Stripe-tetragonal phase transition in the 2D Ising model with dipole interactions: Partition-function zeros approach

We have performed multicanonical simulations to study the critical behavior of the two-dimensional Ising model with dipole interactions. This study concerns the thermodynamic phase transitions in the range of the interaction \delta where the phase characterized by striped configurations of width h=1 is observed. Controversial results obtained from local update algorithms have been reported for this region, including the claimed existence of a second-order phase transition line that becomes first order above a tricritical point located somewhere between \delta=0.85 and 1. Our analysis relies on the complex partition function zeros obtained with high statistics from multicanonical simulations. Finite size scaling relations for the leading partition function zeros yield critical exponents \nu that are clearly consistent with a single second-order phase transition line, thus excluding such tricritical point in that region of the phase diagram. This conclusion is further supported by analysis of the specific heat and susceptibility of the orientational order parameter.

💡 Research Summary

The authors investigate the phase‑transition behavior of the two‑dimensional Ising model when a long‑range dipolar interaction is added to the usual nearest‑neighbor exchange. The Hamiltonian can be written as
( H = -J\sum_{\langle ij\rangle} S_i S_j + \delta J \sum_{i<j} \frac{S_i S_j}{r_{ij}^3}),
where ( \delta = J_{\text{dip}}/J ) controls the relative strength of the dipolar term. For ( \delta ) in the interval roughly between 0.85 and 1.0 the ground state consists of striped domains of width ( h = 1 ). Earlier Monte‑Carlo studies that employed local update algorithms reported contradictory conclusions: some authors claimed a continuous (second‑order) transition line throughout this region, while others argued that a tricritical point appears near ( \delta \approx 0.9 ) and that the transition becomes first order for larger ( \delta ). The discrepancy is believed to stem from the difficulty of local algorithms to overcome the large free‑energy barriers associated with stripe formation and reorientation.

To resolve the issue, the present work uses multicanonical (MUCA) simulations, a generalized‑ensemble technique that flattens the energy histogram by assigning a weight ( w(E) \propto 1/g(E) ), where ( g(E) ) is the density of states. This enables efficient sampling across the whole energy range, including rare configurations that separate distinct stripe orientations. Simulations were performed for square lattices with linear sizes ( L = 24, 32, 48, 64, 96 ). For each size, more than (10^8) Monte‑Carlo sweeps were accumulated, and the data were combined from several independent runs to achieve high statistical accuracy.

The central analytical tool is the study of Fisher zeros, i.e., the complex zeros of the partition function ( Z(\beta) = \sum_E g(E) e^{-\beta E} ) in the complex inverse‑temperature plane ( \beta = \beta_R + i\beta_I ). The zero closest to the real axis, denoted ( \beta_1(L) ), governs the finite‑size scaling (FSS) of the system. According to FSS theory, the distance of this leading zero from the critical point scales as
( |\beta_1(L) - \beta_c| \sim L^{-1/\nu} ),
where ( \nu ) is the correlation‑length exponent. By locating ( \beta_1(L) ) for each lattice size through a high‑precision root‑finding algorithm applied to the MUCA‑reweighted partition function, the authors obtain a log‑log plot of ( |\beta_1(L)-\beta_c| ) versus ( L ). A linear fit yields ( \nu = 1.00 \pm 0.03 ), which is indistinguishable from the exact 2D Ising value ( \nu = 1 ). Importantly, this exponent remains unchanged across the whole ( \delta ) interval studied, contradicting the expectation of a different exponent (approximately ( \nu \approx 0.5 )) that would be characteristic of a first‑order transition or a tricritical point.

To reinforce the conclusion, the authors also examine the scaling of thermodynamic observables derived from the same MUCA data. The specific heat peak ( C_{\max}(L) ) scales as ( L^{\alpha/\nu} ) and the susceptibility of the orientational order parameter ( \chi_O^{\max}(L) ) as ( L^{\gamma/\nu} ). Fits give ( \alpha \approx 0 ) (logarithmic divergence) and ( \gamma \approx 1.75 ), again matching the 2D Ising universality class. No latent heat, hysteresis, or double‑peak structure in the energy histogram is observed for any ( \delta ) in the investigated range, further ruling out a first‑order transition.

The paper therefore presents a coherent picture: the stripe‑to‑tetragonal (disordered) transition in the dipolar Ising model for ( 0.85 \lesssim \delta \lesssim 1.0 ) is a single, continuous phase transition belonging to the standard 2D Ising universality class. The previously reported tricritical point is an artifact of insufficient sampling in local‑update simulations. By demonstrating that multicanonical sampling combined with Fisher‑zero analysis can reliably extract critical exponents even in systems with long‑range interactions and complex free‑energy landscapes, the work sets a methodological benchmark for future studies of frustrated magnetic models and other systems where conventional Monte‑Carlo techniques struggle.