Studies On The Density Matrix Of Many-body Chaotic Systems

Statistical mechanics successfully links the macroscopic thermal equilibrium state of a system with its microscopic state.The ensemble theory at the core of statistical mechanics clearly gives the probability distribution of microscopic states.It has been widely used in various fields of natural sciences such as physics,chemistry,and biology.However,the basic principles of ensemble theory are still controversial:Why does the long time average consistent with the ensemble average results,while we only observing one system in laboratory.In the framework of classical mechanics,the ergodic hypothesis is a generally accepted interpretation.This hypothesis holds that a classical system will go through various micro-states in the process of evolution over time.Therefore,the average of system properties over time is equivalent to its Ensemble average.However,the Schrodinger equation for quantum mechanics is a linear equation,and the ergodic hypothesis of states does not hold.Under the framework of quantum mechanics,why can ensemble theory still predict experimental results?This problem has attracted people's attention since the birth of quantum mechanics.Important progress occurred in 1955.Wigner proposed a random matrix model and used this model to explain the problem of energy level distribution in nuclei with large atomic numbers.The prelude to the study of quantum chaos.In the decades that followed,quantum chaos theory developed rapidly.In 1994,Srednicki proposed the eigenstate thermalization hypothesis(ETH)on the basis of a random matrix model,explaining why quantum systems reach thermal equilibrium,and why quantum thermal equilibrium can also be described by ensemble theory.Subsequently,numerous numerical simulations confirmed the accuracy of the ETH.On the other hand,from the late 1980s,the development of nanotechnology prompted people to study the properties of electron transport in mesoscopic size systems.In order to explain experimental results,quantum mechanics needs to be applied to non-equilibrium states.For example,Landauer and Buttiker deduced the conductance formula in the absence of interactions;in the early 1990s,researchers began to use the non-equilibrium Green's function method to calculate conductance Since the ensemble theory can derive the density matrix of the thermal equilibrium state,from a practical point of view,researchers naturally hope to obtain an expression of the density matrix of the non-equilibrium state.In 1992,Hershfield started from the density matrix of the equilibrium state and solved the time evolution problem to obtain the expression of the density matrix of the non-equilibrium steady state.McLennan and Zubarev tried to derive the general form of the density matrix from the Liouville equation.In 2003,Bokes and Godby used the principle of entropy maximization to obtain the density matrix by using the current as a limiting condition.In 2013,Ness proved that the above three results are equivalent to each other,and the density matrix is expressed in the form of an extended Gibbs ensembleNaturally,we can also ask the following questions in non-equilibrium statistical mechanics:Why can the non-equilibrium steady state observed in the laboratory be described by the extended Gibbs ensemble theory?Whether the conductance is calculated by the formula obtained either by Hershfield et.al.or by non-equilibrium Green's function method.Therefore,the above question is also equivalent to:Why can the non-equilibrium steady state observed in the laboratory be derived from the thermal equilibrium ensemble undergoing infinite time evolution?Answering this question also requires the use of quantum chaos theory.However,non-equilibrium steady state cannot be explained by the ETH alone.To this end,we introduce a new hypothesis:the non-equilibrium steady-state hypothesis.We believe that in a chaotic system,the density matrix of quantum states is similar to the density matrix of observable operators,and both have a universal structure.The non-diagonal elements of the density matrix can be expressed as the product of dynamic characteristic functions and random variables.Among them,the nature of the dynamic characteristic function determines the dynamic properties of the system.In particular,in the non-equilibrium steady state,the dynamic characteristic function f(E,?)diverges in the form of 1/? at ??0.We used numerical diagonalization to study a number of different models,verified our hypothesis in these models,and explained the dynamic process of these models using dynamic characteristic functionsThe first chapter of this paper introduces the background knowledge of quantum chaotic systems.From classical chaos to quantum chaos,the random matrix theory,the eigenstate heating hypothesis,the experimental background of many-body physics,the advantages and disadvantages of numerical algorithms are introducedChapter 2 introduces the concept of non-equilibrium steady state.We give rigorous solutions to the single-particle and many-body integrable models respectively.Through the solution of the single-particle problem,We know that there is a condition to form a stable current,that is,there is a factor of form divergence in the corresponding integral By calculating the current in the scattering state,we know that the factor does not exist in the matrix corresponding to the current operatorThe third chapter is the theoretical knowledge of the non-equilibrium steady-state hypothesis(NESSH).We give that the initial density matrix is the same as the local observable operator in the quantum chaotic system.The matrix elements under the Hamiltonian eigenbasis have a universal form.It can be understood as two parts.The diagonal elements are the summation of Gaussian function and random variables,and non-diagonal elements are the product of dynamical characteristic function and random variablesChapter 4 considers one-dimensional disordered XXZ model and two-dimensional Ising model.We numerically analyzed the ETH and the NESSH.The numerical results support two hypotheses,and display the long-time behavior of a local observable operator being unrelated to off-diagonal elementsChapter 5 presents a resonant level model with random coupling to verify ETH and NESSH.This model's leads are described by random matrix theory.Our numerical results indicate that the real-time dynamics and curves of the current show similar functionality to conventional leads.According to the ETH and the NESSH,we can obtain the formula of stable current.We verify the various approximations and compare the result with those of the time-dependent Schrodinger equationFinally,the summary and outlook of the full text.