Using Majorana spin-12representation for the spin-boson model

Abstract

In this paper we resolve a contradiction between the fact that the method based on the Majorana representation of spin-1/2 is exact and its failure to reproduce the perturbative Bloch-Redfield relaxation rates. Namely, for the spin-boson model, direct application of this method in the leading order allows for a straightforward computation of the transverse-spin correlations, however, for the longitudinal-spin correlations it apparently fails in the long-time limit. Here we indicate the reason for this failure. Moreover, we suggest how to apply this method to allow, nevertheless, for simple and accurate computations of spin correlations. Specifically, we demonstrate that accurate results are obtained by avoiding the use of the longitudinal Majorana fermion, and that correlations of the remaining transverse Majorana fermions can be easily evaluated using an effective Gaussian action. I. INTRODUCTIONApplication of field-theoretical methods to spin systems is hampered by the non-Abelian nature of the spin operators 1 . To circumvent this problem, one can represent spin operators in terms of fermions or bosons and use standard field theory 2 . Several formulations have been suggested, including the Jordan-Wigner 3 and Holstein-Primakoff 4 transformations, the Martin 5 Majorana-fermion and Abrikosov 6 fermion representations as well as the Schwinger-boson 7-10 and slave-fermion 11-16 techniques.Among other representations of spin operators the Majorana-fermion approach has a special property. In most other approaches the necessary extension of the Hilbert space requires taking into account additional constraints on the boson/fermion operators, which place the system to the physical subspace. As it was realized early on 17 and reemphasized recently 18 , for the Majorana representation this complicating step is not needed. Moreover, it was further observed 19,20 that a wide class of spin correlation functions can be reduced to Majorana correlations of the same order, so that the spin correlation functions can be computed very efficiently. This, in particular, includes single-spin problems, like the Kondo problem or the spin-boson problem. In this case one avoids potentially complex vertex corrections to the external vertices, and spin-spin correlation functions are essentially given by single-particle fermionic lines rather than by loop diagrams.Despite these advantages, the Majorana spin-1/2 representation is not used very often. One possible reason is that its application requires careful calculations. In particular, for the spin-boson model, application of this method allows for a straightforward computation of the correlation functions of the transverse spin components 20 , using the lowest-order self-energy. However, for longitudinal-spin correlations, it fails in the long-time limit. Here we indicate the reason for this behavior. Moreover, we suggest an approach that allows one to use the Majorana-fermion representation as a convenient and accurate tool for generic spin correlations. More specifically, we demonstra...

Heat transport in a two-level system driven by a time-dependent temperature

“…We start by performing a polaron transformation, which allows us to rewrite the coupling to the reservoirs in a manner that can be treated using perturbation theory [43][44][45]. Next, we represent the spin operators using Majorana fermions [46,47]. The commutation relations of the spin operators make it difficult to apply Wick's theorem, but we may switch to a Majorana fermion representation, which has proven useful to evaluate spin-spin correlation functions perturbatively [35,45,48].…”

Heat transport in a two-level system driven by a time-dependent temperature

“…We start by performing a polaron transformation, which allows us to rewrite the coupling to the reservoirs in a manner that can be treated using perturbation theory both for weak and strong couplings [35][36][37]. Next, we represent the spin operators using Majorana fermions [38,39]. The commutation relations of the spin operators make it difficult to apply Wick's theorem, but we may switch to a Majorana fermion representation, which has proven useful to evaluate spin-spin correlation functions perturbatively [31,37,40].…”

Energy transfer in the nonequilibrium spin-boson model: From weak to strong coupling

“…Sinceχ ′′ z (ω) is of primary interest , we will mainly focus on the calculation of the Keldysh GFs of σ z in the following, but the methodologies discussed below carry over easily to other Keldysh GFs as well with cautions [37]. We principally work here in the so-called nonadiabatic limit of ∆/ω c ≪ 1.…”