Longitudinal Z-boson polarization and the Higgs boson production cross-section

S. Amoroso1$\star$, for the xFitter Collaboration

1 Deutsches Elektronen-Synchrotron DESY, D 22607, Hamburg

*<EMAIL_ADDRESS>
Proceedings for the XXVIII International Workshop
on Deep-Inelastic Scattering and Related Subjects,
Stony Brook University, New York, USA, 12-16 April 2021
10.21468/SciPostPhysProc.?

## Abstract

We present a study of the LHC (and HL-LHC) potential towards a precise
determination of the gluon parton distribution function of the proton at
intermediate Bjorken-$x$ from measurements of Drell-Yan production. To this
extent, we exploit a clean and theoretically well predicted observable: the
Drell-Yan lepton angular coefficient $A_{0}$, associated with the longitudinal
polarization of the Z boson. Through a detailed numerical analysis using the
open-source xFitter platform, we illustrate how this observable can provide
significant sensitivity over current determinations of the gluon PDF, and a
reduction in the PDF uncertainty on the Higgs boson production cross-section
by a factor of over 50%.

##  1 Introduction

The characterisation of the Higgs sector and the investigation of the
electroweak symmetry breaking mechanism are among the main physics goals of
the LHC and future High Luminosity (HL) LHC. The uncertainties from the
knowledge of the parton distribution functions of the proton constitute one of
the main limitations to fully exploit its physics potential [1]. In
particular, the Higgs cross-section in its dominant gluon-gluon fusion
production channel is extremely sensitive to the gluon PDF at intermediate
Bjorken-$x$ ranges. Currently, the gluon PDF is determined indirectly from
scaling violations in deep-inelastic scattering data, with additional
sensitivity coming from LHC measurements such as jet and top quark production.
These latter two processes are however challenging to measure, and suffer from
theoretical ambiguities making them difficult to predict to high accuracy [2,
3]. In this work [4], we propose a novel determination of the gluon PDF at
intermediate Bjorken-$x$ values using neutral current Drell-Yan data and an
experimentally clean and easy to measure observable, which can be predicted
theoretically to very high accuracy and exhibits a good perturbative
stability. The analysis has been performed using xFitter [5, 6], an open-
source package allowing to extract PDFs and other QCD parameters. It
implements a large number of different datasets, and has been used in a
variety of QCD studies; most recently an extraction of the pion PDF [7] and a
study of the impact of forward-backward asymmetries in Drell-Yan on PDFs [8].

##  2 The $A_{0}$ angular coefficient

We consider Drell-Yan production [9] via $Z/\gamma^{*}$ boson exchange at the
LHC. The cross section summed over the electroweak boson polarizations has the
angular distribution $1+\cos^{2}\theta$ and is sensitive to the gluon PDF for
finite $p_{T}$. However, in the small-$p_{T}$ region where the cross-section
is the largest it is sensitive to large logarithmic corrections, while at
high-$p_{T}$ the missing higher order uncertainties become large. In this
study we focus on the leptonic angular distributions through the angular
coefficient $A_{0}$, the ratio of the longitudinal electroweak boson cross
section to the unpolarized cross section:

$A_{0}(s,M,Y,p_{T})={{2d\sigma^{(L)}/dMdYdp_{T}}\over{d\sigma/dMdYdp_{T}}}.$
(1)

It associated with the $(3\cos^{2}\theta-1)/2$ angular dependence of the
cross-section on the polar angle $\theta^{*}$ [10], with $\theta^{*}$ measured
in the Collins-Soper frame [11]. The longitudinally polarized coefficient
$A_{0}$ in Eq. (1) vanishes in the parton model and receives leading-order
(LO) perturbative QCD contributions at ${\cal O}(\alpha_{s})$. It has been
computed up to ${\cal O}(\alpha_{S}^{3})$ [12], showing a very good
pertubative stability. The $A_{0}$ coefficient is parity-conserving and
sensitive to flavor singlet PDFs. The main sensitivity to the gluon
distribution arises from the region where the change of $A_{0}$ with $p_{T}$
is largest, i.e., around the turn-over point $\partial^{2}A_{0}/\partial
p_{T}^{2}=0$. For dilepton masses near the $Z$-boson peak the turn-over occurs
for boson $p_{T}$ of the order of several ten to 100 GeVs.

The angular coefficients in Drell-Yan, and $A_{0}$, have been measured by many
fixed-target and collider experiments. We consider the $A_{0}$ distribution
from the ATLAS measurement of the angular coefficients in Z boson production
at $\sqrt{s}=8$ TeV [13]. The measurement is performed in 23 $p_{T}$ bins (we
remove the first bins where our fixed-order predictions would not be
adequate), going from 11.4 GeV to 600 GeV, and in three different rapidity
bins: $0<|Y|<1$, $1<|Y|<2$, and $2<|Y|<3.5$. Predictions at NLO in QCD for
finite boson $p_{T}$ (at order ${\cal O}(\alpha^{2})$) have been produced with
mg5_aMC@NLO and aMCfast [14]. A $\chi^{2}$ is constructed, incorporating all
experimental uncertainties and their correlations, as well as PDF
uncertainties. The impact of the $A_{0}$ data on PDFs is then evaluated
through an Hessian profiling technique [15, 16]. We find a good description of
the data, with $\chi^{2}$/ndf of order one, for all modern PDFs: CT18NNLO
[17], NNPDF3.1nnlo [18], ABMP16nnlo [19], and MSHT20nnlo [20]. The reduction
in the PDF uncertainties after the inclusion of the $A_{0}$ data is however
found to be negligible.

##  3 PDF constraints and the Higgs cross-section

###  3.1 Sensitivity to the gluon PDF

We then generate pseudodata for the $A_{0}$ distribution at $\sqrt{s}=13$ TeV
for two projected luminosity scenarios of 300 fb-1 (the expected integrated
luminosity at the end of the LHC Run III) and 3 ab-1 (the design integrated
luminosity of the HL-LHC stage). To do this we extrapolate the statistical
uncertainties, and add a conservative estimate of 0.1% for the dominant
experimental systematic on the lepton momentum scale. We perform a PDF
profiling exercise using the CT18NNLO PDF set, first reduced to 68% CL
coverage, and find that the increased statistics allows for a significant
reduction in the PDF uncertainties. This reduction is particularly large for
the gluon density and for the $u$ and $d$ sea-quark densities coupled to
gluons through QCD evolution, as shown in Fig. 1. The largest sensitivity is
found to come from transverse momenta in the range $p_{T}\sim 50$ GeV, dying
out for $p_{T}>100$ GeV. The gain from a 300 fb-1 measurement is found to
dominate over the 3 ab-1 gain, although an improvement is nonetheless present.

Figure 1: The gluon ($xg$) and gluon/Sea ($xg/\Sigma$) PDFs in the original
CT18NNLO analysis (red) and profiled using $A_{0}$ pseudodata corresponding to
integrated luminosities of 300 fb-1 (blue) and 3 ab-1 (green) in the region 80
GeV $<M<$ 100 GeV and $|Y|<3.5$.

###  3.2 Impact on Higgs phenomenology

The effect of the longitudinally polarized coefficient on the $Q^{2}$ = 104
GeV2 gluon PDF near $x\sim 10^{-2}$ will influence the Higgs boson cross
section. In Fig. 2 we show the gluon-gluon luminosity for the CT18NNLO PDF,
before and after the profiling using $A_{0}$ pseudodata. Most of the reduction
in the uncertainties is found in the Higgs mass range. In the right plot we
show the N3LO gluon-fusion Higgs total cross section and its uncertainty
obtained with ggHiggs [21, 22]. for the CT18NNLO, NNPDF3.1nnlo and MSHT20nnlo
global PDF sets. Notwithstanding the numerical differences, a similar
reduction in the PDF uncertainties is seen for the different sets. We also
show The PDF4LHC15scen1 and 2 projected sets, which include pseudodata based
on complete LHC data sample [23], and which show a smaller, but not
negligible, reduction in uncertainties.

Figure 2: [Left] Ratio of PDF uncertainties for the gluon-gluon luminosity
evaluated at $\sqrt{s}=$13 TeV. The red band shows the uncertainties of the
CT18NNLO PDF set [17], reduced to 68% CL coverage. The blue and green bands
show the impact of including constraints from the $A_{0}$ measurement and
assuming 300 fb-1 and 3 ab-1, respectively. [Right] PDF uncertainties on the
N3LO gluon-gluon fusion Higgs boson cross-section, shown for different PDF
sets before and after the inclusion of the $A_{0}$ measurement constraints.

##  4 Conclusion

We have studied the impact of precise measurements of the $A_{0}$ angular
coefficient in neutral current Drell-Yan on the uncertainties of the gluon PDF
and on the Higgs boson production cross-section in gluon-gluon fusion. We
found that future measurements of $A_{0}$ have the possibility to reduce by
about a factor of two the uncertainty of the gluon PDFs at intermediate
Bjorken-$x$ values. This translates in a large reduction in the PDF
uncertainty on the gluon-fusion Higgs production cross-section. Our results
open a new area of phenomenological studies on connections between the gauge
and Higgs sectors of the SM. Further aspects may be investigated by extending
the analysis to the full structure of lepton angular distributions, including
polarization interferences, and to mass regions far away from the $Z$-boson
peak.

## References

  * [1] M. Cepeda _et al._ ,  _Report from Working Group 2: Higgs Physics at the HL-LHC and HE-LHC_ ,  CERN Yellow Rep. Monogr. 7, 221 (2019),  10.23731/CYRM-2019-007.221,  1902.00134.
  * [2] R. Abdul Khalek _et al._ ,  _Phenomenology of NNLO jet production at the LHC and its impact on parton distributions_ ,  Eur. Phys. J. C 80(8), 797 (2020),  10.1140/epjc/s10052-020-8328-5,  2005.11327.
  * [3] S. Bailey and L. Harland-Lang,  _Differential Top Quark Pair Production at the LHC: Challenges for PDF Fits_ ,  Eur. Phys. J. C 80(1), 60 (2020),  10.1140/epjc/s10052-020-7633-3,  1909.10541.
  * [4] S. Amoroso, J. Fiaschi, F. Giuli, A. Glazov, F. Hautmann and O. Zenaiev,  _Longitudinal Z-Boson Polarization and the Higgs Boson Production Cross Section at the Large Hadron Collider_ (2020),  2012.10298.
  * [5] S. Alekhin _et al._ ,  _HERAFitter_ ,  Eur. Phys. J. C 75(7), 304 (2015),  10.1140/epjc/s10052-015-3480-z,  1410.4412.
  * [6] V. Bertone _et al._ ,  _xFitter 2.0.0: An Open Source QCD Fit Framework_ ,  PoS DIS2017, 203 (2018),  10.22323/1.297.0203,  1709.01151.
  * [7] I. Novikov _et al._ ,  _Parton Distribution Functions of the Charged Pion Within The xFitter Framework_ ,  Phys. Rev. D 102(1), 014040 (2020),  10.1103/PhysRevD.102.014040,  2002.02902.
  * [8] E. Accomando _et al._ ,  _PDF Profiling Using the Forward-Backward Asymmetry in Neutral Current Drell-Yan Production_ ,  JHEP 10, 176 (2019),  10.1007/JHEP10(2019)176,  1907.07727.
  * [9] S. D. Drell and T.-M. Yan,  _Massive Lepton Pair Production in Hadron-Hadron Collisions at High-Energies_ ,  Phys. Rev. Lett. 25, 316 (1970),  10.1103/PhysRevLett.25.316,  [Erratum: Phys.Rev.Lett. 25, 902 (1970)].
  * [10] E. Mirkes and J. Ohnemus,  _Angular distributions of Drell-Yan lepton pairs at the Tevatron: Order $\alpha-s^{2}$ corrections and Monte Carlo studies_,  Phys. Rev. D 51, 4891 (1995),  10.1103/PhysRevD.51.4891,  hep-ph/9412289.
  * [11] J. C. Collins and D. E. Soper,  _Angular Distribution of Dileptons in High-Energy Hadron Collisions_ ,  Phys. Rev. D 16, 2219 (1977),  10.1103/PhysRevD.16.2219.
  * [12] R. Gauld, A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. Glover and A. Huss,  _Precise predictions for the angular coefficients in Z-boson production at the LHC_ ,  JHEP 11, 003 (2017),  10.1007/JHEP11(2017)003,  1708.00008.
  * [13] G. Aad _et al._ ,  _Measurement of the angular coefficients in $Z$-boson events using electron and muon pairs from data taken at $\sqrt{s}=8$ TeV with the ATLAS detector_,  JHEP 08, 159 (2016),  10.1007/JHEP08(2016)159,  1606.00689.
  * [14] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli and M. Zaro,  _The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations_ ,  JHEP 07, 079 (2014),  10.1007/JHEP07(2014)079,  1405.0301.
  * [15] H. Paukkunen and P. Zurita,  _PDF reweighting in the Hessian matrix approach_ ,  JHEP 12, 100 (2014),  10.1007/JHEP12(2014)100,  1402.6623.
  * [16] S. Camarda _et al._ ,  _QCD analysis of $W$\- and $Z$-boson production at Tevatron_,  Eur. Phys. J. C 75(9), 458 (2015),  10.1140/epjc/s10052-015-3655-7,  1503.05221.
  * [17] T.-J. Hou _et al._ ,  _New CTEQ global analysis of quantum chromodynamics with high-precision data from the LHC_ ,  Phys. Rev. D 103(1), 014013 (2021),  10.1103/PhysRevD.103.014013,  1912.10053.
  * [18] R. D. Ball _et al._ ,  _Parton distributions from high-precision collider data_ ,  Eur. Phys. J. C 77(10), 663 (2017),  10.1140/epjc/s10052-017-5199-5,  1706.00428.
  * [19] S. Alekhin, J. Blümlein, S. Moch and R. Placakyte,  _Parton distribution functions, $\alpha_{s}$, and heavy-quark masses for LHC Run II_,  Phys. Rev. D 96(1), 014011 (2017),  10.1103/PhysRevD.96.014011,  1701.05838.
  * [20] S. Bailey, T. Cridge, L. A. Harland-Lang, A. D. Martin and R. S. Thorne,  _Parton distributions from LHC, HERA, Tevatron and fixed target data: MSHT20 PDFs_ ,  Eur. Phys. J. C 81(4), 341 (2021),  10.1140/epjc/s10052-021-09057-0,  2012.04684.
  * [21] M. Bonvini, R. D. Ball, S. Forte, S. Marzani and G. Ridolfi,  _Updated Higgs cross section at approximate N 3LO_,  J. Phys. G 41, 095002 (2014),  10.1088/0954-3899/41/9/095002,  1404.3204.
  * [22] M. Bonvini and S. Marzani,  _Double resummation for Higgs production_ ,  Phys. Rev. Lett. 120(20), 202003 (2018),  10.1103/PhysRevLett.120.202003,  1802.07758.
  * [23] R. Abdul Khalek, S. Bailey, J. Gao, L. Harland-Lang and J. Rojo,  _Towards Ultimate Parton Distributions at the High-Luminosity LHC_ ,  Eur. Phys. J. C 78(11), 962 (2018),  10.1140/epjc/s10052-018-6448-y,  1810.03639.


# Observation of nonlinear fractal higher-order topological insulator

Victor O. Kompanets  Institute of Spectroscopy, Russian Academy of Sciences,
Troitsk, Moscow 108840, Russia  Hua Zhong  Key Laboratory for Physical
Electronics and Devices, Ministry of Education, School of Electronic Science
and Engineering, Xi’an Jiaotong University, Xi’an 710049, China  Yiqi Zhang
<EMAIL_ADDRESS>Key Laboratory for Physical Electronics and Devices,
Ministry of Education, School of Electronic Science and Engineering, Xi’an
Jiaotong University, Xi’an 710049, China  Yaroslav V. Kartashov
<EMAIL_ADDRESS>Institute of Spectroscopy, Russian Academy of
Sciences, Troitsk, Moscow 108840, Russia  Yongdong Li  Key Laboratory for
Physical Electronics and Devices, Ministry of Education, School of Electronic
Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, China
Sergei A. Zhuravitskii  Institute of Spectroscopy, Russian Academy of
Sciences, Troitsk, Moscow 108840, Russia  Quantum Technology Centre, Faculty
of Physics, M. V. Lomonosov Moscow State University, Moscow 119991, Russia
Nikolay N. Skryabin  Institute of Spectroscopy, Russian Academy of Sciences,
Troitsk, Moscow 108840, Russia  Quantum Technology Centre, Faculty of Physics,
M. V. Lomonosov Moscow State University, Moscow 119991, Russia  Ivan V.
Dyakonov  Quantum Technology Centre, Faculty of Physics, M. V. Lomonosov
Moscow State University, Moscow 119991, Russia  Alexander A. Kalinkin
Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow
108840, Russia  Quantum Technology Centre, Faculty of Physics, M. V. Lomonosov
Moscow State University, Moscow 119991, Russia  Sergei P. Kulik  Quantum
Technology Centre, Faculty of Physics, M. V. Lomonosov Moscow State
University, Moscow 119991, Russia  Sergey V. Chekalin  Institute of
Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow 108840, Russia
Victor N. Zadkov  Institute of Spectroscopy, Russian Academy of Sciences,
Troitsk, Moscow 108840, Russia  Faculty of Physics, Higher School of
Economics, Moscow 105066, Russia

###### Abstract

Higher-order topological insulators (HOTIs) are unique materials hosting
topologically protected states, whose dimensionality is at least by a factor
of 2 lower than that of the bulk. Topological states in such insulators may be
strongly confined in their corners that leads to considerable enhancement of
nonlinear processes involving such states. However, all nonlinear HOTIs
demonstrated so far were built on periodic bulk lattice materials. Here we
demonstrate first nonlinear photonic HOTI with the fractal origin. Despite
their fractional effective dimensionality, the HOTIs constructed here on two
different types of the Sierpiński gasket waveguide arrays, may support
topological corner states for unexpectedly wide range of coupling strengths,
even in parameter regions where conventional HOTIs become trivial. We
demonstrate thresholdless solitons bifurcating from corner states in nonlinear
fractal HOTIs and show that their localization can be efficiently controlled
by the input beam power. We observe sharp differences in nonlinear light
localization on outer and multiple inner corners and edges representative for
these fractal materials. Our findings not only represent a new paradigm for
nonlinear topological insulators, but also open new avenues for potential
applications of fractal materials to control the light flow.

Fractals are self-similar structures (i.e., next generation of a fractal can
be constructed by combining copies of its previous generation) widely
represented in universe [1], whose unusual internal composition finds its
manifestation in new physical phenomena observed in solid-state physics,
acoustics, and photonics, to mention just a few areas [2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16]. One of the distinguishing characteristics of a
fractal is its fractional dimension, which can be described by the non-integer
effective Hausdorff dimension ${d_{f}=\log_{\ell}m}$, where $m$ is the number
of previous-generation elements required to construct next-generation fractal,
while $\ell$ is the factor, by which length of the fractal edge would increase
in the next generation. Thus, famous Sierpiński carpet and gasket fractals
have Hausdorff dimensions ${d_{f}=\log_{3}8}$ and ${d_{f}=\log_{2}}3$,
respectively, which reflects their different composition. Fractal systems are
aperiodic, but regular—they are sometimes considered as lacking “bulk” due to
the presence of multiple holes, inner edges and corners. In photonics, such
unusual composition of fractal structures may open new prospects for
manipulation and localization of the light fields in them, including
realization of quantum anomalous transport [8] and flat-band systems [17, 18].
Particularly intriguing problem is the possibility of realization of
topological phases in fractal structures, since the appearance of topological
edge states is tightly connected with the dimensionality of the system and
usually stems from topological properties of its bulk, which can be very
specific in fractal systems.

Photonic systems offer a unique testbed for realization of topologically
nontrivial structures [19, 20, 21], including various types of Chern [22],
Floquet [23], valley-Hall [24], and higher-order [25, 26, 27, 28, 29, 30]
topological insulators. Most of these systems, including HOTIs, were
constructed on structures with periodic bulk. However, very recently it was
shown that unidirectional travelling topological edge states can form in
fractal waveguide arrays with helical channels [12, 13] and in fractal Haldane
model [31] implying that formal absence of insulating bulk in fractals is not
an obstacle for realization of topological phase, and bulk-edge correspondence
[32] is still meaningful for these systems. While fractal HOTIs were proposed
in electronic systems [33, 34] and recently realized in acoustics [14, 15],
higher-order topological states in photonic fractals were never observed so
far. At the same time, first experiments in acoustics hint on a very unusual
manifestation of topological effects in fractal systems [14, 15], connected
with the possibility of localization in their multiple inner corners, and
strong dependence of the parameter range, where fractal system is topological
on fractal generation order, making them clearly distinct from conventional
HOTIs.

Among the advantages of photonic systems in comparison with the electronic and
acoustic ones, is that the former systems can be strongly nonlinear.
Nonlinearity not only offers a convenient knob for controlling localization
and propagation dynamics of the topological excitations, but it is often
crucial for the effects that determine practical applications of such systems
[35], ranging from lasing and harmonic generation to bistability,
nonlinearity-controlled switching and routing with th topologically protected
states. In addition, nonlinearity gives rise to a broad spectrum of
topological edge solitons, inheriting topological protection from their linear
counterparts [36, 37, 38, 39, 40, 41, 42] as demonstrated in [43, 44, 45, 46,
47], and it may create the self-induced topological phases [48, 49, 50].
Nonlinear effects and formation of the unique corner solitons in HOTIs were
recently reported too [51, 52, 53]. Nowadays, there is a considerable interest
to investigation of the nonlinear effects in aperiodic topological photonic
systems, with only a few observations available so far [54]. Fractal
topological photonic systems may thus provide a unique platform for
investigation of such effects.

In this Article, we describe first experimental realization of photonic
fractal HOTI and study the interplay between topological and nonlinear effects
in this aperiodic system, which gives rise to topological corner solitons. To
demonstrate such states we utilize fractal Sierpiński gasket waveguide arrays
of two different types inscribed in fused silica using fs laser writing
technique. Higher-order topological phase is realized due to the controllable
shift of the waveguides that adjusts coupling strengths between sites in the
first and subsequent generations of fractals and is manifested in the
appearance of the corner states of the topological origin. The remarkable
distinctive property of this photonic system is that topological states can
appear not only in the outer, but also in the inner corners of the structure.
Moreover, topological corner states here can exist even for the shifts, where
conventional HOTIs with periodic bulk become trivial. To characterize
topological properties of this aperiodic system we employ real-space
polarization index [55, 56]. Finally, we demonstrate thresholdless fractal
topological corner solitons bifurcating from their linear counterparts and
existing in the forbidden spectral gaps.

Figure 1: Fractal higher-order topological insulators and their linear
spectra. (a) Schematic representation of the third generation $G_{3}$ of the
Sierpiński gasket waveguide arrays with ${r=0.3a}$, $0.5a$, and $0.6a$. Blue
and orange sectors of these arrays represent the first-generation $G_{1}$ and
second-generation $G_{2}$ structures, respectively. Microphotographs of the
corresponding fs-laser written waveguide arrays are presented in the insets.
The orange, green, blue, and magenta ellipses in microphotographs indicate the
representative sites 1, 2, 3, and 4 that will be used below for probing of
excitation dynamics. (b) Eigenvalues $b$ of the stationary linear states of
the fractal array vs distortion parameter $r$. Colored curves represent
localized states, while black ones correspond to the delocalized states. (c)
Intensity distributions shown within ${-23\leq x,y\leq 23}$ window for four
representative eigenstates corresponding to the colored circles in (b). Here
and in figures below the array depth is ${p=5.7}$.

## Results and Discussions

### Fractal arrays, their linear spectra and eigenmodes

We consider two different types of Sierpiński gasket arrays in this Article,
which are further termed case-1 and case-2 arrays. Both arrays are produced
using identical first-generation triangular element $G_{1}$ highlighted with
the blue color in the schematic representation in Fig. 1a, where we show third
generation $G_{3}$ of case-2 structure (for more details of fractal array
construction and results on case-1 structure see Supporting Information).
$G_{n}$ generation of the Sierpiński gasket fractal array is formed by three
$G_{n-1}$ generation structures (for instance, $G_{2}$ elements in Fig. 1a are
highlighted with orange color)—on this reason fractal arrays are self-similar.
In case-2 arrays these previous generations share three common sites, while in
case-1 array there are no such common sites. Consequently, $G_{n}$ generation
includes $3^{n+1}$ sites in case-1 array, and ${3^{n+1}-(3^{n}-3)/2}$ sites in
case-2 array. Due to the method of their construction, fractal arrays posses
multiple holes, inner corners and edges. They are characterized by the
effective Hausdorff dimension ${d_{f}=\log_{2}}3$ that is lower than $2$.
Further we focus on newly designed case-2 arrays, discussing solitons in the
case-1 arrays in Supporting Information. To realize fractal HOTI, we introduce
controllable distortion (via the parameter $r$) into the structure by shifting
the neighboring waveguides in the opposite directions, while keeping spacing
$a$ between the next-nearest-neighbor waveguides constant, as indicated in
Fig. 1a. The examples of undistorted (${r=0.5a}$) and distorted (${r=0.3a}$
and ${r=0.6a}$) arrays are shown in Fig. 1a with the photographs of such fs-
laser written entities in fused silica.

Propagation of light beams in fractal arrays inscribed in focusing cubic
medium is described by the nonlinear Schrödinger equation:

$i\frac{\partial\psi}{\partial
z}=-\frac{1}{2}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)\psi-\mathcal{R}(x,y)\psi-|\psi|^{2}\psi,$ (1)

where $\psi$ is the dimensionless complex amplitude of the light field; $x,y$
and $z$ are the normalized transverse coordinates and propagation distance,
respectively; the function
${\mathcal{R}(x,y)=p\sum_{mn}e^{-(x-x_{m})^{2}/d_{x}^{2}-(y-y_{n})^{2}/d_{y}^{2}}}$
describes the fractal case-2 waveguide array composed of single-mode
elliptical (due to writing process) waveguides with the depth $p$ and widths
$d_{x,y}$ placed in the nodes $(x_{m},y_{n})$ of the Sierpiński gasket grid.
Further, we use arrays with ${p=5.7}$, ${a=6.0}$, and ${d_{x}=0.25}$,
${d_{y}=0.75}$ corresponding to the parameters of the experimental structures
(see Materials and Methods for details of normalization).

We first characterize linear spectrum of such arrays by omitting nonlinearity
in Eq. (1) and calculating the linear eigenmodes of the form
${\psi=u(x,y)e^{ibz}}$, where $u$ describes modal shape, and $b$ is the
propagation constant (eigenvalue). Linear spectrum of fractal array of third
generation $G_{3}$ in the form of dependence of eigenvalues of all supported
modes $b$ on distortion parameter $r$ is presented in Fig. 1b, where colored
curves correspond to the localized states, while black curves correspond to
the extended states. Remarkably, in fractal arrays localized in-gap states are
encountered in both ${r>0.5a}$ and ${r<0.5a}$ regimes, in sharp contrast with
HOTIs with periodic bulk, where localized states appear only for one of these
types of distortion leading to dominance of the inter-cell coupling over the
intra-cell one. Notice that the state shown with magenta curve can exist even
at ${r=0.5a}$. Intensity distributions of four typical localized states
corresponding to circles in Fig. 1b are shown in Fig. 1c. Magenta, green, and
cyan branches correspond to the co-existing outer corner states with different
internal structure, representative namely for the case-2 fractal array. In
contrast, red branch corresponds to the states, where strongly localized spots
appear simultaneously in multiple inner and outer corners, so one can call
them hybrid corner states, as they reflect unique internal composition of the
fractal array (there are several such branches in spectrum with spots only in
the corners that become nearly degenerate for small $r$ values). Outer corner
states are all three-fold degenerate, while their number is not affected by
the fractal generation order $n$. According to the definition of the effective
dimensionality ${d_{e}=\lim_{n\to\infty}{\ln N/\ln N_{l}}}$, where $N$ is the
number of sites occupied by the corner state and ${N_{l}=3\cdot 2^{n-1}+1}$ is
the total number of sites on one outer edge [14, 15], outer corner states are
zero-dimensional, as ${d_{e}\to 0}$ with increase of the fractal generation
order $n$. In contrast, the number of spots in hybrid corner states
${N=(3^{n}+3)/2}$ increases with $n$, so that $d_{e}$ for hybrid state
approaches effective Hausdorff dimensionality $d_{f}=\log_{2}3$ of the
structure. Thus, fractal HOTIs offer unique opportunity to observe photonic
corner states of the different effective dimensionality.

To characterize topological properties of fractal Sierpiński gasket waveguide
arrays one resorts to the real-space polarization index [55, 56]. The
remarkable prediction of this analysis (we glue two Sierpiński gaskets to form
a rhombic structure—see Materials and Methods and Supporting Information for
details) is that the index for both hybrid and outer corner states are $0.5$
in their respective existence domains, meaning that all such corner states are
of topological origin. Similar conclusions can be drawn for the case-1 fractal
arrays, as we also discuss in Supporting Information. Thus, fractal Sierpiński
gasket arrays allow to realize HOTIs in a broader range of distortion
parameters $r$ in comparison with HOTIs based on the periodic kagome and Su-
Schrieffer-Heeger arrays [51, 52], where in our notations corner states are
possible only at ${r<0.5a}$.

### Topological soliton families

Figure 2: Topological solitons in fractal insulators. (a) Power vs propagation
constant for the solitons bifurcating from the linear eigenstates concentrated
around sites 1 (red curve) and 4 (blue curve) in the fractal array with
${r=0.3a}$. Panels (c) and (d) show field modulus distributions corresponding
to the dots in (a). (b) Power vs propagation constant for the solitons
bifurcating from linear eigenstates concentrated around sites 1 (green, cyan
and magenta curves) and 4 (orange curve) in fractal array with ${r=0.6a}$.
Field modulus distributions corresponding to the dots in (b) are shown in
(e)–(h). In (a) and (b) gray regions are associated either with the
eigenvalues of the localized states or with the bands of the delocalized
states.

In the presence of nonlinearity localized corner states in fractal arrays give
rise to the families of bifurcating from them thresholdless topological corner
solitons. Such solitons also have the form ${\psi=u(x,y)e^{ibz}}$, whose
substitution into Eq. (1) yields
${bu=\frac{1}{2}(\partial^{2}_{x}+\partial^{2}_{y})u+\mathcal{R}u+u^{3}}$ that
was solved using Newton method. Now propagation constant $b$ parameterises the
family of solitons, determining their power ${U=\int|\psi|^{2}dxdy}$.

The families of corner solitons in fractal insulators can be very rich. In
Fig. 2 we show representative $U(b)$ dependencies in $G_{3}$ structures. At
${r=0.3a}$, when a set of practically degenerate linear topological modes with
spots localized only on the inner and outer corners exist, topological
solitons can form in any corner of fractal array, see red (outer corner) and
blue (inner corner) families in Fig. 2a bifurcating from corresponding linear
modes (or their linear combination, because for this $r$ there exist a set of
practically degenerate hybrid corner states). In this figure wider gray
regions correspond to the bands occupied by extended states in linear
spectrum, while narrow gray lines correspond to the eigenvalues of linear in-
gap corner states. For the selected $r$ value such solitons are strongly
localized at the low and intermediate powers (see profiles in Fig. 2c and 2d
corresponding to points $1$,$3$ in Figs. 2a). For illustrative purposes here
we superimpose $|\psi|$ distribution for soliton on array profile
$\mathcal{R}$ shown by white ellipses. Importantly, because nonlinearity leads
to shift of the propagation constant within the gap of linear spectrum, one
can control localization degree of corner solitons by increasing their power.
Namely, such solitons gradually extend, developing long tails as their
propagation constant approaches nearest band of extended states, eventually
coupling with them and loosing localization, when $b$ enters into such band
(as illustrated by profiles in the points $2,4$ of Fig. 2a, where soliton
clearly couples with the edge states).

Figure 3: Observation of nonlinear states in fractal array with ${r=0.3a}$.
Output intensity distributions after 10 cm of propagation for the excitation
of sites 1 (a), 2 (b), 3 (c) and 4 (d) in $G_{3}$ fractal array. Figures with
the maroon background show experimental results, while figures with the white
background show results of theoretical simulations. Gray ellipses in all
panels indicate waveguide locations. Input pulse energies $E$ and powers $U$
are indicated on each plot. All distributions are shown within the window
${-23\leq x,y\leq 23}$. Arrows in the experimental panels indicate position of
the input excitation.

Fractal HOTI with ${r=0.6a}$ supports multiple families of topological
solitons with different symmetries forming in outer corners. They are shown in
Fig. 2b with magenta, cyan, and green lines, in accordance with colors of
linear corner states in Fig. 1b, from which these solitons bifurcate. Solitons
inherit symmetry of the corresponding linear states. Thus, in green family
(Fig. 2f, points $7$,$8$) corner maximum is out-of-phase with maxima in two
nearest-neighbour sites; solitons from blue family have empty outer corner
site, while fields in nearest-neighbor sites are out-of-phase (Fig. 2g, points
$9$,$10$); while most localized solitons from magenta branch have nearly equal
intensities in three close sites in the corner (Fig. 2h, points $11$,$12$).
These solitons typically gradually broaden with increase of power and strongly
expand into array when coupling with extended states occurs upon entering of
their propagation constant $b$ into different bands of the linear spectrum.
Notice that extended states occupying multiple sites can also produce
nonlinear families, as illustrated by orange branch in Fig. 2b, see profiles
in Fig. 2e (the power of this state vanishes in the bifurcation point only
because our array is finite). Interestingly, with increase of the power this
extended state couples with the outer corner states (point 6, Fig. 2e).

Despite large variety of corner solitons appearing in fractal HOTI, nearly all
of them are dynamically stable. Their stability was verified by adding small-
scale random perturbation ($5\%$ in the amplitude) into input field
distributions and propagating them in the frames of Eq. (1) over a very long
distances ${z\sim 10^{4}}$. For all branches shown in Figs. 2a and 2b with
solid lines, such perturbations resulted only in small-amplitude oscillations
signalizing on their stability, while decay was observed only for a small part
of the green branch shown with the dashed line.

### Observation of the corner solitons in fractal HOTI

To demonstrate corner solitons in fractal HOTIs we fabricated a set of case-2
$G_{3}$ structures with various distortion parameters ${r=0.3a}$, ${r=0.5a}$
and ${r=0.6a}$ using fs-laser inscription technique (for experiments with
case-1 array see Supporting Information). Arrays were inscribed in
$10~{}\textrm{cm}$-long fused silica samples (see Materials and Methods for
details of fabrication). Exemplary photographs of the inscribed arrays are
presented as insets in Fig. 1a. We selected four representative locations to
study excitation dynamics, as indicated with colored ellipses with numbers 1
(outer corner site), 2 (site on the outer edge), 3 (site on the inner edge
site) and 4 (inner corner site). For excitation we used pulses with the
duration about $280\,\textrm{fs}$ of variable energy $E$ derived from
$1\,\textrm{kHz}$ Ti:sapphire laser to achieve strong nonlinear response (see
Materials and Methods).

Figure 4: Nonlinear dynamics in fractal array with ${r=0.5a}$. Rich nonlinear
dynamics and gradual transition to the localization at the highest power
levels as observed for the excitations of sites 1-4 in $G_{3}$ fractal array.
The arrangement of panels is the same as in Fig. 3. Arrows in experimental
panels indicate position of the input excitation.

First, we investigate the structure with ${r=0.3a}$ that supports hybrid
corner states. In Figs. 3a-3d, we compare experimental output intensity
distributions (images with maroon background) obtained at the different pulse
energies $E$ for the excitation of four above mentioned sites of the array (in
each case, the excited sites are indicated by white arrows) with theoretical
distributions for the different powers $U$ obtained by solving Eq. (1) (images
with white background). Since well-localized thresholdless solitons at
$r=0.3a$ can form in any inner or outer corner of this structure, the
excitation of sites 1 and 4 (Figs. 3a and 3d) yields strongly localized,
practically single-site patterns in both linear (${E=10\,\rm nJ}$) and
nonlinear (e.g., ${E=750\,\rm nJ}$) regimes, confirming the formation of
thresholdless corner solitons. In contrast, when exciting edge sites 2 and 3
(Figs. 3b and 3c), we observe diffraction and dynamic oscillations of power
between close pairs of waveguides even at pulse energies ${E\sim 800\,\rm
nJ}$, indicating that inner and outer edges do not support well-localized
thresholdless in-gap states and significant power levels are needed to achieve
localization at such edges.

Turning to array with ${r=0.5a}$, where spacing between nearest sites is the
same in the entire array, one should take into account that even though
magenta branch of outer corner states already exists in this borderline case,
its localization is rather weak (see state 2 in Fig. 1c), thus efficiency of
its excitation with single-site input is low. On this reason, in Fig. 4a
showing experimental patterns for this structure, the beam with ${E=10\,\rm
nJ}$ launched into site 1 experiences diffraction, even though some fraction
of power clearly remains in the corner waveguide. Diffraction is nearly
suppressed when the pulse energy increases to the moderate value of
${E=250\,\rm nJ}$, while further increase of $E$ results in the excitation of
the outer corner soliton. For excitation of sites 2, 3, and 4 shown in Figs.
4b, 4c and 4d, respectively, one needs substantially higher pulse energies to
achieve comparable degree of localization, while particularly for sites 3 and
4 localization remains weak even for the energies of ${E\sim 750\,\rm nJ}$.

Figure 5: Observation of nonlinear states in fractal array with ${r=0.6a}$.
Formation of the localized thresholdless nonlinear states for the excitation
of sites 1 (a) and 4 (d) and rich nonlinear dynamics with the light switching
between several closely located waveguides for excitation of sites 2 (b) and 3
(c) observed in $G_{3}$ fractal array. Arrows in the experimental panels
indicate position of the input excitation.

Fractal case-2 array with ${r=0.6a}$ supports three types of outer corner
solitons with different symmetries, as shown in Fig. 2b. Among them, the
soliton belonging to cyan branch cannot be excited by focusing beam into site
1, because this site is empty for such nonlinear states (due to their parity).
Therefore, excitation of site 1 is supposed to yield nonlinear combination of
states from magenta and green branches of Fig. 2b. This is what we observe in
Fig. 5a, where for all pulse energies from $10$ to ${750\,\rm nJ}$ light
remains practically confined in three closely spaced the outer corner
waveguides, closely resembling profile of the outer corner soliton and
confirming that in this case it is also thresholdless. For ${r=0.6a}$ no
localized linear states exist in the array, except for states in outer
corners. Consequently, excitations of sites 2 and 3 at all pulse energies
yield broad output distributions, where power oscillates between five
waveguides, and where intensity maximum may not be located in the excited
waveguide (Figs. 5b and 5c). Similarly, excitation of site 4 yields nonlinear
state akin to nontopological delocalized state from orange branch of Fig. 2b,
whose expansion with increase of $E$ is obvious from experimental patterns in
Fig. 5d (compare this state with well-localized inner corner soliton obtained
in the same location at ${r=0.3a}$).

## Conclusion

In summary, we have reported the first example of nonlinear photonic fractal
HOTI that supports a rich variety of topological corner states. The remarkable
new feature of fractal structures considered here is that they possess corner
states (that may have different effective dimensionality) for a very broad
range of distortion parameters, substantially exceeding the range, where
higher-order topological phase emerges in HOTIs built on periodic lattices.
The presence of such states facilitates nonlinear light localization and
resulted in observation of thresholdless corner solitons, in both outer and
inner corners of these structures. Our results extend the class of HOTIs and
highlight new prospects for exploration and practical utilization of nonlinear
phenomena in photonic fractals. They may be used in new designs of topological
lasers that can potentially emit in richer set of states than conventional
higher-order topological lasers (for example, depending on gain landscape and
amplitude, lasing can occur either in outer or in inner corners of the
structure, or in corner states with different parity), in shaping of higher-
harmonic fields in various parametric processes, design of fractal
microresonator networks and quantum interfaces of fractal structures, control
of condensation in light-matter systems with strong coupling, like polariton
condensates in fractal microcavities under resonant or nonresonant pump, and
in many other settings.

## Materials and Methods

### Normalization of parameters in theoretical model

The transverse coordinates $x,y$ in Eq. (1) are normalized to the
characteristic scale ${r_{0}=10\,\mu\textrm{m}}$, the propagation distance $z$
is normalized to the diffraction length ${kr_{0}^{2}\approx
1.14\,\textrm{mm}}$ (corresponding to ${z=1}$), where ${k=2\pi n/\lambda}$ is
the wavenumber in the medium with unperturbed refractive index $n$ (for fused
silica ${n\approx 1.45}$ and the nonlinear refractive index ${n_{2}\approx
2.7\times 10^{-20}\,\rm m^{2}/W}$), and ${\lambda=800~{}\textrm{nm}}$ is the
working wavelength. The array depth ${p=k^{2}r_{0}^{2}\delta n/n}$ is
proportional to the refractive index contrast $\delta n$ of the waveguides.
Thus, in our arrays next-nearest-waveguide distance ${a=6.0}$ corresponds to
${60~{}\mu\textrm{m}}$, waveguide widths ${d_{x}=0.25}$, ${d_{y}=0.75}$
correspond to ${2.5~{}\mu\textrm{m}\times 7.5~{}\mu\textrm{m}}$ wide
elliptical waveguides, sample length of ${10~{}\textrm{cm}}$ corresponds to
${z\approx 88}$, while array depth ${p=5.7}$ corresponds to refractive index
contrast ${\delta n\approx 6.4\times 10^{-4}}$.

### Fs-laser inscription of the waveguide arrays

Fractal waveguide arrays were written in ${10\,\textrm{cm}}$ long fused silica
glass substrate (JGS1). The individual waveguides were inscribed by circularly
polarized beam with central wavelength of $515\,\textrm{nm}$, with pulse
duration of ${230\,\textrm{fs}}$, repetition rate ${1\,\textrm{MHz}}$, and
pulse energy ${270\,\textrm{nJ}}$, focused with an aspheric lens
(${\textrm{NA}=0.3}$) under the sample surface in the depth range from 600 to
$1000\,\mu\textrm{m}$ near the preselected optimal depth of
${800\,\mu\textrm{m}}$. Translation of the sample with respect to the focus
was performed by a high-precision positioner (Aerotech) with a scanning
velocity of ${1\,\textrm{mm/s}}$. Waveguides demonstrate propagation losses
less than ${0.3\,\textrm{dB/cm}}$ at ${\lambda=800\,\textrm{nm}}$.

### Experimental excitation of the waveguide arrays

In experiments, we employed single-waveguide excitations using fs pulses of
variable energy $E$ from ${1\,\rm kHz}$ Ti:sapphire laser at ${800\,\rm nm}$
central wavelength. Initially, short pulses with a ${40\,\rm fs}$ duration and
wide spectrum from a regenerative amplifier system Spitfire HP (Spectra
Physics) first pass through an active beam position stabilization system
(Avesta) and an attenuator, and afterwards are launched into a $4f$ single-
grating stretcher-compressor with a variable slit. Spectra of such pulses are
narrowed by a slit down to ${5\,\rm nm}$, which corresponds to the pulse
duration of ${280\,\rm fs}$. This increase in the pulse duration allows to
prevent optical collapse and strong spectral broadening during pulse
propagation in the waveguides, i.e. it allows to neglect the temporal effects.
The pulses after stretcher compressor were focused into selected waveguides
and the output intensity distributions after propagation in 10 cm sample were
recorded using a Kiralux CMOS camera (Thorlabs). The input peak power in the
waveguide (for each pulse in the $1\,\rm kHz$ sequence) was defined as a ratio
of the input pulse energy $E$ to the pulse duration ${\tau=280\,\rm fs}$.
Taking into account the losses for the matching with the focusing lens the
input power can be evaluated as ${2.5\,\rm kW}$ for each ${1\,\rm nJ}$. For
example, maximal excitation energy of ${E=800\,\rm nJ}$ in experimental
patterns presented here corresponds to the peak power of ${2.0\,\rm MW}$.

### Real-space polarization index

Fractal waveguide arrays are aperiodic structures that are sometimes
considered as structures without bulk due to the method of their construction.
In such structures the appearance of corner states can be associated with the
nonzero real-space polarization, allowing to characterize topological
properties of this system: ${P=-\frac{i}{2\pi}\ln[\rm{det}(S)]}$, where
${S_{m,n}=U_{m}^{\dagger}e^{i2\pi\hat{q}/L}U_{n}}$, $L$ is the length of the
fractal array along the $q$ direction, $\hat{q}$ is the position operator,
$U_{n}$ is the eigenfunction of $n^{\textrm{th}}$ state of the fractal array
obtained with the periodic boundary conditions in the $q$ direction (that is
usually selected along the outer edge of the array). Real-space polarization
for the states in our system can be calculated using tight-binding
approximation. To be able to apply periodic boundary conditions upon
calculation of eigenfunctions $U_{n}$ we considered therhombic structure
composed from two stacked Sierpiński gasket arrays, real and virtual ones,
while neglecting coupling with the virtual and missing sites. It was found
that real-space polarization index is equal to $0.5$ for both hybrid and outer
corner states in their domains of existence in $r$. The detailed information
on this method can be found in the Supporting Information.

## Data availability

The data that support the plots within this paper and other findings of this
study are available from the corresponding author on a reasonable request.

## Code availability

The analysis codes will be made available on a reasonable request.

## Acknowledgements

This work was supported by the research project FFUU-2021-0003 of the
Institute of Spectroscopy of the Russian Academy of Sciences and partially by
the Russian Science Foundation (21-12-00096). Yiqi Zhang and Hua Zhong
acknowledge funding by the National Natural Science Foundation of China
(12074308, 12304370). Sergei A. Zhuravitskii acknowledges support by the
Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”
(22-2-2-26-1).

## Author contributions

Yiqi Zhang and Yaroslav V. Kartashov formulated the problem. Sergei A.
Zhuravitskii, Nikolay N. Skryabin, Ivan V. Dyakonov, and Alexander A. Kalinkin
fabricated the samples. Victor O. Kompanets, and Sergey V. Chekalin performed
experiments. Hua Zhong, Yiqi Zhang and Yongdong Li performed numerical
modeling. Yaroslav V. Kartashov, Sergei P. Kulik, and Victor N. Zadkov
supervised the work. All co-authors took part in discussion of the results and
writing the manuscript.

## Competing interests

The authors declare no competing interests.

## Additional information

Correspondence and requests for materials should be addressed to Yiqi Zhang
and Yaroslav V. Kartashov.

Reprints and permissions information is available at

## References

  * Bunde and Havlin [1994] A. Bunde and S. Havlin, eds.,  _Fractals in science_, 1st ed. (Springer Berlin, 1994).
  * Song, Zhang, and Li [2014] Z.-G. Song, Y.-Y. Zhang, and S.-S. Li, “The topological insulator in a fractal space,” Appl. Phys. Lett. 104, 233106 (2014).
  * He, Liang, and p. Kou [2015] J. He, Y. Liang, and S. p. Kou, “Topological hierarchy insulators and topological fractal insulators,” EPL 112, 17010 (2015).
  * Kempkes  _et al._ [2019] S. N. Kempkes, M. R. Slot, S. E. Freeney, S. J. M. Zevenhuizen, D. Vanmaekelbergh, I. Swart, and C. M. Smith, “Design and characterization of electrons in a fractal geometry,” Nat. Phys. 15, 127–131 (2019).
  * Pai and Prem [2019] S. Pai and A. Prem, “Topological states on fractal lattices,” Phys. Rev. B 100, 155135 (2019).
  * Iliasov, Katsnelson, and Yuan [2020] A. A. Iliasov, M. I. Katsnelson, and S. Yuan, “Hall conductivity of a Sierpiński carpet,” Phys. Rev. B 101, 045413 (2020).
  * Fremling  _et al._ [2020] M. Fremling, M. van Hooft, C. M. Smith, and L. Fritz, “Existence of robust edge currents in Sierpiński fractals,” Phys. Rev. Res. 2, 013044 (2020).
  * Xu  _et al._ [2021] X.-Y. Xu, X.-W. Wang, D.-Y. Chen, C. M. Smith, and X.-M. Jin, “Quantum transport in fractal networks,” Nat. Photon. 15, 703–710 (2021).
  * Liu  _et al._ [2021] C. Liu, Y. Zhou, G. Wang, Y. Yin, C. Li, H. Huang, D. Guan, Y. Li, S. Wang, H. Zheng, C. Liu, Y. Han, J. W. Evans, F. Liu, and J. Jia, “Sierpiński structure and electronic topology in Bi thin films on InSb(111)B surfaces,” Phys. Rev. Lett. 126, 176102 (2021).
  * Ivaki  _et al._ [2022] M. N. Ivaki, I. Sahlberg, K. Pöyhönen, and T. Ojanen, “Topological random fractals,” Commun. Phys. 5, 327 (2022).
  * Manna  _et al._ [2022] S. Manna, C. W. Duncan, C. A. Weidner, J. F. Sherson, and A. E. B. Nielsen, “Anyon braiding on a fractal lattice with a local Hamiltonian,” Phys. Rev. A 105, L021302 (2022).
  * Yang  _et al._ [2020] Z. Yang, E. Lustig, Y. Lumer, and M. Segev, “Photonic Floquet topological insulators in a fractal lattice,” Light Sci. Appl. 9, 128 (2020).
  * Biesenthal  _et al._ [2022] T. Biesenthal, L. J. Maczewsky, Z. Yang, M. Kremer, M. Segev, A. Szameit, and M. Heinrich, “Fractal photonic topological insulators,” Science 376, 1114–1119 (2022).
  * Zheng  _et al._ [2022] S. Zheng, X. Man, Z.-L. Kong, Z.-K. Lin, G. Duan, N. Chen, D. Yu, J.-H. Jiang, and B. Xia, “Observation of fractal higher-order topological states in acoustic metamaterials,” Sci. Bull. 67, 2069–2075 (2022).
  * Li  _et al._ [2022] J. Li, Q. Mo, J.-H. Jiang, and Z. Yang, “Higher-order topological phase in an acoustic fractal lattice,” Sci. Bull. 67, 2040–2044 (2022).
  * Ren  _et al._ [2023a] B. Ren, Y. V. Kartashov, L. J. Maczewsky, M. S. Kirsch, H. Wang, A. Szameit, M. Heinrich, and Y. Zhang, “Theory of nonlinear corner states in photonic fractal lattices,” Nanophoton. 12, 3829–3838 (2023a).
  * Xie  _et al._ [2021a] Y. Xie, L. Song, W. Yan, S. Xia, L. Tang, D. Song, J.-W. Rhim, and Z. Chen, “Fractal-like photonic lattices and localized states arising from singular and nonsingular flatbands,” APL Photon. 6, 116104 (2021a).
  * Song  _et al._ [2023] L. Song, Y. Xie, S. Xia, L. Tang, D. Song, J.-W. Rhim, and Z. Chen, “Topological flatband loop states in fractal-like photonic lattices,” Laser Photon. Rev. 17, 2200315 (2023).
  * Lu, Joannopoulos, and Soljačić [2014] L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8, 821–829 (2014).
  * Ozawa  _et al._ [2019] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
  * Zhang  _et al._ [2023] X. Zhang, F. Zangeneh-Nejad, Z.-G. Chen, M.-H. Lu, and J. Christensen, “A second wave of topological phenomena in photonics and acoustics,” Nature 618, 687–697 (2023).
  * Wang  _et al._ [2009] Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
  * Rechtsman  _et al._ [2013] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
  * Noh  _et al._ [2018a] J. Noh, S. Huang, K. P. Chen, and M. C. Rechtsman, “Observation of photonic topological valley Hall edge states,” Phys. Rev. Lett. 120, 063902 (2018a).
  * Peterson  _et al._ [2018] C. W. Peterson, W. A. Benalcazar, T. L. Hughes, and G. Bahl, “A quantized microwave quadrupole insulator with topologically protected corner states,” Nature 555, 346–350 (2018).
  * Noh  _et al._ [2018b] J. Noh, W. A. Benalcazar, S. Huang, M. J. Collins, K. P. Chen, T. L. Hughes, and M. C. Rechtsman, “Topological protection of photonic mid-gap defect modes,” Nat. Photon. 12, 408–415 (2018b).
  * Mittal  _et al._ [2019] S. Mittal, V. V. Orre, G. Zhu, M. A. Gorlach, A. Poddubny, and M. Hafezi, “Photonic quadrupole topological phases,” Nat. Photon. 13, 692–696 (2019).
  * El Hassan  _et al._ [2019] A. El Hassan, F. K. Kunst, A. Moritz, G. Andler, E. J. Bergholtz, and M. Bourennane, “Corner states of light in photonic waveguides,” Nat. Photon. 13, 697–700 (2019).
  * Xie  _et al._ [2021b] B. Xie, H.-X. Wang, X. Zhang, P. Zhan, J.-H. Jiang, M. Lu, and Y. Chen, “Higher-order band topology,” Nat. Rev. Phys. 3, 520–532 (2021b).
  * Lin  _et al._ [2023] Z.-K. Lin, Q. Wang, Y. Liu, H. Xue, B. Zhang, Y. Chong, and J.-H. Jiang, “Topological phenomena at defects in acoustic, photonic and solid-state lattices,” Nat. Rev. Phys. 5, 483–495 (2023).
  * Li  _et al._ [2023] J. Li, Y. Sun, Q. Mo, Z. Ruan, and Z. Yang, “Fractality-induced topological phase squeezing and devil’s staircase,” Phys. Rev. Res. 5, 023189 (2023).
  * Hatsugai [1993] Y. Hatsugai, “Chern number and edge states in the integer quantum Hall effect,” Phys. Rev. Lett. 71, 3697–3700 (1993).
  * Manna, Nandy, and Roy [2022] S. Manna, S. Nandy, and B. Roy, “Higher-order topological phases on fractal lattices,” Phys. Rev. B 105, L201301 (2022).
  * Brzezińska, Cook, and Neupert [2018] M. Brzezińska, A. M. Cook, and T. Neupert, “Topology in the Sierpiński-Hofstadter problem,” Phys. Rev. B 98, 205116 (2018).
  * Smirnova  _et al._ [2020] D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, “Nonlinear topological photonics,” Appl. Phys. Rev. 7, 021306 (2020).
  * Lumer  _et al._ [2013] Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Self-localized states in photonic topological insulators,” Phys. Rev. Lett. 111, 243905 (2013).
  * Ablowitz, Curtis, and Ma [2014] M. J. Ablowitz, C. W. Curtis, and Y.-P. Ma, “Linear and nonlinear traveling edge waves in optical honeycomb lattices,” Phys. Rev. A 90, 023813 (2014).
  * Leykam and Chong [2016] D. Leykam and Y. D. Chong, “Edge solitons in nonlinear-photonic topological insulators,” Phys. Rev. Lett. 117, 143901 (2016).
  * Kartashov and Skryabin [2016] Y. V. Kartashov and D. V. Skryabin, “Modulational instability and solitary waves in polariton topological insulators,” Optica 3, 1228–1236 (2016).
  * Ablowitz and Cole [2017] M. J. Ablowitz and J. T. Cole, “Tight-binding methods for general longitudinally driven photonic lattices: Edge states and solitons,” Phys. Rev. A 96, 043868 (2017).
  * Ivanov  _et al._ [2020] S. K. Ivanov, Y. V. Kartashov, A. Szameit, L. Torner, and V. V. Konotop, “Vector topological edge solitons in Floquet insulators,” ACS Photon. 7, 735–745 (2020).
  * Zhong  _et al._ [2021] H. Zhong, S. Xia, Y. Zhang, Y. Li, D. Song, C. Liu, and Z. Chen, “Nonlinear topological valley Hall edge states arising from type-II Dirac cones,” Adv. Photon. 3, 056001 (2021).
  * Mukherjee and Rechtsman [2020] S. Mukherjee and M. C. Rechtsman, “Observation of Floquet solitons in a topological bandgap,” Science 368, 856–859 (2020).
  * Mukherjee and Rechtsman [2021] S. Mukherjee and M. C. Rechtsman, “Observation of unidirectional solitonlike edge states in nonlinear Floquet topological insulators,” Phys. Rev. X 11, 041057 (2021).
  * Xia  _et al._ [2020] S. Xia, D. Jukić, N. Wang, D. Smirnova, L. Smirnov, L. Tang, D. Song, A. Szameit, D. Leykam, J. Xu, Z. Chen, and H. Buljan, “Nontrivial coupling of light into a defect: the interplay of nonlinearity and topology,” Light Sci. Appl. 9, 147 (2020).
  * Guo  _et al._ [2020] M. Guo, S. Xia, N. Wang, D. Song, Z. Chen, and J. Yang, “Weakly nonlinear topological gap solitons in Su-Schrieffer-Heeger photonic lattices,” Opt. Lett. 45, 6466–6469 (2020).
  * Kartashov  _et al._ [2022] Y. V. Kartashov, A. A. Arkhipova, S. A. Zhuravitskii, N. N. Skryabin, I. V. Dyakonov, A. A. Kalinkin, S. P. Kulik, V. O. Kompanets, S. V. Chekalin, L. Torner, and V. N. Zadkov, “Observation of edge solitons in topological trimer arrays,” Phys. Rev. Lett. 128, 093901 (2022).
  * Hadad  _et al._ [2018] Y. Hadad, J. C. Soric, A. B. Khanikaev, and A. Alù, “Self-induced topological protection in nonlinear circuit arrays,” Nat. Electron. 1, 178–182 (2018).
  * Zangeneh-Nejad and Fleury [2019] F. Zangeneh-Nejad and R. Fleury, “Nonlinear second-order topological insulators,” Phys. Rev. Lett. 123, 053902 (2019).
  * Maczewsky  _et al._ [2020] L. J. Maczewsky, M. Heinrich, M. Kremer, S. K. Ivanov, M. Ehrhardt, F. Martinez, Y. V. Kartashov, V. V. Konotop, L. Torner, D. Bauer, and A. Szameit, “Nonlinearity-induced photonic topological insulator,” Science 370, 701–704 (2020).
  * Kirsch  _et al._ [2021] M. S. Kirsch, Y. Zhang, M. Kremer, L. J. Maczewsky, S. K. Ivanov, Y. V. Kartashov, L. Torner, D. Bauer, A. Szameit, and M. Heinrich, “Nonlinear second-order photonic topological insulators,” Nat. Phys. 17, 995–1000 (2021).
  * Hu  _et al._ [2021] Z. Hu, D. Bongiovanni, D. Jukić, E. Jajtić, S. Xia, D. Song, J. Xu, R. Morandotti, H. Buljan, and Z. Chen, “Nonlinear control of photonic higher-order topological bound states in the continuum,” Light Sci. Appl. 10, 164 (2021).
  * Arkhipova  _et al._ [2023] A. A. Arkhipova, Y. Zhang, Y. V. Kartashov, S. A. Zhuravitskii, N. N. Skryabin, I. V. Dyakonov, A. A. Kalinkin, S. P. Kulik, V. O. Kompanets, S. V. Chekalin, and V. N. Zadkov, “Observation of $\pi$ solitons in oscillating waveguide arrays,” Sci. Bull. 68, 2017–2024 (2023).
  * Ren  _et al._ [2023b] B. Ren, A. A. Arkhipova, Y. Zhang, Y. V. Kartashov, H. Wang, S. A. Zhuravitskii, N. N. Skryabin, I. V. Dyakonov, A. A. Kalinkin, S. P. Kulik, V. O. Kompanets, S. V. Chekalin, and V. N. Zadkov, “Observation of nonlinear disclination states,” Light Sci. Appl. 12, 194 (2023b).
  * Wheeler, Wagner, and Hughes [2019] W. A. Wheeler, L. K. Wagner, and T. L. Hughes, “Many-body electric multipole operators in extended systems,” Phys. Rev. B 100, 245135 (2019).
  * Kang, Shiozaki, and Cho [2019] B. Kang, K. Shiozaki, and G. Y. Cho, “Many-body order parameters for multipoles in solids,” Phys. Rev. B 100, 245134 (2019).