The $d$-dimensional long-range Ising model, defined by spin-spin interactions decaying with the distance as the power $1/r^{d+s}$, admits a second order phase transition with continuously varying critical exponents. At $s = s_*$, the phase transition crosses over to the usual short-range universality class. The standard field-theoretic description of this family of models is strongly coupled at the crossover. We find a new description, which is instead weakly coupled near the crossover, and use it to compute critical exponents. The existence of two complementary UV descriptions of the same long-range fixed point provides a novel example of infrared duality.
We compute the crossover exponents of all quadratic and cubic deformations of critical field theories with permutation symmetry $S_q$ in $d=6-ε$ (Landau-Potts field theories) and $d=4-ε$ (hypertetrahedral models) up to three loops.We use our results to determine the $ε$-expansion of the fractal dimension of critical clusters in the most interesting cases, which include spanning trees and forests ($q\to0$), and bond percolations ($q\to1$). We also explicitly verify several expected degeneracies in the spectrum of relevant operators for natural values of $q$ upon analytic continuation, which are…
We calculate the fractal dimension $d_{\rm f}$ of critical curves in the $O(n)$ symmetric $(\vec ϕ^2)^2$-theory in $d=4-\varepsilon$ dimensions at 6-loop order. This gives the fractal dimension of loop-erased random walks at $n=-2$, self-avoiding walks ($n=0$), Ising lines $(n=1)$, and XY lines ($n=2$), in agreement with numerical simulations. It can be compared to the fractal dimension $d_{\rm f}^{\rm tot}$ of all lines, i.e. backbone plus the surrounding loops, identical to $d_{\rm f}^{\rm tot} = 1/ν$. The combination $ϕ_{\rm c}= d_{\rm f}/d_{\rm f}^{\rm tot} = νd_{\rm f}$ is the crossover e…
It is well know that systems with an interaction decaying as a power of the distance may have critical exponents that are different from those of short-range systems. The boundary between long-range and short-range is known, however the behavior in the crossover region is not well understood. In this paper we propose a general form for the crossover function and we compute it in a particular limit. We compare our predictions with the results of numerical simulations for two-dimensional long-range percolation.
We calculate the maximal Lyapunov exponent for a bulk system of 256 Lennard-Jones particles in constant energy molecular dynamics simulations deep into the supercritical state. We find that the maximal Lyapunov exponent undergoes a crossover, and that this crossover coincides with the dynamical crossover at the Frenkel line from liquid physics. We explain this crossover in terms of two different contributions to dynamical instability: diffusion in the liquid-like state below the Frenkel line, and collisions in the gas-like state above. These results provide insight into the phase space dynamic…
We study through Monte Carlo simulations and finite-size scaling analysis the nonequilibrium phase transitions of the majority-vote model taking place on spatially embedded networks. These structures are built from an underlying regular lattice over which long-range connections are randomly added according to the probability, $P_{ij}\sim{r^{-α}}$, where $r_{ij}$ is the Manhattan distance between nodes $i$ and $j$, and the exponent $α$ is a controlling parameter [J. M. Kleinberg, Nature 406, 845 (2000)]. Our results show that the collective behavior of this system exhibits a continuous order-di…
Domain-wall dynamics and spatial fluctuations are closely related to each other and to universal features of disordered systems. Experimentally measured roughness exponents characterizing spatial fluctuations have been reported for magnetic thin films, with values generally different from those predicted by the equilibrium, depinning and thermal reference states. Here, we study the roughness of domain walls in GdFeCo thin films over a large range of magnetic field and temperature. Our analysis is performed in the framework of a model considering length-scale crossovers between the reference st…
We study the second-order phase transition in the $d$-dimensional Ising model with long-range interactions decreasing as a power of the distance $1/r^{d+s}$. For $s$ below some known value $s_*$, the transition is described by a conformal field theory without a local stress tensor operator, with critical exponents varying continuously as functions of $s$. At $s=s_*$, the phase transition crosses over to the short-range universality class. While the location $s_*$ of this crossover has been known for 40 years, its physics has not been fully understood, the main difficulty being that the standar…
Dynamical detection of quantum phases and phase transitions (QPT) in quenched systems with experimentally convenient initial states is a topic of interest from both theoretical and experimental perspectives. Quenched from polarized states, longitudinal magnetization decays exponentially to zero in time for the short-range transverse-field Ising model (TFIM) and hence, has a featureless steady state regime, which prevents it from exhibiting dynamical phase transitions of type-I. In this paper, we ask whether the transient regimes of such non-equilibrium processes probed by single-site observabl…
We investigate the influence of the range of interactions in the two-dimensional bond percolation model, by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges, as expressed by the number $z$ of equivalent neighbors. We also consider the $z \to \infty$ limit, i.e., the complete graph case, where percolation bonds are allowed between each pair of sites, and the model becomes mean-field-like. All investigated models with finite $z$ are found to belong to the short-range universality class. There is no evidence of a tricritical point separating the sho…
In surfaces with grainy features, the local roughness $w$ shows a crossover at a characteristic length $r_c$, with roughness exponent changing from $α_1\approx 1$ to a smaller $α_2$. The grain shape, the choice of $w$ or height-height correlation function (HHCF) $C$, and the procedure to calculate root mean-square averages are shown to have remarkable effects on $α_1$. With grains of pyramidal shape, $α_1$ can be as low as 0.71, which is much lower than the previous prediction 0.85 for rounded grains. The same crossover is observed in the HHCF, but with initial exponent $χ_1\approx 0.5$ for fl…
A spectral investigation of criticality and crossover effects in two and three dimensions: Short timescales with small systems in minute random matrices
We investigate the one-dimensional pair contact process with diffusion (PCPD) by extensive Monte Carlo simulations, mainly focusing on the critical density decay exponent $δ$. To obtain an accurate estimate of $δ$, we first find the strength of corrections to scaling using the recently introduced method [S.-C. Park. J. Korean Phys. Soc. {\bf 62}, 469 (2013)]. For small diffusion rate ($d\le 0.5$), the leading corrections-to-scaling term is found to be $\sim t^{-0.15}$, whereas for large diffusion rate ($d=0.95$) it is found to be $\sim t^{-0.5}$. After finding the strength of corrections to sc…
Using numerical diagonalization we study the crossover among different random matrix ensembles [Poissonian, Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE) and Gaussian Symplectic Ensemble (GSE)] realized in two different microscopic models. The specific diagnostic tool used to study the crossovers is the level spacing distribution. The first model is a one dimensional lattice model of interacting hard core bosons (or equivalently spin 1/2 objects) and the other a higher dimensional model of non-interacting particles with disorder and spin orbit coupling. We find that the p…
We study the critical behavior of Sherrington-Kirkpatrick model in transverse field (at finite temperature) using Monte Carlo simulation and exact diagonalization (at zero temperature). We determine the phase diagram of the model by estimating the Binder cumulant. We also determine the correlation length exponent from the collapse of the scaled data. Our numerical studies here indicate that critical Binder cumulant (indicating the universality class of the transition behavior) and the correlation length exponent cross over from their `classical' to `quantum' values at a finite temperature (unl…
The magnetic domain wall motion driven by a magnetic field is studied in (Ga,Mn)As and (Ga,Mn)(As,P) films of different thicknesses. In the thermally activated creep regime, a kink in the velocity curves and a jump of the roughness exponent evidence a dimensional crossover in the domain wall dynamics. The measured values of the roughness exponent zeta_{1d} = 0.62 +/- 0.02 and zeta_{2d} = 0.45 +/- 0.04 are compatible with theoretical predictions for the motion of elastic line (d = 1) and surface (d = 2) in two and three dimensional media, respectively.
Recently, it has been found that an effective long-range interaction is realized among local bistable variables (spins) in systems where the elastic interaction causes ordering of the spins. In such systems, generally we expect both long-range and short-range interactions to exist. In the short-range Ising model, the correlation length diverges at the critical point. In contrast, in the long-range interacting model the spin configuration is always uniform and the correlation length is zero. As long as a system has non-zero long-range interactions, it shows criticality in the mean-field univers…
Motivated by the collective behaviour of biological swarms, we study the critical dynamics of field theories with coupling between order parameter and conjugate momentum in the presence of dissipation. By performing a dynamical renormalization group calculation at one loop, we show that the violation of momentum conservation generates a crossover between a conservative yet IR-unstable fixed point, characterized by a dynamic critical exponent $z=d/2$, and a dissipative IR-stable fixed point with $z=2$. Interestingly, the two fixed points have different upper critical dimensions. The interplay b…
We study the order-disorder transition in two-dimensional incompressible systems of motile particles with alignment interactions through extensive numerical simulations of the incompressible Toner-Tu (ITT) field theory and a detailed finite-size scaling (FSS) analysis. The transition looks continuous in the explored parameter space, but the effective susceptibility exponent $γ/ν$ and the dynamic exponent $z$ exhibit a strong, non-monotonic variation on the system size in the form of double crossovers. At small sizes, mean-field exponents are observed for the homogeneous $k=0$ mode whereas spat…
Recent analyses of wetting in the semi-infinite two dimensional Ising model, extended to include both a surface coupling enhancement and a surface field, have shown that the wetting transition may be effectively first-order and that surprisingly the surface susceptibility develops a divergence described by an anomalous exponent with value $γ_{11}^{\rm eff}=\frac{3}{2}$. We reproduce these results using an interfacial Hamiltonian model making connection with previous studies of two dimensional wetting and show that they follow from the simple crossover scaling of the singular contribution to th…
We study the phase diagram of an elastic interaction model for spin crossover (SC) materials with antiferromagnetic-like short-range interactions. In this model, the interplay between the short-range interaction and the long-range interaction of elastic origin causes complex phase transitions. For relatively weak elastic interactions, the phase diagram is characterized by tricritical points, at which antiferromagnetic (AF) -like and ferromagnetic (F) -like spinodal lines and a critical line merge. On the other hand, for relatively strong elastic interactions, unusual "horn structures," which a…
We numerically study the Loewner driving function U_t of a site percolation cluster boundary on the triangular lattice for p<p_c. It is found that U_t shows a drifted random walk with a finite crossover time. Within this crossover time, the averaged driving function < U_t> shows a scaling behavior -(p_c-p) t^{(ν+1)/2ν} with a superdiffusive fluctuation whereas, beyond the crossover time, the driving function U_t undergoes a normal diffusion with Hurst exponent 1/2 but with the drift velocity proportional to (p_c-p)^ν, where ν= 4/3 is the critical exponent for two-dimensional percolation correl…
We revisit the two-dimensional quantum Ising model by computing renormalization group flows close to its quantum critical point. The low but finite temperature regime in the vicinity of the quantum critical point is squashed between two distinct non-Gaussian fixed points: the classical fixed point dominated by thermal fluctuations and the quantum critical fixed point dominated by zero-point quantum fluctuations. Truncating an exact flow equation for the effective action we derive a set of renormalization group equations and analyze how the interplay of quantum and thermal fluctuations, both no…
An important experimental signature of localization is the low-frequency AC conductivity, which typically vanishes as $ω^ϕ$. The exponent $ϕ= 2$ for Anderson insulators, whereas for many body localized insulators $ϕ$ is a continuously varying exponent $1 \le ϕ\le 2$. In this work, we study the low-frequency AC conductivity of localized superconductors, in which disorder is strong enough to localize all quasiparticles, while remaining weak enough to leave superconductivity intact. We find that while the ac conductivity still follows the general form $σ(ω) \sim ω^ϕ$, the exponent $ϕ$ can be mark…
Local anaxonic neurons with graded potential release are important ingredients of nervous systems, present in the olfactory bulb system of mammalians, in the human visual system, as well as in arthropods and nematodes. We develop a neuronal network model including both axonic and anaxonic neurons and monitor the activity tuned by the following parameters: The decay length of the graded potential in local neurons, the fraction of local neurons, the largest eigenvalue of the adjacency matrix and the range of connections of the local neurons. Tuning the fraction of local neurons, we derive the ph…
It is shown how to embed a critical point in a smooth background equation of state so as to yield the critical exponents and critical amplitude ratios expected of a transition in the same universality class as the liquid-gas phase transition and the 3D Ising model. There are only two independent critical exponents; the relations $α+ 2β+ γ= 2$ and $β(δ- 1) = γ$ arise automatically, as does a new relation between the two critical amplitudes. The resulting equation of state has parameters which may be inferred by hydrodynamic modeling of heavy ion collisions in the Beam Energy Scan II at the Rela…
