Perturbation Expansion for the Magnetization of the Spin ½ Antiferromagnet

Kenan, Richard P.

doi:10.1063/1.1708512

Cited by 15 publications

(15 citation statements)

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“…Equation (6) can now be used to eliminate the Majorana fermion η 0 from the Hamiltonian (1). In practice, it is convenient to use the mixed Majorana-Dirac fermion representation ('drone'-fermion) [23][24][25]31], where two Majoranas are combined into a single complex (Dirac) fermion f = 1 2 (η x + iη y ), and a third Majorana fermion is denoted by η ≡ η z . Using Eqs.…”

Section: A Majorana-dirac Fermion Representationmentioning

confidence: 99%

Fermionic formalism for driven-dissipative multilevel systems

et al. 2020

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We present a fermionic description of non-equilibrium multi-level systems. Our approach uses the Keldysh path integral formalism and allows us to take into account periodic drives, as well as dissipative channels. The technique is based on the Majorana fermion representation of spin-1/2 models which follows earlier applications in the context of spin and Kondo systems. We apply this formalism to problems of increasing complexity: a dissipative two-level system, a driven-dissipative multi-level atom, and a generalized Dicke model describing many multi-level atoms coupled to a single cavity. We compare our theoretical predictions with recent QED experiments and point out the features of a counter-lasing transition. Our technique provides a convenient and powerful framework for analyzing driven-dissipative quantum systems, complementary to other approaches based on the solution of Lindblad master equations.

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Section: A Majorana-dirac Fermion Representationmentioning

confidence: 99%

Fermionic formalism for driven-dissipative multilevel systems

et al. 2020

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“…the drone-fermion representations of Refs. [5,20,34,35]). In this case, the resulting target Hilbert space is 4-dimensional and can be interpreted as two copies of the spin Hilbert space [20,21].…”

Section: The Martin Transformationmentioning

confidence: 99%

Majorana representation for dissipative spin systems

et al. 2015

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The Majorana representation of spin operators allows for efficient field-theoretical description of spin-spin correlation functions. Any N-point spin correlation function is equivalent to a 2N-point correlator of Majorana fermions. For a certain class of N-point spin correlation functions (including "auto" and "pair-wise" correlations) a further simplification is possible, as they can be reduced to N-point Majorana correlators. As a specific example we study the Bose-Kondo model. We develop a path-integral technique and obtain the spin relaxation rate from a saddle point solution of the theory. Furthermore, we show that fluctuations around the saddle point do not affect the correlation functions as long as the latter involve only a single spin projection. For illustration we calculate the 4-point spin correlation function corresponding to the noise of susceptibility. Keywords: Majorana, fermions, dissipation, spin correlatorsSpin systems are notoriously difficult to describe using field-theoretic methods due to non-Abelian nature of the spin operators [1]. Often one tries to circumvent the problem by mapping the spins onto a system of either bosons or fermions, for which a standard field theory can be developed [2]. There is no unique recipe for such an approach. Several formulations have been put forth for solving specific problems, including the Jordan-Wigner The Jordan-Wigner transformation is the only "exact" mapping between the spin-1/2 and fermion operators as it preserves not only the operator algebra but also the dimensionality of the Hilbert space. However, it is a non-local transformation specific to one spatial dimension [2] where it is often applied to Bethe-Ansatz-solvable models or their variations (a generalization to higher dimensions does exist [17,18], but it lacks the simplicity of the original approach). All other mappings suffer from the following two problems: (i) the Hilbert space of the fermionic or bosonic operators (the "target" Hilbert space) is enlarged as compared to the original spin Hilbert space, and (ii) the resulting theory in the fermionic or bosonic representation needs to be treated perturbatively, which often leads to complicated vertex structures (see e.g. [19]). The former issue may be resolved by additional constraints [10,16] or by projecting out unphysical states [6] at the expense of further complications such as the appearance of non-Abelian gauge fields [9,10,13,16].The Majorana-fermion representation, suggested by Martin [5], offers a possibility to avoid both types of problems mentioned above: (i) The target Hilbert space is indeed enlarged, but merely contains two (or more) copies of the original physical spin Hilbert space [20,21]. Matrix elements of physical quantities between different copies of the original Hilbert space vanish and thus the correlation functions may be evaluated directly in the target Hilbert space (this fact is often not fully appreciated; below we justify and further illustrate this statement). (ii) The Martin transformation [5] repr...

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“…We will now study this system using the Majorana representation. We write (5) in terms of Majorana operators to get a quartic expression, and then perform a Hartree-Fock (H-F) decomposition. Thus we write:…”

Section: Hartree-fock Treatment Ground State and Excitationsmentioning

confidence: 99%

“…For s = 1 2 such unconstrained representations can be found. The so called "drone fermion" representation [5,6,1] is one of the possibilities, where we write s + i = a † i φ i , s − i = φ i a i and s z i = a † i a i − 1 2 , where the a's are canonical anticommuting variables, and φ i is a real fermion with φ † = φ and φ 2 = 1. Thus φ is a 'drone' whose only 'job' is to make spins at different sites commute, rather than anticommute.…”

Section: Introductionmentioning

confidence: 99%