Large N - slave fermion techniques have been established as a powerful computational tool to study quantum many-body systems. They have been applied successfully to the study of various systems of strongly correlated electrons in condensed matter physics, particularly to problems where a non-perturbative treatment is required. In the first part of this thesis we use this approach to investigate the Frustrated Quantum Antiferromagnet. We study the Chiral Spin Liquid state of the antiferromagnet at zerotemperature, which has a spin-gap generated by the spontaneous breakdown of time-reversalsymmetry. This state is known as the Chiral Spin State. We determine conditions for the restoration of time-reversal-symmetry in a bilayer antiferromagnet with an interlayer exchange interaction. We show the energetically favored ground state has Chiral Spin Liquids with opposite chiralities on each layer. This constitutes a dynamic mechanism which prevents the observation of broken time-reversal-symmetry in a bilayer system, even though this breakdown may be present on each layer independently. In the second part of this thesis we use the slave-fermion technique to study the physics of magnetic impurities both in Quantum Antiferromagnets (in a flux phase) and in d-wave superconductors. This is a system consisting of a magnetic impurity coupled to either the low lying excitations of a flux phase, or to the normal quasi-particles close to the Fermi surface of a superconductor whose gap function exhibits nodes. We show that both systems display a zero temperature quantum phase transition in the large N approximation. The transition takes place between a weak coupling regime, in which the magnetic impurity is effectively decoupled from the quasi-particles, to a strong coupling phase which exhibits a Ill Kondo effect and the magnetic impurities are screened. The ground state is a singlet. If the system has particle-hole symmetry the impurity appears to be overscreened. This effect is not due to a multichannel character in the system. The physical origin of the quantum phase transition is directly related to the fact that this is a non-marginal Kondo system. This follows from having a quasi-particle band with a density of states that vanishes linearly with the energy in the proximity of the Fermi surface. We show that, due to the presence of nodes in the gap function of a d-wave superconductor, there is a striking analogy between the problem of a Kondo impurity coupled to a flux phase and a magnetic impurity coupled to the superconductor. We consider a clean dx2-y2 superconductor in the presence of a fully quantum mechanical magnetic impurity. It is assumed that the temperature is sufficiently low so as to neglect both phase and amplitude fluctuations of the order parameter. We study the zero temperature behavior and the low temperature and low magnetic field regimes in the vicinity of the quantum phase transition. In the strong coupling phase, we obtain scaling behavior for the chemical potential of the impurity f.J and for the amplitude of the ground state singlet ~- At zero temperature we show that they present different scaling behavior with the distance to critical coupling: EJ scales linearly while ~ scales with a vanishing exponent. Both magnitudes present logarithmic corrections to scaling suggesting that the theory may be at an upper critical dimension. At finite field (zero temperature) and/or at finite temperature (zero field), at the critical coupling the magnetic field or the temperature respectively, play a similar role as the distance to the critical coupling in the scaling formulas. The susceptibility and the specific heat are investigated both at zero temperature, zero field limit and at low temperature, low field regimes. The results are consistent with the existence of perfect screening and a singlet ground state in the strong coupling phase. When particle-hole is an exact symmetry of the system, the impurity appears to be overscreened with a vanishing paramagnetic susceptibility.
Large N - slave fermion techniques have been established as a powerful computational tool to study quantum many-body systems. They have been applied successfully to the study of various systems of strongly correlated electrons in condensed matter physics, particularly to problems where a non-perturbative treatment is required. In the first part of this thesis we use this approach to investigate the Frustrated Quantum Antiferromagnet. We study the Chiral Spin Liquid state of the antiferromagnet at zerotemperature, which has a spin-gap generated by the spontaneous breakdown of time-reversalsymmetry. This state is known as the Chiral Spin State. We determine conditions for the restoration of time-reversal-symmetry in a bilayer antiferromagnet with an interlayer exchange interaction. We show the energetically favored ground state has Chiral Spin Liquids with opposite chiralities on each layer. This constitutes a dynamic mechanism which prevents the observation of broken time-reversal-symmetry in a bilayer system, even though this breakdown may be present on each layer independently. In the second part of this thesis we use the slave-fermion technique to study the physics of magnetic impurities both in Quantum Antiferromagnets (in a flux phase) and in d-wave superconductors. This is a system consisting of a magnetic impurity coupled to either the low lying excitations of a flux phase, or to the normal quasi-particles close to the Fermi surface of a superconductor whose gap function exhibits nodes. We show that both systems display a zero temperature quantum phase transition in the large N approximation. The transition takes place between a weak coupling regime, in which the magnetic impurity is effectively decoupled from the quasi-particles, to a strong coupling phase which exhibits a Ill Kondo effect and the magnetic impurities are screened. The ground state is a singlet. If the system has particle-hole symmetry the impurity appears to be overscreened. This effect is not due to a multichannel character in the system. The physical origin of the quantum phase transition is directly related to the fact that this is a non-marginal Kondo system. This follows from having a quasi-particle band with a density of states that vanishes linearly with the energy in the proximity of the Fermi surface. We show that, due to the presence of nodes in the gap function of a d-wave superconductor, there is a striking analogy between the problem of a Kondo impurity coupled to a flux phase and a magnetic impurity coupled to the superconductor. We consider a clean dx2-y2 superconductor in the presence of a fully quantum mechanical magnetic impurity. It is assumed that the temperature is sufficiently low so as to neglect both phase and amplitude fluctuations of the order parameter. We study the zero temperature behavior and the low temperature and low magnetic field regimes in the vicinity of the quantum phase transition. In the strong coupling phase, we obtain scaling behavior for the chemical potential of the impurity f.J and for the amplitude of the ground state singlet ~- At zero temperature we show that they present different scaling behavior with the distance to critical coupling: EJ scales linearly while ~ scales with a vanishing exponent. Both magnitudes present logarithmic corrections to scaling suggesting that the theory may be at an upper critical dimension. At finite field (zero temperature) and/or at finite temperature (zero field), at the critical coupling the magnetic field or the temperature respectively, play a similar role as the distance to the critical coupling in the scaling formulas. The susceptibility and the specific heat are investigated both at zero temperature, zero field limit and at low temperature, low field regimes. The results are consistent with the existence of perfect screening and a singlet ground state in the strong coupling phase. When particle-hole is an exact symmetry of the system, the impurity appears to be overscreened with a vanishing paramagnetic susceptibility.
Strongly correlated electron systems show a rich variety of interesting physical phenomena. However, their theoretical description poses a highly non-trivial task, and even the simplest models cannot be solved exactly. In this thesis the intricate behavior of the single impurity Anderson model (SIAM) and the Hubbard model is investigated. The main emphasis is laid on two-particle Green functions of these systems. To this end, existing approximation schemes are extended to be capable of treating susceptibilities as well. The quality and range of applicability of these approximations is discussed in detail. The physical pictures behind the phenomena encountered in these systems are introduced. A prominent many-body effect is the emergence of temperature dependent low energy quasiparticles in both models. The occurrence of such excitations is a result of the dynamical screening of local magnetic moments, a characteristic feature of the Kondo effect. It is argued, that within the presented approximations, the low temperature phase of these models can usually be described as a Fermi liquid. As a counterexample, the Hubbard model is shown to exhibit pronounced non-Fermi liquid behavior in situations, where a van Hove singularity in the non-interacting density of states is found in the vicinity of the Fermi level. The magnetic properties of the Hubbard model are discussed in view of the two archetypical pictures of magnetism: local moment magnetism and itinerant Stoner-magnetism. Critical temperatures for antiferromagnetic, ferromagnetic and incommensurate phase transitions are calculated, and the influence of the lattice structure and frustration is examined. In strongly correlated metallic situations, the complicated competition between Kondo-screening and local moment magnetism can lead to a reentrant behavior of the N\'eel temperature. A direct signature of the Kondo effect in the Hubbard model is found in the dynamic magnetic susceptibility, where a localized collective mode emerges at very low temperatures. It is connected to the breakup of a singlet formed by the local moments with their Kondo-screening cloud, and consequently the excitation energy is of the order of the low energy scale T*.
Materials with strongly correlated electrons show a multitude of unconventional phenomena, such as superconductivity or magnetism, that cannot be explained in a single-particle picture and necessitate to take many-body effects into account. A route to explain the origin of these phenomena consists in constructing effective models for these materials that are then solved, often using numerical techniques. Paradigmatic models for two different classes of strongly correlated electron systems are investigated in this thesis: The Hubbard model is a paradigmatic model to describe the Mott metal-insulator transition (MIT) and the Kondo lattice model describes the low-energy physics in heavy fermion compounds. Both models are analyzed numerically using different variations of a quantum cluster technique, the variational cluster approximation (VCA). The first part focuses on the MIT, as it is investigated experimentally in quasi-two-dimensional charge-transfer salts. Its nature and associated universality class are still heavily discussed and both experiments and theoretical approaches come to different conclusions. In this thesis, the Mott transition is investigated on an anisotropic two-dimensional Hubbard model at half-filling. By using a control parameter, which induces this anisotropy, strong evidence for Mott quantum criticality is found in weakly coupled Hubbard chains. The results at zero temperature show that the second-order critical end point Tc of the interaction-driven metal-insulator transition can be tuned down to zero at strong anisotropy. Further results for the antiferromagnetic phase suggest a similar picture and motivate adding the anisotropy as a new axis in the phase diagram to account for a low-temperature critical end point of the Mott transition. The second part of this thesis focuses on unconventional phases that emerge in a paradigmatic model for heavy fermion systems, the Kondo lattice model. Using the VCA, the two-dimensional Kondo lattice model is investigated in the paramagnetic phase as well as in phases with broken symmetry. An antiferromagnetic phase at weak coupling and a phase with Kondo-singlet formation at strong coupling are analyzed. Within the antiferromagnetically ordered region, two phases with different Fermi-surface topology, which are separated by a discontinuous transition, are identified. The model is also tested for s-wave superconductivity as found recently using dynamical mean-field theory, but no indications for robust local pairing are seen within VCA. Instead, nodal d-wave superconductivity is found and analyzed in a large range of couplings and electron fillings, and its interplay with antiferromagnetic order at weak coupling is discussed. This motivates further studies on extended models that might allow for a close comparison with experiments on heavy fermion systems. Understanding the formation of unconventional phases in strongly correlated electron systems is not only of fundamental interest, but it is also essential for designing new functional materials. Although the results of approximate cluster techniques have to be handled with care, the new aspects presented in this thesis pave the way for further investigations in this direction.
In this thesis, several cases of non-equilibrium phenomena and quantum phase transitions in strongly correlated electron systems are analyzed. The unconventional critical behavior near magnetic quantum phase transitions in various heavy-fermion metals has triggered proposals on the breakdown of the Kondo effect at the critical point. In part I, we investigate, within one specific scenario, the fate of such a zero-temperature transition upon coupling of the electronic to lattice degrees of freedom. We study a Kondo-Heisenberg model with volume-dependent Kondo coupling � this model displays both Kondo volume collapse and Kondo-breakdown transitions. Within a large-N treatment, we find that the Kondo breakdown transition remains of second order except for very soft lattices. Finally, we relate our findings to current heavy-fermion experiments. Using non-equilibrium Green�s functions, we derive transport equations for the degrees of freedom participating in the quantum critical region of the Kondo breakdown transition. We discuss conditions under which the transport of electrical charge is described by the independent motion of conduction electrons and auxiliary bosons. Under these conditions, we derive a semiclassical transport equation for the bosons and quantitatively discuss the electrical conductivity of the whole system. Motivated by pressure experiments on FeAs-122 superconductors, in part II we propose a scenario based on local-moment physics to explain salient features at the magnetic phase boundary of CaFe2As2. In this scenario, the low-pressure magnetic phase derives from Fe moments, which become screened in the paramagnetic high-pressure phase. The quantum phase transition can be described as an orbital-selective Mott transition, which is rendered first order by coupling to the lattice. These ideas are illustrated by a suitable mean-field analysis of an Anderson lattice model. An analytical description of non-equilibrium phenomena in interacting quantum systems is rarely possible. In part III we present one example where such a description can be achieved, namely the ferromagnetic Kondo model. In equilibrium, this model is tractable via perturbative renormalization-group techniques. We employ a recently developed extension of the flow-equation method to calculate the non-equilibrium decay of the local magnetization at zero temperature. The flow equations admit analytical solutions which become exact at short and long times, in the latter case revealing that the system always retains a memory of its initial state. Finally, in part IV we analyze the Nernst effect resulting from normal state quasiparticles in the cuprates in presence of various types of translational symmetry breaking. In the electron-doped cuprates, the Nernst signal resulting from a reconstruction of the Fermi surface due to spin density wave order is discussed. An order parameter consistent with the reconstruction of the Fermi surface detected in electron-doped materials is shown to sharply enhance the Nernst signal close to optimal doping. Within a semiclassical treatment, the obtained magnitude and position of the enhanced Nernst signal agrees with Nernst measurements in electron-doped cuprates. In the hole-doped cuprates, we discuss relations between the normal-state Nernst effect and stripe order. We find that Fermi pockets caused by translational symmetry breaking lead to a strongly enhanced Nernst signal with a sign depending on the modulation period of the ordered state and other details of the Fermi surface. This implies differences between antiferromagnetic and charge-only stripes. We compare our findings with recent data from La1.6−xNd0.4SrxCuO4 and YBa2Cu3Oy.
"Strong electron correlation phenomena are ubiquitous, such as unconventional superconductivity and the metal-insulator transitions. Having a proper understanding of strongly correlated electron systems requires a non-perturbative analysis because Fermi liquid theory inevitably breaks down due to the strong correlation. In this thesis, we will study two non-perturbative approaches to tackle strongly correlated electronic system: 1.) multi-dimensional bosonization and 2.) the AdS/CFT correspondence. Multidimensional bosonization is the generalization of bosonization to spatial dimensions larger than one. The bosonized theory is an effective field theory in terms of the bosonic particle-hole fluctuations in different patches of the Fermi surface. The bosonized theory is quadratic and hence can be solved non-perturbatively. The AdS/CFT correspondence is a conjecture that d-dimensional field theory is dual to a d+1 dimensional quantum gravitational theory. Problems in strongly coupled field theory then have an equivalent description using weakly interacting gravitational theory, allowing a non-perturbative analysis. Chapter 1 will serve as an introduction. We will review several experiments which show a breakdown of Fermi liquid theory. Formalisms for multidimensional bosonization and the AdS/CFT correspondence will then be reviewed. In chapter 2, the two-orbital Hubbard model with degenerate d{xz} and d{yz} orbitals are investigated. We apply multidimensional bosonization to solve this problem exactly and discover a z=3 overdamped collective modes that emerges at the orbital-ordering quantum critical point. These modes modify the single-particle density of states and lead to non-Fermi liquid behavior which can provide a possible explanation for the recently observed zero-bias enhancement in the point contact spectroscopy signal on iron pnictides. Chapters 3, 4 and 5 discuss the applications of the AdS/CFT correspondence to model various condensed matter systems. In chapter 3, We consider an interaction term between a bulk spinor field and a gauge field in the Reissner-Nordstr\""om AdS background. When the Pauli interaction term is large enough, a dynamical gap is generated and spectral weight transfer is observed in the spectral density, mimicking the behavior of the Hubbard model. We further consider the finite temperature case and discover that the ratio between the dynamical gap and the critical temperature has the same order as that of VO^2. The Pauli coupling is also studied in the superconducting background. Chapter 4 consider the propagation of a neutral scalar field in the geometry called the electron star. The electron star has Lifshitz scaling with finite dynamical critical exponent at the interior, which is suitable for the modeling of quantum criticality and quantum phase transition with a neutral order parameter, for example for antiferromagnetism. We find that the quantum phase transition has the Berezinski-Kosterlitz-Thouless characteristic and the dynamical critical exponent can change across the quantum critical point. In chapter 5, we construct a gravity dual of the nematic phase by studying the condensation of a spin-two field in the Schwarzschild-AdS background. The condensation of a spin-two field can distort the rotational symmetry of the Fermi surface as long as the spin-two field is coupled to a probe spinor field in the bulk gravity dual."
In this work we consider several exactly solvable models of magnetic impurities in critical quantum antiferromagnetic spin chains and multichannel Kondo impurities. Their ground-state properties are studied, and the finite set of nonlinear integral equations which exactly describe the thermodynamics of the models is constructed. We obtain several analytical low-energy expressions for the temperature, magnetic field, and frequency dependences of important characteristics of exactly solvable disordered quantum spin models and disordered multichannel Kondo impurities with essential many-body interactions. We show that the only low-energy parameter that gets renormalized is the velocity of the low-lying excitations (or the effective crossover scale connected with each impurity); the others appear to be universal. In our study several kinds of strong disorder important for experiments were used. Some of them produce low divergences in certain characteristics of our strongly disordered critical systems (compared with finite values for the homogeneous case or a single impurity). For weak disorder, or for narrow distributions of the local Kondo temperatures, our exact results reveal the presence of Kondo screening of disordered ensembles of magnetic impurities by low-lying excitations of the host. We point out that our results qualitatively coincide with the data of experiments on real disordered quasi-one-dimensional antiferromagnetic systems and with the similar behavior of some heavy metallic alloys.
Strongly correlated electron systems show a rich variety of astonishing physical phenomena. However, the strong interactions make the theoretical description of these systems a highly non-trivial task. Even the simplest models cannot be solved exactly, and one has to reside to approximative solutions, which often cover only certain aspects of the physics contained in these models. In recent years progress has been made in the theoretical description of correlated lattice systems through the mapping of lattice models onto effective impurity models. The most prominent example is the mapping of the Hubbard model onto an effective single impurity Anderson model (SIAM). Within this description both models feature the prominent Kondo effect, which leads to the emergence of low energy quasiparticles. The occurrence of such excitations is a result of the dynamical screening of local magnetic moments and the corresponding formation of a low temperature Fermi liquid phase. On the other hand these theories usually fail in the description of non Fermi liquid behavior, which is for example observed in the normal state of high temperature superconductors and some Heavy-Fermion systems. Strong nonlocal fluctuations drive the dynamics of these systems. In the theoretical treatment, due to the mapping of the lattice problem to an effective single impurity problem, these fluctuations are not adequately described.\\\\ In this thesis a new self-consistent approach for the inclusion of spatial correlations is presented for the Hubbard model. In contrast to existing quantum cluster theories in this field, the approach allows for the simultaneous description of short and long range pair correlations. Due to the intimate connection of this new lattice theory with the two impurity Anderson model (TIAM) a thorough numerical investigation of the TIAM is undertaken in this thesis. The competition of the Kondo effect with the RKKY interactions is studied in great detail. Furthermore two new solvers for the TIAM are introduced, which extend existing non crossing approximations for the SIAM.