Unconventional Quantum Criticality in Long-Range Spin-1 Chains: Insights from Entanglement Entropy and Bipartite Fluctuations

Abstract: We study the ground-state phase diagram of a spin-1 Heisenberg chain with staggered long-range (LR) interactions decaying as $\propto r<sup>{-α}$ using a quantum Monte Carlo approach based on the split-spin representation. This formulation enables efficient large-scale simulations by mapping the spin-1 model onto spin-$1/2$ degrees of freedom with local projection constraints. We resolve the continuous quantum phase transition between the gapped Haldane phase at large $α$ (short-range regime) and a gapless antiferromagnetically ordered Néel phase at small $α$ (LR regime), where the continuous SU(2) symmetry is broken. From finite-size scaling and crossing point analyses, we determine the critical point to be at $α_c = 2.48(2)$ and extract the associated critical exponents, which indicate unconventional criticality. In particular, the transition is found to be nonconformal, characterized by a dynamic exponent $z \neq 1$. We further analyze the scaling of entanglement entropy and bipartite fluctuations across the transition, and determine the corresponding universal scalings in both phases and at criticality.

• The paper locates a single transition at αc=2.48(2) between a gapped Haldane phase and a gapless Néel phase using sign-problem-free QMC, while finding no clear evidence for an intermediate phase.
• The study measures unconventional critical exponents β=0.27(1), ν=1.81(5), and z=0.74(1), showing that the transition is nonconformal and falls outside established one-dimensional WZW universality classes.
• Rényi entropies and bipartite fluctuations distinguish the phases: the Haldane phase obeys an area law, the Néel phase shows power-law growth, and the critical point displays logarithmic scaling despite its nonconformal dynamics.

Model and scope

The paper studies the one-dimensional spin-1 Heisenberg chain with unfrustrated staggered long-range (LR) couplings,

$$H=\sum_{i<j}\frac{(-1)^{j-i+1}}{|j-i|^{\alpha}}\,\mathbf{S}_i\cdot\mathbf{S}_j,$$

using sign-problem-free stochastic series expansion quantum Monte Carlo (QMC) with 1D Ewald summation for the LR tail. The central question is how the Haldane phase, stable at short range, gives way to a gapless Néel phase that breaks continuous SU(2) symmetry once interactions decay slowly enough — a scenario forbidden at short range by the Mermin-Wagner theorem but permitted by sufficiently slow decay. While the $S=1/2$ analogue has been characterized extensively, the spin-1 case had only been touched upon in prior density-matrix renormalization group work, leaving both the location and nature of the critical point open.

Methodologically, the authors map each spin-1 onto two auxiliary spin-1/2 degrees of freedom projected onto the triplet sector via an on-site projector $$P_i = \tfrac34 + \mathbf{s}_{i,1}\cdot\mathbf{s}_{i,2}$$. This split-spin representation embeds the model into standard spin-1/2 SSE machinery: the linked-vertex configuration space acquires one fixed projector vertex per site, while diagonal updates and directed-loop equations reduce to the bounce-free Heisenberg solution. This enables simulations up to $L=1024$, large enough to control finite-size effects that are severe near criticality.

Locating the critical point

Two independent dimensionless observables are used to bracket the transition: the Binder cumulant of the Néel order parameter $m$ and a string-order-parameter (SOP) ratio built from the Fourier transform of $O^{\mathrm{SOP}}(r)$, whose finite-size extension to $r=0,1$ is benchmarked against exact diagonalization. Crossing points between sizes $(L,2L)$ are extrapolated as $\alpha_*(L)-\alpha_c \sim L^{-(1/\nu+\omega)}$, giving $\alpha_c^U = 2.48(2)$ from the Binder cumulant and $S=1/2$0 from the SOP ratio. The agreement within error bars implies that any putative intermediate phase is confined to a narrow window of $S=1/2$1, if it exists at all; the paper adopts $S=1/2$2.

Finite-size scaling collapses of $S=1/2$3 and of the SOP ratio yield $S=1/2$4 with $S=1/2$5 from the order parameter, and $S=1/2$6 from the SOP ratio. An independent crossing-point analysis extracting the correction exponent $S=1/2$7 and the combination $S=1/2$8 gives $S=1/2$9, consistent with the collapse value. These exponents do not match any known 1D universality class — in particular not the SU(2)$$P_i = \tfrac34 + \mathbf{s}_{i,1}\cdot\mathbf{s}_{i,2}$$0 Wess-Zumino-Witten (WZW) class ($$P_i = \tfrac34 + \mathbf{s}_{i,1}\cdot\mathbf{s}_{i,2}$$1), which would require $$P_i = \tfrac34 + \mathbf{s}_{i,1}\cdot\mathbf{s}_{i,2}$$2 and $$P_i = \tfrac34 + \mathbf{s}_{i,1}\cdot\mathbf{s}_{i,2}$$3. The transition is therefore unconventional already at the level of static exponents.

Nonconformal dynamics

Because the Hamiltonian conserves total $$P_i = \tfrac34 + \mathbf{s}_{i,1}\cdot\mathbf{s}_{i,2}$$4, fixed-magnetization sectors can be targeted directly, allowing gap extraction without spin-gap subtraction ambiguities. At criticality the gap scales as $$P_i = \tfrac34 + \mathbf{s}_{i,1}\cdot\mathbf{s}_{i,2}$$5 with $$P_i = \tfrac34 + \mathbf{s}_{i,1}\cdot\mathbf{s}_{i,2}$$6, clearly below the Lorentz-invariant value $$P_i = \tfrac34 + \mathbf{s}_{i,1}\cdot\mathbf{s}_{i,2}$$7. Consistently, the thermodynamic-limit Haldane gap closes as $$P_i = \tfrac34 + \mathbf{s}_{i,1}\cdot\mathbf{s}_{i,2}$$8 with $$P_i = \tfrac34 + \mathbf{s}_{i,1}\cdot\mathbf{s}_{i,2}$$9, matching the product of independently measured exponents. The conclusion is direct: the QCP is not described by any conformal field theory, since conformal invariance requires $L=1024$0. This places the spin-1 Haldane-to-Néel transition in the same nonconformal category as the $L=1024$1 quasi-long-range-order-to-Néel transition, where $L=1024$2 was also reported.

Entanglement entropy scaling

Second Rényi entropies are computed via the replica trick with the nonequilibrium incremental algorithm on half-chain cuts. Three regimes emerge:

Regime	Scaling of $L=1024$3
Néel phase ($L=1024$4)	$L=1024$5, prefactor varies continuously with $L=1024$6
Critical point	Logarithmic, $L=1024$7
Haldane phase ($L=1024$8)	Area law; fitted prefactor extrapolates to zero

Deep in the ordered phase the prefactor approaches the tower-of-states expectation, e.g. $L=1024$9 at $m$0. The striking result concerns the critical prefactor: despite $m$1 and the mismatched static exponents, $m$2 sits remarkably close to the SU(2)$m$3 WZW value $m$4. The authors note this echoes a similar coincidence found in matrix-product-state studies of the $m$5 analogue, suggesting the entanglement content alone does not diagnose the full universality class here — a tension worth emphasizing rather than resolving.

Bipartite fluctuations

Bipartite fluctuations of the conserved $m$6 provide a complementary diagnostic. In the Néel phase, $m$7 grows as a power law $m$8 with an exponent that tracks the linear spin-wave prediction $m$9 well deep in the ordered regime, deteriorating toward criticality. At $O^{\mathrm{SOP}}(r)$0 the scaling crosses over to logarithmic form with coefficient $O^{\mathrm{SOP}}(r)$1; the log ansatz is favored over power-law fitting by reduced chi-square ($O^{\mathrm{SOP}}(r)$2 versus $O^{\mathrm{SOP}}(r)$3). In the Haldane phase, $O^{\mathrm{SOP}}(r)$4 saturates, consistent with an area law. The overall picture — logarithmic entropy, power-law fluctuations in the ordered phase, saturation in the gapped phase — matches spin-wave theory away from the critical region, validating the sublinear magnon dispersion $O^{\mathrm{SOP}}(r)$5 with $O^{\mathrm{SOP}}(r)$6 as the correct low-energy description of the ordered side.

Consistency checks and comparison

A hyperscaling check using $O^{\mathrm{SOP}}(r)$7 with the mean-field estimate $O^{\mathrm{SOP}}(r)$8 yields a residual of about $O^{\mathrm{SOP}}(r)$9, comparable to the $r=0,1$0 case (~0.017), supporting internal consistency of the exponent set. Linear spin-wave theory predicts the onset of Néel order at $r=0,1$1, roughly 10% above the QMC value — the same relative error as for $r=0,1$2 — indicating that semiclassical theory systematically overestimates the stability of magnetic order. A related study combining matrix-product states with series expansions reports consistent physics except for $r=0,1$3: adopting their $r=0,1$4 would worsen the hyperscaling residual to ~0.134, so the discrepancy in $r=0,1$5 between the two works remains an unresolved quantitative issue.

Limitations and open questions

Several caveats qualify the results. First, the exclusion of an intermediate phase rests on the agreement of two crossing estimates within error bars; a narrow intermediate regime cannot be strictly ruled out. Second, the anomalous exponent $r=0,1$6 was not measured directly but taken from the mean-field relation $r=0,1$7, so the hyperscaling test inherits this assumption. Third, the closeness of $r=0,1$8 to the SU(2)$r=0,1$9 WZW value is numerically suggestive but theoretically unexplained given the manifest violation of conformal invariance; whether this reflects an emergent entanglement-level constraint or finite-size coincidence is left open. Finally, the disagreement on $(L,2L)$0 with the complementary MPS-based study indicates that correlation-length exponents near $(L,2L)$1–$(L,2L)$2 remain difficult to pin down, likely due to strong corrections to scaling with $(L,2L)$3.

Conclusion

This work establishes the ground-state phase diagram of the staggered LR spin-1 chain, locating a single unconventional QCP at $(L,2L)$4 between the gapped Haldane phase and a gapless Néel phase. The critical exponents ($(L,2L)$5, $(L,2L)$6, $(L,2L)$7) place the transition outside both the WZW conformal classes and known mean-field regimes, extending the phenomenology of nonconformal LR quantum criticality from $(L,2L)$8 to integer spin. The systematic characterization of Rényi entropy and bipartite fluctuation scalings across all three regimes provides universal benchmarks, and the demonstrated feasibility of large-scale sign-free QMC for higher-spin LR models, together with recent Rydberg-platform realizations of continuous symmetry breaking in LR chains, makes these exponents and entanglement coefficients directly testable experimentally.