Dynamical Critical Exponent Research Articles

Magnetic irreversibility and spin dynamics in nanoparticles of iron-doped europium chromite

The magnetic properties of nanoparticles of Eu1−xFexCrO3 (0≤x≤1) prepared by a combustion reaction technique were investigated. An irreversible behavior in the magnetization was observed for a sample with x=0.10 below the Néel temperature (TN) yielding a complex phase diagram. The irreversibility data was fitted to a de Almeida-Thouless line by using ϕ=3.0 for the critical exponent and a glassy temperature TG of 174.4 K for H≤40 kOe. The dynamics of the spins was investigated by measuring the ac magnetic susceptibility (χac) near TN for frequencies (f) in the range 10−104 Hz. The maximum in χac was found to shift to higher values of T for increasing values of f, yielding ≃0.003 for the shift per decade of f parameter. The Voguel-Fulcher law and a power-law were used for analyzing χac yielding τ0=1.8×10−9 s (=1.6×10−15 s) for the characteristic relaxation time, Ea/kB=39.23 K for the activation energy and TG=165.9 K (=167.5 K) for the Voguel-Fulcher (power-law) model. Moreover, the power-law yielded zν=5.62 for the product of the dynamical critical exponent (ν) with the one associated to the correlation length (z). The dependence of TN with the bonding angles Cr3+−O2−−Cr3+ was investigated for as-prepared samples yielding a trend contrary to the expected. The overall results were accounted for by taking into consideration the microstrain introduced by the sample preparation technique and by the ionic dopping.

Quantum electrodynamics in 2+1 dimensions with quenched disorder: Quantum critical states with interactions and disorder

Quantum electrodynamics in 2+1-dimensions (QED$_3$) is a strongly coupled conformal field theory (CFT) of a U(1) gauge field coupled to $2N$ two-component massless fermions. The $N=2$ CFT has been proposed as a ground state of the spin-1/2 kagome Heisenberg antiferromagnet. We study QED$_3$ in the presence of weak quenched disorder in its two spatial directions. When the disorder explicitly breaks the fermion flavor symmetry from SU($2N$)$\rightarrow$U(1)$\times$SU($N$) but preserves time-reversal symmetry, we find that the theory flows to a non-trivial fixed line at non-zero disorder with a continuously varying dynamical critical exponent $z>1$. We determine the zero-temperature flavor (spin) conductivity along the critical line. Our calculations are performed in the large-$N$ limit, and the disorder is handled using the replica method.

Thermodynamics, phase transitions and Ruppeiner geometry for Einstein\u2013dilaton\u2013Lifshitz black holes in the presence of Maxwell and Born\u2013Infeld electrodynamics

In this paper, we first obtain the higher-dimen-sional dilaton–Lifshitz black hole solutions in the presence of Born–Infeld (BI) electrodynamics. We find that there are two different solutions for the cases of z=n+1 and zne n+1 where z is the dynamical critical exponent and n is the number of spatial dimensions. Calculating the conserved and thermodynamical quantities, we show that the first law of thermodynamics is satisfied for both cases. Then we turn to the study of different phase transitions for our Lifshitz black holes. We start with the Hawking–Page phase transition and explore the effects of different parameters of our model on it for both linearly and BI charged cases. After that, we discuss the phase transitions inside the black holes. We present the improved Davies quantities and prove that the phase transition points shown by them are coincident with the Ruppeiner ones. We show that the zero temperature phase transitions are transitions in the radiance properties of black holes by using the Landau–Lifshitz theory of thermodynamic fluctuations. Next, we turn to the study of the Ruppeiner geometry (thermodynamic geometry) for our solutions. We investigate thermal stability, interaction type of possible black hole molecules and phase transitions of our solutions for linearly and BI charged cases separately. For the linearly charged case, we show that there are no phase transitions at finite temperature for the case zge 2. For z<2, it is found that the number of finite temperature phase transition points depends on the value of the black hole charge and there are not more than two. When we have two finite temperature phase transition points, there is no thermally stable black hole between these two points and we have discontinuous small/large black hole phase transitions. As expected, for small black holes, we observe finite magnitude for the Ruppeiner invariant, which shows the finite correlation between possible black hole molecules, while for large black holes, the correlation is very small. Finally, we study the Ruppeiner geometry and thermal stability of BI charged Lifshtiz black holes for different values of z. We observe that small black holes are thermally unstable in some situations. Also, the behavior of the correlation between possible black hole molecules for large black holes is the same as for the linearly charged case. In both the linearly and the BI charged cases, for some choices of the parameters, the black hole system behaves like a Van der Waals gas near the transition point.

Kibble-Zurek scaling in holography

The Kibble-Zurek (KZ) mechanism describes the generations of topological defects when a system undergoes a second-order phase transition via quenches. We study the holographic KZ scaling using holographic superconductors. The scaling can be understood analytically from a scaling analysis of the bulk action. The argument is reminiscent of the scaling analysis of the mean-field theory but is more subtle and is not entirely obvious. This is because the scaling is not the one of the original bulk theory but is an emergent one that appears only at the critical point. The analysis is also useful to determine the dynamic critical exponent $z$.

Critical and multicritical behaviors of static and complex magnetic susceptibilities for the mean-field Blume-Capel model with a random crystal field

The static and dynamic linear response functions for the spin-1 mean-field Blume-Capel model with a random diluted single-ion anisotropy are derived using the thermodynamics of linear irreversible processes. Thermal dependence of the static susceptibility is studied in detail. In addition, by utilizing the stationary solution of the kinetic equation in existence of sinusoidal external magnetic field, the low frequency thermal behavior of the magnetic absorption and dispersion factors are investigated. Particularly, we have analyzed the behaviors of these quantities near both first- and second-order transition temperatures and multicritical points such as tricritical point, critical endpoint and double critical endpoint. The static and dynamical mean field critical exponents in hydrodynamic limit are also calculated in order to formulate the critical and multicritical behaviors of the magnetic susceptibility. Finally, a comparison of the findings of this study with previous theoretical and experimental studies is presented and it is shown that a good agreement is found with our results.

Correlation functions in theories with Lifshitz scaling

The 2+1 dimensional quantum Lifshitz model can be generalised to a class of higher dimensional free field theories that exhibit Lifshitz scaling. When the dynamical critical exponent equals the number of spatial dimensions, equal time correlation functions of scaling operators in the generalised quantum Lifshitz model are given by a d-dimensional higher-derivative conformal field theory. Autocorrelation functions in the generalised quantum Lifshitz model in any number of dimensions can on the other hand be expressed in terms of autocorrelation functions of a two-dimensional conformal field theory. This also holds for autocorrelation functions in a strongly coupled Lifshitz field theory with a holographic dual of Einstein-Maxwell-dilaton type. The map to a two-dimensional conformal field theory extends to autocorrelation functions in thermal states and out-of-equilbrium states preserving symmetry under spatial translations and rotations in both types of Lifshitz models. Furthermore, the spectrum of quasinormal modes of scalar field perturbations in Lifshitz black hole backgrounds can be obtained analytically at low spatial momenta and exhibits a linear dispersion relation at z = d. At high momentum, the mode spectrum can be obtained in a WKB approximation and displays very different behaviour compared to holographic duals of conformal field theories. This has implications for thermalisation in strongly coupled Lifshitz field theories with z > 1.

Nonequilibrium critical dynamics of the two-dimensional Ashkin-Teller model at the Baxter line.

We investigate the short-time universal behavior of the two-dimensional Ashkin-Teller model at the Baxter line by performing time-dependent Monte Carlo simulations. First, as preparatory results, we obtain the critical parameters by searching the optimal power-law decay of the magnetization. Thus, the dynamic critical exponents θ_{m} and θ_{p}, related to the magnetic and electric order parameters, as well as the persistence exponent θ_{g}, are estimated using heat-bath Monte Carlo simulations. In addition, we estimate the dynamic exponent z and the static critical exponents β and ν for both order parameters. We propose a refined method to estimate the static exponents that considers two different averages: one that combines an internal average using several seeds with another, which is taken over temporal variations in the power laws. Moreover, we also performed the bootstrapping method for a complementary analysis. Our results show that the ratio β/ν exhibits universal behavior along the critical line corroborating the conjecture for both magnetization and polarization.

Critical properties of the contact process with quenched dilution

We have studied the critical properties of the contact process on a square lattice with quenched site dilution by Monte Carlo simulations. This was achieved by generating in advance the percolating cluster, through the use of an appropriate epidemic model, and then by the simulation of the contact process on the top of the percolating cluster. The dynamic critical exponents were calculated by assuming an activated scaling relation and the static exponents by the usual power law behavior. Our results are in agreement with the prediction that the quenched diluted contact process belongs to the universality class of the random transverse-field Ising model. We have also analyzed the model and determined the phase diagram by the use of a mean-field theory that takes into account the correlation between neighboring sites.

Correlation effects and quantum oscillations in topological nodal-loop semimetals

We study the unique physical properties of topological nodal-loop semimetals protected by the coexistence of time-reversal and inversion symmetries with negligible spin-orbit coupling. We argue that strong correlation effects occur at the surface of such systems for relatively small Hubbard interaction $U$, due to the narrow bandwidth of the "drumhead" surface states. In the Hartree-Fock approximation, at small $U$ we obtain a surface ferromagnetic phase through a continuous quantum phase transition characterized by the surface-mode divergence of the spin susceptibility, while the bulk states remain very robust against local interactions and remain non-ordered. At slightly increased interaction strength, the system quickly changes from a surface ferromagnetic phase to a surface charge-ordered phase through a first-order transition. When Rashba-type spin-orbit coupling is applied to the surface states, a canted ferromagnetic phase occurs at the surface for intermediate values of $U$. The quantum critical behavior of the surface ferromagnetic transition is nontrivial in the sense that the surface spin order parameter couple to Fermi-surface excitations from both surface and bulk states. This leads to unconventional Landau damping and consequently a na\"ive dynamical critical exponent $z\!\approx\!1$ when the Fermi level is close to the bulk nodal energy. We also show that, already without interactions, quantum oscillations arise due to bulk states, despite the absence of a Fermi surface when the chemical potential is tuned to the energy of the nodal loop. The bulk magnetic susceptibility diverges logarithmically whenever the nodal loop exactly overlaps with a quantized magnetic orbit in the bulk Brillouin zone. These correlation and transport phenomena are unique signatures of nodal loop states.

Different growth rates for spin and superfluid order in a quenched spinor condensate

In this paper we study the coarsening dynamics of a spinor condensate quenched into an easy-axis ferromagnetic phase by a sudden change in the quadratic Zeeman energy. We show that applying a spin rotation prior to changing the Zeeman energy accelerates the development of local order and reduces heating. We examine the longitudinal spin ordering and the superfluid ordering of the system and show that the respective order parameter correlation functions exhibit dynamic scaling in the late time dynamics. Our results also demonstrate that these two types of order grow at different rates, i.e.~with different dynamic critical exponents. The spin domain area distribution is calculated and is shown to have power law scaling behavior expected from percolation theory.

Shear viscosity at the Ising-nematic quantum critical point in two-dimensional metals

In an isotropic strongly interacting quantum liquid without quasiparticles, general scaling arguments imply that the dimensionless ratio $(k_B /\hbar)\, \eta/s$, where $\eta$ is the shear viscosity and $s$ is the entropy density, is a universal number. We compute the shear viscosity of the Ising-nematic critical point of metals in spatial dimension $d=2$ by an expansion below $d=5/2$. The anisotropy associated with directions parallel and normal to the Fermi surface leads to a violation of the scaling expectations: $\eta$ scales in the same manner as a chiral conductivity, and the ratio $\eta/s$ diverges at low temperature ($T$) as $T^{-2/z}$, where $z$ is the dynamic critical exponent for fermionic excitations dispersing normal to the Fermi surface.

Influence of the ferroelectric quantum critical point on SrTiO3 interfaces

We study a model SrTiO$_3$ interface in which conduction $t_{2g}$ electrons couple to the ferroelectric (FE) phonon mode. We treat the FE mode within a self-consistent phonon theory that captures its quantum critical behavior, and show that proximity to the quantum critical point leads to universal tails in the electron density of the form $n(z) \sim (\lambda+z)^{-2}$, where $\lambda \sim T^{2-d/\mathfrak{z}}$, with $d=3$ the dimensionality and $\mathfrak{z}=1$ the dynamical critical exponent. Implications for the metal-insulator transition at low electron density are discussed.

Quantum criticality in the two-dimensional dissipative quantum XY model

Earlier Monte Carlo calculations on the dissipative two-dimensional XY model are extended in several directions. We study the phase diagram and the correlation functions when dissipation is very small, where it has properties of the classical 3D-XY transition, i.e., one with a dynamical critical exponent $z=1$. The transition changes from $z=1$ to the class of criticality with $z\ensuremath{\rightarrow}\ensuremath{\infty}$ driven by topological defects, discovered earlier, beyond a critical dissipation. We also find that the critical correlations have power-law singularities as a function of tuning the ratio of the kinetic energy to the potential energy for fixed large dissipation, as opposed to essential singularities on tuning dissipation keeping the former fixed. A phase with temporal disorder but spatial order of the Kosterlitz-Thouless form is also further investigated. We also present results for the transition when the allowed Caldeira-Leggett form of dissipation and the allowed form of dissipation coupling to the compact rotor variables are both included. The nature of the transition is then determined by the Caldeira-Leggett form.

Phase separation transition in a nonconserved two-species model.

A one-dimensional stochastic exclusion process with two species of particles, + and -, is studied where density of each species can fluctuate but the total particle density is conserved. From the exact stationary state weights we show that, in the limiting case where density of negative particles vanishes, the system undergoes a phase separation transition where a macroscopic domain of vacancies form in front of a single surviving negative particle. We also show that the phase-separated state is associated with a diverging correlation length for any density and that the critical exponents characterizing the behavior in this region are different from those at the transition line. The static and the dynamical critical exponents are obtained from the exact solution and numerical simulations, respectively.

Finite-temperature scaling close to Ising-nematic quantum critical points in two-dimensional metals

We study finite temperature properties of metals close to an Ising-nematic quantum critical point in two spatial dimensions. In particular we show that at any finite temperature there is a regime where order parameter fluctuations are characterized by a dynamical critical exponent $z=2$, in contrast to $z=3$ found at zero temperature. Our results are based on a simple Eliashberg-type approach, which gives rise to a boson self-energy proportional to $\Omega/\gamma(T)$ at small momenta, where $\gamma(T)$ is the temperature dependent fermion scattering rate. These findings might shed some light on recent Monte-Carlo simulations at finite temperature, where results consistent with $z=2$ were found.

Phase diagram and correlation functions of the two-dimensional dissipative quantum XY model

The two-dimensional dissipative quantum XY model is applicable to the quantum-critical properties of diverse experimental systems, ranging from the superconductor to insulator transitions, ferromagnetic and antiferromagnetic transitions in metals, to the loop-current order transition in the cuprates. We solve the re-expression of this model in terms of topological excitations: vortices and a variety of instantons, by renormalization group methods. The calculations explain the extraordinary properties of the model discovered in Monte-Carlo calculations: the separability of the quantum critical fluctuations (QCF) in space and time, the spatial correlation length proportional to logarithm of the temporal correlation length near the transition from disordered to the fully ordered state, and the occurrence of a phase with spatial order without temporal order. They are intimately related to the flow of the metric of time in relation to the metric of space, i.e. of the dynamical critical exponent $z$. These properties appear to be essential in understanding the strange metallic phase found in a variety of quantum-critical transitions as well as the accompanying high temperature superconductivity.

Observation of quantum Griffiths singularity and ferromagnetism at the superconducting LaAlO3/SrTiO3(110) interface

Diverse phenomena emerge at the interface between band insulators LaAlO3 and SrTiO3, such as superconductivity and ferromagnetism, showing an opportunity for potential applications as well as bringing fundamental research interests. Particularly, the two-dimensional electron gas formed at LaAlO3/SrTiO3 interface offers an appealing platform for quantum phase transition from a superconductor to a weakly localized metal. Here we report the superconductor-metal transition in superconducting two-dimensional electron gas formed at LaAlO3/SrTiO3(110) interface driven by a perpendicular magnetic field. Interestingly, when approaching the quantum critical point, the dynamic critical exponent is not a constant but a diverging value, which is a direct evidence of quantum Griffiths singularity raised from quenched disorder at ultralow temperatures. Furthermore, the hysteretic property of magnetoresistance was firstly observed at LaAlO3/SrTiO3(110) interfaces, which suggests potential coexistence of superconductivity and ferromagnetism.

Dynamic effects of quenched disorder on domain wall motion in magnetic nanowires

The domain wall dynamics in magnetic nanowires is numerically studied with the Landau-Lifshitz-Gilbert equation. Below the Walker breakdown threshold, the domain wall presents a stable propagation, while above the threshold where the retrograde mode dominates, the oscillation period is controlled by the external field and anisotropy. More importantly, the dynamic effects of quenched disorder on the domain wall motion are explored. A continuous pinning-depinning phase transition is detected. The dynamic scaling form is analyzed with the data collapse of the domain wall velocity, and both the static and dynamic critical exponents are extracted.

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.