| The discovery of topological materials has drown great attention to the topological properties of quantum systems.The works of ground state at zero-temperature have been studied thoroughly and systematically,such as topological superconductors,topological insulators,anomalous quantum hall effects and Bose systems.Real physical systems,however,cannot be completely isolated from the environment,and heat exchange is inevitable.In this situation,not only the ground state,but all the eigenstates contribute to the physical properties of systems.Because the system is now in a mixed state,the quantum system must be described by density matrix.We want to study the topological properties of quantum systems at finite temperature and extend the topological invariants of pure states to mixed states.The geometric phase(Berry phase)plays an important role in the study of the topological properties of the system.In recent years,the realization of observing geometric phase experimentally makes it possible to extend the study of topological properties from pure state to mixed state.Now we are studying with the mixed states,a geometric phase should be redefined mathematically.We start with introducing how the German mathematician Uhlmann constructed the connection between two matrices,which is called Uhlmann connection.On this basis,Oiyuela et al.defined the so-called Uhlmann phase,which obtained quantization results in the models of several one-dimensional topological insulators and topological superconductors,providing a method for the study of topological properties at finite temperatures.They also extended their work to twodimensional quantum systems at finite temperatures and obtained a physical quantity they called Uhlmann number,which they considered topologically invariant.We also introduce other researchers’ work.For example,a domestic group modified the definition of the first type of Chern number and obtained the non-quantized physical quantity,which was called nontopological(NT)thermal Uhlmann-Chern number,in the hope of obtaining the topological information of the system.Bardyn et al.generalized the polarization equation of mixed state from the ground state polarization equation suitable for periodic system of Resta,and derived the physical quantity called ”ensemble geometry phase”,or EGP for short.This is another method to study topological phase and topological invariants at finite temperature.It is important to note that EGP is topologically quantized and experimentally measurable.The present work is mainly focused on the two-band model,and there are few researches on the model with more energy levels.Some work has mentioned that the Anderson model shows novel and interesting physical phenomena by transforming between the Dirac Kondo semimetal phase and the topological Kondo insulator phase with the doping amount(chemical potential μregulation).We focuse on the topological properties of Anderson model under weak coupling.In the whole research process of the subject,our research achievements mainly include the following:a.We find that the definition of NT thermal Uhlmann-Chern number is not exact,and there are still some problems in the research results.The reason may be that the definition of external differentiation is not clear,and the detailed derivation process is given.b.Each research method still has limitations.For example,the Uhlmann number defined by Oiyuela et al.in the two-dimensional case is not completely equivalent to the topological invariant of the pure states.We point to applicable situations and try to come up with better solutions.c.Using the numerical method,the Uhlmann number of the Anderson model(weak coupling)was calculated and returned to zero temperature in the limit of zero temperature.And at finite temperature,the effect of temperature regulation is not obvious.How to generalize topological invariants of quantum systems at finite temperature is a challenging but very interesting topic.At the same time,we find that the existing topological framework of pure state at zero temperature is not suitable for finite temperature.When we extend the notion of topological invariants from pure states to mixed states,they are no longer integers.This also makes direct observation experimentally difficult.And the results vary from method to method.We need to understand the topological invariants of quantum systems at finite temperatures from a deeper and more essential perspective. |
