In ferromagnetic spin models above the critical temperature ($T > T_{cr}$) the partition function zeros accumulate at complex values of the magnetic field ($H_E$) with a universal behavior for the density of zeros $ρ(H) \sim | H - H_E |^{\sg}$. The critical exponent $\sg$ is believed to be universal at each space dimension and it is related to the magnetic scaling exponent $y_h$ via $\sg = (d-y_h)/y_h$. In two dimensions we have $y_h=12/5 (\sg = -1/6)$ while $y_h=2 (\sg=-1/2)$ in $d=1$. For the one dimensional Blume-Capel and Blume-Emery-Griffiths models we show here, for different temperatures, that a new value $y_h=3 (\sg =-2/3)$ can emerge if we have a triple degeneracy of the transfer matrix eigenvalues.
We report an extensive Monte Carlo study of critical end point behaviour in a symmetrical binary fluid mixture. On the basis of general scaling arguments, singular behaviour is predicted in the diameter of the liquid-gas coexistence curve as the critical end point is approached. The simulation results show clear evidence for this singularity, as well as confirming a previously predicted singularity in the coexistence chemical potential. Both singularities should be detectable experimentally.
We study the statistics of the work done by globally changing in time with a generic protocol the mass in a free bosonic field theory with relativistic dispersion and the transverse field in the one-dimensional Ising chain both globally and locally. In the latter case we make the system start from the critical point and we describe it in the scaling limit. We provide exact formulas in all these cases for the full statistics of the work and we show that the low energy part of the distribution of the work displays an edge singularity whose exponent does not depend on the specifics of the protoco…
Power-law singularities and critical exponents in n-vector models are considered from different theoretical points of view. It includes a theoretical approach called the GFD (grouping of Feynman diagrams) theory, as well as the perturbative renormalization group (RG) treatment. A non-perturbative proof concerning corrections to scaling in the two-point correlation function of the phi^4 model is provided, showing that predictions of the GFD theory rather than those of the perturbative RG theory can be correct. Critical exponents determined from highly accurate experimental data very close to th…
We have extended, in most cases through 24th order, the series expansions of the dimer density in powers of the activity in the case of bipartite ((hyper)-simple-cubic and (hyper)-body-centered-cubic) lattices of dimensionalities 2<= d <= 7. A numerical analysis of these data yields estimates of the exponents characterizing the Yang-Lee edge-singularities for lattice ferromagnetic spin-models as d varies between the lower and the upper critical dimensionalities. Our results are consistent with, but more extensive and sometimes more accurate than those obtained from the existing dimer series or…
Yang-Lee edge singularities are the branch point of the free energy on the complex plane of physical parameters and were shown to be the simplest universality class of phase transitions. However, the Yang-Lee edge singularities have not been regarded as experimentally observable since they occur at complex physical parameters, which are unphysical. A recent discovery about the relation between partition functions and probe spin coherence makes it experimentally feasible to access the complex plane of physical parameters. However, how to extract the critical point and the critical exponent of Y…
We address a number of outstanding questions associated with the analytic properties of the universal equation of state of the $ϕ^4$ theory, which describes the critical behavior of the Ising model and ubiquitous critical points of the liquid-gas type. We focus on the relation between spinodal points that limit the domain of metastability for temperatures below the critical temperature, i.e., $T < T_{\rm c}$, and Lee-Yang edge singularities that restrict the domain of analyticity around the point of zero magnetic field $H$ for $T > T_{\rm c}$. The extended analyticity conjecture (due to Fonsec…
A class of continuum models with a critical end point is considered whose Hamiltonian ${\mathcal{H}}[ϕ,ψ]$ involves two densities: a primary order-parameter field, $ϕ$, and a secondary (noncritical) one, $ψ$. Field-theoretic methods (renormalization group results in conjunction with functional methods) are used to give a systematic derivation of singularities occurring at critical end points. Specifically, the thermal singularity $\sim|{t}|^{2-α}$ of the first-order line on which the disordered or ordered phase coexists with the noncritical spectator phase, and the coexistence singularity $\si…
Continuum models with critical end points are considered whose Hamiltonian ${\mathcal{H}}[ϕ,ψ]$ depends on two densities $ϕ$ and $ψ$. Field-theoretic methods are used to show the equivalence of the critical behavior on the critical line and at the critical end point and to give a systematic derivation of critical-end-point singularities like the thermal singularity $\sim|{t}|^{2-α}$ of the spectator-phase boundary and the coexistence singularities $\sim |{t}|^{1-α}$ or $\sim|{t}|^β$ of the secondary density $<ψ>$. The appearance of a discontinuity eigenexponent associated with the critical end…
We study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one-dimension. At the critical point, the dynamical exponent is infinite and the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point there are Griffiths-McCoy singularities, characterized by a single, continuously varying exponent, z', which diverges at the critical point, as in one-dimension. Consequently, the zero temperature susceptibility dive…
We study the $\pm J$ transverse-field Ising spin glass model at zero temperature on d-dimensional hypercubic lattices and in the Sherrington-Kirkpatrick (SK) model, by series expansions around the strong field limit. In the SK model and in high-dimensions our calculated critical properties are in excellent agreement with the exact mean-field results, surprisingly even down to dimension $d = 6$ which is below the upper critical dimension of $d=8$. In contrast, in lower dimensions we find a rich singular behavior consisting of critical and Griffiths-McCoy singularities. The divergence of the equ…
We investigate the nonequilibrium phase transition of the disordered contact process in five space dimensions by means of optimal fluctuation theory and Monte Carlo simulations. We find that the critical behavior is of mean-field type, i.e., identical to that of the clean five-dimensional contact process. It is accompanied by off-critical power-law Griffiths singularities whose dynamical exponent $z'$ saturates at a finite value as the transition is approached. These findings resolve the apparent contradiction between the Harris criterion which implies that weak disorder is renormalization-gro…
We show with two numerical examples that the conventional expansion in powers of the field for the critical potential of 3-dimensional O(N) models in the large-N limit, does not converge for values of phi^2 larger than some critical value. This can be explained by the existence of conjugated branch points in the complex phi^2 plane. Pade approximants [L+3/L] for the critical potential apparently converge at large phi^2. This allows high-precision calculation of the fixed point in a more suitable set of coordinates. We argue that the singularities are generic and not an artifact of the large-N …
We study the bipartite entanglement entropy of the two-dimensional (2D) transverse-field Ising model in the thermodynamic limit using series expansion methods. Expansions are developed for the Renyi entropy around both the small-field and large-field limits, allowing the separate calculation of the entanglement associated with lines and corners at the boundary between sub-systems. Series extrapolations are used to extract subleading power laws and logarithmic singularities as the quantum critical point is approached. In 1D, we find excellent agreement with exact results as well as quantum Mont…
The $Q$-state Potts model on the simple-cubic lattice is studied using the zeros of the exact partition function on a finite lattice. The critical behavior of the model in the ferromagnetic and antiferromagnetic phases is discussed based on the distribution of the zeros in the complex temperature plane. The characteristic exponents at complex-temperature singularities, which coexist with the physical critical points in the complex temperature plane for no magnetic field ($H_q=0$), are estimated using the low-temperature series expansion. We also study the partition function zeros of the Potts …
We show that the critical point of the two-dimensional Bose-Hubbard model can be easily found through studies of either on-site atom number fluctuations or the nearest-neighbor two-point correlation function (the expectation value of the tunnelling operator). Our strategy to locate the critical point is based on the observation that the derivatives of these observables with respect to the parameter that drives the superfluid-Mott insulator transition are singular at the critical point in the thermodynamic limit. Performing the quantum Monte Carlo simulations of the two-dimensional Bose-Hubbard…
We introduce a new way of reconstructing the equation of state of a thermodynamic system near a second order critical point from a finite set of Taylor coefficients computed away from the critical point. We focus on the Ising universality class (${\mathbb Z}_2$ symmetry) and show that in the crossover region of the phase diagram it is possible to efficiently extract the location of the nearest thermodynamic singularity, the Lee-Yang edge singularity, from which one can (i) determine the location of the critical point, (ii) constrain the non-universal parameters that maps the equation of state …
The paper gives a systematic analysis of singularities of transition processes in dynamical systems. General dynamical systems with dependence on parameter are studied. A system of relaxation times is constructed. Each relaxation time depends on three variables: initial conditions, parameters $k$ of the system and accuracy $ε$ of the relaxation. The singularities of relaxation times as functions of $(x_0,k)$ under fixed $ε$ are studied. The classification of different bifurcations (explosions) of limit sets is performed. The relations between the singularities of relaxation times and bifurcati…
We study analytically the logarithmic corrections to the critical exponents of the critical behavior of correlation length, susceptibility and specific heat for the temperature and the finite-size scaling behavior, for a generic $ϕ^3$ theory at its upper critical dimension (six). We have also computed the leading correction to scaling as a function of the lattice size. We distinguish the obtained formulas to the following special cases: percolation, Lee-Yang (LY) singularities and $m$-component spin glasses. We have compared our results for the Ising spin glass case with numerical simulations …
