# State Space Geometry of the Spin-1 Antiferromagnetic Heisenberg Chain

James Lambert<EMAIL_ADDRESS>Erik S. Sørensen<EMAIL_ADDRESS>Department of Physics & Astronomy, McMaster University 1280 Main St. W.,
Hamilton ON L8S 4M1, Canada

###### Abstract

We study the phase diagram of the spin-1 antiferromagnetic Heisenberg chain
with uniaxial anisotropy and applied magnetic field in terms of the genuine
multipartite entanglement as witnessed by the mean quantum Fisher information
density. By generalizing the manifold studied in [1] to the many body case for
spin 1, we connect the state space curvature in the vicinity of the ground
state of the Heisenberg chain to the genuine multipartite entanglement. Our
analysis demonstrates that the quantum critical points and SPT phase exhibit
large positive state space curvature. We further show that the entanglement in
the SPT phase is enhanced by the presence of uniaxial anisotropy, and
undiminished in the presence of uniform magnetic fields. The magnon condensate
phase induced by large fields is shown to emanate from the Gaussian critical
point, and exhibits massive multipartite entanglement over a robust region of
the parameter space.

##  I Introduction

The degree to which two probability distributions may be distinguished from
one another in some fixed set of measurements induces a natural notion of
distance on the state space [2]. This notion of distance and the geometrical
structure it gives rise to is sometimes termed _information geometry_ , with
distinguishability being quantified by the Fisher information matrix,

$F_{\mu\nu}\\{p(x|\lambda)\\}=\int\frac{1}{p}\partialderivative{p}{\lambda_{\mu}}\partialderivative{p}{\lambda_{\nu}}\differential
x,$ (1)

where $p(x|\lambda)$ is a probability distribution (which we hereafter refer
to as a state) that depends on some parameterization, $\vec{\lambda}$, of the
state space. $F_{\mu\nu}$ quantifies how distinguishable the distribution
$p(x|\vec{\lambda})$ is from the distribution,
$p(x|\lambda+\differential\lambda)$ [3]. The quantum generalization holds for
both pure [4] and mixed [5] states, and is termed quantum Fisher information
(QFI) with the density matrix $\hat{\rho}$ taking the place of $p(x|\lambda)$,
and the quantum expectation value replacing the integral. The associated
geometrical structure is termed _quantum information geometry_ or simply
_quantum geometry_ (see [6] for a complete introduction to the subject). In
this work we are specifically interested in the case of pure states of spin-S
with unitary parameterizations, for which the QFI matrix is proportional to
the connected covariance [7],

$Q_{\mu\nu}\\{\psi\\}=\frac{A}{S^{2}}\text{Cov}_{\psi}(\Lambda_{\mu},\Lambda_{\nu})$
(2)

where,

$\text{Cov}_{\psi}(\Lambda_{\mu},\Lambda_{\nu})=\langle\Lambda_{\mu}\Lambda_{\nu}\rangle-\langle\Lambda_{\mu}\rangle\langle\Lambda_{\nu}\rangle$
(3)

with $\Lambda_{\mu}$ and $\Lambda_{\nu}$ the hermitian operators that generate
the unitary parameterization for the state $\ket{\psi}$ via,

$\ket{\psi(\lambda)}=e^{-i\lambda\Lambda}\ket{\psi},$ (4)

The constant, $A$, hereafter the _metric constant_ , may be thought of as a
normalization (see Sec. I.1). The QFI matrix can be decomposed into a real,
symmetric, Riemannian metric $g$ [8], and an anti-symmetric symplectic tensor
$\Gamma$ (the Berry curvature [9]),

$Q_{\mu\nu}=g_{\mu\nu}+i\Gamma_{\mu\nu},$ (5)

While the trace of the QFI matrix for a suitable choice of parameterization
obeys the following bound [10, 11],

$\bar{f}\coloneqq\frac{1}{3N}\Tr{Q}>2mA.$ (6)

for states that are $(m+1)$-partite entangled.

In this work, we calculate the QFI matrix associated with the staggered
magnetization operator, and map out the phase diagram of the spin-1
antiferromagnetic Heisenberg chain as a function of a uniform applied field
and the uniaxial anisotropy. Using the metric $g$, hereafter the _quantum
metric_ , we compute the volumes of spheres parameterized by the strength and
orientation of a staggered magnetic field. By computing the scaling of these
spheres with radius, we compute the curvature of the state space in the
vicinity of the ground state, demonstrating that the entanglement in the SPT
Haldane phase and at the critical points appears to be associated with the
presence of positive curvature in the state space, thus demonstrating a
connection between entanglement and state space curvature.

Since the QFI quantifies the distinguishability of probability distributions,
it can be naturally expressed in terms of the dynamical response, of which Eq.
2 is the zero temperature limit [12, 13]. This has allowed for the direct
measurement of GME in recent inelastic neutron scattering experiments [11,
14].

The QFI matrix plays a central role in quantum metrology and parameter
estimation. It also appears in the generalizations of the uncertainty relation
that allow for the formulation of the time-energy uncertainty principle (time
having no associated Hermitian operator and thus being a parameter that must
be inferred by other measurements) [15, 16]. Most importantly for our
purposes, the mean trace of the QFI matrix quantifies the degree to which the
state $\psi$ has genuine multipartite entanglement (GME) [10]. This connection
is especially interesting as it holds even at finite temperatures [17] (we
discuss the details in Sec. III.1). Entries of the QFI matrix have now been
studied in a wide range of models [18, 19, 20, 21, 22, 23, 24, 25]. These
studies have emphasized either parameterizations that are linear in the spin
degrees of freedom, or in the case of [25], parameterizations using non-local
operators that exhibit super extensive scaling in topologically non trivial
phases. The full QFI matrix that would correspond to these parameterizations
has been relatively less explored.

The proof connecting the QFI to GME is for operators linear in the spin
degrees of freedom [10]. This is advantageous from the perspective of response
function experiments, since in a typical scattering experiment one imagines
that that scattering beam is exploring the response of the state to an applied
operator that is often linear in the local degrees of freedom. From a
theoretical perspective, however, we can explore parameterizations
corresponding to a wide range of operators which would not necessarily be
linear in the local degrees of freedom and hence have no established
connection to the GME. For instance, the fidelity susceptibility (FS) [26, 27,
28, 29] is identical to the QFI up to a constant of proportionality [30]. In
this context one considers the overlap in the ground state of a Hamiltonian
depending on a parameter $\lambda$ with the ground state at
$\lambda+\differential\lambda$, where the parameter coupled to $\lambda$ might
be non-local in the spin degrees of freedom. Thus, the FS is not generally
known to have a relationship with the GME. An analogous relationship to
response functions holds true of the FS [31].

In the context where quantum geometry is studied for driving parameters of the
Hamiltonian one tends to talk about the quantum geometric tensor (QGT) [32],
the entries of which are the FS and which, away from level crossings, is the
same object as the QFI matrix defined in Eq. 2. The form of the QFI matrix in
terms of the covariance may thus be thought of as an adiabatic form of the QFI
[7]. In the vicinity of the critical point the QGT and QFI matrix display
different scaling relationships [32, 12]. In this work we focus on the QFI
matrix in this adiabatic form, since determining the entire spectrum is
generally a very hard problem for all but a handful of exactly solvable
models. If, however, the operators generating the transformations are relevant
in a renormalization group sense then both the QFI and QGT will exhibit
divergences at the critical point.

The Hilbert space of a many body system is exponentially large in the number
of degrees of freedom. Computing the QFI matrix for a complete
parameterization of such a space is intractable. Studying quantum geometry in
the many body context therefore relies on the judicious choice of a lower
dimensional slice of the state space. Different systems call for different
choices of parameterization, and we identify two broad categories of approach
in the literature.

First, in the context of topological insulators and band systems, it is
natural to examine the state space from the perspective of the Brillouin zone
geometry of the bands. Such insulators are, after all, characterized by their
integrated Berry curvature, or Chern number [33]. Recently, the Riemannian
component of the QGT, $g_{\mu\nu}$ has been shown to also indicate the
presence of topologically non-trivial band structures via calculations of the
Euler characteristic [23, 34, 35, 36]. This approach has allowed for a general
classification of the Euler characteristic of $N$ level systems from the
intuitive perspective of Bloch-hypersheres (generalizing the familiar 2D Bloch
sphere) [37].

The second approach is to consider a state space manifold composed of some
combination of Hamiltonian parameters that drive the phase transitions, along
with rotations of the local degrees of freedom [26, 38, 27, 39, 32, 40, 41].

The QFI matrix and QGT have been theoretically examined in a wide range of
models, including Andreev states [42], magnetic systems at finite temperature
[39] the Lipkin-Meshkov-Glick model [41], the Dicke model[43], photonic
systems [44], and in quantum dot chains [45]. This in turn has given rise to a
recent surge of experimental investigations in planar micro cavities [46],
superconducting qubits [47], Hall systems [48], and simple qubits formed at
the NV center of diamond [1].

In this work we introduce a generalization, specifically for many body spin
systems, of the Hilbert space manifold considered in [1] to the case of
spin-1. Our construction is directly relevant for inelastic neutron scattering
experiments and relies on a linear operator parameterization, allowing us to
employ the bound in Eq. 6 to detect genuine multipartite entanglement.
Previous studies of the quantum geometry of spin systems tend to emphasize the
geometry of the state space manifold as parameterized by the driving
parameters of the phase transitions. By contrast, the manifold that we
introduce could be applied consistently to any spin system, regardless of the
particular operators that drive the transitions. Using the mean QFI density
for a staggered magnetic field, we first construct a map of the multipartite
entanglement for the spin-1 antiferromagnetic Heisenberg chain with uniaxial
anisotropy, $D$, and applied magnetic field oriented along the $z$ axis,
$B_{z}$ (see Sec. I.2). Subsequently, using the quantum metric, we calculate
the volume of a two dimensional slice of state space parameterized by the
orientation of a perturbatively small staggered magnetic field. By taking the
ratio of the volume of this slice of state space in the vicinity of the ground
state to the volume of a sphere in a flat state space (see II for our
definition of flat), we compute the local curvature in the quantum state
space. The Haldane phase appears to be characterized by the presence of a
large positive curvature, while the trivial insulator phase and Néel phase
appear to be flat.

In the remainder of the introduction we expand our discussion of the QFI and
its quantitative relationship to GME. We also review the relevant details of
the spin-1 antiferromagnetic Heisenberg chain, which we use as a test case. In
Sec. II we introduce our generalization of the construction in [49] and the
notions of quantum volume and quantum curvature. The results of our analysis
of the Heisenberg chain are presented in III with concluding remarks given in
IV.

###  I.1 Quantum Fisher Information

The relationship between the QFI and genuine multipartite entanglement is
derived in [10]. In that work a linear spin operator of the form,

$\Lambda_{\hat{n}_{r}}=\sum_{r}\hat{n}_{r}\cdot\vec{S}_{r},$ (7)

is considered, where $\hat{n}$ is a unit vector whose orientation may, in the
most general case, be site dependent. We call Eq. 7 the _generator_ of the
QFI. In choosing these generators, the key point is that not all generators
relative orientation are equivalent. If the generator commutes with the
Hamiltonian, then the ground state will be an eigenstate of the generator, and
the covariance, and hence the QFI, will be zero. We discuss the implications
of this fact for the spin-1 Heisenberg chain more in Sec. II. The generator
that will detect the greatest QFI is the one for which $[H,\Lambda]$ is
maximal. Often at the critical point this will be the most relevant
operator[12].

Once the orientations of $\hat{n}_{r}$ are fixed on each site $\Lambda$ may be
used to define a family of three operators, corresponding to the three
orthogonal spin directions on each site. Calling the original orientation
$\Lambda_{3}$ (whatever the orientation of $\hat{n}_{r}$ is on a given site),
the other two orientations $\Lambda_{1}$ and $\Lambda_{2}$, can be found by a
suitable global rotation of all of the unit vectors $\hat{n}_{r}$. The
diagonal entries of the $3\times 3$ QFI matrix associated with the generators
in Eq. 7 are given up to the constant $A$ by,

$Q_{\mu\mu}=\frac{A}{S^{2}}(\langle\Lambda_{\mu}^{2}\rangle-\langle\Lambda_{\mu}\rangle^{2}),$
(8)

Once we have chosen our set ${n_{r}}$ and hence our three generators we can
consider the quantum metric at the state $\ket{\psi}$ associated with the
three dimensional parameterization generated by the vector of operators
$\vec{\Lambda}$. The corresponding QFI matrix, $Q$ (see Eq. 2) can be used to
define an intensive, _mean QFI density_ [10]

$\bar{f}=\frac{1}{3N}\Tr(Q),$ (9)

where $Q$ is the QFI matrix and $N$ is the number sites. The threshold for
genuine multipartite entanglement is then given (in the thermodynamic limit)
by $\bar{f}>3mA$ where $A$ is the metric constant. The bound in [10] can be
recovered for the choice of $A=4$ which corresponds to the QFI most used in a
physics context. The trace on the right hand side of Eq. 9 can be arrived at
by considering the operator defined in Eq. 7 and then integrating over all
possible orientations of the unit vectors (for fixed relative orientations).

We now discuss the meaning of the metric constant, $A$. It emerges from the
fact that when Eq. 1 is generalized to the quantum case, one wants to make the
identification $p(x|\vec{\lambda})\rightarrow\hat{\rho}(\vec{\lambda})$, where
$\hat{\rho}$ is a density matrix, and replace the integral over $x$ by a trace
operator. This introduces an ambiguity due to the non commutativity of
$\hat{\rho}$ with $\partial_{\lambda_{\mu}}\hat{\rho}$. Where the classical
Fisher information is the unique, Riemannian, monotone metric on the space of
classical states, the QFI is unique only up to the definition of an operator
monotone function [50]. There is, therefore, a whole family of QFI’s (see [13]
for a list of some common generalizations and their associated operator
monotone functions). In the physics literature, what tends to be called the
QFI corresponds to a special choice of operator monotone function which treats
this ambiguity in the most symmetric possible way. This is done by the
implicit definition of the _symmetric logarithmic derivative_ (SLD),
$\mathcal{L}_{\mu}$, via [51],

$\partial_{\mu}\hat{\rho}=\frac{1}{2}\\{\hat{\rho},\mathcal{L}_{\mu}\\},$ (10)

where $\\{\cdot,\cdot\\}$ is the anti-commutator. The generalization of Eq. 1
is then given by,

$Q_{\mu\nu}=\Tr{\hat{\rho}\mathcal{L}_{\mu}\mathcal{L}_{\nu}},$ (11)

which, in the zero temperature, limit recovers Eq. 2 with the metric constant
$A=4$. In this work we are concerned only with pure states, and in the pure
state limit the difference between various choices of operator monotone
functions amounts to a multiplicative factor that we interpret as a
normalization. However, no choice of operator monotone function gives a metric
constant greater than $4$, and in this sense the SLD choice is the maximal QFI
[52]. In this work we plot $\frac{1}{A}\bar{f}$, and thus avoid the need to
worry about the particular value of the metric constant.

###  I.2 Spin-1 Heisenberg Chain

We focus on the spin-1 antiferromagnetic Heisenberg chain with uniaxial
anisotropy, $D$ and applied magnetic field along the $z$ axis, $B_{z}$,

$H_{0}=\sum_{r}\vec{S}_{r}\cdot\vec{S}_{r+1}+D(S^{z}_{r})^{2}+B_{z}S_{r}^{z},$
(12)

hereafter referred to as the Heisenberg chain. It is a well studied model in
low dimensional magnetism which has been extensively studied [53, 54, 55, 56,
57, 58, 59, 60, 61, 62, 63, 64]. The completely isotropic point ($D=B_{z}=0$)
is an example of a phase with symmetry protected topological (SPT) order [65]
and exhibits a characteristic doubling of the spectrum of the entanglement
Hamiltonian [63]. We call this phase the isotropic phase or the Haldane phase
interchangeably. The ground state in this phase is a singlet with a bulk gap
to a degenerate triplet mode [56] for periodic boundary conditions. In the
presence of an applied magnetic field, the degeneracy of this mode is lifted,
with one (or two depending on the field orientation) of the triplet modes
diminishing in energy until a lower critical field, $B_{z}^{\text{lower}}$
where hybridization with the ground state singlet induces a phase with long
range AFM order that can be interpreted as a BEC phase [57, 66, 67]. We use
the term BEC or magnon BEC to refer to this phase from this point on. Once the
upper critical field $B_{z}^{\text{upper}}$ is attained, the per-site
magnetization saturates and the spins form a classical paramagnet. For large
and positive $D$, there is a Gaussian transition to a so called “Large-D
Phase”, which we refer to as the insulator phase, and for negative $D$ a
transition to a quasi-ordered Néel phase [61, 62].

Figure 1: A schematic of the manifold generated by the applied field
parameterized by $\\{\theta,\phi\\}$, with the corresponding ground state
given by $\ket{\theta,\phi}$. The tangent vector $\partial_{\mu}$ becomes the
generator $\Lambda_{\mu}$.

##  II Methods

###  II.1 Quantum volume and curvature

####  II.1.1 Defining a manifold

In order to investigate the state space geometry in the vicinity of many body
wavefunctions we consider the two-dimensional sub manifold of the state space
that generalizes the result in [49] to a many-body context. In that work, the
manifold we introduce here generated by the operator $\Lambda_{n_{r}}$ defined
in Eq. 7 was examined for just a single site.

When considering the QFI density of a state, one imagines infinitesimal
transformations of the state generated by a hermitian operator. We extend this
concept by perturbing a _source Hamiltonian_ , $H_{0}(x)$ with the generator
defined in Eq. 7,

$\mathcal{H}(x;h,\theta,\phi)=H_{0}(x)+h\Lambda_{h}(\theta,\phi)$ (13)

and considering the manifold in state space defined by the ground state of the
above Hamiltonian for different strengths and orientations of the perturbing
field (see Fig. 1). In state space this manifold can be considered in terms of
slices at fixed values of $h$,

$\Omega_{h}=\\{\ket{h,\theta,\phi}|(\theta,\phi)\in[0,2\pi)\times[0,\pi]\\},$
(14)

with the state manifold (actually a submanifold) given by
$\mathcal{P}=\bigcup_{h>0}\Omega_{h}$.

The spin-1 Heisenberg chain commutes with the uniform magnetization operator,
and so the QFI for the generator with $\hat{n}_{r}$ uniform is zero. In terms
of the QFI, the fully anti-ferromagnetic generator detects the greatest amount
of entanglement [60]. This is easy to see from the equivalence between the QFI
and the equal time structure factor, which, for the AFM Heisenberg chain,
exhibits a peak at $k=\pi$ [58, 59]. From this point on we assume that the
generator is the staggered field operator,

$\Lambda_{h}(\theta,\phi)=\sum_{r}(-1)^{r}\hat{n}\cdot\vec{S}_{r},$ (15)

where $\hat{n}=(\cos(\theta)\sin(\phi),\sin(\theta)\sin(\phi),\cos(\phi))$.
The superscript $\pi$ is omitted in the following discussion. For a given
orientation of the vector $\hat{n}$, the generators of the three orthogonal
transformations are given by

$\Lambda_{\mu}=\partial_{\mu}\mathcal{H}$ (16)

which we hereafter refer to as _tangent operators_. In order to examine the
curvature of the manifold $\mathcal{P}$, we are interested in the scaling of
the volume of the two dimensional submanifolds, $\Omega_{h}$, and thus we are
particularly interested in $\Lambda_{\theta}$ and $\Lambda_{\phi}$. This
manifold is schematically depicted in Fig. 1. $\Lambda_{h}$ is the tangent
operator orthogonal to the sphere. Since we’re keeping $h$ fixed and looking
at the volume of the two dimensional sphere parameterized by $(\theta,\phi)$
we don’t consider the three dimensional metric that would include
$\Lambda_{h}$.

Choosing a manifold in state space parameterized by the orientation and
strength of a magnetic field makes this study distinct from many other
investigations of the many-body quantum state space geometry, which tend to
employ state space manifolds parameterized by the driving parameters of the
phase transitions (see, for example [26, 38, 27, 39, 32, 40, 41], or by
examining the geometry of the momentum bands in Fermion models (see [37] for a
very general treatment of $N$ level systems).

For each manifold in Eq. 14, the real symmetric part of the QFI matrix
corresponding to the generators for transformations in $\theta$ and $\phi$
gives the metric at each point on $\Omega_{h}$,

$g(x;h,\theta,\phi)=\frac{1}{2}(Q+Q^{*}),$ (17)

From the metric, the _quantum volume_ of the state space manifold $\Omega_{h}$
can be calculated,

$V(\lambda;h)=\int_{\mathcal{S}}\sqrt{\det
g}\,\differential\theta\differential\phi$ (18)

where $\mathcal{S}=[0,2\pi)\times[0,\pi)$. Here we use the word volume to
refer to the size of the 2D state space manifold 111Readers familiar with
differential geometry will know that the expression for the volume in Eq. 18
can be stated equivalently in terms of the magnitude of the cross product of
the tangent vectors (or in our case tangent operators),
$\quantity|\Lambda_{\theta}\times\Lambda_{\phi}|$. This can be seen to be
equivalent to the determinant of the quantum metric by considering the
Lagrange identity, which, for vectors in $\mathbb{R}^{3}$, is given by,
$|\vec{a}\times\vec{b}|^{2}=(\vec{a}\cdot\vec{a})(\vec{b}\cdot\vec{b})-(\vec{a}\cdot\vec{b})^{2}$.
Consistency with the definitions in Eq. 18 and Eq. 17 is achieved by
identifying the dot product with the covariance. This is also consistent with
the form of the quantum metric as having matrix elements defined by the inner
products of the tangent operators. It might be tempting to start from the
expression $\quantity|\Lambda_{\theta}\times\Lambda_{\phi}|$ and assume that
the correct notion of the cross product would be achieved by evaluating the
commutator between the operators $\Lambda_{\theta}$ and $\Lambda_{\phi}$. In
fact this is not the case, as taking such a commutator for the case of the
staggered magnetization generator defined in Eq. 15 would yield the uniform
magnetization operator, which would be inconsistent with the definition of the
three tangent operators given in Eq. 16..

In the limit $h\rightarrow 0$ the QFI matrix used to define the mean QFI
density in Eq. 9 can be recovered by considering perturbations in the $x$,
$y$, and $z$ directions as defined by the generators $\Lambda_{\mu}$ with
$\mu\in\\{x,y,z\\}$ discussed in Sec. I.1. The mean QFI density is then
computed by evaluating the three connected covariances in Eq. 8 in the ground
state of the source Hamiltonian.

By contrast, when we wish to compute the quantum volume of the 2D submanifold
depicted schematically in Fig. 1, we evaluate the covariances of the tangent
operators $\Lambda_{\theta}$ and $\Lambda_{\phi}$ for states that are held at
some small, non-zero field $h$ with orientation $(\theta,\phi)$.

####  II.1.2 Volumes in Flat Space

As an example, consider the quantum metric for a space of $S=1/2$ particles
for which there is no energy landscape (i.e. $H_{0}=0$ in Eq. 13). We call a
space where $H_{0}=0$ an _empty_ space, as there is nothing to distinguish
between different spin states except the orientation of the generator. In this
case the generator cannot be treated as a perturbation because there is
nothing to perturb and the ground state manifold is isomorphic to a sphere.
Explicitly for the case of $S=\frac{1}{2}$ the QFI matrix is,

$Q=ANh^{2}\begin{pmatrix}1&-i\sin(\phi)\\\
i\sin(\phi)&\sin^{2}(\phi)\end{pmatrix},$ (19)

where the real symmetric part is the metric for a sphere and integrating the
imaginary asymmetric part simulates a Dirac monopole as the origin as
discussed in [47]. The resulting quantum volume is,

$V_{S=\frac{1}{2}}=4\pi ANh^{2}.$ (20)

(a detailed derivation is given in App. B) this expression can be generalized
to the case of spin-$S$

$V_{\text{P}}=4\pi\left(\frac{AN}{2S}\right)h^{2}.$ (21)

(see App. C). The volume of the quantum paramagnetic has no dependence on the
relative orientations of $\hat{n}_{r}$. Moreover, the volume scales exactly as
$h^{2}$ and thus the local geometry of the empty space is _flat_ , with zero
curvature. In the classical case, the covariance of the generators of
tangential transformations will be zero, since there are no quantum
fluctuations. This can be seen explicitly from Eq. 21 which goes to zero in
the classical limit, $S\rightarrow\infty$. A non-zero quantum volume can be
taken to indicate the presence of quantum fluctuations, since this non-zero
quantum volume is coming from the fluctuations in the spin expectation value
in the direction transverse to the orientation of the generator. Hence the
quantum volume represents the same information as the Heisenberg uncertainty
relations for the spin components and is an intrinsic consequence of the
relationships between the angular momentum components. We will return to this
point in discussing the results for the $S=1$ Heisenberg chain.

####  II.1.3 Quantum Curvature

The notion of state space curvature can be developed by considering the ratio
of the quantum volume of the manifold $\Omega_{h}$ centered on a ground state
$\psi$ relative to the quantum volume in an empty space. The _quantum volume
ratio_ ,

$\displaystyle v_{h}(\psi(\lambda;h))$
$\displaystyle\coloneqq\frac{V(\psi(\lambda;h))}{V_{\text{P}}}$
$\displaystyle=v_{0}-bRh^{2}+\mathcal{O}(h^{3})$ (22)

may be expanded about small values of $h$, where in the quadratic term
determines the curvature of the manifold $\mathcal{P}$. This is essentially
the definition of curvature familiar from classical differential geometry,
with the scalar curvature $R$ controlling the degree to which the scaling of
volumes is enhanced or suppressed in spaces with negative or positive
curvatures respectively. We call $R$ the _quantum curvature_. It is
independent of the metric normalization $A$, and should therefore hold for any
choice of QFI. The constant $b$ is a positive number that depends only on the
dimension of the manifolds used to compute the volumes. For the case of a two
dimensional sphere $b=\frac{1}{12}$. The negative sign can be understood by
imagining drawing a circle on a saddle or on a sphere. On the saddle (a
surface with negative curvature), the circumference of the circle will be
greater than in the Euclidean case, and so the scaling of the circumference
with the radius of the circle will be greater than in the case of flat space,
with the opposite argument going through the sphere (a surface with positive
curvature) [69].

When we speak of the quantum volume ratio of a state $\psi$, we mean the
quantum volume of the spheroidal shell centered at $\psi$ in the state space.

###  II.2 Numerical Methods

All data was collected using an iDMRG algorithm implemented using the ITensor
library [70]. Simulations were performed with bond dimensions of up to $2800$
in the vicinity of the critical points, and optimization was run until
truncation errors not exceeding $10^{-9}$ were achieved. In the BEC phase we
found that convergence was challenging, likely owing to the long range nature
of the correlations that phase [66].

Within the iDMRG, correlation functions were measured to a distance of $1000$
sites starting from the center of an effectively infinite chain. These
correlations were then used to compute the per-site connected covariances
required for the evaluation of the quantum volume.

The sum over $N$ sites is easiest to compute in the iDMRG because we can take
advantage of the translation invariance. The covariances involve a double
summation over the correlation matrix $C_{r_{1},r_{2}}$,
$\sum_{r_{1}=1}^{N}\sum_{r_{2}=1}^{N}C_{r_{1},r_{2}}$. In systems with
translation invariance we can assume that $C_{r_{1},r_{2}}=C_{|r_{1}-r_{2}|}$
and reduce the double sum to $N\sum_{r=1}^{N}C_{r}$. If we had instead
considered a finite system with a boundary, the entire double sum would’ve
been needed. While this is still tractable for non-trivial system sizes when
considering the mean QFI density, evaluating the quantum volumes becomes
challenging in all but the most symmetrical cases.

In order to compute the integral in Eq. 18, we can take advantage of the
symmetries of the source Hamiltonian. We derive (see App. A) the following
formulae for the case of full rotational symmetry and axial symmetry,

$\displaystyle V^{\text{Spherical}}(\lambda;h)$ $\displaystyle=4\pi
Ah^{2}\sqrt{\Xi_{xxyy}}$ (23a) $\displaystyle V^{\text{Axial}}(\lambda;h)$
$\displaystyle=2\pi
Ah^{2}\int_{0}^{\pi}\sqrt{\Xi_{xxyy}\cos^{2}(\phi)\sin^{2}(\phi)-2\Xi_{yyzx}\cos(\phi)\sin^{3}(\phi)+\Xi_{yyzz}\sin^{4}(\phi)}\,\,\differential\phi$
(23b)

Figure 2: Phase Diagram of the mean QFI generated by the staggered
magnetization operator. The Néel and Insulator phases are un-entangled away
from the critical point. The Magnon BEC phase appears as the critical fan of
the Gaussian critical point separating the Haldane phase from the Insulator
phase. Regions of low entanglement in the BEC phase are numerical artifacts
from failed convergence of the iDMRG. When points do converge it is with a
truncation error of no more than $10^{-9}$. The red horizontal line indicates
the slice that is shown and extended in Fig. 3

where in general,

$\Xi_{abcd}(\lambda;h,\theta,\phi)=\mathcal{A}_{ab}\mathcal{A}_{cd}-\mathcal{A}_{ac}\mathcal{A}_{bd}.$
(24)

Here $\mathcal{A}_{ab}$ is the real part of the correlation matrix,

$\mathcal{C}^{ab}=\sum_{r_{1},r_{2}}(-1)^{r_{1}+r_{2}}\text{Cov}_{\psi(\theta,\phi)}(S_{r_{1}}^{a},S_{r_{2}}^{b}),$
(25)

given by $\mathcal{A}=\frac{1}{2}(\mathcal{C}+\mathcal{C}^{*})$ and the
indices $a,b,c,d\in\\{x,y,z\\}$ are taken in the lab frame. The correlation
matrix will depend on the strength of the perturbing field $h$, and also on
the particular values of the Hamiltonian parameters which we here denote as
$\lambda$. In Eqs. 23, the angular dependence in $(\theta,\phi)$ has been
integrated out completely in the spherical case, while in the axial case $\Xi$
will have some dependence on $\phi$. In the axial case the integral over
$\phi$ can be performed numerically with relatively few integration points.
For the data shown in Figs. 5 6 the $\phi$ component was integrated with $50$
equally spaced points. This is relevant for experiments where measurements
would be required at a range of field orientations in order to compute the
quantum volume.

##  III Results and Discussion

###  III.1 Mean QFI Density

Taking Eq. 12 as our source Hamiltonian, the mean QFI density divided by the
metric constant was computed for a patch of the $(D,B_{z})$ parameter space
with $D\in[-0.5,1.5]$ and $B_{z}\in[0,0.5]$. In this section we imagine a
$3\times 3$ QFI matrix where the parameterizations are generated by the fields
in the $x$, $y$, and $z$ directions in the lab frame as discussed in Sec. I.1.
The correlation functions are evaluated for the ground state of the source
Hamiltonian, taking $h=0$ in Eq. 13. The results are given in Fig. 2. We see
robust multipartite entanglement through the Haldane phase indicating at
minimum bipartite entanglement. We find our results to be consistent with the
low temperature single mode approximation employed by the authors in [60].

Taking a slice of the phase diagram along $D=0$ (see Fig. 3), we see that the
multipartite entanglement of the magnon BEC is substantially higher, and
estimate the value of the lower critical field to be
$B_{z}^{\text{lower}}\approx 0.41$, which is consistent with previous studies
[57] which place the value of $B_{z}^{\text{lower}}$ at the Haldane gap [64].
From Fig. 3 the multipartite entanglement can be seen to peak before falling
to the upper critical field of approximately $B_{z}^{\text{upper}}\approx 4$,
beyond which the system enters a classical paramagnetic phase for which the
mean QFI density is nearly zero and the per site magnetization becomes
saturated.

In Fig 2 we see that the BEC phase appears as a fan of genuine multipartite
entanglement emanating from the Gaussian transition to the insulator phase.
The BEC phase can be reached from either the insulator phase or the Haldane
phase. Experimental studies [66] on the material NiCl2-4SC(NH2)2, for which
the ground state is in the insulator phase and which exhibits an additional
easy axis anisotropy, reveal that the long range correlations in the BEC phase
persist up to a finite temperature that grows as a function of the applied
field to a maximal point which occurs at approximately the point where the
average onsite magnetization assumes a value of $\langle M_{z}\rangle/N\approx
0.4$. Owing to the analogous peak in the GME as witnessed by the QFI at the
point where the onsite magnetization assumes the values of $\langle
M_{z}\rangle/N\approx 0.4$, we speculate that this temperature scale may be
related to the massive multipartite entanglement in the ground state. In
particular, the temperature curve for the BEC phase in [66] is of a form that
is very similar to the curve of genuine multipartite entanglement in Fig. 3.

Within the Haldane phase the multipartite entanglement is completely constant
as a function of $B_{z}$, as can been seen in Fig. 3 in the phase labelled H.
This indicates that the Haldane phase entanglement is robust to the presence
of magnetic fields. Coupled with the already established robustness of the
Haldane phase entanglement to finite temperatures [60, 17], our observations
establish Haldane gap anti-ferromagnets as reliable sources of at least bi-
partite entanglement even in the presence of environmental disturbances that
tend to decohere entangled states. This has especially important application
in light of the application of SPT phases to measurement based quantum
computation [71, 72].

Figure 3: QFI associated with staggered magnetization operator as a function
of magnetic field in the $z$ direction (magenta curve) and average per site
magnetization along the $z$ axis (blue curve). Notice that the mean QFI
density is constant in the Haldane phase and nearly zero in the paramagnetic
phase.

In contrast to a homogeneous applied field, any non-zero uniaxial anisotropy
appears to increase the amount of genuine multipartite entanglement as seen by
considering a cut in the state space where $B_{z}=0$ (see Fig. 4). Along this
cut we can see clear divergences in the GME at both the Néel and insulator
transitions. The Haldane phase exhibits two partite entanglement in the
vicinity of the isotropic point where $D=0$, while the other phases are
trivial from a quantum perspective, exhibiting zero GME.

The behaviour of the mean QFI can be contrasted with the fidelity
susceptibility, which, for the Gaussian transition in particular, exhibits a
less pronounced signal (see Fig. 3 of [62]). While the fidelity susceptibility
associated with the Ising transition does exhibit a divergence, this
divergence does not establish the entanglement of the critical point, due to
the reasons discussed in Sec. I.1.

While the Ising transition was already established as a strongly entangled
critical point by the authors in [60], the observation of entanglement at the
Gaussian critical point was not made. This is because the component $Q_{zz}$
of the QFI matrix which was studied by the authors in [60] does not diverge at
the Gaussian transition, and it is rather the components of the QFI matrix in
the directions transverse to the critical point.

Figure 4: QFI associated with staggered magnetization operator as a function
of the uniaxial anisotropy. The Haldane phase exhibits the largest amount of
multipartite entanglement, which is enhanced by non-zero values of the
uniaxial anisotropy.

###  III.2 Quantum Curvature

In this section we consider the 2D submanifold centered at the ground state of
the source Hamiltonian for small, non-zero values of $h$ in Eq. 13. The
correlation functions then in general depend on both $h$ and the orientation
of the field $(\theta,\phi)$. Due to the symmetries of the spin-1 chain
defined in Eq. 12, we can integrate out the dependence on $(\theta,\phi)$ at
the isotropic point, and the dependence on $\theta$ for the case of non-zero
uniaxial anisotropy, using the formulae defined in Eqs. 23. The correlations
are then evaluated as functions of $h$ in the case of spherical symmetry, or
as functions of $h$ and $\phi$ in the case of axial symmetry. In both cases
the correlations will depend on the value of $D$. The volumes of the 2-spheres
for different values of $h$ are sufficient for us to then compute the quantum
curvature defined in Eq. 22. If all correlations $\langle
S_{r_{1}}^{a}S_{r_{2}}^{b}\rangle$ for a given field strength $h$ and
orientation $(\theta,\phi)$ are evaluated with $a,b\in\\{x,y,z\\}$ in the lab
frame then Eq. 23 can be evaluated.

We begin at the isotropic point with $D=B_{z}=0$ and examine the volume ratios
of the ground state of Eq. 13 as function of $h$ that are taken to be small
relative to the exchange coupling. The results of this calculation are shown
in Fig 5. We see that the volume ratio decreases monotonically, indicating
that the manifold $\mathcal{P}$ is asymptotically flat in the limit of large
$h$. Near $h=0$ it is clear that this function is concave down, which, from
Eq. 22 implies that the curvature is positive. The volume ratio being larger
than 1 indicates the presence of a high degree of quantum fluctuation relative
to those implied purely by the uncertainty relation for the spin operators, as
discussed in Sec. II.

Figure 5: Quantum volume ratio for spheres centered at the isotropic point.

By sweeping across values of $D$ from $-1$ to $3$ we can determine how the
curvature of the state space depends on the uniaxial anisotropy. The results
of this analysis are shown in Fig. 6. Taking the second derivative of the
volume ratio as a function of $h$ at $h=0$ gives us the sign of curvature,
indicated by the red (positive curvature) and grey (negative curvature) shaded
regions in Fig. 6). We see that the Haldane phase is characterized by strong
positive curvature, while the Néel and insulator phases are flat with
essentially zero curvature.

Recall from Sec. II that the quantum volume quantifies the fluctuations in the
spin degrees of freedom. In flat space (i.e. spaces where $H_{0}=0$), these
fluctuations come purely from the Heisenberg uncertainty relations. Hence the
quantum volume in a flat space of $N$ spin-$S$ particles tends to zero in the
limit $S\rightarrow\infty$. In the Néel and Insulator phases, we find a
quantum volume ratio $v_{h}>1$, indicating that these phases exhibit enhanced
fluctuations relative to what would be expected from the pure uncertainty
relations. From Fig. 4, we see clearly that neither of these phase are
entangled, (see the inequality in Eq. 6). Hence, quantum phases might have
enhanced fluctuations without necessarily exhibiting any entanglement.

Figure 6: Quantum volume ratio for spheres of perturbatively small radius as a
function of the uniaxial anisotropy $D$. Sphere sizes (applied fields) range
in size from $h=0.0001$ to $h=0.04$. The red shaded region indicates where the
curvature is positive while the grey region indicates where the curvature is
negative, approximately corresponding to the onset of the scaling regime.

Despite these enhanced fluctuations, both the Néel and insulator phases are
flat, with quantum volume ratios that are constant as a function of the field
$h$. By contrast, the Haldane phase, which exhibits the greatest enhancement
in quantum fluctuations of the three, also exhibits a region of large positive
curvature, leading into the critical points where we find a sudden,
discontinuous jump to a region of negative curvature which we interpret as the
onset of the scaling regime. The Haldane phase is the only phase of the three
that exhibits any degree of entanglement, and it appears that the quantum
curvature tends to grow as the entanglement increases, as seen by the
dispersion of the curves at fixed values of $h$ in Fig. 6. The flatness of the
insulator phase also establishes that the curvature in the Haldane phase is
not simply a consequence of the Haldane gap, as the insulator phase is also
gapped but exhibits no quantum curvature.

It seems clear that the quantum curvature is connected to the presence of
entanglement, with the trivial phases appearing completely flat. A phase with
no entanglement would then be expected to have zero curvature, except perhaps
when the critical point is approached. But it is not clear whether or not the
presence of quantum curvature is only incidental in the Haldane phase, and is
in fact due to the nearby quantum critical points which exhibit a massive
divergence in the amount of GME. This would account for the presence of small
amounts of positive curvature in the Néel and insulator phases as the critical
points are approached. In this case the positive curvature in the Haldane
phase is a consequence of the phases existing in between these two critical
points. This introduces the possibility of detecting the presence of quantum
critical points even outside of the scaling regime, without the need to
directly witness the diverging multipartite entanglement at the critical
point. This would be especially useful in inelastic neutron scattering
experiments where resolution issues make it effectively impossible to directly
measure diverging multipartite entanglement [11]. That state space curvature
is induced by the phase boundary implies that quantum critical regions have
long range effects in the state space, influencing not just the geometry of
states in the scaling regime, but also the states that lie in between these
critical points.

It is also possible that state space curvature is generic in the presence of
any amount of entanglement, with entanglement and state space curvature
sharing a relationship that is somewhat analogous to the relationship between
mass and space-time curvature.

Our calculation of the state space curvature can be contrasted with those of
[40, 32, 27] and especially [38, 39]. In these studies, it is the curvature
corresponding to the ground state manifold parameterized by the driving
parameters of the phase transitions themselves that is computed. In
particular, [38] finds that the curvature of the manifold parameterized by a
uniform field along the $z$ axis and the temperature in the XXZ model reveal
different scaling regimes in the vicinity of the quantum critical point.

This is similar to our observation that the state space curvature becomes
negative in the immediate vicinity of the quantum critical point, but our
construction pertains to a state space manifold that might be applied to
transitions where the magnetic field is not a priori the driving parameter of
the phase transition. It is also interesting that there is a discontinuous
jump from a region of positive state space curvature in the Haldane phase and
immediately around the critical points to a region of negative state space
curvature in the quantum critical regime, without any intermediate flat
region. In the grey regions of Fig. 6, it is important to emphasize that the
form of the quantum metric defined in Eq. 2 breaks down, as it does not
account for the non-adiabatic level crossings that occur at the critical
point. A more careful treatment would involve computing the entire spectrum
and applying the form of the quantum metric introduced in [26]. We leave these
questions to future study.

##  IV Conclusion

In this work we have mapped out the phase diagram of the spin-1
antiferromagnetic Heisenberg chain in the space of applied magnetic field and
uniaxial anisotropy (both along the $z$ axis). The five phases of the spin-1
Heisenberg chain considered here have all been shown to have substantially
different behaviour in terms of the mean QFI density. In particular we find
the both the Néel and insulator phases exhibit small but non-zero genuine
multipartite entanglement relative to the SPT Haldane phase. The entanglement
in the SPT phase seems to be greatly enhanced by the presence of unixial
anisotropy, and is undiminished under the application of a uniform magnetic
field. The fact the the multipartite entanglement in the Haldane phase is
robust against the application of magnetic fields and exhibits a finite
temperature plateau up to energies on the scale of the Haldane gap [60, 17]
may have implications from quantum metrology and quantum information, where
state exhibiting robust entanglement are used as a resource in various
measurement and information processing tasks

Once the applied field reaches a strength on the order of the Haldane gap we
find a drastic increase in the multipartite entanglement in the magnon BEC
phase. This condensate is found to emanate from the Gaussian quantum critical
point that is generated by large positive values of the uniaxial anisotropy,
and is similar to the quantum critical fans expected in the finite temperature
region above the quantum critical point. In the BEC phase we find a peak
multipartite entanglement at $B_{z}^{\text{max}}\approx 1.6$. More work should
be done to understand the relationship between the massive amount of
multipartite entanglement in the ground state and the finite temperature
transition points of the BEC phase mapped out in [66].

By computing the quantum metric corresponding to a state space manifold
parameterized by the magnitude and orientation of a small, staggered magnetic
field, we introduced the ideas of quantum volume and quantum curvature, and
computed both for a range of values of the uniaxial anisotropy.

Crucially, we have demonstrated that the Haldane phase in this model is
characterized by a sizeable, positive quantum curvature, while the Néel and
insulator phases are flat. We have identified two possibilities. First, that
the presence of quantum curvature is induced by the presence of quantum
critical points, in which case measurement of such curvature may prove
applicable as a probe of quantum criticality outside of the scaling regime in
neutron scattering experiments. Second, it is possible that massive positive
curvature is an intrinsic feature any GME phase. It is important to emphasize
that the entanglement in the SPT phase is short range in nature [65]. It is
unclear whether or not the state space geometry computed in terms of a linear,
local operator would be sensitive to long range entangled phases, where the
mean QFI density of local operators does not detect GME [19, 25].

One final observation is that the mean QFI density along the $B_{z}=0$ cut
shown in Fig. 4, and the quantum volume ratio in the limit $h\rightarrow 0$,
$v_{0}$ shown as the maximal curve in Fig. 6 appear to be almost exactly
proportional to one another. Furthermore, the curvature appears to grow
proportionately to the mean QFI density up to the onset of the critical
region. This raise the possibility that the GME is a source of curvature in
the state space, analogous to the curvature of space time due to the presence
of mass-energy. We do not present any rigorous proof of the relationship
between $v_{0}$ and $\bar{f}$ in this work, leaving the problem to future
study.

###### Acknowledgements.

This research was supported by NSERC and was enabled in part by support
provided by SHARCNET (www.sharcnet.ca) and the Digital Research Alliance of
Canada (alliancecan.ca). Part of the numerical calculations were performed
using the ITensor library [70].

##  Appendix A Symmetric Formulae for the Quantum Volume

In order to efficiently evaluate Eq. 18 is is crucial that we exploit
symmetries of the source Hamiltonian. This would also be the case for
experimental explorations of the quantum volume ratio. There are two
symmetries that are of particular interest to us. The first is the full
rotational symmetry given by the Euler angles $(\theta,\phi)$. Recall the
rotation operator [73],

$R(\theta,\phi)=e^{i\sum_{r}S^{z}_{r}\theta}e^{i\sum_{r}S^{y}_{r}\phi}$ (26)

where the gauge angle representing an initial rotation about the $z$ axis is
taken to be zero. If this operator represents a symmetry of the source
Hamiltonian, we may generate the state manifold by applying the rotation to
$\mathcal{H}$,

$\mathcal{H}(\lambda;h,\theta,\phi)=R(\theta,\phi)\mathcal{H}(\lambda;h,0,0)R^{-1}(\theta,\phi)$
(27)

Each element of the quantum metric is given by the real symmetric part of the
covariance of the generators, as defined in Eq. 17. Using the particular form
of the generator in Eq. 16 and denoting the real symmetric part of the
covariance matrix $\mathcal{A}$ gives,

$g_{\mu\nu}=\frac{A}{S^{2}}h^{2}d_{\mu}^{a}d_{\nu}^{b}\mathcal{A}^{ab}$ (28)

where $\mathcal{A}=\frac{1}{2}(\mathcal{C}+\mathcal{C}^{*})$, and

$\mathcal{C}^{ab}=\sum_{r_{1},r_{2}}(-1)^{r_{1}+r_{2}}\text{Cov}_{\psi(\theta,\phi)}(S_{r_{1}}^{a},S_{r_{2}}^{b})$
(29)

Under a rotation the covariance matrix transforms

$\text{Cov}_{\psi(\theta,\phi)}(S^{a}_{r_{1}},S^{b}_{r_{2}})\rightarrow
R^{aa^{\prime}}R^{bb^{\prime}}\text{Cov}_{\psi(0,0)}(S_{r_{1}}^{a^{\prime}},S_{r_{2}}^{b^{\prime}})$
(30)

where Einstein summation is used and,

$R=\begin{pmatrix}\cos(\theta)\cos(\phi)&-\sin(\theta)&\cos(\theta)\sin(\phi)\\\
\sin(\theta)\cos(\phi)&\cos(\theta)&\sin(\theta)\sin(\phi)\\\
\sin(\phi)&0&\cos(\phi)\end{pmatrix}$ (31)

Using this transformation we can extract the angular dependence of the metric
for the case of spherical and axial symmetry, and integrate out that
dependence completely, leaving

$V^{\text{Spherical}}(\lambda;h)=4\pi\frac{A}{S^{2}}h^{2}\sqrt{\Xi_{xxyy}}$
(32)

where,

$\Xi_{abcd}=\mathcal{A}_{ab}\mathcal{A}_{cd}-\mathcal{A}_{ac}\mathcal{A}_{bd}.$
(33)

For the axial case we, without loss of generality, choose the case where the
system is symmetric about rotations about the $z$ axis (set $\phi=0$ in Eq.
31). In this case we have,

$V^{\text{Axial}}(\lambda;h)=2\pi\frac{A}{S^{2}}h^{2}\int_{0}^{\pi}\sqrt{\Xi_{xxyy}\cos^{2}(\phi)\sin^{2}(\phi)-2\Xi_{yyzx}\cos(\phi)\sin^{3}(\phi)+\Xi_{yyzz}\sin^{4}(\phi)}\,\,\differential\phi$
(34)

In the axial case we still have to perform a numerical integration with
respect to $\phi$ but good convergence can be achieved using relatively few
slices.

##  Appendix B QFI matrix of a flat space for $S=1/2$

The quantum metric of a flat space can be calculated for N spin-$1/2$
particles by considering the Hamiltonian in Eq. 13 setting $H_{0}=0$. The
tangent operators are given by,

$\Lambda_{\mu}=\frac{h}{2}\vec{d}_{\mu}\cdot\vec{\sigma}$ (35)

Where $d_{\mu}$ is the derivative of the unit vector along the $\mu$
direction.

Substituting these definitions into Eq. 2 (assuming summation over repeated
indices),

$\displaystyle Q_{\mu\nu}$
$\displaystyle=Ah^{2}d_{\mu}^{a}d_{\nu}^{b}\left(\langle\sigma^{a}\sigma^{b}\rangle-\langle\sigma^{a}\rangle\langle\sigma^{b}\rangle\right)$
$\displaystyle=Ah^{2}d_{\mu}^{a}d_{\nu}^{b}C^{ab}$ (36)

Where we’ve defined the connected correlation $C^{ab}$. We can use the algebra
of the Pauli matrices to simplify the expression for the correlations
somewhat,

$\displaystyle C^{ab}$
$\displaystyle=\langle\sigma^{a}\sigma^{b}\rangle-\langle\sigma^{a}\rangle\langle\sigma^{b}\rangle$
$\displaystyle=\delta^{ab}+i\epsilon_{abc}\langle\sigma^{c}\rangle-\langle\sigma^{a}\rangle\langle\sigma^{b}\rangle$
(37)

Now substituting this Eq. 36 gives,

$\displaystyle Q_{\mu\nu}$
$\displaystyle=Ah^{2}\left(\vec{d}_{\mu}\cdot\vec{d}_{\nu}+i(\vec{d}_{\mu}\times\vec{d}_{\nu})\cdot\langle\vec{\sigma}\rangle-
d_{\mu}^{a}d_{\nu}^{b}\langle\sigma^{a}\rangle\langle\sigma^{b}\rangle\right)$
(38)

Now we must evaluate the expectation values of the Pauli matrices. Recall that
these are themselves functions of $(\theta,\phi)$. For the two level case we
can compute the ground state explicitly,

$\displaystyle\ket{\lambda_{-}^{\Delta=0}}$
$\displaystyle=\begin{pmatrix}\sin(\frac{\phi}{2})\\\
-\cos(\frac{\phi}{2})e^{i\theta}\end{pmatrix}$ (39a)

The expectation value of the Pauli vector in the ground state is then,

$\langle\vec{\sigma}\rangle=\begin{pmatrix}-\cos(\theta)\sin(\phi)\\\
-\sin(\theta)\sin(\phi)\\\ -\cos(\phi)\end{pmatrix}=-\hat{n}(\theta,\phi)$
(40)

The metric in Eq. 38 is now given by,

$Q_{\mu\nu}=Ah^{2}\left(\vec{d}_{\mu}\cdot\vec{d}_{\nu}-i(\vec{d}_{\mu}\times\vec{d}_{\nu})\cdot\hat{n}\right)$
(41)

In this form, evaluating the components of the metric proves to be a
straightforward exercise

$Q=Ah^{2}\begin{pmatrix}1&-i\sin(\phi)\\\
i\sin(\phi)&\sin^{2}(\phi)\end{pmatrix}$ (42)

Generalizing this expression to $N$ spins amounts to multiplying by $N$

$Q=ANh^{2}\begin{pmatrix}1&-i\sin(\phi)\\\
i\sin(\phi)&\sin^{2}(\phi)\end{pmatrix}$ (43)

since the connected correlations between sites is zero by construction. Notice
that this expression does not depend on the relative orientation of the
neighbouring sites. For $S=\frac{1}{2}$, the quantum volume (surface area of a
2D sphere in the spin Hilbert space) is then,

$V_{S=\frac{1}{2}}=4\pi ANh^{2}$ (44)

##  Appendix C Quantum volume of a flat space for spin-S

In computing to quantum volume ratio, we want to generalize the volume
computed in Eq. 44 to the case of spin-S. This is easy to do by employing the
formula for the spherically symmetric quantum volume given in Eq. 32

Using Eq. 32 we can compute the quantum volume of a paramagnet. Taking
$H_{0}=0$ in Eq. 13 and denoted the ground state of the spin-S paramagnet as
$\ket{S,-S}$, we can use the raising and lowering operators,

$\displaystyle S^{+}\ket{S,m}$ $\displaystyle=\sqrt{S(S+1)-m(m+1)}\ket{S,m+1}$
$\displaystyle S^{-}\ket{S,m}$ $\displaystyle=\sqrt{S(S+1)-m(m-1)}\ket{S,m-1}$

For the paramagnet the connected correlation between is exactly zero, so we
can evaluate only the onsite correlations in the directions transverse to the
orientation of the field (which we take to be $z$). The expectation values in
$x$ and in $y$ are the same, so we only present the calculation of the former.
For the state that is fully polarized down along the $z$ axis, the only
contribution to $(S^{x})^{2}$ is given by,

$\displaystyle\mathcal{A}^{xx}=\frac{1}{4}\bra{S,-S}S^{-}S^{+}\ket{S,-S}$
$\displaystyle=\frac{S}{2}$ (46)

Substiting this into Eq. 32 gives,

$V_{P}=4\pi\left(\frac{AN}{2S}\right)h^{2}$ (47)

where we have group the spin, S, metric constant $A$, and number of sites $N$
along with the factor of $\frac{1}{2}$ in order to emphasize the familiar form
of the volume of a sphere. The factor $N$ is coming from the fact that
$\mathcal{A}^{ab}$ in $\Xi^{abcd}$ contains a sum over $N$ sites, which for
the the case of $H_{0}=0$ scales linearly in $N$ (since the connected
correlations between sites are zero). We can check and see that for
$S=\frac{1}{2}$ we recover Eq. 44.

## References

  * Yu  _et al._ [2020] M. Yu, P. Yang, M. Gong, Q. Cao, Q. Lu, H. Liu, S. Zhang, M. B. Plenio, F. Jelezko, T. Ozawa, _et al._ , Experimental measurement of the quantum geometric tensor using coupled qubits in diamond, National science review 7, 254 (2020).
  * Fisher [1922] R. A. Fisher, On the mathematical foundations of theoretical statistics, Philosophical transactions of the Royal Society of London. Series A, containing papers of a mathematical or physical character 222, 309 (1922).
  * Johnson [2004] O. Johnson,  _Information theory and the central limit theorem_ (World Scientific, 2004).
  * Wootters [1981] W. K. Wootters, Statistical distance and hilbert space, Physical Review D 23, 357 (1981).
  * Braunstein and Caves [1994] S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Physical Review Letters 72, 3439 (1994).
  * Bengtsson and Życzkowski [2017] I. Bengtsson and K. Życzkowski,  _Geometry of quantum states: an introduction to quantum entanglement_ (Cambridge university press, 2017).
  * Provost and Vallee [1980] J. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Communications in Mathematical Physics 76, 289 (1980).
  * Lesniewski and Ruskai [1999] A. Lesniewski and M. B. Ruskai, Monotone riemannian metrics and relative entropy on noncommutative probability spaces, Journal of Mathematical Physics 40, 5702 (1999).
  * Berry [1984] M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 392, 45 (1984).
  * Hyllus  _et al._ [2012] P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzé, and A. Smerzi, Fisher information and multiparticle entanglement, Physical Review A 85, 022321 (2012).
  * Scheie  _et al._ [2021] A. Scheie, P. Laurell, A. Samarakoon, B. Lake, S. Nagler, G. Granroth, S. Okamoto, G. Alvarez, and D. Tennant, Witnessing entanglement in quantum magnets using neutron scattering, Physical Review B 103, 224434 (2021).
  * Hauke  _et al._ [2016] P. Hauke, M. Heyl, L. Tagliacozzo, and P. Zoller, Measuring multipartite entanglement through dynamic susceptibilities, Nature Physics 12, 778 (2016).
  * Shitara and Ueda [2016] T. Shitara and M. Ueda, Determining the continuous family of quantum fisher information from linear-response theory, Physical Review A 94, 062316 (2016).
  * Laurell  _et al._ [2021] P. Laurell, A. Scheie, C. J. Mukherjee, M. M. Koza, M. Enderle, Z. Tylczynski, S. Okamoto, R. Coldea, D. A. Tennant, and G. Alvarez, Quantifying and controlling entanglement in the quantum magnet cs 2 cocl 4, Physical Review Letters 127, 037201 (2021).
  * Braunstein  _et al._ [1996] S. L. Braunstein, C. M. Caves, and G. J. Milburn, Generalized uncertainty relations: theory, examples, and lorentz invariance, annals of physics 247, 135 (1996).
  * Holevo [2011] A. S. Holevo,  _Probabilistic and statistical aspects of quantum theory_ , Vol. 1 (Springer Science & Business Media, 2011).
  * Gabbrielli  _et al._ [2018] M. Gabbrielli, A. Smerzi, and L. Pezzè, Multipartite entanglement at finite temperature, Scientific reports 8, 1 (2018).
  * Yin  _et al._ [2019] S. Yin, J. Song, Y. Zhang, and S. Liu, Quantum fisher information in quantum critical systems with topological characterization, Physical Review B 100, 184417 (2019).
  * Lambert and Sørensen [2020] J. Lambert and E. S. Sørensen, Revealing divergent length scales using quantum fisher information in the kitaev honeycomb model, Physical Review B 102, 224401 (2020).
  * Yang  _et al._ [2008] S. Yang, S.-J. Gu, C.-P. Sun, and H.-Q. Lin, Fidelity susceptibility and long-range correlation in the kitaev honeycomb model, Physical Review A 78, 012304 (2008).
  * Zheng  _et al._ [2015] Q. Zheng, Y. Yao, and X.-W. Xu, Probing berezinskii–kosterlitz–thouless phase transition of spin-half xxz chain by quantum fisher information, Communications in Theoretical Physics 63, 279 (2015).
  * Li  _et al._ [2013] S.-S. Li, H.-G. Yi, and R.-H. Chen, Spin squeezing and quantum fisher information in the lipkin-meshkov-glick model, International Journal of Theoretical Physics 52, 1175 (2013).
  * Ma  _et al._ [2013] Y.-Q. Ma, S.-J. Gu, S. Chen, H. Fan, and W.-M. Liu, The euler number of bloch states manifold and the quantum phases in gapped fermionic systems, EPL (Europhysics Letters) 103, 10008 (2013).
  * Liu  _et al._ [2013] W.-F. Liu, J. Ma, and X. Wang, Quantum fisher information and spin squeezing in the ground state of the xy model, Journal of Physics A: Mathematical and Theoretical 46, 045302 (2013).
  * Pezze  _et al._ [2017] L. Pezze, M. Gabbrielli, L. Lepori, and A. Smerzi, Multipartite entanglement in topological quantum phases, Physical review letters 119, 250401 (2017).
  * Zanardi and Paunković [2006] P. Zanardi and N. Paunković, Ground state overlap and quantum phase transitions, Physical Review E 74, 031123 (2006).
  * Zanardi  _et al._ [2007a] P. Zanardi, P. Giorda, and M. Cozzini, Information-theoretic differential geometry of quantum phase transitions, Physical review letters 99, 100603 (2007a).
  * Albuquerque  _et al._ [2010] A. F. Albuquerque, F. Alet, C. Sire, and S. Capponi, Quantum critical scaling of fidelity susceptibility, Phys. Rev. B 81, 064418 (2010).
  * Thesberg and Sørensen [2011] M. Thesberg and E. S. Sørensen, General quantum fidelity susceptibilities for the ${J}_{1}$-${J}_{2}$ chain, Phys. Rev. B 84, 224435 (2011).
  * Liu  _et al._ [2014] J. Liu, H.-N. Xiong, F. Song, and X. Wang, Fidelity susceptibility and quantum fisher information for density operators with arbitrary ranks, Physica A: Statistical Mechanics and its Applications 410, 167 (2014).
  * Kolodrubetz  _et al._ [2017] M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, Geometry and non-adiabatic response in quantum and classical systems, Physics Reports 697, 1 (2017).
  * Venuti and Zanardi [2007] L. C. Venuti and P. Zanardi, Quantum critical scaling of the geometric tensors, Physical review letters 99, 095701 (2007).
  * Laughlin [1981] R. B. Laughlin, Quantized hall conductivity in two dimensions, Physical Review B 23, 5632 (1981).
  * Ozawa and Mera [2021] T. Ozawa and B. Mera, Relations between topology and the quantum metric for chern insulators, arXiv preprint arXiv:2103.11582 (2021).
  * Zhang [2022] A. Zhang, Revealing chern number from quantum metric, Chinese Physics B 31, 040201 (2022).
  * Mera and Ozawa [2021] B. Mera and T. Ozawa, Kähler geometry and chern insulators: Relations between topology and the quantum metric, Physical Review B 104, 045104 (2021).
  * Kemp  _et al._ [2021] C. J. Kemp, N. R. Cooper, and F. N. Ünal, Nested spheres description of the n-level chern number and the generalized bloch hypersphere, arXiv preprint arXiv:2110.06934 (2021).
  * Zanardi  _et al._ [2007b] P. Zanardi, L. C. Venuti, and P. Giorda, Bures metric over thermal state manifolds and quantum criticality, Physical Review A 76, 062318 (2007b).
  * Janyszek and Mrugal [1989] H. Janyszek and R. Mrugal, Riemannian geometry and the thermodynamics of model magnetic systems, Physical Review A 39, 6515 (1989).
  * Kolodrubetz  _et al._ [2013] M. Kolodrubetz, V. Gritsev, and A. Polkovnikov, Classifying and measuring geometry of a quantum ground state manifold, Physical Review B 88, 064304 (2013).
  * Gutiérrez-Ruiz  _et al._ [2021] D. Gutiérrez-Ruiz, D. Gonzalez, J. Chávez-Carlos, J. G. Hirsch, and J. D. Vergara, Quantum geometric tensor and quantum phase transitions in the lipkin-meshkov-glick model, Physical Review B 103, 174104 (2021).
  * Klees  _et al._ [2020] R. L. Klees, G. Rastelli, J. C. Cuevas, and W. Belzig, Microwave spectroscopy reveals the quantum geometric tensor of topological josephson matter, Physical Review Letters 124, 197002 (2020).
  * Gutiérrez-Ruiz  _et al._ [2022] D. Gutiérrez-Ruiz, J. Chávez-Carlos, D. Gonzalez, J. G. Hirsch, and J. D. Vergara, Quantum metric tensor of the dicke model: Analytical and numerical study, Physical Review B 105, 214106 (2022).
  * Bleu  _et al._ [2018] O. Bleu, D. Solnyshkov, and G. Malpuech, Measuring the quantum geometric tensor in two-dimensional photonic and exciton-polariton systems, Physical Review B 97, 195422 (2018).
  * Klees  _et al._ [2021] R. L. Klees, J. C. Cuevas, W. Belzig, and G. Rastelli, Ground-state quantum geometry in superconductor–quantum dot chains, Physical Review B 103, 014516 (2021).
  * Gianfrate  _et al._ [2019] A. Gianfrate, O. Bleu, L. Dominici, V. Ardizzone, M. De Giorgi, D. Ballarini, K. West, L. Pfeiffer, D. Solnyshkov, D. Sanvitto, _et al._ , Direct measurement of the quantum geometric tensor in a two-dimensional continuous medium, arXiv preprint arXiv:1901.03219 (2019).
  * Tan  _et al._ [2019] X. Tan, D.-W. Zhang, Z. Yang, J. Chu, Y.-Q. Zhu, D. Li, X. Yang, S. Song, Z. Han, Z. Li, _et al._ , Experimental measurement of the quantum metric tensor and related topological phase transition with a superconducting qubit, Physical review letters 122, 210401 (2019).
  * Gianfrate  _et al._ [2020] A. Gianfrate, O. Bleu, L. Dominici, V. Ardizzone, M. De Giorgi, D. Ballarini, G. Lerario, K. West, L. Pfeiffer, D. Solnyshkov, _et al._ , Measurement of the quantum geometric tensor and of the anomalous hall drift, Nature 578, 381 (2020).
  * Palumbo and Goldman [2018] G. Palumbo and N. Goldman, Revealing tensor monopoles through quantum-metric measurements, Physical review letters 121, 170401 (2018).
  * Morozova and Chentsov [1989] E. A. Morozova and N. N. Chentsov, Markov invariant geometry on state manifolds, Itogi Nauki i Tekhniki. Seriya” Sovremennye Problemy Matematiki. Noveishie Dostizheniya” 36, 69 (1989).
  * Holevo [2003] A. S. Holevo,  _Statistical structure of quantum theory_ , Vol. 67 (Springer Science & Business Media, 2003).
  * Tóth and Petz [2013] G. Tóth and D. Petz, Extremal properties of the variance and the quantum fisher information, Physical Review A 87, 032324 (2013).
  * Botet and Jullien [1983] R. Botet and R. Jullien, Ground-state properties of a spin-1 antiferromagnetic chain, Physical Review B 27, 613 (1983).
  * Botet  _et al._ [1983] R. Botet, R. Jullien, and M. Kolb, Finite-size-scaling study of the spin-1 heisenberg-ising chain with uniaxial anisotropy, Physical Review B 28, 3914 (1983).
  * Peters  _et al._ [2009] D. Peters, I. McCulloch, and W. Selke, Spin-one heisenberg antiferromagnetic chain with exchange and single-ion anisotropies, Physical Review B 79, 132406 (2009).
  * Haldane [1983] F. D. M. Haldane, Continuum dynamics of the 1-d heisenberg antiferromagnet: Identification with the o (3) nonlinear sigma model, Physics letters a 93, 464 (1983).
  * Sørensen and Affleck [1993] E. S. Sørensen and I. Affleck, Large-scale numerical evidence for bose condensation in the s= 1 antiferromagnetic chain in a strong field, Physical review letters 71, 1633 (1993).
  * White and Affleck [2008] S. R. White and I. Affleck, Spectral function for the s= 1 heisenberg antiferromagetic chain, Physical Review B 77, 134437 (2008).
  * Sorensen  _et al._ [1994] E. S. Sorensen, I. Affleck, _et al._ , Equal-time correlations in haldane-gap antiferromagnets, Physical Review B 49, 15771 (1994).
  * Lambert and Sørensen [2019] J. Lambert and E. Sørensen, Estimates of the quantum fisher information in the s= 1 antiferromagnetic heisenberg spin chain with uniaxial anisotropy, Physical Review B 99, 045117 (2019).
  * Chen  _et al._ [2003] W. Chen, K. Hida, and B. Sanctuary, Ground-state phase diagram of s= 1 xxz chains with uniaxial single-ion-type anisotropy, Physical Review B 67, 104401 (2003).
  * Langari  _et al._ [2013] A. Langari, F. Pollmann, and M. Siahatgar, Ground-state fidelity of the spin-1 heisenberg chain with single ion anisotropy: quantum renormalization group and exact diagonalization approaches, Journal of Physics: Condensed Matter 25, 406002 (2013).
  * Pollmann  _et al._ [2010] F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Physical review b 81, 064439 (2010).
  * Nakano  _et al._ [2022] H. Nakano, H. Tadano, N. Todoroki, and T. Sakai, The haldane gap of the s= 1 heisenberg antiferromagnetic chain, Journal of the Physical Society of Japan 91, 074701 (2022).
  * Chen  _et al._ [2011] X. Chen, Z.-C. Gu, and X.-G. Wen, Complete classification of one-dimensional gapped quantum phases in interacting spin systems, Physical review b 84, 235128 (2011).
  * Zvyagin  _et al._ [2007] S. Zvyagin, J. Wosnitza, C. Batista, M. Tsukamoto, N. Kawashima, J. Krzystek, V. Zapf, M. Jaime, N. Oliveira Jr, and A. Paduan-Filho, Magnetic excitations in the spin-1 anisotropic heisenberg antiferromagnetic chain system nicl 2- 4 sc (nh 2) 2, Physical review letters 98, 047205 (2007).
  * Giamarchi  _et al._ [2008] T. Giamarchi, C. Rüegg, and O. Tchernyshyov, Bose–einstein condensation in magnetic insulators, Nature Physics 4, 198 (2008).
  * Note [1] Readers familiar with differential geometry will know that the expression for the volume in Eq. 18 can be stated equivalently in terms of the magnitude of the cross product of the tangent vectors (or in our case tangent operators), $\quantity|\Lambda_{\theta}\times\Lambda_{\phi}|$. This can be seen to be equivalent to the determinant of the quantum metric by considering the Lagrange identity, which, for vectors in $\mathbb{R}^{3}$, is given by, $|\vec{a}\times\vec{b}|^{2}=(\vec{a}\cdot\vec{a})(\vec{b}\cdot\vec{b})-(\vec{a}\cdot\vec{b})^{2}$. Consistency with the definitions in Eq. 18 and Eq. 17 is achieved by identifying the dot product with the covariance. This is also consistent with the form of the quantum metric as having matrix elements defined by the inner products of the tangent operators. It might be tempting to start from the expression $\quantity|\Lambda_{\theta}\times\Lambda_{\phi}|$ and assume that the correct notion of the cross product would be achieved by evaluating the commutator between the operators $\Lambda_{\theta}$ and $\Lambda_{\phi}$. In fact this is not the case, as taking such a commutator for the case of the staggered magnetization generator defined in Eq. 15 would yield the uniform magnetization operator, which would be inconsistent with the definition of the three tangent operators given in Eq. 16.
  * Do Carmo and Flaherty Francis [1992] M. P. Do Carmo and J. Flaherty Francis,  _Riemannian geometry_ , Vol. 6 (Springer, 1992).
  * Fishman  _et al._ [2020] M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations (2020), arXiv:2007.14822 .
  * Else  _et al._ [2012] D. V. Else, I. Schwarz, S. D. Bartlett, and A. C. Doherty, Symmetry-protected phases for measurement-based quantum computation, Physical review letters 108, 240505 (2012).
  * Raussendorf  _et al._ [2022] R. Raussendorf, W. Yang, and A. Adhikary, Measurement-based quantum computation in finite one-dimensional systems: string order implies computational power, arXiv preprint arXiv:2210.05089 (2022).
  * Auerbach [2012] A. Auerbach,  _Interacting electrons and quantum magnetism_ (Springer Science & Business Media, 2012).