Functional Renormalization Group And Its Applications In First Order Phase Transitions

Real physical system usually involves infinite degrees of freedom and the correlation in?teractions between them.Perturbation methods from quantum field theory provides for this a systematic solution method.But when there is no small parameter for expansion,perturbation theory loses its power,and we have to turn to nonperturbative methods.Functional Renormaliza-tion Group(FRG for short)is a nonperturbative method that becomes more and more popular in recent years and can take system's fluctuation effects into account effectively.The core concept of FRG is effective average action,which satisfies an exact functional differential equation--Wettterich flow equation,which has a one loop structure and has the classical action as its initial condition.Traditional renormalization group methods are usually very suitable for the study of second order phase transitions and critical phenomena.But the flexible treatment of the initial conditions makes FRG also very convenient for the study of first order phase transitions,for example those models whose Lagrangian contains explicit symmetry breaking terms.We make full use of this advantage to study those physical problems related with first order phase transitions.We study the phase diagram of two-flavor massless QCD at finite baryon density by apply-ing FRG for a quark-meson model with ?,?,and ? mesons for the first time.The dynamical fluctuations of quarks,?,and ? are included in the flow equations,while the amplitudes of ?fields are also allowed to fluctuate.Previous studies only use mean field methods to study the influence of ? mesons for the phase diagram.We consider for the first time the ? field amplitude fluctuations.At high temperature the effects of the ? field on the phase boundary are qualitative-ly similar to the mean-field calculations;the phase boundary is shifted to the higher chemical potential region.As the temperature is lowered,however,the transition line bends back to the lower chemical potential region,irrespective to the strength of the vector coupling.In our FRG calculations,the driving force of the low temperature first order line is the fluctuations rather than the quark density,and the effects of ? fields have little impact.At low temperature,the effective potential at small ? field is very sensitive to the infrared cutoff scale,and this signifi-cantly affects our determination of the phase boundaries.The critical chemical potential at the tricritical point is affected by the ?-field effects but its critical temperature stays around the sim-ilar value.It is worth noting that the QCD phase diagram critical endpoint is still not found in experiment,and our study has some guidance significance for future Beam Energy Scan(BES for short)experiment.We also study the nematic-isotropic phase transition(NI phase transition for short)of liquid crystals in the framework of FRG for the first time.We solve the Landau-De Gennes model which describes the NI phase transition,and get the complete flow equation satisfied by the effective average potential and also get the simplified partial differential equations satisfied by the "coupling constants".With the aid of Newton-Raphson algorithm,we numerically solve these equations and get the evolution of the effective average potential and clearly see the first order phase transition.Based on these numerical calculations we reinvestigate the famous NI puzzle,i.e.the ex-perimental value(about 1K)of the temperature difference TN1-T*is much smaller than the theoretical one.TNI is the temperature where the first order phase transition occures and T*is the absolute limit temperature of the supercooled phase(isotropic phase),that is the temperature where the supercooled phase disappears.Our final result is 5.85K.We hope that future improve-ments based on this paper would achieve better result,and may serve as a promising approach to the final solution of the NI puzzle.