††thanks: These three authors contributed equally††thanks: These three authors
contributed equally††thanks: These three authors contributed equally

# Chiral polaron formation on the edge of topological quantum matter

Amit Vashisht  Ivan Amelio  Laurens Vanderstraeten  CENOLI, Université Libre
de Bruxelles, CP 231, Campus Plaine, B-1050 Brussels, Belgium  Georg M. Bruun
Department of Physics and Astronomy, Aarhus University, Ny Munkegade, DK-8000
Aarhus C, Denmark  Oriana K. Diessel  ITAMP, Center for Astrophysics, Harvard
& Smithsonian, Cambridge, Massachusetts 02138, USA  Department of Physics,
Harvard University, Cambridge, Massachusetts 02138, USA  Nathan Goldman
<EMAIL_ADDRESS>CENOLI, Université Libre de Bruxelles, CP 231,
Campus Plaine, B-1050 Brussels, Belgium  Laboratoire Kastler Brossel, Collège
de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin
Berthelot, 75005 Paris, France

(August 28, 2024)

###### Abstract

Immersing a mobile impurity in a quantum many-body environment can reveal
fundamental properties of the background medium, hence providing a powerful
probe of quantum matter. This approach is particularly intriguing when
considering media with exotic properties, such as strongly-correlated phases
and topological states of matter. In this work, we study the dressing of a
mobile impurity interacting with a chiral mode, as provided by the edge of
topological quantum matter. The resulting “chiral polaron” is characterized by
an asymmetric spectral function, which reflects the chirality and group
velocity of the topological edge mode and the drag experienced by the mobile
impurity. We first build our theoretical understanding from an effective one-
dimensional chiral model, which captures the hallmark signatures of the chiral
polaron. We then demonstrate how this simple picture extends to realistic
models of integer and fractional Chern insulator states, by adapting tensor-
network methods to polaron spectroscopy. Injecting mobile impurities on the
edge of topological quantum matter is shown to be a powerful tool to probe
exotic edge properties, particularly suitable for cold-atom experiments.

##  I Introduction

Polarons emerge from the interaction of a mobile impurity with the surrounding
many-body medium. Originally introduced to describe the motion of electrons in
polar crystals [1, 2, 3, 4, 5], polarons are key to understanding the
properties of materials, including semiconductors [6], high-temperature
superconductors [7], alkali halide insulators [8], and transition metal oxides
[9].

Experimental studies performed using ultracold atomic gases have demonstrated
the formation of Fermi and Bose polarons, by tuning the impurity-bath
interactions via Feshbach resonances [10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20]. Besides, Fermi [21] and Bose [22] polarons were also realized recently in
transition metal dichalchogenide (TMD) heterostructures, where their formation
sheds light on the optical excitation spectrum of doped monolayer
semiconductors and allows for the study of polaron-polaron interactions [23,
24, 22, 25].

Conventional Fermi and Bose polaron settings involve a mobile impurity
interacting with a cloud of non-interacting or weakly-interacting fermions or
bosons. Despite the simplicity of the medium, the impurity-bath interactions
can lead to rich and correlated many-body states. In particular, whereas most
properties of the Fermi polaron are fairly well understood even for strong
interactions [26], there are many open questions regarding the role of many-
body correlations and bound states for the Bose polaron [27, 28, 29, 30, 31,
32].

Beyond conventional settings, it is appealing to explore polaron formation in
exotic media, such as lattice systems with non-trivial Bloch band properties
(topological properties or quantum geometry), or strongly-correlated phases of
matter. At a fundamental level, the behavior of a mobile impurity interacting
with a non-trivial medium is conceptually interesting, and can potentially
lead to rich impurity-dressing mechanisms and unconventional properties.
Besides, studying the formation of a polaron can provide useful information on
the background medium, such that the polaron can be used as a quantum probe
for exotic states of matter. From a theoretical point of view, the
investigation of polarons in unconventional and strongly-correlated media is
at the dawn, in part due to the complications of treating the bath. So far,
pioneering works have dealt with fractional Chern insulators [33, 34], Fermi
superfluids along the BEC-BCS crossover [35, 36, 37, 38, 39], excitonic
insulators [40], twisted homobilayers [41], the Mott transition [42],
interacting bosons on a lattice [43, 44, 45], kinetic magnetism [46, 47],
many-body topological states in dimerized chains [48], Holstein polarons in
Luttinger liquids [49], Fermi baths with Dirac [50] and flat band dispersion
[51]. We also note a few related works on the dressing of optical excitations
in Mott insulators [52, 53] and fractional quantum Hall systems [54, 55, 56,
57]. Theoretical methods are currently being developed, with variational
approaches complementing numerical tools such as Quantum Monte Carlo [58, 59,
60, 61, 62], exact diagonalization [63] and tensor network [64, 65, 66]
techniques. Very recently, experiments in TMD heterostructures [67, 68, 69,
70, 71] and cold-atom simulators [72, 73, 74, 75] have started to employ
polarons in the study of highly correlated states of matter.

In this context, dropping a mobile impurity into topological quantum matter
has recently emerged as an exciting framework for unconventional polaron
physics. Several studies have investigated the binding of a mobile impurity to
a quasi-hole excitation in fractional quantum Hall (FQH) states [33, 34, 76,
77]: this setting was shown to be ideally suited to probe the underlying many-
body Chern number (a topological invariant associated with the many-body wave
function), but also to characterize and manipulate anyons. Besides, the Hall
drift of a Fermi polaron immersed in the bulk of a Chern insulator was
explored in Refs. [78, 79]. Importantly, these previous works concerned the
behavior of impurities injected in the _bulk_ of topological matter, where the
dressing entirely relies on bulk excitations and bulk properties (e.g. the
Berry curvature); furthermore, topology manifests in _transport or scattering_
scenarios, requiring fine control over individual impurities and bath
excitations (e.g. the anyons).

### Chiral polaron: scope of the paper

Here, we shift gears and explore the formation of polarons on the _edge_ of
topological insulating states of matter, setting the focus on their associated
_spectroscopic_ hallmarks. In this framework, the impurity is mainly dressed
by the low-energy chiral edge modes circulating around the sample, resulting
in a “chiral polaron”, as we further explain below.

For the sake of concreteness, we consider two-dimensional (2D) Chern
insulators, a class of topological insulators characterized by chiral edge
modes. This includes systems of non-interacting fermions, filling Bloch bands
with non-zero Chern numbers [80, 81], but also the so-called _fractional_
Chern insulators, which are strongly-correlated states akin to the FQH states
on a lattice [82]. In the simplest setting, such a topological system displays
a single branch of low-energy edge states, with (quasi-)linear dispersion
relation $E\sim vk$; see the sketch in Fig. 1(a). We then inject a mobile
impurity of mass $M$ on the edge of this sample. Importantly, we assume that
this impurity behaves as a free particle in the absence of coupling to the
topological medium, as illustrated in Fig. 1(b). A chiral polaron then forms
upon activating impurity-bath interactions, as signaled by a characteristic
behavior of the momentum-resolved impurity spectral function, shown
pictorially in Fig. 1(c). As detailed below, this response function displays
two key features: an asymmetric spectrum with respect to momentum
$Q\leftrightarrow-Q$, as a direct consequence of chirality, and a splitting
into two broadened lines for momenta larger than the critical momentum
$Q_{c}\\!=\\!Mv$, reflecting the resonant exchange of energy-momentum between
the impurity and the chiral bath.

This phenomenon is explored in this work, through realistic lattice models of
Chern insulators and fractional Chern insulators. We also introduce and
analyze an effective low-energy model, which correctly captures the key
spectral features of the chiral polaron, determined by the dressing of the
impurity by low-energy edge excitations. Three theoretical approaches are used
to analyse this: exact diagonalisation for small systems, variational wave
function methods, and tensor network techniques, which are particularly well
suited to study the case of fractional Chern insulators, i.e. a strongly-
correlated topological medium. The application of advanced tensor network
methods to the polaron context constitutes another key aspect of this work.

Figure 1:  Chiral dressing of a mobile impurity injected on the edge of a
topological insulator. (a) The edge of the topological insulator provides a
chiral mode, with linear dispersion relation $E\sim vk$. (b) We inject a
mobile impurity of mass M and momentum $Q$ on the edge of the system. In the
absence of interactions with the chiral bath, its dispersion relation is
quadratic (free particle). (c) Upon activating interactions, the impurity is
dressed by the chiral bath. Its spectral function displays a chiral behavior
with strong bath-impurity coupling features at momenta of the order of $Mv$, a
clear signature of the chiral dressing.

Our results show that the chiral polaron can be used as a quantum probe of
topological matter. Indeed, the spectral function that would result from
injecting an impurity in the bulk of the sample would not display these key
chiral features. Similarly, those signatures would not be present when setting
the surrounding medium in a different phase (e.g. a trivial insulator or a
metallic phase). Importantly, the phenomenon is universal, in the sense that
it does not rely on the microscopic details of the underlying system. This is
demonstrated in this work, by analyzing two topological media of very
different nature: a Chern insulator of non-interacting fermions and a
fractional Chern insulator of hardcore bosons, both hosting a single chiral
edge mode. From a practical point of view, our spectroscopic scheme should be
readily implementable in cold atoms, only requiring momentum-resolved (Raman)
injection spectroscopy [19] applied to an impurity that is confined along the
edge of the system.

More generally, this work contributes to an ongoing effort devoted to the
characterization and detection of topological edge modes in ultracold atomic
gases [83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97]. In this
context, previous works explored the spectroscopic characterization of chiral
modes using well-designed probing fields [86, 98, 99], quantized circular
dichroism [100], and quench-dynamics protocols based on applying local
perturbations on the edge of the sample [87, 101, 102, 103]. By injecting a
mobile impurity locally on the edge of a topological system, and by
characterizing the resulting spectroscopic response, the present work
naturally contributes to this ongoing effort.

### Outline

The paper is structured as follows. In Sec. II, we review the general polaron
framework and define the impurity spectral function. Moreover, we discuss the
state-of-the-art of the main theoretical and numerical methods that are used
in this context, with a special attention to tensor-network approaches. In
Sec. III, we study the Fermi polaron spectrum in the case of a non-chiral 1D
lattice, demonstrating the accuracy of tensor-network methods and highlighting
the lattice effects. The key features of the chiral polaron are investigated
in Sec. IV, using an effective 1D chiral model for the bath. The particle-hole
excitation dynamics, as well as the lattice effects are also studied in that
Section. In Sec. V, we eventually study the formation of a chiral polaron on
the edge of a realistic 2D topological (Chern) insulator. Here, the bath
consists of a completely filled Bloch band of non-interacting fermions in the
2D Haldane lattice model. It is demonstrated that the impurity spectrum
inherits the chiral features of the edge modes, provided that the Fermi level
lies in the topological gap and that the impurity is confined on the edge of
the system. The power of tensor-network methods is then fully exploited in
Sec. VI, where the chiral polaron emerges from the interaction with a
(strongly-interacting) fractional Chern insulator. Here, the bath consists of
hardcore bosons in the Harper-Hofstadter model. Finally, we draw our
conclusions and comment on possible perspectives in Sec. VII.

##  II General framework:
polaron spectroscopy and methods

In this work, we consider the general setting of a mobile impurity interacting
with a bath of particles, as described by a Hamiltonian of the form

$\hat{H}=\hat{H}_{\mathrm{bath}}+\hat{H}_{\mathrm{imp}}+\hat{H}_{\mathrm{int}}.$
(1)

Here, $\hat{H}_{\mathrm{bath}}$ denotes the many-body Hamiltonian describing
the bath particles (fermions or bosons) moving on a lattice; its associated
band structure can be trivial or topological, and this Hamiltonian may also
include inter-particle interactions within the bath.

The operator $\hat{H}_{\mathrm{imp}}$ describes the hopping of the mobile
impurity on a subset $\mathcal{E}\subseteq\mathcal{L}$ of the lattice sites
$\mathcal{L}$. For instance, in this work, we will consider restricting the
impurity on the edge of a 2D lattice. It can be written as

$\displaystyle\hat{H}_{\mathrm{imp}}$
$\displaystyle=-J_{\mathrm{I}}\sum_{\langle
i,j\rangle}(\hat{d}_{i}^{\dagger}\hat{d}_{j}{+\,{\rm h.c.}}),$ (2)

where $J_{\mathrm{I}}$ is the impurity hopping strength, $\braket{i,j}$ stands
for a pair of nearest-neighboring sites, and $\hat{d}_{j}^{\dagger}$ denotes
the impurity creation operator at lattice site $j\in\mathcal{E}$. Since we
consider at most one impurity in the bath, our results are independent of its
quantum statistics.

Finally, $\hat{H}_{\mathrm{int}}$ describes the interaction between the
impurity and the bath particles. Considering onsite contact interactions
between the two species, it can be written as

$\hat{H}_{\mathrm{int}}=U\sum_{j\in\mathcal{E}}\hat{c}_{j}^{\dagger}\hat{c}_{j}\hat{d}_{j}^{\dagger}\hat{d}_{j},$
(3)

where $\hat{c}_{j}^{\dagger}$ denotes the creation operator of a bath particle
and $U$ quantifies the interaction strength.

While ground-state properties of the coupled impurity-bath system will be
analyzed below, we note that the most natural observable (in both ultracold
atoms [104] and solid-state experiments [105]) is provided by the spectral
function of the impurity. Our main target will thus concern the computation of
impurity spectra, which can be accessed via injection spectroscopy in cold-
atom setups. Experimentally, this corresponds to the following protocol: One
starts from a state in which the impurity and the bath atoms do not interact
(or only weakly interact). Then, the internal state of the impurity is
resonantly flipped to a hyperfine level characterized by a sizable impurity-
bath interaction. By utilizing Raman transitions, one can impart a finite
momentum to the impurity, which makes it possible to map out its full
momentum-resolved single-particle spectral function, corresponding to the
final observable of the resonant excitation [106].

Mathematically, the polaron spectral function is defined as

$\mathcal{A}(Q,\omega)=-2{\rm
Im}\langle\Psi_{0}|\hat{d}_{Q}\frac{1}{\omega-\hat{H}+E_{\mathrm{bath}}^{0}+i\eta}\hat{d}^{\dagger}_{Q}|\Psi_{0}\rangle,$
(4)

where $|\Psi_{0}\rangle$ is the ground state of the bath Hamiltonian
$\hat{H}_{\mathrm{bath}}$ with energy $E_{\mathrm{bath}}^{0}$, while $\hat{H}$
is the full Hamiltonian including the interaction between the bath particles
and the impurity. The spectral lines are artificially broadened by the
linewidth $\eta$, describing effects such as Fourier broadening of the
experimental injection pulse or a finite lifetime of the impurity. We assume
that the impurity is both injected and detected in the quantum state labeled
by $Q$, which will typically correspond to a momentum quantum number in the
present context. We remark that the spectral function (4) satisfies the
normalization $\int_{-\infty}^{\infty}\text{d}\omega\leavevmode\nobreak\
\mathcal{A}(Q,\omega)/2\pi\\!=\\!1$ for each value of $Q$. Furthermore,
introducing the eigenstates and eigenenergies of the interacting Hamiltonian,
$\hat{H}|n\rangle\\!=\\!E_{\rm int}^{n}|n\rangle$, the spectral function
decomposes as

$\mathcal{A}(Q,\omega)=-2{\rm Im}\sum_{n}\frac{|\langle
n|\hat{d}^{\dagger}_{Q}|\Psi_{0}\rangle|^{2}}{\omega-E_{\rm
int}^{n}+E_{\mathrm{bath}}^{0}+i\eta},$ (5)

where the overlap $|\langle n|d^{\dagger}_{Q}|\Psi_{0}\rangle|^{2}$ is usually
referred to as the oscillator strength of the state $|n\rangle$. In other
words, the many-body states with a large (small) overlap with the injected
state will be bright (dark), respectively. Unless otherwise stated, we set
$\hbar\\!=\\!1$ throughout this work.

### Methods:
From the Chevy ansatz to matrix product states

In the case of a fermionic bath, useful physical insight can be obtained by
restricting the problem to the subspace associated with single particle-hole
excitations on top of the bath’s ground state:

$\hat{\mathcal{H}}_{\rm
Chevy}=\left\\{\hat{d}^{\dagger}_{i}|\Psi_{0}\rangle,\hat{d}^{\dagger}_{i}\hat{c}^{\dagger}_{j}\hat{c}_{k}|\Psi_{0}\rangle\right\\}_{i\in\mathcal{E},jk\in\mathcal{L}}.$
(6)

This method was pioneered by Chevy [107] for an impurity in a non-interacting
Fermi sea, for which it proved to be a surprisingly accurate approximation due
to the destructive interference of higher-order terms with two or more
particle-hole excitations [108, 109, 110]. It was recently adapted to
investigate a number of different scenarios involving interacting fermionic
backgrounds [35, 36, 37, 38, 39, 41, 40].

However, for strongly-correlated environments, more sophisticated numerical
methods are generally required, independently of the presence of the impurity.
To begin with, Quantum Monte Carlo (QMC) methods have proved useful for
studying the ground state of Bose polarons in dipolar atomic clouds [59, 61,
62], as well as weakly-interacting Fermi polarons [58, 60]. Obtaining spectral
information with QMC methods is more challenging, as this requires ill-
conditioned analytical continuation to real frequencies. Moreover, because of
potential sign problems, correlated and/or topological backgrounds can be very
challenging as well.

Then, it was recently shown in Ref. [63] that exact diagonalization (ED) of
small systems can provide valuable insights on the polaron spectra. In this
context, the spectral function can be obtained using Krylov space methods.
While ED will be used in the following to provide benchmarks on small systems,
ED methods become prohibitively expensive as the size of the system is
increased.

In this work, we use tensor-network methods as an unbiased numerical tool for
computing polaron spectral functions. In particular, we employ the variational
class of matrix product states (MPS). Originally, MPS were used as variational
states for approximating ground states of correlated 1D quantum lattice
systems, as in the density matrix renormalization group (DMRG) algorithm [111,
112]. The approximation can be systematically improved by increasing the bond
dimension of the MPS. MPS methods have become so efficient that they are now
also used for simulating 2D lattice systems on geometries such as cylinders or
strips; for that purpose, the 2D lattice is mapped to a chain by snaking
through the system along a 1D path. As a result, local 2D Hamiltonians are
mapped to effective 1D Hamiltonians with longer-range terms, which makes the
simulation substantially more costly and necessitates an exponential scaling
of the MPS bond dimension.

Building on these ground-state algorithms, different approaches were developed
for also accessing the low-energy excited states and spectral functions.
Nowadays, the most efficient approach for computing a spectral function is
through real-time evolution [113]. In the present context of polaron physics,
this amounts to first optimizing a ground-state approximation for the bath
(either using the DMRG [111] algorithm for a finite system, or the iDMRG [114]
or VUMPS [115] algorithms for infinite systems), and applying a local impurity
creation operator on a site $j$ in the system

$\ket{\Psi_{j}(t=0)}=\hat{d}_{j}^{\dagger}\ket{\Psi_{0}}.$ (7)

This state can now be evolved in real time with the full bath-impurity
Hamiltonian, where the time evolved state
$\ket{\Psi_{j}(t)}\\!=\\!e^{-i\hat{H}t}\ket{\Psi_{j}(0)}$ is always
approximated as an MPS. Early algorithms for MPS time evolution [116, 117,
118] relied on the Trotter-Suzuki decomposition, and are therefore unwieldy
for longer-range Hamiltonians – for instance, Hamiltonians involving non-local
interactions or higher-neighbor hopping terms. Fortunately, time evolution
based on the time-dependent variational principle (TDVP) [119, 120] or on
matrix-product operator (MPO) representations of the time-evolution operator
[121, 122] allow us to treat longer-range terms. After the time evolution, one
can measure the dynamical correlation function, which we can Fourier transform
to obtain the spectral function:

$\mathcal{A}(Q,\omega)=\int_{-\infty}^{\infty}\mathrm{d}t\,e^{i\omega
t}\sum_{j^{\prime}}e^{iQ(j^{\prime}-j)}\bra{\Psi_{0}}\hat{d}_{j^{\prime}}\ket{\Psi_{j}(t)}.$
(8)

This method is essentially limited by two factors: (i) Because the
entanglement entropy generically increases under unitary time evolution, the
bond dimension in $\ket{\Psi_{j}(t)}$ would need to increase as a function of
time; in practice, only a limited time window is accessible, which translates
in a limited frequency resolution in the spectral function. (ii) The cylinder
circumference or strip width is limited due to the exponential scaling of the
bond dimension.

While this more advanced methodology has allowed for precise evaluations of
different types of Green’s function and dynamic structure factors [123, 124,
125, 126, 127, 128, 129], for instance in the context of 2D quantum spin
systems or Hubbard models, it has not been fully applied to the study of
polaron formation yet. Indeed, only basic MPS methods – such as time-evolving
block decimation (TEBD) – have been used for calculating polaron spectral
functions, and were limited to 1D systems [64, 66, 65, 130, 131, 132]. In this
work, we show that an extension of state-of-the-art methods can be used to
tackle 2D systems as well; see Appendix A for details. We will apply this
methodology to compute the dynamics of impurities immersed in topological 2D
systems, with and without (strong) interactions in the bath, both in the bulk
and on the edge of the system.

##  III Fermi polaron on a 1D lattice:
the non-chiral case

Before addressing the chiral polaron formation in topological systems, it is
important to build a clear understanding of the impurity dynamics in a 1D,
non-interacting, Fermi sea without any time reversal symmetry breaking. Here,
we focus on lattice systems, motivated by the availability of powerful, quasi-
exact, numerical methods (such as ED and MPS) in this framework. As we will
see, some specific features of the impurity spectra arise from the 1D and
lattice nature of the system. With respect to the existing literature, an
impurity in a 1D Fermi gas has been the subject of a few analytical and
numerical works [133, 134, 64, 135, 136, 65, 137], both on a lattice and in
the continuum (where an effective Tomonaga-Luttinger liquid description is
typically assumed for the background). However, to the best of our knowledge,
a full frequency- and momentum-resolved impurity spectrum has not been
computed using quasi-exact numerical methods, and a detailed account of
lattice effects has remained elusive. For these reasons, we devote this
section to presenting a self-contained description of the momentum-resolved
spectrum of an impurity injected in a 1D lattice Fermi sea.

The Hamiltonian of an impurity immersed in a non-interacting Fermi sea, on a
1D lattice, takes the form of Eq. (1), with the impurity and interaction terms
given by Eqs. (2)-(3) and

$\hat{H}_{\mathrm{bath}}=\hat{H}_{\mathrm{1D}}=-J\sum_{j=0}^{L-1}\left(\hat{c}^{\dagger}_{j}\hat{c}_{j+1}+\hat{c}^{\dagger}_{j+1}\hat{c}_{j}\right),$
(9)

where $J$ is the nearest-neighbor hopping strength of bath particles. In the
following, we assume periodic boundary conditions, $j\\!=\\!0,1,..,L-1$, and
consider $N$ spinless fermions in the system. In momentum space, the
Hamiltonians of the bath fermions and of the impurity read, respectively,

$\displaystyle\hat{H}_{\mathrm{bath}}$
$\displaystyle=\sum_{k}\varepsilon_{\text{bath}}(k)\hat{c}_{k}^{\dagger}\hat{c}_{k}=-2J\sum_{k}\cos(ka)\hat{c}_{k}^{\dagger}\hat{c}_{k},$
(10a) $\displaystyle\hat{H}_{\mathrm{imp}}$
$\displaystyle=\sum_{k}\varepsilon_{\text{I}}(k)\hat{d}_{k}^{\dagger}\hat{d}_{k}=-2J_{\mathrm{I}}\sum_{k}\left(\cos(ka)-1\right)d_{k}^{\dagger}\hat{d}_{k},$
(10b)

with $k\\!=\\!0,1,...,(L-1)\times 2\pi/La$ the quasi-momenta defined modulo
$2\pi/a$, and where $a$ is the lattice constant. We have added the constant
$2J_{\mathrm{I}}$ to the impurity dispersion $\varepsilon_{\text{I}}(k)$, such
that its band minimum has zero energy. If $N$ is odd, the ground state of the
fermionic bath (without the impurity) is simply given by the non-degenerate
Fermi sea state

$|\Psi_{0}\rangle\equiv|\mathrm{FS}\rangle=\prod_{|k|<k_{F}}\hat{c}^{\dagger}_{k}|0\rangle,$
(11)

where $k_{F}\\!=\\!\pi N/La$ is the Fermi momentum.

In the following, we will investigate the energy- and momentum-resolved
impurity spectral function $\mathcal{A}(Q,\omega)$ using three different
methods: ED, MPS and the Chevy ansatz. In the ED case, we will use Krylov
space methods to exactly calculate the impurity spectral function for small
system sizes, yet presenting reasonably weak finite-size effects [63]. Next,
we will use the ED spectra to benchmark the spectral function obtained using
the computationally cheaper MPS and Chevy-ansatz methods.

Figure 2:  Spectral function $\mathcal{A}(Q,\omega)$ of an impurity injected
in a 1D non-interacting Fermi sea, as calculated using ED. The three panels
correspond to different impurity-fermion interaction strengths: (a)
$U\\!=\\!-1.5J$, (b) $U\\!=\\!-5J$, and (c) $U\\!=\\!5J$. We fixed
$J_{\mathrm{I}}=J$, and considered $N\\!=\\!5$ fermions in a 1D lattice with
$L\\!=\\!29$ sites and periodic boundary conditions. The dashed cyan line
indicates the bare impurity dispersion $\varepsilon_{\mathrm{I}}(Q)$, the
green circles correspond to the many-body ground state energy
$E_{\mathrm{GS}}$ at the given $U$, while the cyan triangles denote the many-
body ground state energy $E_{\mathrm{GS}}^{0}$ at $U\\!=\\!0$; the yellow dots
in panel (c) are the energies $E_{\mathrm{RBM}}$ of the two-body repulsively
bound state. The lower energy peak in (a) and (b) at a given $Q$ corresponds
to the attractive polaron (AP) and the higher energy one to the repulsive
polaron (RP).

###  III.1 ED results

The key features of the spectral function for an impurity interacting with the
1D Fermi sea are illustrated in Fig. 2, where we plot $\mathcal{A}(Q,\omega)$
obtained with ED on a ring geometry (periodic boundary condition) with
$L\\!=\\!29$, $N\\!=\\!5$ and $J_{\mathrm{I}}\\!=\\!J$. The three panels
correspond to different impurity-bath interaction strengths, namely weak
attraction $U\\!=\\!-1.5J$ (a), strong attraction $U\\!=\\!-5J$ (b) and strong
repulsion $U\\!=\\!5J$ (c). The bare impurity dispersion
$\varepsilon_{\mathrm{I}}(Q)$ is indicated by the cyan dashed lines.

In panels (a) and (b), one notices a sharp lower branch at small momenta and a
bright upper branch at large momenta, which progressively split as one
increases the interaction strength $|U|$; we will refer to these as the
attractive (AP) and repulsive (RP) polaron branches, respectively.
Interestingly, the RP shows a negative effective mass at small momenta,
whereas the RP peak is very narrow and close to the bare dispersion for
$Q\sim\pi/a$. Instead, in panel (c), one observes that the bare impurity
dispersion is slightly blue-shifted compared to the non-interacting case, and
that it broadens at momenta $|Q|\sim\pi/2a$. Furthermore, a high-energy branch
appears (barely visible in this specific plot), which we ascribe to the
formation of a repulsively bound molecule. Indeed, its position is captured by
the energy $E_{\mathrm{RBM}}$ obtained by solving the two-body problem of one
fermion interacting with one impurity; see the yellow dots in panel (c).

In Fig. 2, we also plot the many-body ground-state energy $E_{\mathrm{GS}}(Q)$
of the coupled impurity-bath system, for each $Q$ sector and for a single
impurity particle; see the green circles. The first peculiarity concerns the
oscillatory behavior of $E_{\rm GS}(Q)$. In fact, this effect was predicted in
Ref. [134], and it is explained as follows: in the limit $L\to\infty$, Umklapp
processes can create a particle-hole excitation of momentum $2k_{F}$ across
the Fermi sea at a negligible energy cost. As a consequence,
$E_{\mathrm{GS}}(Q)$ has to be periodic in $2k_{F}$ in the thermodynamic
limit. In a finite-size system, creating such an excitation costs an energy of
order $1/L$, which lifts the degeneracy. This is also illustrated by the cyan
triangles, which represent $E^{0}_{\mathrm{GS}}(Q)$, the energy of the many-
body ground state at $U\to 0$. The second interesting feature is that the
ground state is mainly dark for momenta larger than $k_{F}$ i.e.
$\mathcal{A}(Q,E_{\mathrm{GS}})\simeq 0$. This is due to the fact that the
ground state at $|Q|>k_{F}$ consists of an impurity with small momentum plus a
few Umklapp particle-hole excitations. These states have negligible overlap
with the non-interacting impurity at $Q$, and are thus dark according to Eq.
(5).

###  III.2 Benchmarking MPS and Chevy methods

Figure 3:  Benchmarking the MPS and Chevy ansatz. Spectral function
$\mathcal{A}(Q,\omega)$ of an impurity injected in a 1D Fermi sea, as computed
using: (a) MPS, and (b) Chevy ansatz calculations. Here we set $U\\!=\\!-5J$
and considered the density $N/L\\!=\\!5/29$, for both the MPS and Chevy
calculations. The dashed cyan line indicates the bare impurity dispersion
$\varepsilon_{I}(Q)$. In panel (c) we compare the impurity spectrum at
$Q\\!=\\!0$ for ED, MPS and the Chevy ansatz. In the ED and MPS results, there
is a slight asymmetry of the AP peak that hints at the absence of quasi-
particle behavior. For the MPS calculation, we used a bath of $N\\!=\\!25$
fermions in a finite chain of length $L\\!=\\!145$, the maximum bond dimension
was fixed at 150 with an SVD cutoff $10^{-12}$ in the DMRG algorithm used to
find an approximation to $|\mathrm{FS}\rangle$, and we performed time-
evolution using TDVP in time steps of length $dt\\!=\\!0.2/J$ with an SVD
cutoff $10^{-8}$, terminating it at a time $t_{f}\\!=\\!25/J$.

We now build on these results, which are exact (although for a relatively
small system), to benchmark the calculation of the spectral function obtained
from MPS time-evolution methods and the variational Chevy ansatz, for a fixed
value of the interaction constant $U\\!=\\!-5J$ and density $N/L\\!=\\!5/29$.

First, we show the MPS results in Fig. 3(a), which are calculated using the
general procedure described in Sec. II. By comparing with the ED results, we
notice that the MPS method gives an excellent approximation to the
translationally-invariant ED calculation performed above keeping the density
$N/L$ fixed; this is further illustrated in Fig. 3(c). The discrepancy in the
spectral function can be attributed to the differing artificial broadening of
the spectral peaks introduced in the calculation of spectral functions: a
Lorentzian broadening was utilized for ED, while a Gaussian one was used for
MPS calculation. However, more importantly, the AP and RP peak positions, and
their spectral ratios are in excellent agreement for the two cases.

Fig. 3(b) shows the spectral function obtained from the Chevy ansatz

$|\Psi^{Q}\rangle=\left(\psi^{Q}\hat{d}^{\dagger}_{Q}+\sum_{kp}{}^{{}^{\prime}}\psi_{kp}^{Q}\,\hat{d}^{\dagger}_{Q-k+p}\hat{c}^{\dagger}_{k}\hat{c}_{p}\right)|\mathrm{FS}\rangle.$
(12)

Here, $\psi^{Q}$ and $\psi_{kp}^{Q}$ are the variational parameters and primed
sums indicate that the summation occurs over momenta fulfilling $k>k_{F}$ and
$p<k_{F}$. The label $Q$ indicates that - thanks to translational invariance -
each total momentum sector is treated separately and is determined by the
momentum at which the impurity is injected. The Schrödinger equations in the
Chevy subspace read

$\displaystyle E\psi^{Q}$
$\displaystyle=\left(\varepsilon_{\mathrm{I}}(Q)+\frac{UN}{L}\right)\psi^{Q}+U\sum_{kp}{}^{{}^{\prime}}\psi_{kp}^{Q},$
(13a) $\displaystyle E\psi^{Q}_{kp}$ $\displaystyle=\left(\varepsilon_{\rm
bath}(k)-\varepsilon_{\rm bath}(p)+\varepsilon_{\rm
I}({Q-k+p})+\frac{UN}{L}\right)\psi_{kp}^{Q}+$
$\displaystyle\quad+U\psi^{Q}+U\sum_{q}(\psi_{k+q,p}^{Q}-\psi_{k,p+q}^{Q}).$
(13b)

These equations are diagonalized numerically for $N\\!=\\!15$ particles and
$L\\!=\\!87$ sites, corresponding to the same density as for the ED
calculation. The spectral function is then obtained from Eq. (5).

Overall, we find that the Chevy approximation accurately describes the
spectral function of the impurity. This might appear surprising, since it is
known that the quasi-particle picture breaks down in 1D, and an orthogonality-
type catastrophe occurs [133, 65]. This is indeed confirmed in panel (c),
where the spectral function obtained from ED, MPS, and the Chevy ansatz are
plotted for $Q\\!=\\!0$. The AP peak obtained through the Chevy ansatz
exhibits a narrow Lorentzian peak corresponding to a well-defined quasi-
particle, and a small detached molecule-hole continuum around
$\omega\simeq-2J$. Instead, in the ED and MPS cases, we find an asymmetric
peak, suggesting a power law decay on the high-energy side, corresponding to a
break-down of the quasi-particle picture. The peak is, however, quite narrow
and well described by the Chevy-polaron ansatz as well as by the MPS.

From this we conclude that, even though the Chevy ansatz does not capture the
conceptually important absence of quasi-particles in 1D, from a practical
level, it does provide a very good estimation of how the impurity spectra
would appear in experiments, where extrinsic broadening of the lines is
present and where resolving the small discrepancies highlighted in panel (c)
would be very challenging.

It is also useful to compare these results with the 2D results reported in
Fig. 7 of Ref. [63], where all secondary peaks are well detached from the
Lorentzian AP peak, in both ED and Chevy calculations. Concerning the RP
branch, one can state that there is excellent agreement between the ED and
Chevy results, and that only very small differences are visible.

##  IV Impurity in a 1D chiral bath: the chiral polaron

We now address the main topic of this work: the properties of a mobile
impurity interacting with a chiral many-body environment.

Chiral modes naturally arise on the edge of 2D systems with broken time-
reversal symmetry, such as quantum Hall states or (fractional) Chern
insulators. In general, these systems are incompressible in the bulk (the
ground state is gapped on the torus), they are characterized by a quantized
Hall conductivity (i.e. a non-zero many-body Chern number), and they display
gapless chiral edge modes [138]. These systems can be realized in various
ways, e.g. by subjecting a 2D material to an external magnetic field, or by
engineering a synthetic gauge field in an ultracold atomic gas [139].

In principle, when immersing a mobile impurity in a quantum-Hall-type
background, one has to consider the dressing of the impurity by the bulk modes
as well as by the edge modes. In a quantum Hall or Chern insulator state, the
Fermi energy falls within a bulk gap, such that the bulk modes are gapped
while the edge modes are gapless. Here, we assume that the impurity is
confined on the edge of the system, such that the coupling to the bulk modes
is strongly reduced, both for energetic reasons and due to the small overlap
of bulk modes with the edge. The validity of this assumption will be analyzed
in Sec. V.

In this Section, we aim at elucidating the key properties and signatures
associated with the dressing of a mobile impurity by the low-energy chiral
edge excitations, neglecting the effects of bulk modes. To do so, we will
introduce a simple toy model describing a single mobile impurity,
characterized by a trivial quadratic dispersion, interacting with a Fermi sea
of chiral, linearly dispersing fermions. In particular, this simplified
setting reveals how the resulting dressing induces characteristic chiral
features in the impurity spectral function.

###  IV.1 Phenomenological 1D chiral model

We model the chiral edge mode with a linear spectrum
$\varepsilon_{\mathrm{bath}}(k)\\!=\\!vk$, where $v$ is the group velocity. In
a 2D quantum-Hall-type system, edge states are located within an energy gap
$\Delta_{\rm gap}$, and thus, one can assume that they have momenta
$k\in[-\Lambda/2,\Lambda/2]$, where the cutoff momentum
$\Lambda\\!=\\!\Delta_{\rm gap}/v$ also defines a typical length scale
$\ell\\!=\\!2\pi/\Lambda$. We point out that shifting this momentum interval
by a constant does not have any physical consequence, as it is a gauge choice.

In a conventional integer quantum Hall system (i.e. non-interacting electrons
subjected to a uniform magnetic field in the continuum), $\Delta_{\rm gap}$
corresponds to the cyclotron frequency $\omega_{B}\\!=\\!eB/m$ and the group
velocity is proportional to the slope of the confining potential at the edge;
here $e$ and $m$ denote the charge and mass of the electron, and $B$ is the
external magnetic field. For a sharp edge, the typical length $\ell$ is of the
order of the magnetic length $\ell_{B}\\!=\\!\sqrt{\hbar/eB}$. In the case of
a Chern insulator (e.g. non-interacting fermions defined on the Haldane
lattice model), the gap is of the order of the hopping strength, and $\ell$ is
of the order of the lattice constant; see also Sec. V.

Based on these observations, we introduce a simplified toy-model Hamiltonian

$\displaystyle\hat{H}_{\mathrm{bath}}$
$\displaystyle\equiv\hat{H}_{\mathrm{chiral}}=\sum_{k}[\varepsilon_{\mathrm{bath}}(k)-\mu]\hat{c}_{k}^{\dagger}\hat{c}_{k},$
(14a) $\displaystyle\hat{H}_{\mathrm{imp}}$
$\displaystyle=\sum_{q}\varepsilon_{\mathrm{I}}(q)\hat{d}_{q}^{\dagger}\hat{d}_{q},$
(14b) $\displaystyle\hat{H}_{\mathrm{int}}$
$\displaystyle=\frac{1}{N_{P}}\sum_{kpq}U_{kp}\hat{c}_{k}^{\dagger}\hat{c}_{p}\hat{d}^{\dagger}_{q+p-k}\hat{d}_{q},$
(14c)

where the fermionic momentum is discretized in $N_{P}$ states,
$k\\!=\\!-\frac{\Lambda}{2}+m\frac{\Lambda}{N_{P}+1}$, with
$m\\!=\\!1,...,N_{P}$, such that
$k\in\left(-\frac{\Lambda}{2},\frac{\Lambda}{2}\right)$. The dispersion of the
impurity is assumed to be parabolic
$\varepsilon_{\mathrm{I}}(q)\\!=\\!q^{2}/2M$, with mass $M$ and momentum
$q\in(-\infty,+\infty)$.

The chemical potential $\mu$ sets the number of occupied edge states in the
non-interacting ground state. Taking $\mu\\!=\\!0$ corresponds to half
filling, with $N\\!=\\!N_{P}/2$ fermionic states occupied (assuming $N_{P}$
even). The corresponding ground state, without the impurity, is a chiral Fermi
sea

$\
|\Psi_{0}\rangle\equiv|\mathrm{FS}\rangle=\prod_{k<0}\hat{c}^{\dagger}_{k}|0\rangle.$
(15)

The interaction term $\hat{H}_{\mathrm{int}}$ conserves the total momentum,
defined as

$\hat{Q}=\sum_{k}k\ \hat{c}_{k}^{\dagger}\hat{c}_{k}+\sum_{q}q\
\hat{d}_{q}^{\dagger}\hat{d}_{q}-\langle\mathrm{FS}|\sum_{k}k\
\hat{c}_{k}^{\dagger}\hat{c}_{k}|\mathrm{FS}\rangle,$ (16)

where we explicitly subtracted a constant such that the total momentum
coincides with the momentum at which the impurity is injected in the Fermi
sea. In principle, realistic interaction terms should take the wavefunctions
of the chiral edge states into account, through the coefficients $U_{kp}$ in
Eq. (14). However, since this is not necessary to capture the essential
features of the chiral polaron formation, we simply set $U_{kp}=U$, i.e. a
constant, in the rest of this Section.

###  IV.2 Chevy ansatz analysis

We calculate the single-particle spectral function of the impurity by
diagonalizing the problem in the truncated Hilbert space provided by the Chevy
ansatz, see Sec. II. This allows one to gain valuable physical insight into
the formation of the polaron through single particle-hole excitations of the
chiral bath 111In two- and three-dimensions, a UV-cutoff $\Lambda$ is required
to regularize the theory where the term
$\frac{1}{N_{P}}\sum_{q^{\prime}}{}^{{}^{\prime}}U_{q^{\prime}\\!q}\psi_{kq}^{Q}$
drops due to regularization when we write $U$ using the Lippmann-Schwinger
equation $\frac{1}{g}=\frac{\mu}{2\pi
a}-\frac{1}{V}\sum_{p}\frac{1}{\varepsilon^{c}_{p}+\varepsilon^{d}_{p}}$.
However, in one dimension, a UV cutoff is not required, and thus, we find a
polaron in one dimension even in the case of microscopic repulsive
interactions.. Momentum conservation is used to treat each total momentum
sector $Q$ independently. Here, it is particularly convenient to measure
energies in units of $Mv^{2}$ and momenta in units of $Mv$, which corresponds,
respectively, to the kinetic energy and momentum of the impurity when moving
at the group velocity of the chiral bath. Hence, this leaves us with only
three parameters: the total momentum $Q/Mv$, the interaction strength
$U/Mv^{2}$, and the cutoff momentum $\Lambda/Mv$ of the chiral modes. In
principle, one does not need a momentum cutoff in 1D, since the energy is
well-defined in the limit $\Lambda\to\infty$. We shall nevertheless retain a
finite $\Lambda$ in our numerical calculations, motivated by the fact that
such a cutoff has a physical origin in the relevant case of a 2D Chern-
insulator, where it is set by the size of the topological bulk gap. Also, we
will study the effects of particle-hole symmetry breaking, which occurs for a
bounded edge mode (i.e. for finite $\Lambda$) whenever $\mu\neq 0$.

Figure 4:  (a) Non-interacting spectrum of the system from Eq. (17), as a
function of the injected momentum $Q$ and for different values of $\delta
k\in[0,\Lambda]$. The vertical dashed lines correspond, from left to right, to
$Q_{c},Q_{s},Q_{\Lambda}$; the pink shaded area indicates the existence of
particle-hole states resonant with the bare impurity (blue curve). Inset:
density of particle-hole states resonant with the bare impurity. (b) Spectral
function $\mathcal{A}(Q,\omega)$ of an impurity in a chiral bath. The dashed
green line indicates the bare impurity dispersion (including the mean-field
shift $U/2$), while the cyan line denotes the energy $E_{\rm GS}(Q)$ of the
ground state, which becomes gradually darker for $Q>Q_{c}$. (c) Momentum
distribution
$\langle\mathrm{GS}(Q)|\hat{d}^{\dagger}_{q}\hat{d}_{q}|\mathrm{GS}(Q)\rangle$
of the impurity in the ground state, illustrating the threshold behavior at
$Q_{c}$. Different colors indicate different momentum sectors $Q$. The
parameters used are $U/Mv^{2}\\!=\\!2$, $\Lambda\\!=\\!5Mv$,
$N_{P}\\!=\\!100$.

We start by analyzing the spectrum of the non-interacting ($U=0$) impurity-
bath system, considering single particle-hole excitations in the chiral bath.
Calling $\delta k$ the momentum of such a particle-hole excitation [see the
sketch in the inset of Fig. 4(a)], the non-interacting spectrum is given by

$E^{\delta k}(Q)=\varepsilon_{\mathrm{I}}(Q-\delta
k)+\varepsilon_{\mathrm{bath}}(\delta k)=\frac{(Q-\delta k)^{2}}{2M}+v\delta
k.$ (17)

This family of curves is represented by thin gray curves in Fig. 4(a), as a
function of the momentum of the injected impurity $Q$ and parameterized by the
transferred momentum $\delta k\in[0,\Lambda]$. Geometrically, the different
(gray) parabolas share a common tangent of slope $v$ (thick black dots).

Three particularly relevant curves are highlighted by the thick blue, green
and red lines. The blue curve is the bare impurity dispersion in the absence
of particle-hole excitations, or, in other terms, at $\delta k=0$. The red
curve corresponds to the maximum momentum transfer $\delta k\\!=\\!\Lambda$,
and it intersects the bare impurity dispersion at
$Q_{\Lambda}\\!=\\!Mv+\Lambda/2$, satisfying
$E^{0}(Q_{\Lambda})\\!=\\!E^{\Lambda}(Q_{\Lambda})$. Finally, the green curve
is obtained for $\delta k\\!=\\!\Lambda/2$, and it meets the blue curve at
$Q_{s}\\!=\\!Mv+\Lambda/4$. The importance of this green curve will become
clear in the next subsection; at this stage, let us simply note that the
expression for $Q_{s}$ provided above only holds at half-filling.

At $U\\!=\\!0$, the impurity spectral function is peaked at the bare impurity
energy, which takes up all the oscillator strength. When the interaction $U$
is switched on, the non-interacting levels will hybridize and the oscillator
strength will be distributed between different many-body states. This
hybridization is enhanced by the existence of resonances at the non-
interacting level, i.e. crossings between the blue and grey curves in Fig.
4(a). This motivates us to inspect the density of single-particle-hole states
at the bare impurity energy $Q^{2}/2M$, which we denote as $\rho(Q)$; see
Appendix B for the precise definition and computation of $\rho(Q)$. This
density of states, which takes the degeneracy of $E^{\delta k}(Q)$ into
account, is plotted in the inset of Fig. 4(a). Importantly, $\rho(Q)$ is found
to be nonzero for $Q_{c}\leq Q\leq Q_{\Lambda}$, with a cusp at
$Q\\!=\\!Q_{s}$; see also the vertical dashed lines and the pink shaded region
in Fig. 4(a). Here, the critical momentum is given by the intuitive relation

$Q_{c}\\!=\\!Mv.$ (18)

Indeed, it derives from the condition $\partial_{\delta k}E^{\delta
k}(Q_{c})|_{\delta k=0}\\!=\\!0$, or equivalently,
$\partial_{Q}\varepsilon_{I}(Q_{c})\\!=\\!v$, which has a clear physical
interpretation: when the impurity is moving at momentum $Q_{c}\\!=\\!Mv$, i.e.
when the velocity of the impurity matches the group velocity of the chiral
background medium, the impurity can resonantly exchange energy-momentum with
the bath.

Building on these analytical arguments, we now move to the most important
result of this Section. In Fig. 4(b), we present the impurity spectral
function $\mathcal{A}(Q,\omega)$, obtained for an interaction strength
$U/Mv^{2}\\!=\\!2$, at half-filling ($\mu=0$); the cutoff is set to
$\Lambda\\!=\\!5Mv$. The most striking feature concerns the asymmetry of the
spectral function, which is distinguished by a pronounced broadening of the
impurity spectral peak in the range delimited by the critical momenta $Q_{c}$
and $Q_{\Lambda}$, and signals the formation of a polaron cloud. Moreover, a
splitting is observed at $Q\\!=\\!Q_{s}$, with a magnitude increasing with
$U$. As previously explained, the interval $[Q_{c},Q_{\Lambda}]$ corresponds
to the existence of particle-hole states that are nearly resonant with the
bare impurity dispersion (dashed green curve). On the contrary, when $Q<Q_{c}$
or $Q>Q_{\Lambda}$, the dressing is strongly suppressed, due to the fact that
scattering processes are non-resonant; in this regime, the spectral function
can thus be accurately determined from the (mean-field shifted) bare-impurity
dispersion in Eq. (17).

It is also insightful to study the ground state of the system as a function of
$Q$. The ground state energy is plotted as a cyan line in Fig. 4(b). For
$Q\lesssim Q_{c}$, the ground state is bright, and its energy follows the bare
impurity dispersion. In contrast, for $Q\gtrsim Q_{c}$, the ground-state
energy disperses linearly according to the chiral-bath group velocity $v$, and
the ground state becomes dark in the impurity spectral function. This linear
dispersion, which can be simply interpreted as the impurity being dragged by
the chiral bath at velocity $v$, can be obtained by minimizing the non-
interacting spectrum $E^{\delta k}(Q)$ in Eq. (17) with respect to $\delta k$.
This simple picture also explains the darkness of the ground state in the
spectral function at large $Q\gg Q_{c}$: injecting an impurity with a very
high velocity $v_{I}\gg v$ into the system has vanishing overlap with the
ground state of the coupled impurity-bath system, drifting at velocity $v$.

This picture is confirmed in Fig. 4(c), where we plot the impurity-momentum
distribution in the many-body ground state,
$\langle\mathrm{GS}(Q)|\hat{d}^{\dagger}_{q}\hat{d}_{q}|\mathrm{GS}(Q)\rangle$,
for five different values of the total momentum $Q$. One finds that the
impurity-momentum distribution has a delta peak at the injected momentum, i.e.
at $q\\!=\\!Q$, with strength decreasing to zero as $Q$ increases; this
signals a slowly vanishing overlap with the bare impurity and results in the
ground state becoming darker. In addition, for $Q>Q_{c}$, the distribution
develops a broader peak pinned at the critical momentum
$q\\!=\\!Q_{c}\\!=\\!Mv$.

We demonstrate the validity of these results beyond the Chevy approximation by
providing an ED benchmark in Appendix C.

###  IV.3 Particle-hole duality

To gain more insight into the physics occurring between the critical points
$Q_{c}$ and $Q_{\Lambda}$, and in particular around $Q_{s}$, we now analyze
the contribution of particle and hole dynamics separately. Here, it will prove
useful to consider imbalanced particle and hole phase spaces, namely, to tune
the chemical potential away from half-filling (i.e. $\nu\\!\neq\\!1/2$).

First, it is instructive to compare the impurity spectral function of a given
impurity-bath configuration with that of its _dual_ , obtained by reversing
the sign of the impurity-bath interaction $U$ and modifying the filling factor
$\nu$ according to:

$\nu\rightarrow 1-\nu,\qquad U\rightarrow-U.$ (19)

As a concrete example, Fig. 5(a) displays the impurity spectral function for a
filling factor $\nu\\!=\\!1/4$ and repulsive interactions $U\\!=\\!2Mv^{2}$,
while Fig. 5(b) shows the dual case, with $\nu\\!=\\!3/4$ and attractive
interactions $U\\!=\\!-2Mv^{2}$. In both cases, the bath cutoff is fixed to
$\Lambda\\!=\\!5Mv$. Importantly, one notices that both spectra are identical,
apart from the mean-field energy shift, which transforms according to $\nu
U\to-U(1-\nu)$. Moreover, the splitting point $Q_{s}$ is shifted to the right
with respect to the half-filling case shown in Fig. 4(b), as we will explain
below.

In order to elucidate the correspondence, or duality, between the two spectra
of Fig. 5, we introduce the particle-hole transformation
$\hat{h}_{k}=\hat{c}_{-k}^{\dagger}$, and we apply it to the Hamiltonian in
Eq. (14). Anti-commuting and redefining the fermionic momenta in the sums, the
transformed Hamiltonian becomes

$\displaystyle\hat{H}$ $\displaystyle=\sum_{k}\left(\varepsilon_{\rm
bath}(k)+\mu\right)\left(1+\hat{h}_{k}^{\dagger}\hat{h}_{k}\right)+\sum_{q}\varepsilon_{\rm
I}(q)\hat{d}_{q}^{\dagger}\hat{d}_{q}$
$\displaystyle\quad-\frac{1}{N_{P}}\left[\sum_{kpq}U_{kp}\hat{h}_{k}^{\dagger}\hat{h}_{p}\hat{d}^{\dagger}_{q+p-k}\hat{d}_{q}+\sum_{kq}U_{kk}\,\hat{d}^{\dagger}_{q}\hat{d}_{q}\right].$
(20)

This differs from the original Hamiltonian through four features: (i) the sign
of $\mu$; (ii) an irrelevant constant, $\sum_{k}(\varepsilon_{\rm
bath}(k)+\mu)$; (iii) the sign of the interaction; (iv) a shift of the
impurity energy in the last term. We remark that this last feature reduces to
a shift of $\frac{1}{N_{P}}\sum_{k}U_{kk}\\!=\\!U$ for a single impurity, if
the interaction is momentum independent. Altogether, this demonstrates that
the Hamiltonian at filling factor $\nu$ and interaction strength $U$ is
equivalent to the particle-hole transformed Hamiltonian at filling factor
$(1-\nu)$ and $-U$, plus a $U$ shift in the impurity energy. This result
provides a simple explanation as to why Fig. 5(a) and Fig. 5(b) are directly
related by an energy shift of $U$.

Figure 5:  Particle-hole duality. Spectral function $\mathcal{A}(Q,\omega)$
for: (a) $U\\!=\\!2Mv^{2}$, at filling $\nu\\!=\\!0.25$; and (b)
$U\\!=\\!-2Mv^{2}$ at filling $\nu\\!=\\!0.75$. The spectra are the same, but
for a vertical shift of $U$, reflecting the particle-hole duality. The bottom
row (c-d) shows the occupation of particle and hole states at energy
$\omega_{\rm bath}$, averaged over all the many-body eigenstates and weighted
by their oscillator strength. The white line denotes the Fermi level.

To investigate the individual roles of the bath particles and holes in the
polaron formation, we analyze the occupation
$\mathcal{P}(\omega_{\mathrm{bath}})$ of particle and hole states (labeled by
their energy $\omega_{\mathrm{bath}}\in[-v\Lambda/2,v\Lambda/2]$), averaged
over all the many-body eigenstates of the impurity-bath system and weighted by
their oscillator strength, for each $Q$ sector. The results are shown in Fig.
5(c-d), for the two cases investigated above; the corresponding formulas are
provided in Appendix B.1.

Let us start with the repulsive case, shown in panel Fig. 5(c). First of all,
we note that the dressing is weak for $Q<Q_{c}\\!=\\!Mv$, since, as previously
discussed, all the bare (single) particle-hole states are off-resonant with
respect to the injected impurity. Increasing $Q$ beyond $Q_{c}$, one finds
that the cloud of holes that dresses the impurity is broad and peaks at the
Fermi level, whereas the particle occupancy peak shifts with the excess
momentum $Q\\!-\\!Mv$, until it reaches its maximum energy at $v\Lambda/2$. In
other words, in this regime, the transferred momentum $\delta k$ is mainly
absorbed by the particle. The particle can absorb at most momentum
$(1-\nu)\Lambda$, and this saturation occurs at the injected momentum
$Q_{s}\\!=\\!Mv+(1-\nu)\Lambda/2$, fulfilling
$E^{0}(Q_{s})=E^{(1-\nu)\Lambda}(Q_{s})$; this results in the splitting shown
in panels Fig. 5(a) and (b). As one increases the injected momentum beyond
$Q_{s}$, the extra momentum goes into the hole cloud, whose occupancy peak
shifts until the lower energy bound $-v\Lambda/2$ is reached at
$Q\\!=\\!Q_{\Lambda}$. Finally, for $Q>Q_{\Lambda}$, all the single particle-
hole states are off-resonant (as already explained above), and the occupancies
are small.

While an analytical understanding remains elusive at this stage, we speculate
that the physical mechanism underlying this behavior (in particular the
asymmetry in the particle and hole behaviors) is that the impurity tends to
bind to holes when $U>0$ and $M>0$.

The whole picture is reversed in the dual case [Eq. (19)], where particle and
hole exchange roles; now the impurity correlates with the particle. As
illustrated in panel Fig. 5(d), the particle-hole duality expressed by Eq.
(20) leads to an occupancy-plot that corresponds to the mirrored image
(vertically mirrored with respect to $\omega_{\mathrm{bath}}\\!=\\!0$) of that
shown in panel Fig. 5(c).

We point out that this particle-hole duality provides a useful tool in the
analysis of other fermionic systems at half-filling, e.g. the Fermi-Hubbard
model [42].

###  IV.4 Lattice effects

Figure 6: Lattice effects. (a) Non-interacting spectrum of the impurity-bath
system as a function of the injected impurity momentum $Q$, parameterized by
the transfer momentum $\delta k\in[0,\Lambda]$. (b) Spectral function of an
impurity moving on the lattice and interacting with a chiral bath, which has a
linear dispersion characterized by its group velocity $v$. Here, we set
$\Lambda a\\!=\\!3$, $v\\!=\\!0.5J_{\mathrm{I}}a$ and the interaction strength
$U\\!=\\!J_{\mathrm{I}}$ for $N_{P}\\!=\\!100$ bath modes at half-filling.

We finally comment on the consequences of adding another length scale in the
problem, associated with the impurity. This can be analyzed, for instance, by
setting the impurity on a 1D lattice. In this case, the impurity length scale
is given by the lattice constant $a$ and its energy scale by the hopping
amplitude $J_{\mathrm{I}}$. The system Hamiltonian would still be described by
Eq. (14), albeit with an impurity dispersion
$\varepsilon_{\mathrm{I}}(q)\\!=\\!-2J_{\mathrm{I}}\cos{(qa)}$, where
$q\in(-\pi/a,\pi/a]$ is the Bloch quasi-momentum defined modulo $2\pi/a$.

The addition of a new scale has some consequences. In particular, if the group
velocity $v$ of the chiral medium is larger than the maximum group velocity
$\max(|\mathrm{d}\varepsilon_{\mathrm{I}}/\mathrm{d}q|)\\!=\\!2J_{\mathrm{I}}a$
allowed for the impurity, then there is no resonant particle-hole state nor
critical momentum $Q_{c}$; in this case, the polaron spectrum will resemble
that of the bare impurity. On the other hand, if $v<2J_{\mathrm{I}}a$, then
one would obtain two critical momenta $Q_{c_{1,2}}$, for which the impurity
group velocity is equal to $v$. These critical momenta are explicitly given by

$Q_{c_{1}}=\frac{1}{a}\sin^{-1}\left(\frac{v}{2J_{\mathrm{I}}a}\right),\quad
Q_{c_{2}}=\frac{\pi}{a}-Q_{c_{1}}.$ (21)

Hence, in this case, one would find two regions of broadening in the first
Brillouin zone, with two corresponding splitting points $Q_{s_{1,2}}$ and
cutoff points $Q_{\Lambda_{1,2}}$. Following the same reasoning as in the
previous subsection, these critical points are given by the roots of the
equation $E^{\delta k}(Q)-\varepsilon_{\mathrm{bath}}(Q)\\!=\\!0$, where
$\delta k\\!=\\!\nu\Lambda$ for splitting points and $\delta k\\!=\\!\Lambda$
for the cutoff points.

These predictions are confirmed in Fig. 6. In panel Fig. 6(a), we identify all
the critical $Q$-points based on the spectrum of the decoupled impurity-bath
system, for $v\\!=\\!0.5J_{I}a$ and $\Lambda\\!=\\!3a^{-1}$. We then show the
impurity spectral function in Fig. 6(b), for an interaction strength
$U\\!=\\!J_{\mathrm{I}}$. One clearly recognizes the main features that were
already present in the parabolic-dispersion case [see Fig. 4(b)], but we also
notice the additional broadening features due to the lattice. We note that
these additional effects extend to negative $Q$ due to the Bloch periodicity.

##  V Realization of a chiral polaron on the edge of a 2D Chern insulator

The previous Section built our understanding of the chiral polaron formation,
through the analysis of a mobile impurity interacting with a 1D chiral bath.
However, in practice, 1D chiral modes can only exist at the boundary of a
higher dimensional topological system. In the next Sections, we consider more
realistic models for the topological chiral bath, both non-interacting and
strongly-interacting.

In this Section, we consider a 2D Chern insulator (CI) of non-interacting
fermions, defined on the Haldane model, which hosts a single chiral edge mode
within the topological bulk gap. As before, we constrain the impurity to move
on the edge of the system, and we numerically calculate its spectral function
using MPS time evolution methods and Chevy ansatz. We also use these methods
to analyze the dressing of the impurity by the topological chiral edge state.

Figure 7: (a) Haldane lattice model, with $A,B$ sublattice sites denoted by
green and red circles, respectively, and nearest neighbour (NN) hoppings by
black lines. The NNN hopping between A lattice sites (red lines) and between B
lattice sites (green lines) are shown for a single plaquette; the sign of the
corresponding Peierls phase $\varphi$ is indicated by the arrows, illustrating
the breaking of time-reversal symmetry. Thick blue lines represent the NN
hopping of an impurity along the lower edge of the lattice. (b) Single-
particle Green’s function in Eq. (23), for the non-interacting Haldane model
in Eq. (22), evaluated along the edge of an infinite strip with a width of 3
unit cells; here we set $J^{\prime}\\!=\\!0.25\,J$ and $\varphi=\pi/2$. The
green dashed line is the linear approximation of the chiral mode dispersion,
with group velocity $v=4Ja/2\pi$.

###  V.1 System Hamiltonian

We consider a bath of non-interacting spinless fermions in the Haldane model
[81], and an impurity confined on the zig-zag edge of the corresponding
honeycomb lattice, as schematically shown in Fig. 7(a) for an infinite strip.

The total Hamiltonian of the system has the general form of Eq. (1), with the
impurity hopping along the edge indicated by the blue chain in Fig. 7(a), and
with contact interactions between the impurity and the surrounding bath of
fermions. The latter are defined on a honeycomb lattice, as described by the
Haldane Hamiltonian [81]

$\begin{split}\hat{H}_{\mathrm{bath}}&\equiv\hat{H}_{\mathrm{Haldane}}=-J\,\sum_{\langle\mathrm{NN}\rangle}\hat{c}_{xy\sigma}^{\dagger}\hat{c}_{x^{\prime}y^{\prime}\sigma^{\prime}}\\\
&\quad-J^{\prime}\,\sum_{\langle\langle\mathrm{NNN}\rangle\rangle}e^{\pm
i\varphi}\,\hat{c}_{xy\sigma}^{\dagger}\hat{c}_{x^{\prime}y^{\prime}\sigma}.\end{split}$
(22)

Nearest-neighbor (NN) and next-nearest-neighbor (NNN) hopping amplitudes are
denoted as $J$ and $J^{\prime}$, respectively. Here,
$\hat{c}_{xy\sigma}^{\dagger}$ ($\hat{c}_{xy\sigma}$) is the creation
(annihilation) operator that creates (destroys) a bath particle at lattice
site $\mathbf{r_{i}}=x\mathbf{a_{1}}+y\mathbf{a_{2}}$, and $\sigma=A,B$
denotes the sublattice index. The sign of the hopping phase $\varphi$ is
indicated by the arrows in Fig. 7(a). Time-reversal symmetry is explicitly
broken and a topological bulk gap opens in the spectrum of the bath provided
that $\varphi\neq 0,\pi$ and $J^{\prime}\neq 0$. In the following, we will fix
the parameters $\phi\\!=\\!\pi/2$ and $J^{\prime}\\!=\\!0.25J$. We will
generally consider a cylindrical geometry, or an infinite strip, as
represented in Fig. 7(a).

Figure 7(b) shows the imaginary part of the single-particle Green’s function
(i.e. the spectral function) of the bath fermions, when evaluated on the edge
of the system,

$\displaystyle\mathcal{A}_{\mathrm{bath}}(k,\omega)$ $\displaystyle=-2{\rm
Im}\sum_{\sigma\sigma^{\prime}}\langle
0|\hat{c}_{k0\sigma}\frac{1}{\omega-\hat{H}_{\mathrm{bath}}+i\eta}\hat{c}_{k0\sigma^{\prime}}^{\dagger}|0\rangle.$
(23)

Here $k\\!=\\!0,1,\ldots,N_{1}-1\times(2\pi/N_{1}a)$ is the quasi-momentum
along $\mathbf{a_{1}}$, $N_{1}$ is the number of unit cells along this
direction, $a\\!=\\!|\mathbf{a_{1}}|$, $\ket{0}$ is the vacuum state, and the
Fourier transform of the field is defined as

$\displaystyle\hat{c}_{ky\sigma}$
$\displaystyle=\frac{1}{\sqrt{N_{1}}}\sum_{x}e^{-ik\mathbf{r}_{i}\cdot\mathbf{a}_{1}}\hat{c}_{xy\sigma}.$
(24)

This operator removes a bath particle with crystal momentum $k$ along the
$x$-direction, at position $y$, on the sublattice $\sigma$. The spectral
function in Eq. (23) can be calculated by diagonalizing Eq. (22) in $k$-space.
The presence of a chiral mode of well-localized edge states is clearly visible
in Fig. 7(b); since the bulk gap is of the order of $4J$, the group velocity
$v$ of the edge mode is approximately $4Ja/2\pi$ (see the green dashed line).

We note that the bulk modes have a small oscillator strength, signaling a weak
overlap between bulk and edge states. Hence, when the impurity-bath
interaction is written in the basis of the bath eigenstates, the coupling to
the edge modes is substantially larger than the coupling to bulk modes; see
Appendix D for details.

We now calculate the momentum-resolved impurity spectral function
$\mathcal{A}(Q,\omega)$ for the coupled impurity-bath system. Here, the
impurity is confined to the edge of the Chern insulator: the bath consists of
a Fermi sea of non-interacting fermions, completely filling the lowest Bloch
band of the Haldane model. To do so, we use both the Chevy approach on a
honeycomb cylinder and the MPS evolution on an infinite strip. We refer to
Appendices A and D, respectively, for details on these methods.

Figure 8: Impurity spectral function in the Haldane model. (a) The non-
interacting spectrum $E^{\delta k}(Q)$ of the system, as a function of the
momentum $Q$ injected by the impurity, and parameterized by the transfer
momentum $\delta k$ used to create a single particle-hole pair (grey curves).
Momentum conservation is relaxed by the Umklapp processes, which carry
momentum $2\pi/a$, giving rise to the $E^{\delta k}_{u}(Q)$ curves (light gray
and dotted lines). Spectral function of the impurity $\mathcal{A}(Q,\omega)$
calculated using: (b) the Chevy ansatz, and (c) the MPS time evolution; here,
we set $U\\!=\\!5J$ and the impurity hopping $J_{\mathrm{I}}\\!=\\!2J$. The
green dotted curve marks the ground state of the system, which is dark for
large momenta. The parameters used in the Haldane model are:
$J^{\prime}=0.25\,J$, and $\varphi=\pi/2$; the Fermi level for the bath
particles is set at $0\,J$. For the MPS calculations, the width of the
infinite strip is 3 units cells; for the Chevy-ansatz calculations, the
cylinder has a height of 3 unit cells and a circumference of 33 unit cells.

###  V.2 Chevy ansatz and MPS results

The impurity spectral functions obtained from the Chevy ansatz and MPS methods
are compared in Fig. 8, for an interaction strength $U\\!=\\!5J$ and an
impurity hopping $J_{\mathrm{I}}\\!=\\!2J$; here, the fermionic bath is
prepared at exactly half-filling. As previously, $Q$ is the momentum injected
by the impurity into the system.

As a technical note, we remark that the spectral function in represented in
the extended zone scheme. This is motivated by the fact that experiments
typically probe the impurity dynamics as a function of momentum rather than
quasi-momentum. In the absence of coupling with the bath, the impurity hops on
the edge of the lattice, with no bias between $A$ and $B$ sites, such that the
natural unit cell for the impurity is half of that of the bath fermions. On
the zig-zag edge, the fermionic density on $A$ sites is smaller than on $B$
sites, hence the impurity momentum is only conserved modulo $2\pi/a$; this is
the analog of Umklapp processes in solids.

In Fig. 8(a), we show the non-interacting spectrum of the system for different
values of the transferred momentum $\delta k$ (gray curves) under the
approximation that the Chern-insulator bath consists of a single linearly
dispersive mode with group velocity $v\\!=\\!4Ja/2\pi$ and cutoff momentum
$\Lambda\\!=\\!2\pi/a$. These non-interacting energy lines are given by two
families of curves $E^{\delta k}(Q)\\!=\\!\varepsilon_{\mathrm{I}}(Q-\delta
k)+v\delta k$ and $E_{u}^{\delta k}(Q)\\!=\\!\varepsilon_{\mathrm{I}}(Q-\delta
k+2\pi/a)+v\delta k$, where the $E_{u}$ energies are the ones enabled by
Umklapp processes and are depicted in a lighter contrast.

The chiral features of the impurity spectral function, direct consequence of
the interaction with the Chern insulator’s chiral edge mode, are clearly
visible in Figs. 8(b) and (c). Starting at $Q\\!=\\!0$, and moving towards the
right, one observes that the impurity behaves as a bare particle until it
reaches the critical momentum
$Q_{c}=(2/a)\mathrm{sin}{}^{-1}(v/J_{\mathrm{I}}a)$, at which point the
impurity starts to resonantly exchange momentum with the bath through the
creation of particle-hole pairs. For $Q>Q_{c}$, we observe the emergence of a
dark ground state (green dots) and the broadening of the spectral lines.
Furthermore, a splitting is present at $Q_{s}$, as the curve $E^{\delta k}$
corresponding to $\delta k\\!=\\!\Lambda/2$ becomes resonant with the bare-
impurity dispersion (see previous Section); this splitting is hardly visible,
due to blurring effects associated with the coupling to the many bulk modes.
On the $Q<0$ side, the impurity behaves as a bare particle for small momenta,
until it reaches $Q\\!=\\!Q_{s}^{u}$, which corresponds to the resonance with
the Umklapp curve $E^{\delta k}_{u}$ at $\delta k\\!=\\!\Lambda/2$.

The identification of these few critical momenta $Q_{c,s}^{(u)}$ in Figs. 8
clearly demonstrates how the spectral features of the dressed impurity can be
linked to resonances in the non-interacting theory, based on a linearly
dispersing chiral mode. In particular, we find that the vertical lines
depicted in Fig. 8, which correspond to the critical momenta predicted by
neglecting the curvature of the edge-mode dispersion in the non-interacting
theory, show good agreement with the impurity spectral function of the fully
interacting impurity-bath system. It is also worth noticing that this spectral
function was calculated using a large interaction strength $U$, larger than
the bulk gap in the bath, such that bulk excitations are potentially relevant;
see also the next paragraph. Altogether, this demonstrates that the mechanism
underlying the formation of the chiral polaron, as well as its related
spectral features, are robust.

In Fig. 15 in Appendix C, we provide further details by filtering out the
contribution of the bulk modes, which have the general effect of blurring the
spectral features at large momenta; this complementary study leads to a
complete identification and classification of all the critical $Q$-points.

###  V.3 Polaron formation on the edge vs in the bulk

Figure 9: Polaron formation on the edge versus in the bulk of a Chern
insulator. The location of the injected impurity is indicated by the blue
links in panels (a) and (b), respectively. The corresponding impurity spectral
functions $\mathcal{A}(Q,\omega)$, calculated using the Chevy ansatz, are
reported in panels (c) and (d). Panels (e) and (g) show the energy-resolved
probability distribution for the creation of a bath particle (empty bars) and
hole (filled blue bars) in the interacting many-body ground state at
$Q\\!=\\!0$. Panel (f) displays the single-particle band spectrum of the
Haldane bath. Here $J_{\mathrm{I}}\\!=\\!2J$ and $U\\!=\\!5J$.

To further verify that the formation of the chiral polaron is indeed due to
the interaction of the impurity with the topological edge mode, we display the
impurity spectral function in Fig. 9 for two distinct cases: (i) when the
impurity is constrained to move along the edge [Fig. 9(c)]; and (ii) when it
is constrained to move along a path within the bulk [Fig. 9(d)]. The blue
lines in panels Fig. 9(a) and (b) signify the impurity path constraints.
Figure 9(d) clearly illustrates that the impurity spectral function is
symmetric whenever the impurity is constrained to a path within the bulk. This
confirms that the asymmetry visible in panel Fig. 9(c) originates from the
chiral edge mode. This is further illustrated in panels Fig. 9(e) and (g),
where we plot the probability of particle (empty bars) and hole (filled bars)
excitations at energy $\omega$, for the Chevy wavefunction of the ground state
at $Q\\!=\\!0$. In panel Fig. 9(f), we plot the corresponding particle (empty
circles) and hole (filled circles) bands. Figure 9(e) shows that, for an
impurity constrained to move on the edge, the polaron dressing cloud mostly
consists of low-energy excitations within the chiral branch (i.e. modes
between $\pm J$ approximately); in contrast, when confining the impurity in
the bulk, the result in Fig. 9(g) shows that low-energy excitations are highly
suppressed, resulting in a symmetrical spectral function, reminiscent of the
non-chiral bath case discussed in Sec. III.

In Appendix E.2, we also demonstrate that the chirality of the edge mode only
clearly manifests itself in the spectral function $\mathcal{A}(Q,\omega)$ if
the Fermi level of the bath lies within the topological gap, i.e. when the
bath forms a genuine Chern insulator. Indeed, when the bath is prepared in a
metallic phase, the impurity spectral function only displays weak asymmetries
and its features are difficult to predict. In this sense, one concludes that
polaron spectroscopy is sensitive to the topological nature of the surrounding
many-body medium.

##  VI Polaron on the edge of a fractional Chern insulator

As a final application of our chiral polaron framework, we study the impurity
dynamics on the edge of a fractional Chern insulator (FCI). Similar to the
case of the non-interacting Chern insulator (Haldane model) discussed in Sec.
V, we expect that the impurity couples to the chiral, linearly dispersing edge
mode of the FCI, and that the general features found in the previous Sections
extend to this strongly-correlated-bath setting. However, since the bath is
indeed an interacting, strongly correlated system, and since we do not have
any analytical access to the ground state of the bath, the Chevy method cannot
be used and we entirely resort to MPS calculations. The numerical results
presented in this Section confirm that the chiral polaron picture persists
even in the presence of strong quantum correlations in the bath.

Figure 10: Fractional Chern insulator on a strip. (a) The density profile
along the vertical direction of the strip, imprinted by the harmonic confining
potential. Note that the sum of the densities on a rung is exactly one, as
explicitly encoded in the infinite MPS representation. (b) Dispersion of the
lowest-energy neutral mode of the FCI, the color indicating the average
position along the $y$ direction with respect to the center of the system. The
bulk gap is around $0.25J$ for this system configuration (see text).

We consider the Hofstadter-Bose-Hubbard model on a square lattice, which is
known [141, 142, 143, 144] to host a stable Laughlin-type FCI ground state at
filling factor $\nu\\!=\\!1/2$, for large enough interactions. Here, the
filling factor $\nu$ refers to the filling of the lowest-energy (single-
particle) Hofstadter band; for convenience, we will consider a bath of hard-
core bosons in such a band. This Laughlin-type FCI ground state is known to
host a single chiral edge mode on its boundary [145, 98].

In our calculations, we consider an infinite strip geometry in order to have a
gapless edge mode with a continuous momentum. On the strip, the most
convenient gauge choice for the vector potential is
$\vec{A}\\!=\\!(y\varphi,0)$, such that the Hamiltonian is invariant with
respect to translations in the horizontal direction:

$\hat{H}_{\text{FCI}}=-J\sum_{xy}\left(e^{i\varphi
y}\hat{c}_{x,y}^{\dagger}\hat{c}_{x+1,y}+\mathrm{h.c.}\right)\\\
-J\sum_{xy}\left(\hat{c}_{x,y}^{\dagger}\hat{c}_{x,y+1}+\mathrm{h.c.}\right)-\sum_{xy}V(y)\,\hat{n}_{xy}\;.$
(25)

with $\hat{n}_{xy}\\!=\\!\hat{c}_{xy}^{\dagger}\hat{c}_{xy}$. The $x$ label
runs over the infinite direction of the strip, whereas
$y\\!=\\!0,\dots,N_{y}-1$; in the following we consider a strip of thickness
$N_{y}\\!=\\!11$. We set the magnetic flux per plaquette at the value
$\varphi\\!=\\!\pi/2$, and impose that there is, on average, one boson per
rung in the ground state resulting in a filling $\nu\\!=\\!1/2$ of the lowest
Hofstadter band.

It is well-known in the literature that a suitable confining potential is
crucial for observing a linearly dispersing chiral edge mode in such FQH
systems [146, 145, 147, 148, 98]. In particular, we expect that a hard-wall
confinement will gap out the edge mode [147, 148, 149, 98]. Instead, we use a
harmonic potential of the form $V(y)=\mu(y-y_{0})^{2}$, where $y_{0}$ is the
center of the strip. We use the value $\mu/J\\!=\\!0.01$ for the strength of
the potential, which was shown in Ref. [98] to yield a clear signature of the
chiral branch.

Figure 11:  Impurity spectral function for an impurity injected in a
fractional Chern insulator. The impurity is constrained to move: (a) on the
edge ($y\\!=\\!2$); (b) in the bulk ($y\\!=\\!5$). The spectral function is
computed by MPS time evolution methods on the infinite strip geometry. Here,
we used $J_{\mathrm{I}}\\!=\\!0.05J$ and $U\\!=\\!J$. The vertical line
depicts the critical momentum $Q_{c}\\!=\\!Mv$.

Given this infinite-strip geometry with harmonic potential, we optimize an MPS
approximation for the ground state. In Fig. 10(a), we observe a clear plateau
of the boson density $\braket{\hat{n}_{xy}}$ around the value $n=1/8$ (dashed
black line). We have thus realized an FCI “droplet” on this infinite strip
geometry.

Next, using a variational MPS approach for the low-lying excited states in 2D
correlated systems [150], we obtain an estimate of the different excitation
energies in this system. First of all, by placing the system on an infinite
cylinder with a circumference of eight sites, we find that the bulk gap is
about $\Delta_{\text{bulk}}\approx 0.25J$. Secondly, on the strip geometry
with harmonic potential, we find an upper bound for the edge gap
$\Delta_{\text{edge}}<0.01J$, and an effective edge velocity $v\approx 0.06J$.

The localization of the edge mode can be further diagnosed by computing the
spectral weight of the low-energy excited state as

$s_{y}(k_{x})=\bra{\Phi_{k_{x}}}\sum_{x}e^{ik_{x}x}(\hat{n}_{xy}-\braket{\hat{n}_{xy}})\ket{\Psi_{0}},$
(26)

where $\ket{\Psi_{0}}$ is the ground state in the bath, and
$\ket{\Phi_{k_{x}}}$ is the excited state wavefunction with momentum $k_{x}$,
both in their variational MPS representation. From this quantity, one can
estimate the average position of the edge mode as

$\langle
y\rangle\\!=\\!\sum_{y}y\left|s_{y}\right|^{2}/\sum_{y}\left|s_{y}\right|^{2}.$
(27)

The dispersion and the average position of the edge mode are plotted in Fig.
10(b). We find an approximately linear dispersion for a significant portion of
the Brillouin zone, where the edge mode is mostly located between sites
$y\\!=\\!2,3$ for positive momenta; for negative momenta, the edge mode is
localized at the other side of the strip. Around momentum zero, the dispersion
bends off and the edge localization disappears. We attribute this behavior
around the origin either to a coupling of the edge modes across the strip, to
the finite trapping frequency, or to artifacts in the variational MPS
calculations. We leave a more quantitative analysis of the FCI edge spectrum
on an infinite strip geometry for future work.

Finally, we immerse a mobile impurity into the FCI droplet, which we restrict
to move only horizontally on one row of the strip with hopping strength
$J_{I}$. The total Hamiltonian of the system is given by

$\hat{H}=\hat{H}_{\text{FCI}}-J_{I}\sum_{x}\left(\hat{d}_{x,y}^{\dagger}\hat{d}_{x+1,y}+\mathrm{h.c.}\right)\\\
+U\sum_{x}\left(\hat{c}_{x,y}^{\dagger}\hat{c}_{x,y}\right)\left(\hat{d}_{x,y}^{\dagger}\hat{d}_{x,y}\right)$
(28)

where $y$ denotes the location of the impurity along the $y$-axis on the
strip. We take the parameters $J_{I}\\!=\\!0.05J$ and $U\\!=\\!J$, which are
small enough to expect that the impurity mainly couples to the edge mode,
provided that $y$ is chosen to be on the edge of the system. We again use
time-dependent MPS methods to compute the polaron spectral function, and the
results are shown in Fig. 11 for two choices of the impurity location.

In line with the results of the previous Sections, one can retrieve the key
characteristics of the chiral polaron in Fig. 11(a), which corresponds to the
case of an impurity injected on the edge of the FCI strip. First, we observe
that the spectral weight becomes asymmetric due to the coupling to the chiral
edge mode. Second, we observe a splitting of the spectral function around the
critical momentum $Q_{c}\\!=\\!Mv$, at which the group velocity of the
impurity equals the velocity of the FCI edge mode. In contrast, when the
impurity is constrained to move within the bulk of the system, one does not
find such a chiral response and the spectral function is symmetric; see Fig.
11(b). This demonstrates that the chiral polaron characteristics observed in
Fig. 11(a) indeed result from the coupling of the impurity to the chiral mode
located on the edge of the FCI droplet.

This study illustrates how polaron spectroscopy can be used to probe and
characterize the edge-mode structure of strongly-correlated topological
matter.

##  VII Conclusions and outlook

This work introduced the chiral polaron, a mobile impurity dressed by a bath
of chiral fermions. We hereby summarize our main results and list a series of
open questions and perspectives.

First, we have provided a complete calculation of the spectral properties of
1D Fermi polarons on a non-chiral chain, benchmarking the accuracy of the
Chevy-ansatz and MPS approaches against exact diagonalization. Then, we have
introduced an effective 1D model for the chiral polaron, revealing its unique
spectral properties inherited from the chirality of the bath. In particular,
the impurity propagation was shown to be strongly modified whenever its group
velocity becomes comparable to the one of the chiral modes. The ground state,
corresponding to the impurity being dragged by the chiral bath, then becomes
dark. The role of particle and hole processes was analyzed, and a duality
relation was established. Building on this strong theoretical understanding,
we considered a realistic 2D Chern-insulator setting provided by the Haldane
model, which displays a chiral edge mode propagating along the boundary of the
system. In this setting, the impurity spectrum inherits the chirality of the
edge mode, in spite of contributions stemming from the bulk modes, and
provided that the impurity is confined on the edge of the system. Importantly,
these signatures of the chiral polaron disappear when the impurity is injected
in the bulk of the system, or when the bath is prepared in a metallic phase.
In this sense, the chiral polaron can be used as a practical probe of the
topological chiral edge modes characterizing topological insulating states of
matter. Finally, we used MPS methods to reveal the formation of a chiral
polaron on the edge of a fractional Chern insulator, a strongly-correlated
topological state of matter. We remark that the adaptation of advanced tensor
network methods to the study of polaron formation constitutes an important
outcome of this work.

Our results are relevant to cold-atom experiments, where the momentum-resolved
polaron spectral function can be measured through Raman injection spectroscopy
[19, 106], and where important efforts are devoted to the realization of
topological insulating states of matter [151], including fractional Chern
insulator states [152]. Moreover, the chiral dressing of excitons or electrons
could also be relevant in recent solid-state realizations of (fractional)
Chern insulators [153, 154, 155, 156, 82, 157], even though the momentum-
resolved spectral function might not be easily accessible in these settings.

On the theoretical side, this work leaves a number of outstanding questions
and possible extensions. These concern the detailed mechanism underlying
chiral polaron formation in fractional Chern insulators; the role of
interactions among the constituents of the bath; and the case of multiple
chiral edge modes in non-Abelian fractional Chern insulators. Other intriguing
directions of research include the addition of magnetic impurities in spin
Hall insulators; the possibility of generating effective polaron-polaron
interactions mediated by the chiral bath [158]; and the emergence of novel
many-body phases at the edges or surfaces of topological insulators.

## Acknowledgements

We are grateful to Leonardo Mazza, Alberto Nardin, Botao Wang, Nader Mostaan,
Cécile Repellin and Fabian Grusdt for stimulating discussions. Work in
Brussels is supported by the ERC Grant LATIS, the EOS project CHEQS and the
Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS). Computational
resources have been provided by the Consortium des Équipements de Calcul
Intensif (CÉCI), funded by the Fonds de la Recherche Scientifique de Belgique
(F.R.S.-FNRS) under Grant No. 2.5020.11 and by the Walloon Region. O. K. D
acknowledges financial support from the NSF through a grant for ITAMP (Award
No: 2116679) at Harvard University. GMB acknowledges the Danish National
Research Foundation through the Center of Excellence CCQ (Grant no. DNRF156).

##  Appendix A MPS simulations

In this Appendix, we give the technical details for our numerical MPS
simulations.

###  A.1 Setups and initial states

In this work we have considered models on different lattices and with
different boundary conditions: a purely 1D chain, a hexagonal lattice on an
infinite 2D strip and a square lattice on an infinite cylinder and strip. In
the latter cases, we map the 2D geometry to an effective 1D one by winding
around the smallest dimension. As a result, we end up with an infinite 1D
model with an $N$-site unit cell, but the mapping to an effective 1D geometry
induces long-range terms in the Hamiltonian. Besides the geometry, we specify
the local Hilbert space. On each site, we have a spinless fermionic or bosonic
degree of freedom corresponding to the bath. In addition, we enlarge the local
space by an impurity degree of freedom – no need to specify the statistics for
the impurity, because there is always at most one particle in the system. For
the 2D geometries, the impurity is restricted to the sites on a 1D slice of
the system (typically, the edge). The different geometries and local spaces
are represented graphically in Fig. 12.

For these effective 1D models, we can now propose MPS ansätze for
approximating the ground state. In the finite cases, we use finite MPS with a
different tensor on each site, whereas the infinite strip requires the use of
a translationally invariant MPS, repeating itself within each $N$-site unit
cell. We use the DMRG algorithm [111, 112] to variationally optimize the
tensors in the finite MPS, whereas we use the VUMPS algorithm [115] to
optimize the tensors in the unit cell of the infinite MPS. In all cases we
exploit the internal symmetries of the model, namely:

  * •

$\mathrm{U}(1)_{\rm bath}$: charge conservation of the fermionic bath

  * •

$\mathrm{U}(1)_{I}$: charge conservation of the impurity

The $\mathrm{U}(1)_{\rm bath}$ symmetry is used to impose the filling of the
bath particles. For the finite case, we impose the total number of bath
particles by specifying the total charge of all the MPS tensors. In the
infinite case, we impose the number of bath particles per unit cell.
Initially, there are no impurities in the system, so all MPS tensors only have
non-zero entries in trivial sectors of the $\mathrm{U}(1)_{I}$ symmetry.

Figure 12: The different geometries considered in this work: a 1D chain, the
hexagon lattice on an infinite strip, and the square lattice on an infinite
strip. Dotted lines correspond to physical hopping links, while the solid bond
with an arrow indicates the 1D MPS representation, winding along the 2D
lattice; notice that the two external (physical) legs on the first line denote
the presence of the impurity degrees of freedom. The last rows correspond to
an infinite MPS.

###  A.2 Impurity spectral function through time evolution with MPS

In this work we use time dependent MPS methods. First we optimize a ground
state MPS for the bath $\ket{\Psi_{0}}$ with ground state energy
$E^{0}_{\mathrm{bath}}$, apply a local impurity creation operator and perform
time evolution of the interacting Hamiltonian, to obtained the evolved
perturbed state

$\ket{\Psi_{j}(t)}=e^{-i(\hat{H}-E^{0}_{\mathrm{bath}})t}\hat{d}_{j}^{\dagger}\ket{\Psi_{0}}.$
(29)

After each time step, we can compute the correlator straightforwardly

$C(j,j^{\prime},t)=\bra{\Psi_{0}}\hat{d}_{j^{\prime}}\ket{\Psi_{j}(t)}.$ (30)

as a function of $j^{\prime}$. In order to resolve the spectral function, we
can first transform to momentum space

$\mathcal{A}_{j}(Q,t)=\sum_{j^{\prime}}e^{iQ(j-j^{\prime})}C(j,j^{\prime},t).$
(31)

One can numerically evolve up to time $T$, which is in practice limited by the
bond dimension reachable with the available computational resources, since
entanglement increases in time. Therefore, the Fourier transformation to
frequency space requires us to introduce a window function

$\mathcal{A}_{j}(Q,\omega)=\int_{-T}^{T}\mathcal{A}_{j}(Q,t)e^{i\omega
t}\exp\left(-\frac{t^{2}}{\alpha T^{2}}\right)$ (32)

where the parameter $\alpha$ can be tuned to give an optimal balance between
the final frequency resolution and spurious oscillations.

###  A.3 Time evolution in infinite systems

The idea is that we start from the ground state of the infinite system, and
approximate the time-evolved state [Eq. 29] as a window of time-dependent MPS
tensors around site $j$, embedded in the infinite MPS to the left and to the
right of the window:

$\ket{\Psi_{j}(t)}\approx\ket{\Psi_{\text{wMPS}}(\\{A^{l}_{i}\\},\\{A^{r}_{i}\\},\\{X_{i}(t)\\})}$
(33)

with

$\ket{\Psi_{\text{wMPS}}(\\{A^{l}_{i}\\},\\{A^{r}_{i}\\},\\{X_{i}(t)\\})}=\\\
\;\vbox{\hbox{\includegraphics[scale={0.5},page=6]{./Figures/mps_diagrams.pdf}}}\;$
(34)

Only the tensors $\\{X_{1},\dots,X_{M}\\}$ are taken to be time-dependent. In
each time step, given a set of window tensors $X_{i}(t)$ we can determine the
tensors $X_{i}(t+dt)$ variationally by optimizing the squared fidelity

$\Big{|}\hat{U}(dt)\ket{\Psi_{\text{wMPS}}(\\{A^{l}_{i}\\},\\{A^{r}_{i}\\},\\{X_{i}(t)\\})}-\\\
\ket{\Psi_{\text{wMPS}}(\\{A^{l}_{i}\\},\\{A^{r}_{i}\\},\\{X_{i}(t+dt)\\})}\Big{|}^{2}.$
(35)

Here, $\hat{U}(dt)=e^{-i(\hat{H}-E^{0}_{\mathrm{bath}})dt}$ is approximated by
a matrix product operator (MPO) [121, 122]. Since the window tensors are a
string of MPS tensors, a simple sweeping algorithm works very well for
optimizing this fidelity in each time step.

We can dynamically grow the size of the time-dependent window in order to
capture the spreading of the perturbation as time evolves. In the current
context of the impurity problem, we can simply monitor the probability that
the impurity has reached the edge of the window, and increase the window size
when this probability has reached a certain threshold value.

A crucial advantage of this setup is that we can exploit translation
invariance. Indeed, the site $j$ around which the window-MPS is centered is
arbitrary, implying that we can just translate the full window-MPS to obtain
the time-evolved state centered around site $j^{\prime}$,

$\ket{\Psi_{j^{\prime}}(t)}=\hat{T}^{j^{\prime}-j}\ket{\Psi_{j}(t)}.$ (36)

Using this property, we can split the time evolution operator in the time-
dependent correlation function:

$C(j,j^{\prime},t)=\braket{\Psi_{j^{\prime}}(-t/2)}{\Psi_{j}(t/2)}.$ (37)

In practice, we can run two parallel simulations for the forward and backward
time evolutions, and compute the shifted overlaps to obtain the correlator.
This trick allows us to reach twice the total simulation time with the same
resources.

###  A.4 Impurity dynamics with long-range hopping

The standard time-evolution MPS algorithms struggle to make the impurity hop,
because most bond expansion methods rely on two-site updates.

To make the problem transparent, let us explicitly keep track of the
$\mathrm{U}(1)_{I}$ charge on the different legs in the time evolved MPS. In
the ground state there are no impurity particles, so on the virtual level of
the MPS tensors there are only contributions with a zero $\mathrm{U}(1)_{I}$
charge. When we insert the impurity particle at site $j$, we therefore obtain
the initial MPS

$\;\vbox{\hbox{\includegraphics[scale={0.5},page=1]{./Figures/mps_diagrams.pdf}}}\;.$
(38)

We have explicitly denoted the local charge sectors $(0,1)$ on the physical
legs where the impurity is allowed to hop, whereas on the other sites we only
have the $0$ sector. After a certain time, the impurity particle should be
able to move to other sites $j^{\prime}$ in the lattice. This means that the
time evolved MPS will contain contributions of the form

(39)

We need to transfer the non-trivial charge through the virtual level of the
MPS from site $j$ (where the non-trivial charge is encoded in the external
leg) to site $j^{\prime}$ (where the physical particle resides). As time
progresses, the particle travels further through the system, so that the MPS
tensors need to be extended progressively with non-trivial $\mathrm{U}(1)_{I}$
sectors. A two-site scheme, however, will never be able to generate these non-
trivial sectors. If one would try to generate such a non-trivial charge sector
on the virtual level between sites $j-1$ and $j$ through a two-site update,
one would perform the following decomposition

$\;\vbox{\hbox{\includegraphics[scale={0.5},page=4]{./Figures/mps_diagrams.pdf}}}\;=\;\vbox{\hbox{\includegraphics[scale={0.5},page=5]{./Figures/mps_diagrams.pdf}}}\;.$
(40)

One immediately sees that a non-trivial charge sector cannot be supported
through a two-site update, because the physical space on site $j-1$ does not
support an impurity particle. Simple bond expansion schemes based on two-site
updates will therefore not work for simulating the impurity dynamics in this
context.

However, since we know exactly which sectors we need to add to the virtual
level of the MPS, there is an easy fix for this problem: instead of using a
two-site scheme, we just initialize the MPS tensors within a certain window
around site $j$ with these additional charge sectors, and perform a
variational one-site scheme over the extended MPS tensors. Put differently, we
just add the right charge sectors in the MPS tensors by hand, and use a
variational scheme to optimize the enlarged MPS tensors.

##  Appendix B Density of single particle-hole states

Figure 13:  Density of states $\rho(Q,Q^{2}/2M)$ containing one particle-hole
excitation in the $Q$ momentum sector and at kinetic energy resonant with the
non-interacting impurity at momentum $Q$. The bath cutoff $\Lambda$ determines
the support as well as the kinks in $\rho$.

The density of states in the one particle-hole subspace at a given bare energy
$\omega$ and in the $Q$ momentum sector is given by

$\rho(Q,\omega)=\int dk\ n(k)\
\delta\left(\omega-\frac{(Q-k)^{2}}{2M}-vk\right),$ (41)

where we introduced the density $n(k)$ of particle-hole pairs with momentum
$k$. In other words, $k$ is the momentum which is absorbed by the bath via
creation of a particle-hole pair. More precisely,

$n(k)=\frac{1}{Z}\times\begin{cases}k/\Lambda_{1}&{\rm for\
}k\in[0,\Lambda_{1}]\\\ 1&{\rm for\ }k\in[\Lambda_{1},\Lambda_{2}]\\\
\frac{\Lambda-k}{\Lambda-\Lambda_{2}}&{\rm for\
}k\in[\Lambda_{2},\Lambda]\end{cases},$ (42)

where
$\Lambda_{1}=\min(\nu\Lambda,(1-\nu)\Lambda),\Lambda_{2}=\max(\nu\Lambda,(1-\nu)\Lambda)$,
and $Z=(\Lambda+\Lambda_{2}-\Lambda_{1})/2$ imposes the normalization $\int
dk\ n(k)=1$.

Evaluated at the bare impurity energy, the density of states reads

$\rho\left(Q,\frac{Q^{2}}{2M}\right)=\frac{n(2Q-2Mv)}{Q-Mv},$ (43)

and it is shown in Fig. 13 for $\Lambda=4Mv$ and $|\nu-1/2|=1/4$. The density
of states is zero for $Q\leq Q_{c}=Mv$ or $Q\geq Q_{\Lambda}=Mv+\Lambda/2$,
while it displays two kinks at $Q_{1}=Mv+\Lambda_{1}/2$ and
$Q_{2}=Mv+\Lambda_{2}/2$. Notice that the $Q_{s}$ splitting point defined in
the main text is equal to $Q_{2}$ for $U>0,\nu<1/2$ or $U<0,\nu>1/2$, and to
$Q_{1}$ in the other cases.

###  B.1 Occupation of particle-hole states

The energy resolved particle-hole occupations reported in Fig. 5(c,d) are
defined in terms of the Chevy expansion coefficients as follows:

$\mathcal{P}(Q,\omega_{\rm bath})=\mathcal{P}_{e}(Q,\omega_{\rm
bath})+\mathcal{P}_{h}(Q,\omega_{\rm bath}),$ (44)

with

$\mathcal{P}_{e}(Q,\omega_{\rm
bath})=\sum_{nkp}|\psi^{Qn}|^{2}|\psi^{Qn}_{kp}|^{2}\delta(\varepsilon_{\rm
bath}(k)-\omega_{\rm bath}),$ (45) $\mathcal{P}_{h}(Q,\omega_{\rm
bath})=\sum_{nkp}|\psi^{Qn}|^{2}|\psi^{Qn}_{kp}|^{2}\delta(\varepsilon_{\rm
bath}(p)-\omega_{\rm bath}),$ (46)

where $n$ labels the many-body eigenstates and $|\psi^{Qn}|^{2}$ expresses the
fact that we are weighting them by their oscillator strength. The idea behind
$P$ is to identify which particle and hole states contribute the most to the
bright lines in the impurity spectral function.

##  Appendix C 1D chiral polaron: ED benchmark

Figure 14: (a) Spectral function of an impurity, with a parabolic dispersion,
interacting with a 1D chiral bath calculated using ED for interaction strength
$U=Mv^{2}$ and $\Lambda=4Mv$ at half-filling with the number of bath modes
$N_{P}=20$. Cyan solid curve marks the ground state energy $E_{\mathrm{GS}}$.
(b) Comparison of spectral function calculated using ED (solid curves) and
Chevy ansatz (dashed curves) for the pivotal $Q$-points indicated by dotted
gray lines in panel (a) and $Q=0$.

In this appendix, we benchmark the calculation of impurity spectral function
$A(Q,\omega)$ with a parabolic dispersion in Sec.IV by exactly diagonalizing
the Hamiltonian of the system (Eq.14) in the full Hibert space using Krylov
subspace methods. Since the chiral dispersion needs to be represented in a
momentum space basis, the interaction term in the Hamiltonian is much denser
than the very sparse real space matrices used for the conventional Fermi
polaron in Section III which highly restricts the accessible Hilbert space
sizes. It includes all the higher order processes as opposed to including only
one particle-hole pair creation process used in Chevy ansatz. In Fig. 14(a) we
present results for $N=10$ fermions at half filling, meaning $N_{P}=20$ with
the interaction strength $U=Mv^{2}$ and bath momentum cutoff $\Lambda=4Mv$. In
panel (b) we can clearly see a strong agreement between the predictions of the
Chevy ansatz and the results obtained from exact diagonalization in the
spectral peak positions apart from the minute finite size effects in the ED
calculation. Here, we once again see that the influence of higher order
particle-hole pair creation processes on the impurity spectral function is
very minimal and justifies our use of Chevy ansatz and kinematic arguments
based on three-particle elastic scattering to explain the physical origins of
the pivotal $Q$-points in the impurity spectrum.

##  Appendix D Details of the Chevy ansatz calculation in the Haldane bath
case

The Hamiltonian of the system has the general form of Eq. 1 where the bath
Hamiltonian is given by Eq. (22) and for an impurity constrained to hop along
the edge (blue lines in Fig. 7(a)) of the honeycomb lattice, its Hamiltonian
reads

$\hat{H}_{\mathrm{imp}}=-J_{\mathrm{I}}\sum_{x}\bigg{[}\left(\hat{d}_{x0A}^{\dagger}+\hat{d}_{x+1,0,A}^{\dagger}\right)\hat{d}_{x0B}+\\\
+\hat{d}_{x0B}^{\dagger}\left(\hat{d}_{x0A}+\hat{d}_{x+1,0,A}\right)-2\,\sum_{\sigma}\hat{d}^{\dagger}_{x0\sigma}\hat{d}^{\dagger}_{x0\sigma}\bigg{]},$
(47)

where $\hat{d}_{xy\sigma}^{\dagger}$ is the impurity creation operator at
lattice position
$\mathbf{r_{i}}=x\mathbf{a_{1}}+y\mathbf{a_{2}}+\mathbf{\delta}^{\sigma}$,
with $x\in\\{0,N_{1}-1\\}$ and $y$ constrained to zero, and $J_{\mathrm{I}}$
is the impurity hopping strength. The third term is added to set the energy
minimum of the impurity dispersion at zero. This is obtained by going to the
quasi-momentum representation

$\displaystyle\hat{H}_{\mathrm{imp}}$
$\displaystyle=-2J_{\mathrm{I}}\sum_{k}\left[\cos{\left(\frac{k}{2}\right)}-1\right]\bigg{[}\hat{d}_{k0A}^{\dagger}\hat{d}_{k0B}+\hat{d}_{k0B}^{\dagger}\hat{d}_{k0A}\bigg{]}.$
(48)

where
$\hat{d}_{ky\sigma}=\frac{1}{\sqrt{N_{1}}}\sum_{x}e^{-ik\mathbf{r}_{i}\cdot\mathbf{a}_{1}}\hat{d}_{xy\sigma}$
and $q\in(-\pi/a,\pi/a]$ is the quasi-momentum. This yields two bands of
energy

$\displaystyle\varepsilon_{\mathrm{I}}^{\nu}(q)$
$\displaystyle=-2J_{\mathrm{I}}\bigg{[}\nu\cos\left(\frac{qa}{2}\right)-1\bigg{]},$
(49)

where $\nu=\pm$ is the band label. Notice that, for a non-interacting
impurity, the folding of the cosine impurity dispersion in the halved
Brillouin zone is purely formal, having distinguished the equivalent
sublattices $A$ and $B$. However, when the interaction with the bath is turned
on, it will be clear that $A$ and $B$ sites are inequivalent, e.g. they have
different fermionic density. In the following, it is convenient to keep the
indices $\sigma$, instead of using the representation with $\nu$. The
impurity-fermion interaction is contact-like and, keeping into account the
confinement of the impurity on $y=0$ sites, it reads

$\displaystyle\hat{H}_{\mathrm{int}}$
$\displaystyle=U\sum_{x\sigma}\hat{c}_{x0\sigma}^{\dagger}\hat{c}_{x0\sigma}\hat{d}_{x0\sigma}^{\dagger}\hat{d}_{x0\sigma}.$
(50)

The Chevy approach, which we detail below, is very similar in spirit to the
one explained in the main text for the 1D Fermi polaron. The only nontrivial
technical aspect is the implementation of Bloch theorem, which requires
dealing with band and quasi-momentum indices. The Haldane Hamiltonian on a
cylinder can be diagonalized in quasi-momentum space to yield
$\hat{H}_{\mathrm{Haldane}}=\sum_{k\alpha}\varepsilon_{\mathrm{bath}}^{\alpha}(k)\,\hat{c}_{k\alpha}^{\dagger}\hat{c}_{k\alpha}$,
where $\alpha=0,1\ldots,2N_{2}-1$ is the band index and
$\hat{c}_{k\alpha}^{\dagger}=\sum_{y\sigma}\phi^{k}_{\alpha,y\sigma}\hat{c}_{ky\sigma}^{\dagger}$
creates a fermion in the eigenstate of energy
$\varepsilon_{\mathrm{bath}}^{\alpha}(k)$ and wavefunction
$\phi^{k}_{\alpha,y\sigma}$. The ground state of the bath is the Fermi sea
state

$|\Psi_{0}\rangle\equiv|\mathrm{FS}\rangle=\prod_{\varepsilon_{\mathrm{bath}}^{\alpha}(k)<\epsilon_{\mathrm{F}}}\hat{c}^{\dagger}_{k\alpha}|0\rangle,$
(51)

where $\epsilon_{\rm F}$ is the Fermi energy. In this basis, the impurity-bath
interaction reads

$\displaystyle\hat{H}_{\mathrm{int}}$
$\displaystyle=\sum_{\begin{subarray}{c}kk^{\prime}q,\\\
\alpha\beta\sigma\end{subarray}}U_{kk^{\prime}}^{\alpha\beta\sigma}\,\hat{c}_{k\alpha}^{\dagger}\hat{c}_{k^{\prime}\beta}\,\hat{d}_{q0\sigma}^{\dagger}\hat{d}_{q+k-k^{\prime},0,\sigma},$
(52)

where the interaction matrix elements are defined as

$\displaystyle U_{kk^{\prime}}^{\alpha\beta\sigma}$
$\displaystyle=\frac{U}{N_{1}}(\phi_{\alpha,0\sigma}^{k})^{\star}\
\phi_{\beta,0\sigma}^{k^{\prime}}.$ (53)

In other words, the interaction matrix elements are determined by the
localization of the Haldane eigenstates on the edge. One can now project the
Hamiltonian on the truncated Hilbert space of single particle-hole pair
excitations in the symmetry sector of injection quasi-momentum $q$, where the
states take the Chevy ansatz form

$\displaystyle|\Psi^{q}\rangle$
$\displaystyle=\Bigg{(}\sum_{\sigma}\psi_{\sigma}^{q}\,\hat{d}_{q0\sigma}^{\dagger}+$
$\displaystyle\quad+\sum_{kk^{\prime}\sigma}\sum_{\begin{subarray}{c}\alpha\notin\mathrm{FS},\\\
\beta\in\mathrm{FS}\end{subarray}}\psi_{kk^{\prime}\alpha\beta\sigma}^{q}\,\hat{d}_{q+k^{\prime}-k,0,\sigma}^{\dagger}\hat{c}_{k\alpha}^{\dagger}\hat{c}_{k^{\prime}\beta}\Bigg{)}|\mathrm{FS}\rangle.$
(54)

The Schrödinger equations in this subspace read

$\displaystyle E\psi_{kk^{\prime}\alpha\beta\sigma}^{q}$
$\displaystyle=\left[\varepsilon_{\mathrm{bath}}^{\alpha}(k)-\varepsilon_{\mathrm{bath}}^{\beta}(k^{\prime})+E^{\sigma}_{\rm
MF}\right]\psi_{kk^{\prime}\alpha\beta\sigma}^{q}$
$\displaystyle\quad-2J_{I}\cos\left(\frac{q-k+k^{\prime}}{2}\right)\bigg{[}\delta_{\sigma,A}\psi_{kk^{\prime}B\alpha\beta}^{q}$
$\displaystyle\quad+\delta_{\sigma
B}\psi_{kk^{\prime}A\alpha\beta}^{q}\bigg{]}+\sum_{m\alpha^{\prime}}\,U_{km}^{\alpha\alpha^{\prime}\sigma}\psi_{mk^{\prime}\alpha^{\prime}\beta\sigma}^{q}\,$
$\displaystyle\quad-\sum_{m^{\prime}\beta^{\prime}}U_{m^{\prime}k^{\prime}}^{\beta^{\prime}\beta\sigma}\psi_{k,m^{\prime}\alpha\beta^{\prime}\sigma}^{q}+U_{kk^{\prime}}^{\alpha\beta\sigma}\,\psi_{\sigma}^{q}$
(55a) $\displaystyle E\psi_{\sigma}^{q}$
$\displaystyle=E^{\sigma}_{\mathrm{MF}}\psi_{\sigma}^{q}-2J_{I}\cos\left(\frac{q}{2}\right)\Big{[}\psi_{B}^{q}\delta_{\sigma,A}+\psi_{A}^{q}\delta_{\sigma,B}\Big{]}$
$\displaystyle+\sum_{kk^{\prime}\alpha\beta}\quad
U_{k^{\prime}k}^{\beta\alpha\sigma}\psi_{kk^{\prime}\alpha\beta\sigma}^{q},$
(55b)

where crucially the Hartree term $E^{\sigma}_{\rm
MF}=\sum_{m\gamma}\,U_{mm}^{\gamma\gamma\sigma}$ depends on the sublattice
$\sigma$, since at the edge of the system the density is not uniform on $A$
and $B$ sites. This term can induce Umklapp scattering, which results in the
small gaps visible at $Q=\pm\pi/a$ in the spectra of Fig. 8, as well as in the
emergence of the $Q_{s}^{u}$ point. Note that we have subtracted the energy of
Fermi sea
$E_{\mathrm{bath}}^{0}=\sum_{k,\beta\in\mathrm{FS}}\varepsilon_{\mathrm{bath}}^{\beta}(k)$.

Figure 15: (a) Non-interacting spectrum of the system $E^{\delta k}(Q)$
calculated using the effective chiral model with the linear bath spectrum
momentum cutoff $\Lambda=\pi/a$ and group velocity $v=4Ja/2\pi$. The Umklapp
scattering enabled spectrum $E^{\delta k}_{u}(Q)$ is depicted in lighter
shade. (b) Impurity spectral function calculated using Chevy ansatz and
neglecting all the interaction processes which lead to creation of particles
or holes with $|\omega_{\mathrm{bath}}|>J$ (upper right inset). Here
$J_{\mathrm{imp}}=2J$, $U=5J$ and the Haldane parameters are same as in Sec. V
Figure 16:  Variation in spectral function of impurity wrt Fermi level and
sign of interaction strength using Chevy ansatz. Columns correspond to
variation in Fermi level of the bath as can be inferred from panel (a)-(c)
which depicts the Haldane spectrum with Fermi level at $-2J$ (a), $0J$ (b) and
$2J$ (c). $\mathcal{A}(Q,\omega)$ calculated using Chevy ansatz with repulsive
(in (d)-(f)) and attractive (in (g)-(i)) on-site interaction between the
impurity and the bath particles. Here $J_{\mathrm{imp}}=2J$ and the parameters
used in Haldane model are the same as in Sec. V.

Importantly, we consider the impurity spectral function with momentum defined
in the extended-zone scheme. This is because, in an experiment, the non-
interacting impurity will be prepared with momentum $Q\in(2\pi/a,2\pi/a]$,
defined with respect to the 1D chain with equivalent $A$ and $B$ sublattices
(as opposed to the quasi-momentum $q$). In other words, in the extended-zone
scheme the dispersion of the bare impurity manifests itself as an unfolded,
conventional cosine. More precisely, the spectral function reads

$\mathcal{A}(Q,\omega)=-2{\rm
Im}\langle\Psi_{0}|\hat{d}_{Q}\frac{1}{\omega-\hat{H}+E_{\mathrm{bath}}^{0}+i\eta}\hat{d}^{\dagger}_{Q}|\Psi_{0}\rangle,$
(56)

where
$\hat{d}_{Q}=\frac{1}{\sqrt{2N_{1}}}\sum_{x\sigma}e^{-iQa(x+\frac{1}{2}\delta_{\sigma
B})}\hat{d}_{x0\sigma}$. For $Q=q$ in the first Brillouin zone, one simply has
$\hat{d}_{Q}=\frac{\hat{d}_{q0A}+\hat{d}_{q0B}}{\sqrt{2}}$, while, for
$Q=q+2\pi/a$, the sign of the $B$ contribution is flipped,
$\hat{d}_{Q}=\frac{\hat{d}_{q0A}-\hat{d}_{q0B}}{\sqrt{2}}$. With this in mind,
the spectral function is computed by numerically diagonalizing the Chevy-
Schödinger equations and invoking Eq. 5.

##  Appendix E Role of the Haldane bulk modes

Below we try to better understand the role of the bulk modes first by
artificially excluding them from the dressing of the impurity, and then by
moving the Fermi level from the topological gap into the Haldane bands, in
order to explore metallic states of the bath.

###  E.1 Exclusion of the bulk modes

Since in Fig. 8, the impurity spectral function at large $Q$ values becomes
blurred because of interaction with the non-localized bulk modes, it becomes
difficult to isolate all the crucial $Q$-points which one can obtain from the
effective chiral model. Hence, in Fig. 15 we filter out all the interaction
processes in the Chevy ansatz calculation which lead to creation of an
electron or hole with energy modulus greater than $J$ (black shaded area in
the bath spectrum plot on top right of panel (a)) for the same value of
$J_{\mathrm{imp}}=2J$ and $U=5J$ at half-filling, though in this case the bath
cutoff momentum is reduced to $\pi/a$. The resulting non-interacting spectrum
$E^{\delta k}(Q)$ is shown in panel (a) (gray curves) and the Umklapp
scattering enabled spectrum $E^{\delta k}_{u}(Q)=E^{\delta k}(Q+2\pi/a)$ in
lighter shade. All the crucial $Q$-points are marked by the vertical gray
dashed lines and are calculated by the intersection of green and red curves
(solid and dotted) with the blue solid curve (conventional impurity
dispersion).

###  E.2 Investigation of metallic phases

While in the main text we set the Fermi level at zero energy and considered
the half-filled Haldane model, it is interesting to consider the scenario
where the bulk bands are doped. In this case, one does not have a topological
insulator, but a metal with broken time-reversal symmetry. In Fig. 16 this
setting is studied for three fillings, corresponding to hole-doping, half-
filling and particle-doping, as sketched in panels (a-c). The Fermi energies
are set to $-2J,0,2J$, respectively. The corresponding spectral functions are
computed in panels (d-f) for repulsive $U$ and in (g-i) for attractive $U$.
While some asymmetry in $Q$ is present also in the metallic phases, in the
topological insulator phase one has the clearest asymmetry, with features
clearly related to the ones studied in Sec. IV. Another interesting remark is
that the particle-hole symmetry of the Haldane model at half-filling results
in identical spectra at opposite $U$ values, but for a mean-field shift of
$U$. This is a consequence of the particle-hole duality analyzed in Sec. IV.3
and is visible in panels (e) and (h).

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