Research On Lieb-Robinson Bounds In One-dimensional Non-uniform Electronic System

According to the principle of constancy of light velocity in special relativity,the speed of information propagation is limited and cannot exceed the speed of light.In timelike space,two events are related;in spacelike space,two events are independent.In non-relativistic quantum theory,although the speed of information propagation can be infinite in theory,in 1972,Lieb and Robinson found that the speed of information propagation was limited in the study of short-range interaction quantum many-body system.An effective linear "light cone":|x| ?v|t| appears in space-time,the boundary x of "light cone"is called Lieb-Robinson bounds(LRB),the propagation speed v is called Lieb-Robinson velocity.Based on LRB,we can have a better understanding of many equilibrium and non-equilibrium phenomena,including entanglement growth,entropy yield after quenching,exponential decay of ground state correlation function,generation of isochronal correlation function and so on.In 2012,Cheneau et al.observed LRB experimentally for the first time in the Bose-Hubbard model by manipulating cold atoms.Then,LRB was observed in experiments by manipulating magnetic atoms,polar molecules,trapped ions and Rydberg atoms.In addition,LRB has been applied to quantum computing.The initial consideration of LRB is the uniform system.Later,the effect of inhomo-geneity on LRB was considered in short-range interaction systems.In 2007,Osborne et al.found that when the magnetic field obeys Cauchy distribution,a logarithmic "light cone" can be obtained by approximating the two-point time-dependent correlation func-tion.In 2014,Damanik et al.found that when the magnetic field obeys the quasi-periodic sequence,a new singular "light cone":|x| ?v|t|? can be obtained when the magnetic field is large.The original proof of LRB needed to ensure that the interaction in the system was limited.Later,LRB was extended to power-law decaying long-range inter-action systems,in this case,the limited causal region is not conical.For example,in a long-range interaction system in which the interaction decays by 1/r?,when ? is less than or equal to the spatial dimension d,there is no LRB-like result.In other word-s,for ?<d,the behavior of long-range interaction systems often differs from that of short-range interaction systems.Therefore,the study of LRB in different physical sys-tems has become a hot topic in condensed matter physics,statistical physics,quantum information and other fields.In this paper,we mainly study LRB in one-dimensional semi-occupied non-interacting nearest-neighbor transition electronic systems and LRB in one-dimensional semi-occupied non-interacting long-range transition electronic systems with decaying long-range transition rate of 1/r?.First,We study LRB in one-dimensional semi-occupied nearest-neighbor transition electron systems without interaction.By calculating two-point time-dependent correlation fulction<c1(?)(t)ck(t)>,we find that when on-site potential is uniform,the propagation of the correlation function presents a linear "light cone",which verifies the conclusions of Lieb and Robinson;In the case of on-site potential obeys binary disordered distribution,there exists LRB in space-time,and the boundary x is proportional to the logarithm of t:x ?? log it,this shows that a localized logarithmic "light cone" appears in space-time.With the increase of t,the propagation speed of the correlation function will eventually reach zero.In the study of the relationship between the coefficient ? and the potential strength VA,we find that ? decreases nonlinearly with the increase of VA in the case of binary disorder;When on-site potential is quasi-periodic,we find that the boundary x is proportional to t?:x ? t?,the propagation speed of the correlation function is proportional to f?-1,this shows that the propagation speed of correlation function will decrease with the increase of t.In addition,in the study of the relationship between the coefficient ? and potential strength VA,we find that in the quasi-periodic case,with the increase of VA,? first decreases nonlinearly and then tends to be flat.We also find that when the value of VA is the same,the ? in the first class of generalized Fibonacci quasi-periodic case is bigger than that in the second class of generalized Fibonacci quasi-periodic case.This shows that the propagation speed of the correlation function in the first class of generalized Fibonacci quasi-periodic case is faster than that in the second class of generalized Fibonacci quasi-periodic case.Second,we study LRB in one-dimensional semi-occupied non-interacting long-range transition electron system with a decay rate of 1/r?.By calculating two-point time-dependent correlation function<c1(?)(t)ck(t)>,We find that when on-site potential is uniform,for ?=0.75 and ?=1.5,the correlation function can be propagated to the whole space-time,and there is no LRB.But for ?=3,similar to short-range electronic transition system,a linear "light cone" appears in space-time;When on-site potential obeys binary disordered distribution,for ?=0.75 and ?=1.5,the results are similar,and there is no LRB.But for ?=3,we find similar to short-range electronic transition system,LRB exists in space-time,and a logarithmic "light cone" is obtained;When on-site potential is quasi-periodic,the results obtained under two quasi-periodic conditions are similar,for ?=0.75 and ?=1.5,there is no LRB.But for ?=3,we find a new result:the correlation function can be propagated linearly,and the conclusion is different from that in the case of short-range transition.Our research plays an important role in understanding LRB in inhomogeneous systems.