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We study a $mathcal PT$-symmetric scalar Euclidean field theory with a complex action, using both theoretical analysis and lattice simulations. This model has a rich phase structure that exhibits pattern formation in the critical region. Analytical results and simulations associate pattern formation with tachyonic instabilities in the homogeneous phase. Monte Carlo simulation shows that pattern morphologies vary smoothly, without distinct microphases. We suggest that pattern formation in this model may be regarded as a form of arrested spinodal decomposition. We extend our theoretical analysis to multicomponent $mathcal PT$-symmetric Euclidean scalar field theories and show that they give rise to new universality classes of local field theories that exhibit patterned behavior in the critical region. QCD at finite temperature and density is a member of the $Z(2)$ universality class when the Polyakov loop is used to distinguish confined and deconfined phases. This suggests the possibility of the formation of patterns of confined and deconfined matter in QCD in the critical region in the $mu-T$ plane.
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Partial differential equations (PDE) have been widely used to reproduce patterns in nature and to give insight into the mechanism underlying pattern formation. Although many PDE models have been proposed, they rely on the pre-request knowledge of physical laws and symmetries, and developing a model to reproduce a given desired pattern remains difficult. We propose a novel method, referred to as Bayesian modelling of PDE (BM-PDE), to estimate the best PDE for one snapshot of a target pattern under the stationary state. We show the order parameters extracting symmetries of a pattern together with Bayesian modelling successfully estimate parameters as well as the best model to make the target pattern. We apply BM-PDE to nontrivial patterns, such as quasi-crystals, a double gyroid and Frank Kasper structures. Our method works for noisy patterns and the pattern synthesised without the ground truth parameters, which are required for the application toward experimental data.
We investigate the effects of stochastic interactions on hydrodynamic correlation functions using the Schwinger-Keldysh effective field theory. We identify new stochastic transport coefficients that are invisible in the classical constitutive relations, but nonetheless affect the late-time behaviour of hydrodynamic correlation functions through loop corrections. These results indicate that classical transport coefficients do not provide a universal characterisation of long-distance, late-time correlations even within the framework of fluctuating hydrodynamics.
We present a general formalism which allows us to derive the evolution equations describing one-dimensional (1D) and isotropic 2D interfacelike systems, that is based on symmetries, conservation laws, multiple scale arguments, and exploits the relevance of coarsening dynamics. Our approach becomes especially significant in the presence of surface morphological instabilities and allows us to classify the most relevant nonlinear terms in the continuum description of these systems. The formalism applies to systems ranging from eroded nanostructures to macroscopic pattern formation. In particular, we show the validity of the theory for novel experiments on ion plasma erosion.
We consider the two-dimensional classical XY model on a square lattice in the thermodynamic limit using tensor renormalization group and precisely determine the critical temperature corresponding to the Berezinskii-Kosterlitz-Thouless (BKT) phase transition to be 0.89290(5) which is an improvement compared to earlier studies using tensor network methods.
We study perturbations that break gauge symmetries in lattice gauge theories. As a paradigmatic model, we consider the three-dimensional Abelian-Higgs (AH) model with an N-component scalar field and a noncompact gauge field, which is invariant under U(1) gauge and SU(N) transformations. We consider gauge-symmetry breaking perturbations that are quadratic in the gauge field, such as a photon mass term, and determine their effect on the critical behavior of the gauge-invariant model, focusing mainly on the continuous transitions associated with the charged fixed point of the AH field theory. We discuss their relevance and compute the (gauge-dependent) exponents that parametrize the departure from the critical behavior (continuum limit) of the gauge-invariant model. We also address the critical behavior of lattice AH models with broken gauge symmetry, showing an effective enlargement of the global symmetry, from U(N) to O(2N), which reflects a peculiar cyclic renormalization-group flow in the space of the lattice AH parameters and of the photon mass.
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