Role of a spin-flip scatterer in a magnetized Luttinger liquid
M.A.N Ara´joa,b , J. Berakdarc, V.K. Dugaevb,d and V. R. Vieirab,e
u

´
´
Departamento de F´
ısica, Universidade de Evora, Evora, Portugal
b
CFIF, Instituto Superior T´cnico, Lisbon, Portugal
e
c
Insitut f¨r Physik, Martin-Luther-Universit¨t Halle-Wittenberg,
u
a
Heinrich-Damerow-Str. 4, 06120 Halle, Germany
d
Department of Mathematics and Applied Physics, Rzesz´w University of Technology,
o
Al. Powsta´c´w Warszawy 6, 35-959 Rzesz´w, Poland and
n o
o
e
Departamento de F´
ısica, Instituto Superior T´cnico, Lisbon, Portugal
e

arXiv:0707.0784v1 [cond-mat.str-el] 5 Jul 2007

a

We study the spin-dependent scattering of charge carriers in a magnetized one dimensional Luttinger liquid from a localized non-homogeneous magnetic field, which might be brought about by the
stray field of magnetic tip near a uniform liquid, or by a transverse domain wall (DW) between two
oppositely magnetized liquids. From a renormalization group treatment of the electron interactions
we deduce scaling equations for the transmission and reflection amplitudes as the bandwidth is progressively reduced to an energy scale set by the temperature. The repulsive interactions dictate two
possible zero temperature insulator fixed points: one in which electrons are reflected in the same
spin channel and another where the electron spin is reversed upon reflection. In the latter case, a
finite spin current emerges in the absence of a charge current at zero temperature and the Friedel
oscillations form a transverse spiraling spin density. Adding a purely potential scattering term has
no effect on the fixed points of a uniformly magnetized liquid. For a DW we find that the introduction of potential scattering stabilizes the spin-flip insulator phase even if the single-particle spin-flip
scattering produced by the DW is arbitrarily weak. The potential can be induced externally, e.g.
by a local gate voltage or a constriction, providing a means for controlling the transport properties
of the wire.
PACS numbers: 73.63.Nm, 71.10.Pm, 75.70.Cn, 75.75.+a

I.

INTRODUCTION

Transport properties of magnetic nanowires are currently in the focus of intense research in view of possible applications in magneto-electronic devices1,2,3,4,5 .
Particularly interesting are magnetic nanowires with a
localized topological magnetic disorder which acts as a
spin-dependent scatterer of charge carriers. Examples of
this situations are magnetized wires with domain walls
or magnetic microvortices that can be well controlled externally by a magnetic field or by an electric current.
Here, we address the problem of scattering of interacting electrons from a single defect in a magnetic nanowire,
assuming that the nanowire is thin enough so that the
electron energy spectrum is one-dimensional. A similar
problem in the nonmagnetic case attracted a lot of attention in the past6,7 because of the key role of electronelectron interactions leading to substantial renormalization of the scattering amplitudes.
Recently8 , we treated the problem of the spin dependent scattering in a short transverse domain wall (DW)
separating two oppositely magnetized regions in a 1D
wire. In particular, we considered the case of a DW
whose extension is comparable to the carriers’ de Broglie
wavelength, in which case the influence of scattering and
interactions is particularly strong. On the other hand,
this limitation implies severe demands on an experimental realization restricting possible systems to magnetic
semiconductor nanowires, such as (Ga,Mn)As9 . A further complication is brought about by magnetic impurities in the semiconductor that cause scattering in the

region of the domain wall10 . If the DW is pinned by
a constriction, the constriction itself behaves as a pure
potential scatterer.
There is yet another possibility for inducing localized
magnetic imperfections in a wire: the stray magnetic field
of a nearby tip of a spin-polarized scanning tunneling
microscope (STM) in the T-shape geometry11 illustrated
in Fig.1. In this case, the wire itself may be magnetically
homogeneous and free of impurities.
For magnetic nanowires the scattering involves at least
two channels and different components of the transmission amplitudes may well be renormalized in different
ways. As shown below, this results in some interesting
effects such as a spin-flip insulator state. The possibility of these effects has been demonstrated recently for
scattering from a magnetic transverse domain wall8,12 .
In the case of a homogeneous magnetization of the wire,
possible types of defects can be classified as purely potential defects (i.e., a local perturbation that affects equally
the spin up and down electrons), local variation of the
magnitude of the magnetization, and a local variation
of the magnetization direction. The first two cases are
straightforwardly treated since they can be analyzed in
terms of potential scattering in each of the spin channels.
The problem tackled here is the case of a local variation
of the magnetization direction, when both spin-flip and
non-spin-flip scattering amplitudes are renormalized, by
electron interactions, in different ways.
Technically, we need to address the scattering of electrons from a localized transverse magnetic field. The
single-particle scattering (either pure potential or spin

2
dependent) can be treated exactly. We then address the
role of electron interactions perturbatively, to first order, resulting in the well-known logarithmic divergences.
The divergent terms are then circumvented by a poor
man’s scaling approach that yields a set of renormalization equations for the scattering amplitudes.
The two problems have similar fixed point solutions at
zero temperature (T = 0). Considering repulsive spin
dependent interactions, two types of insulator may arise,
depending on the interaction parameters: the electrons
may be 100% reflected with or without spin reversal. In
the former case there is a spin current without charge current and the spin current exerts a torque on the magnetic
tip. For the case of a DW we find the counter-intuitive effect that a non-spin-flip insulator phase is obtained if the
DW’s transverse field is weak. But by adding a purely
potential scattering term, one is able to drive the DW to
the spin-flip insulator phase. The potential itself can be
externally imparted, e.g. by a constriction or by a local
gate voltage, much in the same way as demonstrated by
recent experiments13 .
Section II introduces the electronic Hamiltonian for
the problems depicted in Figure 1 and the single-particle
solutions. Section III gives the renormalization group
equations for the scattering amplitudes in each problem,
which are a consequence of the electron interactions. Section IV contains a discussion and summary.

II.

In the case where the wire has two opposite ferromagnetic domains separated by a thin transverse DW, as
shown in Figure 1(b), the single particle Hamiltonian is:
h
¯ 2 d2
ˆ
+¯ V δ(z)−JM (z)ˆz −¯ λδ(z)ˆx . (2)
h
σ h¯
σ
HDW = −
2m dz 2
where the term ¯ λδ(z)ˆx now produces spin-flip scatterh¯
σ
ing due to the x component of the magnetization in the
DW:
h¯
¯ λ = −J

∞

Mx (z)dz
−∞

and the longitudinal magnetization M (z) = −M0 for
z < 0 and M (z) = M0 for z > 0. In this case the
longitudinal magnetization produces a Zeeman potential
step which scatters electrons more strongly as compared
to the magnetic tip problem above.
In both models, a single spin-majority electron may be
transmitted either preserving or reversing its spin with
amplitudes t↑ or t′ , respectively. A spin minority electron
↑
has transmission amplitudes t↓ or t′ . The reflection am↓
plitudes in the same (opposite) spin channel are denoted
′
by r↑ (r↑ ) for spin majority and spin minority electrons,
respectively.

MODEL HAMILTONIAN

We consider a ferromagnetic metallic wire close to a
magnetic tip as shown in Figure 1(a). The latter produces an effective magnetic field which acts as a spinflip scatterer in a localized region of the wire. The wire
defines the easy (ˆ) axis with uniform magnetization
z
M = M z . The effective magnetic field B⊥ (z)ˆ + B|| (z)ˆ
ˆ
x
z
due to the influence of the tip which is placed nearly perpendicular to the wire affects a small region near z = 0.
The conduction electron spin in the wire is Zeeman coupled to this magnetic field. We treat the conduction
electrons as one-dimensional and write the single particle
Hamiltonian as:
h
¯ 2 d2
ˆ
H0 = −
+¯ V δ(z)−JM0 σz −¯ λδ(z)ˆx −¯ λ′ δ(z)ˆz .
h
ˆ h
σ h
σ
2m dz 2
(1)
′
The terms ¯ λ( ) δ(z)ˆx , describe spin scattering produced
h
σ
by the the magnetic tip:
∞

h
¯λ = µ

B⊥ (z)dz ,
−∞
∞

h
¯ λ′ = µ

B|| (z)dz ,
−∞

and µ is the electron’s magnetic moment. We also allow
for a purely potential scattering term, V , that may be
present.

FIG. 1: (a) Magnetic tip produces a nearly transverse magnetic field in a localized region of a uniformly magnetized
metallic wire; (b) a transverse DW in the wire also acts as a
transverse field.

We wish to consider the effect of electron interactions
on the scattering amplitudes. The interactions can be
described by the g-ology model14 :
ˆ
Hint = g1,α,β
+ g2,α,β

dk1 dq † ˆ†
b
a
ˆ
b
ak +q,β ˆk1 −q,α
ˆ
(2π)2 k1 ,α k2 ,β 2
dk1 dq † ˆ† ˆ
a
ˆ
b
bk +q,β ak1 −q,α . (3)
ˆ
(2π)2 k1 ,α k2 ,β 2

The couplings g1 and g2 describe back and forward scattering processes between opposite moving electrons, respectively, and are positive if the interactions are repulsive. 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 g1↑ , g1↓

3
g1⊥ and g2↑ , g2↓ , g2⊥ . The g2 processes imply zero momentum transfer whereas the g1 processes involve a 2kF
momentum transfer. We therefore expect g2 larger than
g1 for a finite spatial range of the repulsive interactions.
Our treatment of the interactions follows the same
method invented in Ref7 and developed in Ref8 : the
corrections to the scattering amplitudes are calculated
ˆ
to first order in the perturbation Hint , leading to logarithmic divergent terms if the electron is near the Fermi
level. Using a poor man’s scaling procedure, the divergent terms lead to a renormalization of the scattering
amplitudes as the bandwidth, D, is progressively reduced
from its initial value, D0 , to D = T . The fixed points are
attained at temperature T = 0.

drσ
∗
= gσ |rσ |2 rσ + rσ t′2 − rσ
σ
dξ
∗
∗
′
′
+ g−σ r−σ tσ t−σ + r−σ r−σ rσ
+ 2g⊥

′∗ ′
′∗
rσ r−σ rσ + rσ t−σ t′
σ

,

′
drσ
′
∗
= gσ |rσ |2 rσ + rσ t′ tσ
σ
dξ
∗
′
∗
′
+ g−σ r−σ tσ t−σ + r−σ r−σ rσ

+ g⊥

′∗
′∗ ′2
′∗
r−σ r−σ rσ + r−σ rσ + rσ t′ t′
−σ σ
′∗
′
+ r−σ t2 − rσ ) .
σ

III.

SCALING EQUATIONS

We write down the renormalization group (RG) equations for the scattering amplitudes, in each problem, using the variable ξ = log (D/D0 ) which will be integrated
from 0 to log (T /D0 ), corresponding to the fact that the
bandwidth is progressively reduced from D0 to T . We
also define:
g2σ − g1σ
g2⊥
gσ =
,
g⊥ =
.
(4)
4hvσ
2h(v↓ + v↑ )
where v↑ (v↓ ) denotes the Fermi velocity of spin majority
(minority) particles.
The RG equations result from a first order perturbative treatment of the interactions. The total scattering
(transmission or reflection) amplitude is the sum over
all virtual scattering processes involving the electronelectron interaction once8 . The latter occurs in the form
of Bragg back-scattering caused by the 2kF Hartree-Fock
potential caused by the Fermi sea Friedel oscillations. So,
the simplest process is just one Bragg reflection produced
by the Friedel oscillation. The other possible process involve three virtual scattering events: the electron first
collides with the barrier, then with the Friedel oscillation and finally with the barrier again.
A.

Thin DW

A detailed derivation of the RG equations for this case
has been given elsewhere8 and we merely reproduce them
here:
dtσ
∗ ′
= gσ |rσ |2 tσ + rσ rσ t′
σ
dξ
∗
′
+ g−σ |r−σ |2 tσ + r−σ rσ t′
−σ
′∗ ′
′∗
′∗
+ g⊥ 2r−σ rσ tσ + rσ r−σ t′ + rσ rσ t′
σ
−σ

dt′
σ
∗
′
= 2gσ |rσ |2 t′ + 2g−σ r−σ r−σ tσ
σ
dξ
′∗
′∗ ′
+ 2g⊥ r−σ rσ tσ + rσ r−σ t′ ,
σ

,(5)

(6)

(7)

(8)

The Wronskian theorem8,15 for the scattering problem
establishes the following relations between the scattering
amplitudes:
|rσ |2 + |t′ |2 +
σ

v−σ
′
|rσ |2 + |tσ |2 = 1 ,
vσ

v−σ ′∗
rσ r−σ + t∗ t′
σ −σ = 0 ,
vσ
v−σ ′∗ ′
∗
′
rσ t−σ + t∗ r−σ = 0 ,
rσ t−σ + t′∗ r−σ +
σ
σ
vσ
v−σ
′∗
Re[t′∗ rσ ] +
Re[rσ tσ ] = 0 ,
σ
vσ
′
′
vσ t−σ = v−σ tσ ,
vσ r−σ = v−σ rσ .
∗ ′
rσ r−σ + t′∗ t−σ +
σ

(9)
(10)
(11)
(12)
(13)

The initial non-interacting values (ξ = 0) of the scattering amplitudes near the Fermi level are obtained from
the Hamiltonian (2):
2vσ (vσ + v−σ + 2iV )
= rσ + 1 ,
2
¯
(vσ + v−σ + 2iV ) + 4λ2
¯
4iλvσ
′
=
= rσ .
2
¯
(vσ + v−σ + 2iV ) + 4λ2

tσ =

(14)

t′
σ

(15)

In the absence8 of potential scattering (V = 0), tσ (ξ)
′
and rσ (ξ) are real while t′ (ξ) and rσ (ξ) are pure imagiσ
nary. In this case, two types of insulator fixed points were
′
found8 : the ordinary insulator with tσ = t′ = rσ = 0,
σ
|rσ | = 1, which is attained if g⊥ < (g↑ + g↓ ) /2; the ”spin′
flip reflector” with tσ = t′ = rσ = 0, |rσ | = v↑ /v↓ ,
σ
which is attained for g⊥ > (g↑ + g↓ ) /2 and not small
¯
λ/v↑ .
Our aim here is to analyze the effect of the purely po¯
tential term V . For small λ/v↑ and V = 0, the DW flows
to the zero temperature fixed point with t = t′ = r′ = 0
and r↑ = −r↓ = 1. The particles are reflected in the
¯
same spin channel because the spin-flip term λ/v↑ is not
strong enough and the Zeeman potential step −JM (z)ˆz
σ
in (2) is dominant. For finite potential scattering, V , all
the amplitudes become complex. We may study the RG
flow near this fixed point by linearizing equation (8) for
′
σ =↑ in the small quantity r↑ :

4

′
dr↑
′
′∗
≈ (g↑ + g↓ − g⊥ ) r↑ + g⊥ r↑ r↓ r↑ ,
dξ

(16)

which, for r↑ r↓ = −1, gives the scaling of the real and
imaginary parts as
′
Re[r↑ ] ∝ e(g↑ +g↓ −2g⊥ )ξ ,

′
Im[r↑ ] ∝ e(g↑ +g↓ )ξ ,

(17)

′
implying that Im[r↑ ] scales to zero as T → 0 (or ξ →
′
−∞), while Re[r↑ ] is relevant for g⊥ > (g↑ + g↓ ) /2 and
′
irrelevant if g⊥ < (g↑ + g↓ ) /2. Since the real part of r↑
is due to the finiteness of V , we see that the potential
scattering will enhance the spin-flip reflection processes
′
if g⊥ > (g↑ + g↓ ) /2. For such interactions, the rσ = 0
fixed point is in fact unstable to any small V and the
system will flow to the other fixed point with rσ = 0 and
′
|r↑ | = v↑ /v↓ . This is shown in Figure 2.

′
to first order both in the interactions g and r↑ : (i) the
electron is Bragg reflected back, in the ↓-spin channel,
by the Friedel oscillations of spin-down electrons with a
′
probability amplitude g⊥ r↑ ; (ii) the electron is first reflected by the DW with amplitude r↑ , then it is spin-flip
back-scattered by the ↑-spin Friedel oscillation with an
′∗
amplitude g⊥ r↑ , and finally reflected by the DW with
′
amplitude r↓ . Since r↑ r↓ = −1 and rσ is pure imaginary, process (ii) has exactly symmetrical amplitude to
process (i). So, for V = 0, the two processes interfere
destructively, resulting in the absence of spin-flip reflection. If V = 0, the amplitudes in processes (i) and (ii)
acquire different phases and no longer cancel each other.
In this case there is a reflection with spin reversal, with
1
amplitude proportional to g⊥ , and if g⊥ > 2 (g↑ + g↓ )
the system will flow to the spin flip reflector phase. A
¯
strong λ leads to the fixed point with r↑ = r↓ = 1 and is
this case there is no destructive interference.

B.

Magnetic tip

The derivation of the RG equations for the magnetic
tip problem follows exactly the same method as for the
DW. In fact, the scaling equations for this case can simply be written down by inspection of the same Feynman
diagrams:
dtσ
∗
′
= 2gσ |rσ |2 tσ + 2g−σ r−σ rσ t′
−σ
dξ
′∗ ′
′∗
+ 2g⊥ r−σ rσ tσ + r−σ rσ t′ ,
σ
FIG. 2: Zero temperature phase diagram of a DW for g⊥ >
(g↑ + g↓ ) /2. Electron reflection is accompanied with spin re¯
versal everywhere except at small λ and V = 0.

¯
Considering stronger λ/v↑ , g⊥ < (g↑ + g↓ ) /2 and V =
0 the zero temperature fixed point has t = t′ = r′ = 0
and r↑ = r↓ = 1. Near this fixed point, equation (16)
yields:
′
Re[r↑ ] ∝ e(g↑ +g↓ )ξ ,

′
Im[r↑ ] ∝ e(g↑ +g↓ −2g⊥ )ξ .

(18)

′
The potential V , responsible for Re[r↑ ], is, therefore, irrelevant. This fixed point is stable for interactions obeying g⊥ < (g↑ + g↓ ) /2 and unstable otherwise, thereby
taking the system to the spin-flip reflector fixed point.
For general finite V , the interactions determine whether
the fixed point has |r| = 1, for g⊥ < (g↑ + g↓ ) /2, or
′
|r↑ | = v↑ /v↓ when g⊥ > (g↑ + g↓ ) /2.
We can provide a physical explanation for the behavior
¯
of the DW shown in Figure 2 for small λ. If the DW’s
¯ is zero, the renormalized reflection
transverse field (λ)
amplitudes r↑ and r↓ tend to 1 and -1, respectively. There
is no reflection with spin flip as there is no spin-flip scattering term. Now consider an incident electron from the
left with spin up. What is the probability amplitude of a
¯
spin-flip reflection for small λ? There are two processes

(19)

dt′
σ
∗ ′
= gσ |rσ |2 t′ + rσ rσ tσ
σ
dξ
∗
′
+ g−σ r−σ rσ t−σ + |r−σ |2 t′
σ
′∗ ′
′∗
′∗
+ g⊥ 2r−σ rσ t′ + rσ rσ t−σ + rσ r−σ tσ ,(20)
σ

drσ
∗
= gσ |rσ |2 rσ + rσ t2 − rσ
σ
dξ
∗
∗
′
′
+ g−σ r−σ t′ t′ + r−σ r−σ rσ
σ −σ
′∗ ′
′∗
+ 2g⊥ rσ r−σ rσ + rσ tσ t′
−σ

,

(21)

′
drσ
′
∗
= gσ |rσ |2 rσ + rσ t′ tσ
σ
dξ
′
∗
+ g−σ |r−σ |2 rσ + r−σ t−σ t′
σ

+ g⊥

′∗
′∗ ′2
′∗
rσ r−σ rσ + r−σ rσ + rσ t−σ tσ
′∗
′
+ rσ t′ t′ −rσ ) .
σ −σ

(22)

The relations between the scattering amplitudes which
follow from the Wronskian theorem are:
v−σ
′
|rσ |2 + |tσ |2 +
|rσ |2 + |t′ |2 = 1 , (23)
σ
vσ

5
v−σ ′∗
(rσ r−σ + t′∗ t−σ ) = 0 ,
σ
vσ
v−σ ′∗
∗
′
rσ t′ + t∗ r−σ +
(rσ t−σ + t′∗ r−σ ) = 0 ,
−σ
σ
σ
vσ
v−σ
∗
′∗
Re[rσ tσ ] +
Re[rσ t′ ] = 0 ,
σ
vσ
′
′
′
′
vσ t−σ = v−σ tσ ,
vσ r−σ = v−σ rσ .

∗ ′
rσ r−σ + t∗ t′ +
σ −σ

(24)
(25)
(26)
(27)

The initial scattering amplitudes are the (noninteracting) solutions to (1):
vσ [v−σ + i (V − λ′ )]
[vσ + i (V + λ′ )] · [v−σ + i (V − λ′ )] + λ2
= rσ + 1 ,
(28)
iλvσ
′
=
= rσ .
[vσ + i (V + λ′ )] · [v−σ + i (V − λ′ )] + λ2
(29)

tσ =

t′
σ

In this problem, the zero temperature fixed point
is entirely determined by the interactions: if g⊥ <
(g↑ + g↓ ) /2 the system will flow to r′ = 0, |r| = 1;
if g⊥ > (g↑ + g↓ ) /2 the system will flow to r = 0,
′
|r↑ | = v↑ /v↓ .
IV.

DISCUSSION AND SUMMARY

Estimates of the g couplings have been made for the
Hubbard model in a magnetic field16 , where the inequality g2⊥ > 1 (g2↑ + g2↓ ) was obtained in the strong cou2
pling regime (U > t). Using the results for the renormalized Fermi velocities16 it is seen that a regime with
g⊥ > 1 (g↑ + g↓ ) is possible, as well as the opposite in2
equality.
In the spin-flip reflector fixed point, the spinor wave
function for an incident spin majority electron from the
left has the asymptotic behavior:
ψ(z < 0) ≃

eik↑ z
′
r↑ e−ik↓ z

,

(30)

for which the spatial distribution of the spin vector obeys
′
r↑ e−i(k↑ +k↓ )z = eiφ(z) tan

θ(z)
,
2

(31)

where φ and θ are the spherical angles of the spin
′
vector17 . Since r↑ has modulus 1 and a finite phase,

1
2

3

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then θ(z) = π/2 implying that the conduction electron spin is normal to the wire and winds around it
in ”circularly polarized” fashion with azimuthal angle
φ(z) = − (k↑ + k↓ ) z for z < 0. For z > 0, φ(z) changes
sign. Thus, the resulting spin density Friedel oscillations
in the Luttinger liquid have such a structure and decay
with distance as a power law.
Our RG method, being perturbative in the electron interactions, is valid when the latter are weak. But it allows
for the scattering barrier to be treated exactly, whatever
its strength, and hence predict the fixed point to which
a particular barrier will flow once the interactions are
turned on.
In summary, we considered the magneto-transport
properties a magnetized one dimensional Luttinger liquid
containing a localized disorder of the magnetization. We
addressed two cases depending on the origin of this disorder: 1) a stray magnetic field of an STM tip disturbing
the magnetization beneath it, or 2) a transverse domain
wall (DW). Our theoretical analysis relies on a renormalization group treatment of the electron interactions that
provides us with scaling equations for the transmission
and reflection amplitudes as the bandwidth is progressively reduced. We find two T = 0 insulator fixed points:
one in which the carriers are reflected in the same spin
channel and another where the carrier’s spin is reversed
upon reflection leading to a finite spin current without a
charge current. We also found the role of pure potential
scattering in driving the DW to the spin-flip insulator
phase when the transverse field of the DW is arbitrarily
weak.

Acknowledgments

Discussions with P.D. Sacramento are gratefully acknowledged. This research was supported by Portuguese
program POCI under Grant No. POCI/FIS/58746/2004,
German DFG under Grant No. SSP 1165, BE216/32, Polish Ministry of Science and Higher Education as
a research project in 2006–2009, and the STCU Grant
No. 3098 in Ukraine. M.A.N.A. would like to thank the
hospitality of the Max-Planck Institut f¨ r Mikrostruku
turphysik, Halle (Germany).

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