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Lattice effects on Laughlin wave functions and parent Hamiltonians

机译：格子对劳夫林波函数和父母哈密顿量的影响

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摘要

We investigate lattice effects on wave functions that are lattice analogs of bosonic and fermionic Laughlin wave functions with number of particles per flux v = 1/q in the Landau levels. These wave functions are defined analytically on lattices with μ particles per lattice site, where μ may be different than v. We give numerical evidence that these states have the same topological properties as the corresponding continuum Laughlin states for different values of q and for different fillings μ. These states define, in particular, particle-hole symmetric lattice fractional quantum Hall states when the lattice is half filled. On the square lattice it is observed that for q ≤4 this particle-hole symmetric state displays the topological properties of the continuum Laughlin state at filling fraction v = 1/q, while for larger q there is a transition towards long-range ordered antiferromagnets. This effect does not persist if the lattice is deformed from a square to a triangular lattice, or on the kagome lattice, in which case the topological properties of the state are recovered. We then show that changing the number of particles while keeping the expression of these wave functions identical gives rise to edge states that have the same correlations in the bulk as the reference lattice Laughlin states but a different density at the edge. We derive an exact parent Hamiltonian for which all these edge states are ground states with different number of particles. In addition this Hamiltonian admits the reference lattice Laughlin state as its unique ground state of filling factor 1/q. Parent Hamiltonians are also derived for the lattice Laughlin states at other fillings of the lattice, when μ≤1/q or μ≥ 1 - 1/q and when q = 4 also at half filling.
机译：我们研究了波函数的晶格效应，这些波函数是Bosonic和Fermionic Laughlin波函数的晶格类似物，在Landau层中每个通量的粒子数量v = 1 / q。这些波函数是在每个晶格位点上有μ个粒子的晶格上解析定义的，其中μ可能不同于v。我们提供了数值证据，表明这些状态与q值不同和填充不同的连续统Laughlin状态具有相同的拓扑特性。 μ。当晶格被半填充时，这些状态尤其定义了粒子-孔对称晶格分数量子霍尔状态。在方格上可以观察到，对于q≤4，该粒子-孔对称态在填充分数v = 1 / q时表现出连续Laughlin态的拓扑性质，而对于更大的q，则向长程有序反铁磁体过渡。如果晶格从正方形变形为三角形晶格，或者在kagome晶格上变形，则这种效果不会持续，在这种情况下，状态的拓扑特性得以恢复。然后，我们表明，在保持这些波函数的表达式相同的同时更改粒子数会导致边缘状态在体积上与参考晶格Laughlin状态具有相同的相关性，但在边缘处具有不同的密度。我们导出一个精确的父哈密顿量，其所有这些边缘状态都是具有不同数量粒子的基态。此外，该哈密顿量也将参考晶格的劳克林状态作为其填充因子1 / q的唯一基态。当μ≤1/ q或μ≥1-1/ q时，以及在半填充时q = 4时，也可以为晶格的其他Laughlin状态导出母体哈密顿量。

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  • 来源
    《Physical review》 |2016年第24期|245104.1-245104.16|共16页
  • 作者

    Ivan Glasser; J. Ignacio Cirac; German Sierra; Anne E. B. Nielsen;

  • 作者单位

    Max-Planck-Institut fuer Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany;

    Max-Planck-Institut fuer Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany;

    Instituto de Fisica Teorica, UAM-CSIC, Madrid, Spain;

    Max-Planck-Institut fuer Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany,Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark,Max-Planck-Institut fuer Physik komplexer Systeme, D-01187 Dresden, Germany;

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