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Transmission of correlated electrons through sharp domain walls in magnetic nanowires: a renormalization group approach

The transmission of correlated electrons through a domain wall in a ferromagnetic one dimensional system is studied theoretically in the limit of a domain wall width smaller or comparable to the electron Fermi wavelength. The domain wall gives rise to both potential and spin dependent scattering of the charge carriers. Using a poor man's renormalization group approach for the electron-electron interactions, we obtain the low temperature behavior of the reflection and transmission coefficients. The results show that the low-temperature conductance is governed by the electron correlations, which may suppress charge transport without suppressing spin current. The results may account for a huge magnetoresistance associated with a domain wall in ballistic nanocontacs.


I. INTRODUCTION
Domain walls (DWs), i.e. the boundaries separating different domains of homogeneous magnetizations 1 are recently a subject of extensive theoretical and experimental investigations. This renewed interest in DWs is stimulated by their possible applications in magnetic logic elements and other nanoelectronics and spintronics devices. Two effects associated with DWs are of particular interest. The first one is the way a DW affects electronic transport, i.e., the associated magnetoresistance. The crucial point here is that the influence of a single DW on the resistance can be controlled by an external magnetic field 2,3 . The second effect concerns the influence of electric current on the DW behavior (DW motion, magnetic switching) 4 , which allows controlling of DWs by means of an electric field.
Recent advances in experimental techniques have made possible the determination of the resistance of a single DW in submicron structured samples 2,5,6,7,8,9,10,11 . The results on the single domain resistance are different in magnitude and sometimes differ also in the sign. In the case where the DW width is large on the scale set by the Fermi wavelength, λ F , of the carriers, the theory of the DW contribution to electrical resistance is well-established 12,13,14,15,16,17 . The spin of the electron moving across the wall changes its orientation quasiadiabatically (or even adiabatically for very thick DWs). However, the DWs formed at nanoconstrictions may be atomically sharp 18,19,20,21 and the spin of an electron crossing the wall does not change quasi-adiabatically. Accordingly, a completely different approach to the transport theory through DWs is required. This is particularly true for ferromagnetic semiconductors which are considered to be most promising for spintronic applications 22 . Indeed, recent experiments on magnetic nanostructures and nanowires indicate that the presence of DWs may result in a magnetoresistance (MR) as large as several hundreds 8,23 or even thousands 24,25 of percents, as opposed to the case of thick on scale of λ F (or adiabatic) DWs in bulk metallic ferromagnets. In the ballistic regime, the theoretical treatments towards explaining this effect 26,27,28,29,30,31 rely on the assumption that the DW is sharp enough to be treated as a spin dependent scatterer for the charge carriers. The success of these theories in explaining the extraordinary large MR is moderate, in particular for metallic ferromagnets such as Ni where some features of the physics governing the behaviour of MR are still unclear 32 .
Another feature of the DWs created at nanoconstrictions is their small lateral size (cross-section of the constriction). This small size limits the number of quantum channels active in transport to a few ones or even to a single one. Consequently, the constriction behaves as a one-or quasi-one-dimensional system. In such a case, the role of electron-electron interactions may be crucial 33 for understanding the behavior and basic transport characteristics of the DWs formed at nanoconstrictions. It is well-established that electronic correlations in a onechannel wire result in a non-Fermi-liquid behavior -thus forming a Luttinger liquid 34,35 . It is also known that an impurity present in the 1D Luttinger liquid suppresses the linear conductivity, which vanishes even for a weak impurity scattering potential 36,37 . This can be traced back to a vanishing density of states at the Fermi level.
At finite applied voltages the transport through the wire does not vanish due to the nonlinearity of the currentvoltage characteristics 36 . Since a sharp domain wall acts in a one-channel wire as a localized spin-dependent scattering center, one can expect a strong influence of electron correlations on the MR at low temperatures.
To confirm this theoretically one could use bosonization techniques 38,39 .
However, we will follow another route based on the "poor man's" renormalization method 40,41,42 . In our case, the DW scatters both the charge and spin of the carriers. As shown below, our scheme allows us to obtain results for the renormalized transmission and reflection coefficients in terms of the uncorrelated spin-dependent ones (i.e., in terms of the reflection and transmission coefficients of the wall in the absence of electron-electron interactions). The uncorrelated quantities can be obtained from other schemes, such as the Hartree-Fock or density-functional theory (within local density approximation) and then used as an input in our results to obtain renormalized transmission through the DW. Hence, our approach -in combination with numerical (effective single particle) methods -offers a new possibility to understand the material-dependent MR associated with a DW creation (destruction), and possibly to resolve some controversy concerning huge magnetoresistance in some ballistic nanocontacts.
The paper is organized as follows. In Section II we introduce the problem and the non-interacting scattering states for a sharp domain wall. In Section III we use perturbation theory in the electron-electron interaction to calculate corrections to the scattering amplitudes. We obtain the renormalization group differential equations for the scattering amplitudes. In Section IV we describe the zero temperature fixed points predicted by the scaling equations and the power law behavior of the reflection and transmission coefficients of the DW as T → 0. In Section V we discuss the relevance of our findings to realistic physical systems and summarize our results.

II. MODEL
We consider a magnetized system with electrons being constrained to move in one dimension while being exchange-coupled locally to the space varying magnetization, M(r). The wire itself defines the easy (z) axis, and a domain wall centered at z = 0 separates two regions with opposite magnetizations, M z (z → ±∞) = ±M 0 . Assuming M(r) to lie in the xz plane, and the domain wall to be thinner than the Fermi wavelength, we write the single-particle Hamiltonian as: where the termhλδ(z)σ x describes spin scattering produced by the M x (z) component, 43 and V is a potential scattering term. Single electron wavefunctions are spinors with components χ σ (z) satisfying the condition (2) The electron's wavevector in each domain is related to the energy E by The electron gas in the negative semi-axis (z < 0) is predominantly ↑-spin. An electron incident from the left with the momentum k and spin ↑ (or ↓) can be transmitted to the positive semi-axis while preserving its spin, but the energy conservation requires the momentum to change from k to k − (or k + ) defined by: If the transmission occurs with spin reversal, the momentum k is not changed. We label the states through the incident wave, so that describes a scattering state with a wave incident from z = −∞ with spin ↑ and momentum k > 0. Reflection amplitudes of a spin σ electron with or without spin reversal are denoted by r ′ σ and r σ , respectively. The same convention applies to the transmission amplitudes t ′ σ , t σ . The transmitted wave corresponding to Eq. (5) is and the scattering amplitudes are given by: where we have defined the velocities v (±) =hk (±) /m. The scattering state corresponding to a wave incident from the left with ↓-spin is: and the corresponding amplitudes are: The expressions for the scattering states corresponding to the waves incident from +∞ are: and where we consider k > 0. We shall henceforth denote by ǫ(±p, ↑) the eigenenergy of a scattering state with momentum +p (or −p) incident from the left (or right). The scattering amplitudes satisfy some general relations that can be found from a generalization of the Wronskian theorem 44 to spinor wavefunctions. We provide such relations in Appendix A.
In order to deal with the electron interactions, it is convenient to rewrite the scattering states in second quantized form, making use of right (â qσ ) and left (b qσ ) moving plane-wave states.
The operators for the scattering states with electrons incident from the left (ĉ k,σ ) are: and the operators for scattering states with electrons incident from the right (d k,σ ) are: where 0 denotes a positive infinitesimal and the ksubscript σ = ±1. By inverting these equations, we obtain the plane wave operators as linear combinations of the scattering-state operators: The electron interactions are introduced in the Hamiltonian through the term: The coupling constants g 1 and g 2 describe back and forward scattering processes between opposite moving electrons, respectively. The Greek letters denote here the spin indices, and the summation over repeated indices is implied. Because the Fermi momentum depends on spin, we allow for the dependence of g on the spins of the interacting particles. We therefore distinguish between g 1↑ , g 1↓ g 1⊥ and g 2↑ , g 2↓ , g 2⊥ . The forward scattering process between particles which move in the same direction will not affect the transmission amplitudes, although it will renormalize the Fermi velocity 40 . This effect is equivalent to an effective mass renormalization and the electrons with different spin orientations may turn out to have different effective masses, in which case our calculations remain valid, as shown in Appendix B.

III. SCALING EQUATIONS
The corrections to the transmission amplitudes will be calculated to first order in the perturbationĤ int . It has been shown in Ref. [40] that the corrections diverge logarithmically near the Fermi level. These divergences will later be dealt with in a poor man's renormalization procedure.
Let us consider the Matsubara propagator, where ... 0 denotes the average in the non-interacting Fermi sea. The propagator for non-interacting electrons is then given by: The transmission amplitude appears associated with the denominator p − p ′ − − i0 which, for the variable p, gives a pole in the upper half plane. The meaning of this pole is that the transmitted particle is a right-mover in the z > 0 half-axis. Our strategy is to calculate the first order correction term (inĤ int ) to G, which will have the same form as the second term in (20), so that the amplitude correction, δt ↑ (p ′ ), can be read off from the result. We now explain the procedure in some detail.
We begin by considering the first order expansion for G in the coupling g 1↑ . For simplicity, we shall henceforth omit the subscript "0" in the brackets, since we will be dealing with the non-interacting Fermi sea, unless otherwise stated. From Wick's theorem we get the first order correction to the propagator in equation (20) as where the internal momenta k 1(2) , q and time τ ′ are to be integrated over and the time ordering T τ is implicit. There are also two other Wick paired terms at instant τ ′ of the form â † (τ ′ )â(τ ′ ) and b † (τ ′ )b(τ ′ ) . We have omitted these terms in Eq.(21) because they will not be logarithmically divergent: the divergences arise from electron reflection by the Friedel oscillations in the Fermi sea 40 . Such reflection processes appear in equation (21) To calculate â † k1,↑b k1−q,↑ we make use of the expression (17) forb k1−q,↑ . The contour integration over k 1 eliminates the terms containing poles in the same half-plane. Fermi sea averages, such as â † k1,↑ĉ k,↑ and â † k1,↑d k,↑ , can be calculated in the same way as in Eq. (20). The result is: where f (±Q ↑) denotes the Fermi occupation number of the state ψ ±Q,↑ . In order to calculate the propagator − T τ a p,↑ (τ )b † k2,↑ we again expandb † k2,↑ using Eq. (17), and then with help of Eq. (A7) we obtain The presence of two different energy poles can be understood from the fact thatâ qσ (orb qσ ) represents a plane wave running over the entire z axis and its energy cannot be the same on both sides of the domain wall because of the energy dependence on spin. Using Eqs. (22) and (23) we can calculate the first term in Eq. (21) as The analytic continuation of the Green's function frequency, iω → ω + i0, gives the retarded Green's function. The frequency denominator (iω − ǫ(−Q 2 ↑)) −1 yields a principal Cauchy part plus a delta function part. The latter isolates the energy pole at ǫ(−Q 2 ↑) = ω and we choose ω = ǫ(p ′ ↑) ⇒ Q 2 = p ′ − . We shall only retain this delta function part. Therefore, we put Q 2 = p ′ − in the integrand and, by comparing with (20), we conclude that the contribution of the first perturbative term in Eq. (21) to the transmission amplitude is given by where v F − now denotes the Fermi velocity corresponding to the minority spin Fermi momentum k F − . A logarithmic divergence appears as p ′ → k F − .
The above discussion describes the calculation method. We now need to calculate all the first order terms in the interactions g 1αβ and g 2αβ . The diagrammatic representation of G (1) is shown in Fig. 2. The horizontal lines represent the electron being scattered by the Hartree-Fock potential of the Fermi sea (of scattering states).
Consider, for instance, the upper left diagram: an electron, initially in state c p ′ ,↑ close to the Fermi level, passes through the barrier as a right-mover (â particle) and then interacts with the Fermi sea (on the positive z semi-axis). The electron is reflected (fromâ tob particle) while exchanging momentum q with the Fermi sea. Finally, it is reflected by the barrier again, becoming a spin-up right mover with momentum p. A logarithmic divergence occurs if the polarized Fermi sea can provide exactly the momentum that is required to keep the electron always near the Fermi level during the intermediate virtual steps. Concerning the spin dependence of the interaction parameters, we distinguish between g 1↑ , g 2↑ , which describe interaction between spin majority particles (that is spin-↑ on the right and spin-↓ on the left of the barrier) and g 1↓ , g 2↓ , which describe interaction between spin minority particles (that is spin-↓ on the right and spin-↑ on the left of the barrier). We use g 1⊥ , g 2⊥ to denote interaction between particles with opposite spins. According to the physical interpretation of the Feynman diagrams just given above, we always know on which side of the barrier the interaction with the Fermi sea (closed loop in the diagram) is taking place. It can be seen that the g 1⊥ terms are proportional to log |k F + − k F − |, so they do not diverge. The logarithmic divergence would be restored in a spin degenerate system (k F + = k F − ). This can be understood from the diagrams in Fig. 2 as follows: the electron with spin α is reflected by a polarized Fermi sea with spin −α. The momentum provided by the Fermi sea is 2k F −α , while the momentum required by the electron is 2k F α . The g 2⊥ terms produce logarithmic divergences that would not exist in the absence of spin-flip scattering (t ′ = r ′ = 0). Introducing the Fermi level velocities v F ± for majority or minority spin particles, we write the diverging contributions to δt ↑ (p ′ ) as where Q ± is related to Q as in Eq. (4). In order to apply the poor man's renormalization method, it is preferable to transform the momentum integrations in Eq. (26) into energy integrals. In order to do this, we linearize the spectrum near the Fermi level as: where energy of the scattered electron is ǫ ′ and the energies ǫ(ǫ ′ ) < 0 are measured with respect to the Fermi level. The linearization is assumed to be valid within an energy range D around the Fermi level. The Q−integrals appearing in equation (26) can now be written as: The scattering amplitudes with ↑ (↓) spin index are always associated with the momentum p ′ (p ′ − ). Therefore, we shall henceforth omit the momentum argument p ′ (p ′ − ) of the scattering amplitudes. The divergent perturbative correction, δt ↑ , is proportional to log(|ǫ ′ |/D), For the calculation of δt ′ ↑ (p ′ ) and δt ↓ (p ′ − ), the propagators we need to consider are − T τâp,↓ (τ )ĉ † p ′ ,↑ and − T τâp,↓ (τ )ĉ † p ′ − ,↓ , respectively. The perturbation theory is analogous to that described above. In order to obtain t ′ ↓ (p ′ − ) we consider the propagator − T τâp,↑ (τ )ĉ † p ′ − ,↓ . The logarithmically divergent perturbative terms can be dealt with using a renormalization procedure: we reduce the bandwidth D by removing states in a narrow strip δD near the band edge and work again the perturbation theory in the new bandwidth D − δD. The effect of removal of the band edge states must be compensated by adopting a new value of t ↑ for the new bandwidth. Applying this procedure step by step yields successive renormalizations of t ↑ . A differential equation is obtained by noting that the perturbation theory result, t ↑ + δt ↑ , remains invariant as D is reduced: We introduce now a variable ξ = log(D/D 0 ) which will be integrated from 0 to log(|ǫ ′ |/D 0 ), corresponding to the fact that the bandwidth is progressively reduced from D 0 to |ǫ ′ | (which will eventually be taken as temperature: |ǫ ′ | = T ) and the scaling differential equations for the transmission amplitudes become: In order to obtain the perturbative correction to the reflection amplitude r ↑ (p ′ ) we consider the propagator: In this case, there is a process where the incoming electron from the left is reflected back by the Hartree potential without even crossing the domain wall. The corresponding term comes from the Wick pairing term and gives a contribution to δr ↑ (p ′ ) equal to The differential equation for r ↑ (p ′ ) is and the differential equation for r ′ ↑ (p ′ ) is

IV. FIXED POINTS
The parameters of the model, which enter the scaling equations are: and the ratio v F − /v F + . The results can be presented in terms of the transmission coefficients, defined by and the reflection coefficients By definition, these coefficients refer to the respective currents divided by the incident current.
A. Insulator fixed points We have made a numerical study of the scaling equations. The non-interacting domain wall described in Section II provides the initial scattering parameters for our numerical scaling. Below we describe analytically the scaling behavior close to the fixed points we have found.
For repulsive interactions (g ↑ , g ↓ , g ⊥ > 0) the system flows to insulator fixed points. For a moderate to large λ/v F + (larger than about 0.1) all the transmission amplitudes, t σ and t ′ σ , vanish faster than any reflection amplitude as T → 0. We may then rewrite the scaling equations neglecting the small transmission amplitudes. The scaling equation for r ↑ , for instance, becomes where we used Eq. (A10). The Wronskian relation (A6), allowing for complex reflection amplitudes, shows that The charge conservation condition is satisfied solely by the reflections, from which we easily conclude that |r ↑ | = |r ↓ | at the fixed point. Then, Eq. (46) may be rewritten as Consider now the scaling equation (36). For r ′ ↑ in case of negligible transmissions we have The Wronskian relation (A6), allowing for complex reflection amplitudes, tell us that and we recast Eq. (50) as In the derivation of (49) and (52) the only assumption made was that the transmission amplitudes are negligibly small. The reflection amplitudes may be, in general, complex and are still renormalized after the transmissions became negligible. Now we see that Eqs. (49) and (52) predict that the phases of the complex numbers r ↑ , r ′ ↑ are unchanged during scaling. The two fixed points we may consider correspond to r ↑ approaching 0, or |r ↑ | approaching 1 along a constant phase line in the complex plane.
The situation |r ↑ | → 0 requires 2g ⊥ − g ↑ − g ↓ > 0 and, by charge conservation we have |r ′ ↑ | → v F + /v F − . Upon integrating (52) with ξ ranging from 0 to log(T /D 0 ), the amplitude r ′ ↑ will vary from its initial value r ′ ↑,0 to r ′ ↑ (T ). Using the definition (45) for the reflection coefficient, we write The domain wall becomes insulating. It reflects all incident electrons while reversing their spin. Therefore, such a DW may be considered as a perfect spin-flip reflector at zero temperature. In order to find the low T behavior of transmissions we put r ↑ = r ↓ = 0 in equations (30)- (33) and obtain Figure 3 shows numerical solutions to the scaling equations, where the system is flowing to this fixed point.
In the regime where g ↑ +g ↓ −2g ⊥ > 0 we have R ′ ↑ (T ) → 0, R ↑ (T ) → 1. So, the domain wall reflects all incident electrons while preserving their spin. From Eqs. (30)- (33) for the transmission amplitudes we obtain: If g ↑ + g ↓ − 2g ⊥ = 0 then both R ′ ↑ (T ) and R ↑ (T ) tend to finite values. Such a regime is illustrated in Fig. 4. In this case, Eqs. (30), (31) and (33) with constant reflection amplitudes become a linear (in t ↑ , t ′ ↑ , t ′ ↓ ) algebraic 3 × 3 system. The eigenvalues of the matrix give three temperature exponents and each transmission amplitude will be a linear combination of the three powers of T . For decreasing temperature, there may be a crossover from one exponent to the other and the lowest exponent dominates as T → 0.
For smaller values of λ/v F + (smaller than about 0.1) in the Hamiltonian (1), the system flows to a fixed point, where r ′ ↑ vanishes faster than the transmissions and |r σ | → 1. The transmission amplitudes still scale to zero as in Eqs. (55). The scaling equation (36) for r ′ ↑ can be linearized in r ′ ↑ by neglecting the second order terms in t, t ′ , and considering that |r σ | → 1, from which we obtain We see that the exponent for |r ′ ↑ (T )| is not bigger than the exponents in (55). Now, in the scaling equation (35) for r ↑ we cannot neglect the terms containing transmission amplitudes on the right hand side. One can easily see that the g ↑ term becomes g ↑ (|r ↑ | 2 − 1)r ↑ , which is of the same order of magnitude as the other terms. Consequently, the scaling behavior derived in Eqs. (49) and (52) does not apply here, since it was assumed there that the transmissions were smaller than r ′ σ . The behavior of r σ as T → 0 can be found from the charge conservation condition: in Fig. 5, where t ′ ↓ is seen to initially flow very fast to zero. The explanation is the following: for small λ in Eqs. (7) and (8), the noninteracting domain wall has t ↑ > 1, r ↑ > 0 and t ↓ < 1, r ↓ < 0. Also t ′ = r ′ is small. The scaling equation for t ′ ↓ becomes The first term on the right-hand side is positive and much larger than the second one, so t ′ ↓ tends fast to zero and disappears from the equations. The equation for t ′ ↑ is The first term on the right is negative while the second is smaller because of small initial t ′ ↑ . Then, t ′ ↑ initially grows as can be seen in Fig. 5.

B. Transparent barrier fixed points
Zero temperature fixed points corresponding to a transparent domain wall can be achieved when the interaction constants are all negative, i.e., for attractive electron interaction. Although we do not expect such a situation to occur in realistic physical systems, we describe below the fixed points for the case V = 0 in the model Hamiltonian (1). For moderate to strong λ/v F + in the model (7)-(8), the zero temperature values 1 ≥ |t ′ ↑ | = |t ′ ↓ | > |t ↑ | depend on the initial parameters. Smaller λ/v F + enhances t ↑ relative to t ′ σ . The reflection coefficients vanish under scaling as powers of temperature. The corresponding exponents can be obtained after linearizing (for small reflections) the scaling equations (35)- (36). The resulting 3 by 3 matrix contains the finite limiting values of the transmission amplitudes and its eigenvalues give the temperature exponents for the vanishing reflection amplitudes. Figure 6 shows an example of this behavior.
If some of the interaction constants are positive and the others negative, the situation becomes more complex. Below we describe several possible situations.
The system flows to the fixed point r ↑ = r ↓ = −1 with all other amplitudes vanishing. The low-T behavior of the transmission can be easily found by inserting the fixed point reflections into Eqs. (30)- (33): (58) The scaling equation for r ′ ↑ , neglecting second order terms in the scattering amplitudes, takes the form: so that we must have g ↑ + g ↓ − 2g ⊥ > 0 in order for r ′ ↑ → 0. 2. The case g ↑ , g ↓ < 0, g ⊥ > 0 The system flows to the perfect spin-flip reflector fixed point |r ′ ↑ | = v F + /v F − with all other amplitudes vanishing, in accordance with the condition g ↑ +g ↓ −2g ⊥ < 0 derived earlier. 3. The case g ↑ > 0, g ↓ < 0 For a negative or small positive g ⊥ , the system flows to a fixed point where |t ′ ↓ | = 1, r ↑ = −1. The wall transmits all spin-↓ particles with a spin-flip and reflects all spin-↑ particles. From Eqs. (30)- (33) we see that the exponents for the transmission amplitudes are: After linearizing Eq. (36) for r ′ ↑ in small amplitudes, we have which requires g ↑ > g ⊥ for vanishing r ′ ↑ . If g ↑ − g ⊥ is small, the small quantities neglected in the right-handside of Eq. (61) become important. Therefore, this fixed point holds for g ↑ − g ⊥ above some small quantity. Linearizing Eq. (34) for r ↑ in small amplitudes, we find which tends to zero since g ↓ < 0. For larger g ⊥ the system flows to the spin-flip reflector fixed point (|r ′ The situation is analogous to the previous one. For negative or small positive g ⊥ the system flows to a fixed point where |t ′ ↑ | = 1, r ↓ = −1 with all the others vanishing. The wall transmits all spin-↑ particles with a flip and reflects all spin-↓ particles. From equations (30)- (33) we see that the exponents for the transmission amplitudes are: Linearizing equation (36) for r ′ ↑ in small amplitudes, we have: which requires g ↓ > g ⊥ in order for r ′ ↑ to vanish. If g ↓ −g ⊥ is small, the small quantities neglected in the right-handside of Eq. (64) will become important. Therefore, this fixed point holds for g ↓ − g ⊥ above some small quantity. For larger g ⊥ the system flows to the spin-flip reflector

V. DISCUSSION AND SUMMARY
Lateral ferromagnetic semiconductor wires with nanoconstrictions make it possible to achieve the limit of sharp domain walls 25 . It has been shown that the constriction itself does not cause significant reflection of the incident waves because it only produces a semiclassical potential 46 . We may estimate the parameter λ of our model (1) by assuming that M (z) = M 0 cos θ(z)ˆ z + M 0 sin θ(z)ˆ x with cos θ(z) = tanh(z/L) 38 , where L is the width of the domain wall. We then find implying that The condition for the domain wall to be smaller than the Fermi wavelength is Lk F + < 2π. The smaller Fermi wavelength is that of majority spin electrons, k F + . On the other hand, for small Lk F + the barrier becomes a poor spin-flip scatterer. The ratio v F − /v F + depends on the polarization degree of the electron system. We consider now a one-channel system. In a nonmagnetic system there is a single Fermi momentum k F for up and down electrons, and the Fermi energy is E F = h 2 k 2 F /(2m). Once the system becomes magnetized, the two new Fermi momenta, k F ± , satisfy the particle conservation condition, and the spin-up and spin-down Fermi surfaces must correspond to the same energy, where ∆E/2 = JM 0 is the Zeeman shift of the bands. From this we get so that the ratio k F − /k F + is Inserting (69) into Eq. (66) we obtain The full polarization limit is k F − = 0 and k F + = 2k F , meaning that ∆E/4E F = 1, and then equation (71) gives Typical values for a non fully polarized system are E F = 90 meV and ∆E = 30 meV 25 . In this case we have v F − /v F + = 0.84 and equation (71) gives Therefore, if Lk F + is smaller than about 2π, the system can flow to any of the fixed points described above, especially the ones described in Section IV A. The lateral quantization may produce several channels. The higher channels have larger Fermi wavelength and larger ∆E/4E F , so they can be in the spin-flip reflector fixed point. If a channel of high energy is fully spin polarized, then it corresponds to λ/v F + = 0.79 Lk F + . But the possibility of inter-channel scattering arises. This could be due to the two following reasons: (i) electron-electron interactions (such would require a modification of our theory to allow for inter-channel scattering); (ii) the impurity scattering. For the latter to be negligible we need the electron mean free path (not the transport mean free path) to be larger than the size of the constriction.
In summary, we have studied the effect of electronelectron interactions on the transmission through a domain wall in a ferromagnetic wire in the regime in which the wall width is smaller than the Fermi wavelength. Applying a renormalization technique to the logarithmically divergent perturbation, we obtained the scaling equations for the scattering amplitudes. The T = 0 fixed points were identified corresponding to: (i) perfectly insulating wall (with or without complete spin reversal), and (ii) transparent wall. Both repulsive and attractive interactions were considered. We have estimated physical parameters for a domain wall model which may be realized in physical systems. Such estimates suggest that realistic systems can display the behavior predicted in the vicinity of the fixed points we have found.

APPENDIX B: FORMULATION FOR SPIN DEPENDENT ELECTRON EFFECTIVE MASSES
Electron interactions such as g 4 and g 2 , which describe forward scattering between particles moving in the same direction, may produce renormalization of the electron's effective mass. 45 The latter could depend on spin orientation because the Fermi surfaces of spin-up and spin-down electrons are different. These effects can be taken into account from the beginning by rewriting the Hamiltonian (1) in a more general form, where, in the kinetic energy term, we allow for a position and spin dependent effective mass tensor,m(z). The tensor may take the form: where Θ(z) denotes the Heaviside function.
The appropriate mass values must be used in Eqs.
(2)-(4). The expressions for the scattering eigenstates and transmission amplitudes given in the main text remain unchanged if we take into account that the velocities must be calculated considering the renormalized masses.