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Strong effective photon-photon interactions mediated by atom-photon couplings have been routinely achievable in QED setups for some time now. Recently, there have been several proposals to push the physics of interacting photons into many- body distributed architectures. The essential idea is to coherently couple together arrays of QED resonators, such that photons can hop between resonators while interacting with each other inside each resonator. These proposed structures have attracted intense theoretical attention while simultaneously inspiring experimental efforts to realise this novel regime of strongly-correlated many-body states of light. A central challenge of both theoretical and practical importance is to understand the physics of such coupled resonator arrays (CRAs) beyond equilibrium, when unavoidable (or sometimes even desired) photon loss processes are accounted for. This thesis presents several studies whose purpose can roughly be divided in two aims. The first part studies just what constitutes a valid physical and computational representation of non-equilibrium driven-dissipative CRAs. Addressing these ques- tions constitutes essential groundwork for further investigations of CRA phenomena, as numerical experiments are likely to guide and interpret near-future experimen- tal array observations. The relatively small body of existing work on CRAs out of equilibrium has often truncated their full, rich physics. It is important to establish the effects and validity of these approximations. To this end we introduce powerful numerical algorithms capable of efficiently simulating the full dynamics of CRAs, and use them to characterise the non-equilibrium steady states of arrays reached under the combined influence of dissipation and pumping. Having established the rigour necessary to realistically describe CRAs, we exam- ine two novel phenomena observable in near-future small arrays. Firstly we relate a counter-intuitive ‘super bunching’ in the statistics of photons emitted from arrays engineered to demonstrate strong effective photon-photon repulsion at the single and two-photon level, to an interplay between the underlying eigen-structure and details of the non-equilibrium operation. Secondly we characterise a dynamical phenomenon in which domains of ‘frozen’ photons remain trapped in sufficiently nonlinear arrays. Finally we present a preliminary characterisation of a previously unexplored phase diagram of arrays under coherent two-photon pumping. Com- petition between the coherence injected by the pumping, photon interactions and delocalisation processes lead to interesting new physical signatures.

Orientador: Eduardo Miranda / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-10-31T13:24:18Z (GMT). No. of bitstreams: 1 HoyosNeto_JoseAbel_D.pdf: 1434769 bytes, checksum: 70acbb99e5c8d9636d4209b0919b56ca (MD5) Previous issue date: 2005 / Resumo: O propósito desse trabalho é estudar o papel da desordem em sistemas de spins fortemente interagentes de baixa dimensionalidade. Do ponto de vista teórico, cadeias de spin são extremamente atrativas por apresentarem uma nova física de baixas energias que vem da competição entre o ordenamento magnético e as .utuações quânticas. A introdução de desordem, onipresente no contexto experimental, é um elemento que pode desestabilizar as fases puras dando origem a uma nova física. Essa é a motivação principal do estudo de seu papel. Neste trabalho nós estudamos 4 sistemas de spins antiferromagnéticos desordenados:(i ) as escadas de spins-1/2 dos tipos 2 pernas e zig-zag, (ii ) as cadeias isotrópicas de spins SU(N), (iii ) a cadeia anisotrópica de spins SU(4), e (iv ) revisitamos a cadeia de spins-1/2. O estudo destes sistemas foi realizado aplicando generalizações da técnica do grupo de renormalização no espaço real para desordem forte. No caso do primeiro sistema, nós mostramos que as escadas de spins sempre renormalizam em cadeias de spins muito bem conhecidas. A escada de 2 pernas renormaliza para uma cadeia de spins-1/2 dimerizada antiferromagnética desordenada e, portanto, possui duas fases. Para dimerização forte ou equivalentemente desordem fraca, o sistema se encontra na fase de Haldane onde há um "gap" e a desordem é irrelevante. Para dimerização fraca ou equivalentemente desordem forte, o "gap" de Haldane se fecha e o sistema se encontra numa fase de Griffiths onde as quantidades termodinâmicas são controladas por um expoente não universal denominado expoente dinâmico z . Em contraste, a escada zig-zag renormaliza ou para uma cadeia de spins-1/2 antiferromagnética desordenada ou para uma cadeia de spins com acoplamentos ferro e antiferromagnéticos desordenada. Se a desordem e a frustração são suficientemente fracas, a escada renormaliza para a primeira cadeia, caso contrário esta pertence à mesma classe de universalidade da segunda. Além disso, relacionamos o expoente dinâmico da cadeia de spins com acoplamentos ferro e antiferromagnéticos com a distribuição de ponto fixo desses acoplamentos. Finalmente, através de argumentos simples, consideramos dizimações de acoplamentos correlacionados. Nesse caso, torna-se bem claro que a frustração é responsável pelo surgimento de acoplamentos ferromagnéticos que põem a escada na bacia de atração do ponto fixo das cadeias com acoplamentos ferro e antiferromagnéticos. Com relação à cadeia SU(N), desenvolvemos uma generalização do método do grupo de renormalização para desordem forte para uma cadeia isotrópica antiferromagnética de spins que pertencem à representações irredutíveis totalmente anti-simétricas do grupo SU(N), com N maior ou igual a 2. Conseguimos resolver as equações de fluxo analiticamente e descobrimos que essas cadeias pertencem a uma nova classe de universalidade cujos pontos fixos são de desordem infinita e, por tal motivo, nossos resultados se tornam assintoticamente exatos. Próximo a esses pontos fixos, os expoentes característicos são universais, i. e., independentes da desordem inicial da cadeia, e dependem somente do posto N do grupo de simetria. Devido às similaridades entre as regras de aglomeração de spins quando da dizimação de uma cadeia de spins com acoplamentos ferro e antiferromagnéticos e da dizimação da cadeia isotrópica de spins SU(N) no limite N ® µ , fomos capazes de calcular analiticamente, através de expansões de 1/N , a função correlação da primeira cadeia.Com relação à cadeia de spins SU(4), modificamos a generalização do método de grupo de renormalização para levar em conta a anisotropia dos acoplamentos. Conseguimos determinar o diagrama de fases através de cálculos analíticos e numéricos. Todos os pontos fixos encontrados são universais e de desordem infinita, entretanto, os expoentes característicos dependem de uma maneira não trivial da anisotropia do sistema. Por fim, revisitamos a cadeia de spins-1/2 antiferromagnética. Calculamos a amplitude da função de correlação média e a relacionamos com a amplitude da entropia de emaranhamento da mesma. Além disso, argumentamos em favor da universalidade dessas amplitudes / Abstract: The purpose of this thesis is the study of the role of quenched disorder in low-dimensional strongly interacting quantum spin systems. From the theoretical point of view, spin chains are extremely attractive due to their unconventional behavior that originates in the competition between magnetic ordering and quantum fluctuations. The introduction of disorder, ubiquitous in experimental realizations, is an element that can destabilize the clean phases giving rise to new physical behavior. That is the main motivation of this study. In this thesis, we study 4 random antiferromagnetic spin systems: (i ) the spin-1/2 two-leg and zigzag ladders, (ii ) the isotropic SU(N) spin chains, (iii ) the anisotropic SU(4) spin chain, and (iv ) we also revisit the spin-1/2 chain. For such a task, we use generalizations of the strong disorder real-space renormalization group method. Concerning the first systems, we show that the ladders are always renormalized to well-known spin chains. The two-leg ladder is renormalized to a random dimerized antiferromagnetic spin-1/2 chain, hence exhibiting two phases. For strong dimerization or equivalently weak disorder the system is in the gapful Haldane phase where disorder is irrelevant. Otherwise, the Haldane gap closes and the system is driven into a nonuniversal Griffiths phase, where the thermodynamical quantities are controled by the dynamical exponent z. In contrast, the zigzag ladder is renormalized either to a random antiferromagnetic spin-1/2 chain or to a random spin chain with both ferro- and antiferromagnetic couplings. If the randomness and frustration are sufficiently weak, the ladder is renormalized to the former chain, but otherwise it belongs to the same universality class of the latter one. In addition, we related the dynamical exponent of the ferro- and aniferromagnetic spin chain with its fixed point coupling constant distributions. Moreover, through simple qualitative arguments, we determined the phase diagram of the zigzag ladder with correlated disorder. That calculation clearly showed that frustration is responsible for the appearance of ferromagnetic couplings, which place the system in the basin of attraction of the fixed point of the ferro- and antiferromagnetic spin chains. With respect to theSU(N) spin chain, we developed a generalization of the strong-disorder renormalization group method to the case of an antiferromagnetic isotropic spin chain whose spins belong to the totally antisymmetric irreducible representations of the SU(N) group, with N greater than or equal to 2. We solved the flow equations analytically and found that such chains belong to a new universality class whose fixed point distributions are characterized by infinite disorder, rendering our results asymptotically exact. The characteristic exponents of these fixed points are universal, i. e., independent of the bare disorder, and depend only on the symmetry group rank. Due to the similarities of the spin clustering rules between the ferro- and antiferromagnetic spin chain and the isotropic SU(N) spin chain in the limit of N ® µ, we were able to analytically calcu- late, through a 1/N expansion, the mean correlation function of the former chain. In the case of the SU(4) spin chain, we modified the generalization of the renormalization group method to take into account the coupling anisotropy. We determined the phase diagram through analytical and numerical calculations. All fixed points found are universal and of infinite-randomness type. However, the characteristic exponents depend in a nontrivial fashion on the anisotropy. Finally, we revisited the antiferromagnetic spin-1/2 chain. We calculated the amplitude of the mean correlation function and related it with the amplitude of the entanglement entropy of the chain. In addition, we gave arguments in favor of the universality of these amplitudes / Doutorado / Física da Matéria Condensada / Doutor em Ciências

In this thesis we consider various applications the gauge/gravity duality to study transport in strongly coupled systems. The main content is organized in three parts. In the first part we investigate the interrelation between dimensionality and strength of interactions. It is known that the dynamics of systems in Condensed Matter and General Relativity simplify for high dimensionality. Therefore, in this limit of large dimensionality, analytic results are usually possible. We study the dependence of the conductivity and the entanglement entropy on the space-time dimensionality in two different models of holographic superconductors: one dual to a quantum critical point with spontaneous symmetry breaking, and the other modelled by a charged scalar that condenses at a sufficiently low temperature in the presence of a Maxwell field. In the large dimensionality limit we obtain explicit analytical results for the conductivity at zero temperature and the entanglement entropy. Our results suggest that, as dimensionality increases, the condensate interactions become weaker. In the second part we first investigate the Drude weight and the related Mazur-Suzuki (MS) bound in a broad variety of strongly coupled field theories with a gravity dual at nonzero temperature and chemical potential. We show that the MS bound, which in the context of Condensed Matter provides information on the integrability of the theory, is saturated in Einstein-Maxwell-dilaton (EMd) and R-charged backgrounds. We then explore EMd theories with U(1) spontaneous symmetry breaking, and gravity duals of non-relativistic field theories, in which the MS bound is not saturated. Finally, we study the effect of a weak breaking of translational symmetry and we show that the MS bound sets a lower bound on the DC conductivity for a given scattering time. In the last part, we study asymptotically anti de Sitter Brans-Dicke (BD) backgrounds as effective models of metals with a varying coupling constant. We show that, for translational invariant backgrounds, the zero-frequency conductivity (dc conductivity) deviates from the universal result of EMd models. Once translational symmetry is broken, the shear viscosity to entropy ratio is always lower than the Kovtun-Son-Starinets bound, in line with other gravity backgrounds with momentum relaxation. In the BD models studied, we observed insulating like features in the dc conductivity. However, the module and argument of the optical conductivity at intermediate frequencies are not consistent with cuprates experimental results, even assuming several channel of momentum relaxation. We have also included the research carried out in the first year of the PhD as appendices. The topics studied in these appendices lie outside the main framework of this thesis.

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.

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.

Cette thèse vise à utiliser le t-J Hamiltonian de la corrélation forte pour mieux comprendre la micro-fonctionnalité des scénarios de matériau condensé. Un des problèmes qui existe depuis longtemps est que pour ce type de modèle comme Hubbard Hamiltonian ou t-J Hamiltonian avec une corrélation forte ne peut pas être résolu complètement analytiquement. Par conséquent, quand on aborde ces modèles, il est important de les exploiter de façon numérique. Dans cette thése, nous utiliserons la manière qui s'appelle "Renormalized Mean-Field Theory"(RMFT) pour le t-J Hamiltonian. Grâce à M. Gutzwiller, ce que nous devons faire est simplement de chiffrer des paramètres qui incluent l'influence de la corrélation électronique et de les mettre avant chaque partie du Hamiltonian. Après ce calcul, nous calculerons l'Hamiltonian du champ moyen de manière standard. Ceci sera notre façon principale pour aborder des questions physiques. Ensuite, nous l'appliquerons sur deux systèmes. Le premier est la mystique de supraconducteur à haute température. Après sa découverte il y a 30 ans, on ne peut pas encore définir une théorie pour expliquer sa micromécanique de manière appropriée. Cependant, avec des équipements avancés, on peut faire des expériences correctement et obtenir des résultats exacts. Ces preuves nous facilitent l'élaboration d'une bonne théorie, même s'il est aussi très difficile d'inclure tous les phénomènes ensemble. Nous avons obtenu des résultats et par rapport aux expériences, ils sont similaires qualitativement. Nous montrerons les détails dans le texte. Le deuxième système qui nous intéresse est le mouvement d'électron dans un champ magnétique fort. Le papillon d'Hofstadter et son modèle, l'Hamiltonian de Harper-Hofstadter ont obtenu un grand succès à décrire la mécanique d'électrons libres aux treillis. Donc il est ainsi intéressant de se demander ce qu'il se passera si nous remplaçons des électrons libres avec ceux qui s'interagissent. D'ailleurs, t-J Hamiltonian s'utilise comme bon modèle à le découvrir. Nous allons comparer nos résultats avec ceux de la diagonalisation exacte. Nous proposerons des découvertes intéressantes qui désormais seront réalisées par l'expérience d'atome froide. / This thesis is aiming in utilizing the strongly correlated t-J Hamiltonian for better understanding the microscopic pictures of certain condensed matter scenario. One of the long existing issues in the Hubbard model and its extreme version, t-J model, lies in the fact that there is not an analytical way of solving them. Therefore, when dealing with these models, numerical approaches become very crucial. In this thesis, we will present one of the methods called renormalized mean-field theory (RMFT) and exploit it upon the t-J model. Thanks to the concept proposed by Gutzwiller, all we have to do is to try to include the correlation of electrons, which is mainly the most difficult part, with several renormalization factors. After obtaining the correct form of these factors, we can apply the routine mean-field theory in solving for the Hamiltonian, which is the principle methodology throughout this thesis. Next, the physical systems that we are interested in consist of two parts. The mystery of High-Tc superconductivity comes first. After 30 years of its discovery, people still cannot settle down a complete microscopic theory in describing this exotic phenomenon. However, with more and more experimental equipment with higher accuracy nowadays, lots of behavior of copperoxide superconductor (also known as cuprate) have been revealed. Those discoveries can definitely help us better understand its microscopic mechanism. Therefore, from the theoretical side, to compare the calculated data with experiments leads us to know whether our theory is on the right track or not. We have produced tons of data and made a decent comparison which will be shown in the main text. The second system we are curious about is the mechanism of electrons under magnetic field. The Hofstadter butterfly along with its Hamiltonian, the Harper-Hofstadter model has achieved great success in describing free electrons' movement with lattice present. Thus, it will be also interesting to ask the question: what will happen if the electrons are correlated. Our RMFT for t-J Hamiltonian, by adding an additional phase in the hopping term, happens to serve as a great preliminary model for answering this question. We will compare the results of ours with our collaborators, who solved this model by a different approach, the exact diagonalization(ED). Together with our calculations, we proposed several discoveries which might be realized by the cold atom experiments in the future.