Majorana representation for dissipative spin systems

# Majorana representation for dissipative spin systems

P. Schad Yu. Makhlin B.N. Narozhny G. Schön A. Shnirman Institut für Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany Landau Institute for Theoretical Physics, acad. Semyonov av., 1a, 142432, Chernogolovka, Russia Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Russia National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe shosse 31, 115409 Moscow, Russia Institut für Theoretische Festkörperphysik, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany Institut für Nanotechnologie, Karlsruhe Institute of Technology, D-76021 Karlsruhe, Germany
###### Abstract

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 correlators
journal: Annals of Physics

Spin systems are notoriously difficult to describe using field-theoretic methods due to non-Abelian nature of the spin operators altland (). 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 Tsvelik (). There is no unique recipe for such an approach. Several formulations have been put forth for solving specific problems, including the Jordan-Wigner JordanWigner () and Holstein-Primakoff Holstein () transformations, the Martin Martin () Majorana-fermion and Abrikosov Avella12 () fermion representations, as well as the Schwinger-boson schwinger (); arovas (); read (); wang () and slave-fermion affleck (); marston (); andrei (); dagotto (); wen (); nayak () techniques.

The Jordan-Wigner transformation is the only “exact” mapping between the spin- 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 Tsvelik () where it is often applied to Bethe-Ansatz-solvable models or their variations (a generalization to higher dimensions does exist fradkin (); huerta (), 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. ZawFaz ()). The former issue may be resolved by additional constraints wang (); nayak () or by projecting out unphysical states Avella12 () at the expense of further complications such as the appearance of non-Abelian gauge fields read (); wang (); nayak (); andrei ().

The Majorana-fermion representation, suggested by Martin Martin (), 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  Spencer68 (); Shnirman03 (). 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 Martin () represents the spin operators in terms of bilinear combinations of Majorana fermions. The resulting Majorana theory appears to be interacting (similarly to the situation with the Jordan-Wigner transformation JordanWigner () and other fermionic representations affleck (); marston (); andrei (); dagotto (); wen (); nayak ()) and -point spin correlation functions are equivalent to -fermion correlators. However, a certain class of -point spin correlation functions can be reduced to -point Majorana correlators Mao03 (); Shnirman03 (). In this case, complicated vertex structures do not arise. This significant simplification applies to correlation functions which involve only a single spin (“auto-correlation” functions) or an even number of spin operators for each spin (“pair-wise” correlations). In the present paper we generalize this approach to higher-order correlation functions and develop a technique for practical calculations based on the Keldysh formalism Kamenev ().

We illustrate our general arguments by a specific example. For a model of paramagnetic spins coupled to an Ohmic bath dutta () we determine the four-point correlation function describing the noise of susceptibility Schad14 (). This quantity is closely related to the recently measured inductance fluctuations in SQUIDs SendelbachMcDermott2 (). In addition, it is a direct measure of non-Gaussian fluctuations that are relevant also in other physical contexts weissman88 (); weissman93 (); Burnett14 (). The noise of susceptibility is distinct from the better-known four-point correlator “second noise”, which always comprises a significant Gaussian contribution kogan (); beckspruit (); SeidlerSolin ().

The paper is divided into two major parts. In Section 1 we introduce the Majorana representation of spin operators and address general issues related to this approach. In Subsection 1.2 we discuss the fact that spin-spin correlation functions can be directly calculated as correlations of the Majorana fermions despite the enlargement of the fermion Hilbert space. In Subsection 1.3 we demonstrate the simplification of the theory for the case of auto- and pair-wise-correlations. In Subsections 1.4 and 1.5 we formulate of the above correlators within the Keldysh path-integral approach.

In Section 2 we apply our general arguments to the problem of the noise of higher-order spin correlators in the Bose-Kondo model. Here we introduce the path-integral formalism in the Matsubara representation and show that certain higher-order spin correlators can be calculated in the saddle-point approximation. Further technical details are provided in the Appendices. In A, we present the traditional diagrammatic perturbation theory. In B we justify the saddle-point approximation used in Section 2. Finally, in C we analyze a remarkable gauge freedom in our model.

## 1 Majorana Representation for Spin Operators

### 1.1 The Martin transformation

In this paper we focus on the following Majorana representation of the spin- operators introduced by Martin in 1959 Martin () in the framework of generalized classical dynamics:

 ^Sα=−i2ϵαβγ^ηβ^ηγ ,^Sx=−i^ηy^ηz ,^Sy=−i^ηz^ηx ,^Sz=−i^ηx^ηy . (1)

The Majorana operators obey the Clifford algebra

 {^ηα,^ηβ}=δαβ ,^η2α=1/2 , (2)

where denotes the anticommutator. The Majorana representation has been used in a variety of physical contexts, cf. Refs. Berezin77 (); Tsvelik (); Mao03 (); Shnirman03 (). (We remark that another normalization of the Majorana operators, , used in Ref. Shnirman03 () only changes some numerical prefactors at intermediate stages of the calculation.)

The above representation with the real Majorana fermion operators,

 ^η†=^η, (3)

perfectly reproduces the SU algebra of the operators

 [^Sα,^Sβ]=iϵαβγ^Sγ, (4)

and explicitly preserves the spin-rotation symmetry.

Applying standard field-theoretical methods to fermionic systems implies the existence of a Fock space. In a faithful representation of a spin , the dimensionality of the fermionic Fock space should coincide with the dimensionality of the Hilbert space of the spin. For example, the Jordan-Wigner transformation represents a system of spins- with the -dimensional Hilbert space in terms of fermions with the -dimensional Fock space. However, each Jordan-Wigner fermion may be expressed in terms of two Majorana fermions. In contrast, the Martin transformation (1) represents each spin in terms of three Majorana fermions and hence does not preserve the dimensionality of the spin Hilbert space.

One method of dealing with this issue is to express the Majorana (or “real fermion”) -operators in terms of ”complex” or ”Dirac” fermions (this is a common slang used to distinguish usual fermions from Majoranas; these fermions do not necessarily obey the Dirac equation) and use their respective standard Fock spaces. This requires an even number of Majorana fermions. The simplest possibility is to add one auxiliary Majorana (cf. the drone-fermion representations of Refs. Martin (); Kenan66 (); Doniach67 (); Spencer68 ()). In this case, the resulting target Hilbert space is -dimensional and can be interpreted as two copies of the spin Hilbert space Spencer68 (); Shnirman03 (). To make the spin-space isotropy even more explicit, one could add (and pair) an extra Majorana to each of the three ; this, however, results in an -dimensional Hilbert space, equivalent to four copies of the spin Hilbert space. In the following subsection we demonstrate that regardless of the construction of the Hilbert space, spin correlation functions coincide with Majorana-fermion correlators.

### 1.2 Equivalence of spin and Majorana correlation functions

Now we show explicitly that correlation functions of spin- operators can be directly computed in the Majorana representation (1) (regardless of the explicit construction of the Majorana Hilbert space). Indeed, the equation (1) gives a representation of SU(2) with . Thus, it is necessarily a direct sum of a certain number of irreducible, two-dimensional spin- representations, i.e, we obtain an integer number of copies of the spin (the actual number is determined by the particular choice of the number of auxiliary Majorana operators). The spin operators (1) do not switch between the copies, thus any trace of an operator built out of spin operators (1) is given by the number of copies times the trace in a two-dimensional spin space. This leads directly to the desired result as we discuss in more detail below.

Let us remind the reader that any function of the spin- operators (i.e., of Pauli matrices) is in fact a linear function. This follows from the following relation for the spin- operators

 ^Sα^Sβ=i2ϵαβγ^Sγ+14δαβ. (5)

For an arbitrary number of operators describing the same spin this implies that

 ^Sα1…^Sαn=aα^Sα+a0,a0,ax,ay,az∈C, (6)

where the complex numbers , , , and depend on the sequence . Since spin operators are traceless, the trace over the -dimensional () spin Hilbert space of the above combination of spin operators is given by the constant term :

 TrS{^Sα1…^Sαn}=dSa0(α1,...,αn),dS=2. (7)

The Martin representation (1) preserves the commutation relations as well as the relation (5), and hence the equality (6) remains valid in the Majorana representation (where the spin operators should be understood as Majorana bilinears). In particular, the coefficients remain the same. The trace over the Majorana Hilbert space can be performed by noting that Majorana bilinears are traceless due to their anticommutation properties. Again [cf. Eq. (7)], the only remaining contribution is given by the constant term :

 TrM{^Sα1…^Sαn}=dMa0(α1,...,αn), (8)

where is the dimension of the Majorana Hilbert space. Thus, tracing an arbitrary product of the spin- operators over the spin and Majorana Hilbert spaces yields the same result up to a numerical factor, determined by the dimensionalities of the Hilbert spaces).

Similar arguments were put forth in Ref. Spencer68 () in the context of the drone-fermion representation. The results of this Subsection were implied in Refs. Mao03 (); Shnirman03 () and given without proof in Ref. Schad14 ().

The above statement can be readily generalized to an arbitrary ensemble of spins. Indeed, any function of spin operators is still linear in the components of each spin. If the operators on the left-hand side of Eq. (6) describe more than one spin, then on the right-hand side additional terms appear, which contain all possible products of operators related to different spins (for example, in the case of two spins the right-hand side reads ). However, all such additional terms are still traceless, and hence the only change in Eqs. (7) and (8) will be in the constants and .

Consider now a real-time spin auto-correlation function:

 (9)

where is the non-normalized Gibbs density matrix. If in addition the spins are coupled to other degrees of freedom denoted collectively by (this is also described by ), then following the line of arguments presented above we can write , where and are matrices in the space. Moreover, is the reduced density matrix describing the rest of the system (the -degrees of freedom). In this case, the partition function may be written as

 Tr{^ρ}=TrXTrS{^ρ}=TrXTrS{^ρ0(X)+^Sα^ρα(X)}=dsTrX{^ρ0(X)} . (10)

Now, each Heisenberg spin operator is related to the Schrödinger operators (i.e., the Pauli matrices) by the time evolution operator . The latter can be expanded similarly to the density matrix. Thus we can write , where and are matrices in -space. As a result, we can formally perform the trace over the spin variables in the auto-correlation function (9) and find

 (11)

In the Majorana representation the structure of the above equations remains the same. The averages can be formally calculated similarly to Eq. (8). As a result, we arrive at the expression

 (12)

which is identical to Eq. (11). The generalization to the case of general (multi-spin) correlation function is straightforward (see above). Thus we have demonstrated that spin correlation functions can be calculated with the help of the Majorana representation (1) without any projection onto ”physical” states:

 (13)

The operators on the right-hand of Eq. (13) represent the Majorana bilinears  (1).

### 1.3 Simplified representation for autocorrelation functions

The arguments presented in the previous Section show that any -point spin correlation function may be represented in terms of a -point correlation function of Majorana fermions. Here we demonstrate that for a certain class of correlation functions this correspondence can be significantly simplified Mao03 (); Shnirman03 (); Coleman95 ().

Let us rewrite Eq. (1) as follows

 ^Sα=^Θ^ηα,^Θ=−2i^ηx^ηy^ηz,^Θ2=1/2. (14)

One can easily verify that the operator commutes with all three Majorana operators . Consequently, this operator also commutes with any Hamiltonian expressed in terms of , and thus the corresponding Heisenberg operator is time-independent. The averaged product of a pair of spin operators can now be represented as follows

 ⟨^Sα(t))^Sβ(t′)⟩M=⟨^Θ^ηα(t)^Θ^ηβ(t′)⟩=12⟨^ηα(t)^ηβ(t′)⟩. (15)

Thereby a two-point spin correlation function reduces to a two-point (rather than four-point) Majorana-fermion correlation.

Unfortunately, the above relation cannot be directly generalized to time-ordered correlators (or Green’s functions) due to the fact that spin and Majorana operators are influenced by time ordering in different ways. Explicitly, a time-ordered average of two spin operators is given by

 ⟨T{^Sα(t)^Sβ(t′)}⟩M≡⎧⎪⎨⎪⎩⟨^Sα(t)^Sβ(t′)⟩,t>t′⟨^Sβ(t′)^Sα(t)⟩,tt′⟨^ηβ(t′)^ηα(t)⟩,t

where the latter expression differs from the time-ordered average of the two Majorana-fermion operators by the absence of the minus sign in the lower line.

While this problem can be circumvented by introducing Green’s functions of the operator Cabrera13 (), we consider here a simpler approach. The missing sign can be compensated for with the help of an auxiliary Majorana fermion that anti-commutes with the three operators , i.e., and . Since any Hamiltonian will be expressed in terms of bilinears of , the operator commutes with the Hamiltonian and, thus, is time independent. We keep, however, its formal time argument in order to be able to treat time-ordered operator products correctly. We notice that

 ⟨T{im(t)ηα(t)im(t′)ηβ(t′)}⟩≡{⟨im(t)ηα(t)im(t′)ηβ(t′)⟩,t>t′⟨im(t′)ηβ(t′)im(t)ηα(t)⟩,tt′⟨ηβ(t′)ηα(t)⟩,t

Thus we arrive at the identity

 ⟨T{^Sα(t)^Sβ(t′)}⟩M=⟨T{i^m(t)^ηα(t)i^m(t′)^ηβ(t′)}⟩. (17)

Here the -point spin correlator is again expressed in terms of a -point Majorana correlation function. However, in contrast to the direct application of the Martin transformation (1), here two of the Majorana operators and do not have any dynamical properties and only serve the purpose of writing the time ordering (16) in a compact form.

Similarly, we can use the auxiliary operator to express higher-order correlation functions. For a -point spin correlator we find

 (18)

The correlation functions (17) and (18) are in fact auto-correlation functions in the sense that they involve operators describing the same spin. Clearly, the simplification (15) cannot be extended to different spins since the operators and anti-commute and . Therefore, at the level of -point correlation functions the simplification described in this section applies to auto-correlation functions only. Generalizing this technique to higher-order correlation functions, we can compute autocorrelators, such as the -point function (18), as well as “pair-wise” correlators comprised of pairs of operators for each spin, such as

Other correlators, such as , have to be computed by different methods.

### 1.4 Spin correlation functions in the Keldysh formalism

Real-time correlation functions at finite temperatures can be conveniently computed within the Keldysh formalism altland (); Kamenev (). The calculation amounts to finding the generating functional  Kamenev () and then taking the derivative with respect to the source fields.

For a spin system, the generating functional may be defined as follows

 Zλ=∫D[…]exp⎧⎪⎨⎪⎩iS0+i∫Cdt(λclαSqα+λqαSclα)⎫⎪⎬⎪⎭, (19)

where denotes the appropriate measure of integration whereas represents the action of the model under consideration. In particular one could choose to integrate over the SU group manifold and would represent then the appropriate Haar measure altland (). In this paper we choose a more straightforward method of integrating over real Grassmann variables representing the Majorana operators of (1). In (19) are the source fields. The superscripts and refer to the “classical” and “quantum” variables Kamenev () that are defined as the sum and difference of the corresponding fields belonging to the upper () and lower () branch of the Keldysh contour

 Scl(q)α=1√2(Suα±Sdα),λcl(q)α=1√2(λuα±λdα).

The ”classical” source term defined in this way describes the physical probing field, , while the ”quantum” term is only needed to construct the correlation function and is set to zero at the end of the calculation.

Taking the derivative of the functional (19) with respect to the source fields , one finds the spin correlation functions. In particular, the one-point function defines the magnetization

 √2Mα=⟨Sα,cl(t)⟩=−iδZλδλq(t)∣∣∣λq=0. (20)

The spin susceptibility is given by a -point function

 χα,β(t,t′)=i⟨TKSα,cl(t)Sβ,q(t′)⟩=−iδ2Zλδλβ,cl(t′)δλα,q(t)∣∣∣λq=0, (21)

while the noise spectrum kogan () is determined by a different -point correlator, .

Below in Section  2, we will be interested in a specific -point function that determines the experimentally accessible noise of susceptibility Schad14 (); SendelbachMcDermott2 ()

 Cχ(t1,t′1,t2,t′2)=−⟨TKSα,cl(t1)Sα,q(t′1)Sα,cl(t2)Sα,q(t′2)⟩=−δ4Zλδλα,cl(t′2)δλα,q(t2)δλα,cl(t′1)δλα,q(t1)∣∣∣λq=0, (22)

and we focus on the case of identical spin indices.

### 1.5 Correlation functions of Majorana fermions in the path-integral representation

Let us now reformulate the mapping between the spin and Majorana fermion operators discussed above in Sections  1.2 and 1.3 in the language of the Keldysh functional integrals. In the operator language we have established the correspondence (13) between the spin -point functions and Majorana -point functions. In special cases (namely, for autocorrelation functions and ‘pair’ correlators) we found simpler relations (17) and (18).

For usual (Dirac) fermions the path integral approach is based on the concept of coherent states Kamenev (); altland (), which are eigenstates of the fermionic annihilation operators. An annihilation operator can only be constructed from two Majorana fermions. In many-body problems, this is typically achieved by either “halving” deBoer96 (); bastianelli (); shankar () or “doubling” deBoer96 (); bastianelli (); Nilsson13 () the number of Majorana operators. Keeping in mind applications to systems with a small number of degrees of freedom (such as the single-spin problem discussed below), we adopt the doubling procedure, where we add a free Majorana fermion to each of the operators introduced by the Martin transformation (1). The interaction part of the Keldysh action can then be formulated in terms of the Grassmann variables corresponding to the Majorana operator in the previous sections. The additional Grassmann variables appear only in the “free” quadratic part of the action and are decoupled from the physical system under consideration.

The Majorana-fermion -point correlation function on the right-hand side of Eq. (13), or rather its time-ordered counterpart, can be obtained either by differentiations of the generating functional with respect to Grassmann source fields each coupled to a single Grassmann variable , or by differentiations with respect to -number source fields coupled to pairs of Grassmann variables . These pairs should then be chosen to represent the spin components according to Eq. (1) as in (19). For the autocorrelation functions or the pair correlators we would like to use the simplified correspondence, i.e., Eqs. (17) and (18). Here the auxiliary Majorana fermion can be represented either by one of the free Grassmann variables used to construct the functional integral, or by one out of yet another pair of Grassmann variables that are added to the theory specifically for the purpose of computing the correlators (17) and (18). The source fields can again be chosen as either Grassmann variables or -numbers. In the explicit calculation below we chose the latter option. Of course, physical results are independent of these technical details.

## 2 Spin correlators in the Bose-Kondo model

In this section we apply the general conclusion reached in the previous section to a specific example. For simplicity of the presentation we use, first, the Matsubara technique, while the final results are formulated in real time in the frame of the Keldysh formalism.

As a model we choose a zero-field spin-isotropic Bose-Kondo model (see Ref. ZarandDemlerBoseKondo (); ZhuSi () and references therein). This model appears in various physical contexts, including spin glasses and liquids GeorgesParcolletSachdev2000 (); GeorgesParcolletSachdev2001 (). In the context of noise a similar model is known as the Dutta-Horn model dutta (), where a large number of independent spins are coupled each to their own bath. In the Majorana representation introduced above (1), the model Hamiltonian reads

 H=→S→X+HB=−i2Xαϵαβγηβηγ+HB , (23)

where is the Hamiltonian of the bosonic bath controlling the free dynamics of . The latter may be characterized by a Matsubara correlation function . The function can be written as

 Π(iωm)=Λ∫−Λdxπρ(|x|)signxx−iωm , (24)

where is the bath spectral density and . In the Ohmic case considered here, . For , this gives . One can perform an RG procedure by integrating out energies of the bath between and . As a result the coupling constant is rescaled. The RG differential equation reads (see, e.g., Ref. ZarandDemlerBoseKondo (); ZhuSi ()). One can supplement this with the scaling equation for the quasi-particle weight , which could be important PhysRevB.61.15152 () as we are interested in Green’s functions of the Majorana fermionic operators. For , the renormalization effects are not important as long the the temperature is high . Here we assume this to be the case. Thus we can safely reduce the cutoff to a value of the order of temperature.

### 2.1 Matsubara path integral

We use the Matsubara imaginary-time technique (, ). The partition function (, for brevity we omit the source fields) reads

 Z =∫D[→X]D[ηα]exp⎧⎪⎨⎪⎩iSB+i1/T∫0dτ[12ηα(τ)i∂τηα(τ)+12Xα(τ)ϵαβγηβ(τ)ηγ(τ)]⎫⎪⎬⎪⎭ . (25)

Here is the free bosonic action. The first step is to average over the fluctuations of , yielding

 Z (26)

the matrix is of the form

 M =⎛⎜⎝011101110⎞⎟⎠. (27)

Next we decouple the quartic Majorana-interaction in a different channel. To this end we rearrange

 iSM,int[ηα] =−14∫dτdτ′MαβΠ(τ−τ′)ηα(τ)ηβ(τ)ηα(τ′)ηβ(τ′) =14∫dτdτ′Π(τ−τ′)[ηα(τ)ηα(τ′)]Mαβ[ηβ(τ)ηβ(τ′)] . (28)

We now employ the Hubbard-Stratonovich transformation by introducing fields . These fields inherit the symmetry of Majorana propagators, therefore . The new effective action reads

 iS[ηα,Σα]= ∫dτdτ′⎛⎝12ηα(τ)(G−1α)ττ′ηα(τ′)−14Σα(τ,τ′)M−1αβΣβ(τ,τ′)Π(τ−τ′)⎞⎠ . (29)

The Majorana Green’s function in (29) is

 (G−1α)ττ′ =−δ(τ−τ′)∂τ′−Σα(τ,τ′) . (30)

The function is positive and non-zero. The standard form reads

 Π(τ−τ′)=gπT2sin2(πT(τ−τ′)) . (31)

It is cut off at short times, , leading to the maximal value of order . We use the divergent form keeping the regularization and renormalization in mind. Diagonalizing , we choose the eigenmodes with positive eigenvalues to be real and eigenmodes with negative eigenvalues to be imaginary, such that the overall sign of the action term in the exponent is negative, and the functional integral over converges. In other words, we choose , where the real part is ‘diagonal’, , and the imaginary part is orthogonal to it, (i.e., and describes two degenerate modes); with this choice the 3D integral over and converges. The redecoupled action (29) is again quadratic in Majorana Grassmann variables , which allows us to integrate them out and to obtain an effective action of -fields:

 iS[Σα]=12∑α={x,y,z}Trlog(G−1α)−14∫dτdτ′Σα(τ,τ′)M−1αβΣβ(τ,τ′)Π(τ−τ′) . (32)

We can now identify the saddle point and fluctuations of the effective -action. The saddle-point solution is found by expanding . The linear order in vanishes for

 Σ0α(τ−τ′) =Π(τ−τ′)MαβG0β(τ−τ′), (33)

where . A straightforward calculation, in which we disregard the broadening of on the r.h.s. of Eq. (33), leads to

 Σ0α(iϵn)=−Λ∫−Λdxπgxcoth[βx/2]x−iϵn . (34)

Upon the analytic continuation and for , we obtain the retarded self-energy

 ΣR0α(ϵ→0)=−2igT=−iΓ . (35)

Here we recognize to be the (Korringa) relaxation rate. In terms of the eigenmodes this solution means , whereas .

### 2.3 Fluctuations

To study the fluctuations we expand the trace-log-term in (32) to second order in . The action reads

 iSδΣ = −14∑α∫dτ1dτ2dτ3dτ4G0,α(τ1−τ2)δΣα(τ2,τ3)G0,α(τ3−τ4)δΣα(τ4,τ1) (36) − 14∫dτdτ′δΣα(τ,τ′)M−1αβδΣβ(τ,τ′)Π(τ−τ′) .

The Fourier transform of is introduced via

 δΣα(ϵn,νm)=∫dτdτ′eiνm(τ+τ′)eiϵn(τ−τ′)δΣα(τ,τ′) . (37)

The direct analysis shows that one of and must be fermionic and the other bosonic, so that both and are fermionic. Then we obtain

 iSδΣ = −T24∑α∑ϵn,νmG0,α(ϵn+νm)G0,α(ϵn−νm)δΣα(ϵn,νm)δΣα(ϵn,−νm) (38) − T34∑αβ∑ϵ1ϵ2νA(ϵ1+ϵ2)δΣα(ϵ1,ν)M−1αβδΣβ(ϵ2,−ν) .

Here

 A(iωm)=1/T∫0dτeiωmτΠ(τ)=1gπT21/T∫0dτeiωmτsin2(πTτ)=12gπT3(δm,0−12δm,1−12δm,−1) . (39)

 G0,α(iϵn)=1iϵn+iΓsign(ϵn) . (40)

Since is necessarily fermionic, we have . Thus, the first term of (38) cannot compete with the second one which is proportional to . This important observation allows us to disregard the first term of (38) and essentially all the contributions of the second and higher orders, originating from the trace-log term of (32). This in turn simplifies calculations of the higher-order spin correlators in the next section.

The argument above for the smallness of the first term of (38) is based on the discreteness of the Matsubara fermionic frequencies. Ultimately, we are interested in real times and the behavior in various frequency ranges, including low frequencies ; hence one should be careful with the estimates. The expressions above indicate that for the Green functions in the first term of (38) might become of order . This, in turn, might imply that the (prefactor of in the) first term of (38) scales with and dominates over the second term . To clarify the situation in the low-frequency range we perform a direct Keldysh calculation in B. We conclude that the first term of (38) does not become large in the domain of low real frequencies. Thus, the second term of (38) dominates.

### 2.4 Averaging over fluctuations

The knowledge of the propagator of the -fluctuations allows one to construct a new perturbative series, starting at the fixed-point solution given in Eq. (35). The new perturbative expansion for the Green function is based on the following series:

 ^Gα(τ,τ′)=^G0,α(τ,τ′)+∫dτ1dτ2^G0,α(τ,τ1)δ^Σα(τ1,τ2)^G0,α(τ2,τ′)+∫dτ1dτ2dτ3dτ4^G0,α(τ,τ1)δ^Σα(τ1,τ2)^G0,α(τ2,τ3)δ^Σα(τ3,τ4)^G0,α(τ4,τ′)+... (41)

This series is to be averaged over the fluctuations to obtain . However, on the right-hand side the spin index is the same in all the terms, essentially because the saddle-point Green function is diagonal in spin space. Taking into account the dominance of the second term in the action (38), we conclude that

 ⟨δ^Σα(…)δ^Σα(…)⟩∝Mαα=0 , (42)

because the diagonal entries of the matrix are zeros (note that in Eq. (42) the contributions of the real and imaginary -components cancel each other, cf. the discussion above Eq. (32)). Thus, the fluctuation-averaged Green function coincides with the saddle-point solution

 ⟨^Gα(t,t′)⟩δΣ=^G0,α(t,t′) . (43)

Moreover, for averages of Green-function products with the same spin index we can substitute all with their saddle-point values due to the relation (42); this simplifies calculations considerably. In the next section we employ this convenient property.

### 2.5 Correlation functions

As a first example, we calculate the diagonal spin susceptibility

 χ(t,t′)=i⟨TKSα,cl(t)Sα,q(t′)⟩=−i⟨TK[^m(t)^ηα(t)]cl[^m(t′)^ηα(t′)]q⟩ . (44)

The only contribution to is given by the diagram in Fig. 1.

We immediately arrive at the standard result

 χ′′(ω)=12Γtanhω2Tω2+Γ2ω≪T≈14TωΓω2+Γ2 . (45)

Notice that no vertex corrections appear in our case, and the precision of this calculation rests solely on the precision, with which the self-energy is evaluated.

We further demonstrate the power of our approach by calculating one of the higher-order spin correlators. We evaluate the connected part of the 4-th order correlator (22), which is related to noise of spin susceptibility Schad14 (); SendelbachMcDermott2 (). Using (18) we obtain

 Cχ(t1,t′1,t2,t′2) = −⟨TKSα,cl(t1)Sα,q(t′1)Sα,cl(t2)Sα,q(t′2)⟩ (46) = −⟨TK[^m(t1)^ηα(t1)]cl[^m(t′1)^ηα(t′1)]q[^m(t2)^ηα(t2)]cl[^m(t′2)^ηα(t′2)]q⟩ .

The discussion in the previous section implies that the connected part of this correlator is given by the six diagrams depicted in Fig. 2. Here, the double solid lines stand for the saddle-point Green functions , whereas the dashed lines represent the trivial correlators of the conserved quantity . As shown in the previous section, we may use the saddle-point Green functions, since the contribution of the -fluctuations vanishes, cf. (42).

For completeness, we show the result, which is put into context and interpreted in detail elsewhere Schad14 ().

 Cconnχ(ν,ω1,ω2)=iΓ28T2ω1+ω2+2iΓ(ω1+iΓ)(ω2+iΓ)(ω1+ν+iΓ)(ω2−ν+iΓ).

If a spin correlation function involves different spin components, one can no longer rely on the saddle-point contributions. One example is the correlator

 ⟨TKSα,cl(t1)Sα,q(t′1)Sβ,cl(t2)Sβ,q(t′2)⟩, (47)

related to the correlations of susceptibilities in different directions. The non-zero off-diagonal fluctuations contribute to this correlator, and thus additional diagrams have to be considered. This is, however, beyond the scope of the present paper.

## 3 Conclusion

In this paper we have demonstrated the efficiency of the Majorana representation of the spin- operators (1). We have shown that the representation (1) allows for a direct calculation of spin correlation functions in terms of Majorana fermions despite the fact that their Hilbert space is enlarged as compared to that of the spins. The precise construction of the Majorana Hilbert space was shown to be irrelevant as far as the correlation functions are concerned.

The Majorana representation is particularly efficient in the case of auto-correlation functions (or “pair-wise” correlation functions). Such -spin functions can be represented in terms of -point Majorana correlators, which significantly simplifies calculations. In particular complicated vertex structures do not appear.

As an example we have revisited the well known Bose-Kondo model. We have developed the Keldysh path-integral approach and have shown that the spin relaxation and susceptibility are efficiently described within the saddle-point approximation. Moreover we have shown that correlation functions containing a single spin projection can also be efficiently calculated at the saddle-point. In particular we have evaluated a -spin correlation function corresponding to the noise of susceptibility.

It would be interesting to apply our approach to a wider range of physical problems, for example, to a sub-Ohmic Bose-Kondo model describing physics of spin glasses GeorgesParcolletSachdev2000 (); GeorgesParcolletSachdev2001 (); ZarandDemlerBoseKondo (); ZhuSi ().

## 4 Acknowledgements

We acknowledge discussions with A. Mirlin and J. Clarke. This research was funded by the Intelligence Advanced Research Projects Activity (IARPA) through the US Army Research Office, the German Science Foundation (DFG) through Grants No. SCHO 287/7-1 and No. SH 81/2-1, the German-Israeli Foundation (GIF), and EU Network Grant Inter- NoM.

## Appendix A Perturbation theory

Given the Majorana representation for spins (1), the identities (17), (18), and the Hamiltonian (23), one could calculate correlation functions within perturbation theory by diagrammatic expansion. Here the zeroth-order Hamiltonian consists of the bath part only. The dressed Majorana Green function is self-consistently constructed, using the lowest-order self-energy. The lowest-order self-energy diagram is depicted in Fig. 3, it contains the “free” Majorana Green function ,

 GR/Af,α(ω)=(ω±i0)−1, (48) ΣRα(ω)=−∫dΩ2π(ΠK(ω+Ω)GAf,α(Ω)+ΠR(ω+Ω)GKf,α(Ω))=−2igT(1+O(ωT)), (49) GRα(ω)=(ω−ΣRα(ω))−1=(ω+iΓ)−1,Γ=2gT. (50)

Here the Keldysh components of the bath correlation functions read

 ^Πabαβ(t,t′)=δαβ⟨TK^Xα,a(t)^Xα,b(t′)⟩=^Πab(t,t′), (51) (52)

where , the spin indices , the Keldysh indices , and the ”classical” and ”quantum” operators .

Typically, this method is used to calculate 1- or 2-point correlation functions, e.g., magnetization or susceptibility. In lower orders Majorana propagators are never connected by a bosonic line. Such an element only appears in higher orders or in higher correlation functions as discussed in Schad14 (). An example is shown in Fig. 4. Here it is in principle necessary to consider more complex diagrams including, e.g., ladders of bosonic propagators. In the perturbation theory with a vanishing unperturbed spin Hamiltonian it is by no means justified to simply neglect these diagrams.

As an illustration let us consider renormalization of the interaction line. The dressed interaction carries four times (or frequencies), the Keldysh indices , and the spin indices ; diagrammatically it is depicted by

 ^Γab,cdαβ,γδ=\parbox[c]86.4pt\includegraphics[width=86.724pt,clip=true]GammaRenInt.pdf. (53)

For demonstration, we pick one of several possible components of that carries identical spin indices on the left (as well as on the right) and the Keldysh indices (21,21), which correspond to the retarded-retarded component, that is . The partially dressed interaction is obtained from the Dyson-type equation, depicted in Fig. 5, by summation of the leading contributions in the small- expansion. These are constructed out of the bosonic correlator combined with Green’s functions such that the upper and lower halves of the complex plane each contain one of the Green-functions poles. These leading contributions are of the same order in as the bare bosonic line, thus we might suspect a strong renormalization of the interaction line. This is, however, not the case due to a cancellation. Taking into account both contributions depicted in Fig. 5 we obtain

 ΓRRαβ(ω1,ω2,ν)=ΠK(ν)Mαβ−∫dΩ2πΠK(ν−Ω)MαγGRγ(ω1+Ω)GRγ(ω2−Ω)ΓRRγβ(ω1,ω2,Ω)+∫dΩ2πΠK(ν−Ω)MαγGRγ(ω1+Ω)GRγ(ω2−Ω)ΓRRγβ(ω1,ω2,−Ω−ω1+ω2) (54)

The Green functions are calculated within the approximation , which is justified and consistent in the high-temperature regime . Let us assume that the leading term in does not depend on the third frequency and splits . If this is the case, the integrals in the second and third term become equal, and we find that the renormalized interaction coincides with the bare one:

 ΓRRαβ(ω1,ω2)=ΠK(0)Mαβ. (55)

The problem in this perturbative approach is that the non-perturbed spin Hamiltonian is zero (the non-perturbed Hamiltonian consists of the bath Hamiltonian only). In other words, the spin-bath interaction is not weak as compared to the energy scale of the unperturbed spin dynamics. One might therefore expect that higher-order diagrams are important. Having observed a particular cancellation we cannot in principle exclude other important higher-order contributions. Instead of evaluating multiple higher-order diagrams we choose the path-integral approach in the following section allowing for a more straightforward analysis.

## Appendix B Majorana path integral in the Keldysh representation

In Section 2 we have developed a path-integral technique, which allowed us to obtain the self-energy (Korringa relaxation rate) as a saddle-point solution. We have argued that the fluctuations around this saddle point can sometimes be disregarded. Namely, this is the case for a correlation function involving only one spin component. This conclusion was based on the smallness of the higher-than-linear contributions to the trace-log term of the action (32). We have shown this using the small parameter for the fermionic Matsubara frequencies. To be able to treat also low real frequencies, we provide here the Keldysh version of the path-integral calculation.

For the model considered here, (23), the partition function reads

 Z =∫D[→X]D[η