PHYSICAL REVIEW B 88, 184419 (2013)

Dynamics of magnetic moments coupled to electrons and lattice oscillations

B. Mera,1,2,* V. R. Vieira,2 and V. K. Dugaev2,3,4

1Physics of Information Group, Instituto de Telecomunicacoes, 1049-001 Lisbon, Portugal2Centro de Fısica das Interaccoes Fundamentais, Instituto Superior Tecnico, Universidade Tecnica de Lisboa, Avenida Rovisco Pais 1,

1049-001 Lisboa, Portugal3Department of Physics, Rzeszow University of Technology, aleja Powstancow Warszawy 6, 35-959 Rzeszow, Poland

4Institut fur Physik, Martin-Luther-Universitat Halle-Wittenberg, Heinrich-Damerow-Strasse 4, 06120 Halle, Germany(Received 16 July 2012; revised manuscript received 24 October 2013; published 19 November 2013)

Inspired by the models of Rebei and Parker [Phys. Rev. B 67, 104434 (2003)] and [Rebei, Hitchon and ParkerPhys. Rev. B 72, 064408 (2005)], we study a physical model which describes the behavior of magnetic momentsin a ferromagnet. The magnetic moments are associated to 3d electrons which interact with conduction-bandelectrons and with phonons. We study each interaction separately and then collect the results, assuming that theelectron-phonon interaction can be neglected. For the case of the spin-phonon interaction, we study the derivationof the equations of motion for the classical spin vector and find that the correct behavior, as given by the Brownequation for the spin vector and the Bloch equation, using the results obtained by Garanin [Phys. Rev. B 55,3050 (1997)] for the average over fluctuations of the spin vector, can be obtained in the high-temperature limit.At finite temperatures, we show that the Markovian approximation for the fluctuations is not correct for timescales below some thermal correlation time τth. For the case of electrons we work a perturbative expansion of theFeynman-Vernon influence functional. We find the expression for the random field correlation function whichexhibits the properties of the electron bath; namely, we observe Friedel oscillations at small temperatures. Thecomposite model (as well as the individual models) is shown to satisfy a fluctuation-dissipation theorem for alltemperature regimes if the behavior of the coupling constants of the phonon-spin interaction remain unchangedwith the temperature. The equations of motion are derived.

DOI: 10.1103/PhysRevB.88.184419 PACS number(s): 75.10.Jm, 76.60.Es

I. INTRODUCTION

The discovery of the Giant Magnetoresistance effect in1988, for which Grunberg and Fert were awarded the NobelPrize in Physics in 2007, motivated very intense scientificresearch of the magnetization dynamics at the nanometer scaleand led to the birth of a new field of research now calledspintronics.1–3 Spintronic devices usually have nanometer-scale sizes, can operate at high frequencies (∼1 GHz), andhave a wide range of applications which go from the creationof small-dimension (<1 μm) microwave frequency generatorsto the improvement of magnetic storage devices.

To successfully design high-frequency devices one needsto develop substantially the theoretical comprehension ofmagnetization dynamics at the appropriate scales.4–6 Thecomplete understanding of magnetization dynamics at thenanoscale can be probably achieved only by theorizing fromfirst principles implying a full quantum-mechanical treatment.In particular, if one wants to describe a spin system farfrom equilibrium, one needs to use the methods of quantumopen systems far from equilibrium, namely the Keldysh7 orLindblad8 formalisms.

It has been shown9 that the linear coupling interactionof a spin with a bosonic bath allows for the existence ofwhite noise in the equation of motion which, under someparticular conditions regarding the density of states of thebath, adopts the form of the Landau-Lifshitz-Gilbert-Brownequation.10–12 Also, it has been shown that if the spin vectorsatisfies a Landau-Lifshitz equation10 supplemented withwhite noise, then the magnetic moment as the average overthe fluctuations of the spin satisfies, in the limit of lowtemperatures, the Landau-Lifshitz equation10 and, in the limit

of high temperatures, the Landau-Lifshitz-Bloch13 equation.14

Collecting these results with the known result on formalequivalence between Landau-Lifshitz and Landau-Lifshitz-Gilbert equations allows to conclude that the interactionwith phonons (or any other bosonic bath satisfying certainconditions) can be responsible for the motion of magneticmoments as described by the Landau-Lifshitz-Bloch equation,which, in fact, gives a good description of the physical situationat high temperatures.15 A quantum field theoretical treatmentof the s-d interaction of conduction electrons and spins16 hasshown that in the semiclassical limit, the magnetization obeysa generalized Landau-Lifshitz equation.

The need to increase the speed of storage of informationin magnetic media and the limitations associated with thegeneration of magnetic field pulses by an electric currentrequire the research for ways of controlling magnetizationby other means than external magnetic fields. In 1996,subpicosecond demagnetization in ferromagnetic nickel wasachieved using a 60-fs laser in the experiments of Beaurepaireet al.17 Manipulating magnetization with ultrashort (of theorder of a femtosecond) laser pulses is now a major researchchallenge because at such time scales it might be possible toreverse the magnetization faster than within half a precessionalperiod.18 Because of this, it is of fundamental importanceto understand the time evolution of magnetic moments athigh temperatures and time scales approaching femtoseconds.Recent reviews on the state of the art of ultrafast spin dynamicsand prospects are given, for example, by Kyrilyuk et al.18 andZhang et al.19

The ultrashort laser pulses are expected to strongly cou-ple lattice oscillations, that is, phonons, and/or conduction

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B. MERA, V. R. VIEIRA, AND V. K. DUGAEV PHYSICAL REVIEW B 88, 184419 (2013)

electrons to the spins. This suggests that we may consideran effective microscopic theory for the system in which thefundamental interactions are spin-phonon, spin-electron, andelectron-phonon type. In this work we introduce and developa theory to model such physical system in which, for the sakeof simplicity, we neglect the electron-phonon interaction.

This paper is divided into three major sections. In thefirst section we consider a generalized version of the modelof Rebei and Parker9 that modeled the interaction of aspin-j and a bosonic bath, now written to describe theinteraction between a spin-j and a bath of phonons whichare assumed to be spin 1. The second section considersa model of a ferromagnet, inspired by the work of Rebeiet al.,16 assuming the interaction of a magnetic system of 3d

electrons, represented by a collection of spin-j vectors, with4s conduction electrons as a bath. In both sections we use thepath-integral representation for coherent-state matrix elementsof the reduced density matrix associated with the system. Inthe case of phonons, since we assume a linear interaction, theycan be exactly integrated out. After obtaining the effectiveaction appearing in the phase of the path integral we usea stationary phase approximation and obtain an equationof motion for the classical spin vector. We then display arandom field in the equations of motion through a Hubbard-Stratonovich transformation in the path-integral expression.The high-temperature limit is discussed and the Landau-Lifshitz-Gilbert-Brown and Landau-Lifshitz-Bloch equationsare recovered in this limit. The finite-temperature case isconsidered.

In the case of conduction electrons interacting with spins,the same procedure of integrating out the bath is not possibledue to the nonlinearity of the interaction. Nevertheless, we usea Hubbard-Stratonovich transformation and obtain an expan-sion for the effective action of the system which we truncate atthe second order of interaction. We relate this approximationto the Lindblad equation8 for the reduced density matrixobtained through the Born-Markov approximation.20 After thatwe arrive at an explicit expression of the electron contributionto the bath noise correlation function and present some plots ofthis function which manifest the physical phenomena involved.

The third section compiles the results of the first andsecond sections in a single model assuming a quantum systemcomposed of a spin-j vector field interacting with a bathof spin- 1

2 electrons and spin-1 phonons. Finally, we list ourconclusions.

II. SPIN-PHONON THEORY

We consider first the system of a single spin-j particleinteracting with a bath of spin-1 phonons which can havelongitudinal and transverse polarizations.

The total Hamiltonian is written in the form

H = HS + HR + Hi , (1)

presenting, respectively, the Hamiltonians for spin, bath, andinteraction between them.

We can introduce quantum operators associated to the spin-j particle Sα which are Hermitian operators satisfying the

angular momentum commutation relations

[Sα,Sβ] = iεαβγ Sγ , (2)

where εαβγ is the Levi-Civita tensor, and the repeated indicesare assumed to be summed from now on, unless otherwisestated. These operators constitute a representation of the Liealgebra of SU(2) in the (2j + 1)-dimensional vector spaceassociated to the irreducible representation labeled by j . Wealso introduce the creation and annihilation operators for thephonons, a†

μ(k), aμ(k), which are labeled by the polarizationindex μ = −1, 0, or 1 and by the momentum index k. Theysatisfy the Weyl algebra

[aμ(k),a†ν(k′)] = δμν(2π )3δ3(k − k′). (3)

The spin operators and the phonon operators commute.The system Hamiltonian is that of a spin-j particle in the

presence of an external magnetic field, denoted by H,

HS = −SαHα = −S · H. (4)

We write the phonon Hamiltonian in the form

HR =∑μ,k

ωμ(k)a†μ(k)aμ(k), (5)

where ωμ(k) is the phonon energy. In the following we assumethat transverse modes have the same energy, ω−1(k) = ω1(k).

The Hermitian Hamiltonian for the linear spin-phononcoupling in the macrospin approximation21 is

Hi = −∑

k

(H ∗αμ(k)a†

μ(k)Sα + Hαμ(k)Sαaμ(k)). (6)

Here the SU(2) invariance is explicit provided that the operator

H phα :=

∑k

(H ∗αμ(k)a†

μ(k) + Hαμ(k)aμ(k)) (7)

transforms as a vector under an SU(2) transformation associ-ated with the spin-j particle and consequently Hi will behaveas a scalar. We can then write the interaction Hamiltonian as

Hi = −SαH phα = −S · H

ph. (8)

Writing the system reduced density matrix with a coherent-state path-integral representation, one arrives, under theKeldysh formalism, at an effective action [see Eq. (A30)] inthe closed time path which is a functional of S, the classicalfield, and D, associated to the quantum fluctuations. The detailsof this calculation are presented in Appendix A. Performinga Hubbard-Stratonovich transformation on the part of theeffective action which couples two D(t) fields, as in Ref. 9,we obtain the equations of motion with a random field ξ (t).The real part of the correlation function of this random field is

Re{〈ξα(t1)ξβ(t2)〉} = Re{〈ξα(0)ξβ(t1 − t2)〉}

= Re

{ ∑k,μ

H ∗αμ(k)e−iωμ(k)(t1−t2)

× (1 + 2n(ωμ(k)))Hβμ(k)

}. (9)

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The last expression can be rewritten as

Re

{∑k,μ

∫ ∞

0dωδ(ω − ωμ(k))H ∗

αμ(k)Hβμ(k)

× e−iωμ(k)(t1−t2) coth

(βωμ(k)

2

) }. (10)

Let us assume that Hαμ(k) = Hα(ωμ(k)). In this case, wecan further write, interchanging the sum and integral andusing the property f (ωk)δ(ωk − ω) = f (ω)δ(ωk − ω) of theδ distribution,

Re

{∫ ∞

0dω

∑k,μ

δ(ω − ωμ(k))H ∗α (ω)Hβ(ω)e−iω(t1−t2)

× coth

(βω

2

) }(11)

or, equivalently,

Re

{ ∫ ∞

0dω ρ(ω)H ∗

α (ω)Hβ(ω)e−iω(t1−t2)

× coth

(βω

2

) }, (12)

where we defined the density of states

ρ(ω) =∑k,μ

δ(ω − ωμ(k)). (13)

In the high-temperature limit, the expression (12) reads

2kBT Re

{∫ ∞

0ρ(ω)

H ∗α (ω)Hβ(ω)

ωe−iω(t1−t2)dω

}. (14)

Assuming a linear dispersion relation for longitudinal andtransverse phonons, we obtain

ω0(k) = clk, ω±1(k) = ctk, (15)

and if we take the continuum limit for the bath, we arrive atthe density of states

ρ(ω) =∑k,μ

δ(ω − ωμ(k))

= V∑

μ

∫d3k

(2π )3δ(ω − ωμ(k))

= V

2π2

(1

c3l

+ 2

c3t

)ω2 = ρ0

(ω

ω0

)2

, (16)

where V is the volume of the reservoir and ρ0 = ρ(ω0) is thedensity of states evaluated at some frequency ω0 taken as areference. This means that if we want to recover the Gaussianrandom field in the limit of high temperatures we must ensurethat

Re{H ∗α (ω)Hβ(ω)} ∝ 1

ω. (17)

In particular, one must have

Re{H ∗α (ω)Hβ(ω)}ρ(ω)π

ω= αGδαβ, (18)

so that

Re{〈ξα(t)ξβ(t ′)〉} = δαβ2αGkBT δ(t − t ′). (19)

The constant αG is a parameter which gives the intensity ofthe Gilbert damping term in the equation of motion. The lastcondition constrains the form of Hα(ω). If we define

zα =(

πρ0

αGω20

)1/2

ω1/2Hα(ω), (20)

then the above condition reads

z∗αzβ + z∗

βzα = 2δαβ, (21)

which means that

zα = eiϕα (22)

and that the phases must satisfy

ϕα = ϕβ + (2n + 1)π

2(23)

for all α different from β and with n being an integer. Theform of Hα(ω) is, thus, constrained to be

Hα(ω) =(

πρ0

αGω20

)−1/2

ω−1/2eiϕα (24)

if the familiar fluctuation-dissipation theorem is satisfied.Assuming the above dependence for Hα(ω) for arbitrary

temperatures, we find that the noise correlation function canbe written as

Re{〈ξα(t)ξβ(0)〉} = αGδαβ

∫ ∞

0

dω

πω coth

(βω

2

)cos ωt

= αGδαβϕ(t), (25)

where we define the generalized function ϕ:

ϕ(t) =∫ ∞

0

dω

πω coth

(βω

2

)cos ωt, (26)

which characterizes the noise correlation function. The studyof such a generalized function or distribution can be found, forinstance, in Ref. 22.

To study the random field correlation function we noticethat the above integral can be expressed as

ϕ(t) =∫ ∞

0

dω

π

[coth

(βω

2

)ω − ω

]cos ωt

+ 1

2

(∫ ∞

−∞

dω

2π|ω|e−iωt

). (27)

Here the first integral is convergent and can be computed as∫ ∞

0

dω

π

[coth

(βω

2

)ω − ω

]cos ωt

= 1

2π

{1

t2−

(π

β

)2

cosech2

[(π

β

)t

]}. (28)

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B. MERA, V. R. VIEIRA, AND V. K. DUGAEV PHYSICAL REVIEW B 88, 184419 (2013)

The second integral can be computed if we regularize it withan exponential cutoff,∫ ∞

−∞

dω

2π|ω|e−iωt →

∫ ∞

−∞

dω

2π|ω|e−iωt e−λω = 1

π

λ2 − t2

(λ2 + t2)2

= 1

π

(− 1

t2+ λ2(λ2 + 3t2)

t2(λ2 + t2)2

), λ → 0+.

(29)

Thus, formally, we can write

ϕ(t) = − π

2β2cosech2

[(π

β

)t

], (30)

which is a well-known result.23 The correlation exhibits athermal correlation time arising from the behavior of ϕ in thelimit of |t | → ∞,

ϕ(t) → −2π (kBT )2 exp(−2πkBT |t |)= −2π (kBT )2 exp

(− |t |

τth

), (31)

where τth = (2πkBT )−1. In the case of arbitrarily smalltemperatures, the correlation time becomes very large [in fact,as seen from Eq. (30) the correlation function becomes inversequadratic in |t |].

It is important to note that the long time behavior of thecorrelations is negative. One must study the full expression forϕ for finite cutoff λ to understand what is happening in thislimit. We have∫ ∞

−∞dt Re{〈ξα(t)ξβ(0)〉} = 2αGkBT δαβ (32)

and what really happens is that, in fact, the cutoff-dependentpositive term is usually larger than the asymptotic negativeterm, except at zero temperature when the two effects canceleach other.

This behavior is most easily understood if we notice that theintegral in the definition of ϕ, Eq. (26), can be also written as

ϕ(t) = kBTd

dtcoth(πkBT t), (33)

and take into account, as pointed out by Ford andO’Connel,24,25 that the correct formula for the derivative ofthe hyperbolic cotangent is, in fact,

d coth x

dx= −csch2x + 2δ(x). (34)

The results are identical in both approaches, but the second ismore economical.

The existence of a thermal correlation time, which instandard units is τth = h/kBT = (1.27 × 10−12 s)/T , impliesthat the approximation of a Markovian description for therandom field is only valid for time scales which are longerthan τth or in the limit of high temperatures. Thus, in theproblem of magnetization control using ultrashort laser pulsesit might be not a good approximation to consider the processas Markovian. Equation (32) is remarkable because it is amanifestation of a fluctuation-dissipation theorem.

We also note that, with our assumptions regarding the con-stants Hα(ω) [that is, that Hαμ(k) = Hα(ωμ(k)) and Eq. (24)valid in all temperature regimes], we can always recover the

Landau-Lifshitz-Gilbert form of the equations of motion inthe limit of D → 0. To see it we just need to note that, usingEq. (A30), with D → 0,

δSeff

δD(s)

∣∣∣∣D=0

=∫ t

t0

dt ′K(s − t ′)S(t ′), (35)

where the matrix K arises from the response components ofthe closed time path Green’s function [see Eq. (A27)] and hasmatrix elements

Kαβ(t) = 2i∑k,μ

Re{H ∗αμ(k)Hβμ(k)}

× e−iωμ(k)t θ (−t). (36)

If we use all these assumptions, the last expression becomes

Gαβ(t) = 2αGδαβθ (−t)d

dtδ(t), (37)

which yields the result

δSeff

δD(s)

∣∣∣∣D(s)=0

= αGS(s), (38)

i.e., the Gilbert damping term, so that the equation of motionfor the spin becomes always the Landau-Lifshitz-Gilbertequation in the limit in which the random field is neglected.In the case of high temperatures we also recover the Landau-Lifshitz-Gilbert-Brown equation with the random field andin view of the results of Garanin14 we are able to recoverthe Landau-Lifshitz-Bloch equation for the average over spinfluctuations. For this purpose we can neglect the term in theequation of motion which couples two spin fields and a randomfield.14 This is valid in the limit of small fluctuations.

III. SPIN-ELECTRON THEORY

In this model the system can be viewed as a set of spinson a lattice, each spin written in the form of a generalspin-j representation of the SU(2) group, modeling 4d-typeelectrons in a magnetic medium. We consider the reservoirto be composed of conduction electrons. The interactionHamiltonian is an s-d-type interaction of the conductionelectrons with the spin vector field.

The model Hamiltonian we consider here has the form ofEq. (1), where

HS = −∑

i

Hi · Si − 1

2

∑ij

Jij Si · Sj , Jij � 0, (39)

or, in the momentum representation,

HS = −∑

k

H(−k) · S(k) − 1

2

∑k

J (k)S(−k) · S(k). (40)

The other terms are given by

HR =∑k,α

ε(k)c†α(k)cα(k) − λ∑

k

s(−k) · S(k)

−∑

k

s(−k) · h(k), (41)

where ε(k) is the energy of the conduction electrons, h(k)denotes the magnetic field felt by the electrons, and s(k)

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DYNAMICS OF MAGNETIC MOMENTS COUPLED TO . . . PHYSICAL REVIEW B 88, 184419 (2013)

denotes the composite operator

s(k) = 1

2

∑k′,α,β

c†α(k′ − k)(σ )αβ cβ(k′), (42)

which is the momentum representation of the spin densityoperator of conduction electrons. In the above equation,σ = (σ1,σ2,σ3)T is the vector whose components are thePauli matrices. The Latin indices, {i,j,k, . . .}, refer to spaceindices and the Greek indices, {α,β, . . .}, refer to spinindices. Furthermore, the operators cα(k), c†α(k) are electronannihilation and creation operators, which satisfy the algebra

{cα(k1),c†β(k2)} = δαβ(2π )3δ3(k1 − k2), (43)

{cα(k1),cβ(k2)} = {c†α(k1),c†β(k2)} = 0. (44)

We now consider the path-integral representation of thereduced density matrix associated with this system, as we didin Sec. II. As so, we are going to use the basis of coherentstates for the system and the reservoir (in this case we haveto use Grassmann variables for fermions), so that the states ofthe Hilbert space are written in the form

|S〉 ⊗ ||γ 〉, (45)

where

|S〉 =∏

k

|S(k)〉 =∏

k

D(S(k))|0〉, (46)

||γ 〉 =∏α,k

||γα(k)〉 =∏α,k

exp(c†α(k)γα(k))|0〉. (47)

The states |S〉 are to be defined in such a way that they satisfy〈S|S(k)|S〉 = S(k)〈S|1S |S〉. Let us now consider the state

|S′〉 =∏

i

(1+|ζ (Si)|2)−j exp(ζ (Si)S−,i)|ψ0〉, (48)

where |ψ0〉 is the tensor product of highest weightstates of the spin-j representation and ζ (Si) =tan(θi/2)eiϕi denotes the stereographic projection ofSi = j (sin θi cos ϕi, sin θi sin ϕi, cos θi)T through the northpole. Clearly, this state satisfies

〈S′|Si |S′〉 = Si〈S′|1S |S′〉 = Si , (49)

and we can now replace in the last equation Si and Si by theirFourier representations, yielding

〈S′|∑

k

eik·xi S(k)|S′〉 =∑

k

eik·xi S(k) (50)

or ∑k

eik·xi 〈S′|(S(k) − S(k))|S′〉 = 0. (51)

Since this relation is valid for all xi , we identify |S′〉 with |S〉because

〈S′|S(k)|S′〉 = S(k)〈S′|1S |S′〉. (52)

Assuming that at time t0 the spins and electron systems aredecoupled and the reduced density matrix of the bath, at thattime, takes the form of a Boltzmann factor with HamiltonianHR and inverse temperature β, as given by Eq. (A2), we easily

find, in analogy with Sec. II, replacing α → γ and takingspecial care because the γ ’s are Grassmann variables, that theexpression for the propagator is the same as Eq. (A13) but nowin I [S1,S2] one must sum over all spin-j degrees of freedomand the Feynman-Vernon functional is given by

F[S1,S2] =∫

dμ(γ0)dμ(γ1)dμ(γ2) k(−γ†0 ,t ; γ1,t0|S1)

×Z−1R k(γ †

1 , − iβ; γ2,0|0) × k∗(γ †0 ,t ; γ2,t0|S2),

(53)

where dμ(γ ) = ∏α,k d2γα(k) exp(−γ ∗

α (k)γα(k)) and the ker-

nel k(γ †f ,tf ; γi,ti |S) is defined in the same way as in Eq. (A19)

(again, taking special care because Grassmann variablesanticommute). The minus sign in the first kernel in Eq. (53) isdue to the antiperiodic conditions on the trace formula usingfermionic coherent states.

Unlike in the case of the phonons, we cannot compute theGaussian integrals exactly. We would like then to do someexpansion depending on the parameter λ. In order to achievethat we perform a Hubbard-Stratonovich transformation andthen expand to second order a determinant resulting from thefunctional integrals. To do so, one can define an auxiliarybilinear form

(G−1)αβ(t − t ′,k − k′; S)

= − 12δ(t − t ′)[σ αβ · h(k − k′) + λσ αβ · S(t,k − k′)].

(54)

With this definition, it is clear that we can do the Hubbard-Stratonovich transformation of the form

exp

(−i

∫ t

t0

∫ t

t0

dsds ′ ∑kk′,αβ

γ ∗α (s,k)(G−1)αβ

× (s − s ′,k − k′; S)γβ(s ′,k′))

=∫ ∏

α,k D2ζα(k)

det(−iG)[S]exp

(i

∫ t

t0

∫ t

t0

dsds ′

×∑

kk′,αβ

ζ ∗α (s,k)(G)αβ(s − s ′,k − k′; S)ζβ(s ′,k′)

+ i

∫ t

t0

∫ t

t0

ds∑k,α

ζ ∗α (s,k)γα(s,k) + γ ∗

α (s,k)ζα(s,k)

).

(55)

By doing this, we achieve a linear coupling between the γ ’sand the ζ ’s, which allows us to compute the Gaussian integralsin the γ ’s. Replacing this transformation in the expression forthe influence functional and computing the Gaussian integralsassociated with the γ ’s, we arrive at∫ ∏

α,k D2ζ1,α(k)

det(−iG)[S1]

∏α,k D2ζ2,α(k)

det(iG)[S2]

× exp

(i

∫ t

t0

∫ t

t0

dsds ′ ∑kk′,αβ

(ζ ∗1,α(s,k) ζ ∗

2,α(s,k))

× (�(s − s ′,k − k′)αβ)

(ζ1,β (s ′,k′)ζ2,β (s ′,k′)

) ), (56)

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B. MERA, V. R. VIEIRA, AND V. K. DUGAEV PHYSICAL REVIEW B 88, 184419 (2013)

where

(�(t,k)αβ) =(

(G11)αβ(t,k) + (G)αβ(t,k; S1) (G12)αβ(t,k)

(G21)αβ(t,k) (G22)αβ(t,k) − (G)αβ(t,k; S2)

), (57)

in which((G11)αβ(t,k) (G12)αβ(t,k)

(G21)αβ(t,k) (G22)αβ(t,k)

)

= i(2π )3δ3(k)δαβe−iε(k)t

((1 − f (ε(k)))θ (t) − f (ε(k))θ (−t) −f (ε(k))

1 − f (ε(k)) (1 − f (ε(k)))θ (−t) − f (ε(k))θ (t)

), (58)

with f (x) = (ex + 1)−1 being the Fermi-Dirac distribution.The functional integral is readily evaluated to be

det

{[−i

(G11 + G(S1) G12

G21 G22 − G(S2)

)]

×[i

(G−1(S1) 0

0 −G−1(S2)

)] }= det(1 + S), (59)

where

S =(G11G

−1(S1) −G12G−1(S2)

G21G−1(S1) −G22G

−1(S2)

). (60)

The determinant of Eq. (59) should be understood, of course,in the functional sense. We can write

det(1 + S) = exp{Tr[log(1 + S)]}

= exp

(∑k

(−1)k+1

kTrSk

), (61)

where the trace is also understood in the functional sense. Inthe way it is written, this produces an expansion in powers ofthe matrix elements of S and consequently an expansion inthe parameter λ.

The first term of the expansion is easily found to be zero andif we keep only terms of second order in the matrix elementsof S we obtain, in the Keldysh representation of the fields asdefined by Eq. (A25),

log [det (1 + S)]

= −1

2TrS2 + O(S3)

= −λ∑

k

∫ t

t0

∫ t

t0

dt1dt2(Sg)(t1 − t2,k)θ (t2 − t1)

× h(−k) · D(t2,k)

− λ2∑

k

∫ t

t0

∫ t

t0

dt1dt2(Sg)(t1 − t2,k)θ (t2 − t1)

× S(t1, − k) · D(t2,k)

− λ2

4

∑k1

∫ t

t0

∫ t

t0

dt1dt2(Pg)(t1 − t2,k)D(t1, − k) ·

× D(t2,k) + O(S3), (62)

where

g(t,k) =∑

k′e−i(ε(k′)−ε(k′−k))t

×(1 − f (ε(k′)))f (ε(k′ − k)), (63)

and

(Pφ)(t,k) = 12 (φ(t,k) + φ(−t, − k)),

(64)(Sφ)(k) = 1

2 (φ(t,k) − φ(−t, − k)),

are the symmetrizer and the antisymmetrizer operators asso-ciated with the time and momentum variables, respectively.This makes it clear that the part leading to dissipation willbe the term coupling two D fields since it will introduce animaginary term in the action. This is easy to see because theFourier transform of an even (odd) function is real (imaginary)if the function is real.

The effective action coming from the electron contributionthus obtained can be compactly written as

−1

2

∑k

∫ t

t0

dt1

∫ t

t0

dt2

(√2Sα(t1, − k)

1√2Dα(t1, − k)

)

× σ1Gαβ(t1 − t2,k)σ1

( √2Sβ(t2,k)

1√2Dβ(t2,k)

), (65)

where

iGαβ(t,k) =(

(Pg)(t,k) −(Sg)(t,k)θ (t)

(Sg)(t,k)θ (−t) 0

)

:= i

(GK (t,k) GR(t,k)

GA(t,k) 0

). (66)

We now prove, for consistency, that the following fluctuationdissipation theorem holds:

GK (ω,k) = (GR(ω,k) − GA(ω,k)) coth

(βω

2

). (67)

Computing the time Fourier transform of (Pg)(t,k), we endup with

(Pg)(ω,k) = 1

2

∑k′

2πδ(ω − ε(k − k′) + ε(k′))

× [(1 − f (ε(k′))f (ε(k − k′))− (1 − f (ε(k − k′)))f (ε(k′))]. (68)

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We now recall the identity

f (ε)(1 − f (ε − ω)) = n(ω)(f (ε − ω) − f (ε)), (69)

where n(ω) is the Bose-Einstein distribution, and we observethat we can write

(1 − f (ε − ω))f (ε) + (1 − f (ε))f (ε − ω)

= (1 − f (ε − ω))f (ε) + f (ε − ω)

− f (ε)f (ε − ω) + f (ε) − f (ε)

= 2(1 − f (ε − ω))f (ε) + f (ε − ω) − f (ε)

= 2n(ω)(f (ε − ω) − f (ε)) + (f (ε − ω) − f (ε))

= (2n(ω) + 1)(f (ε − ω) − f (ε))

= coth

(βω

2

)(f (ε − ω) − f (ε)). (70)

Now using f (ω)δ(ω − ω0) = f (ω0)δ(ω − ω0), we find

(Pg)(ω,k) = coth

(βω

2

)1

2

∑k′

2πδ(ω − ε(k − k′) + ε(k′))

× [f (ε(k′)) − f (ε(k − k′))]. (71)

Since

GR(t,k) − GA(t,k) = i(Sg)(t,k), (72)

we compute the time Fourier transform of (Sg)(t,k), obtaining

(Sg)(ω,k) = 1

2

∑k′

2πδ(ω − ε(k − k′) − ε(k′))

×[f (ε(k − k′))(1 − f (ε(k′)))− (1 − f (ε(k − k′)))f (ε(k′))]

= 1

2

∑k′

2πδ(ω − ε(k − k′) − ε(k′))

× [f (ε(k − k′)) − f (ε(k′))]

= −1

2

∑k′

2πδ(ω − ε(k − k′) − ε(k′))

×[f (ε(k′)) − f (ε(k − k′))], (73)

which implies

(Pg)(ω,k) = − coth

(βω

2

)(Sg)(ω,k), (74)

so that Eq. (67) holds as we originally claimed.One can, analogously to what we did in Sec. II, introduce

a random field using a Hubbard-Stratonovich transformation.This time the correlation function in position space reads

〈ξα(t,x)ξβ(0)〉 = δαβ

1

2[(F−1 ◦ GK ](t,x)

= δαβ

λ2

2[(F−1 ◦ P)g](t,x)

= δαβ

λ2

2Re{[F−1g](t,x)}, (75)

where F−1 denotes the inverse Fourier transform associatedwith the variable k.

There is another fluctuation dissipation theorem whichholds for this particular effective theory which we prove in

the following. In the case of Brownian motion, the timeintegral of the noise correlation function is proportional tothe temperature with the constant of proportionality being thedamping constant. We prove that also here, in the case of aspin system interacting with a bath of conduction electrons,a relation of the same type holds. In order to prove this, weintegrate the correlation function of Eq. (75),∫ ∞

−∞ds

∫d3x〈ξα(s,x)ξβ(0)〉

= δαβ

λ2

2Re

{∫ ∞

−∞ds

∫d3x[F−1g](s,x)

}. (76)

Now we rewrite the integral appearing in the above expressionby replacing the expression for g(t,k) of Eq. (63),∫ ∞

−∞ds

∫d3x[F−1g](s,x)

=∫ t

t0

ds

∫d3x

∑kk′

ei[k·x−i(ε(k′)−ε(k′−k))s]

× (1 − f (ε(k′)))f (ε(k′ − k)). (77)

Identifying the Dirac δ’s appearing in the above expression,we find ∑

kk′(2π )3δ3(k)δ(ε(k′) − ε(k′ − k))

× (1 − f (ε(k′)))f (ε(k′ − k)), (78)

or, if we define the excitation energy ωk′,k = ε(k′ − k) − ε(k′),we get ∑

kk′(2π )3δ3(k)δ(ωk′,k)

× (1 − f (ε(k′)))f (ε(k′) + ωk′,k). (79)

The above formula has to be treated with some care. Recallingthe identity of Eq. (69), we obtain∑

kk′(2π )3δ3(k)δ(ωk′,k)

× n(ωk′,k)(f (ε(k′)) − f (ε(k′) + ωk′,k)). (80)

Thanks to the δ function associated with the excitation energy,we only need the integrand evaluated at ωk′,k = 0. Clearly, theexpression is ill determined because the Bose-Einstein distri-bution diverges and the difference of Fermi-Dirac distributionsgoes to zero. To proceed we have to expand the integrand forsmall ωk′,k. Observing that f ′(ε) = −f (ε)(1 − f (ε)) we canTaylor expand the part containing Fermi-Dirac distributionsso that we obtain

n(ωk′,k)(f (ε(k′)) − f (ε(k′) + ωk′,k))

= 1

βωk′,k[ωk′,kf (ε(k′))(1 − f (ε(k′)))]

+O(ωk′,k)

= 1

βf (ε(k′))(1 − f (ε(k′))) + O(ωk′,k). (81)

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B. MERA, V. R. VIEIRA, AND V. K. DUGAEV PHYSICAL REVIEW B 88, 184419 (2013)

This means that our sum becomes simply

1

β

∑kk′

(2π )3δ3(k)δ(ωk′,k)

× f (ε(k′))(1 − f (ε(k′))). (82)

If we sum over k′ the only contribution of the sum will be theone in which ωk′,k = 0, or ε(k′ − k) = ε(k). Assuming thatε(k) is a power-law function of |k|, this implies that the onlyterm surviving is the one with k′ = 0. This reasoning gives

1

βf (ε(0))(1 − f (ε(0)))

∑k

(2π )3δ3(k)

= 1

βf (ε(0))(1 − f (ε(0))) = 1

4β. (83)

Thus, the final result is∫ ∞

−∞ds

∫d3x〈ξα(s,x)ξβ(0)〉

=∫ ∞

−∞ds

∫d3xRe{〈ξα(s,x)ξβ(0)〉}

= 2α′GkBT δαβ, (84)

where we have defined α′G = λ2/16. The above formula

shows that a fluctuation-dissipation theorem holds. Althougha fluctuation-dissipation theorem is satisfied in the exacttreatment,26,27 this is an important verification since in ourtreatment we were forced to make the lowest-order nonpertur-bative approximation which is equivalent to a linear responsetheory. In the next two sections, we explore the relevance ofthe result obtained in Eq. (75) for the electron contributionto the noise correlation function by showing its relationwith the Lindblad equation obtained under the Born-Markovapproximation, by computing its exact analytical form andstudying it numerically.

A. Relation to the Lindblad equation

In the Born-Markov approximation one can derive the so-called Lindblad equation for the reduced density matrix of thesystem in the interaction picture, ρS(t),20

∂ρS

∂t= −i

⎡⎣ ∑

ω,k,k′,αβ

�αβ(ω,k,k′)(Sα(ω,k))†Sβ(ω,k′),ρS

⎤⎦

+∑

ω,k,k′,α,β

�αβ(ω,k,k′)[Sβ (ω,k′)ρS(Sα(ω,k))†

− 1

2

{(Sα(ω,k))†Sβ(ω,k′),ρS

} ], (85)

in which

Sα(ω,k) :=∑

E−E′=ω

PE′ Sα(k)PE, (86)

where PE = |E〉〈E| are projectors onto energy eigenstates(HS|E〉 = E|E〉) of the system Hamiltonian. The tensors i�αβ

and �αβ are, respectively, the anti-Hermitian and twice the

Hermitian part of the tensor

Jαβ(ω,k,k′) :=∫ ∞

0ds Cαβ(s,k,k′)eiωs, (87)

Cαβ(s,k,k′) := λ2Tr{ρR(sα(k,t))†sβ(k′,t − s)}, (88)

in which sα(t,k) are the compound operators associated tothe spin of the conduction-band electrons in the interactionrepresentation. A simple application of Wick’s theorem atfinite temperature to Eq. (88) shows that Cαβ(t,k,k′) isdiagonal in momentum and spin variables, and the diagonalcomponents are precisely the Fourier transform of the electroncorrelator, i.e.,

Cαβ(t,k,k′) = λ2

2δαβ(2π )3δ3(k − k′)g(t,k), (89)

in which g(t,k) is the same quantity defined in Eq. (63)which naturally appeared while expanding the effective actioncoming from the conduction electrons to second order inthe self-energy. In the Markovian approximation, the two-point correlation function of the bath fields which appearsin the interaction Hamiltonian has all the relevant physicalinformation to describe the dynamics of the system densitymatrix. Namely, the tensor �αβ is responsible for the so-calledLamb shift which appears in the energies of the unperturbedsystem20 and the tensor �αβ is related to the rates in whichdissipative processes occur. From Eq. (89) we find thatthe correlation function of the noise originating from theinteraction of the conduction electrons with the spin, Eq. (75),has a very deep physical meaning: it carries all the physicalinformation on the shifts of the system energies and the ratesof the nonconservative processes due to the interaction withthe bath in the Markovian approximation. By measuring it wecan, thus, extract the relevant physical properties of the systemand the interaction in the Markovian limit.

B. Analyzing the electron contribution for 〈ξα(t,x)ξβ (0)〉In this section, we derive the exact analytical form of the

electron contribution to the noise correlator obtained givenin Eq. (75) and study it numerically. To make contact withthe notation used in the literature for the Keldysh-Schwingerclosed-time-path Green’s function formalism,23,28 we showthat the electron contribution to the correlator 〈ξα(t,x)ξβ(0)〉is proportional to the Keldysh component of the polarizationinsertion in lowest order. To see that this is the case, we startwith the formula for the Keldysh component of the polarizationtensor, which is given by28

�K (t,k) ∝∑

k′[GK (t,k − k′)GK (−t,k′)

+GR(t,k − k′)GA(−t,k′) + GA(t,k − k′)GR(−t,k′)],

where

GR(t,k) = −iθ (t)e−iε(k)t ,

GA(t,k) = +iθ (−t)e−iε(k)t ,

GK (t,k) = −i(1 − 2f (ε(k)))e−iε(k)t .

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DYNAMICS OF MAGNETIC MOMENTS COUPLED TO . . . PHYSICAL REVIEW B 88, 184419 (2013)

These last expressions yield

�K (t,k) ∝∑

k′e−i(ε(k−k′)−ε(k′))t [f (ε(k − k′))(1 − f (ε(k′)))

+ (1 − f (ε(k − k′)))f (ε(k′))]

Now, since

g(t, − k)

=∑

k′e−i[ε(−k−k′)−ε(k′)]t (1 − f (ε(−k − k′)))f (ε(k′))

=∑

k′e−i[ε(k′−k)−ε(−k′)]t (1 − f (ε(k′ − k)))f (ε(−k′))

=∑

k′e−i[ε(k−k′)−ε(k′)]t (1 − f (ε(k − k′)))f (ε(k′))

= g(t,k),

then

g(−t, − k) =∑

k′ei[ε(k−k′)−ε(k′)]t (1 − f (ε(k − k′)))f (ε(k′)),

×∑

q

ei[ε(q)−ε(q+k)]t (1 − f (ε(q)))f (ε(q + k)),

g(−t, − k)

=∑

k′e−i[ε(−k−k′)−ε(k′)]t f (ε(k − k′))(1 − f (ε(k′))).

In the last two derivations we used the fact that the conductionelectrons spectrum is even so that ε(k) = ε(−k). So now wecan finally check that

�K (t,x) ∝∑

k

eik·x(Pg)(t,k)

= 1

2

∑k

[eik·xg(t,k) + e−ik·xg(−t,k)]

= 1

2

∑k

eik·xg(t,k) + c.c. = Re(F−1 ◦ g)(t,x),

as claimed.One can use the free thermal Green’s function to write this

expression in a more tractable way. The thermal polarizationinsertion operator is given by

D0(xτ ; x′τ ′) := G0(xτ ; x′τ ′)G0(x′τ ′; xτ ), (90)

where G0(xτ ; x′τ ′) = G0(x − x′,τ − τ ′),

G0(x,τ ) =∑

k

eik·xe−ε(k)τ f (ε(k)), τ < 0, (91)

G0(x,τ ) = −∑

k

eik·xe−ε(k)τ

× (1 − f (ε(k))) , τ > 0 (92)

is the thermal free electron propagator29 in which f (x) =(exp(βx) + 1)−1 is the Fermi-Dirac distribution. If we allowτ = it , the electron contribution to the correlator is found to be

〈ξα(x,t)ξβ(0)〉|el = −δαβ

λ2

2Re{D0(x,τ = it)}, t > 0. (93)

In terms of Feynman diagrams in the imaginary time formalismthis is given by the graph of Fig. 1. The calculations are shown

FIG. 1. Bubble diagram. The free one-particle Green’s functiononly depends on r = |x|, so space inversion does not affect it.

in detail in Appendix C. The results are

G0(x, − it) = − μ

πrβ

∞∑n=0

e−

√(2n+1)μπ

βr

× cos

[√(2n + 1)μπ

βr + i(2n + 1)πt

],

(94)

and

−G0(| − x| = |x| = r,it)

=(

μi

2πt

)3/2

exp

(ir2μ

2t

)

+ μ

πrβ

∞∑n=0

e−

√(2n+1)μπ

βr

× cos

[√(2n + 1)μπ

βr − i(2n + 1)πt

], (95)

where μ is the effective mass of the electrons in the parabolicapproximation. The plots in Fig. 2 give us the qualitativebehavior of the electron correlator.

We can see that this noise correlation function, just bylooking at t = 0 (when the time separation of the two noisefields is zero), is much more textured than the usual white noisecorrelation function as it inherits the natural Fermi-statisticsstructure. In particular, for small temperatures it manifests thephenomenon of electron screening because the spin acts as animpurity and, because of the sharpness of the Fermi level, onlyelectrons near the Fermi level can scatter with the spin.

This is seen by the oscillatory behavior, Friedel oscillations,at these small temperatures. This shows that, in fact, one canuse ferromagnets with impurities to probe properties of theFermi surface of those ferromagnets just by measuring noisecorrelation functions in the laboratory. Later on, at higher tem-peratures, this phenomenon is highly suppressed. At any tem-perature the correlation is damped with distance, with dampingconstants dictated by ∼√

μ/β ∼ √μT . At r → 0 the corre-

lation grows as 1/r2, for finite temperatures. This dampingconstant can be measured at each temperature and this allowsone to estimate an effective mass for the conduction electrons.

IV. COMPOSITE MODEL

We now collect the results from Secs. II and III in a theorymodeling the interaction of a spin vector field with phonons

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B. MERA, V. R. VIEIRA, AND V. K. DUGAEV PHYSICAL REVIEW B 88, 184419 (2013)

FIG. 2. (Color online) Electron contribution to the noise corre-lation function (minus the real part of the polarization insertion) attime t = 0 as a function of relative distance r and temperature of theelectron bath, T . Here we have truncated the series of the Green’sfunction in the 20th term. Natural units were used.

and electrons. It is straightforward to see that the spin-phononinteraction of Eq. (6) generalizes to

Hi = −∑

k

(H ∗αμ(k)a†

μ(k)Sα(k)

+Hαμ(k)Sα(−k)aμ(k)) (96)

in order to account for a spin vector field (each space positionnow has an independent spin-j degree of freedom). The totalHamiltonian is now given by the Hamiltonian of Sec. IIIplus the phonon reservoir Hamiltonian and the interactionHamiltonian.

Assuming again the factorization of the initial density ma-trix, the Feynman-Vernon functional of this model factorizesinto the product of two functionals, one associated to thephonons and the other to the electrons:

F[S1,S2] = Fp[S1,S2]Fe[S1,S2]. (97)

The functional associated with the electrons is approximatedby the exponential of the effective action of Eq. (62) and the

one associated with the phonons is given by

Fp[S,D] = exp

{− i

2

∫ t

t0

dt1

∫ t

t0

dt2

×∑

k

(√2Sα(t1,−k)

1√2Dα(t1,−k)

)

× σ1Gαγ (t1 − t2,k)σ1

( √2Sγ (t2,k)

1√2Dγ (t2,k)

)}, (98)

where

i

2Gαγ (t,k) =

∑μ

H ∗αμ(k)e−iωμ(k)t

×(

1 + 2n(ωμ(k)) θ (t)−θ (−t) 0

)Hγμ(k)

=(

(�β)αγ (t,k) (�)αγ (t,k)θ (t)−(�)αγ (t,k)θ (−t) 0

).

(99)

Regarding the introduction of a random field into theequations of motion, now its correlations are given by the sumof two terms. The one arising from the electrons was alreadycalculated in Sec. III. The contribution from the phonons isgiven by

〈ξα(t,x)ξβ(0)〉phonons =∑k,μ

ei[k·x−ωμ(k)t]

×H ∗αμ(k)Hβμ(k) coth

(βωμ(k)

2

).

(100)

Assuming that Hαμ(k) = Hα(ωμ(k)), this last expression canbe written as

〈ξα(t,x)ξβ(0)〉phonons =∫ ∞

0dωρ(ω,x)e−iωμ(k)t

×H ∗α (ωμ(k))Hβ(ωμ(k)) coth(

βω

2),

(101)

in which

ρ(ω,x) =∑k,μ

δ(ω − ωμ(k))eik·x. (102)

If we make the following additional assumption that

ρ(ω,x) = ρ(ω)δ3(x), (103)

then the discussion of the simplified model of Sec. II appliesto this part of the correlation function and we have for the fullcorrelation function

Re{〈ξα(t,x)ξβ(0)〉} = δαβ(αGϕ(t)δ3(x)

+ λ2

2Re{[F−1g](t,x)}), (104)

in which we have applied the considerations we made in Sec. IIregarding the coupling constants. The discussion of the validityof the fluctuation-dissipation theorem is the same as in the endof Secs. II and III. It is clearly satisfied by both contributions.

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V. CONCLUSIONS

Even though the model considered is very complex, wewere able to obtain very interesting results. In the limit ofhigh temperatures, the magnetic moments associated to the3d electrons feel a random field which has two contributions:one from the interaction with the phonons which, with someadditional assumptions regarding coupling constants, can bereduced to the one predicted by Brown;11 and the other onewhich comes from the interaction with the conduction elec-trons. In addition to an external magnetic field, the magneticmoments also feel an effective magnetic field associated withtheir interaction with electrons. This effective magnetic field,given by Eq. (B13), is related to the magnetic field feltby the electrons, h(k), and explicitly manifests the Fermi-Dirac statistics of these particles because it is weighted bythe function (Sg)(t,k) = (1/2)[

∑k′(1 − f (ε(k′)))f (ε(k′ −

k))e−i[ε(k′)−ε(k′−k)]t − (t → −t,k → −k)], where f is theFermi-Dirac distribution. In the limit in which the interac-tion with conduction electrons is small compared with theinteraction with phonons, maxμ |γμ|/λ 1, one can obtainthe Landau-Lifshitz-Bloch equation from the interaction withphonons using the results of Rebei et al.9 and Garanin,14 whichis a remarkable fact.

The limit of high temperatures should be further investi-gated in the case of the contribution given by the conductionelectrons since it is not clear from the expression of the randomfield correlation function that it will have Brown’s form,i.e., white noise, under some assumption of the conductionelectrons’ energy spectrum and density of states in such alimit. If this is the case, then the Gilbert damping constantis given by the sum of two terms, one from the phonons andthe other from the electrons. The one given by the conductionelectrons appears to be independent of the electron density ofstates and is equal to λ2/16.

The case of finite temperatures is described by two gener-alized functions which are essentially the Fourier transformsof some given functions characterizing the type of interaction,the statistics, and the density of states of the bath degrees offreedom.

The part of the correlation function of the random fieldwhich comes from the interaction with the phonons yieldsan intimate relation between friction (the associated Gilbertconstant) and the random field fluctuations; see Eq. (32). Thevalidity of the Markovian approximation is measured in termsof the thermal correlation time τth. For time scales shorterthan this, the Markovian approximation for the stochasticfield fails. In the case of the contribution for the randomfield correlation function given by the electrons, we see thatEq. (84) manifests the existence of a fluctuation-dissipationtheorem. This is consistent with the expansion done beingsimply a linear-response theory. Thus the theory consideredhere satisfies a general fluctuation-dissipation theorem whichrelates the random field fluctuations to the friction constantswhich measure the effect of the interaction of the spin withelectrons and phonons.

The electron contribution to the noise correlation functionsexhibits the properties of the Fermi level of the electrons.These properties are better seen at low temperatures in whichFriedel oscillations appear naturally (see Fig. 2). By measuring

noise correlation functions one is indeed probing the Fermisurface of the ferromagnet in study. We have also shownthat at any temperature the correlation decays as ∼√

μT ,meaning that one can also probe the effective mass, μ, for theconduction electrons by measuring this correlation functionat different temperatures. This shows that, by studying thenoisy dynamics of impurities in a ferromagnet (representedby a spin-j field), one can extract physical properties of theferromagnetic material.

Our model is more general than the so-called three-temperature model considered and validated in experimentsof Beaurepaire et al.17 This is because we do not assume thatthe spins, electrons, and phonons are thermalized. Instead, weconsider that at an initial time t0 they are thermalized and thedensity matrix decouples at that time, but the time evolutioncouples the various systems and, thus, considering individualtemperatures at each instant has no precise meaning withinthis model.

We believe that the proposed model can lead to a correctqualitative description of the ultrafast dynamics of magneticmoments in the case of laser-induced excitations at time scalesshorter than the picosecond.

ACKNOWLEDGMENTS

B.M. thanks the support from Project IT-PQuantum, as wellas from Fundacao para a Ciencia e a Tecnologia (Portugal),namely through programme POC-TI/PO-CI/PT-DC and DP-PMI and Project Nos. PEst-OE/EGE/UI0491/2013, and PEst-OE/EEI/LA0008/2013, and PTDC/EEA-TEL/103402/2008QuantPrivTel, partially funded by FEDER (EU), and fromthe EU FP7 projects LANDAUER (Project No. GA 318287)and PAPETS (Project No. GA 323901). This work was alsosupported by the National Science Center in Poland as aresearch project.

APPENDIX A: EFFECTIVE ACTION FOR THESPIN-PHONON THEORY

Following Rebei and Parker,9 we consider now the reduceddensity matrix of the system in the model of Sec. II,

ρS(t) = TrR{U(t ; t0)ρ(t0)U †(t ; t0)}, (A1)

where U(t ; t0) = Tt exp(−i∫ t

t0H(s)ds) and Tt is the time

ordering operator.We assume that at t = t0 the spin and bath are decoupled,

and the bath is in an equilibrium state with temperature T .Then we have

ρ(t0) = ρS(t0) ⊗ ρR(t0),(A2)

ρR(t0) = Z−1R exp(−βHR),

where ZR = Tr{exp(−βHR)} is the partition function for thebath, and we denoted β = 1/kBT .

For the reduced density matrix we choose a path-integralrepresentation using the basis of coherent states in the Hilbertspace of the physical system. For the phonons we use theholomorphic representation in which the coherent states aredefined by

||α〉 =∏

k

exp(a†μ(k)αμ(k))|0〉, (A3)

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B. MERA, V. R. VIEIRA, AND V. K. DUGAEV PHYSICAL REVIEW B 88, 184419 (2013)

which satisfies the following property:

aμ(k)||α〉 = αμ(k)||α〉. (A4)

These states provide a decomposition of the identity (of thereservoir)

1R =∫

dμ(α)||α〉〈α||,

dμ(α) =∏k,μ

d2αμ(k)

πe−α∗

μ(k)αμ(k). (A5)

For the spin j we use the spin coherent states,30–32

|S〉 = (1+|ζ (S)|2)−j exp(ζ (S)S−)|jj 〉, (A6)

with

ζ (S) = eiϕ tanθ

2, (A7)

S = (Sα) = j (sin θ cos ϕ, sin θ sin ϕ, cos θ )T , (A8)

where S± = S1 ± iS2 and |jj 〉 is the highest weight vectorof the representation of the SU(2) group labeled by j . Thesestates have the property

〈S|Sα|S〉 = Sα (A9)

and provide a decomposition of the identity operator in theform

1S =∫

dμ(S) |S〉 〈S| ,

dμ(S) = 2j + 1

4πδ(S2 − j 2)d3S. (A10)

The properties (A4) and (A9) allow us to establish a correspon-dence between classical and quantum quantities. We considerpath-integral representations which make use of coherent-state matrix elements which can be computed trivially, usingEq. (A4) and its Hermitian conjugate, and Eq. (A9), for linearterms in the spin operators. A general prescription to obtain(spin) coherent-state matrix elements of operators is given inRef. 30, where a holomorphic definition of the (spin) coherentstates is used.

Matrix elements of the reduced density matrix ρ(t) with thespin coherent states are

ρS(Sf ,Si ,t) = 〈Sf |ρS(t)|Si〉= 〈Sf |TrR{U(t ; t0)ρ(t0)U †(t ; t0)}|Si〉. (A11)

They are related to the density matrix at time t = 0 throughthe propagator J (Sf ,Si ,t ; S2,S1,t0), given by

ρS(Sf ,Si ,t) =∫

dμ(S1)dμ(S2)J (Sf ,Si ,t ; S2,S1,t0)

× ρS(S1,S2,t0). (A12)

The propagator has a path-integral representation of the form

J (Sf ,Si ,t ; S2,S1,t0)

=∫ Sf

S1

Dμ(S′1)

∫ Si

S2

Dμ(S′2) exp(iI [S′

1,S′2])F[S′

1,S′2],

(A13)

where

I [S1,S2] := SWZ[S1] − SWZ[S2]

+ SS[S1] − SS[S2], (A14)

in which SWZ is the Wess-Zumino action,

SWZ[S] = j

∫ t

t0

dt

∫ 1

0du n · (∂un × ∂tn), (A15)

where

n(u,t) = (sin(uθ ) cos ϕ, sin(uθ ) sin ϕ, cos(uθ ))T (A16)

is a map which continuously deforms the constant curve givenby n0(t) = n0 = (0,0,1)T (the north pole) into the curve n(t) =S(t)/j through a great circle, a geodesic, in S2. We see thatthe Wess-Zumino action, given by Eq. (A15), gives the areaenclosed by the path traced by the spin vector and the twogreat circles from the north pole of the sphere to the endpointsof that path. We also have

SS[S] = −∫ t

t0

dt〈S(t)|HS(t)|S(t)〉 =∫ t

t0

dt S(t) · H,

(A17)

and F[S1,S2] is the Feynman-Vernon influence functionalwhich can be written as

F[S1,S2] =∫

dμ(α0)dμ(α1)dμ(α2) k(α†0,t ; α1,t0|S1)

×Z−1R k(α†

1,−iβ; α2,0|0) × k∗(α†0,t ; α2,t0|S2),

(A18)

where we defined the kernel

k(α†f ,tf ; αi,ti |S) =

∫D2α exp

[1

2(α†

f α(tf ) + α(ti)†αi)

]

× exp

[i

∫ tf

ti

dtL

], (A19)

in which

iL = 1

2

(dα†

dtα − α† dα

dt

)− i[HR(α†(t),α(t))

+Hi(α†(t),α(t),S(t))] (A20)

and

HR = 〈α(t)||HR(t)||α(t)〉〈α(t)||1R||α(t)〉 , (A21)

Hi = 〈S(t)|〈α(t)||Hi(t)||α(t)〉|S(t)〉〈α(t)||1R||α(t)〉 . (A22)

Note that in the formulas presented above we adopted vectornotations α = (αμ(k)) and α† = (α∗

μ(k)).The Feynman-Vernon functional can be computed exactly

because the integrals are Gaussian. After some algebra weobtain

F[S1,S2] = exp

{−i

[ ∫ t

t0

∫ t

t0

dt1dt2(S1,α(t1) S2,α(t1))

× σ3Gαβ(t1 − t2)σ3

(S1,β (t2)

S2,β (t2)

) ]}, (A23)

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DYNAMICS OF MAGNETIC MOMENTS COUPLED TO . . . PHYSICAL REVIEW B 88, 184419 (2013)

where

iGαβ (t) =∑k,μ

H ∗αμ(k)e−iωμ(k)t

((n(ωμ(k)) + 1)θ (t) + n(ωμ(k))θ (−t) n(ωμ(k))

n(ωμ(k)) + 1 (n(ωμ(k)) + 1)θ (−t) + n(ωμ(k))θ (t)

)Hβμ(k),

(A24)

in which n(ω) = (eβω − 1)−1 is the Bose-Einstein distributionfunction.

We can introduce the fields associated with the Keldyshrepresentation(

Sα(t)

Dα(t)

)=

(12 (Sα,+(t) + Sα,−(t))

Sα,+(t) − Sα,−(t)

), (A25)

where the field S(t) is associated with the classical spin andD(t) with quantum and thermal fluctuations. The Feynman-Vernon functional in terms of these fields is

F[S,D] = exp

{−i

∫ t

t0

dt1

∫ t

t0

dt2

(√2Sα(t1)

1√2Dα(t1)

)

× σ1Gαβ(t1 − t2)σ1

( √2Sβ(t2)

1√2Dβ(t2)

) }, (A26)

where

iGαβ (t) =∑k,μ

H ∗αμ(k)e−iωμ(k)t

×(

1 + 2n(ωμ(k)) θ (t)

−θ (−t) 0

)Hβμ(k). (A27)

In the stationary phase approximation we take the variationof the action phase in the path integral to be zero. Thecorresponding equations of motion are

D(t) = S(t) × δSeff

δS(t)+ D(t) × δSeff

δD(t), (A28)

S(t) = S(t) × δSeff

δD(t)+ 1

4D(t) × δSeff

δS(t), (A29)

where

Seff[S,D] =∫ t

t0

dt D(t) · H − i logF[S,D]. (A30)

APPENDIX B: EQUATIONS OF MOTION

The equations of motion in momentum space of the fulltheory resulting from taking the variation of the effective action(in which we consider the truncated expansion of Sec. III)appearing in the path-integral representation of the spin densitymatrix are

D(s,k) =∑

p

[S(s,k − p) × W(s,p) + D(s,k − p)

× (T(s,p) + Heff(s,p))], (B1)

S(s,k)

=∑

p

[S(s,k − p) × (Jk2S(s,p) + T(s,p) + Heff(s,p))

+ 1

4D(s,k − p) × (Jk2D(s,p) + W(s,p))

],

(B2)

in which

T(s,−k) = T(S)(s,−k) + T(D)(s,−k), (B3)

T(S)(s,−k) = T(S)ph (s,−k) + T(S)

sd (s,−k), (B4)

T(D)(s,−k) = T(D)ph (s,−k) + T(D)

sd (s,−k), (B5)

T(S)ph (s,−k) = i

∫ t

t0

ds ′[�(s − s ′,−k) − �(s ′ − s,k)]

× θ (s − s ′)S(s ′,−k), (B6)

T(S)sd (s,−k) = iλ2

∫ t

t0

ds ′(Sg)(s − s ′,−k)

× θ (s − s ′)S(s ′,−k), (B7)

T(D)ph (s,−k) = i

∫ t

t0

ds ′[�β(s − s ′,−k) + �β(s ′ − s,k)]

× D(s ′,−k), (B8)

T(D)sd (s,−k) = i

λ2

2

∫ t

t0

ds ′(Pg)(s − s ′,k)D(s ′,−k), (B9)

W(s,−k) = Wph(s,−k) + Wsd(s,−k), (B10)

Wph(s,−k) = −i

∫ t

t0

ds ′[�(s − s ′,−k) − �(s ′ − s,k)]

× θ (s ′ − s)D(s ′,−k), (B11)

Wsd(s,−k) = −iλ2∫ t

t0

ds ′(Sg)(s − s ′,−k)θ (s ′ − s)

× D(s ′,−k), (B12)

Heff(s,−k) = H(−k) + iλ

∫ t

t0

ds ′(Sg)(s − s ′,k)θ (s − s ′)

× h(−k′), (B13)

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B. MERA, V. R. VIEIRA, AND V. K. DUGAEV PHYSICAL REVIEW B 88, 184419 (2013)

where the indices “ph” and “sd” emphasize that these termscome from the interaction with phonons or with electrons,respectively. In the above expression the long-wavelengthapproximation has been taken for the Heisenberg exchangeterm in the action so that it becomes a diffusive term in theequation of motion in which J is some constant measuring theinteraction strength.

APPENDIX C: COMPUTATION OF FREE THERMALELECTRON PROPAGATORS

Consider t > 0 and let us compute

G0(x,−τ = −it) =∑

k

eik·xeiε(k)t f (ε(k)). (C1)

This can be done by writing the sum on k as an integral(∑

k → ∫d3k/(2π )3),∫

d3k

(2π )3eik·xeiε(k)t f (ε(k)). (C2)

Now we consider that the conduction electrons have aquadratic spectrum of the form ε(k) = k2/2μ, where μ isan effective mass. Using spherical coordinates, we see that theangular integrals are trivially computed and we end up with

− 1

(2π )2r

d

dr

∫ ∞

−∞dk

exp[i(kr + ε(k)t)]

1 + exp[βε(k)]. (C3)

To compute ∫dk

ei[kr+ε(k)t]

1 + eβε(k), (C4)

we complexify the integral with the contour in the upper half-plane being a semicircle, with radius R → ∞, containing thepoles. The poles are given by solving the algebraic equation

βε(kn) = (2n + 1)iπ, n ∈ Z. (C5)

Near each pole,

exp

(βk2

2μ

)≈ −β

μkn(k − kn) − 1. (C6)

The solution is∫dk

ei[kr+ε(k)t]

1 + eβε(k)= −2πiμ

β

∞∑n=−∞

′(kn)−1ei[knr+ε(kn)t], (C7)

where in the summation the prime is there to indicate thatwe are only summing for values of kn which have a positiveimaginary part. Plugging this back into the expression for theGreen’s function yields

G0(x, − it) = − μ

2πrβ

∞∑n=−∞

′ei[knr+ε(kn)t], (C8)

where the sum is restricted to those kn satisfying Eq. (C5), or

βk2n

2μ= i(2n + 1)π, n ∈ Z (C9)

having a positive imaginary part. Solving this equation gives

kn = ±(1 + i)

√(2n + 1)μπ

β. (C10)

Suppose 2n + 1 > 0; then the solution with a positive imagi-nary part is given by

kn = (1 + i)

√(2n + 1)μπ

β; (C11)

now if 2n + 1 < 0, then

kn = (−1 + i)

√|2n + 1|μπ

β. (C12)

For fixed |2n + 1| we have the two contributions

e

[i(1+i)

√|2n+1|μπ

βr+i×i|ε(kn)|t

]+ e

[i(−1+i)

√|2n+1|μπ

βr−i×i|ε(kn)|t

], (C13)

which gives, after simple algebra,

2e−

√|2n+1|μπ

βr cos

[√|2n + 1|μπ

βr + i|2n + 1|πt

]. (C14)

So the sum becomes

G0(x,−it) = − μ

πrβ

∞∑n=0

e−

√(2n+1)μπ

βr

× cos

[√(2n + 1)μπ

βr + i(2n + 1)πt

].

(C15)

The other inverse Fourier transform gives

−G0(|−x| = |x| = r,it)

=(

μi

2πt

)3/2

exp

(ir2μ

2t

)

+ μ

πrβ

∞∑n=0

e−

√(2n+1)μπ

βr

× cos

[√(2n + 1)μπ

βr − i(2n + 1)πt

], (C16)

where the first term comes from the (inverse) Fourier transformof the exponential of the conduction-band energy, which is aGaussian integral.

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DYNAMICS OF MAGNETIC MOMENTS COUPLED TO . . . PHYSICAL REVIEW B 88, 184419 (2013)

*bruno.mera@ist.utl.pt1I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323(2004).

2J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Zutic, ActaPhys. Slovaca 57, 563 (2007).

3C. Chappert, A. Fert, and F. Nguyen van Dau, Nat. Mater. 6, 813(2007).

4J. Y. Bigot, M. Vomir, and E. Beaurepaire, Nat. Phys. 5, 515 (2009).5B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. Fahle,T. Roth, M. Cinchetti, and M. Aeschlimann, Nat. Mater. 9, 259(2010).

6M. G. Munzenberg, Nat. Mater. 9, 184 (2010).7L. V. Keldysh, Sov. Phys. JETP 1018 (1965).8G. Lindblad, Commun. Math. Phys. 48, 119 (1976).9A. Rebei and G. J. Parker, Phys. Rev. B 67, 104434 (2003).

10L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153(1935).

11W. F. Brown Jr., Phys. Rev. 130, 1677 (1963).12T. L. Gilbert, Phys. Rev. 100, 1243 (1955).13F. Bloch, Phys. Rev. 70, 460 (1946).14D. A. Garanin, Phys. Rev. B 55, 3050 (1997).15O. Chubykalo-Fesenko, U. Nowak, R. W. Chantrell, and D. Garanin,

Phys. Rev. B 74, 094436 (2006).16A. Rebei, W. N. G. Hitchon, and G. J. Parker, Phys. Rev. B 72,

064408 (2005).17E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Phys. Rev.

Lett. 76, 4250 (1996).

18A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731(2010).

19Spin Dynamics in Confined Magnetic Structures I, 1st ed., editedby B. Hillebrands and K. Ounadjela (Springer-Verlag, Berlin,2002).

20H. P. Breuer and F. Petruccione, The Theory of Open QuantumSystems (Oxford University Press, Oxford, 2002).

21J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 72, 014446(2005).

22C. W. Gardiner and P. Zoller, Quantum Noise, 2nd ed. (Springer-Verlag, Berlin, 2000).

23J. Rammer, Quantum Field Theory of Non-equilibrium States(Cambridge University Press, Cambridge, U.K., 2007).

24G. W. Ford and R. F. O’Connel, Nature (London) 380, 113 (1996).25G. W. Ford and R. F. O’Connel, J. Phys. A: Math. Gen. 35, 4183

(2002).26R. Kubo and N. Hashitsume, Prog. Theor. Phys. Suppl. 46, 210

(1970).27K. Miyazaki and K. Seki, J. Chem. Phys. 108, 7052 (1998).28A. Kamenev and A. Levchenko, Adv. Phys. 58, 197 (2009).29A. L. Fetter and J. D. Walecka, Quantum Theory of Many-particle

Systems (McGraw-Hill, San Francisco, 1971).30V. R. Vieira and P. D. Sacramento, Ann. Phys. 242, 188 (1995).31V. R. Vieira and P. D. Sacramento, Nucl. Phys. B 448, 331

(1995).32A. Perelomov, Generalized Coherent States and Their Applications

(Springer-Verlag, Berlin, 1985).

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