| High temperature superconductivity is one of the most important research fields in condensed matter physics. Despite the extensive investigations in the past two decades, the microscopic pairing mechanism and the highly anomalous normal state of high temperature superconductors remain mysterious. One important reason for this situ-ation is that high temperature superconductor has a very complicated phase diagram, which contains several quantum phase transitions in a superconducting dome at zero temperature. Depending on doping and other parameters, its ground state, could exhibit antiferromagnetic order, superconducting order, nematic order, or stripe order. Due to quantum fluctuations at quantum critical points where fluctuations are divergent, these orders are not independent, but have important influence on each other. It is helpful to capture critical behaviors of physical quantities by studying these quantum phase transitions, and hence it would be instructive to understand many anomalous behaviors of high-temperature superconductors and reveal the microscopic mechanism.In this dissertation, we focus on several quantum phase transitions, including ne-matic phase transition in d-wave high temperature superconductors at zero temperature by employing renormalization group method. We study the disorder effects on the ne-matic quantum critical behavior and especially on the flow of fermion velocities. A renormalization group analysis shows that random mass and random gauge field are both irrelevant and thus do not change the fixed point of extreme velocity anisotropy. However, the marginal interaction due to random chemical potential destroys this fixed point and makes the nematic phase transition unstable. In additional, we investigate the competition between d-wave superconductivity and nematic order in high-Tc supercon-ductor at nematic critical point, and examine the role played by gapless fermionic de-grees of freedom beyond the Hertz-Millis theory by using renormalization group. We find that the gapless fermions can play an important role and should be carefully in-cluded in the theoretical description of competing orders. In case the fermionic degree of freedom are entirely neglected, the competitive interaction between two bosonic or- der parameters is strongly relevant, and can lead to runaway behavior. However, these properties are fundamentally changed once the dynamics of fermions are taken into account. At the nematic quantum critical point where an extreme fermion velocity anisotropy occurs, the superconducting and nematic order parameters are decoupled from each other. Consequently, the phase transitions are continuous, and d-wave su-perconductivity can coexist with nematic order homogeneously.Beside the application of the renormalization group to the nematic phase tran-sition, we extend the renormalization group method to deal with the unconventional behavior of massless Dirac fermions due to interaction with a U(1) gauge field in three time-space dimensions. We find the massless Dirac fermions behave quite differently at finite and zero chemical potential. At zero chemical potential, there is no fermion ve-locity renormalization. At finite chemical potential, the longitudinal gauge interaction becomes short-ranged, but the transverse gauge interaction remains long-ranged and leads to singular velocity renormalization. An explicit calculation shows that a finite anomalous dimension of velocity is generated and gives rise to unconventional prop-erties in some physical quantities, including specific heat, DOS, and compressibility. Furthermore, we derive an efficient and unbiased method for computing order param-eters in correlated electron systems with competing instabilities by combing the func-tional renormalization group method with the mean-field theory. The method would capture fluctuations driven instabilities. And then we employ this method to study the competition between antiferromagnetism and superconductivity in the ground state of the two-dimensional Hubbard model. |
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