Research On Practice Teaching Of Ideological And Political Theory Course In Our Universities

The most mysterious part of quantum physics is the entanglement: two spatial separated local measurement will instantaneously affect each other.The seemingly superluminal phenomenon actually reveal the non-locality of the quantum states,which is the fundamental framework of mordern quantum information and communications.How to understand and quantify the entanglement is a long-standing and intriguing branch of research in quantum physics.Staring from Bell’s inequality,for a long time people try to answer the three basic questions about entanglement: 1)its characterization,2)its manipulation,3)its quantification.In this thesis,we are focusing on the entropy of entanglement,which is an important measure of the entanglement.As a fundamental quantity of quantum physics,people also use entanglement entropy to study other interesting physics,such as: black hole entropy,critical phenomenon of low temperature physics and holographical principle,etc.In some sense,entanglement entropy build a connection between condense matter physics,quantum information theory and high energy physics.On on hand,entanglement entropy as en order parameter of phase transition of 1D Ising model,i.e.,entanglement entropy divergent at the critical point.On the other hand,it can be equally seen as a partition function of some CFT on the Riemann surfaces.For this subject,most study were carried out in perturbative string theory.Like multiloop amplitude in quantum field theory,in string theory,higher genus amplitude plays the same role.Although for genus g > 2,the string amplitude is very hard to calculate,but luckily we are facing a rather special kind of Riemann surfaces,in such cases,the tools invented in string theory are already enough and quite handy.It should be mentioned that,entanglement entropy is even related to AdS/CFT conjecture.Pepole already found out that the entanglement entropy in two dimensional spacetime is the same as the length of geodesic in Ad S 3.Besides applications in physics,CFT on Riemann surfaces is highly related to some interesting physics such as moonshine theory.The studying subject of the theis is a complex free boson compactified on a torus.There are three different parts:(1)We first consider the boson lives on a infinite line,in such system there is a subsystem A consisted of one or more intervals,the question is that how to find out the entangment entropy of the subsystem A? In early study of these questions,people only consider the most simple case: A only contains one interval.It is simple because all one need to do is to introduce the so-called twist fields and calculate the two point function of them.Also for one interval case,the entanglement entropy has no instanton contribution.Then the entanglement entropy of two intervals has also been studied.It turns out that it is a much harder problem for two reasons: Firstly,one has to compute the four point function of the twist fields,which can not be fixed by the global conform group;secondly,the existence of the two intervals change the topology of the worldsheet drastically,therefore one has to consider the instanton contribution from nontrivial windings.Indeed,the instanton term has been missed at first,but it was corrected some years later.One of the purpose of this theis is to generalize the results to the cases of arbitrary intervals.However the old method is not that efficient.The most obstacle is that there are redundant classical solutions,it is getting worse and worse when the number of intervals becomes larger.Thus,by noticing that these kind of question is the same as calculating the partition function on the branch covering of CP1,which can be represented by a special kind of singular algebraic curve.Thus by borrowing the knowledge of algebraic curve,I directly find all the independent classical solutions,and then find the R′enyi entanglement entropy for arbitrary intervals.(2)The second part of the thesis is to study such a physical problem: Consider the scalar lives on a finite circle,there is a subsysterm A consisted of two disjoint intervals,then the question is that how to find the R′enyi entanglement entropy at finite temperature.Before the work was done,people already calculate the same problem for only one interval.It turns out the problem is much more complicated than that of one interval.To check the results,we also study some non-trivial properties of the entanglement entropy,such as T-dualiy.In order to compare the earlier results,the low temperature expansion has been studied.In the infinite system limit,the leading term of low temperature expansion is agreed with earlier results,and the first thermo correction is also agreed.(3)The third part of the thesis is to generalize the results of the second part to arbitrary intervals.Unlike the first part,for finite system and finite temperature,the corresponding Riemann surface is quite different: it is a branch covering of the torus rather than CP1.For such surfaces,one can not expect to find a algebraic description.Therefore,in order to find the classical solutions,one need to find a new way to construct the canonical basis of the first homology and the first co-homology.In this thesis,an efficient way how to find out the basis has been proposed,then the R′enyi entanglement entropy of arbitrary intervals in a finite system at finite temperature has been given.