We study numerically the one-dimensional Kondo and Hund lattices consisting of localized spins interacting antiferromagnetically or ferromagnetically with the itinerant electrons, respectively. Using the density-matrix renormalization group we find, for both models and in the small coupling regime, the existence of new magnetic phases where the local spins order forming ferromagnetic islands coupled antiferromagnetically. Furthermore, by increasing the interaction parameter |J| we find that this order evolves toward the ferromagnetic regime through a spiral-like phase with longer characteristic wavelengths. These results shed new light on the zero temperature magnetic phase diagram for these models.9317720

Alascio, B

Avignon, M

Garcia, D J

Hallberg, K

Physical Review Letters

English

Repositorio da Producao Cientifica e Intelectual da Unicamp

VOLUME 93, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S
week ending
22 OCTOBER 2004Spin Order in One-Dimensional Kondo and Hund Lattices
D. J. Garcia
Instituto de Fisica Gleb Wataghin, Unicamp, Campinas (6165), São Paulo, Brasil
Instituto Balseiro and Centro Atómico Bariloche, CNEA, 8400 San Carlos de Bariloche, Argentina
K. Hallberg and B. Alascio
Instituto Balseiro and Centro Atómico Bariloche, CNEA, 8400 San Carlos de Bariloche, Argentina
M. Avignon
Laboratoire d’Etudes des Propriétés Electroniques des Solides,* CNRS, BP 166, 38042 Grenoble Cedex 9, France
(Received 14 April 2004; published 20 October 2004)177204-1We study numerically the one-dimensional Kondo and Hund lattices consisting of localized spins
interacting antiferromagnetically or ferromagnetically with the itinerant electrons, respectively. Using
the density-matrix renormalization group we find, for both models and in the small coupling regime,
the existence of new magnetic phases where the local spins order forming ferromagnetic islands coupled
antiferromagnetically. Furthermore, by increasing the interaction parameter jJj we find that this order
evolves toward the ferromagnetic regime through a spiral-like phase with longer characteristic wave-
lengths. These results shed new light on the zero temperature magnetic phase diagram for these models.
DOI: 10.1103/PhysRevLett.93.177204 PACS numbers: 75.10.–b, 75.30.Mb, 75.40.MgThe interplay between charge and spin degrees of free-
dom in strongly correlated systems has triggered enor-
mous interest in recent years due to the rich variety of
phases found in a plethora of compounds. Charge and spin
superstructures with a doping dependent wave vector
were found, for example, in La2xSrxNiO4 using neutron
scattering [1] and electron diffraction [2]. Stripe forma-
tion together with incommensurate spin fluctuations in
high-Tc superconductors can also be regarded as a mani-
festation of similar phenomena [3] as well as the charge
and spin ordering found in many of the doped manganese
perovskites. Another large group of compounds, the
heavy-fermion materials, presents various types of
ground states including antiferromagnetically ordered
states, the normal heavy-fermion state, as well as super-
conducting and insulating phases. Heavy-fermion sys-
tems and Kondo insulators are typical examples of
systems in which the interactions between conduction
electrons and quantum localized spins are essential
[4,5]. Their physical properties result from an antiferro-
magnetic coupling J between these two types of particles,
the so-called Kondo lattice model (KLM). The corre-
sponding Hamiltonian has the well-known form
H  t
X
hi;ji;

cyi
cj
  J
X
i
~Si  ~
i: (1)
The first term represents the conduction electron hop-
ping between nearest-neighbor sites cyi
 (ci
) being stan-
dard creation (annihilation) operators. In the second term
the exchange interaction J is antiferromagnetic (J < 0),
and ~
i 
1
2
P

;
0c
y
i
 ~

0ci
0 ( ~

0 are Pauli matrices).
It is interesting to note that, in recent years, the same
model Hamiltonian with ferromagnetic coupling (J > 0)
has been considered to contain the basic physics of man-0031-9007=04=93(17)=177204(4)$22.50 ganites exhibiting the ‘‘colossal’’ magnetoresistance ef-
fect [6–9]. In this case, both localized spins and itinerant
electrons originate from manganese d states. The system
is assumed to contain essentially Mn4	 ions with three
localized t2g orbitals represented as local spins ~Si and
additional itinerant electrons in the eg orbital. Because of
the strong Hund coupling the spin of the eg electron is
constrained to be parallel to the local spin on that site.
Hund’s rule together with the hopping term gives rise to
the ‘‘double-exchange’’ (DE) interaction that favors fer-
romagnetic ordering of the local spins [10]. In recent
literature this model is often referred to as the ferromag-
netic Kondo lattice; however, to avoid confusion with the
Kondo model, we call it the Hund model (HM). The DE
mechanism requires only that the system is away from
half filling and is independent of the sign of J [11].
We will study the Hamiltonian (1) for both antiferro-
magnetic and ferromagnetic (FM) couplings, considering
S  1=2 localized quantum spins. To this end we use the
density-matrix renormalization group (DMRG)[12] with
open boundary conditions for chains of different sizes.We
implemented the finite version of the DMRG algorithm
reaching chains sizes of 36 sites (the discarded weight
being less than 105). The different phases are charac-
terized through the local spin-spin correlation functions
and its Fourier transform, the following spin structure
factor:
S
q 
1
L
X
i;j
eiq
RjRih ~Si  ~Sji;
where L is the number of sites in the system and n is the
number of conduction electrons per site.2004 The American Physical Society 177204-1
0
3
6
9
3)
(a) J/t=-10
(c) J/t=-1.5 (d) J/t=10
(b) J/t=20
VOLUME 93, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S
week ending
22 OCTOBER 2004In Fig. 1 we present the phase diagram of the one-
dimensional Kondo and Hund models which we propose
from our numerical results for several commensurate
fillings (marked with dashed lines), improving previous
results [5,13–18]. As can be seen, except for a scale factor,
both phase diagrams present great similarities. The half-
filled case n  1 is pathological in both models whose
ground state is very different from the n  1 case: char-
acterized by a spin gap, however, with a different behav-
ior as a function of jJj in the Kondo and Hund cases, it has
been referred to as a ‘‘spin-liquid phase’’[19]. For the
Hund model it scales to the S  1 chain with a Haldane
gap. The transition to the FM phase in the Kondo model
coincides with the previous work mentioned. However,
for the Hund model we find that the border lies at slightly
larger values of J as compared with Ref. [14] due to the
larger system sizes considered here. We also include the
phase separated regime [14].
At low jJj a phase qualified as ‘‘paramagnetic’’ in the
KLM and ‘‘incommensurate’’ in the HM had been iden-
tified with exact diagonalization and DMRG [13–
15,18,20,21]; this phase is, however, much less under-
stood than the ferromagnetic phase. In these references
the local spin-spin correlations are calculated and S
q
shows a peak at the wave number corresponding to 2kF of
the conduction electrons. However, no scaling was per-
formed and the existence of what we call ‘‘island phases’’
(see below) was missed. Recently, the existence of a ‘‘spin
dimerized’’ phase has been reported [22] for the Kondo
model at quarter-filling (n  1=2), through the order
parameter D
i  h ~Si  ~Si	1i. The spin structure is of
the island-type    ""##    similar to the one we have
identified previously for the DE-superexchange model
[23]. Our present calculations reproduce the results for
D
i. However, the absence of a spin gap [22] and our
results on the long distance spin-spin correlation func-
tions suggest a nonspontaneously broken translation sym-H
und
K
ondo
Spin  liquid
2/3
n
J/t
1/3 1/2 10
−3.0
−1.0
−2.0
−0.5
−1.5
5
10
15
20
25
"S
pi
ra
l"
FM
FM
Island
PS
"Spiral"
FIG. 1. Magnetic phase diagram for the KLM and HM. The
various phases are described in the text. The circles indicate the
crossover values.
177204-2metry state compatible with the generalization of the
Lieb-Schultz-Mattis theorem [24]. We find that similar
structures are present for other commensurate fillings
n  1=4, 1=3, 2=3, and 3=4 as well. However, we do not
find that the ‘‘island’’ phases stable at small jJj transform
directly into the FM phase [22]. Instead, they evolve
toward the FM regime through an intermediate spiral-
like phase with a wave vector q that changes from q 
2kF  n (island) to q  0 (FM) as jJj=t increases. A
spiral phase has been proposed analytically in a semi-
classical model showing a similar evolution toward the
FM phase [25].
Let us first present the quarter-filled case n  1=2
(Fig. 2). For the Hund model and large J, i.e., J=t * 13,
the FM phase is clearly identified with a peak of S
q at
q  0, and the total spin (considering only localized
spins) is ST  Ns=2, where Ns is the number of localized
spins in the system. Each electron forms a triplet state
S  1 with the localized spins and all localized spins are
ferromagnetically ordered. For intermediate values of J
(6 & J & 10) the FM phase gives rise to a ‘‘spiral’’ phase
(Fig. 3), characterized by two broad peaks located at
incommensurate values of q which evolve inwards (the
momentum of the peak grows from q  0 to larger values
and another peak at a symmetric point with respect to
q   moves towards smaller values). When J decreases
further (J=t & 6) the spin structure transforms into an
island structure with a more defined peak of S
q at q 
n  =2 as shown in Fig. 2(f) indicating a four-site
periodicity. This phase remains stable down to very small
values of J=t; it has total spin ST  0 and zero spin gap.0
1
2
S(
q
0 0.5 1 1.5
q/π
0
1
2
3
0 0.5 1 1.5 2
0 1/36 1/24 1/12
1/L
1
2
3
4
S(
q*
)
J=10
J=2
J=-0.5
J=-1.5
(e) J/t=-0.5
(f) J/t=2
(g)
FIG. 2. Spin structure factor S
q for the quarter-filled case
(n  1=2) for the Kondo (left panel) and Hund (right panel)
models. From top to bottom: the ferromagnetic, spiral, and
island phases, appearing as jJj is decreased. The lower panel
shows the size dependence of the peak weight S
q in island
(full symbols) and spiral phases (empty symbols).
177204-2
VOLUME 93, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S
week ending
22 OCTOBER 2004Similar results are obtained in the Kondo case (J < 0).
The island-type phase is conserved for a small Kondo
coupling (J=t * 1:3) and the ferromagnetic phase is
recovered for J=t & 1:6 [Figs. 2(a), 2(c), and 2(e) ].
In the strong Kondo coupling limit the ferromagnetic
state differs from the Hund case in the sense that now,
the conduction electrons form singlet states (S  0) with
the localized spins, so that the system contains Ns=2
singlets and Ns=2 unpaired localized spins. The unpaired
spins are ferromagnetically coupled and the total spin is
now ST  Ns
1 n=2  Ns=4. In the ferromagnetic
phase, our results for the ground-state energy compare
well with those obtained with the effective Hamiltonian
in the strong coupling limit (jJj=t ! 1) [26] i.e., J=8
2t= in the Hund case and 3J=8 t= in the Kondo case.
Triplet states in the Hund case conserve the hopping t
while singlet states in the Kondo case acquire a reduced
hopping t=2.
The weight of the peak of the spin correlation S
q for
the island (and ferromagnetic) phases increases with the
chain sizes indicating a quasi-long-range order in this
case. Instead, for the spiral-like phase, this weight re-
mains constant, indicating a short-range correlation. A
similar behavior for these phases is found for the other
fillings considered.
In order to visualize the order in real space we show in
Fig. 3 the spin-spin correlation functions for the island
and spiral phases. For the former case, the correlations
change sign quite abruptly every two sites indicating that
the structure is effectively of the island-type ""## men-
tioned above with quasi-long-range ordering. The
nearest-neighbor correlation h ~S
i  ~S
i	 1i changes
abruptly from site to site, with a slight border effect.
Here we reproduce the results of Ref. [22]. In the latter
case, instead, the spiral order is clearly identified and
differs qualitatively from the island order. The nearest--0.4
0
0.4
<
S(
i)
. S
(L
/2
)>
-0.4
0
<
S(
i)
. S
(i
+
1)
>
-0.4
0
0.4
<
S(
i)
. S
(L
/2
)>
9 18 270
i
-0.4
0
<
S(
i)
. S
(i
+
1)
>
9 18 27 360
i
(d) J/t=10(c) J/t=-1.5
(a) J/t=-0.5 (b) J/t=2
(a’) (b’)
(c’) (d’)
Island
Spiral
FIG. 3. Real-space local spin-spin correlations for n  1=2
and the Kondo (left) and Hund (right) models. The qualitative
difference between the island (top panel) and spiral (lower
panel) phases can be seen.
177204-3neighbor correlation h ~S
i  ~S
i	 1i remains ferromag-
netic over the chain, with a small oscillation of period
two suggesting a dominant spiral-like state with a rem-
nant of the island order, which diminishes with J.
Similar island phases are clearly evidenced for n 
1=3 at intermediate values of the coupling for both mod-
els (0:5 & J=t & 1). Typical examples are for J=t  1
and J=t  0:3 [see Figs. 4(c) and 4(d)]. S
q shows a
clear peak at q  n  =3 and the spin correlation in
real space presents an island structure of three ferromag-
netic spins coupled antiferromagnetically between is-
lands, basically """### (not shown). The ferromagnetic
phase as discussed above is clearly recovered for J=t &
1 and J=t * 5. In the intermediate region (for values of
J between these intervals) we find again the spiral phase
[Figs. 4(a) and 4(b)].
A similar behavior occurs for n  2=3 as shown in
Figs. 4(e)–4(h) . The island structure of the type ""#""#
is clearly identified for 1 & J=t & 10 in S
q with a
peak at q  2=3. This island structure has a ferrimag-
netic character [27] and we suggest that this configuration
corresponds to the new ‘‘ferromagnetic’’ phase reported
in Ref. [28]. For 15 & J & 21 we observe a spiral phase.
In the Kondo model we find that the island phase trans-
forms into the FM regime through another intermediate
phase, in the region 3 & J & 1, with a double wave-
length (see below).
To illustrate the crossover between the different phases
we plot in Fig. 5 the behavior of the wave vector where the
spin structure factor is maximum, q, as a function of J
for several fillings. Here it is clearly seen that for small jJj
the island phases show up with their characteristic wave
vector q  2kF  n. This phase is stable for a certain
region until the spiral phase takes over for larger jJj. The
wave vector of the spiral phase decreases to zero as jJj
increases, leading finally to the FM phase at sufficiently
large values of the interaction parameter. We observe that0
1
2
3
4
0
1
2
3
0
1
2
3
S(
q)
0 0.5 1 1.5
q/π
0
1
2
3
0 0.5 1 1.5 2
(b) n=1/3 J/t=4
(d) n=1/3 J/t=1(c) n=1/3 J/t=-0.3
(a) n=1/3 J/t=-0.88
(e) n=2/3 J/t=-2 (f) n=2/3 J/t=20
(g) n=2/3 J/t=-0.5 (h) n=2/3 J/t=2
FIG. 4. Spin structure factors for both Kondo (left) and Hund
(right) models showing the spiral [(a),(b) and (e),(f)] and island
[(c),(d) and (g),(h)] phases for different fillings.
177204-3
-1.5 -1.0 -0.5 0.0
0
0.2
0.4
q*
/π
4.0 8.0
-2.0 -1.0 0.0
0
0.2
0.4
q*
/π
10.0 20.0
-4.0 -3.0 -2.0 -1.0 0.0
0
0.2
0.4
0.6
q*
/π
20.0 40.0
J/t
Kondo Hund
n=1/3
n=1/2
n=2/3
Island
Sp. Spiral
FM
FM
q =2π/6
q =2π/4*
q =2π/3*
*
FIG. 5. Evolution of the momentum where the spin structure
factor is maximum as a function of the interaction strength J
covering both models for different fillings.
VOLUME 93, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S
week ending
22 OCTOBER 2004this behavior is qualitatively similar to the ‘‘unwinding’’
of the spiral phase towards the FM order obtained by
Fazekas and Müller-Hartmann [25]. For n  2=3 in the
Kondo case we find that q remains fixed at q  =3 in
the region labeled as ‘‘polaronic liquid’’ in Ref. [28]. In
this case, from the real-space spin-spin correlations it is
difficult to distinguish clearly between a spiral and an
island of type """### but with short-range order.
In conclusion, we have studied numerically, using the
DMRG, the Kondo, and Hund models for localized spins
interacting with itinerant electrons. In addition to the
ferromagnetic phase at large jJj, we find the existence
of island and spiral phases within the ‘‘paramagnetic
regime’’ in these models. Both phases differ qualitatively
as seen in different correlation functions. The island
phase has quasi-long-range order and zero spin gap.
Furthermore, we show how the ground state evolves
from the low jJj=t island 2kF phase to the FM regime
through a spiral-like phase at intermediate couplings. By
carefully analyzing the finite-size scaling, we conclude
that all phases obtained are not due to Friedel oscillations
of the open boundaries. Based on the results for commen-
surate fillings we suggest the phase diagram shown in
Fig. 1. We point out that the difference in the scale of J
between the Kondo and Hund models has to be a conse-
quence of the quantum nature of the localized spins, so
that a complete understanding of the phase diagram re-
quires a full quantum description of the Hamiltonian.
D. J. G. acknowledges discussions with E. Miranda and
J. C. Xavier. We also thank C. Batista for fruitful com-
ments. We acknowledge partial support from the
Collaboration Program between France and Argentina
CNRS-PICS 1490, CONICET-PICT, PIP (CONICET)
437/98, and Fundación Antorchas Project 14116-168. We
acknowledge full (D. J. G. and K. H.) and partial (B. A.)177204-4support by CONICET.*Associated with Université Joseph Fourier.
[1] S. Hayden et al., Phys. Rev. Lett. 68, 1061 (1992).
[2] C. H. Chen and S-W. Cheong, Phys. Rev. Lett. 76, 4042
(1996).
[3] J. M. Tranquada et al., Nature (London) 375, 561 (1995).
[4] A. C. Hewson, The Kondo Problem to Heavy Fermions
(Cambridge University Press, Cambridge, 1993).
[5] H. Tsunetsugu, M. Sigrist, and K. Ueda, Rev. Mod. Phys.
69, 809 (1997).
[6] J. M. D. Coey, M.Viret, and S. von Molnar, Adv. Phys. 48,
167 (1999).
[7] Colossal Magnetoresistive Oxides, edited by Y. Tokura,
Advances in Condensed Matter Science Vol. 2 (Gordon
and Breach Science Publishers, Amsterdam, 2000).
[8] Physics of Manganites, edited by T. A. Kaplan and S. D.
Mahanti, Fundamental Materials Research (Kluwer
Academic/Plenum Publishers, New York, 1999).
[9] E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. 344, 1
(2001).
[10] C. Zener, Phys. Rev. 82, 403 (1951).
[11] P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675
(1955).
[12] Density Matrix Renormalization, Lectures Notes in
Physics, edited by I. Peschel, X. Wang, M. Kaulke, and
K. Hallberg (Springer-Verlag, Berlin, New York, 1999);
S. R. White, Phys. Rev. Lett. 69, 2863 (1992).
[13] S. Yunoki et al., Phys. Rev. Lett. 80, 845 (1998).
[14] E. Dagotto et al., Phys. Rev. B 58, 6414 (1998).
[15] S. Caprara and A. Rosengren, Europhys. Lett. 39, 55
(1997).
[16] J. Riera, K. Hallberg, and E. Dagotto, Phys. Rev. Lett. 79,
713 (1997).
[17] G. Honner and M. Gulacsi, Phys. Rev. B 58, 2662 (1998).
[18] N. Shibata and K. Ueda, J. Phys. Condens. Matter 11, R1
(1999).
[19] H. Tsunetsugu et al., Phys. Rev. B 46, 3175 (1992).
[20] S. Moukouri and L. G. Caron, Phys. Rev. B 52, 15 723
(1995).
[21] J. C. Xavier, E. Novais, and E. Miranda, Phys. Rev. B 65,
214406 (2002).
[22] J. C. Xavier, R. G. Pereira, E. Miranda, and I. Affleck,
Phys. Rev. Lett. 90, 247204 (2003).
[23] D. J. Garcia, K. Hallberg, C. D. Batista, M. Avignon, and
B. Alascio, Phys. Rev. Lett. 85, 3720 (2000); D. J. Garcia
et al., Phys. Rev. B 65, 134444 (2002).
[24] M. Yamanaka, M. Oshikawa, and I. Affleck, Phys. Rev.
Lett. 79, 1110 (1997).
[25] P. Fazekas and E. Müller Hartmann, Z. Phys. B 85, 285
(1991).
[26] D. Garcı́a, K. Hallberg, M. Avignon, and B. Alascio, in
Highlights in Condensed Matter Physics, edited by
A. Avela, R. Citro, C. Noce, and M. Salerno, AIP
Conf. Proc. No. 695 (AIP, Melville, NY, 2003), p. 304.
[27] Preliminary calculations in this region lead to nonzero
values of the total spin.
[28] I. P. McCulloch et al., Phys. Rev. B 65, 052410 (2002).177204-4


Spin Order In One-dimensional Kondo And Hund Lattices.

http://repositorio.unicamp.br/jspui/bitstream/REPOSIP/196012/1/pmed_15525121.pdf

Spin Order In One-dimensional Kondo And Hund Lattices.

Abstract

Similar works

Full text

Available Versions

Repositorio da Producao Cientifica e Intelectual da Unicamp