We investigate the self-organization of strongly interacting particles confined in 1D and 2D. We consider hardcore bosons in spinless Hubbard lattice models with short range interactions. We show that, many-body orders with topological characteristics emerge, at different energy bands separated by large gaps. These topological orders manifest in the way the particles organize in real space to form states with different energy. Each of these states contains topological defects/condensations whose Euler characteristic can be used as a topological number to categorize states belonging to the same energy band. We provide analytical formulas for this topological number and the full energy spectrum of the system for both sparcly and densely filled systems. Furthermore, we discuss the connection with the Gauss-Bonnet theorem of differential geometry, by using the curvature generated in real space by the particle structures. Our result is a demonstration of how topological orders can arise in many-body systems with simple underlying rules, without considering the spin or long-range microscopic interactions.
We investigate the self-organization of strongly interacting particles confined in 1D and 2D. We consider hardcore bosons in spinless Hubbard lattice models with short range interactions. We show that, many-body orders with topological characteristics emerge, at different energy bands separated by large gaps. These topological orders manifest in the way the particles organize in real space to form states with different energy. Each of these states contains topological defects/condensations whose Euler characteristic can be used as a topological number to categorize states belonging to the same energy band. We provide analytical formulas for this topological number and the full energy spectrum of the system for both sparcly and densely filled systems. Furthermore, we discuss the connection with the Gauss-Bonnet theorem of differential geometry, by using the curvature generated in real space by the particle structures. Our result is a demonstration of how topological orders can arise in many-body systems with simple underlying rules, without considering the spin or long-range microscopic interactions.
Abstract We investigate the self-organization of point-particles with short-range interactions modeled via simple 1D and 2D Hubbard-like models. We show how various properties emerge such as, boson-like ordering leading to topological structures in real space, via the clustering of the particles at discrete energy levels, which can be analyzed using a network/graph mathematical language. We calculate analytically the number of clusters, the correlations between them and topological numbers like the Euler characteristic, deriving different organizational schemes and entanglement entropy scaling laws. All calculations are performed for an arbitrary number of particles N and energy of the system E . Our results demonstrate how orders with diverse topological/geometric and entanglement features, emerge in strongly interacting many-body systems, that follow minimal microscopic interaction rules.
Many-body phenomena were always an integral part of physics comprising of collective behaviors through self-organization, in systems consisting of many components and degrees of freedom. We investigate the collective behaviors of strongly interacting particles confined in one dimension. We show that many-body orders with topological characteristics can be found at the Mott insulator limit for hardcore bosons, at different fillings, without considering the spin degree of freedom or long-range microscopic interactions. These orders have unique properties like weak or strong quantum correlations (entanglement), quantified by the entanglement entropy, edge excitations/modes and gapped energy spectrum with highly degenerate ground state, bearing resemblance to topologically ordered phases of matter.
We study two-particle states in a Su-Schrieffer-Heeger (SSH) chain with periodic boundary conditions and nearest-neighbor (NN) interactions. The system is mapped into a problem of a single particle in a two-dimensional (2D) SSH lattice with potential walls along specific edges. The 2D SSH model has a trivial Chern number but a non-trivial Zak's phase, the one-dimensional (1D) topological invariant, along specific directions of the lattice, which allow for the presence of topological edge states. Using center-of-mass and relative coordinates, we calculate the energy spectrum of these two-body states for strong interactions and find that, aside from the expected appearance of doublon bands, two extra in-gap bands are present. These are identified as bands of topological states localized at the edges of the internal coordinate, the relative distance between the two particles. As such, the topological states reported here are intrinsically many-body in what concerns their real space manifestation, having no counterpart in single-particle states derived from effective models. Finally, we compare the effect of Hubbard interactions with that of NN interactions to show how the presence of the topological bound states is specific to the latter case.
Quantum dynamics underlies macroscopic systems exhibiting some kind of ordering, such as superconductors, ferromagnets and crystals. Even large scale structures in the Universe and ordering in biological systems appear to be the manifestation of microscopic dynamics ruling their elementary components. The scope of this book is to answer questions such as : how it happens that the mesoscopic/macroscopic scale and stability characterizing those systems are dynamically generated out of the microscopic scale of fluctuating quantum components; how quantum particles coexist and interact with classically behaving macroscopic objects, e.g. vortices, magnetic domains and other topological defects. The quantum origin of topological defects and their interaction with quanta is a crucial issue for the understanding of symmetry breaking phase transitions and structure formation in a wide range of systems from condensed matter to cosmology. Deliberately not discussing other important problems, primarily renormalization problems, this book provides answers to such questions in a unitary, self-consistent physical and mathematical framework, which makes it unique in the panorama of existing texts on a similar subject. Crystals, ferromagnets and superconductors appear to be macroscopic quantum systems, i.e. their macroscopic properties cannot be explained without recourse to the underlying quantum dynamics. Recognizing that quantum field dynamics is not confined to the microscopic world is one of the achievements of this book, also marking its difference from other texts. The combined use of algebraic methods, and operator and functional formalism constitutes another distinctive, valuable feature
Chiral topological orders formed in correlated fermion systems have been widely explored. However, the mechanism of how they emerge from interacting fermions is still unclear. Here, we propose a susceptibility condition. Under this condition, we show that chiral topological orders can spontaneously take place in correlated fermion systems. The condition leads to a low-energy effective theory of bosons with strong frustration, mimicking flat-band systems. The frustration then melts the long-range orders and results in topological orders with time-reversal symmetry breaking. We apply the theory to strongly correlated semiconductors doped to the metallic phase. An excitonic topological order with semionic excitations and a chiral excitonic edge state is revealed. We also discuss the application to frustrated magnets. The theory predicts a chiral spin-liquid state, which is numerically confirmed by our tensor network calculations. These results demonstrate an indicator for chiral topological orders, which bridges the existing gap between interacting fermions and correlated topological matter.
We discuss a protocol for growing states with topological order in interacting many-body systems using a sequence of flux quanta and particle insertion. We first consider a simple toy model, the superlattice Bose-Hubbard model, to explain all required ingredients. Our protocol is then applied to fractional quantum Hall systems in both, continuum and lattice. We investigate in particular how the fidelity, with which a topologically ordered state can be grown, scales with increasing particle number N. For small systems, exact diagonalization methods are used. To treat large systems with many particles, we introduce an effective model based on the composite fermion description of the fractional quantum Hall effect. This model also allows to take into account the effects of dispersive bands and edges in the system, which will be discussed in detail.