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State Space Geometry of the Spin-1 Antiferromagnetic Heisenberg Chain

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, 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.

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, where p(x|λ) is a probability distribution (which we hereafter refer to as a state) that depends on some parameterization, λ, of the state space. F µν quantifies how distinguishable the distribution p(x| λ) is from the distribution, p(x|λ + dλ) [3]. The quantum generalization holds for both pure [4] and mixed [5] states, and is termed quantum Fisher information (QFI) with the density matrixρ taking the place of p(x|λ), 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], where, with Λ µ and Λ ν the hermitian operators that generate the unitary parameterization for the state |ψ via, While the trace of the QFI matrix for a suitable choice of parameterization obeys the following bound [10,11], Tr{Q} > 2mA. (5) 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 ψ 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 A). 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 λ with the ground state at λ + dλ, where the parameter coupled to λ 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 [12,32]. 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 µν 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 [26,27,32,[38][39][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. 5 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 B). 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.

A. 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, is considered, wheren is a unit vector whose orientation may, in the most general case, be site dependent. We call Eq. 6 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, Λ] is maximal. Often at the critical point this will be the most relevant operator [12].
Once the orientations ofn r are fixed on each site Λ may be used to define a family of three operators, corresponding to the three orthogonal spin directions on each site. Calling the original orientation Λ 3 (whatever the orientation ofn r is on a given site), the other two orientations Λ 1 and Λ 2 , can be found by a suitable global rotation of all of the unit vectorsn r . The diagonal entries of the 3 × 3 QFI matrix associated with the generators in Eq. 6 are given up to the constant A by, Once we have chosen our set n r and hence our three generators we can consider the quantum metric at the state |ψ associated with the three dimensional parameterization generated by the vector of operators Λ. The corresponding QFI matrix, Q (see Eq. 2) can be used to define an intensive, mean QFI density [10] 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) byf > 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. 8 can be arrived at by considering the operator defined in Eq. 6 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| λ) →ρ( λ), whereρ is a density matrix, and replace the integral over x by a trace operator. This introduces an ambiguity due to the non commutativity of ρ with ∂ λµρ . 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), L µ , via [51], where {·, ·} is the anti-commutator. The generalization of Eq. 1 is then given by, 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 1 Af , and thus avoid the need to worry about the particular value of the metric constant.

B. 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 , 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 lower z 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 upper z 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].

II. METHODS
A. Quantum volume and curvature

Defining a manifold
In order to investigate the state space geometry in the vicinity of many body wavefunctions we consider the twodimensional 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 Λ nr defined in Eq. 6 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. 6, 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, with the state manifold (actually a submanifold) given by P = h>0 Ω h . The spin-1 Heisenberg chain commutes with the uniform magnetization operator, and so the QFI for the generator withn 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 = π [58,59]. From this point on we assume that the generator is the staggered field operator, wheren = (cos(θ) sin(φ), sin(θ) sin(φ), cos(φ)). The superscript π is omitted in the following discussion. For a given orientation of the vectorn, the generators of the three orthogonal transformations are given by which we hereafter refer to as tangent operators. In order to examine the curvature of the manifold P, we are interested in the scaling of the volume of the two dimensional submanifolds, Ω h , and thus we are particularly interested in Λ θ and Λ φ . This manifold is schematically depicted in Fig. 1. Λ 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 (θ, φ) we don't consider the three dimensional metric that would include Λ 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 [26,27,32,[38][39][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. 13, the real symmetric part of the QFI matrix corresponding to the generators for transformations in θ and φ gives the metric at each point on Ω h , From the metric, the quantum volume of the state space manifold Ω h can be calculated, where S = [0, 2π) × [0, π). Here we use the word volume to refer to the size of the 2D state space manifold [68].
In the limit h → 0 the QFI matrix used to define the mean QFI density in Eq. 8 can be recovered by considering perturbations in the x, y, and z directions as defined by the generators Λ µ with µ ∈ {x, y, z} discussed in Sec. I A. The mean QFI density is then computed by evaluating the three connected covariances in Eq. 7 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 Λ θ and Λ φ for states that are held at some small, non-zero field h with orientation (θ, φ).

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. 12). 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 = 1 2 the QFI matrix is, 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, (a detailed derivation is given in App. B) this expression can be generalized to the case of spin-S (see App. C). The volume of the quantum paramagnetic has no dependence on the relative orientations ofn 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. 20 which goes to zero in the classical limit, S → ∞. 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.

Quantum Curvature
The notion of state space curvature can be developed by considering the ratio of the quantum volume of the manifold Ω h centered on a ground state ψ relative to the quantum volume in an empty space. The quantum volume ratio, may be expanded about small values of h, where in the quadratic term determines the curvature of the manifold 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 = 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 ψ, we mean the quantum volume of the spheroidal shell centered at ψ in the state space.

B. 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 r1,r2 , N r1=1 N r2=1 C r1,r2 . In systems with translation invariance we can assume that C r1,r2 = C |r1−r2| and reduce the double sum to N N r=1 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. 17, 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, 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, Here A ab is the real part of the correlation matrix, given by A = 1 2 (C+C * ) and the indices a, b, c, d ∈ {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 λ. In Eqs. 22, the angular dependence in (θ, φ) has been integrated out completely in the spherical case, while in the axial case Ξ will have some dependence on φ. In the axial case the integral over φ can be performed numerically with relatively few integration points. For the data shown in Figs. 5 6 the φ 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.

A. Mean QFI Density
Taking Eq. 11 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 ∈ [−0.5, 1.5] and B z ∈ [0, 0.5]. In this section we imagine a 3 × 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 A. The correlation functions are evaluated for the ground state of the source Hamiltonian, taking h = 0 in Eq. 12. 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 lower z ≈ 0.41, which is consistent with previous studies [57] which place the value of B lower z 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 upper z ≈ 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 NiCl 2 -4SC(NH 2 ) 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 M z /N ≈ 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 M z /N ≈ 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 [17,60], 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].
In contrast to a homogeneous applied field, any nonzero 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 A.
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.

B. 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. 12. The correlation functions then in general depend on both h and the orientation of the field (θ, φ). Due to the symmetries of the spin-1 chain defined in Eq. 11, we can integrate If all correlations S a r1 S b r2 for a given field strength h and orientation (θ, φ) are evaluated with a, b ∈ {x, y, z} in the lab frame then Eq. 22 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. 12 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 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. 21 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.
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 → ∞. 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. 5). Hence, quantum phases might have enhanced fluctuations without necessarily exhibiting any entanglement.
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 [27,32,40] 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 [17,60] 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 max z ≈ 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 → 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 andf in this work, leaving the problem to future study.

ACKNOWLEDGMENTS
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]. In order to efficiently evaluate Eq. 17 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 (θ, φ). Recall the rotation operator [73], 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 H, Each element of the quantum metric is given by the real symmetric part of the covariance of the generators, as defined in Eq. 16. Using the particular form of the generator in Eq. 15 and denoting the real symmetric part of the covariance matrix A gives, where A = 1 2 (C + C * ), and Under a rotation the covariance matrix transforms where Einstein summation is used and, 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 where, For the axial case we, without loss of generality, choose the case where the system is symmetric about rotations about the z axis (set φ = 0 in Eq. A5). In this case we have, Ξ xxyy cos 2 (φ) sin 2 (φ) − 2Ξ yyzx cos(φ) sin 3 (φ) + Ξ yyzz sin 4 (φ) dφ (A8) In the axial case we still have to perform a numerical integration with respect to φ 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. 12 setting H 0 = 0. The tangent operators are given by, Where d µ is the derivative of the unit vector along the µ direction. Substituting these definitions into Eq. 2 (assuming summation over repeated indices), 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, Now substituting this Eq. B2 gives, Now we must evaluate the expectation values of the Pauli matrices. Recall that these are themselves functions of (θ, φ). For the two level case we can compute the ground state explicitly, The expectation value of the Pauli vector in the ground state is then, The metric in Eq. B4 is now given by, In this form, evaluating the components of the metric proves to be a straightforward exercise Generalizing this expression to N spins amounts to multiplying by N Q = AN h 2 1 −i sin(φ) i sin(φ) sin 2 (φ) (B9) 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 = 1 2 , the quantum volume (surface area of a 2D sphere in the spin Hilbert space) is then, 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. B10 to the case of spin-S. This is easy to do by employing the formula for the spherically symmetric quantum volume given in Eq. A6 Using Eq. A6 we can compute the quantum volume of a paramagnet. Taking H 0 = 0 in Eq. 12 and denoted the ground state of the spin-S paramagnet as |S, −S , we can use the raising and lowering operators, 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, Substiting this into Eq. A6 gives, where we have group the spin, S, metric constant A, and number of sites N along with the factor of 1 2 in order to emphasize the familiar form of the volume of a sphere. The factor N is coming from the fact that A ab in Ξ 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 = 1 2 we recover Eq. B10.