Optical-lattice implementation scheme of a bosonic topological model with fermionic atoms

Nielsen, Anne E. B.; Sierra, Germȧn; Cirac, J. I.

doi:10.1103/physreva.90.013606

Cited by 9 publications

(8 citation statements)

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“…However in some cases it has turned out that states constructed from correlators of conformal fields had very high overlaps with ground states of local Hamiltonians [31,36,42,43]. This has lead to a protocol to implement one of these states in experiments [31,57]. In this section we show that there is a local Hamiltonian for which the ground state is close to the q ( 1, 1) 1 η = = CFT state in one and in two dimensions and that this result is also true for the q ( 2, 1) 1 2 η = = CFT state in one dimension.…”

Section: Local Hamiltoniansmentioning

confidence: 99%

Exact parent Hamiltonians of bosonic and fermionic Moore–Read states on lattices and local models

et al. 2015

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We introduce a family of strongly-correlated spin wave functions on arbitrary spin-1 2 and spin-1 lattices in one and two dimensions. These states are lattice analogues of Moore-Read states of particles at filling fraction q 1 , which are non-Abelian fractional quantum Hall states in 2D. One parameter enables us to perform an interpolation between the continuum limit, where the states become continuum Moore-Read states of bosons (odd q) and fermions (even q), and the lattice limit. We show numerical evidence that the topological entanglement entropy stays the same along the interpolation for some of the states we introduce in 2D, which suggests that the topological properties of the lattice states are the same as in the continuum, while the 1D states are critical states. We then derive exact parent Hamiltonians for these states on lattices of arbitrary size. By deforming these parent Hamiltonians, we construct local Hamiltonians that stabilize some of the states we introduce in 1D and in 2D.

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Section: Local Hamiltoniansmentioning

confidence: 99%

Exact parent Hamiltonians of bosonic and fermionic Moore–Read states on lattices and local models

et al. 2015

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“…Some numerical studies have systematically explored the interaction effects within TFB models, and have firmly established the existence of FCI/FQAH states [8,9,[11][12][13]. Recently, there have been various proposals of material and coldatom realization schemes for such FCI/FQAH systems [14][15][16][17][18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Wave functions for fractional Chern insulators in disk geometry

Luo

Wang

et al. 2015

New J. Phys.

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Recently, fractional Chern insulators (FCIs), also called fractional quantum anomalous Hall (FQAH) states, have been theoretically established in lattice systems with topological flat bands. These systems exhibit similar fractionalization phenomena to the conventional fractional quantum Hall (FQH) systems. Using the mapping relationship between the FQH states and the FCI/FQAH states, we construct the many-body wave functions of the fermionic FCI/FQAH states in disk geometry with the aid of the generalized Pauli principle (GPP) and Jack polynomials. Compared with the ground state by the exact diagonalization method, the wave-function overlap is higher than 0.97, even when the Hilbert space dimension is as large as 3×10 6 . We also use the GPP and the Jack polynomials to construct edge excitations for the fermionic FCI/FQAH states. The quasi-degeneracy sequences of fermionic FCI/FQAH systems reproduce the prediction of the chiral Luttinger liquid theory, complementing the exact diagonalization results with larger lattice sizes and more particles. OPEN ACCESS RECEIVED exclusion principle corresponds to the Fermi statistics, the GPP corresponds to the fractional statistics of anyons.Recently, finding the suitable trial wave functions of FCI/FQAH states has become an outstanding problem. It is conjectured that there is a one-to-one mapping between IQH and CI/QAH states [10,[34][35][36][37]. In other words, the single-particle orbitals of an LL can be mapped to the ones in a TFB. Based on one-dimensional (1D) maximally localized Wannier functions, Qi et al have used a mapping between the FQH and the FCI/FQAH states to construct generic wave functions of FCI/FQAH states in cylinder geometry with these 1D Wannier functions [10,[34][35][36][37]. In terms of this mapping relationship, the Haldane pseudo-potential [26] for these FCI/ FQAH states can be constructed [38,39] through a suitable gauge choice for the Wannier functions. An improved prescription has been adopted to construct variational wave functions of FCI/FQAH states in torus geometry by utilizing gauge-fixed (non-maximally) localized Wannier states [40][41][42]. From another aspect, conventional FQH states can also be obtained for 2D lattices analytically by using conformal field theory [43][44][45]. In contrast to the above analytical or semi-analytical approaches, we here pursue a very direct yet effective purely numerical prescription to construct FCI/FQAH states in disk geometry, without any variational parameter or adjustable gauge freedom, but just utilizing the powerful GPP and the Jacks structure of FQH states, and the information from exact numerical single-particle orbitals of TFB models.In this paper, we exploit the single-particle wave functions of CI/QAH states (in the Kagomé model [5, 46] and the Haldane [3, 9] model) with TFB parameters in disk geometry, and explore the polynomial structure of the continuum Laughlin wave functions [25] to establish the many-body wave functions of the FCI/FQAH states. We firstly construct finite-size lattices in...

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“…One of the local Hamiltonians has, in addition, been used to propose an implementation scheme for a FQH lattice model in ultracold atoms in optical lattices. 13,51 Quasi-holes.-In the following, I use the Metropolis Monte Carlo algorithm to investigate important properties of the quasi-holes in the models. I shall consider q = 3 and the square lattice in Fig.…”

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confidence: 99%

Anyon braiding in semianalytical fractional quantum Hall lattice models

Nielsen

2015

Phys. Rev. B

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It has been demonstrated numerically, mainly by considering ground state properties, that fractional quantum Hall physics can appear in lattice systems, but it is very difficult to study the anyons directly. Here, I propose to solve this problem by using conformal field theory to build semi-analytical fractional quantum Hall lattice models having anyons in their ground states, and I carry out the construction explicitly for the family of bosonic and fermionic Laughlin states. This enables me to show directly that the braiding properties of the anyons are those expected from analytical continuation of the wave functions and to compute properties such as internal structure, size, and charge of the anyons with simple Monte Carlo simulations. The models can also be used to study how the anyons behave when they approach or even pass through the edge of the sample. Finally, I compute the effective magnetic field seen by the anyons, which varies periodically due to the presence of the lattice.PACS numbers: 05.30. Pr, 03.65.Fd, 11.25.Hf The discovery of the fractional quantum Hall (FQH) effect 1 revealed the existence of phases of matter that are fundamentally different from previously known phases. These phases have attracted much attention both because new physics is needed to describe them 2 and because their properties are interesting for quantum computing 3 . With the aim of getting a deeper understanding of the effect and find more robust and controllable ways to realize it experimentally, much effort is currently being put into exploring under which conditions the effect occurs. A major result in this direction is the discovery that FQH physics can be realized in lattice systems, 4-16 which opens up doors towards investigating the effect under new parameter regimes, maybe even room temperature 17 . One of the special features of FQH states is the possibility to create anyons. Anyons are particle-like excitations that have a more complicated exchange statistics than bosons and fermions. By now, more techniques have been developed that allow one to determine which types of anyons can be created in a system by looking only at the properties of the ground states, 14,18-29 and these techniques have been used to demonstrate the FQH nature of the above mentioned lattice models. The techniques do, however, have limitations in that they do not provide information about, e.g., what the size and internal structure of the anyons are, how the anyons can be created and moved around, and the details of braiding operations. In order to describe these features, one needs to study the anyons directly, but this is very difficult for lattice FQH models, where only the Hamiltonian (or at best the Hamiltonian and the anyon free ground state 6,11,12,15,30,31 ) is known analytically (although test computations can be done for very small systems 32 ). In this article, I provide a solution to these problems by proposing to construct lattice FQH models that have anyons in their ground states and for which both the Hamiltonian and the gr...

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Optical-lattice implementation scheme of a bosonic topological model with fermionic atoms

Cited by 9 publications

References 56 publications

Exact parent Hamiltonians of bosonic and fermionic Moore–Read states on lattices and local models

Exact parent Hamiltonians of bosonic and fermionic Moore–Read states on lattices and local models

Wave functions for fractional Chern insulators in disk geometry

Anyon braiding in semianalytical fractional quantum Hall lattice models

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