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arXiv:1707.06722 · Journal of the Physical Society of Japan

Many-Body Chern Numbers of ν = 1/3 and 1/2 States on Various Lattices

Koji Kudo, Toshikaze Kariyado, Yasuhiro Hatsugai

Published Sep 4, 2017• 10 citations
arXivHTMLPDF

Abstract

For various two dimensional lattices such as honeycomb, kagome, and square-octagon, gauge conventions (string gauge) realizing minimum magnetic fluxes that are consistent with the lattice periodicity are explicitly given. Then many-body interactions of lattice fermions are projected into the Hofstadter bands to form pseudopotentials. By using these pseudopotentials, degenerate many-body ground states are numerically obtained. We further formulate a scheme to calculate the Chern number of the ground state multiplet by the pseudopotentials. For the filling factor of the lowest Landau level, $ν=1/3$, a simple scaling form of the energy gap are numerially obtained and the ground state is unique except the three-fold topological degeneracy. This is a quantum liquid, which can be lattice analogue of the Laughlin state. For the $ν=1/2$ case, validity of the composite fermion picture is discussed in relation to the existence of the Fermi surface. Effects of disorder are also described.

DOI
10.7566/jpsj.86.103701

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For various two dimensional lattices such as honeycomb, kagome, and square-octagon, gauge conventions (string gauge) realizing minimum magnetic fluxes that are consistent with the lattice periodicity are explicitly given. Then many-body interactions of lattice fermions are projected into the Hofstadter bands to form pseudopotentials. By using these pseudopotentials, degenerate many-body ground states are numerically obtained. We further formulate a scheme to calculate the Chern number of the ground state multiplet by the pseudopotentials. For the filling factor of the lowest Landau level, ν=1/3ν=1/3ν=1/3, a simple scaling form of the energy gap are numerially obtained and the ground state is unique except the three-fold topological degeneracy. This is a quantum liquid, which can be lattice analogue of the Laughlin state. For the ν=1/2ν=1/2ν=1/2 case, validity of the composite fermion picture is discussed in relation to the existence of the Fermi surface. Effects of disorder are also described.
The Harper-Hofstadter model provides a fractal spectrum containing topological bands of any integer Chern number C. We study the many-body physics that is realized by interacting particles occupying Harper-Hofstadter bands with |C|>1. We formulate the predictions of Chern-Simons or composite fermion theory in terms of the filling factor ν, defined as the ratio of particle density to the number of single-particle states per unit area. We show that this theory predicts a series of fractional quantum Hall states with filling factors ν=r/(r|C|+1) for bosons, or ν=r/(2r|C|+1) for fermions. This series includes a bosonic integer quantum Hall state in |C|=2 bands. We construct specific cases where a single band of the Harper-Hofstadter model is occupied. For these cases, we provide numerical evidence that several states in this series are realized as incompressible quantum liquids for bosons with contact interactions.
We analyze the two-body spectrum within the Hofstadter-Hubbard model on a square lattice through an exact variational ansatz and study the topological properties of its low-lying two-body bound-state branches. In particular, we discuss how the Hofstadter-Hubbard butterfly of the two-body branches evolves as a function of onsite interactions and how to efficiently calculate their Chern numbers using the Fukui-Hatsugai-Suzuki approach. Our numerical results are fully consistent with the simple picture that appears in the strong-coupling limit, where the attraction between fermions forms a composite boson characterized by an effective hopping parameter and an effective magnetic-flux ratio.
Strongly interacting bosons that live in a lattice with degeneracy in its lowest-energy band experience frustration that can prevent the formation of a Bose-Einstein condensate. Such systems form an ideal playground to investigate spin-liquid behavior. We use the variational principle and the Chern-Simons technique of fermionization of hard-core bosons on the kagome lattice to find that below lattice filling fraction $\ensuremath{\nu}=\frac{1}{3}$ the system favors a topologically ordered chiral spin-liquid state that is gapped in bulk, spontaneously breaks time-reversal symmetry, and supports massless chiral bosonic edge mode. We construct the many-body variational wave function of the state and show that the corresponding energy coincides with the energy of the flat band. This result proves that the ground state of the system cannot stabilize a Bose condensate below $\ensuremath{\nu}=\frac{1}{3}$. The fermionization and variational scheme we outline apply to any non-Bravais lattice. We distinguish between the roles played by the Chern-Simons gauge field in lattices with a flat band and those exhibiting a moatlike dispersion (which is degenerate along a closed contour in the reciprocal space). We also suggest experimental probes to differentiate the proposed ground state from a condensate.
Frustrated spin systems on Kagome lattices have long been considered to be a promising candidate for realizing exotic spin liquid phases. Recently, there has been a lot of renewed interest in these systems with the discovery of experimental materials such as Volborthite and Herbertsmithite that have Kagome like structures. In this thesis I will focus on studying frustrated spin systems on the Kagome lattice using a spin-1/2 antiferromagnetic XXZ Heisenberg model in the presence of an external magnetic field as well as other perturbations. Such a system is expected to give rise to magnetization platueaus which can exhibit topological characteristics in certain regimes. We will first develop a flux-attachment transformation that maps the Heisenberg spins (hard-core bosons) onto a problem of fermions coupled to a Chern-Simons gauge field. This mapping relies on being able to define a consistent Chern-Simons term on the lattice. Using this newly developed mapping we analyse the phases/magnetization plateaus that arise at the mean-field level and also consider the effects of adding fluctuations to various mean-fi eld states. Along the way, we show how to discretize an abelian Chern-Simons gauge theory on generic 2D planar lattices that satisfy certain conditions. We find that as long as there exists a one-to-one correspondence between the vertices and plaquettes defined on the graph, one can write down a discretized lattice version of the abelian Chern-Simons gauge theory. Using the newly developed flux attachment transformation, we show the existence of chiral spin liquid states for various magnetization plateaus for certain range of parameters in the XXZ Heisenberg model in the presence of an external magnetic field. Speci cally, in the regime of XY anisotropy the ground states at the 1/3 and 2/3 plateau are equivalent to a bosonic fractional quantum Hall Laughlin state with filling fraction 1/2 and that the 5/9 plateau is equivalent to the first bosonic Jain daughter state at filling fraction 2/3. Next, we also consider the effects of several perturbations: a) a chirality term, b) a Dzyaloshinskii-Moriya term, and c) a ring-exchange type term on the bowties of the kagome lattice, and inquire if they can also support chiral spin liquids as ground states. We find that the chirality term leads to a chiral spin liquid even in the absence of an uniform magnetic field, with an effective spin Hall conductance of 1/2 in the regime of XY anisotropy. The Dzyaloshinkii-Moriya term also leads a similar chiral spin liquid but only when this term is not too strong. An external magnetic field when combined with some of the above perturbations also has the possibility of giving rise to additional plateaus which also behave like chiral spin liquids in the XY regime. Under the in influence of a ring-exchange term we find that provided its coupling constant is large enough, it may trigger a phase transition into a chiral spin liquid by the spontaneous breaking of time-reversal invariance. Finally, we also present some numerical results based on some exact diagonalization studies. Here, we specifically focus on the 2/3-magnetization plateau which we previously argued should be a chiral spin liquid with a spin hall conductance of 1/2 . Such a topological state has a non-trivial ground state degeneracy and it excitations are described by semionic quasiparticles. In the numerical analysis, we analyse the ground state degeneracy structure on various Kagome clusters of different sizes. We compute modular matrices from the resultant minimally entangled states as well as the Chern numbers of various eigenstates all of which provide strong evidence that the 2/3-magnetization plateau very closely resembles a chiral spin liquid state with the expected characteristics.
We construct generalized Hofstadter models that possess ``color-entangled'' flat bands and study interacting many-body states in such bands. For a system with periodic boundary conditions and appropriate interactions, there exist gapped states at certain filling factors for which the ground-state degeneracy depends on the number of unit cells along one particular direction. This puzzling observation can be understood intuitively by mapping our model to a single-layer or a multilayer system for a given lattice configuration. We discuss the relation between these results and the previously proposed ``topological nematic states,'' in which lattice dislocations have non-Abelian braiding statistics. Our study also provides a systematic way of stabilizing various fractional topological states in C>1C>1C>1 flat bands and provides some hints on how to realize such states in experiments.
We propose to use generic Chern numbers for a characterization of topological insulators. It is suitable for a numerical characterization of low dimensional quantum liquids where strong quantum fluctuations prevent from developing conventional orders. By twisting parameters of boundary conditions, the non-Abelian Chern number are defined for a few low lying states near the ground state in a finite system, which is a ground state multiplet with a possible (topological) degeneracy. We define the system as a topological insulator when energies of the multiplet are well separated from the above. Translational invariant twists up to a unitary equivalence are crutial to pick up only bulk properties without edge states. As a simple example, the setup is applied for a two-dimensional XXZXXZXXZ-spin system with an ising anisotropy where the ground state multiplet is composed of doubly almost degenerate states. It gives a vanishing Chern number due to a symmetry. Also Chern numbers for the generic fractional quantum Hall states are discussed shortly.