This thesis is organized in two independent parts, in which I study two different aspects of high Tc superconductivity. The first part begins with an introduction aimed to briefly introduce some relevant experimental and theoretical works performed in recent years, that have helped us to think about cuprates the way we do now. Afterwards, I introduce Landau Fermi liquid theory in a standard text book way. The question of validity of Fermi liquid theory in 2-d is then raised and investigated by searching for singularities in Landau's /-function. I show that the interaction function between two quasiparticles whose momenta approach each other near a curved point of the Fermi surface, contains a 1-d singularity not strong enough to change the Fermi liquid behavior. On the other hand, inflection points provide 2-d singularities that have to be taken seriously in Fermi liquid considerations. I then introduce nearly antiferromagnetic Fermi liquid theory (NAFL), which is a phenomenological theory proposed to describe high Tc systems. I mainly focus on the self-consistency of the theory in calculations. I criticize the theory on the basis of overlooking the vertex corrections in the strong coupling calculations of the transition temperature Tc. I calculate the first vertex correction for an optimally doped system and show that it is of the same order of magnitude as the bare vertex. Migdal's theorem is therefore not valid and Eliashberg formalism is not applicable to this situation. The same conclusion is obtained even after inclusion of the quasiparticle residue Z to the calculation. The sign of the vertex correction is then considered. I show that the positive sign of the vertex correction for the optimally doped system requires a phase transition of some sort as the doping is decreased. Part II of the thesis is devoted to the vortex lattice properties of high Tc superconductors. I establish a method to study vortex lattice properties of d-wave superconductors based on a generalization of the London model. The method has the advantage of simplicity as well as having very few free parameters (one at most) compared to other methods. The generalized London free energy is obtained from an s-d mixing Ginsburg-Landau free energy and also from the microscopic theory of Gorkov. The generalized London equation is found to be analytic at high temperatures. At very low temperatures however nonanalyticities arise as a result of the nodes on the superconducting gap. I then present the results of our calculations of some measurable quantities, such as the vortex lattice geometry and the effective penetration depth (as denned in μSR experiments). Comparison between our results and different experimental data is then performed. Especially our prediction for the magnetic field dependence of the effective penetration depth at T = 0, which is recently observed in //SR experiments with excellent agreement, is discussed.
This thesis is organized in two independent parts, in which I study two different aspects of high Tc superconductivity. The first part begins with an introduction aimed to briefly introduce some relevant experimental and theoretical works performed in recent years, that have helped us to think about cuprates the way we do now. Afterwards, I introduce Landau Fermi liquid theory in a standard text book way. The question of validity of Fermi liquid theory in 2-d is then raised and investigated by searching for singularities in Landau's /-function. I show that the interaction function between two quasiparticles whose momenta approach each other near a curved point of the Fermi surface, contains a 1-d singularity not strong enough to change the Fermi liquid behavior. On the other hand, inflection points provide 2-d singularities that have to be taken seriously in Fermi liquid considerations. I then introduce nearly antiferromagnetic Fermi liquid theory (NAFL), which is a phenomenological theory proposed to describe high Tc systems. I mainly focus on the self-consistency of the theory in calculations. I criticize the theory on the basis of overlooking the vertex corrections in the strong coupling calculations of the transition temperature Tc. I calculate the first vertex correction for an optimally doped system and show that it is of the same order of magnitude as the bare vertex. Migdal's theorem is therefore not valid and Eliashberg formalism is not applicable to this situation. The same conclusion is obtained even after inclusion of the quasiparticle residue Z to the calculation. The sign of the vertex correction is then considered. I show that the positive sign of the vertex correction for the optimally doped system requires a phase transition of some sort as the doping is decreased. Part II of the thesis is devoted to the vortex lattice properties of high Tc superconductors. I establish a method to study vortex lattice properties of d-wave superconductors based on a generalization of the London model. The method has the advantage of simplicity as well as having very few free parameters (one at most) compared to other methods. The generalized London free energy is obtained from an s-d mixing Ginsburg-Landau free energy and also from the microscopic theory of Gorkov. The generalized London equation is found to be analytic at high temperatures. At very low temperatures however nonanalyticities arise as a result of the nodes on the superconducting gap. I then present the results of our calculations of some measurable quantities, such as the vortex lattice geometry and the effective penetration depth (as denned in μSR experiments). Comparison between our results and different experimental data is then performed. Especially our prediction for the magnetic field dependence of the effective penetration depth at T = 0, which is recently observed in //SR experiments with excellent agreement, is discussed.
"This thesis consists of three Chapters. In the first Chapter, based on a planar unconventional Fermi liquid model, we present several results on the optimally doped and overdoped cuprate superconductors. For the normal state, we have analytically derived a linear in temperature and energy scattering rate for the carriers. This scattering rate yields necessarily a linear in temperature resistivity and a linear in 1/energy optical conductivity. The linearity of the scattering rate requires that the interacting Fermi liquid has strong peaks in its density of states (van-Hove singularities in 2 dimensions) near the chemical potential fl· Our results are backed by self-consistent Baym-Kadanoff (BK) numerical calculations, which also show that the interactions tend to pin the van- Hove singularities close to fl· We show that the low energy dependence of the Millis-Monien- Pines susceptibility XMMP can have a fermionic origin. We obtain particularly high transition temperatures Tc from our BK-Eliashberg scheme by introducing an ansatz for the susceptibility of the carriers. We postulate that the latter is enhanced in an additive manner due to the weak antiferromagnetic order of the Cu02 planes. Thus we have obtained a dx2_y2 gap with Tc > 120°K for nearest neighbor hopping t = 250me V. In the second Chapter we propose a novel mechanism for giant corrections to the transport quantities, including positive giant magnetoresistance, due to the presence of paramagnons in a weakly disordered metal. At low temperature and for finite impurity spin scattering, for a certain value, predicted from our theory, of the material-dependent paramagnon interaction, the total conductivity becomes highly sensitive to the orbital effects of a finite magnetic field. This is attributed to certain microscopic processes, otherwise negligibly small, which can be enhanced by a resonance factor, emanating from· the spin-density channel. We obtain very good agreement between this theory and four (4) different positive giant magnetoresistance experiments, including very recent ones, while making specific material-dependent predictions. We emphasize that our theory broadens the 'conventional wisdom' viewpoint that weak disorder would only generate small corrections to transport quantities. In the third Chapter we examine the effects of interface roughness and/or planar impurity doping in a superlattice, under the assumption that a weak disorder description is adequate. We find that these two types of disorder are equivalent, and that they can be viewed as effective ""bulk"" disorder, with anisotropic diffusion coefficients. This study can offer quantitative insight to transport properties of multilayers and devices, which contain inadvertently structural disorder at the interfaces."
This thesis studies two modern mean field theories of the t-J model for high Tc superconductors, namely the Slave-Boson Mean Field Theory and the Spinon-Dopon Mixing Mean Field Theory, from the view point of the traditional Fermi Liquid language. This study mainly focuses on whether the two mean field theories are consistent with Fermi Liquid behaviour in the underdoped regime of the superconducting and pseudogap phases of the t-J model. Historically, experimental results indicate that the superconducting phase can be well described by the conventional Fermi liquid-like quasiparticle picture; while the pseudogap phase exhibits non-trivial non-Fermi Liquid properties. This work aims at interpreting the pseudogap phase in a generalized Fermi Liquid framework and explaining why the simple quasiparticle picture works so well in the superconducting phase even in this strongly correlated superconducting system. Within the Gaussian approximation, we formally derived the Landau interaction in Fermi liquid theory, which serves as the key to classify the behaviour of the quasiparticles in different phases. It is found that the Slave-Boson Mean Field Theory describes the superconducting to the pseudogap phase transition as a transition with an interesting Landau parameter F1s changing from 1+F1s~x to 1+F1s=0, reflecting the occurrence of a quasiparticle instability. A new phase is proposed and its experimental consequences are discussed. On the other hand, we also find that the Spinon-Dopon Mixing Mean Field Theory can be understood in a conventional Fermi Liquid framework; even though quasiparticles in the theory carry larger charges. The issue of superfluid density is also clarified.
The U(1) slave-boson mean-field theory of the t - J model has been used by many authors as a starting point for the theoretical analysis of high - Tc cuprate superconductors. With suitable refinements, the theory can explain a lot of qualitative features of cuprates. However, the theory does not produce a correct description of the low-energy quasi-particle properties in the under-doped regime. There exist several possible schemes to resolve this difficulty, among them, one generalizes the U(1) theory to SU(2) formulation and the other introduces confinement between spinons and holons. In this thesis, we generalize the Landau transport equations of the U(1) slave-boson mean-field theory of the t - J model to the SU(2) formulation. We first generalize the SU(2) slave-boson mean-field theory to the time-dependent regime. After eliminating the boson and constraint fields exactly in the linear-response regime at zero-temperature, we obtain a set of transport equations for quasi-particles, which has a similar structure as the Landau transport equations for Fermi-liquid superconductors, with all the Landau parameters explicitly given except that besides the usual effective interaction in the particle-hole channel in Fermi-liquid theory, we find an effective interaction in the particle-particle channel, which reflects the SU(2) symmetry between particles and holes. The transport equation is SU(2) invariant at half-filling. Around q⃗ = 0, the electron density couples only to the transverse part of the external field and the quasi-particle charge is still of order x. Besides the SU(2) formulation approach, we also investigate spinon-holon confinement in the U(1) slave-boson theory. The optical and thermal conduc-tivities of U (1) slave-boson theory with confinement are calculated. For optical conductivity, we consider only the contributions from static impurities. The qualitative behavior of our results agrees with experimental results. However, to explain the experiments quantitatively, we need to study further the con-tributions from phonons. We found a phenomenological effective Hamiltonian for the quasi-particle Green's functions in the U(1) slave-boson theory with spinon-hoion confinement. Starting from this effective Hamiltonian, we exam-ine the thermal conductivity within the theory. In particular, we compare the thermal conductivity for four different situations: (i) the ordinary d-wave su-perconducting state with four Dirac Fermi points; (ii) the normal metal state with a half-pocket Fermi surface; (iii) the spinon-holon confinement supercon-ducting state with a half-pocket Fermi surface; (iv) BCS superconducting state with a half-pocket Fermi surface.
New insights into the strongly correlated Fermi liquid state of the SU(N) infinite-U Hubbard and t-J models are presented in connection to the metallic state of unbroken symmetry of the copper oxide planes. The nature of single particle and collective excitations is analyzed and the Landau Fermi Liquid parameters are derived. The one-particle spectral function in the Hubbard valence band is obtained with both Hubbard and Gutzwiller-like features. A simple interpretation of the large incoherent background is given in terms of a superposition of quasiparticles and a collective mode describing the propagation of a bare hole in a large U system. The spin exchange interactions produce very low lying collective modes describing the propagation of staggered spin chirality which soften at incommensurate wavevectors. The quasiparticles are strongly scattered by the low lying chirality fluctuations resulting in deviations from canonical Fermi liquid behavior at low energy scales. Of the same origin, the quasiparticle self energy is found to be strongly k-dependent. It leads to corrections to the dispersion and enhancement of the quasiparticle lifetime in the hole-like region of the interacting Fermi surface. The combined effects are sufficient to produce a sign change in the Hall coefficient evaluated for parameters relevant to the copper oxide planes. We show that these models also exhibit a zero temperature fluctuation driven transition from Fermi liquid to non Fermi liquid phase without the onset of magnetic ordering below a critical doping δc=1/4N, whereby providing a microscopic mechanism for the destruction of the Fermi liquid coherence by strong longitudinal gauge fluctuations which represent the effects of the infinite-U constraint.
The topic of this thesis is the functional renormalization group. We discuss some approximations schemes. Thereafter we apply these approximations to study different fields of condensed matter physics. Generally we have to evaluate an infinite set of vertex functions describing the scattering of particles. These vertex functions get renormalized away from their bare values governed by an infinite hierarchy of flow equations. We cannot expect to actually solve these equations but have to apply a couple of approximations. The aim is to somehow separate relevant contributions from irrelevant ones. One possible scheme opens up if we rescale fields and vertices. Here "relevance" is used in a quantitative way to describe the scaling behaviour of vertices close to a fixed point of the RG. One disadvantage of describing the system in terms of infinitely many vertices is that the majority of these vertices we have to evaluate are not of interest to us. In most cases we are just looking for the self-energy or the two-particle effective interaction. However there might be contributions to the flow of these vertices that are generated by irrelevant vertices. We generally assume that we can express irrelevant vertices in terms of the relevant and marginal ones. Then in turn it should be possible to write the contributions of these irrelevant vertices to the flow of relevant and marginal ones in terms of relevant and marginal vertices as well. We show how this can be achieved by what we term the adiabatic approximation. We now consider weakly interacting bosons at the critical point of Bose-Einstein condensation. As the transition takes place at a finite temperature this temperature defines an effective ultraviolet cut-off. For the investigation of physical properties that depend on momenta smaller than this cut-off it is therefore sufficient to describe the system by a classical field theory. Our central topic here is the self-energy of the bosons and we are able to evaluate it with the full momentum dependence. For small momenta it approaches a scaling form and as the momentum is gradually increased we observe a crossover to the perturbative regime. As a test for the reliability of our expression for the selfenergy we investigate the interaction induced shift of the critical. Our results compare quite satisfactory to the best available estimates for this shift. For the anomalous dimension our approach predicts the correct order of magnitude however with a considerable error. As an improvement we include more vertices into our calculations. Here we observe that our fixed point estimates indeed approach the best known results but this convergence is quite weak. We turn toward systems of interacting fermions. The formulation of the functional renormalization group implicitly requires knowledge of the true Fermi surface of the full interacting system. In general however we can just calculate it a-posteriori from the self-energy. The requirement to flow into a fixed point can be translated into a fine-tuning of the frequency/momentum independent part r_0 of the rescaled 2-point function. We show how this bare value is related to the momentum dependent effective interaction along the complete trajectory of the RG. On the other hand r_0 expresses the difference between the bare and the true Fermi surface. Putting both equations together results into an exact selfconsistency equation for the Fermi surface. We apply our self-consistency equation above to tackle the problem of finding the true Fermi surface of interacting fermions in low dimensions. The most simple non-trivial model with an inhomogeneous Fermi surface is a system of two coupled metallic chains. The process of interband backward scattering leads to a smoothing of the Fermi surface. Of special interest is if the Fermi momenta of the two bands collapse into just one value. We propose the term confinement transition for this behaviour. We bosonize the interband backward scattering by means of a Hubbard-Stratonovich transformation and treat our system as a single channel problem. This bosonization together with the adiabatic approximation allows us to investigate the system even at strong coupling. Within a simple one-loop treatment our method predicts a confinement transition at strong coupling. However taken vertex renormalizations into account we observe that this confinement is destroyed by fluctuations beyond one-loop. Actually we observe how the confined phase can be stabilized by the inclusion of interband umklapp scattering. Thereafter we consider the physically more relevant case of a two-dimensional system of infinitely many coupled metallic chains. Here the Fermi surface consists of two disconnected weakly curved sheets. We are able to repeat the calculations we have performed for our toy model. Within a self-consistent 2-loop calculation indeed signs for a confinement transition at finite coupling strength emerge.
Strongly correlated systems show a variety of intriguing properties and have been extensively studied over the years. In this thesis two strongly correlated systems are discussed: the two-dimensional antiferromagnet and the Luttinger liquid. The Luttinger liquid has attracted much interest as a paradigm of non-Fermi liquid behavior. In the first part of the thesis, basic properties of Fermi liquids are briefly described while those of Luttinger liquids are considered in more detail. The differences in spectral properties of Luttinger liquids and Fermi liquids are discussed, as are the consequences for photoemission experiments. In this context Luttinger liquids with boundaries, treated in Papers I, II, and III, are introduced. The exact single-electron Green's function at finite temperature is calculated with bosonization methods, for both periodic and finite open-boundary systems. The corresponding local spectral density is analyzed. For a system with open boundaries, the well-known zero temperature bulk behavior always crosses over to a boundary dominated regime for small energies, while at finite temperature the low energy behavior becomes quadratic. For a periodic finite-size system we also calculate the Coulomb-blockade-like oscillations and study their dependence on interaction and temperature. The two-dimensional antiferromagnet has attracted much attention due to its connection to the undoped phase of the ceramic high-temperature superconductors. In the second part of the thesis including Papers IV and V, the two-dimensional Heisenberg antiferromagnet defined on the triangular and the honeycomb lattices is studied. We use the Schwinger-boson mean-field theory to investigate the stability of the ground state with respect to frustration. For the honeycomb lattice model we find an increased Néel stability as compared to the classical case. The dynamic structure factor is considered for various temperatures and the spin-wave velocity and the susceptibility are presented as functions of frustration for spin S=1/2. In the case of the intrinsically frustrated triangular lattice model we obtain, in the limit of infinite spin, results in complete agreement with the linear spin-wave theory. In contrast to the latter, however, two of the three modes acquire a mass for finite spin. The origin of this effect is discussed.