Parent Hamiltonian for the non-Abelian chiral spin liquid

Greiter, Martin; Schroeter, Darrell F.; Thomale, Ronny

doi:10.1103/physrevb.89.165125

Cited by 81 publications

(81 citation statements)

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“…(2) is frustration free, i.e., there are states satisfying the zero mode condition Eq. (14). The question we wish to answer is how this fact can be understood in terms of algebraic properties of the operators Q m R , Q m R † .…”

Section: 54mentioning

confidence: 99%

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Algebraic approach to the study of zero modes of Haldane pseudopotentials

Chen

Seidel

2015

Phys. Rev. B

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We consider lattice Hamiltonians that arise from putting Haldane pseudo-potentials into a second quantized or "guiding-center-only" form. These are fascinating examples for 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 pseudo-potentials can be understood from a polynomial-free, intrinsically lattice point of view is also suggested by the fact that these pseudo-potentials 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.

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Section: 54mentioning

confidence: 99%

“…This simply follows from the zero mode condition (14), together with the observation that [Q m R , c r ] = 0. Hence we can always generate new zero modes from old ones by acting with destruction operators c r .…”

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confidence: 94%

Algebraic approach to the study of zero modes of Haldane pseudopotentials

Chen

Seidel

2015

Phys. Rev. B

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“…Despite preserving spin SU(2) symmetry, the Kalmeyer-Laughlin CSL shares basic properties of quantum Hall states, such as a bulk gap and chiral edge states 12,15,16 . While it was shown later that the CSL is not realized in the Heisenberg antiferromagnet on the triangular lattice, a few models have been proposed [17][18][19][20] for which the CSL state is an exact ground state. However, the question remained as to whether the CSL can be realized in more realistic spin models.…”

Section: Introductionmentioning

confidence: 99%

Chiral spin liquids in arrays of spin chains

2015

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We describe a coupled-chain construction for chiral spin liquids in two-dimensional spin systems. Starting from a one-dimensional zigzag spin chain and imposing SU(2) symmetry in the framework of non-Abelian bosonization, we first show that our approach faithfully describes the low-energy physics of an exactly solvable model with a three-spin interaction. Generalizing the construction to the two-dimensional case, we obtain a theory that incorporates the universal properties of the chiral spin liquid predicted by Kalmeyer and Laughlin: charge-neutral edge states, gapped spin-1/2 bulk excitations, and ground state degeneracy on the torus signalling the topological order of this quantum state. In addition, we show that the chiral spin liquid phase is more easily stabilized in frustrated lattices containing corner-sharing triangles, such as the extended kagome lattice, than in the triangular lattice. Our field theoretical approach invites generalizations to more exotic chiral spin liquids and may be used to assess the existence of the chiral spin liquid as the ground state of specific lattice systems.

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“…While readily implemented in a language of wave functions, this method of obtaining a CSL state in two steps-writing it out in the continuum and then projecting it onto the lattice-is not available in a description in terms of Hamiltonians, such as our CWC. In all known constructions [10][11][12][13] of parent Hamiltonians of the KL state, P and T violation is implemented through a three-spin interaction of the form S i (S j × S k ), where i, j, and k are three lattice sites on a plaquette.…”

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confidence: 99%

“…The approach was subsequently expanded to yield different classes of such trial Hamiltonians [11], where the latest and more generic versions are more short-ranged than the initial microscopic models: they involve 2-body and 3-body spin interactions which can be deduced from the explicit construction of appropriate annihilation operators [12] or null operators in conformal field theory [13]. In particular, non-Abelian chiral spin liquids with level k parafermionic spin excitations have been proposed [12,14], which nurture the hope for alternative scenarios of topological quantum computation in frustrated magnets and Mott regimes of alkaline earth atoms deposited in optical lattices [15,16].Since their discovery, CSLs have been appreciated as a realisation of a bosonic Laughlin state at Landau level filling fraction ν = 1/2 on a spin lattice. Naturally, the CSL of Refs.…”

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confidence: 99%

Coupled-wire construction of chiral spin liquids

et al. 2015

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We develop a coupled wire construction of chiral spin liquids. The starting point are individual wires of electrons in the Mott regime that are subject to a Zeeman field and Rashba spin-orbit coupling. Suitable spin-flip couplings between the wires yield an Abelian chiral spin liquid state which supports spinon excitations above a bulk gap, and chiral edge states. The approach generalizes to non-Abelian chiral spin liquids at level k with parafermionic edge states.PACS numbers: 71.10. Pm, 75.10.Kt, Introduction.-The experimental discovery [1] and conceptual understanding [2] of the fractional quantum Hall effect (FQHE) had a tremendous impact on contemporary research of strongly correlated electron systems. In particular, it triggered interest in topologically ordered quantum states of matter, which since then have persisted as a predominant focus. Following up on an idea by D. H. Lee, Kalmeyer and Laughlin [3,4] proposed the chiral spin liquid [5,6] (CSL) as a fractionally quantized Hall liquid for bosonic spin flip operators acting on a spin-polarized reference state. Fractionalization of charge for FQHE relates to fractionalization of spin for the CSL, which supports S = 1/2 spinons obeying halfFermi statistics [7]. The CSL has been an invaluable seed for new concepts such as topological order [8], providing a direct perspective on the fundamental relations between FQHE, spin liquids, and superconductivity [5,9]. Despite its high relevance as a paradigm formulated via wave functions, the first Hamiltonian for which the CSL is the (aside from topological degeneracies) unique ground state was only identified two decades after the liquid had been proposed [10]. The approach was subsequently expanded to yield different classes of such trial Hamiltonians [11], where the latest and more generic versions are more short-ranged than the initial microscopic models: they involve 2-body and 3-body spin interactions which can be deduced from the explicit construction of appropriate annihilation operators [12] or null operators in conformal field theory [13]. In particular, non-Abelian chiral spin liquids with level k parafermionic spin excitations have been proposed [12,14], which nurture the hope for alternative scenarios of topological quantum computation in frustrated magnets and Mott regimes of alkaline earth atoms deposited in optical lattices [15,16].Since their discovery, CSLs have been appreciated as a realisation of a bosonic Laughlin state at Landau level filling fraction ν = 1/2 on a spin lattice. Naturally, the CSL of Refs. 3 and 4 can be defined on any lattice [17,18], which becomes mathematically transparent via the generalized Perelomov identity [19] for lattices with a primitive unit cell. Some of these motifs have later reappeared in the field of fractional Chern insulators [20][21][22]. As

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Parent Hamiltonian for the non-Abelian chiral spin liquid

Cited by 81 publications

References 61 publications

Algebraic approach to the study of zero modes of Haldane pseudopotentials

Algebraic approach to the study of zero modes of Haldane pseudopotentials

Chiral spin liquids in arrays of spin chains

Coupled-wire construction of chiral spin liquids

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