Unique Translation between Hamiltonian Operators and Functional Integrals

Gollisch, Tim; Wetterich, C.

doi:10.1103/physrevlett.86.1

Cited by 14 publications

(20 citation statements)

References 23 publications

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“…All expressions in the Hamiltonian should therefore be symmetrized whenever powers of non-commutating operators are encountered 18,19,20 . Only after symmetrization we may replace the field operators by numbers.…”

Section: Spin-wave Interactionsmentioning

confidence: 99%

Quantum Heisenberg antiferromagnets in a uniform magnetic field: Nonanalytic magnetic field dependence of the magnon spectrum

et al. 2008

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We re-examine the 1/S-correction to the self-energy of the gapless magnon of a D-dimensional quantum Heisenberg antiferromagnet in a uniform magnetic field h using a hybrid approach between 1/S-expansion and non-linear sigma model, where the Holstein-Primakoff bosons are expressed in terms of Hermitian field operators representing the uniform and the staggered components of the spin-operators [N. Hasselmann and P. Kopietz, Europhys. Lett. 74, 1067 (2006)]. By integrating over the field associated with the uniform spin-fluctuations we obtain the effective action for the staggered spin-fluctuations on the lattice, which contains fluctuations on all length scales and does not have the cutoff ambiguities of the non-linear sigma model. We show that in dimensions D ≤ 3 the magnetic field dependence of the spin-wave velocityc−(h) is non-analytic in h 2 , withc−(h)−c−(0) ∝ h 2 ln |h| in D = 3, andc−(h) −c−(0) ∝ |h| in D = 2. The frequency dependent magnon self-energy is found to exhibit an even more singular magnetic field dependence, implying a strong momentum dependence of the quasi-particle residue of the gapless magnon. We also discuss the problem of spontaneous magnon decay and show that in D > 1 dimensions the damping of magnons with momentum k is proportional to |k| 2D−1 if spontaneous magnon decay is kinematically allowed.

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Section: Spin-wave Interactionsmentioning

confidence: 99%

Quantum Heisenberg antiferromagnets in a uniform magnetic field: Nonanalytic magnetic field dependence of the magnon spectrum

et al. 2008

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“…Since the coherent state formalism is widely used in field theory and statistical physics, one would have to view any suggestion that it may produce incorrect results with deep concern. In order to support their finding, GW study the simple one-dimensional nonlinear oscillator described by the Hamiltonian H ma y a 1 l 2 a y a y aa (1) and calculate the thermal expectation value ͗ p 2 ͘ both in conventional quantum mechanics and the proposed new functional integral approach ͓ p i͑a y 2 a͒͞ p 2 ͔. A comparison of both calculations indeed supports the proposed correction term.…”

Section: Comment On "Unique Translation Between Hamiltonian Operatorsmentioning

confidence: 97%

“…In a recent Letter [1], Gollisch and Wetterich (GW) show that a careful treatment of discretization errors in a phase-space path integral formulation of quantum mechanics leads to a correction term as compared to the standard form based on coherent states. Since the coherent state formalism is widely used in field theory and statistical physics, one would have to view any suggestion that it may produce incorrect results with deep concern.…”

Section: Comment On "Unique Translation Between Hamiltonian Operatorsmentioning

confidence: 99%

Comment on “Unique Translation between Hamiltonian Operators and Functional Integrals”

Weyrauch¹,

Schreiber²

2002

Phys. Rev. Lett.

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“…(14). General symmetry arguments [8] suggest that this suppression survives when we take into account the renormalization of the staggered spin fluctuations by the ferromagnetic ones.…”

mentioning

confidence: 99%

“…Since we shall later use the phase space path integral [13] to eliminate the ferromagnetic fluctuations, we symmetrizeĤ 4 whenever the vertices involve non-commuting operators [14]. The final result iŝ…”

mentioning

confidence: 99%

Spin-wave interactions in quantum antiferromagnets

Hasselmann

Kopietz

2006

Europhys. Lett.

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PACS. 75.10.Jm -Quantized spin models. PACS. 75.30.Ds -Spin waves.Abstract. -We study spin-wave interactions in quantum antiferromagnets by expressing the usual magnon annihilation and creation operators in terms of Hermitian field operators representing transverse staggered and ferromagnetic spin fluctuations. In this parameterization, which was anticipated by Anderson in 1952, the two-body interaction vertex between staggered spin fluctuations vanishes at long wavelengths. We derive a new effective action for the staggered fluctuations only by tracing out the ferromagnetic fluctuations. To one loop order, the renormalization group flow agrees with the nonlinear-σ-model approach.The spin-wave approach to quantum Heisenberg antiferromagnets, which was pioneered by Anderson [1] and Kubo [2] more than half a century ago, is still one of the most powerful methods to calculate the low-temperature properties of ordered magnets. In this approach, the components of the quantum-mechanical spin-operators S i are expressed in terms of canonical boson operators b i using the Holstein-Primakoff [3] or the Dyson-Maleev [4] transformation. The spin-S antiferromagnetic Heisenberg Hamiltoniancan then be written as a bosonic many-body HamiltonianĤ = −DN JS 2 +Ĥ 2 +Ĥ int , where the quadratic part isĤThe spins are assumed here to be localized at the sites r i of a D-dimensional hypercubic lattice with lattice spacing a. The lattice is bipartite, with sublattices labelled A and B. We also assume J ij = J > 0 for all pairs of nearest neighbors and J ij = 0 otherwise. The interactionĤ int involves at least four boson operators and higher powers of the small parameter 1/S, so that for large S it is reasonable to treatĤ int perturbatively. However, as we discuss in more detail below, in momentum space the vertices ofĤ int have a complicated non-analytic structure for small momenta [5][6][7], so that the expected supression of the interaction between long-wavelength Goldstone modes [8] is not manifest in this approach. On the other hand, in the calculation of physical quantities that are dominated by staggered spin-fluctuations the leading momentum-dependence of the vertices eventually cancels [9].c EDP Sciences

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Unique Translation between Hamiltonian Operators and Functional Integrals

Cited by 14 publications

References 23 publications

Quantum Heisenberg antiferromagnets in a uniform magnetic field: Nonanalytic magnetic field dependence of the magnon spectrum

Quantum Heisenberg antiferromagnets in a uniform magnetic field: Nonanalytic magnetic field dependence of the magnon spectrum

Comment on “Unique Translation between Hamiltonian Operators and Functional Integrals”

Spin-wave interactions in quantum antiferromagnets

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