Spin Dynamics from Majorana Fermions

Abstract

Using the Majorana fermion representation of spin-1/2 local moments, we show how it is possible to directly read off the dynamic spin correlation and susceptibility from the one-particle propagator of the Majorana fermion. We illustrate our method by applying it to the spin dynamics of a nonequilibrium quantum dot, computing the voltage-dependent spin relaxation rate and showing that, at weak coupling, the fluctuation-dissipation relation for the spin of a quantum dot is voltagedependent. We confirm the voltage-dependent Curie susceptibility recently found by Parcollet and Hooley [Phys. Rev. B 66, 085315 (2002)].PACS numbers: 03.65. Ca, 72.15.Qm, 73.63.Kv, 76.20.+q The mathematical difficulties of representing spins in many body physics have long been recognized. The essence of the problem is that spin operators are nonabelian: they do not obey Wick's theorem and an expectation value of the product of many spin operators cannot be decomposed into products of two-operator expectation values, even within a free theory.A conventional response to this difficulty is to represent spins as bilinears of fermions [1] or as bosons [2]. One of the disadvantages of these approaches is that the Hilbert space of the fermions or bosons needs to be restricted by the application of constraints [3,4,5]. Another difficulty is the "vertex problem", which arises in the context of spin dynamics and spin relaxation. Once the spins are represented as bilinears, the spin-spin correlation functions are represented by two-particle Green's functions. The calculation of these quantities requires a knowledge of both the four-leg vertex and the single-particle Green's function. Typically, the vertex is simply neglected, or treated in a very approximate fashion.An alternative approach is to take advantage of the anticommuting properties of Pauli matrices, writing the spin operator in terms of Majorana fermions [6,7,8,9,10],where η = (η 1 , η 2 , η 3 ) is a triplet of Majorana fermions which satisfy {η a , η b } = δ ab . This representation does not require the imposition of a constraint: the fact that S 2 = 3/4 follows directly from the operator properties of the Majorana fermions. In this letter, we show how this representation also solves the vertex problem. To demonstrate this, we employ an alternative derivation [11] of the Majorana spin representation. Consider a spin-1/2 operator S with dynamics described by a Hamiltonian H. Let us now introduce a single Majorana fermion Φ which lives in a completely different Hilbert space, commuting with S and H. It follows that Φ is a fermionic constant of motion, dΦ/dt = −i[H, Φ] = 0: an object of fixed magnitude Φ 2 = 1/2 which anticommutes with all other fermion operators. We may now identify η in (1) with the operator identityWe may confirm that {η a , η b } = δ ab using the anticommuting algebra of spin-1/2 operators {S a , S b } = 1 2 δ ab . Furthermore, using the SU(2) algebra of spins, η × η = 2 S × S = 2i S, from which (1) follows immediately. As a last step in the derivation, we note tha...

Transport properties of fully screened Kondo models

“…correlation function of the spin-1/2 model to the one-particle Green function in an effective description in terms of Majorana fermions [89,90]. For large frequencies we recover the perturbative result [73,85] …”

Quantum Phase Transition in the Spin Boson Model

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

“…Noting the crucial property of the MFR σ α (τ )σ β = η α (τ )η β [16,17,19], if we introduce the Keldysh GFs of Majorana-fermions…”

“…Velizhanin and coworkers [13] work out * Electronic address: cqw@fudan.edu.cn expressions for GFs using the Redfield approximation, however their results violate the energy conservation. In order to preserve conservation laws, a Majorana-Fermion diagrammatic method (MFDM) [16][17][18][19] is further used in calculating GFs [15]. Compared with the QME, the NEGF can be applied to the intermediate coupling regime, but the strong coupling regime is still beyond its scope.…”