Enhance Long-Range Ordered (EnLRO)

• EnLRO is a finite-size scaling regime in continuous-spin models where the magnetization increases with system size within the true long-range ordered phase.
• Studies reveal clear distinctions between EnLRO and ReLRO through measurable crossovers and changes in correlation functions, such as σ_c≈1.575 in the 1D XY model.
• This operational classification of finite-size behavior offers practical insights for numerical simulations and experimental interpretations in low-dimensional magnetism.

Searching arXiv for recent and related papers on EnLRO and long-range ordered regimes. Enhance Long-Range Ordered (EnLRO) denotes a scaling regime identified within the true long-range ordered phase of low-dimensional continuous-spin models with algebraically decaying couplings $J(r)\sim r^{-\sigma}$, rather than a separate thermodynamic phase in the usual sense. In the formulation introduced for long-range XY and Heisenberg models, EnLRO is defined operationally by the finite-size behavior of the magnetization: inside the ordered phase, the magnetization $M$ increases with system size $L$, in contrast to the Reduce Long-Range Ordered (ReLRO) regime where $M$ decreases with $L$ but approaches a nonzero thermodynamic limit (Ding et al., 14 Jul 2025). The concept refines the standard picture of order in the non-Mermin-Wagner-Hohenberg regime $D<\sigma<2D$, where true finite-temperature long-range order is allowed by long-range interactions despite the absence of such order in the corresponding short-range models (Ding et al., 14 Jul 2025).

1. Definition and conceptual scope

EnLRO was introduced in the study of classical one- and two-dimensional XY and Heisenberg models with pair interactions decaying as $r^{-\sigma}$ (Ding et al., 14 Jul 2025). The paper’s central claim is that the true long-range ordered phase is not scaling-uniform: depending on $\sigma$ and $T$, finite systems can approach the same ordered thermodynamic limit in qualitatively different ways. In EnLRO, $M$ increases with $M$0; in ReLRO, $M$1 decreases with $M$2; near the crossover, $M$3 can be nonmonotonic, with a turning scale $M$4 (Ding et al., 14 Jul 2025).

The distinction is explicitly defined as a finite-size scaling property within the ordered phase, not as a confirmed thermodynamic phase transition. The authors state that “The crossover between EnLRO and ReLRO does not constitute a true phase transition but rather reflects a change in the scaling behavior of the magnetization with system size” (Ding et al., 14 Jul 2025). They also note that it remains unclear whether the distinction is “merely a crossover or if it is a phase transition for which the order parameter has yet to be identified” (Ding et al., 14 Jul 2025). Accordingly, EnLRO is best understood as a regime classification internal to the ordered phase rather than a separate equilibrium phase.

This usage is narrower than the broader phrase “enhanced long-range order,” which has appeared in other contexts to describe order promoted by long-range couplings, geometry, or structured interactions. For example, long-range couplings can induce true $M$5 order in a one-dimensional quantum Heisenberg chain with an all-to-all $M$6 term (Li et al., 2021), and a quasi-two-dimensional short-range XY architecture can generate anisotropic long-range order when auxiliary planes enter a Berezinskii-Kosterlitz-Thouless critical phase (Hu et al., 24 Jun 2025). These works concern mechanisms that enhance or induce long-range order, but they do not use “EnLRO” as the specific scaling-regime label introduced in (Ding et al., 14 Jul 2025).

2. Model class and mathematical definition

The EnLRO/ReLRO framework was formulated for classical continuous-spin systems with long-range couplings of the form $M$7 (Ding et al., 14 Jul 2025). The generic Hamiltonian is written as

$M$8

with periodic boundary conditions and shortest-image distances (Ding et al., 14 Jul 2025). For the XY model this becomes

$M$9

and for the Heisenberg model

$L$0

with the component terms given explicitly in the source paper (Ding et al., 14 Jul 2025).

The regime of interest is the non-Mermin-Wagner-Hohenberg window

$L$1

where true long-range order with finite magnetization at finite $L$2 exists (Ding et al., 14 Jul 2025). The region $L$3 is non-extensive and was not analyzed, while for $L$4 the system lies in the Mermin-Wagner-Hohenberg regime, where only quasi-long-range order or short-range order is expected (Ding et al., 14 Jul 2025).

The magnetization per site is defined for the XY model as

$L$5

and for the Heisenberg model as

$L$6

with

$L$7

(Ding et al., 14 Jul 2025).

Within this framework, EnLRO is defined by the sign of the finite-size magnetization flow: $L$8 whereas ReLRO is characterized by the opposite trend, while still retaining a nonzero thermodynamic-limit magnetization (Ding et al., 14 Jul 2025). This makes EnLRO a finite-size scaling phenomenon embedded in the ordered phase.

3. Physical mechanism and finite-size interpretation

The physical intuition behind EnLRO in (Ding et al., 14 Jul 2025) is based on competition between long-range alignment and thermal disorder. Decreasing $L$9 strengthens the effective reach of the interaction, so each spin is influenced by more distant aligned spins. In the ordered phase, this can make a larger system effectively more ordered than a smaller one because the cumulative alignment effect of long-range couplings outweighs the additional thermal disorder channels. This is the meaning of “enhanced” in EnLRO (Ding et al., 14 Jul 2025).

For the one-dimensional XY model at low $M$0, the paper interprets EnLRO in terms of suppressed spin-wave excitation. A thermal deflection $M$1 costs more energy at smaller $M$2 because the perturbed spin is coupled to many aligned neighbors. The authors interpret the resulting lower acceptance of large-angle fluctuations as suppression of spin-wave excitation and shortening of spin-wave lifetime, favoring EnLRO (Ding et al., 14 Jul 2025). At larger $M$3, thermal fluctuations are less strongly opposed, and the system falls in ReLRO.

At higher temperatures within the ordered phase, the same work proposes an additional mechanism involving local defects. In the 1D XY model, a bond defect is defined by

$M$4

The authors argue that defects can proliferate in an intermediate parameter window and partition the system into subsystems, thereby blocking long-distance propagation of spin-wave disorder. Under those conditions, increasing system size can increase magnetization, producing a temperature-driven ReLRO $M$5 EnLRO crossover inside the ordered phase (Ding et al., 14 Jul 2025). This suggests that EnLRO is not restricted to the lowest temperatures or smallest $M$6, but can also arise from a nontrivial interplay of spin waves and defects.

A plausible implication is that EnLRO is especially relevant to experimentally accessible finite systems, since it concerns how ordering manifests before the thermodynamic limit is reached. This implication is consistent with the paper’s emphasis on finite-size trends but is interpretive rather than explicitly stated.

4. Diagnostics and numerical identification

The identification of EnLRO in (Ding et al., 14 Jul 2025) is primarily operational and finite-size based. Four diagnostics are emphasized.

First, the sign of the size dependence of the magnetization serves as the direct criterion: EnLRO corresponds to magnetization increasing with $M$7, ReLRO to magnetization decreasing with $M$8 (Ding et al., 14 Jul 2025).

Second, at fixed low temperature, crossings of $M$9 for different system sizes define a crossover point $L$0 (Ding et al., 14 Jul 2025). In the 1D XY model at $L$1, the common crossing gives

$L$2

(Ding et al., 14 Jul 2025).

Third, near the boundary between EnLRO and ReLRO, the magnetization can be nonmonotonic in $L$3, and the turning scale $L$4 diverges as $L$5 approaches the crossover from the EnLRO side (Ding et al., 14 Jul 2025). This divergence is treated as evidence of a crossover in scaling behavior.

Fourth, the authors distinguish the two regimes through the real-space correlation function

$L$6

for the XY model (Ding et al., 14 Jul 2025). In ReLRO they report

$L$7

while in EnLRO they report

$L$8

with both forms approaching a nonzero constant $L$9 (Ding et al., 14 Jul 2025). The distinction is therefore not between ordered and disordered behavior, but between algebraic-plus-constant and exponential-plus-constant relaxation inside the ordered phase.

In the 1D XY model at $D<\sigma<2D$0 and $D<\sigma<2D$1, $D<\sigma<2D$2 is reported to fit the power-law form well, whereas $D<\sigma<2D$3 is fit well by the exponential form. The extracted $D<\sigma<2D$4 shows a marked discontinuous change around

$D<\sigma<2D$5

which the authors interpret as evidence of a clear scaling change, though not proof of a true phase transition (Ding et al., 14 Jul 2025).

The ordered–disordered transition itself is analyzed separately by standard finite-size scaling using susceptibility,

$D<\sigma<2D$6

Binder-like cumulants,

$D<\sigma<2D$7

their derivatives, and the scaling relations

$D<\sigma<2D$8

$D<\sigma<2D$9

(Ding et al., 14 Jul 2025). These formulas characterize the ordered–disordered transition, not the EnLRO–ReLRO crossover itself.

5. Phase structure and quantitative crossover results

The phase diagrams presented in (Ding et al., 14 Jul 2025) place EnLRO within the low-temperature true long-range ordered region of the non-Mermin-Wagner-Hohenberg regime. The organization is as follows: for $r^{-\sigma}$0, the region is non-extensive and not studied; for $r^{-\sigma}$1, the system has a true long-range ordered phase at low $r^{-\sigma}$2, internally divided into EnLRO and ReLRO; at higher $r^{-\sigma}$3, a disordered phase appears; and for $r^{-\sigma}$4, the system enters the Mermin-Wagner-Hohenberg regime with quasi-long-range or short-range order (Ding et al., 14 Jul 2025).

The reported low-temperature crossover locations are summarized below.

Model	Non-MWH window	Low-$r^{-\sigma}$5 crossover
1D XY	$r^{-\sigma}$6	$r^{-\sigma}$7
2D XY	$r^{-\sigma}$8	$r^{-\sigma}$9
1D Heisenberg	$\sigma$0	$\sigma$1
2D Heisenberg	$\sigma$2	$\sigma$3

The near-equality of the XY and Heisenberg crossover values is presented as evidence that the EnLRO/ReLRO distinction is a generic feature of continuous-spin models with long-range interactions (Ding et al., 14 Jul 2025). The authors emphasize common continuous rotational symmetry rather than a universal set of critical exponents.

The XY phase diagrams exhibit a further temperature dependence of the crossover line. In the 1D XY model, the line begins near $\sigma$4 at low $\sigma$5, stays nearly constant up to $\sigma$6, then rises with temperature. In the interval

$\sigma$7

heating produces the sequence

$\sigma$8

(Ding et al., 14 Jul 2025). Similarly, in the 2D XY model the low-$\sigma$9 crossover is near

$T$0

and the line increases toward about $T$1 with temperature, yielding the same sequence in the interval

$T$2

(Ding et al., 14 Jul 2025).

For the order–disorder transition, the paper reports in the 1D XY model that the correlation-length exponent differs significantly across the two ordered scaling sectors: at $T$3 on the EnLRO side,

$T$4

whereas at $T$5 on the ReLRO side,

$T$6

(Ding et al., 14 Jul 2025). This suggests that the ordered–disordered critical behavior sampled from the two sectors can differ substantially, although the paper does not claim a universal exponent set for EnLRO.

6. Relation to other routes to enhanced long-range order

The term EnLRO is specific to the crossover/scaling regime identified in long-range continuous-spin models (Ding et al., 14 Jul 2025), but related papers document other mechanisms by which long-range order is enhanced, induced, or qualitatively restructured.

In a one-dimensional quantum Heisenberg chain with nearest-neighbor $T$7 exchange supplemented by an infinite-range $T$8 interaction,

$T$9

an infinitesimal nonzero infinite-range coupling is reported to stabilize true $M$0 long-range order by regularizing the infrared divergence that would otherwise destroy continuous-symmetry breaking in one dimension (Li et al., 2021). This is enhancement of long-range order by interaction range, but it is a phase-inducing mechanism rather than an internal scaling distinction of the ordered phase.

A different mechanism appears in a short-range quasi-2D XY model formed by one vertical plane intersected by a family of parallel planes. There, the onset of anisotropic long-range order in the vertical plane occurs exactly when the auxiliary planes enter a Berezinskii-Kosterlitz-Thouless critical phase, so that criticality itself enhances ordering along the common intersection direction (Hu et al., 24 Jun 2025). The resulting ordered phase has long-range correlations along one direction and critical correlations perpendicular to it (Hu et al., 24 Jun 2025). This suggests a broader theme: long-range order can be enhanced not only by explicit long-range couplings but also by geometry plus critical mediation.

In a two-leg quantum spin ladder with staggered long-range interactions decaying as $M$1, decreasing $M$2 strengthens nonfrustrating long-range antiferromagnetic exchange and drives a direct transition from a rung-dimer phase to a Néel-ordered phase at

$M$3

(Yang et al., 2020). This again concerns enhancement or induction of long-range order by interaction range, but not the EnLRO/ReLRO subdivision within an already ordered phase.

These comparisons indicate that EnLRO belongs to a family of phenomena in which long-range interactions or structured couplings qualitatively modify ordered states. The specific novelty of EnLRO lies in identifying distinct finite-size scaling sectors inside the ordered phase itself.

7. Interpretation, significance, and open issues

The principal significance of EnLRO is that it refines the conventional binary description of low-dimensional long-range spin systems as simply ordered or disordered. In the non-Mermin-Wagner-Hohenberg region, long-range interactions do more than permit true order where short-range models cannot order; they also alter how finite systems approach the ordered thermodynamic limit (Ding et al., 14 Jul 2025). This makes EnLRO especially pertinent to numerical simulations, mesoscopic realizations, and any setting where accessible system sizes remain finite.

A common misconception would be to treat EnLRO and ReLRO as distinct equilibrium phases separated by a proved thermodynamic transition. The supporting paper does not establish that claim. Its demonstrated result is a crossover in scaling behavior of the magnetization with system size, supported by crossings of $M$4, divergence of $M$5, and a change in the form of the correlation function (Ding et al., 14 Jul 2025). Whether a sharper thermodynamic distinction exists is left open.

Another possible misunderstanding is that EnLRO means the thermodynamic-limit magnetization itself diverges or becomes anomalously large. The definition is narrower: both EnLRO and ReLRO lie inside a true long-range ordered phase with finite thermodynamic-limit magnetization, and the distinction concerns the finite-size flow toward that limit (Ding et al., 14 Jul 2025).

Several open questions follow directly from the published results. The most immediate is whether the EnLRO–ReLRO boundary is purely a crossover or conceals a transition with an as-yet unidentified order parameter (Ding et al., 14 Jul 2025). A second issue is the degree of universality of the correlation-function distinction between algebraic-plus-constant and exponential-plus-constant forms within the ordered phase. The reported near-coincidence of $M$6 values in XY and Heisenberg models suggests genericity across continuous-spin systems (Ding et al., 14 Jul 2025), but no full universality theory is established.

A plausible implication is that EnLRO provides a useful language for classifying finite-size ordering behavior across a broader class of long-range models than those explicitly studied. This suggests possible links to quantum long-range magnets, geometrically assisted ordered phases, and experimentally engineered finite systems, but such extensions remain inferential unless demonstrated explicitly.

In its strict original usage, therefore, Enhance Long-Range Ordered denotes the part of the true long-range ordered phase in which larger finite systems become more ordered as they grow, as measured by an increasing magnetization with system size, within low-dimensional continuous-spin models with algebraically decaying long-range couplings (Ding et al., 14 Jul 2025). Its importance lies in exposing internal structure within the ordered phase that is invisible to a purely thermodynamic ordered/disordered classification.