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arXiv:1502.04815 · Physical Review B

Growing quantum states with topological order

Fabian Letscher, Fabian Grusdt, Michael Fleischhauer

Published May 26, 2015• 7 citations
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Abstract

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.

DOI
10.1103/physrevb.91.184302

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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 NNN. 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.
We suggest a scheme for the preparation of highly correlated Laughlin states in the presence of synthetic gauge fields, realizing an analogue of the fractional quantum Hall effect in photonic or atomic systems of interacting bosons. It is based on the idea of growing such states by adding weakly interacting composite fermions along with magnetic flux quanta one by one. The topologically protected Thouless pump ("Laughlin's argument") is used to create two localized flux quanta and the resulting hole excitation is subsequently filled by a single boson, which, together with one of the flux quanta, forms a composite fermion. Using our protocol, filling 1/2 Laughlin states can be grown with particle number N increasing linearly in time and strongly suppressed number fluctuations. To demonstrate the feasibility of our scheme, we consider two-dimensional lattices subject to effective magnetic fields and strong on-site interactions. We present numerical simulations of small lattice systems and also discuss the influence of losses.
We study NNN-component interacting particles (hardcore bosons and fermions) loaded in topological lattice models with SU(N)\mathrm{SU}(N)SU(N)-invariant interactions based on exact diagonalization and density matrix renormalization group method. By tuning the interplay of interspecies and intraspecies interactions, we demonstrate that a class of SU(N)\mathrm{SU}(N)SU(N) fractional quantum Hall states can emerge at fractional filling factors $\ensuremath{\nu}=N/(N+1)$ for bosons [$\ensuremath{\nu}=N/(2N+1)$ for fermions] in the lowest Chern band, characterized by the nontrivial fractional Hall responses and the fractional charge pumping. Moreover, we establish a topological characterization based on the K\mathbf{K}K matrix and discuss the close relationship to the fractional quantum Hall physics in topological flat bands with Chern number NNN.
Multicomponent quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, leads us to new emergent topological orders. Here, we report the theoretical discovery of fractional quantum hall effect of strongly correlated Bose-Fermi mixtures classified by the K=(m11n)\mathbf{K}=\left(\begin{array}{cc}m& 1\\ 1& n\end{array}\right)K=(m1​1n​) matrix (even mmm for boson and odd nnn for fermion), using topological flat band models. Utilizing the state-of-the-art exact diagonalization and density-matrix renormalization group methods, we build up the topological characterization based on three inherent aspects: (i) topological $(mn\ensuremath{-}1)$-fold ground-state degeneracy equivalent to the determinant of the K\mathbf{K}K matrix; (ii) fractionally quantized topological Chern number matrix equivalent to the inverse of the K\mathbf{K}K matrix; and (iii) two parallel-propagating chiral edge branches with level counting 1,2,5,10 consistent with the conformal field theory description.
Topologically-ordered matter is a novel quantum state of matter observed only in a small number of physical systems, notably two-dimensional electron systems exhibiting fractional quantum Hall effects. It was recently proposed that a simple form of topological matter may be created in interacting systems of rotating ultra-cold atoms. We describe ensemble measurements on small, rotating clusters of interacting bosonic atoms, demonstrating that they can be induced into quantum ground states closely analogous to topological states of electronic systems. We report measurements of inter-particle correlations and momentum distributions of Bose gases in the fractional quantum Hall limit, making comparison to a full numerical simulation. The novel experimental apparatus necessary to produce and measure properties of these deeply entangled quantum states is described.
We study the many-body ground states of three-component quantum particles in two prototypical topological lattice models under strong intercomponent and intracomponent repulsions. At band filling $\ensuremath{\nu}=3/4$ for hard-core bosons, we demonstrate the emergence of the three-component fractional quantum Hall (FQH) effect characterized by the K\mathbf{K}K matrix, through an exact diagonalization study of fourfold quasidegenerate ground states with a robust spectrum gap and the combined density-matrix renormalization group calculation of fractional drag charge pumping. Further, we formulate the topological characterization of FQH states of three-component Bose-Fermi mixtures at various fillings by the K\mathbf{K}K matrix. Finally, we discuss the possible generalization of our approach to identify non-Abelian three-component spin-singlet FQH states.
We develop a method to characterize topological phase transitions for strongly correlated Hamiltonians defined on two-dimensional lattices based on the many-body Berry curvature. Our goal is to identify a class of quantum critical points between topologically nontrivial phases with fractionally quantized Hall (FQH) conductivity and topologically trivial gapped phases through the discontinuities of the many-body Berry curvature in the so-called flux Brillouin zone (fBZ), the latter being defined by imposing all possible twisted boundary conditions. For this purpose, we study the finite-size signatures of several quantum phase transitions between fractional Chern insulators and charge-ordered phases for two-dimensional lattices by evaluating the many-body Berry curvature numerically using exact diagonalization. We observe degeneracy points (nodes) of many-body energy levels at high-symmetry points in the fBZ, accompanied by diverging Berry curvature. We find a correspondence between the number and order of these nodal points, and the change of the topological invariants of the many-body ground states across the transition, in close analogy with Weyl nodes in noninteracting band structures. This motivates us to apply a scaling procedure, originally developed for noninteracting systems, for the Berry curvature at the nodal points. This procedure offers a useful tool for the classification of topological phase transitions in interacting systems harboring FQH like topological order.