Reduce Long-Range Ordered (ReLRO) in Spin Models

• Reduce Long-Range Ordered (ReLRO) is defined in continuous-spin models as a state where magnetization decreases with system size but converges to a finite limit.
• Finite-size scaling diagnostics such as magnetization, susceptibility, and Binder cumulants reveal the ReLRO–EnLRO crossover, highlighting non-thermodynamic phase transition behavior.
• In the Falicov–Kimball framework, ReLRO emerges as a fractionalized state with a gapped single-particle spectrum while maintaining gapless charge and spin responses.

Reduce Long-Range Ordered (ReLRO) most directly denotes, in long-range continuous-spin models, a regime inside the true long-range ordered phase in which the magnetization decreases with system size $L$, but still tends toward a finite value in the thermodynamic limit. In that setting, ReLRO is contrasted with Enhance Long-Range Ordered (EnLRO), where the magnetization increases with $L$, and the distinction is a crossover in finite-size scaling behavior rather than a thermodynamic phase transition. A second usage appears in the Falicov–Kimball literature, where ReLRO designates a fractionalized long-range ordered phase whose single-particle spectrum is gapped while charge and spin responses remain gapless. This suggests that the term is used to identify ordered states in which some conventional manifestation of long-range order is reduced, reweighted, or reorganized, without eliminating the ordered phase itself (Ding et al., 14 Jul 2025, Tran, 2018).

1. Continuous-spin definition and ordered-phase splitting

The clearest definition of ReLRO is given for continuous-spin long-range models with couplings decaying as

$J(r)\sim r^{-\sigma},$

where smaller $\sigma$ means interactions remain strong over longer distances, while larger $\sigma$ weakens those long-range couplings and makes thermal disorder relatively more important. The main result is that, inside the non-MWH regime $D<\sigma<2D$, the ordered phase is not uniform but is split into two distinct scaling regimes: EnLRO, where the magnetization increases with $L$, and ReLRO, where the magnetization decreases with $L$ but still approaches a finite thermodynamic-limit value. The paper emphasizes that this EnLRO–ReLRO distinction is not a thermodynamic phase transition in the usual sense; it is a crossover in finite-size scaling behavior within the ordered phase itself (Ding et al., 14 Jul 2025).

The physical content of this distinction is the competition between long-range alignment from the $r^{-\sigma}$ interaction and thermal fluctuations that scramble the spins. In EnLRO, long-range coupling is strong enough that larger systems become more ordered, so increasing $L$ enhances magnetization. In ReLRO, thermal disturbances and finite-size effects suppress magnetization more strongly at smaller sizes; as $L$0 grows, magnetization decreases toward its thermodynamic value. Near the crossover, the finite-size trend can become non-monotonic, with magnetization first decreasing and then increasing with $L$1, indicating a change in which mechanism dominates.

The same work contrasts this ordered, non-MWH behavior with the MWH regime $L$2, where thermal fluctuations destroy true long-range order and only quasi-long-range order or short-range order remains. For the 1D XY and Heisenberg models this corresponds to $L$3 for the non-MWH side, and for the 2D models to $L$4.

2. Finite-size scaling diagnostics

ReLRO is identified through a finite-size scaling program based on magnetization $L$5, susceptibility $L$6, Binder cumulants $L$7, and derivatives such as $L$8 and $L$9. The susceptibility and Binder cumulants are defined as

$J(r)\sim r^{-\sigma},$0

$J(r)\sim r^{-\sigma},$1

The scaling analysis uses

$J(r)\sim r^{-\sigma},$2

$J(r)\sim r^{-\sigma},$3

Here $J(r)\sim r^{-\sigma},$4 is the correlation-length critical exponent, $J(r)\sim r^{-\sigma},$5 is the susceptibility exponent, and $J(r)\sim r^{-\sigma},$6 is the thermodynamic critical temperature. The maxima of $J(r)\sim r^{-\sigma},$7, $J(r)\sim r^{-\sigma},$8, and $J(r)\sim r^{-\sigma},$9 are used to fit $\sigma$0, $\sigma$1, and $\sigma$2 (Ding et al., 14 Jul 2025).

Within this framework, the ReLRO–EnLRO distinction appears primarily in the system-size dependence of $\sigma$3. In EnLRO, $\sigma$4 grows with $\sigma$5; in ReLRO, $\sigma$6 shrinks with $\sigma$7. In the 1D XY case, the slope of magnetization versus $\sigma$8 is fitted by

$\sigma$9

and the crossover is identified by the sign change of $\sigma$0. This makes the crossover operationally detectable even though it is not a thermodynamic singularity.

Correlation functions provide a second diagnostic. For the 1D XY model, the spin-spin correlator

$\sigma$1

is reported to have different functional forms in the two regimes. In ReLRO it is well described by

$\sigma$2

while in EnLRO it is closer to

$\sigma$3

The fitted correlation length $\sigma$4 changes sharply around the crossover near $\sigma$5–$\sigma$6, which supports the distinction between the two scaling regimes. A recurring conceptual point is that true long-range order does not imply a single universal finite-size trend.

3. Crossover values and genericity across models

At low temperature $\sigma$7, the crossover is extracted from magnetization curves $\sigma$8 crossing for different system sizes. The reported values are:

Model	Dimension	Crossover $\sigma$9
XY	1D	$D<\sigma<2D$0
XY	2D	$D<\sigma<2D$1
Heisenberg	1D	$D<\sigma<2D$2
Heisenberg	2D	$D<\sigma<2D$3

The near-identity of the 1D XY and 1D Heisenberg values, and the close 2D values around $D<\sigma<2D$4, is used to argue that the EnLRO–ReLRO crossover is a generic feature in continuous spin models with long-range interactions (Ding et al., 14 Jul 2025).

The interpretation given for this genericity is tied to the common continuous rotational symmetry of the XY and Heisenberg models and to the shared competition between long-range alignment, spin-wave excitations, and local defect formation. At low temperature, reducing $D<\sigma<2D$5 strengthens long-range alignment and suppresses spin-wave disorder. At higher temperature, local defects can break up spin-wave propagation, which can paradoxically allow larger systems to sustain more effective order. This is the basis for the claim that the ordered phase contains distinct scaling subregimes even though it remains a single thermodynamic phase.

A common misconception is to treat the crossover as an additional phase boundary. The analysis states explicitly that it is not an order-to-order phase transition in the thermodynamic sense. Rather, it identifies two finite-size scaling regimes inside the same ordered phase. In that respect, ReLRO is best understood as a statement about how order manifests under finite-size scaling, not as a statement that order has vanished.

4. Fractionalized long-range ordered states

A different use of ReLRO appears in the symmetric three-component Falicov–Kimball model studied within dynamical mean-field theory. The model is designed so that a local Coulomb term is effectively reduced, in a thermodynamic sense, to either repulsive or attractive Hubbard physics depending on an Ising-like local variable $D<\sigma<2D$6. In the DMFT solution, the local Green’s function becomes a weighted mixture of two effective Hubbard problems,

$D<\sigma<2D$7

so the model literally interpolates between repulsive and attractive local physics through the thermodynamic weights $D<\sigma<2D$8 (Tran, 2018).

At high temperature, the homogeneous phase has $D<\sigma<2D$9 and $L$0. In the strong-coupling regime, the single-particle density of states opens a gap, but the charge compressibility and the spin susceptibility remain finite. Below the critical temperature $L$1, the system develops long-range order, with charge-ordered and antiferromagnetic solutions. In the ordered phase, the single-particle density of states is always gapped, but in the strong-interaction regime the charge and spin excitations remain gapless. This coexistence of an ordered, gapped one-particle spectrum with finite charge compressibility and spin susceptibility is the central signature of the fractionalized long-range ordered phase that the data identify with ReLRO.

The same work emphasizes that this is not conventional one-dimensional spin-charge separation. Rather, the electron’s physical response decomposes into distinct charge and spin objects whose low-energy properties differ from those of the electron itself. The single-particle excitation is gapped, yet $L$2 and $L$3 do not inherit that gap in the strong-coupling regime. In the ordered state, both the charge compressibility and spin susceptibility collapse onto universal scaling functions of temperature,

$L$4

with empirical fits of the form

$L$5

In this usage, ReLRO denotes not a finite-size crossover of magnetization, but an ordered state whose low-energy electron physics is reduced into separate charge and spin responses.

5. Adjacent low-dimensional and finite-size phenomena

Closely related studies show that long-range order in reduced dimensions can be anisotropic, induced by geometry, or only apparently suppressed at finite size. In a short-range quasi-two-dimensional XY model defined on one vertical plane intersected by a set of parallel planes, a long-range ordered phase emerges when the parallel planes develop a Berezinskii–Kosterlitz–Thouless critical phase. The resulting order is anisotropic: $L$6 lines in the vertical plane become truly long-range ordered, while $L$7 lines remain critical. For $L$8, $L$9 and $L$0 approach nonzero constants, whereas $L$1 is compatible with a logarithmic decay. The ordered phase also exhibits Goldstone-mode physics, with $L$2 in the finite-size corrections (Hu et al., 24 Jun 2025).

A different route to long-range order appears in the 1D spin-$L$3 Heisenberg $L$4 chain supplemented by an infinite-range $L$5 interaction generated by coupling the spins to a single cavity mode. There, the cavity-mediated term stabilizes a genuine $L$6-broken $L$7 phase with $L$8, and the paper argues that even an infinitesimally weak infinite-range interaction can induce this order. The mechanism is that the global coupling suppresses the low-energy phase fluctuations that would otherwise destroy continuous symmetry breaking in 1D. The phase diagram contains an Ising ferromagnet, a Tomonaga–Luttinger liquid, and a true $L$9 long-range ordered phase, the latter distinguished by logarithmic violation of the area law and an effective central charge $r^{-\sigma}$0 (Li et al., 2021).

Finite-size reduction without thermodynamic elimination is especially clear in the study of 2D Ising ferromagnets with power-law couplings $r^{-\sigma}$1 in the weak long-range regime $r^{-\sigma}$2. After a quench to $r^{-\sigma}$3, long-range interactions suppress striped metastable states in finite systems, especially for smaller $r^{-\sigma}$4, but in the thermodynamic limit their occurrence probabilities are consistent with the short-range case. For $r^{-\sigma}$5, the final state is always ordered, and equilibration occurs at earlier times with an increase in the strength of the interactions. This provides a useful contrast: an apparent reduction of metastability in finite systems need not correspond to a genuine thermodynamic suppression (Agrawal et al., 2021).

6. Methods and broader conceptual setting

Several methodological and dynamical works situate ReLRO-like questions within a broader program of describing ordered phases that are difficult to capture with standard tools. In non-perturbative linked-cluster expansions for ordered quantum phases, the central difficulty is that finite clusters treated in isolation are not aware of the symmetry-broken phase they should belong to. The proposed solution is to embed each cluster in an ordered reference environment, which generates effective edge-fields such as

$r^{-\sigma}$6

The field $r^{-\sigma}$7 is optimized either by a self-consistent fixpoint condition $r^{-\sigma}$8 or by choosing a local minimum of the ground-state energy per site $r^{-\sigma}$9. The inclusion of edge-fields is reported to regularize NLCEs in ordered phases and yield convergent data sequences (Ixert et al., 2016).

Momentum-space continuous similarity transformations address the opposite problem: the quantum melting of ordered phases. For Néel-ordered and columnar antiferromagnets, the flow equation

$L$0

is truncated by scaling dimension with all operators up to $L$1 retained, and the ground state is isolated by a $L$2-generator. Long-range order is monitored through the sublattice magnetization and through the behavior of $L$3. The practical marker of quantum melting is the loss of flow stability; the last convergent coupling is taken as the estimate of the critical point. The method locates the ordered-phase boundaries to within a few percent, but the associated critical exponents remain quantitatively unreliable (Hering et al., 2024).

A dynamical extension appears in long-range interacting Hamiltonian systems with pair potential $L$4, where a finite-time non-Hamiltonian perturbation applied during early evolution produces states that are macroscopically stationary but microscopically highly ordered in phase space. These states are characterized by rings or circles in $L$5 space, discrete energy peaks, and coherent periodic or quasi-periodic motion, with the entanglement parameter

$L$6

reported around $L$7, far from the equilibrium value $L$8. No measurable evolution away from the ordered state is observed up to $L$9 for $L$00 (Joyce et al., 2017).

Taken together, these results place ReLRO within a wider research vocabulary about ordered phases that are not exhausted by the simple dichotomy of ordered versus disordered. In the continuous-spin setting, ReLRO is a finite-size scaling regime inside true long-range order. In the Falicov–Kimball setting, it is an ordered but fractionalized state with gapless collective responses. Adjacent work on anisotropic ordering, cavity-induced order, finite-size suppression of metastability, regularized cluster expansions, and quantum melting shows that modern treatments of long-range order increasingly distinguish between thermodynamic order, finite-size trends, excitation structure, and dynamical stability.