# Mott insulators with boundary zeros

N. Wagner  Institut für Theoretische Physik und Astrophysik, Universität
Würzburg, 97074 Würzburg, Germany  L. Crippa  Institut für Theoretische Physik
und Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat,
Universität Würzburg, 97074 Würzburg, Germany  A. Amaricci  CNR-IOM
DEMOCRITOS, Istituto Officina dei Materiali, Consiglio Nazionale delle
Ricerche, Via Bonomea 265, 34136 Trieste, Italy  P. Hansmann  Department of
Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058, Erlangen,
Germany  M. Klett  Max-Planck-Institut für Festkörperforschung, Heisenbergstr.
1, 70569 Stuttgart, Germany  E. J. König  Max-Planck-Institut für
Festkörperforschung, Heisenbergstr. 1, 70569 Stuttgart, Germany  T. Schäfer
Max-Planck-Institut für Festkörperforschung, Heisenbergstr. 1, 70569
Stuttgart, Germany  D. Di Sante  University of Bologna, Bologna, Italy and
CCQ-Flatiron Institute, New York, NY, USA  J. Cano  Department of Physics and
Astronomy, Stony Brook University, Stony Brook, New York 11974, USA and CCQ-
Flatiron Institute, New York, NY, USA  A. J. Millis  Columbia University, New
York, NY, USA and CCQ-Flatiron Institute, New York, NY, USA  A. Georges
Collège de France, Paris, France and CCQ-Flatiron Institute, New York, NY, USA
G. Sangiovanni  Institut für Theoretische Physik und Astrophysik and Würzburg-
Dresden Cluster of Excellence ct.qmat, Universität Würzburg, 97074 Würzburg,
Germany

The topological classification of electronic band structures is based on
symmetry properties of Bloch eigenstates of single-particle Hamiltonians. In
parallel, topological field theory has opened the doors to the formulation and
characterization of non-trivial phases of matter driven by strong electron-
electron interaction. Even though important examples of topological Mott
insulators have been constructed, the relevance of the underlying non-
interacting band topology to the physics of the Mott phase has remained
unexplored. Here, we show that the momentum structure of the Green’s function
zeros defining the “Luttinger surface” provides a precise topological
characterization of the Mott phase surprisingly related to the one of the
single-particle electronic dispersion. Considerations on the zeros lead to the
prediction of new phenomena: a topological Mott insulator with an inverted gap
for the bulk zeros must possess gapless zeros at the boundary, which behave as
a form of “topological antimatter” annihilating conventional edge states.
Placing band and Mott topological insulators in contact produces distinctive
observable signatures at the interface, revealing the otherwise
spectroscopically elusive Green’s function zeros. The theoretical description
of topological order in physical systems has progressed along the parallel
routes of band- and quantum field-theory [1, 2, 3, 4, 5, 6]. The former, based
on single-particle Hamiltonians, offers a clear explanation of the origin of
topological invariants but is limited to the realm of weakly-interacting
perturbation theory. The latter, making use of the Green’s function formalism,
encompasses a wider range of phases, such as topological Mott insulators [7,
8, 9], at the cost of a higher theoretical complexity and a greater
computational effort. This more sophisticated approach is, however, necessary
since single-particle wave functions are no longer eigenvectors of the
interacting many-electron Hamiltonian and a full-fledged group theory-based
analysis of all possible two-body terms is far too impractical.

Mott insulators (MIs) are characterized by an interaction-driven gap opening
occurring without explicit breaking of an underlying symmetry or long-range
ordering. Phenomena of this kind, which are intrisically non-perturbative in
the coupling constant [10], result in gaps even when globally robust crossings
of bands such as Weyl cones or topological boundary states [11, 12, 13, 14,
15] are present. In these cases [16, 17] as well as in MIs arising from
conventional topological insulators (TIs) with an inverted gap [18, 19], a key
question is if and how the topological information encoded in the non-
interacting electronic dispersion survives and reemerges after the Mott
transition.

Topological gapless excitations can occur in the context of MI quantum spin
liquids concomitant with non-trivial entanglement of the fractionalized ground
state [20] allowing to circumvent the necessity of a Luttinger’s theorem-
imposed Fermi sea volume [21, 22]. Strongly correlated counterparts to bulk
semimetals disclose a relation to lattice symmetries in the case of non-
symmorphic space groups [23, 24, 25]. Topological Luttinger invariants have
been formulated specifically for such systems [26], broadening the conceptual
perimeter of the Luttinger surface in a MI [27]. Further, a discrete symmetry-
breaking has been proposed to be associated to a Mott-like transition [28]
considering an interaction term diagonal in momentum space [29]. Recently, the
existence of massless quasiparticles approaching a pseudogapped Luttinger
surface has been demonstrated [30].

Figure 1: Bulk and edge zeros in strongly correlated topological phases. a-b,
MIs originating at large values of $U$ from a semimetal with protected
crossings and, c-d, from a conventional non-interacting TI. In both situations
the non-interacting dispersion represents the energy-momentum location of the
Green’s function poles (light blue surfaces). In the corresponding Mott
phases, we find dispersive bulk zeros (red surfaces) inside the gap between
the lower and upper Hubbard bands (dark blue). In the case of the semimetallic
bandstructure, b, the bulk zeros display a symmetry-protected crossing. For
the topological insulating bandstructure, d, the bulk zeros form instead an
inverted gap, inherited from the non-interacting topological band structure.
Therefore, gapless edge zeros appear, as indicated by the red dashed lines.
Note the opposite spin direction associated to the edge zeros of the TMI
w.r.t. to that of the edge states of the TI.

Here, we search for symmetry-protected gapless modes despite the presence of a
hard Mott gap and for experimentally observable fingerprints thereof. We
present analytic as well as numerical evidence that the zeros of the Green’s
functions have a dispersion in momentum space that can be topologically
classified exactly as the corresponding non-interacting single-particle band
structure. This observation provides us with a clean signature of the
topological nature of Mott insulating states: first of all, MIs arising from
symmetry-protected semimetals must have gapless bulk zeros and are thereby
topologically distinct. Second, the zeros of a MI originating from a gapped
bandstructure are also gapped. When this bulk gap is of inverted character,
exotic gapless zeros localized at the boundaries must exist. We give proof of
their existence and propose how to detect them in an experiment, relying on
the intrinsically incoherent state that forms when a pole and a boundary zero
annihilate.

Momentum dispersion of Green’s function zeros – A pole of the single-particle
propagator close to the real-frequency axis describes a conventional
quasiparticle excitation of the system. A zero eigenvalue of $G({\bf
k},\omega)$, with ${\bf k}$ and $\omega$ indicating crystal momentum and
complex frequency respectively, corresponds instead to a divergence of the
self-energy $\Sigma({\bf k},\omega)$ and causes the opening of a Mott gap [31,
32, 33, 34]. For an isolated atom, the pole of $\Sigma$ is momentum-
independent. Moving away from this extreme limit, information on the lattice
dispersion enters into the picture. Deep in the Mott phase, we show that the
self-energy acquires a remarkably simple form:

$\displaystyle\Sigma({\bf
k},\omega)=\frac{U^{2}/4}{\omega+\widetilde{H_{0}}({\bf k})}.$ (1)

$\widetilde{H_{0}}({\bf k})$ indicates the non-interacting tight-binding
Hamiltonian with renormalized parameters. In the Supplementary Information we
extensively describe how to obtain these effective parameters in a controlled
way, by relating them to the spatially non-local spin-density correlators
${\langle n_{i\sigma}n_{j\sigma^{\prime}}\rangle-\langle
n_{i\sigma}\rangle\langle n_{j\sigma^{\prime}}\rangle}$. These are only found
to be finite beyond local mean-field theories [35], as discussed previously
for single-orbital models [36], in the pseudogap phase [37], to prove the
breakdown of Luttinger’s theorem in a MI [27] as well as in the context of
magnetically ordered phases [38] and doped spin liquids [39].

In this expression for the self-energy, it is important to note the $+$ sign
in front of $\widetilde{H_{0}}({\bf k})$, which is opposite to that in the
denominator of Green’s functions. Eq. 1 holds for single- as well as multi-
orbital Hamiltonians and correctly predicts that on-site and ${\bf
k}$-dependent terms are differently affected by the renormalization (see
Supplementary Information). The same form of the self-energy is obtained in
“failed” (quantum disordered) superconductors or spin density-wave systems, in
which fermions couple to a strongly fluctuating boson which prevents long-
range ordering. In the latter case, for instance, the $+$ sign is a
consequence of $H_{0}({\bf k}+{\bf Q})=-H_{0}({\bf k})$ appearing in the
denominator [38]. This observation suggests an intimate connection between Eq.
1 and paradigmatic quantum spin liquids, such as the resonating valence bond
theory [40] (a failed superconductor), and recent theories proposing
topological order for the pseudogap phase [41] by means of failed
antiferromagnetism. In the present work we explore the surprising implications
of Eq. 1 for the topology of strongly correlated electronic systems. The link
between Green’s function zeros and interacting topology has been pioneered by
Gurarie [42] and their role has been considered for the topological
classification of various systems [43, 44, 45] A simple connection between the
topological properties of the zeros and the microscopic non-interacting
Hamiltonian is however lacking. In Fig. 1 we illustrate two prototypical cases
predicted by Eq. 1: panels (a)-(b) show a symmetry-protected semimetal with
bulk bands that meet at some momentum. If the bandwidth is finite, such system
in three dimensions turns into a MI at a critical interaction strength [16].
As long as the interaction does not break any symmetry that protects the cones
of the non-interacting dispersion, Eq. 1 dictates that the zeros of $G$ will
display a crossing: Analogously to the non-interacting case, a gap in the
zeros can only be opened when two “zero-nodes” meet in momentum space, due to
the renormalization of the parameters. This finding offers a particularly
transparent explanation of the Mott transition in a Dirac or Weyl semimetal
[17, 16]: even if a gap between poles of $G$ opens, the protected linear
crossing is in fact not lifted: it just occurs between spectroscopically
invisible zeros.

The second implication regards a TI with an inverted bulk gap and boundary
Green’s function poles, as shown in Fig. 1c-d, turning into a topological Mott
insulator (TMI) at large interaction strengths. The bulk zeros of $G$
responsible for the opening of the Mott gap again obey Eq. 1: depending on the
renormalization of $\widetilde{H_{0}}({\bf k})$, the zeros can acquire an
inverted gapped dispersion. The sketch also illustrates that the predicted
gapless zeros are spatially localized at the boundaries. Their dispersion
inside the bulk gap follows the renormalized one of the edge modes in the
corresponding non-interacting TI with opposite sign.

Figure 2: Poles and zeros in an infinite chain. Spectral representation of the
determinant of $G^{R}(k,\omega)$ on the real axis highlighting poles (dark
blue) and zeros (dark red). In a, this is shown for a weakly interacting
SSH+$U$ with hopping parameters $v=0.2$ and $w=0.5$ in units of $t$, i.e. on
the topologically non-trivial side. The numerical solution is obtained with
cluster dynamical mean-field theory (CDMFT), whose details can be found in
Methods. In the Supplementary Information we show also QMC solutions of longer
SSH+$U$ chains. b, In the strongly interacting limit, in addition to the
Hubbard bands at about $\pm 3t$, dispersive zeros (in red) are visible inside
the spectral gap. The dashed black lines show a fit of analytical formula of
Eq. 1 to the position of the zeros of the CDMFT Green’s function.

Infinite and finite SSH+U chains – Firstly we test the validity of Eq. 1 with
the example of a one-dimensional Su-Schrieffer–Heeger (SSH) model [46] with
periodic boundary conditions. We supplement it by a local Hubbard $U$ in order
to induce the Mott phase. For an infinite SSH+$U$ chain at small values of
$U$, most of the spectral weight $A({\bf k},\omega)$ resembles the non-
interacting eigenvalues $|v+\operatorname{e}^{-ik}w|$ (blue bands in Fig. 2a).
At large interactions instead, a gap of order $U$ is sustained by low-energy
divergencies of $\Sigma({\bf k},\omega)$ (red dispersive features in Fig. 2b).
This result, which is in agreement with previous literature [43, 47], allows
us to compare the momentum dispersion of the zeros of $G$ with those predicted
by Eq. 1, shown by the black dashed lines in Fig. 2b. The renormalized
dispersion of the zeros perfectly describes our numerical result (see
Methods), confirming that Eq. 1 captures the essence of the momentum-dependent
self-energy of a MI.

Figure 3: Zero/pole annihilation in an SSH+$U$ finite chain. a, Exact solution
for a 12-site non-interacting SSH chain in the topologically non-trivial phase
($v=0.1$ and $w=1.0$). Zero-dimensional states are visible at the two ends of
the chain with some finite propagation inside the chain. We visualize this by
showing site-resolved eigenvector components, weighted with the corresponding
eigenvalues (see Methods). b, Exact solution for a 12-site SSH+$U$ chain in
the topological phase with the same $v$ and $w$ but a large value of the on-
site interaction $U$, i.e. in the Mott phase. The ends of the chain now host
two zeros inside the Mott gap (of order 4 in units of $t$). The topological
invariant of these two chains is the same but the interacting chain has no
edge pole, rather only zeros that are nicely visualized via the weights. c,
Interface (i.e. $f=1$) between the two situations above; the position of the
interface is indicated with a dashed, vertical line. In order to align the
chemical potential inside the global bulk gap, we subtract the Hartree-shift
on the interacting chain.

Since our focus is on boundary zeros, we need to validate Eq. 1 also in the
case of an open SSH+$U$ chain. With a finite number of sites, the momentum
dependence in the denominator is replaced by the real-space matrix
representation of $\widetilde{H_{0}}({\bf k})$. At $U=0$ and for nonzero
winding number of the SSH Hamiltonian, $G$ possesses zero-energy poles at the
two ends of the chain. This is signaled by the two blue states on the left-
and right-most sites of the chain in Fig. 3a. For large $U$, we solve the
finite chain with exact diagonalization (ED) and obtain zeros at the ends of
the chain instead, shown in red in Fig. 3b. Interestingly, the topological
nature of the gap of the zeros, analogously to what happens for the poles of
$G$, implies the existence of two “in-gap” zeros at the boundary of the
system. In the Supplementary Information we compare these ED results with
those acquired using Eq. 1, demonstrating that the analytic formula is also
well suited to describe the non-local many-body features of a MI for finite
size systems. To address the case in which an edge pole and an edge zero get
spatially close to one another we look at the system shown in Fig. 3c. The two
non-trivial SSH chains ($U$=0 on the right and finite $U$ on the right) are
connected by a hopping in the center, that can be switched on and off via a
parameter dubbed $f$. Fig. 3c shows the fate of pole and zero at the two ends
which meet at the center of the new chain. They hybridize giving rise to a
finite spectral weight which appears at the interface (sites 6 and 7). This is
an incoherent “bad metal”, i.e. neither a peak nor a vanishing spectral
weight, resulting from the pole/zero annihilation. The solution obtained in
Fig. 3 is fully compatible with symmetry and topological requirements of
having two gapless modes at the two ends of the chain, due to the
interchangeable role of poles and zeros. From a spectroscopic point of view,
instead, it is highly unexpected as the new chain, seen as a whole, has in
fact only one “detectable” edge state, at the left end. Its partner at the
right end, is the dual zero which of course would not be visible in a
tunneling experiment.

Two-dimensional topological Mott insulator – In two dimensions we find that
Eq. 1 continues to give an excellent description of the dispersion of the
zeros of $G$ in MIs. A detailed analysis including a comparison with numerical
quantum cluster methods for different real-space geometries, can be found in
the Supplementary Information. Analogously to the one-dimensional case, we
define a TMI through the bulk-boundary correspondence in an extended sense: if
the zeros of $G$ have a topologically inverted gap in the bulk, then gapless
zeros have to appear at the edge. In contrast, conventional MIs never display
gapless zeros, regardless whether one looks at their interior or at the
boundary.

Figure 4: Evolution of wave packet along the edge. a, A wave packet is
initialized from the leftmost part of the edge of a QSH (green part, see
inset) and let evolve towards the interface with a trivial Mott insulator (see
inset). Upon entering the blue interface region, no appreciable change of the
wave packet is observed, except for some small loss of its relative weight due
to reflections and scattering from the walls of the two-dimensional structure.
b, The left-moving wave packet evolving along the edge now enters the red
region where a topological Mott insulator is placed on the other side. The
edge zeros of the TMI hybridize with the edge mode of the QSH and determine
the immediate collapse of the wave packet. This sudden loss of weight can be
used as a detection protocol for the presence of boundary zeros on the Mott
insulating side. In both cases the time evolution is calculated via the
Green’s function of a 40$\times$40 slab where the interaction is taken into
account via the analytic self-energy in Eq.1.

Compared to zero-dimensional poles/zeros of the SSH+$U$, the gapless edge
zeros – living inside the gap of the bulk zeros of a TMI – acquire a
dispersion w.r.t. the momentum parallel to the edge. This implies that the
pole/zero annihilation gets even more intriguing than that shown in Fig. 3c.
In the following we therefore analyze a two-dimensional heterostructure
between a conventional quantum spin Hall (QSH) system and a TMI, as
illustrated in Fig. 4b. The two parts have a segment of the edge in common and
we ask what the consequences are on the helical modes when they travel in this
region. We also consider two additional cases: a “benchmark”, in which the two
sides are completely disconnected ($f$=0) and one where the TMI is replaced by
a trivial Mott (Fig. 4a). We initialize a wave packet at the very left of the
QSH side and let it evolve in time towards the interface with the TMI. The
time evolution is governed by the full Green’s function of this hybrid 2D
interacting system, where the interaction is included using the analytic
formula of Eq.1. As shown in Fig. 4b as well as in the movie in the
Supplementary Information, the wave packet is well defined only up to the
interface with the TMI (marked in red in the inset to Fig. 4b). As soon as it
enters this region, the edge state becomes immediately incoherent and loses
spectral weight. In the other two cases ($f$=0 and trivial MI) the propagation
proceeds undisturbed, as standard topological arguments predict (Fig. 4a and
movie in the Supplementary Information).

We have thus described an experimental probe sensitive to the otherwise
invisible zeros. Based on the dynamics of the wave packet and its clearcut
distinct coherence, the propagation represents a perfect way to detect the
presence of the boundary zeros on the TMI side. We have checked that when the
wave packet starts to lose weight, there is no component of the QSH edge state
that tries to circumvent the TMI part. This is strikingly different from what
would instead happen if we were to replace a portion of the QSH with a trivial
system. In that case the edge state of the QSH would go around the trivial
region. Here instead, the QSH loses the helical state even though nothing has
happened to its topological properties.

Operatively, one can first quantify the renormalization of parameters of a
single-particle tight-binding model for a given material, as outlined in the
Supplementary Information. Then, Eq. 1 can be used to predict the nature of
the corresponding MI. Importantly, the renormalization of the parameters in
$\widetilde{H_{0}}({\bf k})$ can in principle be in both directions: the non-
trivial region on the Mott side can be smaller or larger than that of
$H_{0}({\bf k})$, i.e. one can generate a TMI from a trivial band structure or
get vice-versa a trivial Mott from a QSH, depending on the specific model and
type of interaction.

Should a Mott material be non-trivial according to Eq. 1, one can then exploit
the annihilation phenomenon discussed in the last part of this work to
unambiguously tell whether or not zeros exist at its boundary. These results
open interesting perspectives both in connection to the theory of topological
order and in the characterization of protected bulk and surface features in
the realm of non-Hermitian physics with strong correlation.

##  I Methods

Cluster-DMFT. The numerical results for the bulk systems presented in this
work are obtained within the framework of cluster-DMFT, an extension of
Dynamical Mean-Field Theory capable of grasping nonlocal correlations. The
Exact Diagonalization results have been obtained using a cluster extension of
the EDIPack code [48], where the SSH model is mapped to a finite “cluster
impurity”, consisting of two or three interacting dimers, coupled to a finite
bath. This is structured as a number of non-interacting clusters replicating
the one-particle hopping matrix of the impurity, each site of which is coupled
to the corresponding impurity one. The intra-replica hopping amplitudes and
bath-impurity couplings are adjusted self-consistently. Two such replicas have
been used for the 2-dimers case, and 1 replica for the 3-dimer case. The BHZ
model is solved through an asymmetric cluster impurity consisting of two sites
along the $x$ direction, coupled to two bath replicas. Benchmark tests with a
3x1-sites cluster plus 1 bath replica and a 2$\times$2 cluster plus 1 replica
have confirmed analogous results, the latter restoring the symmetry of the
model though at the cost of a dramatically increased computational time.

Single-shot calculation for finite chains. The finite size SSH effects have
been obtained through a single-shot exact diagonalization of an impurity
cluster of 6 dimers decoupled from any bath. For the quantum Monte Carlo
calculations a single-shot solution of the impurity problem, again decoupled
from any bath, has been acquired via the use of a continuous time quantum
Monte Carlo solver based on the interaction expansion (CT-INT)[49]. For every
QMC calculation a statistic of 50 million Monte Carlo cycles is used.

Determination of the finite chain zeros. In order to spatially resolve the
zeros of real-space Green’s function (cfr. Fig. 3) we look at weighted
eigenvector components, i.e. the quantity

$w_{i}=\sum_{j}\big{\lvert}\psi_{i}^{(j)}E_{j}\Big{\rvert}$ (2)

where $\psi_{i}^{(j)}$ is the i-th element of the j-th eigenvector and $E_{j}$
is the j-th eigenvalue.
Slab wavepacket evolution. For the 2D slab calculations the interaction is
taken into account using the analytic formula for the self-energy (Eq. 1). The
time evolution of a wave packet at location $r$ and time $t$ is given by

$\psi(r,t)=\int dr^{\prime}G(r,r^{\prime},t)f(r_{0}-r^{\prime})$ (3)

where $f(r_{0}-r^{\prime})$ is a gaussian centered about $r_{0}$.

###### Acknowledgements.

Acknowledgments – N.W. is supported by the SFB 1170 Tocotronics, funded by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID
258499086. L.C. acknowledges financial support from the DFG through the
Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum
Matter–ct.qmat (EXC 2147, project-id 390858490). G.S. acknowledges support
from the DFG through FOR 5249-449872909 (Project P05). We gratefully
acknowledge the Gauss Center for Supercomputing e.V. (www.gauss-center.eu) for
funding this project by providing computing time on the GCS Supercomputer
SuperMUC at Leibniz Supercomputing Center (www.lrz.de). Part of the numerical
calculations were carried out using the Julia programming language [50]. J.C.
acknowledges support from the Air Force Office of Scientific Research under
Grant No. FA9550-20-1-0260. A.J.M. was supported in part by Programmable
Quantum Materials, an Energy Frontier Research Center funded by the U.S.
Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES),
under Award No. DE-SC0019443. The research leading to these results has
received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie Grant Agreement No.
897276. The Flatiron Institute is a division of the Simons Foundation.

## References

  * Bradlyn  _et al._ [2017] B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature 547, 298 (2017).
  * Schnyder  _et al._ [2008] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Physical Review B 78, 195125 (2008).
  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Reviews of Modern Physics 83, 1057 (2011).
  * Qi  _et al._ [2008] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Physical Review B 78, 195424 (2008).
  * Wang  _et al._ [2010] Z. Wang, X.-L. Qi, and S.-C. Zhang, Topological Order Parameters for Interacting Topological Insulators, Physical Review Letters 105, 256803 (2010).
  * Fu and Kane [2006] L. Fu and C. L. Kane, Time reversal polarization and a Z 2 adiabatic spin pump, Physical Review B 74, 195312 (2006).
  * Raghu  _et al._ [2008] S. Raghu, X.-L. Qi, C. Honerkamp, and S.-C. Zhang, Topological Mott Insulators, Physical Review Letters 100, 156401 (2008).
  * Wen  _et al._ [2010] J. Wen, A. Rüegg, C.-C. J. Wang, and G. A. Fiete, Interaction-driven topological insulators on the kagome and the decorated honeycomb lattices, Physical Review B 82, 075125 (2010).
  * Pesin and Balents [2010] D. Pesin and L. Balents, Mott physics and band topology in materials with strong spin–orbit interaction, Nature Physics 6, 376 (2010).
  * Gunnarsson  _et al._ [2017] O. Gunnarsson, G. Rohringer, T. Schäfer, G. Sangiovanni, and A. Toschi, Breakdown of Traditional Many-Body Theories for Correlated Electrons, Physical Review Letters 119, 056402 (2017).
  * Bradlyn  _et al._ [2016] B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser, R. J. Cava, and B. A. Bernevig, Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals, Science 353, aaf5037 (2016).
  * Muechler  _et al._ [2016] L. Muechler, A. Alexandradinata, T. Neupert, and R. Car, Topological Nonsymmorphic Metals from Band Inversion, Physical Review X 6, 041069 (2016).
  * Young  _et al._ [2012] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, Dirac Semimetal in Three Dimensions, Physical Review Letters 108, 140405 (2012).
  * Kim  _et al._ [2015] Y. Kim, B. J. Wieder, C. Kane, and A. M. Rappe, Dirac Line Nodes in Inversion-Symmetric Crystals, Physical Review Letters 115, 036806 (2015).
  * Wieder  _et al._ [2016] B. J. Wieder, Y. Kim, A. Rappe, and C. Kane, Double Dirac Semimetals in Three Dimensions, Physical Review Letters 116, 186402 (2016).
  * Wagner  _et al._ [2021] N. Wagner, S. Ciuchi, A. Toschi, B. Trauzettel, and G. Sangiovanni, Resistivity Exponents in 3D Dirac Semimetals From Electron-Electron Interaction, Physical Review Letters 126, 206601 (2021).
  * Braguta  _et al._ [2016] V. V. Braguta, M. I. Katsnelson, A. Y. Kotov, and A. A. Nikolaev, Monte Carlo study of Dirac semimetals phase diagram, Physical Review B 94, 205147 (2016).
  * Budich  _et al._ [2012] J. C. Budich, R. Thomale, G. Li, M. Laubach, and S.-C. Zhang, Fluctuation-induced topological quantum phase transitions in quantum spin-hall and anomalous-hall insulators, Phys. Rev. B 86, 201407 (2012).
  * Amaricci  _et al._ [2015] A. Amaricci, J. Budich, M. Capone, B. Trauzettel, and G. Sangiovanni, First-Order Character and Observable Signatures of Topological Quantum Phase Transitions, Physical Review Letters 114, 185701 (2015).
  * Savary and Balents [2017] L. Savary and L. Balents, Quantum spin liquids: a review, Reports on Progress in Physics 80, 016502 (2017).
  * Oshikawa [2000] M. Oshikawa, Topological Approach to Luttinger’s Theorem and the Fermi Surface of a Kondo Lattice, Physical Review Letters 84, 3370 (2000).
  * Paramekanti and Vishwanath [2004] A. Paramekanti and A. Vishwanath, Extending Luttinger’s theorem to Z 2 fractionalized phases of matter, Physical Review B 70, 245118 (2004).
  * Watanabe  _et al._ [2015] H. Watanabe, H. C. Po, A. Vishwanath, and M. Zaletel, Filling constraints for spin-orbit coupled insulators in symmorphic and nonsymmorphic crystals, Proceedings of the National Academy of Sciences 112, 14551 (2015).
  * Kimchi  _et al._ [2013] I. Kimchi, S. A. Parameswaran, A. M. Turner, F. Wang, and A. Vishwanath, Featureless and nonfractionalized Mott insulators on the honeycomb lattice at 1/2 site filling, Proceedings of the National Academy of Sciences 110, 16378 (2013).
  * Ware  _et al._ [2015] B. Ware, I. Kimchi, S. A. Parameswaran, and B. Bauer, Topological crystalline Bose insulator in two dimensions via entanglement spectrum, Physical Review B 92, 195105 (2015).
  * Parameswaran [2019] S. A. Parameswaran, Topological ‘Luttinger’ invariants for filling-enforced non-symmorphic semimetals, Journal of Physics: Condensed Matter 31, 104001 (2019).
  * Rosch [2007] A. Rosch, Breakdown of Luttinger’s theorem in two-orbital Mott insulators, The European Physical Journal B 59, 495 (2007).
  * Huang  _et al._ [2022] E. W. Huang, G. L. Nave, and P. W. Phillips, Discrete symmetry breaking defines the Mott quartic fixed point, Nature Physics 10.1038/s41567-022-01529-8 (2022).
  * Morimoto and Nagaosa [2016] T. Morimoto and N. Nagaosa, Weyl Mott Insulator, Scientific Reports 6, 19853 (2016).
  * Fabrizio [2022] M. Fabrizio, Emergent quasiparticles at Luttinger surfaces, Nature Communications 13, 1561 (2022).
  * Stanescu and Kotliar [2006] T. D. Stanescu and G. Kotliar, Fermi arcs and hidden zeros of the Green function in the pseudogap state, Physical Review B 74, 125110 (2006).
  * Stanescu  _et al._ [2007] T. D. Stanescu, P. Phillips, and T.-P. Choy, Theory of the Luttinger surface in doped Mott insulators, Physical Review B 75, 104503 (2007).
  * Sakai  _et al._ [2009] S. Sakai, Y. Motome, and M. Imada, Evolution of Electronic Structure of Doped Mott Insulators: Reconstruction of Poles and Zeros of Green’s Function, Physical Review Letters 102, 056404 (2009).
  * Sakai  _et al._ [2010] S. Sakai, Y. Motome, and M. Imada, Doped high- T c cuprate superconductors elucidated in the light of zeros and poles of the electronic Green’s function, Physical Review B 82, 134505 (2010).
  * Rohringer and Toschi [2016] G. Rohringer and A. Toschi, Impact of nonlocal correlations over different energy scales: A dynamical vertex approximation study, Physical Review B 94, 125144 (2016).
  * Pairault  _et al._ [2000] S. Pairault, D. Sénéchal, and A.-M. Tremblay, Strong-coupling perturbation theory of the Hubbard model, The European Physical Journal B 16, 85 (2000).
  * Pudleiner  _et al._ [2016] P. Pudleiner, T. Schäfer, D. Rost, G. Li, K. Held, and N. Blümer, Momentum structure of the self-energy and its parametrization for the two-dimensional Hubbard model, Physical Review B 93, 195134 (2016).
  * Altshuler  _et al._ [1998] B. L. Altshuler, A. V. Chubukov, A. Dashevskii, A. M. Finkel’stein, and D. K. Morr, Luttinger theorem for a spin-density-wave state, Europhysics Letters (EPL) 41, 401 (1998).
  * Konik  _et al._ [2006] R. M. Konik, T. M. Rice, and A. M. Tsvelik, Doped spin liquid: Luttinger sum rule and low temperature order, Phys. Rev. Lett. 96, 086407 (2006).
  * Anderson [1987] P. W. Anderson, The resonating valence bond state in la2cuo4 and superconductivity, science 235, 1196 (1987).
  * Scheurer  _et al._ [2018] M. S. Scheurer, S. Chatterjee, W. Wu, M. Ferrero, A. Georges, and S. Sachdev, Topological order in the pseudogap metal, Proceedings of the National Academy of Sciences 115, E3665 (2018).
  * Gurarie [2011] V. Gurarie, Single-particle Green’s functions and interacting topological insulators, Physical Review B 83, 085426 (2011).
  * Manmana  _et al._ [2012] S. R. Manmana, A. M. Essin, R. M. Noack, and V. Gurarie, Topological invariants and interacting one-dimensional fermionic systems, Physical Review B 86, 205119 (2012).
  * Wang and Zhang [2012] Z. Wang and S.-C. Zhang, Simplified Topological Invariants for Interacting Insulators, Physical Review X 2, 031008 (2012).
  * Muechler [2020] L. Muechler, Topological classification of molecules and chemical reactions with a perplectic structure, Phys. Rev. B 101, 045123 (2020).
  * Su  _et al._ [1980] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Soliton excitations in polyacetylene, Physical Review B 22, 2099 (1980).
  * Yoshida  _et al._ [2014] T. Yoshida, R. Peters, S. Fujimoto, and N. Kawakami, Characterization of a Topological Mott Insulator in One Dimension, Physical Review Letters 112, 196404 (2014).
  * Amaricci  _et al._ [2022] A. Amaricci, L. Crippa, A. Scazzola, F. Petocchi, G. Mazza, L. de Medici, and M. Capone, EDIpack: A parallel exact diagonalization package for quantum impurity problems, Computer Physics Communications 273, 108261 (2022).
  * Parcollet  _et al._ [2015] O. Parcollet, M. Ferrero, T. Ayral, H. Hafermann, I. Krivenko, L. Messio, and P. Seth, Triqs: A toolbox for research on interacting quantum systems, Computer Physics Communications 196, 398 (2015).
  * Bezanson  _et al._ [2017] J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, Julia: A Fresh Approach to Numerical Computing, SIAM Review 59, 65 (2017).
  * Emanuel [2008] G. Emanuel,  _Continuous-Time Quantum Monte Carlo Algorithms for Fermions_ , Ph.D. thesis, ETH Zurich (2008).
  * Gull  _et al._ [2011] E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Continuous-time Monte Carlo methods for quantum impurity models, Reviews of Modern Physics 83, 349 (2011).
  * Haule [2007] K. Haule, Quantum Monte Carlo impurity solver for cluster dynamical mean-field theory and electronic structure calculations with adjustable cluster base, Physical Review B 75, 155113 (2007).
  * Tomczak [2007] J. M. Tomczak, Spectral and Optical Properties of Correlated Materials (2007).
  * Fu and Kane [2007] L. Fu and C. L. Kane, Topological insulators with inversion symmetry, Physical Review B 76, 045302 (2007).
  * Yoshida  _et al._ [2012] T. Yoshida, S. Fujimoto, and N. Kawakami, Correlation effects on a topological insulator at finite temperatures, Physical Review B 85, 125113 (2012).
  * Ando [2013] Y. Ando, Topological Insulator Materials, Journal of the Physical Society of Japan 82, 102001 (2013).

## Supplementary Material: Mott insulators with boundary zeros

This Supplementary Material is structured as follows: The first section (I)
describes the derivation of Eq.(1) of the main text, i.e.
$\Sigma(k,\omega)=\frac{U^{2}/4}{\omega+\widetilde{H_{0}}(k)}$. The second
section (II) gives further details about the numerical calculations for the 2D
TI slab calculations. Additional results and comparisons between the analytic
self-energy formula and numerical calculations are shown in the third section
(III).

##  I High-frequency expansion

###  I.1 Derivation of the expansion

In order to derive the high-frequency expansion we start by expanding the
Green’s function. Following references [51, 52, 53], the Matsubara Green’s
function $G_{\eta\rho}(i\omega_{n},k)$ (where $\eta$ and $\rho$ describe the
orbital and spin degrees of freedom, $\omega_{n}$ is the Matsubara frequency
and $k$ is crystal momentum) can be expressed using the Fourier transform of
$G(\tau,k)$:

$G_{\eta\rho}(i\omega_{n},k)=\int_{0}^{\beta}e^{i\omega_{n}\tau}G_{\eta\rho}(\tau,k).$
(4)

Here $\tau$ is the imaginary time and $\beta$ is the inverse temperature. This
expression can be rewritten as

$G_{\eta\rho}(i\omega_{n},k)=\frac{1}{i\omega_{n}}\int_{0}^{\beta}(\partial_{\tau}e^{i\omega_{n}\tau})G_{\eta\rho}(\tau,k).$
(5)

Now, integration by parts yields

$G_{\eta\rho}(i\omega_{n},k)=\frac{1}{i\omega_{n}}\left(-G_{\eta\rho}(\beta^{-},k)-G_{\eta\rho}(0^{+},k)-\int_{0}^{\beta}e^{i\omega_{n}\tau}(\partial_{\tau}G_{\eta\rho}(\tau,k))\right).$
(6)

This procedure can be repeated for the second term leading to

$\displaystyle G_{\eta\rho}(i\omega_{n},k)$
$\displaystyle=-\frac{1}{i\omega_{n}}\left(G_{\eta\rho}(\beta^{-},k)+G_{\eta\rho}(0^{+},k)\right)+\frac{1}{(i\omega_{n})^{2}}\left(\partial_{\tau}G_{\eta\rho}(\beta,0)+\partial_{\tau}G_{\eta\rho}(0,k)\right)$
(7)
$\displaystyle\quad+\frac{1}{(i\omega_{n})^{3}}\int_{0}^{\beta}(\partial_{\tau}e^{i\omega_{n}\tau})(\partial_{\tau}^{2}G_{\eta\rho}(\tau,k)).$

Carrying out this procedure repeatedly we arrive at

$\displaystyle
G_{\eta\rho}(i\omega_{n},k)=\frac{1}{i\omega_{n}}\sum_{j=0}^{\infty}(-1)^{j+1}\frac{\left(\partial_{\tau}^{j}G_{\eta\rho}(\beta^{-},k)+\partial_{\tau}^{j}G_{\eta\rho}(0^{+},k)\right)}{(i\omega_{n})^{j}}=\frac{1}{i\omega_{n}}\sum_{j=0}^{\infty}\frac{(-1)^{j+1}C_{j}^{\eta\rho}}{(i\omega_{n})^{j}}.$
(8)

The derivatives of $G_{\eta\rho}(\tau,k)$ can be computed using

$G_{\eta\rho}(\tau,k)=-\expectationvalue{c_{\eta}(k,\tau)c^{\dagger}_{\rho}(k,0)}=-\frac{\operatorname{Tr}(e^{-\beta
H}e^{H\tau}c_{\eta}(k,0)e^{-H\tau}c^{\dagger}_{\rho}(k,0))}{\operatorname{Tr}(e^{-\beta
H})}.$ (9)

We get

$\displaystyle
C_{0}^{\eta\rho}=(G_{\eta\rho}(0^{+},k)+G_{\eta\rho}(\beta^{-},k))$
$\displaystyle=-1$ (10) $\displaystyle C_{1}^{\eta\rho}$
$\displaystyle=(\partial_{\tau}G(0^{+},k)+\partial_{\tau}G(\beta^{-},k))=-\expectationvalue{\anticommutator{\commutator{H}{c_{\eta}}}{c_{\rho}^{\dagger}}}$
(11)

and in general

$C^{\eta\rho}_{i}=-\expectationvalue{\anticommutator{\underbrace{[H,\dots\commutator{H}{c_{\eta}(k)]}}_{\text{i
commutators}}}{c^{\dagger}_{\rho}(k)}}.$ (12)

The Green’s function is given by

$G(i\omega_{n},k)=\left(i\omega_{n}-\varepsilon(k)-\Sigma(i\omega_{n},k)\right)^{-1}$
(13)

where, depending on the degrees of freedom, all quantities can be matrices.
Inverting the previous formula yields the self-energy

$\Sigma(i\omega_{n},k)=i\omega_{n}-\varepsilon(k)-\frac{1}{G(i\omega_{n},k)}=i\omega_{n}-\varepsilon(k)-\frac{i\omega_{n}}{1-\sum_{j=1}^{\infty}(-1)^{j}\frac{C_{j}}{(i\omega_{n})^{j}}}.$
(14)

In the limit of $i\omega_{n}\rightarrow\infty$ the last term can be seen as
the sum of a geometric series. Therefore:

$\displaystyle\Sigma(i\omega_{n},k)$
$\displaystyle=i\omega_{n}-\varepsilon(k)-(i\omega_{n})\left(1+\sum_{n=1}^{\infty}(-1)^{n}\frac{C_{n}}{(i\omega_{n})^{n}}+\left(\sum_{n=1}^{\infty}(-1)^{n}\frac{C_{n}}{(i\omega_{n})^{n}}\right)^{2}+\dots\right)$
(15)
$\displaystyle=-\varepsilon(k)-(i\omega_{n})\left(\sum_{n=1}^{\infty}(-1)^{n}\frac{C_{n}}{(i\omega_{n})^{n}}+\left(\sum_{n=1}^{\infty}(-1)^{n}\frac{C_{n}}{(i\omega_{n})^{n}}\right)^{2}+\dots\right).$

Expanding in orders of $\frac{1}{\omega}$ we arrive at the following
expression for the self-energy:

$\displaystyle\Sigma(i\omega,k)$
$\displaystyle=-\varepsilon(k)+\mu+C_{1}(k)-\frac{C_{2}(k)+C_{1}^{2}(k)}{i\omega}$
(16)
$\displaystyle\quad+\frac{C_{3}(k)+C_{1}(k)C_{2}(k)+C_{2}(k)C_{1}(k)+C^{3}_{1}(k)}{i\omega^{2}}+\mathcal{O}\left(\left(\frac{1}{i\omega}\right)^{3}\right).$

Note that $C_{i}$ are in general matrices. We can identify the poles of the
self-energy by comparing with the expansion

$\frac{A}{i\omega-B}=\frac{A}{i\omega}+\frac{AB}{i\omega^{2}}+\mathcal{O}\left(\left(\frac{1}{i\omega}\right)^{3}\right).$
(17)

Close to the atomic limit, $B$ is small. This will be justified below but is
also apparent when comparing with the atomic limit self-energy
$\Sigma=\frac{U^{2}}{4i\omega}$. This means that higher order corrections are
not only suppressed by powers of $\frac{1}{i\omega}$ but also by increasing
powers of $B$. For the Hubbard dimer discussed in section I.4 we have checked
that the expansions of Eq. 16 and Eq.17 coincide up to fourth order in
$\frac{1}{i\omega}$.

###  I.2 Application to the Hubbard-model

In order to arrive at Eq.(1) of the main text we have to compute the
commutators in Eq. (12) for the Hubbard model.
We consider a multi-orbital case with local Hubbard repulsion of strength $U$:

$H=\sum_{kab\sigma}(\varepsilon_{k}^{ab\sigma\sigma^{\prime}}-\mu\delta_{ab}\delta_{\sigma\sigma^{\prime}})c_{ka\sigma}^{\dagger}c_{kb\sigma^{\prime}}+\frac{U}{2}\sum_{ia\sigma}n_{ia\sigma}n_{ia-\sigma}$
(18)

where $\varepsilon_{k}^{ab\sigma\sigma^{\prime}}$ is the dispersion as a
function of momentum $k$ with orbital indices $a$ and $b$ as well as spin
degrees of freedom $\sigma$ and $\sigma^{\prime}$. The chemical potential is
given by $\mu$, $i$ indicates a lattice site and $n$ is the number operator.
We call the on-site, $k$-independent terms of the dispersion
$\varepsilon_{\text{on-site}}^{ab\sigma\sigma^{\prime}}\delta_{ab}$.
Up to second order in $\frac{1}{\omega}$ ($\omega$ being either $i\omega_{n}$
or $\omega+i\delta$), equation (16) leads to

$\Sigma^{ab\sigma\sigma^{\prime}}(k,\omega)=\frac{U}{2}\delta_{ab}\delta_{\sigma\sigma^{\prime}}+\frac{U^{2}}{4}\frac{1}{\omega}\delta_{ab}\delta_{\sigma\sigma^{\prime}}+\frac{U^{2}}{4}\frac{\chi^{ab\sigma\sigma^{\prime}}(k)}{\omega^{2}}$
(19)

for the half-filled case. $\chi^{ab\sigma\sigma^{\prime}}(k)$ is given by

$\displaystyle\chi^{ab\sigma\sigma^{\prime}}(k)$
$\displaystyle=-\varepsilon_{k}^{ab\sigma\sigma^{\prime}}+\frac{2}{N}\sum_{q}\varepsilon_{q}^{ab\sigma\sigma^{\prime}}\delta_{ab}\delta{\sigma\sigma^{\prime}}+\frac{2}{N}\sum_{q}\varepsilon_{q}^{ab\sigma\sigma^{\prime}}\delta_{ab}\delta_{\sigma,-\sigma^{\prime}}$
(20)
$\displaystyle-\frac{4}{N^{2}}\sum_{qlj}\varepsilon_{q}^{ab\sigma\sigma^{\prime}}e^{i(k-q)(r_{l}-r_{j})}\expectationvalue{c^{\dagger}_{la-\sigma}c^{\dagger}_{jb-\sigma^{\prime}}c_{la-\sigma}c_{jb-\sigma^{\prime}}}$
$\displaystyle-\frac{4}{N^{2}}\sum_{qlj}\varepsilon_{q}^{ab\sigma\sigma^{\prime}}e^{i(k-q)(r_{l}-r_{j})}\expectationvalue{c^{\dagger}_{la-\sigma}c^{\dagger}_{jb\sigma^{\prime}}c_{la\sigma}c_{jb-\sigma^{\prime}}}$

where $N$ is a normalization factor. Corrections to this expression depend
either on the double occupations or on the expectation value of hopping terms.
As shown in section I.3 these are much smaller then the expectation values in
$\chi^{ab\sigma\sigma^{\prime}}(k)$.
For on-site terms (with $a=b$) the expectation values in the last two terms
vanish and we arrive at

$\chi^{ab\sigma\sigma^{\prime}}_{\text{on-site}}=-\varepsilon_{\text{on-
site}}^{ab\sigma\sigma^{\prime}}\delta_{ab}+\frac{2}{N}\sum_{q}\varepsilon_{q}^{ab\sigma\sigma^{\prime}}\delta_{ab}\delta_{\sigma\sigma^{\prime}}+\frac{2}{N}\sum_{q}\varepsilon_{q}^{ab\sigma\sigma^{\prime}}\delta_{ab}\delta_{\sigma,-\sigma^{\prime}}$
(21)

which under the assumption that all diagonal $q$-dependent terms in
$\varepsilon_{q}^{ab\sigma\sigma^{\prime}}$ sum to zero reduces to

$\chi^{ab\sigma\sigma^{\prime}}_{\text{on-site}}=\varepsilon_{\text{on-
site}}^{ab\sigma\sigma^{\prime}}\delta_{ab}.$ (22)

This is consistent with what one would expect for on-site terms. Those can be
exactly described within the atomic limit, leading only to constant shifts of
poles and zeros respectively. Going beyond the atomic limit we therefore would
not expect any renormalization.
For the remaining terms we have to evaluate the expectation values instead:

$\displaystyle\chi_{\text{rest}}^{ab\sigma\sigma^{\prime}}(k)$
$\displaystyle=-\varepsilon_{k}^{ab\sigma\sigma^{\prime}}-\frac{4}{N^{2}}\sum_{qlj}\varepsilon_{q}^{ab\sigma\sigma^{\prime}}e^{i(k-q)(r_{l}-r_{j})}\expectationvalue{c^{\dagger}_{la-\sigma}c^{\dagger}_{jb-\sigma^{\prime}}c_{la-\sigma}c_{jb-\sigma^{\prime}}}$
(23)
$\displaystyle\quad-\frac{4}{N^{2}}\sum_{qlj}\varepsilon_{q}^{ab\sigma\sigma^{\prime}}e^{i(k-q)(r_{l}-r_{j})}\expectationvalue{c^{\dagger}_{la-\sigma}c^{\dagger}_{jb\sigma^{\prime}}c_{la\sigma}c_{jb-\sigma^{\prime}}}.$

The prefactor in front of the expectation values in Eq. (23) restricts $j$ and
$l$ to sites connected by a hopping in the non-interacting dispersion. This
can be seen by defining $\alpha=l-j$ and explicitly writing the dispersion as
a sum over all hopping directions $\beta$:

$\displaystyle\frac{4}{N^{2}}\sum_{q,l,j}\varepsilon_{q}^{ab\sigma\sigma^{\prime}}e^{i(k-q)(r_{l}-r_{j})}f^{ab\sigma\sigma^{\prime}}_{lj}$
$\displaystyle=\frac{4}{N}\sum_{q,\alpha,\beta}t^{ab\sigma\sigma^{\prime}}_{\beta}e^{iqd_{\beta}}e^{i(k-q)d_{\alpha}}f^{ab\sigma\sigma^{\prime}}_{\alpha}$
(24)
$\displaystyle=4\sum_{\beta}t^{ab\sigma\sigma^{\prime}}_{\beta}e^{ikd_{\beta}}f^{ab\sigma\sigma^{\prime}}_{\beta}$

where $f$ stands for either one of the expectation values in Eq. (23) and
$d_{\alpha}$ indicates the distance between two sites with indices differing
by $\alpha$. Here it is assumed that $d_{\beta}\neq 0$ or $a\neq b$, otherwise
the corresponding expectation values vanish (as already discussed for the on-
site terms).
Considering the restrictions for $l$ and $j$, the first expectation value can
be rewritten as
${\expectationvalue{c^{\dagger}_{la-\sigma}c^{\dagger}_{jb-\sigma^{\prime}}c_{la-\sigma}c_{jb-\sigma^{\prime}}}=-\expectationvalue{n_{la-\sigma}n_{jb-\sigma^{\prime}}}}$.
Now we define the correlation function
$\Delta^{ab\sigma\sigma^{\prime}}_{lj\sigma}\equiv\Delta^{ab\sigma\sigma^{\prime}}_{l-j}$
which fulfills

$\expectationvalue{n_{l,a,-\sigma}n_{j,b,-\sigma^{\prime}}}=\expectationvalue{n_{l,a,-\sigma}}\expectationvalue{n_{j,b,-\sigma^{\prime}}}-\Delta^{ab\sigma\sigma^{\prime}}_{lj\sigma}=\frac{1}{4}-\Delta^{ab\sigma\sigma^{\prime}}_{lj\sigma}.$
(25)

As shown in section I.3 the second expectation value can be expressed using
the same correlation function:

$\expectationvalue{c^{\dagger}_{la-\sigma}c^{\dagger}_{jb\sigma^{\prime}}c_{la\sigma}c_{jb-\sigma^{\prime}}}=2\Delta^{ab\sigma\sigma^{\prime}}_{lj\sigma}.$
(26)

Inserting the results into Eq. (23) we get

$\chi_{\text{rest}}^{ab\sigma\sigma^{\prime}}(k)=-12\sum_{\beta}t^{ab\sigma\sigma^{\prime}}_{\beta}e^{ikd_{\beta}}\Delta^{ab\sigma\sigma^{\prime}}_{\beta}$
(27)

which is just the renormalized non-interacting dispersion. As will be shown in
the section below for certain limits of the parameters the correlation
function can be determined analytically, yielding

$\Delta^{ab\sigma\sigma^{\prime}}_{\beta}=\frac{1}{4}\frac{(t^{ab\sigma\sigma^{\prime}})^{2}\beta}{U}$
(28)

which leads to a renormalization factor of
$3\frac{(t^{ab\sigma\sigma^{\prime}})^{2}\beta}{U}$, consistent with the the
results of Pairault et al. [36] for a single-orbital cosine-dispersion.
The full self-energy is given by

$\Sigma_{ab\sigma\sigma^{\prime}}(\omega,k)=\frac{U}{2}\delta_{ab}+\left(\frac{U^{2}/4}{\omega-\chi_{\text{rest}}(k)-\chi_{\text{on-
site}}}\right)_{ab\sigma\sigma^{\prime}}=\frac{U}{2}\delta_{ab}+\left(\frac{U^{2}/4}{\omega+\widetilde{H_{0}}(k)}\right)_{ab\sigma\sigma^{\prime}}$
(29)

where $\widetilde{H_{0}}(k)$ has been identified with
$-\chi_{\text{rest}}(k)-\chi_{\text{on-site}}$. Note that
$\chi_{\text{rest}}(k)$ has the opposite sign in respect to the non-
interacting Hamiltonian terms while $\chi_{\text{on-site}}$ has the same sign.

###  I.3 Correlation Values

As discussed above the result of the high-frequency expansion depends
crucially on the correlation value
$\expectationvalue{n_{i\sigma}n_{j\sigma}}$. In order to justify the above
result for the analytic expression for the self-energy, in this section we
provide numerical and analytic evidence that
$\expectationvalue{n_{i\sigma}n_{j\sigma}}$ does indeed behave as assumed in
the previous section.

#### Expansion about the atomic limit

In order to estimate $\Delta_{lj}$ we use an expansion of the thermal
expectation value
$\expectationvalue{\hat{O}}=\frac{1}{Z}\sum_{n}\bra{n}\exp(-\beta H+\beta\mu
N)\hat{O}\ket{n}$. The chemical potential term commutes with $H$ and we can
ignore it. After rewriting the exponential function as a series,

$\exp(-\beta H)=\sum_{x}\frac{(-\beta
H)^{x}}{x!}=\sum_{x}\frac{(-\beta)^{x}}{x!}(H_{U}+H_{t})^{x},$ (30)

we restrict ourselves to the lowest order terms in $t$:

$(H_{U}+H_{t})^{x}=H_{U}^{x}+\sum_{a+b=x-1}H_{U}^{a}H_{t}H_{U}^{b}+\sum_{a+b+c=x-2}H_{U}^{a}H_{t}H_{U}^{b}H_{t}H_{U}^{c}+\mathcal{O}(H_{t}^{3}).$
(31)

The $U$ and $t$ subscripts correspond to the interaction and hopping parts of
the Hamiltonian respectively. For ${\hat{O}=n_{l\sigma}n_{j\sigma}}$ the first
order in $t$ vanishes and we get

$\exp(-\beta H)=\sum_{x}\frac{(-\beta
U)^{x}}{x!}\left(h_{U}^{x}+\Theta(x-2)\frac{t^{2}}{U^{2}}\sum_{a+b+c=x-2}h_{U}^{a}h_{t}h_{U}^{b}h_{t}h_{U}^{c}+\mathcal{O}(t^{4})\right)$
(32)

where $H_{U}=Uh_{U}$ and $H_{t}=th_{t}$ and the second term contributes only
for $x\geq 2$. We can now make use of eq. (32) in the expression for the
expectation value, with particular attention posed to the effect of the series
of operators defined by $h_{U}$, $h_{t}$ and $\hat{O}$ on the number of
doubly-occupied sites in the system. In this sense, it can be shown that at
large $\beta U$ the contribution of each state to the sum is suppressed in a
way proportional to the number of double occupations. In first approximation,
therefore, we are allowed to restrict our further analysis to states with no
double occupations. The final step is to determine the contributing states in
the trace, by counting how many hopping processes defined by Eq. (32) are
allowed for a given electronic configuration. The result depends on the
relation between $i$ and $j$:

$\expectationvalue{n_{l\sigma}n_{j\sigma}}=\frac{1}{4}\frac{1+\frac{t^{2}}{U^{2}}\beta
U(\frac{z}{2}N-\delta_{<lj>})}{1+\frac{t^{2}}{U^{2}}\beta U\frac{z}{2}N}$ (33)

where $\delta_{<lj>}$ is non-zero only if $l$ and $j$ are neighbors (or, more
generally, if there is a hopping term connecting $l$ and $j$). The factor
$\frac{z}{2}N$ and the correction $\delta_{<lj>}$ comes from the number of
states with non-zero contribution in the sum over all possible states. Here
$z$ is the coordination number and $N$ is the number of sites. We evaluate
this expression for small $t$ keeping in mind that the number of sites $N$ is
large. Eq. (33) can then be seen as the expansion of

$\frac{1}{4}\left(\frac{1+\frac{t^{2}}{U^{2}}\beta
U(\frac{z}{2}-\frac{\delta_{<lj>}}{N})}{1+\frac{t^{2}}{U^{2}}\beta
U\frac{z}{2}}\right)^{N}\approx\frac{1}{4}\left(1-\frac{t^{2}}{U^{2}}\beta
U\frac{\delta_{<lj>}}{N}\right)^{N}\approx\frac{1}{4}-\frac{1}{4}\frac{t^{2}}{U}\beta\delta_{<lj>}.$
(34)

Thus we can infer that
$\Delta_{lj}=\frac{1}{4}\frac{t^{2}}{U}\beta\delta_{<lj>}$. This result is
restricted to $\frac{t^{2}\beta}{U}\ll 1$ and $\beta U\gg 1$ due to the
assumptions needed in the expansion. For the expectation value of the spin-
hopping term
$\expectationvalue{c^{\dagger}_{la-\sigma}c^{\dagger}_{jb\sigma^{\prime}}c_{la\sigma}c_{jb-\sigma^{\prime}}}$
a similar analysis can be done. For the second-order term the two hopping
terms coming from the expansion in Eq. (32) have to compensate the spin-
hopping. This is only possible if $l$ and $j$ are connected by a hopping in
the non-interacting dispersion, yielding
$\expectationvalue{c^{\dagger}_{la-\sigma}c^{\dagger}_{jb\sigma^{\prime}}c_{la\sigma}c_{jb-\sigma^{\prime}}}=2\frac{1}{4}\frac{t^{2}}{U}\beta\delta_{<lj>}$
where the factor of $2$ is due to the possibility to interchange the two
hopping terms.

#### Numerical results

In order to check the analytic results we used exact diagonalization (ED)
calculations for a one dimensional chain with periodic boundary conditions. As
shown in Figure 5 the dependence of the correlation value on $t$, $\beta$ and
$U$ predicted by the analytic calculations of the previous section is very
well reproduced (for small $t$, small $\beta$ and large $U$ respectively).
Further the double occupation and the expectation value of a hopping term are
much smaller, justifying to neglect corrections including these expressions in
the derivation for the self-energy.

Figure 5: ED results for different expectection values as a function of
hopping $t$, inverse temperature $\beta$ and interaction $U$. For the leftmost
plot $\beta=10$ and $U=100$; for the center plot $t=1$ and $U=100$; and for
the right hand plot $t=1$ and $\beta=100$.

###  I.4 Application to the Hubbard dimer

Applying the analytic self-energy formula to a finite system means replacing
the $k$-dependence with the full matrix structure of the Hamiltonian $H_{0}$
in real space:

$\Sigma(\omega)=\frac{U^{2}/4}{\omega+\alpha H_{0}}.$ (35)

Here, the renormalization is taken into account by the prefactor $\alpha$.
For the Hubbard dimer the full Hamiltonian H is given by

$H=-t\sum_{\sigma}\left(c_{1\sigma}^{\dagger}c_{2\sigma}+c_{2\sigma}^{\dagger}c_{1\sigma}\right)+U\sum_{i=1,2}n_{i\uparrow}n_{i\downarrow}-\frac{U}{2}\sum_{i=1,2;\sigma}n_{i\sigma}.$
(36)

In the following the high-frequency approach is tested against the exact
solution for the Hubbard dimer, both by applying Eq.(16) and by explicitly
solving the high-frequency expansion for the dimer, i.e. solving Eq.(12) with
the full dimer Hamiltonian.
Furthermore it is possible to describe the interface of two SSH models using a
dimer where U is restricted to one site.

#### Exact Solution

At zero temperature the exact self-energy is given by [54]

$\displaystyle\Sigma_{11}$
$\displaystyle=\frac{U^{2}}{4}\left(\frac{\omega}{\omega^{2}-9t^{2}}\right)$
(37) $\displaystyle\Sigma_{12}$
$\displaystyle=\frac{U^{2}}{4}\left(\frac{3t}{\omega^{2}-9t^{2}}\right),$

Expanding in orders of $\frac{1}{\omega}$ leads to

$\displaystyle\Sigma_{11}$
$\displaystyle=\frac{U^{2}}{4\omega}+\frac{9U^{2}t^{2}}{4\omega^{3}}+\mathcal{O}\left(\frac{1}{\omega^{5}}\right)$
(38) $\displaystyle\Sigma_{12}$
$\displaystyle=\frac{3U^{2}t}{4\omega^{2}}+\frac{27U^{2}t^{3}}{4\omega^{4}}+\mathcal{O}\left(\frac{1}{\omega^{6}}\right).$

#### Analytic Formula

With

$H_{0}=\begin{pmatrix}0&-t\\\ -t&0\\\ \end{pmatrix}$ (39)

and $\alpha=3$ (due to $\expectationvalue{n_{i\sigma}n_{j\sigma}}=0$ leading
to $\Delta_{ij}=\frac{1}{4}$) Eq.(35) yields the same self-energy as the exact
solution.

#### Explicit High-Frequency Expansion

For the high-frequency expansion expectation values have to be evaluated (Eq.
12). This can also be done by explicitly carrying out the calculations for the
dimer Hamiltonian. Then it is enough to know that the ground state has the
form

$\ket{\psi_{0}}=\gamma\ket{\uparrow\downarrow}-\gamma\ket{\downarrow\uparrow}+\beta\ket{0\uparrow\downarrow}+\beta\ket{\uparrow\downarrow
0}$ (40)

with $\gamma^{2}+\beta^{2}=\frac{1}{2}$. Using this assumptions, we can
confirm that the first four orders in $\frac{1}{\omega}$ are equivalent to Eq.
(38).

#### Hubbard dimer as interface

By restricting the Hubbard interaction to one site and changing the coupling
between the sites we can model the interface between two SSH chains (one
interacting, one non-interacting) in the atomic limit. For the analytic
formula the interaction is restricted to one site by setting all other
elements of the self-energy to zero after the inversion:

$\Sigma(\omega)=\begin{pmatrix}\frac{U^{2}}{4}\frac{1}{\omega-\frac{\alpha^{2}t^{2}}{\omega}}&0\\\
0&0\\\ \end{pmatrix}.$ (41)

As shown in Figure 6 the agreement with ED results is perfect. This simple
example captures the zero/pole annihilation upon coupling the two atoms.
Furthermore the difference between the ”absence of weight” and a zero becomes
clear: the weight of annihilated zeros and poles is orders of magnitude larger
than the zeros but still orders of magnitude smaller than the poles.

Figure 6: Uncoupled Hubbard dimer (left) and coupled Hubbard dimer (right)
with $U$ only on site 1.

##  II Numerical Calculations

###  II.1 Slab calculations using the analytic formula

Figure 7: Slab geometry consisting of an interface between two topological
insulators, one with $U=0$, the other in the Mott phase. Along the gray edge
of the interface there is no coupling between the two sides, while a finite
coupling exists along the black edge. The blue and red lines indicate the edge
states and zeros respectively and are only shown for one spin species, the
other having all directions inverted. At the interface the edge channels have
the same direction which is a consequence of the plus sign in the denominator
of Eq. (1) in the main text.

For the slab calculations we use a geometry as depicted in Fig. 7. The right
hand side is a non-interacting topological insulator, whereas the the left
hand side is either a trivial or topological Mott insulator. We compute the
Green’s function using

$G=\left(\omega\mathbb{1}-H_{0}-\Sigma(\omega)\right)^{-1}$ (42)

where the non-interacting Hamiltonian is given by a BHZ model[55, 56, 57]:

$H_{0}=\sum_{i\alpha\sigma}M\tau^{z}_{\alpha\alpha}c^{\dagger}_{i\alpha\sigma}c_{i\alpha\sigma}-\sum_{i\alpha\beta\mu\sigma}t^{\alpha\beta}_{\mu\sigma}c^{\dagger}_{i+\mu\alpha\sigma}c_{i\beta\sigma}$
(43)

with

$t_{\mu\sigma}=\begin{pmatrix}t&\sigma\frac{i}{2}\lambda
e^{i\sigma\theta_{\mu}}\\\ \sigma\frac{i}{2}\lambda
e^{-i\sigma\theta_{\mu}}&&-t\end{pmatrix}$ (44)

where $\mu$ runs over the the neighbors ($\pm x$,$\pm y$), $\theta_{\mu}$ is
the angle between the x-axis and direction of the neighbor and $\sigma$ is the
spin. The self-energy is computed using the analytic formula (Eq. (29)). In
this case we do not have a k-dependence. Instead the full matrix $H_{0}$ (with
orbital, spin and spatial degrees of freedom) is taken into account. The
renormalization of the non-local terms is set to one for simplicity. Due to
the stability of topological edge states, the exact parameters of the system
aren’t important as long as there is no topological phase transition.
Therefore we can essentially choose the parameter almost arbitrarily as long
as we stay in the topological phase because our aim is to see the effect of
the interplay of edge poles and zeros. The $M$-parameters are chosen with
opposite signs for the two slab sides in order to ensure that the zeros and
poles can annihilate. The restriction of $U$ to one side of the slab is
implemented by setting the components of the self-energy corresponding to a
non-interacting site to zero after the inversion.

At the interface the coupling is set to zero on the gray part and to one on
the black part. The wave packet is initialized (restricted to one spin
species) using a gaussian on the uncoupled egde (gray):

$\psi(r,t)=\frac{1}{N}\int
dr^{\prime}G(r,r^{\prime},t)e^{-\left((r_{0,x}-r^{\prime}_{x})^{2}/s_{x}^{2}+(r_{0,y}-r^{\prime}_{y})^{2}/s_{y}^{2}\right)/2}.$
(45)

Here $r$ and $r^{\prime}$ are the slab coordinates and $r_{0}$ is the point
where the packet is initialized with variance $s_{x}$ and $s_{y}$ in $x$\- and
$y$-direction respectively. $N$ is a normalization factor.

##  III Additional results

###  III.1 Orbital and Spin Character of the Zeros

Analogously to the non-interacting band structure the zeros also retain an
orbital/spin structure which is a consequence of the analytic self-energy
formula Eq. (29). This is illustrated in Fig. 8, showing the inverted
(standard) gap in the zeros for a topological insulator (band insulator).
Analogously, the zeros also have a spin character as shown in Fig. 9. As a
consequence of the plus sign in the denominator of Eq. 29, their spin
character is interchanged between non-interacting and Mott system.

Figure 8: Orbital character of the zeros obtained using the analytic formula
Eq. (29) for a BHZ-model in the topological (left) and trivial (right) phase
for $k_{y}=0$, $t=0.5$ and $\lambda=0.3$ (see also Eq. (46) in section III.4).
Figure 9: BHZ model with finite size in $y$-direction and periodic boundary
conditions in $x$-directions. The first row shows the poles and zeros for non-
interacting and Mott Green’s function. The interaction is included by using
the analytic formula Eq.(29). The lower rows show the spin-momentum locking of
the edge poles and zeros respectively for the two edges of the model. The
second column shows data where $M$ has the opposite sign compared to the non-
interacting model in order to have the edge states/zeros at the same momenta.

###  III.2 QMC vs ED

As a check of the ED calculations we compare the results for a finite SSH
chain with quantum Monte Carlo (QMC) (Figure 10). In order to avoid an
analytic continuation of the QMC data, we use the ED results on the imaginary
axis. As the figure is showing we find a very good agreement between the two
methods and also with the results obtained using the analytic formula.
Especially these results are showing that the analytic approach is viable (at
least qualitatively) for a wide range of temperatures, starting from the zero
temperature limit where ED is capable to describe the system, to larger
temperatures, accessible using QMC. With QMC we are also able to investigate
much larger chains. The results are shown in Fig.11, confirming that our
results are stable also for large systems.

Figure 10: Comparison between ED (a), QMC (b) and the analytic self-energy
formula (c) for a non-trivial finite SSH chain on the Matsubara axis. The
interaction is $U=4$ and the hopping parameters are $v=0.1$ and $w=1.0$. The
QMC results are for $\beta=10$, the ED calculations are at zero temperature
(where the spacing of the Matsubara frequencies is set to be equivalent to
$\beta=1000$). For the analytic formula the spacing of the Matsubara
frequencies is set $\beta=100$ and the renormaliazion of the parameters to
$3$.

Figure 11: QMC results for a SSH chain with 48 sites for $U=4$. The inverse
temperature is given by $\beta=5$ and $\beta=10$ respectively. The SSH model
is in the topological phase with parameters $v=0.1$ and $w=1.0$.

###  III.3 Comparison of analytic formula and numerical results for an
interface between SSH chains

Figure 12 shows that we find a very good agreement between ED and the analytic
formula for the self-energy (Eq. 1 in the main text) for the coupled and
uncoupled interface of two SSH chains (one in the Mott phase, the other non-
interacting), that have been discussed in the main text in Figure 3. The
renormalization parameter for the analytic formula has been chosen as
$\alpha=3$.

Figure 12: Comparison between ED (top) and the analytic formula (Eq.(29))
(down) for uncoupled SSH chains (left) and coupled SSH chains (right). The
right chain in each plot is for $U=0$ and the left chain for $U=4$.

###  III.4 Comparison of analytic formula and numerical results for a bulk BHZ
model

We tested the analytic formula for a bulk BHZ model given by

$H_{0}(k)=(M-2t(\cos(k_{x})+\cos(k_{y}))\tau_{z}+\lambda\sin(k_{x})\tau_{x}+\lambda\sin(ky)\tau_{y}$
(46)

where $\tau$ describes the orbital degree of freedom. In comparing the
analytical dispersion with the CDMFT data, we have to be careful in including
the effects the asymmetric cluster has on the results: computational memory
constraints render a complete CDMFT simulation for a 4-site cluster (8 energy
levels per spin, to be supplemented by at least as many bath levels) difficult
to perform to convergence. We can however grasp the correctness of our
analytical formula by considering a $2\times 1$-site cluster, which breaks the
$C_{2}$-symmetry of the square lattice. In this case, the self-energy will
have nonlocal components in the $x$-direction, but not in the $y$ direction,
where the problem is effectively single-site. Hence, we expect the effective
dispersion of the zeros to be comparable to an equation of type (46), once the
terms depending on $k_{y}$ have been completely neglected:

$H_{0}^{2\times 1}(k)=(M-2t\cos(k_{x}))\tau_{z}+\lambda\sin(k_{x})\tau_{x}.$
(47)

This will provide a gapped dispersion for $M=0$, and a gap closing at $M=1$
around the high-symmetry point $\Gamma$ and around $M=-1$ around high-symmetry
point $X$.
We fit the zeros of the CDMFT results with Eq. (47) using $t$ and $\lambda$ as
fitting parameters, taking the signs due to the renormalization (Eq.(29)) into
account. The crystal field-splitting $M$ is not renormalized in the analytic
self-energy formula and is therefore kept constant during the fit. As shown in
Fig. 13 the two approaches are in very good agreement. The renormalization of
the parameters $t$ and $\lambda$ is about $0.5$ which explains the shifting of
the topological transition from $M=1$ in the non-interacting case to $M\approx
0.5$. Note that for CDMFT we use a Kanamori-type interaction instead of the
only-U case, which is used in the derivation of the analytic formula. In the
only-U case the parameter $\lambda$ is strongly suppressed, which is
consistent with the high-frequency expansion, since the corresponding
correlation values $\expectationvalue{n_{i}n_{j}}$ remain uncorrelated. This
suggests that the analytic formula can also be applied to more complicated
interactions as long as the correct behavior of the correlation values is
taken into account.

Finally, we address the claim, mentioned in the main text, that in absence of
perturbations that provoke the opening of gapless Dirac or Weyl points in the
noninteracting Hamiltonian the zeros of the Green’s function of the
interacting system in the Mott phase remain gapless. We can verify this claim
by making again use of the 2D BHZ model of eq. (46), which in the $M=0$ case
has a gapless dispersion at the $X$ and $Y$ high-symmetry points. As
previously mentioned, a comprehensive analysis of the topological phase
diagram for the BHZ model using a $2\times 2$ cluster through CDMFT is
hampered by the size of the impurity problem. Nevertheless, a converged result
for selected M values is still within reach. In Fig.14 we show the results
after 10 DMFT steps for a $2\times 2$ cluster for $M=0$ as well as $M=0.2$. As
for the $2\times 1$ cluster in the first part of this section we fit the zeros
with the non-interacting dispersion using $t$ and $\lambda$ as fitting
parameters. The resulting renormalization of the parameters is again given by
a factor of about $0.5$.

Figure 13: Position of the zeros of G of a modified BHZ model (using a
$2\times 1$ cluster, see text) obtained using CDMFT ($U=8$) and fits with the
analytic formula for the self-energy. The dashed lines show the non-
interacting band dispersion.

Figure 14: Position of the zeros for a BHZ model obtained using CDMFT ($U=8$)
with a 2$\times$2 cluster and fit with the analytic formula for the self-
energy. Dashed lines show the non-interacting dispersion.

## References

  * Bradlyn  _et al._ [2017] B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature 547, 298 (2017).
  * Schnyder  _et al._ [2008] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Physical Review B 78, 195125 (2008).
  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Reviews of Modern Physics 83, 1057 (2011).
  * Qi  _et al._ [2008] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Physical Review B 78, 195424 (2008).
  * Wang  _et al._ [2010] Z. Wang, X.-L. Qi, and S.-C. Zhang, Topological Order Parameters for Interacting Topological Insulators, Physical Review Letters 105, 256803 (2010).
  * Fu and Kane [2006] L. Fu and C. L. Kane, Time reversal polarization and a Z 2 adiabatic spin pump, Physical Review B 74, 195312 (2006).
  * Raghu  _et al._ [2008] S. Raghu, X.-L. Qi, C. Honerkamp, and S.-C. Zhang, Topological Mott Insulators, Physical Review Letters 100, 156401 (2008).
  * Wen  _et al._ [2010] J. Wen, A. Rüegg, C.-C. J. Wang, and G. A. Fiete, Interaction-driven topological insulators on the kagome and the decorated honeycomb lattices, Physical Review B 82, 075125 (2010).
  * Pesin and Balents [2010] D. Pesin and L. Balents, Mott physics and band topology in materials with strong spin–orbit interaction, Nature Physics 6, 376 (2010).
  * Gunnarsson  _et al._ [2017] O. Gunnarsson, G. Rohringer, T. Schäfer, G. Sangiovanni, and A. Toschi, Breakdown of Traditional Many-Body Theories for Correlated Electrons, Physical Review Letters 119, 056402 (2017).
  * Bradlyn  _et al._ [2016] B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser, R. J. Cava, and B. A. Bernevig, Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals, Science 353, aaf5037 (2016).
  * Muechler  _et al._ [2016] L. Muechler, A. Alexandradinata, T. Neupert, and R. Car, Topological Nonsymmorphic Metals from Band Inversion, Physical Review X 6, 041069 (2016).
  * Young  _et al._ [2012] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, Dirac Semimetal in Three Dimensions, Physical Review Letters 108, 140405 (2012).
  * Kim  _et al._ [2015] Y. Kim, B. J. Wieder, C. Kane, and A. M. Rappe, Dirac Line Nodes in Inversion-Symmetric Crystals, Physical Review Letters 115, 036806 (2015).
  * Wieder  _et al._ [2016] B. J. Wieder, Y. Kim, A. Rappe, and C. Kane, Double Dirac Semimetals in Three Dimensions, Physical Review Letters 116, 186402 (2016).
  * Wagner  _et al._ [2021] N. Wagner, S. Ciuchi, A. Toschi, B. Trauzettel, and G. Sangiovanni, Resistivity Exponents in 3D Dirac Semimetals From Electron-Electron Interaction, Physical Review Letters 126, 206601 (2021).
  * Braguta  _et al._ [2016] V. V. Braguta, M. I. Katsnelson, A. Y. Kotov, and A. A. Nikolaev, Monte Carlo study of Dirac semimetals phase diagram, Physical Review B 94, 205147 (2016).
  * Budich  _et al._ [2012] J. C. Budich, R. Thomale, G. Li, M. Laubach, and S.-C. Zhang, Fluctuation-induced topological quantum phase transitions in quantum spin-hall and anomalous-hall insulators, Phys. Rev. B 86, 201407 (2012).
  * Amaricci  _et al._ [2015] A. Amaricci, J. Budich, M. Capone, B. Trauzettel, and G. Sangiovanni, First-Order Character and Observable Signatures of Topological Quantum Phase Transitions, Physical Review Letters 114, 185701 (2015).
  * Savary and Balents [2017] L. Savary and L. Balents, Quantum spin liquids: a review, Reports on Progress in Physics 80, 016502 (2017).
  * Oshikawa [2000] M. Oshikawa, Topological Approach to Luttinger’s Theorem and the Fermi Surface of a Kondo Lattice, Physical Review Letters 84, 3370 (2000).
  * Paramekanti and Vishwanath [2004] A. Paramekanti and A. Vishwanath, Extending Luttinger’s theorem to Z 2 fractionalized phases of matter, Physical Review B 70, 245118 (2004).
  * Watanabe  _et al._ [2015] H. Watanabe, H. C. Po, A. Vishwanath, and M. Zaletel, Filling constraints for spin-orbit coupled insulators in symmorphic and nonsymmorphic crystals, Proceedings of the National Academy of Sciences 112, 14551 (2015).
  * Kimchi  _et al._ [2013] I. Kimchi, S. A. Parameswaran, A. M. Turner, F. Wang, and A. Vishwanath, Featureless and nonfractionalized Mott insulators on the honeycomb lattice at 1/2 site filling, Proceedings of the National Academy of Sciences 110, 16378 (2013).
  * Ware  _et al._ [2015] B. Ware, I. Kimchi, S. A. Parameswaran, and B. Bauer, Topological crystalline Bose insulator in two dimensions via entanglement spectrum, Physical Review B 92, 195105 (2015).
  * Parameswaran [2019] S. A. Parameswaran, Topological ‘Luttinger’ invariants for filling-enforced non-symmorphic semimetals, Journal of Physics: Condensed Matter 31, 104001 (2019).
  * Rosch [2007] A. Rosch, Breakdown of Luttinger’s theorem in two-orbital Mott insulators, The European Physical Journal B 59, 495 (2007).
  * Huang  _et al._ [2022] E. W. Huang, G. L. Nave, and P. W. Phillips, Discrete symmetry breaking defines the Mott quartic fixed point, Nature Physics 10.1038/s41567-022-01529-8 (2022).
  * Morimoto and Nagaosa [2016] T. Morimoto and N. Nagaosa, Weyl Mott Insulator, Scientific Reports 6, 19853 (2016).
  * Fabrizio [2022] M. Fabrizio, Emergent quasiparticles at Luttinger surfaces, Nature Communications 13, 1561 (2022).
  * Stanescu and Kotliar [2006] T. D. Stanescu and G. Kotliar, Fermi arcs and hidden zeros of the Green function in the pseudogap state, Physical Review B 74, 125110 (2006).
  * Stanescu  _et al._ [2007] T. D. Stanescu, P. Phillips, and T.-P. Choy, Theory of the Luttinger surface in doped Mott insulators, Physical Review B 75, 104503 (2007).
  * Sakai  _et al._ [2009] S. Sakai, Y. Motome, and M. Imada, Evolution of Electronic Structure of Doped Mott Insulators: Reconstruction of Poles and Zeros of Green’s Function, Physical Review Letters 102, 056404 (2009).
  * Sakai  _et al._ [2010] S. Sakai, Y. Motome, and M. Imada, Doped high- T c cuprate superconductors elucidated in the light of zeros and poles of the electronic Green’s function, Physical Review B 82, 134505 (2010).
  * Rohringer and Toschi [2016] G. Rohringer and A. Toschi, Impact of nonlocal correlations over different energy scales: A dynamical vertex approximation study, Physical Review B 94, 125144 (2016).
  * Pairault  _et al._ [2000] S. Pairault, D. Sénéchal, and A.-M. Tremblay, Strong-coupling perturbation theory of the Hubbard model, The European Physical Journal B 16, 85 (2000).
  * Pudleiner  _et al._ [2016] P. Pudleiner, T. Schäfer, D. Rost, G. Li, K. Held, and N. Blümer, Momentum structure of the self-energy and its parametrization for the two-dimensional Hubbard model, Physical Review B 93, 195134 (2016).
  * Altshuler  _et al._ [1998] B. L. Altshuler, A. V. Chubukov, A. Dashevskii, A. M. Finkel’stein, and D. K. Morr, Luttinger theorem for a spin-density-wave state, Europhysics Letters (EPL) 41, 401 (1998).
  * Konik  _et al._ [2006] R. M. Konik, T. M. Rice, and A. M. Tsvelik, Doped spin liquid: Luttinger sum rule and low temperature order, Phys. Rev. Lett. 96, 086407 (2006).
  * Anderson [1987] P. W. Anderson, The resonating valence bond state in la2cuo4 and superconductivity, science 235, 1196 (1987).
  * Scheurer  _et al._ [2018] M. S. Scheurer, S. Chatterjee, W. Wu, M. Ferrero, A. Georges, and S. Sachdev, Topological order in the pseudogap metal, Proceedings of the National Academy of Sciences 115, E3665 (2018).
  * Gurarie [2011] V. Gurarie, Single-particle Green’s functions and interacting topological insulators, Physical Review B 83, 085426 (2011).
  * Manmana  _et al._ [2012] S. R. Manmana, A. M. Essin, R. M. Noack, and V. Gurarie, Topological invariants and interacting one-dimensional fermionic systems, Physical Review B 86, 205119 (2012).
  * Wang and Zhang [2012] Z. Wang and S.-C. Zhang, Simplified Topological Invariants for Interacting Insulators, Physical Review X 2, 031008 (2012).
  * Muechler [2020] L. Muechler, Topological classification of molecules and chemical reactions with a perplectic structure, Phys. Rev. B 101, 045123 (2020).
  * Su  _et al._ [1980] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Soliton excitations in polyacetylene, Physical Review B 22, 2099 (1980).
  * Yoshida  _et al._ [2014] T. Yoshida, R. Peters, S. Fujimoto, and N. Kawakami, Characterization of a Topological Mott Insulator in One Dimension, Physical Review Letters 112, 196404 (2014).
  * Amaricci  _et al._ [2022] A. Amaricci, L. Crippa, A. Scazzola, F. Petocchi, G. Mazza, L. de Medici, and M. Capone, EDIpack: A parallel exact diagonalization package for quantum impurity problems, Computer Physics Communications 273, 108261 (2022).
  * Parcollet  _et al._ [2015] O. Parcollet, M. Ferrero, T. Ayral, H. Hafermann, I. Krivenko, L. Messio, and P. Seth, Triqs: A toolbox for research on interacting quantum systems, Computer Physics Communications 196, 398 (2015).
  * Bezanson  _et al._ [2017] J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, Julia: A Fresh Approach to Numerical Computing, SIAM Review 59, 65 (2017).
  * Emanuel [2008] G. Emanuel,  _Continuous-Time Quantum Monte Carlo Algorithms for Fermions_ , Ph.D. thesis, ETH Zurich (2008).
  * Gull  _et al._ [2011] E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Continuous-time Monte Carlo methods for quantum impurity models, Reviews of Modern Physics 83, 349 (2011).
  * Haule [2007] K. Haule, Quantum Monte Carlo impurity solver for cluster dynamical mean-field theory and electronic structure calculations with adjustable cluster base, Physical Review B 75, 155113 (2007).
  * Tomczak [2007] J. M. Tomczak, Spectral and Optical Properties of Correlated Materials (2007).
  * Fu and Kane [2007] L. Fu and C. L. Kane, Topological insulators with inversion symmetry, Physical Review B 76, 045302 (2007).
  * Yoshida  _et al._ [2012] T. Yoshida, S. Fujimoto, and N. Kawakami, Correlation effects on a topological insulator at finite temperatures, Physical Review B 85, 125113 (2012).
  * Ando [2013] Y. Ando, Topological Insulator Materials, Journal of the Physical Society of Japan 82, 102001 (2013).