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...

We analyze out-of-equilibrium fluctuations in a driven spin system and relate them to the noise of spin susceptibility. In the spirit of the linear response theory we further relate the noise of susceptibility to a 4-spin correlation function in equilibrium. We show that, in contrast to the second noise (noise of noise), the noise of susceptibility is a direct measure of non-Gaussian fluctuations in the system. We develop a general framework for calculating the noise of susceptibility using the Majorana representation of spin-1/2 operators. We illustrate our approach by a simple example of non-interacting spins coupled to a dissipative (Ohmic) bath.PACS numbers: 85.25. Dq, 85.25.Am Noise in electronic circuits provides information about the microscopic structure of the system complementary to that obtained from linear response transport measurements 1,2 . For electronic circuits, the standard JohnsonNyquist noise is intimately related to dissipative processes with typical time scales of the order of picoseconds. At low frequencies it is "white", i.e. frequencyindependent. In contrast, the ubiquitous 1/f noise is related to slow processes, e.g., to slow rearrangements of impurities or the internal dynamics of two-level systems 3 . Its power spectrum is commonly described by the Hooge's law 4,5 , S V (f ) ∝ V 2 /f , where V is the average observed voltage. This suggests that this noise could only be observed out of equilibrium. But this was shown not to be the case by Voss and Clarke 6,7 , who measured the low-frequency fluctuations of the mean-square Johnson voltage in equilibrium (i.e., the second noise or noise of noise) and showed that these fluctuations possess a 1/f -like spectrum.Motivated by these experiments, Beck and Spruit 8 calculated the variance of the Johnson-Nyquist noise and showed that it comprised two contributions. The first one could be interpreted as arising from resistance fluctuations with a 1/f spectrum. The second, with a white spectrum, is intrinsic to any Gaussian fluctuating quantity. Consequently the equilibrium 1/f noise could only be observed at very low frequencies.From a technical point of view the variance of noise is described by a four-point correlation function 1,8 . Such objects appear also in other physical contexts. For example, Weissman 9,10 has proposed to distinguish the droplet and hierarchical models of spin glasses by the properties of the second noise, which can be expressed in terms of a particular four-spin correlation function.More recently, the problem of 1/f noise has attracted much attention in the field of superconducting quantum devices. Flux noise measurements initially performed with relatively large SQUIDs showed the 1/f behavior 11 .In the last decade similar effect has been observed in nanoscale quantum circuits 12-23 . Remarkably, the noise magnitude appears to be "universal", i.e., of the same order of magnitude for a wide range of device sizes. This noise is believed to originate from an assemblage of spins localized at the surface or interface lay...

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...