# Ergodic inclusions in many body localized systems

Luis Colmenarez<EMAIL_ADDRESS>Institute for Quantum
Information, RWTH Aachen University, 52056 Aachen, Germany  Institute for
Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, 52428 Jülich,
Germany  Max Planck Institute for the Physics of Complex Systems, Nöthnitzer
Straße 38, 01187-Dresden, Germany  David J. Luitz<EMAIL_ADDRESS>Institute of Physics, University of Bonn, Nußallee 12, 53115 Bonn, Germany
Wojciech De Roeck<EMAIL_ADDRESS>KU Leuven, 3001 Leuven,
Belgium

(August 28, 2024; August 28, 2024)

###### Abstract

We investigate the effect of ergodic inclusions in putative many-body
localized systems. We consider the random field Heisenberg chain, which is
many-body localized at strong disorder and we couple it to an ergodic bubble,
modeled by a random matrix Hamiltonian. Recent theoretical work suggests that
localized systems are unstable to ergodic bubbles, driving the delocalization
transition. We tentatively confirm this by numerically analyzing the response
of the on-site purities to the insertion of the bubble. For a range of
intermediate disorder strengths, this response decays very slowly, or not at
all, with increasing distance to the bubble. This suggests that at those
disorder strengths, the system is delocalized in the thermodynamic limit.
However, artefacts in the numerics cannot be ruled out.

##  I Introduction

The discovery that noninteracting particles in a disorder potential can become
completely immobile by P. W. Anderson in 1958 Anderson (1958) has created a
new field of study, which became enormously active in the last decade after it
was predicted that this localized phase could persist in the presence of
interactions, leading to a _perfect insulator_ at any temperature Fleishman
and Anderson (1980); Gornyi  _et al._ (2005); Basko  _et al._ (2006);
Oganesyan and Huse (2007); Imbrie (2016a, b). While it is difficult to realize
such systems in condensed matter experiments due to the presence of phonons,
many-body localized (MBL) systems were realized in synthetic quantum matter
Schreiber  _et al._ (2015); Lüschen  _et al._ (2017a); Smith  _et al._ (2016).
It was first suggested that the interplay of interaction and disorder gives
rise to a non-equlibrium phase transition between a thermal phase at weak
disorder, which satisfies the eigenstate thermalization hypothesis (ETH)
Deutsch (1991); Srednicki (1994); Rigol  _et al._ (2008); D’Alessio  _et al._
(2016) and a many-body localized phase at strong disorder. Such a transition
would be unparalleled in equilibrium Nandkishore and Huse (2015); Vosk  _et
al._ (2015); Abanin and Papić (2017); Imbrie  _et al._ (2017); Luitz and Lev
(2017); Agarwal  _et al._ (2017); Alet and Laflorencie (2018); Abanin  _et
al._ (2019). A large body of theoretical work now supports the picture that
the many-body localized phase is characterized by an emergent complete set of
quasi-local integrals of motion Serbyn  _et al._ (2013a); Huse  _et al._
(2014), which are fully consistent with the observed phenomenology of area law
entanglement in all many-body eigenstates Bauer and Nayak (2013); Luitz
(2016); Yu  _et al._ (2016) as well as with the logarithmic post-quench
entanglement production Chiara  _et al._ (2006); Žnidarič  _et al._ (2008);
Bardarson  _et al._ (2012); Serbyn  _et al._ (2013b).

Recently, however, the stability of the insulating phase in the thermodynamic
limit has been put into question, generating a debate on MBL as a phase of
matter Abanin  _et al._ (2021); Panda  _et al._ (2020); Ghosh and Žnidarič
(2022); Kiefer-Emmanouilidis  _et al._ (2021); Luitz and Lev (2020);
Morningstar  _et al._ (2022); Sels (2022); Sels and Polkovnikov (2021, 2023);
Sierant  _et al._ (2020a); Šuntajs  _et al._ (2020a, b); Sierant  _et al._
(2020b); Sierant and Zakrzewski (2022). Some works suggest that the critical
disorder might be way higher than what was expected Morningstar  _et al._
(2022); Sels (2022); Weiner  _et al._ (2019); Crowley and Chandran (2022);
Doggen  _et al._ (2018), others even predict an infinite critical disorder in
the thermodynamic limit Sels and Polkovnikov (2021, 2023); Kiefer-
Emmanouilidis  _et al._ (2021); Šuntajs  _et al._ (2020b); Evers and Bera
(2023). As a matter of fact, quantum simulations have shown MBL signatures in
systems with few tens of particles Schreiber  _et al._ (2015); Smith  _et al._
(2016); Kondov  _et al._ (2015); Gong  _et al._ (2021), thus today’s
experiments can only access a MBL regime that asymptotically, in time and
system size, may or may not thermalize. One central aspect in this debate is
the delocalization transition mechanism and the crossover behavior in finite
systems, whose understanding is still incomplete. In the vicinity of the
transition, anomalously slow dynamics was observed Bar Lev  _et al._ (2015);
Agarwal  _et al._ (2015); Luitz  _et al._ (2016); Lüschen  _et al._ (2017b);
Lezama  _et al._ (2019), which was related to anomalous thermalization
behavior Luitz (2016); Luitz and Bar Lev (2016); Roy  _et al._ (2018);
Colmenarez  _et al._ (2019); Žnidarič  _et al._ (2016) and a theoretical
description based on rare insulating inclusions was proposed Gopalakrishnan
_et al._ (2016); Potter  _et al._ (2015); Vosk  _et al._ (2015). It remains
however unclear how this picture can be reconciled with the observation of
slow dynamics in quasiperiodic potentials Lüschen  _et al._ (2017b); Lev  _et
al._ (2017); Luitz and Lev (2017). It was also proposed that the many-body
resonances are driving the slow dynamics in this regime Long  _et al._ (2022).

Figure 1: System setup: A spin chain (blue) is coupled to an ergodic bubble
(red). The spins at site 0 and $L-1$ are both coupled to the bubble with
coupling strength $g$, such that both ”ends” of the chain are symmetric. In
order to make the bubble perfectly thermal, its Hamiltonian is given by a
random GOE matrix, while the spin chain corresponds to a XXZ spin $1/2$ model
in the presence of a random field coupled to local $\hat{S}_{i}^{z}$ operators
and nearest neighbour interactions.

It was pointed out in Ref. De Roeck and Huveneers (2017) that many-body
localized systems are unstable under certain conditions towards thermal
inclusions by a mechanism dubbed “avalanche” De Roeck and Imbrie (2017);
Thiery  _et al._ (2018), and such a transition as a function of the
localization length was confirmed numerically in idealized models Luitz  _et
al._ (2017); Crowley and Chandran (2020), also tailored to address
implications for the instability of MBL in higher dimensions De Roeck and
Huveneers (2017); Ponte  _et al._ (2017); Potirniche  _et al._ (2019). Current
activities now focus on the identification of this mechanism driving the
transition in more realistic models Goihl  _et al._ (2019); Sels and
Polkovnikov (2023); De Tomasi  _et al._ (2021); Tu  _et al._ (2023a, b) and
experiments Rubio-Abadal  _et al._ (2019); Léonard  _et al._ (2023). This is
challenging and so far direct evidence for the avalanche mechanism in standard
MBL models is still lacking. In this work, we directly address the issue of
avalanches in a standard MBL model and consider a thermal inclusion coupled to
a disordered spin chain.

##  II Model

We study the random field XXZ chain, modeled by the Hamiltonian
$H_{\text{XXZ}}$ (“the chain”) coupled to an ergodic (or “thermal”) bubble
with Hamiltonian $\hat{\mathsf{R}}_{0}$. The setup for the model is
illustrated in Fig. 1. The total Hamiltonian is of the form

$\hat{H}=\hat{H}_{\text{XXZ}}\otimes\hat{\mathsf{1}}+\hat{\mathsf{1}}\otimes\hat{\mathsf{R}}_{0}+gH_{\text{coupling}}.$
(1)

Here the first factor of the tensor product refers to the chain and the second
factor refers to the thermal bubble. The XXZ Hamiltonian acts on $L$ spins
labelled by $i=0,\ldots L-1$;

$\begin{split}\hat{H}_{\text{XXZ}}&=J\sum_{i=0}^{L-2}\left[\tfrac{1}{2}\left(\hat{S}_{i}^{+}\hat{S}_{i+1}^{-}+\hat{S}_{i}^{-}\hat{S}_{i+1}^{+}\right)+\Delta\hat{S}_{i}^{z}\hat{S}_{i+1}^{z}\right]\\\
&+\sum_{i=0}^{L-1}h_{i}\hat{S}_{i}^{z}.\end{split}$ (2)

where we have taken the random fields $h_{i}\in[-W,W]$ drawn from a box
distribution. We consider the isotropic point $\Delta=1$. All parameters are
measured in $J=1$ units. The coupling term reads:

$\begin{split}\hat{H}_{\text{coupling}}&=(\hat{S}^{z}_{0}\otimes\hat{\mathsf{1}})\cdot(\hat{\mathsf{1}}\otimes\hat{\mathsf{R}}_{1})\cdot(\hat{S}^{z}_{L-1}\otimes\hat{\mathsf{1}})\\\
&+[(\hat{S}^{+}_{0}\otimes\hat{\mathsf{1}})\cdot(\hat{\mathsf{1}}\otimes\hat{\mathsf{R}}_{2})\cdot(\hat{S}^{-}_{L-1}\otimes\hat{\mathsf{1}})+h.c.]\end{split}$
(3)

where the matrices $\hat{\mathsf{R}}_{l},l=0,1,2$ are independent random
matrices from the Gaussian Orthogonal Ensemble (GOE), with a scaled bandwidth
defined by

$\mathsf{R}_{l}=\frac{\beta}{2}\left(\mathsf{A}+\mathsf{A}^{T}\right)\in\mathbb{R}^{n_{\text{GOE}}\times
n_{\text{GOE}}},\quad\mathsf{A}_{ij}=\text{norm(0,1)}.$ (4)

where norm(0,1) are normal random variables with zero mean and unit variance.
The dimension $n_{\text{GOE}}$ of the random matrices controls the power of
the ergodic bubble, here we use $n_{\text{GOE}}=3,4,5,6,8$. $\beta$ is a real
variable that controls the band width of the random matrices and is chosen
such that the level mixing (reflected in the overall gap ratio) is maximal for
the largest possible range of parameters (see SM Sec. VI.1 for more details).

Note that the ergodic bubble is coupled to spins $0$ and $L-1$, thus closing
the chain into a ring. Therefore, both ends of the chain are symmetric and the
longest distance from the bubble corresponds to the spin situated at the
middle of the chain $i=\lfloor L/2\rfloor$. We also note that the coupling in
Eq. (3) of the bubble to the chain is chosen such that all terms preserve the
total spin $\hat{S}_{z}=\sum_{i=0}^{L-1}\hat{S}_{i}^{z}$ in the chain. This
allows us to restrict our study to the largest sector of the Hilbert space,
given by $S_{z}=0$ for $L$ even and $S_{z}=1$ for $L$ odd, with a Hilbert
space of dimension
$\text{dim}\left(\mathcal{H}\right)=n_{\text{GOE}}\binom{L}{\lfloor
L/2\rfloor}$. We perform massively parallel shift-invert diagonalization Luitz
_et al._ (2015); Pietracaprina  _et al._ (2018) for computing eigenstates
close to the exact spectral center of each sample given by
$(E_{\text{max}}+E_{\text{min}})/2$, with ($E_{\text{min}}$)$E_{\text{max}}$
being the (anti)ground state energy. We confront results for our system with a
thermal bubble, Eq. (1) with the isolated XXZ chain Eq. (2).

Numerical protocol We consider two types of observables. Firstly, we compute
the $r_{n}$-parameter of adjacent energy gaps in the middle of the spectrum
given by
$r_{n}=\text{min}(\delta_{n},\delta_{n+1})/\text{max}(\delta_{n},\delta_{n+1})$
Oganesyan and Huse (2007), with $\delta_{n}=E_{n+1}-E_{n}$. Here
$E_{n-1},E_{n},E_{n+1}$ are consecutive energy levels of the XXZ chain, and
the system with a bubble, respectively. This parameter is useful to
distinguish the ergodic and localized phase in a simple way.

Second, we use the single site purity $\gamma_{i}=\mathrm{Tr}(\rho^{2}_{i})$,
where $\rho_{i}=\mathrm{Tr}_{\\{L-i\\}}(|n\rangle\langle n|)$ is the reduced
density matrix on a single site $i$ for a given eigenvector $|n\rangle$. Due
to the $U(1)$ symmetry of the model, $\gamma_{i}$ can be conveniently
expressed via the matrix element $\langle n|S^{z}_{i}|n\rangle$ as (a brief
analysis of these matrix elements is presented in SM Sec. VI.2 )

$\gamma_{i}=2\langle n|S^{z}_{i}|n\rangle^{2}+\dfrac{1}{2}.$ (5)

In an ergodic system, the ETH and random matrix theory predict that
$\gamma_{i}-1/2\sim d^{-1/2}$ where $d$ is the Hilbert space dimension, at
least for states $|n\rangle$ chosen at maximal entropy D’Alessio  _et al._
(2016). The matrix elements $\langle n|S^{z}_{i}|n\rangle$ by themselves show
qualitative signatures of thermalization, see SM Sec. VI.2 for a short
discussion.

In models where ergodicity is induced from boundary effects, as it presumably
happens in the case investigated here, one should be more careful and write
$\gamma_{i}-1/2\sim d^{-1/2}_{\text{eff},i}$ where $d_{\text{eff},i}$ is the
_effective dimension_ , see Luitz (2016) and SM. In contrast, in a localized
system, we expect $\gamma_{i}$ to depend substantially on the state
$|n\rangle$ and the disorder realization, and we expect the average
$\bar{\gamma}_{i}$ to be given by a volume-independent value, tending to $1$
as $W\rightarrow\infty$. For each disorder realization, we use 50…100
eigenstates $|n\rangle$ and for each set of model parameters at least 2000
disorder realizations of the fields $h_{i}$ and bubble Hamiltonian
$\hat{R}_{l}$.

Figure 2: Average gap ratio Oganesyan and Huse (2007) $\overline{r}$ for the
disordered field XXZ spin chain of size $L$ coupled to a thermal bubble of
size $n_{\text{GOE}}=4$ (left) and the same system _without_ bubble and chain
length $L+2$ (right). Inset: Pair wise crossing of gap ratio $\overline{r}(W)$
as function of $1/L$ for both systems.

##  III Shift of MBL transition in the interacting chain

In an infinitely long disordered chain, it should not matter whether we add a
thermal bubble or not, because thermal Griffiths regions acting as ergodic
patches are expected to be present anyhow. In a chain of moderate size, we
expect that a substantial fraction of samples appears to be localized simply
by lack of ergodic regions. This would lead to a shift of the apparent
critical disorder value $W_{c}$ for short chains if one compares the isolated
chain to our model where we add a bubble by hand. In Fig. 2, we compare the
disorder averaged gap ratio $\overline{r}$ of adjacent energy gaps in the
middle of the spectrum given of the isolated XXZ chain (right) to XXZ chain-
bubble system (left) as a function of disorder strength $W$ for different
system sizes $L$. Using the crossing of $\overline{r}$ of size $L$ and $L+2$
as a proxy for the apparent critical point at this length scale, the inset
shows that it is indeed the case that the crossings appear at slightly larger
disorder strengths in the presence of the bubble. However, due to large
statistical uncertainty and significant finite size effects, this crossing
analysis is not a reliable way to pin down the critical point.

##  IV Change of local thermality in interacting chains

Figure 3: Purity $\gamma_{i}$ as function of the distance from the bubble for
different disorder strengths $W$. Left (Right) panel shows system size $L=16$
($L=18$) and $n_{\text{GOE}}=4$. Horizontal shaded areas correspond to the
$\gamma_{\text{XXZ}}$ in the absence of the bubble with $L+2$ chain length
compared to the bubble-chain system. Figure 4: Purity $\gamma_{i}$ at fixed
distance $i=5$ from the bubble plotted as function of system size $L$ for
different disorder strengths $W$. Left panel: XXZ spin chain of length $L$
with bubble of size $n_{\text{GOE}}=4$. Right panel: XXZ Heisenberg chain of
size $L$. Average is taken over disorder realizations and eigenstates.  Figure
5: Average purity correlation $C(r)$ as function of the distance from the
bubble for disorder strengths $W=4.4,5.2,6.0,8.0$. Dashed lines correspond to
bubble-less system of size $L+2$ while solid lines to bubble-full system of
size $L$

We investigate the one-site purities $\gamma_{i}$ as a measure of local
thermality. Throughout this paper, we consistently use the labels “localized”
and “thermal” as inferred from the crossing point of $\overline{r}$ (see Fig.
2), i.e. with the critical point at $W\approx 3.5-4$, even though recent works
locate the transition at larger $W$. Inspecting the average purities in the
presence and absence of the thermal bubble (see Fig. 3) is clear that the
effect of the bubble at long distance is rather small and subtle. Therefore,
it is useful to first map out what the theory of quantum avalanches predicts.

Theoretical background For a simple avalanche model of our setup, we assume
the existence of an _apparent_ localization length $\xi$, describing the
system in absence of the bubble, and a number $p$ giving the probability that
the bubble can kickstart a thermalization process in its near vicinity, see
Ref. Crowley and Chandran (2020) for a detailed discussion and arguments why
$p\ll 1$ in our setup. A simple model, based on an unrealistic dichotomy
between ergodicity and localization, leads (see SM) to the following very
rough prediction:

$(\bar{\gamma}_{i}-1/2)\approx(1-p)(1-e^{-|i|/\xi})(\bar{\gamma}_{\mathrm{XXZ}-1/2})+pd^{-1/2}_{\text{eff},i}$
(6)

with $\bar{\gamma}_{i}$ the average purity at site $i$ and
$\bar{\gamma}_{\mathrm{XXZ}}$ the average purity in the bubble-less system.
For $p=0$, we recover a perfectly localized system, whereas for $p=1$, the
system is ergodic but the average purity still increases with $i$, because the
effective dimension $d_{\text{eff},i}$ depends on the distance to the bubble.
This effective dimension is defined as
$d_{\text{eff},i}=e^{-2i/\xi}d_{\mathrm{therm}}$ with $d_{\mathrm{therm}}$ the
total Hilbert space dimension of the thermal region, given by
$d_{\mathrm{therm}}=n_{\text{GOE}}^{\frac{1}{1-\xi/\xi_{*}}}$ if $\xi<\xi_{*}$
and $d_{\mathrm{therm}}=n_{\text{GOE}}2^{L}$ if $\xi>\xi_{*}$, where
$\xi_{*}=1/\log 2$ is the critical localization length. According to the above
formula Eq. (6), there are two regimes in which
$\bar{\gamma}_{\mathrm{XXZ}}-\bar{\gamma}_{i}$ does not decay to zero, or only
very slowly.

  * I)

$\xi>\xi_{*}$. Here, the bubble thermalizes a fraction $p$ of samples,
resulting in a shift $\bar{\gamma}_{\mathrm{XXZ}}-\bar{\gamma}_{i}$ which
remains finite as $i\to\infty$, even though it decreases due to the decrease
of $d_{\text{eff},i}$.

  * II)

$\xi$ approaches $\xi_{*}$ from below. Then
$\bar{\gamma}_{\mathrm{XXZ}}-\bar{\gamma}_{i}\to 0$ at large $i$, but the
decay is arbitrarily slow when $\xi\to\xi_{*}$, because the decay of
$d_{\text{eff},i}$ is arbitrarily slow.

In practice, I) and II) are of course hard to distinguish.
Large distance behaviour In Fig. 3, we indeed see a sign of influence of the
bubble that does not, or only very slowly, decay with distance from the
bubble. We observe that the purity in the presence of the bubble seems to tend
to an asymptotic value that is significantly lower than the value for the
bubble-less system. This seems to be the case up to disorder strength
$W=5.6-6$, after which the signal, i.e. the difference
$\bar{\gamma}_{\mathrm{XXZ}}-\bar{\gamma}_{i}$, becomes comparable to the
error bars. We will analyze this further below, but let us first note that it
is far from clear whether our system sizes are large enough to speculate about
the limit $i\to\infty$. Indeed, in Fig. 4, we investigate the dependence of
the purity at fixed distance on increasing system size $L$. We see that for
disorder values at and above $W=5.2$, the purity $\gamma_{5}$ is actually
increasing with increasing system size. This is an effect that is not even
accounted for in our model equation and it should be interpreted as a sign
that finite-size effects are still important (see SM material Sec. VI.3 for
further analysis). Another way to look for thermalization induced by the
bubble is through correlations between different sites of the chain. In that
spirit we look at statistical correlations between two purities $\gamma_{i}$
and $\gamma_{j}$. Those are defined as

$C(r)=\dfrac{\overline{(\gamma_{0}-\bar{\gamma}_{0})(\gamma_{r}-\bar{\gamma}_{r})}}{\sigma_{0}\sigma_{r}},$
(7)

where $\gamma_{i}-\bar{\gamma}_{i}$ captures the purity fluctuations around
its mean and
$\sigma^{2}_{i}=\overline{(\gamma_{i}-\overline{\gamma_{i}})^{2}}$ is the
variance of the purity at site $i$. The quantity $C(r)$ is the Pearson
correlation coefficient between purities at site next to the bubble ($i=0$)
and sites at distance $r$ from that spin with $r=0,1,...,L/2$. This quantity
for different disorder and system size is shown in Fig. 5. Interestingly, for
$W=4.4$ the bubble seems to be enhancing correlations compare to the bubble-
less case. This effect is even growing with system size and it persists for
$W=5.2$ and, up to some extent, for $W=6.0$.

Since the signals that we observe, the difference
$\bar{\gamma}_{\mathrm{XXZ}}-\bar{\gamma}_{i}$ and the correlations in Fig. 5,
are rather small, there is the concern that they might be caused by some
artefact. For example, even in a well-localized system, in the absence of any
avalanches, the purity-purity correlation $C(r)$ from Eq. (7) has a non-zero
limit as $r\to\infty$ of order $\frac{C}{L^{2}}$ with
$C\propto\frac{1}{W^{2}}$ in the high-disorder limit $W\to\infty$, see
Appendix for more details. While we do not see any clear mechanism how this
might pollute our analysis, we find it hard to exclude it.

##  V Conclusion

We have studied the effect of ergodic inclusions modeled by local random
matrices in disordered spin chains of up to 20 sites. We report little or no
drift of the thermal-to-localized transition in the average level spacing
compare to the bubble less-case, which was also observed in similar settings
Goihl  _et al._ (2019). We also investigate long distance effects of the
bubble by looking at functions of the local magnetization expectation value,
in this case the purity and its fluctuations. There, we have numerically
witnessed a potentially divergent, long wavelength, response of an apparently
localized chains (as estimated by ED studies like Luitz  _et al._ (2015)) to
an ergodic bubble. The effect is weak and not unambiguous, but it is
compatible with the avalanche theory proposed in earlier works. Importantly,
such weak correlations may be influenced by spurious long-range correlations
due to the $U(1)$ symmetry (see Sec. VI.5 in SM), although we can not confirm
this from our numerics.

Recent studies also point out weak but persisting correlations produced by
many-body resonances in the same disorder regime we study Morningstar  _et
al._ (2022); Crowley and Chandran (2022). Our results suggest that the bubble
is effectively enhancing such weak correlations when the system looks well
localized in average. However it is yet not clear how to directly relate the
observed signal with the many-body resonances.

###### Acknowledgements.

L.C. gratefully acknowledges funding by the U.S. ARO Grant No.
W911NF-21-1-0007. All statements of fact, opinion or conclusions contained
herein are those of the authors and should not be construed as representing
the official views or policies of the US Government. This project was
supported by the Deutsche Forschungsgemeinschaft (DFG) through the cluster of
excellence ML4Q (EXC 2004, project-id 390534769). We further acknowledge
support from the QuantERA II Programme that has received funding from the
European Union’s Horizon 2020 research innovation programme (GA 101017733),
and from the Deutsche Forschungsgemeinschaft through the project DQUANT
(project-id 499347025).

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##  VI Supplementary material

###  VI.1 Parameters of the model

In this model two systems, the spin chain and the bubble are coupled via
$H_{\mathrm{coupling}}$ given by Eq. (3), where Eq. (4) shows the definition
of the random matrices that described the bubble Hamiltonian. For coupling
efficiently both systems their spectra should have comparable band widths
which allows maximum possible mixing of their eigenstates. The parameter
$\beta$ controls the average band width of the random matrices, then an
optimal choice of $\beta$ enhances the mixing of the two systems and therefore
maximizes the probabilities of the avalanche to happen. In order to do so, the
average gap ratio $\langle r\rangle$ is computed for different values of band
width $\beta$ at several disorder strength (Fig. 6). We see that $\langle
r\rangle$ always has maxima and they happen to be almost at the same $\beta$.
Choosing $\beta$ such that the average gap ratio is always maximum ensures
that the spin chain and the bubble are strongly coupled. For $1.0<W<4.0$ and
$n_{\text{GOE}}=3,4,6,8$ we found that $\beta=0.55,0.6,0.55,0.5$ respectively
satisfy this requirement. It could be that at $W>4.0$ the band width we chose
is de-tuned from the optimal value by a bit, however we do not expect this to
change the main results since the maxima blunt and the curves become flatter
when disorder is increased.

Figure 6: Average gap ratio $\langle r\rangle$ as function of random matrix
band width $\beta$ for different $n_{\text{GOE}}={3,4,6,8}$. The black dashed
line stands for $\beta=0.55,0.6,0.55,0.5$ respectively the chosen values in
our simulations that cut off the plateus where the spin chain-bubble mixing is
maximal for the relevant parameters.

###  VI.2 Local magnetization

Figure 7: Distribution of local magnetization for $\mathrm{XXZ}$ chain with
(color lines) and without bubble (black lines) for system size $L=20$ and
$L=18$ respectively (bubble size $n_{\text{GOE}}=4$). Each color denotes the
spin at site $i$ from the bubble (see Fig. 1 for exact layout)

In this section we present the distribution of local magnetization in the
system with and without bubble (see Fig. 7). There we confirm that at low
disorder the bubble has no effect beyond a small shift to “more Guassian”
distributions. At disorder $W=4.0$, which is supposed to be in the critical
region, we see that the localized peaks at $\langle S^{z}\rangle=\pm 1$ loose
weight in favor of the a central “thermal” peak in the presence of the bubble.
The latest suggest that an important fraction of samples thermalizes in the
presence of the bubble. Unfortunately we could not quantify the fraction of
thermalizing samples in a more rigorous manner. A similar but more detailed
analysis of distributions was done in Ref.Luitz  _et al._ (2017) where non-
interacting localize bits were coupled to a ergodic bubble. Unfortunately
finite size effects are stronger in our current setup, thus avoiding
trustworthy conclusions from such distributions.

###  VI.3 Additional data

Figure 8: Purity $\gamma_{i}$ average over disorder realization and
eigenstates as function of bubble size $n_{\text{GOE}}$, several disorder
strengths $W$ and different distances $i$ from the bubble. The black line
corresponds to $\gamma_{\text{XXZ}}$ in absence of the bubble ($L=14$). The
chain length $L=14$ is fixed. $\beta=\\{0.55,0.6,0.58,0.55,0.5\\}$ is tuned to
be the optimal value for each bubble size $n_{\text{GOE}}=\\{3,4,5,6,8\\}$

.

Figure 9: Upper panel: difference
$\overline{\gamma}_{XXZ}-\overline{\gamma}_{i}$ for different disorder
strength, $L=18$ and $n_{\text{GOE}}=4$. Dashed lines are fittings
$\gamma_{0}+Ae^{-i/\xi}$, with $\gamma_{0}$ as the limiting value
$i\rightarrow\infty$ . Lower panel: $\gamma_{0}$ as function of $W$ for system
sizes $L=16,18$. In both panels, $\overline{\gamma}_{XXZ}$ is computed using
$L+2$ chain sites compared to the shown system sizes.

In this section we present a further analysis on the purity $\gamma_{i}$ on
localized $\mathrm{XXZ}$ chains with and without bubble. Although the system
sizes are small and quantitative results are not conclusive, we find
instructive to sketch the qualitative signals of thermalization due to the
presence of the bubble.

In Fig. 8, we study the limit where the bubble size $n_{\text{GOE}}$ growth
while the length of the chain is kept fixed. The most visible feature in these
plots is that the purity $\bar{\gamma}_{i}$ decreases when a bubble is coupled
to the chain, with bigger bubbles having a larger effect. This trend is
obvious and expected, because coupling an ergodic bubble can, on average, only
increase the thermality. Hence, this trend does not give us any non-trivial
information on the nature of the ergodic-to-MBL transition.

To go beyond this, we note first that at very large distance $i\to\infty$, one
expects that $\bar{\gamma}_{i}-\overline{\gamma}_{\mathrm{XXZ}}\to 0$
independently of whether the systems is localized or ergodic, simply because
in a very large system, arbitrarily large bubbles will occur naturally.
Numerically, at disorder strengths $W<7$, we observe however that the
difference $\overline{\gamma}_{XXZ}-\bar{\gamma}_{i}$ seems to decay very
slowly, if at all. In Fig. 9, we remark that this function is actually
reasonably fitted by an exponential decay law added to a nonzero constant
$\delta_{0}$. For $W\geq 7$, the deviation of $\delta_{0}$ from $0$ is no
longer statistically meaningful. It appears hence that for disorder strengths
$W<7$, there is a effect of the bubble at large distances (see discussion in
the main text). This effect is weak, for example $5\%$ at $W=5.2$, but;
however due to the smallness of the systems considered we can not rule out
attributing this signal to finite size effects.

###  VI.4 Quantum avalanche theory

Let us recall the avalanche theory De Roeck and Huveneers (2017) for a
disordered spin chain coupled to a thermal inclusion. If the chain is isolated
from the bubble, it is assumed to be many-body localized, forming local
integrals of motion (LIOM) Serbyn  _et al._ (2013a); Huse  _et al._ (2014). We
assume that the LIOMs can be described by a localization length $\xi$, whose
meaning is that a LIOM centered at some site of the chain is coupled with
strength $e^{-\ell/\xi}$ (operator norm of the coupling) to sites at distance
$\ell$. The ergodic bubble is assumed to be perfectly thermal and is described
by a Hilbert space of dimension $n_{\text{GOE}}$.

The logic of the theory is based on an iterative application of perturbation
theory: We start with the ergodic bubble and couple in the first step only the
LIOMs which have the largest overlap (and therefore coupling) with the bubble.
Those are the LIOMs centered at the sites adjacent to the ergodic bubble.
There are two of those sites because we consider a periodic chain. If the
coupling is non-resonant, we declare that the adjacent LIOMs have not been
thermalized, and the analysis stops there. If the coupling is resonant, then
we declare the adjacent spins thermalized and we imagine that the ergodic
region has now dimension $4n_{\text{GOE}}$ instead of $n_{\text{GOE}}$, i.e.
the LIOMs have been _absorbed_ by the ergodic bubble. This is now repeated,
proceeding further and further away from the original bubble, until either the
adjacent LIOMs are non-resonant, or until all LIOMs have been thermalized.

Let us make this quantitative. We assume that the bubble has thermalized all
LIOM’s within a distance $\ell$, leading to a delocalized (ergodic) space with
dimension $d_{\mathrm{therm}}=n_{\text{GOE}}4^{\ell}$. We will now only keep
the overall scaling of the relevant terms to obtain a criterion for
thermalization: We couple the next LIOM with strength of order
$e^{-\ell/\xi}$. We use random matrix theory to compute a typical matrix
element of such a coupling and the typical level spacing is estimated simply
as $\propto d_{T}^{-1}$. We then get for the quantity distinghuishing
resonance from non-resonance:

$\frac{\text{matrix element}}{\text{level spacing}}\sim
e^{-\ell/\xi}d_{\mathrm{therm}}^{1/2}=e^{-\ell/\xi}(d_{G}4^{\ell})^{1/2}$ (8)

The thermalization proceeds until this ratio is of order $1$, hence up to
$\ell=\ell_{c}$ given by

$\ell_{c}=(1/\xi-1/\xi_{*})^{-1}\tfrac{1}{2}\log
n_{\text{GOE}},\qquad\xi^{-1}_{*}=\log 2,$

where $\xi_{*}$ is the critical localization length. This formula shows that
$\ell_{c}$ is finite if $\xi<\xi_{*}$ and infinite otherwise. Due to the final
length of the system, the correct distance up to which LIOMS are thermalized
is of course the minimum of $\ell_{c}$ and $L/2$. The Hilbert space dimension
of the thermal region is then, for $\xi<\xi_{*}$,

$d_{\mathrm{therm}}=n_{\text{GOE}}4^{\ell_{c}}=n_{\text{GOE}}^{\frac{1}{1-\xi/\xi_{c}}}$

Taking also here into account the finite size, we arrive at the expression
given in the main text.

Let us now briefly comment on the _effective dimension_ which was mentioned in
the main text. This concept was introduced in Luitz  _et al._ (2017) as
follows. The spectral form factor of local observables at site $i$ have a
width that is way smaller than of the order of local energy scales, as it is
in a normal many-body system. Instead, this width is of order of the mean
level spacing times the effective dimension
$d_{\text{eff},i}=e^{2i/\xi}d_{\mathrm{therm}}$. This formula was also
verified numerically in Luitz  _et al._ (2017). It is the smallness of this
width, or, equivalently, the fact that $d_{\text{eff},i}$ is way smaller than
the total Hilbert space dimension, that gives rise to the extremely slow
dynamics in this system, as one goes far away from the ergodic bubble.

Finally, we return to equation (6), which can be viewed as a consequence of
the above discussion. In the case (occuring with probability $1-p$) the
ergodic bubble did not kick-start an avalanche, i.e. when the coupling to the
first LIOM was too weak to thermalize it, we naively approximate the influence
of the ergodic bubble on the purity as
$(\bar{\gamma}_{i}-1/2)\approx(1-e^{-|i|/\xi})(\bar{\gamma}_{\mathrm{XXZ}-1/2})$.
This simply reflects the fact that the purity at site $i$ is influenced by the
delocalized states in the ergodic bubble when $|i|/\xi$ is not too large. If
an avalanche was indeed started, we can use the theory described above. The
purity is computed from local observables and hence it can be computed
assuming that one has an ergodic system with dimension $d_{\text{eff},i}$.
Some simple random matrix theory leads then to the conclusion that this purity
is given by $d_{\text{eff},i}^{-1/2}$.

###  VI.5 Purity correlations

We compute the purity perturbatively in the high-disorder limit. This will
confirm the existence of a residual purity-purity correlation at large
distance.

####  VI.5.1 Perturbative expression for purity

We focus on the standard MBL model

$\displaystyle
H=\sum_{i}h_{i}\sigma^{z}_{i}+J(\sigma^{+}_{i}\sigma^{-}_{i+1}+hc)+J_{z}\sigma^{z}_{i}\sigma^{z}_{i+1}.$
(9)

Let $\eta$ be classical configurations $\eta_{i}=\pm 1$ and let
$h_{i}\in[-W,W]$. We assume that we are in the MBL phase and the eigenstates
connected to the classical configurations. We will actually do a perturbative
expansion in $J/W$ (locator expansion) that disregards resonant regions. It
yields an expression for the eigenstates given by

$\displaystyle|\Psi(\eta)\rangle=e^{iA}|\eta\rangle,$ (10)

where $A=\sum_{i}A_{i,i+1}$ is a sum of local Hermitian terms, determined by
the condition

$\displaystyle\langle\eta|A_{i,i+1}|\eta^{\prime}\rangle=\frac{1}{E^{(0)}(\eta^{\prime})-E^{(0)}(\eta)}\langle\eta|J(\sigma^{+}_{i}\sigma^{-}_{i+1}+h.c.)|\eta^{\prime}\rangle$
(11)

Depending on $\eta$, there is either zero or one $\eta^{\prime}$ such that
this matrix element is non-zero. We compute the unique non-zero matrix element
as

$\displaystyle\langle\eta|A_{i,i+1}|\eta^{\prime}\rangle=\frac{J(\eta_{i+1}-\eta_{i})}{4(h_{i+1}-h_{i})+2J_{z}(\eta_{i+2}-\eta_{i-1})}$
(12)

Now we compute

$\displaystyle\langle\Psi(\eta)|\sigma_{i}^{z}|\Psi(\eta)\rangle$
$\displaystyle=$
$\displaystyle\langle\eta|\left(\sigma_{i}^{z}+i[A,\sigma_{i}^{z}]-\tfrac{1}{2}[A,[A,\sigma_{i}^{z}]]\right)|\eta\rangle$
(14) $\displaystyle+O((J/W)^{3}).$

It is easy to see that the first order term $[A,\sigma_{i}^{z}]$ does not
contribute to the expectation value so we calculate the second order term as
well, which will be of order $(J/W)^{2}$. This suggests that we should have
considered also the second order in the expression for $A$, but this is
fortunately not the case. This is because the term linear in the second order
$A$ will again be a commutator $[A^{(2)},\sigma^{z}_{i}]$, whose expectation
value will vanish, just as it did for the first order $A$. The result is hence

$\displaystyle
Z_{i}:=-\langle\eta|\left(\tfrac{1}{2}[A,[A,\sigma_{i}^{z}]]+\ldots\right)|\eta\rangle$
$\displaystyle=\sum_{\eta^{\prime}}(\eta^{\prime}_{i}-\eta_{i})|\langle\eta|A|\eta^{\prime}\rangle|^{2}$
(15)

which can be computed to give

$\displaystyle Z_{i}$ $\displaystyle=$
$\displaystyle-2\eta_{i}\left[\frac{J(\eta_{i}-\eta_{i+1})}{h_{i+1}-h_{i}+2J_{z}(\eta_{i+2}-\eta_{i-1})}\right]^{2}+$
(16)
$\displaystyle-2\eta_{i}\left[\frac{J(\eta_{i}-\eta_{i-1})}{h_{i}-h_{i-1}+2J_{z}(\eta_{i+1}-\eta_{i-2})}\right]^{2}$

We have then

$\displaystyle\langle\Psi(\eta)|\sigma_{i}^{z}|\Psi(\eta)\rangle\approx\eta_{i}+Z_{i}$
(17)

Note that $Z_{i}$ is such that the left-hand side is indeed in $[-1,1]$, as it
should.

Now from this we can compute the purity

$\displaystyle
P_{i}(\eta)=|\langle\Psi(\eta)|\sigma_{i}^{z}|\Psi(\eta)\rangle|^{2}\approx
1+2\eta_{i}Z_{i}+Z_{i}^{2}$ (18)

To order $(J/W)^{2}$, we have then

$\displaystyle P_{i}(\eta)$ $\displaystyle=$ $\displaystyle
1-2\left[\frac{J(\eta_{i}-\eta_{i+1})}{h_{i+1}-h_{i}+2J_{z}(\eta_{i+2}-\eta_{i-1})}\right]^{2}+$
(19)
$\displaystyle-2\left[\frac{J(\eta_{i}-\eta_{i-1})}{h_{i}-h_{i-1}+2J_{z}(\eta_{i+1}-\eta_{i-2})}\right]^{2}$

This is a well-function of $\eta$ and this ends the calculation of the purity.

####  VI.5.2 Correlation of purity

Let us now for simplicity disregard the interaction, setting $J_{z}=0$ and use
the fact that

$\displaystyle(\eta_{i}-\eta_{i+1})^{2}=2(1-\eta_{i}\eta_{i+1})$ (20)

Then

$\displaystyle P_{i}(\eta)$ $\displaystyle=$ $\displaystyle
1-(\tfrac{2J}{(h_{i+1}-h_{i})})^{2}(1-\eta_{i}\eta_{i+1})-$ (21)
$\displaystyle(\tfrac{2J}{(h_{i-1}-h_{i})})^{2}(1-\eta_{i}\eta_{i-1})$

Now we are ready to calculate

$\displaystyle\langle P_{i}(\eta)P_{j}(\eta)\rangle-\langle
P_{i}(\eta)\rangle\langle P_{j}(\eta)\rangle$ (22)

when $i,j$ are far apart. This correlation will be of the form

$\displaystyle
C\frac{J^{2}}{W^{2}}\left(\langle\eta_{i}\eta_{i+1}\eta_{j}\eta_{j+1}\rangle-\langle\eta_{i}\eta_{i+1}\rangle\langle\eta_{j}\eta_{j+1}\rangle\right)$
(23)

where we replaced the disorders by some typical value. The expression between
brackets is now computed explicitly, yielding the value $2/L^{2}$.


# A group theoretic description of the $\kappa$-Poincaré Hopf algebra

Michele Arzano<EMAIL_ADDRESS>Dipartimento di Fisica “E. Pancini”,
Università di Napoli Federico II, I-80125 Napoli, Italy
INFN, Sezione di Napoli,
Complesso Universitario di Monte S. Angelo,
Via Cintia Edificio 6, 80126 Napoli, Italy  Jerzy Kowalski-Glikman
<EMAIL_ADDRESS>University of Wrocław, Faculty of Physics
and Astronomy
Maksa Borna 9, 50-204 Wrocław, Poland
National Centre for Nuclear Reserach,
Pasteura 7, 02-093 Warsaw, Poland

###### Abstract

It is well known in the literature that the momentum space associated to the
$\kappa$-Poincaré algebra is described by the Lie group
$\mathsf{A}\mathsf{N}(3)$. In this letter we show that the full
$\kappa$-Poincaré Hopf algebra structure can be obtained from rather
straightforward group-theoretic manipulations starting from the Iwasawa
decomposition of the of the $\mathsf{SO}(1,4)$ group.

A key question in the field of quantum gravity is whether there exists a
regime in which phenomena induced by the quantum properties of the space-time
geometry could possibly be observed. The first proposals for probing the
quantum nature of space-time date to about twenty years ago Amelino-
Camelia:1999hpv  and over the years the study of quantum gravity phenomenology
has developed into a full fledged research program (see Addazi:2021xuf  for an
up-to-date review). Models of deformed relativistic kinematics based on
$\kappa$-deformations of the Poincaré algebra Lukierski:1991pn ;
Lukierski:1992dt ; Lukierski:1993wx ; Lukierski:1993wxa ; Majid:1994cy ;
Kosinski:1994br  (see also Arzano:2021scz ) have been among the key players in
the list of test theories which could lead to potentially observable effects
Amelino-Camelia:2008aez ; Amelino-Camelia:2000cpa . Motivation for considering
such models comes from studies of quantum gravity in 2+1 dimensions
Freidel:2005me ; Arzano:2013sta ; Arzano:2014ppa ; Cianfrani:2016ogm  and
there are reasons to believe that $\kappa$-deformed relativistic symmetries
might be relevant in describing at least some regimes of quantum gravity in
the physical 3+1 dimensional case Amelino-Camelia:2003ezw ; Freidel:2003sp .
There is also an intrinsic interest in the study of such models since they
provide self-consistent mathematical models of relativistic symmetries and
kinematics with a built-in observer independent energy scale with non-trivial
structures subject to subtle physical interpretations Amelino-Camelia:2001rtw
; Agostini:2006nc ; Arzano:2007ef ; Arzano:2007nx ; Arzano:2008bt ;
Arzano:2010jw ; Arzano:2014cya ; Arzano:2014jfa ; Arzano:2017uuh ;
Arzano:2018gii .

Historically, the $\kappa$-Poincaré algebra was derived using the tools of the
theory of quantum groups as a contraction of the Hopf algebra
$\mathsf{SO}_{q}(3,2)$ Lukierski:1991pn ; Lukierski:1992dt , with the aim of
providing a model which would play a role in Planck-scale physics analogous to
that played by the Poincaré algebra in quantum field theory. A few years later
Majid:1994cy  this algebra was re-derived and brought into the modern form
with the help of the general concept of ‘matched pair of Lie algebras/groups’.
In this paper the relation of $\kappa$-Poincaré algebra with non-commutative
geometry proposed in Lukierski:1993wxa  was confirmed. In all these
developments advanced techniques of the theory of Hopf algebras were used.

A notable property of models based on $\kappa$-deformed symmetries is that
they are characterized by curved momentum space being the manifold of the Lie
group $\mathsf{A}\mathsf{N}(3)$, which geometrically is a submanifold of de
Sitter space Kowalski-Glikman:2002oyi ; Kowalski-Glikman:2003qjp . The aim of
the present letter is to show that the whole Hopf algebra structure of the
$\kappa$-Poincaré algebra can be derived from simple group theoretical
arguments concerning the action of the Lorentz group $\mathsf{SO}(1,3)$ on
$\mathsf{A}\mathsf{N}(3)$ as described from the Iwasawa decomposition of
$\mathsf{SO}(1,4)$ of which both $\mathsf{SO}(1,3)$ and
$\mathsf{A}\mathsf{N}(3)$ are subgroups. We derive $\kappa$-Poincaré Hopf
algebra in a language, which is easily understandable to physicists, using
only basic group theoretical ingredients. This construction is motivated by,
and is an extension of, the notes of S. Nowak and one of the present authors
Kowalski-Glikman:2004lyj . Although the result of our construction is not new,
it is the construction itself that is novel, somewhat unexpected, and will
likely turn out to be very useful. This is particularly significant in the
context of physical applications of models based on the $\kappa$-Poincaré
algebra given their relevance for the field of quantum gravity phenomenology
(see the recent review Addazi:2021xuf ).

The results derived here will be crucially important in a forthcoming paper,
in which we propose a resolution of the long standing puzzle of multiparticle
states in $\kappa$-deformed quantum field theory. For related investigations
see Sitarz1997 , Majid:2006xn , Gubitosi:2011hgc  and also Stachura:2015muw ,
Stachura:2018nzh . All the details of the Hopf-algebraic aspect of our
investigations can be found in the monograph Majid:1996kd .

Finally, let us remark that, as it is customary in many physical applications,
we are mostly interested here in investigating explicitly the properties of
infinitesimal symmetry transformations, although the many properties of finite
transformations are also implicit here.

The starting point of our considerations, as mentioned above, is the Iwasawa
decomposition of a Lie algebra and the associted Lie group Iwasawa  (see also,
eg., Vilenkin ). We will be concerned here with the five-dimensional Lorentz
group $\mathsf{SO}(1,4)$. Its Lie algebra $\mathfrak{so}(1,4)$ can be uniquely
decomposed into the following direct sum of its subalgebras

$\mathfrak{so}(1,4)=\mathfrak{k}\oplus\mathfrak{a}\oplus\mathfrak{n}$ (1)

where $\mathfrak{k}$ denotes the Lorentz algebra $\mathfrak{so}(1,3)$, the
algebra $\mathfrak{a}$ is generated by an element111In the consideration below
we choose to display explicitly the dimensionful scale of deformation
$\kappa$, so that all the functions and operators have the canonical physical
dimensions. Of course one could use the units in which $\kappa=1$, but then
the physical meaning of the resulting formulas would be less transparent.

$X^{0}=-\frac{i}{\kappa}\,\left[\begin{array}[]{ccc}0&0&1\\\ 0&\mathbb{0}&0\\\
1&0&0\end{array}\right]$ (2)

where $\mathbb{0}$ is the $3\times 3$ zero-matrix, while the algebra
$\mathfrak{n}$ is generated by matrices of the form

$X^{i}=\frac{i}{\kappa}\,\left[\begin{array}[]{ccc}0&\epsilon_{i}&0\\\
(\epsilon_{i})^{T}&\mathbb{0}&(\epsilon_{i})^{T}\\\
0&-\epsilon_{i}&0\end{array}\right]$ (3)

where $\epsilon_{i}$ are unit vectors in the direction $i$
($\epsilon_{1}=(1,0,0)$, ect), while $T$ denotes transposition. These matrices
can be seen as generators of the product algebra $\mathfrak{an}$ with the
following commutators

$[X^{0},X^{i}]=\frac{i}{\kappa}X^{i}\,,\quad[X^{i},X^{j}]=0\,.$ (4)

This algebra, with the generators interpreted as coordinates of a non-
commutative generalization of four-dimensional Minkowski space-time, are known
in the literature as the $\kappa$-Minkowski non-commutative space-time
Majid:1994cy .

An arbitrary element of the group $G\in\mathsf{SO}(1,4)$ can be uniquely
decomposed into the product of three elements

$G=\mathsf{K}\mathsf{A}\mathsf{N}$ (5)

where $\mathsf{K}$ is an element of the Lorentz group $\mathsf{SO}(1,3)$,
$\mathsf{A}$ is a group element generated by ${X^{0}}$ while $\mathsf{N}$
belongs to the group generated by $X^{i}$. A generic element of the product
group $\mathsf{A}\mathsf{N}$ (often denoted by $\mathsf{A}\mathsf{N}(3)$ to
stress its dimensionality, in this note for brevity we will use the simpler
notation $\mathsf{A}\mathsf{N}$) can be thus written in terms of the matrices
(2) and (3)

$\displaystyle g=e^{ip_{i}X^{i}}e^{ip_{0}X^{0}}\,.$ (6)

The coefficients $p_{\mu}$ of these exponentials are coordinates on the group
manifold which can be seen as functions on the group $\mathsf{A}\mathsf{N}$:
$p_{\mu}=p_{\mu}(g)$. Using the matrix representation of the generators (2),
(3) the group element (6) can be computed to have the form

$g=\frac{1}{\kappa}\left(\begin{array}[]{ccc}\tilde{P}_{4}&\kappa\mathbf{P}/P_{+}&P_{0}\\\
\mathbf{P}&\kappa\mathbbm{1}&\mathbf{P}\\\
\tilde{P}_{0}&-\kappa\mathbf{P}/P_{+}&P_{4}\\\ \end{array}\right)$ (7)

where $\mathbbm{1}$ is the unit $3\times 3$ matrix and the non-trivial entries
of the matrix are related to the coordinates $p_{\mu}$ via

$\displaystyle P_{0}$
$\displaystyle=\kappa\,\sinh\frac{p_{0}}{\kappa}+\frac{\mathbf{p}{}^{2}}{2\kappa}e^{{p_{0}}/{\kappa}}$
$\displaystyle P_{i}$ $\displaystyle=p_{i}e^{{p_{0}}/{\kappa}}$ $\displaystyle
P_{4}$
$\displaystyle=\kappa\,\cosh\frac{p_{0}}{\kappa}-\frac{\mathbf{p}{}^{2}}{2\kappa}e^{{p_{0}}/{\kappa}}$
$\displaystyle P_{+}$ $\displaystyle=P_{0}+P_{4}=\kappa\,e^{{p_{0}}/{\kappa}}$
(8)

and

$\displaystyle\tilde{P}_{0}$
$\displaystyle=\kappa\,\sinh\frac{p_{0}}{\kappa}-\frac{\mathbf{p}{}^{2}}{2\kappa}e^{{p_{0}}/{\kappa}}$
$\displaystyle\tilde{P}_{4}$
$\displaystyle=\kappa\,\cosh\frac{p_{0}}{\kappa}+\frac{\mathbf{p}{}^{2}}{2\kappa}e^{{p_{0}}/{\kappa}}\,.$
(9)

It is easily seen that the coordinates $(P_{0},\mathbf{P},P_{4})$ are not
independent as they satisfy the constraint

$-P_{0}^{2}+\mathbf{P}^{2}+P_{4}^{2}=\kappa^{2}$ (10)

which defines a four-dimensional de Sitter space with cosmological constant
equal to $\kappa^{-2}$ embedded in a five-dimensional Minkowski space. The
inequality $P_{+}>0$ restricts us to half of the manifold and thus the
momentum space Lie group $\mathsf{A}\mathsf{N}$ is described by half of de
Sitter space.

The four-momenta associated to the coordinates $(p_{0},\bf{p})$ are the
eigenvalues of so called “bicrossproduct basis” Majid:1994cy ; Borowiec:2009vb
of generators of translations for the $\kappa$-Poincaré algebra
$\mathcal{P}^{B}_{\mu}$ acting on a non-commutative plane wave (6) Amelino-
Camelia:1999jfz . More precisely, we define a one-particle state to be
parametrized by the group element, $|g\rangle$, then

$\mathcal{P}_{\mu}^{B}\,|g\rangle=p_{\mu}(g)\,|g\rangle\,,$ (11)

where $p_{\mu}(g)$ are the coefficients $p_{0},p_{i}$ in (6).

The embedding coordinates $(P_{0},\bf{P})$ are instead associated to the so-
called “classical basis” of the $\kappa$-Poincaré algebra Kosinski:1994br  and
whose generators of translations $\mathcal{P}^{C}_{\mu}$ are such that

$\mathcal{P}_{\mu}^{C}\,|g\rangle=P_{\mu}(g)\,|g\rangle\,,\quad\mathcal{P}_{4}^{C}(g)\,|g\rangle=P_{4}(g)\,|g\rangle\,$
(12)

where $P_{\mu}(g)$ and $P_{4}(g)$ are the entries in the last column of the
matrix (7). Therefore, the operators
$\mathcal{P}_{\mu}^{C}(g),\mathcal{P}_{4}^{C}(g)$ compute the last column of
the matrix (7). The first four entries of this column are coordinates on
momentum space $\mathsf{A}\mathsf{N}(3)$ and the last, as we will see in a
moment, is the Lorentz-invariant $\kappa$-Poincaré Casimir. We will use this
fact frequently in what follows.

As we will soon see, the commutator structure of the generators of the
$\kappa$-Poincaré algebra in this basis is just the ordinary Poincaré one. The
generators $\mathcal{P}_{\mu}^{B}$ and $\mathcal{P}_{\mu}^{C}$ are directly
related to derivatives of the non-commutative differential calculi, see
Sitarz:1994rh ; Freidel:2007hk ; Rosati:2021sdf ; Arzano:2009ci .

We will now show how the non-abelian nature of the momentum space leads to a
non-abelian composition of four-momenta. To this end, let us consider elements
of $\mathsf{SO}(1,4)$ belonging to the $\mathsf{A}\mathsf{N}$ subgroup (for
which the first factor in (5) is the identity $\mathsf{K}=\mathbbm{1}$, of the
form (6). Using the commutation relations (4) or the explicit matrix form of
$\mathsf{A}\mathsf{N}$ elements one can show that

$e^{ip_{i}X^{i}}e^{ip_{0}X^{0}}=e^{ip_{0}X^{0}}e^{ip_{i}e^{p_{0}/\kappa}X^{i}}$
(13)

and thus for the product of two group elements we have

$\displaystyle
gh=e^{ip_{i}X^{i}}e^{ip_{0}X^{0}}e^{iq_{i}X^{i}}e^{iq_{0}X^{0}}=e^{i\left(p_{i}+q_{i}\,e^{-p_{0}/\kappa}\right)X^{i}}e^{i(p_{0}+q_{0})X^{0}}$
(14)

which determines the non-abelian addition of four-momenta

$p\oplus q=\left(p_{0}+q_{0},\mathbf{p}+\mathbf{q}e^{-p_{0}/\kappa}\right)\,.$
(15)

These deformed composition rules for four-momenta reflect a non trivial action
of translation generators on tensor product representations Arzano:2021scz .

Now let us define the map $\Delta(P^{B}_{\mu})$, known in Hopf algebra
language as the coproduct, which defines the action of translations (momenta)
on the a two particles state, being an element of the tensor product of two
one-particle Hilbert spaces

$\displaystyle\Delta(\mathcal{P}_{\mu}^{B})\,|g\rangle\otimes|h\rangle\equiv
p_{\mu}(gh)\,|g\rangle\otimes|h\rangle=(p\oplus
q)_{\mu}\,|g\rangle\otimes|h\rangle$ (16)

which with the help of (15) can be solved to give

$\displaystyle\Delta(\mathcal{P}^{B}_{i})=\mathcal{P}^{B}_{i}\otimes\mathbbm{1}+e^{-{\mathcal{P}^{B}_{0}}/{\kappa}}\otimes
P^{B}_{i}\,,\quad\Delta(\mathcal{P}^{B}_{0})=\mathcal{P}^{B}_{0}\otimes\mathbbm{1}+\mathbbm{1}\otimes\mathcal{P}^{B}_{0}.$
(17)

This is the standard way one derives the co-product for the deformed
translation generators of the $\kappa$-Poincaré algebra and can be found in
various works in the literature.

Returning to the $\mathsf{A}\mathsf{N}(3)$ group, for the inverse group
element one has

$\displaystyle\left(e^{ip_{i}X^{i}}e^{ip_{0}X^{0}}\right)^{-1}=e^{-ip_{i}e^{p_{0}/\kappa}X^{i}}e^{-ip_{0}X^{0}}$
(18)

from which we can define the following map on four-momenta

$\ominus p\equiv\left(-p_{0},-\mathbf{p}\,e^{p_{0}/\kappa}\right)$ (19)

which is the non-abelian analogue of the minus operation on usual four-vectors
ensuring that $p\oplus(\ominus(p))=(\ominus(p))\oplus p=0$.

The inverse group element relation (19) make it possible to define another
important concept of the antipode denoted by $S$ and defined by the condition

$\displaystyle
S(\mathcal{P}_{\mu}^{B}))|g\rangle=\mathcal{P}^{B}_{\mu}|g^{-1}\rangle$ (20)

which can be solved to give

$\displaystyle S(\mathcal{P}^{B}_{0})=-\mathcal{P}^{B}_{0}\,,\quad
S(\mathcal{P}^{B}_{i})=-\mathcal{P}^{B}_{i}e^{\mathcal{P}^{B}_{0}/\kappa}\,.$
(21)

Let us pause here to explain the physical meaning of the notions introduced
above. As we saw, the coproduct tells us how to consistently define the action
of a symmetry generator (in the case above, the momentum) on multiparticle
states. The (co)-associativity of the coproduct guarantees that the action on
two particles state can be generalized to the action on all multiparticle
states. The antipode instead provides an information about the action of the
inverse symmetry generators. In the case of generators different from momenta
(mathematically, functions on the group) to compute the antipode one has to
use a more general construction, which we will explain when discussing Lorentz
transformations.

Let us now consider an alternative, more straightforward derivation of the
coproduct which makes use of the product of two momentum matrices (7)

$gh=\frac{1}{\kappa^{2}}\,\left(\begin{array}[]{ccc}\tilde{P}_{4}&\kappa\mathbf{P}/P_{+}&P_{0}\\\
\mathbf{P}&\kappa\mathbbm{1}&\mathbf{P}\\\
\tilde{P}_{0}&-\kappa\mathbf{P}/P_{+}&P_{4}\\\
\end{array}\right)\left(\begin{array}[]{ccc}\tilde{Q}_{4}&\kappa\mathbf{Q}/Q_{+}&Q_{0}\\\
\mathbf{Q}&\kappa\mathbbm{1}&\mathbf{Q}\\\
\tilde{Q}_{0}&-\kappa\mathbf{Q}/Q_{+}&Q_{4}\\\ \end{array}\right)\,.$ (22)

Remembering that $P_{\mu}(gh)$ reads off the first four components of the
fifth column of the resulting matrix on the right hand side we get the
following non-abelian composition of momenta $P_{\mu}(g)\oplus
P_{\mu}(h)\equiv P_{\mu}(gh)$:

$\displaystyle P_{0}\oplus Q_{0}$
$\displaystyle=\frac{1}{\kappa}\,\tilde{P}_{4}\,Q_{0}+\frac{\mathbf{P}\cdot\mathbf{Q}}{P_{+}}+\frac{1}{\kappa}\,P_{0}\,Q_{4}=\frac{\kappa}{P_{+}}Q_{0}+\frac{\mathbf{P}\cdot\mathbf{Q}}{P_{+}}+\frac{1}{\kappa}\,P_{0}\,Q_{+}$
(23) $\displaystyle\mathbf{P}\oplus\mathbf{Q}$
$\displaystyle=\frac{1}{\kappa}\,\mathbf{P}\,Q_{+}+\mathbf{Q}$ (24)

Using the defining property

$\displaystyle\Delta(\mathcal{P}_{\mu}^{C})\,|g\rangle\otimes|h\rangle\equiv
P_{\mu}(gh)\,|g\rangle\otimes|h\rangle=(P\oplus
Q)_{\mu}\,|g\rangle\otimes|h\rangle$ (25)

we thus find

$\displaystyle\Delta(\mathcal{P}^{C}_{0})$
$\displaystyle=\kappa\,\left(\mathcal{P}^{C}_{+}\right)^{-1}\otimes\mathcal{Q}^{C}_{0}+\sum_{i}\left(\mathcal{P}^{C}_{+}\right)^{-1}\mathcal{P}^{C}_{i}\otimes\mathcal{Q}^{C}_{i}+\frac{1}{\kappa}\,\mathcal{P}^{C}_{0}\otimes\mathcal{Q}^{C}_{+}$
(26) $\displaystyle\Delta(\mathcal{P}^{C}_{i})$
$\displaystyle=\frac{1}{\kappa}\,\mathcal{P}^{C}_{i}\otimes\mathcal{Q}^{C}_{+}+\mathcal{Q}^{C}_{i}\otimes\mathbbm{1}\,.$
(27)

From the condition $P_{\mu}\oplus(\ominus P_{\mu})=0$ one can also easily
derive

$\displaystyle\ominus P_{0}$ $\displaystyle=-P_{0}+\mathbf{P}^{2}/P_{+}$ (28)
$\displaystyle\ominus\mathbf{P}$ $\displaystyle=-\kappa\mathbf{P}/P_{+}\,.$
(29)

so that

$\displaystyle S(\mathcal{P}^{C})_{0}$
$\displaystyle=-\mathcal{P}^{C}_{0}+(\mathcal{P}^{C}_{i})^{2}\left(\mathcal{P}^{C}_{+}\right)^{-1}$
(30) $\displaystyle S(\mathcal{P}^{C})_{i}$
$\displaystyle=-\kappa\mathcal{P}^{C}_{i}\left(\mathcal{P}^{C}_{+}\right)^{-1}\,.$
(31)

Let us now turn to unveiling the Hopf algebra structure of the Lorentz sector,
i.e. the coproduct and antipodes for the generators of boosts and rotations.
To this end we must first spell out the action of the Lorentz group on the
group manifold momentum space $\mathsf{A}\mathsf{N}$. This can be done by
noticing that the Iwasawa decomposition guarantees that given the Lorentz
group element $K$ and $g$ belonging to $\mathsf{A}\mathsf{N}$ group, there are
unique $K^{\prime}_{g}$ and $g^{\prime}$ satisfying the equality

$K\,g=g^{\prime}\,K^{\prime}_{g}\,.$ (32)

Since $K$ is a Lorentz transformation we define $g^{\prime}$ to be the
Lorentz-transformed $g$, and we write

$K\,e^{ip_{i}X^{i}}e^{ip_{0}X^{0}}\,K^{\prime}_{g}{}^{-1}=e^{ip^{\prime}_{i}X^{i}}e^{ip^{\prime}_{0}X^{0}}$
(33)

where the four-momenta $(p_{0}^{\prime},p_{i}^{\prime})$ are defined to be
resulting from the Lorentz action on $(p_{0},p_{i})$. For infinitesimal
Lorentz transformation we assume that

$K\approx\mathbbm{1}+i\xi^{a}\mathfrak{K}_{a}\,,\quad
K^{\prime}_{g}\approx\mathbbm{1}+i\xi^{a}\,h_{a}^{b}(g)\mathfrak{K}_{b}$ (34)

where $\mathfrak{K}_{a}$ are the generators of Lorentz algebra
$\mathfrak{so}(1,3)$, $\mathfrak{K}_{a}=(N_{i},M_{i})$ with $N_{i}$, $M_{i}$
denoting the generators of boost and rotations, and $h_{a}^{b}(g)$ is a
momentum-dependent matrix that we will explicitly derive below. As we will see
this function is responsible for the effect of ‘back-reaction’ of momenta on
Lorentz transformations on products of momentum group elements, making the
effective infinitesimal parameter of Lorentz transformation momentum
dependent, as noticed in Majid:2006xn  (see also Gubitosi:2011hgc ). We will
return to this point below.

In order to derive the explicit form of infinitesimal Lorentz transformation
on four-momenta we focus on the the parametrization of the group element (7)
in terms of coordinates $P_{\mu}$. For definiteness, and without loss of
generality we consider a boost in the 1-direction. In matrix form equation
(32) can be written explicitly as

$\displaystyle\left(\begin{array}[]{ccccc}1&\xi&0&0&0\\\ \xi&1&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&1&0\\\ 0&0&0&0&1\\\
\end{array}\right)\left(\begin{array}[]{ccccc}\tilde{P}_{4}&\kappa
P_{1}/P_{+}&\kappa P_{2}/P_{+}&\kappa P_{3}/P_{+}&P_{0}\\\
P_{1}&\kappa&0&0&P_{1}\\\ P_{2}&0&\kappa&0&P_{2}\\\ P_{3}&0&0&\kappa&P_{3}\\\
\tilde{P}_{0}&-\kappa P_{1}/P_{+}&-\kappa P_{2}/P_{+}&-\kappa
P_{3}/P_{+}&P_{4}\\\ \end{array}\right)$ (45)
$\displaystyle=\left(\begin{array}[]{ccccc}\tilde{P}^{\prime}_{4}&\kappa(P_{1}/P_{+})^{\prime}&\kappa(P_{2}/P_{+})^{\prime}&\kappa(P_{3}/P_{+})^{\prime}&P^{\prime}_{0}\\\
P^{\prime}_{1}&\kappa&0&0&P^{\prime}_{1}\\\
P^{\prime}_{2}&0&\kappa&0&P^{\prime}_{2}\\\
P^{\prime}_{3}&0&0&\kappa&P^{\prime}_{3}\\\
\tilde{P}_{0}&-\kappa(P_{1}/P_{+})^{\prime}&-\kappa(P_{2}/P_{+})^{\prime}&-\kappa(P_{3}/P_{+})^{\prime}&P_{4}\\\
\end{array}\right)\left(\begin{array}[]{ccccc}1&\bar{\xi}_{1}&\bar{\xi}_{2}&\bar{\xi}_{3}&0\\\
\bar{\xi}_{1}&1&\bar{\rho}_{3}&-\bar{\rho}_{2}&0\\\
\bar{\xi}_{2}&-\bar{\rho}_{3}&1&\bar{\rho}_{1}&0\\\
\bar{\xi}_{3}&\bar{\rho}_{2}&-\bar{\rho}_{1}&1&0\\\ 0&0&0&0&1\\\
\end{array}\right)$ (56)

In the last matrix representing
$\mathbbm{1}+i\xi^{a}h_{a}^{b}(p)\mathfrak{K}_{b}$ the entries $\bar{\xi}_{i}$
and $\bar{\rho}_{i}$ are, respectively, the parameters of the infinitesimal
boost and rotation, which by force of Iwasawa decomposition are uniquely
defined. As we pointed out above, the embedding coordinate functions
$P_{\mu}(g)$ simply read the first four entries of the fifth column of the
matrix (7). We thus have the key property

$P_{\mu}(g^{\prime})=P_{\mu}(K\,g\,K_{g}^{{}^{\prime}-1})=P_{\mu}(K\,g)$ (57)

which simply tells us that infinitesimally $P^{\prime}_{\mu}=P_{\mu}+\delta
P_{\mu}$ with

$P_{0}^{\prime}=P_{0}+\xi P_{1}\,,\quad P_{1}^{\prime}=P_{1}+\xi P_{0}\,,\quad
P_{2}^{\prime}=P_{2}\,,\quad P_{3}^{\prime}=P_{3}\,,\quad
P_{4}^{\prime}=P_{4}$ (58)

and therefore $P_{\mu}$ transforms as a ordinary Lorentz vector. We thus have
that the commutators between the generators of boosts and the generators of
translations of the classical basis of the $\kappa$-Poincaré algebra are just
the undeformed ones

$\left[\mathcal{N}_{i},\mathcal{P}^{C}_{j}\right]=i\,\delta_{ij}\mathcal{P}^{C}_{0},\,\,\,\left[\mathcal{N}_{i},\mathcal{P}^{C}_{0}\right]=i\
\mathcal{P}^{C}_{i}\,.$ (59)

Although in what follows we will only need the infinitesimal transformation,
it is easy to see that the structure of the matrices in (56) remains the same
in the case of finite transformations so that, for the final boost along the
first axis we obtain

$\displaystyle P_{0}^{\prime}=\cosh\xi P_{0}+\sinh\xi P_{1}\,,\quad
P_{1}^{\prime}=\cosh\xi P_{1}+\sinh\xi P_{0}\,,\quad
P_{2}^{\prime}=P_{2}\,,\quad P_{3}^{\prime}=P_{3}\,,\quad
P_{4}^{\prime}=P_{4}$ (60)

Using (A group theoretic description of the $\kappa$-Poincaré Hopf algebra) we
easily derive the boost action on the bicrossproduct coordinates $p_{\mu}$
which determine the following non-linear commutators between generators of
boosts and bicrossproduct generators of translations

$\left[\mathcal{N}_{i},\mathcal{P}^{B}_{j}\right]=i\,\delta_{ij}\left({\frac{\kappa}{2}}\left(1-e^{-2\mathcal{P}^{B}_{0}/\kappa}\right)+\frac{(\mathcal{P}^{B}_{i})^{2}}{2\kappa}\right)-i\,\frac{1}{\kappa}\,\mathcal{P}^{B}_{i}\mathcal{P}^{B}_{j},\,\,\,\left[\mathcal{N}_{i},\mathcal{P}^{B}_{0}\right]=i\
\mathcal{P}^{B}_{i}\,.$ (61)

Repeating the steps leading to (58) one finds that infinitesimal rotations act
in the ordinary way on the four-momenta $P_{\mu}$ and through the coordinate
transformation (A group theoretic description of the $\kappa$-Poincaré Hopf
algebra) it is immediate to see that also the bicrossproduct momenta $p_{\mu}$
transform in the standard way and thus we have the commutators

$[\mathcal{M}_{i},\mathcal{P}^{B}_{j}]=i\,\epsilon_{ijk}\mathcal{P}^{B}_{k},\quad[\mathcal{M}_{i},\mathcal{P}^{B}_{0}]=0\,.$
(62)

From (58) we can also see that $P_{4}$ is a Lorentz scalar and thus it must be
related to the Casimir opertator. Indeed from this observation and from (10)
we see that

$P^{2}_{0}-\mathbf{P}^{2}=P_{4}^{2}-\kappa^{2}$ (63)

is an invariant and indeed, since in the classical basis the algebra remains
invariant, we have that the classical translation generators can be combined
in the standard Casimir operator

$\mathcal{C}=(\mathcal{P}^{C}_{0})^{2}-(\vec{\mathcal{P}}^{C})^{2}\,.$ (64)

With simple algebraic manipulations we have thus derived the important result
that in the bicrossproduct basis boosts act non-linearly on four-momenta
(while rotations transform momenta in the ordinary way) while momenta defined
by embedding coordinates on the $\mathsf{A}\mathsf{N}$ group manifold
transform under the ordinary action of boosts and rotations. As we
anticipated, the $\kappa$-Poincaré algebra in the classical basis associated
to the translation generators $\mathcal{P}^{C}_{\mu}$ closes an ordinary
Poincaré algebra; the non-abelian nature of four-momenta affects only the co-
products and antipodes of the algebra generators. In the remainder of this
letter we show how to derive co-products and antipodes for the generators of
boosts and rotations using the tools introduced so far.

Before continuing let us determine the matrix $h_{a}^{b}(g)$ in (34). To this
end we now consider the fourth columns of the matrices on both sides of (56),

$\displaystyle\left(\begin{array}[]{c}\kappa P_{3}/P_{+}\\\ \xi\kappa
P_{3}/P_{+}\\\ 0\\\ \kappa\\\ -\kappa
P_{3}/P_{+}\end{array}\right)=\left(\begin{array}[]{c}\bar{\xi}_{3}\,\tilde{P}^{\prime}_{4}-\bar{\rho}_{2}\kappa(P_{1}/P_{+})^{\prime}+\bar{\rho}_{1}\kappa(P_{2}/P_{+})^{\prime}+\kappa(P_{3}/P_{+})^{\prime}\\\
\bar{\xi}_{3}P_{1}^{\prime}-\bar{\rho}_{2}\kappa\\\
\bar{\xi}_{3}P_{2}^{\prime}+\bar{\rho}_{1}\kappa\\\
\bar{\xi}_{3}P_{3}^{\prime}+\kappa\\\
\bar{\xi}_{3}\,\tilde{P}^{\prime}_{0}+\bar{\rho}_{2}\kappa(P_{1}/P_{+})^{\prime}-\bar{\rho}_{1}\kappa(P_{2}/P_{+})^{\prime}-\kappa(P_{3}/P_{+})^{\prime}\end{array}\right)$
(75)

from which we get the conditions $\bar{\xi}_{3}=0$ and $\bar{\rho}_{1}=0$.
Analogously, form the third column we derive $\bar{\xi}_{2}=0$, so that the
only remaining boost parameter is $\bar{\xi}_{1}\equiv\bar{\xi}$. Further,
comparing the third and the fourth columns on both sides we find that

$\displaystyle\bar{\rho}_{3}=\xi\,\frac{P_{2}}{P_{+}}\,,\quad\bar{\rho}_{2}=-\xi\,\frac{P_{3}}{P_{+}}$
(76)

Finally, we compare the second columns, and from the second rows we find the
equation

$\xi\kappa\frac{P_{1}}{P_{+}}+\kappa=\bar{\xi}P_{1}^{\prime}+\kappa$

Since $P_{1}^{\prime}=P_{1}+\xi P_{0}$, working in the linear order we derive

$\displaystyle\bar{\xi}=\frac{\kappa}{P_{+}}\,\xi$ (77)

From (76) and (77) we deduce the form of the matrix
$i\xi^{a}h_{a}^{b}(g)\mathfrak{K}_{b}$ for $\xi^{a}=(\xi,0,0)$

$\displaystyle
i\xi^{a}h_{a}^{b}(g)\mathfrak{K}_{b}=\frac{1}{P_{+}}\,\xi\left(\begin{array}[]{ccccc}0&\kappa&0&0&0\\\
\kappa&0&P_{2}&P_{3}&0\\\ 0&-P_{2}&0&0&0\\\ 0&-P_{2}&0&0&0\\\ 0&0&0&0&0\\\
\end{array}\right)$ (83)

This can be easily generalized to the case of a boost in an arbitrary
direction, with infinitesimal parameters $\xi^{i}$. The matrix in this case
will consist of a combination of the infinitesimal boost
$\bar{\xi}^{i}\,N_{i}$ and infinitesimal rotation $\bar{\rho}^{i}\,M_{i}$ with

$\displaystyle\bar{\xi}^{i}=\xi^{i}\frac{\kappa}{P_{+}}=\xi^{i}\,e^{-p_{0}/\kappa}$
(84)

and

$\displaystyle\bar{\rho}^{i}=\epsilon_{j}{}^{ki}\,\xi^{j}\,\frac{P_{k}}{P_{+}}=\frac{1}{\kappa}\,\epsilon_{j}{}^{ki}\,\xi^{j}\,p_{k}\,.$
(85)

Notice that if instead of boost we considered an infinitesimal rotation on the
left hand side of (56) we would find the same rotation on the right hand side,
which shows that the matrix $h_{i}^{b}(g)$ for rotations is the identity.

Let us now turn to the co-product of Lorentz generators. To this end, we look
at the action of the Lorentz group on the product of two
$\mathsf{A}\mathsf{N}$ group elements $g$ and $h$. Using the Iwasawa
decomposition again we have

$\displaystyle Kgh=(gh)^{\prime}K^{\prime}_{gh}$ (86)

which we can rewrite as

$\displaystyle(gh)^{\prime}=KgK_{g}^{{}^{\prime}-1}\,K_{g}\,hK_{gh}^{{}^{\prime}-1}=g^{\prime}\,K^{\prime}_{g}\,hK_{gh}^{{}^{\prime}-1}$
(87)

and from which it is clear that $(gh)^{\prime}\neq g^{\prime}\,h^{\prime}$,
i.e., in physical terms, that the Lorentz group action on momentum space is
not Leibnizian. Using the property (57) we have

$P_{\mu}((gh)^{\prime})=P_{\mu}(g^{\prime}\,K^{\prime}_{g}\,hK_{gh}^{{}^{\prime}-1})=P_{\mu}(g^{\prime}K^{\prime}_{g}\,h)$
(88)

where the last equality comes from the property (57). We see that, unlike the
case of Lorentz transformations of single group elements, now the matrix
$K^{\prime}_{g}$ plays a role in determining how the Lorentz boosted momentum
obtained by a product of two group elements can be expressed in terms of the
transformations of the individual factors.

In order to determine the coproduct for boost generator let us again consider
an infinitesimal boost matrix $K=\mathbbm{1}+\xi{N}_{1}$ and write equation
(87) in the matrix form

$\displaystyle(gh)^{\prime}=g^{\prime}\,K^{\prime}_{g}\,hK_{gh}^{{}^{\prime}-1}$
$\displaystyle=\frac{1}{\kappa^{2}}\left(\begin{array}[]{ccc}\tilde{P}^{\prime}_{4}&\kappa\left(\mathbf{P}/P_{+}\right)^{\prime}&P^{\prime}_{0}\\\
\mathbf{P}^{\prime}&\kappa\mathbbm{1}&\mathbf{P}^{\prime}\\\
\tilde{P}^{\prime}_{0}&-\kappa\left(\mathbf{P}/P_{+}\right)^{\prime}&P^{\prime}_{4}\\\
\end{array}\right)\,\left(\begin{array}[]{ccccc}1&\frac{\kappa}{P_{+}}\xi&0&0&0\\\
\frac{\kappa}{P_{+}}\xi&1&\frac{P_{2}}{P_{+}}\xi&\frac{P_{3}}{P_{+}}\xi&0\\\
0&-\frac{P_{2}}{P_{+}}\xi&1&0&0\\\ 0&-\frac{P_{3}}{P_{+}}\xi&0&1&0\\\
0&0&0&0&1\\\
\end{array}\right)\left(\begin{array}[]{ccc}\tilde{Q}_{4}&\kappa\mathbf{Q}/Q_{+}&Q_{0}\\\
\mathbf{Q}&\kappa\mathbbm{1}&\mathbf{Q}\\\
\tilde{Q}_{0}&-\kappa\mathbf{Q}/Q_{+}&Q_{4}\\\
\end{array}\right)\,K_{gh}^{{}^{\prime}-1}$ (100)

Keeping only the terms up to the linear order in $\xi$ this equation can be
rewritten as

$\displaystyle(gh)^{\prime}=\left[gh+\xi\left(\left(N_{1}g\right)\left(h\right)+\left(\frac{\kappa}{P_{+}}g\right)\left(N_{1}h\right)+\epsilon_{1jk}\left(\frac{P_{j}}{P_{+}}g\right)\left(M_{k}h\right)\right)\right]K_{gh}^{{}^{\prime}-1}$
(101)

As we know, the matrix $K_{gh}^{{}^{\prime}-1}$ does not influence the
momenta, so its explicit form is not relevant. The formula (101) can be easily
generalized to the case of $K=\mathbbm{1}+\xi^{i}{N}_{i}$. From (101) we can
immediately read off the following expression for the co-product of boosts in
the classical basis

$\Delta^{C}\mathcal{N}_{i}=\mathcal{N}_{i}\otimes\mathbbm{1}+\frac{\kappa}{\mathcal{P}^{C}_{+}}\otimes\mathcal{N}_{i}+\epsilon_{ij}{}^{k}\,\frac{\mathcal{P}^{C}_{j}}{\mathcal{P}^{C}_{+}}\otimes\mathcal{M}_{k}\,.$
(102)

Equation (102) can be rewritten in the bicroproduct basis

$\Delta^{B}\mathcal{N}_{i}=\mathcal{N}_{i}\otimes\mathbbm{1}+e^{-\mathcal{P}^{B}_{0}/\kappa}\otimes\mathcal{N}_{i}+\frac{1}{\kappa}\,\epsilon_{ijk}\,\,\mathcal{P}^{B}_{j}\otimes\mathcal{M}_{k}\,.$
(103)

One can repeat the same procedure for infinitesimal rotations. In this case
however, as noticed above, the matrix $h^{a}_{a}(g)$ reduces to the identity
matrix and thus the generators of rotations act following the the usual
Leibniz rule i.e. the have a trivial co-product

$\Delta\mathcal{M}_{i}=\mathcal{M}_{i}\otimes\mathbbm{1}+\mathbbm{1}\otimes\mathcal{M}_{i}\,.$
(104)

Finally, let us consider the antipode of Lorentz generators. While the co-
product determines the way the generators of Lorentz transformations act on
the product to two $\mathsf{A}\mathsf{N}$ group elements, the antipode tells
what the inverse Lorentz generator is. Consider (101) for $g\mapsto g^{-1}$,
$h\mapsto g$

$\displaystyle\mathbbm{1}=(g^{-1}g)^{\prime}=\left[\mathbbm{1}+\xi^{i}\left(\left(N_{i}g^{-1}\right)\left(g\right)+\left(\frac{P_{+}}{\kappa}g^{-1}\right)\left(N_{i}g\right)-\epsilon_{ijk}\left(P_{j}g^{-1}\right)\left(M_{k}g\right)\right)\right]K_{g^{-1}g}^{{}^{\prime}-1}$
(105)

and we see that the term proportional to $\xi^{i}$ must vanish.

Let us define the antipode of Lorentz generator by the formula

$\displaystyle N_{i}g^{-1}$ $\displaystyle\equiv(g^{-1})S(N_{1})\,,\quad
M_{k}g^{-1}\equiv(g^{-1})S(M_{k})$ (106)

substituting this to (105) we get

$\displaystyle
S(N_{i})+\frac{P_{+}}{\kappa}N_{i}-\epsilon_{ijk}\,{P_{j}}M_{k}=0$ (107)

Repeating the same procedure for the $\mathbbm{1}=g^{-1}g$ we derive

$\displaystyle
N_{i}+\frac{\kappa}{P_{+}}S(N_{i})+\epsilon_{ijk}\,\frac{P_{j}}{P_{+}}S(M_{k})=0$
(108)

Solving (107) we find expression for $S(N)$ and substituting the solution to
we derive $S(M)$. In terms of $\mathcal{N}_{i}$ and $\mathcal{M}_{i}$ we have

$\displaystyle
S^{C}(\mathcal{N}_{i})=-\frac{\mathcal{P}^{C}_{+}}{\kappa}\mathcal{N}_{i}+\frac{1}{\kappa}\epsilon_{ijk}\mathcal{P}^{C}_{j}\mathcal{M}_{k}$
(109)

and for rotations

$\displaystyle S^{C}(\mathcal{M}_{i})=-\mathcal{M}_{i}\,.$ (110)

The corresponding expressions in the bicrossproduct basis are

$\displaystyle S^{B}(\mathcal{N}_{i})$
$\displaystyle=-e^{\mathcal{P}^{B}_{0}/\kappa}\left(\mathcal{N}_{i}-\frac{1}{\kappa}\,\epsilon_{ijk}\mathcal{P}^{B}_{j}\mathcal{M}_{k}\right)$
(111) $\displaystyle S^{B}(\mathcal{M}_{i})$
$\displaystyle=-\mathcal{M}_{i}\,.$ (112)

It should be stressed that it is the matrix $h^{a}_{b}(g)$ (83) that captures
the essence of all nontrivial structures associated with coproducts and
antipodes of the $\kappa$-Poincaré algebra. It is also responsible for the
emergence of what was called in Majid:2006xn  and Gubitosi:2011hgc  the ‘back-
reaction of the momenta on the Lorentz sector’. Indeed, the first factor
$e^{-p_{0}/\kappa}N_{i}$ in (84) is the infinitesimal counterpart of the back-
reaction found in these papers.

To conclude, in this note we showed that the whole of the Hopf structure of
$\kappa$-Poincaré algebra can be straightforwardly deduced from the theory of
Iwasawa decomposition of the five-dimensional Lorentz group
$\mathsf{SO}(1.4)$. We hope that this construction will help, among others, to
to understand better the relations between $\kappa$-deformation and quantum
gravity in four and three dimensions, generalizing the results of Kowalski-
Glikman:2008fix  and Trzesniewski:2017sqa .

## Acknowledgment

This paper is inspired by unpublished notes by Sebastian Nowak and JKG
Kowalski-Glikman:2004lyj . We thank Andrea Bevilacqua and Giacomo Rosati for
their helpful comments on the early version of the manuscript. For JKG, this
work was supported by funds provided by the National Science Center, project
number 2019/33/B/ST2/00050. This work contributes to the European Union COST
Action CA18108 Quantum gravity phenomenology in the multi-messenger approach.

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