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arXiv:2604.20831

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

Justin Tim-Lok Chau, Jiarui Zhao, Nicolas Laflorencie, Zi Yang Meng

cond-mat.str-elquant-phSubmitted Apr 22, 2026• Announced Apr 23, 2026• 0 citations• Revised Jul 7, 2026 (v2)
v1 · Apr 22, 2026v2 · Jul 7, 2026v3 · Jul 22, 2026
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Abstract

We study the ground-state phase diagram of a spin-1 Heisenberg chain with staggered long-range (LR) interactions decaying as $\propto r^{-α}$ 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 $α_c = 2.49(1)$ and extract the associated critical exponents, which indicate unconventional criticality. In particular, the transition is found to be nonconformal, characterized by a dynamical 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.

Comments
6+10 pages, 5+7 figures

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•Dec 29, 2020•1 citation
We study the ground-state phase diagram of a spin-1 Heisenberg chain with staggered long-range (LR) interactions decaying as ∝r−α\propto r^{-α}∝r−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-1/21/21/2 degrees of freedom with local projection constraints. We resolve the continuous quantum phase transition between the gapped Haldane phase at large ααα (short-range regime) and a gapless antiferromagnetically ordered Néel phase at small ααα (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 αc=2.49(1)α_c = 2.49(1)αc​=2.49(1) and extract the associated critical exponents, which indicate unconventional criticality. In particular, the transition is found to be nonconformal, characterized by a dynamical exponent z≠1z \neq 1z=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.
Long-range interactions can fundamentally reshape the low-energy properties of low-dimensional quantum matter, altering both continuous symmetry breaking and topological phenomena. However, their impact on the quantum criticality separating these regimes remains poorly understood. We determine the ground-state phase diagram and critical properties of the spin-one Heisenberg chain with single-ion anisotropy and staggered antiferromagnetic power-law interactions, using matrix-product state (MPS) calculations complemented by high-order series expansions (pCUT+MC). Such long-range, non-frustrated interactions circumvent the Hohenberg-Mermin-Wagner theorem, thereby stabilizing continuous symmetry breaking (CSB) phases in direct competition with the Haldane phase. We map out the resulting phase diagram and analyze the entanglement entropy scaling behavior in the U(1) and SU(2) CSB phases, finding logarithmic corrections beyond the short-range, universal contributions expected from linearly dispersed Goldstone modes. We further characterize all critical boundaries through finite-size scaling of either the entanglement entropy or the staggered magnetization. In particular, the large-D-to-U(1)-CSB transition exhibits unconventional, continuously varying critical exponents as a function of the long-range decay exponent with a strong dependence on the imposed boundary conditions leading to distinct finite-size scalings for sufficiently long-range potentials. Remarkably, the Haldane-to-U(1)-CSB transition likewise displays unconventional quantum criticality with distinct continuously varying critical exponents. Our work positions this model as a target for near-term atomic platforms with tunable long-range couplings and exhibiting natural single-ion anisotropy, offering a minimal playground for exploring the interplay between long-range interactions, continuous symmetry breaking, and topology.
The study of quantum criticality and entanglement in systems with long-range (LR) interactions is still in its early stages, with many open questions remaining. In this work, we investigate critical exponents and scaling of entanglement entropies (EE) in the LR bilayer Heisenberg model using large-scale quantum Monte Carlo (QMC) simulations and the recently developed nonequilibrium increment algorithm for measuring EE. By applying modified (standard) finite-size scaling (FSS) above (below) the upper critical dimension and field theory analysis, we obtain precise critical exponents in three regimes: the LR Gaussian regime with a Gaussian fixed point, the short-range (SR) regime with Wilson-Fisher (WF) exponents, and a LR non-Gaussian regime where the critical exponents vary continuously from LR Gaussian to SR values. We compute the Rényi EE both along the critical line and in the Néel phase and observe that as the LR interaction is enhanced, the area-law contribution in EE gradually vanishes both at quantum critical points (QCPs) and in the Néel phase. The log-correction in EE arising from sharp corners at the QCPs also decays to zero as LR interaction grows, whereas the log-correction for Néel states, caused by the interplay of Goldstone modes and restoration of the symmetry in a finite system, is enhanced as LR interaction becomes stronger. We also discuss relevant experimental settings to detect these nontrivial properties in critical behavior and entanglement information for quantum many-body systems with LR interactions.
We study spin-1/21/21/2 chains with long-range power-law decaying unfrustrated (bipartite) Heisenberg exchange Jr∝r−αJ_r \propto r^{-α}Jr​∝r−α and multi-spin interactions QQQ favoring a valence-bond solid (VBS) ground state. Employing quantum Monte Carlo techniques and Lanczos diagonalization, we analyze order parameters and excited-state level crossings to characterize quantum states and phase transitions in the (α,Q)(α,Q)(α,Q) plane. For weak QQQ and sufficiently slowly decaying Heisenberg interactions (small ααα), the system has a long-range-ordered antiferromagnetic (AFM) ground state, and upon increasing ααα there is a continuous transition into a quasi long-range ordered (QLRO) critical state of the type in the standard Heisenberg chain. For rapidly decaying long-range interactions, there is transition between QLRO and VBS ground states of the same kind as in the frustrated J1J_1J1​-J2J_2J2​ Heisenberg chain. Our most important finding is a direct continuous quantum phase transition between the AFM and VBS states - a close analogy to the 2D deconfined quantum-critical point. In previous 1D analogies the ordered phases both have gapped fractional excitations, and the critical point is a conventional Luttinger Liquid. In our model the excitations fractionalize upon transitioning from the AFM state, changing from spin waves to deconfined spinons. We extract critical exponents at the AFM-VBS transition and use order-parameter distributions to study emergent symmetries. We find emergent O(444) symmetry of the O(333) AFM and scalar VBS order parameters. Thus, the order parameter fluctuations exhibit the covariance of a uniaxially deformed O(444) sphere (an "elliptical" symmetry). This unusual quantum phase transition does not yet have any known field theory description, and our detailed results can serve to guide its construction. We discuss possible experimental realizations.
The characterization of the quantum phase transition in a lowdimensional system has attracted a considerable amount of attention in quantum manybody systems. As one of the fundamental models in quantum magnetism, spin-1 models have richer phase diagrams and show more complex physical phenomena. In the spin-1 antiferromagnetic XXZ model, the Haldane phase and the Nel phase are the gapped topologic phases which cannot be characterized by the local order parameters. To characterize the nature in such phases, one has to calculate the non-local long range order parameters. Normally, the non-local order parameter in the topological phase is obtained from the extrapolation of finite-sized system in numerical study. However, it is difficult to extract the critical exponents with such an extrapolated non-local order parameter due to the numerical accuracy. In a recently developed tensor network representation, i.e., the infinite matrix product state (iMPS) algorithm, it was shown that the non-local order can be directly calculated from a very large lattice distance in an infinite-sized system rather than an extrapolated order parameter in a finite-sized system. Therefore, it is worthwhile using this convenient technique to study the non-local orders in the topological phases and characterize the quantum criticalities in the topological quantum phase transitions. In this paper, by utilizing the infinite matrix product state algorithm based on the tensor network representation and infinite time evolving block decimation method, the quantum entanglement, fidelity, and critical exponents of the topological phase transition are investigated in the one-dimensional infinite spin-1 bond-alternating XXZ Heisenberg model. It is found that there is always a local dimerization order existing in the whole parameter range when the bond-alternative strength parameter changes from 0 to 1. Also, due to the effect of the bond-alternating, there appears a quantum phase transition from the long-rang ordering topological Nel phase to the local ordering dimerization phase. The von Neumann entropy, fidelity per lattice site, and order parameters all give the same phase transition point at c = 0.638. To identify the type of quantum phase transition, the central charge c 0.5 is extracted from the ground state von Neumann entropy and the finite correlation length, which indicates that the phase transition belongs to the two-dimensional Ising universality class. Furthermore, it is found that the Nel order and the susceptibility of Nel order have power-law relations to |-c|. From the numerical fitting of the Nel order and its susceptibility, we obtain the characteristic critical exponents ' = 0.236 and ' = 0.838. It indicates that such critical exponents from our method characterize the nature of the quantum phase transition. Our critical exponents from the iMPS method can provide guidance for studying the properties of the phase transition in quantum spin systems.
The phase diagram of the quantum spin-12\frac{1}{2}21​ antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XXZ chain was obtained by Haldane using bosonization techniques [Haldane, Phys. Rev. B 25, 4925 (1982); Phys. Rev. B 26, 5257 (1982)]. It supports three distinct phases for $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$, i.e., a gapless algebraic spin-liquid phase, a gapped long-range ordered Neel phase, and a gapped long-range ordered dimer phase. Even though the Neel and dimer phases are not related hierarchically by a pattern of symmetry breaking, it was shown that they meet along a line of quantum critical points with a U(1) symmetry and central charge c=1c=1c=1. Here, we extend the analysis made by Haldane on the quantum spin-12\frac{1}{2}21​ antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XYZ chain using both bosonization and numerical techniques. We show that there are three Neel phases and the dimer phase that are separated from each other by six planes of phase boundaries realizing Gaussian criticality when $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$. We also show that each long-range ordered phase harbors topological point defects (domain walls) that are dual to those across the phase boundary in that a defect in one ordered phase locally binds the other type of order around its core. By using the bosonization approach, we identify the critical theory that describes simultaneous proliferation of these dual point defects, and show that it supports an emergent U(1) symmetry that originates from the discrete symmetries of the XYZ model. To confirm this numerically, we perform exact diagonalization and density-matrix renormalization-group calculations and show that the critical theory is characterized by the central charge c=1c=1c=1 with critical exponents that are consistent with those obtained from the bosonization approach. Furthermore, we generalize the field-theoretic description of direct continuous phase transition to higher dimensions, especially in d=3d=3d=3, by using a nonlinear sigma model (NLSM) with a topological term. In particular, we propose the $\ensuremath{\pi}$-flux phase on the cubic lattice with local quartic interactions as a platform for direct continuous phase transition and deconfined criticality. We discuss possible phase diagrams for the $\ensuremath{\pi}$-flux phase on the cubic lattice with these quartic interactions from the renormalization flow of NLSMs.
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