Tricritical Scaling Research Articles

On the order of the QCD chiral phase transition for different numbers of quark flavours

The nature of the QCD chiral phase transition in the limit of vanishing quark masses has remained elusive for a long time, since it cannot be simulated directly on the lattice and is strongly cutoff-dependent. We report on a comprehensive ongoing study using unimproved staggered fermions with Nf ∈ [2, 8] mass-degenerate flavours on Nτ ∈ {4, 6, 8} lattices, in which we locate the chiral critical surface separating regions with first-order transitions from crossover regions in the bare parameter space of the lattice theory. Employing the fact that it terminates in a tricritical line, this surface can be extrapolated to the chiral limit using tricritical scaling with known exponents. Knowing the order of the transitions in the lattice parameter space, conclusions for approaching the continuum chiral limit in the proper order can be drawn. While a narrow first-order region cannot be ruled out, we find initial evidence consistent with a second-order chiral transition in all massless theories with Nf ≤ 6, and possibly up to the onset of the conformal window at 9 ≲ {N}_{mathrm{f}}^{ast } ≲ 12. A reanalysis of already published mathcal{O} (a)-improved Nf = 3 Wilson data on Nτ ∈ [4, 12] is also consistent with tricritical scaling, and the associated change from first to second-order on the way to the continuum chiral limit. We discuss a modified Columbia plot and a phase diagram for many-flavour QCD that reflect these possible features.

Numerical simulations study of a spin-1 Blume–Emery–Griffiths model on a square lattice

We use the Monte Carlo simulation technique to study the critical behavior of a three-state spin model, with bilinear and biquadratic nearest-neighbor interactions, known as the Blume–Emery–Griffiths model (BEG), in a square lattice. In order to characterize this model, we study the phase diagram, in which we identify three different phases: ferromagnetic, paramagnetic and quadrupolar, the later with one sublattice filled with spins and the other with vacancies. We perform our studies by using two algorithms: Metropolis update (MU) and Wang–Landau (WL). The critical scaling behavior of the model is complementary studied by applying results obtained by using both algorithms, while tricritical points and the tricritical scaling behavior is analyzed by means of WL measuring the joint density of states and using the method of field mixing in conjunction with finite-size scaling. Furthermore, motivated by the decoupling between spins observed within the quadrupolar phase, we further generalize the BEG model in order to study the behavior of the system by adding a next nearest neighbour (NNN) interaction between spins. We found that by increasing the strength of the (ferromagnetic) NNN interaction, a new ferromagnetic phase takes over that contains both quadrupolar and ferromagnetic order.

QCD chiral phase transition from noninteger numbers of flavors

Attempts to extract the order of the chiral transition of QCD at zero chemical potential, with two dynamical flavors of massless quarks, from simulations with progressively decreasing pion mass, have remained inconclusive because of their increasing numerical cost. In an alternative approach to this problem, we consider the path integral as a function of continuous number ${N}_{\mathrm{f}}$ of degenerate quarks. If the transition in the chiral limit is first order for ${N}_{\mathrm{f}}\ensuremath{\ge}3$, a second-order transition for ${N}_{\mathrm{f}}=2$ then requires a tricritical point in between. This, in turn, implies tricritical scaling of the critical boundary line between the first-order and crossover regions as the chiral limit is approached. Noninteger numbers of fermion flavors are easily implemented within the staggered fermion discretization. Exploratory simulations at $\ensuremath{\mu}=0$ and ${N}_{\mathrm{f}}=2.8$, 2.6, 2.4, 2.2, 2.1, on coarse ${N}_{\ensuremath{\tau}}=4$ lattices, indeed show a smooth variation of the critical mass mapping out a critical line in the ($m$, ${N}_{\mathrm{f}}$) plane. For the smallest masses, the line appears consistent with tricritical scaling, allowing for an extrapolation to the chiral limit.

Field-driven quantum phase transitions in S=12 spin chains

We study the magnetization process of a 1D extended Heisenberg model, the $J$-$Q$ model, as a function of an external magnetic field. In this model, $J$ represents the traditional antiferromagnetic Heisenberg exchange and $Q$ is the strength of a competing four-spin interaction. Without external field, this system hosts a twofold-degenerate dimerized (valence-bond solid) state above a critical value $q_c\approx 0.85$ where $q\equiv Q/J$. The dimer order is destroyed and replaced by a partially polarized translationally invariant state at a critical field value. We find magnetization jumps (metamagnetism) between the partially polarized and fully polarized state for $q>q_{\rm min}$, where we have calculated $q_{\rm min}=2/9$ exactly. For $q>q_{\rm min}$ two magnons (flipped spins on a fully polarized background) attract and form a bound state. Quantum Monte Carlo studies confirm that the bound state corresponds to the first step of an instability leading to a finite magnetization jump for $q>q_{\rm min}$. Our results show that neither geometric frustration nor spin-anisotropy are necessary conditions for metamagnetism. Working in the two-magnon subspace, we also find evidence pointing to the existence of metamagnetism in the unfrustrated $J_1$-$J_2$ chain ($J_1>0$, $J_2<0$), but only if $J_2$ is spin-anisotropic. We also investigate quantum-critical scaling near the transition into the fully polarized state for $q\le q_{\rm min}$ at $T>0$. While the expected `zero-scale-factor' universality is clearly seen for $q=0$ and $q\ll q_{\rm min}$; closer to $q_{\rm min}$ we find that extremely low temperatures are required to observe the asymptotic behavior, due to the influence of the tricritical point at $q_{\rm min}$, which leads to a cross-over at a temperature $T^*(q)$ between logarithmic tricritical scaling and zero-scale-factor universality, with $T^*(q)\to 0$ when $q\to q_{\rm min}$.

Area-width scaling in generalised Motzkin paths

We consider a generalised version of Motzkin paths, where horizontal steps have length ℓ, with ℓ being a fixed positive integer. We first give the general functional equation for the area-width generating function of this model. Using a heuristic ansatz, we then derive the area-width scaling behaviour in terms of a scaling function in one variable for the special cases of Dyck, (standard) Motzkin and Schröder paths, before generalising our approach to arbitrary ℓ. We then rigorously derive the tricritical scaling of Schröder paths by applying the generalised method of steepest descents to the known exact solution for their area-width generating function. Our results show that for Dyck and Schröder paths, the heuristic scaling ansatz reproduces the rigorous results.

Discontinuous percolation

I present recent progress in understanding abrupt connectivity transitions, i.e. percolation-type models exhibiting first order phase transitions. The introduction of bridge bonds and their tricritical scaling allows obtaining exact expressions for the appearance of a jump in the order parameter. Another exact result can be obtained through the Potts model using Gliozzi's scheme. Finally also hybrid transitions are discussed.

First-order phase transition and tricritical scaling behavior of the Blume-Capel model: A Wang-Landau sampling approach.

We investigate the tricritical scaling behavior of the two-dimensional spin-1 Blume-Capel model by using the Wang-Landau method of measuring the joint density of states for lattice sizes up to 48×48 sites. We find that the specific heat deep in the first-order area of the phase diagram exhibits a double-peak structure of the Schottky-like anomaly appearing with the transition peak. The first-order transition curve is systematically determined by employing the method of field mixing in conjunction with finite-size scaling, showing a significant deviation from the previous data points. At the tricritical point, we characterize the tricritical exponents through finite-size-scaling analysis including the phenomenological finite-size scaling with thermodynamic variables. Our estimation of the tricritical eigenvalue exponents, yt=1.804(5), yg=0.80(1), and yh=1.925(3), provides the first Wang-Landau verification of the conjectured exact values, demonstrating the effectiveness of the density-of-states-based approach in finite-size scaling study of multicritical phenomena.

Phase structure of QCD for heavy quarks

We investigate the nature of the deconfinement and Roberge-Weiss transition in the heavy quark regime for finite real and imaginary chemical potential within the functional approach to continuum QCD. We extract the critical phase boundary between the first order and cross-over regions, and also explore tricritical scaling. Our results confirm previous ones from finite volume lattice studies.

Towards designing robust coupled networks.

Natural and technological interdependent systems have been shown to be highly vulnerable due to cascading failures and an abrupt collapse of global connectivity under initial failure. Mitigating the risk by partial disconnection endangers their functionality. Here we propose a systematic strategy of selecting a minimum number of autonomous nodes that guarantee a smooth transition in robustness. Our method which is based on betweenness is tested on various examples including the famous 2003 electrical blackout of Italy. We show that, with this strategy, the necessary number of autonomous nodes can be reduced by a factor of five compared to a random choice. We also find that the transition to abrupt collapse follows tricritical scaling characterized by a set of exponents which is independent on the protection strategy.

Tricritical Point in Explosive Percolation

The suitable interpolation between classical percolation and a special variant of explosive percolation enables the explicit realization of a tricritical percolation point. With high-precision simulations of the order parameter and the second moment of the cluster size distribution a fully consistent tricritical scaling scenario emerges yielding the tricritical crossover exponent $1/\varphi_t=1.8\pm0.1$.

Tricritical Point in Explosive Percolation

The suitable interpolation between classical percolation and a special variant of explosive percolation enables the explicit realization of a tricritical percolation point. With high-precision simulations of the order parameter and the second moment of the cluster size distribution a fully consistent tricritical scaling scenario emerges yielding the tricritical crossover exponent 1/φ(t)=1.8 ± 0.1.

Constraining the QCD Phase Diagram by Tricritical Lines at Imaginary Chemical Potential

We present unambiguous evidence, from lattice simulations of QCD with three degenerate quark species, for two tricritical points in the (T, m) phase diagram at fixed imaginary chemical potential μ/T = iπ/3 mod2π/3, one in the light and one in the heavy mass regime. These represent the boundaries of the chiral and deconfinement critical lines continued to imaginary μ, respectively. It is demonstrated that the shape of the deconfinement critical line for real chemical potentials is dictated by tricritical scaling and implies the weakening of the deconfinement transition with real chemical potential. The generalization to nondegenerate and light quark masses is discussed.

Polarized superfluids near their tricritical point

We obtain a variety of predictions for the properties of population-imbalanced (or polarized) fermionic superfluids near their tricritical point. In the vicinity of the high-symmetry tricritical point, observable quantities such as the cloud shape, heat capacity, local polarization, and correlation length should exhibit distinct behavior arising from the tricritical scaling laws, as well as logarithmic corrections to scaling reflecting the marginal nature of interactions.

Asymptotic Behavior of the Magnetization Near Critical and Tricritical Points via Ginzburg–Landau Polynomials

The purpose of this paper is to prove connections among the asymptotic behavior of the magnetization, the structure of the phase transitions, and a class of polynomials that we call the Ginzburg–Landau polynomials. The model under study is a mean-field version of a lattice spin model due to Blume and Capel. It is defined by a probability distribution that depends on the parameters β and K, which represent, respectively, the inverse temperature and the interaction strength. Our main focus is on the asymptotic behavior of the magnetization m(β n ,K n ) for appropriate sequences (β n ,K n ) that converge to a second-order point or to the tricritical point of the model and that lie inside various subsets of the phase-coexistence region. The main result states that as (β n ,K n ) converges to one of these points (β,K), \(m(\beta_{n},K_{n})\sim \bar{x}|\beta -\beta_{n}|^{\gamma}\rightarrow 0\) . In this formula γ is a positive constant, and \(\bar{x}\) is the unique positive, global minimum point of a certain polynomial g. We call g the Ginzburg–Landau polynomial because of its close connection with the Ginzburg–Landau phenomenology of critical phenomena. For each sequence the structure of the set of global minimum points of the associated Ginzburg–Landau polynomial mirrors the structure of the set of global minimum points of the free-energy functional in the region through which (β n ,K n ) passes and thus reflects the phase-transition structure of the model in that region. This paper makes rigorous the predictions of the Ginzburg–Landau phenomenology of critical phenomena and the tricritical scaling theory for the mean-field Blume–Capel model.

Tricritical Directed Percolation

We consider a modification of the contact process incorporating higher-order reaction terms. The original contact process exhibits a non-equilibrium phase transition belonging to the universality class of directed percolation. The incorporated higher-order reaction terms lead to a non-trivial phase diagram. In particular, a line of continuous phase transitions is separated by a tricritical point from a line of discontinuous phase transitions. The corresponding tricritical scaling behavior is analyzed in detail, i.e., we determine the critical exponents, various universal scaling functions as well as universal amplitude combinations.

Rectangular vesicles in three dimensions

The phase diagram of a model of rectangular vesicles in three dimensions is examined. The scaling of the generating function is determined in the area–perimeter, volume–area and volume–perimeter ensembles. The results are interpreted within the framework of tricritical scaling, and the crossover exponents associated with the transitions are determined. We identify three phases in the phase diagram, a needle phase, a disc phase and a cubical phase. These phases are separated by three curves of transitions that meet in a multicritical point.

Inflating square and rectangular lattice vesicles

In this paper, two simple models of lattice vesicles in two and in three dimensions are examined. Lattice vesicles have a rich mathematical nature, and the combinatorial and scaling properties of square and rectangular lattice vesicle models are investigated and then generalized to three dimensions. The phase diagram of the vesicles also includes a multicritical point, and scaling of the generating functions of these models are studied to test the models for tricritical scaling behaviour. In the models of square vesicles in the square lattice, and cubical vesicles in the cubic lattice, the tricritical scaling forms are explicitly confirmed. However, rectangular vesicles in the square lattice, and rectangular box vesicles in the cubic lattice, appear not to conform to the expected tricritical scaling forms.

Statistical mechanics of directed models of polymers in the square lattice

Directed square lattice models of polymers and vesicles have received considerable attention in the recent mathematical and physical sciences literature. These are idealized geometric directed lattice models introduced to study phase behaviour in polymers, and include Dyck paths, partially directed paths, directed trees and directed vesicles models. Directed models are closely related to models studied in the combinatorics literature (and are often exactly solvable). They are also simplified versions of a number of statistical mechanics models, including the self-avoiding walk, lattice animals and lattice vesicles. The exchange of approaches and ideas between statistical mechanics and combinatorics have considerably advanced the description and understanding of directed lattice models, and this will be explored in this review. The combinatorial nature of directed lattice path models makes a study using generating function approaches most natural. In contrast, the statistical mechanics approach would introduce partition functions and free energies, and then investigate these using the general framework of critical phenomena. Generating function and statistical mechanics approaches are closely related. For example, questions regarding the limiting free energy may be approached by considering the radius of convergence of a generating function, and the scaling properties of thermodynamic quantities are related to the asymptotic properties of the generating function. In this review the methods for obtaining generating functions and determining free energies in directed lattice path models of linear polymers is presented. These methods include decomposition methods leading to functional recursions, as well as the Temperley method (that is implemented by creating a combinatorial object, one slice at a time). A constant term formulation of the generating function will also be reviewed. The thermodynamic features and critical behaviour in models of directed paths may be informative about the underlying properties that determine phase diagrams for wider classes of models, including physical models of polymers. Of particular interest are adsorption and collapse transitions in models of polymers and copolymers. The properties of thermodynamic quantities in those models are described by tricritical scaling. This is reviewed for directed path models, and the generating function approaches can be used to apply tricritical scaling to models of adsorbing, inflating and collapsing directed lattice paths. Critical exponents for a variety of models can be obtained in this manner, and with it a better understanding, and a classification, of the models.

Interacting columns: generating functions and scaling exponents

A model of self-interacting columns, related to models of partially directed walks and to histogram polygons, is considered. The generating function of this model is found in terms of q-deformed Bessel functions using a functional recursion scheme. A transition, related to a deflation-inflation transition seen in staircase polygons, or to a rough-smooth transition in a solid-on-solid model, is found, and its scaling exponents are found in the context of a tricritical scaling analysis. If the columns are also interacting with the horizontal axis, then the inflated or smooth phase is also found to undergo an adsorption transition. A special point exists where the model is critical with respect to both an adsorption transition and a deflation-inflation transition.

The statistical mechanics of vesicles

Partially convex polygons, Exact solution, q-series asymptotics, Scaling function. I discuss two-dimensional lattice models of closed fluctuating membranes, or vesicles (11. The underlying mathematical model is that of self-avoiding polygons and their enumeration by perime- ter and area.. Physically, this model is of interest as its phase diagram contains a point at which tricritical scaling is observed [2]. By adding the constraint of partial directedness, one gets combinatorial models related to the partitioning of integers. These models are solvable in the sense that an explicit expression for the generating function can be given [3,4]. The derivation of this generating function involves a nonlinear functional equation, and the solution is given in terms of alternating q-series. The asymptotic analysis of these generating functions can be done with the help of a new contour integral representation. This leads to an explicit calculation of the scaling behaviour around the critical point in terms of the Airy function [5]. In particular, one gets a uniform asymptotic expansion of the involved q-series as