit cells and open boundary along $y$ direction. In the interacting model,
we consider the Green’s function
$[G_{d}(\omega,k_{x},k_{z})]_{y_{1}\alpha_{1},y_{2}\alpha_{2}}$ that is a
$300\times 300$ matrix, where $y_{i}$ is the coordination of unit-cell along
the $y$ direction and $\alpha_{i}$ is the sublattice and spin index. Through
diagonalizing the Green’s function matrix, we obtain the eigenvalues
$\\{g_{i}(\omega,k_{x},k_{z})\\}_{i}$ and eigenvectors
$\\{[v_{i}(\omega,k_{x},k_{z})]_{y\alpha}\\}_{i}$. For a local self-energy,
$[v_{i}(\omega,k_{x},k_{z})]_{y\alpha}$ becomes $\omega$ independent and
describes the real-space distribution of the corresponding mode. Numerically,
we identify an edge mode that decays exponentially in the real space as shown
in Fig. S8A. The corresponding spectral function
$\text{Im}[g_{i}(\omega,k_{x},k_{z})]/\pi$ is shown in the main text, Fig. 3A.

### J. Hall conductivities

In this section, we describe the calculation of the spin and valley Hall
conductivities. Within the DMFT approach, in which vertex corrections are
formally suppressed, the in-plane spin Hall conductivity of each $k_{z}$:
$\sigma^{s}(k_{z})$, can be expressed as

$\displaystyle\sigma_{xy}^{s}(k_{z})=\frac{1}{4\pi^{2}}\int_{k_{x},k_{y}}\sum_{m,n\in\uparrow}\tilde{\Omega}_{mn}(k_{x},k_{y},k_{z})-\frac{1}{4\pi^{2}}\int_{k_{x},k_{y}}\sum_{m,n\in\downarrow}\tilde{\Omega}_{mn}(k_{x},k_{y},k_{z})$
(S9)

where $m,n$ are band indices. The first (second) terms sum over bands with
spin up (down). The full bulk contribution is obtained via
$\sigma_{xy}^{s}=\int_{k_{z}}\sigma_{xy}^{s}(k_{z})$. In addition,

$\displaystyle\tilde{\Omega}_{mn}(\bm{k})$ $\displaystyle=$ $\displaystyle
i\lim_{\nu\rightarrow
0}\frac{d}{d(i\nu)}\frac{1}{\beta}\sum_{i\omega}\frac{(V_{\bm{k}}^{\dagger}\partial_{k^{x}}t_{\bm{k}}V_{\bm{k}})_{nm}(V_{\bm{k}}^{\dagger}\partial_{k^{y}}t_{\bm{k}}V_{\bm{k}})_{mn}}{[i\omega+i\nu-\epsilon_{\bm{k},n}-\Sigma_{c}(i\omega)][i\omega-\epsilon_{\bm{k},m}-\Sigma_{c}(i\omega)]}$
$\displaystyle=$ $\displaystyle i\lim_{\nu\rightarrow
0}\frac{d}{d(i\nu)}\frac{1}{\beta}\sum_{i\omega}\frac{[A^{x}_{\bm{k},mn}A^{y}_{\bm{k},nm}-A^{x}_{\bm{k},nm}A^{y}_{\bm{k},mn}](\epsilon_{\bm{k},n}-\epsilon_{\bm{k},m})^{2}}{[i\omega+i\nu-\epsilon_{\bm{k},n}-\Sigma_{c}(i\omega)][i\omega-\epsilon_{\bm{k},m}-\Sigma_{c}(i\omega)]}\,.$

$V_{k}$, $\\{\epsilon_{\bm{k},n}\\}_{n}$ are the eigenvectors and eigenvalues
(i.e. the Bloch bands) of the hopping matrix $t_{\bm{k}}$.
$A^{\mu}_{mn,\bm{k}}=i\langle
u_{m,\bm{k}}|\partial_{k^{\mu}}|u_{n,\bm{k}}\rangle$ is a Berry connection.

Similarly, we calculate valley Hall conductivity $\sigma^{\nu}$ of 2D kagome
lattice via

$\displaystyle\sigma^{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\int_{|\bm{k}-K|<C}\sum_{m,n}\tilde{\Omega}_{mn}(\bm{k})-\frac{1}{4\pi^{2}}\int_{|\bm{k}-K^{\prime}|<C}\sum_{m,n}\tilde{\Omega}_{mn}(\bm{k}),$
(S10)

where the $\bm{k}$ integration is taken in a finite region near $K/K^{\prime}$
with cutoff $C=\pi/2$. In the calculation, we introduce a small mass term to
the Dirac point in order to generate a finite Berry curvature distribution.
From top to bottom, we label the bands by 1, 2, 3. In the noninteracting
limit, the lowest band that is below the Fermi energy contributes a positive
$\sigma^{\nu}$. The top two bands are above the Fermi energy near
$K(K^{\prime})$ and, thus the corresponding contributions
$\tilde{\Omega}_{12}$ and $\tilde{\Omega}_{21}$ vanish. Once we turn on the
hybridization $V$, dispersive modes develop near the Fermi energy and generate
nonzero $\tilde{\Omega}_{12}$ and $\tilde{\Omega}_{21}$. We can extract the
contributions from $\tilde{\Omega}_{12}$ and $\tilde{\Omega}_{21}$ and define
$\sigma^{\nu}_{12}$

$\displaystyle\sigma^{\nu}_{12}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\int_{|\bm{k}-K|<\Lambda}(\tilde{\Omega}_{12}(\bm{k})+\tilde{\Omega}_{21}(\bm{k}))-\frac{1}{4\pi^{2}}\int_{|\bm{k}-K^{\prime}|<\Lambda}(\tilde{\Omega}_{12}(\bm{k})+\tilde{\Omega}_{21}(\bm{k})).$

$\sigma^{\nu}_{12}$ is part of $\sigma^{\nu}$ that is directly generated by
the interactions. The evolution of $\sigma^{\nu}$ and $\sigma^{\nu}_{12}$ as a
function of hybridization is shown in Fig. S8B.

### K. Non-linear Hall effect

We now turn to the non-linear Hall effect in the non-Fermi liquid phase. Using
the Green’s function approach (?), the non-linear Hall conductivity is

$\displaystyle\chi_{xyy}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\int_{\bm{k},\epsilon}\text{Im}\bigg{\\{}\text{Tr}\bigg{[}\partial_{\epsilon}f(\epsilon)u_{x}\partial_{\epsilon}G_{c}(\epsilon-i0^{+})u_{y}G_{c}(\epsilon-i0^{+})u_{y}G(\epsilon+i0^{+})$
(S11)
$\displaystyle+\partial_{\epsilon}f(\epsilon)u_{x}\partial_{\epsilon}G_{c}(\epsilon-i0^{+})u_{yy}G_{c}(\epsilon+i0^{+})+f(\epsilon)u_{x}\partial_{\epsilon}^{2}G_{c}(\epsilon-i0^{+})u_{yy}$
$\displaystyle+2f(\epsilon)u_{x}\partial_{\epsilon}[G_{c}(\epsilon-i0^{+})u_{y}G_{c}(\epsilon-i0^{+})]u_{y}G_{c}(\epsilon+i0^{+})\bigg{]}\bigg{\\}},$

where $f(\epsilon)$ is the Fermi-Dirac function, and
$u_{\mu_{1}\mu_{2}...\mu_{n}}=\partial^{n}_{k_{\mu_{1}}k_{\mu_{2}}...k_{\mu_{n}}}t_{\bm{k}}$.
Each term contains a product of three Green’s functions with two derivatives
of $\epsilon$. In the asymptotic region $c_{IR}<\omega<<T_{K}$ where scaling
behaviors appear, $G_{c}(\epsilon)\sim g_{0}+g_{1}\sqrt{|\epsilon|}$. $c_{IR}$
acts as an infrared cutoff. The integral behaves as
$\sim\epsilon^{-3/2}d\epsilon$. (Note that at least one derivative is acting
on the Green’s function.) Transforming to the temperature behaviors, we would
expect $\sim 1/\sqrt{T}$ in the scaling region $c_{IR}<T<<T_{K}$. To more
explicitly observe this $\sim 1/\sqrt{T}$ behaviors, we consider the Green’s
function near the Weyl node $\tilde{\bm{k}}$

$\displaystyle G_{c}(\omega,\bm{k}+\tilde{\bm{k}})^{-1}\approx[\omega-
h_{0}-\Sigma_{c}(\omega)]\mathbb{I}-h_{x}\sigma^{x}-h_{y}\sigma^{y}-h_{z}\sigma^{z},$

where $\Sigma_{c}(\omega)$ is the self-energy. $\sigma^{x,y,z}$ are Pauli
matrices in the $2\times 2$ subspace that holds the Weyl node.
$h_{0},h_{x},h_{y},h_{z}$ are functions of wave vector. At the Weyl node,
$h_{0}(\tilde{\bm{k}})=E_{0},h_{x}(\tilde{\bm{k}})=h_{y}(\tilde{\bm{k}})=h_{z}(\tilde{\bm{k}})=0$.
The eigenvalues of the Green’s function matrix are

$\displaystyle g_{\pm}^{-1}=\omega-\epsilon_{\pm}-\Sigma_{c}(\omega)$
$\displaystyle\epsilon_{\pm}=h_{0}\pm\sqrt{h_{x}^{2}+h_{y}^{2}+h_{z}^{2}},$

where $\Sigma_{c}(\omega)$ is the self-energy and has non-Fermi liquid
behavior. We also introduce the velocity matrix:
$v_{\mu}^{\alpha\gamma}=(V^{\dagger}\partial_{k^{\mu}}t_{\bm{k}}V)_{\alpha\gamma}$,
where
$t_{\bm{k}}=h_{0}\mathbb{I}+h_{x}\sigma^{x}+h_{y}\sigma^{y}+h_{z}\sigma^{z}$
is the hopping matrix and $V$ is the matrix formed by the eigenvectors of the
hopping matrix.

Here, we focus on the intrinsic contributions from the Berry curvature (?).
However, all the other contributions produce the same $1/\sqrt{T}$ behavior.
The intrinsic contributions reads (from the first term of Eq. S11),

$\displaystyle\chi^{in}_{xyy}=-\frac{1}{2\pi}\int_{\bm{k}}\int
d\epsilon\frac{\partial
f(\epsilon)}{\partial\epsilon}\text{Im}\bigg{\\{}\text{Tr}\bigg{[}$
$\displaystyle
v_{x}^{+-}v_{y}^{-+}v_{y}^{++}\partial_{\epsilon}g_{-}(\epsilon+i0^{+})g_{+}(\epsilon+i0^{+})g_{+}(\epsilon-i0^{+})$
$\displaystyle+$ $\displaystyle
v_{x}^{-+}v_{y}^{++}v_{y}^{+-}\partial_{\epsilon}g_{+}(\epsilon+i0^{+})g_{+}(\epsilon+i0^{+})g_{-}(\epsilon-i0^{+})\bigg{]}\bigg{\\}}.$

Due to the pre-factor $\frac{\partial f(\epsilon)}{\partial\epsilon}$, it is
sufficient to focus on the low-energy contribution. Since the self-energy
behaves as $\text{Im}[\Sigma_{c}]\sim a+b\sqrt{|\omega|}$, approximately we
have

$\chi^{in}_{xyy}=-\frac{1}{2\pi}\int_{k}\int_{\epsilon}\frac{\partial
f(\epsilon)}{\partial\epsilon}\Omega_{xy}^{+}v_{y}^{++}[a_{0}+\frac{a_{1}}{\sqrt{|\epsilon|}}+o(\epsilon)]$

where
$a_{1}\approx\pi(\rho_{0}\frac{\sqrt{h_{x}^{2}+h_{y}^{2}+h_{z}^{2}}}{h_{0}})^{3}\frac{1}{\sqrt{\Lambda}}$
and $\Lambda$ is the low energy scale of $\sqrt{|\omega|}$ behaviors and
$\rho_{0}$ is the density of state near zero Fermi energy. After integrating
over $\epsilon$, we have

$\chi^{in}_{xyy}=\frac{1}{\sqrt{T}}\cdot\bigg{[}\frac{-1.34}{2\pi}\int_{k}\Omega_{xy}^{+}v_{y}^{++}a_{1}\bigg{]}+...$
(S12)

where
$\Omega_{xy}^{+}=i\langle\partial_{x}u_{+}|\partial_{y}u_{+}\rangle-i\langle\partial_{y}u_{+}|\partial_{x}u_{+}\rangle$
is the Berry curvature.

### L. Candidate materials

We have identified three candidate materials, PrBi (?, ?, ?), PrFe4P12 (?, ?),
and UPt3Au2 (?), to realize the correlated topological phases predicted here.
They crystallize in the space groups $\\#225$, $\\#204$, and $\\#216$,
respectively, which all allow for a $\Gamma_{3}$ non-Kramers doublet ground
state of the Pr3+ or U4+ $4f^{2}$ wave function. The former two are
centrosymmetric and candidate correlated Dirac semimetals, whereas the latter
is noncentrosymmetric and thus a candidate correlated Weyl semimetal.

In PrBi, evidence for a $\Gamma_{3}$ ground state was provided by fitting a
crystal electric field (CEF) model to temperature-dependent magnetic
susceptibility data; the assignment was supported by a much better agreement
of the specific heat data with this than other possible CEF level schemes (?).
A low charge carrier concentration was evidenced from quantum oscillation
experiments (?). For PrFe4P12, key evidence is summarized in Fig. S9. A non-
magnetic CEF ground state doublet is suggested by the antiferroquadrupolar
ordering at low temperatures (panel A) (?). Strong Kondo coupling was
evidenced by a large Sommerfeld coefficient determined from specific heat data
(?), and is confirmed by the large $A$ coefficient of the Fermi liquid $T^{2}$
form seen in electrical resistivity data under applied pressure and magnetic
field (inset of panel B). Under finite pressure, the material shows an upturn
of the electrical resistivity with decreasing temperature, i.e., it ceases to
be metallic. We thus anticipate that PrFe4P12 under finite pressure and PrBi
under ambient pressure may fulfill the theoretical requirements for
topological two-channel Kondo systems. Finally, UPt3Au2 shows a moderately
enhanced Sommerfeld coefficient of 60 mJ/mol/K2, and an only very weakly
temperature-dependent electrical resistivity (?) (see Fig. S10), that may
point to the presence of Kondo interaction in a reduced carrier concentration
setting.

### M. Space group symmetry constraints in the candidate materials

As discussed in the main text, PrBi (?, ?) and PrFe4P12 (?, ?) have space
groups $\\#225$ and $\\#204$, respectively. Whereas UPt3Au2 (?), which has a
broken inversion symmetry, has space group $\\#216$. These space groups allow
for Dirac or Weyl points. For example, in PrBi, if two bands with the crossing
point on the $\Gamma$-$X$ line have different eigenvalues under the
$4^{+}_{100}$ rotational symmetry, it would be a symmetry-protected Dirac
node; this symmetry-based argument is manifested in the DFT calculations
(albeit without treating correlation effects), in which a band crossing occurs
on the $\Gamma$-$X$ line (?). Similarly, the $3^{+}_{111}$ rotational symmetry
could protect Dirac nodes on the $\Gamma$-$H$ line in PrFe4P12. Likewise, the
$2_{100}$ rotational symmetry could protect Weyl nodes on the $X$-$W$ line of
UPt3Au2.

### N. DFT calculations

The theoretical electronic-structure calculations for PrBi were carried out
using the Vienna ab initio simulation package (VASP) (?, ?) with the modified
Becke-Johnson potential (?, ?) for the generalized gradient approximation of
the exchange-correlation functional. The pseudo-potentials in the projector
augmented-wave approximation (?) included $p$ and $s$ valence electrons for Bi
atoms, while for Pr the $f$ electrons were frozen in the core. The spin-orbit
interaction was also considered in the calculation by the second variation
method (?). The self-consistent calculation of the ground state density was
performed in a $11\times 11\times 11$ Monkhorst-Pack grid with an energy
cutoff of $600$ eV.

## Supplementary Figures

Figure S1: Lattice structure and the noninteracting dispersion. (A) The 3D
kagome lattice. (B) The first Brillouin zone of the 3D kagome lattice, where
red and blue dots denote the Weyl nodes and Dirac nodes, respectively. (C) The
first Brillouin zone of the 2D kagome lattice, where the blue dots denote the
Dirac nodes. (D) Dispersion of the 2D kagome lattice. (E), (F) Dispersion of
the 3D kagome lattice along two high-symmetry lines of the Brillouin zone.
Here $0$ marks the Fermi energy of the conduction electrons in the absence of
the Kondo hybridization. The relevant bands are marked by their eigenvalues of
the $C_{3z}$ symmetry, where $\eta=e^{i2\pi/6}$.

Figure S2: Breakdown of Landau quasiparticles. (A) The self-energy
($\Sigma_{c}$) shows a $\sqrt{|\omega|}$-dependence at low temperatures (shown
here for $T/T_{K}=10^{-3}$) in the 3D kagome lattice. To isolate the frequency
dependence, the zero frequency contribution has been subtracted in the plot:
$\Delta\text{Im}[\Sigma_{c}](\omega)=\text{Im}[\Sigma_{c}](\omega)-\text{Im}[\Sigma_{c}](0)$.
(B) Temperature dependence of the electrical resistivity along the $x$
direction in the 3D kagome lattice. Both the $\sqrt{|\omega|}$ and $\sqrt{T}$
dependencies characterize the breakdown of quasiparticles.

Figure S3: Dynamical scaling and power-law behavior. (A), (B) spectral
functions of $\rho_{f}$ and $\rho_{b}$ in the 3D kagome lattice at various
temperatures in unit of $T_{K}$. (C) and (D) show the power-law dependence of
the the pseudo-particle spectral functions for the models on the 3D and 2D
kagome lattices, respectively. The spectral function $\rho_{b}$ is shown to
depend on frequency in a power-law form, $\omega^{-1/2}$. The power-law form
operates for about two decades of frequency. The deviation at high frequencies
captures the effect of the ultraviolet energy cutoff. That at low frequencies
reflects a nonzero (albeit very low) temperature where the calculation is
performed ($T=4\times 10^{-5}$ $T_{K}$).

Figure S4: Emergence of a pole in the self-energy.  Fitting of the self-energy
using Eq. S7. Note that $g(\omega)$, which describes the asymptotic behavior
of the self-energy, takes the form $a+c\sqrt{|\omega|}$. The blue solid line
shows the imaginary part of the self-energy calculated in the 3D kagome
lattice at $T/T_{K}=10^{-3}$. The orange dash line is the fitted curve. We
observe a good consistency between the two. (B) Evolution of the fitted width
$\gamma$ as a function of temperature in the 3D kagome lattice. It saturates
to a nonzero value at temperatures low compared to $T_{K}$.

Figure S5: The imaginary part of the self-energy. Shown are the results for
the 2D kagome lattice (A) and cubic lattice (B). The calculations are done at
the temperature $T/T_{K}=10^{-3}$ in both cases.

Figure S6: Band reconstruction from the self-energy pole. (A) and (B) show the
spectral functions of the $d$ and $c$ electrons in the 2D kagome lattice. (C)
and (D) show the corresponding results in the 3D kagome lattice. All
calculations are done at $T/T_{K}=10^{-3}$. We can observe band reconstruction
near the Fermi energy in both cases.

Figure S7: Emergence of non-Fermi liquid Dirac nodal excitations in the 2D
model. (A) The $d$ electron spectral functions ($\rho_{d}$) at various
$\bm{k}$ points (marked by the red bar on the right) along high symmetry line
in the 2D kagome lattice. Solid and dashed lines denote two dispersive modes
from two eigenvalues of the Green’s function. The blue and orange dots label
the positions of the spectral peaks, which meet at a Dirac point. Each
$\rho_{d}$ curve has been shifted vertically to avoid overlapping. (B),(C)
illustrate the spectral functions of the two Dirac-point-bearing branches.
Here, the dashed line denotes the energy spectral peaks. The red bar marks the
cut of wave vectors in the Brillouin zone along which the spectral functions
are shown in panel A. The calculations are done at the temperature
$T/T_{K}=10^{-3}$ in all cases. Temperature dependence of the electrical
resistivity along the $x$ direction (D) and valley Hall conductivity (E). Both
quantities depend on temperature in a $\sqrt{T}$ manner.

Figure S8: Edge state and the valley Hall effect.  (A) Exponential decay of
the edge-mode eigenvector $v(k_{x}=4\pi/3,k_{z}=0)_{\alpha y}$ in the 3D
kagome model. $R$ denotes the distance to the edge. (B) Valley Hall
conductivity as a function of $V$ in the 2D kagome model with fixed
$V/\epsilon_{f}=-0.5$ at $T/T_{K}=10^{-3}$.

A B

Figure S9: Candidate two-channel Kondo material PrFe4P12. (A) Temperature-
dependent electrical resistivity at ambient pressure and at 2 GPa and 2.4 GPa,
revealing that the transition temperature $T_{\rm Q}$ of the phase with
antiferroquadrupolar order is successively suppressed with increasing pressure
(see inset). At lower temperatures, a resistivity upturn is observed below
$T_{\rm MI}$. (B) Temperature-dependent electrical resistivity at 3.3 GPa, for
several fixed magnetic fields up to 6 T. The non-metallic state is readily
suppressed by magnetic field. At 6 T, PrFe4P12 has metallized and exhibits
heavy fermion behavior, as evidenced by the strongly enhanced $T^{2}$
prefactor $A=13.7\,\mu\Omega$cm/K2 which, assuming the validity of the
Kadowaki-Woods relation, corresponds to a Sommerfeld coefficient of about 1
J/mol/K2, in good agreement with specific heat experiments (?). From Ref. ?.

Figure S10: Candidate two-channel Kondo material UPt3Au2. (A) Temperature-
dependent electrical resistivity for several UPt5-xAux samples. The weak
temperature dependence of UPt3Au2, together with the large resistivity values
at low temperatures, suggests that it is not metallic. From Ref. ?.

Figure S11: DFT results in PrBi. The bandstructure of the $spd$ conduction
electrons of PrBi, from $f$-core DFT calculations. The three yellow dots mark
the Dirac points The Dirac points in the non-$f$ conduction electron states
appear far away from the Fermi energy, and support the lattice symmetry
argument given in this SM, Sec. M.