A model is proposed of a dynamic associated with general phase changes. We regard a system exhibiting various collective phenomena near the phase change point, where a macroscopic variable X approaches a transition point Xc, as a strongly interacting many-cluster system. The cluster means that of unstable phase locally appearing as fluctuations in the overwhelmingly large area of stable phase. The variable X is temperature in many cases. The present hypothesis enables us to find a form of the propagator of the cluster-excitation, whose linewidth is shown to be a homogeneous function of the wave number and the characteristic length diverging as |X-Xc|2/3 for X →;Xc except for Xc = absolute zero on the temperature axis. A hydrodynamics in the interacting many-cluster system yields an energy density wave which leads to the thermal conductivity diverging as |X-Xc|-1/3 for X →;Xc. When the transition point is absolute zero on the temperature axis T, the temperature dependence of the characteristic length, the specific heat and the thermal conductivity is of the form T-1/3, T and T7/3, respectively, for T →0.
A model is proposed of a dynamic associated with general phase changes. We regard a system exhibiting various collective phenomena near the phase change point, where a macroscopic variable X approaches a transition point Xc, as a strongly interacting many-cluster system. The cluster means that of unstable phase locally appearing as fluctuations in the overwhelmingly large area of stable phase. The variable X is temperature in many cases. The present hypothesis enables us to find a form of the propagator of the cluster-excitation, whose linewidth is shown to be a homogeneous function of the wave number and the characteristic length diverging as |X-Xc|2/3 for X →;Xc except for Xc = absolute zero on the temperature axis. A hydrodynamics in the interacting many-cluster system yields an energy density wave which leads to the thermal conductivity diverging as |X-Xc|-1/3 for X →;Xc. When the transition point is absolute zero on the temperature axis T, the temperature dependence of the characteristic length, the specific heat and the thermal conductivity is of the form T-1/3, T and T7/3, respectively, for T →0.
The critical phenomenon of the second order phase transition is considered for temperatures above the critical point. A general phenomenological approach is developed. The generalized free energy F[{η q }: T ] is introduced. It is a functional of the Fourier component η q of the local order parameter η( r ). A general property of the correlation function <η q η - q > is established rigorously. The equilibrium free energy and the wave-number dependent susceptibility χ q are expressed in terms of the linked clusters, large clusters corresponding to highr order fluctuation of the order parameter. The general rule for calculating the contribution of each cluster is given. The rule is shown to be very simple. It is found that the specific heat is expressed as a sum of the temperature derivatives of χ q 's. As an example of the application of the linked cluster method, the susceptibility is calculated in the vicinity of the critical point by summing up the clusters that are most divergent when approaching the criticat point. The linked cluster expansion developed here could be a clue to the whole mystery of the singularity at the critical point.
Phase transitions occur when a macroscopic number of local degrees of freedom coherently change their behavior. In ground states of quantum many-body systems, phase transitions due to quantum fluctuations are observed as non-analytic behaviors of order parameters, such as magnetization, as functions of a conjugate parameter, such as the magnetic field. However, as soon as thermal fluctuations are present, these effects are believed to disappear for local observables. We show that this is not necessarily the case: order parameters may still show non-analytic behaviors within their dynamics. With the example of the Ising model and using methods based on hydrodynamic fluctuations, we evaluate the exact order-parameter correlation time, in space-time directions of all velocities, in equilibrium states at nonzero temperature. We reveal non-analytic behaviors of spin correlation times as functions of the magnetic field, velocity, and temperature. As a function of the magnetic field, they occur at values that continuously approach that of the zero-temperature equilibrium transition point as the velocity is decreased and reach it within the light cone, where we obtain a new, temperature-independent logarithmic divergence characterizing the collective dynamics. Thus, collective effects induced by quantum fluctuations persist within the dynamics of local observables.
Pairing and topological phases in cold atoms with long range interactions Chapter 1 Introduction The systematic study and classification of phase transitions has become a popular research area in physics over the past decades. The most familiar phenomena associated to the concept of phase transition are the ones giving rise to macroscopic changes in the properties of strongly correlated many-body systems due to thermal fluctuations. The most prominent examples of this kind of behaviour are the ferromagnetic-paramagnetic transition in lattice spin models and the liquid-gas transition. On the other hand, when dealing with quantum-mechanical systems changes of matter from one state to another can arise as the result of the variation of parameters other than temperature. In particular, when a given many-body quantum problem is studied at zero temperature, thereby suppressing thermal fluctuations, it is the interplay between kinetic energy and interactions that can cause the system to jump into different phases as the model parameters are varied. Phase transitions occurring at zero temperature are widely known as quantum phase transitions. In the framework of the physics of phase transitions, the role of dimensionality is a noticeable one. The major challenge in this perspective is the treatment of low dimensional systems, where standard mean-field treatments and ordinary perturbation theory are known to fail due to the enhancement of fluctuations. The goal of describing collective behaviours in reduced dimensionality is usually pursued in one-dimensional (1D) systems, which turn out to be simple enough to be studied in an effective way through both numerical and analytical techniques, while exhibiting a rich and deeply fascinating phenomenology. As a matter of fact, since particles confined to one dimension cannot avoid interactions with each other, the typical low energy excitations of the system are represented by collective density waves, which substitute nearly free quasiparticle excitations characterizing the behaviour of three-dimensional Fermi liquids in the description of the effects of the interplay between kinetic fluctuations and interactions. On the other hand, the tools employed in the activity of tackling 1D manybody problems are efficient and well developed. The analytical techniques aremainly based on the bosonization technique, which allows for the reformulation of strongly correlated 1D systems onto free bosonic theories, which can then be analyzed by means of standard path integral techniques. Meanwhile, the density matrix renormalization group (DMRG) algorithm represents the state-of-theart numerical machinery in the study of 1D physics, since it allows to efficiently extract the low energy properties of the model Hamiltonian and consequently characterize the corresponding phase diagram by means of the behaviour of properly chosen order parameters. As a final remark pointing out the relevance of the research effort in the direction of a better understanding of 1D many-body quantum systems, it is worth noticing that 1D systems do not represent only a playground for theorists, but have been experimentally realized by means of setups with cold atoms trapped in optical lattices, where the experimentalist is able to tune the model parameters by varying the characteristic features of the trapping laser beams and meanwhile has a comfortable access to the most relevant observables.
We formulate a theory for the transfer of energy from collective degrees of freedom to internal degrees of freedom when the former belong to the adiabatic large-amplitude regime. The framework is that of an extended mean-field theory including dissipation, with the following salient features: (i) The Born-Oppenheimer approximation is extended to finite excitation energy by the introduction of the concept of thermal state, that is pure with respect to collective coordinates and a (square root of a) density matrix with respect to internal coordinates. (ii) By assuming that these states form an approximate complete set for application of the Kerman-Klein method, the self-consistent theory of large amplitude collective motion is extended to finite excitation energy. (iii) The mean-field limit is taken, and further study is carried out in the basis of natural orbitals that diagonalizes the one-particle density matrix. In the approximation maintained uniformly in this work, the equations of motion for the orbitals are shown to be of Hamiltonian form; to these are conjoined the master equations for the time rate of change of the occupation numbers. The latter are studied in two extreme limits. In the collisionless limit, dissipation still arises from the response of the mean field to the collective motion (one-body friction). At the opposite extreme, collisions are assumed to be so effective as to force the system always to be in a state of local (constrained) equilibrium. (iv) We review a procedure by which the noncollective variables in the Hamiltonian may be eliminated, leading to equations of motion for the collective variables with dissipative terms that in general depend on the history of the system. The limit of instantaneous friction can be justified in an adiabatic approximation. (v) A decoupling procedure is developed for deriving the form of these equations of motion from the mean-field theory, with the ultimate aim of obtaining the macroscopic parameters that appear in the classical equations of motion for the collective variables. This procedure generalizes that developed in extensive previous work on large amplitude collective motion at zero temperature. (vi) Associating the dissipative equations of motion for the collective variables with one of the assumptions concerning the rates of change of the occupation numbers provides a description of the relaxation of a system initially perturbed from a state of dynamic equilibrium.
Studying the collective pairing phenomena in a two-component Fermi gas, we predict the appearance near the transition temperature Tc of a well-resolved collective mode of quadratic dispersion. The mode is visible both above and below Tc in the system's response to a driving pairing field. When approaching Tc from below, the phononic and pair-breaking branches, characteristic of the zero temperature behavior, reduce to a very low energy-momentum region when the pair correlation length reaches its critical divergent behavior ξpair∝∣Tc−T∣−1/2; elsewhere, they are replaced by the quadratically-dispersed pairing resonance, which thus acts as a precursor of the phase transition. In the strong-coupling and Bose-Einstein Condensate regime, this mode is a weakly-damped propagating mode associated to a Lorentzian resonance. Conversely, in the BCS limit it is a relaxation mode of pure imaginary eigenenergy. At large momenta, the resonance disappears when it is reabsorbed by the lower-edge of the pairing continuum. At intermediate temperatures between 0 and Tc, we unify the newly found collective phenomena near Tc with the phononic and pair-breaking branches predicted from previous studies, and we exhaustively classify the roots of the analytically continued dispersion equation, and show that they provided a very good summary of the pair spectral functions.
We develop a mean-field theory of the metallic phase near the many-body localization (MBL) transition, using the observation that a system near the MBL transition should become an increasingly slow heat bath for its constituent parts. As a first step, we consider the properties of a many-body localized system coupled to a generic ergodic bath whose characteristic dynamical time scales are much slower than those of the system. As we discuss, a wide range of experimentally relevant systems fall into this class; we argue that relaxation in these systems is dominated by collective many-particle rearrangements, and compute the associated time scales and spectral broadening. We then use the observation that the self-consistent environment of any region in a nearly localized metal can itself be modeled as a slowly fluctuating bath to outline a self-consistent mean-field description of the nearly localized metal and the localization transition. In the nearly localized regime, the spectra of local operators are highly inhomogeneous and the typical local spectral linewidth is narrow. The local spectral linewidth is proportional to the dc conductivity, which is small in the nearly localized regime. This typical linewidth and the dc conductivity go to zero as the localized phase is approached, with a scaling that we calculate, and which appears to be in good agreement with recent experimental results.