Quantum phase transitions in a spin-1 antiferromagnetic chain with long-range interactions and modulated single-ion anisotropy

We study the phase diagram of spin-1 antiferromagnetic chain with isotropic antiferromagnetic interactions decaying with a power-law $\propto r^{-\alpha}$ ($\alpha\ge 1$) accompanied by modulated single-ion anisotropy. Employing the techniques of the density-matrix renormalization group, effects of long-range interactions and single-ion anisotropy on a variety of correlations are investigated. In order to check the consistency, the fidelity susceptibilities are evaluated across quantum phase transitions. The quantum critical points are faithfully detected and orders of phase transitions are determined. The correlation-length critical exponent is extracted from scaling functions of the fidelity susceptibility. The presence of long-range interactions leads to quantitative change of the phase boundaries and reduces the order of phase transition under certain conditions. A direct first-order transition between the periodic N\'eel phase and the large-$D$ phase occurs for slowly decaying antiferromagnetic interactions.


I. INTRODUCTION
The Haldane phase [1,2] of the one-dimensional (1D) antiferromagnetic (AF) spin-1 (S = 1) chain has received continued attention as it is closely related to the breaking of a hidden Z 2 ⊗ Z 2 symmetry [3] through a nonlocal unitary transformation. In fact, long-range order is absent in the isotropic 1D spin-1 chain with short-range exchange interaction but the ground state is characterized by a finite spectral gap and exponentially decaying AF spin correlations. The thoroughly studied gapped phase was conjectured by Haldane [1,2], and was confirmed in a series of experimental and theoretical papers [4][5][6][7][8]. An important benchmark of the Haldane phase is the occurrence of the nonlocal string order in the isotropic AF S = 1 Heisenberg chain [9]. At the same time, the string order is a manifestation of the topological hidden order, as pointed out by Kennedy and Tasaki [3].
Frustration may be introduced in the AF spin-1 chain by other interactions, and as a result novel types of order could emerge. For instance, Affleck, Lieb, Kennedy, and Tasaki (AKLT) proposed [10] an extended model of an S = 1 chain with additional biquadratic spin interaction, whose ground state is exactly solvable and gives the phase depicted in Fig. 1(a)]. The fourfold ground state degeneracy arises due to two free edge S = 1/2 spins for a chain with an open boundary condition and provides crucial insight into symmetry-protected topological (SPT) order. Such a general bilinear-biquadratic Hamiltonian leads to a dimerised spin-1 chain for a range of parameters [10]. Note that the biquadratic spin-spin interaction, ∝ (S i · S i+1 ) 2 , arises naturally within the S = 1 Heisenberg model describing the strong coupling theory of the * wlyou@nuaa.edu.cn iron pnictides [11][12][13].
Yet, perhaps the simplest frustrated model for the AF spin-1 chain includes both AF nearest neighbor and nextnearest neighbor interactions-it has been shown [14] that the latter dilutes the AKLT phase by singlet bond correlations [15]. Here we go further and analyze the frustrated spin-1 chain with infinite range AF correlations which decay algebraically in presence of the superposition of uni-form and alternating uniaxial single ion anisotropy. We show below that such a model gives an interesting competition of spin correlations with those encountered in the Haldane phase.
We emphasize that the long-range interactions can not be considered as perturbations and play essential role in the critical phenomena [41][42][43][44][45][46][47][48][49]. Recently two of us considered a chain with anisotropic long-range decaying interactions and investigated its QPTs [50]. Interesting phase diagrams were established, with the long-range interactions of the z component resulting in a Wigner crystal phase, and the transversal one resulting in a symmetry broken phase. The present work addresses primarily the QPTs in the spin-1 chains with long-range interactions and modulated single-ion anisotropy. Using a combination of the density-matrix renormalization group (DMRG) calculations and spin-wave analysis, various correlations for a spin-1 chain with long-range interactions were presented. The concept of fidelity susceptibility has been successfully used to classify QPTs in the spin-1 AF chain in the past [24,25,[51][52][53][54][55][56][57][58][59]. We employ here the fidelity susceptibility and establish a rich phase diagram of spin-1 chain by changing the uniform and alternating single-ion anisotropy.
The remainder of this paper is organized as follows. We introduce the S = 1 Heisenberg model with long-range interactions and alternating single-ion anisotropy in Sec. II. The details of numerical methods and information metric are also introduced. In Sec. III, effects of long-range interactions and modulated single-ion anisotropy are investigated. Using fidelity susceptibility, the phase diagrams for various cases are determined. The characteristics of each phase are interpreted by diverse correlations and effective Hamiltonians in various limiting cases. The discussion and summary are presented in the last Sec. IV.

A. Frustrated Heisenberg chain
The Hamiltonian of the spin-1 Heisenberg chain of length of N sites with long-range interactions and modulated single-ion anisotropy is given by where S i is the spin-1 operator on site i. We consider spin states using the following notation |↑ , |0 , and |↓ for the single-site states with S z i = 1, 0, and −1, respectively. The interactions J ij between two spins decay algebraically with distance r = |i − j| ≥ 1, i.e., as and we take α ≥ 1. The parameters D 1 and D 2 stand for uniform and alternating components of uniaxial single-ion anisotropy, respectively. For convenience, we take J = 1 for nearest neighbors (|i − j| ≡ 1) and use open boundary conditions in this study. Note that for D 1 = D 2 one finds single-ion anisotropy only at every second site.

B. Order parameter and spin correlations
According to Ginzburg-Landau scenario, a well defined order parameter is a vital ingredient for characterizing the nature of a phase. In order to characterize the QPTs, the two-point correlations and the nonlocal string order parameter (SOP), are defined. The difficulty in defining suitable order parameters in miscellaneous phases therein motivate us to adopt instead widely accepted information measures. As a Riemannian metric in the parameter space, quantum fidelity susceptibility is intimately related to quantum fluctuations and dissipative responses of the system. Consider the Hamiltonian H(λ) with a set of external parameters, λ = {λ 1 , λ 2 , · · · , λ κ }, where κ is the dimension of the parameter space. The quantum geometric tensor describes the geometric structure upon projecting the dynamics onto the (non-degenerate) ground state ψ 0 (λ), given by The imaginary part of the geometric tensor Im{χ µ,ν } is related to the Berry curvature, and the real part Re{χ µ,ν } is dubbed as quantum fidelity susceptibility which measures the change rate of the distance between the two closest states as a driving parameter λ µ is modulated [60]. The quantum geometric tensor has been experimentally extracted by using a superconducting qubit [61], coupled qubits in diamond [62], and exciton-polaritons in a planar microcavity [63].
As an information metric in the Hilbert space, quantum fidelity susceptibility has a gravity dual with the spatial volume of the Einstein-Rosen bridge in anti-de Sitter (AdS) space [64]. The sensitivity is greatly enhanced especially for the system at the quantum criticality comparing with that away from the critical region [65]. The divergence of χ µ,µ (5) (in what follows we use an abbreviation χ for this quantity) can directly signal a QPT and locate the quantum critical points. Quantum fidelity susceptibility has been proved to play a role of universal order parameter in identifying the QPTs [66][67][68].
Both of these observations require us to calculate the ground state or the reduced density matrix of the system. Based on matrix product states, the finite-size DMRG technique was adopted [69][70][71]. In the numerics we keep up to m = 300 eigenstates during the procedure of basis truncation and the number of sweeps is n = 50. These conditions guarantee that the simulation converges sufficiently fast and the truncation error is smaller than 10 −9 . Remarkably for the AF Heisenberg model with nearest neighbor interactions (not shown for a general case), numerical studies revealed that the Haldane phase creeps in between the periodic Néel phase, the Néel phase and the large-D phase. This limit is reached with increasing value of α from a finite value-then the Haldane phase is squeezed, and a direct QPT from the large-D phase to the periodic Néel phase occurs for large D 1 through a 1 st order QPT. The critical line of the periodic Néel to large-D transition will occur at D 1 = D 2 for sufficiently large single-ion anisotropy D 1 .

A. Phase diagrams
A tricritical point emerges at moderate D 1 for α = 1, see Fig. 2(c). However, there is still no direct transition between the Néel phase and the periodic Néel phase, which emerges from the Haldane state by lowering specific Néel-type spin configurations compatible with the spatial modulation of single-site anisotropy. A narrow Haldane phase would survive in the neighborhood of the line D 1 + D 2 = c 0 , i.e., at D 2 = −D 1 + c 0 , where c 0 is a constant and depends on the parameter α.
To better understand such rich phase diagram of Fig. 2 that results from the interplay of long-range AF couplings and alternating single-ion anisotropies, we first consider the cases with either D 2 = 0 or D 1 = 0, see Fig. 3. For α = ∞ (the AF Heisenberg model) , the system reduces to a spin-1 chain with nearest-neighbor interaction, and it is in the Haldane phase at the isotropic point (D 1 = 0 and D 2 = 0), which is composed of the superposition of states with hidden nonlocal AF order. At D 2 = 0 one finds two QPTs by increasing the value of D 1 , from the Néel phase through Haldane phase to the large-D phase, see Fig. 3(a). By going upwards vertically at D 1 = 0, there is only one QPT for increasing D 2 , from the Haldane phase to the periodic Néel phase, see Fig. 3(b). Figure 4 shows spin-spin correlation functions for the different phases found in the phase diagram of Fig. 2. In Fig. 4(a) one observes that both spin correlation functions, C z 1,i and C x 1,i , vanish fast over a separation of just a few sites, while the nonlocal O z 1,i (4) converges quickly to 0.452 (0.358) for odd (even) site i, manifesting the existence of the Haldane phase. The odd-even effect follows from the open boundary condition. At D 1 = 2 and D 2 = 0 one finds the large-D phase, see Fig. 4(b).
Some specific AF order will be favored by varying the single-site anisotropy. Changing the sign of D 1 , one finds the Néel phase at D 1 = −2, see Fig. 4 S z 2i−1 ≈ ±1 and those on the even sites are confined to S z 2i = 0. In Fig. 4(d) this state is shown for D 1 = 0 and D 2 = 4. Indeed, two-spin correlations C z 1,i oscillate periodically between the values being close to −1, 0, and 1, and C x 1,i ≈ 0. To elucidate the dominated configurations, the effective interactions between S z 2i−1 and S z 2i+1 as the first-order perturbation in J plays the leading role, which for a chain of even length N is of the form, The crucial point here is the AF coupling J 13 between next nearest neighbor spins at odd sites which triggers the periodic Néel phase, being |...↑0↓0↑0↓ ... [cf. Fig. 1(d)];

B. Spin correlations and fidelity susceptibility
The phase transition between the Haldane phase and the periodic Néel phase can be accurately determined by the analysis of the fidelity susceptibility. The results for D 1 = 0 and α = 1 are shown in Fig. 5(a). With increasing D 2 , a peak of the ground-state fidelity susceptibility is observed at D * 2 1.84, which signals approaching to the transition in the thermodynamic limit, N → ∞. Further evidence for identifying the QPT is provided by the results of the second derivative of the ground-state energy density, (d 2 e/dD 2 2 ), which is shown in Fig. 5(b). We thus confirm that the Haldane-to-periodic-Néel transition is a QPT of 2 nd order.
According to the finite-size scaling theory [72], the position of the maximal points of the fidelity susceptibility can be fitted by the following formula: where b is a constant given by the critical exponent ν, b ≡ 1/ν, and D c 2 is the quantum critical point in the thermodynamic limit. Accordingly, the scaling of the extremal points of the fidelity susceptibility gives rise to D c 2 = 1.845, b 1 1, as is shown in the inset of Fig. 5(a). For a 2 nd order QPT, it is shown that the fidelity susceptibility at the peak point for finite size N behaves as where µ is the critical adiabatic dimension. The critical exponent ν of the correlation length can be obtained from µ, ν = 2/µ. As is disclosed in Fig.  6(a), the linear dependence of log-log plot suggests that µ = 1.94 and ν = 1.03. This illustrates that the Haldaneto-periodic-Néel QPT belongs to the Ising universality class [73,74]. We recall that the phase diagram of the model Hamiltonian (1) for D 1 = 0 is shown in Fig. 3(a), and the whole critical line corresponds to 2 nd order QPT. For the AF Heisenberg model (at α = ∞), the Haldaneto-periodic-Néel transition occurs at D 2 3.30 [20,58]. It is seen that the critical point D c1 2 drops when α decreases. Analogously, we consider the uniform case with D 2 = 0. Although the AF long-range interactions are frustrated, the system would be reminiscent of Haldane phase when D 1 = 0 and α ≤ 1 [47]. The system is in the Haldane phase for α = ∞ and −0.31 < D 1 < 0.99 [19,21,25,51]. For D 1 −J, all spins are restricted to be in the states S z i = ±1. When α = ∞, the effective coupling between the nearest neighbor spins are obtained within the first order perturbation theory in J as [20], In this regard, the system is in the |↑↓↑↓ -type Néel phase for a strong easy axis single-site anisotropy D 1 < −0.31. In the Néel phase, the two-spin correlations C z 1,i (−1) i−1 and C x 1,i , decay exponentially over the distance between two spins, as shown in Fig. 4(c). In the opposite limit, i.e., with D 1 J, spins will be confined to be in the state | S z i | = 0, annotating that the system enters the large-D phase for a sufficiently large easy plane anisotropy, D 1 > 0.99. In the large-D phase, the correlations C z 1,i , C x 1,i , and O z 1,i vanish, see Fig. 4(b). In order to determine the phase boundary with high accuracy, we also calculated the fidelity susceptibility for α = 1 and D 2 = 0 of the ground state for a system size N up to 120. The ground state fidelity susceptibility per site χ D1,D1 /N is plotted as a function of the parameter D 1 for different sizes N in Fig. 7(a). Two peaks in the fidelity susceptibility can be seen and they both increase with increasing system size. This implies the divergence of fidelity susceptibility in the thermodynamic limit, which suggests the occurrence of two successive QPTs. We can conclude that the left peak indicates the Néel-to-Haldane transition and the right peak indicates the Haldane-tolarge-D transition. We then plot the location of the maximum fidelity susceptibility as a function of 1/N and show the numerical fits in the inset of Fig. 7(a). We obtain that D c1 1 = −0.688, b 1 1.05, and D c2 1 = 0.805, b 2 1.00 according to Eq. (7).
Further evidence which indicates QPTs is provided by the second derivative of the ground state energy density (d 2 e/dD 2 1 ) in Fig. 7(b). One observes that two peaks of (d 2 e/dD 2 1 ) become more pronounced for increasing system size N , which means that both of the phase transitions are of 2 nd order. As is shown in Fig. 6(b), the linear dependence of the log-log plot gives rise to µ = 1.93 and ν = 1.03 for the Haldane-to-Néel QPT with α = 1, D 2 = 0. The Haldane-to-Néel phase transition belongs to the Ising universality class for all values of α. However, it was reported that the Haldane-to-large-D transition may be of higher order larger than two for the nearest neighbor model (α = ∞) [25]. The long-range interaction would then reduce the order of the transition. The values of critical points of Haldane-to-large-D and Haldane-to-Néel shift to a lower value when α decreases, as is shown in Fig. 3(b).
After considering the special cases, we then speculate that both the easy-axis and easy-plane D-terms coexist in the Heisenberg chain. The correlation functions and the SOP for D 2 = 3 are plotted as a function of D 1 with α = 1 in Fig. 8(a). One finds C z 1,N/2−1 1 and C z 1,N/2 −1 when D 1 < −2.5. After surpassing the critical point, the z-component correlations vanish. When D 1 > −1.5, the correlation C z 1,N/2−1 rebounds but C z 1,N/2 remains vanishing, which means that the system enters the periodic Néel phase. Furthermore, C z 1,N/2−1 vanishes suddenly at the periodic-Néel-to-large-D transition point.
We also studied the SOP to characterize the phase in the range −2.5 < D 1 < −1.5 in Fig. 8(b). It is noted that the SOP O z 1,i ≈ |C z 1,i | in the Néel phase. In order to determine the phase boundaries, we adopt O z 1,N/2−1 − |C z 1,N/2−1 | to identify the Haldane phase, in which the long-distance correlation C z 1,i disappears in the Haldane phase [75].

IV. CONCLUSIONS
In this paper, we investigated the quantum phase transitions in the one-dimensional spin-1 chains with long-range antiferromagnetic interactions decaying with a power law and modulated single-ion anisotropy by using the density-matrix renormalization group technique. Together with the short-range correlations and the nonlocal string order parameter, the ground-state fidelity susceptibility was employed to determine the phase diagram and critical phenomena.
The presence of long-range interactions increased the difficulty of simulating the system numerically. Nevertheless, we provided compelling evidence for the phase transitions and critical lines in the thermodynamic limit. We identified four phases including the Haldane, large-D, the |↑↓↑↓ Néel phase, and the |↑0↓0 periodic Néel phase. The appearance of long-range interactions modi-fies the phase boundaries and the order of phase transition comparing with their counterparts with short-range interactions and leads to a direct 1 st order transition between periodic Néel and large-D phase. However, a narrow Haldane phase survives between the periodic Néel and the Néel phase as the long-range interactions increase.
In summary, employing the scaling functions of the fidelity susceptibility gives a numerically economical way of obtaining accurately the correlation-length critical exponents. We find that both the Haldane-to-periodic-Néel and Haldane-to-Néel quantum phase transitions are of 2 nd order and are classified for the model in Eq. (1) as belonging to the Ising universality class. The local correlations are capable of characterizing those phases as topological trivial ones, while only the nonlocal string order parameter can identify the topological phase. More precisely, the difference between the string order parameter and the two-point spin correlation, O z 1,N/2−1 − |C z 1,N/2−1 |, can exclusively detect the Haldane phase. We remark that the order of the Haldane-to-large-D transition is higher than two for the nearest neighbor model (α = ∞) and will drop to two as α decreases. Further studies of the spin ordered phases at the presence of long-range interactions is experimentally challenging and could also bring about some novel types of phase transitions.