The phase diagram of a novel two-dimensional frustrated Ising model with both anti-ferromagnetic and ferromagnetic couplings is studied using Tensor-Network Renormalization-Group techniques. This model can be seen as two anti-ferromagnetic Ising replicas coupled by non-local spin-spin interactions, designed in such a way that the continuum limit matches that of the still debated J1 -J2 model and induces a marginal critical behavior. Our model has the advantage of having more symmetries than the J1 -J2 model and of allowing a more straightforward implementation of Tensor-Network Renormalization-Group algorithms We demonstrate the existence of two transition lines, featuring both first and second-order regimes. In the latter, the central charge and the critical exponents are shown to be compatible with the Ashkin-Teller universality class. This picture is consistent with that given by Monte Carlo simulations of the J1 -J2 model but not with recent studies with Tensor-Network techniques.
We analyze the phase transition of the frustrated
J1-
J2 Ising model with antiferromagnetic nearest- and strong next-nearest-neighbor interactions on the square lattice. Using extensive Monte Carlo simulations we show that the nature of the phase transition for $1/2<{J}_{2}/{J}_{1}\ensuremath{\lesssim}1$ is not of the weakly universal type---as commonly believed---but we conclude from the clearly doubly peaked structure of the energy histograms that the transition is of weak first order. Motivated by these results, we analyze the phase transitions via field-theoretic methods; i.e., we calculate the central charge of the underlying field theory via transfer-matrix techniques and present, furthermore, a field-theoretic discussion on the phase-transition behavior of the model. Starting from the conformally invariant fixed point of two decoupled critical Ising models (
J1=0), we calculate the effect of the nearest-neighbor coupling term perturbatively using operator product expansions. As an effective action we obtain the Ashkin-Teller model.
We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions
J1<0 (nearest neighbor, ferromagnetic) and
J2>0 (second neighbor, antiferromagnetic). The striped phase breaks a
Z4 symmetry and is stabilized at low temperatures for
g=J2/∣J1∣>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The
J1-
J2 model has a line of very weak first-order phase transitions in the whole region
1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from
g=g∗ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point
g∗ in the
J1-
J2 model. The continuous transitions near
g∗ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of
g>1/2. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.
By using extensive tensor network calculations, we map out the phase diagram of the frustrated J_{1}-J_{2} Ising model on the square lattice. In particular, we focus on the cases with controversy in the phase diagram, especially the stripe transition in the regime g=|J_{2}/J_{1}|>1/2 (J_{2}>0,J_{1}<0). While recent studies claimed that the phase transition is of first order when 1/2<g<g^{*} (with the smallest g^{*} being 0.67), our simulations suggest that if there is such a first-order region, it is smaller than those found in earlier studies by other methods. Combining with the analysis of critical properties, we provide evidence that the classical J_{1}-J_{2} model evolves continuously from two decoupled Ising models (g→∞ with central charge c=1) to a point belonging to the tricritical Ising universality class (with c=0.7) as g decreases to g^{*}≃0.54.
The phase diagram of a two-dimensional frustrated Ising model with both anti-ferromagnetic and ferromagnetic couplings is studied using tensor network renormalization group techniques. This model can be seen as two antiferromagnetic Ising replicas coupled by nonlocal spin-spin interactions, designed in such a way that the continuum limit matches that of the still-debated J_{1}-J_{2} model and induces a marginal critical behavior. Our model has the advantage of having more symmetries than the J_{1}-J_{2} model and of allowing a more straightforward implementation of tensor network renormalization group algorithms. We demonstrate the existence of two transition lines, featuring both first- and second-order regimes. In the latter, the central charge and the critical exponents are shown to be compatible with the Ashkin-Teller universality class. This picture is consistent with that given by Monte Carlo simulations of the J_{1}-J_{2} model but not with recent studies with tensor network techniques.
The Ising model on fractal lattices has attracted renewed interest, yet a quantitative understanding of how the critical temperature depends on the Hausdorff dimension (D_{H}) and connectivity (Q) remains limited. We study the ferromagnetic Ising model on the one-parameter "legs-extension" family of fractal lattices originally introduced by Genzor et al. and apply the higher-order tensor renormalization group (HOTRG) method, following the tensor-network construction developed for these geometries in earlier work. By tuning a single structural parameter, we access lattices with D_{H} decreasing from 1.792 to 1.260 and connectivity Q from 0.50 to 0.131, and obtain high-precision critical temperatures T_{c} and magnetization exponents β with carefully quantified uncertainties from systematic bond-dimension and fitting-window analyses, finding that β is strongly and almost exponentially suppressed as D_{H} is reduced. Along this one-dimensional trajectory in the (D_{H},Q) plane, T_{c} decreases monotonically with D_{H} and, when plotted against Q, exhibits a robust near-linear behavior within this lattice family over almost an order of magnitude in connectivity. A finite-temperature phase transition persists throughout the explored range down to our lowest Hausdorff dimension, D_{H}≈1.260, implying that the lower critical dimension of this family, if it exists, lies below this value. Comparison with zero-order and first-order mean-field theory shows that HOTRG systematically incorporates fluctuation effects beyond mean-field approximations. The tabulated T_{c} and β values thus provide a stringent benchmark for theories of criticality on fractal lattices and a quantitative reference for future numerical and analytical studies on other complex geometries.