Critical behavior and crossover scaling in the Light-Heavy model

Abstract: The Light-Heavy (LH) model involves two species of particles (light and heavy) coupled with a fluctuating surface (described by tilts). The dynamics include the inherent diffusion of the particles (or tilts) as well as the drive provided by the tilts (or particles). When the two are of similar magnitude, the system lies in the unscaled (uLH) regime, while a significantly weaker drive leads to the scaled (sLH) regime. In the unscaled limit, the model exhibits an order-disorder transition characterized by the fluctuation-dominated phase ordering (FDPO). In this state, interestingly the dynamics is driven by multiple modes, giving rise to dynamic clusters. Away from the critical regime the disordered phase retains vestiges of FDPO behavior on length scales smaller than the correlation length. We examine this local FDPO-like behavior by using a scaling function that links the off-critical and critical regimes. We next turn to the scaled model and show that the multi-mode dynamics present in the unscaled regime is replaced by dynamics that is effectively controlled by a single dominant mode in the scaled regime. Concurrently, the two-point correlations change from the $\mathcal{O}(1)$ FDPO form to an anomalous long-range form that decays as $1/\sqrt{L}$. Drawing on the analogy with the sABC model, where similar anomalous correlations appear at criticality, we derive an analytical expression for the two-point correlation function using the same approach used for that model.

• The paper demonstrates that unscaled Light–Heavy dynamics retain local FDPO behavior in the disordered phase below a correlation length scaling as ξ ∼ Δ⁻⁰·⁸, with crossover collapse supporting a mixed-order transition.
• The paper shows that scaling the drive as 1/L replaces multi-mode FDPO with single-mode criticality, where the first Fourier mode scales as L⁻¹⁄⁴ and real-space correlations have a 1/√L cosine form.
• The paper derives the scaled-model critical correlations through fluctuating hydrodynamics and a Fokker–Planck reduction, while simulations confirm the result and reveal the failure of the independent-interval approximation.

The Light–Heavy (LH) model couples two conserved species—light and heavy particles—to a fluctuating surface encoded by up/down tilts, with each species biasing the dynamics of the other. This paper by Prakash, Barma, and Ramola (2608.18016) addresses how the critical behavior of this model changes when the drive is scaled down with system size, and how FDPO-like signatures persist off criticality in the unscaled regime. The central findings are that (i) in the unscaled LH (uLH) model, fluctuation-dominated phase ordering (FDPO) phenomenology survives locally within the disordered phase on scales below a correlation length $\xi \sim \Delta^{-\nu}$ with $\nu \simeq 0.8$; and (ii) in the scaled LH (sLH) model, the multi-mode FDPO is replaced by a single-mode critical state of the sABC type, with anomalous correlations decaying as $1/\sqrt{L}$, for which an exact analytical form is derived.

The two regimes of the LH model

The model consists of spin variables $\sigma_j = \pm 1$ for particles (heavy/light) on integer sites and $\tau_{j+1/2} = \pm 1$ for tilts on half-integer sites. Setting all biases $a, b, b'$ to zero decomposes the dynamics into two independent SSEPs; the drift terms are what couple the subsystems. Two regimes are distinguished:

• Unscaled LH: drift and diffusion are both $\mathcal{O}(1)$. The phase diagram contains three ordered phases (SPS, IPS, FPS), a disordered phase, and an FDPO line at $b + b' = 0$. Ordered phases show strong particle separation with interfaces only a few sites wide.
• Scaled LH: biases scale as $\mathcal{O}(1/L)$ while diffusion remains $\mathcal{O}(1)$, so local dynamics are diffusive and an exact fluctuating hydrodynamic description applies. The phase diagram simplifies to ordered, disordered, and critical phases, with the critical locus shifted from $\nu \simeq 0.8$0 by $\nu \simeq 0.8$1.

A particle–tilt duality holds when $\nu \simeq 0.8$2: exchanging particles and tilts maps $\nu \simeq 0.8$3. A direct consequence, confirmed by Monte Carlo data with cusp exponent $\nu \simeq 0.8$4 and Edwards–Wilkinson scaling ($\nu \simeq 0.8$5), is the existence of an FDPO of tilts along $\nu \simeq 0.8$6, $\nu \simeq 0.8$7, dual to the well-known particle FDPO. No dual exists for the KPZ-type particle FDPO since there is no parameter $\nu \simeq 0.8$8 in the tilt dynamics.

Local FDPO in the disordered phase

In conventional critical phenomena, subsystems smaller than the correlation length behave as if at criticality. The paper asks whether the analogous statement holds for FDPO: does the correlation function obey

$\nu \simeq 0.8$9

in the homogeneous phase? Simulations confirm this: deep in the disordered regime the correlations are exponential and size-independent, but near the critical line they collapse onto the FDPO master curve when rescaled by $1/\sqrt{L}$0. Because neither scaling collapse nor direct fits determine $1/\sqrt{L}$1 unambiguously (the prefactor $1/\sqrt{L}$2 absorbs unknown constants), the authors instead use the distance $1/\sqrt{L}$3 from the critical locus as control parameter and derive the crossover scaling form

$1/\sqrt{L}$4

with $1/\sqrt{L}$5, $1/\sqrt{L}$6, $1/\sqrt{L}$7 producing good data collapse for $1/\sqrt{L}$8 up to 8192. The implication is twofold: FDPO-like behavior is a local property of the disordered phase near criticality, and the diverging $1/\sqrt{L}$9 combined with the discontinuous jump of the first Fourier mode across the transition implies that the disorder–order transition mediated by FDPO is of mixed order—consistent with the TIDSI model result and supporting the conjecture that FDPO generally accompanies mixed-order transitions.

The authors note explicitly that these results were obtained along one approach trajectory to criticality governed by KPZ-type surface fluctuations; the EW-controlled neighborhood and other trajectories remain unanalyzed.

Single-mode criticality in the scaled model

In the uLH model, Fourier modes scale as $\sigma_j = \pm 1$0 with $\sigma_j = \pm 1$1 (EW) or $\sigma_j = \pm 1$2 (KPZ), reflecting multi-mode dynamics and macroscopic dynamic clusters. In the sLH model the structure collapses: no mode stays finite; the first mode decays as $\sigma_j = \pm 1$3 while higher modes decay as $\sigma_j = \pm 1$4. Consequently the real-space correlations lose their $\sigma_j = \pm 1$5 amplitude and decay as $\sigma_j = \pm 1$6—anomalous long-range behavior of exactly the type found at the critical point of the scaled ABC model [Gerschenfeld2012].

Analytical derivation of the critical correlations

Working on the $\sigma_j = \pm 1$7 plane (the analysis is explicitly restricted to this subspace), the authors project the exact fluctuating hydrodynamic equations onto Fourier modes. Dominance of the first mode and critical slowing down allow a controlled truncation: leading-order projection relates $\sigma_j = \pm 1$8 linearly to $\sigma_j = \pm 1$9; second-mode amplitudes are quadratic in $\tau_{j+1/2} = \pm 1$0; and a next-to-leading-order treatment, after eliminating correction terms via a judicious linear combination, yields a closed Langevin equation for a single complex variable,

$\tau_{j+1/2} = \pm 1$1

with noise strength $\tau_{j+1/2} = \pm 1$2. Under rescaling this reduces to a universal overdamped Langevin equation equivalent to a Fokker–Planck equation in the complex plane, whose rotationally invariant stationary solution gives, at criticality,

$\tau_{j+1/2} = \pm 1$3

Monte Carlo simulations agree with this prediction for both particle and tilt correlations, including the dependence of the amplitude on $\tau_{j+1/2} = \pm 1$4 (using the nearest-neighbor correlation as proxy for the regular part). The cosine form contrasts sharply with the cusp singularity of FDPO, and the vanishing amplitude means the critical state has no macroscopic clusters.

Failure of the Independent Interval Approximation

The cluster size distribution at sLH criticality is exponential, $\tau_{j+1/2} = \pm 1$5, rather than the power law underlying FDPO. A naive IIA calculation would then predict $\tau_{j+1/2} = \pm 1$6, contradicting both the analytical result and simulations. The failure indicates that cluster sizes are not independent random intervals in the scaled model—a structural breakdown of the approximation scheme that works in the unscaled case.

Porod's law in the ordered phases

FDPO violates Porod's law through its cusp singularity. The paper verifies that genuine phase-separated states do not share this violation: strong phase separation in the uLH model obeys Porod's law trivially, and, less obviously, the sLH ordered phase also exhibits linear decay of $\tau_{j+1/2} = \pm 1$7 outside an interfacial core whose width itself scales with $\tau_{j+1/2} = \pm 1$8—consistent with the broad interfaces expected from the hydrodynamic description.

Limitations and open questions

Several restrictions qualify the results. The analytical correlation function is derived only on the $\tau_{j+1/2} = \pm 1$9 subspace and is representative rather than fully general. The crossover scaling analysis covers a single trajectory toward the FDPO line (KPZ-controlled); the EW-controlled critical region and other trajectories are left open. The determination of $a, b, b'$0 via scaling collapse is not strictly independent, motivating the $a, b, b'$1-based reformulation. Finally, whether the mixed-order character inferred here generalizes to all FDPO transitions remains a conjecture supported so far by the TIDSI analogy.

Conclusion

This work establishes a sharp dichotomy in the LH model between its unscaled and scaled limits: multi-mode FDPO with $a, b, b'$2 cusp-singular correlations versus single-mode criticality with $a, b, b'$3 cosine-form anomalous correlations, derived analytically via a Fokker–Planck reduction paralleling the sABC treatment. It also demonstrates that FDPO phenomenology persists locally within the disordered phase below a diverging correlation length, captured by a crossover scaling function, and argues that the associated transition is of mixed order. The results position the LH model alongside WASEP and sABC as systems where drive scaling qualitatively reshapes the thermodynamic limit, and identify the generality of the FDPO–mixed-order connection as the principal open question.