Dynamics Of First-order Phase Transitions Research Articles

Structural dynamics of first-order phase transition in giant magnetocaloric La(Fe,Si)13: The free energy landscape

Maximizing the performance of magnetic refrigerators and thermomagnetic energy harvesters is imperative for their successful implementation and can be done by maximizing their operation frequency. One of the features delimiting the frequency and efficiency of such devices is the phase transition kinetics of their magnetocaloric/thermomagnetic active material. While previous studies have described the magnetic component governing the kinetics of the magnetovolume phase transition in La(Fe,Si)13 giant magnetocaloric materials, a comprehensive description of its structural component has yet to be explored. In this study, in situ synchrotron X-ray diffraction is employed to describe the structural changes upon magnetic field application/removal. Long magnetic field dependent relaxation times up to a few hundred seconds are observed after the driving field is paused. The phase transition is found to be highly asymmetric upon magnetic field cycling due to the different Gibbs energy landscapes and the absence of an energy barrier upon field removal. An exponential relationship is found between the energy barriers and the relaxation times, suggesting the process is governed by a non-thermal activation over an energy barrier process. Such fundamental knowledge on first-order phase transition kinetics suggests pathways for materials optimization and smarter design of magnetic field cycling in real-life devices.

Delocalization-localization dynamical phase transition of random walks on graphs.

We consider random walks evolving on two models of connected and undirected graphs and study the exact large deviations of a local dynamical observable. We prove, in the thermodynamic limit, that this observable undergoes a first-order dynamical phase transition (DPT). This is interpreted as a "coexistence" of paths in the fluctuations that visit the highly connected bulk of the graph (delocalization) and paths that visit the boundary (localization). The methods we used also allow us to characterize analytically the scaling function that describes the finite-size crossover between the localized and delocalized regimes. Remarkably, we also show that the DPT is robust with respect to a change in the graph topology, which only plays a role in the crossover regime. All results support the view that a first-order DPT may also appear in random walks on infinite-size random graphs.

Anomalous scaling and first-order dynamical phase transition in large deviations of the Ornstein-Uhlenbeck process.

We study the full distribution of A=∫_{0}^{T}x^{n}(t)dt,n=1,2,⋯, where x(t) is an Ornstein-Uhlenbeck process. We find that for n>2 the long-time (T→∞) scaling form of the distribution is of the anomalous form P(A;T)∼e^{-T^{μ}f_{n}(ΔA/T^{ν})} where ΔA is the difference between A and its mean value, and the anomalous exponents are μ=2/(2n-2) and ν=n/(2n-2). The rate function f_{n}(y), which we calculate exactly, exhibits a first-order dynamical phase transition which separates between a homogeneous phase that describes the Gaussian distribution of typical fluctuations, and a "condensed" phase that describes the tails of the distribution. We also calculate the most likely realizations of A(t)=∫_{0}^{t}x^{n}(s)ds and the distribution of x(t) at an intermediate time t conditioned on a given value of A. Extensions and implications to other continuous-time systems are discussed.

An elementary proof for dynamical scaling for certain fractional non-homogeneous Poisson processes

Dynamical scaling is an asymptotic property typical for the dynamics of first-order phase transitions in physical systems and related to self-similarity. Based on the integral-representation for the marginal probabilities of a fractional non-homogeneous Poisson process introduced by Leonenko et al. (2017) and generalizing the standard fractional Poisson process, we prove the dynamical scaling under fairly mild conditions. Our result also includes the special case of the standard fractional Poisson process.

Theory of classical metastability in open quantum systems

We present a general theory of classical metastability in open quantum systems. Metastability is a consequence of a large separation in timescales in the dynamics, leading to the existence of a regime when states of the system appear stationary, before eventual relaxation toward a true stationary state at much larger times. In this work, we focus on the emergence of classical metastability, i.e., when metastable states of an open quantum system with separation of timescales can be approximated as probabilistic mixtures of a finite number of states. We find that a number of classical features follow from this approximation, for the manifold of metastable states, long-time dynamics between them, and symmetries of the dynamics. Namely, those states are approximately disjoint and thus play the role of metastable phases, the relaxation toward the stationary state is approximated by a classical stochastic dynamics between them, and weak symmetries correspond to their permutations. Importantly, the classical dynamics is observed not only on average but also at the level of individual quantum trajectories: We show that time coarse-grained continuous measurement records can be viewed as noisy classical trajectories, while their statistics can be approximated by that of the classical dynamics. Among others, this explains how first-order dynamical phase transitions arise from metastability. Finally, to verify the presence of classical metastability in a given open quantum system, we develop an efficient numerical approach that delivers the set of metastable phases together with the effective classical dynamics. Since the proximity to a first-order dissipative phase transition manifests as metastability, the theory and tools introduced in this work can be used to investigate such transitions---which occur in the large size limit---through the metastable behavior of many-body systems of moderate sizes accessible to numerics.

Bubble quenches in the AdS/BCFT model

In this paper we construct time-dependent solutions of three-dimensional gravity in anti--de Sitter space dual to systems with boundaries (BCFTs), following the AdS/BCFT prescription. Such solutions can be discussed in the context of the dynamics of first-order phase transitions, or more generally, in the description of quantum quenches. As an example, we apply the holographic model to calculate the dynamics of the entanglement entropy of a local quench corresponding to a nucleation of a Euclidean bubble. As in the known $1+1$ conformal field theory examples of local cut and glue quenches, the holographic entanglement entropy grows logarithmically with time with the correct universal coefficient. However, in the bubble quench, the behavior is different at late times. The AdS/BCFT model exhibits the light-cone spreading of correlations and saturation at late times. We also find an analytical formula for the entropy at finite temperature. In the latter case, the initial logarithmic growth is followed by the linear law at intermediate times.

Fate of false vacua in holographic first-order phase transitions

Using the holographic correspondence as a tool, we study the dynamics of first-order phase transitions in strongly coupled gauge theories at finite temperature. Considering an evolution from the large to the small temperature phase, we compute the nucleation rate of bubbles of true vacuum in the metastable phase. For this purpose, we find the relevant configurations (bounces) interpolating between the vacua and we compute the related effective actions. We start by revisiting the compact Randall-Sundrum model at high temperature. Using holographic renormalization, we compute the derivative term in the effective bounce action, that was missing in the literature. Then, we address the full problem within the top-down Witten-Sakai-Sugimoto model. It displays both a confinement/deconfinement and a chiral symmetry breaking/restoration phase transition which, depending on the model parameters, can happen at different critical temperatures. For the confinement/deconfinement case we perform the numerical analysis of an effective description of the transition and also provide analytic expressions using thick and thin wall approximations. For the chiral symmetry transition, we implement a variational approach that allows us to address the challenging non-linear problem stemming from the Dirac-Born-Infeld action.

Dynamical phase coexistence in the Fredrickson–Andersen model

We analyse a first-order dynamical phase transition that takes place in the Fredrickson–Andersen (FA) model. We construct a two-dimensional spin system whose thermodynamic properties reproduce the dynamical large deviations of the FA model and we analyse this system numerically, comparing our results with finite-size scaling theory. This allows us to rationalise recent results for the FA model, including the exponential divergence of its susceptibility at phase coexistence. We also discuss a simple interfacial model that reproduces quantitatively the behaviour of the FA model at coexistence.

Classical discrete time crystals

The spontaneous breaking of time-translation symmetry in periodically driven quantum systems leads to a new phase of matter: discrete time crystals (DTC). This phase exhibits collective subharmonic oscillations that depend upon an interplay of non-equilibrium driving, many-body interactions, and the breakdown of ergodicity. However, subharmonic responses are also a well-known feature of classical dynamical systems ranging from predator-prey models to Faraday waves and AC-driven charge density waves. This raises the question of whether these classical phenomena display the same rigidity characteristic of a quantum DTC. In this work, we explore this question in the context of periodically driven Hamiltonian dynamics coupled to a finite-temperature bath, which provides both friction and, crucially, noise. Focusing on one-dimensional chains, where in equilibrium any transition would be forbidden at finite temperature, we provide evidence that the combination of noise and interactions drives a sharp, first-order dynamical phase transition between a discrete time-translation invariant phase and an activated classical discrete time crystal (CDTC) in which time-translation symmetry is broken out to exponentially-long time scales. Power-law correlations are present along a first-order line which terminates at a critical point. We analyze the transition by mapping it to the locked-to-sliding transition of a DC-driven charge density wave. Our work points to a classical limit for quantum time crystals, and raises several intriguing questions concerning the non-equilibrium universality class of the CDTC critical point.

Large deviations and dynamical phase transitions in stochastic chemical networks

Chemical reaction networks offer a natural nonlinear generalization of linear Markov jump processes on a finite state-space. In this paper, we analyze the dynamical large deviations of such models, starting from their microscopic version, the chemical master equation. By taking a large-volume limit, we show that those systems can be described by a path integral formalism over a Lagrangian functional of concentrations and chemical fluxes. This Lagrangian is dual to a Hamiltonian, whose trajectories correspond to the most likely evolution of the system given its boundary conditions. The same can be done for a system biased on time-averaged concentrations and currents, yielding a biased Hamiltonian whose trajectories are optimal paths conditioned on those observables. The appropriate boundary conditions turn out to be mixed so that, in the long time limit, those trajectories converge to well-defined attractors. We are then able to identify the largest value that the Hamiltonian takes over those attractors with the scaled cumulant generating function of our observables, providing a nonlinear equivalent to the well-known Donsker-Varadhan formula for jump processes. On that basis, we prove that chemical reaction networks that are deterministically multistable generically undergo first-order dynamical phase transitions in the vicinity of zero bias. We illustrate that fact through a simple bistable model called the Schlögl model, as well as multistable and unstable generalizations of it, and we make a few surprising observations regarding the stability of deterministic fixed points and the breaking of ergodicity in the large-volume limit.

A first-order dynamical transition in the displacement distribution of a driven run-and-tumble particle

We study the probability distribution P(XN = X,N) of the total displacement XN of an N-step run and tumble particle on a line, in the presence of a constant nonzero drive E. While the central limit theorem predicts a standard Gaussian form for near its peak, we show that for large positive and negative X, the distribution exhibits anomalous large deviation forms. For large positive X, the associated rate function is nonanalytic at a critical value of the scaled distance from the peak where its first derivative is discontinuous. This signals a first-order dynamical phase transition from a homogeneous ‘fluid’ phase to a ‘condensed’ phase that is dominated by a single large run. A similar first-order transition occurs for negative large fluctuations as well. Numerical simulations are in excellent agreement with our analytical predictions.

Large fluctuations of a Kardar-Parisi-Zhang interface on a half line: The height statistics at a shifted point.

We consider a stochastic interface h(x,t), described by the 1+1 Kardar-Parisi-Zhang (KPZ) equation on the half line x≥0 with the reflecting boundary at x=0. The interface is initially flat, h(x,t=0)=0. We focus on the short-time probability distribution P(H,L,t) of the height H of the interface at point x=L. Using the optimal fluctuation method, we determine the (Gaussian) body of the distribution and the strongly asymmetric non-Gaussian tails. We find that the slower-decaying tail scales as -sqrt[t]lnP≃|H|^{3/2}f_{-}(L/sqrt[|H|t]) and calculate the function f_{-} analytically. Remarkably, this tail exhibits a first-order dynamical phase transition at a critical value of L, L_{c}=0.60223⋯sqrt[|H|t]. The transition results from a competition between two different fluctuation paths of the system. The faster-decaying tail scales as -sqrt[t]lnP≃|H|^{5/2}f_{+}(L/sqrt[|H|t]). We evaluate the function f_{+} using a specially developed numerical method which involves solving a nonlinear second-order elliptic equation in Lagrangian coordinates. The faster-decaying tail also involves a sharp transition which occurs at a critical value L_{c}≃2sqrt[2|H|t]/π. This transition is similar to the one recently found for the KPZ equation on a ring, and we believe that it has the same fractional order, 5/2. It is smoothed, however, by small diffusion effects.

Quantum critical detector: amplifying weak signals using discontinuous quantum phase transitions.

We propose a quantum critical detector (QCD) to amplify weak input signals. Our detector exploits a first-order discontinuous quantum-phase-transition and exhibits giant sensitivity (χ ∝ N2) when biased at the critical point. We propose a model consisting of spins with long-range interactions coupled to a bosonic mode to describe the time-dynamics in the QCD. We numerically demonstrate dynamical features of the first order (discontinuous) quantum phase transition such as time-dependent quantum gain in a system with 80 interacting spins. We also show the linear scaling with the spin number N in both the quantum gain and the corresponding signal-to-quantum noise ratio during the time evolution of the device. Our work shows that engineering first order discontinuous quantum phase transitions can lead to a device application for metrology, weak signal amplification, and single photon detection.

Landau theory of the short-time dynamical phase transitions of the Kardar-Parisi-Zhang interface.

We study the short-time distribution P(H,L,t) of the two-point two-time height difference H=h(L,t)-h(0,0) of a stationary Kardar-Parisi-Zhang interface in 1+1 dimension. Employing the optimal-fluctuation method, we develop an effective Landau theory for the second-order dynamical phase transition found previously for L=0 at a critical value H=H_{c}. We show that |H| and L play the roles of inverse temperature and external magnetic field, respectively. In particular, we find a first-order dynamical phase transition when L changes sign, at supercritical H. We also determine analytically P(H,L,t) in several limits away from the second-order transition. Typical fluctuations of H are Gaussian, but the distribution tails are highly asymmetric. The tails -lnP∼|H|^{3/2}/sqrt[t] and -lnP∼|H|^{5/2}/sqrt[t], previously found for L=0, are enhanced for L≠0. At very large |L| the whole height-difference distribution P(H,L,t) is time-independent and Gaussian in H,-lnP∼|H|^{2}/|L|, describing the probability of creating a ramplike height profile at t=0.

Full counting statistics approach to the quantum non-equilibrium Landauer bound

We develop the full counting statistics of dissipated heat to explore the relation with Landauer’s principle. Combining the two-time measurement protocol for the reconstruction of the statistics of heat with the minimal set of assumptions for Landauer’s principle to hold, we derive a general one-parameter family of upper and lower bounds on the mean dissipated heat from a system to its environment. Furthermore, we establish a connection with the degree of non-unitality of the system’s dynamics and show that, if a large deviation function exists as stationary limit of the above cumulant generating function, then our family of lower and upper bounds can be used to witness and understand first-order dynamical phase transitions. For the purpose of demonstration, we apply these bounds to an externally pumped three level system coupled to a finite sized thermal environment.

Finite-Size Scaling of a First-Order Dynamical Phase Transition: Adaptive Population Dynamics and an Effective Model.

We analyze large deviations of the time-averaged activity in the one-dimensional Fredrickson-Andersen model, both numerically and analytically. The model exhibits a dynamical phase transition, which appears as a singularity in the large deviation function. We analyze the finite-size scaling of this phase transition numerically, by generalizing an existing cloning algorithm to include a multicanonical feedback control: this significantly improves the computational efficiency. Motivated by these numerical results, we formulate an effective theory for the model in the vicinity of the phase transition, which accounts quantitatively for the observed behavior. We discuss potential applications of the numerical method and the effective theory in a range of more general contexts.

Phase transitions in large deviations of reset processes

We study the large deviations of additive quantities, such as energy or current, in stochastic processes with intermittent reset. Via a mapping from a discrete-time reset process to the Poland–Scheraga model for DNA denaturation, we derive conditions for observing first-order or continuous dynamical phase transitions in the fluctuations of such quantities and confirm these conditions on simple random walk examples. These results apply to reset Markov processes, but also show more generally that subleading terms in generating functions can lead to non-analyticities in large deviation functions of ‘compound processes’ or ‘random evolutions’ switching stochastically between two or more subprocesses.

Dynamical phase transitions as a resource for quantum enhanced metrology

We consider the general problem of estimating an unknown control parameter of an open quantum system. We establish a direct relation between the evolution of both system and environment and the precision with which the parameter can be estimated. We show that when the open quantum system undergoes a first-order dynamical phase transition the quantum Fisher information (QFI), which gives the upper bound on the achievable precision of any measurement of the system and environment, becomes quadratic in observation time (cf. "Heisenberg scaling"). In fact, the QFI is identical to the variance of the dynamical observable that characterises the phases that coexist at the transition, and enhanced scaling is a consequence of the divergence of the variance of this observable at the transition point. This identification allows to establish the finite time scaling of the QFI. Near the transition the QFI is quadratic in time for times shorter than the correlation time of the dynamics. In the regime of enhanced scaling the optimal measurement whose precision is given by the QFI involves measuring both system and output. As a particular realisation of these ideas, we describe a theoretical scheme for quantum enhanced phase estimation using the photons being emitted from a quantum system near the coexistence of dynamical phases with distinct photon emission rates.

First-Order Dynamical Phase Transitions

Recently, dynamical phase transitions have been identified based on the nonanalytic behavior of the Loschmidt echo in the thermodynamic limit [Heyl etal., Phys. Rev. Lett. 110, 135704 (2013)]. By introducing conditional probability amplitudes, we show how dynamical phase transitions can be further classified, both mathematically, and potentially in experiment. This leads to the definition of first-order dynamical phase transitions. Furthermore, we develop a generalized Keldysh formalism which allows us to use nonequilibrium dynamical mean-field theory to study the Loschmidt echo and dynamical phase transitions in high-dimensional, nonintegrable models. We find dynamical phase transitions of first order in the Falicov-Kimball model and in the Hubbard model.

Finite-size effects in a mean-field kinetically constrained model: dynamical glassiness and quantum criticality

On the example of a mean-field Fredrickson–Andersen kinetically constrained model, we focus on the known property that equilibrium dynamics take place at a first-order dynamical phase transition point in the space of time realizations. We investigate the finite-size properties of this first-order transition. By discussing and exploiting a mapping of the classical dynamical transition—an argued glassiness signature—to a first-order quantum transition, we show that the quantum analogy can be exploited to extract finite-size properties, which in many respects are similar to those in genuine mean-field quantum systems with a first-order transition. We fully characterize the finite-size properties of the order parameter across the first-order transition.