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Bulletin of the American Physical Society

Fractional Quantum Hall Effect from Hilbert Space Algebra and Frustration-free Hamiltonians

Ying-Hai Wu, Bo Yang, Zlatko Papić

Published Mar 17, 2021• 0 citations
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We study frustration free Hamiltonians of fractional quantum Hall (FQH) states from the point of view of the matrix-product-state (MPS) representation of their ground and excited states. There is a wealth of solvable models relating to FQH physics, which, however, is mostly derived and analyzed from vantage point of first quantized analytic clustering properties. In contrast, one obtains long-ranged frustration free lattice models when these Hamiltonians are studied in an orbital basis, which is the natural basis for the MPS representation of FQH states. The connection between MPS-like states and frustration free parent Hamiltonians is the central guiding principle in the construction of solvable lattice models, but thus far, only for short range Hamiltonians and MPS of finite bond dimension. The situation in the FQH context is fundamentally different. Here we expose the direct link between the infinite-bond-dimension MPS structure of Laughlin-CFT states and their parent Hamiltonians. While focussing on the Laughlin state, generalizations to other CFT-MPS will become transparent.

•Nov 19, 2021•0 citations
The study of frustration-free Hamiltonians and their relation to finite bond-dimension matrix product states (MPS) has a long tradition. However, fractional quantum Hall (FQH) states do not quite fit into this theme since the known MPS representations of their ground states have infinite bond dimensions, which considerably obscures the relations between such MPS representations and the existence of frustration-free parent Hamiltonians. This is related to the fact that the latter necessarily are of infinite range in the orbital basis. Here, we present a theorem tailored to establishing the existence of frustration-free parent Hamiltonians in such a context. We explicitly demonstrate the utility of this theorem in the context of non-Abelian Moore-Read FQH states but argue the applicability of this theorem to transcend considerably beyond the realm of conformal-field-theory-derived MPSs or quasi-one-dimensional Hilbert spaces.
We show that model states of fractional quantum Hall fluids at all experimentally detected plateaus can be uniquely determined by imposing translational invariance with a particular scheme of Hilbert space truncation. The truncation is based on classical local exclusion conditions, motivated by constraints on physical measurements. The scheme allows us to identify filling factors, topological shifts, and clustering of topological quantum fluids universally without resorting to microscopic Hamiltonians. This prompts us to propose the notion of emergent commensurability as a fundamental property for many known fractional quantum Hall (FQH) states, which allows us to predict families of new FQH states that can be realized in principle. We also discuss the implications of certain missing states proposed from other phenomenological approaches, and suggest that the physics of the FQH effect could fundamentally arise from the algebraic structure of the Hilbert space in a single Landau level.
We study frustration free Hamiltonians of fractional quantum Hall (FQH) states from the point of view of the matrix-product-state (MPS) representation of their ground and excited states. There is a wealth of solvable models relating to FQH physics, which, however, is mostly derived and analyzed from vantage point of first quantized analytic clustering properties. In contrast, one obtains long-ranged frustration free lattice models when these Hamiltonians are studied in an orbital basis, which is the natural basis for the MPS representation of FQH states. The connection between MPS-like states and frustration free parent Hamiltonians is the central guiding principle in the construction of solvable lattice models, but thus far, only for short range Hamiltonians and MPS of finite bond dimension. The situation in the FQH context is fundamentally different. Here we expose the direct link between the infinite-bond-dimension MPS structure of Laughlin-CFT states and their parent Hamiltonians. While focussing on the Laughlin state, generalizations to other CFT-MPS will become transparent.