# Hilbert space fragmentation in a long-range system

Gianluca Francica, Luca Dell’Anna  Dipartimento di Fisica e Astronomia e
Sezione INFN, Università degli Studi di Padova, via Marzolo 8, 35131 Padova,
Italy

###### Abstract

We study the role of the interaction range on the Hilbert-space fragmentation
and many-body scar states considering a spin-1/2 many-body Hamiltonian
describing a generalized Fredkin spin chain. We show that both scar states and
weak fragmentation of the Hilbert space survive for almost any range of the
coupling. Moreover, when the interaction range is small enough, there are
sectors with definite symmetries such that the ratio between the dimension of
their largest fragment and their dimension decays algebraically with the
system size. Finally we investigate the effects of such structures of the
Hilbert space on the out-of-equilibrium dynamics, triggered by certain initial
states, characterized by either local persistent oscillations or non-uniform
stationary profile of the magnetization.

##  I Introduction

A closed system given in a certain initial state can thermalize, i.e., the
expectation values of local observables at long times tend to the expectation
values calculated with respect to a thermal state which does not depend on the
details of the initial state, except for the energy.

Thermalization can be explained by using the eigenstate thermalization
hypothesis (ETH) Deutsch91 ; Srednicki94 . In simple terms, an energy
eigenstate satisfies the ETH if the reduced state of a small part of the
system is a thermal state. Thus, the matrix element of any local operator
expressed in a basis of energy eigenstates is a smooth function of the energy,
and, if diagonal, is determined by its average value in an appropriate Gibbs
density matrix Rigol08 ; Polkovnikov11 ; DAlessio16 . There are cases where
ETH is violated (see, e.g., the reviews Refs. Papic21 ; Moudgalya22 ). In
general, ETH is said to be strongly satisfied if every eigenstate satisfies
ETH, and weakly satisfied if almost all eigenstates satisfy ETH except a zero
measure set of them. Roughly speaking, in both cases a randomly chosen initial
state with a narrow energy distribution will thermalize after long times,
however if ETH is only satisfied weakly we can always find some initial states
having narrow energy distributions which do not thermalize.

We expect that for a non-integrable system without any symmetries the time
evolution of a random initial state generates the full Hilbert space and ETH
is strongly satisfied. We recall that integrable systems have a level
statistics which follows a Poisson distribution, whereas non-integrable
systems have a level statistics which can be described by a random matrix
theory ensemble DAlessio16 . However, there are some exceptions, i.e., cases
where ETH is not satisfied, for instance, some integrable systems having an
extensive number of conserved quantities, which, in one spatial dimension, can
be exactly solved by a Bethe ansatz procedure. In this case the long time
expectation values are given by a so-called generalized Gibbs ensemble
Essler16 . There are also some interacting systems exhibiting many-body
localization, in the presence of strong disorder or quasiperiodicity, where
again an extensive number of conserved quantities can be constructed (see,
e.g., the reviews Refs. Nandkishore15 ; Abanin19 ). In these cases both strong
and weak ETH are violated.

In a chaotic system, ETH can be only weakly satisfied in the presence of
quantum many-body states called scars states (see, e.g., Ref. Serbyn21  for a
recent review), which form a zero measure set of highly excited eigenstates
that do not satisfied ETH and thus ’scar’ the spectrum of ETH-satisfying
eigenstates. Furthermore, strong ETH is not satisfied if there is
fragmentation of the Hilbert space Sala20 ; Khemani20 , namely there is a
number of dynamically disconnected invariant subspaces, called Krylov
subspaces, which exponentially grows with the size of the system. We can have
weak (strong) fragmentation if there is (is not) a dominant subspace and weak
ETH is (is not) satisfied. Moreover, Krylov-restricted ETH can be satisfied
leading to surprising behavior, e.g., thermal expectation values of
observables which are spatially non-uniform Moudgalya21 . Fragmentation is
known to appear in dipole-moment or center-of-mass conserving systems Sala20 ;
Khemani20 ; Moudgalya21 ; Pai19 ; Rakovszky20 ; Morningstar20 , in other
systems like the t-V model DeTommasi19 , certain one-dimensional models with
strict confinement Yang20 ; Chen21 ; Bastianello22 , PXP models Mukherjee21 ;
Mukherjee221 , in the presence of frustration Lee21 ; Hahn21  and in polar
lattice gases Li21 . Furthermore, spin chains such as the Fredkin Langlett21 ,
Motzkin Richter22 , Pair-Flip Moudgalya222 , folded XXZ Zadnik21 , and
Temperley-Lieb Read07 ; Moudgalya222  spin chains, can exhibit fragmentation.
Although there are several systems that show fragmentation only in one
dimension Moudgalya222 , it can occur also in systems with dimension larger
than one Khudorozhkov22 ; Chattopadhyay23 .  Moreover, many-body scars
resulting from Hilbert space fragmentation have been explored in Refs. scar1 ;
scar2 ; scar3 ; scar4 ; scar5 .

To explain how ETH implies thermalization, we consider a non-degenerate
spectrum of a Hamiltonian $H$, i.e., $E_{i}\neq E_{j}$ whenever $i\neq j$, and
an initial state $|\psi_{0}\rangle$ with a narrow energy distribution with
mean value $E=\langle\psi_{0}|H|\psi_{0}\rangle$, so that
$\sum_{i:E_{i}\approx E}p_{i}\approx 1$, where $p_{i}=|\langle
E_{i}|\psi_{0}\rangle|^{2}$ and $|E_{i}\rangle$ is the eigenstate of $H$ with
eigenvalue $E_{i}$. The time-evolved state is
$|\psi_{t}\rangle=e^{-iHt}|\psi_{0}\rangle$, and the time-averaged density
matrix is

$\bar{\rho}=\lim_{\tau\to\infty}\frac{1}{\tau}\int_{0}^{\tau}|\psi_{t}\rangle\langle\psi_{t}|=\sum_{i}p_{i}|E_{i}\rangle\langle
E_{i}|\,,$ (1)

so that it is equal to the so-called diagonal ensemble.  If there is
thermalization the long-time expectation value of an observable is determined
by $\bar{\rho}$. Since the initial state has a narrow energy distribution, we
get

$\bar{\rho}\approx\sum_{i:E_{i}\approx E}p_{i}|E_{i}\rangle\langle E_{i}|\,.$
(2)

In principle, this state can depend on the initial state details through the
distribution $p_{i}$. Anyway, if the states $|E_{i}\rangle$ satisfy the ETH,
they are thermal, and for $E_{i}\approx E$ the reduced states of a small part
of the system are thermal states with inverse temperature
$\beta_{E_{i}}\approx\beta_{E}$, and of course $\bar{\rho}$ locally is
approximately equal to a thermal state with inverse temperature $\beta_{E}$.
More precisely, ETH postulates that the matrix elements of a local observable
$O$ with respect to the basis of the energy eigenstates, $O_{mn}=\langle
E_{m}|O|E_{n}\rangle$, are

$O_{mn}=O(\bar{E})\delta_{m,n}+e^{-S(\bar{E})/2}f_{O}(\bar{E},\omega)R_{mn}$
(3)

where $O(\bar{E})$ is a smooth function of the energy
$\bar{E}=(E_{m}+E_{n})/2$, corresponding to the expectation value with respect
to the microcanonical ensemble with energy $\bar{E}$, while
$\omega=E_{m}-E_{n}$, $S(\bar{E})$ is the equilibrium entropy of the system,
so that the temperature is given by $\beta_{E}=\partial_{E}S(E)$,
$f_{O}(\bar{E},\omega)$ is some observable-dependent function and $R_{mn}$ is
some random variable with zero mean and unit variance. We note that the
entropy $S(E)$ is generally peaked at the middle of the spectrum, so that for
those energies the effective temperature is very large and the average
$O(\bar{E})$ is given by a local completely mixed state and $e^{-S(E)/2}\sim
1/\sqrt{\mathcal{D}}$, where $\mathcal{D}$ is the dimension of the Hilbert
space (almost all states are contained near the entropy peak in the limit of
large system size). Equation (3) is, then, equivalent to a random matrix
theory description DAlessio16 .

In this paper, we focus on a longer-range generalization of the Fredkin chain
studied in Ref. Langlett21 . In principle, these kind of models can be
realized experimentally by cold atoms in optical lattices, and a three-spin
interaction can be designed as in Ref. Pachos04 . For the nearest neighbor
case, strong fragmentation and scars can coexist, while what happens by
increasing the interaction range was so far an open question. It was supposed
that longer range interactions could be detrimental for fragmentation.
Interestingly, we find that weak fragmentation and scar states survive for
long-range interactions.

##  II Model

Let us now introduce the following Hamiltonian

$H=\sum_{\ell=1}^{L-3}d_{\ell}\sum_{i=2}^{L-\ell-1}\left({\cal
P}^{\uparrow}_{i-1}\otimes{\cal S}_{i,i+\ell}-{\cal S}_{i,i+\ell}\otimes{\cal
P}^{\downarrow}_{i+\ell+1}\right)\,,$ (4)

where we defined the projectors

$\displaystyle{\cal P}^{\sigma}_{i}$ $\displaystyle=$
$\displaystyle|\sigma_{i}\rangle\langle\sigma_{i}|,$ (5) $\displaystyle{\cal
S}_{i,j}$ $\displaystyle=$ $\displaystyle|s_{i,j}\rangle\langle s_{i,j}|$ (6)

with $\sigma=\uparrow,\downarrow$, and the singlet state

$|s_{i,j}\rangle=\big{(}|\uparrow_{i}\downarrow_{j}\rangle-|\downarrow_{i}\uparrow_{j}\rangle\big{)}/\sqrt{2}.$
(7)

Differently from the Fredkin chain DellAnna16 ; Salberger16 , the minus sign
in the second term produces an interference in the Fredkin moves, and the
sector of the Dyck paths is fragmented in many disconnected subspaces. The
nearest neighbor case, i.e., $d_{\ell>1}=0$, has been investigated in Ref.
Langlett21 , and exhibits Hilbert space fragmentation and scar states. Here,
we aim to study how a longer range interaction affects those peculiar
behaviors. Hence, we consider an interaction of range $r$, i.e., $d_{\ell}\neq
0$ for $\ell\leq r$ and $d_{\ell}=0$ for $\ell>r$. Of course, $r\leq L-3$, and
for $r=L-3$ we get the fully connected case.

Let us introduce the relevant symmetries. The total magnetization
$S_{z}=\sum_{i=1}^{L}\sigma^{z}_{i}$ is a symmetry for any range $r$, i.e.,
$[H,S_{z}]=0$, while the domain-wall number
$n_{\textrm{dw}}=\sum_{i=1}^{L-1}\sigma^{z}_{i}\sigma^{z}_{i+1}$ does not
commutate with the Hamiltonian for $r>2$, and it is a symmetry only for $r\leq
2$, where $\sigma^{\alpha}_{i}$ with $\alpha=x,y,z$ are the Pauli matrices on
the site $i$. Furthermore, for any $r$, there are the two symmetries
$S_{1}=\sigma^{z}_{1}$ and $S_{L}=\sigma^{z}_{L}$, which fix the spins at the
boundaries. Of course, there is also the parity symmetry
$P=\otimes_{i=1}^{L}\sigma^{z}_{i}$. We note that there is also a chiral
symmetry represented by the operator $C=\otimes_{i=1}^{L}\sigma^{x}_{i}$ that
anticommutates with the Hamiltonian, so that the spectrum is symmetric with
respect to zero energy.

##  III Hilbert space fragmentation

The Hilbert space $\mathcal{H}$ can be decomposed into dynamically
disconnected Krylov subspaces as

$\mathcal{H}=\bigoplus_{n=1}^{K}\mathcal{K}_{n}\,,$ (8)

where the Krylov subspace $\mathcal{K}_{n}$ is spanned by $|\psi_{n}\rangle$,
$H|\psi_{n}\rangle$, $H^{2}|\psi_{n}\rangle$ and so on, for a certain so-
called root state $|\psi_{n}\rangle$ which is a product state having defined
symmetries. If there are only conventional symmetries, e.g., $S_{z}$, the
number $K$ of Krylov subspaces grows at most polynomially upon increasing the
system size $L$. In contrast, if there is fragmentation of the Hilbert space,
then $K\sim e^{cL}$ in the thermodynamic limit $L\to\infty$. There are Krylov
subspaces with dimension one as well as exponentially large subspaces (see
Appendix A). The one-dimensional Krylov subspaces are called frozen states.
For our model the total number of frozen states is $N_{froz}=12+2k$ for a size
$L=4+k$ with $k=0,1,\ldots,r-1$, whereas for $L>3+r$ an exponential growth
starts, i.e., $N_{froz}\sim a_{r}^{L-r-3}$ as $L\to\infty$, where $a_{r}$
decreases upon increasing $r$ (see Fig. 1).

Figure 1: $N_{froz}$ as a function of $L$ for different values of $r$. The
solid lines are obtained by fitting the points, for $L\geq 3+r$, with the
function $\propto a^{L}$.

Concerning the total number of Krylov subspaces, we get $K=30+6(k-2)$, if
$k\geq 2$, or $K=14+8k$, if $k<2$, for a size $L=4+k$ with $k=0,1,\ldots,r-1$.

Let us calculate the total number of sectors with definite symmetries,
$S_{z}$, $S_{1}$ and $S_{L}$, for $r\geq 3$. We note that, in this case, we
get 4 distinct combinations of the boundaries spins, and, in the bulk, there
are $L-1$ sectors with defined magnetization, then we have $4(L-1)$ sectors
with defined symmetries. As a result, for a size $L=4+k$ we get a number of
sectors equal to $12+4k$ which is always smaller than the total number of
Krylov subspaces for any range $r$. We deduce, therefore, that there is a
fragmentation of the Hilbert space (with an exponential number of Krylov
subspaces) for any range $r$ such that $r/L\to b$ as $L\to\infty$, with
$b\in[0,1)$. Thus, there is almost always fragmentation, except when the range
is $r=L-c_{r}(L)$, for any number $c_{r}(L)\geq 3$ such that $c_{r}(L)/L\to
0$, i.e., the range is asymptotically linear in the size with a linear
coefficient equal to one. In this case, the number of Krylov subspaces will
grow slower than exponential.

In order to characterize the fragmentation of a sector of dimension $D$
defined by certain symmetries, we consider the largest Krylov subspace with
dimension $D_{max}$, belonging to this sector. For instance, if the symmetries
are $S_{z}$, $S_{1}$ and $S_{L}$, as in the case with $r\geq 3$, the dimension
of the sector is given by $D=\binom{L-2}{(L+S_{z}-S_{1}-S_{L})/2-1}$. We can
get, therefore, sectors with dimension $D$ which grows exponentially with $L$,
e.g., for $S_{z}\sim 0$, or polynomially, e.g., linearly for $S_{z}=\pm(L-2)$
and $S_{1}=S_{L}=\pm 1$. If there is also the symmetry $n_{\textrm{dw}}$, as
in the cases with $r\leq 2$, $D$ has a different expression. For example for
$L$ even, namely $L=2+2k$, and $S_{z}=0$, $S_{1}=1$, $S_{L}=-1$, we get
$n_{\textrm{dw}}=L-1-2n_{k}$, where the kink number is $n_{k}=1+2x$, with
$x=0,1,2,\ldots,(L-2)/2$, and the dimension $D=\binom{k}{x}^{2}$. Given $D$
and $D_{max}$, we have the so-called strong fragmentation or weak
fragmentation if $D_{max}/D\to 0$ or $D_{max}/D\to 1$, respectively, in the
thermodynamic limit Sala20 .

The case $r=1$ has been investigated in Ref. Langlett21  where it has been
shown that there are sectors of dimension $D$ such that the dimension of their
larger Krylov subspace $D_{max}$ grows exponentially, still the ratio
$D_{max}/D\to 0$, vanishes exponentially for $L\to\infty$, revealing strong
fragmentation.

For $r=2$, if there is fragmentation of a sector with dimension $D$, that
sector is fragmented in frozen states plus at most one dominant larger Krylov
subspace. In detail, the sector will be the direct sum of all the
$N^{\prime}_{froz}$ frozen states in that sector and the complementary
subspace having dimension $D_{c}=D-N^{\prime}_{froz}$. One can check that, for
any fragmented sector of dimension $D$, the quantity $D_{c}-D_{max}$ is zero
(there is only one Krylov subspace with dimension larger than one, since
$D_{max}\geq 1$ and if $D_{c}\neq 0$ then necessarily $D_{c}>1$, by
definition) or -1 (all the Krylov subspaces are frozen states, namely
$D_{c}=0$ and $D_{max}=1$). Actually, when $D-D_{max}-N^{\prime}_{froz}=0$ we
find always only one frozen state in the sector, i.e., $N^{\prime}_{froz}=1$,
thus getting only two Krylov subspaces in the sector, and, therefore, we get
weak fragmentation, since $D_{max}\simeq D$ for large $L$.Furthermore, we can
get a kind of strong fragmentation, i.e., $D_{max}/D\to 0$ as $L\to\infty$,
when $D=N^{\prime}_{froz}>1$. This situation occurs, for instance, in the
sectors with dimension $D=L-4$ (i.e., which grows linearly with $L$), and for
$S_{z}=\pm(L-6)$, $n_{dw}=L-5$, $S_{1}=S_{L}=\pm 1$, which are the largest
sectors among those with dimension $D=N^{\prime}_{froz}>1$. In this case
$D_{max}/D\sim 1/L\to 0$ as $L\to\infty$, i.e., the ratio does not tend to
zero exponentially. We note that this kind of non-exponential strong
fragmentation is also present for $r=1$.

In contrast, for $3\leq r\leq L-3$, we do not find strong fragmentation in any
sector. In particular, we find that for the sectors such that
$D-D_{max}-N^{\prime}_{froz}=-1$ we get $N^{\prime}_{froz}=1$, i.e., the
sector is just a frozen state ($D=D_{max}=1$), therefore, trivially there is
not fragmentation. However, for any $r\geq 3$, there are sectors of dimension
$D$ with definite symmetries such that $D_{max}<D$ and $D_{max}/D\to 1$ as $L$
increases (see Fig 2). In conclusion we observe, therefore, that, for $r>3$,
there is at most weak fragmentation if the total number of Krylov subspaces
goes as $K\sim e^{cL}$.  These findings are pictorially summarized by Fig. 3.

Figure 2: $D_{max}/D$ as a the function of $L$ for different values of $r$. We
consider the sector with $S_{z}=0$ or $1$ for $L$ even or odd, $S_{1}=1$ and
$S_{L}=-1$.

Figure 3: A pictorial representation of the fragmentation. In Fig. (a) we
represent the sectors with definite symmetries as regions containing certain
states represented by points (the largest sector with $100$ points is obtained
for $L=12$, $S_{z}=0$, $n_{\textrm{dw}}=1$, $S_{1}=-S_{L}=1$ and $r\leq 2$.
For $r>2$ the largest sectors have different symmetries although we
pictorially represent them in the same way). The sectors will be fragmented in
Krylov subspaces (regions with gray boundary lines in Figs. (b,c,d)). For
$r=1$ (see Fig. (b)), there are sectors fragmented in frozen states, with
dimensions linearly dependent on $L$, and sectors fragmented in an exponential
number of Krylov subspaces, with the largest Krylov subspace with dimension
$D_{max}$ which grows exponentially with $L$, so that $D_{max}/D\to 0$. For
$r=2$ (see Fig. (c)), there are still sectors fragmented in frozen states,
with a dimension linearly dependent on $L$, and sectors fragmented in one
frozen state and one dominant large Krylov subspace with a dimension $D_{max}$
such that $D_{max}/D\to 1$. For $r>2$ (see Fig. (d)), there are fragmented
sectors containing a dominant large Krylov subspace with a dimension $D_{max}$
such that $D_{max}/D\to 1$.

##  IV Quantum chaos

Let us consider the energy level spacing statistics of the Krylov subspaces.
We focus on the average energy level spacing ratio for the eigenenergies in
these subspaces, i.e., ${\sf
r}_{i}=\min(\delta_{i},\delta_{i+1})/\max(\delta_{i},\delta_{i+1})$, where
$\delta_{i}=E_{i}-E_{i+1}$ is the gap between adjacent energy levels
Oganesyan07 . The distribution of ${\sf r}_{i}$ has been studied in Ref.
Atas13 . If the energy spectrum has a Gaussian-orthogonal ensemble level
statistics, then the average value is $\langle{\sf r}_{i}\rangle\approx 0.53$
for $L\to\infty$. In contrast, if the level statistics is Poissonian, then
$\langle{\sf r}_{i}\rangle\approx 0.39$. Differently from the fragmentation,
the level spacing statistics depends on the specific values of the non-zero
couplings $d_{\ell}$. We consider a power-law decay
$d_{\ell}=1/\ell^{\alpha}$. For $r=3$, $L=16$, in the largest Krylov subspace
of the sector defined by $S_{z}=0$, $S_{1}=1$ and $S_{L}=-1$, we notice that,
for large $\alpha$ (short range), $\langle{\sf r}_{i}\rangle$ is consistent
with an integrable system, while decreasing $\alpha$, the system becomes more
and more chaotic, and, for small $\alpha$ (long-range), $\langle{\sf
r}_{i}\rangle$ is consistent with the Wigner-Dyson distribution (see Fig. 4).

Figure 4: $\langle r_{i}\rangle$ as a function of $\alpha$. We consider $r=3$,
$L=16$ and the largest Krylov subspace in the sector with $S_{z}=0$, $S_{1}=1$
and $S_{L}=-1$.

##  V Scar states

Given a Krylov subspace $\mathcal{K}_{n}$, we consider the state

$|\Psi_{n}\rangle=\mathcal{N}_{n}\sum_{\alpha}|\psi^{(n)}_{\alpha}\rangle\,,$
(9)

where $|\psi^{(n)}_{\alpha}\rangle$,  with $\alpha=1,\dots,D_{n}$, are the
states of the computational basis (namely, all the states of the form
$|s_{1}\rangle|s_{2}\rangle\cdots|s_{L}\rangle$ with
$s_{i}=\uparrow,\downarrow$) which belong to the Krylov subspace
$\mathcal{K}_{n}$ with dimension $D_{n}$, and
$\mathcal{N}_{n}=1/\sqrt{D_{n}}$. Noticing that
$\otimes_{i=1}^{L}(|\uparrow_{i}\rangle+|\downarrow_{i}\rangle)$ is equal to
the sum of all the states in the computational basis, we get

$\otimes_{i=1}^{L}(|\uparrow_{i}\rangle+|\downarrow_{i}\rangle)=\sum_{n}\frac{1}{\mathcal{N}_{n}}|\Psi_{n}\rangle\,.$
(10)

Moreover, $H\otimes_{i=1}^{L}(|\uparrow_{i}\rangle+|\downarrow_{i}\rangle)=0$,
then

$\sum_{n}\frac{1}{\mathcal{N}_{n}}H|\Psi_{n}\rangle=0$ (11)

and since $H|\Psi_{n}\rangle\in\mathcal{K}_{n}$, all the states
$H|\Psi_{n}\rangle$ are linearly independent then $H|\Psi_{n}\rangle=0$ for
every $n$. We calculate the entanglement entropy of $|\Psi_{n}\rangle$ for a
block of length $l$, which is defined as
$S(l)=-\text{Tr}\left\\{\rho_{[1,l]}\ln\rho_{[1,l]}\right\\}$, where
$\rho_{[1,l]}$ is the reduced density matrix of the block. As shown in Fig. 5,
for the state $|\Psi_{n}\rangle$ in one of the largest Krylov subspaces, we
find that, for $l<L/2$, the entropy $S(l)$ increases slower than a linear
function of $l$.

Figure 5: Entanglement entropy $S(l)$ of a block of length $l$ for $L=16$ and
different $r$. We consider the state $|\Psi_{n}\rangle$, Eq. (9), that is in
the largest Krylov subspace of the sector with $S_{z}=0$, $S_{1}=1$,
$S_{L}=-1$ (for $r=1$ we have, in addition, $n_{\textrm{dw}}=-3$, and for
$r=2$, $n_{\textrm{dw}}=1$).

In contrast,  as shown in Fig. 6, for the eigenstates with energies in the
middle of the spectrum, the ETH predicts that the entropy is extensive (for
energies in the middle of the spectrum the microcanonical ensemble has a very
large temperature and from ETH we can give a random-matrix theory description,
so that the entropy is given by the Page formula Page93  restricted to the
subspace and it is extensive).

Figure 6: Entanglement entropy $S(l)$ of a block of length $l$ for $L=12$,
$r=3$ and $d_{\ell\leq r}=1$. We consider all the eigenstates in the largest
Krylov subspace of the sector with $S_{z}=0$, $S_{1}=1$, $S_{L}=-1$, having
energy in the middle of the spectrum. The red line corresponds to the scar
state $|\Psi_{n}\rangle$. The blue lines correspond to the remaining states at
zero energy, which show a thermal behavior, namely scales approximately
linearly.

Let us consider $r\geq 3$, which can host at most weak fragmentation. As shown
in Fig. 7, there are isolated eigenstates below the ETH-like curve, indicating
the presence of further scar states. Considering one of those states, the
entanglement entropy $S(L/2)$ grows at most logarithmically with $L$, as shown
in Appendix B, therefore ETH is violated.

Figure 7: Entanglement entropy $S(L/2)$ of all the energy eigenstates in the
largest Krylov subspace in the sector with $S_{z}=0$, $S_{1}=1$ and
$S_{L}=-1$, with $L=16$, $r=3$ and $d_{\ell\leq r}=1$. The diamond purple
point corresponds to the scar state $|\Psi_{n}\rangle$, the square red point
corresponds to a scar state having a large overlap with the state
$|\psi_{s}\rangle$ in Eq. (13).

##  VI Out-of-equilibrium dynamics

In order to study the out-of-equilibrium dynamics we consider $r\geq 3$, with
$d_{\ell}=1$ for any $\ell\leq r$, and even $L/2$. We will focus our attention
to the largest Krylov subspace in the sector defined by $S_{z}=0$, $S_{1}=1$
and $S_{L}=-1$. Let us consider the following initial state

$|\psi_{l}\rangle=|\uparrow\rangle^{\otimes
l}\otimes|\uparrow\downarrow\rangle^{\otimes
L/2-l}\otimes|\downarrow\rangle^{\otimes l}\,,$ (12)

which either belongs to the largest Krylov subspace or it is a frozen state,
depending on the value of $l$. In detail, the initial state belongs to the
largest Krylov subspace for $1\leq l\leq L/2-2$ while it is a frozen state for
$l=0$ and $l=L/2-1$. We calculate the time-evolution of the entanglement
entropy of half-chain, $S(L/2,t)$, starting from the initial state in Eq.
(12). As shown in Fig. 8, typically, initially, at early time, the
entanglement entropy remains approximately null for a while before it starts
growing. In Ref. Langlett21 , this behavior is explained by the long time
needed for the spin up at site $l$ and the spin down at site $L-l+1$ to
propagate towards the center of the chain before they scatter. However, in
contrast to this behavior, we found that for the special case with $l=L/2-3$
the entropy starts growing immediately, see Fig. 8 for $r=3$ (we check that
the same behavior occurs for $r\geq 3$). At later time the entropy grows
linearly, therefore, the propagation of the excitations is ballistic.

Figure 8: Time-evolution of the entanglement entropy $S(L/2,t)$, with the
initial states $|\psi_{l}\rangle$, for different values of $l$, and $r=3$,
$L=16$.

The average value of the energy of $|\psi_{l}\rangle$ is
$E=\langle\psi_{l}|H|\psi_{l}\rangle=0$ which is in the middle of the
spectrum, then from the ETH we expect that the time average of the reduced
density matrix of a small part of the system tends to a thermal state with
infinite temperature, so that the magnetization
$\langle\sigma^{z}_{j}(t)\rangle$ tends to zero for $1<j<L$ at long times,
while the boundary spins do not evolve. In Fig. 9 we study the time evolution
of the magnetization $\langle\sigma_{j}^{z}(t)\rangle$.

Figure 9: (Left panel) Time-evolution of the magnetization
$\langle\sigma_{j}^{z}(t)\rangle$, with the initial state $|\psi_{l}\rangle$,
Eq. (12), with $l=5$ and $L=16$, for $r=3$. (Right panel) Magnetization
profile at different long times.

Figure 10: (Left panel) Time-evolution of the magnetization
$\langle\sigma_{j}^{z}(t)\rangle$ with the initial state $|\psi_{s}\rangle$,
Eq. (13), for $L=16$ and $r=3$.  (Right panel) Time-evolution of the
magnetization $\langle\sigma_{j}^{z}(t)\rangle$ at the site $j=L/2-3$.

We note that at long times the magnetization tends to a stationary spatially
non-uniform profile. The weak fragmentation can contribute to this behavior:
although Krylov-restricted ETH is weakly satisfied in the dominant Krylov
subspace, the initial state does not involve the eigenstates belonging to the
non-dominant Krylov subspaces giving a non-uniform expectation value; this
because the magnetization profile for the eigenstates in the non-dominant
Krylov subspaces is non-uniform having predominantly an inversion of the spin
directions (in the bulk) with respect to the stationary profile. We note,
indeed, that the magnetization profile for the completely mixed state
restricted to the dominant Krylov subspace is
$\langle\sigma^{z}_{i}\rangle=A_{i}/D_{max}$ in the bulk, where $A_{i}$ is
predominantly positive (negative) in the left (right) part of the chain,
whereas $\langle\sigma^{z}_{i}\rangle$ is zero in the bulk for the completely
mixed state restricted to the full sector with definite symmetries.  However,
since the fragmentation is weak, $A_{i}/D_{max}$ is small compared to the
long-time magnetization value observed (since $A_{i}\sim(D-D_{max})\rightarrow
0$). Thus, we deduce that the scars will strongly contribute to the non-
uniformity of the magnetization. In particular, the scar $|\Psi_{n}\rangle$
has also the magnetization $\langle\sigma^{z}_{i}\rangle=A_{i}/D_{max}$, so
that the main contribution predominantly comes from other scar states.
Moreover, due to the scar states we can have initial states which do not
exhibit thermalization. This behavior is very prominent, for $r=3$, using the
following initial state

$|\psi_{s}\rangle=|\uparrow\rangle^{\otimes
L/2-4}\otimes|\downarrow\rangle\otimes|\uparrow\rangle^{\otimes
4}\otimes|\downarrow\rangle^{\otimes L/2-1}\,,$ (13)

which belongs to the largest Krylov subspace and has a large overlap with a
high energy scar state. For $L=16$, the energy is
$E=\langle\psi_{s}|H|\psi_{s}\rangle=24$ and the magnetization
$\langle\sigma_{j}^{z}(t)\rangle$ stays almost equal to its initial value,
except for the site $L/2-3$ at which the magnetization shows persistent
oscillations, see Figs. 10.  This behavior is due to the fact that the
magnetization of the scar state differs from the state $|\psi_{s}\rangle$ only
locally, near the site $L/2-3$.

##  VII Conclusions

Fragmentation of the Hilbert space is believed to exist for interactions
involving few sites. Here, we investigated the fate of fragmentation and scar
states upon increasing the range of interaction by considering a modified
long-range Fredkin spin chain which exhibits fragmentation due to
interference. We prove that there is fragmentation, i.e., there is an
exponential number of Krylov subspaces in the thermodynamic limit, for any
range which is not asymptotically linear in the size $L$ with a linear
coefficient equal to one.  The nearest neighbor case $r=1$ has been
investigated also in Ref. Langlett21 , where it is shown that there are scars
and strong fragmentation with the standard exponential decay. Here, we find
that for $r=1$ the strong fragmentation can also exhibit an algebraic decay of
the ratio $D_{max}/D\sim 1/L$ in some sectors. By increasing the range, for a
range $r=2$, we have both weak and strong fragmentation. Interestingly, in
this case, the strong fragmentation exhibits only the algebraic decay of the
ratio $D_{max}/D\sim 1/L$. For a range $r\geq 3$ there is only weak
fragmentation. Moreover, we show that scar states exist at any range of the
interaction inside non-integrable Krylov subspaces. Finally, we investigate
the effects of such structures of the Hilbert space on the out-of-equilibrium
dynamics which is characterized by either local persistent oscillations or
non-uniform stationary profile of the magnetization.  It might be interesting
investigating the out-of-equilibrium dynamics through Krylov complexity as
done in Ref. Nandy .

#### Note added:

After our work was completed, we became aware of the work Stephen  where an
example of fragmentation, in two dimensions, robust to long-range
perturbations is reported.

### Acknowledgements

The authors acknowledge financial support from the project BIRD 2021
”Correlations, dynamics and topology in long-range quantum systems” of the
Department of Physics and Astronomy, University of Padova, and from the
European Union-Next Generation EU within the ”National Center for HPC, Big
Data and Quantum Computing” (Project No. CN00000013, CN1 Spoke 10 - Quantum
Computing).

##  Appendix A Calculation of the dimension of the largest Krylov subspace and
the number of frozen states

Given an operator $H$ and a state $|\psi\rangle$, the corresponding Krylov
subspace is generated by the vectors $|\psi\rangle$, $H|\psi\rangle$,
$H^{2}|\psi\rangle$, and so on. In detail, only the first $n$ vectors are
linearly independent, for a certain integer $n$, which is the dimension of the
Krylov subspace. A way to calculate $n$, is based on applying iteratively $H$
and using the Gram-Schmidt process at each step to orthonormalize the set of
vectors, so that, at the $(n+1)$-th step, one has a null vector. Let us
consider a Krylov subspace $\mathcal{K}_{n}$ obtained from a root state
$|\psi_{n}\rangle$.  We consider all the product states
$|\psi^{(n)}_{\alpha}\rangle$ in the computational basis generated by applying
powers of $H$. We have that $|\psi^{(n)}_{\alpha}\rangle\in\mathcal{K}_{n}$,
then since $|\psi^{(n)}_{\alpha}\rangle$, for $\alpha=1,\ldots,D_{n}$, are
$D_{n}$ linear independent states, we have that
$D_{n}\leq\text{dim}\mathcal{K}_{n}$. In order to prove that
$D_{n}=\text{dim}\mathcal{K}_{n}$, we can proceed ad absurdum. Let us assume
that $D_{n}<\text{dim}\mathcal{K}_{n}$ for some $n$. We have that
$\mathcal{H}=\bigoplus_{n}\mathcal{K}_{n}$ so that
$\text{dim}\mathcal{H}=\sum_{n}\text{dim}\mathcal{K}_{n}$. Since all the
states $|\psi^{(n)}_{\alpha}\rangle$ form the computational basis of
$\mathcal{H}$, we get $\text{dim}\mathcal{H}=\sum_{n}D_{n}$, then
$\text{dim}\mathcal{H}=\sum_{n}D_{n}<\sum_{n}\text{dim}\mathcal{K}_{n}$, which
is a contradiction. This means that $\mathcal{K}_{n}$ is generated by the
product states $|\psi^{(n)}_{\alpha}\rangle$ for $\alpha=1,\ldots,D_{n}$. A
simple method to determinate the dimension $D_{max}$ of the largest Krylov
subspace is to apply $H$ to every product state belonging to the sector of
dimension $D$. On the other hand, to determinate the number $N_{froz}$ of
Krylov subspaces of dimension one, called frozen states, we apply $H$ to all
the product states of the full Hilbert space $\mathcal{H}$, and we count the
number of those product states which are eigenstates of $H$.

##  Appendix B Entropy of a scar state

Let us show that the entanglement entropy of the scar state in Eq. (9) grows
at most logarithmically with $L$. For simplicity we consider the following
state, which, for large even $L$ and $S_{z}=0$ well approximates a scar state
with $E=0$,

$|\Psi\rangle=\frac{1}{\sqrt{N(L)}}\sum_{1\leq i_{1}<i_{2}<\cdots<i_{L/2}\leq
L}\sigma^{+}_{i_{1}}\sigma^{+}_{i_{2}}\cdots\sigma^{+}_{i_{L/2}}|\downarrow\rangle^{\otimes
L}\,,$ (14)

where $\sigma^{+}=|\uparrow\rangle\langle\downarrow|$ and
$N(L)=\binom{L}{L/2}$. A string of local spins, like
$\uparrow\uparrow\downarrow\ldots$ can be mapped to a lattice path by
attaching a diagonal line $\diagup$ to $\uparrow$, and $\diagdown$ to
$\downarrow$, from left to right. The state $|\Psi\rangle$ can be expressed as
the sum over all the paths $w$ starting from $(0,0)$ and ending at $(L,0)$

$|\Psi\rangle=\frac{1}{\sqrt{N(L)}}\sum_{w}|w\rangle\,.$ (15)

To calculate the reduced density matrix $\rho_{1,L/2}$, we consider the paths
$w_{y}$ from $(0,0)$ to $(L/2,y)$ and the paths $z_{y}$ from $(L/2,y)$ to
$(L,0)$, where $y=-L/2,-L/2+2,\ldots,L/2$, thus we can write

$|\Psi\rangle=\frac{1}{\sqrt{N(L)}}\sum_{y}\sum_{w_{y}}\sum_{z_{y}}|w_{y}\rangle\otimes|z_{y}\rangle\,.$
(16)

We can calculate the partial trace by using the basis $\\{|z_{y}\rangle\\}$,
thus we get

$\rho_{[1,L/2]}=\frac{1}{N(L)}\sum_{y}N_{y}(L)\sum_{w_{y}}\sum_{w^{\prime}_{y}}|w_{y}\rangle\langle
w^{\prime}_{y}|\,,$ (17)

where $N_{y}(L)=\binom{L/2}{L/4+y/2}$ is the number of paths $w_{y}$. Then,
the eigenvalues of $\rho_{[1,L/2]}$ are $\lambda_{y}=(N_{y}(L))^{2}/N(L)$, and
the entanglement entropy is

$S(L/2)=-\sum_{y}\lambda_{y}\ln\lambda_{y}\simeq\ln L\,.$ (18)

To derive this asymptotic formula we use the Stirling’s approximation, so that

$\lambda_{y}\simeq\frac{1}{\sqrt{L}}\frac{e^{Lf(x_{y})}}{1-x_{y}^{2}}\,,$ (19)

where $x_{y}=2y/L$ and $f(x)=-(1-x)\ln(1-x)-(1+x)\ln(1+x)$. For $L\to\infty$
we get

$S(L/2)\simeq
L\int_{-1}^{1}\frac{1}{\sqrt{L}}\frac{e^{Lf(x)}}{1-x^{2}}\ln\left(\frac{1}{\sqrt{L}}\frac{e^{Lf(x)}}{1-x^{2}}\right)dx\,.$
(20)

Using the Laplace method, since $f(x)$ has only one maximum at $x=0$, we get
$S(L/2)\simeq\ln L\,$.

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