The Hubbard model in the strong coupling limit is considered in two dimensions. A bosonization method based on the extention of the Jordan-Wigner transformation to particles with spin and no double occupancy is introduced. We find that at T=0 a Fermi liquid exists and superconductivity appears. For a finite temperature and finite doping a normal state populated with vortices exists. As a result, we have a normal state with separated charge- and spin-density-wave excitations. Formally those excitations can be described with the help of the Luttinger-liquid representation. In the second part we parametrize the Fermi surface by a set of N points. Around each point we describe the Fermi surface excitations by holons and spinons and find a set of N-coupled Luttinger channels. At T=0 we have space-time tunneling of electric charges which destroys the Luttinger liquid in favor of a Fermi liquid. At T\ensuremath{\ne}0 or LT\ensuremath{\rightarrow}\ensuremath{\infty} (L is the size of the system) tunneling of electric charges might be prohibited giving rise to gapless excitations. In the last part we consider the case with a commensurate frozen flux in the ground state. As a result we find that the excitations are semions.
The Hubbard model in the strong coupling limit is considered in two dimensions. A bosonization method based on the extention of the Jordan-Wigner transformation to particles with spin and no double occupancy is introduced. We find that at T=0 a Fermi liquid exists and superconductivity appears. For a finite temperature and finite doping a normal state populated with vortices exists. As a result, we have a normal state with separated charge- and spin-density-wave excitations. Formally those excitations can be described with the help of the Luttinger-liquid representation. In the second part we parametrize the Fermi surface by a set of N points. Around each point we describe the Fermi surface excitations by holons and spinons and find a set of N-coupled Luttinger channels. At T=0 we have space-time tunneling of electric charges which destroys the Luttinger liquid in favor of a Fermi liquid. At T\ensuremath{\ne}0 or LT\ensuremath{\rightarrow}\ensuremath{\infty} (L is the size of the system) tunneling of electric charges might be prohibited giving rise to gapless excitations. In the last part we consider the case with a commensurate frozen flux in the ground state. As a result we find that the excitations are semions.
Using the space-time curve representation, a bosonization scheme is constructed which includes density excitations around each point on the Fermi surface (FS) and space-time tunneling of charges which belong to different FS points. The space-time tunneling reproduces the two-dimensional Jordan-Wigner transformation. In two dimensions, since the tunneling involves two electrons, the total Jordan-Wigner phase maps boson into boson. Therefore, the tunneling corresponds to instantons in 2e bosonic systems. Using this formalism, a spinless fermion model with short-range repulsive interactions is investigated. We find two components: sound-wave excitations in the normal direction of the FS reproduce the Fermi liquid (FL); the second component corresponds to the cooperon channel, violates charge conservation, and gives rise to two-particle coherent tunneling. As a result, the FL phase is destroyed at temperature T=0, and a tunneling gap for charge 2e is formed with angular momentum l\ensuremath{\ne}0. We apply our formalism to the U(1) gauge field coupled to a spinless fermion and find that at T\ensuremath{\ne}0 the FL is again destroyed. We also make some remarks concerning the t-J model.
This thesis deals with different aspects of high temperature superconductivity in hole-doped cuprates. We assume the t - J model to describe the CuO2 planes of cuprates and we use a spin-charge gauge approach with spin-charge separation to describe holes in terms of a spinless fermion carrying the charge (holon) and a neutral boson carrying spin 1/2 (spinon), coupled by a slave-particle gauge field. In this framework we consider the effects of the presence of a finite density of incoherent holon pairs in the normal state as a precursor of superconductivity. We show that it is possible to take such pairs into account through a strongly direction dependent spectral weight for holons, which suppresses quasi-particles starting from the anti-nodal directions, and a wave function renormalization for the slave particle gauge field. In this way we prove that, when temperature is reduced, the formation of holon pairs causes both the deviation from linearity of the resistivity and the deviation from the constant value of the Knight shift. Moreover we point out the need for a negative next-nearest-neighbor hopping term ((t - t' - J model) to get a good continuum limit and to evaluate the Knight shift. These results are obtained through a Green's function, which follows naturally from the formalism and analytically interpolates between a Fermi liquid-like behavior and a d-wave superconductor when the coherence length of the holon pair order parameter is increased. The system preserves a finite Fermi surface until the superconducting transition, where it reduces to four nodes. Finally when, at lower temperatures, both the holon pairs and the RVB spinon pairs condense we enter the superconducting phase. In this phase we study the magnetic excitations showing how to extend to spin waves the hourglass-shape dispersion, found for spinons near the antiferromagnetic vector, if a suitable local mechanism of attraction between spinon and anti-spinon is assumed. The resulting spin wave Green's function, whose imaginary part is directly comparable with experiments, generates the hourglass with a finite gap between the two branches. Since the U(1) slave-particle gauge field gains mass in the superconducting phase via Anderson-Higgs mechanism, we propose the necessary mechanism of attraction comes from the unbroken Z2 subgroup of the U(1) gauge group.
We discuss certain physical contents of the two-dimensional Luttinger liquid state using the spin and charge current backflow in strongly correlated electron systems. Also we attempt a description of the Luttinger liquid state in the slave boson formalism. Conventional slave boson mean field theory does not possess an important property expected of a Luttinger liquid, namely a pseudo fermi surface for holons. This arises from the bose condensation of holes. We attempt to rectify this by discussing the physics of the fermionic backflow associated with the hole motion. We argue that the fermionic backflow suppresses the Bose condensation. To understand this we introduce the notion of ‘Statistics Transfer’-the medium transfer certain statistics to the slave particle (e's). A variational wave function in which boson condensation is manifestly absent is introduced. The form of the wave function is justified by the concept of statistics transfer. The spinon and holon excitations both exhibiting pseudo fermi surfaces are constructed.
A fermion-spin transformation is used to implement the charge-spin separation, and is developed to study the low-dimensional t-J model. In this approach, the charge and spin degrees of freedom of the physical electron are separated, and the charge degree of freedom is represented by a spinless fermion while the spin degree of freedom is representd by a hard-core boson. The on-site local constraint for single occupancy is satisfied even in the mean-field approximation and the sum rule for the physical electron is obeyed. This approach can be applied to both one and two-dimensional systems. In the one-dimensional case, the spinon as well as the physical electron behave like Luttinger liquids. We have obtained a gapless charge and spin excitation spectrum, a good ground state energy, and a reasonable electron-momentum distribution within the mean-field approximation. The correct exponents of the correlation functions and momentum distribution are also obtained if the squeezing effect and rearrangement of the spin configurations are taken into account. In the two-dimensional case, within the mean-field approximation the magnetized flux state with a gap in the spinon spectrum has the lowest energy at half filling. The antiferromagnetic long-range order is destroyed by hole doping of the order \ensuremath{\sim}10-15 % for t/J=3-5 and a disordered flux state with gapless spinon spectrum becomes stable. The calculated specific heat is roughly consistent with observed results on copper oxide superconductors. The possible phase separation is also discussed at the mean-field level.
The t-J model in two dimensions is bosonized using a set of N, coupled two-dimensional Fermi-surface patches. Ignoring tunneling between the patches, the coherent tunneling of holes and the superfluid phase are suppressed. Within this scheme the system remains in the normal phase when temperature T\ensuremath{\rightarrow}0. The main feature of this construction is the absence of screening of the dissipative transversal gauge field generated by the spinons. This dissipative gauge field is responsible for the non-Fermi-liquid behavior, which is manifested in the free energy and single-particle Green function. The deviation from Fermi-liquid behavior is due to the U(1) gauge field, and at long distances a new exponent due to the holes is identified. Experimental consequences are discussed. \textcopyright{} 1996 The American Physical Society.
The Fermi liquid theory is a central concept in modern condensed matter physics used to describe conventional metals. A state of this universality class consists of well-defined quasiparticles, which occupy a finite number of states within a Fermi volume defined in momentum space. According to the Luttinger’s theorem, such a state encloses a Fermi volume proportional to the electron density modulo the filled bands if no symmetry is broken. In the past decades strong deviations from Fermi liquid theory have been observed in the scaling behavior of thermodynamic observables in different materials, among which are for instance cuprates and iron pnictides. This thesis is about two prototypical systems that can not be described by Fermi liquid theory. The first part of this work investigates two-dimensional effective models, which are in a Mott insulating state at half-filling and have a finite conductivity when doped with holes. In such a Mott insulating state, the charge carriers are strongly localized due to the Coulomb repulsion. Hence, strong correlations are assumed to have a strong impact on the formation of the ground state, also in the regime of small hole concentration. In a first project we have used a so-called spinon dopon mean-field theory to represent the two-dimensional Fermi-Hubbard model with strong on-site repulsive interaction in effective degrees of freedom, in which the holes can be embedded into a quantum spin liquid. The corresponding SU(2) invariant ground state belongs to the class of fractionalized Fermi liquids. In a second project we investigate a quantum dimer model, an effective model based on a Hilbert space spanned by short range singlets and bound states of holes and spins. The focus here is on the calculation of the hole-part of the electron spectral function by using exact diagonalization and its comparison with two analytic methods, a diagrammatic computation based on the Bethe-Salpeter equation and a so-called two-mode approximation. The electron spectral function shows a similar analytic form in momentum space between nodal and antinodal point when compared to results from photoemission spectroscopy experiments on cuprates. Furthermore, in a subsequent work we calculate the exact ground state of the quantum dimer model along a certain parameter line. In order to analyze the behavior of the elctron spectral function when increasing the density of holes, we investigate the Fermi-Hubbard model in a current project using a dynamical mean-field approach. To solve the 4-site cluster impurity problem, we use a numerical renormalization group approach. However, numerical limitations force us to restrict the analysis to spin-polarized baths. According to the spectral data, the system is similar to the SU(2) invariant case at half-filling in a Mott insulating state and posseses a momentum-selective energy gap at finite doping. The topology of the Fermi surface shows a Lifshitz transition when increasing the hole concentration. Here, the curvature of the Fermi surface changes from electron- to hole-like. For comparison we apply the dynamical mean-field theory also to the quantum dimer model and observe that the electron spectral functions at finite doping are qualitatively similar to that of the two-dimensional Fermi-Hubbard model. The second half of the work is about Tomonaga-Luttinger liquid theory, which is used to describe the low-energy effective degrees of freedom of one-dimensional systems. Here, we first discuss a conceptional extension of the operator-based bosonization theory for one-dimensional systems. This extension is especially suited for inhomogeneous one-dimensional systems. First, we investigate a one-dimensional system with a local interation potential and compute an exact solution of the single particle propagator at T = 0. The critical exponent of the single particle propagator has an unconventional form as a function of the microscopic Tomonaga-Luttinger parameters, which is not covered by the original Luttinger paradigm postulated by F. Duncan M. Haldane. In a second project on one-dimensional systems, we study the impact of scattering processes among bosonic low-energy excitations on the thermalization process. Such scattering processes are irrelevant on large length scales, however strongly affect the dynamics. In our analytic analysis we focus on a experimental setup, where a one-dimensional Bose gas is instantaneously splitted in two identical, however strongly correlated, halves of one-dimensional electronic systems. In the following, the corresponding non-equilibrium state runs through multiple regimes in time, such as a metastable prethermalization regime. However, above a certain threshold in time such scattering processes cause an effective thermalization of the system. In order to demonstrate this, we compute the kinetic equation in the Keldysh field integral formalism from a diagrammatic expansion based on a self-consistent Born approximation.