Quantization Of Two-dimensional Massive Dirac Billiards

A fundamental phenomenon in traditional field of nonrelativistic quantum chaos is quantum scarring in classically chaotic systems,in which the quantum eigen-states tend to concentrate on certain unstable periodic orbits.Recent years have witnessed the discovery of scars in relativistic quantum systems,especially in sol-id state systems described by the massless Dirac equation,where the unique class of chiral scars arise,which has no counterpart in nonrelativistic quantum systems.Are nonrelativistic quantum scars and chiral scars in massless Dirac fermion sys-tems two completely different entities or two different aspects of the same thing?Given the special role of mass in the relativistic quantum systems,namely the infin-ity mass corresponding to non-relativistic limit,in this dissertation we innovatively introduce mass into the Dirac billiards and mainly study the quantization of mas-sive Dirac billiard.From the aspects of quantum behavior such as length spetral and energy level-spacing statistics,we prove the unification of the non-relativistic quantum scars and relativistic chiral scars from the view of phase.In chapter I,we introduce several methods of semi-classical quantization in quantum chaos and the concepts of non-relativistic quantum scars and relativistic chiral scars,expecially focus on the development and improvement of quantization conditions.Then we enumerate the methods of energy level-spacing statistics in quantum billiards to characterize the fluctuation properties of quantum systems and give theoratical results through random matrix theory.These concepts are the important basis for this dissertation to analyze the influence of mass on the behavior of classically chaotic quantum systems.In chapter ?,We consider a Dirac fermion with spin 1/2 and mass m which move in a two-dimensional closed billiard constrainted by the infinite mass poten-tial and give the boundary condition.In order to provide abundant data support for scar statistics,we extend the conformal mapping method to the mass Dirac system,which is a semi-analytical method for solving eigenvalues and eigenstates.Then through the propagation behavior of plane wave solution in mass Dirac billiards,we find a non-trivial phase associated with boundary reflections.A detailed anal-ysis reveals two types of phases constituting the non-trivial phase:primary and dynamical phase,where the former is independent of the mass and generated natu-rally by the spinor reflections and the latter is dependent on mass and wave number and effectively connected with geometric phase.We have clearly described the mass Dirac billiards,which have a natural advantage on studying the relationship between non-relativistic quantum scars and relativistic chiral scars.In chapter III,we reveal the origin and behavior of dynamical phase in detail,which varies continuously from 0 to ?(for odd periodic orbits)or 2?(even periodic orbits)with the increase of energy level.Then show its indispensable role in the quantization condition which can help to predict the emergence and properties of scars.Through comparising the datas of scar states with the theoretical results,we deduce the generally semiclassical quantization condition with a particular focus on bridging the two opposite limits:the massless case and the large mass regime where the equation degenerates effectively to the Schrodinger equation.According to the analysis of length spectrum,with the increase of mass,the appearance for peaks of odd orbits and the abnormal disappearance for those of even orbits can more intuitively reflect the role of dynamical phase difference in transforming the chiral scars and in bridging the relativistic and nonrelativistic quantum scars.In chapter ?,we qualitatively study the influence of mass on time-reversal symmetry.In the energy level statistics,it is intuitive to see that with the increase of mass,time-reversal symmetry is restored as evidenced by the fluctuation property of energy spectrum transiting from GUE distribution to GOE.Spin polarization at the boundary is another effective tool to manifestation time-reversal symmetry.The expansion coefficient for the spinor at the boundary on the eigenstates of the tangential spin operator,and the spatial distribution of current quantificationally reflect the degree to restoration of time-reversal symmetry with mass.Finally,in chapter V,we summarize the whole paper,and make a brief introduction and prospect of the mass Dirac billiards with discrete symmetry.