Decades after the discovery of high-temperature superconductivity, understanding the effects of doping a Mott insulator remains a topic of great relevance, since the fundamental excitations in the normal state are not known, nor the precise origin of the pairing mechanism. In particular, the nature of the pseudogap phase at finite temperatures, above the superconducting dome, or the redistribution of spectral weight observed in experiments indicate that our knowledge is incomplete and theories may be missing some key ingredients.Due to the two-dimensionality of the problem and the limitations imposed by the Mermin-Wagner theorem, the numerical study of these questions is remarkably difficult: most treatments are either limited to small or quasi one-dimensional systems at zero temperature, or simulations may be carried out at finite temperature, completely missing the possibility for long-range order. In this dissertation, we tackle paradigmatic problems in condensed matter from a fundamentally new perspective. We numerically study interacting chains and ladders introducing RKKY-like SU(2)-symmetric long-range interactions that allow one to realize actual spontaneous symmetry breaking and true antiferromagnetic order. We shed new light on problems such as the role of magnetic order on quasi-particle formation and fractionalization, and pairing instabilities in generalized Hubbard-like models. In particular, we address questions of a very fundamental nature: (i) How do spinons evolve into magnons and what are their signatures in the excitation spectrum? (ii) how do coherent quasi-particles form in the presence of long range order? Will we still observe signatures of spin-charge separation in the spectrum? (iii) what is the interplay between long-range antiferromagnetic order and pairing? Our work offers new insight onto the role played by long-range antiferromagnetism in quasi-particle formation and pairing in the context of new toy-models that realize much of the phenomenology of their higher dimensional counterparts.--Author's abstract
Decades after the discovery of high-temperature superconductivity, understanding the effects of doping a Mott insulator remains a topic of great relevance, since the fundamental excitations in the normal state are not known, nor the precise origin of the pairing mechanism. In particular, the nature of the pseudogap phase at finite temperatures, above the superconducting dome, or the redistribution of spectral weight observed in experiments indicate that our knowledge is incomplete and theories may be missing some key ingredients.Due to the two-dimensionality of the problem and the limitations imposed by the Mermin-Wagner theorem, the numerical study of these questions is remarkably difficult: most treatments are either limited to small or quasi one-dimensional systems at zero temperature, or simulations may be carried out at finite temperature, completely missing the possibility for long-range order. In this dissertation, we tackle paradigmatic problems in condensed matter from a fundamentally new perspective. We numerically study interacting chains and ladders introducing RKKY-like SU(2)-symmetric long-range interactions that allow one to realize actual spontaneous symmetry breaking and true antiferromagnetic order. We shed new light on problems such as the role of magnetic order on quasi-particle formation and fractionalization, and pairing instabilities in generalized Hubbard-like models. In particular, we address questions of a very fundamental nature: (i) How do spinons evolve into magnons and what are their signatures in the excitation spectrum? (ii) how do coherent quasi-particles form in the presence of long range order? Will we still observe signatures of spin-charge separation in the spectrum? (iii) what is the interplay between long-range antiferromagnetic order and pairing? Our work offers new insight onto the role played by long-range antiferromagnetism in quasi-particle formation and pairing in the context of new toy-models that realize much of the phenomenology of their higher dimensional counterparts.--Author's abstract
We present a numerical study of competing orders in the 1D $t\text{\ensuremath{-}}J$ model with long-range RKKY-like staggered spin interactions. By circumventing the constraints imposed by Mermin-Wagner's theorem, this Hamiltonian can realize long-range N\'eel order at half filling. We determine the full phase diagram as a function of the exchange and particle density using the density matrix renormalization group (DMRG) method. We show that pairing is disfavored and the AFM insulator and metallic phases are separated by a broad regime with phase segregation, before spin-charge separation re-emerges at low densities. Upon doping, interactions induce a confining potential that binds holons and spinons into full fledged fermionic quasiparticles in a range of parameters and densities. We numerically calculate the photoemission spectrum of the model, showing the appearance of a coherent quasiparticle band splitting away from the holon-spinon continuum with a width determined by J that survives at finite doping. Comparison with analytical results using the self-consistent Born approximation (SCBA) and by solving the spinon-holon problem offer insight into the internal structure of the quasiparticles and help us explain the different features in the spectrum. We discuss how this simple toy model can teach us about the phenomenology of its higher-dimensional counterpart.
Quasi-one-dimensional quantum models are ideal for theoretically exploring the physical phenomena associated with strong correlations. In this thesis we study three examples where strong correlations play an important role in the static or dynamic properties of the system. Firstly, we examine the behaviour of a doped fermionic two-leg ladder in which umklapp interactions are present. Such interactions arise at special band fillings and can be induced by the formation of charge density wave order in an array of two-leg ladders with long-range (three-dimensional) interactions. For the umklapp which arises from the half-filling of one of the bands, we show that the low-energy theory has a number of phases, including a strong coupling regime in which the dominant fluctuations are superconducting in nature. These superconducting fluctuations carry a finite wave vector – they are the one-dimensional analogue of Fulde-Ferrell-Larkin-Ovchinnikov superconductivity. In a second example, we consider a quantum spin model which captures the essential one-dimensional physics of CoNb2O6, a quasi-one-dimensional Ising ferromagnet. Motivated by high-resolution inelastic neutron scattering experiments, we calculate the dynamical structure in the paramagnetic phase and show that a small misalignment of the transverse field can lead to quasi-particle breakdown – a surprising broadening in the single particle mode observed in experiment. Finally, we study the out-of-equilibrium dynamics of a model with tuneable integrability breaking. When integrability is broken by the presence of weak interactions, we show that the system relaxes to a non-thermal state on intermediate time scales, the so-called “prethermalization plateau”. We describe the approximately stationary behaviour in this regime by constructing a generalised Gibbs ensemble with charges deformed to leading order in perturbation theory. Expectation values of these charges are time-independent, but interestingly the charges do not commute with the Hamiltonian to leading order in perturbation theory. Increasing the strength of the integrability breaking interactions leads to behaviour compatible with thermalisation. In each case we use a combination of perturbative analytical calculations and non-perturbative numerical computations to study the problem at hand.
This thesis is devoted to the study of strongly correlated phases of quantum matter in two spatial dimensions. In presence of strong interactions, the electron can split up into separate spin and charge degrees of freedom. The diagnosis of such fractionalized excitations in recent experiments, as well as finding new probes for their detection, comprise the two major themes of this thesis.\nThe first part of the thesis, comprising chapters 2-6, discusses the relevance of fractionalization to the pseudogap phase of the underdoped high Tc cuprate superconductors. This is motivated by transport experiments that show a violation of Luttinger's theorem, possible only in the presence of topological order that can arise naturally as a consequence of fractionalization. Chapters 2 and 3 focus on phenomenological models with bosonic charge carriers and fermionic spin carriers, and investigate transport properties as well as various confinement transitions to symmetry broken phases. Chapters 4, 5 and 6 deal with a different framework of electron fractionalization, where the charge carriers are fermionic and spin carriers are bosonic. We find that these models can better explain the thermal and electrical transport properties in chapter 5. In chapters 6 and 7, we show how such models arise naturally from quantum fluctuations of antiferromagnetism, and can simultaneously intertwine the anti-nodal spectral gap with the discrete broken symmetries that are observed in the cuprates. \nThe second part of the thesis, comprising chapters 7 and 8, proposes new experimental probes to study the behavior of insulating two dimensional quantum magnets. Aided by geometric frustration and strong quantum fluctuations, they may realize a spin liquid ground state which can be extremely difficult to detect via conventional probes. Chapter 7 discusses how novel spin-transport probes from the spintronics community can be used to detect fractionalized excitations in a quantum spin liquid. Chapter 8 studies how the temperature dependent anisotropy of in-plane versus out-of-plane thermal conductivities can serve as an explicit signature of fractionalization in layered two-dimensional materials.
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 introduce a slave-fermion formulation in which to study the charge dynamics of the half-filled Hubbard model on the square lattice. In this description, the charge degrees of freedom are represented by fermionic holons and doublons and the Mott-insulating characteristics of the ground state are the consequence of holon–doublon bound-state formation. The bosonic spin degrees of freedom are described by the antiferromagnetic Heisenberg model, yielding long-ranged (Néel) magnetic order at zero temperature. Within this framework and in the self-consistent Born approximation, we perform systematic calculations of the average double occupancy, the electronic density of states, the spectral function and the optical conductivity. Qualitatively, our method reproduces the lower and upper Hubbard bands, the spectral-weight transfer into a coherent quasiparticle band at their lower edges and the renormalisation of the Mott gap, which is associated with holon–doublon binding, due to the interactions of both quasiparticle species with the magnons. The zeros of the Green function at the chemical potential give the Luttinger volume, the poles of the self-energy reflect the underlying quasiparticle dispersion with a spin-renormalised hopping parameter and the optical gap is directly related to the Mott gap. Quantitatively, the square-lattice Hubbard model is one of the best-characterised problems in correlated condensed matter and many numerical calculations, all with different strengths and weaknesses, exist with which to benchmark our approach. From the semi-quantitative accuracy of our results for all but the weakest interaction strengths, we conclude that a self-consistent treatment of the spin-fluctuation effects on the charge degrees of freedom captures all the essential physics of the antiferromagnetic Mott–Hubbard insulator. We remark in addition that an analytical approximation with these properties serves a vital function in developing a full understanding of the fundamental physics of the Mott state, both in the antiferromagnetic insulator and at finite temperatures and dopings.
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.