Dissertations / Theses on the topic 'Kagome lattices'

La recherche de phases magnétiques exotiques de la matière qui fondent même à T=0 uniquement sous l'action des fluctuations quantiques a été long et ardu, à la fois théoriquement et expérimentalement. La percée est venue récemment avec la découverte de l'Herbertsmithite, un composé formant un réseau kagome parfait avec des moments magnétiques de spin-1/2. Des expériences pionnières, mêlant des mesures de NMR, µSR et de diffusion de neutrons, ont montré une absence totale de gel ou d'ordre des moments magnétiques de spin, fournissant ainsi une forte signature d'une phase paramgnétique quantique. Théoriquement, l'Herbertsmithite est extrêmement bien modélisé par le modèle de Heisenberg quantique antiferromagnétique pour des spins-1/2 sur le réseau kagome, problème qui n'a pas été résolu jusqu'à présent. Plusieurs méthodes approximatives numériques et analytiques ont donné différents états fondamentaux, allant des liquides de spins Z2 gappés et un liquide de spins exotique algébrique U(1) de Dirac aux liquides de spins chiraux et les cristaux à liaisons de valence. Dans cette thèse, le problème est traité dans le cadre d'une approche particule-esclave fermionique, à savoir le formalisme des fermions de Schwinger SU(2). Il est conclu qu'un liquide de spins sans gap algébrique de Dirac a l'énergie variationnelle la plus basse et peut en fait constituer un vrai état fondamental physique de liquide de spins. Une implémentation sophistiquée de méthodes numériques de pointes comme le Monte-Carlo variationnel, le Monte-Carlo fonctions de Green et l'application de pas Lanczos dans un schéma variationnel ont été utilisés. Il est montré que contrairement à la croyance habituelle, le liquide de spins de Dirac U(1) projeté en "2+1" dimensions est remarquablement robuste par rapport à une large classe de perturbations, incluant les liquides de spins topologiques Z2 et les cristaux à liaisons de valence. De plus, l'application de deux pas Lanczos sur la fonction d'onde du liquide de spins de Dirac U(1) montre que son énergie est compétitive avec celles proposées pour les liquides de spins topologiques Z2. Ce résultat, combiné avec les indications expérimentales qui pointent vers un liquide de spins sans gap pour l'Herbertsmithite, appuie l'affirmation que le vrai état fondamental de ce modèle est en fait un liquide de spins algébrique de Dirac.

The extended kagome lattice, composed of alternating kagome and triangular layers, provides a novel geometry for frustrated magnetism. In this thesis, we study the properties of Heisenberg spins with nearest-neighbour antiferromagnetic interactions on this lattice. In common with many other models of frustrated magnets, this system has highly degenerate classical ground states. It is set apart from other examples, however, by the strong interlayer correlations between triangular layer spins. We study the implications of such correlations in both the statics and dynamics. We characterise classical ground states using a flux picture for a single layer of kagome spins, a theoretical description that sets geometrical bounds on correlations. We quantify the divergent but sub-extensive ground state degeneracy by a Maxwellian counting argument, and verify this calculation by analysing the energy eigenvalues of numerical ground states. We explore the ground state connectedness but do not reach firm conclusions on this issue. We use the self-consistent Gaussian approximation (SCGA) to calculate static spin correlations at finite temperature. The results of these calculations agree well with elastic neutron scattering experiments. We derive an expression for the effective interlayer interaction between kagome spins by integrating out the triangular lattice spins. We use linear spinwave theory to compute the spin excitation spectrum numerically. This shows encouraging similarity with inelastic neutron scattering data on a single-crystal YBaCo$_4$O$_7$ sample, for a wide range of wavevector and frequency. This agreement shows that our spin model is a reasonable description of the physics, and suggests that this numerical technique might be useful for other geometrically frustrated magnets. We study the dynamics analytically using the stochastic SCGA recently developed for the pyrochlore lattice. For technical reasons, we apply this technique on a related model, the stacked kagome lattice, rather than on the extended kagome lattice itself. From this we find slow relaxation at low temperature, with a rate ~ T² compared to the faster ~ T scaling for the pyrochlore. Strikingly, in simulations of the dynamics on the extended kagome lattice by numerical integration of the semiclassical equations of motion, we find two different relaxation rates. Kagome layer spins relax more quickly than the triangular layer spins, having ~ T.

Kyoto University (京都大学)
0048
新制・課程博士
博士(人間・環境学)
甲第8668号
人博第111号
12||113(吉田南総合図書館)
新制||人||27(附属図書館)
UT51-2001-A756
京都大学大学院人間・環境学研究科文化・地域環境学専攻
(主査)教授 後藤 喬雄, 教授 前川 覚, 教授 冨田 博之
学位規則第4条第1項該当

Within condensed matter physics, systems with strong electronic correlations give rise to fascinating phenomena which characteristically require a physical description beyond a one-electron theory, such as high temperature superconductivity, or Mott metal-insulator transitions. In this thesis, a class of strongly correlated electron systems is considered. These systems exhibit fractionally charged excitations with charge +e/2 or -e/2 in two dimensions (2D) and three dimensions (3D), a consequence of both strong correlations and the geometrical frustration of the interactions on the underlying lattices. Such geometrically frustrated systems are typically characterized by a high density of low-lying excitations, leading to various interesting physical effects. This thesis constitutes a study of a model of spinless fermions on the geometrically frustrated kagome lattice. Focus is given in particular to the regime in which nearest-neighbour repulsions V are large in comparison with hopping t between neighbouring sites, the regime in which excitations with fractional charge occur. In the classical limit t = 0, the geometric frustration results in a macroscopically large ground-state degeneracy. This degeneracy is lifted by quantum fluctuations. A low-energy effective Hamiltonian is derived for the spinless fermion model for the case of 1/3 filling in the regime where |t| << V . In this limit, the effective Hamiltonian is given by ring-exchange of order ~ t^3/V^2, lifting the degeneracy. The effective model is shown to be equivalent to a corresponding hard-core bosonic model due to a gauge invariance which removes the fermionic sign problem. The model is furthermore mapped directly to a Quantum Dimer model on the hexagonal lattice. Through the mapping it is determined that the kagome lattice model exhibits plaquette order in the ground state and also that fractional charges within the model are linearly confined. Subsequently a doped version of the effective model is studied, for the case where exactly one spinless fermion is added or subtracted from the system at 1/3 filling. The sign of the newly introduced hopping term is shown to be removable due to a gauge invariance for the case of hole doping. This gauge invariance is a direct result of the bipartite nature of the hole hopping and is confirmed numerically in spectral density calculations. For further understanding of the low-energy physics, a derivation of the model gauge field theory is presented and discussed in relation to the confining quantum electrodynamic in two dimensions. Exact diagonalization calculations illustrate the nature of the fractional charge confinement in terms of the string tension between a bound pair of defects. The calculations employ topological symmetries that exist for the manifold of ground-state configurations. Dynamical calculations of the spectral densities are considered for the full spinless fermion Hamiltonian and compared in the strongly correlated regime with the doped effective Hamiltonian. Calculations for the effective Hamiltonian are then presented for the strongly correlated regime where |t| << V . In the limit g << |t|, the fractional charges are shown to be effectively free in the context of the finite clusters studied. Prominent features of the spectral densities at the Gamma point for the hole and particle contributions are attributed to approximate eigenfunctions of the spinless fermion Hamiltonian in this limit. This is confirmed through an analytical derivation. The case of g ~ t is then considered, as in this case the confinement of the fractional charges is observable in the spectral densities calculated for finite clusters. The bound states for the effectively confined defect pair are qualitatively estimated through the solution of the time-independent Schroedinger equation for a potential which scales linearly with g. The double-peaked feature of spectral density calculations over a range of g values can thus be interpreted as a signature of the confinement of the fractionally charged defect pair. Furthermore, the metal-insulator transition for the effective Hamiltonian is studied for both t > 0 and t < 0. Exact diagonalization calculations are found to be consistent with the predictions of the effective model. Further calculations confirm that the sign of t is rendered inconsequential due to the gauge invariance for g in the regime |t| << V . The charge-order melting metal-insulator transition is studied through density-matrix renormalization group calculations. The opening of the energy gap is found to differ for the two signs of t, reflecting the difference in the band structure at the Fermi level in each case. The qualitative nature of transition in each case is discussed. As a step towards a realization of the model in experiment, density-density correlation functions are introduced and such a calculation is shown for the plaquette phase for the effective model Hamiltonian at 1/3 filling in the absence of defects. Finally, the open problem of statistics of the fractional charges is discussed.

Strongly correlated systems on geometrically frustrated lattices can stabilize a large number of interesting phases that includes a wide array of novel Mott insulators in both bosonic and electronic systems. Charge fluctuations in a Mott insulator are suppressed due to strong mutual interaction among the particles. The presence of frustration is of particular importance as the physics it offers is often rich, unexpectedly complicated, and continues to raise many open questions. The thesis elucidates some of these issues on a kagome lattice where strong interactions among the particles in the Mott phase impose non-trivial local constraints depending on the filling fraction on the lattice. These Mott insulators, in addition to featuring unusual magnetic and/or charge ordering, can also harbor topologically ordered states of quantum matter, e.g., resonating valence bond liquids realized in certain quantum dimer models on non-bipartite lattices. The dimer models can be regarded as low-energy effective theories for different types of bosonic models in the strong-coupling limit. Exploring this connection is a central theme of this thesis with the aim of realizing novel strongly correlated ground states. Past studies of these models have revealed the existence of various ordered and disordered phases with distinct signatures. Among these low-energy phases, the presence of a stable topological liquid at a particular point, known as Rokhsar-Kivelson point, in the phase diagram is notable. The classical versions of the dimer model are also known to have garnered a vast interest in various fields ranging from problems of pure mathematical origin to ones in physical chemistry as well as statistical physics. Pioneered by Kasteleyn, several analytical works came forward to exactly calculate the partition function of the problem from which other physical observables can be derived. Classical numerical methods are extensively applied to these models to verify the analytical predictions. We introduce a new classical algorithm here to compute the correlation functions of a classical dimer model on a square (bipartite) and a triangular (non-bipartite) lattice based on a tensor network construction. The method, called tensor network renormalization group, turns out to be a powerful tool for simulating short-ranged gapped systems as inferred from our results benchmarked against the classical Monte-Carlo technique and compared with past analytical studies. One should note that the quantum dimer model at the Rokhsar-Kivelson point can also be described as an infinite temperature canonical ensemble of classical dimers because of the particular structure of the ground state which is an equal weight superposition in the configuration manifold. The geometry of the lattice plays a pivotal role in deciding the nature of the phases that arise in the dimer models. Many physical properties of the dimer liquid phase can be extracted in the simple classical setting which certainly allows for a deep understanding of the classical models to be developed. The liquid phase is gapped on non-bipartite lattices and gapless on bipartite lattices, which is reflected in the decay of correlation functions with spatial distances. In general on non-bipartite lattices, the topological nature of the dimer liquid is characterized by a Z2 topological order which survives even when the model is perturbed away from the Rokhsar-Kivelson point. Stability of this liquid phase not only depends on the lattice geometries but notably on dimer concentrations also. In this context, we focus on a particular variant of the dimer model on a triangular lattice which is known as the quantum fully packed loop model. The model is composed of nonintersecting closed loops made of dimers and governed by the same Hamiltonian as the quantum dimer model. The loop model provides an effective low-energy description of a strongly correlated bosonic system at 1/3 filling on the kagome lattice. The corresponding Bose-Hubbard Hamiltonian consists of nearest-neighbor hopping and all possible repulsive interactions within a hexagonal plaquette. Conspicuous features of the zero-temperature phase diagram for this model include (i) presence of a stable Z2 liquid even without any Rokhsar-Kivelson potential term (in distinction to the standard quantum dimer model), and (ii) an unconventional phase transition from the liquid phase to a novel crystalline phase that has nematic order (dubbed lattice nematic). For a deeper understanding of the physics, a mapping to an Ising gauge theory is presented. The gauge theoretic description provides a useful way to predict the nature of the quantum phase transition to lie in the O(3) universality class. Finally a fermionic model at the same 1/3 filling is considered in which the ground state exhibits a number of exotic local orderings resulting from the spin-charge interplay of electrons. The Hamiltonian comprises nearest-neighbor hopping, strong on-site Coulomb interaction, and repulsive interaction terms only between nearest-neighbors. In the strong correlation limit, this fermionic problem maps to a two-color fully packed loop model – a model in which the loop segments carry an additional quantum number as color on a honeycomb lattice. The effective theory is governed by coherent three-particle ring exchanges and nearest-neighbor antiferromagnetic spin exchanges. The competition between these two leads to a phase diagram composed of a novel plaquette ordered state (known as the plaquette phase) that undergoes phase transition to a new kind of charge ordered state which we call a short loop phase. From our numerical analysis, we conclude that the plaquette phase features an unusual antiferromagnetic order with gapless spin excitations while the charge-ordered state is subjugated by spin fluctuations of localized electrons arranged in small hexagonal loops on the kagome lattice.

Topological insulators are electronic materials that behave like an ordinary insulator in their bulk but have robust conducting states on their edge. Besides, in some materials the band structure presents completely flat bands, a special feature leading to strong interactions effects. In this thesis we present a study of the edge states of three particular two-dimensional models presenting flat bands: the honeycomb-Kagome, the $\alpha$--graphyne and a ligand decorated honeycomb-Kagome lattice models. We extend earlier work done on these lattice models by focusing on the topological nature of the edge states involving flat bands. We start by giving a review of the band structure theory and the tight-binding approximation. We then present several main topics in two-dimensional topological insulators such as the notion of topological invariants, the Kane-Mele model and the bulk-edge correspondence. Using these theoretical concepts we study the band structure of these lattices firstly without taking into account the spin and spin-orbit interations. We finally add these interactions to get their bulk band structures as well as the edge states. We observe how these spin-orbit interactions relieve degeneracies and allow for the emergence of edge states of topological nature. Since the lattices studied have an arrangement based on the honeycomb-Kagome lattice, two-dimensional materials having the structures of these lattices can be designed assembling metal ions and organic ligands. Therefore the results obtained could be used as a first hint to create new two-dimensional materials presenting topological properties.
Topologiska isolatorer är elektroniska material som uppför sig som en vanlig isolator i sin bulk men har robusta ledande stater på kanten. Dessutom presenterar bandstrukturen i vissa material helt platta band, en speciell egenskap som leder till starka interaktionseffekter. I denna avhandling presenterar vi en studie av kanttillstånden för tre speciella tvådimensionella modeller som presenterar platta band: bikakan-Kagome, $\alpha$-grafynen och en liganddekorerad honungskaka-Kagome modeller. Vi utökar tidigare arbete med dessa gittermodeller genom att fokusera på den topologiska karaktären hos kanttillstånd som innefattar platta band. Vi börjar med att ge en genomgång av bandstruktursteorin och den tätt bindande approximationen. Vi presenterar sedan flera huvudämnen i tvådimensionella topologiska isolatorer såsom begreppet topologiska invarianter, Kane-Mele modellen och bulk-kant korrespondensen. Med hjälp av dessa teoretiska begrepp studerar vi bandstrukturen för dessa gitter först utan att ta hänsyn till spinnen och spinnsorbital interaktioner. Vi lägger sedan till dessa interaktioner för att få sina bulkbandstrukturer såväl som kanttillstånden. Vi observerar hur dessa spinnsorbital interaktioner lindrar degenerationer och möjliggör uppkomsten av kanttillstånd av topologisk naturen. Eftersom de undersökta gitterna har ett arrangemang baserat på honungskaka-Kagome gitteren, kan tvådimensionella material med strukturerna hos dessa gitter utformas genom att montera metalljoner och organiska ligander. Därför kan de erhållna resultaten användas som en första ledtråd för att skapa nya tvådimensionella material med topologiska egenskaper.

Neste trabalho estudamos o modelo da rede de Kondo em uma rede kagome, buscando uma maior compreensão dos efeitos da frustração geométrica em sistemas de férmions pesados. Para tanto, fizemos uma aproximação de campo médio no hamiltoniano do sistema que serve para todas as fases do sistema. Analisamos inicialmente o caso não magnético. Obtemos neste limite as energias eletrônicas e as funções de Green necessárias ao cálculo numérico autoconsistente das ocupações e do parâmetro de Kondo. Os resultados encontrados estão em concordância qualitativa com trabalhos publicados em outras geometrias. A seguir analisamos o caso magnético, onde introduzimos uma aproximação suplementar, a qual é compatível com a de campo médio já considerada e, em princípio, existente apenas em sistemas com frustração geométrica. Realizamos cálculos autoconsistentes através de somas sobre as frequências de Matsubara. Os resultados mostram que não há coexistência entre ordem magnética e efeito Kondo, além de haver a supressão do antiferromagnetismo com o aumento de temperatura e variações no preenchimento de bandas.
In this work we study the Kondo Lattice model for the kagome lattice, in order to understand better the effects of geometrical frustration in heavy-fermion systems. In this context, we consider a mean field scheme valid for all the system’s phases. Firstly, we analyzed the nonmagnetic case. In this approximation the electron energies and spectral functions are reachable, then we use the density of states to calculate the occupations selfconsistently. Our results are qualitatively compared with previous works in other geometries. In the second part we introduce an approximation for magnestism, which takes into account the mean field scheme considered and the presence of geometrical frustration. Self-consistent calculations are done through the frequencies summation method. Our results show that the magnetism is supressed when the temperature is increased or the band filling deviates from half-filling. Besides, the coexistence of magnetic order and Kondo effect is not observable.

Thesis: S.M., Massachusetts Institute of Technology, Department of Physics, 2010.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 67-69).
An ideal spin-% kagomé lattice has been a long sought material. This system is characterized by strong magnetic frustration and is a likely candidate for a spin-liquid ground state. The spin-liquid state was originally proposed to exist in the parent compounds of the high temperature superconducting cuprates as originally proposed by Anderson. However, the lack of ideal samples have hampered experimental tests of the theories. A few years ago, a kagomé lattice material called herbertsmithite (ZnCu3(OH)6Cl2) has been successfully synthesized and studied. Since then, many experiments have been performed which have produced a lot of new guidance for our theoretical understanding of this frustrated magnetic system. However, single crystals are crucial for further progress. We have successfully produced high quality single crystals ZnCu3(OH)6Cl2 . These crystals are large enough for measurements, such as x-ray diffraction, magnetism, heat capacity, neutron scattering, thermal conductivity, muon-scattering and optical measurement. In this thesis, I will summarize the current state of knowledge for herbertsmithite and its family, the single crystal growth technique, and characterization of the resulting samples. A discussion of further directions of growth and measurement is at the end.
by Tianheng Han.
S.M.

Nous etudions trois reseaux frustres geometriquement. En utilisant l'hamiltonien de heisenberg quantique antiferromagnetique sur le reseau pyrochlore, nous donnons des arguments qui caracterisent ce systeme comme un liquide de spins ou le gap est tres faible. Les delafossites dopes (la,y)cuo#2#. #6#6 sont interpretes comme des reseaux kagome. En considerant differents modeles (xy, hubbard, t-j), nous concluons que le compose au lanthane est certainement metallique alors que celui a l'yttrium doit etre isolant. Nous caracterisons dans les deux cas (la ou y) les proprietes magnetiques. Enfin, le compose uni#4b est decrit comme un reseau de anderson triangulaire. En derivant un modele effectif, nous reproduisons son etat fondamental et prevoyons son comportement sous champ.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2010.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 121-128).
Quantum spin liquids, which are quantum ground states of interacting spin systems that break no symmetries, have long been a fascination among the theoretical condensed matter community. After years of experimental searches, several promising candidates finally emerged, including herbertsinithite ZnCu3 (OH)6 Cl2 , which can be modeled as a spin-1/2 kagome lattice. Theoretically, the U(1) Dirac spin liquid (U(1) DSL) state is shown to be a plausible description of the system, and previous works have indicate that this particular quantum spin liquid state may enjoy a host of interesting properties, such as the power-law decay of correlation functions, the existence of spin-1/2 excitations known as the spinon, and the existence of an emergent U(1) gauge field. In this thesis, after the relevant motivation and background information are discussed, I shall present my work on the spin-1/2 kagome lattice that built upon the U(1) DSL state. First, I shall present the theoretical study of Raman scattering in the U(1) DSL state, which shows that in all symmetry channels the Raman intensity profiles contain broad continua that display power-law behaviors at low energy, which can be attributed to the excitations of spinon-antispinon pairs. In, addition, for the A2g channel, the Raman profile also contains a characteristic 1/o singularity, which arise from an excitation of the emergent U(1) gauge field. The possibility of more clearly observing the signature of this U(1) emergent gauge field in resonant inelastic X-ray scattering (RIXS) is also discussed. Next, I shall consider the case when the spin-1/2 kagome lattice is subjected to an external magnetic field, in which a state with an additional uniform amount of gauge flux of top of the U(1) DSL ansatz, which results in the formation of Landau levels in the spinon spectrum, is shown to be energetically favorable. Unlike the usual quantum Hall system, the Landau level state is shown to contain a gapless S2 density mode, which in turns indicate that system is XY ordered in the plane perpendicular to the applied magnetic field. Third, I shall consider the case in which the spin-1/2 kagome lattice is hole-doped. Similar to the B-field case, a Landau-level state is shown to be energetically favorable, in which a gapless charge density mode is shown to exists, and which through the Anderson-Higgs mechanism causes the system to become a superconductor. This resulting superconductor is then shown to be exotic, in the sense that it contains minimal vortices having a flux of hc/4e, as well as bosonic quasiparticles that have semionic mutual statistics. The thesis concludes with a short summary and outlook.
by Wing-Ho Ko.
Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Physics, 2014.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 191-198).
The geometry of the kagome lattice leads to exciting novel magnetic behavior in both ferromagnetic and antiferromagnetic systems. The collective spin dynamics were investigated in a variety of magnetic materials featuring spin-1/2 and spin-1 moments on kagome lattices using neutron scattering and thermodynamic probes. Both ferromagnetic and antiferromagnetic systems were studied. Cu(1,3-bdc) is an organometallic material, where the Cu2+ ions form a ferromagnetic S = 1/2. kagomé system. Synthesis techniques were developed to produce -mg-sized deuterated single crystals, and ~2,000 crystals were partially coaligned to create a sample for neutron scattering measurements. Elastic neutron scattering measurements show the existence of long range magnetic ordering below T = 1.77 K. Integrated Bragg peak intensities were analyzed to determine the structure of ordered magnetic moments. Inelastic neutron scattering measurements show the magnon dispersion spectrum, which consists of a flat high energy band and two dispersive, lower energy bands. The application of a magnetic field perpendicular to the kagome plane opens gaps between these three bands and distorts the flatness of the highest energy band. The system was modelled as a nearest-neighbor Heisenberg ferromagnet with Dzyaloshinskii-Moriya(DM) interaction. The model dispersion and scattering structure factor were calculated and fit to the data to precisely determine the strengths of the nearest-neighbor coupling and DM interaction. The observed manon band structure is a bosonic analog to the band structure of the topological insulator systems. Antiferromagnetic kagome systems can exhibit novel magnetic ground states such as quantum spin liquids and spin nematics. Thermodynamic measurements were performed on the antiferromagnetic kagome materials MgxCu₄-x(OH)₆ Cl₂ , featuring S = 1/2 moments. These measurements reveal magnetic ordering at low values of x that is suppressed with increasing x. At x = 0.75, this ordering is not fully suppressed, but susceptibility and specific heat measurements reveal behavior similar to that of the quantum spin liquid candidate herbertsmithite. Thermodynamic and neutron scattering measurements were performed on the kagome lattice material BaNi₃(OH)₂(VO₄)₂, which features S = 1 moments. These measurements reveal competing interactions, which result in a spin glass ordering transition.
by Robin Michael Daub Chisnell.
Ph. D.

Kyoto University (京都大学)
0048
新制・課程博士
博士(人間・環境学)
甲第9151号
人博第133号
12||135(吉田南総合図書館)
新制||人||32(附属図書館)
UT51-2001-K358
京都大学大学院人間・環境学研究科文化・地域環境学専攻
(主査)教授 前川 覚, 教授 後藤 喬雄, 教授 冨田 博之
学位規則第4条第1項該当

Includes bibliographical references (p. 203-222).
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, February 2008.
(cont.) The former represents a continuous planar rotational symmetry corresponding to the SO(2) symmetry, while the latter is a discrete symmetry associated with the Z2 symmetry. Depending on which measurements are performed, the critical behavior of the system can belong to either SO(2) or Z2 universality classes with two distinct critical temperatures; one is associated with the spontaneous breaking of the Z2 symmetry, and the other corresponds to a topological order (BKT transition) due to vortex- antivortex binding. The former occurs at a slightly higher temperature than the latter. Neutron scattering measurements show a signature of the BKT transition, while specific heat measurements show a feature of the 2D Ising transition. Above TN, the in-plane spin gap vanishes, and the system retains the SO(2) symmetry when measured with neutron scattering. On the other hand, specific heat measurements show a feature of the 2D Ising transition, since the underlying symmetry of the spin Hamiltonian is the time-reversal or Z2 symmetry.
The collective behavior of interacting magnetic moments can be strongly influenced by the topology of the underlying lattice. In geometrically frustrated spin systems, interesting spin dynamics and chiral correlations may develop that are related to the spin arrangement on triangular plaquettes. We report studies of the spin-wave excitations and spin chirality on a two-dimensional geometrically frustrated lattice. Our new chemical synthesis methods allow us to produce large single crystal samples of KFe3(OH)6 (SO4)2, an ideal kagom6 lattice antiferromagnet. The spin-wave excitations have been measured using high-resolution inelastic neutron scattering. We directly observe a flat mode which corresponds to a lifted "zero energy mode," verifying a fundamental prediction for the kagome lattice. A simple Heisenberg spin Hamiltonian provides an excellent fit to our spin-wave data. The antisymmetric Dzyloshinskii-Moriya interaction is the primary source of anisotropy and explains the low-temperature magnetization and spin structure. In addition, combined thermodynamic and neutron scattering measurements reveal that the phase transition to the ordered ground-state is unusual. At low temperatures, application of a magnetic field induces a transition between states with different non-trivial spin- textures. The transition indicated by the sudden increase in the magnetization arises as the spins on alternating layers, which are previously oppositely canted due to the ferromagnetic interplane coupling, rotate 1800 to align the canting moment along the c-axis. These observations are consistent with the ordering induced by the Dzyloshinskii-Moriya interaction. Elastic neutron scattering measurements in high field verify the 180' spin rotation at the transition. The critical behavior in jarosite cannot be categorized by any known universality classes. We propose a scenario where both 2D XY and 2D Ising symmetries are present.
by Kittiwit Matan.
Ph.D.

La déstabilisation de l'ordre antiferromagnétique de Néel au profit de nouvelles phases quantiques à température nulle à deux dimensions est envisageable grâce au phénomène de frustration magnétique. Le modèle théorique de spins Heisenberg S=1/2 répartis sur le réseau bidimensionnel frustré kagome, constitué de triangles joints uniquement par leurs sommets, est susceptible de stabiliser des phases quantiques originales de liquides de spin, qui ne présentent aucune brisure de symétrie à T = 0. Cette thèse a été consacrée à l'étude expérimentale de deux types de composés de spins S=1/2 (Cu2+) à géométrie kagome à l'aide de techniques spectroscopiques locales, la RMN et la μSR, ainsi que de mesures thermodynamiques (susceptibilité magnétique, chaleur spécifique). Dans Mg-herbertsmithite, la frustration est générée par une interaction d'échange premiers voisins antiferromagnétique J et est responsable d'un comportement liquide de spin jusqu'à des températures de l'ordre de J/10000. Par rapport au composé isostructural antérieur, Zn-herbertsmithite, nous avons montré qu'il possédait des propriétés physiques similaires tout en permettant une caractérisation fine du taux de défauts de substitutions Cu/Mg. Nos expériences réalisées à partir d'échantillons contrôlés permettent d'étudier finement l'origine des plateaux de relaxation observés en μSR à basse température en lien avec l'existence des défauts de spins interplans. La kapellasite et l'haydéite possèdent des interactions ferromagnétiques (J1) et antiferromagnétiques (Jd), offrant la possibilité d'explorer le diagramme de phases générées par la compétition de ces interactions sur le réseau kagome. Pour la kapellasite, nos mesures de μSR démontrent le caractère liquide de spin jusqu'à T ≈ J1/1000. La dépendance en température de la susceptibilité magnétique sondée par RMN du 35Cl ainsi que de la chaleur spécifique permettent d'évaluer le rapport Jd/J1 = 0.85, qui localise classiquement son fondamental au sein d'une phase originale de spins non coplanaires à 12 sous-réseaux appelée cuboc2. Les interactions présentes dans l'haydéite localisent son fondamental au sein de la phase ferromagnétique, en bon accord avec nos mesures qui indiquent une transition partielle à caractère ferromagnétique à T = 4 K. Cette étude confirme la pertinence du réseau kagome frustré pour la stabilisation de phases quantiques originales et démontre l'existence d'une nouvelle phase liquide de spin sur ce réseau, distincte de celle attendue pour des spins couplés antiferromagnétiquement.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2009.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Cataloged from student submitted PDF version of thesis.
Includes bibliographical references (p. 201-214).
The magnetic properties of the geometrically frustrated quantum magnets clinoatacamite, Cu2(OH)3Cl, and herbertsmithite, ZnCu3(OH)6Cl2, are studied by means of neutron scattering measurements as well as specific heat, susceptibility, and mag-netization measurements. These materials are studied to investigate the nature of the ground state of the spin-1 2 kagomé lattice antiferromagnet, as such a system is considered ideal for the emergence of spin liquid physics. Clinoatacamite, a distorted kagomé lattice antiferromagnet with weak inter-plane coupling, undergoes a Néel or- dering transition at TN ~/~ 6.2 K and shows evidence of a static local moment in the disordered phase below 18 K. Our experiments suggest two-dimensional Ising fluctuations at the Néel transition. A proposed spin ordering model is developed that suggests an order structure below TN and two-dimensional short range order of the kagomé plane spins up to 18 K. The inelastic spectrum is analyzed in terms of spin waves in an ordered kagomé lattice antiferromagnet with a Dzyaloshinskii-Moriya interaction. Herbertsmithite is the first structurally perfect spin- 1 2 kagomé lattice antiferromagnet. Susceptibility, specific heat, and neutron scattering measurements show no sign of any spin freezing or transition to a long range ordered state down to 50 mK. The data shows magnetic excitations extending adjacent to the ground state, suggesting the lack of any measurable spin gap. Several hypotheses are explored as possible explanations for the apparent lack of a spin gap.
(cont.) Dynamic susceptibility data display an unusual scaling relation, suggesting proximity to a quantum critical point. In sum, a wide range of data suggest that herbertsmithite displays a disordered gapless spin liquid ground state.
by Joel Strader Helton.
Ph.D.

Uměle vytvořené systémy spinového ledu jsou vhodným nástrojem pro zkoumání neobvyklých jevů, které se v přírodě dají jen těžko pozorovat. Speciálním případem umělého spinového ledu je kagome mřížka, která umožňuje zkoumat kolektivní chování spinů v látce. Tento systém má řadu předpovězených exotických magnetických fází, které zatím nebyly změřeny a prozkoumány v reálném prostoru. V rámci této práce se zabýváme úpravou kagome mřížky tak, aby mohla být využita ke zkoumání exotických stavů v reálném prostoru. Experimenty provedené na naší upravené mřížce ukazují, že jsme schopni detekovat nízko i vysoko energiové stavy, a tedy, že námi navržená úprava kagome mřížky je vhodná pro zkoumání exotických stavů v reálném prostoru.

In the vast realm of inorganic materials, the Cu2+-containing cuprates form one of the richest classes. Due to the combined effect of crystal-field, covalency and strong correlations, all undoped cuprates are magnetic insulators with well-localized spins S=1/2, whereas the charge and orbital degrees of freedom are frozen out. The combination of the spin-only nature of their magnetism with the unique structural diversity renders cuprates as excellent model systems. The experimental studies, boosted by the discovery of high-temperature superconductivity in doped La2CuO4, revealed a fascinating variety of magnetic behaviors observed in cuprates. A digest of prominent examples should include the spin-Peierls transition in CuGeO3, the Bose-Einstein condensation of magnons in BaCuSi2O6, and the quantum critical behavior of Li2ZrCuO4. The magnetism of cuprates originates from short-range (typically, well below 1 nm) exchange interactions between pairs of spins Si and Sj, localized on Cu atoms i and j. Especially in low-dimensional compounds, these interactions are strongly anisotropic: even for similar interatomic distances |Rij|, the respective magnetic couplings Jij can vary by several orders of magnitude. On the other hand, there is an empirical evidence for the isotropic nature of this interaction in the spin space: different components of Si are coupled equally strong. Thus, the magnetism of cuprates is mostly described by a Heisenberg model, comprised of Jij(Si*Sj) terms. Although the applicability of this approach to cuprates is settled, the model parameters Jij are specific to a certain material, or more precisely, to a particular arrangement of the constituent atoms, i.e. the crystal structure. Typically, among the infinite number of Jij terms, only several are physically relevant. These leading exchange couplings constitute the (minimal) microscopic magnetic model. Already at the early stages of real material studies, it became gradually evident that the assignment of model parameters is a highly nontrivial task. In general, the problem can be solved experimentally, using elaborate measurements, such as inelastic neutron scattering on large single crystals, yielding the magnetic excitation spectrum. The measured dispersion is fitted using theoretical models, and in this way, the model parameters are refined. Despite excellent accuracy of this method, the measurements require high-quality samples and can be carried out only at special large-scale facilities. Therefore, less demanding (especially, regarding the sample requirements), yet reliable and accurate procedures are desirable. An alternative way to conjecture a magnetic model is the empirical approach, which typically relies on the Goodenough-Kanamori rules. This approach links the magnetic exchange couplings to the relevant structural parameters, such as bond angles. Despite the unbeatable performance of this approach, it is not universally applicable. Moreover, in certain cases the resulting tentative models are erroneous. The recent developments of computational facilities and techniques, especially for strongly correlated systems, turned density-functional theory (DFT) band structure calculations into an appealing alternative, complementary to the experiment. At present, the state-of-the-art computational methods yield accurate numerical estimates for the leading microscopic exchange couplings Jij (error bars typically do not exceed 10-15%). Although this computational approach is often regarded as ab initio, the actual procedure is not parameter-free. Moreover, the numerical results are dependent on the parameterization of the exchange and correlation potential, the type of the double-counting correction, the Hubbard repulsion U etc., thus an accurate choice of these crucial parameters is a prerequisite. In this work, the optimal parameters for cuprates are carefully evaluated based on extensive band structure calculations and subsequent model simulations. Considering the diversity of crystal structures, and consequently, magnetic behaviors, the evaluation of a microscopic model should be carried out in a systematic way. To this end, a multi-step computational approach is developed. The starting point of this procedure is a consideration of the experimental structural data, used as an input for DFT calculations. Next, a minimal DFT-based microscopic magnetic model is evaluated. This part of the study comprises band structure calculations, the analysis of the relevant bands, supercell calculations, and finally, the evaluation of a microscopic magnetic model. The ground state and the magnetic excitation spectrum of the evaluated model are analyzed using various simulation techniques, such as quantum Monte Carlo, exact diagonalization and density-matrix renormalization groups, while the choice of a particular technique is governed by the dimensionality of the model, and the presence or absence of magnetic frustration. To illustrate the performance of the approach and tune the free parameters, the computational scheme is applied to cuprates featuring rather simple, yet diverse magnetic behaviors: spin chains in CuSe2O5, [NO]Cu(NO3)3, and CaCu2(SeO3)2Cl2; quasi-two-dimensional lattices with dimer-like couplings in alpha-Cu2P2O7 and CdCu2(BO3)2, as well as the 3D magnetic model with pronounced 1D correlations in Cu6Si6O18*6H2O. Finally, the approach is applied to spin liquid candidates --- intricate materials featuring kagome-lattice arrangement of the constituent spins. Based on the DFT calculations, microscopic magnetic models are evaluated for herbertsmithite Cu3(Zn0.85Cu0.15)(OH)6Cl2, kapellasite Cu3Zn(OH)6Cl2 and haydeeite Cu3Mg(OH)6Cl2, as well as for volborthite Cu3[V2O7](OH)2*2H2O. The results of the DFT calculations and model simulations are compared to and challenged with the available experimental data. The advantages of the developed approach should be briefly discussed. First, it allows to distinguish between different microscopic models that yield similar macroscopic behavior. One of the most remarkable example is volborthite Cu3[V2O7](OH)2*2H2O, initially described as an anisotropic kagome lattice. The DFT calculations reveal that this compound features strongly coupled frustrated spin chains, thus a completely different type of magnetic frustration is realized. Second, the developed approach is capable of providing accurate estimates for the leading magnetic couplings, and consequently, reliably parameterize the microscopic Hamiltonian. Dioptase Cu6Si6O18*6H2O is an instructive example showing that the microscopic theoretical approach eliminates possible ambiguity and reliably yields the correct parameterization. Third, DFT calculations yield even better accuracy for the ratios of magnetic exchange couplings. This holds also for small interchain or interplane couplings that can be substantially smaller than the leading exchange. Hence, band structure calculations provide a unique possibility to address the interchain or interplane coupling regime, essential for the magnetic ground state, but hardly perceptible in the experiment due to the different energy scales. Finally, an important advantage specific to magnetically frustrated systems should be mentioned. Numerous theoretical and numerical studies evidence that low-dimensionality and frustration effects are typically entwined, and their disentanglement in the experiment is at best challenging. In contrast, the computational procedure allows to distinguish between these two effects, as demonstrated by studying the long-range magnetic ordering transition in quasi-1D spin chain systems. The computational approach presented in the thesis is a powerful tool that can be directly applied to numerous S=1/2 Heisenberg materials. Moreover, with minor modifications, it can be largely extended to other metallates with higher value of spin. Besides the excellent performance of the computational approach, its relevance should be underscored: for all the systems investigated in this work, the DFT-based studies not only reproduced the experimental data, but instead delivered new valuable information on the magnetic properties for each particular compound. Beyond any doubt, further computational studies will yield new surprising results for known as well as for new, yet unexplored compounds. Such "surprising" outcomes can involve the ferromagnetic nature of the couplings that were previously considered antiferromagnetic, unexpected long-range couplings, or the subtle balance of antiferromagnetic and ferromagnetic contributions that "switches off" the respective magnetic exchange. In this way, dozens of potentially interesting systems can acquire quantitative microscopic magnetic models. The results of this work evidence that elaborate experimental methods and the DFT-based modeling are of comparable reliability and complement each other. In this way, the advantageous combination of theory and experiment can largely advance the research in the field of low-dimensional quantum magnetism. For practical applications, the excellent predictive power of the computational approach can largely alleviate designing materials with specific properties.

碩士
國立清華大學
物理研究所
82
Critical exponents and some amplitude combinations of self- avoiding walks and polygons are believed to be universal. The main intension of this research is to check whether those constants on planar lattice are universal by studing the kagome lattice. We write a Fortran program to enumerate the self- avoiding polygon, the mean square radius of gyration of self- avoiding polygon and the moments of the area of self-avoiding polygon simultaneously. We apply some methods of series analysis on those series to get the values of the estimation. The universal constants as metioned above have not been rigorously proved but their values have been predicted. Some estimation values are very close to the predicted values but some not.

碩士
國立清華大學
物理研究所
82
We have calculated exactly the total number Cn of n-step self- avoiding walks (SAWs) and the mean- square end-to-end distance n of n-step SAWs on the kagome lattice up to 30 steps; the corresponding mean-square radius of gyration n and the mean-square distance of a monomer from the origin n of n-step SAWs are calculated exactly up to 28 steps. From these data we estimated several critical parameters, including the amplitude ratios and the critical exponents. We estimate the amplitude ratios F/C, G/C and F/G (C for n, F for n, G for n) by Meir's method. Our numerical results are in agreement with the results of Caracciolo et al obtained from the Monte Carlo study of SAWs on the square lattice. The critical exponents γ and ν are estimated by the first order differential approximants and agree with the results of Nienhuis.

Within condensed matter physics, systems with strong electronic correlations give rise to fascinating phenomena which characteristically require a physical description beyond a one-electron theory, such as high temperature superconductivity, or Mott metal-insulator transitions. In this thesis, a class of strongly correlated electron systems is considered. These systems exhibit fractionally charged excitations with charge +e/2 or -e/2 in two dimensions (2D) and three dimensions (3D), a consequence of both strong correlations and the geometrical frustration of the interactions on the underlying lattices. Such geometrically frustrated systems are typically characterized by a high density of low-lying excitations, leading to various interesting physical effects. This thesis constitutes a study of a model of spinless fermions on the geometrically frustrated kagome lattice. Focus is given in particular to the regime in which nearest-neighbour repulsions V are large in comparison with hopping t between neighbouring sites, the regime in which excitations with fractional charge occur. In the classical limit t = 0, the geometric frustration results in a macroscopically large ground-state degeneracy. This degeneracy is lifted by quantum fluctuations. A low-energy effective Hamiltonian is derived for the spinless fermion model for the case of 1/3 filling in the regime where |t| << V . In this limit, the effective Hamiltonian is given by ring-exchange of order ~ t^3/V^2, lifting the degeneracy. The effective model is shown to be equivalent to a corresponding hard-core bosonic model due to a gauge invariance which removes the fermionic sign problem. The model is furthermore mapped directly to a Quantum Dimer model on the hexagonal lattice. Through the mapping it is determined that the kagome lattice model exhibits plaquette order in the ground state and also that fractional charges within the model are linearly confined. Subsequently a doped version of the effective model is studied, for the case where exactly one spinless fermion is added or subtracted from the system at 1/3 filling. The sign of the newly introduced hopping term is shown to be removable due to a gauge invariance for the case of hole doping. This gauge invariance is a direct result of the bipartite nature of the hole hopping and is confirmed numerically in spectral density calculations. For further understanding of the low-energy physics, a derivation of the model gauge field theory is presented and discussed in relation to the confining quantum electrodynamic in two dimensions. Exact diagonalization calculations illustrate the nature of the fractional charge confinement in terms of the string tension between a bound pair of defects. The calculations employ topological symmetries that exist for the manifold of ground-state configurations. Dynamical calculations of the spectral densities are considered for the full spinless fermion Hamiltonian and compared in the strongly correlated regime with the doped effective Hamiltonian. Calculations for the effective Hamiltonian are then presented for the strongly correlated regime where |t| << V . In the limit g << |t|, the fractional charges are shown to be effectively free in the context of the finite clusters studied. Prominent features of the spectral densities at the Gamma point for the hole and particle contributions are attributed to approximate eigenfunctions of the spinless fermion Hamiltonian in this limit. This is confirmed through an analytical derivation. The case of g ~ t is then considered, as in this case the confinement of the fractional charges is observable in the spectral densities calculated for finite clusters. The bound states for the effectively confined defect pair are qualitatively estimated through the solution of the time-independent Schroedinger equation for a potential which scales linearly with g. The double-peaked feature of spectral density calculations over a range of g values can thus be interpreted as a signature of the confinement of the fractionally charged defect pair. Furthermore, the metal-insulator transition for the effective Hamiltonian is studied for both t > 0 and t < 0. Exact diagonalization calculations are found to be consistent with the predictions of the effective model. Further calculations confirm that the sign of t is rendered inconsequential due to the gauge invariance for g in the regime |t| << V . The charge-order melting metal-insulator transition is studied through density-matrix renormalization group calculations. The opening of the energy gap is found to differ for the two signs of t, reflecting the difference in the band structure at the Fermi level in each case. The qualitative nature of transition in each case is discussed. As a step towards a realization of the model in experiment, density-density correlation functions are introduced and such a calculation is shown for the plaquette phase for the effective model Hamiltonian at 1/3 filling in the absence of defects. Finally, the open problem of statistics of the fractional charges is discussed.

Strongly correlated systems on geometrically frustrated lattices can stabilize a large number of interesting phases that includes a wide array of novel Mott insulators in both bosonic and electronic systems. Charge fluctuations in a Mott insulator are suppressed due to strong mutual interaction among the particles. The presence of frustration is of particular importance as the physics it offers is often rich, unexpectedly complicated, and continues to raise many open questions. The thesis elucidates some of these issues on a kagome lattice where strong interactions among the particles in the Mott phase impose non-trivial local constraints depending on the filling fraction on the lattice. These Mott insulators, in addition to featuring unusual magnetic and/or charge ordering, can also harbor topologically ordered states of quantum matter, e.g., resonating valence bond liquids realized in certain quantum dimer models on non-bipartite lattices. The dimer models can be regarded as low-energy effective theories for different types of bosonic models in the strong-coupling limit. Exploring this connection is a central theme of this thesis with the aim of realizing novel strongly correlated ground states. Past studies of these models have revealed the existence of various ordered and disordered phases with distinct signatures. Among these low-energy phases, the presence of a stable topological liquid at a particular point, known as Rokhsar-Kivelson point, in the phase diagram is notable. The classical versions of the dimer model are also known to have garnered a vast interest in various fields ranging from problems of pure mathematical origin to ones in physical chemistry as well as statistical physics. Pioneered by Kasteleyn, several analytical works came forward to exactly calculate the partition function of the problem from which other physical observables can be derived. Classical numerical methods are extensively applied to these models to verify the analytical predictions. We introduce a new classical algorithm here to compute the correlation functions of a classical dimer model on a square (bipartite) and a triangular (non-bipartite) lattice based on a tensor network construction. The method, called tensor network renormalization group, turns out to be a powerful tool for simulating short-ranged gapped systems as inferred from our results benchmarked against the classical Monte-Carlo technique and compared with past analytical studies. One should note that the quantum dimer model at the Rokhsar-Kivelson point can also be described as an infinite temperature canonical ensemble of classical dimers because of the particular structure of the ground state which is an equal weight superposition in the configuration manifold. The geometry of the lattice plays a pivotal role in deciding the nature of the phases that arise in the dimer models. Many physical properties of the dimer liquid phase can be extracted in the simple classical setting which certainly allows for a deep understanding of the classical models to be developed. The liquid phase is gapped on non-bipartite lattices and gapless on bipartite lattices, which is reflected in the decay of correlation functions with spatial distances. In general on non-bipartite lattices, the topological nature of the dimer liquid is characterized by a Z2 topological order which survives even when the model is perturbed away from the Rokhsar-Kivelson point. Stability of this liquid phase not only depends on the lattice geometries but notably on dimer concentrations also. In this context, we focus on a particular variant of the dimer model on a triangular lattice which is known as the quantum fully packed loop model. The model is composed of nonintersecting closed loops made of dimers and governed by the same Hamiltonian as the quantum dimer model. The loop model provides an effective low-energy description of a strongly correlated bosonic system at 1/3 filling on the kagome lattice. The corresponding Bose-Hubbard Hamiltonian consists of nearest-neighbor hopping and all possible repulsive interactions within a hexagonal plaquette. Conspicuous features of the zero-temperature phase diagram for this model include (i) presence of a stable Z2 liquid even without any Rokhsar-Kivelson potential term (in distinction to the standard quantum dimer model), and (ii) an unconventional phase transition from the liquid phase to a novel crystalline phase that has nematic order (dubbed lattice nematic). For a deeper understanding of the physics, a mapping to an Ising gauge theory is presented. The gauge theoretic description provides a useful way to predict the nature of the quantum phase transition to lie in the O(3) universality class. Finally a fermionic model at the same 1/3 filling is considered in which the ground state exhibits a number of exotic local orderings resulting from the spin-charge interplay of electrons. The Hamiltonian comprises nearest-neighbor hopping, strong on-site Coulomb interaction, and repulsive interaction terms only between nearest-neighbors. In the strong correlation limit, this fermionic problem maps to a two-color fully packed loop model – a model in which the loop segments carry an additional quantum number as color on a honeycomb lattice. The effective theory is governed by coherent three-particle ring exchanges and nearest-neighbor antiferromagnetic spin exchanges. The competition between these two leads to a phase diagram composed of a novel plaquette ordered state (known as the plaquette phase) that undergoes phase transition to a new kind of charge ordered state which we call a short loop phase. From our numerical analysis, we conclude that the plaquette phase features an unusual antiferromagnetic order with gapless spin excitations while the charge-ordered state is subjugated by spin fluctuations of localized electrons arranged in small hexagonal loops on the kagome lattice.

碩士
國立中山大學
物理學系研究所
103
Since the single layer graphene are observed by Andre Geim and Konstantin Novoselov, the quantum phenomena emerging from graphene as attracted many physicists’s attentions in the two dimensional lattice structure. One of the important quantum phenomena is the unconventional quantum Hall effect which the Hall plateau (Chern number pattern) does not show successive integers but only odd integers. However, there are some other problems to calculate the quantum Hall conductivity, such as the determination of the Chern number pattern from a specific lattice structure and the reduction of the computational time. Motivated by these issues, I use Yasuhiro Hatsugai’s topological approaches to investigate the quantum Hall effect of other lattices. I use Hatsugai’s method to calculate the quantum Hall conductivity in two different lattices: Kagomé and star lattices. I find that the Dirac-like regions of quantum Hall conductivity also exists in both Kagomé and star lattices, which is similar to graphene. Unlike the graphene, the star lattice shows complicated Chern number structure in the electron-hole like regions. Hopefully, the Chern number pattern of the star lattice predicted by us can be determined by experiments in the near future.