# Existence of two distinct valence bond solid states in the dimerized
frustrated ferromagnetic $J_{1}$-$J_{1}^{\prime}$-$J_{2}$ chain

Jędrzej Wardyn  Faculty of Physics, University of Warsaw, Pasteura 5, PL-02093
Warsaw, Poland  Institute for Theoretical Solid State Physics, IFW Dresden,
01171 Dresden, Germany  Satoshi Nishimoto  Institute for Theoretical Solid
State Physics, IFW Dresden, 01171 Dresden, Germany  Department of Physics,
Technical University Dresden, 01069 Dresden, Germany  Cliò Efthimia Agrapidis
<EMAIL_ADDRESS>Faculty of Physics, University of Warsaw, Pasteura
5, PL-02093 Warsaw, Poland

###### Abstract

We study the frustrated and dimerized ferromagnetic-antiferromagnetic
$J_{1}$-$J_{1}^{\prime}$-$J_{2}$ chain using the density-matrix
renormalization group method. Based on numerical calculations of the second
derivative of energy, spin gap, spin-spin correlations, string order parameter
(SOP), and entanglement spectrum (ES), we obtain the ground-state phase
diagram for a wide range of $J_{1}^{\prime}/J_{1}$ and $J_{2}/|J_{1}|$ values.
This phase diagram reveals a ferromagnetic phase and two distinct valence-
bond-solid (VBS) phases. The first VBS phase, referred to as
$\mathcal{D}_{3}$-VBS, is typified by the formation of valence bonds between
third-neighbor spin-1/2’s, persisting as a continuation from the
$J_{1}^{\prime}/J_{1}=1$ limit. Alternatively, the second VBS phase, referred
to as mixed-VBS, exhibits a coexistence of both second- and third-neighbor
valence bonds, interpreted as a continuation from the $J_{1}^{\prime}/J_{1}=0$
case. Remarkably, both VBS states are identified as being of Haldane-type,
marked by a finite SOP and 2-fold ES degeneracy. Unexpectedly, our analysis
uncovers a significant enhancement of the valence bond stability at the
boundary of the two VBS phases. This study provides the first empirical
demonstration of a nontrivial quantum phase transition between different
topological VBS states in spin-1/2 chains. Moreover, we find that the ground
state of the relevant quasi-one-dimensional material LiCuSbO4 is classified as
the $\mathcal{D}_{3}$-VBS state. Collectively, these results mark a
substantial stride forward in our comprehension of quantum phase transitions
and topological states.

##  I  Introduction

The interplay of magnetic frustration and low-dimensionality can lead to
exotic states of matter in spin chains due to extensive quantum fluctuations
[1, 2]. A simple representative of such systems is the one-dimensional (1D)
$J_{1}$-$J_{2}$ model, where $J_{1}$ and $J_{2}$ represent the nearest-
neighbor and next-nearest-neighbor magnetic interactions, respectively. This
model has facilitated understanding the magnetic behavior of various
materials, including certain quasi-1D magnetic insulators like copper-based
compounds. While the signs of the interactions can be influenced by a plethora
of factors – including the crystal structure, electronic configuration as well
as the nature of the magnetic ions and their ligands – they can be broadly
categorized as follows [3]: In corner-shared cuprates, both $J_{1}$ and
$J_{2}$ interactions are antiferromagnetic (AFM) because the superexchange
interaction between the copper ions is mediated through an oxygen atom at the
corner. Whereas in edge-shared cuprates, the nearest-neighbor $J_{1}$
interaction is typically ferromagnetic (FM), while the next-nearest-neighbor
$J_{2}$ interaction remains AFM. The FM $J_{1}$ interaction comes about due to
the direct overlap of the copper and oxygen orbitals along the edge.
Therefore, the 1D $J_{1}$-$J_{2}$ model is straightforward enough to be
studied theoretically and numerically, but it offers a good starting point to
comprehend the intricate physics in real materials, especially regarding
frustration and magnetic ordering.

The ground state of the AFM-AFM $J_{1}$-$J_{2}$ chain at $J_{2}\gtrsim 0.241$
is identified as a valence bond solid (VBS) state, as defined by the Majumdar-
Ghosh picture [4]. In contrast, recent findings suggest that the ground state
of the FM-AFM $J_{1}$-$J_{2}$ chain exhibits a distinct type of VBS state,
characterized by the formation of a valence bond dimer between third-
neighboring spin-1/2’s, referred to as the $\mathcal{D}_{3}$-VBS state [5].
This state is especially intriguing as it represents a topologically non-
trivial phase, bearing similarities with the Haldane [6, 7, 8, 9] or Affleck-
Kennedy-Lieb-Tasaki (AKLT)-like VBS state [10, 11]. The emergence of this
state can be intuitively anticipated due to the emergence of effective spin-1
degrees of freedom, resulting from the spontaneous formation of nearest-
neighbor FM dimers of spin-1/2’s.

The current work extends the rudimentary chain model to consider the dimerized
frustrated FM $J_{1}$-$J_{1}^{\prime}$-$J_{2}$ chain. This model introduces a
further complication of different types of nearest-neighbor interactions,
$J_{1}$ and $J_{1}^{\prime}$ ($|J_{1}|>|J_{1}^{\prime}|$), which alternate in
a dimerized pattern. The presence of this structural FM alternation was indeed
predicted in the edge-shared cuprate LiCuSbO4 [12]. However, it is largely
unknown how the ground state of this model changes as a function of
dimerization and frustration. From a theoretical perspective, the role of such
alternating interactions may even simplify the VBS issue, given that the
effective spin-1 degrees of freedom are explicitly created on the $J_{1}$
bond. This is equivalent to the formation of a spin-triplet dimer on the
$J_{1}$ bond. In the strong dimer limit $J_{1}^{\prime}/J_{1}=0$, the
$J_{1}$-$J_{1}^{\prime}$-$J_{2}$ chain can be effectively mapped onto the
spin-1 Heisenberg chain [13, 14] (see also Sec. II). This indicates that the
ground state is no longer the $\mathcal{D}_{3}$-VBS one at
$J_{1}^{\prime}/J_{1}=0$. Thus, a crossover or phase transition is feasible
between the $\mathcal{D}_{3}$-VBS state at $J_{1}^{\prime}/J_{1}=1$ and a
symmetry-protected Haldane state at $J_{1}^{\prime}/J_{1}=0$.

Prompted by this, we examine the $J_{1}$-$J_{1}^{\prime}$-$J_{2}$ chain using
the density-matrix renormalization group (DMRG) algorithm [15, 16]. We first
detect the presence of phase boundary by observing the second derivative of
the ground-state energy as a function of the parameters. We then calculate the
spin gap and string order parameter (SOP) to confirm the presence of a VBS
state. Earlier, some of the present authors had estimated the spin gap for
this system [17]. However, in certain specific parameter regions, it was
somewhat underestimated due to insufficient management of the edge state,
since the VBS structure was not well established. Consequently, it is
revisited in this work. Lastly, local spin-spin correlations and the
entanglement spectra (ES) are considered to further validate the estimated
phase boundary.

The remainder of the paper is organized as follows: Section II elaborates on
the model and details the applied numerical method. Section III presents the
ground-state phase diagram based on the numerical results. Section IV is
devoted to summarizing the study and discussing the findings.

##  II Model and Methods

The Hamiltonian for the $J_{1}$-$J_{1}^{\prime}$-$J_{2}$ chain is given by:

$H=J_{1}\sum_{i=\text{even}}\mathbf{S}_{i}\cdot\mathbf{S}_{i+1}+J_{1}^{\prime}\sum_{i=\text{odd}}\mathbf{S}_{i}\cdot\mathbf{S}_{i+1}+J_{2}\sum_{i}\mathbf{S}_{i}\cdot\mathbf{S}_{i+2}$
(1)

where $\mathbf{S}_{i}$ are Heisenberg spin-1/2 operators at site $i$, $J_{1}$
and $J_{1}^{\prime}$ are FM alternating nearest neighbor interactions, and
$J_{2}$ is the AFM next-nearest-neighbor coupling. The lattice of our model is
depicted in Fig. 1. In our analysis, we parametrize the model via the
dimerization parameter $\beta=J_{1}^{\prime}/J_{1}$ and the frustration
parameter $\alpha=J_{2}/|J_{1}|$.

Figure 1: (a) Lattice structure of the $J_{1}$$-$$J_{1}^{\prime}$$-$$J_{2}$
chain. Snapshots of (b) the $\mathcal{D}_{3}$-VBS, (c) the mixed-VBS, and (d)
symmetry-protected $J_{1}^{\prime}=0$ Haldane state, where valence bonds are
symbolized by red ellipses and emergent spins-1’s are represented by bold
lines. The lattices illustrated in (b-d) are topologically equivalent ladder
representations of the lattice in (a). The $J_{1}^{\prime}$ on both edges
(indicated by dashed lines) are set to be zero in our calculations (refer to
the main text for further details).

At $\beta=0$ (Fig. 1(d)), our system (1) can be effectively mapped onto an
$S=1$ Heisenberg chain [13, 14] since two spins coupled by $J_{1}$, i.e., a
rung in the ladder representation, form a spin-triplet pair via three $S^{z}$
states: $\ket{1}_{i,i+1}=\ket{\uparrow}_{i}\ket{\uparrow}_{i+1}$,
$\ket{0}_{i,i+1}=(\ket{\uparrow}_{i}\ket{\downarrow}_{i+1}+\ket{\downarrow}_{i}\ket{\uparrow}_{i+1})/\sqrt{2}$,
$\ket{-1}_{i,i+1}=\ket{\downarrow}_{i}\ket{\downarrow}_{i+1}$ for $S^{z}=1$,
$0$, and $-1$ states, respectively. Therefore, the effective model for
$\beta=0$ is

$H_{\rm eff}=\frac{J_{2}}{2}\sum_{i}\tilde{\bf S}_{i}\cdot\tilde{\bf
S}_{i+1}-\frac{J_{1}}{4}L,$ (2)

where $\tilde{\bf S}_{i}$ is a spin-1 operator defined as a resultant spin
$\tilde{\bf S}_{i}={\bf S}_{2i}+{\bf S}_{2i+1}$.

We study finite-size systems with lengths up to $L=600$ sites using the DMRG
method. Open boundary conditions (OBC) are applied unless specified otherwise.
As confirmed below, our system is marked by a Haldane-like VBS state
throughout the entire range of the gapped phase. Hence, following Ref. 5, we
explicitly control the edge states of our system by imposing
$J_{1}^{\prime\text{edge}}=0$ at both chain edges [18, 19, 20, 21, 22]. This
effectively eliminates the degeneracy of edge spins and facilitates the
correct computation of the bulk spin gap (see also App. A). We keep up to
$m=5500$ density-matrix eigenstates in the renormalization procedure,
resulting in an error $\epsilon/L\sim 10^{-8}$ for fixed system size.
Consequently, we can perform a reliable finite-size scaling analysis
extrapolating to the thermodynamic limit.

##  III Results

###  III.1 Ground-state phase diagram

It has been established that our system (1) is in a FM phase for
$\alpha<\beta/[2(1+\beta)]$ and in a singlet gapped phase for
$\alpha>\beta/[2(1+\beta)]$ [17]. However, the magnetic structure of the
gapped phase has not been determined yet except at $\beta=1$ and $\beta=0$.
When $\beta=1$, the ground state is a VBS state with valence bond formation
between third-neighbor spin-1/2’s due to the order-by-disorder mechanism,
leading to a spontaneous breaking of the translational symmetry [Fig. 1(b)]
[5]. This specific state is designated as the $\mathcal{D}_{3}$-VBS state.
Whereas at $\beta=0$, the ground state is interpreted as the so-called
(symmetry-protected) Haldane state since the system is effectively mapped onto
a spin-1 Heisenberg chain (2) [13, 14, 23, 24]. This state translates into the
formation of a valence bond on either the diagonal pair or $J_{2}$ bond in the
ladder representation of our system [Fig. 1(d)]. This state is referred as the
mixed-VBS state in the sense of coexisting different types of valence bonds.

Figure 2: (a) Example of second partial derivative of the ground-state energy
as a function of $\beta$ for fixed $\alpha=0.42$. (b) Examples of finite-size
scaling analysis for the peak positions in the second derivative. The linear
scaling is achieved by plotting the peak positions as a function of $1/L^{2}$.
The numbers enclosed within squares correspond to the peaks in (a). The
presence of two red lines is due to the multiple peak feature by finite-size
effect (see text).

Our findings indicate that the gapped region within the range $0<\beta<1$ is
predominantly characterized by the $\mathcal{D}_{3}$-VBS or the mixed-VBS
phase. These are considered as natural extensions of the limiting cases at
$\beta=1$ and $\beta=0$. Note that, unlike for the $\beta=0$ limit, only one
direction of the diagonal valence bond is allowed in the mixed-VBS state at
$\beta>0$ because the mirror symmetry of the ladder is broken when
$J_{1}^{\prime}$ is finite [Fig. 1(c)].

We have determined the phase boundary through detection of peaks in the second
derivative of the ground-state energy. For this analysis, periodic chains with
up to 48 sites are used. In Fig. 2(a) we plot the second derivative as a
function of $\beta$. Even though a discernible peak at a specific $\beta$
value for a fixed system size is noticeable, the peak position seems to vary
depending on the system size. Additionally, multiple peaks appear in the
parameter region adjacent to the FM instability when plotted against $\beta$
(not shown here). This multiple peak feature is typically interpreted as
finite-size effects arising from discrete quantum numbers relative to the
system size. Nevertheless, we can achieve a reasonable extrapolation of the
peak positions to a singular value in the thermodynamic limit, by performing a
finite-size scaling analysis of the second-derivative peaks using a linear
function as illustrated in Fig. 2(b). Note that in this case there are two
scaling lines for $L=4n$ due to the multiple peak feature. Furthermore, a more
accurate convergence of extrapolated values in the thermodynamic limit may be
achieved if we employ a polynomial function instead of a linear one. We also
find that other than the above peaks we see a system-size independent single
peak indicating the FM and $\mathcal{D}_{3}$-VBS phase boundary.

Figure 3: Ground-state phase diagram of the $J_{1}$-$J_{1}^{\prime}$-$J_{2}$
chain in the $\alpha$-$\beta$ plane. The FM phase line was determined via
spin-wave theory in Ref. 17. The boundary between the $\mathcal{D}_{3}$-VBS
and the mixed-VBS phases is determined via the peak position of the second
derivative of the ground-state energy. At $\beta=1$ the system is in the
$\mathcal{D}_{3}$-VBS phase for $\alpha>1/4$, as indicated by the green line.
The black square indicates the parameter set of LiCuSbO4.

The resultant $\alpha$-$\beta$ phase diagram is shown in Fig. 3, where the
width of the error bar is estimated based on the variations in the
extrapolated values derived from the preceding finite-size scaling analysis.
The green and yellow areas correspond to the $\mathcal{D}_{3}$-VBS and the
mixed-VBS phases, respectively. Note that the gapped phase at $\beta=1$ is in
the $\mathcal{D}_{3}$-VBS at any $\alpha(>1/4)$ [5]. The two VBS phases can be
generally distinguished based on the ratio of spin-spin correlations between
the second- and third-neighbor spin-1/2’s. Specifically, the system is
typified by the $\mathcal{D}_{3}$-VBS phase (or the mixed-VBS phase) when the
valence bond between the third neighbors is more robust (or weaker) than that
between the second neighbors (see Sec. III.4). However, this trend is less
obvious as $\beta$ decreases, and the strength of the second- and third-
neighbor valence bonds becomes equivalent in the limit of $\alpha=\beta=0$. An
additional intriguing aspect is the identification of a hidden AFM order
validated by a finite SOP for both VBS states [18] (see Sec. III.3).
Consequently, both VBS states can be characterized as Haldane-like VBS states.
This is further confirmed by the analysis of their entanglement spectra (ES)
[see Sec. III.5].

###  III.2 Spin gap

Figure 4: Spin gap in the $\alpha$-$\beta$ plane in terms of (a) $|J_{1}$| and
(b) $J_{2}$. Color bars shows the magnitude of the gap. The green lines denote
the boundaries between the FM, the $\mathcal{D}_{3}$-VBS, and the mixed-VBS
phases.

We start by computing the spin gap as a function of $\alpha$ and $\beta$.
Since the whole gapped region is characterized by a Haldane-like VBS state, we
specifically eliminate the ground-state degeneracy due to the edge states by
setting $J^{\prime 1,\mathrm{edge}}_{1}=0$ at both ends of the chain [21, 5,
18, 19]. By evaluating the ground-state energy $E_{0}(S^{z}=0)/L$ in the
thermodynamic limit, we have confirmed that this choice of edge structure does
not affect the bulk ground-state properties (see Appendix A). With the
degeneracy of edge spins removed, we can determine the spin gap as the energy
difference between the singlet ground state and the first excited triplet
state:

$\displaystyle\Delta=\lim_{L\to\infty}\Delta(L),$
$\displaystyle\Delta(L)=E_{0}(L,S^{z}=1)-E_{0}(L,S^{z}=0),$ (3)

where $L$ is the susyem size and $E_{0}(L,S^{z})$ is the ground-state energy
in the $S^{z}$ spin sector for a fixed system size $L$. Examples of finite-
size scaling for $\Delta(L)$ are shown in Appendix B. The extrapolated spin
gap $\Delta$ in the thermodynamic limit is plotted in Fig. 4. As the mapping
to the spin-1 model (2) is the most efficacious at $\beta=0$, the gap is
expected to increase with approaching $\beta=0$.

Let us first see the gap at $\beta=0$. When we consider the gap in terms of
$|J_{1}|$, it increases almost linearly with $\alpha$, reflecting the
effective exchange coupling $J_{2}/2$ in (2). On the other hand,
$\Delta/J_{2}$ increases slowly with decreasing $\alpha$. This is because the
mapping to (2) is generally good for any $\beta=0$ but is exact only in the
limit of $\alpha=\beta=0$. Consequently, the gap saturates at
$\Delta/J_{2}=0.2045$, which is that for the simple $S=1$ Heisenberg chain
with AFM exchange interaction $J_{2}/2$, with approaching $\alpha=\beta=0$
[17]. Given this, one might naively anticipate the gap to reduce with
increasing $\beta$ due to the potential obstruction of the effective spin-1
formation. However, it is worth noting that the gap has a maximum around the
boundary between the $\mathcal{D}_{3}$-VBS and the mixed-VBS phases with fixed
$\beta$. This behavior is more evident when taking the gap in units of $J_{2}$
[Fig. 4(b)] than in unit of $|J_{1}|$ [Fig. 4(a)]. Nevertheless, since the
peak is relatively sharp, its position remains nearly unaffected by the energy
unit $|J_{1}|$ or $J_{2}$. This suggests that frustration is maximized around
the phase boundary where the two VBS phases highly compete. The emergence of
this peak significantly validates the existence of a phase transition.

We should also note that the present gap, particularly within the small
$\alpha$ and large $\beta$ region, is somewhat larger than the one previously
estimated in Ref. 17. In other words, the gap was previously underestimated
without explicit lifting the degeneracy of edge spins, meaning the free spin
state at the system edges within the VBS structure was not adequately defined.
This validates that the $\mathcal{D}_{3}$-VBS is a kind of Haldane state.

###  III.3 String order parameter

Figure 5: String order parameter
$\langle\mathcal{O}^{z}_{\mathrm{string}}\rangle$ in the $\alpha$-$\beta$
plane. The red lines denote the boundaries between the FM, the
$\mathcal{D}_{3}$-VBS, and the mixed-VBS phases.

As discussed above, it appears that the entire gapped phase may be
characterized by a Haldane-type VBS state, which is topologically nontrivial.
Thus, we proceed to calculate the SOP which serves a good indicator to
evaluate the the stability of such a VBS state. This quantity was originally
defined for spin-1 systems [25]. However, for spin-1/2 systems, the SOP can be
suitably redefined as [5, 26, 27]

$\displaystyle\mathcal{O}^{z}_{\mathrm{string}}=$
$\displaystyle\lim_{|j-k|\to\infty}(-1)^{\frac{j-k-2}{2}}\langle(S^{z}_{k}+S^{z}_{k+1})$
(4) $\displaystyle\prod_{l=k+2}^{j-1}S^{z}_{l}(S^{z}_{j}+S^{z}_{j+1})\rangle,$

where $S^{z}_{j}$ is the $z$-component of the spin-1/2 at site $j$. In our
model (1), the dimerization introduced by $J_{1}^{\prime}<J_{1}$ breaks the
translational symmetry of the chain. Hence, our system has a stronger FM
coupling between the even-odd sites than between the odd-even sites. This
leads to the emergence of effective spins-1 degrees of freedom on each $J_{1}$
bond. Therefore, $j$ and $k$ in Eq. (III.3) must be even. Indeed, we have also
confirmed that if $j$ and $k$ are taken to be odd, the value of
$\mathcal{O}^{z}_{\mathrm{string}}$ is always extrapolated to zero in the
thermodynamic limit. Since OBC are applied here, we take $k=L/4$ and $j=3L/4$
to perform a systematic finite-size scaling with sufficiently minimizing the
influence of Friedel oscillations.

The extrapolated values of the SOP are plotted in Fig. 5. We find that the SOP
is finite throughout the gapped region of the phase diagram. Moreover, the
trend of $\mathcal{O}^{z}_{\mathrm{string}}$ in the $\alpha$-$\beta$ space is
very similar to that of $\Delta/J_{2}$ as seen in Fig. 4(b). Specifically, it
diminishes when moving from the bottom to the top right corner, and it has
relatively large value around the boundary of the two VBS phases.
Intriguingly, the SOP around the phase boundary can exceed
$\mathcal{O}^{z}_{\mathrm{string}}=0.3743$ for the $S=1$ Heisenberg chain
[15]: For instance, at $(\alpha,\beta)=(0.2,0.4)$
$\mathcal{O}^{z}_{\mathrm{string}}$ is $0.4101$ which is even closer to
$\mathcal{O}^{z}_{\mathrm{string}}=4/9=0.4444\cdots$ for the perfect VBS state
of the AKLT model [10]. This finding illustrates a possible enhancement of
valence bond formation by frustration. This enhancement of the SOP at the
phase boundary is reflected as a minimization of the lowest level in the ES
(see Sec. III.5).

###  III.4 Existence of two distinct VBS states

Thus far, the existence of a Haldane-type VBS state has been confirmed in the
whole gapped region of the $\alpha$-$\beta$ parameter space. However, the
exact arrangement of the valence-bond formation remains to be elucidated. It
is known that our system is in a Haldane mixed-VBS state when $\beta=0$,
wherein the valence bonds are formed along both the $J_{2}$ couplings and the
diagonal directions of the ladder representation [Fig. 1(d)]. On the other
hand, when $\beta=1$, our system exhibits a unique $\mathcal{D}_{3}$-VBS
state, characterized by valence bonds forming exclusively along third
neighbors in a direction dictated by spontaneous FM dimerization [5]. What
remains to be determined is how these two VBS phases develop in the parameter
region $0<\beta<1$, and whether there exists a possible additional VBS phase
in any region of the $\alpha$-$\beta$ plane.

Figure 6: Ratio between the second- and third-neighbor spin-spin correlations
in the $\alpha$-$\beta$ plane. The red lines denote the boundaries between the
FM, the $\mathcal{D}_{3}$-VBS, and the mixed-VBS phases.

Given that we have verified that the SOP (with assuming only $J_{1}$ rungs to
exhibit spin-1 degrees of freedom) is finite within the whole gapped region,
all valence bonds are likely to be formed between neighboring $J_{1}$ rungs.
Considering the ladder representation, a connection between neighboring
$J_{1}$ rungs involves the nearest-, second-, and third-neighbor bonds in the
original chain. However, the formation of nearest-neighbor valence bonds is
disregarded except for $\beta=0$ as it corresponds to a FM $J_{1}^{\prime}$
coupling. Therefore, we can focus only on the possibility of the second-
and/or third-neighbor valence bond formations. A simple approach to determine
the structure of a VBS state is to calculate the dimerization order parameter
defined by
$\mathcal{D}_{\delta}=\lim_{L\to\infty}|\langle\mathbf{S}_{i-\delta}\cdot\mathbf{S}_{i}\rangle-\langle\mathbf{S}_{i}\cdot\mathbf{S}_{i+\delta}\rangle|$
where $\delta$ assigns the length of the considered valence bond. Since the
dimerization between $J_{1}$ and $J_{1}^{\prime}$ is explicitly introduced in
our system (1) for $0\leq\beta<1$, $\mathcal{D}_{\delta}$ is always finite for
odd $\delta$. This order parameter can also detect spontaneous dimerization.
We thus calculate $\mathcal{D}_{2}$ for the second-neighbor bonds and find it
to be always zero in the thermodynamic limit (not shown). This suggests that a
VBS state composed solely of second-neighbor valence bond formations does not
exist. Accordingly, we conclude that the gapped region at $0<\beta<1$ is
covered by either the $\mathcal{D}_{3}$-VBS or the mixed-VBS phases.

To ascertain how the two VBS phases are distributed within the
$\alpha$-$\beta$ plane, we evaluate the relative strength of second- and
third-neighbor spin-spin correlations. We define the ratio as:

$\displaystyle\mathcal{R}$ $\displaystyle=\lim_{L\to\infty}\mathcal{R}(L),$
$\displaystyle\mathcal{R}(L)$
$\displaystyle=\left|\frac{\langle\mathbf{S}_{L/2}\cdot\mathbf{S}_{L/2+2}\rangle}{\langle\mathbf{S}_{L/2}\cdot\mathbf{S}_{L/2+3}\rangle}\right|.$
(5)

This is measured at the center of the chain to minimise the effect of Friedel
oscillations. In general, a larger correlation does not always imply valence
bond formation. Nevertheless, in this context, the relative strength
$\mathcal{R}$ should provide a good indication as to the transfer between the
$\mathcal{D}_{3}$-VBS and the mixed-VBS phases since it represents a
contiguous change between the leg and the diagonal valence bonds within each
plaquette in the ladder representation, rather than a complete redistribution
of valence bonds.

When $\mathcal{R}>1$ ($\mathcal{R}<1$) the AFM spin-spin correlation between
second-neighbor (third-neighbor) spins exceeds that between third-neighbor
(second-neighbor) spins. Note that both second- and third-neighbor spin-spin
correlations are always AFM in the gapped region. Intuitively, a spin-singlet
formation on a bond with a larger AFM spin-spin correlation is energetically
favorable. Hence, we may expect our system to be in the $\mathcal{D}_{3}$-VBS
state for $\mathcal{R}<1$ and in the mixed-VBS state for $\mathcal{R}>1$. Our
results for $\mathcal{R}$ are plotted in Fig. 6. It exhibits a similar
behavior to that of the gap $\Delta/J_{2}$ [Fig. 4(b)] and of the SOP [Fig.
5]. Indeed, we observe $\mathcal{R}\sim 1$ at the boundary between the two VBS
phases, indicating the competition between second- and third-neighbor valence
bonds. This seems to reasonably explain the phase transition for $0<\beta<1$.

It should be noted, however, that this approach is not universally reliable in
detecting the valence bonds structure. For example, at $\beta=1$,
$\mathcal{R}$ is larger than 1 and simply increases with $\alpha$ within the
$\mathcal{D}_{3}$-VBS phase. For larger values of $\alpha$, our system can be
regarded as weakly coupled AFM chains. In such cases, avoiding the formation
of the second-neighbor valence bonds could be possible due to large
fluctuation. Even so, an unresolved issue remains: Do the physical quantities
asymptotically approach the undimerized limit $\beta=1$, or does this limit
serve as a singular point? We believe that insights into this question could
be gained from an examination of the Berry phase in the $\alpha$-$\beta$
plane, specifically near the $\beta=1$ line. Nonetheless, this goes beyond the
scope of our current work.

###  III.5 Entanglement Spectra

Figure 7: Plots of the entanglement spectrum (ES) of a system with PBC and
size $L=82$ (a) as a function of $\alpha$ for $\beta=0.6$, (b) as a function
of $1-\beta$ for $\alpha=0.35$. The green region denotes
$\mathcal{D}_{3}$-VBS, while the yellow denotes the mixed-VBS state. Numbers
indicate the degeneracy of the state.

Entanglement related quantities are often used to study topological properties
of VBS states including the Haldane state [28]. Having confirmed that both VBS
states in our phase diagram are classified as of the Haldane-type, it is
interesting to confirm their topological nature by looking at the low-lying
eigenstates of the ES [29, 30, 31, 32]. In our calculations, we consider the
ES obtained by partitioning the system into two subsystems $A$ and $B$.
Defining $\xi_{\lambda}$ in the Schmidt decomposition of the ground state
$\ket{\psi}$ as

$\displaystyle\ket{\psi}=\sum_{\lambda}e^{-\xi_{\lambda}/2}\ket{\lambda}_{A}\ket{\lambda}_{B},$
(6)

where $\ket{\lambda}_{A}$ ($\ket{\lambda}_{B}$) is the orthonormal basis for
the subsystem $A$ ($B$), we can interpret the ES as the energy spectrum of the
entanglement Hamiltonian $\mathcal{H}_{e}$ defined as
$e^{-\mathcal{H}_{e}}=\rho_{A}=\mathrm{Tr}_{B}\ket{\psi}\bra{\psi}=\sum_{\lambda}e^{-\xi_{\lambda}}\ket{\lambda}_{A}\bra{\lambda}_{A}$.
If the system size is sufficiently larger than the correlation length, the ES
can be described by the two virtual edge states; the ES therefore represents
the gapless edge modes in its lowest-lying eigenstate. Since spin-spin
correlations decays exponentially in the VBS phase, the correlation length is
expected to be quite short.

We now consider periodic chains with system size $L=4n+2=2(2n+1)$ and divide
them in half. Since the two subsystems consist of an odd number of sites, we
can directly detect the emergent spin-1/2 edge states, if they exist. Our
results for the ES with size $L=82$ are shown in Fig. 7. We clearly see 2-fold
degeneracy of the lowest-lying eigenstate. In our dimerized system, each of
the subsystems contains only one free spin-1/2 at one edge. Accordingly, non-
entangled two spin-1/2’s with total $S^{z}=0$ lead to the observed 2-fold
degeneracy. Since this picture is valid for both the $\mathcal{D}_{3}$-VBS and
the mixed-VBS states, the 2-fold degeneracy is unchanged through the phase
boundary. It is worth mentioning that, as seen in Fig. 7(a), the lowest-lying
level in the ES reaches a minimum around the phase boundary. This is a
reflection of the stability of the VBS state as validated by the SOP.
Moreover, this supports the validity of the determined VBS structures. We also
find that the second eigenstate is 6-fold degenerate, with four states located
in the $S^{z}=0$ sector and the remaining two at $S^{z}=\pm 1$. More details
are discussed in App. C.

Note that the $\mathcal{D}_{3}$-VBS state exhibits 4-fold degeneracy in the
lowest-lying eigenstate for the undimerized case $\beta=1$ [5]. This is
because there are two possible ground states with spontaneous FM dimerization.
As a result, the free spin-1/2 can be positioned at both edges of each
subsystem. The presence of this 2-fold degeneracy at both cut edges
cumulatively leads to the 4-fold degeneracy. More details are given in App. C.

###  III.6 Evolution of the gap with explicit dimerization

Figure 8: Spin gap as function of $(J_{3}^{\prime}/|J_{1}|)^{2/3}$ for (a)
$(\alpha,\beta)=(0.3,0.8)$, (b) $(0.3,0.1)$, and (c) $(0.5,0.5)$. Inset: The
same quantity as a function of $J_{3}^{\prime}/|J_{1}|$. Solid lines are
linear fits and dotted lines are guides for eyes.

Based on the above results, we have found that the gapped region in the
ground-state phase diagram is divided into two kinds of VBS phases, namely,
the $\mathcal{D}_{3}$-VBS and the mixed-VBS phases. Nevertheless, the
$\mathcal{D}_{3}$-VBS phase is confined to only a narrow range near the FM
instability except for the $\beta=1$ line. To substantiate this observation,
following Ref. 5, we explore the dependence of the spin gap on an additional
AFM third-neighbor exchange interaction $J_{3}^{\prime}$. The modified
Hamiltonian is written as

$H^{\prime}=H+J_{3}^{\prime}\sum_{i=\mathrm{even}}\mathbf{S}_{i}\cdot\mathbf{S}_{i+3},$
(7)

where $J_{3}^{\prime}(>0)$ is added only for even $i$ in Fig. 1. This third-
neighbor bond is associated with the valence bond in the $\mathcal{D}_{3}$-VBS
state. If the system is in the $\mathcal{D}_{3}$-VBS state which is a kind of
spin-Peierls state, the spin gap is expected to increase as
$\Delta-\Delta(J_{3}^{\prime}=0)\propto J_{3}^{\prime\frac{2}{3}}$ at small
$J_{3}^{\prime}$ [33].

In Fig. 8(a) we plot the spin gap as a function of
$(J_{3}^{\prime}/|J_{1}|)^{2/3}$ for the $\mathcal{D}_{3}$-VBS state at
$(\alpha,\beta)=(0.3,0.8)$. We can see a power-law behavior of the spin gap
with exponent $2/3$ at $0<J_{3}^{\prime}\lesssim 0.002$. We thus confirm that
the $\mathcal{D}_{3}$-VBS state can exist even if our system is away from
$\beta=1$. On the other hand, the gap behavior clearly deviates from
$\Delta-\Delta(J_{3}^{\prime}=0)\propto J_{3}^{\prime\frac{2}{3}}$ for the
mixed-VBS states at $(\alpha,\beta)=(0.3,0.1)$ and $(0.5,0.5)$, as shown in
Fig. 8(b,c). Note that these $(\alpha,\beta)$ points are in the mixed-VBS
phase, but not very far from the boundary to the $\mathcal{D}_{3}$-VBS phase.
For these points, the spin gap increases linearly with $J_{3}^{\prime}$, i.e.,
$\Delta-\Delta(J_{3}^{\prime}=0)\propto J_{3}^{\prime}$. This implies that the
lowest spin excitation is given by the collapse of a spin-singlet pair on the
second-neighbor bond. This is because the second-neighbor bond involves a
direct coupling by AFM $J_{2}$ as well as an indirect coupling via FM $J_{1}$
and AFM $J_{3}^{\prime}$, i.e, a superexchange path through $J_{1}$ and
$J_{3}^{\prime}$, so that the second-neighbor exchange interaction is
effectively linearly enhanced by $J_{3}^{\prime}$. Morevoer, this also means
that the valence bond between the third-neighbor spins is stronger than that
between the second-neighbor spins in the mixed-VBS phase, and their strengths
become equivalent in the limit of $\alpha=\beta=0$.

Furthermore, it is interesting to consider the difference value
$\delta\Delta/\delta J_{3}^{\prime}$: We obtain $0.0367$, $0.3633$, and
$0.8757$ for $(\alpha,\beta)=(0.3,0.1)$, $(0.5,0.5)$, and $(0.3,0.8)$,
respectively. Since the valence bonds are formed only on the $J_{3}^{\prime}$
bonds in the $\mathcal{D}_{3}$-VBS phase, the increase of $J_{3}^{\prime}$
directly contributes to the gap increase. The different values are obtained
via the gap increase from $J_{3}^{\prime}=0$ to $0.001$. More detailed data is
shown in App. D.

##  IV Summary and discussion

We studied the frustrated dimerized FM-AFM $J_{1}$-$J_{1}^{\prime}$-$J_{2}$
chain using the DMRG method. We obtained the ground-state phase diagram for a
wide range of $J_{1}^{\prime}/J_{1}$ and $J_{2}/|J_{1}|$ based on the results
for the second derivative of energy, spin gap, spin-spin correlations, SOP,
and ES. The phase diagram consists of a FM and two gapped VBS phases. These
VBS phases are characterized as the $\mathcal{D}_{3}$-VBS and the mixed-VBS
phase. In the $\mathcal{D}_{3}$-VBS state, valence bonds are formed between
third-neighbor spin-1/2’s. The mixed-VBS state, on the other hand, exhibits a
coexistence of both second- and third-neighbor valence bonds. Remarkably,
these VBS states are identified as being of the Haldane-type, characterized by
a finite value of the SOP and the 2-fold degeneracy of the ES. It is
remarkable that the formation of valence bonds is significantly stabilized at
the boundary between the two VBS phases. Moreover, at this boundary, the gap
and SOP are maximized, while the lowest-lying eigenstate of the ES is
minimized. These observations underscore the critical role of frustration in
the emergence of topological states in our system.

Finally, we briefly discuss the relevance to the experimental observations for
LiCuSbO4. For this compound, density-functional theory (DFT) calculations have
estimated the effective exchange parameters to be $J_{1}\approx-160$K,
$J^{\prime}_{1}\approx 90$K, and $J_{2}\approx 37.6$K [12]. This set of
parameters corresponds to $(\alpha,\beta)=(0.235,0.5625)$ in our notation,
putting this material in the $\mathcal{D}_{3}$-VBS phase in our ground-state
phase diagram as shown in Fig. 3. Intriguingly, the possibility of exotic
phenomena, such as multipolar states in the presence of a magnetic field, has
been experimentally suggested [12, 34]. The presence of multipolar physics in
1D systems under an applied field has been discussed only within the uniform
$J_{1}$-$J_{2}$ chain case ($\beta=1)$[35, 36, 37]. We thus believe that the
effect of dimerization on the multipolar physics is a question of interest and
should be the subject of future studies.

The present study represents the first demonstration of a nontrivial quantum
phase transition between topological VBS states in spin-1/2 chains, though the
possibility of phase transitions through a complete reconstruction of valence
bond structure have been suggested in 2D systems [38, 39, 40], 1D systems with
alternating different spins [41, 42, 43], and higher spin systems [44, 45]. It
is very exciting that quasi-1D materials such as LiCuSbO4
[$(\alpha,\beta)=(0.235,0.5625)$] and Rb2Cu2Mo3O12
[$(\alpha,\beta)=(0.37,0.65)$] [46, 17] may be situated near the boundary
between two topological phases, and that transitions between topological
ordered states could potentially be observed experimentally through controls
such as pressure and stretching.

## Acknowledgements

We thank Krzysztof Wohlfeld for fruitful discussion. This work was supported
by Narodowe Centrum Nauki (NCN, Poland) under Project Nos.
2021/40/C/ST3/00177. This research was carried out with the support of the
Interdisciplinary Center for Mathematical and Computational Modeling at the
University of Warsaw (ICM UW) under grants No. G81-4 and No. G93-1613. This
project is funded by the German Research Foundation (DFG) via the projects A05
of the Collaborative Research Center SFB 1143 (project-id 247310070). The code
to reproduce the data and figures presented in this manuscript is available at
Ref. [47].

##  Appendix A Special open boundary conditions

Figure 9: Finite-size scaling of the ground state energy per site $E_{0}/L$
for $J_{1}^{\prime\mathrm{edge}}$=0 (blue) and
$J_{1}^{\prime\mathrm{edge}}=J^{\prime}_{1}$ (orange) with (a) $\alpha$ = 0.8,
$\beta$=0.2 and (b) $\alpha$ = 0.4, $\beta$=0.6.

In this study, we applied special open boundary conditions where the
$J_{1}^{\prime}$ bonds at the chain edges are set to be zero, i.e.,
$J^{\prime\mathrm{edge}}_{1}=0$. This setting enables us to explicitly define
the emergent spin-1 formations on the $J_{1}$ bonds, thereby offering us a
mechanism to exert demonstrable control over the edge states of our Haldane-
type VBS states. As a result, we are able to correctly calculate bulk physical
quantities such as the spin gap as well as the SOP in the thermodynamic limit.
In the absence of this setting, these quantities could be underestimated due
to a potential confluence of two symmetry broken states. More details are
discussed in Ref. 5. To substantiate our hypothesis that the choice of
boundary conditions does not affect the bulk ground state, we calculate the
ground-state energies for the cases where $J^{\prime\mathrm{edge}}_{1}=0$ and
$J^{\prime\mathrm{edge}}_{1}=J_{1}^{\prime}$. Fig. 9 shows a comparison of the
energy per site ($E/L$) between these two scenarios for two sets of parameters
($\alpha$, $\beta$). For both parameter sets, it is observed that $E/L$ for
$J^{\prime\mathrm{edge}}_{1}=0$ and
$J^{\prime\mathrm{edge}}_{1}=J_{1}^{\prime}$ seem to be extrapolated to the
same value in the thermodynamic limit. The difference in $E/L$ at the
thermodynamic limit between the two boundary conditions is found to be within
the range of numerical zero, i.e., $\Delta E/L=10^{-8}-10^{-7}$. We thus
conclude that the bulk ground state remains unaffected by the choice of
$J^{\prime\mathrm{edge}}_{1}$.

##  Appendix B Finite-size scaling of the spin gap

Figure 10: Examples of finite-size scaling analysis for the spin gap
$\Delta(L)/|J_{1}|$, where open chains with lengths $L=80$, $100$, $120$,
$140$, and $160$ are used.

In Fig. 10, we show some examples of finite-size scaling analysis of the spin
gap for various $(\alpha,\beta)$ values: $(0.6,0.2)$ (blue line), $(0.5,0.5)$
(green line), $(0.6,0.9)$ (red line), and $(0.2,0.1)$ (black line). In most
cases, the scaling can be reasonably performed by fitting the data with a
quadratic function $\Delta(L)=a/L^{2}+b/L+\Delta$, where $a$ and $b$ are
fitting parameters. In certain instances, particularly when the VBS state is
quite robust, an alternative fitting function
$\Delta(L)=a/L^{3}+b/L^{2}+\Delta$ is used. In such cases, the spin-spin
correlation tends to decay rapidly with distance, leading to a fast
convergence of the spin gap with respect to the system size. The parameter set
$(\alpha,\beta)=(0.6,0.9)$ is representative of such a case.

##  Appendix C Additional information about the entanglement spectrum

Figure 11: (a) Depiction of the partition of a periodic chain with 14 sites,
where bold and thin lines represent $J_{1}$ and $J^{\prime}_{1}$ interactions,
respectively. The blue ellipses represent the emergent spin-1 sites, and the
dashed-line ellipse denotes a fractionalized spin-1 site resulting from the
partitioning. (b) Low-lying ES as a function of total $S^{z}$ of the subsystem
for $(\alpha,\beta)=(0.3,0.8)$ and $L=82$. The numbers next to each point
indicate the degree of degeneracy. (c) Level of the lowest-lying 2-fold
degenerate eigenstate in the ES as a function of $\alpha$ for $L=22$ (black),
$L=26$ (gray), and $L=82$ (red) at $\beta=0.6$. Dashed vertical lines mark the
position of the minimum for each $L$.

We here provide additional information about our calculations for the ES. In
our analysis for the ES the system size is chosen to be $L=4n+2=2(2n+1)$. Fig.
11(a) illustrates how the bipartite cutting acts on our periodic system when
the ES is calculated. Indeed, the cut goes through two inequivalent bonds: one
$J_{1}$ bond corresponding to the emerging spin-1 site, and one
$J_{1}^{\prime}$ bond. Thus, the cut creates two spin-1/2 edge states, one for
each subsystem, leading to the 2-fold degeneracy of the lowest-lying
eigenstate in the ES, as shown in Fig. 7.

In Fig. 11(b) the low-lying ES for $(\alpha,\beta)=(0.3,0.6)$ and $L=82$ is
plotted as a function of total $S^{z}$ of the subsystem. we see that the
6-fold degeneracy in the second level is resolved to 4-fold degeneracy at
$S^{z}=0$ and the remaining 2-fold degeneracy at $S^{z}=\pm 1$. The states at
$S^{z}=\pm 1$ may come from a possible spin-triplet components of each
subsystem.

Moreover, as stated in the main text, the phase boundary between the two VBS
states aligns with the minimum of the lowest-lying 2-fold degenerate
eigenstate in the ES. To demonstrate this, we examine the behavior of the ES
in the vicinity of the phase transition between the two VBS states. In Fig.
11(c), we plot the lowest-lying level of the ES as a function of $\alpha$ for
system sizes $L=22$, $26$, and $82$ at a fixed $\beta=0.6$. It is apparent
that the position of the minimum promptly approaches the phase boundary as the
system size increases. This may be interpreted as the weight of the lowest-
lying density-matrix eigenstate becoming more pronounced than those of the
higher eigenstates at the phase boundary due to the robustness of the VBS
state.

Figure 12: Relative increase of the spin gap with $J_{3}^{\prime}$ in the
$\alpha$-$\beta$ plane. The size of the circles is proportional to the
relative increase of the gap at $J_{3}^{\prime}=0.001|J_{1}|$. Results are
superimposed to the phase diagram for clarity.

##  Appendix D Spin gap increase with $J^{\prime}_{3}$

As discussed in section III.6, the spin gap exhibits a linear increase with
the third-neighbor AFM interaction $J^{\prime}_{3}$ within the mixed-VBS
phase, and a 2/3 power-law relationship within the mixed-VBS phase. This
section elaborates further on the disparate rates of increase observed in
these two VBS phases. Fig. 12 presents the relative percentage increase of the
spin gap when comparing $\Delta(J_{3}^{\prime}=0)$ and
$\Delta(J_{3}^{\prime}=0.001|J_{1}|)$. The data reveals that the overall
increase in the mixed-VBS region is of the order of $\sim 1\%$. The growth of
the spin gap is smaller if we increase the frustration parameter $\alpha$,
while the opposite is true for the parameter $\beta$, with exceptions close to
the critical line. Among the considered points, we see the most significant
gap increase of $23.5\%$ for $(\alpha,\beta)=(0.3,0.95)$. This increase can be
ascribed to the third-neighbors valence bond geometry in the
$\mathcal{D}_{3}$-VBS region. In the $\mathcal{D}_{3}$-VBS phase, the valence
bonds are formed only on the $J_{3}^{\prime}$ bonds, so that the increase of
$J_{3}^{\prime}$ directly enhances the spin gap. Whereas in the mixed-VBS
phase, the spin gap is roughly estimated as an energy needed to break a
second-neighbor valence bond. Thus, the increase of $J_{3}^{\prime}$ just
indirectly affects the spin gap.

## References

  * Balents [2010] L. Balents, Spin liquids in frustrated magnets, Nature 464, 199 (2010).
  * Moessner and Ramirez [2006] R. Moessner and A. P. Ramirez, Geometrical frustration, Physics Today 59, 24 (2006).
  * Kuzian [2023] R. Kuzian, Methods of modeling of strongly correlated electron systems, Nanomaterials 13, 238 (2023).
  * Majumdar and Ghosh [1969] C. K. Majumdar and D. K. Ghosh, On next-nearest-neighbor interaction in linear chain. I, J. Math. Phys. 10, 1388 (1969).
  * Agrapidis  _et al._ [2019] C. E. Agrapidis, S.-L. Drechsler, J. van den Brink, and S. Nishimoto, Coexistence of valence-bond formation and topological order in the Frustrated Ferromagnetic $J_{1}$-$J_{2}$ Chain, SciPost Phys. 6, 019 (2019).
  * Haldane [1983a] F. Haldane, Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the ${O}(3)$ nonlinear sigma model, Phys. Lett. A 93, 464–468 (1983a).
  * Haldane [1983b] F. D. M. Haldane, Nonlinear field theory of large-spin Heisenberg antiferromagnets: Semiclassically quantized solitons of the one-dimensional easy-axis Néel state, Phys. Rev. Lett. 50, 1153–1156 (1983b).
  * Botet and Jullien [1983] R. Botet and R. Jullien, Ground-state properties of a spin-1 antiferromagnetic chain, Phys. Rev. B 27, 613–615 (1983).
  * Renard  _et al._ [1988] J. P. Renard, M. Verdaguer, L. P. Regnault, W. A. C. Erkelens, J. Rossat-Mignod, J. Ribas, W. G. Stirling, and C. Vettier, Quantum energy gap in two quasi-one-dimensional ${S}$=1 Heisenberg antiferromagnets (invited), J. of Appl. Phys. 63, 3538–3542 (1988).
  * Affleck  _et al._ [1987] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59, 799–802 (1987).
  * Affleck  _et al._ [1988] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Valence bond ground states in isotropic quantum antiferromagnets, Comm. Math. Phys. 115, 477–528 (1988).
  * Grafe  _et al._ [2017] H.-J. Grafe, S. Nishimoto, M. Iakovleva, E. Vavilova, L. Spillecke, A. Alfonsov, M.-I. Sturza, S. Wurmehl, H. Nojiri, H. Rosner, J. Richter, U. K. Rößler, S.-L. Drechsler, V. Kataev, and B. Büchner, Signatures of a magnetic field-induced unconventional nematic liquid in the frustrated and anisotropic spin-chain cuprate LiCuSbO4, Sci. Rep. 7, 6720 (2017).
  * Hida [1991] K. Hida, Haldane gap in the spin$-1/2$ double chain Heisenberg antiferromagnet -numerical diagonalization and projector Monte Carlo study-, J. Phys. Soc. Jpn. 60, 1347–1354 (1991).
  * Watanabe  _et al._ [1993] H. Watanabe, K. Nomura, and S. Takada, ${S}=1/2$ quantum Heisenberg ladder and ${S}=1$ Haldane phase, J. Phys. Soc. Jpn. 62, 2845–2860 (1993).
  * White and Huse [1993] S. R. White and D. A. Huse, Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic ${S}=1$ Heisenberg chain, Phys. Rev. B 48, 3844 (1993).
  * Schollwöck [2005] U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005).
  * Agrapidis  _et al._ [2017] C. E. Agrapidis, S.-L. Drechsler, J. van den Brink, and S. Nishimoto, Crossover from an incommensurate singlet spiral state with a vanishingly small spin gap to a valence-bond solid state in dimerized frustrated ferromagnetic spin chains, Phys. Rev. B 95, 220404 (2017).
  * Oshikawa [1992] M. Oshikawa, Hidden $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry in quantum spin chains with arbitrary integer spin, J Phys. Condens. Matter 4, 7469–7488 (1992).
  * Kennedy and Tasaki [1992a] T. Kennedy and H. Tasaki, Hidden $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry breaking in Haldane-gap antiferromagnets, Phys. Rev. B 45, 304–307 (1992a).
  * Kennedy and Tasaki [1992b] T. Kennedy and H. Tasaki, Hidden symmetry breaking and the Haldane phase in ${S}=1$ quantum spin chains, Comm. Math. Phys. 147, 431–484 (1992b).
  * Kennedy [1990] T. Kennedy, Exact diagonalisations of open spin-1 chains, J Phys. Condens. Matter 2, 5737 (1990).
  * Pollmann  _et al._ [2012] F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Symmetry protection of topological phases in one-dimensional quantum spin systems, Phys. Rev. B 85, 075125 (2012).
  * Gu and Wen [2009] Z.-C. Gu and X.-G. Wen, Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order, Phys. Rev. B 80, 155131 (2009).
  * Tasaki [2020] H. Tasaki,  _Physics and Mathematics of Quantum Many-Body Systems_ (Springer International Publishing, Springer-Verlag, Berlin Heidelberg, 2020).
  * den Nijs and Rommelse [1989] M. den Nijs and K. Rommelse, Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains, Phys. Rev. B 40, 4709–4734 (1989).
  * Kohmoto and Tasaki [1992] M. Kohmoto and H. Tasaki, Hidden ${S}=1/2$ quantum spin chain with bond alternation, Phys. Rev. B 46, 3486–3495 (1992).
  * Hida [1992] K. Hida, Crossover between the Haldane-gap phase and the dimer phase in the spin$-1/2$ alternating Heisenberg chain, Phys. Rev. B 45, 2207–2212 (1992).
  * Pollmann  _et al._ [2010] F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Phys. Rev. B 81, 064439 (2010).
  * Li and Li [2008] Y.-C. Li and S.-S. Li, Quantum phase transitions in the ${S}=1/2$ distorted diamond chain, Phys. Rev. B 78, 184412 (2008).
  * Fidkowski [2010] L. Fidkowski, Entanglement spectrum of topological insulators and superconductors, Phys. Rev. Lett. 104, 130502 (2010).
  * Li and Haldane [2008] H. Li and F. D. M. Haldane, Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum hall effect states, Phys. Rev. Lett. 101, 010504 (2008).
  * Vidal  _et al._ [2003] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90, 227902 (2003).
  * Uhrig  _et al._ [1999] G. Uhrig, F. Schönfeld, M. Laukamp, and E. Dagotto, Unified quantum mechanical picture for confined spinons in dimerized and frustrated spin chains, Eur. Phys. J. B 7, 67 (1999).
  * Bosiočić  _et al._ [2017] M. Bosiočić, F. Bert, S. E. Dutton, R. J. Cava, P. J. Baker, M. Požek, and P. Mendels, Possible quadrupolar nematic phase in the frustrated spin chain LiCuSbO4: An nmr investigation, Phys. Rev. B 96, 224424 (2017).
  * Hikihara  _et al._ [2008] T. Hikihara, L. Kecke, T. Momoi, and A. Furusaki, Vector chiral and multipolar orders in the spin-1/2 frustrated ferromagnetic chain in magnetic field, Phys. Rev. B 78, 144404 (2008).
  * Sudan  _et al._ [2009] J. Sudan, A. Lüscher, and A. M. Läuchli, Emergent multipolar spin correlations in a fluctuating spiral: The frustrated ferromagnetic spin-1/2 Heisenberg chain in a magnetic field, Phys. Rev. B 80, 140402 (2009).
  * Mourigal  _et al._ [2012] M. Mourigal, M. Enderle, B. Fåk, R. K. Kremer, J. M. Law, A. Schneidewind, A. Hiess, and A. Prokofiev, Evidence of a bond-nematic phase in LiCuVO4, Phys. Rev. Lett. 109, 027203 (2012).
  * Vishwanath  _et al._ [2004] A. Vishwanath, L. Balents, and T. Senthil, Quantum criticality and deconfinement in phase transitions between valence bond solids, Phys. Rev. B 69, 224416 (2004).
  * Slagle and Xu [2014] K. Slagle and C. Xu, Quantum phase transition between the Z2 spin liquid and valence bond crystals on a triangular lattice, Phys. Rev. B 89, 104418 (2014).
  * Liao  _et al._ [2022] Y. D. Liao, X. Y. Xu, Z. Y. Meng, and Y. Qi, Dirac fermions with plaquette interactions. iii. SU(N) phase diagram with Gross-Neveu criticality and first-order phase transition, Phys. Rev. B 106, 155159 (2022).
  * Tonegawa  _et al._ [1998] T. Tonegawa, T. Hikihara, M. Kaburagi, T. Nishino, S. Miyashita, and H.-J. Mikeska, Ground-state and thermodynamic properties of the quantum mixed spin-1/2-1/2-1-1 chain, J. Phys. Soc. Jpn 67, 1000 (1998).
  * Li  _et al._ [2006] S.-B. Li, Z.-X. Xu, J.-H. Dai, and J.-B. Xu, Entanglement and quantum phase transitions in quantum mixed spin chains, Phys. Rev. B 73, 184411 (2006).
  * Veríssimo  _et al._ [2023] L. M. Veríssimo, M. S. Pereira, J. Strečka, and M. L. Lyra, Topological quantum phase transition in a mixed-spin Heisenberg tetramer chain with alternating spin-1/2 and spin-5/2 dimers, J. Magn. Magn. Mater. 571, 170595 (2023).
  * Kitazawa and Nomura [1997] A. Kitazawa and K. Nomura, Phase transitions of ${S}=3/2$ and ${S}=2$ XXZ spin chains with bond alternation, J. Phys. Soc. Jpn 66, 3379 (1997).
  * Miyakoshi  _et al._ [2016] S. Miyakoshi, S. Nishimoto, and Y. Ohta, Entanglement properties of the Haldane phases: A finite system-size approach, Phys. Rev. B 94, 235155 (2016).
  * Hase  _et al._ [2004] M. Hase, H. Kuroe, K. Ozawa, O. Suzuki, H. Kitazawa, G. Kido, and T. Sekine, Magnetic properties of ${\mathrm{rb}}_{2}{\mathrm{cu}}_{2}{\mathrm{mo}}_{3}{\mathrm{o}}_{12}$ including a one-dimensional spin-$1/2$ heisenberg system with ferromagnetic first-nearest-neighbor and antiferromagnetic second-nearest-neighbor exchange interactions, Phys. Rev. B 70, 104426 (2004).
  * Wardyn  _et al._ [2023] J. Wardyn, C. E. Agrapidis, and S. Nishimoto, Existence of two distinct valence bond solid states in the dimerized frustrated ferromagnetic ${J}_{1}$-${J}^{\prime}_{1}$-${J}_{2}$ chain, zenodo.7997521 10.5281/ZENODO.7997521 (2023).