$\det\left(f_{i+1}\left(\tau_{k-j-1},\tau_{k-j},\tau_{k-j+1}\right)\right)_{i\in\left[n-2\right],j\in\left[n-2\right]}=0.$
(66)

###### Proof.

Since $\forall k\in{\mathbb{N}}^{*}$,
${\boldsymbol{x}}_{1:2}\left(t_{3k-3}\right)={\boldsymbol{x}}_{1:2}\left(t_{3k}\right)=x_{{\mathrm{m}}2}{\boldsymbol{e}}_{2}$,
it can be proved that the control $u$ in $\left(t_{3k-3},t_{3k}\right)$ is

$\begin{dcases}u\left(t\right)\equiv
u_{3k-2}=-{u_{\mathrm{m}}},&t\in\left(t_{3k-3},t_{3k-2}\right),\\\
u\left(t\right)\equiv
u_{3k-1}={u_{\mathrm{m}}},&t\in\left(t_{3k-2},t_{3k-1}\right),\\\
u\left(t\right)\equiv
u_{3k}=-{u_{\mathrm{m}}},&t\in\left(t_{3k-1},t_{3k}\right).\end{dcases}$ (67)

Among them, $t_{3k-1}=\frac{3t_{3k}+t_{3k-3}}{4}$ and
$t_{3k-2}=\frac{t_{3k}+3t_{3k-3}}{4}$. Hence, $\Delta
u_{3k-1}=2{u_{\mathrm{m}}}$, $\Delta u_{3k}=-2{u_{\mathrm{m}}}$, and $\Delta
u_{3k+1}=0$.

$\forall N\in{\mathbb{N}}^{*}$, the augmented switching law between
${\boldsymbol{x}}\left(t_{0}\right)$ and ${\boldsymbol{x}}\left(t_{3N}\right)$
is denoted as ${\mathcal{L}}_{N}$, which requires that $\forall
k\in\left[N-1\right]$,
${\boldsymbol{x}}_{3k,1:2}=x_{{\mathrm{m}}2}{\boldsymbol{e}}_{2}$, and
${\boldsymbol{x}}_{3N}={\boldsymbol{x}}\left(t_{3N}\right)$. According to
Corollary 1, the Jacobian matrix ${\boldsymbol{J}}$, except the last column,
is (62). $\forall k\in\left[N-1\right]$, the $3k$-th columns and the
$\left(2k-1\right)$-th rows can be eliminated, and the coefficients of each
column, i.e., $\pm 2{u_{\mathrm{m}}}$, can be normalized, resulting in
${\boldsymbol{J}}\leftrightarrow{\boldsymbol{J}}_{1}$. Upon left
multiplication by (63) and right multiplication by
$\left(\delta_{j=i}-\delta_{j=i+1}\right)_{i\in\left[N\right],j\in\left[N\right]}^{3}$,
it has ${\boldsymbol{J}}\leftrightarrow{\boldsymbol{J}}^{\prime}$. Considering
the rank of columns $\left(k-n+3\right)$ through $k$ of
${\boldsymbol{J}}^{\prime}$, (66) holds. ∎

######  Remark.

According to Corollary 3, the chattering phenomenon can be induced by
$x_{2}\leq x_{{\mathrm{m}}2}$ only if $\lim_{k\to\infty}\tau_{k}=0$ and
$\tau_{k}>0$ when applying (66). When given initial values, (66) is a
polynomial equation w.r.t. $\tau_{k}$. It is evident that $n=3$ does not hold
since it has $\tau_{k}=2\tau_{k-1}-\tau_{k-2}$. Through more refined
derivation, it can be rigorously proven that $n=4$ does not hold. The proof is
lengthy and omitted due to space limitation, whose approach is similar to our
previous work [18]. In other words, the chattering phenomenon induced by
$x_{2}\leq x_{{\mathrm{m}}2}$ can only occur in a 5th order problem or higher.
One can try to determine whether the chattering phenomenon can occur in an
$n$-th order problem for $n\geq 5$ based on the proposed Corollary 3.

######  Remark.

It is challenging to obtain the recursive equation (66) for problem (53a) of
high order based on the traditional costate-based necessary condition [14]
since the behaviors of costates are significantly complex for high order
problems.

Although Theorem 3 holds under the assumption of the non-existence of the
chattering phenomenon, Corollary 4 provides an example to apply the proposed
necessary condition to the case the chattering phenomenon occurs, where a sub-
arc without chattering of the optimal trajectory is investigated.

###  VI-D Simulation Experiment

Figure 2: Results of the simulation experiment. (a) The plot of the local
minimal $t_{\mathrm{f}}^{\prime}-t_{\mathrm{f}}$ and $t_{4}^{\prime}-t_{4}$.
(b) The original trajectory and the optimized trajectory. Among them,
${\boldsymbol{x}}_{0}=\left(0.75,-0.375,2,9\right)$,
${{\boldsymbol{x}}_{\mathrm{f}}}=\left(0.25,0.5,-2,-5\right)$,
${\boldsymbol{x}}_{\mathrm{m}}=\left(1,1.5,4,20\right)$, ${u_{\mathrm{m}}}=1$.

A simulation experiment is conducted to verify the proposed state-centric
necessary condition. As shown in Fig. 2(b), our previous work [8] provided a
4th order suboptimal trajectory
${\boldsymbol{x}}={\boldsymbol{x}}\left(t\right)$, $u=u\left(t\right)$,
$t\in\left[t,t_{\mathrm{f}}\right]$. The augmented switching law of the
original trajectory is
${\mathcal{L}}=\underline{01}\overline{0}\underline{2}\overline{01}\underline{0}\overline{01}\underline{0}\overline{0}$.
According to Corollary 3, $N-\sum_{i=1}^{N}\left|{\mathcal{S}}_{i}\right|=6>4$
implies that ${\boldsymbol{x}}={\boldsymbol{x}}\left(t\right)$ is not optimal.
Let ${\boldsymbol{H}}$ be the equality constraints induced by $\mathcal{L}$.
Note that
${\mathrm{rank}}\frac{\partial{\boldsymbol{H}}}{\partial{\boldsymbol{t}}}=9$
and
$\frac{\partial{\boldsymbol{H}}}{\partial{\boldsymbol{t}}}\in{\mathbb{R}}^{9\times
11}$ except the 4th and 11th columns is full row rank. Then, disturb $t_{4}$
into $t_{4}^{\prime}$ and search for the minimum terminal time
$t_{\mathrm{f}}^{\prime}$ subject to the constraints of (53a). As shown in
Fig. 2(a), $t_{\mathrm{f}}^{\prime}$ achieves local minimum
$t_{\mathrm{f}}-0.0061$ at $t_{4}^{\prime}=t_{4}+0.0208$, resulting in the
optimized trajectory ${\boldsymbol{x}}^{\prime}$ shown in Fig. 2(b). It can be
observed in Fig. 2(b4) that $x_{3}^{\prime}$ is tangent to
$-x_{{\mathrm{m}}3}$, while $x_{3}>-x_{{\mathrm{m}}3}$ holds in the original
trajectory. In other words, the original trajectory fails to fully utilize the
feasible set, and the extreme of ${t_{\mathrm{f}}}^{\prime}$ activates a new
constraint; hence, ${\boldsymbol{t}}^{\prime}$ cannot be further optimized
under fixing ${\mathcal{L}}$. Among them,

$\begin{dcases}{\boldsymbol{t}}=\left(1.7500,2.1562,3.1562,4.5342,5.5342,\right.\\\
\quad\left.6.0342,6.5221,7.0100,7.2226,8.9165,9.8604\right),\\\
{\boldsymbol{t}}^{\prime}=\left(1.7500,2.1562,3.1562,4.5550,5.5550,\right.\\\
\quad\left.6.2023,6.6621,7.1218,7.2166,8.9104,9.8543\right).\end{dcases}$ (68)

Therefore, ${\boldsymbol{x}}^{\prime}$ achieves a $0.6\%$ reduction in the
terminal time compared to ${\boldsymbol{x}}$. The proposed state-centric
necessary condition is verified through the simulation experiment. More
efficient optimization algorithms based on the proposed state-centric
necessary condition can be developed in future works.

##  VII Conclusions

This paper has set out to establish an innovative theoretical framework for
the augmented switching law for time-optimal control of controllable linear
systems and its applications. The proposed augmented switching law represents
the input control and the feasibility of the Bang-Singular-Bang trajectory in
a compact form. Specifically, the equality and inequality constraints induced
by the augmented switching law represent a sufficient and necessary condition
for the feasibility of the Bang-Singular-Bang trajectory. Based on the
augmented switching law, a given feasible trajectory can be disturbed with the
fixing augmented switching law and feasibility guarantees, resulting in a
state-centric necessary condition for optimal control since any disturbed
feasible trajectory should have a strictly longer terminal time than the
optimal terminal time due to the uniqueness of optimal control. The proposed
necessary condition states that the Jacobian matrix induced by the augmented
switching law must not be full row rank. High order conditions can be derived
based on the local optimality of the optimal trajectory. Different from the
traditional necessary condition based on adjoining methods, the developed
necessary condition requires only state and control information without
dependence on costate information.

The proposed state-centric necessary condition is applied to the optimal
control for chain-of-integrators systems with full box constraints. Under the
condition that every arc is strictly feasible except for the constraints
active during the whole arc, the proposed state-centric necessary condition
implies that the number of arcs should be no greater than the sum of the order
of constrained arcs plus the order of the problem. Considering the chattering
phenomenon induced by the 2nd order state constraints, this paper proves a
recursive equation for the junction time before the chattering limit point.
The above two conclusions are challenging to prove by traditional costate-
based necessary conditions, which demonstrates the effectiveness of the
proposed state-centric necessary conditions.

## Acknowledgment

This work was supported by the National Key Research and Development Program
of China under Grant 2023YFB4302003.

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|  Yunan Wang (S’22) received the B.S. degree in mechanical engineering, in
2022, from the Department of Mechanical Engineering, Tsinghua University,
Beijing, China. He is currently working toward the Ph.D. degree in mechanical
engineering. His research interests include optimal control, trajectory
planning, toolpath planning, and precision motion control. He was the
recipient of the Best Conference Paper Finalist at the 2022 International
Conference on Advanced Robotics and Mechatronics, and 2021 Top Grade
Scholarship for Undergraduate Students of Tsinghua University.
---|---
|  Chuxiong Hu (S’09-M’11-SM’17) received his B.S. and Ph.D. degrees in
Mechatronic Control Engineering from Zhejiang University, Hangzhou, China, in
2005 and 2010, respectively. He is currently an Associate Professor (tenured)
at Department of Mechanical Engineering, Tsinghua University, Beijing, China.
From 2007 to 2008, he was a Visiting Scholar in mechanical engineering with
Purdue University, West Lafayette, USA. In 2018, he was a Visiting Scholar in
mechanical engineering with University of California, Berkeley, CA, USA. His
research interests include precision motion control, high-performance
multiaxis contouring control, precision mechatronic systems, intelligent
learning, adaptive robust control, neural networks, iterative learning
control, and robot. Prof. Hu was the recipient of the Best Student Paper
Finalist at the 2011 American Control Conference, the 2012 Best Mechatronics
Paper Award from the ASME Dynamic Systems and Control Division, the 2013
National 100 Excellent Doctoral Dissertations Nomination Award of China, the
2016 Best Paper in Automation Award, the 2018 Best Paper in AI Award from the
IEEE International Conference on Information and Automation, and 2022 Best
Paper in Theory from the IEEE/ASME International Conference on Mechatronic,
Embedded Systems and Applications. He is now an Associate Editor for the IEEE
Transactions on Industrial Informatics and a Technical Editor for the
IEEE/ASME Transactions on Mechatronics.
---|---
|  Yujie Lin received the B.S. degree in mathematics and applied mathematics,
in 2023, from the Department of Mathematical Sciences, Tsinghua University,
Beijing, China. He is currently working toward the Ph.D. degree in
mathematics, at Qiuzhen College, Tsinghua University. His research interests
include 3&4-dimensional topology and knot theory.
---|---
|  Zeyang Li received the B.S. degree in mechanical engineering in 2021, from
the School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai,
China. He is currently working toward the M.S. degree in mechanical
engineering with the Department of Mechanical Engineering, Tsinghua
University, Beijing, China. His current research interests include the areas
of optimal control and reinforcement learning.
---|---
|  Shize Lin received the B.S. degree in mechanical engineering, in 2020, from
the Department of Mechanical Engineering, Tsinghua University, Beijing, China,
where he is currently working toward the Ph.D. degree in mechanical
engineering. His research interests include robotics, motion planning and
precision motion control.
---|---
|  Suqin He received the B.S. degree in mechanical engineering from Department
of Mechanical Engineering, Tsinghua University, Beijing, China, in 2016, and
the Ph.D. degree in mechanical engineering from the Department of Mechanical
Engineering, Tsinghua University, Beijing, China, in 2023. His research
interests include multi-axis trajectory planning and precision motion control
on robotics and CNC machine tools. He was the recipient of the Best Automation
Paper from IEEE Internal Conference on Information and Automation in 2016.
---|---


# Gifts from long-range interaction: Emergent gapless topological behaviors in
quantum spin chain

Sheng Yang  Institute for Advanced Study in Physics and School of Physics,
Zhejiang University, Hangzhou 310058, China  Hai-Qing Lin<EMAIL_ADDRESS>Institute for Advanced Study in Physics and School of Physics, Zhejiang
University, Hangzhou 310058, China  Xue-Jia Yu<EMAIL_ADDRESS>Department
of Physics, Fuzhou University, Fuzhou 350116, Fujian, China  Fujian Key
Laboratory of Quantum Information and Quantum Optics, College of Physics and
Information Engineering, Fuzhou University, Fuzhou, Fujian 350108, China

###### Abstract

Topology in condensed matter physics is typically associated with a bulk
energy gap. However, recent research has shifted focus to topological phases
without a bulk energy gap, exhibiting nontrivial gapless topological
behaviors. In this letter, we explore a cluster Ising chain with long-range
antiferromagnetic interactions that decay as a power law with the distance.
Using complementary numerical and analytical techniques, we demonstrate that
long-range interactions can unambiguously induce an algebraic topological
phase and a topological Gaussian universality, both of which exhibit
nontrivial gapless topological behaviors. Our study not only provides a
platform to investigate the fundamental physics of quantum many-body systems
but also offers a novel route toward searching for gapless topological phases
in realistic quantum simulators.

_Introduction._ —Locality is a fundamental concept in quantum many-body
physics, implying that the Hamiltonian consists of a sum of terms with local
interactions [1, 2, 3, 4]. However, describing many-body systems in terms of
local interactions is often an approximation and not always a reliable one
[5]. Long-range (LR) power-law interactions ($1/r^{\alpha}$) are typical non-
local interactions that are both of fundamental and practical importance,
appearing ubiquitously in nature and in numerous experimental systems of
current interest, such as trapped-ion [6, 7], nitrogen-vacancy centers [8, 9],
and programmable Rydberg atom arrays [10, 11]. On the theoretical side, LR
interactions are widespread and lead to qualitatively new physics, including
modifications to the Mermin-Wagner [12, 13] and Lieb-Schultz-Mattis theorems
[14, 15], the breakdown of the entanglement area law for gapped phases or the
conformal symmetry at criticality [16, 17, 18, 19], the emergence of new Lieb-
Robinson bounds [20, 21], as well as modifying the critical behavior at phase
transitions [22, 23, 24, 25, 26, 27, 28, 29].

On a different front, symmetry-protected topological (SPT) phases, as a
typical class of quantum matter beyond the Landau paradigm, have attracted
much attention in the past few decades [30, 31, 32, 33, 34]. Although the bulk
gap plays a crucial role in defining SPT phases, recent progress [35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56,
57, 58, 59, 60, 61, 62] has revealed that many key features of topological
properties persist even with nontrivial coupling between the boundary and the
gapless bulk. This extension is termed _gapless topological phases_ [41, 44,
47, 46], which exhibit nontrivial gapless topological behaviors, including
symmetry-protected degenerate edge modes, power-law decay in string order
parameters [47], nontrivial conformal boundary conditions [51, 52], and
universal bulk-boundary correspondence in the entanglement spectrum [53].

Figure 1: Summary of the main results. Below the phase diagram is a schematic
of the LR cluster Ising chain, where the blue dashed line (green-filled
ellipsoid) represents the LR AFM interaction (cluster interaction). The
ground-state phase diagram of the Hamiltonian (1) exhibits AFM, SPT, and
algebraic SPT phases, along with symmetry-enriched Ising and topological
Gaussian (yellow dashed line) universality classes. The red line represents a
new gapped SPT phase with two-fold degeneracy. The blue markers are determined
by the Binder ratio crossings, and the colored background indicates the
effective central charge $c_{\text{eff}}$ defined in the main text. The gray
region ($\alpha<0.1$) is beyond our computation ability.

From an experimental point of view, it is inevitable to consider the influence
of LR interactions when implementing nontrivial topological phases in state-
of-the-art quantum simulators. To date, various studies [63, 64, 65] have
demonstrated qualitative changes of physical properties under LR interactions
in gapped topological phases, including the emergence of new massive edge
modes [16, 66, 17] and novel bulk-boundary correspondence [67, 68]. However,
it is not immediately clear whether LR interactions can have a significant
impact on gapless topological phases. This is because the gapless bulk is
usually sensitive to LR interactions, while topological properties remain
robust against them. Therefore, it is valuable to address the following
intriguing questions: how does the gapless topological phase respond to LR
interactions, and do LR interactions induce new phases or phase transitions
with nontrivial gapless topological behaviors? If so, how to understand the
underlying mechanism?

To conclusively address the aforementioned issues, in this letter, we focus on
investigating the simplest type of gapless topological phase—symmetry-enriched
or topological nontrivial quantum critical points (QCPs). Notably, we
concentrate on a one-dimensional (1D) cluster Ising model featuring LR
antiferromagnetic (AFM) interactions, which is amenable to powerful density
matrix renormalization group (DMRG) simulations [69, 70, 71, 72, 73]. We
conduct comprehensive simulations to investigate the ground-state phase
diagram as a function of the cluster interaction strength $\lambda$ and the LR
power exponent $\alpha$. For large $\alpha\textgreater 1.0$, the ground state
manifests as an AFM (cluster SPT) phase for $\lambda$ below (above) a critical
value $\lambda_{\rm c}$, separating a symmetry-enriched Ising critical point.
Conversely, for sufficiently small $\alpha\lesssim 1.0$, the cluster SPT phase
gradually crossovers to a different SPT phase characterized by algebraic
correlations. In the limit $\alpha=0$, the ground state is a gapped
topological phase with two-fold degenerate edge modes for any
$\lambda\textgreater 0$. Meanwhile, the universality class of the phase
transition gradually changes from topological Ising to the topological
Gaussian universality as $\alpha$ decreases to $0$. Finally, we provide an
intuitive physical picture to elucidate the numerical observations.

_Model and method._ —We consider the following LR interacting Hamiltonian for
spins on a lattice of length $L$ (see Fig. 1)

$\begin{split}H_{\text{LRCI}}=\sum_{i<j}\frac{(1-\lambda)}{d_{ij}^{\alpha}}\sigma^{z}_{i}\sigma^{z}_{j}+\lambda\sum_{j}\sigma^{z}_{j-1}\sigma^{x}_{j}\sigma^{z}_{j+1}\,,\end{split}$
(1)

where $\vec{\sigma_{i}}=(\sigma^{x}_{i},\sigma^{y}_{i},\sigma^{z}_{i})$
represents the Pauli matrix on site $i$. The parameter $\alpha$ is the power
exponent of the LR interaction, and $d_{ij}$ is the distance between sites $i$
and $j$, which is $|i-j|$ and $\min(|i-j|,L-|i-j|)$ for open (OBC) and
periodic (PBC) boundary conditions, respectively. It is noted that the LR AFM
interactions compete with each other and the ground-state energy is still an
extensive quantity linear with $L$, therefore, the implementation of a Kac
rescaling is not necessary in our case (see Sec. I of Supplementary Materials
(SM)). Notably, for $\alpha\to\infty$, Eq. (1) reduces to the nearest neighbor
short-range (SR) cluster Ising model, which can be exactly solved via the
Jordan-Wigner transformation [74, 75, 76, 77, 78]. The parameter $\lambda$
governs the competition between the Ising and cluster interactions, driving
the system toward different phases, including AFM and cluster SPT phases [74].
The phase transition between them belongs to the symmetry-enriched Ising
universality class, featuring topological-protected edge modes with two-fold
degeneracy [44, 47, 54, 53]. Conversely, for $\alpha=0$, the all-to-all
interacting model can be solved via the Holstein-Primakoff transformation,
leading to a gapped topological phase with two-fold degenerate edge modes for
any $\lambda\textgreater 0$ (see Sec. VII of SM for details).

For general $\alpha$, Eq. (1) is no longer exactly solvable and we can only
resort to numerical simulations. In this work, we employ a large-scale DMRG
simulation to study the global ground-state phase diagram of the LR model. The
numerical details are introduced in Sec. I of the SM. We focus on the model
for both sufficiently strong and weak LR interactions, exploring the exotic
phases and phase transitions arising from the interplay between topology and
LR interactions.

Figure 2: The double logarithmic plot of the connected spin-spin correlation
$|C_{\rm c}^{zz}(r)|$ (a) and the SPT string order parameter $\mathcal{S}(r)$
(b) versus the lattice distance $r$ for $\alpha=0.3,1.8,3.0$ with fixed
$\lambda=0.55$ . The bulk entanglement spectrum (c) and magnetization profile
(d) within the typical algebraic SPT region for $\alpha=0.3$ and
$\lambda=0.55$ . $m$ counts the spectrum from the lowest-lying levels and $i$
labels the site position. Figure 3: The Binder ratio $R_{4}$ for AFM order
versus cluster interaction strength $\lambda$ for $\alpha=3.0$ (a) and $0.6$
(c). The effective central charge as a function of $\lambda$ for $\alpha=3.0$
(b) and $0.6$ (d). The simulated system size is $L=96$ under PBC for (b) and
(d).

_Quantum phase diagram._ —Before presenting the details of the DMRG results,
we first summarize our main findings. A schematic phase diagram with varying
LR power $\alpha$ and cluster interaction strength $\lambda$ is shown in Fig.
1. For large $\alpha\textgreater 1.0$, the ground state can exhibit AFM and
cluster SPT phases separated by a symmetry-enriched Ising critical point. As
$\alpha$ decreases, the LR quantum fluctuation is enhanced, resulting in
exotic phases and phase transitions. In the limit $\alpha=0$, the LR
interaction becomes all-to-all interactions. We can solve the model via the
Holstein-Primakoff transformation and show that the ground state is a gapped
topological phase exhibiting two-fold degenerate edge modes for any
$\lambda\textgreater 0$ (the red line in Fig. 1). For sufficiently small
$\alpha\lesssim 1.0$, we find that the connected spin-spin correlation (string
order parameter) changes from a hybrid behavior—exponential at short distances
and algebraic (constant) at long distances—to purely algebraic decay for
$\lambda>\lambda_{\rm c}$. This suggests that the gapped SPT phase gradually
crossovers to a new algebraic SPT phase (define below). Notably, the entire
critical line exhibits robust two-fold degenerate edge modes, and the
universality class gradually changes from topological Ising with central
charge $c=1/2$ to the topological Gaussian universality class with effective
central charge $c_{\rm eff}=1$ as $\alpha$ approaches $0$, indicating that LR
interactions can induce a new topologically nontrivial universality class.

_Algebraic SPT phase._ —To investigate the effect of LR interactions on
quantum phases, we compute the connected spin-spin correlation function and
SPT string order parameter, as defined in Sec. IV and V of the SM. It is
important to note that LR interactions do not significantly affect the AFM
long-range order for $\lambda<\lambda_{\rm c}$, as demonstrated in Sec. V of
the SM and previous works [17]. Therefore, in this work, we primarily focus on
the influence of LR interactions on topological nontrivial QCPs and the gapped
SPT phase.

In the SR limit $\alpha\rightarrow\infty$, our numerical results show AFM and
cluster SPT phases for $\lambda<\lambda_{\rm c}=0.5$ and $\lambda>\lambda_{\rm
c}$, respectively, in agreement with previous studies [74, 75, 54] (see Sec.
III of the SM). In the presence of LR interactions, the simulation also
verifies the stability of the gapped SPT phase. Fig. 2 (a) and (b) depict a
double logarithmic plot of the connected spin-spin correlation function
$|C^{zz}_{\rm c}(r)|$ and string order parameter $\mathcal{S}(r)$ versus
lattice distance $r$ for various $\alpha$ with fixed $\lambda=0.55$ within the
gapped SPT region, revealing anomalous behaviors of the correlation function.
Specifically, the connected spin-spin correlation displays a hybrid behavior
that is exponential decay at short distances and algebraic decay at long
distances for large $\alpha\textgreater 1$ and purely algebraic decay at all
distances for sufficiently small $\alpha\lesssim 1$. On the other hand, the
string order parameter displays “long-range order” and purely algebraic decay
for large and small $\alpha$, respectively (see Sec. V of the SM for more
results). These properties are similar to critical states in SR interacting
systems, but with a finite bulk energy gap $\Delta_{\rm b}$, as demonstrated
in Sec. IV of the SM for typical values of $\lambda\textgreater\lambda_{\rm
c}$ where the cluster interaction dominates.

To explicitly demonstrate the robustness of topological features for
$\lambda>\lambda_{\rm c}$, we present the bulk entanglement spectrum (as
defined in Sec. III of the SM) and the magnetization profile
$\langle\sigma^{z}_{i}\rangle$ for particular values of $\alpha$. For
$\alpha\rightarrow\infty$, the ground state corresponds to the gapped SPT
phase at $\lambda>\lambda_{\rm c}$, as evidenced by the four-fold degeneracy
in the bulk entanglement spectrum and the obvious magnetization near the
boundaries [34, 53, 41, 51, 52]. For finite $\alpha$, the four-fold degeneracy
in the lowest-lying entanglement spectrum and the edge magnetization remain
robust against LR interactions, as depicted in Fig. 2 (c) and (d) for
$\alpha=0.3$ and $\lambda=0.55$ as an example. Consequently, within the
sufficiently small $\alpha\lesssim 1$ regime, in addition to the existence of
four-fold degenerate edge modes and a finite bulk energy gap, the ground state
also displays algebraically decaying string and spin-spin correlations
simultaneously. Therefore, the two key properties of the gapless topological
phase, a degenerate entanglement spectrum and a power-law decay string order
parameter (see Sec. IV and V in the SM for details), are both observed. Hence,
we term this phase as _algebraic SPT_ (the lower right corner of the phase
diagram in Fig. 1), exhibiting nontrivial gapless topological behaviors even
when the bulk is gapped. However, we want to emphasize that the algebraic
behavior of the correlation functions solely arises from the LR interaction
and does not relate to any gaplessness.

_Topological Gaussian universality._ —After exploring the ground-state phase
diagram, we now investigate the quantum phase transition between AFM and SPT
phases for typical values of $\alpha$. To determine the transition point, we
compute the Binder ratio for the AFM order [79], $R_{4}=\frac{1}{2}(3-\langle
O^{4}_{\text{AFM}}\rangle/\langle O_{\text{AFM}}^{2}\rangle^{2})$ where
$O_{\rm AFM}=\frac{2}{L}\sum\nolimits_{i=L/4+1}^{3L/4}(-1)^{i}\sigma_{i}^{z}$.
In the long-range ordered phase, $R_{4}$ increases with the system size $L$.
At the critical point, $R_{4}$ has a zero scaling dimension and is independent
of $L$. In Fig. 3 (a) and (c), we present the results of $R_{4}$ versus
$\lambda$ for $\alpha=3.0,0.6$, respectively (see Sec. III of the SM for other
$\alpha$); the crossing points pin down the AFM-SPT QCPs. We find the critical
value $\lambda_{\rm c}$ of the transition decreases upon decreasing LR power
exponent $\alpha$. Intuitively, as the LR interaction decays more slowly
(smaller $\alpha$), the Ising part of the Hamiltonian becomes more frustrated
and the cluster part will be more favored. We notice that our approach does
not allow us to extract the critical point when $\alpha\lesssim 0.1$, since
within our numerical results, Binder ratio with different $L$ do not cross at
a single point in this region. Furthermore, we find that the conformal
symmetry is preserved and broken for large and sufficiently small $\alpha$,
respectively, as shown in Sec. VI of the SM. Nevertheless, we can still define
an effective central charge via $S(l)=\frac{c_{\rm
eff}}{3}\ln[L/\pi\sin{(l\pi/L)}]+S_{0}$ where $S(l)$ is the entanglement
entropy for an interval of length $l$, for the non-conformal region to
characterize the universality class of the LR model [80] (see Sec. II of the
SM for details). Fig. 3 (b) and (d) exhibit $c_{\rm eff}$ as a function of
$\lambda$ with fixed $\alpha$ displaying a sharp peak at the critical point,
which is consistent with the results given by the Binder ratio in Fig. 3 (a)
and (c). In the following, the topological features of the critical line will
be studied at the critical points determined from $c_{\rm eff}$ since the
finite-size effect is generally less severe under PBC compared with OBC.

More importantly, since the critical point of the SR cluster Ising model
exhibits nontrivial topological properties such as degenerate edge modes, it
is natural to examine whether these properties are robust against the LR
interaction. Specifically, we demonstrate the two-fold topological degeneracy
by computing the bulk entanglement spectrum at the critical point for
$\alpha=3.0$ and $0.3$, respectively, as shown in Fig. 4 (a) and (b).
Similarly, Fig. 4 (c) depicts that the critical edge mode is stable even for
sufficiently small $\alpha=0.3$ (see Sec. III of the SM for other $\alpha$).
This indicates that the entire critical line in the phase diagram exhibits
nontrivial topological properties, which are also verified by the anomalous
scaling of the edge mode energy splitting (see Sec. VI of the SM for details).
More remarkably, we present the results of the effective central charge
$c_{\text{eff}}$ versus LR power exponent $\alpha$ in Fig. 4 (d). The result
unambiguously shows that within the numerical accuracy, the value of
$c_{\text{eff}}$ tends to $1$ as $\alpha\approx 0.1$, corresponding to a
Gaussian universality with nontrivial gapless topological behaviors. Hence, we
term it as _topological Gaussian universality_ , as depicted by the yellow
dashed line in the phase diagram in Fig. 1. Despite the numerical difficulties
for $\alpha<0.1$, fortunately, for $\alpha=0$, the 1D geometry of the system
is completely lost and this limit case can be studied analytically (See Sec.
VII of the SM for details). Consequently, we find that the LR interaction
induces a new gapped topological phase with a two-fold degenerate edge mode,
which occupies the entire interaction region ($\lambda\textgreater 0$) for
$\alpha=0$ (the red solid line in Fig. 1).

Figure 4: The bulk entanglement spectrum $\xi_{m}$ at the critical points for
$\alpha=3.0$ (a) and $\alpha=0.3$ (b). The index $m$ counts the spectrum from
the lowest-lying levels. (c) The magnetization profile at the critical point
for a sufficiently small $\alpha=0.3$ . (d) The effective central charge along
the critical line as a function of $\alpha$ for a fixed system size $L=96$
under PBC.

_Discussion._ —To gain a physical understanding of the numerical results
presented above, we first focus on the effect of the LR interaction on the
bulk. The Hamiltonian (1) can be transformed into the LR transverse field
Ising model by the unitary $U=\prod_{i}CZ_{i,i+1}$. Under the renormalization
group flow, the LR interaction will flow to operators in the infrared
continuum field theory and correspond to both a local part $S_{L}$ and a non-
local part $S_{H}$. Depending on $\alpha$, the non-local part can be
irrelevant for large $\alpha$ or relevant for small $\alpha$ [5, 28] at the
criticality. When the non-local part flows strong, the conformal symmetry will
not emerge in the infrared. Away from the criticality, the competition between
the local and non-local parts contributes to the hybrid behavior in the
correlation function for any finite $\alpha$, as argued in [65, 81]. Since the
effective central charge is defined via the entanglement entropy, the non-
local interaction will create more entanglement between the disjoint regions.
Therefore, less $\alpha$ will increase $c_{\text{eff}}$ [82], eventually
reaching $1.0$ as $\alpha$ approaches $0$ [81].

It is instructive to understand the case when $\alpha=0$, where the LR
interaction becomes all-to-all interaction. In this case, the system becomes
effectively a ($0+1$)D system and has a large site permutation symmetry. As a
result, all the correlation functions
$\langle\sigma_{i}^{z}\sigma_{j}^{z}\rangle$ are constants depending only on
$\lambda$. The gap size remains constant, corresponding to dynamical exponent
$z=0$. Thus, the ground state represents a gapped topological phase
characterized by two-fold degenerate edge modes, as illustrated in the next
paragraph.

To verify the stability of critical edge modes against LR interactions, let’s
consider the open finite chain with size $L$. The local operators
$\sigma_{1}^{z},\sigma_{L}^{z}$ commute with the Hamiltonian, and the four-
fold degenerate ground states are given by,

$\left|\uparrow_{L}\uparrow_{R}\right>\pm\left|\downarrow_{L}\downarrow_{R}\right>,\quad\left|\uparrow_{L}\downarrow_{R}\right>\pm\left|\downarrow_{L}\uparrow_{R}\right>\,,$
(2)

where $L(R)$ denotes the leftmost (rightmost) site. These four edge modes are
robust in the SPT phase with LR interactions except for $\alpha=0$. For
$\alpha=0$, the all-to-all interaction contains the term
$\sigma_{1}^{z}\sigma_{L}^{z}$ which will favor the antiferromagnetic
combination
$\left|\uparrow_{L}\downarrow_{R}\right>\pm\left|\downarrow_{L}\uparrow_{R}\right>$.
For general $\alpha$, when driving the system to the critical line, the Ising
interaction no matter SR or LR will generate the
$\sigma_{1}^{z}\sigma_{L}^{z}$ coupling, which results in the two-fold
degenerate ground state,
$\left|\uparrow_{L}\downarrow_{R}\right>\pm\left|\downarrow_{L}\uparrow_{R}\right>$.
In short, the critical line with varying $\alpha$ exhibits robust two-fold
degenerate edge modes, regardless of the LR interaction. The SPT phases with
finite $\alpha$ have four-fold degenerate edge modes, but it is lifted to two-
fold degenerate edge modes when the LR interaction is all-to-all (see Sec. VII
of the SM for details).

_Concluding remarks._ —In summary, we have conducted large-scale numerical
simulations to demonstrate the ground-state properties of a spin-1/2 chain
with LR interactions, unveiling various exotic phases and phase transitions as
a function of the cluster interaction strength $\lambda$ and the LR power
exponent $\alpha$. For large $\alpha\textgreater 1.0$, the phase diagram
exhibits the AFM long-range order, the gapped SPT phase, and a symmetry-
enriched Ising criticality between them. Conversely, the results for
sufficiently small but finite values of $\alpha\lesssim 1.0$ are particularly
intriguing. The conneted spin-spin correlation function and string order
parameter gradually change to purely algebraic decay for
$\lambda\textgreater\lambda_{\rm c}$, implying the emergence of an algebraic
SPT phase with nontrivial gapless topological behaviors. Furthermore, in the
limit $\alpha=0$, the entire interaction region is dominated by a distinct
gapped topological phase with two-fold degenerate edge modes. More remarkably,
the universality class gradually changes from topological Ising to the
topological Gaussian universality as $\alpha$ approaches $0$, indicating that
LR interactions can induce a new topological universality class.
Generalization of these findings for higher dimensions is a fascinating and
challenging direction for future study. From the perspective of experimental
realizations, LR interacting spin chains hold the potential for implementation
in state-of-the-art quantum circuit platforms [83, 84]. It has been proposed
that highly entangled quantum matter and unconventional phase transitions can
be observed in quantum circuits [85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95],
which may be useful to demonstrate our findings experimentally (details are
discussed in Sec. VIII of the SM). Hence, our findings open a new avenue
toward exploring and understanding gapless topological phases of matter.

Acknowledgement: We thank Da-Chuan Lu, Ruizhe Shen and Youjin Deng for helpful
discussions. Numerical simulations were carried out with the ITENSOR package
[96] on the Kirin No.2 High Performance Cluster supported by the Institute for
Fusion Theory and Simulation (IFTS) at Zhejiang University. X.-J.Yu is
supported by a start-up grant XRC-23102 of Fuzhou University. This work is
also supported by MOST 2022YFA1402701.

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##  I Supplemental Material for Gifts from long-range interaction: Emergent
gapless topological behaviors in quantum spin chain

###  I.1 Section I: Density matrix renormalization group algorithm and
numerical setups

To determine the ground state of the long-range (LR) interacting cluster Ising
model introduced in the main text, we conducted extensive density matrix
renormalization group (DMRG) simulations [69, 70, 71, 72] using the matrix
product state (MPS) representation [73]. Over the past few decades, DMRG has
emerged as a powerful tool for studying one-dimensional (1D) quantum many-body
systems with local interactions. However, when dealing with 1D quantum systems
featuring LR interactions, the ground state no longer necessarily adheres to
the so-called entanglement area law [97], which can hinder an efficient MPS
representation. In our specific scenario, the ground-state phase diagram
(refer to Fig. 1 in the main text) exhibits area-law or logarithmic entangled
phases (as well as critical points). Despite the complexity induced by LR
interactions, a faithful MPS representation remains feasible by employing a
sufficiently large MPS bond dimension for finite-size systems.

In practice, we simulated finite systems of an even total length, i.e., $L\in
2\mathbb{Z}^{+}$, under open or periodic boundary conditions with a maximum
MPS bond dimension set to $512$. Once the MPS energy has converged up to the
order $10^{-8}$, the DMRG sweeping route would be stopped; the resulting MPS
can be treated as a faithful representation of the true ground state. It is
also noted that the initial MPS is chosen as one of the Néel states for two
reasons: i) for open boundary conditions, there is no spin-flip operator
$\sigma^{x}$ in the Hamiltonian acting on the leftmost ($L$) or rightmost
($R$) boundary site, it is thus necessary to set the two boundary spins to
anti-parallel directions, namely, $\left|\uparrow_{L}\downarrow_{R}\right>$ or
$\left|\downarrow_{L}\uparrow_{R}\right>$, so that the calculation is
converged to the correct result. In Fig. 5 (a), we have verified that the
doubly degenerate ground states indeed have two anti-parallel directed
boundary spins via the calculation of the exact diagonalization (ED). Of
course, one can also choose the product state
$\prod_{i=1}^{L}\left|+\right>_{i}$, in which
$\sigma_{i}^{x}\left|+\right>_{i}=\left|+\right>_{i}$, as the starting point.
The product state respects the global symmetry
$G=\prod_{i=1}^{L}\sigma_{i}^{x}$ and finally converges to a superposition
$\left|\uparrow_{L}\downarrow_{R}\right>+\left|\downarrow_{L}\uparrow_{R}\right>$.
Compared to the former choice, the latter one usually requires a larger MPS
bond dimension to achieve the same accuracy level. ii) for periodic boundary
conditions, the main quantity under investigation is the entanglement entropy
and its spectrum. More specifically, the nontrivial topological property of
the critical point is partly identified by the double degeneracy of the
lowest-lying level of the entanglement spectrum. If the MPS is initialized to
be the product state $\prod_{i=1}^{L}\left|+\right>_{i}$, a doubly degenerate
entanglement spectrum would be obtained within the anti-ferromagnetic (AFM)
phase (spontaneously symmetry breaking happens only in the thermodynamic
limit). To have a clear distinction between the AFM phase, cluster symmetry-
protected-topological (SPT) phase, and the nontrivial critical point, it is
useful to artificially break the $G$ symmetry at the beginning. By
initializing the MPS as one of the Néel states, the resulting degeneracy of
the lowest-lying level of the entanglement spectrum would be 1, 2, and 4 for
the AFM phase, topological critical point, and cluster SPT phase,
respectively.

Additionally, we would like to note that it is common to include a Kac factor
as a coefficient of the LR ferromagnetic interaction to preserve the
Hamiltonian extensive. In the present scenario, however, the LR interaction is
anti-ferromagnetic (or frustrated). It means it is impossible to
simultaneously minimize all LR terms $\sigma_{i}^{z}\sigma_{j}^{z}/d_{ij}$,
therefore, the ground-state energy of the LR cluster Ising Hamiltonian can
still be an extensive quantity which is linear with the size $L$. As
illustrated in Fig. 6, we have numerically verified a clear linear dependence
of the ground-state energy $E_{g}$ with respect to the system size $L$ for
sufficiently small $\alpha\lesssim 1.0$ . Based on this observation, the Kac
factor is not necessary in our case.

Finally, we want to demonstrate that an MPS bond dimension $\chi=512$ is
sufficiently large in our studies. Since the half-chain entanglement entropy
$S_{\rm Half}$ is usually the quantity slowest to converge in DMRG
calculations, we have also tested the convergence of $S_{\rm Half}$ with
respect to the bond dimension $\chi$ as illustrated in Fig. 5 (b) and (c). It
is confirmed that $\chi=512$ can give converged reliable results.

To explore the interesting physics brought by the LR interactions, we have
calculated various quantities, such as the connected spin-spin correlation,
the energy gap, the entanglement spectrum, and so on, which we detail in the
following sections.

Figure 5: (a) ED solution of the first four lowest-lying energy levels of the
LR cluster Ising model for $L=8$, $10$, and $12$ along the $\alpha=0.5$ line
under open boundary conditions. The ground state is doubly degenerate with the
two boundary spins pointing to anti-parallel directions, namely,
$\left|\uparrow_{L}\downarrow_{R}\right>$ and
$\left|\downarrow_{L}\uparrow_{R}\right>$. The convergence of the half-chain
entanglement entropy $S_{\rm Half}$ with respect to the maximum MPS bond
dimension $\chi$ for typical system sizes under (b) open and (c) periodic
boundary conditions, respectively. An MPS bond dimension $\chi=512$ is
sufficiently large for the results to be converged in DMRG calculations. Here,
the simulated parameter is $\alpha=0.5$ and $\lambda=0.253$ . Figure 6: The
ground-state energy $E_{g}$ of the LR cluster Ising model under the open
boundary condition as a function of the system size for typical values of
$\alpha$ with (a) $\lambda=0.1$ and (b) $\lambda=0.6$, respectively. The open
markers are numerical data and the dashed lines are linear least-squares
fittings.

###  I.2 Section II: Logarithmic entanglement entropy and effective central
charge

It is now well accepted that quantum entanglement plays an essential role in
our understanding of quantum many-body physics. In particular, the low-energy
eigenstates of a gapped Hamiltonian with local interactions are known to obey
the entanglement area law [97], indicating that the entanglement entropy for
an interval of a 1D chain is constant regardless of the subsystem size $l$,
namely, $S(l)\equiv-\text{tr}[\rho_{l}\log(\rho_{l})]\sim\text{const}$, where
$\rho_{l}$ is the reduced density matrix of the subsystem. The entanglement
area law explains why MPS works so well for solving 1D quantum many-body
problems.

On the other hand, the entanglement entropy also provides a convenient way to
extract universal characteristics of quantum criticality. For a (1+1)D
critical system described by a conformal field theory (CFT), the central
charge $c$ of the underlying CFT can be determined from the universal
logarithmic scaling of the entanglement entropy via the famous formula [98,
99, 100]

$S(l)=\frac{c}{3}\log{\left[\frac{L}{\pi}\sin\left(\frac{\pi
l}{L}\right)\right]}+S_{0}\,,$ (3)

where $l$ the interval length, $L$ the system total size, and $S_{0}$ a non-
universal constant; $L/\pi\sin(\pi l/L)$ is also called the chord length. The
distinct scaling behaviors of $S(l)$ shown in the gapped phase and at the
critical point give us an effective way to pin down the location of quantum
critical points.

In the present work, we simulated systems of size $L=96$ under periodic
boundary conditions to calculate $S(l)$ where $l\leq L/2$; a least-squares
fitting according to Eq. (3) is then performed to extract the coefficient of
the logarithmic scaling. By scanning through the whole ground-state phase
diagram, in addition to the critical points, we also found the entanglement
entropy display a logarithmic scaling with the chord length within the gapped
SPT phase for sufficiently small $\alpha\lesssim 1$; see Fig. 7 (d) and (e)
for two representative examples. This implies that the LR interaction can
break down the entanglement area law obeyed by gapped phases. However, the
violation of entanglement area law is only observed within the SPT phase; the
AFM phase still satisfies the area law. Since the breakdown of the area law is
found to be logarithmic, we can thus define an effective central charge
$c_{\rm eff}$ to characterize the entanglement scaling behaviors within the
whole phase diagram. It is also worth to emphasize that the value of $c_{\rm
eff}$ extracted from the formula (3) does not necessarily relate to a CFT. In
our case, $c_{\rm eff}$ corresponds to the true central charge of an Ising CFT
only along the critical line with $\alpha\gtrsim 1$; for $\alpha\lesssim 1$,
the conformal symmetry breaks down with a dynamical exponent $z<1$ and the
central charge is not well-defined in the low-energy field theories describing
the critical points. At last, Fig. 7 gives some representative results to
support the observation made above; the critical point $\lambda_{\rm c}$ for a
given $\alpha$ used in Fig. 7 (a), (b), and (c) is identified by the peak
position of $c_{\rm eff}$ as a function of $\lambda$ along the fixed $\alpha$
line [see Fig. 3 (b) and (d) in the main text for examples].

Figure 7: The entanglement entropy shows a logarithmic scaling with the chord
length (a)-(c) along the critical line and (d)-(e) within the gapped SPT phase
for sufficiently small $\alpha\lesssim 1$. (f) summarizes the effective
central charge as a function of $\alpha$ along $\lambda=0.55$ and $0.65$
lines. Purple circles represent DMRG data and the dashed lines are least-
squares fittings according to Eq. (3). The simulated system size is $L=96$
under the periodic boundary condition for all $\alpha$ and $\lambda$.

###  I.3 Section III: Numerical results of critical and topological properties
at quantum critical points for other LR power exponents $\alpha$

In the main text, we presented extensive numerical results for $\alpha=3.0$
and $0.6$, which reveal nontrivial topological features at quantum critical
points between the AFM and cluster SPT phases. In this section, to further
corroborate our conclusions and investigate how the topological stability of
the critical line evolves when decreasing $\alpha$, we performed simulations
to study the model with other LR power exponents $\alpha$. As mentioned above,
the critical point $\lambda_{c}$ can be determined by the peak position of the
effective central charge $c_{\rm eff}$ as a function of $\lambda$ along the
fixed $\alpha$ line. Here, we employed another commonly used quantity, the
Binder ratio [101], to provide evidence that there remains to be a phase
transition between the AFM and cluster SPT phases as $\alpha$ decreases.
Moreover, the critical point obtained independently from the Binder ratio can
serve as a cross-check for the correctness of $\lambda_{c}$ determined from
the effective central charge.

The Binder ratio is a useful tool in the study of different types of quantum
phase transitions. With the definition of the AFM order parameter

$O_{\rm
AFM}=\frac{2}{L}\sum\nolimits_{i=L/4+1}\nolimits^{3L/4}(-1)^{i}\sigma_{i}^{z}$
(4)

where we have restricted the sum over the middle region of the chain to reduce
boundary effects, the corresponding Binder ratio is given by

$R_{4}=\frac{1}{2}\left(3-\frac{\langle O_{\rm AFM}^{4}\rangle}{\langle O_{\rm
AFM}^{2}\rangle^{2}}\right)\,.$ (5)

This observable has a vanishing scaling dimension and can therefore provide an
unbiased estimation of the critical point. In addition, we calculated the
long-range order relevant to the cluster SPT phase, which equals the product
of cluster terms [47],

$O_{\rm
SPT}=\sum\nolimits_{i=L/4+1}\nolimits^{3L/4}\sigma_{i-1}^{z}\sigma_{i}^{x}\sigma_{i+1}^{z}=\sigma_{L/4}^{z}\sigma_{L/4+1}^{y}\left(\prod\nolimits_{j=L/4+2}\nolimits^{3L/4-1}\sigma_{j}^{x}\right)\sigma_{3L/4}^{y}\sigma_{3L/4+1}^{z}\,.$
(6)

Figure 8: The Binder ratio of the AFM order $R_{4}$ (the upper panel), the AFM
order parameter $\langle{O_{\rm AFM}}\rangle$ (the middle panel), and the SPT
string order parameter $\langle{O_{\rm SPT}}\rangle$ (the lower panel) as a
function of $\lambda$ are shown for (a1-a3) $\alpha\rightarrow\infty$, (b1-b3)
$\alpha=1.8$, (c1-c3) $\alpha=0.8$, and (d1-d3) $\alpha=0.3$, with system
sizes $L=60$, $72$, $84$, and $96$ under open boundary conditions. The
critical points determined through the Binder ratio crossings are marked by
black dashed vertical lines in the upper panel.

As displayed in the upper panel of Fig. 8, the Binder ratio of different
system size $L$ intersect with each other at a single point giving an
estimation of the critical point $\lambda_{c}$ consistent with the one
obtained from the effective central charge (see Fig. 1 in the main text for
details). Similarly, the results of $|\langle{O_{\rm AFM}}\rangle|$ and
$|\langle{O_{\rm SPT}}\rangle|$ explicitly show that the ground states are AFM
and cluster SPT phases for $\lambda<\lambda_{c}$ and $\lambda>\lambda_{c}$,
respectively. The quantum phase transition between the two phases is
ubiquitously present at all values of $\alpha$; the critical point shifts to a
smaller value as expected due to the enhancement of anti-ferromagnetic
frustrations as $\alpha$ decreases.

Figure 9: The bulk entanglement spectrum $\\{\xi_{m}\\}$ for fixed system size
$L=96$ under periodic boundary conditions at the critical line is displayed
for (a) $\alpha\rightarrow\infty$, (b) $\alpha=2.0$, (c) $\alpha=1.0$, (d)
$\alpha=0.7$, and (e) $\alpha=0.5$ . The index $m$ counts the spectrum from
the lowest-lying levels and only the first $20$ levels are presented in the
plot. The red boxes highlight the two-fold degeneracy in the entanglement
spectrum. Figure 10: The magnetization profile
$\langle{\sigma_{i}^{z}}\rangle$ under open boundary conditions at the
critical points (determined by the effective central charge $c_{\rm eff}$) is
shown for (a) $\alpha\rightarrow\infty$, (b) $\alpha=3.0,2.0,1.0$, (c)
$\alpha=0.7$, and (d) $\alpha=0.5$ .

To further verify the topological features of the critical line, we also
investigate the bulk entanglement spectrum (under periodic boundary
conditions) and magnetization profile (under open boundary conditions) at the
critical point for various $\alpha$. The critical points explored here are
determined by the effective central charge $c_{\rm eff}$. By partitioning the
whole spin chain into two equal parts, i.e., $A=(1,\dots L/2)$ and
$B=(L/2+1,\dots L)$, the so-called entanglement Hamiltonian $\tilde{H}_{A}$ is
related to the half-chain reduced density matrix
$\rho_{A}=\text{tr}_{B}(\left|\psi\right>\\!\left<\psi\right|)$ via
$\tilde{H}_{\rm A}\equiv-\ln{\rho_{A}}$, where $\left|\psi\right>$ is the
ground-state wavefunction. The entanglement spectrum $\\{\xi_{m}\\}$ is then
the eigenvalues of the entanglement Hamiltonian sorted in ascending order.
According to the universal bulk-boundary correspondence, the degeneracy of the
lowest-lying level in the entanglement spectrum can faithfully reflect the
existence of edge modes living on the boundary. As evidenced by Figs. 9 and
10, the profile $\langle{\sigma_{i}^{z}}\rangle$ shows an obvious
magnetization near the open boundary and the lowest-lying level of the
entanglement spectrum exhibits a clear double degeneracy, both indicating that
the topological properties of the critical line remain stable against the LR
AFM interactions.

###  I.4 Section IV: Finite bulk energy gap in the algebraic SPT phase

In the main text, we have presented the results of the connected spin-spin
correlation function and the nonlocal SPT string order parameter, which
exhibit purely algebraic decay behaviors at all distances for sufficiently
small $\alpha\lesssim 1.0$ deep in the cluster SPT phase; it corresponds to
the lower right part of the phase diagram in the main text, which is termed
algebraic SPT phase there. Here, we would like to clarify that this algebraic
decay behavior is caused by the LR AFM interaction and is not related to a
critical phase. To show that the algebraic SPT phase is gapped, it is
straightforward to check the existence of a finite bulk energy gap in the
thermodynamic limit, which we detail below.

Before we show the numerical results, it is helpful to understand the lowest-
lying energy spectrum in the short-range cluster SPT phase ($\alpha\to\infty$
and $1/2<\lambda\leq 1$). If we consider the limit $\lambda=1$, the ground-
state manifold under the open boundary condition would be four-fold degenerate
coming from the edge modes on the boundary. Since the two $\sigma_{1}^{z}$ and
$\sigma_{L}^{z}$ on the boundary commutes with the Hamiltonian, we can label
these four states by $\left|\uparrow_{L}\uparrow_{R}\right>$,
$\left|\uparrow_{L}\downarrow_{R}\right>$,
$\left|\downarrow_{L}\uparrow_{R}\right>$, and
$\left|\downarrow_{L}\downarrow_{R}\right>$, where $L$ ($R$) denotes the left
(right) boundary. By including a finite Ising interaction but still within the
gapped SPT phase ($1/2<\lambda<1$), the AFM interaction will penalize the FM
states splitting the original four-fold degenerate manifold into two doubly
degenerate manifolds, i.e.,
$\\{\left|\uparrow_{L}\downarrow_{R}\right>,\left|\downarrow_{L}\uparrow_{R}\right>\\}$
and
$\\{\left|\uparrow_{L}\uparrow_{R}\right>,\left|\downarrow_{L}\downarrow_{R}\right>\\}$;
the former one has a slightly lower energy compared with the latter one. It is
noted that the small gap opened by the Ising interaction is finite only for
finite-size systems, the four-fold ground-state degeneracy is exact (restored)
in the thermodynamic limit.

Now we can consider the case of finite $\alpha$ and verify the finite bulk
energy gap within the cluster SPT phase even for very small $\alpha\lesssim
1.0$ . To this end, we have to calculate the first $6$ lowest-lying energy
levels of the system to obtain the correct bulk energy gap; in this case, we
initialized the MPS randomly in the DMRG simulations. By sorting the obtained
energy levels in ascending order, we found a doubly degenerate energy
spectrum, $E_{1,2}<E_{3,4}<E_{5,6}$, as expected. More specifically, the
energy difference between $E_{1}$ and $E_{3}$ is much smaller than the one
between $E_{3}$ and $E_{5}$. Based on this observation, the bulk energy gap
can be defined separately by $\Delta_{\rm b}^{1}\equiv E_{5}-E_{1}$ and
$\Delta_{\rm b}^{2}\equiv E_{5}-E_{3}$. As illustrated in Fig. 11,
$\Delta_{\rm b}^{1}$ and $\Delta_{\rm b}^{2}$ are extrapolated to the same
value in the thermodynamic limit. It means that the first $4$ lowest-lying
energy levels are degenerate in the $L\to\infty$ limit and there does exist a
finite bulk energy gap in the algebraic SPT phase even when the LR interaction
is very strong.

Figure 11: The bulk energy gap defined by $\Delta_{\rm b}^{1}\equiv
E_{5}-E_{1}$ or $\Delta_{\rm b}^{2}\equiv E_{5}-E_{3}$ as a function of $1/L$
for (a) $\alpha=0.5,\lambda=0.60$ and (b) $\alpha=0.3,\lambda=0.60$ . The
dashed lines are least-squares fittings according to $\Delta_{\rm
b}(L)=a/L^{b}+c$. Figure 12: The log-log plot of the SPT string order
parameter $\mathcal{S}(r)$ and the PM string order parameter $O_{\rm PM}(r)$
versus lattice distance $r$ at $\alpha=0.3$ and $\lambda=0.55$, respectively,
for LR cluster Ising model (the left panel) and LR transverse field Ising
chain (the right panel). Simulated system size is $L=200$ and only middle
sites are considered to reduce boundary effects. The dashed lines are least-
squares fittings according to $O(r)=a/r^{\gamma}$, where $O(r)$ is
$\mathcal{S}(r)$ or $O_{\rm PM}(r)$.

In addition, we would like to mention that a similar phenomena has been
observed in the LR transverse field Ising model, where there is a gapped
paramagnetic (PM) phase featuring algebraic decaying correlations [17].
Although the algebraic SPT phase found here shares some similarities with the
gapped PM2 phase reported in Ref. [17], it is worth emphasizing that the
algebraic SPT phase exhibits nontrivial gapless topological behaviors, such as
the topological degeneracy in the bulk entanglement spectrum and localized
magnetization near the boundaries. To have a direct comparison, we also
computed two distinct types of nonlocal string orders

$\begin{split}&\mathcal{S}(r=|m-n|)=\langle\sigma^{z}_{n-1}\sigma^{y}_{n}\left(\prod\nolimits_{j=n+1}\nolimits^{m-1}\sigma^{x}_{j}\right)\sigma^{y}_{m}\sigma^{z}_{m+1}\rangle\,,\\\
&O_{\text{PM}}(r=|m-n|)=\langle\prod\nolimits_{j=n}\nolimits^{m}\sigma^{x}_{j}\rangle\,.\end{split}$
(7)

The former can be seen as the order parameter for the SPT phase, while the
latter is the order parameter for the topologically trivial PM phase. In the
short-range limit, i.e., $\alpha\to\infty$, $\mathcal{S}(r)$ ($O_{\rm PM}(r)$)
is a long-range order within the SPT (PM) phase, while it decays exponentially
fast in the PM (SPT) phase. By including a strong enough LR interaction
($\alpha\lesssim 1.0$), it is found that both $\mathcal{S}(r)$ and $O_{\rm
PM}(r)$ become a quasi-long-range order exhibiting a power-law dependence on
the string length, $\sim 1/r^{\gamma}$, whether the ground state is in the SPT
or PM phase. However, the nontrivial gapless topological behaviors can be
revealed through the slowest decay of the string operator, known as the
symmetry-flux operator in the context of Ref.[47]. In our case, the
$\mathcal{S}(r)$ is charged under time-reversal symmetry, while the
$O_{\text{PM}}(r)$ is not. Therefore, as evidenced by Fig. 12,
$\mathcal{S}(r)$ has a significantly smaller power exponent compared with
$O_{\rm PM}(r)$ in the algebraic SPT phase. While in the PM2 phase, the
situation is different, and $O_{\rm PM}(r)$ decays much slower than
$\mathcal{S}(r)$. We note that the chosen parameters $\alpha=0.3$ and
$\lambda=0.55$ correspond to the algebraic SPT phase in the LR cluster Ising
chain and to the PM2 phase in the LR transverse field Ising chain. The direct
comparison between the two models unequivocally reveals the topological
distinction between the algebraic SPT phase found here and the PM2 phase
reported in Ref. [17].

###  I.5 Section V: Asymptotic behavior of correlation functions

Figure 13: (a) In the AFM phase,
$|{\langle{\sigma_{i}^{z}\sigma_{i+r}^{z}}}\rangle|\sim O(1)$ indicates the
long-range AFM order, and $\mathcal{S}(r)\sim\exp(-r/\xi)$ decays
exponentially. (b) In the SPT phase, the string order parameter
$\mathcal{S}(r)$ changes from a long-range order $\sim O(1)$ to a power-law
decay behavior as $\alpha$ decreases below $1.0$ with fixed $\lambda=0.55$ .
(c) The connected spin-spin correlation $|C_{\rm c}^{zz}(r)|$ for
$\lambda=0.55$, showing a hybrid exponential and power-law decaying behavior
when $\alpha\gtrsim 1.0$ and a purely algebraic decaying behavior when
$\alpha\lesssim 1.0$ . The black dashed lines are least-squares fittings
according to $|C_{\rm c}^{zz}(r)|\sim 1/r^{\gamma_{\rm s}}$. (d) The power
exponent $\gamma_{\rm s}$ as a function of $\alpha$, respectively, along the
$\lambda=0.55$ (red circles) and $\lambda=0.65$ (purple triangles) lines.
Simulated system size is $L=200$ under the open boundary condition. The
finite-size effect is severe when $\alpha\lesssim 0.1$ .

In this section, we examine various types of correlation functions in the
presence of LR interactions. Specifically, we consider the connected spin-spin
correlation function

$C_{\rm
c}^{zz}(r)=\langle\sigma^{z}_{i}\sigma^{z}_{i+r}\rangle-\langle\sigma^{z}_{i}\rangle\langle\sigma^{z}_{i+r}\rangle$
(8)

and the nonlocal SPT string order parameter $\mathcal{S}(r)$ [refer to Eq. (7)
for its definition]. In the practical simulations, we have chosen $i=L/4$, and
$r\in[1,L/2]$ to reduce the boundary effect.

As elaborated in the main text and above sections, the LR cluster Ising model
exhibits a continuous phase transition between the gapped AFM and SPT phases
by tuning the cluster interaction strength $\lambda$, irrespective of the
magnitude of the LR interaction. In the AFM regime (for $\lambda$ below the
critical point), we computed the spin-spin correlation function
$\langle\sigma_{i}^{z}\sigma_{i+r}^{z}\rangle$ with respect to the lattice
distance $r$ from large to small $\alpha$, as depicted in Fig. 13 (a). The
result shows that the spin-spin correlation function exhibits an AFM long-
range order, i.e., $|\langle\sigma_{i}^{z}\sigma_{i+r}^{z}\rangle|\sim O(1)$,
for any $\alpha\textgreater 0$. Conversely, the nonlocal SPT order parameter
always exhibits an exponential decay, $\mathcal{S}(r)\sim\text{exp}(-r/\xi)$,
in which $\xi$ can be seen as the correlation length. For a given $\lambda$,
e.g., $\lambda=0.1$ here, by decreasing the LR power $\alpha$, we can expect a
growth of the correlation length because the critical point is effectively
approached. This explains the slower decaying rate of $\mathcal{S}(r)$ for a
smaller $\alpha$. Hence, the LR interaction does not significantly affect the
AFM long-range order for $\alpha\textgreater 0$, which is why we do not focus
on this region in this work.

However, motivated by Refs. [17, 19], we are particularly interested in the
gapped SPT regime (for $\lambda$ above the critical point). As shown in Fig.13
(b) and (c), we found that for large $\alpha\gtrsim 1.0$, the SPT string order
parameter exhibits a long-range order, while the connected spin-spin
correlation displays an exponential decay at short distances and a power-law
decay at longer distances. Conversely, for sufficiently small $\alpha\lesssim
1.0$, both the connected spin-spin correlation function and the string order
parameter decay algebraically at all distances; for this reason, we named this
phase the algebraic SPT as mentioned in the main text. In addition, we
examined the power exponent of the connected spin-spin correlation $|C_{\rm
c}^{zz}(r)|\sim 1/r^{\gamma_{\rm s}}$ as a function of $\alpha$ along a fixed
$\lambda$ line. The result shown in Fig. 13 (d) demonstrates that there is a
clear crossover as $\alpha$ decreases below $\alpha_{\rm c}\approx 1.0$; the
estimated value of $\gamma_{\rm s}$ is approximately proportional to $\alpha$
when $\alpha>1.0$, while it is roughly a constant when $\alpha<1.0$ . It is
noted that the change of the behavior of $C_{\rm c}^{zz}(r)$ is merely induced
by the LR interaction. Considering that we are actually moving away from the
critical point (line) as $\alpha$ decreases with $\lambda$ fixed, it is still
possible that $\lambda$ can have an effect on the concrete values of
$\gamma_{\rm s}$. As exhibited in Fig. 13 (d), the results of the power
exponent $\gamma_{\rm s}$ along $\lambda=0.55$ and $0.65$ lines have no
obvious difference, which excludes this possibility directly.

###  I.6 Section VI: Edge mode splitting and the breakdown of the conformal
symmetry at the topological nontrivial critical points

In this section, we provide further evidence to identify the topological
properties of the LR quantum critical points in the phase diagram. In the
short-range limit $\alpha\rightarrow\infty$, the model reverts to the nearest
neighbor cluster Ising model, which exhibits a symmetry-enriched quantum
critical point at $\lambda=0.5$ with nontrivial topologically protected edge
modes [47, 102]. In particular, the edge modes at the critical point display
an unconventional algebraic finite-size splitting, scaling as $1/L^{14}$,
which has attracted a great interest among theoretical physicists. According
to Ref. [47], this splitting can be revealed by adding a symmetry-preserving
boundary perturbation,
$g(\sigma_{1}^{x}\sigma_{2}^{z}\sigma_{3}^{z}+\sigma_{L-2}^{z}\sigma_{L-1}^{z}\sigma_{L}^{x})$,
where $g$ is a small quantity. This perturbation can flip the boundary spins
and couple the two edge modes, $\left|\uparrow_{L}\downarrow_{R}\right>$ and
$\left|\downarrow_{L}\uparrow_{R}\right>$, leading to a finite-size splitting
within the two-fold degenerate ground-state manifold. To say that the
degeneracy of the edge modes is stable, it is necessary that the edge mode
splitting is smaller than the bulk energy gap; in other words, the algebraic
power needs to be larger than the dynamical exponent $z$. For the short-range
case, the power of the algebraic edge mode splitting is $14$ which is much
larger than $z=1$, establishing the stability of the edge modes coexisting
with the critical bulk fluctuations.

Figure 14: The finite-size scaling of the bulk energy gap $\Delta_{\rm b}$ at
the critical points is depicted for typical values of $\alpha$: (a)
$\alpha=\infty$, (b) $\alpha=3.0$, (c) $\alpha=2.0$, (d) $\alpha=1.0$, (e)
$\alpha=0.7$, and (f) $\alpha=0.5$ . The critical points $\lambda_{\rm c}$ are
determined by the peak position of $c_{\rm eff}$ for $\alpha\geq 1$; for
$\alpha=0.7$ and $\alpha=0.5$, the critical point is determined by the
observation of a linear behavior of $\Delta_{\rm b}$ in the log-log plot. The
blue crosses in (a)-(d) are obtained from ED and the open markers are DMRG
data. The red dashed lines are least-squares fittings according to
$\Delta_{\rm b}\sim 1/L^{z}$ . Figure 15: The finite-size scaling of the edge
mode splitting $\Delta_{\rm e}$ by including a small boundary perturbation
$0.1(\sigma_{1}^{x}\sigma_{2}^{z}\sigma_{3}^{z}+\sigma_{L-2}^{z}\sigma_{L-1}^{z}\sigma_{L}^{x})$
at the critical points is depicted for typical values of $\alpha$: (a)
$\alpha=\infty$, (b) $\alpha=3.0$, (c) $\alpha=2.0$, (d) $\alpha=1.0$, (e)
$\alpha=0.7$, and (f) $\alpha=0.5$ . The blue crosses are obtained from ED
calculations and the green open circles are DMRG data. The red dashed lines
are least-squares fittings according to $\Delta_{\rm e}=a/L^{b}$ .

Now, a natural question to ask is whether the algebraically localized edge
modes at the critical point can exist and be robust with the LR interactions?
To answer this question, we first need to extract the dynamical exponent $z$,
which enters the finite-size scaling of the critical bulk gap, $\Delta_{\rm
b}\equiv(E_{3}-E_{1})\sim 1/L^{z}$. For a conformal invariant quantum critical
point, we have $z=1$. As illustrated in Fig. 14, we have performed a standard
finite-size scaling analysis for the bulk gap $\Delta_{\rm b}$ at the critical
points identified by the effective central charge $c_{\rm eff}$ (see Sec. II
for details). It is found that the dynamical exponent $z$ remains to be $1.0$
within the numerical accuracy for large $\alpha\gtrsim 1.0$, suggesting that
the conformal symmetry can be preserved when the power-law LR interaction is
introduced. The perfect power-law scaling observed in Fig. 14 (a)-(d) also
verifies the high accuracy of the critical points located by the effective
central charge $c_{\rm eff}$. Conversely, for small $\alpha<1.0$, we found
that the critical point obtained from $c_{\rm eff}$ suffers from a small
finite-size effect, and the scaling of the bulk gap deviates slightly from a
power-law behavior. However, the peak position of $c_{\rm eff}$ still provides
a good estimation of the critical point. As shown in Fig. 14 (e)-(f), a
perfect power-law scaling of $\Delta_{\rm b}$ can be found at $\lambda=0.290$
for $\alpha=0.70$ and $\lambda=0.245$ for $\alpha=0.50$, respectively. The
result unequivocally demonstrates that the dynamical exponent $z$ is smaller
than $1$ when $\alpha<1$, implying a breakdown of the conformal symmetry.

To investigate the properties of the doubly degenerate edge modes at the
critical points, we added the boundary perturbation mentioned above with a
strength $g=0.1$ and computed the edge mode splitting as a function of $1/L$
for typical values of $\alpha$, as displayed in Fig. 15. The result shows that
the edge mode splitting always exhibits an algebraic decay regardless of the
strength of the LR interactions. More importantly, by comparing with the
dynamical exponent $z$ obtained in Fig. 14, we can find that the power of the
algebraic edge mode splitting is always larger than the corresponding
dynamical exponent $z$ at least for the $\alpha\gtrsim 0.5$ considered here.
The calculation for $\alpha<0.5$ is more challenging because an accurate
determination of the critical point requires to simulate systems of larger
size. It is noted that the power exponent obtained in Fig. 15 can be effected
by finite-size effects. For example, the power exponent in the short-range
limit is predicted to be $14$ in the thermodynamic limit, and there is a
finite-size correction $\sim 1/L^{16}$ [47] leading to a slightly larger
estimation given in Fig. 15 (a) for example. As the concrete form of the
finite-size correction is unknown for finite values of $\alpha$, we can not
perform a systematic extrapolation to the $L\to\infty$ limit. However, we can
expect that the finite-size effect can bring about $(15.4-14)/14=10\%$
estimation error for finite $\alpha$ (especially for $\alpha\lesssim 1$
because larger system sizes are used in the least-squares fittings). In
summary, the result indicates that even with LR interactions, the
algebraically localized edge modes persist at the critical point. This is a
typical feature of topologically nontrivial critical points and provides
further evidence to confirm the conclusion we drawn in the main text.

###  I.7 Section VII: Details of analytical calculation for $\alpha=0$

#### Ground state degeneracy in the SPT phase and critical point

In this section, we explain the degeneracy found in the entanglement spectrum.
The degeneracy in entanglement spectrum of periodic chain corresponds to the
degeneracy in the energy spectrum of open chain. In the following, we consider
an open finite chain with size $L$. Deep in the SPT phase, $\lambda=1$, the
local operators $\sigma_{1}^{z}$, $\sigma_{L}^{z}$ commute with the
Hamiltonian results in 4-fold ground state degeneracy,

$\left|\uparrow_{L}\uparrow_{R}\right>\pm\left|\downarrow_{L}\downarrow_{R}\right>,\quad\left|\uparrow_{L}\downarrow_{R}\right>\pm\left|\downarrow_{L}\uparrow_{R}\right>,$
(9)

These 4 degenerate states correspond to the 4-fold degeneracy found in the
entanglement spectrum.

At the critical point, the long-range Ising interaction generates the AFM
interaction between the left boundary and right boundary. It favors

$\left|\uparrow_{L}\downarrow_{R}\right>\pm\left|\downarrow_{L}\uparrow_{R}\right>$
(10)

and lifts the other two states with a $\sim 1/L^{z}$ gap, where $z$ is the
dynamical exponent. Therefore, the critical point only hosts 2-fold degenerate
ground states.

We note that the the gapped phase under the $\alpha\rightarrow 0$ limit also
hosts 2-fold degenerate ground states. Since the all-to-all Ising interaction
contains the coupling between the leftmost and rightmost boundary spins and
favors (10), while lifts the other two states with an order $1$ gap.

#### Properties in $\alpha\rightarrow 0$ limit

In this paragraph, we explain the emergence of a new _gapped_ SPT phase with
two-fold degenerate edge modes in the $\alpha\rightarrow 0$ limit. In the
$\alpha\rightarrow 0$ limit, the first term in the Hamiltonian (1) is all-to-
all Ising interaction,

$H_{\infty}=\frac{(1-\lambda)}{N}\sum_{i<j}\sigma_{i}^{z}\sigma_{j}^{z}+\lambda\sum_{j}\sigma^{z}_{j}\sigma^{x}_{j+1}\sigma^{z}_{j+2}$
(11)

where $N$ is the total sites. The spectrum of the model is unchanged when
applying unitary transformations. This model can be further mapped to
infinite-range transverse field Ising model by conjugating the Hamiltonian
with $U=\prod_{i}CZ_{i,i+1}$, where,

$CZ_{i,i+1}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&-1\end{pmatrix}$ (12)

The transformed Hamiltonian is,

$\displaystyle H^{\prime}_{\infty}$
$\displaystyle=U^{\dagger}H_{\infty}U=\frac{(1-\lambda)}{N}\sum_{i<j}\sigma_{i}^{z}\sigma_{j}^{z}+\lambda\sum_{j}\sigma^{x}_{j}$
(13)
$\displaystyle=\frac{(1-\lambda)}{2N}(\sigma^{z}_{tot})^{2}+\lambda\sigma^{x}_{tot}-\frac{(1-\lambda)}{8}$
(14)

where $\sigma^{a}_{tot}=\sum_{i}\sigma_{i}^{a}$. The algebraic SPT phase is
mapped to the symmetric paramagnetic phase in the infinite-range transverse
field Ising model. The infinite-range transverse field Ising model is
effectively a 0+1d quantum mechanics model of spin $\frac{N}{2}$. The total
spin $S(S+1)$ is a good quantum number, and $0\leq S\leq\frac{N}{2}$.

In the paramagnetic phase $\lambda>\lambda_{c}$, we use the Holstein-Primakoff
transformation,

$\sigma_{tot}^{x}=S-a^{\dagger}a,\quad\sigma_{tot}^{z}-i\sigma_{tot}^{y}=\sqrt{2S-a^{\dagger}a}a,\quad\sigma_{tot}^{z}+i\sigma_{tot}^{y}=a^{\dagger}\sqrt{2S-a^{\dagger}a}.$
(15)

where $[a,a^{\dagger}]=1$. Since the low energy properties of the system are
dominated by $S$ closed to $\frac{N}{2}$, we can approximate the Holstein-
Primakoff transformation to be,

$\sigma_{tot}^{x}=S-a^{\dagger}a,\quad\sigma_{tot}^{z}=\sqrt{\frac{S}{2}}(a+a^{\dagger}).$
(16)

Then the Hamiltonian (14) becomes,

$H^{\prime}_{\infty}=\frac{(1-\lambda)}{4N}(a+a^{\dagger})^{2}+\lambda(S-a^{\dagger}a)-\frac{(1-\lambda)}{8}$
(17)

One can easily find the spectrum is given by,

$E_{S,n}=-(-\lambda)(S+\frac{1}{2})+(n+\frac{1}{2})(-\lambda)\sqrt{1-\frac{(1-\lambda)S}{\lambda
N}}-\frac{(1-\lambda)}{8}$ (18)

The gap between ground state and the first excited is a constant which
indicates the $z=0$.

In the $\alpha\rightarrow 0$ limit, all the two point correlation functions
are equal $\langle\sigma_{i}^{z}\sigma_{j}^{z}\rangle=\text{const}.$, while
the constant varies with $\lambda$. This is related to the fact that system
has a large site permutation symmetry, and it is effectively a $0+1d$ system.

###  I.8 Section VIII: Experimental implementation

In this section, we discuss the potential experimental realization of our
model. We note that Refs [83, 84] propose that generic LR spin models can be
implemented in a quantum circuit. Specifically, we can transform the 1D spin
chain into an IBM quantum qubit device suitable for quantum circuits. To
investigate the ground state of our model, we employ the Quantum Imaginary
Time Evolution (QITE) method [103]. In our circuit design, the non-unitary
process of QITE is effectively realized by coupling the main circuit,
constructed from the sites, with a single ancilla qubit.

First, we prepare the non-unitary dynamics $U_{N}=e^{-\beta H}$ of our LR spin
system with $N$ qubits. This step is programmable and can be achieved through
the tensor network method. Thus, the LR Ising coupling and cluster interaction
are numerically incorporated into our target non-unitary operation $U_{N}$.
Subsequently, this non-unitary $U_{N}$ is numerically embedded in a larger
unitary process $U_{N+1}$ of $(N+1)$ qubits [84]. It’s worth noting that
there’s no need to physically implement the LR Ising coupling and cluster
interaction terms in our model; what needs to be realized is this extended
unitary $U_{N+1}$. This setup enables us to perform final post-selection on
the ancilla qubit, leading to effective QITE on the physical system. For
practical implementation on a quantum computer, i.e., realizing the unitary
$U_{N+1}$, we can utilize variational circuit optimization. Here, we prepare a
parameterized circuit $V$ with relatively short depth and then employ
variational optimization to approximate our target circuit: $U_{N+1}\approx
V$.