We consider the thermal phase transition from a paramagnetic to 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. [PRL 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 ± 0.01. Thereafter, the transitions from g = g * to g → ∞ are continuous and can be fully mapped, using universality arguments, to the critical line of the well known Ashkin-Teller model from its 4-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 AshkinTeller 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 to be 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 results are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological bench-marks for weakly first-order behaviors and Ashkin-Teller criticality.
Recently, significant progress has been made in (2 þ 1)-dimensional conformal field theories without supersymmetry. In particular, it was realized that different Lagrangians may be related by hidden dualities; i.e., seemingly different field theories may actually be identical in the infrared limit. Among all the proposed dualities, one has attracted particular interest in the field of strongly correlated quantum-matter systems: the one relating the easy-plane noncompact CP 1 model (NCCP 1 ) and noncompact quantum electrodynamics (QED) with two flavors (N ¼ 2) of massless two-component Dirac fermions. The easyplane NCCP 1 model is the field theory of the putative deconfined quantum-critical point separating a planar (XY) antiferromagnet and a dimerized (valence-bond solid) ground state, while N ¼ 2 noncompact QED is the theory for the transition between a bosonic symmetry-protected topological phase and a trivial Mott insulator. In this work, we present strong numerical support for the proposed duality. We realize the N ¼ 2 noncompact QED at a critical point of an interacting fermion model on the bilayer honeycomb lattice and study it using determinant quantum Monte Carlo (QMC) simulations. Using stochastic series expansion QMC simulations, we study a planar version of the S ¼ 1=2 J-Q spin Hamiltonian (a quantum XY model with additional multispin couplings) and show that it hosts a continuous transition between the XY magnet and the valence-bond solid. The duality between the two systems, following from a mapping of their phase diagrams extending from their respective critical points, is supported by the good agreement between the critical exponents according to the proposed duality relationships. In the J-Q model, we find both continuous and first-order transitions, depending on the degree of planar anisotropy, with deconfined quantum criticality surviving only up to moderate strengths of the anisotropy. This explains previous claims of no deconfined quantum criticality in planar two-component spin models, which were in the stronganisotropy regime, and opens doors to further investigations of the global phase diagram of systems hosting deconfined quantum-critical points.
We study the challenging thermal phase transition to stripe order in the frustrated square-lattice Ising model with couplings J(1) < 0 (nearest-neighbor, ferromagnetic) and J(2) > 0 (second-neighbor, antiferromagnetic) for g = J(2)/|J(1| > 1/2. Using Monte Carlo simulations and known analytical results, we demonstrate Ashkin-Teller criticality for g ≥ g*; i.e., the critical exponents vary continuously between those of the 4-state Potts model at g = g* and the Ising model for g → ∞. Thus, stripe transitions offer a route to realizing a related class of conformal field theories with conformal charge c = 1 and varying exponents. The transition is first order for g < g* = 0.67 ± 0.01, much lower than previously believed, and exhibits pseudo-first-order behavior for |g* ≤ g ~1.
Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average per-capita growth rate of populations. For sedentary populations in a spatially homogeneous yet temporally variable environment, a simple model of population growth is a stochastic differential equation dZt = μZtdt + σ ZtdWt, t ≥ 0, where the conditional law of Zt+Δt − Zt given Zt = z has mean and variance approximately zμΔt and z2σ2Δt when the time increment Δt is small. The long-term stochastic growth rate limt→∞ t−1 log Zt for such a population equals μ−σ22. Most populations, however, experience spatial as well as temporal variability. To understand the interactive effects of environmental stochasticity, spatial heterogeneity, and dispersal on population growth, we study an analogous model Xt=false(Xt1,…,Xtnfalse), t ≥ 0, for the population abundances in n patches: the conditional law of Xt+Δt given Xt = x is such that the conditional mean of Xt+normalΔti−Xti is approximately [xiμi +∑j (xj Dji − xi Dij)]Δt where μi is the per capita growth rate in the ith patch and Dij is the dispersal rate from the ith patch to the jth patch, and the conditional covariance of Xt+normalΔti−Xti and Xt+normalΔtj−Xtj is approximately xixjσijΔt for some covariance matrix Σ = (σij). We show for such a spatially extended population that if St=Xt1+⋯+Xtn denotes the total population abundance, then Yt = Xt /St, the vector of patch proportions, converges in law to a random vector Y∞ as t → ∞, and the stochastic growth rate limt→∞ t−1 log St equals the space-time average per-capita growth rate ∑iμi𝔼false[Y∞jfalse] experienced by the population minus half of the space-time average temporal variation 𝔼false[∑i,jσijY∞iY∞jfalse] experienced by the population. Using this characterization of the stochastic growth rate, we derive an explicit expression for the stochastic growth rate for populations living in two patches, determine which choices of the dispersal matrix D produce the maximal stochastic growth rate for a freely dispersing population, derive an analytic approximation of the stochastic growth rate for dispersal limited populations, and use group theoretic techniques to approximate the stochastic growth rate for populations living in multi-scale landscapes (e.g. insects on plants in meadows on islands). Our results provide fundamental insights into “ideal free” movement in the face of uncertainty, the persistence of coupled sink populations, the evolution of dispersal rates, and the single large or several small (SLOSS) debate in conservation biology. For example, our analysis implies that even in the absence of density-dependent feedbacks, ideal-free dispersers occupy multiple patches in spatially heterogeneous environments provided environmental fluctuations are sufficiently strong and sufficiently weakly correlated across space. In contrast, for diffusively dispersing populations living in similar environments, intermediate dispersal rates maximize their stochastic growth rate.
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