Dynamic phase transition in classical Ising models

Quantum states beyond thermodynamic equilibrium represent fascinating and cutting-edge research. However, the behavior of dynamical quantum phase transitions in complex open quantum systems remains poorly understood. Here, using state-of-the-art numerical approaches, we show that by quenching the qubits-oscillator coupling in a dissipative two-qubit Rabi model, the system undergoes dynamical quantum phase transitions. These transitions are characterized by kinks in the Loschmidt echo rate function at parameter values close to a thermodynamic quantum phase transition and are associated with distinct entanglement features. The two classes of critical phenomena depend on qubit interactions and entanglement, revealing different behaviors of the critical exponent of the first kink of the Loschmidt echo for interacting versus non-interacting qubits. This research enhances our understanding of non-equilibrium quantum systems and offers potential applications in quantum sensing and metrology, as it examines how dynamical transitions can enhance the sensitivity of the Loschmidt echo to the quench parameters.

Phase transitions are a common phenomenon in condensed matter and act as a critical degree of freedom that can be employed to tailor the mechanical or electronic properties of materials. Understanding the fundamental mechanisms of the thermodynamics and kinetics of phase transitions is, thus, at the core of modern materials design. Conventionally, studies of phase transitions have, to a large extent, focused on pristine bulk phases. However, realistic materials exist in a complex form; their microstructures consist of different point and extended defects. The presence of defects impacts the thermodynamics and kinetics of phase transitions, but has been commonly ignored or treated separately. In recent years, with the significant advances in theoretical and experimental techniques, there has been an increasing research interest in modeling and characterizing how defects impact or even dictate phase transitions. The present review systematically discusses the recent progress in understanding the kinetics of defect-characterized phase transitions, derives the key mechanisms underlying these phase transitions, and envisions the remaining challenges and fruitful research directions. We hope that these discussions and insights will help to inspire future research and development in the field.

Chapter 2 - Thermodynamics and Kinetics of Phase Transitions

We study the slow quenching dynamics (characterized by an inverse rate, $\tau^{-1}$) of a one-dimensional transverse Ising chain with nearest neighbor ferromagentic interactions across the quantum critical point (QCP) and analyze the Loschmidt overlap {measured using the subsequent temporal evolution of the final wave function (reached at the end of the quenching) with the final time-independent Hamiltonian}. Studying the Fisher zeros of the corresponding generalized "partition function", we probe non-analyticities manifested in the rate function of the return probability known as dynamical phase transitions (DPTs). In contrast to the sudden quenching case, we show that DPTs survive {in the subsequent temporal evolution following the quenching across two critical points of the model for a sufficiently slow rate; furthermore, an interesting "lobe" structure of Fisher zeros emerge.} We have also made a connection to topological aspects studying the dynamical topological order parameter ($\nu_D(t)$), as a function of time ($t$) {measured from the instant when the quenching is complete. Remarkably, the time evolution of $\nu_D(t)$ exhibits drastically different behavior following quenches across a single QCP and two QCPs. } {In the former case, $\nu_D (t)$ increases step-wise by unity at every DPT (i.e., $\Delta \nu_D =1$). In the latter case, on the other hand, $\nu_D(t)$ essentially oscillates between 0 and 1 (i.e., successive DPTs occur with $\Delta \nu_D =1$ and $\Delta \nu_D =-1$, respectively), except for instants where it shows a sudden jump by a factor of unity when two successive DPTs carry a topological charge of same sign.

We present two approaches to the dynamics of a quench-induced phase transition in the quantum Ising model. One follows the standard treatment of thermodynamic second order phase transitions but applies it to the quantum phase transitions. The other approach is quantum, and uses Landau-Zener formula for transition probabilities in avoided level crossings. We show that predictions of the two approaches of how the density of defects scales with the quench rate are compatible, and discuss the ensuing insights into the dynamics of quantum phase transitions.

In this study, a controllable thermo-induced phase transition process was first observed in the aqueous solution of backbone-thermoresponsive hyperbranched polyethers. According to the variable temperature UV-visible spectrophotometry, the phase transition process was extremely slow as temperature was set around lower critical solution temperature (LCST). Fluorescence tests showed that this slow phase transition behavior was related to the gradual thermally-induced dehydration of hyperbranched polyethers. Based on this unique behavior, the dynamics of phase transition was conveniently investigated by the temperature-jump method. Results showed that the dynamics of phase transition can be described by single exponential functions, and the dynamic parameters depended on temperature. Dynamic laser scattering and transmission electron microscopy revealed that the hyperbranched polyether self-assembled into micelles below its LCST, while the temperature-dependent aggregation occurred and complex micelles with larger size formed above LCST. Furthermore, the gradual aggregation process was in accordance with the rate-limited colloidal aggregation mechanism. Our research can contribute to the clarification of the dynamics and mechanism of this typical gradual phase transition process of backbone-thermoresponsive hyperbranched polymers, as well as their self-assembly and aggregation mechanisms.

Dynamically order–disorder transition in the kinetic Ising model on a triangular lattice driven by a time dependent magnetic field

We generalize the simplest kinetically constrained model of a glass-forming liquid by softening kinetic constraints, allowing them to be violated with a small rate. We demonstrate that this model supports a first-order dynamical (space-time) phase transition between active (fluid) and inactive (glass) phases. The first-order phase boundary in this softened model ends in a finite-temperature dynamical critical point, which may be present in natural systems. In this case, the glass phase has a very large but finite relaxation time. We discuss links between the dynamical critical point and quantum phase transitions, showing that dynamical phase transitions in d dimensions map to quantum transitions in the same dimension, and hence to classical thermodynamic phase transitions in d + 1 dimensions.

Being inspired by the results of recent experimental work ‘Phys. Rev. E 102 022 804’, we perform a series of Monte Carlo simulations to investigate the dynamic critical properties of a binary magnetic alloy system composed of two different components with different spin moments with the chemical formula . In the first part of the work, we concentrate our attention on the similarities between dynamic critical behavior and thermodynamic phase transition properties of the model. Finite-size scaling analyses suggest that the proposed binary alloy model exhibits 2D Ising universality class. In the second part, we examine the dynamic critical behavior in the presence of a magnetic bias field for which a metamagnetic sideband behavior originates in dynamic susceptibility curves. Our results yield that these metamagnetic anomalies survive for a wide range of magnetic atom concentration p A , and the strength of the anomalies observed in the field-dependent dynamic susceptibility varies very slowly with the T/T c ratio of the binary magnetic alloy.

We have studied the dynamical evolution of the shock in a neutron star (NS). The conversion of nuclear to quark matter (QM) is assumed to take place at the shock discontinuity. The density and pressure discontinuity is studied both spatially and temporally as it starts near the center of the star and moves toward the surface. Polytropic equations of state (EoS), which mimic original nuclear and QM EoS, are used to study such dynamical phase transition (PT). Solving relativistic hydrodynamic equations for a spherically symmetric star, we study the PT, assuming a considerable density discontinuity near the center. We find that as the shock wave propagates outward, its intensity decreases with time; however, the shock velocity peaks up and reaches a value close to that of light. Such fast shock velocity indicates rapid PT in NS taking place on a timescale of some 10s of microseconds. Such a result is quite interesting, and it differs from previous calculations that the PT in NSs takes at least some 10s of milliseconds. Rapid PT can have significant observational significance, because such fast PT would imply rather strong gravitational wave (GW) signals that are rather short lived. Such short-lived GW signals would be accompanied with short-lived gamma-ray bursts and neutrino signals originating from the neutrino and gamma-ray generation from the PT of nuclear matter to QM.

The kinetics of the ferroelectric phase transition between modulated phases in Rb2ZnCl4 at about 190K has been investigated by time-resolved neutron scattering. Observing the time dependence of satellite spectra, it is found that the structural changes associated with the field-induced transformation proceed on a millisecond time scale. Under the influence of fast cycling strong electric fields unusual non-equilibrium states are obtained which are modulated in space as well as in time and characterised by the presence of internal mechanical strains.

Dynamic phase transitions in the kinetic spin-1 Blume–Capel model on the Bethe lattice

Characterizing quantum phase transitions through quantum correlations has\nbeen deeply developed for a long time, while the connections between dynamical\nphase transitions (DPTs) and quantum entanglement is not yet well understood.\nIn this work, we show that the time-averaged two-mode entanglement in the spin\nspace reaches a maximal value when it undergoes a DPT induced by external\nperturbation in a spin-orbit-coupled Bose-Einstein condensate. We employ the\nvon Neumann entropy and a correlation-based entanglement criterion as\nentanglement measures and find that both of them can infer the existence of\nDPT. While the von Neumann entropy works only for a pure state at zero\ntemperature and requires state tomography to reconstruct, the experimentally\nmore feasible correlation-based entanglement criterion acts as an excellent\nproxy for entropic entanglement and can determine the existence of entanglement\nfor a mixed state at finite temperature, making itself an excellent indicator\nfor DPT. Our work provides a deeper understanding about the connection between\nDPTs and quantum entanglement, and may allow the detection of DPT via\nentanglement become accessible as the examined criterion is suitable for\nmeasuring entanglement.\n

Interacting many-body systems that are driven far away from equilibrium can exhibit phase transitions between dynamically emerging quantum phases, which manifest as singularities in the Loschmidt echo. Whether and under which conditions such dynamical transitions occur in higher-dimensional systems with spontaneously broken continuous symmetries is largely elusive thus far. Here, we study the dynamics of the Loschmidt echo in the three dimensional O(N) model following a quantum quench from a symmetry breaking initial state. The O(N) model exhibits a dynamical transition in the asymptotic steady state, separating two phases with a finite and vanishing order parameter, that is associated with the broken symmetry. We analytically calculate the rate function of the Loschmidt echo and find that it exhibits periodic kink singularities when this dynamical steady-state transition is crossed. The singularities arise exactly at the zero-crossings of the oscillating order parameter. As a consequence, the appearance of the kink singularities in the transient dynamics is directly linked to a dynamical transition in the order parameter. Furthermore, we argue, that our results for dynamical quantum phase transitions in the O(N) model are general and apply to generic systems with continuous symmetry breaking.

We analyze the critical dynamics of the order-disorder phase transition through the pseudo-spin quantum and classical Ising models. The findings derived by the Green-function equation-motion method and realized through Tyablikov's and Tserkovnikov's schemes, on the one hand, and the Thompson–Silva renormalization group approach, on the other hand are discussed. The slowing-down Δ and the dynamical z critical exponents are predicted for the relaxation (Δ=5/4, z=2) and resonant (Δ=5/16, z=1/2) motion regimes for three-dimensional Ising models. The results obtained give support to the idea that the classical and quantum d=2, 3 Ising models at nonzero critical temperature belong to the same universality class.