# Quantum Fisher Information and multipartite entanglement in spin-1 chains

Federico Dell’Anna  Sunny Pradhan  Dipartimento di Fisica e Astronomia
dell’Università di Bologna, I-40127 Bologna, Italy  INFN, Sezione di Bologna,
I-40127 Bologna, Italy  Cristian Degli Esposti Boschi  INFN, Sezione di
Bologna, I-40127 Bologna, Italy  CNR-IMM, Sezione di Bologna, via Gobetti 101,
40129, Bologna, Italy  Elisa Ercolessi  Dipartimento di Fisica e Astronomia
dell’Università di Bologna, I-40127 Bologna, Italy  INFN, Sezione di Bologna,
I-40127 Bologna, Italy

###### Abstract

In this paper, we study the ground state Quantum Fisher Information (QFI) in
one-dimensional spin-$1$ models, as witness to Multipartite Entanglement. The
models addressed are the Bilinear-Biquadratic model, the most general
isotropic $SU(2)$-invariant spin-$1$ chain, and the XXZ spin-$1$ chain, both
with nearest-neighbor interactions and open boundary conditions. We show that
the scaling of the QFI of strictly non-local observables can be used for
characterizing the phase diagrams and, in particular, for studying topological
phases, where it scales maximally. Analysing its behavior at the critical
phases we are also able to recover the scaling dimensions of the order
parameters both for local and string observables. The numerical results have
been obtained by exploiting the Density Matrix Renormalization Group algorithm
and Tensor Network techniques.

##  I Introduction

In addition to be a crucial resource for quantum-enhanced metrology [1] and
quantum computation [2], entanglement has been used to characterize quantum
phases and quantum phase transitions (QPTs) in many-body models, particularly
for low-dimensional systems, and has been important also to uncover exotic
states of matter like topological spin liquids [3] or to describe many-body
localization [4].

Bipartite entanglement has been the primary focus in the literature [5], with
the area law [6] serving as a benchmark for relating the amount of
entanglement between two partitions of a quantum many-body system to the
surface area between the blocks [7, 8]. It has been proved [9] that the ground
state of some spin chains should exhibit _Multipartite Entanglement_ (ME), but
somehow this topic has received less attention [9], despite the fact that
many-body quantum states are far more complex than what can be captured with
bipartite entanglement only.

A possible estimator of multipartite entanglement is _Quantum Fisher
Information_ (QFI), a quantity which is introduced in the context of the
problem of phase estimation in metrology [10] and is of use in the study of
the sensitivity of atomic interferometers beyond the shot-noise limit [11].
The QFI associated to local operators has recently been used to observe ME in
models exhibiting Ginzburg-Landau-type quantum phase transitions [12] and in
spin systems such as the Ising and XY models [12, 13], where ME is expected to
diverge at criticality. It has been pointed out, however, that the use of
local operators in this method fails to detect ME at topological quantum
phases and transitions. To address this issue, QFI-based methods need to be
extended to include also non-local operators, as first outlined in [14, 15].

In this paper we are going to study the ME in two paradigmatic spin-1 systems
with nearest-neighbor interactions: the _Bilinear-Biquadratic (BLBQ) model_
and the _XXZ model_ , two models with a rich phase diagram which exhibit a
topological Haldane phase. More specifically, we will show that QFI of local
and _non-local_ order parameters (such as string-order parameters [16]) are
able to classify all phases of the model as well as to give us information
about universal critical exponents at phase transitions.

The paper is structured as follows. In Sec. II, we briefly review ME and QFI,
and their relationship. In Sec. III we discuss the BLBQ model; after
describing its phase diagram, we analyze the scaling of the QFI with respect
to some selected operators. The same is done in the last Sec. (IV) for the XXZ
model. A summary of the obtained results is discussed in the conclusions in
Sec. V, with some possible outlooks for future research.

##  II Quantum Fisher Information and Multipartite entanglement

In this section we concisely review the concepts of ME and QFI, elucidating
their relationship [17, 1].

A pure state of $N$ particles is $k$-_producible_ if can be written as:

$\ket{\psi_{k-\text{prod}}}=\bigotimes_{l=1}^{M}\ket{\psi_{l}}$ (1)

where $\ket{\psi_{l}}$ is a state with $N_{l}\leq k$ particles and $M$ is the
number of parties in which it is possible to split up the state so that
$\sum_{l=1}^{M}N_{l}=N$.

A state is $k$-_entangled_ if it is $k$-_producible_ but not
$(k-1)$-_producible_. Therefore, a $k$-particle entangled state can be written
as a product $\ket{\psi_{k}}$ which contains at least one state
$\ket{\psi_{l}}$ of $N_{l}=k$ particles which does not factorize further. So,
in this notation, a state $\ket{\psi_{1-\text{ent}}}$ is fully separable while
a state $\ket{\psi_{N-\text{ent}}}$ is maximally entangled. This definitions
can be extended to mixed states via convex combination.

QFI is a fundamental quantity in the context of _phase estimation_ and is
crucial to prove that entanglement can increase the sensitivity of an
interferometer beyond the shot noise up to the Heisenberg limit. QFI for a
general observable $\hat{O}$ and a mixed probe state
$\rho=\sum_{i}p_{i}\outerproduct{\phi_{i}}{\phi_{i}}$, with $p_{i}>0$ and
$\sum_{i}p_{i}=1$, is given by

$F_{\text{Q}}[\rho,\hat{O}]=2\sum_{i,i^{\prime}}\frac{(p_{i}-p_{i^{\prime}})}{p_{i}+p_{i^{\prime}}}\absolutevalue{\matrixelement{\phi_{i}}{\hat{O}}{\phi_{i^{\prime}}}}^{2}.$
(2)

In the case of a pure state $\ket{\psi}$ the QFI has a simple expression and
is directly proportional to the variance of the operator:

$F_{\text{Q}}[\ket{\psi},\hat{O}]=4(\Delta\hat{O})^{2}\equiv
4(\expectationvalue*{\hat{O}^{2}}-\expectationvalue*{\hat{O}}^{2}).$ (3)

For separable states, $\rho_{\text{sep}}$ the
$F_{\text{Q}}[\rho_{\text{sep}},\hat{O}]$ is bounded from above [11]:

$F_{\text{Q}}[\ket{\psi}_{\text{sep}},\hat{O}]\leq
N(\lambda_{\text{max}}-\lambda_{\text{min}})$ (4)

where $\lambda_{\text{max}}$ and $\lambda_{\text{min}}$ are the maximum and
minimum eigenvalue of $\hat{O}$. This is not a fundamental limit, since it can
be surpassed by using proper entangled states. Indeed, for general probe pure
states $\ket{\psi}$ of $N$ particles, we have [11, 18]

$F_{\text{Q}}[\ket{\psi},\hat{O}]\leq
N^{2}(\lambda_{\text{max}}-\lambda_{\text{min}})^{2},$ (5)

where the equality is saturated by only maximally entangled states. This gives
the Heisenberg limit in phase estimation and quantum interferometer theory.

There is a direct relationship between ME and QFI, as it has been show in
[17]. For any $k$-producible states $\ket{\psi}_{k-\text{prod}}$ of $N$
particles the QFI is bounded by

$F_{\text{Q}}[\ket{\psi}_{k-\text{prod}},\hat{O}]\leq sk^{2}+r^{2}$ (6)

where $s=\left\lfloor N/k\right\rfloor$ (the integer part of $N/k$) and
$r=N-sk$. Therefore, a violation of (6) will indicate a $(k+1)$-particle
entanglement. The quantity $F_{\text{Q}}$ in (6) has been rescaled by a factor
$(\lambda_{\text{max}}-\lambda_{\text{min}})^{2}$, which in the case of spin-1
operators is equal to 4. By a straightforward calculation is possible to see
that this bound is saturated by the product of $s$ GHZ states of $k$ particles
and a GHZ state with the remaining $r$ particles :

$\ket{\psi}=\bigotimes^{s}_{i}\left(\frac{\ket{\lambda_{\text{max}}}^{\otimes
k}+\ket{\lambda_{\text{min}}}^{\otimes k}}{\sqrt{2}}\right)_{i}\times\\\
\times\left(\frac{\ket{\lambda_{\text{max}}}^{\otimes
r}+\ket{\lambda_{\text{min}}}^{\otimes r}}{\sqrt{2}}\right)$ (7)

If we introduce the QFI density

$f_{\text{Q}}[\ket{\psi},\hat{O}]\equiv F_{\text{Q}}[\ket{\psi},\hat{O}]/N,$
(8)

then (6) can immediately be read as

$f_{\text{Q}}[\ket{\psi}_{k-\text{prod}},\hat{O}]\leq k,$ (9)

where, for simplicity, we put the term $s=N/k$. It has been proved that
$f_{\text{Q}}>1$ is a sufficient condition for multipartite entanglement [18].

In this paper, the observable we consider are constructed by using the spin-1
operators $S_{i}^{\alpha}$, where $\alpha=x,z$, and their _non-local_
counterparts $\widetilde{S}^{\alpha}$. The latter are defined as follows:

$\begin{split}\widetilde{S}^{x}_{j}=S_{j}^{x}\left(e^{i\pi\sum_{l>j}S_{l}^{x}}\right),\\\
\widetilde{S}^{z}_{j}=\left(e^{i\pi\sum_{l<j}S_{l}^{z}}\right)S_{j}^{z}.\end{split}$
(10)

These operators have been obtained by applying a non-local unitary
transformation on the spin degrees of freedom. For more details regarding the
origin of this transformation we refer to the discussion about the AKLT model
in Appendix A.

##  III Bilinear-Biquadratic model

Figure 1: Phase diagram of BLBQ model with some remarkable points: $\theta=0$
the AF Heisenberg model, $\theta=\arctan(-1/3)$ the AKLT point and,
$\theta=\pm\pi/4$ the Takhtajan-Babujian models and Lai-Sutherland
respectively.

In this section we consider the _Bilinear-Biquadratic_ (BLBQ) model on a chain
of $N$ sites:

$H=J\sum_{i=1}^{N}\quantity[\bm{S}_{i}\cdot\bm{S}_{i+1}-\beta(\bm{S}_{i}\cdot\bm{S}_{i+1})^{2}],$
(11)

where $\bm{S}_{i}=(S_{i}^{x},S_{i}^{y},S_{i}^{z})$ is the spin-1 operator for
site $i$, $J$ is the nearest-neighbor coupling and $\beta$ is a real parameter
expressing the ratio between the bilinear and biquadratic terms. This is the
most general $SU(2)$-invariant isotropic spin-$1$ Hamiltonian with nearest-
neighbor interactions only. It is sometimes convenient to rewrite the
Hamiltonian (11) as

$H=J^{\prime}\sum_{i=1}^{N}\quantity[\cos(\theta)\bm{S}_{i}\cdot\bm{S}_{i+1}-\sin(\theta)(\bm{S}_{i}\cdot\bm{S}_{i+1})^{2}]$
(12)

where we have set $J=J^{\prime}\cos(\theta)$ and $\beta=\tan(\theta)$, with
the angular parameter $\theta$ vary in $[-\pi,\pi]$. By fixing $J^{\prime}=1$,
the phase diagram can be drawn by letting the angular parameter $\theta$ vary
in $[-\pi,\pi]$, as shown in Fig. 1.

In the following we will describe the phase of the BLBQ model and some
remarkable points.

###  III.1 Phase Diagram

The _Haldane phase_ corresponds to the region
$-\frac{\pi}{4}<\theta<+\frac{\pi}{4}$: here the system is massive, with a
unique ground state and exponentially decaying correlation functions [19]. We
recognize the antiferromagnetic Heisenberg model for $\theta=0$ [20, 21, 22].
For $\theta=-\arctan{(\frac{1}{3})}$ we recover the AKLT model, whose ground
state is a Valence-Bond State (VBS), in which each spin-1 is thought of as
made of two $1/2$-spins that couple with the spins of neighboring sites in a
singlet (entangled) state. A pictorial image of the AKLT state for a six sites
chain is given in the upper panel of Fig. 2.

The ground state has an exact description as a Matrix Product State, which is
very useful for performing exact calculations. In particular, it can be shown
that the local correlation functions have and exponential decay (see Appendix
A).

Figure 2:  Example of states of the BLBQ model: every site (light blue oval)
contains two spin-$1/2$ particles (blue dots) and each singlet state of
spin-$1/2$ particles is represented with a thick black line. _Upper panel._
Entangled pair structure of the AKLT’s ground state in the VBS representation.
_Lower panel._ An example of spin-1 dimer state in the VBS representation with
six sites.

The _Dimer phase_ corresponds to $\pi/4<\theta<3\pi/4$: the system has a two-
fold degenerate ground state and a small excitation gap [23]. The degeneracy
is due to the broken translation symmetry since neighboring spins tend to be
coupled in pairs. A good approximation of the ground state in the whole phase
is given by the the dimer state [16]:

$\ket{d}_{\pm}=\bigotimes_{i=1}^{L/2}\frac{1}{\sqrt{3}}\Big{(}\ket{+}_{2i}\ket{-}_{2i\pm
1}+\\\ +\ket{-}_{2i}\ket{+}_{2i\pm 1}-\ket{0}_{2i}\ket{0}_{2i\pm 1}\Big{)}$
(13)

which is shown in the lower panel of Fig. 2.
Haldane and Dimer phases are separated by the so-called _Takhtajan-Babujian
critical point_ , for $\theta=\pi/4$. Here the Hamiltonian is integrable by
means of Bethe Ansatz technique [24, 25] and its universality class is that of
a $SU(2)_{k}$ Wess-Zummino-Witten conformal field theory with $k=2$ and
therefore with central charge $c=3/2$ [26].

In the range $-\pi/2<\theta<-\pi/4$, there is another antiferromagnetic phase,
called the _Trimer Phase_ since the ground state tends to be invariant under
translations of three sites. This is a gapless phase [27]. At $\theta=-\pi/4$,
it is separated from the Haldane phase by a continuous phase transition. This
point corresponds to the so-called _Lai-Sutherland model_ , which has an
enhanced symmetry to $SU(3)$, the Hamiltonian being equivalent to

$\sum_{i=1}^{N-1}\bm{S}_{i}\cdot\bm{S}_{i+1}+(\bm{S}_{i}\cdot\bm{S}_{i+1})^{2}=\frac{N}{3}+\frac{1}{2}\sum_{i=1}^{N-1}\sum_{a=1}^{8}\lambda_{i}^{a}$
(14)

where $\lambda^{a}$ are the Gell-Mann matrices, the eight generators of
$SU(3)$ algebra. It is in the universality class of the $SU(3)_{k}$ Wess-
Zummino-Witten conformal field theory with $k=1$ [26, 28].

Here we will not consider the last phase present in Fig. 1, nameley the
ferromagnetic phase, which corresponds to an ordered and separable ground
state.

(a) Haldane phase. (b) Dimer and trimer phase. (c) QFI density at $N=30$.

Figure 3:  Scaling behaviours of the QFI density $f_{\text{Q}}$ at different
points of phase diagram using $\widetilde{O}^{z}$ in (16): (a) in the Haldane
phase $f_{\text{Q}}$ grows linearly, with the highest slope in correspondence
of the AKLT point $\beta=-1/3$; (b) in the dimer and trimer phase
$f_{\text{Q}}$ grows logarithmically; (c) values of $f_{\text{Q}}$ as a
function of $\beta$ with system size $N=30$; the blue circle is the AKLT point
where $f_{\text{Q}}$ is maximal, while the two blue squares correspond to the
phase-transition points.

The BLBQ model has a hidden symmetry (see Appendix), that forces to introduce
_non-local order parameters_ (NLOPs) [16] to classify all phases. NLOPs -which
are also called String Order Parameters- are defined as follows:

$\widetilde{C}^{\alpha}=\lim_{r\to\infty}\bigg{\langle}S^{\alpha}_{0}\bigg{(}\prod_{k=2}^{r-1}e^{i\pi
S_{k}^{\alpha}}\bigg{)}S^{\alpha}_{r}\bigg{\rangle}$ (15)

where $\alpha=x,y,z$. The NLOPs $\widetilde{C}^{\alpha}$ have a non-zero
expectation value only in the Haldane phase.

In the following we will examine both the expectation value and the QFI of the
nonlocal operator:

$\widetilde{O}^{z}\equiv\sum_{j=1}^{N}\widetilde{S}_{j}^{z}\;\;,\;\;\widetilde{S}^{z}_{j}\equiv\left(e^{i\pi\sum_{l<j}S_{l}^{z}}S_{j}^{z}\right)$
(16)

evaluated on the ground state $|\psi\rangle$ in the different phases of the
BLBQ model. With some algebra one finds:

$\expectationvalue*{\widetilde{O}^{z}}=\sum_{l=1}^{N}\Big{\langle}\bigg{(}\prod_{j=1}^{l-1}\Omega(j)\bigg{)}S_{l}^{z}\Big{\rangle}$
(17)

and

$\begin{split}\expectationvalue*{(\widetilde{O}^{z})^{2}}=\sum_{l=1}^{N}\expectationvalue*{(S^{z}_{l})^{2}}-2\sum_{l<m}\Big{\langle}S^{z}_{l}\bigg{(}\prod_{j=l+1}^{m-1}\Omega(j)\bigg{)}S^{z}_{m}\Big{\rangle},\end{split}$
(18)

where we have used

$\Omega(l)=e^{i\pi
S_{l}^{z}},\quad\Omega^{2}(l)=\mathbb{I}\quad\text{and}\quad
S_{l}^{z}\Omega(l)=-S_{l}^{z}.$ (19)

This expressions are used to calculate the FQI

$F_{Q}\big{[}\ket{\psi},\widetilde{O}^{z}\big{]}=\bigg{[}\bra{\psi}(\widetilde{O}^{z})^{2}\ket{\psi}-\bra{\psi}\widetilde{O}^{z}\ket{\psi}^{2}\bigg{]}$
(20)

which coincides with (3) but for the factor $4$ that we have neglected since
we are dealing with spin-1 operator with
$\lambda_{\text{max}}=-\lambda_{\text{min}}=1$.

###  III.2 Numerical results

To rewrite (18), it is useful to define the following $N\times N$ matrix:

$M=\left(\begin{smallmatrix}\expectationvalue*{(S_{1}^{z})^{2}}&\expectationvalue*{S_{1}^{z}S_{2}^{z}}&\expectationvalue*{S_{1}^{z}\Omega(2)S_{3}^{z}}&\cdots&\expectationvalue*{S_{1}^{z}\Omega(2)\cdots\Omega(N-1)S_{N}^{z}}\\\
0&\expectationvalue*{(S_{2}^{z})^{2}}&\expectationvalue*{S_{2}^{z}S_{3}^{z}}&\cdots&\expectationvalue*{S_{2}^{z}\Omega(3)\cdots\Omega(N-1)S_{N}^{z}}\\\
0&0&\expectationvalue*{(S_{3}^{z})^{2}}&\cdots&\cdots\\\
\cdots&\cdots&\cdots&\cdots&\cdots\\\
0&0&0&\cdots&\expectationvalue*{S_{N-1}^{z}S_{N}^{z}}\\\
0&0&0&\cdots&\expectationvalue*{(S_{N}^{z})^{2}}\end{smallmatrix}\right),$

where each matrix element $M_{ij}$ is given by

$M_{ij}=\begin{cases}\expectationvalue*{S_{i}^{z}\,\Omega(i+1)\cdots\Omega(j-1)\,S_{j}^{z}}&\text{if
$i\leq j$}\\\ 0&\text{otherwise.}\end{cases}$ (21)

Similarly, for the term (17) we can define the $N$-dimensional vector:

$V=\left(\expectationvalue*{S_{1}^{z}},\expectationvalue*{\Omega(1)S_{2}^{z}},\dots,\expectationvalue*{\Omega(1)\cdots\Omega(N-1))S_{N}^{z}}\right),$
(22)

such that $\expectationvalue*{\widetilde{O}^{z}}$ turns out to be the sum of
all its elements.

In this way, the QFI can be written as

$F_{\text{Q}}\quantity[\ket{\psi},\widetilde{O}^{z}]=\sum^{N}_{i=1}M_{ii}-2\sum^{N-1}_{i=1}\sum^{N}_{j>i}M_{ij}-\quantity(\sum_{i=1}^{N}V_{i})^{2},$
(23)

Simulations to compute the elements of $M$ and $V$ can be easily implemented
numerically. The states can be represented with Matrix Product States (MPSs)
and the ground states can be obtained with the DMRG algorithm. The numerical
simulations have been done using the ITensor library [29, 30, 31] and the DMRG
computations have been performed with bond dimensions up to $\chi=300$ and
truncation error cutoff set to $10^{-12}$, for a higher precision.

In order to investigate the scaling of the QFI density
$f_{\text{Q}}=F_{\text{Q}}/N$, we have looked for a function of the form
$q+bN^{\delta}$ (for the Haldane and critical points) or $q+b\ln N$ (for the
dimer and trimer phases), for system sizes up to $N=120$. However, when the
data showed a particularly flat trend, we have fitted $f_{\text{Q}}$ against a
constant function, in order to minimize the standard error on the parameters.

The results of the numerical calculations are summarized in Table 2 for the
Haldane phase and in Table 2 for the Dimer and Trimer phases. The fit and
their errors are computed using standard methods, like the one provided by
Mathematica [32].

To analyze these results, let us start from the AKLT point where the ground
state is known exactly. To calculate the QFI analytically we can exploit Lemma
2.6 of [19], extended to a string observable. Let $O$ be an observable and $N$
the system’s size; then for any $l\leq N$ such that the support of $O$ is
contained in $l$ we have

$\lim_{N\to\infty}\frac{\expectationvalue*{O}{\Omega^{N}_{\alpha\beta}}}{\innerproduct*{\Omega^{N}_{\alpha\beta}}{\Omega^{N}_{\alpha\beta}}}=\frac{\sum_{\alpha,\beta}\expectationvalue*{O}{\Omega^{l}_{\alpha\beta}}}{\sum_{\alpha,\beta}\innerproduct*{\Omega^{l}_{\alpha\beta}}{\Omega^{l}_{\alpha\beta}}}$
(24)

where $\ket*{\Omega^{N}_{\alpha\beta}}$ is one of the four ground states of
the AKLT model (see Appendix A). This gives us an operational way to
analytically calculate the terms of the QFI on the infinite volume ground
state from (23) for a finite chain. It turns out that the each diagonal term
is equal to $2/3$ while each of the $N(N-1)/2$ off diagonal terms quickly
approach to $-4/9$ (i.e. the value of NLOP (15) defined in the asymptotic
limit) when $N$ becomes larger. As the last addend in (23) is negligible, the
QFI density for a system of $N$ sites scales linearly as:

$f_{\text{Q}}(\ket{\psi_{\text{AKLT}}},\widetilde{O}^{z})\simeq\frac{2}{9}+\frac{4}{9}N$
(25)

as confirmed by numerical results in Table 2. The same argument holds for the
Heisenberg point where the asymptotic value of its NLOP is known to be $\simeq
0.36$ [33]. Furthermore, we observe that the QFI keeps a linear scaling in the
whole Haldane phase, as shown in Fig. 3a. One can notice that the slope of the
curves progressively decreases as we move away from the AKLT point.

When moving outside the Haldane phase, the linear scaling in the dimer and
trimer phase becomes sublinear, as it can be seen in Fig. 3b. In the dimer
phase, the numerical results can be compared with the analytical calculations
performed on the dimer state (13) which can be consider a good approximation,
as mentioned in Sec. III. The resulting QFI density
$f_{\text{Q}}(\ket{d},\widetilde{O}^{z})$ yields $4/3$, corresponding to a
$2$-partite entanglement structure, which is expected from the state (13)
being a two-sites product state. Then, assuming that $\widetilde{O}^{z}$ is a
good choice for the whole dimer phase, we can appreciate how good this
approximation is in the different points of this phase, by comparing the
various scaling with the exact value $4/3$. As we show in Table 2 and Fig. 3b,
we get that a good function that fits the data is of the form $q+b\log{N}$,
with $b$ that progressively decreases when $\beta$ goes to infinity.

We want to stress the crucial difference between the Haldane phase and the
dimer and trimer ones. From the point of view of QFI criterion, the
multipartite entanglement structure, in other words the $k$ in (9), grows
linearly with the system size in the Haldane phase while in the other two
phases the $k$ grows sub-linearly. This may suggests that the ground state in
the Haldane phase may not be factorizable in blocks of finite length in the
thermodynamic limit, and this can be shown using only _non-local operators_.
However we cannot have direct information on the exact value of $k$ using only
$\widetilde{O}^{z}$, because we cannot be sure that this is the operator
saturating the ground state QFI.

Let us now analyse the scaling behaviour at the transition points $\beta=\pm
1$. The spin-spin correlations are asymptotically given by the fundamental WZW
primary fields, leading to the prediction that, in an infinite system, the
dominant antiferromagnetic correlations decay as a power law:

$\langle
S^{\alpha}_{0}S^{\alpha}_{r}\rangle\sim\frac{(-1)^{r}}{\absolutevalue{r}^{\eta}}$
(26)

where $\eta=2\Delta$ and the scaling dimension $\Delta=h+\bar{h}$ can be
obtained from the primary field scaling dimension for a general $SU(n)$ level
$k$ WZW model [34]:

$h=\bar{h}=\frac{n^{2}-1}{2n(n+k)}$ (27)

As we said in the previous sections, $\beta=\pm 1$ ($\theta=\pm\pi/4$) are
described by $SU(2)_{2}$ and $SU(3)_{1}$ conformal theories which means that
their values of $\eta$ are equal to $3/4$ and $4/3$ respectively. We recover
this power-law scaling of correlators both for string and local operators as
we show in Fig. 4.

Takhtajan-Babujian ($\beta=1$)
Lai-Sutherland ($\beta=-1$)

Figure 4:  Power-law decay of correlation functions by using both string (in
blue) and local (in green) operators in the Takhtajan-Babujian (upper panel)
and Lai-Sutherland (lower panel) models. The dots are the computed value,
while the grey line is obtained fit.

For $\beta=1$, the numerical data display small oscillations between $N$ even
and odd, due to the double degeneracy that emerges in the dimer phase. To
increase the accuracy of the fitting, we have decided to consider only the
odd-numbered sites, this however does not modify the value of the exponents in
the thermodynamic limit since these oscillations tend to zero as $N$
increases.

As shown in [12], the QFI density of one-dimensional models at the critical
point is supposed to scale as $f(O^{\alpha})\sim N^{\delta_{Q}}$ (up to a non-
universal prefactor and subleading corrections) with
$\delta_{Q}=1-2\Delta^{\alpha}$, where $\Delta^{\alpha}$ the scaling dimension
of the operator $O^{\alpha}$. We can recover this result from our approach and
numerical data as well. Indeed, considering that the first sum in (23) goes as
$\sim N$ (so it brings just a constant contribution in $f$) and neglecting $V$
(because we are at the critical point), the only relevant contribution is
given by the sum of the off-diagonal terms in the $M$ matrix. Exploiting the
(26) in the continuum limit, we get:

$\sum_{r^{\prime}=1}^{N-1}\sum_{r>r^{\prime}}^{N}\langle
S^{\alpha}_{r^{\prime}}S^{\alpha}_{r}\rangle\longrightarrow\int_{1}^{N}dr^{\prime}\int_{r^{\prime}}^{N}\frac{dr}{r^{\eta}}\sim
N^{2-\eta}$ (28)

so that:

$f_{\text{Q}}(O^{\alpha})\sim N^{1-2\Delta^{\alpha}}$ (29)

The same holds for string operators up to a non-universal prefactor and
subleading corrections. It is evident now why we get the expected numerical
value $\delta\simeq\delta_{Q}=1-2\Delta=1/4$ for the string magnetization, as
reported in Table 2. A similar reasoning can be put forward for the
calculation of $f_{\text{Q}}(O^{z}_{\text{st}})$ of the local staggered
magnetization operator along $z$-axis, defined as

$O^{z}_{\text{st}}=\sum_{j=1}^{N}(-1)^{j}S_{j}^{z}.$ (30)

Our numerical results for the calculation of the QFI density for
$O^{z}_{\text{st}}$ yield: $q=$-3.770\pm 0.002$$, $b=$3.201\pm 0.001$$ and
$\delta=$0.244\pm 0.001$$. Thus we are able to read the critical exponent of
the operator from its QFI.

At the Lai-Sutherland point $\beta=-1$, the numerical data display small
oscillations with a periodicity of three sites, due to the trimer
configuration that merges for $\beta<-1$. Unfortunately, from the data we
observe what is mostly probable a flat trend, but we are not able to
distinguish a linear fit from a one that decreases exponentially or, like it
should be in this case, as a power law with a negative exponent. We believe
that prefactors and subleading terms that depends on $N$ might contribute to
mask the predicted behaviour at criticality.

BLBQ model, Haldane phase
---
$f_{\text{Q}}(\ket{\psi_{\beta}},\widetilde{O}^{z})=q+bN^{\delta}$
$\beta$ | $q$ | $b$ | $\delta$
$-1/3$ | $0.225\pm 0.003$ | $0.4441\pm 0.0001$ | $1.0002\pm 0.0001$
$0$ | $0.35\pm 0.05$ | $0.355\pm 0.002$ | $1.002\pm 0.003$
$1/3$ | $1.122\pm 0.009$ | $0.197\pm 0.004$ | $0.9999\pm 0.0005$
$2/3$ | $1.55\pm 0.04$ | $0.111\pm 0.02$ | $0.997\pm 0.001$
$1$ | $-3.632\pm 0.004$ | $3.132\pm 0.002$ | $0.252\pm 0.001$

Table 1: Numerical values of the fitting parameters at different points in the
Haldane phase of the BLBQ model, where the QFI density has been fitted against
a power law.

BLBQ model
---
$f(\ket{\psi_{\beta}},\widetilde{O}^{z})=q+b\ln N$
Dimer phase | Trimer phase
$\beta$ | $q$ | $b$ | $\beta$ | $q$ | $b$
$2$ | $0.81\pm 0.04$ | $0.58\pm 0.01$ | $-2$ | $0.98\pm 0.06$ | $0.19\pm 0.01$
$4$ | $1.03\pm 0.01$ | $0.405\pm 0.003$ | $-4$ | $1.08\pm 0.06$ | $0.12\pm 0.01$
$8$ | $1.34\pm 0.05$ | $0.24\pm 0.01$ | $-8$ | $1.11\pm 0.05$ | $0.09\pm 0.01$
$\infty$ | $1.39\pm 0.04$ | $0.18\pm 0.01$ |  |  |

Table 2: Numerical values of the fitting parameters at different points in the
dimer and trimer phases of the BLBQ model, where the QFI density has been
fitted against a logarithm function.

##  IV XXZ spin-1 model

###  IV.1 Phase diagram

The XXZ spin-1 chain is a well-studied quantum system that exhibits an
interesting phase diagram as a function of the anisotropy parameter $J_{z}$.
It has the following Hamiltonian:

$H=\sum_{i=1}^{N-1}J_{xy}(S^{x}_{i}S^{x}_{i+1}+S^{y}_{i}S^{y}_{i+1})+J_{z}(S^{z}_{i}S^{z}_{i+1})$
(31)

where we take $J_{xy}=1$ and let $J_{z}$ vary. It can also be considered as a
particular case of the so-called $\lambda-D$ model [16], that includes also an
isotropy term of the form $\sum_{i=1}^{N}D(S^{z}_{i})^{2}$.

The quantum phase diagram of this Hamiltonian has been extensively studied
[35]. It includes the _Haldane phase_ for $0<J_{z}\sim 1$. A second-order
phase transition occurs from the Haldane phase to an antiferromagnetic (AF)
phase that belongs to the same universality class of the 2D Ising model with
central charge $c=1/2$. Various numerical techniques, including Monte-Carlo
[36] and DMRG [37, 38], have determined the critical value:
$J_{z}^{(\text{IS})}=1.186$. A _Berezinskii-Kosterlitz-Thouless_ (BKT)
transition occurs at $J_{z}^{(\text{BKT})}=0$ between the Haldane phase and a
gapless disordered XY phase ($-1<J_{z}<0$). The value of
$J_{z}^{(\text{BKT})}$ is theoretically predicted to be exactly zero, using
bosonization techniques [39]. Numerically, this has been verified via finite-
size scaling [40, 41] and DMRG [37]. The entire XY phase (including the BKT
transition point) is a critical phase which has conformal symmetry with
central charge $c=1$. Finally, at $J_{z}=-1$, a first-order phase transition
from the XY phase to a ferromagnetic (F) phase takes place [35, 38, 42]. We
will not examine in detail such ferromagnetic phase in the following.

Figure 5:  Trend of QFI densities in the phase diagram of the XXZ model of
size $N=30$, with the string magnetizations $\widetilde{O}^{z}$,
$\widetilde{O}^{x}$, and the local magnetization $O^{x}_{\text{st}}$. From
left to right: ferromagnetic (F) phase, XY gapless phase, Haldane phase and
antiferromagnetic (AF) phase. The critical points are located at $J_{z}=-1$,
$J_{z}^{(\text{BKT})}=0$ and $J_{z}^{(\text{IS})}=1.186$.

###  IV.2 Numerical results

(a) Staggered magnetization $O^{x}_{\text{st}}$ (b) String operator
$\widetilde{O}^{x}$ (c) String operator $\widetilde{O}^{z}$

Figure 6:  Scaling behaviours of the QFI density of different operators at
some points of interest: (a) staggered magnetization $O^{x}_{\text{st}}$
(shades of red); (b) $x$-string operator $\widetilde{O}^{x}$ (shades of
green); (c) $z$-string operator $\widetilde{O}^{z}$ (shades of blue). Notice
the abrupt change in behaviour for the string operators $\widetilde{O}^{x}$
and $\widetilde{O}^{z}$ from $J=1$ to $J=1.186$, which can also be seen in
Fig. 5.

Given the symmetries of the Hamiltonian, we consider the scaling behaviour of
the QFI density of local and string operators along the $x$ and $z$ axes,
including the staggered ones. The ones that show an extensive scaling, at
least in some phases of the model, are the following:

$\widetilde{O}^{z}=\sum_{i=1}^{N}\widetilde{S}_{i}^{z},\quad\widetilde{O}^{x}=\sum_{i=1}^{N}\widetilde{S}_{i}^{x},\quad
O^{x}_{\text{st}}=\sum_{i=1}^{N}(-1)^{i}S_{i}^{x},$ (32)

where, as usual, the operators with the tilde symbol are string operators.
Similarly to the previous section, the numerically computed QFI density
$f_{\text{Q}}$ is fitted against the function $f_{\text{Q}}=q+bN^{\delta}$, or
with a constant if the data presents an extremely flat trend.

In Fig. 5 we plot the shapes of the QFI densities of the operators (32) in the
different phases of the model for a chain with $N=30$ sites. The results of
the fitting of the scaling with $N$ are given in Tables 5, 5 and 5 and some
details of the scaling are reported in Fig. 5. Let’s analyze each operator
below.

The operator $O^{x}_{\text{st}}$ takes its maximal value close to the F-XY
transition point and then decreases progressively moving toward the Haldane
phase. In particular, analyzing its scaling with $N$ (see Fig. 6a and Table
5), $f_{\text{Q}}$ reveals a power-law behaviour in the XY phase with the
coefficient $\delta=0.8376\pm 0.0001$ at $J_{z}=-1/2$ which gradually reduces
(e.g. $\delta=0.7574\pm 0.0002$ at $J_{z}=0$) until it vanishes for
$J_{z}\gtrsim 1$.

Regarding the string operators (see Tables 5 and 5), it is possible to observe
that $f_{\text{Q}}(\widetilde{O}^{x})$ has a power-law scaling in the whole XY
phase (including $J_{z}=0$) where the $f_{\text{Q}}(\widetilde{O}^{z})$
appears to be almost flat, ($\delta=0.138\pm 0.003$). In the Haldane phase,
the FQI for both these operators shows a linear scaling ($\delta\simeq 1$)
with a slope that increases with $J_{z}$, reaching the maximal values at
$J_{z}=0.8$ and $J_{z}=1$ respectively. For $J_{z}=1$ we recover the
Heisenberg model where both have the same scaling coefficients as expected in
an isotropic point.

The data on QFI can be used to extract information about the critical
exponents of relevant operators at phase transition points and about
correlation functions in general. At the critical point $J_{z}^{(\text{IS})}$,
we predict that the scaling dimension of the order parameter is $\Delta=1/8$,
in accordance with the universality of the 2D Ising model, since
$\delta=1-2\Delta\simeq 3/4$. This holds true for the string order operator
$\widetilde{O}^{x}$, see Table 5, and the local staggered magnetization
$O^{z}_{\text{st}}$. The latter is defined similarly to $O^{x}_{\text{st}}$ in
(32), for which we obtained $\delta=0.76\pm 0.01$.

More generally, we can consider the asymptotic behaviour of local staggered
and string correlation functions

$\begin{split}C^{\alpha}_{\text{st}}(r)&=(-1)^{r}\expectationvalue*{S_{0}^{\alpha}S_{r}^{\alpha}},\\\
\widetilde{C}^{\alpha}(r)&=\left\langle
S^{\alpha}_{0}\bigg{(}\prod_{k=2}^{r-1}e^{i\pi
S_{k}^{\alpha}}\bigg{)}S^{\alpha}_{r}\right\rangle\end{split}$ (33)

which are known to have the following behavior for large $r$ in the (massive)
Haldane phase [43]:

$C^{\alpha}=a_{0}\frac{e^{-\frac{r}{a_{1}}}}{\sqrt{r}},\qquad\widetilde{C}^{\alpha}=a_{2}+a_{0}\frac{e^{-\frac{r}{a_{1}}}}{r^{2}}$
(34)

where $a_{0}$, $a_{1}$ and $a_{2}$ are fitting parameters and $\alpha=x,z$ as
usual, while at the transition point, they scale algebraically:

$C^{z}=\widetilde{C}^{x}=\frac{a_{0}}{r^{1/4}},\quad
C^{x}=a_{0}\frac{e^{-\frac{r}{a_{1}}}}{r^{1/4}},\quad\widetilde{C}^{z}=a_{2}+\frac{a_{0}}{r^{2}}.$
(35)

The data reported in the Tables 5, 5 and 5 and Fig. 6 of the our fitting
parameters of $f_{\text{Q}}$ are in agreement with these theoretical
predictions. In order to understand the results, two comments are necessary.

The first one is that in the Haldane phase and at the critical points the only
relevant contribution to the QFI density is due to (18), i.e. the $M$ matrix
made by the spin-spin correlators. The second one is that, as we said
previously for the BLBQ, from our data it is not possible to distinguish the
flat scaling of $f_{\text{Q}}$ from an exponential or power-law decay with
$\delta<0$. Then, considering the correlations (34) and (35), we can
understand that for string operators in the Haldane phase, the elements
$M_{ij}$ are going to approach $a_{2}$. This leads to a $f_{\text{Q}}$ that
scales linearly, with the slope $b\simeq a_{2}$. From our computations we get
$\delta$ equal to $0.757\pm 0.001$ and $0.727\pm 0.001$ for
$O_{\text{st}}^{z}$ and $\widetilde{O}^{x}$, respectively, which is comparable
to $1-\eta$ as expected.

Finally, when $-1<J_{z}<0$, the system is in the XY phase. In this extended
area of critically, also called “critical fan”, the Hamiltonian can be
replaced by the Hamiltonian of a Gaussian model [44], which admits two primary
operators with conformal dimensions:

$\Delta_{1}=\frac{1}{8},\qquad\Delta_{2}=\frac{1}{4}\chi(J_{z}),$ (36)

where $\chi$ is a function of the coupling $J_{z}$ such that $\chi(0)=1/2$ and
$\chi(-1)=0$. The explicit form of the function $\chi$ depends on the details
about how the lattice model can be mapped to the Gaussian model at
criticality. This means that there exists one operator for which the critical
index $\delta$ of QFI densities will be constantly $3/4$ and one with varying
between $3/4$ and $1$, respectively. We identify such operators with
$\widetilde{O}^{x}$ and $O^{x}_{\text{st}}$, respectively, as it suggested by
the data of Tables 5 and 5: at $J_{z}=0$ the values of their fitting
parameters are extremely close to each other and close to $0.75$; moving
toward $J_{z}=-1/2$, $f_{\text{Q}}(\widetilde{O}^{x})$ remains fixed to a
similar value ($\delta=0.745\pm 0.002$) while
$f_{\text{Q}}(O^{x}_{\text{st}})$ has $\delta=0.8376\pm 0.0001$ and the latter
continues to increase as suggested by Fig. 5.

XXZ model, staggered magnetization
---
$f\big{(}\ket{\psi_{(J_{z})}},O^{x}_{\text{st}}\big{)}=q+bN^{\delta}$
$J_{z}$ | $q$ | $b$ | $\delta$
$-1/2$ | $0.231\pm 0.003$ | $0.7041\pm 0.0002$ | $0.8376\pm 0.0001$
$0$ | $0.138\pm 0.003$ | $0.797\pm 0.001$ | $0.7574\pm 0.0002$
$1/2$ | $-3.5\pm 0.5$ | $2.8\pm 0.004$ | $0.43\pm 0.01$
$1$ | $3.77\pm 0.06$ |  |
$1.186$ | $2.199\pm 0.009$ |  |

Table 3: Numerical values of the fitting parameters of the QFI density for the
staggered magnetization $O^{x}_{\text{st}}$ at different point of the XXZ
model.

XXZ model, $x$-string operator
---
$f(\ket{\psi_{(J_{z})}},\widetilde{O}^{x})=q+bN^{\delta}$
$J_{z}$ | $q$ | $b$ | $\delta$
$-1/2$ | $0.11\pm 0.04$ | $0.829\pm 0.009$ | $0.745\pm 0.002$
$0$ | $0.131\pm 0.001$ | $0.7992\pm 0.0002$ | $0.7570\pm 0.0001$
$1/2$ | $7.2\pm 0.3$ | $1.26\pm 0.03$ | $0.996\pm 0.002$
$1$ | $0.16\pm 0.05$ | $0.356\pm 0.004$ | $1.008\pm 0.002$
$1.186$ | $2.74\pm 0.05$ | $1.65\pm 0.01$ | $0.727\pm 0.001$

Table 4: Numerical values of the fitting parameters of the QFI density for the
$x$-string operator $\widetilde{O}^{x}$ at different points of the XXZ model.

XXZ model, $z$-string operator
---
$f(\ket{\psi_{(J_{z})}},\widetilde{O}^{z})=q+bN^{\delta}$
$J_{z}$ | $q$ | $b$ | $\delta$
$-1/2$ | $0.989\pm 0.003$ |  |
$0$ | $-3.1\pm 0.1$ | $5.8\pm 0.1$ | $0.138\pm 0.003$
$1/2$ | $1.21\pm 0.04$ | $0.058\pm 0.004$ | $1.03\pm 0.01$
$1$ | $0.16\pm 0.05$ | $0.356\pm 0.004$ | $1.008\pm 0.002$
$1.186$ | $0.489\pm 0.007$ |  |

Table 5: Numerical values of the fitting parameters of the QFI density for the
$z$-string operator $\widetilde{O}^{z}$ at different points of the XXZ model.

##  V Conclusions and outlooks

In this paper we have shown how QFI is able to detect multipartite
entanglement (ME) in spin-1 chains with short range interactions. A key aspect
in these calculations is the use of string operators whereas the QFI relative
to local operators fails to detect ME especially in the topological phases of
these models, i.e the Haldane phase. For the BLBQ model, given the symmetries
of the Hamiltonian, we chose the string magnetization along $z$ and obtained
an extensive behavior in the topological phase, signaling the divergence of ME
with the system size. The same applies to the Haldane phase of XXZ model as
well.

In the dimer and trimer phases we found a sublinear behaviour; in particular
for the dimer phase we also propose to use QFI density to estimate how well
the 2-sites product state is approximating the various ground states in this
phase. Furthermore, we recover the expected power-law scaling of the QFI
density for these 1D models in the critical phases. In fact, by knowing the
critical exponent $\eta$ of the correlators or the scaling dimension $\Delta$
of the operator with which the QFI is calculated, it is possible to predict
how $f_{\text{Q}}$ will scale at these critical points: $\delta=1-2\Delta$.

From numerical simulation we obtained $\delta\simeq 0.25$ in the Takhtajan-
Babujian point of BLBQ model and $\delta\simeq 0.75$ in the AF-Haldane
transition point of XXZ model as expected. Throughout the “critical fan” (XY
phase) of the XXZ model, we observe a power-law behavior of $f_{\text{Q}}$
with two different trends of $\delta$: one fixed at the constant value of
$3/4$ (string operator $\widetilde{O}^{x}$), the other varying between $3/4$
and $1$ (staggered magnetization $O^{x}_{\text{st}}$) in analogy to what was
done in [44].

In the light of these promising results, it would be interesting to
investigate whether it is feasible to use it for systems with more complicated
degrees of freedom, such as models with higher symmetry groups [45] or with
long range interactions [46].

###### Acknowledgements.

The authors would like to thank D. Vodola and S. Tibaldi for the helpful
discussions. The work is partially supported by INFN through the project
QUANTUM. E.E. is also supported by the the QuantERA 2020 Project QuantHEP.

##  Appendix A The AKLT model

The AKLT model is the projection point at $\beta=-1/3$, where the Hamiltonian
can be expressed as a sum over the projection operators $P_{j}(i,i+1)$. Each
projector acts on a pair of interacting spins for a given value of the total
spin $j=0,1,2$. Thus, it can be written as:

$H_{\text{AKLT}}=-\frac{2}{3}NJ+2J\sum_{i=1}^{N}P_{2}(i,i+1)$ (37)

where

$P_{2}(i,i+1)=\frac{1}{3}+\frac{1}{2}\quantity(\bm{S}_{i}\cdot\bm{S}_{i+1}+\frac{1}{3}(\bm{S}_{i}\cdot\bm{S}_{i+1})^{2}).$
(38)

As shown in [19], the system can be thought of as made up of two spin-$1/2$
variables for each site. By introducing the _valence bond basis_ , it is
possible to build the ground state, called a _valence bond solid_ (VBS), so
that in the chain there is always a bond between two neighboring spins (see
upper panel of Fig. 2).

The VBS state $\ket{\text{VBS}}$ satisfies

$P_{2}(i,i+1)\ket{\text{VBS}}=0\qquad\forall i.$ (39)

In the spin-$1/2$’s computational basis $\psi_{1}=\ket{0}$,
$\psi_{2}=\ket{1}$, we can construct an orthogonal basis for the $s=1$ state
space, by taking the symmetrized tensor products:

$\psi_{\alpha\beta}=\frac{1}{\sqrt{2}}\quantity(\psi_{\alpha}\otimes\psi_{\beta}+\psi_{\beta}\otimes\psi_{\alpha})$
(40)

Then, in order to contract a pair of spin-$1/2$’s to form a singlet, we use
the Levi-Civita tensor of rank two:

$\Omega_{\alpha\beta}=\epsilon^{\gamma\delta}\psi_{\alpha\gamma}\otimes\psi_{\delta\beta},$
(41)

where the indices $\alpha$ and $\beta$ refer to the outer spin-$1/2$’s. It is
now easy to generalize the construction for a chain of length $N$:

$\Omega_{\alpha\beta}=\epsilon^{\beta_{1}\alpha_{2}}\cdots\epsilon^{\beta_{N-1}\alpha_{N}}\psi_{\alpha\beta_{1}}\otimes\psi_{\alpha_{2}\beta_{2}}\otimes\cdots\otimes\psi_{\alpha_{N}\beta}.$
(42)

The AKLT model has exponentially decaying correlations and this applies to the
whole Haldane phase. In fact, this can be shown by computing the two-point
correlation function in the limit $N\to\infty$, which yields:

$\lim_{N\to\infty}\bra{\Omega}S_{0}^{a}S_{r}^{b}\ket{\Omega}=\delta^{ab}(-1)^{r}\;\frac{4}{3}\;3^{-r}.$
(43)

showing, as anticipated, an exponentially decaying correlation function with
correlation length $\xi=\ln(3)^{-1}$. Therefore, one may conclude that there
is no order in this phase but, as we will see, a different kind of hidden
order is actually there. We are going to show this fact on the valence bond
state.

As it can be easily understood from Fig. 2, in a finite chain the ground state
of AKLT model is four-fold degenerate due to the effective free spin-$1/2$’s
at the boundaries. Let us write the ground state of AKLT as $\Phi_{\sigma}$,
where $\sigma$ is a string of $+$’s, $-$’s and $0$’s so that $\Phi_{\sigma}$
can be expressed as a tensor product of a single site states $\ket{+}$,
$\ket{-}$ and $\ket{0}$. If the first spin-$1/2$ of the chain is in the
$\ket{\uparrow}$ state, then for the first site we cannot have a $\ket{-}$
state but only $\ket{+}$ or $\ket{0}$. In the latter case we still must have
the first non-zero character to be a $+$ in $\sigma$ in order to satisfy the
construction of the valence bond state. It can be verified that there has to
be the same number of $+$’s and $-$’s alternating all along the $\sigma$
string, with no further restrictions on the number of $0$’s between them.

Therfore, a typical allowed state $\Phi_{\sigma}$ in the AKLT model could look
like this:

$\Phi_{\sigma}=\ket{000+-0+-+0-+0-+-0}$ (44)

A look at (44) reveals that is a sort of Néel order (antiferromagnetic order)
if we ignore the $0$’s. Still, we cannot predict what two spins in two distant
sites will be, as we have no control on the number of the $0$’s. Indeed there
is no _local_ order parameter that can be found to be non-zero in the Haldane
phase and that can be used to distinguish this phase from the others. But,
there is actually a _non-local_ order parameter, the string order parameter,
that is able to reveal the hidden order of the Haldane phase.

In order to see how we can arrive at its definition, let us to introduce the
non-local unitary transformation

$U=\prod_{k=1}^{N}\prod_{j=1}^{k-1}\exp\bigg(i\pi
S^{z}_{j}S^{x}_{k}\bigg{missing}),$ (45)

where $N$ is the number of sites, such that Consider a typical AKLT state
$\Phi_{\sigma}$, for example (44). On this state, the operator $U$ acts as

$U\Phi_{\sigma}=(-1)^{z(\sigma)}\Phi_{\bar{\sigma}},$ (46)

where $z(\sigma)$ is the number of $0$ characters in odd sites and
$\bar{\sigma}$ is the new transformed string. It is defined as follows:

  * •

if $\sigma_{i}=+$ (or $-$) and the number of non zero characters to the left
of the site $i$ is odd then $\bar{\sigma}_{i}=-$ (or $+$).

  * •

otherwise $\sigma_{i}=\bar{\sigma}_{i}$

where $\sigma_{i}$ is the $i$-th character of the string $\sigma$. In
particular, if we apply this transformation on the allowed state (44), it
becomes:

$U\Phi_{\sigma}=\ket{000++0+++0++0+++0}.$ (47)

Then this unitary transformation aligns all the non-zero spins i.e. if the
first non-zero character is $+$ (or $-$) all the other non-zero characters
become $+$ (or $-$). It is also evident that $U^{-1}=U$.

Under the action of $U$, the spin operators transform as follows:

$\begin{split}\widetilde{S}^{x}_{j}&=US_{j}^{x}U^{\dagger}=S_{j}^{x}\left(e^{i\pi\sum_{l>j}S_{l}^{x}}\right),\\\
\widetilde{S}^{y}_{j}&=US_{j}^{y}U^{\dagger}=\left(e^{i\pi\sum_{l<j}S_{l}^{z}}\right)S_{j}^{y}\left(e^{i\pi\sum_{l>j}S_{l}^{x}}\right),\\\
\widetilde{S}^{z}_{j}&=US_{j}^{z}U^{\dagger}=\left(e^{i\pi\sum_{l<j}S_{l}^{z}}S_{j}^{z}\right).\end{split}$
(48)

Notice that the local operators have been mapped onto non-local operators, as
they contain a sum of spin operators acting on different sites. This is not
surprising, given that $U$ itself is a non-local unitary transformation.

It is reasonable to expect that also the local Hamiltonian $H$ is mapped onto
a non-local one $\widetilde{H}=UHU^{-1}$, but it turns out that
$\widetilde{H}$ is still, in fact, local:

$\widetilde{H}=J\sum_{j}\big{[}h_{j}+\beta(h_{j})^{2}\big{]},$ (49)

where

$h_{j}=-S^{x}_{j}S^{x}_{j+1}+S_{j}^{y}e^{i\pi\left(S_{j}^{z}+S^{x}_{j+1}\right)}S^{y}_{j+1}-S^{z}_{j}S^{z}_{j+1}$
(50)

The transformed Hamiltonian $\widetilde{H}$ still has the same symmetries of
$H$, but they may not be local anymore. Actually, the only local symmetry of
$H$ is related to its invariance under rotations of $\pi$ about each
coordinate axis. This symmetry group is equivalent to
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$: indeed the product of two
$\pi$-rotations about two different axes produce a $\pi$\- rotation about the
third one.

It is possible to prove [19] that at the AKLT point the transformed
Hamiltonian has four ground states, which are product states and break such
symmetry. These four degenerate ground states of $H_{\text{AKLT}}$ converge to
a single ground state in the infinite volume limit. The same is not true for
the ground states of $\widetilde{H}_{\text{AKLT}}$, as they converge to four
distinct states in the infinite volume limit, even though the two Hamiltonians
are related by a unitary transformation. In a sense, the non-locality of the
transformation $U$ does not guarantee a one-to-one correspondence between the
ground states in the infinite volume limit.

Finally, we can understand the role of the string order parameter (48). In
fact, it is straightforward to verify that

$S^{\alpha}_{0}\bigg{(}\prod_{k=2}^{r-1}e^{i\pi
S_{k}^{\alpha}}\bigg{)}S^{\alpha}_{r}=-U^{-1}S_{0}^{\alpha}S_{r}^{\alpha}U.$
(51)

This shows that the NLOPs in (15) reveal the ferromagnetic order in the
language of the non-local spins (48) or, equivalently, the breaking of the
hidden symmetry in the original system. Such a symmetry breaking holds in the
whole Haldane phase, not just the AKLT model. Indeed, in the dimer phase the
symmetry is completely unbroken and the string order parameter (15) will
vanish for every $\alpha$.

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