京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18772号 / 理博第4030号 / 新制||理||1581(附属図書館) / 31723 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 川上 則雄, 教授 田中 耕一郎, 教授 松田 祐司 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

In recent decades enormous progress has been made in studying the geometrical structure of the quantum state space. Far from an abstraction, this geometric struc- ture is defined operationally in terms of the distinguishability of states connected by parameterizations that can be controlled in a laboratory. This geometry is manifest in the kinds of response functions that are measured by well established experimen- tal techniques, such as inelastic neutron scattering. In this thesis we explore the properties of the state space geometry in the vicinity of the ground state of two paradigmatic models of low dimensional magnetism. The first model is the spin-1 anti-ferromagnetic Heisenberg chain, which is a central example of symmetry pro- tected topological physics in one dimension, exhibiting a non-local string order, and symmetry protected short range entanglement. The second is the Kitaev honeycomb model, a rare example of an analytically solvable quantum spin liquid, characterized by long range topological order. In Chapter 2 we employ the single mode approximation to estimate the genuine multipartite entanglement in the spin-1 chain as a function of the unaxial anisotropy up to finite temperature. We find that the genuine multipartite entanglement ex- hibits a finite temperature plateau, and recove the universality class of the phase transition induced by negative anisotropy be examining the finite size scaling of the quantum Fisher information. In Chapter 4 we map out the zero temperature phase diagram in terms of the QFI for a patch of the phase space parameterized by the anisotropy and applied magnetic field, establishing that any non-zero anisotropy en- hances that entanglement of the SPT phase, and the robustness of the phase to finite temperatures. We also establish a connection between genuine multipartite entanglement and state space curvature. In Chapter 3 we turn to the Kitaev honeycomb model and demonstrate that, while the QFI associated to local operators remains trivial, the second derivative of such quantities with respect to the driving parameter exhibit divergences. We characterize the critical exponents associated with these divergences. / Thesis / Doctor of Philosophy (PhD) / Systems composed of many bodies tend to order as their energy is reduced. Steam, a state characterized by the complete disorder of the constituent water molecules, condenses to liquid water as the temperature (energy) decreases, wherein the water molecules are organized enough for insects to walk atop them. Water freezes to ice, which is so ordered that it can hold sleds and skaters. Quantum mechanics allows for patterns of organization that go beyond the solid-liquid-gas states. These patterns are manifest in the smallest degrees of freedom in a solid, the electrons, and are responsible for fridge magnets and transistors. While quantum systems still tend to order at lower energies, they are characterized by omni-present fluctuations that can conceal hidden forms of organization. One can imagine that the states of matter live in a vast space, where each point represents a different pattern. In this thesis we show that by probing the geometry of this space, we can detect hidden kinds of order that would be otherwise invisible to us.

Neste trabalho exploramos o emaranhamento em sistemas de muitos férmions. Para o estudo de sistemas inomogêneos, propusemos uma aproximação de densidade local (LDA) para a entropia de emaranhamento de um único sítio com o restante do sistema e uma LDA para o emaranhamento entre blocos de sítios. Analisamos as contribuições universal e não-universal do emaranhamento entre blocos e obtivemos uma expressão para o termo não-universal. Usando o modelo de Hubbard unidimensional, investigamos o emaranhamento em nanoestruturas eletrônicas, quantificando o emaranhamento de um único sítio com relação ao restante da cadeia via entropia de emaranhamento. Para o modelo de Hubbard homogêneo estudamos o comportamento do emaranhamento em função da densidade, da magnetização, da interação eletrônica e de campos magnéticos externos. Encontramos que o emaranhamento é sensível às fases metálica, isolante e supercondutora. Observamos um platô de emaranhamento na região do gap de spin e verificamos que susceptibilidade magnética e emaranhamento estão intrinsecamente relacionados. Obtendo as energias e densidades do modelo de Hubbard inomogêneo através da Teoria do Funcional da Densidade e usando nossa proposta LDA para a entropia de emaranhamento, exploramos o comportamento do emaranhamento na presença de diversas inomogeneidades: superredes, impurezas e confinamento harmônico. Verificamos que o emaranhamento sempre diminui com a inomogeneidade, embora os efeitos de cada inomogeneidade sejam completamente diferentes. Encontramos uma relação entre energias de troca e correlação, de Hartree e cinética, capaz de prever quantitativamente o emaranhamento em função de qualquer das inomogeneidades. / In this work we investigated entanglement in many-fermions systems. To explore inhomogeneous systems we proposed a local density approximation (LDA) for the single-site entanglement entropy. We analysed the universal and nonuniversal contributions to block-block entanglement and obtained an expression for the nonuniversal term. We employ a description in terms of the one-dimensional Hubbard model to investigate the entanglement in electronic nanostructures and to quantify the single-site entanglement with respect to the rest of the chain by means of the entanglement entropy. For the homogeneous Hubbard model we studied the entanglement behavior as a function of density, magnetization, electronic interaction and external magnetic fields. We found that the entanglement is sensitive to the metallic, insulating and superconducting phases. We observed an entanglement plateau in the region of the spin gap and verified that magnetic susceptibility and entanglement are intrinsically related. Energies and densities of the inhomogeneous Hubbard model, obtained from Density Functional Theory, combined with our proposal of an LDA for the entanglement entropy, were used to explore the behavior of the entanglement entropy in the presence of several inhomogeneities: superlattices, impurities and harmonic confinement. We verified that entanglement always decreases with the inhomogeneity, although the effect of each inhomogeneity is completely different. For the same model we found a relation of exchange-correlation, Hartree and kinetic energies, able to predict quantitatively the entanglement as a function of any inhomogeneity.

Neste trabalho exploramos o emaranhamento em sistemas de muitos férmions. Para o estudo de sistemas inomogêneos, propusemos uma aproximação de densidade local (LDA) para a entropia de emaranhamento de um único sítio com o restante do sistema e uma LDA para o emaranhamento entre blocos de sítios. Analisamos as contribuições universal e não-universal do emaranhamento entre blocos e obtivemos uma expressão para o termo não-universal. Usando o modelo de Hubbard unidimensional, investigamos o emaranhamento em nanoestruturas eletrônicas, quantificando o emaranhamento de um único sítio com relação ao restante da cadeia via entropia de emaranhamento. Para o modelo de Hubbard homogêneo estudamos o comportamento do emaranhamento em função da densidade, da magnetização, da interação eletrônica e de campos magnéticos externos. Encontramos que o emaranhamento é sensível às fases metálica, isolante e supercondutora. Observamos um platô de emaranhamento na região do gap de spin e verificamos que susceptibilidade magnética e emaranhamento estão intrinsecamente relacionados. Obtendo as energias e densidades do modelo de Hubbard inomogêneo através da Teoria do Funcional da Densidade e usando nossa proposta LDA para a entropia de emaranhamento, exploramos o comportamento do emaranhamento na presença de diversas inomogeneidades: superredes, impurezas e confinamento harmônico. Verificamos que o emaranhamento sempre diminui com a inomogeneidade, embora os efeitos de cada inomogeneidade sejam completamente diferentes. Encontramos uma relação entre energias de troca e correlação, de Hartree e cinética, capaz de prever quantitativamente o emaranhamento em função de qualquer das inomogeneidades. / In this work we investigated entanglement in many-fermions systems. To explore inhomogeneous systems we proposed a local density approximation (LDA) for the single-site entanglement entropy. We analysed the universal and nonuniversal contributions to block-block entanglement and obtained an expression for the nonuniversal term. We employ a description in terms of the one-dimensional Hubbard model to investigate the entanglement in electronic nanostructures and to quantify the single-site entanglement with respect to the rest of the chain by means of the entanglement entropy. For the homogeneous Hubbard model we studied the entanglement behavior as a function of density, magnetization, electronic interaction and external magnetic fields. We found that the entanglement is sensitive to the metallic, insulating and superconducting phases. We observed an entanglement plateau in the region of the spin gap and verified that magnetic susceptibility and entanglement are intrinsically related. Energies and densities of the inhomogeneous Hubbard model, obtained from Density Functional Theory, combined with our proposal of an LDA for the entanglement entropy, were used to explore the behavior of the entanglement entropy in the presence of several inhomogeneities: superlattices, impurities and harmonic confinement. We verified that entanglement always decreases with the inhomogeneity, although the effect of each inhomogeneity is completely different. For the same model we found a relation of exchange-correlation, Hartree and kinetic energies, able to predict quantitatively the entanglement as a function of any inhomogeneity.

Studies on strongly correlated electron systems over decades have allowed physicists to discover unusual properties such as spin density waves, ferromagnetic and antiferromagnetic states with unusual ordering of spins and orbitals, and Mott insulating states, to name a few. In this thesis, the focus will be on the specific heat property of these materials exhibiting novel electronic ground states in the presence and absence of a field. The purpose of these measurements is to characterize the phase transitions into these states and the low energy excitations in these states. From measurements at the phase transitions, one can learn about the amount of order involved [i.e. entropy: ΔS = ∫Δc p/T dT], while measurements at low temperatures illuminate the excitation spectrum. In order to study the thermodynamic properties of the materials at their phase transitions, a high sensitive technique, ac-calorimetry was used. The ac-calorimeter, workhorse of our low dimensional materials lab, is based on modulating the power that heats the sample and measuring the temperature oscillations of the sample around its mean value. The in-house ac-calorimetry set up in our lab has the capability to produce a quasi-continuous readout of heat capacity as a function of temperature. A variety of single crystals were investigated using this technique and a few among them are discussed in my thesis. Since many of the crystals that are studied by our group are magnetically active, it becomes useful for us to also study them in the presence of a moderate to high magnetic field. This motivated me to design, develop, and build a heat capacity probe that would enable us to study the crystals in the presence of non-zero magnetic fields and at low temperatures. The probe helped us not only to revisit some of the studied materials and to draw firm conclusions on the previous results but also is vital in exploring the untouched territory of novel materials at high magnetic fields (~ 14 T).

Ultracold atoms in optical lattices can be used to model condensed matter systems. They provide a clean, tuneable system which can be engineered to reach parameter regimes that are not accessible in condensed matter systems. Furthermore, they provide different techniques for probing the properties of these systems. This thesis presents an experimental and theoretical study of ultracold atoms in optical lattices for quantum simulation of two-dimensional systems.The first part of this thesis describes an experiment with a Bose-Einstein condensate of 87Rb loaded into a two-dimensional optical lattice. The beams that generate the optical lattice are controlled by acousto-optic deflection to provide a flexible optical lattice potential. The use of a dynamic ‘accordion’ lattice with ultracold atoms, where the spacing of the lattice is increased in both directions from 2.2 to 5.5 μm, is described. This technique allows an experiment such as quantum simulations to be performed with a lattice spacing smaller than the resolution limit of the imaging system, while allowing imaging of the atoms at individual lattice sites by subsequent expansion of the optical lattice. The optical lattice can also be rotated, generating an artificial magnetic field. Previous experiments with the rotating optical lattice are summarised, and steps to reaching the strongly correlated regime are discussed. The second part of this thesis details numerical techniques that can be used to describe strongly correlated two-dimensional systems. These systems are challenging to simulate numerically, as the exponential growth in the size of the Hilbert space with the number of particles means that they can only be solved exactly for very small systems. Recently proposed correlator product states [Phys. Rev. B 80, 245116 (2009)] provide a numerically efficient description which can be used to simulate large two-dimensional systems. In this thesis we apply this method to the two-dimensional quantum Ising model, and the Bose-Hubbard model subject to an artificial magnetic field in the regime where fractional quantum Hall states are predicted to occur.

Strong electron correlation lies at the root of many quantum collective phenomena observed in solids, including high Tc superconductivity. Theoretically, the problem of many interacting electrons is difficult to treat, however, and a microscopic understanding of strongly correlated systems remains one of the foremost challenges in modern physics. A particularly clean realisation of this general problem is found in magnetic systems, where theory and experiment are both well developed and complementary. The role of the chemist in this endeavour is to provide model experimental systems to both inspire new developments in theory and to confirm existing predictions. This thesis aims to demonstrate aspects of both synthesis and physical characterisation of such model systems, with particular emphasis on materials which exhibit unusual quantum ground states due to a combination of reduced dimensionality, low spin, and geometric frustration. Four materials are considered: The first among these is a new material, KTi(SO4)2·(H2O), which was prepared using a hydrothermal route, and characterised by magnetic susceptibility, specific heat, and high field magnetisation measurements. Fitting exact diagonalisation and series expansion results to these data imply that KTi(SO4)2·(H2O)is a long-sought experimental realization of the S = 1/2 Heisenberg frustrated (J1 − J2) chain model in the dimerised regime of the phase diagram. The anhydrous analogue of KTi(SO4)2·(H2O), KTi(SO4)2, was also investigated, and found by magnetic neutron scattering to exemplify the S = 1/2 Heisenberg anisotropic triangular lattice model in the 1D chain limit. The final two materials discussed are the naturally occurring minerals volborthite and herbertsmithite, both thought to realise the S = 1/2 Heisenberg kagome antiferromagnet model. Diffuse and inelastic magnetic neutron scattering experiments, however, indicate that the kagome physics are partially destroyed by defects in the former and lattice distortion in the latter.

Neutron scattering measurements were used to investigate the magnetic and crystal structure and magnetic excitations of three compounds characterized as low-dimensional quantum magnets. The materials are frustrated systems with low spin quantum number. The first was a powder sample of AgNiO₂. The Ni ions form a triangular lattice antiferromagnet in which, according to the published crystal structure, both the orbital order and magnetic couplings are frustrated. However, it is shown here that there was a small distortion of the crystal structure at 365 K, which is proposed to result from charge disproportionation and this relieves the orbital frustration. The magnetic structure was investigated and, below 20 K, the triangular lattice of electron-rich Ni sites was observed to order into antiferromagnetic stripes. Investigations of the magnetic excitations showed that the main dispersions were within the triangular plane, indicating a strong two-dimensionality. The dispersion was larger along the stripes than between the stripes of collinear spins. The second material investigated was CoNb₂O₆, a quasi Ising-like ferromagnet. It was studied with a magnetic field applied transverse to the Ising direction. The magnetic field introduced quantum fluctuations which drove a phase transition at a field comparable to the main exchange interaction. The phase diagram of the magnetic order was mapped outs and a transition from an ordered phase to a paramagnetic phase was identified at high field. This low-temperature high-field phase transition was further investigated by inelastic neutron scattering measurements to observe the change in the energy gap and magnetic excitation spectrum on either side of the transition. The spectrum had two components in the ordered phase and had sharp magnon modes in the paramagnetic phase. The third material was the spin-half layered antiferromagnet CuSb₂O₆. It has a square lattice of Cu²⁺ ions in which the main interaction is across only one diagonal of the square. The magnetic structure was studied by neutron scattering with a field applied along the direction of the zero-field ordered moment. A spin-flop was observed at low field and there was evidence for a high-field transition. The magnetic excitation spectrum was unusual in that it had an intense resonance at 13 meV at the magnetic Brillouin zone boundary.

Thesis advisor: Ziqiang Wang / Exotic phases and associated phase transitions in low dimensions have been a fascinating frontier and a driving force in modern condensed matter physics since the 80s. Due to strong correlation effect, they are beyond the description of mean-field theory based on a single-particle picture and Landau's symmetry-breaking theory of phase transitions. These new phases of matter require new physical quantities to characterize them and new languages to describe them. This thesis is devoted to the study on exotic phases of correlated electrons in two spatial dimensions. We present the following efforts in understanding two-dimensional exotic phases: (1) Using Zn vertex algebra, we give a complete classification and characterization of different one-component fractional quantum Hall (FQH) states, including their ground state properties and quasiparticles. (2) In terms of a non-unitary transformation, we obtain the exact form of statistical interactions between composite fermions in the lowest Landau level (LLL) with v=1/(2m), m=1,2... By studying the pairing instability of composite fermions we theoretically explains recently observed FQHE in LLL with v=1/2,1/4. (3) We classify different Z2 spin liquids (SLs) on kagome lattice in Schwinger-fermion representation using projective symmetry group (PSG). We propose one most promising candidate for the numerically discovered SL state in nearest-neighbor Heisenberg model on kagome lattice}. (4) By analyzing different Z2 spin liquids on honeycomb lattice within PSG classification, we find out the nature of the gapped SL phase in honeycomb lattice Hubbard model, labeled sublattice pairing state (SPS) in Schwinger-fermion representation. We also identify the neighboring magnetic phase of SPS as a chiral-antiferromagnetic (CAF) phase and analyze the continuous phase transition between SPS and CAF phase. For the first time we identify a SL called 0-flux state in Schwinger-boson representation with one (SPS) in Schwinger-fermion representation by a duality transformation. (5) We show that when certain non-collinear magnetic order coexists in a singlet nodal superconductor, there will be Majorana bound states in vortex cores/on the edges of the superconductor. This proposal opens a window for discovering Majorana fermions in strongly correlated electrons. (6) Motivated by recent numerical discovery of fractionalized phases in topological flat bands, we construct wavefunctions for spin-polarized fractional Chern insulators (FCI) and time reversal symmetric fractional topological insulators (FTI) by parton approach. We show that lattice symmetries give rise to different FCI/FTI states even with the same filling fraction. For the first time we construct FTI wavefunctions in the absence of spin conservation which preserve all lattice symmetries. The constructed wavefunctions also set up the framework for future variational Monte Carlo simulations. / Thesis (PhD) — Boston College, 2011. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

Quantum phase transitions (QPT) on the border of magnetism have provided a fertile hunting ground for the discovery of new states of matter, for example; the marginal Fermi Liquid and non Fermi Liquid states as well high T$_C$ cuprate and magnetically mediated superconductivity. In this thesis I present work on three materials in which it may be possible to tune the system through a magnetic QPT with the application of hydrostatic pressure. Although the details of the underlying physics are different in each of the materials, they are linked by the possibility of finding new states on the border of magnetism. Applying hydrostatic pressure, we have suppressed the ferromagnetic (FM) transition in metallic Fe$_2$P to very low temperature and to a potential QPT. Counter-intuitive broadening of the magnetic hysteresis leading up to the FM-AFM QPT may well be a crucial clue as to the nature of the model needed to understand this phase transition. A sharp increase in the quasi-particle scattering cross-section as well as the residual resistivity accompany a departure from the quadratic temperature dependence of the resistivity. This possible deviation from Fermi liquid behaviour is stable over a significant range of temperature. The unexplained upturn in the resistivity of CeGe that accompanies the AFM transition was studied under pressure. Pressure increased the residual resistivity as well as decreasing the relative size of the upturn, but had a moderate effect on the Neel temperature. The insensitivity of the N$\acute e$el temperature to pressure has been compared to its relative sensitivity to applied feld. The existence of the upturn and its evolution with pressure and applied feld can reasonably be argued to be due to the details of the electron band structure in the system. By applying pressure we have drastically reduced the resistivity of the insulating antiferromagnet NiPS$_3$. Concurrent work on FePS$_3$ has shown metallisation under pressure. It seems reasonable to speculate that NiPS$_3$ may also metallise at higher pressure. The energy gap is narrowed in both materials as pressure is increased. Magnetisation measurements have revealed a low temperature upturn indicating some possible ferromagnetic component or proximity to another magnetic state. A peak in the magnetisation is also seen at 45K in zero-feld cooled measurements. Both of these features point to a system with a complex magnetic ground state.