Low dimensional magnetic models.-Recently several new materials have been discovered in which the localized magnetic moments are coupled in a way resembling ladders.The energy spectrum and the behaviour of these systems depends strongly on the value of the spin and the number of legs of the ladder.We studied the phase diagram of such ladder models using the density matrix renormalization group (DMRG).First in a rather technical work we have checked the accuracy of the DMRG procedure on the example of the Ising model in transverse field, as the simplest quantum mechanical spin problem, and have shown how the so-called truncation error is related to the real error in the energy values.With this background we have determined the phase diagram of a two leg ladder model including also plaquette couplings, and have found indication for the existence of several different massive phases.One-dimensional fermionic models.-We investigated in detail the model defined through the relativistic limit of the one dimensional attractive Hubbard model.This model possesses both massive and massless dressed particles which are described just like the particles in the Hubbard chain, by Bethe-Ansatz type equations.We analysed the eigenstates of the system.We have found that the limiting model has higher symmetry than the original Hubbard chain; it acquires in addition also the chiral symmetry.We determined the scattering properties of the particles, and also the conformal properties of the field corresponding to the massless sector.All these results confirm, that the scaling limit of the Hubbard chain can be identified with the chiral invariant Gross-Neveu model.Theory of dissipative motion of heavy particles.-We studied the low temperature behaviour of two level systems (TLS) interacting with the conduction electrons in amorphous metals.To approach the temperature region below the Kondo temperature we generalized the model to electron spin degeneracies: N f > 2. The solution of the scaling equations written in N f 2 order confirmed that the only stable fixed point of the system is equivalent to the Kondo-model of spin 1/2, moreover we found the limits of this equivalency.We also studied what conditions can make a commutative TLS model -not showing Kondo-effect in simple casesto be non-commutative.The asymmetry of the electron band even in leading order, the splitting of the levels in higher orders result in logarithmic divergencies in the couplings, which were originally invariant in the scaling.
Low dimensional magnetic models.-Recently several new materials have been discovered in which the localized magnetic moments are coupled in a way resembling ladders.The energy spectrum and the behaviour of these systems depends strongly on the value of the spin and the number of legs of the ladder.We studied the phase diagram of such ladder models using the density matrix renormalization group (DMRG).First in a rather technical work we have checked the accuracy of the DMRG procedure on the example of the Ising model in transverse field, as the simplest quantum mechanical spin problem, and have shown how the so-called truncation error is related to the real error in the energy values.With this background we have determined the phase diagram of a two leg ladder model including also plaquette couplings, and have found indication for the existence of several different massive phases.One-dimensional fermionic models.-We investigated in detail the model defined through the relativistic limit of the one dimensional attractive Hubbard model.This model possesses both massive and massless dressed particles which are described just like the particles in the Hubbard chain, by Bethe-Ansatz type equations.We analysed the eigenstates of the system.We have found that the limiting model has higher symmetry than the original Hubbard chain; it acquires in addition also the chiral symmetry.We determined the scattering properties of the particles, and also the conformal properties of the field corresponding to the massless sector.All these results confirm, that the scaling limit of the Hubbard chain can be identified with the chiral invariant Gross-Neveu model.Theory of dissipative motion of heavy particles.-We studied the low temperature behaviour of two level systems (TLS) interacting with the conduction electrons in amorphous metals.To approach the temperature region below the Kondo temperature we generalized the model to electron spin degeneracies: N f > 2. The solution of the scaling equations written in N f 2 order confirmed that the only stable fixed point of the system is equivalent to the Kondo-model of spin 1/2, moreover we found the limits of this equivalency.We also studied what conditions can make a commutative TLS model -not showing Kondo-effect in simple casesto be non-commutative.The asymmetry of the electron band even in leading order, the splitting of the levels in higher orders result in logarithmic divergencies in the couplings, which were originally invariant in the scaling.
Low dimensional magnetic models.-We continued our study of the physical properties of the so-called magnetic ladder systems.We have considered anisotropic two-leg ladder models with S=1/2 and S=1 spins on the rungs, using the density matrix renormalization group (DMRG) method.We have generalized the numerical procedure to allow to take into account the symmetries of the models in order to reduce the dimension of the matrix to be diagonalized or to achieve better accuracy.We have determined the boundaries of the extended region in the parameter space, where the gapped Haldane phase exists.In our study of the scaling limit of the one dimensional XXZ Heisenberg chain we have found, that the limiting model possesses spin 1/2 massive particles, and possibly can be identified with the Sine-Gordon chain.One-dimensional fermionic models.-We continued the study of the models defined through the relativistic limit of the Hubbard chain by investigating the scaling limit of the one dimensional attractive Hubbard model at less than half filling.Like in the half filled band case, also this model possesses both massive and massless dressed particles.Now, unlike the half filled case, only the massive particles are described by Bethe Ansatz type equations, nevertheless the massless excitations still fit into the framework of a conform field theory.Theory of dissipative motion of heavy particles.-We studied the low temperature behaviour of two level systems (TLS) interacting with the conduction electrons in various models and approximations.We continued the study of the region below the Kondo temperature by the use of the model with electron spin degeneracy N f > 2. The only stable fixed point and the exponents of the leading irrelevant operators are equivalent to those of the spin 1/2 Kondo model, but there are essentially more irrelevant operators in our case.We studied the role of the electron-hole symmetry in the low temperature behaviour of TLS, moreover the differences between the scaling methods starting from path integrals and Matsubara technique.We showed that the breaking of the electron-hole symmetry results in electron assisted tunneling, but the contribution cannot bring the system to the two-channel Kondo fixed point.
Low dimensional magnetic models.-The one-dimensional Heisenberg model is known to be soluble by the Bethe Ansatz.The spectrum of a ferromagnet consists of magnons and bound state of magnons.The problem of finding the bound state of a macroscopically large number of magnons poses a special problem.These states are described by macroscopically long string solutions of the Bethe Ansatz equations.To find such solutions numerically we developed a program by which even chains of several hundred sites can be treated.Our results show that, as expected, the structure of the macroscopically long strings is not of the Bethe-type.Recently several new materials have been discovered in which the localized magnetic moments are situated in a chain like structure.The energy spectrum and the behaviour of these systems depends strongly on the value of the spin and the number of coupled chains.We started to study these models using the density matrix renormalization group.One-dimensional fermionic models.-The Hubbard model is the simplest model of the interacting electron system.It is usually formulated on a discrete lattice.Sometimes it is more appropriate to work in the continuum limit.We have shown, that the continuum limit of the Hubbard chain, in which the particle number per site is kept constant leads to a relativistic model describing both finite and zero mass particles.By constructing the relativistic limit of the half-filled band we have shown, that also the relativistic particles are described by Bethe-Ansatz type equations, while in the relativistic limit of the non-half-filled chain this is true only for the massive particles, and the equations of the massless particles are of somewhat different structure.Theory of dissipative motion of heavy particles.-We studied the low energy fixed point structure of an N f -flavor two level system (TLS) model, where the spin of the conduction electrons is replaced by a flavor quantum number, f= 1, ... , N f .Using the multiplicative renormalization group we constructed the (1/N f ) 2 order scaling equations and we analyzed their fixed point structure.We showed that the fixed point structure of this Hamiltonian is equivalent to that of the N f -channel Kondo model and we found that the exponent of the leading irrelevant operators is also the same.However, for a TLS an infinite number of leading irrelevant operators appear which give a contribution to the different physical quantities but do not change their critical exponents.
This dissertation discussed four low-dimensional systems: Hubbard ladder and its 2D extension, anisotropic Hubbard ladder, one-dimensional two-orbital t-J model, and Kondo model. All of them are of strongly correlated. Doped cuprates, as strongly correlated systems, have attracted a lot of attention in the condensed matter physics community, not just from the purely theoretical perspective, but also due to the novel applications beyond the conventional semiconductor paradigm. They display complicated phase diagram, defying the conventional rigid band theory, shaking our common sense and driving us to wrestle with many-body physics. With the aid of advanced experimental techniques such as ARPES, its various exotic electronic structure, such as waterfalls, hole pockets, Fermi arcs and kinks have been unveiled and remain open to interpretation. To attack such complicated problems, people usually build various models to mimic the CuO2 planes, which is the common feature of various cuprate compounds and is believed to be mainly responsible for their complicated properties, especially the high temperature superconductivity. In Chapter 3, we simulated the spectral properties of the 2- leg single-band Hubbard ladder and its 2-dimensional extension at and away from half-lling using the numerical tool: the combination of Cluster Perturbation Theory (CPT) with the time-dependent density matrix renormalization group method (tDMRG) as a solver. The main features of the spectrum can be described with a mean-eld dispersion, with kinks and pseudogap traced back to scattering between spin and charge degrees of freedom. As a bonus, we nd that the spectrum of hubbard ladder involves spin-polaron, the bound state of spinon and holon. Interacting one-dimensional electron systems generically behave dierent from higher dimensional systems in the sense that its low-energy excitations are described by Luttinger liquid theory as collective bosonic modes carrying spin and charge separately, the so-called spinons and holons, rather than the coherent quasi-particles in Fermi liquid theory. This phenomenon is known as spin-charge separation, an intrinsic property of 1D system. It can be thought of as a particular case of electron fractionalization. When multiple 1D chains get coupled, how the picture changes? In Chapter 4, we study the spectral function of the anisotropic 2-leg Hubbard ladder, one of the simplest multiple-chain systems. The spectrum can be understood as the scattering resultants of spinon and holons: As the inter-chain hopping is increased, the continuum in the bonding channel moves to higher energies and spinon and holon branches merge into a single coherent quasi-particle band. Simultaneously, the spectrum undergoes a crossover from a regime with two minima at incommensurate values of kx (a Mott insulator), to one with a single minimum at kx = (a band insulator). In addition, the spectrum contains a continuum of scattering states consisting of a triplon and a polaron, accompanied by the formation of bound states with higher spin 3 2 , ferromagnetic \spin bags" in which the hole can move freely. To analyze the formation of quasi-particle, we study the time-evolution of charge and spin degrees of freedom in real space after a hole is injected, which indicates that incoherent holons and spinons can emerge into polarons, that can propagate coherently, after a characterstic time . Multiorbital systems can involve Mott and Hund physics, and are rich in various phenomena, including orbital selective Mott transitions, spin-orbital separations, and pairing density wave due to the complex interplay between spin, charge and orbital degrees of freedom. Therefore, it can capture many unique features in real strongly correlated systems, such as iron pnictides and ruthenates, and thus attract a lot of investigations. In Chapter 5, we comprehensively study a one-dimensional two-orbital model at and below quarter lling, which realizes a number of unconventional phases, by means of DMRG and some analytical calculations. In particular, we nd that the system supports an extensive density wave in which excitons quasicondense with nite center-of-mass momentum. In this phase, excitons behave as hardcore bosons without charge order. In addition, excitons can pair to form biexcitons in a state that is close to a charge density wave instability. Finally, a genuine orbital-selective paired state (metallic plus spin-gapped superconductor) can be realized when pairing dominates over the interorbital repulsion. Kondo problem, as one of the most studied and best understood correlated systems, has been studied by various techniques, but the internal structure of its ground state has two deeply con icting pictures: At strong coupling limit, the system is essentially composed of a scattering center, which is a tightly bound singlet formed by the impurity and a localized electron at r0, and the remaining conduction electrons that are decoupled to the impurity and remain almost \free". At weak enough couplings, the singlet extends to the entire volume and the impurity will couple mostly to one electron at the Fermi level. In chapter 6, we provide a coherent picture to unveil the puzzle in the context of quantum information theory by deconstructing the Kondo singlet, both in real and energy space, and measuring the spin correlations between the impurity and the invididual free electron eigenstate based on two dierent measures. As a bonus, we can identify \projected natural orbitals" that can be used as a \good" approximate wavefunctions for simulations, and a quasi-order parameter that determines the crossover from weak to strong coupling regimes and furthermore oers the means to measure the Kondo screening length K. This screening length might be used to characterize the \size" of the Kondo cloud.
Topological Kondo insulators are strongly correlated materials where itinerant electrons hybridize with localized spins, giving rise to a topologically nontrivial band structure. Here, we use nonperturbative bosonization and renormalization-group techniques to study theoretically a one-dimensional topological Kondo insulator, described as a Kondo-Heisenberg model, where the Heisenberg spin-1/2 chain is coupled to a Hubbard chain through a Kondo exchange interaction in the p-wave channel (i.e., a strongly correlated version of the prototypical Tamm-Schockley model). We derive and solve renormalization-group equations at two-loop order in the Kondo parameter, and find that, at half filling, the charge degrees of freedom in the Hubbard chain acquire a Mott gap, even in the case of a noninteracting conduction band (Hubbard parameter U=0). Furthermore, at low enough temperatures, the system maps onto a spin-1/2 ladder with local ferromagnetic interactions along the rungs, effectively locking the spin degrees of freedom into a spin-1 chain with frozen charge degrees of freedom. This structure behaves as a spin-1 Haldane chain, a prototypical interacting topological spin model, and features two magnetic spin-1/2 end states for chains with open boundary conditions. Our analysis allows us to derive an insightful connection between topological Kondo insulators in one spatial dimension and the well-known physics of the Haldane chain, showing that the ground state of the former is qualitatively different from the predictions of the naive mean-field theory.
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.
We investigate the magnetic order and related strongly correlated effects in a one-dimensional Ising-Kondo lattice with transverse field. This model is the anisotropic limit of the conventional isotropic Kondo lattice model, in the sense that the itinerant electrons interact with the localized magnetic moments via only longitudinal Kondo exchange. Adopting the numerical density-matrix renormalization group method, we map out the ground-state phase diagram in various parameter spaces. Depending on the Kondo coupling and filling number, three distinct phases, including a metallic paramagnetic, a metallic ferromagnetic, and a gapped spin-density wave phase, are obtained. The spin-density wave is characterized by an ordering wave vector, which coincides with the nesting wave vector of the Fermi surface. This makes the corresponding magnetic transition a spin analog of the Peierls transition occurring in the one-dimensional metal. Moreover, by analyzing the momentum distribution function and charge correlation function, the conduction electrons are shown to behave like free spinless fermions in the ferromagnetic phase. We finally discuss the effect of the repulsive Hubbard interaction between conduction electrons. Our paper enriches the Kondo physics and deepens the current understanding of the heavy fermion compounds.