# Rare collapse of fermionic quasiparticles upon coupling to local bosons

Piotr Wrzosek  Institute of Theoretical Physics, Faculty of Physics,
University of Warsaw, Pasteura 5, PL-02093 Warsaw, Poland  Adam Kłosiński
Institute of Theoretical Physics, Faculty of Physics, University of Warsaw,
Pasteura 5, PL-02093 Warsaw, Poland  Krzysztof Wohlfeld  Institute of
Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5,
PL-02093 Warsaw, Poland  Cliò Efthimia Agrapidis<EMAIL_ADDRESS>Institute of Theoretical Physics, Faculty of Physics, University of Warsaw,
Pasteura 5, PL-02093 Warsaw, Poland

###### Abstract

We study the stability of the fermionic quasiparticle in a fermion-boson model
on a Bethe lattice, with fermions interacting with local bosons by Peierls
coupling. We solve the problem by mapping it onto a non-interacting chain with
site-dependent potential. We show that, despite a finite number of bosonic
excitations costing zero energy, it is hard for the quasiparticle to
completely collapse. The quasiparticle disappearance becomes easier with an
increase in: (i) the total number of bosons with zero energy, and (ii) the
relative strength of the coupling between bosons and fermions. The postulated
model can, among other things, be applied to study systems in which fermions
are introduced into antiferromagnetic (or antiferro-orbital) domains
surrounded by ferromagnetic (or ferro-orbital) ordered states. This might take
place in the overdoped cuprates or upon doping manganese or vanadium oxides.
Finally, we show how this model leads to an in-depth understanding of the
onset of quasiparticles in the 1D and 2D $t$-$J^{z}$ model.

##  I Introduction

One of the most standard approaches to tackle a quantum many-body system is to
approximate it by weakly-interacting long-lived quasiparticles [1]. This is a
very successful picture as destroying a quasiparticle turns out to be a far
more complex task than naively expected. One of the first physical system
posited to exhibit quasiparticle decay was 4He: already in 1959 Pitaevskii
suggested that the phononic quasiparticle was allowed to decay into a two-
roton continuum [2, 3]. Yet, it is nowadays believed that the quasiparticle
does not enter the two-roton continuum in 4He [3] and instead the decay is
exponentially avoided due to strong interactions [4, 5]. A somewhat similar
situation takes place in some non-collinearly ordered magnets with small spin.
Here, the decay of magnon quasiparticles is also not observed [6, 4]. It is
then only for large spins and weaker magnon interactions that the magnon decay
takes place and can be observed [7, 8]. Nevertheless, these decays do not seem
to overdamp the magnons in realistic spin models [9].

To search for the total quasiparticle collapse, often referred to as non-Fermi
liquid or ‘unparticle’ physics [10, 11] and realised in the high-temperature
superconducting cuprates (cf. [12, 13]), one has to go to more exotic models.
These typically concern fermions coupled to a gapless mobile boson with a very
particular type of fermion-boson coupling [14]. The latter occurs when the
gapless bosons are Goldstone modes and the coupling to fermions does not
vanish in the limit of low energy-momentum transfer [15, 14]. However, as
suggested in Ref. [14], such a coupling is relatively rare in nature (its best
realisation being a nematic Fermi liquid). Another, probably more common,
route is to explore interactions between massless gauge bosons [16] and
fermions [17, 18, 19, 20] which can lead to non-Fermi liquid behavior in two
dimensions (2D). Such couplings may play a vital role in quantum Hall systems
[21], spin liquids [22] or quantum critical systems such as heavy fermions
[23].

In this paper we explore yet another route to quasiparticle extinction. To
this end, we adhere to a fermion-boson model with a Peierls-type coupling
between the two particle species. The primary difference with respect to all
of the cases above is that the bosons in our model are immobile, i.e., local,
and the lattice translational symmetry is explicitely broken. On the other
hand, also for the here-studied model, it is important that a finite number of
bosons can become massless (cost zero energy). While such a model is
interesting per se, we believe it to be of relevance to, for instance, systems
in which fermions are introduced into antiferromagnetic (or antiferro-orbital)
domains surrounded by ferromagnetic (or ferro-orbital) ordered states. This
might take place in overdoped cuprates [24, 25, 26, 27, 28, 29, 24, 30] or
upon doping manganese or vanadium oxides [31, 32, 33, 34, 35].

All of the results obtained in the paper follow from exact analytical
diagonalisation of the Hamiltonian. This enables us to address the issue of
quasiparticle stability in an unbiased way. The main result is that a complete
quasiparticle collapse is in general possible in the fermion-boson model –
though only in a small portion of the model parameter space, making it a rare
occurence. The quasiparticle disappearance is allowed once a certain number of
local bosonic excitations cost zero energy. This decay becomes easier with an
increase in: (i) the total number of bosons with zero energy, and (ii) the
relative strength of the coupling between bosons and fermions.

Finally, we uncover an interesting relation between the model introduced in
this paper and the well-studied problem of a mobile hole introduced in the
one-dimensional (1D) or 2D Ising antiferromagnet – as given by the 1D or 2D
$t$–$J^{z}$ model [36, 37, 15, 38, 39, 40, 41, 42, 43, 44, 45, 46]. It turns
out that these models, which always support a quasiparticle solution, are a
specific realisation of the class of fermion-boson models considered in this
paper. This means that the hole in the Ising antiferromagnet always forming a
quasiparticle solution is not the generic one.

The paper is organised as follows: in Sec. II we introduce the fermion-boson
Hamiltonian and the methods used to perform the calculations, including
mapping the interacting model to a non-interacting chain with the same
spectral properties; in Sec. III we study the effect of the coupling to the
impurity-like bosons, i.e. bosons which all cost zero energy except for one
particular site, on the appearance of a quasiparticle; then in Sec. IV we
extend our study to string-like local bosons, i.e. bosons which have finite
energy for a whole range of sites. Within Secs. III and IV, we also show how
our results relate to the $t$-$J^{z}$ model on a 1D and 2D Bethe lattice,
respectively. Lastly, we discuss the results and draw our conclusions in Sec.
V.

##  II Models

###  II.1 Fermion-boson model

We consider an interacting Hamiltonian on a Bethe lattice with coordination
number $z$,

$\displaystyle\mathcal{H}=-t$ $\displaystyle\sum_{\langle
i,j\rangle}\left[h_{i}^{\dagger}h_{j}\left(a_{i}+a_{j}^{\dagger}\right)\right.+\left.h_{j}^{\dagger}h_{i}\left(a_{j}+a_{i}^{\dagger}\right)\right]$
(1) $\displaystyle+\sum_{i}J_{i}a_{i}^{\dagger}a_{i},$

where $h_{i}$ are fermion annihilation operators and $a_{i}$ are hard-core
bosons annihilation operators at site $i$, $t$ is the Peierls coupling between
fermions and bosons, and $J_{i}$ is the on-site boson potential. The model
lives in a restricted Hilbert space including states with the constraint

$\displaystyle n_{a_{i}}+n_{h_{i}}\leq 1,$ (2)

with $n_{a_{i}}$ and $n_{h_{i}}$ being the number operator at site $i$ for
bosons and fermions respectively.

Figure 1: Example of the distance labelling on a Bethe lattice with $z=3$. The
numbering corresponds to the distance $d_{i}$ from the root of the Bethe
lattice, labelled as $0$.

We chose one arbitrary site for the root of the Bethe lattice. We label this
site with index 0 and we will call it an origin of the lattice. Let $d_{i}$
stand for the distance of site $i$ to the origin of the lattice, calculated in
the number of edges in the graph of the lattice that separate the site i from
site 0, as shown in Fig. 1. With this, we impose a geometrical constraint on
the shape of the on-site boson potentials $J_{i}$ that renders equivalent all
the branches of the Bethe lattice (starting at site 0),

$d_{i}=d_{j}\Rightarrow J_{i}=J_{j}.$ (3)

In this work we are interested in the single-fermion spectral function

$\mathcal{A}(\omega)=-\frac{1}{\pi}\text{Im}\mathcal{G}(\omega+i\delta),$ (4)

where the local single-particle Green’s function $\mathcal{G}$ is given by:

$\mathcal{G}(\omega+i\delta)=\left\langle\varnothing\left|h_{0}\frac{1}{\omega-\mathcal{H}+i\delta}h^{\dagger}_{0}\right|\varnothing\right\rangle,$
(5)

where $\lvert\varnothing\rangle$ denotes the vacuum state for the particles in
Eq. (1) and the fermion $h$ is created at site $0$ of the Bethe lattice. Here
the local Green’s function describes the motion of a single fermion in the
considered Hamiltonian (1), for it describes the case when a single fermion is
added at the site labelled as 0 of the Bethe lattice, then propagates via
Hamiltonian (7), and finally it is annihilated at site 0. Note that we choose
to probe the system using the local Green’s function, for the model we
consider is anisotropic in real space (through, among other features, the
site-dependent boson energies $J_{i}$).

###  II.2 Mapping to a non-interacting model

Since we are interested in the spectral function (4), we can restrict
ourselves to the subspace of states that contribute to it [see below]. Within
this subspace, the above model in Eq. (1) can be mapped to a single spinless
fermion moving on an otherwise empty 1D chain with nearest-neighbour hopping
and an external potential. The hopping amplitudes are equal everywhere other
than between the 0-th site at the center of the chain and its two neighbors
and the chain is placed in a site-dependent external potential that is
symmetric around the 0-th site. The mapping is done via

$V_{i}=\sum_{d_{n}=0}^{i-1}J_{n}\quad{\rm
and}\quad\left\\{\begin{array}[]{c}\tau_{0}=t\sqrt{z}\\\
\tau=t\sqrt{z-1}\end{array}\right.$ (6)

and the non-interacting Hamiltonian is given by

$\displaystyle H$
$\displaystyle=\sum^{\infty}_{i=-\infty}\;\left[-\tau\left(c^{\dagger}_{i+1}c_{i}+\text{h.c.}\right)+\;V_{|i|}\;n_{i}\right]$
(7)
$\displaystyle+\left(\tau-\frac{\tau_{0}}{\sqrt{2}}\right)\left(c^{\dagger}_{1}c_{0}+c_{0}^{\dagger}c_{-1}+\text{h.c.}\right),$

where $c^{\dagger}_{i}(c_{i})$ creates (annihilates) a spinless fermion at
site $i$, $n_{i}=c^{\dagger}_{i}c_{i}$ is the fermion density operator at site
$i$, $\tau$ and $\tau_{0}$ are the hopping amplitudes and $V_{|i|}$ is an
external potential taken to be symmetric around the origin $i=0$, where the
fermion is originally introduced in the system. Without loss of generality we
assume that the hopping amplitude $\tau>0$.

Since we are interested in the single-fermion spectral function (4), we need
to define the relevant quantity also for the non-interacting model:

$\displaystyle A(\omega)=-\frac{1}{\pi}\text{Im}{G}(\omega+i\delta).$ (8)

where the corresponding single-particle Green’s function is

$G(\omega+i\delta)=\left\langle\varnothing\left|c_{0}\frac{1}{\omega-{H}+i\delta}c^{\dagger}_{0}\right|\varnothing\right\rangle.$
(9)

Now, $\lvert\varnothing\rangle$ denotes the vacuum for $c$ fermions, i.e. the
empty 1D chain.

Model (7-8) is an effective model resulting from the mapping of model (1-4),
but it can also be seen as a model describing a fermion in a 1D crystal (or
optical lattice) with a particular pattern of impurities given rise to the
potential $V_{|i|}$, cf. [47] for a recent work on a related problem.

###  II.3 Solution

Our goal is to calculate the spectral function defined in Eq. (8). To this end
we choose the basis $\mathcal{B}$ in the following manner: We start with the
initial state $\lvert 0\rangle\equiv c^{\dagger}_{0}\lvert\varnothing\rangle$,
which corresponds to a spinless fermion located at site $i=0$. From this state
we construct states ‘reachable’ by the propagator $1/(\omega-{H}+i\delta)$ by
repeated application of the Hamiltonian on our initial state, namely
$H^{n}\lvert 0\rangle$, for some $n\in\mathbb{N}$. This leads to basis
$\mathcal{B}$ defined as

$\begin{split}\lvert 0\rangle&=c^{\dagger}_{0}\lvert\varnothing\rangle,\\\
\lvert
i\rangle&=\frac{1}{\sqrt{2}}\left(c^{\dagger}_{-i}+c^{\dagger}_{i}\right)\lvert\varnothing\rangle.\end{split}$
(10)

In this basis, the Hamiltonian (7) takes the form
$H=\sum_{i,j}h_{i,j}|i\rangle\langle j|$. Note that we can restrict the
Hilbert subspace to $\mathcal{B}$, since all neglected states, i.e. the anti-
symmetric ones $\lvert
i\rangle_{-}=\frac{1}{\sqrt{2}}\left(c^{\dagger}_{-i}-c^{\dagger}_{i}\right)\lvert\varnothing\rangle$,
give zero contribution to the spectral function [48].

An essential feature of the introduced basis $\mathcal{B}$ is that the
Hamiltonian matrix $[h_{i,j}]$ becomes tridiagonal,

$[h_{i,j}]=\begin{bmatrix}V_{0}&-\tau_{0}&&&\\\ -\tau_{0}&V_{1}&-\tau&&\\\
&-\tau&V_{2}&-\tau&\\\ &&-\tau&V_{3}&\ddots\\\ &&&\ddots&\ddots\\\
\end{bmatrix}.$ (11)

This yields a simple formula for the Green’s function in the form of a
continued fraction,

$G(\omega)=\frac{1}{\omega-V_{0}-\Sigma(\omega)}.$ (12)

Let us denote

$\displaystyle\Gamma(\omega)$ $\displaystyle=\frac{\tau^{2}}{\omega-
V_{n}-\Gamma(\omega)}$ (13) $\displaystyle\Omega_{i<n}=$ $\displaystyle\omega-
V_{i}\quad\text{and}\quad\Omega_{n}=\frac{\tau^{2}}{\Gamma(\omega)},$ (14)

where $i\geq 0$ denotes the distance from the origin. Then, we can write the
expression for the self-energy for a generic symmetric potential centered
around $i=0$:

$\displaystyle\Sigma(\omega)=\frac{\tau_{0}^{2}}{\Omega_{1}-\frac{\tau^{2}}{\Omega_{2}-\frac{\tau^{2}}{\Omega_{3}-\ldots}}}=\frac{\tau_{0}^{2}}{\tau^{2}}\operatornamewithlimits{\mathcal{K}}_{i=1}^{n}\left(\frac{\tau^{2}}{\Omega_{i}}\right).$
(15)

In what follows, we are interested in the properties of the state contributing
to the spectral function $A(\omega)$ at the lowest energy (denoted as
$\omega_{QP}$). In particular, the central question is whether this state is a
discrete one (i.e. a bound state) or is part of a continuum. As in the
fermion-boson model this question corresponds to the issue of quasiparticle
stability, we denote the spectral weight carried by the discrete state at
$\omega_{QP}$ with the quasiparticle spectral weight $a_{\mathrm{QP}}$. In
terms of the quasiparticle energy $\omega_{QP}$, this is given by:

$\displaystyle a_{\mathrm{QP}}$
$\displaystyle(\tau,\tau_{0},V)=\lim_{\omega\to\omega_{\mathrm{QP}}}\left(\omega-\omega_{\mathrm{QP}}\right)\;G(\omega)=$
(16)
$\displaystyle=\lim_{\omega\to\omega_{\mathrm{QP}}}\frac{1}{1-\frac{d}{d\omega}\Sigma(\omega)}$

Hence, to obtain the expression for the quasi-particle residue we calculate
the derivative of the self-energy with respect to the frequency $\omega$,

$\displaystyle\frac{d}{d\omega}\Sigma(\omega)$
$\displaystyle=\frac{\tau_{0}^{2}}{\tau^{2}}\frac{d}{d\omega}\operatornamewithlimits{\mathcal{K}}_{i=1}^{n}\left(\frac{\tau^{2}}{\Omega_{i}}\right)=$
(17)
$\displaystyle=\frac{\tau_{0}^{2}}{\tau^{4}}\sum_{j=1}^{n}\left(-1\right)^{j+1}\prod_{i=1}^{j}\left[\operatornamewithlimits{\mathcal{K}}_{l=i}^{n}\left(\frac{\tau^{2}}{\Omega_{l}}\right)\right]^{2}\frac{d\Omega_{j}}{d\omega},$

where $\frac{d}{d\omega}\Omega_{i<n}=1$ and
$\frac{d}{d\omega}\Omega_{n}=\frac{d}{d\omega}\left(\frac{\tau^{2}}{\Gamma(\omega)}\right)$.

Note that $\omega_{\mathrm{QP}}$ in (16) depends on the specific form of the
potential $V_{|i|}$, so we cannot provide a general form for this quantity at
this point.

##  III Impurity-like bosons

We consider a class of fermion-boson models with impurity-like bosons. To this
end, we assume that the site-dependent bosonic energy $J_{n}$ in Eq. (1) takes
the form

$J_{n}=\begin{cases}J/2&\mbox{if }d_{n}=0\\\ 0&\mbox{if }d_{n}\neq
0\end{cases},$ (18)

i.e., all but one boson in the Bethe lattice are massless. This may look as a
quite unphysical regime – however, as shown below, it may find its realisation
in a number of physical systems.

The above case corresponds to considering a point potential at site $n=0$
(i.e. at the position where we introduce the spinless fermion in the system)
(7)

$V_{n}=\begin{cases}0&\mbox{if }n=0\\\ V&\mbox{if }n\neq 0\end{cases},$ (19)

in the non-interacting model given in Eq. (7).

The other parameters of the two Hamiltonians, namely the coordination number
$z$ in (1) and, correspondingly, the $\tau_{0}/\tau$ ratio in (7) are left
free. Since the two models are equivalent, we present the solution in terms of
the easier-to-solve non-interacting model (7).

###  III.1 Solution

Figure 2: The quasiparticle spectral weight $a_{\mathrm{QP}}(V)$ for different
values of ${\tau_{0}}/{\tau}$ as a function of the depth of the point
potential $V$ [cf. Eq. (19)]. Note that the lines start at $V>0$ for any
${\tau_{0}}/{\tau}<\sqrt{2}$. Dashed lines shows values of $\tau_{0}/\tau$ for
which the mapping to the Bethe lattice coordination number $z$ leads to a
fractional value of the latter.

In what follows we leave the ratio $\tau_{0}/\tau$ as a parameter in the
system. However, we assume that $\tau_{0}\leq\sqrt{2}\tau$. This is due to an
earlier study [48] which shows that larger values of $\tau_{0}$ always
stabilise a quasiparticle solution. In fact, $\tau_{0}/\tau=\sqrt{2}$ is a
limiting case which corresponds to a non-interacting isotropic chain [see Eq.
(7)] and, as shown below, is the case for which the quasiparticle solution is
always present, independently of the value of the point potential.

The above assumption leads to the following expression for the self-energy
(15),

$\Sigma(\omega)=\left\\{\begin{array}[]{lr}\frac{\tau_{0}^{2}}{\tau^{2}}\left(\frac{\omega-V}{2}+\sqrt{\left(\frac{\omega-V}{2}\right)^{2}-\tau^{2}}\right),&\>\omega\leq
V\\\
\frac{\tau_{0}^{2}}{\tau^{2}}\left(\frac{\omega-V}{2}-\sqrt{\left(\frac{\omega-V}{2}\right)^{2}-\tau^{2}}\right),&\>\omega>V\\\
\end{array}\right..$ (20)

Moreover, following the general solution presented above in Sec. II.3, we
calculate the analytic formula for the quasiparticle spectral weight
$a_{\mathrm{QP}}$ for the potential given in Eq. (19). In terms of the
quasiparticle energy $\omega_{\mathrm{QP}}$ we have

$\displaystyle a_{\mathrm{QP}}$
$\displaystyle(\tau,\tau_{0},V)=\lim_{\omega\to\omega_{\mathrm{QP}}}\left(\omega-\omega_{\mathrm{QP}}\right)\;G(\omega)=$
(21)
$\displaystyle=\lim_{\omega\to\omega_{\mathrm{QP}}}\frac{1}{1-\frac{d}{d\omega}\Sigma(\omega)}=$
$\displaystyle=\left[1-\frac{1}{2}\frac{\tau_{0}^{2}}{\tau^{2}}\left(1+\frac{\omega_{\mathrm{QP}}-V}{\sqrt{(\omega_{\mathrm{QP}}-V)^{2}-4\tau^{2}}}\right)\right]^{-1},$

where

$\omega_{\mathrm{QP}}=\left\\{\begin{array}[]{lr}\frac{\tau_{0}^{2}}{2}\left(\frac{V}{\tau_{0}^{2}-\tau^{2}}-\frac{\sqrt{V^{2}+4\tau_{0}^{2}-4\tau^{2}}}{|\tau_{0}^{2}-\tau^{2}|}\right)&\tau_{0}>\tau,0\\\
-\frac{1}{V}&\tau_{0}=\tau>0\\\
\frac{\tau_{0}^{2}}{2}\left(\frac{V}{\tau_{0}^{2}-\tau^{2}}+\frac{\sqrt{V^{2}+4\tau_{0}^{2}-4\tau^{2}}}{|\tau_{0}^{2}-\tau^{2}|}\right)&0<\tau_{0}<\tau\\\
0&\tau_{0}=0\end{array}\right..$ (22)

All of the above solutions come from the $\omega\leq V$ branch in (20). We
plot the quasiparticle weight $a_{\mathrm{QP}}$ as a function of $V/\tau$ for
different values of $\tau_{0}/\tau$ in Fig. 2 and notice that the value of the
potential $V$ at which the quasiparticle weight is finite shifts away from
zero with tuning $\tau_{0}/\tau$ away from $\sqrt{2}$. We then determine for
which depth of the potential $V$ the quasiparticle weight $a_{\mathrm{QP}}$
becomes finite as the hopping ratio $\tau_{0}/\tau$ changes. This leads to the
definition of the critical value of $V^{*}$

$V^{*}(\tau,\tau_{0})=\left\\{\begin{array}[]{lr}\frac{2\tau^{2}-\tau_{0}^{2}}{|\tau|}&\tau_{0}>0\\\
-\infty&\tau_{0}=0\\\ \end{array}\right..$ (23)

We can then also calculate the spectral function (8) by using the self-energy
in Eq. (20) and substituting in Eq. (12). We show an example of the spectral
function for the model (7)-(19) with $\tau_{0}/\tau=\sqrt{3/2}$ in Fig. 3.
From Eq. (23), we find $V^{*}=0.5\tau$. Indeed, no discrete energy branch is
visible for $V\leq V^{*}$, $V^{*}$ being the critical value of the point
potential for which the discrete state appears.

Figure 3: (a) Spectral function $A(\omega)$ as a function of the point-
potential depth $V/\tau$ [cf. Eq. (19)] with ${\tau_{0}}/{\tau}=\sqrt{3/2}$.
The right panel shows the spectral function $A(\omega)$ at $V=0.5\tau$: for
this value no discrete energy peak is visible, since for
$\tau_{0}/\tau=\sqrt{3/2}$ this corresponds to the critical value $V^{*}$ and
the quasiparticle appears only for $V>0.5\tau$. (b) Schematic representation
of the model at the vertical cut shown in (a) with
${\tau_{0}}/{\tau}=\sqrt{3/2}$ and $V=0.5\tau$. The different hopping around
the origin is drawn in red, the gray area represents the shape of the
potential, the black filled circle represents the moving fermion and the
dashed circle its starting position.

Let us now go back to the fermion-boson interacting Hamiltonian (1). We are
considering a boson potential as given by (18), while we can vary the Bethe
lattice coordination number $z$. Substituting (6) in (23), we find the
critical value of the boson potential $J^{*}$

$J^{*}(t,z)=\frac{t(z-2)}{\sqrt{z-1}}.$ (24)

Note that $z$ can only take integer values equal or larger than two, so that
we lose part of the solution available when considering the non-interacting
chain (7). Condition (24) implies that, also for the interacting fermion-boson
model on a Bethe lattice with coordination number $z$ (1), there are two
distinct regimes: one with a well defined fermionic quasiparticle for
$J>J^{*}$, and one with no quasiparticle for $J\leq J^{*}$.

###  III.2 Special limit

Figure 4: Spectral function $A(\omega)$ as a function of the potential depth
$V/\tau$ [cf. Eq. (19)] for the ‘special limit’ $\tau_{0}/\tau=\sqrt{2}$. The
right panel shows the spectral function $A(\omega)$ at $V=0.5\tau$: for this
value there is a clear discrete energy state split from a continuum, since for
$\tau_{0}/\tau=\sqrt{2}$ we have the critical $V^{*}=0$ and the quasiparticle
appears for any $V>0$. (b) Schematic representation of the model at the
vertical cut shown in (a) with ${\tau_{0}}/{\tau}=\sqrt{2}$ and $V=0.5\tau$.
In this case, all of the hoppings in the non-interacting chain are equivalent.
The gray area represents the shape of the potential, the black filled circle
represents the moving fermion and the dashed circle its starting position.

Fig. 2 shows the existence of a special value $\tau_{0}/\tau=\sqrt{2}$ for
which the discrete state exists for any value $V>0$. We then consider the
spectral function $A(\omega)$ for this case (see Fig. 4). Indeed, a discrete
energy branch is present for all values $V>0$. As shown in Fig. 2(b), taking
$\tau_{0}/\tau=\sqrt{2}$ corresponds to considering a chain with equal
hoppings on all bonds, i.e., the hopping from the origin is not distinct from
the others anymore. In terms of the fermion-boson model (1), this translates
to the existence of a special value of the coordination number $z$ at which
the quasiparticle is present for any value $J>0$: $z=2$ (corresponding to
$\tau_{0}/\tau=\sqrt{2}$). This corresponds to a 1D interacting fermion-boson
model with a boson point potential. The latter system is equivalent to a well-
known interacting model: the $t$-$J^{z}$ chain with one single hole (see
below).

###  III.3 Physical realization

Figure 5: Dynamics of a single hole on a 1D chain $t$-$J^{z}$ (Bethe lattice
with $z=2$). The black solid circle represents the hole in the current state,
the dashed circle is the hole at its initial creation site. The cartoon shows
the system after few hole hoppings have occured.

$z=2$ case We have seen how a critical value of the relative strength of the
fermion-boson coupling $J^{*}/t$ determines the existence or suppression of a
quasiparticle and we have determined a critical case for the coordination
number $z=2$ at which the quasiparticle exists for any $J/t>0$. For this
special case, we now show how we recover the physics of one of the most
studied interacting models in condensed matter physics: the $t$-$J^{z}$ chain
with one single hole. Indeed, we will show how its spectral function and of
the fermion-boson model (1) (with $z=2$) are the same.

The Hamiltonian of the 1D $t\text{-}J^{z}$ model with a single hole reads

$\mathcal{H}_{t\text{-}J^{z}}=-t\sum_{\langle
i,j\rangle,\sigma}\left(\tilde{c}_{i\sigma}^{{\dagger}}\tilde{c}_{j\sigma}\\!+\\!\mathrm{H.c.}\\!\right)+J^{z}\sum_{\langle
i,j\rangle}\left(S_{i}^{z}S_{j}^{z}\\!-\tfrac{1}{4}\,\tilde{n}_{i}\tilde{n}_{j}\right),$
(25)

where $\tilde{c}_{i\sigma}$ annihilates an electron at site $i$ in the
constrained Hilbert space without double occupancy, $t$ is the nearest
neighbor hopping parameter, $S^{z}_{i}$ is the $z$-component of the spin-1/2
operator at site $i$,
$\tilde{n}_{i}=\tilde{c}_{i\sigma}^{\dagger}\tilde{c}_{i\sigma}$ is the
electron density operator at site $i$ and $J^{z}$ is the antiferromagnetic
exchange coupling. Rather than using the electron and spin operators, we can
rewrite the Hamiltonian (25) in terms of hole and magnon operators [49]. The
transformation starts with the rotation of all spins on one sublattice, which
in turn allows for the introduction of holes and magnons in terms of the
following transformations:

$\displaystyle\tilde{c}_{i\uparrow}^{\dagger}$ $\displaystyle=h_{i}P_{i},$
$\displaystyle\tilde{c}_{i\uparrow}=P_{i}h_{i}^{\dagger},$ (26)
$\displaystyle\tilde{c}_{i\downarrow}^{\dagger}$
$\displaystyle=h_{i}a_{i}^{\dagger},$
$\displaystyle\tilde{c}_{i\downarrow}=h_{i}^{\dagger}a_{i},$ $\displaystyle
S_{i}^{z}$
$\displaystyle=\frac{1}{2}-a_{i}^{\dagger}a_{i}-\frac{1}{2}h_{i}^{\dagger}h_{i},$
$\displaystyle\tilde{n}_{i}=1-h_{i}^{\dagger}h_{i},$ $\displaystyle P_{i}$
$\displaystyle=\sqrt{1-a_{i}^{\dagger}a_{i}}$

where $h_{i}$ is a fermion operators that annihilates a hole and $a_{i}$ is a
bosonic operator annihilating a magnon at site $i$. The kinetic energy now
reads:

$\displaystyle\mathcal{H}_{t}=\mathcal{P}\Big{\\{}-t\sum_{\langle i,j\rangle}$
$\displaystyle\left[h_{i}^{\dagger}h_{j}\left(a_{i}+a_{j}^{\dagger}\right)+h_{j}^{\dagger}h_{i}\left(a_{j}+a_{i}^{\dagger}\right)\right]\Big{\\}}\mathcal{P},$
(27)

with $\mathcal{P}$ being the global action of the projection operators $P_{i}$
and acting by projecting out states that do not satisfy
$n_{a_{i}}+n_{h_{i}}\leq 1$. The potential energy reads

$\displaystyle\mathcal{H}_{J^{z}}$
$\displaystyle=E_{0}+\frac{J^{z}}{2}\sum_{\langle
i,j\rangle}\left[a_{i}^{\dagger}a_{i}+a_{j}^{\dagger}a_{j}-2a_{i}^{\dagger}a_{i}a_{j}^{\dagger}a_{j}\right.+$
(28)
$\displaystyle+\left.h_{i}^{\dagger}h_{i}+h_{j}^{\dagger}h_{j}-h_{i}^{\dagger}h_{i}a_{j}^{\dagger}a_{j}-h_{j}^{\dagger}h_{j}a_{i}^{\dagger}a_{i}-h_{i}^{\dagger}h_{i}h_{j}^{\dagger}h_{j}\right].$

Note that this transformation is exact, provided that one considers initial
states with no more than two magnons per site, a subset of the full bosonic
Hilbert space. Of course, all eigenstates of (25) belong to this subspace and
any state in this subspace is confined to it when time evolution is applied.

It is straightforward to check that, up to a constant energy shift, the
Green’s function of a single hole is exactly the same for the $t$-$J^{z}$
Hamiltonian $\mathcal{H}_{t}+\mathcal{H}_{J^{z}}$, given by Eqs. (27) and
(28), and for the interacting boson-fermion model $\mathcal{H}$ in Eq. (1):

$\displaystyle\mathcal{G}(\omega+i\delta)$
$\displaystyle=\left\langle\varnothing\left|h_{0}\frac{1}{\omega^{\prime}+i\delta-\mathcal{H}_{t}-\mathcal{H}_{J^{z}}}h^{\dagger}_{0}\right|\varnothing\right\rangle$
$\displaystyle=\left\langle\varnothing\left|h_{0}\frac{1}{\omega+i\delta-\mathcal{H}}h^{\dagger}_{0}\right|\varnothing\right\rangle,$
(29)

where $|\varnothing\rangle$ denotes the vacuum state for holes and magnons,
$\omega^{\prime}=\omega-E$.

To show the validity of the equivalence in Eq. (III.3), let us start with the
hole operators $h_{i}$. For a single hole, the hole-hole interaction term
$h_{i}^{\dagger}h_{i}h_{j}^{\dagger}h_{j}$ vanishes. Moreover, the summation
over the hole number operators $h_{i}^{\dagger}h_{i}$ yields a constant. Thus
these terms can be hidden in the constant energy shift $E$ included in
$\omega^{\prime}$.

For other terms, the equivalence depends on the chosen initial state $\lvert
0\rangle=h^{{\dagger}}_{0}\lvert\varnothing\rangle$. Here, magnons can only be
created by the moving hole and there is no separate magnon dynamics. If the
initial state contains magnons or if magnons are allowed to move, equation Eq.
(III.3) does not have to be fulfilled.

States that belong to a linear combination arising from repeated application
of the Hamiltonian $\mathcal{H}_{t}^{n}\lvert 0\rangle$, with
$\mathcal{H}_{t}$ given by Eq. (27), for any $n$ are reachable by the
$\mathcal{G}$ (or $\mathcal{H}_{t}$) operator, defined in Eq. (III.3), from
the initial state $\lvert 0\rangle$. On the other hand, any reachable state
belongs to $\mathcal{H}_{t}^{n}\lvert 0\rangle$ for some $n$. Hence, a chain
of magnons connects the hole with the site where it was initially created (see
Fig. 5). Moreover, there are no other magnons in this state, since magnons can
only be created by the hole dynamics.

Now we need to consider three distinct cases for magnon chains of length
$n=0$, $n=1$ and $n>1$. For $n=0$, the magnon number operator gives $0$ and
thus $\mathcal{H}=\mathcal{H}_{t}+\mathcal{H}_{J^{z}}$ up to a shift by a
constant energy $E$. For $n=1$, only the hole-magnon interaction and the cost
of the creation of a single magnon have to be taken into account (the hole
cost can be incorporated into the constant energy shift $E$). It is easy to
check that the cost of creating a magnon in a 1D chain is ${J^{z}}$, and that
there will always be only one single hole-magnon interaction (even for longer
chains) contributing an energy $-{J^{z}}/{2}$, so that the total energy
contribution of creating the first magnon is ${J^{z}}/{2}$ and for states with
one magnon we again recover $\mathcal{H}=\mathcal{H}_{t}+\mathcal{H}_{J}^{z}$.
Creating more magnons in the 1D chain costs no energy, since the energy
required to create a magnon is exactly cancelled by the magnon-magnon
interaction between neighboring bosons [45]. Hence, we show that indeed
$\mathcal{H}=\mathcal{H}_{t}+\mathcal{H}_{J^{z}}$ for the whole class of
reachable states.

Indeed, it is well known that a fermionic quasiparticle appears for any value
$J^{z}>0$ in the $t$-$J^{z}$ chain with one single hole, agreeing with our
results that a quasiparticle is present for any $J\neq 0$ in the fermion-boson
model (1) for $z=2$ with a bosonic point potential acting on the site at which
the hole is originally introduced, as given in (18) (see Fig. 5) and with the
equivalence $J\equiv J^{z}$. Thus, the spectral function shown in Fig. 4 for
the limiting value $\tau_{0}/\tau=\sqrt{2}$ is the same as that for the 1D
$t$-$J^{z}$ with one single hole, where $V/\tau$ is replaced by $J/2t$.

$z>2$ case For a point potential acting on a Bethe lattice with $z>2$, we can
draw a parallel to lightly doped systems with mobile charges. In fact, when
doping ions are introduced in an atomic crystal, they affect the crystal
structure so that an effective potential is introduced at the point of origin
of the hole. This corresponds to $J_{n}$ given by (18). If we consider the
quasi-2D Bethe lattice as a first approximation for 2D systems, we can claim
that for lightly doped systems, such that dopants are sparse and distant
enough to not affect each other, a quasiparticle will not always stabilize
when impurities are introduced in fermionic systems, rather it will depend on
the strength of the local potential around the impurity. A similar situation
to this case is also present in the intermediate state of resonant inelastic
X-ray scattering (RIXS) on transition metal oxides (TMOs) [50, 51]. Here,
introduction of a core hole results in a $3d^{10}$ configuration on the $d$
orbitals, which can be treated as a mobile hole. In turn, this mobile hole is
affected by the core-hole potential on the site where it was originally
introduced. For TMOs with relatively small magnetic exchange, our model (1)
would then be a good first approximation of the fundamental physics happening
at the intermediate state of RIXS, meaning a quasiparticle would stabilize
only for large enough values of core-hole potential.

##  IV String-like bosons

In this section, we consider a different shape of the bosonic potential
$J_{n}$ so that not one but several bosons have finite energy. More
specifically, we choose a model in which a finite number of bosons, clustered
around the central site of the Bethe lattice, have the same nonzero energy.
Without loss of generality, we fix the coordination number $z=3$ in (1). This
choice corresponds to a string-like potential on the Bethe lattice and to
setting the hopping ratio $\tau_{0}/\tau=\sqrt{3/2}$ in the non-interacting
model (7).

Thus, we consider the on-site boson energies $J_{n}$ in Eq. (1) as given by

$J_{n}=\begin{cases}{J}&\mbox{if }d_{n}=0\\\ \frac{J}{2}&\mbox{if
}0<d_{n}<l\\\ 0&\mbox{if }d_{n}\geq l\end{cases},$ (30)

which translates in the non-interacting Hamiltonian Eq. (7) to:

$V_{n}=\begin{cases}0&\mbox{if }n=0\\\ V+\frac{V}{2}(n-1)&\mbox{if }0<|n|<l\\\
V+\frac{V}{2}(l-1)&\mbox{if }|n|\geq l\end{cases}.$ (31)

Again, we present the solution in terms of the non-interacting model (7)-(31).

###  IV.1 Solution

Figure 6: Spectral function $A(\omega)$ for a string-like potential [cf. (31)]
and varying half-width of the potential well $l$ for
${\tau_{0}}/{\tau}=\sqrt{3/2}$ and the (right panel) potential parameter
$V=0.5\tau$. A transition from a solution with no quasiparticles to one with
multiple quasiparticles is observed. (b) Vertical cuts of panel (a) at
different values of the potential half-width $l$. (b1) $l=1$, no discrete
state peak is present, (b2) $l=6$ two discrete state peaks are present in the
spectrum as well as a continuum, (b3) $l=12$ several discrete peaks are
visible as well as a higher energy continuum of states, (b4) $l=18$ several
discrete peaks make up the spectrum, similar to the known ladder-spectrum. (c)
Schematic of the model in a string potential with defects [cf. (31)]. The
different hopping around the origin is drawn in red, the gray area shows the
altered string potential, the black filled circle represents the moving
fermion and the dashed circle its starting position, we explicitly shown the
half-width of the potential $l$.

We plot the spectral function for the string-like potential (31) and
$\tau_{0}/\tau=\sqrt{3/2}$ in Fig. 6. As the relevant parameter in this case
we use the half-width of the potential $l$, while the potential parameter is
set to be $V=0.5\tau$.

To better show the progression of the spectral function $A(\omega)$ as a
function of $l$, we present $A(\omega)$ for four values of $l$ in Fig. 6(b).
For $l=1$ (Fig. 6(b1)), no discrete state is present. Note that this is the
same situation shown in the vertical cut of Fig. 3(a), since $l=1$ corresponds
to a point potential. Now we focus on increasing $l$. At $l=6$, already two
distinct discrete states are present, together with a relatively large
continuum of states (cf. Fig. 6(b2)). When $l=12$, the spectral function shows
several consecutive peaks, but a continuum persists for larger energies
$\omega/\tau$ (cf. Fig. 6(b3)). Finally, for $l=18$, we see a spectrum
composed only of consecutive sharp peaks, i.e., discrete states, similar to a
ladder spectrum. This situation keeps on as $l$ goes to infinity. It is then
natural to consider the special limit of a full string potential covering the
whole lattice.

###  IV.2 Special Limit

Figure 7: Spectral function $A(\omega)$ in a string potential [cf. (33)] for
$\tau_{0}/\tau=\sqrt{3/2}$. The right panel shows the spectral function
$A(\omega)$ at $V=0.5\tau$: the spectral function takes the shape of the so-
called ladder spectrum and contains only discrete (i.e. quasiparticle-like)
peaks. (b) Schematic of the model at the vertical cut drawn in (a). The
different hopping around the origin is drawn in red, the gray area shows the
string potential, the black filled circle represents the moving fermion and
the dashed circle its starting position. Note that the string potential
completely covers the infinite chain.

We now consider the limit $\l\to\infty$. This results in a finite on-site
bosonic energy $J_{n}$ for $every$ site $n$ on the Bethe lattice in Eq. (1):

$J_{n}=\begin{cases}J&\mbox{if }d_{n}=0\\\ \frac{J}{2}&\mbox{if
}d_{n}>0,\end{cases}$ (32)

i.e., all bosons are massive. This situation corresponds to a potential
$V_{n}$ in the non-interacting Hamiltonian Eq. (7) having a perfect string
(i.e. discrete linear) character:

$V_{n}=\begin{cases}0&\mbox{if }n=0\\\ V+\frac{V}{2}(n-1)&\mbox{if }n\neq
0.\end{cases}$ (33)

In Fig. 7, we show the spectral function for such a model as a function of the
potential parameter $V/\tau$. There is an important difference in the physics
depicted in Fig. 6 and Fig. 7. In the first case, we plot the spectral
function in terms of the half-width of the potential $l$, i.e., in terms of
how many sites are actually affected by it and we fix the potential parameter
to be $V=0.5\tau$. In the second case, we show the spectral function evolution
as a function of the potential energy parameter $V/\tau$, similarly to what
done previosuly for the case of a point potential. Indeed, we observe that
this case is very different from the previously considered case of a point
potential, since there exist at least one discrete energy state for any value
of $V$. However, as the potential increases, we see that more discrete states
appear in the system, in contrast to the point potential case were only one
discrete state would stabilize.

The stark contrast to the case of impurity-like bosons [cf. Sec. III] is
underlined when we fix the value of $V$ and plot $A(\omega)$: a structure with
several consecutive delta-like peaks appear, [right panel of Fig. 7(a)]. This
structure is the well-known ladder-spectrum of [36, 15]. In the language of
the fermion-boson model this means that for any value of the finite bosonic
energy $J$ the spectral function of model (7) with (32) solely consists of
quasiparticles, cf. [36, 15].

Note that at least one discrete state emerges for any value of $V$, in
contrast with the finite string potential case of Fig. 6 for which for any
fixed value of V, we have a continuum until a critical value $l^{*}$ is
reached, meaning the discrete state appears only when the potential ‘covers’
enough sites around the origin, where the fermion is introduced. This number
depends on the parameter $V$ as well as the hopping ratio $\tau_{0}/\tau$. In
terms of the interacting fermion-boson model, this translates to the
possibility of having quasiparticle decay in case only some bosons around the
origin of the Bethe lattice are massive – with the rest being massless.
Nonetheless, to reach the ladder spectrum form when $l$ is finite, the
potential needs to cover a relatively large number of lattice sites. Hence, it
is rather easy to destabilise the ladder spectrum and recover some kind of
state continuum.

In the language of the interacting fermion-boson model this implies that,
whereas in the case presented in Fig. 7 we are tuning the strength of the
boson-fermion coupling $t/J$ (strictly speaking $J/t$ in Fig. 7), in the case
of Fig. 6 we are tuning the number of massless bosons $\propto l$ present in
our system.

###  IV.3 Physical realization

Figure 8: Dynamics of a single hole on a Bethe lattice with $z=3$. The black
solid circle represents the hole in the current state, the dashed circle is
the hole at its initial creation site. The cartoon shows the system after few
hole hoppings have occurred.

Perfect string potential The spectrum in Fig. 7 immediately brings to mind the
one of the $t$-$J^{z}$ model on a Bethe lattice with $z>2$ [36, 15]. While the
relation between the fermion-boson model in Eq. (1) and the $t$-$J^{z}$ model
on a Bethe lattice was studied earlier in [40, 45, 46], for completeness let
us describe it in detail here. It is straightforward to generalize Eq. (25) to
any $z>2$. The subsequent consideration also apply, especially the equivalence
of the Green’s functions shown in Eq. (III.3), with the shape of the bosonic
potential as given in (32). More generally, equality (III.3) holds true for
any coordination number $z$ and the choice of $J_{n}$

$\begin{split}J_{0}&=\frac{J^{z}}{2}(z-1),\\\ J_{n\neq
0}&=\frac{J^{z}}{2}(z-2)d_{n}\end{split}$ (34)

and the site $n=0$ is the site at which the hole is originally introduced in
our lattice (see Fig. 8). While the consideration about the energy shift due
to the presence of a hole stay the same, generalizing the energy equivalence
for the cases when magnons are present in (III.3) requires some attention,
since now we are considering a Bethe lattice with $z>2$ and there are more
neighboring sites. However, the analysis of states reachable through
$\mathcal{H}_{t}$ still holds true, so that the motion of the hole on a Bethe
lattice still results in a chain of magnons connecting the hole to the site
where it was initially created. Again, there are no other magnons in the
system, since only the hole dynamics can result in magnon creation. Thus, the
magnons and the hole form a 1D-like chain on the Bethe lattice branch.

It is again necessary to consider three distinct cases of magnon chains of
length $m=0$, $m=1$ and $m>1$. For the $m=0$ case, the same conclusions as in
the $z=2$ case apply, so that
$\mathcal{H}=\mathcal{H}_{t}+\mathcal{H}_{J^{z}}$. However, we now generalize
the energy cost of creating a magnon to include any value of $z$. This cost is
found to be ${zJ^{z}}/{2}$. Again, there will always be only one single hole-
magnon interaction (even for longer effective chains) contributing an energy
$-{J^{z}}/{2}$. Thus, setting the cost of the first magnon to
$J_{0}=({J^{z}}/{2})(z-1)$, we see that also for states with one magnon we
have $\mathcal{H}=\mathcal{H}_{t}+\mathcal{H}_{J}^{z}$. For $m>1$ we need to
take into account the magnon-magnon interaction. Since magnons form a 1D-like
chain, then number of such interactions will be $m-1$ and each one of them
will contribute an energy $-J^{z}$, cf. [46]. This lowers the energy of every
magnon beyond the first one, so that choosing
$J_{n>0}=({J^{z}}/{2})(z-2)d_{n}$ we show that indeed
$\mathcal{H}=\mathcal{H}_{t}+\mathcal{H}_{J^{z}}$ for the whole class of
reachable states.

Note that, when $z=3$, we recover (32). We can conclude that Fig. 7 exactly
shows the spectral function for a single hole moving on a Bethe lattice with
$z=3$ hosting the $t$-$J^{z}$ model, depicted in Fig. 8.

String-potential with defects The most accurate realisation of a fermion-boson
system with bosons subject to a string potential with defects as considered by
Eq. (30) is as follows. Let us assume that we have a system which consists of
two kinds of subsystems: an Ising ferromagnet and an Ising antiferromagnet.
Next, we put these two subsystems on a Bethe lattice in such a way that the
antiferromagnet surrounds the origin of the Bethe lattice and the ferromagnet
starts $l$ sites away from the origin. Finally, once we probe such a system
with the local fermion Green’s function, by putting a mobile hole into the
origin of the Bethe lattice, then such a problem is described exactly by model
(1)-(4) with the potential (30). This is because: (i) the hole in the
antiferromagnetic subsystem is described by (1) with the constant bosonic
energies $J_{n}$ given by Eq. (32) – see discussion immediately above, (ii)
the hole in the ferromagnetic subsystem can freely move without introducing
spin flips – which in the language of model (1) means that the hole excites
magnons with zero-energy, i.e. $J_{n}=0$ for $d_{n}>l$ on a Bethe lattice.

Figure 9: An example of an antiferromagnetic domain of size $d=3$ embedded in
a ferromagnet on a Bethe lattice with $z=3$.

The aforementioned ferromagnetic-antiferromagnetic system can be realised by
considering antiferromagnetic domains of size $\propto l$ immersed in a
ferromagnet, as sketched in Fig. 9. Such a situation, to some extent, may
occur in the overdoped cuprates. In this case the tendency to ferromagnetism
upon doping [24, 25, 26, 27] and diminishing antiferromagnetic correlations
with doping [28, 29] may, in the first-order quantum phase transition scenario
[24] or phase separation scenario [30], locally lead to the onset of
antiferromagnetic domains within the ferromagnetic background. Naturally,
there are several differences between the cuprate models (such as doped
$t$–$J$ or Hubbard) and the model considered here – the main ones concern the
Heisenberg (and not Ising) spin exchange, the finite number of holes (and not
just one single hole), and the 2D square lattice (and not Bethe lattice with
$z>2$). Nevertheless, we believe that these differences are not important
enough to fully hinder the applicability of the current study to the cuprate
problem. Thus, the interesting insight gained from the problem under study
here is that, if the antiferromagnetic domains are very small and the system
is ‘almost’ ferromagnetic, a fermion inserted into the antiferromagnetic
subsystem may not form a well-defined quasiparticle (especially if the
antiferromagnetic exchange is small). On the other hand, if the
antiferromagnetic domains are relatively large (compared to the ferromagnetic
ones), the quasiparticle may be stabilised relatively easily. This
counterintuitive conclusion can be tested experimentally by STM studies of the
overdoped cuprates.

A somewhat similar situation can occur in correlated systems with orbital
degrees of freedom and incommensurate filling – such as doped manganites or
vanadates [31, 32, 33, 34, 35]. In this case hole doping induces transition
from the alternating orbital to the ferro-orbital state. Thus one may expect
that upon moderate doping, and assuming the onset phase separation, domains
with alternating orbital order would become surrounded by the ferro-orbital
ordered state. Then, depending on the relative size of such alternating
orbital domains, the quasiparticle decay may (small domains) or may not (large
domains) happen . Note that in the orbital case we are closer to model Eq. (1)
with (30) than in the cuprate case described above, since the orbital degrees
of freedom interact in correlated systems in an almost Ising manner [52].

One can also think of other realisations of such a string potential with
defects. This may for instance occur once a hole is introduced in the
antiferromagnetic side of a ferromagnetic-antiferromagnetic interface (or
alternatively on the alternating orbital side of a ferro-orbital-alternating
orbital interface). Here, however, a detailed investigation is needed, for the
topology of such a problem is quite different: in this case one of the
subsystems is not entirely surrounded by the other one. Nevertheless, the
intuition gained by the current study suggests that also in this case the
quasiparticle could decay if the size of the ferromagnetic (or ferro-orbital)
subsystem is significantly larger than the antiferromagnetic (alternating
orbital) one. Finally, one also speculates that there may exist also other
physical systems with fermions coupled to zero-energy bosons in one subsystem
and to bosonic excitations with finite energy in another. This might be the
case in a rather exotic situation in which one of the subsystems contains
condensed bosons in real space and the other one is a ‘normal’ state with
bosonic excitations costing finite energy.

##  V Discussion & Conclusions

In this work, we have considered an interacting fermion-boson model with
Peierls coupling between fermions and local bosons. We have solved it
analytically by mapping it to an impurity-like non-interacting chain. The
fermion-boson coupling introduced in model (1) can take several forms, but we
have restricted ourselves to two important cases: the impurity-like and
string-like bosons (future studies of other cases are encouraged). We have
shown how in these two cases it is possible to destabilize the fermionic
quasiparticle either by tuning the relative strength of the fermion-boson
coupling or by increasing the number of zero-energy bosons:

When considering impurity-like bosons in Sec. III, one can tune the relative
strength of the fermion-boson coupling $t/J$ such that no quasiparticle
appears in the system for $J\leq J^{*}$ (for fixed $t=1$), with $J^{*}$
depending on the Bethe lattice coordination number $z$. On the other hand, for
$J>J^{*}$ the quasiparticle is stabilised. Overall, the critical value $J^{*}$
increases with the coordination number $z$ – but $z=2$ is a limiting case with
$J^{*}=0$, i.e., we always obtain a quasiparticle solution (see below). Note
that for the impurity-like bosons the number of zero-energy bosons cannot be
tuned (it equals the number of lattice sites minus one, for the bosons have a
hard core). Thus, it is interesting to observe that, despite the coupling to
an overall huge number of massless bosons, a fermionic quasiparticle is still
stable once $J>J^{*}$.

For string-like bosons as in Sec. IV, the situation is more complex, for we
can tune here both the number of zero-energy bosons as well as the coupling
between fermions and bosons. First, just as for the impurity-like bosons, an
increase in the relative fermion-boson coupling strength $t/J$ can destabilize
the quasiparticle. Second, the stability of a quasiparticle depends on the
number of bosons with zero-energy. On the one hand, if there are no zero-
energy bosons, the hole is affected by a discrete linear potential (string
potential), since each created boson costs energy. In this case all of the
eigenstates of the system are of quasiparticle-type and we obtain the so-
called ladder spectrum (see below). On the other hand, the ladder spectrum of
the perfect string-potential is rather fragile, since a finite number of
bosons with zero-energy will result in a decrease in the number of
quasiparticles, the emergence of an energy continuum and, for the critical
number of these bosons, the onset of a completely incoherent spectrum. The
latter takes place for a relatively large number of massless bosons and strong
fermion-boson coupling $t/J$.

Lastly, we have mapped the fermionic Green’s function of the fermion-boson
model in specific limits to that of the single hole in the $t$-$J^{z}$ model.
We have shown that: (i) the fermion-boson model with impurity-like bosons and
the coordination number $z=2$ corresponds to the 1D $t$–$J^{z}$ model; (ii)
the fermion-boson model with string-like bosons subject to a perfect string
potential and with coordination number $z>2$ corresponds to the quasi-2D
(Bethe lattice) $t$–$J^{z}$ model. Note that these two particular limits carry
a quasiparticle solution for any finite value of the model parameters, as
discussed above and as well-known from the extensive $t$–$J^{z}$ model
literature, cf. [36, 37, 15, 38, 39, 40, 41, 42, 43, 44, 45, 46].

Finally, it is possible to slightly modify these particular limits of the
fermion-boson model to destabilise the quasiparticle solution (thus, e.g. in
1D one can scale the first hopping of the hole in the $t$–$J^{z}$ model; in 2D
one can add an Ising ferromagnetic interface next to the Ising
antiferromagnet). Nevertheless, it is a stunning observation of this work that
the parameter range for which the quasiparticle decay happens is small when
compared to the range for which it is stable – and this is despite the
ubiquitous presence of zero-energy bosons in the system.

Examples of boson-fermion systems in which the quasiparticle collapse may
happen include systems with a low concentration of impurities and mobile
fermions (for instance created in the intermediate state of RIXS experiment).
Another example concerns fermions introduced to antiferromagnetic domains
immersed in the ferromagnetic background or alternating orbital domains
immersed in the ferro-orbital background. Rather counterintuitively, the
quasiparticle extinction in these cases is more likely once the ferro-ordered
state dominates. Such a situation might take place in overdoped cuprates – or
other doped transition metal oxides with orbital degrees of freedom (e.g.
manganites or vanadates). Further experimental and theoretical studies are
needed to verify the latter proposal.

## Acknowledgements

This work was supported by Narodowe Centrum Nauki (NCN, Poland) under Project
Nos. 2016/22/E/ST3/00560 and 2021/40/C/ST3/00177.

For the purpose of Open Access, the author has applied a CC-BY public
copyright licence to any Author Accepted Manuscript (AAM) version arising from
this submission.

The code to reproduce the data and figures presented in this manuscript is
available at Ref. [53].

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