Non-Abelian Braiding of Lattice Bosons

Kapit, Eliot; Ginsparg, Paul; Mueller, Erich J.

doi:10.1103/physrevlett.108.066802

Cited by 54 publications

(46 citation statements)

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“…In a recent paper, braiding statistics of lattice bosons were numerically demonstrated using the long range hopping model. Instead of the usual exact diagonalization calculations which rely on conserved quantities to limit the size of the Hilbert space, projection to the relevant lattice states can be achieved using the long range hopping model [12]. It would be interesting to extend this method to lattices with non-abelian gauge fields [23].…”

Section: Discussionmentioning

confidence: 99%

“…First, hopping strengths in an optical lattice can be modified by tailoring the lattice potential or by dynamical hopping methods [8], and an optical lattice with long range hopping can experimentally realize fractional quantum Hall states. Second, as a theoretical development, this model bridges the gap between continuum and lattice models for quantum Hall physics, and makes it possible to simulate quantum Hall effects with a new method [12]. Hence, it is important to investigate and generalize this long range hopping model, both to understand the underlying physics in greater detail and to search for models that can be experimentally implemented.…”

Section: Introductionmentioning

confidence: 99%

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Landau levels in lattices with long-range hopping

Atakisi

Oktel

2013

Phys. Rev. A

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In the presence of a periodic potential Landau levels (LLs) are broadened, forming a barrier for accurate simulation of fractional quantum Hall effect using cold atoms in optical lattices. Recently, it has been shown that the degeneracy of the lowest Landau level (LLL) can be restored in a tight binding lattice, if a particular form of long range hopping is introduced [E. Kapit and E. J. Mueller, Phys. Rev. Lett. 105, 215303 (2010)]. In this paper, we investigate three problems related to such quantum Hall parent Hamiltonians in lattices. First, we show that there are infinitely many long range hopping models in which a massively degenerate manifold is formed by lattice discretizations of wavefunctions in the continuum LLL. We then give a general method to construct such models, which is applicable to not only the LLL but also higher LLs. We use this method to give an analytic expression for the hoppings that restore the LLL, and an integral expression for the next LL. We also consider whether the space spanned by discretized LL wavefunctions is as large as the space spanned by continuum wavefunctions and find the constraints on magnetic field for this condition to be satisfied. Finally, using these constraints and first Chern numbers, we identify the bands of the Hofstadter butterfly that correspond to continuum LLs. * oktel@fen.bilkent.edu.tr 1 arXiv:1309.5335v1 [cond-mat.quant-gas]

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Landau levels in lattices with long-range hopping

Atakisi

Oktel

2013

Phys. Rev. A

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“…If the energy U n is sufficiently large, or if additional flux quanta are inserted into the system, then fractional quasiholes will be pinned at the impurity sites, as they exactly eliminate the energy cost of the impurities. By adiabatically moving these potentials around the lattice, non-abelian quasiholes can be braided to perform quantum logic gates [2,28,[32][33][34][35][36][37].…”

Section: K = 3: Fibonacci and Anyon Braidingmentioning

confidence: 99%

Three- and four-body interactions from two-body interactions in spin models: A route to Abelian and non-Abelian fractional Chern insulators

Kapit

Simon

2013

Phys. Rev. B

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We describe a method for engineering local k + 1-body interactions (k = 1, 2, 3) from two-body couplings in spin-1 2 systems. When implemented in certain systems with a flat single-particle band with a unit Chern number, the resulting many-body ground states are fractional Chern insulators which exhibit abelian and non-abelian anyon excitations. The most complex of these, with k = 3, has Fibonacci anyon excitations; our system is thus capable of universal topological quantum computation. We then demonstrate that an appropriately tuned circuit of qubits could faithfully replicate this model up to small corrections, and further, we describe the process by which one might create and manipulate non-abelian vortices in these circuits, allowing for direct control of the system's quantum information content. The search for non-abelian anyons has become one of the most important developments in quantum condensed matter physics. Non-abelian anyons are exotic collective modes of gapped topological quantum systems, defined by the unique property that when a pair of identical anyons are adiabatically exchanged, the system's wavefunction rotates between different degenerate states [1][2][3][4][5][6][7][8][9][10][11]. These states cannot be distinguished by local operations, and since the creation or destruction of anyons is suppressed by an energy gap, quantum information encoded with anyons will be topologically protected against noise. So far, the most promising systems [12][13][14] for observing non-abelian anyons are superconducting heterostructures and the 2d electron gas at filling ν = 5/2, and numerous other systems exist in the literature.Arguing for the existence of non-abelian anyons is often extremely difficult. While Majorana zero modes can be derived straightforwardly at a mean-field level in superconducting heterostructures [4], most other realistic models are not amenable to standard techniques such as perturbation theory or quantum Monte Carlo. Much (though not all) of our theoretical justification for nonabelian anyon states comes from either small-system numerical studies or from considering Hamiltonians with k + 1-body interactions (with k > 1) instead of ordinary 2-body interactions, which are rarely realistic for physical systems.We here propose a new set of models which can simulate k + 1-body interactions through realistic 2-body interactions and thus have non-abelian anyon ground states. These models are constructed by modifying the lattice in models with complex hopping and a flat Chern band [15][16][17][18][19][20] so that each site is replaced by a vertex containing a cluster of k spin-1 2 degrees of freedom, with tuned interactions between the spins at each vertex, but not between spins at different vertices. The low-energy manifold of the resulting lattice mimics that of particles hopping on a lattice with a single site per vertex and a hard core local k +1-body interaction. The ground states of these models [21][22][23][24] are generalizations of the ReadRezayi wavefunctions [3], the most comple...

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“…the Fermi statistics of the proton causing the rotational levels of H 2 to depend on whether the spins of the nuclei are in singlet or triplet state). Our method differs substantially from proposals to measure the fractional braiding statistics of quasiholes [7][8][9], notably by not requiring local time-dependent potentials for the adiabatic manipulation of the positions of the quasiholes.We have in mind a fast rotating gas of identical bosonic atoms, initially in a single internal (hyperfine) state ⇑, and confined to a quasi-2D layer with oscillator length a z . The gas is subjected to a tight circularly symmetric harmonic trap of frequency ω 0 , with ω 0 V 0 , in which V 0 ≡ 2 π as az ω 0 is a characteristic interaction energy for atoms with scattering length a s .…”

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confidence: 99%

Signatures of Fractional Exclusion Statistics in the Spectroscopy of Quantum Hall Droplets

Cooper

Simon

2015

Phys. Rev. Lett.

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We show how spectroscopic experiments on a small Laughlin droplet of rotating bosons can directly demonstrate Haldane fractional exclusion statistics of quasihole excitations. The characteristic signatures appear in the single-particle excitation spectrum. We show that the transitions are governed by a "many-body selection rule" which allows one to relate the number of allowed transitions to the number of quasihole states on a finite geometry. We illustrate the theory with numerically exact simulations of small numbers of particles.PACS numbers: 73.43.Cd, 05.30.Pr One of the most dramatic features of strongly correlated phases is the emergence of quasiparticle excitations with unconventional quantum statistics. The archetypal example is the fractional, "anyonic", quantum statistics predicted for the quasiparticles of the fractional quantum Hall phases [1,2]. While experiments on semiconductor devices have shown that these quasiparticles have fractional charges [3][4][5], a direct observation of the fractional statistics has remained lacking.In this Letter we show how precision spectroscopy measurements of rotating droplets of ultracold atoms could be used to demonstrate the Haldane fractional exclusion statistics [6] of quasiholes in the Laughlin state of bosons. By involving only spectroscopic signatures of the rotating droplet, our proposal plays to the strengths of atomic physics experiments. We show that evidence of the fractional exclusion statistics appears in counting the numbers of lines in the radio-frequency (RF) absorption spectrum. In this sense, the method is conceptually similar to classic evidence of quantum statistics, as appearing in the rotational levels of homonuclear diatomic molecules (e.g. the Fermi statistics of the proton causing the rotational levels of H 2 to depend on whether the spins of the nuclei are in singlet or triplet state). Our method differs substantially from proposals to measure the fractional braiding statistics of quasiholes [7][8][9], notably by not requiring local time-dependent potentials for the adiabatic manipulation of the positions of the quasiholes.We have in mind a fast rotating gas of identical bosonic atoms, initially in a single internal (hyperfine) state ⇑, and confined to a quasi-2D layer with oscillator length a z . The gas is subjected to a tight circularly symmetric harmonic trap of frequency ω 0 , with ω 0 V 0 , in which V 0 ≡ 2 π as az ω 0 is a characteristic interaction energy for atoms with scattering length a s . Hence, the interactions leave the particles in the lowest Landau level (LLL) [10].In addition, we shall consider a weak quartic potential -weak compared to both ω 0 and V 0 -for reasons to be described below. Specifically, we shall consider an initial state of N i atoms which has been spun up to the angular momentum L i = N i (N i − 1). Then, for the case of contact repulsive interactions relevant in typical cold gas experiments, the groundstate is the (exact) ν = 1/2 Laughlin state. Furthermore, for the case of contact interactions, the...

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Non-Abelian Braiding of Lattice Bosons

Cited by 54 publications

References 56 publications

Landau levels in lattices with long-range hopping

Landau levels in lattices with long-range hopping

Three- and four-body interactions from two-body interactions in spin models: A route to Abelian and non-Abelian fractional Chern insulators

Signatures of Fractional Exclusion Statistics in the Spectroscopy of Quantum Hall Droplets

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