Crossover in the Ordered Phase in the Non-Mermin-Wagner-Hohenberg Regime of Spin Models with Long-Range Coupling

Technical Summary: Crossover in the Ordered Phase in the Non-Mermin-Wagner-Hohenberg Regime of Spin Models with Long-Range Coupling

Problem Statement
Classical spin models with long-range interactions decaying as $r^{-\sigma}$ exhibit complex phase behaviors that differ significantly from short-range counterparts. While the Mermin-Wagner-Hohenberg (MWH) theorem prohibits true long-range order (LRO) in low-dimensional systems with short-range interactions, the presence of long-range interactions allows for true LRO in the "non-MWH regime" ($D < \sigma < 2D$, where $D$ is spatial dimension). Previous research has largely focused on the transition between ordered and disordered phases or specific values of $\sigma$. However, a comprehensive investigation into the scaling behaviors within the true long-range ordered phase has been lacking. Specifically, it remains unclear whether the magnetization in this regime monotonically converges to a finite value or exhibits distinct scaling regimes dependent on system size and interaction range.

Methodology
The authors employed Monte Carlo simulations using the single-update Metropolis algorithm to study one-dimensional (1D) and two-dimensional (2D) classical XY and Heisenberg models with long-range coupling $J/|i-j|^\sigma$.

• Models: Simulations were conducted on 1D chains and 2D square lattices with periodic boundary conditions. The Hamiltonian includes interactions decaying as $r^{-\sigma}$.
• Simulation Parameters: System sizes ($L$) ranged up to $L=4000$ (1D) and $L=120$ (2D). Simulations included $10^5$ warm-up sweeps followed by $5 \times 10^6$ sampling sweeps. Special care was taken at low temperatures ($T < 0.1$) to mitigate metastability by restricting proposed spin flips.
• Observables: The study analyzed energy ($E$), magnetization ($M$), magnetic susceptibility ($\chi$), and magnetic cumulants (Binder ratios, $U_n$). Crucially, the authors utilized the first derivatives of cumulants and magnetization with respect to inverse temperature ($\beta$) to locate phase transitions more precisely than traditional intersection methods.
• Scaling Analysis: Finite-size scaling (FSS) was applied to extract critical exponents ($\nu, \gamma$) and critical temperatures ($T_c$). The authors specifically examined the dependence of magnetization on $\ln L$ and the behavior of spin-spin correlation functions $G(r)$ to distinguish between different ordered regimes.

Key Contributions and Results

1. Identification of Distinct Scaling Regimes (EnLRO and ReLRO):
   The primary finding is that the true long-range ordered phase is not uniform but comprises two distinct scaling regimes:

   • Enhanced Long-Range Order (EnLRO): In this regime, magnetization $M$ increases with system size $L$. This occurs at smaller $\sigma$ values (stronger long-range interactions).
   • Reduced Long-Range Order (ReLRO): In this regime, magnetization $M$ decreases with system size $L$ (though it remains finite), occurring at larger $\sigma$ values (weaker long-range interactions).
   • Crossover: A crossover between these regimes occurs at a critical $\sigma_c$. Near this point, $M$ exhibits non-monotonic behavior with $L$ (decreasing then increasing).

2. Quantification of Crossover Points:

   • 1D XY Model: The crossover occurs at $\sigma_c \approx 1.575$ at low temperatures ($T=0.01$). The crossover temperature increases with $\sigma$.
   • 2D XY Model: The crossover occurs at $\sigma_c \approx 3.2$ at low temperatures.
   • Heisenberg Models: Similar crossovers were observed in 1D and 2D Heisenberg models at $\sigma_c \approx 1.575$ and $\sigma_c \approx 3.22$, respectively. The authors suggest this similarity indicates the phenomenon is a generic feature of continuous spin models with long-range interactions, likely due to shared continuous rotational symmetry.

3. Correlation Function Behavior:

   • In the ReLRO regime, spin-spin correlation functions decay algebraically ($G(r) \sim r^{-\kappa}$) to a non-zero value.
   • In the EnLRO regime, the decay is better described by an exponential function ($G(r) \sim k_1 + k_2 e^{-r/\xi}$) approaching a non-zero constant. The correlation length $\xi$ shows a discontinuous change near $\sigma_c$.

4. Phase Diagrams and Critical Exponents:
   Comprehensive phase diagrams in the $T-\sigma$ plane were constructed for both 1D and 2D models. The authors extracted critical exponents $\nu$ and $\gamma$ for the order-to-disorder transitions. For instance, in the 1D model at $\sigma=1.1$, $\nu \approx 1.525$, while at $\sigma=1.8$, $\nu \approx 2.65$. The critical temperature $T_c$ diverges as $\sigma \to D$.

5. Mechanism of Crossover:
   The paper proposes a mechanism based on the competition between spin alignment, spin-wave excitations, and local defects:

   • Low Temperature: Decreasing $\sigma$ suppresses spin-wave excitation and propagation, driving the system from ReLRO to EnLRO.
   • High Temperature: Local defects (which break the chain or lattice into subsystems) block the propagation of spin waves. As system size increases, defects proliferate faster than spin-wave disturbances, effectively isolating ordered regions and enhancing local order, leading to EnLRO.

Significance and Claims
The paper claims to refine the understanding of classical spin models in the non-MWH regime by identifying that the ordered phase contains distinct scaling behaviors previously overlooked. The authors posit that the distinction between EnLRO and ReLRO is a generic feature of continuous spin models.

The study highlights that while the EnLRO-ReLRO transition is a change in scaling behavior rather than a traditional thermodynamic phase transition (with a latent heat or symmetry breaking), it represents a significant shift in the system's response to finite-size effects. The authors note that while it is currently unclear if this crossover constitutes a true phase transition with a yet-to-be-identified order parameter, characterizing these scaling differences provides valuable reference points for experimental systems, such as Rydberg atom arrays, which operate in finite-size regimes where these effects are prominent. The work bridges the gap between theoretical models and the finite-size constraints observed in current experimental realizations of long-range interacting systems.