Equilibrium Phase Transitions Research Articles

Fluctuations in a model ferromagnetic film driven by a slowly oscillating field with a constant bias

We present a numerical and theoretical study that supports and explains recent experimental results on anomalous magnetization fluctuations of a uniaxial ferromagnetic film in its low-temperature phase, which is forced by an oscillating field above the critical period of the associated dynamic phase transition (DPT) [P. Riego, P. Vavassori, A. Berger, Phys. Rev. Lett. 118, 117202 (2017)]. For this purpose, we perform kinetic Monte Carlo simulations of a two-dimensional Ising model with nearest-neighbor ferromagnetic interactions in the presence of a sinusoidally oscillating field, to which is added a constant bias field. We study a large range of system sizes and supercritical periods and analyze the data using a droplet-theoretical description of magnetization switching. We find that the period-averaged magnetization, which plays the role of the order parameter for the DPT, presents large fluctuations that give rise to well-defined peaks in its scaled variance and its susceptibility with respect to the bias field. The peaks are symmetric with respect to zero bias and located at values of the bias field that increase toward the field amplitude as an inverse logarithm of the field oscillation period. Our results indicate that this effect is independent of the system size for large systems, ruling out critical behavior associated with a phase transition. Rather, it is a stochastic-resonance phenomenon that has no counterpart in the corresponding thermodynamic phase transition, providing a reminder that the equivalence of the DPT to an equilibrium phase transition is limited to the critical region near the critical period and zero bias.

Spontaneous collective synchronization in the Kuramoto model with additional non-local interactions

In the context of the celebrated Kuramoto model of globally-coupled phase oscillators of distributed natural frequencies, which serves as a paradigm to investigate spontaneous collective synchronization in many-body interacting systems, we report on a very rich phase diagram in presence of thermal noise and an additional non-local interaction on a one-dimensional periodic lattice. Remarkably, the phase diagram involves both equilibrium and non-equilibrium phase transitions. In two contrasting limits of the dynamics, we obtain exact analytical results for the phase transitions. These two limits correspond to (i) the absence of thermal noise, when the dynamics reduces to that of a non-linear dynamical system, and (ii) the oscillators having the same natural frequency, when the dynamics becomes that of a statistical system in contact with a heat bath and relaxing to a statistical equilibrium state. In the former case, our exact analysis is based on the use of the so-called Ott-Antonsen ansatz to derive a reduced set of nonlinear partial differential equations for the macroscopic evolution of the system. Our results for the case of statistical equilibrium are on the other hand obtained by extending the well-known transfer matrix approach for nearest-neighbor Ising model to consider non-local interactions. The work offers a case study of exact analysis in many-body interacting systems. The results obtained underline the crucial role of additional non-local interactions in either destroying or enhancing the possibility of observing synchrony in mean-field systems exhibiting spontaneous synchronization.

On the glass transition of the one-component metallic melts

In this paper, the conditions for one-component metallic melts vitrification by quenching from a liquid state were formulated. It is shown that the tendency to the glass formation drastically increases with the temperature of melting. The maximum glass layer thickness and the associated cooling rates along with the vitrification temperatures was determined for Al, Cu, and Ni melts deposited on the Cu substrate. The results are in agreement with the available experimental data. Based on analytical solution of the impinging droplet solidification, the numerical value of the early-introduced asymptotic Ω criterion, which separates equilibrium and non-equilibrium phase transitions, was determined. Good agreement between the calculated and experimental values of the thickness of the splats shows that Ω criterion indeed predicts a priori a scenario of solidification.

A non-equilibrium Ising model of turbulence

ABSTRACTWe introduce a model of interacting lattices at different resolutions driven by the two-dimensional Ising dynamics with a nearest-neighbor interaction. We study this model both with tools borrowed from equilibrium statistical mechanics as well as non-equilibrium thermodynamics. Our findings show that this model keeps the signature of the equilibrium phase transition. The critical temperature of the equilibrium models corresponds to the state maximizing the entropy and delimits two out-of-equilibrium regimes, one satisfying the Onsager relations for systems close to equilibrium and one resembling convective turbulent states. Since the model preserves the entropy and energy fluxes in the scale space, it seems a good candidate for parametric studies of out-of-equilibrium turbulent systems.

Experimental Determination of Dynamical Lee-Yang Zeros.

Statistical physics provides the concepts and methods to explain the phase behavior of interacting many-body systems. Investigations of Lee-Yang zeros-complex singularities of the free energy in systems of finite size-have led to a unified understanding of equilibrium phase transitions. The ideas of Lee and Yang, however, are not restricted to equilibrium phenomena. Recently, Lee-Yang zeros have been used to characterize nonequilibrium processes such as dynamical phase transitions in quantum systems after a quench or dynamic order-disorder transitions in glasses. Here, we experimentally realize a scheme for determining Lee-Yang zeros in such nonequilibrium settings. We extract the dynamical Lee-Yang zeros of a stochastic process involving Andreev tunneling between a normal-state island and two superconducting leads from measurements of the dynamical activity along a trajectory. From the short-time behavior of the Lee-Yang zeros, we predict the large-deviation statistics of the activity which is typically difficult to measure. Our method paves the way for further experiments on the statistical mechanics of many-body systems out of equilibrium.

Microscopic Theory of Resistive Switching in Ordered Insulators: Electronic versus Thermal Mechanisms.

We investigate the dramatic switch of resistance in ordered correlated insulators when they are driven out of equilibrium by a strong voltage bias. Microscopic calculations on a driven-dissipative lattice of interacting electrons explain the main experimental features of resistive switching (RS), such as the hysteretic I-V curves and the formation of hot conductive filaments. The energy-resolved electron distribution at the RS reveals the underlying nonequilibrium electronic mechanism, namely Landau-Zener tunneling, and also justifies a thermal description in which the hot-electron temperature, estimated from the first moment of the distribution, matches the equilibrium-phase transition temperature. We discuss the tangled relationship between filament growth and negative differential resistance and the influence of crystallographic structure and disorder in the RS.

Observation of a Dissipative Phase Transition in a One-Dimensional Circuit QED Lattice

Condensed matter physics has been driven forward by significant experimental and theoretical progress in the study and understanding of equilibrium phase transitions based on symmetry and topology. However, nonequilibrium phase transitions have remained a challenge, in part due to their complexity in theoretical descriptions and the additional experimental difficulties in systematically controlling systems out of equilibrium. Here, we study a one-dimensional chain of 72 microwave cavities, each coupled to a superconducting qubit, and coherently drive the system into a nonequilibrium steady state. We find experimental evidence for a dissipative phase transition in the system in which the steady state changes dramatically as the mean photon number is increased. Near the boundary between the two observed phases, the system demonstrates bistability, with characteristic switching times as long as 60 ms -- far longer than any of the intrinsic rates known for the system. This experiment demonstrates the power of circuit QED systems for studying nonequilibrium condensed matter physics and paves the way for future experiments exploring nonequilbrium physics with many-body quantum optics.

The influence of composition and heat treatment on the phase composition and mechanical properties of ML19 magnesium alloy

The samples of ML19 magnesium alloy with composition, wt. %: (0,1÷0,6)Zn–(0,4÷1,0)Zr–(1,6÷2,3)Nd–(1,4÷2,2)Y was investigated. The influence of Nd, Y, Zn and Zr on the equilibrium phase transitions temperatures and phase composition using the Thermo-Calc software is established. The Scheil–Gulliver solidification model was also used. We show the significant liquidus temperature increase if zirconium content in alloy is higher than (0,8–0,9) wt.%. Thus, the higher temperature of melting is required (more than 800 °C). This is undesirable if melting in a steel crucibles. The change of equilibrium fractions of phases at different temperatures in ML19 magnesium alloy with a minimum and maximum amount of alloying elements are calculated. A microstructures of the alloys with different amount of the alloying elements in as-cast and heat-treated condition has been studied using scanning electron microscopy (SEM). We investigate the concentration profile of Nd, Y, Zn and Zr in the dendritic cell of as-cast alloy. An amount of neodymium and zinc on the dendritic cell boundaries is increased. High concentration of yttrium is observed both in center and on the boundaries of the dendritic cell. High zirconium concentration mainly observed in the center of the dendritic cells. A small amount of yttrium is also present in a zirconium particles. These particles acting as the nucleation sites for the magnesium solid solution (Mg) during the solidification. The effect of aging temperature (200 and 250 °C) on the hardness of a samples after quenching was studied. Aging at 200 °C provides a higher hardness. Investigated the change of the hardness quenched samples during the aging at 200 °C. The maximum hardness is observed in samples aged for 16-20 hours. The two-stage solution heat treatment for 2 h at 400 °C and 8 h at 500 °C with water quenching and aging at 200 °C for 16 h was performed. This heat treatment enable us to get tensile strength 306 ± 8 MPa and yield strength 161 ± ± 1 MPa with elongation 8,7 ± 1,6 %.

A model of order–order transition in titanium monoxide TiO1.0

A new class of defect structures has been proposed, in which point defects of the crystal lattice occupy the sites of two different superstructures simultaneously. The formation of these structural modifications is due to the order‐order equilibrium phase transition of the second order that does not proceed completely. Using atomic‐vacancy ordering in titanium monoxide TiO1.0 as an example, we have studied the permissible ratio between the long‐range order parameters in the structural modification formed by a combination of monoclinic M5X5(mon) (space group C2/m (A2/m)) and cubic M5X5(cub) (space group ) superstructures. With the use of thermodynamic calculations it was shown that the proposed structural modification is equilibrium and should exist instead of the supposed high‐temperature cubic phase Ti5O5(cub).

Nonequilibrium variational cluster perturbation theory: Quench dynamics of the quantum Ising model

We introduce a variational implementation of cluster perturbation theory (CPT) to address the dynamics of spin systems driven out of equilibrium. We benchmark the method with the quantum Ising model subject to a sudden quench of the transverse magnetic field across the transition or within a phase. We treat both the one-dimensional case, for which an exact solution is available, as well the two-dimensional one, for which has to resort to numerical results. Comparison with exact results shows that the approach provides a quite accurate description of the real-time dynamics up to a characteristic time scale $\tau$ that increses with the size of the cluster used for CPT. In addition, and not surprisingly $\tau$ is small for quenches across the equilibrium phase transition, but can be quite larger for quenches within the ordered or disordered phases.

Superradiant Phase Transition in a Superconducting Circuit in Thermal Equilibrium.

We propose a superconducting circuit that shows a superradiant phase transition (SRPT) in thermal equilibrium. The existence of the SRPT is confirmed analytically in the limit of an infinite number of artificial atoms. We also perform a numerical diagonalization of the Hamiltonian with a finite number of atoms and observe an asymptotic behavior approaching the infinite limit as the number of atoms increases. The SRPT can also be interpreted intuitively in a classical analysis.

Cluster Mean-Field Approach to the Steady-State Phase Diagram of Dissipative Spin Systems

We show that short-range correlations have a dramatic impact on the steady-state phase diagram of quantum driven-dissipative systems. This effect, never observed in equilibrium, follows from the fact that ordering in the steady state is of dynamical origin, and is established only at very long times, whereas in thermodynamic equilibrium it arises from the properties of the (free) energy. To this end, by combining the cluster methods extensively used in equilibrium phase transitions to quantum trajectories and tensor-network techniques, we extend them to nonequilibrium phase transitions in dissipative many-body systems. We analyze in detail a model of spin-1=2 on a lattice interacting through an XYZ Hamiltonian, each of them coupled to an independent environment that induces incoherent spin flips. In the steady-state phase diagram derived from our cluster approach, the location of the phase boundaries and even its topology radically change, introducing reentrance of the paramagnetic phase as compared to the single-site mean field where correlations are neglected. Furthermore, a stability analysis of the cluster mean field indicates a susceptibility towards a possible incommensurate ordering, not present if short-range correlations are ignored.

Phase Transitions in Equilibrium and Non-Equilibrium Models on Some Topologies

On some regular and non-regular topologies, we studied the critical properties of models that present up-down symmetry, like the equilibrium Ising model and the nonequilibrium majority vote model. These are investigated on networks, like Apollonian (AN), Barabási–Albert (BA), small-worlds (SW), Voronoi–Delaunay (VD) and Erdös–Rényi (ER) random graphs. The review here is on phase transitions, critical points, exponents and universality classes that are compared to the results obtained for these models on regular square lattices (SL).

Laser transition in the thermodynamic limit for identical emitters in a cavity

The laser--phase-transition analogy is considered for a system of identical emitters interacting with a cavity mode. As in the case of equilibrium phase transitions, the thermodynamic limit is shown to be an essential condition for the appearance of an abrupt transition, with a well-defined threshold. The proof is discussed first in the simpler formalism of the rate equation and then at the higher level of the master equation for probabilities. The latter is derived from the von Neumann--Lindblad evolution of the density operator. In the thermodynamic limit and for two-level emitters the master equation becomes a two-dimensional dynamic flow with two possible fixed points, related to the normal and the lasing regimes, respectively. The master equation analysis confirms the threshold value deduced from the rate equation, but also provides further evidence of lasing by showing that all photonic autocorrelation functions converge to unity.

Phase transitions and order in two-dimensional generalized nonlinear σ models.

We study phase transitions and the nature of order in a class of classical generalized O(N) nonlinear σ models (NLS) constructed by minimally coupling pure NLS with additional degrees of freedom in the form of (i) Ising ferromagnetic spins, (ii) an advective Stokesian velocity, and (iii) multiplicative noises. In examples (i) and (ii), and also (iii) with the associated multiplicative noise being not sufficiently long-ranged, we show that the models may display a class of unusual phase transitions between stiff and soft phases, where the effective spin stiffness respectively diverges and vanishes in the long wavelength limit at two dimensions (2D), unlike in pure NLS. In the stiff phase, in the thermodynamic limit the variance of the transverse spin (or, the Goldstone mode) fluctuations are found to scale with the system size L in 2D as lnlnL with a model-dependent amplitude, which is markedly weaker than the well-known lnL dependence of the variance of the broken symmetry modes in models that display quasi-long-range order in 2D. Equivalently, for N=2 at 2D the equal-time spin-spin correlations decay in powers of inverse logarithm of the spatial separation with model-dependent exponents. These transitions are controlled by the model parameters those couple the O(N) spins with the additional variables. In the presence of long-range noises in example (iii), true long-range order may set in 2D, depending upon the specific details of the underlying dynamics. Our results should be useful in understanding phase transitions in equilibrium and nonequilibrium low-dimensional systems with continuous symmetries in general.

Edge exponents in work statistics out of equilibrium and dynamical phase transitions from scattering theory in one-dimensional gapped systems

I discuss the relationship between edge exponents in the statistics of work done, dynamical phase transitions, and the role of different kinds of excitations appearing when a nonequilibrium protocol is performed on a closed, gapped, one-dimensional system. I show that the edge exponent in the probability density function of the work is insensitive to the presence of interactions and can take only one of three values: $+1/2,\phantom{\rule{0.16em}{0ex}}\ensuremath{-}1/2$, and $\ensuremath{-}3/2$. It also turns out that there is an interesting interplay between spontaneous symmetry breaking or the presence of bound states and the exponents. For instantaneous global protocols, I find that the presence of the one-particle channel creates dynamical phase transitions in the time evolution.

Dynamical Symmetries and Causality in Non-Equilibrium Phase Transitions

Dynamical symmetries are of considerable importance in elucidating the complex behaviour of strongly interacting systems with many degrees of freedom. Paradigmatic examples are cooperative phenomena as they arise in phase transitions, where conformal invariance has led to enormous progress in equilibrium phase transitions, especially in two dimensions. Non-equilibrium phase transitions can arise in much larger portions of the parameter space than equilibrium phase transitions. The state of the art of recent attempts to generalise conformal invariance to a new generic symmetry, taking into account the different scaling behaviour of space and time, will be reviewed. Particular attention will be given to the causality properties as they follow for co-variant n-point functions. These are important for the physical identification of n-point functions as responses or correlators.

Separating Exchange Splitting from Spin Mixing in Gadolinium by Femtosecond Laser Excitation.

Employing spin-, time-, and energy-resolved photoemission spectroscopy, we present the first study on the spin polarization of a single electronic state after ultrafast optical excitation. Our investigation concentrates on the majority-spin component of the d-band-derived Gd(0001) surface state d(z(2))(↑). While its binding energy shows a rapid Stoner-like shift by 90 meV with an exponential time constant of τ(E)=0.6±0.1 ps, the d(z(2))(↑) spin polarization remains nearly constant within the first picoseconds and decays with τ(S)=15±8 ps. This behavior is in clear contrast to the equilibrium phase transition, where the spin polarization vanishes at the Curie temperature.

Scaling and Universality at Dynamical Quantum Phase Transitions.

Dynamical quantum phase transitions (DQPTs) at critical times appear as nonanalyticities during nonequilibrium quantum real-time evolution. Although there is evidence for a close relationship between DQPTs and equilibrium phase transitions, a major challenge is still to connect to fundamental concepts such as scaling and universality. In this work, renormalization group transformations in complex parameter space are formulated for quantum quenches in Ising models showing that the DQPTs are critical points associated with unstable fixed points of equilibrium Ising models. Therefore, these DQPTs obey scaling and universality. On the basis of numerical simulations, signatures of these DQPTs in the dynamical buildup of spin correlations are found with an associated power-law scaling determined solely by the fixed point's universality class. An outlook is given on how to explore this dynamical scaling experimentally in systems of trapped ions.

Critical behavior at a dynamic vortex insulator-to-metal transition.

An array of superconducting islands placed on a normal metal film offers a tunable realization of nanopatterned superconductivity. This system enables investigation of the nature of competing vortex states and phase transitions between them. A square array creates the eggcrate potential in which magnetic field-induced vortices are frozen into a vortex insulator. We observed a vortex insulator-vortex metal transition driven by the applied electric current and determined critical exponents that coincided with those for thermodynamic liquid-gas transition. Our findings offer a comprehensive description of dynamic critical behavior and establish a deep connection between equilibrium and nonequilibrium phase transitions.