Phase Transition With Trivial Quantum Criticality In An Anisotropic Weyl Semimetal

The phase transition is one of the core concepts in condensed matter physics,of which the quantum critical phase transition is especially significant that has been at-tracting more and more attentions from both theoretical and experimental physicists.One of the striking features of the quantum critical phase transition is that the cor-relation length diverges when approaching the critical point from above,along with which several thermodynamical parameters will also diverge,surprisingly and fortu-nately,these quantum critical properties can be described by some universal critical exponents that are independent of the detailed materials.Of a series of the quantum critical behaviors,the superconductivity is definitely one of the most significant and magical phenomena.To unveil its mysterious mask,generations of physicists have dedicated all their energy and time.Over the past one century,physicists have developed several theories to characterize this magical phe-nomenon,such as the Ginzburg-Landau and BCS theory,which have achieved great success.Landau Fermi liquid theory is usually applied to study the weakly interacting elec-trons in metals.Due to large amounts of electron-hole excitations the Coulomb inter-action is screened so that the long-ranged coupling reduces to be short-ranged.Con-sequently the strong Coulomb coupling alters to an effective weak interaction and the electrons are renormalized to the quasiparticles,which is the core concept of Landau Fermi liquid theory.At the mean field approximation level,the fluctuation corrections are all neglected so that only the average values of operators are remaining,which gives us an overview of the critical phenomenon,meanwhile,we lose lots of useful and key information about the critical property.Actually fluctuations may have a significant impact on the critical behavior so that the mean field theory may breakdown.One practical and powerful tool is proposed to remedy several drawbacks in the mean field approximation,the Wilson renormalization group theory,which allows us to consider the quantum fluctuation over the average value.In language of the renormal-ization group theory,the result given by mean field theory is equivalent to the zeroth order approximation.By continuously integrating out an infinitesimal shell of high-momentum and doing iteratively rescaling transformations,the Wilson renormalization group method provides us a way to trace the flow of quantities along with the changing of energy level,based on which one can predict the final fate of the system.One extensively studied system is the Weyl semimetal,in which the semimetal-superconductor phase transition is usually expected to occur.With similar anticipation,in this thesis we primarily concentrate upon the quantum critical behavior of one special three-dimensional anisotropic Weyl semimetal whose momentum components inside the x-y plane are linear and the one along the z direction is quadratic.Following the conventional method,we first perform a mean field approximation to determine the critical point,after which we apply the Wilson renormalization group theory to study the quantum critical behavior.In strikingly contrast to the previous published result that some nontrivial quantum critical phenomena occur,we find that in our present system the quantum fluctuation does not qualitatively modify the low energy property so that we can still offer a well description of the system by the free fermion model.We therefore find a concrete system with trivial quantum criticality.