Time-dependent Mean-field Theory Research Articles

A simple theory for quantum quenches in the ANNNI model

In a recent numerical study by Haldar et al. [Phys. Rev. X 11, 031062] it was shown that signatures of proximate quantum critical points can be observed at early and intermediate times after certain quantum quenches. Said work focused mainly on the case of the axial next-nearest neighbour Ising (ANNNI) model. Here we construct a simple time-dependent mean-field theory that allows us to obtain a quantitatively accurate description of these quenches at short times, which for reasons we explain remains a fair approximation at late times (with some caveats). Our approach provides a simple framework for understanding the reported numerical results as well as fundamental limitations on detecting quantum critical points through quench dynamics. We moreover explain the origin of the peculiar oscillatory behaviour seen in various observables as arising from the formation of a long-lived bound state.

Criticality and rigidity of dissipative discrete time crystals in solids

We consider a dissipative quantum Ising model periodically driven by a train of $\pi$-pulses and investigate dissipative discrete time crystals (DTCs) in solids. In this model, the interaction between the spins spontaneously breaks the discrete time translation symmetry, giving rise to a dissipative DTC, where two ferromagnetic states are switched alternately by each pulse. We microscopically describe the generic dissipation due to thermal contact to an equilibrium heat bath using the Bloch-Redfield equation. In contrast to other DTC studies, this dissipation stabilizes, rather than destroys, the DTC order without fine-tuning as long as the temperature is low enough. Invoking the time-dependent mean-field theory and solving self-consistently the periodic drive, dissipation, and DTC order parameter, we investigate the nonequilibrium DTC phase transition and determine the critical exponents, including a dynamical one. We also find phase transitions without equilibrium counterpart: a nontrivial interplay of the periodic drive and dissipation gives rise to reentrant DTC transition when changing the pulse interval at a fixed temperature. Besides, to demonstrate the rigidity of the DTC, we consider imperfect $\pi$-pulses, showing that the DTC is robust against the small imperfections and finding that discrete time quasicrystals (DTQC) can appear for the larger imperfections. Together with experimental proposals in magnetic materials, our results pave the ways for realizing the DTC and for uncovering nonequilibrium critical phenomena in real solid-state materials.

Dynamics of the price behavior in stock markets: A statistical physics approach

We study the time evolution of stock markets using a statistical physics approach. We consider an ensemble of agents who sell or buy a good according to several factors acting on them: the majority of the neighbors, the market atmosphere, the variation of the price and some specific measure applied at a given time. Each agent is represented by a spin having a number of discrete states q or continuous states, describing the tendency of the agent for buying or selling. The market atmosphere is represented by a parameter T which plays the role of the temperature in physics: low T corresponds to a calm market, high T to a turbulent one. We show that there is a critical value of T, say Tc, where strong fluctuations between individual states lead to a disordered situation in which there is no majority: the numbers of sellers and buyers are equal, namely the market clearing. The specific measure, by the government or by an economic organization, is parameterized by H applied on the market at the time t1 and removed at the time t2. We have used Monte Carlo simulations to study the time evolution of the price as functions of those parameters. In particular we show that the price strongly fluctuates near Tc and there exists a critical value Hc above which the boosting effect remains after H is removed. Our model replicates the stylized facts in finance (time-independent price variation), volatility clustering (time-dependent dynamics) and persistent effect of a temporary shock. The second part of the paper deals with the price variation using a time-dependent mean-field theory. By supposing that the sellers and the buyers belong to two distinct communities with their characteristics different in both intra-group and inter-group interactions, we find the price oscillation with time. Results are shown and discussed.

Newtonian dynamics of imaginary time-dependent mean field theory

A Time Dependent Hartree-Fock (TDHF) based classical model is applied to sub-barrier fusion reactions using the Feynman Path Integral Method (FPIM). The fusion cross-sections and modified astrophysical S*-factors are calculated for the 12C+12C reactions and compared to direct and indirect experimental results. Different channels cross-sections are estimated from the statistical decay of the compound nucleus. A good agreement with the direct data is found. We suggest a complementary observable given by the (imaginary) action A easily derived from theory and experiments. When properly normalized by the action in the Gamow limit it has an upper value of 1 at zero beam energies. It becomes negative at the Coulomb barrier which is Vcb=5.05±0.05MeV from direct data and Vcb=5.5MeV from model calculations.

Investigation of the magnetic phase diagram of the 2D ferrimagnetic J 1–J 2 model

The zero-temperature phase diagram and spin dynamics of the 2D ferrimagnetic J 1–J 2 model with (S 1, S 2) = (1/2, 1) are investigated using the time-dependent cluster mean-field theory (t-CMFT). The t-CMFT enables the investigation of the quantum-mechanical as well as semi-classical phase diagram and spin dynamics by control of the entanglement. For the characterization of the ferrimagnetic system, the magnetization, the energy per atom, the cluster quantum states and the von Neumann entropy have been determined.

Higgs-like modes in two-dimensional spatially indirect exciton condensates

Higgs-like modes in condensed-matter physics have drawn attention because of analogies to the Higgs bosons of particle physics. Here we use a microscopic time-dependent mean-field theory to study the collective mode spectra of two-dimensional spatially indirect exciton (electron-hole pair) condensates, focusing on the Higgs-like modes, i.e., those that have a large weight in electron-hole pair amplitude response functions. We find that in the low exciton density (Bose-Einstein condensate) limit, the dominant Higgs-like modes of spatially indirect exciton condensates correspond to adding electron-hole pairs that are orthogonal to the condensed pair state. We comment on the previously studied Higgs-like collective excitations of superconductors in light of this finding.

Multicharged-ion–water-molecule collisions in a classical-trajectory time-dependent mean-field theory

A recently proposed classical-trajectory dynamical screening model for the description of multiple ionization and capture during ion--water-molecule collisions is extended to incorporate dynamical screening on both the multicenter target potential and the projectile ion. Comparison with available experimental data for $\mathrm{He}{}^{2+}$ $+$ $\mathrm{H}{}_{2}\mathrm{O}$ collisions at intermediate energies (10--150 keV/u) and $\mathrm{Li}{}^{3+}$ $+$ $\mathrm{H}{}_{2}\mathrm{O}$ at higher energies (100--850 keV/u) demonstrates the importance of both screening mechanisms. The question of how to deal with the repartitioning of the capture flux into allowed capture channels is addressed. The model also provides insights for data on highly charged projectile ions $(\mathrm{C}{}^{6+},$ $\mathrm{O}{}^{8+},$ $\mathrm{Si}{}^{13+})$ in the MeV/u range where the question of saturation effects in net ionization was raised in the literature.

Ultrafast Electron Correlations and Memory Effects at Work: Femtosecond Demagnetization in Ni.

Experimental observations of the ultrafast (less than 50fs) demagnetization of Ni have so far defied theoretical explanations particularly since its spin-flipping time is much less than that resulting from spin-orbit and electron-lattice interactions. Through the application of an approach that benefits from spin-flip time-dependent density-functional theory and dynamical mean-field theory, we show that proper inclusion of electron correlations and memory (time dependence of electron-electron interaction) effects leads to demagnetization at the femtosecond scale, in good agreement with experimental observations. Furthermore, our calculations reveal that this ultrafast demagnetization results mainly from spin-flip transitions from occupied to unoccupied orbitals implying a dynamical reduction of exchange splitting. These conclusions are found to be valid for a wide range of laser pulse amplitudes. They also pave the way for abinitio investigations of ultrafast charge and spin dynamics in a variety of quantum materials in which electron correlations may play a definitive role.

Electron correlations and memory effects in ultrafast electron and hole dynamics in VO2

By applying an approach based on time-dependent density functional theory and dynamical mean-field theory (TDDFT+DMFT) we examine the role of electron correlations in the ultrafast breakdown of the insulating M1 phase in bulk VO2. We consider the case of a spatially homogeneous ultrafast (femtosecond) laser pulse perturbation and present the dynamics of the melting of the insulating state, in particular the time-dependence of the excited charge density. The time-dependence of the chemical potential of the excited electron and hole subsystems shows that even for such short times the dynamics of the system is significantly affected by memory effects—the time-resolved electron–electron interactions. The results pave the way for obtaining a microscopic understanding of the ultrafast dynamics of strongly-correlated materials.

Viscosity effects for ion transport in electrochemical systems

Ion transport in a fluid medium is often described by the Poisson-Nernst-Planck (PNP) equation, a time-dependent mean-field theory that ignores the ion excluded-volume effects and electrostatic correlations, and a number of theoretical procedures have been recently proposed to alleviate the mean-field approximation. In this work, we investigate the charging kinetics of an electric double layer (EDL) of organic electrolytes using the dynamic density functional theory (DDFT). Unlike the PNP equation, DDFT predicts a nonlinear dependence of the surface charge density on the charging voltage. For systems with low solvent viscosity, the collective behaviors caused by inertial effects lead to an oscillatory variation of the surface charge density, occurrences in the charging process of the electrochemical system. Both the strength and frequency of charge oscillation are affected by the solvent viscosity and the degree of EDL overlap.

An average stochastic approach to two-body dissipation in finite fermion systems

We present average stochastic time-dependent Hartree–Fock (ASTDHF) as a theory for efficient handling of incoherent dynamical correlations in finite fermion systems at fully quantum-mechanical level. The basic time evolution is given by time-dependent mean-field theory. The two-body collisions beyond mean-field propagation are evaluated stochastically in an incoherent ensemble of mean-field states as was done in the previously developed STDHF. The new, and time saving, feature in ASTDHF is that it employs one common mean field throughout the whole ensemble. A detailed presentation of ASTDHF is given and extensive comparison with STDHF and other approaches to dissipation are discussed. ASTDHF is found to be a very efficient approach for well bound systems where mean-field fluctuations remain small.

On the inclusion of dissipation on top of mean-field approaches

We discuss extensions of time-dependent mean-field theories such as time-dependent local density approximation (TDLDA) in order to include incoherent dynamical correlations, which are known to play a key role in far-off equilibrium dynamics. We focus here on the case of irradiation dynamics in clusters and molecules. The field, still largely unexplored, requires quantum approaches which represents a major formal and computational effort. We present several approaches we have investigated to address such an issue. We start with time-dependent current-density functional theory (TDCDFT), known to provide damping in the linear regime and explore its capability far-off equilibrium. We observe difficulties with the scaling of relaxation times with deposited energy. We next briefly discuss semi-classical approaches which deliver kinetic equations applicable at sufficiently large excitation energies. We then consider a first quantum kinetic equation at the level of a simplified, though rather elaborate in its content, relaxation time approximation (RTA). Thanks to its sophistication, the method allows us to address numerous realistic irradiation scenarios beyond the usual domain of reliability of such theories. We demonstrate in particular the key role played by dense spectral regions in the impact of dissipation in the response of the irradiated system. RTA nevertheless remains a phenomenological approach which calls for more fundamental descriptions. This is achieved by a stochastic extension of mean field theory, coined stochastic time dependent Hartree–Fock (STDHF), which provides an ensemble description of far-off equilibrium dynamics. The method is equivalent to a quantum kinetic equation complemented by a stochastic collision term. STDHF clearly leads to proper thermalization behaviors in 1D test systems considered here. It remains limited by its ensemble nature which requires possibly huge ensembles to properly sample small transition rates. An alternative approach, coined average STDHF (ASTDHF), consists in overlooking mean field fluctuations of STDHF. ASTDHF provides a robust tool, properly matching STDHF when possible and allowing extension to realistic dynamical scenarios in full 3D. It can also be used in open systems to explore, as done in RTA, the competition between ionization and dissipation.

Enhanced energy absorption through dissipation in metal clusters

The paper investigates the dissipation in the electronic dynamics of metallic clusters using a theoretical description in terms of time-dependent mean-field theory with an extension to include electron-electron collisions beyond mean field in terms of a relaxation-time approximation (RTA). The topic is demonstrated on two examples: first, the damping of the all-dominant surface plasmon mode in metallic clusters becoming manifest in the spectral width of the mode and in its lifetime (when viewed in the time domain), and second, the influence of dissipation on the energy absorption of the cluster from an external laser field. The example of laser excitation reveals a dramatic enhancement of energy absorption because dissipation suppresses significantly induced photon emission which is else-wise a process limiting energy intake. The enhancement effect becomes the more important the longer the duration of the laser pulse.

Spin-selective thermalization plateau in the mass-imbalanced Hubbard model

We use time-dependent non-equilibrium dynamical mean-field theory with weak-coupling auxiliary-field continuous time quantum Monte Carlo as an impurity solver to study the thermalization behavior of the mass-imbalanced single-band Hubbard model after a quench of the Coulomb interaction from the non-interacting limit to a finite positive value. By contrast, when the Coulomb interaction in our model is increased under equilibrium conditions, the quasi-particle weight for spin-up and spin-down (the mass imbalance) electrons approach zero simultaneously, indicating the absence of an orbital selective Mott transition. Our out-of-equilibrium study suggests that there exists a time window where an orbital selective dynamical phase transition occurs. The dynamical phase transition is characterized by the relaxation behavior of energy (kinetic and Coulomb interaction) and the spin-resolved momentum-dependent occupation. To make connection with possible experiments, we calculate the spin-resolved two-time optical conductivity, which confirms the orbital-selective thermalization plateau. We find the time window for the orbital selective dynamical phase transition grows as the mass imbalance increases.

Time-dependent mean-field determination of the excitation energy in transfer reactions: Application to the reaction U238 on C12 at 6.14 MeV/nucleon

The internal excitation of nuclei after multi-nucleon transfer is estimated by using the time-dependent mean-field theory. Transfer probabilities for each channel as well as the energy loss after re-separation are calculated. By combining these two informations, we show that the excitation energy distribution of the transfer fragments can be obtained separately for the different transfer channels. The method is applied to the reaction involving a $^{238}$U beam on a $^{12}$C target, which has recently been measured at GANIL. It is shown that the excitation energy calculated with the microscopic theory compares well with the experimental observation, provided that the competition with fusion is properly taken into account. The reliability of the excitation energy is further confirmed by the comparison with the phenomenological HIPSE model at higher center of mass energies.

Light cone in the two-dimensional transverse-field Ising model in time-dependent mean-field theory

We investigate the propagation of a local perturbation in the two-dimensional transverse-field Ising model with a time-dependent application of the mean-field theory based on the BBGKY hierarchy. We show that the perturbation propagates through the system with a finite velocity and that there is a transition from Manhattan to Euclidian metric, resulting in a light cone with an almost circular shape at sufficiently large distances. The propagation velocity of the perturbation defining the front of the light cone is discussed with respect to the parameters of the Hamiltonian and compared to exact results for the transverse-field Ising model in one dimension.

A collisional extension of time-dependent Hartree–Fock

We propose a collisional extension of time-dependent mean-field theories on the basis of a recently proposed stochastic extension of mean-field dynamics (stochastic time-dependent Hartree–Fock, STDHF). The latter theory is unfortunately too involved to envision practical applications in realistic systems in the near future and is thus bound to model systems. It is also hard to explore moderate to low energies with STDHF, because of vanishing transition probabilities that are impossible to sample properly. For such moderately excited situations covering small fluctuations, we compactify sampling by employing the same average mean field for all STDHF trajectories. The new approach, coined average STDHF (ASTDHF), ignores the fluctuations of the mean field but still accounts correctly for the collisional correlations responsible for dissipative features on top of mean-field dynamics. We detail the main features of the new approach in relation to existing equations, in particular quantum kinetic theories. The new theory is directly connected to STDHF, both formally and practically. We thus discuss in detail how the two approaches are related to each other. We apply the new scheme to illustrative examples taking as benchmark STDHF dynamics in 1D. ASTDHF provides results that are in remarkable agreement with the more elaborate STDHF. It makes it a promising approach to deal with dissipative dynamics in finite quantum systems, because of its moderate cost allowing applications in realistic systems and the possibility of exploring any excitation energy range where collisional correlations are expected to play a role.

Microscopic derivation of the quadrupole collective Hamiltonian for shape coexistence/mixing dynamics

Assuming that the time-evolution of the self-consistent mean field is determined by five pairs of collective coordinate and collective momentum, we microscopically derive the collective Hamiltonian for low-frequency quadrupole modes of excitation. We show that the five-dimensional collective Schrödinger equation is capable of describing large-amplitude quadrupole shape dynamics seen as shape coexistence/mixing phenomena. We focus on basic ideas and recent advances of the approaches based on the time-dependent mean-field theory, but relations to other time-independent approaches are also briefly discussed.

Theory of two-dimensional spatially indirect equilibrium exciton condensates

We present a theory of bilayer two-dimensional electron systems that host a spatially indirect exciton condensate when in thermal equilibrium. Equilibrium bilayer exciton condensates (BXCs) are expected to form when two nearby semiconductor layers are electrically isolated, and when the conduction band of one layer is brought close to degeneracy with the valence band of a nearby layer by varying bias or gate voltages. BXCs are characterized by spontaneous inter-layer phase coherence and counterflow superfluidity. The bilayer system we consider is composed of two transition metal dichalcogenide monolayers separated and surrounded by hexagonal boron nitride. We use mean-field-theory and a bosonic weakly interacting exciton model to explore the BXC phase diagram, and time-dependent mean-field theory to address condensate collective mode spectra and quantum fluctuations. We find that a phase transition occurs between states containing one and two condensate components as the layer separation and the exciton density are varied, and derive simple approximate expressions for the exciton-exciton interaction strength which we show can be measured capacitively.

The low-energy quadrupole mode of nuclei

The phenomenological classification of collective quadrupole excitations by means of the Bohr–Hamiltonian (BH) is reviewed with focus on signatures for triaxility. The variants of the microscopic BH derived by means of the Adiabatic Time-Dependent Mean Field theory from the Pairing-plus-quadrupole–quadrupole interaction, the Shell Correction Method, the Skyrme Energy Density Functional, the Relativistic Mean Field Theory and the Gogny interaction are discussed and applications to concrete nuclides reviewed. The Generator Coordinate Method for the five-dimensional quadrupole deformation space and first applications to triaxial nuclei are presented. The phenomenological classification in the framework of the Interacting Boson Model is discussed with a critical view on the boson number counting rule. The recent success in calculating the model parameters by mapping the mean field deformation energy surface on the bosonic one is discussed and the applications listed. A critical assessment of the models is given with focus on the limitations due to the adiabatic approximation. The Tidal Wave approach and the Triaxial Projected Shell Model are presented as practical approaches to calculate spectral properties outside the adiabatic region.