Generalized Perturbation Theory in Operator Form

Primas, Hans

doi:10.1103/revmodphys.35.710

Cited by 207 publications

(92 citation statements)

References 11 publications

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“…In each case discussed below, we begin by using standard degenerate perturbation theory [9] in λ to obtain the low-energy effective Hamiltonian that encodes the 'slow' dynamics induced by the H 1 term within the ground state manifold of H 0 . Terms in this expansion are naturally organized according to the power (n λ ) of λ they carry, and by the number (n b ) of different bonds on which constituents of H 1 act.…”

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confidence: 99%

Magnetization Plateaus and Sublattice Ordering in Easy-Axis Kagome Lattice Antiferromagnets

2008

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We study kagome lattice antiferromagnets where the effects of easy axis single-ion anisotropy (D) dominates over the Heisenberg exchange J. For S ≥ 3/2, virtual quantum fluctuations help lift the extensive classical degeneracy. We demonstrate the presence of a one-third magnetization plateau for a broad range of magnetic fields J 3 /D 2 B JS along the easy axis. The fully equilibriated system at low temperature on this plateau develops an unusual nematic order that breaks sublattice rotation symmetry but not translation symmetry-however, extremely slow dynamics associated with this ordering is expected to lead to glassy freezing of the system on intermediate time-scales. Geometrically frustrated magnets, which are characterized by a large number of symmetry unrelated classical ground states, display a wealth of new phenomena. In contrast to their unfrustrated counterparts, in which the low temperature ordering is largely determined by classical energetics even for small spin length S, quantum effects can play a crucial role in frustrated magnets, precisely because the classical energetics fails to pick a unique state. Of particular interest are those systems in which a broad cooperative paramagnetic regime [1] at intermediate temperatures gives way at low temperature to a variety of novel ordered and liquid phases [2] arising from the quantum fluctuations of spins. Even when magnetic order is selected, the resulting patterns are often complex, and are promising candidates for realizing multiferroic properties [3].The Kagome lattice composed of corner sharing triangles is one of the the most frustrated lattice arrangements possible, and one that occurs commonly in nature. Ground states of the antiferromagnetic Heisenberg model on this lattice typically involve coplanar arrangement of spins, and are extensively degenerate in the limit of classical spins [4]. Despite much work on the role of quantum fluctuations in lifting this degeneracy, several fundamental questions regarding the ground state of low spin Kagome antiferromagnets (eg. S = 1/2, 1, 3/2 . . . ) remain to be decisively settled [2].In some cases however, collinear spin arrangements are preferred, for example in the presence of strong single-ion anisotropy that leads to an easy axis. This occurs in the recently studied Nd 3 Ga 5 SiO 14 (NGS) compound [5,6], where Nd 3+ (total angular momentum J = 9/2) ions form a kagome antiferromagnet in which the common c axis is the easy axis at low temperature (below 33K). The classical ground states of such collinear spins are also extensively degenerate. In contrast to the isotropic case, the effect of quantum fluctuations in selecting the low temperature state can be studied in a much more controlled fashion, and this is the focus of our work here.Our results are readily stated. First, the selection mechanism depends strongly on the spin length S. Below a critical spin length S c = 3/2, quantum transitions between different classical ground states dominate the degeneracy splitting. This 'kinetic energy' dominat...

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confidence: 99%

Magnetization Plateaus and Sublattice Ordering in Easy-Axis Kagome Lattice Antiferromagnets

2008

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“…(8) we obtain an alternative expression, viz, Primas [21] considered an order by order expansion of thisH using the Baker-CampbellHausdorff formula and commutator functions to determine S, a technique also used in many other settings in which a transformation is in a Lie group, see, e.g., Ref. [22] and references therein.…”

Section: B Geometry Of the Model Spacementioning

confidence: 99%

Geometry of effective Hamiltonians

Kvaal

2008

Phys. Rev. C

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We give a complete geometrical description of the effective Hamiltonians common in nuclear shell model calculations. By recasting the theory in a manifestly geometric form, we reinterpret and clarify several points. Some of these results are hitherto unknown or unpublished. In particular, commuting observables and symmetries are discussed in detail. Simple and explicit proofs are given, and numerical algorithms are proposed, that improve and stabilize common methods used today.

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“…It is natural to chooseF ≡ 0 or PĤĜ = 0. This is not equal to the "off-diagonal" conditionPĤ 0Ĝ P = 0 traditionally used in canonical perturbation theory 5 . BecauseLPĤFPĤ =PĤLPĤF , the superprojectorPĤ is itself insensitive to this choice.…”

Section: Application Of This Expression To the Hamiltonianĥ And The Imentioning

confidence: 99%

“…Such transformations may be handled conveniently using superoperator formalism 5,13 . Here we will outline only some basics of it.…”

Section: Basic Perturbation Superoperatorsmentioning

confidence: 99%

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Kato expansion in quantum canonical perturbation theory

Nikolaev¹

2016

Journal of Mathematical Physics

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This work establishes a connection between canonical perturbation series in quantum mechanics and a Kato expansion for the resolvent of the Liouville superoperator.Our approach leads to an explicit expression for a generator of a block-diagonalizing Dyson's ordered exponential in arbitrary perturbation order. Unitary intertwining of perturbed and unperturbed averaging superprojectors allows for a description of ambiguities in the generator and block-diagonalized Hamiltonian.We compare the efficiency of the corresponding computational algorithm with the efficiencies of the Van Vleck and Magnus methods for high perturbative orders.

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Generalized Perturbation Theory in Operator Form

Cited by 207 publications

References 11 publications

Magnetization Plateaus and Sublattice Ordering in Easy-Axis Kagome Lattice Antiferromagnets

Magnetization Plateaus and Sublattice Ordering in Easy-Axis Kagome Lattice Antiferromagnets

Geometry of effective Hamiltonians

Kato expansion in quantum canonical perturbation theory

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