The understanding of matter does not only require a precise analysis of individual characteristics of the fundamental constituents, but also the study of their collective behavior.Solid state physics, and in particular research in strongly correlated electron systems, deals with such collective phenomena.The properties of the electrons, charge and spin quantum numbers, do not on their own explain the rich world of states of matter, however mutual interactions lead to various states or phases of matter.This work focuses on phases in quantum spin systems.On the basis of the Heisenberg model one (1D) and two dimensional (2D) spin lattices are investigated.In a first part the bilayer Heisenberg Model and the 2D Kondo necklace model are studied.Both models exhibit a quantum phase transition between an ordered and disordered phase.For small interlayer couplings the models have antiferromagnetic long-range order, which is destroyed by increasing the coupling.However, even in the disordered phase the models show global features in terms of collective excitations called magnons.Coming from the disordered phase the gapped magnons condense at a critical coupling and form the order of the ordered phase.Here, another interesting question arises, namely that of the coupling of a single doped hole to such critical fluctuations.A self-consistent Born approximation predicts that the doped hole couples to the magnons such that the quasiparticle residue vanishes at the quantum critical point.In this work the delicate question about the fate of the quasiparticle residue across the quantum phase transition is also tackled by means of large scale quantum Monte Carlo simulations.Furthermore the dynamics of a single hole doped in the magnetic background is investigated.
The understanding of matter does not only require a precise analysis of individual characteristics of the fundamental constituents, but also the study of their collective behavior.Solid state physics, and in particular research in strongly correlated electron systems, deals with such collective phenomena.The properties of the electrons, charge and spin quantum numbers, do not on their own explain the rich world of states of matter, however mutual interactions lead to various states or phases of matter.This work focuses on phases in quantum spin systems.On the basis of the Heisenberg model one (1D) and two dimensional (2D) spin lattices are investigated.In a first part the bilayer Heisenberg Model and the 2D Kondo necklace model are studied.Both models exhibit a quantum phase transition between an ordered and disordered phase.For small interlayer couplings the models have antiferromagnetic long-range order, which is destroyed by increasing the coupling.However, even in the disordered phase the models show global features in terms of collective excitations called magnons.Coming from the disordered phase the gapped magnons condense at a critical coupling and form the order of the ordered phase.Here, another interesting question arises, namely that of the coupling of a single doped hole to such critical fluctuations.A self-consistent Born approximation predicts that the doped hole couples to the magnons such that the quasiparticle residue vanishes at the quantum critical point.In this work the delicate question about the fate of the quasiparticle residue across the quantum phase transition is also tackled by means of large scale quantum Monte Carlo simulations.Furthermore the dynamics of a single hole doped in the magnetic background is investigated.
Solid state physics, a very broad field in natural science, aims to explain physical properties of materials modeled as a many-body ensemble of the order of ∼ 10^23 interacting particles. For interacting metallic systems, one of the most powerful tools for unraveling the consequences of the collective interaction is the Fermi liquid theory. While still being an approximation, it excellently models many materials especially in the low-temperature and low-energy regime by treating themany-body problem as being equivalent to effectively non-interacting ”particles”, which is done by introducing the concept of quasiparticles. During the last few decades, several exotic phases of matter have been discovered, which originate from strong Coulomb interactions between the electrons, whose description is beyond the Fermi liquid theory. This thesis addresses the question of the validity of Fermi liquid theory once a single hole is introduced into an antiferromagnet with completely localized spins due to the strong interactions. Here, the most-studied case concerns the so-called 2D t − J model, which hosts a well-defined quasiparticle, the spin polaron. Focusing on a more tractable case, the t − Jz model without quantum spin fluctuations, we study the stability of the spin polaron solution once the nextnearest neighbor hopping t′ is introduced into the system. We focus on observations at the Γ-point and the X-point of systems with fixed Jz = 0.4t and varying t′ ∈ [−0.5t, 0.5t]. The model is numerically solved using exact diagonalization on finite-size lattices of up to 32 sites. A study of the quasiparticle weight under finite-size scaling reveals an “anomalous” regime at the Γ-point fort′ ≤ −0.3t, indicated by a vanishing quasiparticle weight. In order to characterize the low-energy dynamics within this “anomalous” regime, we investigated the spin correlation centered around the hole, as well as the magnon number distribution of a single-doped 20-site system. Based on the results, we present a preliminary picture of the antiferromagnetic ground state doped with one hole within that “anomalous” regime (t′ ≤ −0.3t) that deviates from the standard spin polaron as the cloud of magnetic fluctuations, typically localized around the hole, delocalizes over the whole finite-size lattice. Consequently, such systems show a weakly suppressed antiferromagnetic configuration under single-hole doping.
We study single-hole dynamics in the bilayer Heisenberg and Kondo necklace models. Those models exhibit a magnetic order-disorder quantum phase transition as a function of the interlayer coupling ${J}_{\ensuremath{\perp}}$. At strong coupling in the disordered phase, both models have a single-hole dispersion relation with band maximum at $\mathbit{p}=(\ensuremath{\pi},\ensuremath{\pi})$ and an effective mass at this $\mathbit{p}$ point which scales as the hopping matrix element t. In the Kondo necklace model, we show that the effective mass at $\mathbit{p}=(\ensuremath{\pi},\ensuremath{\pi})$ remains finite for all considered values of ${J}_{\ensuremath{\perp}}$ such that the strong-coupling features of the dispersion relation are apparent down to weak coupling. In contrast, in the bilayer Heisenberg model, the effective mass diverges at a finite value of ${J}_{\ensuremath{\perp}}$. This divergence of the effective mass is unrelated to the magnetic quantum phase transition, and at weak coupling the dispersion relation maps onto that of a single hole doped in a planar antiferromagnet with band maximum at $\mathbit{p}=(\ensuremath{\pi}∕2,\ensuremath{\pi}∕2)$. We equally study the behavior of the quasiparticle residue in the vicinity of the magnetic quantum phase transition both for a mobile and static hole. In contrast to analytical approaches, our numerical results do not unambiguously support the fact that the quasiparticle residue of the static hole vanishes in the vicinity of the critical point. The above results are obtained with a generalized version of the loop algorithm to include single-hole dynamics on lattice sizes up to $20\ifmmode\times\else\texttimes\fi{}20$.
In this thesis we apply recently developed, as well as sophisticated quantum Monte Carlo methods to numerically investigate models of strongly correlated electron systems on honeycomb structures. The latter are of particular interest owing to their unique properties when simulating electrons on them, like the relativistic dispersion, strong quantum fluctuations and their resistance against instabilities. This work covers several projects including the advancement of the weak-coupling continuous time quantum Monte Carlo and its application to zero temperature and phonons, quantum phase transitions of valence bond solids in spin-1/2 Heisenberg systems using projector quantum Monte Carlo in the valence bond basis, and the magnetic field induced transition to a canted antiferromagnet of the Hubbard model on the honeycomb lattice. The emphasis lies on two projects investigating the phase diagram of the SU(2) and the SU(N)-symmetric Hubbard model on the hexagonal lattice. At sufficiently low temperatures, condensed-matter systems tend to develop order. An exception are quantum spin-liquids, where fluctuations prevent a transition to an ordered state down to the lowest temperatures. Previously elusive in experimentally relevant microscopic two-dimensional models, we show by means of large-scale quantum Monte Carlo simulations of the SU(2) Hubbard model on the honeycomb lattice, that a quantum spin-liquid emerges between the state described by massless Dirac fermions and an antiferromagnetically ordered Mott insulator. This unexpected quantum-disordered state is found to be a short-range resonating valence bond liquid, akin to the one proposed for high temperature superconductors. Inspired by the rich phase diagrams of SU(N) models we study the SU(N)-symmetric Hubbard Heisenberg quantum antiferromagnet on the honeycomb lattice to investigate the reliability of 1/N corrections to large-N results by means of numerically exact QMC simulations. We study the melting of phases as correlations increase with decreasing N and determine whether the quantum spin liquid found in the SU(2) Hubbard model at intermediate coupling is a specific feature, or also exists in the unconstrained t-J model and higher symmetries.
The study of various types of orders and phases of matter in magnetic systems is extremely popular theme in condensed matter physics. Not only frequently applicable in real world, quantum magnetism provides a fruitful and at the same time frequently feasible framework to search for new phases of matter. In this work we study quantum and associated classical phase transitions of spin systems by combinations of numerical and analytical methods.In Chapter 1 we study quantum paramagnets that are a generalization of antiferromagnetic AKLT [2] construction for the spin group SU(N). We map the so called simplex solid state ansatz wavefunction into a partition function of the classical system on the same lattice using coherent state representation. Then we study phase transitions of that classical models via classical Monte Carlo methods to determine properties of the original quantum simplex solid state [3].In Chapter 2 we are interested in the O(2) model in (2+1)-dimensional system with the dilution, i.e. when some sites are removed from the system. This relativistic O(2) model originates from the description of a transition between superfluid and Mott insulator phases. We study critical properties of the transition using a recently invented [4] worm algorithm.In Chapter 3 we are studying large-N expansion for the symplectic spin group Sp(N). When the original SU(2) spin model proves to be too difficult to be solved, for example at low temperatures, one of the alternative ways is to consider a system based on a more general spin group, for example SU(N) or Sp(N). Results often remain relevant for the original SU(2) group. We are using the mean-field theory approach to find phase transitions associated with the cubic Sp(N) system at large N.Another fundamental question of interest is how different systems behave when they are periodically driven by some external field. Such systems are called Floquet systems, and we will study them in Chapter 4. We will consider interacting and non-interacting systems, that might have a non-conventional type of ordering, topological order.
In this thesis we apply recently developed, as well as sophisticated quantum Monte Carlo methods to numerically investigate models of strongly correlated electron systems on honeycomb structures. The latter are of particular interest owing to their unique properties when simulating electrons on them, like the relativistic dispersion, strong quantum fluctuations and their resistance against instabilities. This work covers several projects including the advancement of the weak-coupling continuous time quantum Monte Carlo and its application to zero temperature and phonons, quantum phase transitions of valence bond solids in spin-1/2 Heisenberg systems using projector quantum Monte Carlo in the valence bond basis, and the magnetic field induced transition to a canted antiferromagnet of the Hubbard model on the honeycomb lattice. The emphasis lies on two projects investigating the phase diagram of the SU(2) and the SU(N)-symmetric Hubbard model on the hexagonal lattice. At sufficiently low temperatures, condensed-matter systems tend to develop order. An exception are quantum spin-liquids, where fluctuations prevent a transition to an ordered state down to the lowest temperatures. Previously elusive in experimentally relevant microscopic two-dimensional models, we show by means of large-scale quantum Monte Carlo simulations of the SU(2) Hubbard model on the honeycomb lattice, that a quantum spin-liquid emerges between the state described by massless Dirac fermions and an antiferromagnetically ordered Mott insulator. This unexpected quantum-disordered state is found to be a short-range resonating valence bond liquid, akin to the one proposed for high temperature superconductors. Inspired by the rich phase diagrams of SU(N) models we study the SU(N)-symmetric Hubbard Heisenberg quantum antiferromagnet on the honeycomb lattice to investigate the reliability of 1/N corrections to large-N results by means of numerically exact QMC simulations. We study the melting of phases as correlations increase with decreasing N and determine whether the quantum spin liquid found in the SU(2) Hubbard model at intermediate coupling is a specific feature, or also exists in the unconstrained t-J model and higher symmetries.
Phase transitions have been an active area of research in statistical mechanics for almost a century and have recently been integrated into quantum mechanics. Many phenomena such as superconductivity and unconventional magnetism are understood to arise from exotic quantum phases and at points describing quantum phase transitions. A detailed understanding of these phase transitions requires numerical simulations of models which benchmark realistic models against theoretical frameworks. The topic of this thesis is the implementation of Quantum Monte Carlo simulation, which is a powerful technique to understand quantum condensed matter, in interesting models to illustrate novel phenomena in magnetic systems. The novel features of condensed matter systems described in this thesis consist of emergent symmetries at critical points, interesting dynamical features of such systems and the drastic effects of defects in spin systems used in the field of adiabatic quantum computing. Emergent symmetries are shown by condensed matter systems especially at critical points and are features which cannot be shown by individual or a small number of spins. Examples of this in one and two dimensions are presented in an early chapter of this thesis. In addition to this, spin systems can show excitations which have an interesting spatial structure as a consequence of restricted dynamics which only allow the excitations to spread in a particular region. This is presented in the context of a simple model in the following chapter along with numerical support. The following chapter contains a description of adiabatic quantum computing along with a particular model which we study. The phase transition and the effects on the performance of adiabatic quantum computing are studied in this context.