Emergence of Objectivity for Quantum Many-Body Systems

We examine the emergence of objectivity for quantum many-body systems in a setting without an environment to decohere the system’s state, but where observers can only access small fragments of the whole system. We extend the result of Reidel (2017) to the case where the system is in a mixed state, measurements are performed through POVMs, and imprints of the outcomes are imperfect. We introduce a new condition on states and measurements to recover full classicality for any number of observers. We further show that evolutions of quantum many-body systems can be expected to yield states that satisfy this condition whenever the corresponding measurement outcomes are redundant.     

New Exact Ground States for One-Dimensional Quantum Many-Body Systems

We consider one-dimensional quantum many-body systems with pair interactions in external fields and (re)investigate the conditions under which exact ground-state wave functions of product type can be found. Contrary to a claim in the literature that an exhaustive list of such systems is already known, we show that this list can still be enlarged considerably. In particular, we are able to calculate exact ground-state wave functions for a class of quantum many-body systems with Ax ^–2+Bx ² interaction potentials and external potentials given by sixth-order polynomials.     

Aspects of Entanglement in Quantum Many-Body Systems

Knowledge of the entanglement properties of the wave functions commonly used to describe quantum many-particle systems can enhance our understanding of their correlation structure and provide new insights into quantum phase transitions that are observed experimentally or predicted theoretically. To illustrate this theme, we first examine the bipartite entanglement contained in the wave functions generated by microscopic many-body theory for the transverse Ising model, a system of Pauli spins on a lattice that exhibits an order-disorder magnetic quantum phase transition under variation of the coupling parameter. Results for the single-site entanglement and measures of two-site bipartite entanglement are obtained for optimal wave functions of Jastrow-Hartree type. Second, we address the nature of bipartite and tripartite entanglement of spins in the ground state of the noninteracting Fermi gas, through analysis of its two- and three-fermion reduced density matrices. The presence of genuine tripartite entanglement is established and characterized by implementation of suitable entanglement witnesses and stabilizer operators. We close with a broader discussion of the relationships between the entanglement properties of strongly interacting systems of identical quantum particles and the dynamical and statistical correlations entering their wave functions.     

Nonlinear Field Theories and Non-Gaussian Fluctuations for Near-Critical Many-Body Systems

This review article outlines a number of efforts made over the past several decades to understand the physics of near critical many-body systems. Beginning with the phenomenological theories of Landau and Ginzburg the paper discusses the two main routes adopted in the past. The first approach is based on statistical calculations while the second investigates the underlying nonlinear field equations. In the last part of the paper we outline a generalisation of these methods which combines classical and quantum properties of the many-body systems studied.     

On Ground State Correlations in Nuclear Many-Body Systems

The ground state correlation in model systems with different interaction strength and pardcle numbers has been studied. Nunnerical results tell us that if the ground state shape is quite stable, the ground state correlation can be approximately explained by the zero-point vibration based on the stadc ground state. It is rather important to carry out first the static self consistent field calculation. But around the critical point where the monopole deformation begins to occur, behaviors of the ground state conflation become very complicate and sensitive to the variation of controlling parameters. lt seems to indicate that particular attentions should be paid to ground state compilations in further studies of light nuclei near the drip line.     

Interaction-Induced Characteristic Length in Strongly Many-Body Localized Systems

A complete set of local integrals of motion(LIOM) is a key concept for describing many-body localization(MBL),which explains a variety of intriguing phenomena in MBL systems.For example,LIOM constrain the dynamics and result in ergodicity violation and breakdown of the eigenstate thermalization hypothesis.However,it is difficult to find a complete set of LIOM explicitly and accurately in practice,which impedes some quantitative structural characterizations of MBL systems.Here we propose an accurate numerical method for constructing LIOM,discover through the LIOM an interaction-induced characteristic length +,and prove a ‘quasi-productstate' structure of the eigenstates with that characteristic length + for MBL systems.More specifically,we find that there are two characteristic lengths in the LIOM.The first one is governed by disorder and is of Andersonlocalization nature.The second one is induced by interaction but shows a discontinuity at zero interaction,showing a nonperturbative nature.We prove that the entanglement and correlation in any eigenstate extend not longer than twice the second length and thus the eigenstates of the system are the quasi-product states with such a localization length.     

Quantifying Spontaneously Symmetry Breaking of Quantum Many-Body Systems

Spontaneous symmetry breaking is related to the appearance of emergent phenomena, while a non-vanishing order parameter has been viewed as the sign of turning into such symmetry-breaking phase. We study the spontaneous symmetry breaking in the conventional superconductor and Bose–Einstein condensation with a continuous measure of symmetry by showing that both the many-body systems can be mapped into the many spin model. We also formulate the underlying relation between the spontaneous symmetry breaking and the order parameter quantitatively. The degree of symmetry stays unity in the absence of the two emergent phenomena, while decreases exponentially at the appearance of the order parameter which indicates the inextricable relation between the spontaneous symmetry and the order parameter.     

The Convergence of Cluster Expansion for Continuous Systems with Many-Body Interaction

The existence of a unique thermodynamic state for dilute classical systems is proved for a class of multi-particle potentials under ordinary assumptions of stability and integrability. Thus we do not use the cumbersome conditions of regularity needed in previous publications for the many-body analysis. The method relies on the Poisson measure representation and cluster expansion for distribution functions.     

Exact Stationary Solution of Fokker-Planck Equation and Generalized Potential for Non-Equilibrium Systems Without Detailed Balance

An exact analogy is approached between systems in thermal equilibrium and those far from equilibrium which can be the cases without detailed balance. The analogy is based on the requirement that a given drift in the Fokker-Planck equation can be decomposed into two parts, one of which is divergence-free and the other can be derived from a potential which is invariant along the direction of the first part. If the conditions are fulfilled the Fokker-Planck equation changes in to a standard Poisson equation. The relations of this requirement to other conditions are diecussed. As a concrete example, the stationary Fokker-Planck equation for optical bistability is solved by using"this method.     

Conditional Stability of Solitary-Wave Solutions for Generalized Compound KdV Equation and Generalized Compound KdV-Burgers Equation

In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travellingwave form satisfies some special conditions.     

Conditional Stability of Solitary-Wave Solutions for Generalized Compound KdV Equation and Generalized Compound KdV-Burgers Equation

In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travelling wave form satisfies some special conditions.     

On a Kinetic Equation for Coalescing Particles

Existence of global weak solutions to a spatially inhomogeneous kinetic model for coalescing particles is proved, each particle being identified by its mass, momentum and position. The large time convergence to zero is also shown. The cornestone of our analysis is that, for any nonnegative and convex function, the associated Orlicz norm is a Liapunov functional. Existence and asymptotic behaviour then rely on weak and strong compactness methods in L¹ in the spirit of the DiPerna-Lions theory for the Boltzmann equation.     

Strong and Weak Chaos in Weakly Nonintegrable Many-Body Hamiltonian Systems

We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology.     

Solitary Wave Solutions for Generalized Rosenau-KdV Equation

In this work, we study the generalized Rosenau-KdV equation. We shall use the sech-ansätze method to derive the solitary wave solutions of this equation.     

Traveling Wave Solutions for Generalized Bretherton Equation

This paper studies the Generalized Bretherton equation using trigonometric function method including the sech-function method, the sine-cosine function method, and the tanh-function method, and He's semi-inverse method (He's variational method). Various traveling wave solutions are obtained, revealing anintrinsic relationship among the amplitude, frequency, and wave speed.     

Generalized Variable-Coefficient KP Equation

The variable-coefficient generalizations of thecelebrated KP equation (GvcKPs) are realistic models forvarious physical and engineering situations. In thisnote, the application of symbolic computation and the truncated Painleve expansion leads toan auto-Backlund transformation and soliton-typedsolutions to a type of the GvcKPs.     

Generalized Hitchin Systems and the Knizhnik–zamolodchikov–bernard Equation on Ellipic Curves

The Knizhnik–Zamolodchikov–Bernard (KZB) equation on an elliptic curve with a marked point is derived by classical Hamiltonian reduction and further quantization. We consider classical Hamiltonian systems on a cotangent bundle to the loop group L(GL(N, C)) extended by the shift operators, to be related to the elliptic module. After reduction, we obtain a Hamiltonian system on a cotangent bundle to the moduli of holomorphic principle bundles and an elliptic module. It is a particular example of generalized Hitchin systems (GHS) which are defined as Hamiltonian systems on cotangent bundles to the moduli of holomorphic bundles and to the moduli of curves. They are extensions of the Hitchin systems by the inclusion the moduli of curves. In contrast with the Hitchin systems, the algebra of integrals are noncommutative on GHS. We discuss the quantization procedure in our example. The quantization of the quadratic integral leads to the KZB equation. We present an explicit form of higher quantum Hitchin integrals which, upon reducing from GHS phase space to the Hitchin phase space, gives a particular example of the Beilinson–Drinfeld commutative algebra of differential operators on the moduli of holomorphic bundles.