Dynamical Critical Exponent Research Articles

Indications for a critical point in the phase diagram for hot and dense nuclear matter

Two-pion interferometry measurements are studied for a broad range of collision centralities in Au+Au (sNN=7.7–200 GeV) and Pb+Pb (sNN=2.76 TeV) collisions. They indicate non-monotonic excitation functions for the Gaussian emission source radii difference (Rout−Rside), suggestive of reaction trajectories which spend a fair amount of time near a soft point in the equation of state (EOS) that coincides with the critical end point (CEP). A Finite-Size Scaling (FSS) analysis of these excitation functions, provides further validation tests for the CEP. It also indicates a second order phase transition at the CEP, and the values Tcep∼165 MeV and μBcep∼95 MeV for its location in the (T,μB)-plane of the phase diagram. The static critical exponents (ν≈0.66 and γ≈1.2) extracted via the same FSS analysis, place this CEP in the 3D Ising model (static) universality class. A Dynamic Finite-Size Scaling analysis of the excitation functions, gives the estimate z∼0.87 for the dynamic critical exponent, suggesting that the associated critical expansion dynamics is dominated by the hydrodynamic sound mode.

Critical initial-slip scaling for the noisy complex Ginzburg–Landau equation

We employ the perturbative fieldtheoretic renormalization group method to investigate the universal critical behavior near the continuous non-equilibrium phase transition in the complex Ginzburg–Landau equation with additive white noise. This stochastic partial differential describes a remarkably wide range of physical systems: coupled nonlinear oscillators subject to external noise near a Hopf bifurcation instability; spontaneous structure formation in non-equilibrium systems, e.g., in cyclically competing populations; and driven-dissipative Bose–Einstein condensation, realized in open systems on the interface of quantum optics and many-body physics, such as cold atomic gases and exciton-polaritons in pumped semiconductor quantum wells in optical cavities. Our starting point is a noisy, dissipative Gross–Pitaevski or nonlinear Schrödinger equation, or equivalently purely relaxational kinetics originating from a complex-valued Landau–Ginzburg functional, which generalizes the standard equilibrium model A critical dynamics of a non-conserved complex order parameter field. We study the universal critical behavior of this system in the early stages of its relaxation from a Gaussian-weighted fully randomized initial state. In this critical aging regime, time translation invariance is broken, and the dynamics is characterized by the stationary static and dynamic critical exponents, as well as an independent ‘initial-slip’ exponent. We show that to first order in the dimensional expansion about the upper critical dimension, this initial-slip exponent in the complex Ginzburg–Landau equation is identical to its equilibrium model A counterpart. We furthermore employ the renormalization group flow equations as well as construct a suitable complex spherical model extension to argue that this conclusion likely remains true to all orders in the perturbation expansion.

Point to point continuum percolation in two dimensions

The outcome of the classic percolation approach is several power-law curves with some universal (critical) exponents. Here, the universality means that these power laws as well as their critical exponents, which control the global properties of a system, are independent of its details. Classic percolation considers the connectivity between two lines and two faces at opposite sides of a system in 2- and 3D problems, respectively; whereas, in practice (e.g. hydrocarbon formations), production and injection wells are represented by points (in 2D areal models) and lines (in 3D models). This study presents the results of Monte Carlo simulations of a 2D percolation model wherein the connection locations (i.e. wells) are represented by two points, called point-to-point (P2P) connectivity. The main contribution is to find the percolation threshold as well as the geometrical and dynamical critical exponents of a continuum percolation system with a P2P connection, which is closer to reality in some applications. The result shows that in comparison to classical percolation, some critical exponents definitely changes in the P2P connection.

Horizon as critical phenomenon

We show that renormalization group(RG) flow can be viewed as a gradual wave function collapse, where a quantum state associated with the action of field theory evolves toward a final state that describes an IR fixed point. The process of collapse is described by the radial evolution in the dual holographic theory. If the theory is in the same phase as the assumed IR fixed point, the initial state is smoothly projected to the final state. If in a different phase, the initial state undergoes a phase transition which in turn gives rise to a horizon in the bulk geometry. We demonstrate the connection between critical behavior and horizon in an example, by deriving the bulk metrics that emerge in various phases of the U(N) vector model in the large N limit based on the holographic dual constructed from quantum RG. The gapped phase exhibits a geometry that smoothly ends at a finite proper distance in the radial direction. The geometric distance in the radial direction measures a complexity : the depth of RG transformation that is needed to project the generally entangled UV state to a direct product state in the IR. For gapless states, entanglement persistently spreads out to larger length scales, and the initial state can not be projected to the direct product state. The obstruction to smooth projection at charge neutral point manifests itself as the long throat in the anti-de Sitter space. The Poincare horizon at infinity marks the critical point which exhibits a divergent length scale in the spread of entanglement. For the gapless states with non-zero chemical potential, the bulk space becomes the Lifshitz geometry with the dynamical critical exponent two. The identification of horizon as critical point may provide an explanation for the universality of horizon. We also discuss the structure of the bulk tensor network that emerges from the quantum RG.

Alternative method to characterize continuous and discontinuous phase transitions in surface reaction models

In this paper we revisited the Ziff-Gulari-Barshad model to study its phase transitions and critical exponents through time-dependent Monte Carlo simulations. We use a method proposed recently to locate the nonequilibrium second-order phase transitions and that has been successfully used in systems with defined Hamiltonians and with absorbing states. This method, which is based on optimization of the coefficient of determination of the order parameter, was able to characterize the continuous phase transition of the model, as well as its upper spinodal point, a pseudocritical point located near the discontinuous phase transition. The static critical exponents β,ν_{∥}, and ν_{⊥}, as well as the dynamic critical exponents θ and z for the continuous transition point, were also estimated and are in excellent agreement with results found in literature.

Interaction-Induced Dirac Fermions from Quadratic Band Touching in Bilayer Graphene.

We revisit the effect of local interactions on the quadratic band touching (QBT) of the Bernal honeycomb bilayer model using renormalization group (RG) arguments and quantum MonteCarlo (QMC) simulations. We present a RG argument which predicts, contrary to previous studies, that weak interactions do not flow to strong coupling even if the free dispersion has a QBT. Instead, they generate a linear term in the dispersion, which causes the interactions to flow back to weak coupling. Consistent with this RG scenario, in unbiased QMC simulations of the Hubbard model we find compelling evidence that antiferromagnetism turns on at a finite U/t despite the U=0 hopping problem having a QBT. The onset of antiferromagnetism takes place at a continuous transition which is consistent with (2+1)D Gross-Neveu criticality. We conclude that generically in models of bilayer graphene, even if the free dispersion has a QBT, small local interactions generate a Dirac phase with no symmetry breaking and that there is a finite-coupling transition out of this phase to a symmetry-broken state.

Coarsening and thermalization properties of a quenched ferromagnetic spin-1 condensate

We examine the dynamics of a quasi-two-dimensional spin-1 condensate in which the quadratic Zeeman energy q is suddenly quenched to a value where the system has a ferromagnetic ground state. There are two distinct types of ferromagnetic phases, i.e. a range of q values where the magnetization prefers to be in the direction of the external field (easy-axis), and a range of q values where it prefers to be transverse to the field (easy-plane). We study the quench dynamics for a variety of q values and show that there is a single dynamical critical exponent to characterize the scale invariant domain growth for each ferromagnetic phase. For both quenches we give simple analytic models that capture the essential scale invariant dynamics, and correctly predict the exponents. Because the order parameter for each phase is different, the natures of the domains and the relevant topological defects in each type of coarsening is also different. To explore these differences we characterize the fractal dimension of the domain walls, and the relationship of polar-core spin vortices to the domains in the easy-plane phase. Finally, we consider how the energy liberated from the quench thermalizes in the easy-axis quench. We show that local equilibrium is established in the spin waves on moderate time scales, but continues to evolve as the domains anneal.

Threshold of coexistence and critical behavior of a predator-prey stochastic model in a fractal landscape

We investigate a stochastic lattice model describing a predator-prey system in a fractal scale-free landscape, mimicked by the fractal Sierpinski carpet. We determine the threshold of species coexistence, that is, the critical phase boundary related to the transition between an active state, where both species coexist and an absorbing state where one of the species is extinct. We show that the predators must live longer in order to persist in a fractal habitat. We further performed a finite-size scaling analysis in the vicinity of the absorbing-state phase transition to compute a set of stationary and dynamical critical exponents. Our results indicate that the transition belongs to the directed percolation universality class exhibited by the usual contact process model on the same fractal landscape.

Hyperscaling violation at the Ising-nematic quantum critical point in two-dimensional metals

Understanding optical conductivity data in the optimally doped cuprates in the framework of quantum criticality requires a strongly-coupled quantum critical metal which violates hyperscaling. In the simplest scaling framework, hyperscaling violation can be characterized by a single non-zero exponent $\theta$, so that in a spatially isotropic state in $d$ spatial dimensions, the specific heat scales with temperature as $T^{(d-\theta)/z}$, and the optical conductivity scales with frequency as $\omega^{(d-\theta-2)/z}$ for $\omega \gg T$, where $z$ is the dynamic critical exponent. We study the Ising-nematic critical point, using the controlled dimensional regularization method proposed by Dalidovich and Lee (Phys. Rev. B {\bf 88}, 245106 (2013)). We find that hyperscaling is violated, with $\theta =1$ in $d=2$. We expect that similar results apply to Fermi surfaces coupled to gauge fields in $d=2$.

Neutron spin-echo study of the critical dynamics of spin-52antiferromagnets in two and three dimensions

We report a neutron spin-echo study of the critical dynamics in the $S=5/2$ antiferromagnets MnF$_2$ and Rb$_2$MnF$_4$ with three-dimensional (3D) and two-dimensional (2D) spin systems, respectively, in zero external field. Both compounds are Heisenberg antiferromagnets with a small uniaxial anisotropy resulting from dipolar spin-spin interactions, which leads to a crossover in the critical dynamics close to the N\'eel temperature, $T_N$. By taking advantage of the $\mu\text{eV}$ energy resolution of the spin-echo spectrometer, we have determined the dynamical critical exponents $z$ for both longitudinal and transverse fluctuations. In MnF$_2$, both the characteristic temperature for crossover from 3D Heisenberg to 3D Ising behavior and the exponents $z$ in both regimes are consistent with predictions from the dynamical scaling theory. The amplitude ratio of longitudinal and transverse fluctuations also agrees with predictions. In Rb$_2$MnF$_4$, the critical dynamics crosses over from the expected 2D Heisenberg behavior for $T\gg T_N$ to a scaling regime with exponent $z = 1.387(4)$, which has not been predicted by theory and may indicate the influence of long-range dipolar interactions.

Prethermalization and thermalization of a quenched interacting Luttinger liquid

We study the relaxation dynamics of interacting, one-dimensional fermions with band curvature after a weak quench in the interaction parameter. After the quench, the system is described by a non-equilibrium initial state, which relaxes towards thermal equilibrium, featuring prethermal behavior on intermediate time and length scales. The model corresponds to the class of interacting Luttinger Liquids, which extends the quadratic Luttinger theory by a weak integrability breaking phonon scattering term. In order to solve for the non-equilibrium time evolution, we use kinetic equations for the phonon densities, exploiting the resonant but subleading character of the phonon interaction term. The interplay between phonon scattering and the quadratic Luttinger theory leads to the emergence of three distinct spatio-temporal regimes for the fermionic real-space correlation function. It features the crossover from a prequench to a prethermal state, finally evolving towards a thermal state on increasing length and time scales. The characteristic algebraically decaying real-space correlations in the prethermalized regime become modulated by an amplitude, which, as an effect of the interactions, is decaying in time according to a stretched-exponential, while in the thermal regime exponentially decaying real-space correlations emerge. The asymptotic thermalization dynamics is governed by energy transport over large distances from the thermalized to the non-thermalized regions, which is carried out by macroscopic, dynamical slow modes. This is revealed in an algebraic decay of the system's effective temperature. The numerical value of the associated exponent agrees with the dynamical critical exponent of the Kardar-Parisi-Zhang universality class, which shares with the present interacting Luttinger Liquid the conservation of total energy and momentum.

Tensor-network algorithm for nonequilibrium relaxation in the thermodynamic limit.

We propose a tensor-network algorithm for discrete-time stochastic dynamics of a homogeneous system in the thermodynamic limit. We map a d-dimensional nonequilibrium Markov process to a (d+1)-dimensional infinite tensor network by using a higher-order singular-value decomposition. As an application of the algorithm, we compute the nonequilibrium relaxation from a fully magnetized state to equilibrium of the one- and two-dimensional Ising models with periodic boundary conditions. Utilizing the translational invariance of the systems, we analyze the behavior in the thermodynamic limit directly. We estimated the dynamical critical exponent z=2.16(5) for the two-dimensional Ising model. Our approach fits well with the framework of the nonequilibrium-relaxation method. Our algorithm can compute time evolution of the magnetization of a large system precisely for a relatively short period. In the nonequilibrium-relaxation method, one needs to simulate dynamics of a large system for a short time. The combination of the two provides a different approach to the study of critical phenomena.

Particle size-dependent superspin glass behavior in random compacts of monodisperse maghemite nanoparticles

Dense random assemblies made from highly monodisperse γ-Fe2O3 nanoparticles with sizes ranging from 6.2 to 11.5 nm have been investigated by DC and AC magnetometry. It is found that all assemblies undergo superspin glass phase transitions. The superspin glass phase transition temperature is strongly dependent on the particle size and the nature of the interparticle interaction. However the transition from superparamagnet to superspin glass, as evidenced by the shape of the ac-susceptibility curves and the dynamic critical exponents associated with the transition, is similar in all systems.

Causality and quantum criticality in long-range lattice models.

Long-range quantum lattice systems often exhibit drastically different behavior than their short-range counterparts. In particular, because they do not satisfy the conditions for the Lieb-Robinson theorem, they need not have an emergent relativistic structure in the form of a light cone. Adopting a field-theoretic approach, we study the one-dimensional transverse-field Ising model with long-range interactions, and a fermionic model with long-range hopping and pairing terms, explore their critical and near-critical behavior, and characterize their response to local perturbations. We deduce the dynamic critical exponent, up to the two-loop order within the renormalization group theory, which we then use to characterize the emergent causal behavior. We show that beyond a critical value of the power-law exponent of the long-range couplings, the dynamics effectively becomes relativistic. Various other critical exponents describing correlations in the ground state, as well as deviations from a linear causal cone, are deduced for a wide range of the power-law exponent.

Tuneability and criticality in a three-dimensional stacked molecular system

With the changing of physical properties through chemical substitution in mind, this work shows theapplicability of this approach to organic based spin-Peierls (SP) systems. To demonstrate this we have useda well known system, potassium TCNQ, that undergoes an SP transition atTSP=396 K. Simply substitutingprotons for fluorine, using TCNQF4, shows a decrease in the coupling strength between TCNQ anions where thespin-Peierls transition is dramatically reduced in temperature, with KTCNQF4showing aTSPat approximately160 K, which is due to changes of the electron spin density across the molecule. Muon spin relaxation is a moresuitable technique for determining the magnetic properties of these systems and measurements were conducted onboth KTCNQ and KTCNQF4in order to study the behavior of the SP transition. This has highlighted and shownthat both transitions exhibit the same behavior and so are indeed similar; however, we have also succeeded inbeing able to tune the transition. Estimates of the dynamic critical exponents from both samples are obtained fromthe muon data and are found to be∼0.33, corresponding to a three-dimensional (3D) antiferromagnetic systemimplying that spin fluctuations associated with the SP state are correlated not just along stacks of dimerizedTCNQ anions. This result extends the understanding of the SP phase transition in the KTCNQ system beyondthat determined from purely structural studies.

Static and dynamical critical behavior of the monomer–monomer reaction model with desorption

We studied in this work the monomer–monomer reaction model on a linear chain. The model is described by the following reaction: A+B→AB, where A and B are two monomers that arrive at the surface with probabilities yA and yB, respectively, and we have considered desorption of the monomer B with probability α. The model is studied in the adsorption controlled limit where the reaction rate is infinitely larger than the adsorption rate. We employ site and pair mean-field approximations as well as static and dynamical Monte Carlo simulations. We show that the model exhibits a continuous phase transition between an active steady state and an A-absorbing state, when the parameter yA is varied through a critical value, which depends on the value of α. Monte Carlo simulations and finite-size scaling analysis near the critical point are used to determine the static critical exponents β and ν⊥ and the dynamical critical exponents ν∥ and z. The results found for the monomer–monomer reaction model with B desorption, in the linear chain, are different from those found by E. V. Albano (Albano, 1992) and are in accordance with the values obtained by Jun Zhuo and Sidney Redner (Zhuo and Redner, 1993), and endorse the conjecture of Grassberger, which states that any system undergoing a continuous phase transition from an active steady state to a single absorbing state, exhibits the same critical behavior of the directed percolation universality class.

Universal Coarsening Dynamics of a Quenched Ferromagnetic Spin-1 Condensate.

We demonstrate that a quasi-two-dimensional spin-1 condensate quenched to a ferromagnetic phase undergoes universal coarsening in its late time dynamics. The quench can be implemented by a sudden change in the applied magnetic field and, depending on the final value, the ferromagnetic phase has easy-axis (Ising) or easy-plane (XY) symmetry, with different dynamical critical exponents. Our results for the easy-plane phase reveal a fractal domain structure and the crucial role of polar-core spin vortices in the coarsening dynamics.

Exact holographic mapping in free fermion systems

In this paper, we shall perform a detailed analysis of the Exact Holographic Mapping first introduced in arXiv:1309.6282, which was proposed as an explicit example of holographic duality between quantum many-body systems and gravitational theories. We obtain analytic results for free-fermion systems that not only confirm previous numerical results, but also elucidate the exact relationships between the various physical properties of the bulk and boundary systems. Our analytic results allow us to study the asymptotic properties that are difficult to probe numerically, such as the near horizon regime of the black hole geometry. We shall also explore a few interesting but hitherto unexplored bulk geometries, such as that corresponding to a boundary critical fermion with nontrivial dynamic critical exponent. Our analytic framework also allows us to study the holographic mapping of some of these boundary theories in dimensions 2+1 or higher.

The connection between statics and dynamics of spin glasses

We present results of numerical simulations on a one-dimensional Ising spin glass with long-range interactions. Parameters of the model are chosen such that it is a proxy for a short-range spin glass above the upper critical dimension (i.e. in the mean-field regime). The system is quenched to a temperature well below the transition temperature and the growth of correlations is observed. The spatial decay of the correlations at distances less than the dynamic correlation length agrees quantitatively with the predictions of a static theory, the ‘metastate’, evaluated according to the replica symmetry breaking (RSB) theory. We also compute the dynamic exponent z (T ) defined by and find that it is compatible with the mean-field value of the critical dynamical exponent for short range spin glasses.

Fermion loops and improved power-counting in two-dimensional critical metals with singular forward scattering

We analyze general properties of the perturbation expansion for two-dimensional quantum critical metals with singular forward scattering, such as metals at an Ising nematic quantum critical point and metals coupled to a U(1) gauge field. We derive asymptotic properties of fermion loops appearing as subdiagrams of the contributing Feynman diagrams -- for large and small momenta. Substantial cancellations are found in important scaling limits, which reduce the degree of divergence of Feynman diagrams with boson legs. Implementing these cancellations we obtain improved power-counting estimates that yield the true degree of divergence. In particular, we find that perturbative contributions to the boson self-energy are generally ultraviolet convergent for a dynamical critical exponent $z<3$, and divergent beyond three-loop order for $z \geq 3$.