The ubiquitous phase diagram of the cuprate superconductors has a phase where the density of states still has a gap above the superconducting transition temperature. Due to the presence of a gap without any other superconducting characteristics, this phase is known as the pseudogap (PG) state. Emery and Kivelson proposed that in this state the superconducting order parameter still has a finite amplitude, however the quantal phase is disordered in such a way that it does not possess long range order. Using these ideas, Franz and Tesanovic developed a theory to explain the pseudogap phase that is identical to three dimensional quantum electrodynamics (QED3 ) apart from an intrinsic anisotropy. Amazingly, this theory can easily explain the antiferromagnetic region. The main purpose of this thesis is twofold. First, to explore some finite temperature properties of QED3 . We do this by calculating a Bosonic polarization bubble at finite temperature in the Matsubara formalism. Secondly we wish to derive some experimentally testable predictions of QED3 . To this end we calculate the modification to the Patrick Lee universal conductivity in the pseudogap state.
We consider a lattice version of the ``negative-
U′′ Hubbard model in three dimensions, to study the crossover from a superconducting phase transition (dominated by phase fluctuation) to a regime where the amplitude of the superconducting order parameter controls the critical temperature. Starting from a fermionic Hamiltonian, we give a microscopic derivation of the effective action in terms of the relevant physical fields: the modulus and phase of the superconducting order parameter. Furthermore, by employing to the resulting coarse--grained quantum
XY model the phase fluctuation algebra between number and phase operators (given by the Euclidean group
E2), we map the ``phase-only'' action onto a solvable quantum spherical model. We establish the self--consistent theory (involving both fermionic and bosonic degrees of freedom), and calculate the superconducting phase coherence transition temperature
Tc as a function of the coupling strength
∣U∣, exhibiting a maximum where the behavior crosses over from BCS to Bose-Einstein (BE) condensation. We examine a pseudogap in the normal state which emerges naturally as a precursor of superconductivity at the second characteristic temperature
Tg due to a state with bound pairs but without long--range phase coherence. Furthermore, we demonstrate that the reduced gaps of the model $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{g}\ensuremath{\sim}4$ is almost independent on the pairing strength
∣U∣/t, whereas $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{c}$ increases with increasing
∣U∣---in striking analogy with the phenomenology of the high--temperature cuprate superconductors when interpreted in terms of the BCS-BC crossover scenario.
I study the effect of Coulomb interaction on superconducting order in a
d-wave lattice superconductor at
T=0 by considering the superconducting saddle point in the two-dimensional $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U$ model with a repulsion
U. The theory of low-energy superconducting phase fluctuations around this saddle point is derived in terms of the effective hard-core bosons (representing the density of spin-up electrons and the phase of the order parameter), interacting with the fluctuating density of spin-down electrons. Whereas the saddle-point value of the superconducting gap is found to continuously increase towards half filling, the phase stiffness at
T=0 has a maximum, and then decreases with further underdoping. Right at half filling the phase stiffness vanishes for large
U. This argues that the pseudogap phenomenon of the type observed in cuprates is in principle possible without a development of any competing order, purely as a result of growing correlations in the superconducting state. Implications for the finite temperature superconducting transition and the effects of static disorder are discussed qualitatively.
A version of non‐compact QED in 2+1 dimensions (QED3), with anisotropies introduced by differing Fermi and gap velocities in the spatial directions has been proposed as an effective theory of the SDW‐pseudogap phase transition in underdoped cuprates at T=0. Preliminary simulations using lattice gauge theory techniques suggest that matters are a little more complicated than previously thought; in particular, measurements of the fermion propagator in Landau gauge indicate that one should be cautious in inferring properties of this form of QED3 from those of isotropic QED3.
The phase diagram of the superconducting cuprates is a manifestation of quantum phase transition (QPT). There is still controversy if the quantum fluctuations generating the pseudogap region is a precursor to the superconducting region or they are competing phases. By modelling the basic possible Hamiltonian for this class of materials as a nonlinear dynamic system, using doping as the tuning parameter, it is possible to obtain a visualized phase diagram which captures fairly the phase diagram of the superconducting cuprates. The numerical results depict that as we doped from slight to optimal, there are three quantum critical points (QCPs): the first at the QPT from the Mott insulator and antiferromagnetic (AFM) phase to the pseudogap at slight doping, the second at the QPT from the pseudogap to the superconducting phase at minimum doping and the third is a hidden QPT due to collapse of the pseudogap around optimal doping. Therefore, the results obtained here in the context of nonlinear dynamic visualization shed new light on the study of the phase diagram of the high TC superconducting cuprates.
Bosonization of Cooper pairs and novel Bose-liquid superconductivity and superfluidity in high-
Tc cuprates and other systems
The criteria for bosonization of Cooper pairs and novel Bose-liquid superconductivity and superfluidity in high-
Tc cuprates and other pseudogap matters are formulated by using the uncertainty principle and the boson mean field theory. We argue that the BCS-type s- or d-wave superconductivity occurring in the fermionic limit of Cooper pairs (which exist in ordinary metals and heavily overdoped cuprates with large Fermi energies) is not characteristic of underdoped to overdoped cuprates with low Fermi energies. The superconducting order parameter in high-
Tc cuprates and other pseudogap matters cannot be determined as the BCS-like (s- or d-wave) gap. We show that the unconventional superconductivity/superfluidity occurring in the bosonic limit of Cooper pairs exists in low Fermi energy systems where the bosonic Cooper pairs are formed at a pseudogap temperature T* above the uperconducting/superfluid transition temperature
Tc and then such bosons condenses into a Bose superfluid at
Tc. The pair condensation of attracting bosons occurs at
Tc and then their single particle condensation sets in at
Tc∗ lower than
Tc (in three dimensions (3D)) or at T=0 (in two dimensions (2D)). The coherent single particle and pair condensates of bosons exist as two distinct superfluid phases below
Tc. By solving the mean field equations for high-
Tc cuprates, the novel superconducting states (i.e., a vortex-like state existing below the temperature
Tv=Tc2D lower than T* but higher than
Tc=Tc3D and two superconducting phases below
Tc) and their properties characterized by the boson superfluid stiffness are determined and compared with the key experimental findings. The novel Bose-liquid superconductivity and superfluidity exists also in other exotic superconductors/superfluids, including quantum liquids, atomic Fermi gases and low-density nuclear matter.
We study the effect of gapless quasiparticles in a d-wave superconductor on the
T=0 end point of the Kosterlitz-Thouless transition line in underdoped high-temperature superconductors. Starting from a lattice model that has gapless fermions coupled to three-dimensional
XY phase fluctuations of the superconducting order parameter, we propose a continuum field theory to describe the quantum phase transition between the d-wave superconductor and the spin-density-wave insulator. Without fermions, the theory reduces to the standard Higgs scalar electrodynamics (HSE), which is known to have the critical point in the inverted
XY universality class. Extending the renormalization group calculation for the HSE to include the coupling to fermions, we find that the qualitative effect of fermions is to increase the portion of the space of coupling constants where the transition is discontinuous. The critical exponents at the stable fixed point vary continuously with the number of fermion fields N, and we estimate the correlation length exponent $(\ensuremath{\nu}=0.65)$ and the vortex field anomalous dimension $({\ensuremath{\eta}}_{\ensuremath{\Phi}}=\ensuremath{-}0.48)$ at the quantum critical point for the physical case
N=2. The stable critical point in the theory disappears for the number of Dirac fermions
N>Nc, with ${N}_{c}\ensuremath{\approx}3.4$ in our approximation. We discuss the relationship between the superconducting and the chiral (SDW) transitions, and point to some interesting parallels between our theory and the Thirring model.