Algebraic approach to the study of zero modes of Haldane pseudopotentials

Chen, Li; Seidel, Alexander

doi:10.1103/physrevb.91.085103

Cited by 23 publications

(42 citation statements)

References 77 publications

(77 reference statements)

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“…Indeed, only recently the second quantized presentation of FQH Hamiltonians has become a subject of interest in its own right. 27,36,37,59 For one thing, it can be argued that this approach more readily gives access to spectral properties at finite energies. 60 For another, the second quantized approach seems to be effective also in unravelling the zero mode structure of special Hamiltonians, as the present example demonstrates.…”

Section: Discussionmentioning

confidence: 99%

“…(In all aspects, such an operator-based approach has been constructed by some of us previously for the n = 1 case related to the 1/3-Laughlin state, and in fact for all the Laughlin states. 27,36,37 We will comment more on the situation below. )…”

Section: Quantized Operators Q (I)mentioning

confidence: 99%

“…Such an approach has been carried out earlier by some of us 27,36,37 for the Laughlin states and their parent Hamiltonians. One attractive feature of this approach is its re-…”

Section: Zero Mode Generatorsmentioning

confidence: 99%

“…Moreover, in Ref. 36 we have succeeded in constructing a microscopic operator that, when acting on the smallest angular momentum zero modes in the n = 1 (Laughlin) case, leads to the corresponding zero mode with the total particle number increased by 1. This can be interpreted as a microscopic realization of the operator of the edge theory that raises the quantity N 0 + N 1 .…”

Section: Zero Mode Generatorsmentioning

confidence: 99%

“…27,36,37 In particular, generalization to more than two-body Hamiltonians is certainly possible. An obvious direction for future exploration is the case of larger n in Eq.…”

mentioning

confidence: 99%

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Jain-2/5 parent Hamiltonian: Structure of zero modes, dominance patterns, and zero mode generators

2017

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We analyze general zero mode properties of the parent Hamiltonian of the unprojected Jain-2/5 state. We characterize the zero mode condition associated to this Hamiltonian via projection onto a four-dimensional two-particle subspace for given pair angular momentum, for the disk and similarly for the spherical geometry. Earlier numerical claims in the literature about ground state uniqueness on the sphere are substantiated on analytic grounds, and related results are derived. Preference is given to second quantized methods, where zero mode properties are derived not from given analytic wave functions, but from a "lattice" Hamiltonian and associated zero mode conditions. This method reveals new insights into the guiding-center structure of the unprojected Jain-2/5 state, in particular a system of dominance patterns following a "generalized Pauli principle", which establishes a complete one-to-one correspondence with the edge mode counting. We also identify one-body operators that function as generators of zero modes.

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Section: Discussionmentioning

confidence: 99%

Section: Quantized Operators Q (I)mentioning

confidence: 99%

“…Such an approach has been carried out earlier by some of us 27,36,37 for the Laughlin states and their parent Hamiltonians. One attractive feature of this approach is its re-…”

Section: Zero Mode Generatorsmentioning

confidence: 99%

Section: Zero Mode Generatorsmentioning

confidence: 99%

“…27,36,37 In particular, generalization to more than two-body Hamiltonians is certainly possible. An obvious direction for future exploration is the case of larger n in Eq.…”

mentioning

confidence: 99%

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Jain-2/5 parent Hamiltonian: Structure of zero modes, dominance patterns, and zero mode generators

2017

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Quantum Algorithms for Quantum Chemistry and Quantum Materials Science

et al. 2020

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As we begin to reach the limits of classical computing, quantum computing has emerged as a technology that has captured the imagination of the scientific world. While for many years, the ability to execute quantum algorithms was only a theoretical possibility, recent advances in hardware mean that quantum computing devices now exist that can carry out quantum computation on a limited scale. Thus, it is now a real possibility, and of central importance at this time, to assess the potential impact of quantum computers on real problems of interest. One of the earliest and most compelling applications for quantum computers is Feynman’s idea of simulating quantum systems with many degrees of freedom. Such systems are found across chemistry, physics, and materials science. The particular way in which quantum computing extends classical computing means that one cannot expect arbitrary simulations to be sped up by a quantum computer, thus one must carefully identify areas where quantum advantage may be achieved. In this review, we briefly describe central problems in chemistry and materials science, in areas of electronic structure, quantum statistical mechanics, and quantum dynamics that are of potential interest for solution on a quantum computer. We then take a detailed snapshot of current progress in quantum algorithms for ground-state, dynamics, and thermal-state simulation and analyze their strengths and weaknesses for future developments.

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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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Algebraic approach to the study of zero modes of Haldane pseudopotentials

Cited by 23 publications

References 77 publications

Jain-2/5 parent Hamiltonian: Structure of zero modes, dominance patterns, and zero mode generators

Jain-2/5 parent Hamiltonian: Structure of zero modes, dominance patterns, and zero mode generators

Quantum Algorithms for Quantum Chemistry and Quantum Materials Science

Wave functions for fractional Chern insulators in disk geometry

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