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arXiv:1409.3569 · Physical Review B

Algebraic approach to the study of zero modes of Haldane pseudopotentials

Li Chen, Alexander Seidel

Published Feb 5, 2015• 21 citations
arXivHTMLPDF

Abstract

We consider lattice Hamiltonians that arise from putting Haldane pseudopotentials into a second quantized or ``guiding-center-only'' form. These are fascinating examples of frustration-free lattice Hamiltonians. This is so since, even though their highest density zero energy ground states, the Laughlin states, are known to have matrix-product structure (with unbounded bond dimension), the frustration-free character of these lattice Hamiltonians seems obscure, unless one goes back to the original first quantized picture of analytic lowest Landau level wave functions. This step involves putting back additional degrees of freedom associated with dynamical momenta, and one wonders whether the addition of these degrees of freedom is truly necessary to recognize the frustration-free character of the underlying lattice Hamiltonian. Fundamentally, these degrees of freedom have nothing to do with spectrum of a ``guiding-center-only'' Hamiltonian. Moreover, such constructions are unfamiliar and not available in the study of simpler (finite range) frustration-free lattice Hamiltonians with matrix-product ground states (of finite bond dimension). That the zero mode properties of ``lattice versions'' of pseudopotentials can be understood from a polynomial-free, intrinsically lattice point of view is also suggested by the fact that these pseudopotentials are constructed from an algebra of reasonably simply looking operators. Here we show that zero mode properties, and hence the frustration-free character, of these lattice Hamitlonians can be understood as a consequence of algebraic structures that these operators are part of. We believe that our results will deepen insights into parent Hamiltonians of matrix-product states with infinite bond dimensions, as could be of use, especially, in the study of fractional Chern insulators.

DOI
10.1103/physrevb.91.085103

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We consider lattice Hamiltonians that arise from putting Haldane pseudopotentials into a second quantized or guiding-center-only'' form. These are fascinating examples of frustration-free lattice Hamiltonians. This is so since, even though their highest density zero energy ground states, the Laughlin states, are known to have matrix-product structure (with unbounded bond dimension), the frustration-free character of these lattice Hamiltonians seems obscure, unless one goes back to the original first quantized picture of analytic lowest Landau level wave functions. This step involves putting back additional degrees of freedom associated with dynamical momenta, and one wonders whether the addition of these degrees of freedom is truly necessary to recognize the frustration-free character of the underlying lattice Hamiltonian. Fundamentally, these degrees of freedom have nothing to do with spectrum of a guiding-center-only'' Hamiltonian. Moreover, such constructions are unfamiliar and not available in the study of simpler (finite range) frustration-free lattice Hamiltonians with matrix-product ground states (of finite bond dimension). That the zero mode properties of ``lattice versions'' of pseudopotentials can be understood from a polynomial-free, intrinsically lattice point of view is also suggested by the fact that these pseudopotentials are constructed from an algebra of reasonably simply looking operators. Here we show that zero mode properties, and hence the frustration-free character, of these lattice Hamitlonians can be understood as a consequence of algebraic structures that these operators are part of. We believe that our results will deepen insights into parent Hamiltonians of matrix-product states with infinite bond dimensions, as could be of use, especially, in the study of fractional Chern insulators.
The study of model Hamiltonians, whose exact ground states are well known quantum Hall states, have been investigated solely in the literature from the point of view of first quantized approach in the last few decades. However, a second quantized approach to such Hamiltonians has not been taken in the same measure. In this dissertation, we study Haldane pseudopotential, which is the parent Hamiltonian of 1/M Laughlin state, and Trugman-Kivelson pseudopotential, which is the parent Hamiltonian of 2/5 unprojected Jain state. We find that the study of these Hamiltonians in their second quantized forms not only reproduces all properties of their zero modes already known in first quantized approach, but also sheds lights on general studies of such structures of these Hamiltonians as frustration free lattice Hamiltonians.
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 the 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 MPSs 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--conformal field theory (CFT) states and their parent Hamiltonians. While focusing on the Laughlin state, generalizations to other CFT-MPSs will become transparent.
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.
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 introduce one-dimensional lattice models with exact matrix-product ground states describing the fractional quantum Hall (FQH) states in Laughlin series (given by filling factors $\ensuremath{\nu}=1/q$) on torus geometry. Surprisingly, the exactly solvable Hamiltonian has the same mathematical structure as that of the pseudopotential for the Laughlin wave function, and naturally derives the general properties of the Laughlin wave function such as the Z2{Z}_{2}Z2​ properties of the FQH states and the fermion-boson relation. The obtained exact ground states have high overlaps with the Laughlin states and well describe their properties. Using the matrix product method, density functions and correlation functions are calculated analytically. Especially, obtained entanglement spectra reflect gapless edge states as was discussed by Li and Haldane.
We devise local lattice models whose ground states are \emph{ideal fractional Chern insulators}---Abelian and non-Abelian topologically ordered states characterized by exact ground state degeneracies at any finite size and infinite entanglement gaps. Most saliently, we construct exact parent Hamiltonians for two distinct families of bosonic lattice generalizations of the Zk\mathcal{Z}_kZk​ parafermion quantum Hall states: (i) color-entangled fractional Chern insulators at band filling fractions ν=k/(C+1)\nu=k/(\mathcal{C}+1)ν=k/(C+1) and (ii) nematic states at ν=k/2\nu=k/2ν=k/2, where C\mathcal{C}C is the Chern number of the lowest band in our models. In spite of a fluctuating Berry curvature, our construction is partially frustration free: the ground states reside entirely within the lowest band and exactly minimize a local (k+1)(k+1)(k+1)-body repulsion term by term. In addition to providing the first known models hosting intriguing states such as higher Chern number generalizations of the Fibonacci anyon quantum Hall states, the remarkable stability and finite-size properties make our models particularly well-suited for the study of novel phenomena involving e.g. twist defects and proximity induced superconductivity, as well as being a guide for designing experiments.