Stripe phases — possible ground state of the high-Tc superconductors
Based on the mean-field method applied either to the extended single-band Hubbard model or to the single-band Peierls-Hubbard Hamiltonian we study the stability of both site-centered and bond-centered charge domain walls. The difference in energy between these phases is found to be small. Therefo...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
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| Цитувати: | Stripe phases — possible ground state of the high-Tc superconductors / M. Raczkowski, A.M. Oles, R. Fresard // Физика низких температур. — 2006. — Т. 32, № 4-5. — С. 411-429. — Бібліогр.: 96 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859745053932519424 |
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| author | Raczkowski, M. Oles, A.M. Fresard, R. |
| author_facet | Raczkowski, M. Oles, A.M. Fresard, R. |
| citation_txt | Stripe phases — possible ground state of the high-Tc superconductors / M. Raczkowski, A.M. Oles, R. Fresard // Физика низких температур. — 2006. — Т. 32, № 4-5. — С. 411-429. — Бібліогр.: 96 назв. — англ. |
| collection | DSpace DC |
| container_title | Физика низких температур |
| description | Based on the mean-field method applied either to the extended single-band Hubbard model or
to the single-band Peierls-Hubbard Hamiltonian we study the stability of both site-centered and
bond-centered charge domain walls. The difference in energy between these phases is found to be
small. Therefore, moderate perturbations to the pure Hubbard model, such as next nearest neighbor
hopping, lattice anisotropy, or coupling to the lattice, induce phase transitions, shown in the
corresponding phase diagrams. In addition, we determine for stable phases charge and magnetization
densities, double occupancy, kinetic and magnetic energies, and investigate the role of a finite
electron-lattice coupling. We also review experimental signatures of stripes in the superconducting
copper oxides.
|
| first_indexed | 2025-12-01T20:34:54Z |
| format | Article |
| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5, p. 411–429
Stripe phases — possible ground state
of the high-Tc superconductors
Marcin Raczkowski1,2, Andrzej M. Oles� 1, and Raymond Fresard� 2
1 Marian Smoluchowski Institute of Physics, Jagellonian University
Reymonta 4, PL-30059 Krak�w, Poland
E-mail: A.M.Oles@fkf.mpg.de
2 Laboratoire CRISMAT, UMR CNRS-ENSICAEN(ISMRA) 6508,
6 Bld. du Mar�chal Juin Bld., F-14050 Caen, France
Received November 1, 2005
Based on the mean-field method applied either to the extended single-band Hubbard model or
to the single-band Peierls-Hubbard Hamiltonian we study the stability of both site-centered and
bond-centered charge domain walls. The difference in energy between these phases is found to be
small. Therefore, moderate perturbations to the pure Hubbard model, such as next nearest neigh-
bor hopping, lattice anisotropy, or coupling to the lattice, induce phase transitions, shown in the
corresponding phase diagrams. In addition, we determine for stable phases charge and magnetiza-
tion densities, double occupancy, kinetic and magnetic energies, and investigate the role of a finite
electron-lattice coupling. We also review experimental signatures of stripes in the superconduct-
ing copper oxides.
Pacs: 71.10.Fd, 71.27.+a, 74.25.–q, 74.72.–h
Keywords: high-Tc superconductivity, Hubbard model, stripe phases.
1. Introduction
Since the discovery of high-temperature supercon-
ductivity by Bednorz and M�ller [1], the unusual
physical properties of the copper oxides have stimu-
lated theorists and have led to the appearance of many
new ideas [2]. One of the especially appealing new
pictures that has emerged is the instability towards a
novel type of coexisting incommensurate (IC) charge
and magnetic order, i.e., stripe phase. As a rare event
in the theory of high temperature superconductivity,
the theory preceeded here the experiment and the exis-
tence of stripe phases was predicted on the basis of
Hartree–Fock (HF) calculations in the two-band
model for CuO2 planes of layered La2�xSr xCuO4
(LSCO) [3], before their experimental confirmation.
This instability persists as well in the effective sin-
gle-band Hubbard model [4–7]. All these calculations
yielded solutions with a phase separation manifested
in formation of nonmagnetic lines of holes, one-dimen-
sional (1D) domain walls or stripes, which separate
antiferromagnetic (AF) domains of opposite phases.
Such states result from the competition between the
superexchange interaction, which stabilize the AF
long-range order in the parent Mott insulator, and the
kinetic energy of doped holes. Indeed, the magnetic
energy is gained when electrons occupy the neighbor-
ing sites and their spins order as in the N�el state,
whereas the kinetic energy is gained when the holes
can move and the AF order is locally suppressed along
a domain wall (DW). Thus, a stripe phase provides
the best compromise between the superexchange pro-
moting the AF order and the kinetic energy of doped
holes.
However, the debate on the microscopic origin of
the stripe instability is far from closed. Two main sce-
narios, based on a Ginzburg–Landau free energy, for
the driving mechanism of the stripe phase have been
discussed [8,9]. In the first one, stripes are
charge-density waves with large periodicity arising
from the Fermi surface (FS) instability with the tran-
sition being spin driven [3]. A general feature of such
© Marcin Raczkowski, Andrzej M. Oles� , and Raymond Fr�sard, 2006
an instability is a gap/pseudogap which opens up pre-
cisely on the FS. Hence, the spacing between DWs is
equal to 1/x, with x denoting doping level so as to
maintain a gap/pseudogap on the FS. In this scenario
spin and charge order occur at the same temperature or
charge stripe order sets in only after spin order has de-
veloped.
An alternative scenario comes from the Coulomb-
frustrated phase separation suggesting that stripe for-
mation is charge driven. Indeed, using the Ising
model, it has been shown that the competition be-
tween long range Coulomb interactions and short
range attraction between holes leads to formation of
stripes [10]. In this case Ginzburg–Landau consider-
ations lead to an onset of charge order prior to spin or-
der as the temperature is lowered. However, the above
analysis does not take into account spin fluctuations
which might be crucial for the nature of the phase
transition by precluding the spins from ordering at the
charge-order temperature [11]. Moreover, the conjec-
ture that long range Coulomb forces are required to
stabilize stripe phases has been challenged by the
studies of the t–J model, in which the DW structures
were obtained without such interactions [12].
In order to investigate the influence of strong elec-
tron correlations due to large on-site Coulomb repul-
sion U at Cu ions, several methods have been em-
ployed to study the stripe phases which go beyond
the HF approximation, such as: density matrix re-
normalization group (DMRG) [12,13], Slave-Boson
approximation (SBA) [14–16], variational local an-
satz approximation [17], Exact Diagonalization (ED)
of finite clusters [18], analytical approach based on
variational trial wave function within the string pic-
ture [19], dynamical mean field theory (DMFT)
[20,21], Cluster perturbation theory (CPT) [22], and
quantum Monte Carlo (QMC) [23,24]. In spite of this
huge effort, it remains unclear whether DWs are cen-
tered on rows of metal atoms, hereafter named site-
centered (SC) stripes, or if they are centered on rows
of oxygen atoms bridging the two neighboring metal
sites, the so-called bond-centered (BC) stripes, and
even calculations performed on larger clusters did not
yield a definite answer [25]. Therefore, the purpose of
this paper is to study the stability of both structures
based on the mean-field method applied either to the
extended single-band Hubbard model or the sin-
gle-band Peierls-Hubbard Hamiltonian which in-
cludes the so-called static phonons [26]. For stable
phases we determine charge and magnetization densi-
ties, double occupancy, kinetic and magnetic energies,
and investigate the role of a finite electron-lattice cou-
pling.
2. Experimental signatures of stripes
Experimentally, stripe phases are most clearly de-
tected in insulating compounds with a static stripe or-
der, but there is growing evidence of fluctuating stripe
correlations in metallic and superconducting materi-
als. The most direct evidence for stripe phases in
doped antiferromagnets has come from neutron scat-
tering studies in which charge and spin modulations
are identified by the appearance of some IC Bragg
peaks, in addition to those which correspond to the
crystal structure. However, sometimes sufficiently
large crystals are not available for such experiments,
and one has to resort to other methods capable of
probing local order. These methods include nuclear
magnetic resonance (NMR), nuclear quadruple reso-
nance (NQR), muon spin rotation (�SR), scanning
tunneling microscopy (STM), and transmission elec-
tron microscopy (TEM). Furthermore, angle-resolved
photoemission spectroscopy (ARPES), angle-inte-
grated photoemission spectroscopy (AIPES), as well
as x-ray photoemission (XPS) and ultraviolet
photoemission (UPS) spectroscopies all provide essen-
tial information about conspicuous changes in the
electronic structure when stripe structure sets in.
Finally, a distinct imprint of the 1D spin-charge mod-
ulation on transport properties should be detectable as
the in-plane anisotropy of the resistivity and the Hall
coefficient RH .
The abundance of the current evidence on various
types of stripe order as well as the recent ARPES re-
sults on the spectral weight of the cuprate supercon-
ductors is contained in the review articles by Kivelson
et al. [27], and by Damascelli et al. [28]. Historically,
the first compelling evidence for both magnetic and
charge order in the cuprates was accomplished in a neo-
dymium codoped compound La2� �x yNd ySr xCuO4
(Nd-LSCO). For y � 0 4. and x � 012. , Tranquada et al.
[29,30] found that the magnetic scattering is not char-
acterized by the two-dimensional (2D) AF wave vec-
tor (1 2 1 2/ /, ), but by IC peaks at the wave vectors
( , )1 2 1 2/ /� � with � � 0118. . Moreover, inspired by
the pioneering works demonstrating that the stag-
gered magnetization undergoes a phase shift of � at the
charge DWs [3–7], the authors found additional
charge order peaks ( , )�2 0� , precisely at the expected
position 2 0 236� � . . Interestingly, this doping corre-
sponds to a local minimum in the doping dependence
of the superconducting temperature Tc in Nd-LSCO
[31], suggesting that the static stripes are responsible
for this anomalous depression of superconductivity.
However, it may well be that the apparent correlation
is entirely accidental and therefore the role of stripes
in superconductivity remains an open question [2].
412 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
Marcin Raczkowski et al.
Unfortunately, in early studies Tranquada et al.
[32] detected only magnetic IC peaks at higher doping
levels x � 015. and x � 0 2. . Nevertheless, systematic
NQR studies of Nd-LSCO revealed the presence of ro-
bust charge stripe order throughout the entire super-
conducting regime of doping 0 07 0 25. .� �x [33]. Also
in a more recent study, both charge and spin
superlattice peaks at x � 015. were found recently
in the neutron diffraction experiments by Wakimoto
et al. [34].
In fact, the reason why static stripes could be de-
tected in this compound is a structural transition from
the low temperature orthorhombic (LTO) to the low
temperature tetragonal (LTT) phase, induced by the
substitution for La ions by isovalent Nd ions. This, in
turn, provides a pinning potential for dynamic stripes
and stabilizes the charge order. Evidence of a similar
pinning potential has also been found both in the �SR
and NQR studies of La2� �x yEu ySr xCuO4
(Eu-LSCO) with y � 0 2. [35,36]. Moreover, the con-
nection between the LTT phase and the appearance
of charge and spin stripe order has been clearly
demonstrated both in the neutron scattering and
x-ray diffraction studies on La2� �x yBa ySr xCuO4
(Ba-LSCO) with y /� 1 8 [37,38]. Finally, static
IC charge ( , )2 2 0� � and magnetic ( , )1 2 1 2/ /� �
peaks have been detected within the LTT phase of
La2�xBa xCuO4 (LBCO) with x /� 1 8 [39]. The po-
sition of the peaks and the established
incommensurability � � 0118. are exactly the same as
those obtained by Tranquada et al. [30] for Nd-LSCO.
Notably, the peaks that correspond to charge order ap-
pear always at somewhat higher temperature than the
magnetic ones, indicating that the stripe order is
driven by the charge instability.
Let us now discuss the experimental evidence of
slowly fluctuating stripes in La2�xSr xCuO4. The
main difference between the Ba and Sr codoped sys-
tem is the fact that the latter undergoes a structural
phase transition from the high-temperature tetragonal
(HTT) phase to the LTO phase. As a consequence, in
the superconducting regime x � 0 06. , the LSCO sys-
tem exhibits purely dynamic magnetic correlations
which give rise to IC peaks at the wave vector
( , )1 2 1 2/ /� � specified in tetragonal lattice units
2�/atetra . In their seminal inelastic neutron scattering
studies, Yamada et al. [40] established a remarkably
simple relation � � x for 0 06 012. .� �x , followed by a
lock-in effect at � � 1 8/ for larger x.
In contrast, in the insulating spin-glass regime of
LSCO x � 0 06. , quasielastic neutron scattering experi-
ments with the main weight at zero frequency demon-
strate that IC magnetic peaks are located at the wave
vectors ( , )1 2 2 1 2 2/ / / /� �� � [41–43]. This phe-
nomenon has often been interpreted as the existence of
static diagonal stripes, even though no signatures of a
charge modulation were observed. Another possible
explanation is the formation of a short ranged spiral
order as its chirality also breaks the translational sym-
metry of the square lattice by a clockwise or anti-
clockwise twist [44]. Remarkably, even though the
spin modulation changes from a diagonal to verti-
cal/horizontal one, i.e., along Cu–O bonds, at x
around 0.06, � follows the doping x reasonably well
over the entire range 0 03 012. .� �x , as shown in
Fig. 1. In fact, just for x � 0 06. , both diagonal
( . )� � 0 053 and vertical/horizontal (� � 0 049. ) IC spin
modulations have been found to coexist [43]. In a
stripe model this corresponds to a constant density of
0.5 (0.7) holes per Cu atom in the DWs in the verti-
cal/horizontal (diagonal) stripe phases, respectively,
because of the difference in Cu spacings in the two ge-
ometries, i.e., a aortho tetra� 2 . In contrast, in the
narrow region 0 02 0 024. .� �x , IC magnetic peaks are
located at the wave vector ( , )1 2 2 1 2 2/ / / /� �� �
with � � x corresponding to a constant charge of one
hole/Cu ion along a diagonal DW [45–47]. However,
below x � 0 02. , this does not hold anymore and the
incommensurability gets locked with the value
� � 0 014. .
Stripe phases-possible ground state of the high-Tc superconductors
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 413
0.12
0.08
0.04
0 0.05 0.10 0.15 0.20 0.25
x
Nd-LSCO
LSCO
LCO
Zn-LSCO
YBCO
ktetra
ktetra
htetra
htetra
hortho
hortho
2
2
1/2
1/2
1/2
1/2
Fig. 1. Summary of experimental data illustrating the
doping dependence of incommensurability � in the
cuprates. Results have been obtained by different groups:
Nd-LSCO (Refs. 29–32); LSCO (Refs. 40–43,45–47);
LCO (Ref. 51); Zn-LSCO (Refs. 52,53); YBCO (Refs.
54,55). In LSCO, � has been defined as a distance from
the IC peak position to the AF wave vector (1 2 1 2/ /, ) ei-
ther in the orthorhombic (x � 006. ) or tetragonal (x � 006. )
notation (see insets), whereas at x � 006. , both definitions
are used due to the coexistence of diagonal and parallel to
the Cu–O bonds spin modulations.
Unfortunately, any concomitant charge ordering
has not yet been detected in LSCO. Nevertheless, by
comparing the data based on the wipeout effect of
63Cu NQR charge order parameter in LSCO with
the ones obtained from charge stripe compounds as
(Nd,Eu,Ba)-LSCO, Hunt et al. [48] concluded that
a similar stripe instability exists in LSCO over
the whole underdoped superconducting region
1 16 1 8/ x /� � . It is also worth mentioning that a
very compiling evidence for its existence has been es-
tablished in the measurements of the in-plane resistiv-
ity and the dynamical infrared conductivity aniso-
tropy [49,50].
Experimental detection of IC magnetic peaks in the
LTO phase of LSCO suggests that the LTT structure
is not essential for the appearance of stripes. This con-
jecture has been confirmed in experiments on the oxy-
gen doped La2CuO4� � (LCO) with the orthorhombic
crystal structure [51]. It is also supported by the
evidence for static IC magnetic peaks in another
orthorhombic compound La2�xSr xCu1�yZn yO4
(Zn-LSCO) with y up to 0 03. , even though attempts to
observe the charge order peaks were unsuccessful
[52,53]. In fact, Zn substitution pins the stripe fluctu-
ations similarly to the rare-earth elements. However,
in contrast to the latter, it does not induce a structural
transition to the LTT phase, but provides randomly
distributed pinning centers that promote meandering
of stripes and correspondingly broadens IC peaks.
An important question is whether charge stripes ap-
pear solely in monolayered lanthanum compounds
or if they are a generic feature of all the cuprates.
The latter conjecture seems to be supported by in-
elastic neutron scattering experiments on bilayered
YBa2Cu3O6� � (YBCO) compounds that have identi-
fied the presence of IC spin fluctuations throughout
its entire superconducting regime [54]. In fact, as the
doped charge is nontrivially distributed between the
CuO2 planes and CuO chains, it is very difficult to
determine the precise doping level x in the CuO2
sheet of YBCO. Nevertheless, systematic studies by
Dai et al. [54] have shown that the incommensur-
ability in YBCO increases initially with doping but it
saturates faster than in LSCO, i.e., already at x � 01.
with the value � � 01. . Unfortunately, there is no any
compelling explanation that would account for such a
different behavior of � in both systems. Eventually,
charge order peaks have been observed in YBCO635.
but in spite of several attempts, no static charge order
could be detected in YBCO65. and YBCO66. so far
[55].
Furthermore, although some neutron scattering ex-
periments have been performed on Bi2Sr2CaCu2O8� �
(BSCCO) sample, the sample has only produced weak
evidence of the IC structure [56]. In contrast, Fourier
transform of the recent STM data has revealed some
IC peaks corresponding to a four-period modulation of
the local density of states along the Cu–O bond direc-
tion, which may imply the existence of stripes [57].
Nevertheless, definite answer pertinent to the appear-
ance of stripes in all the cuprates remains still unset-
tled and further experiments are required to reach an
unambiguous conclusion, even though the summary of
the experimental data illustrating the doping depen-
dence of the incommensurability � in cuprates, de-
picted in Fig. 1, includes an array of compounds.
Tendency towards phase separation is also a start-
ing point to understand the doping evolution of the
electronic structure in LSCO and Nd-LSCO. For ex-
ample, ARPES spectra measured at the X � ( , )� 0
point in LSCO show that even though the data are
solely characterized by a single high binding energy
feature in the insulating regime, upon increasing dop-
ing one observes a systematic transfer of spectral
weight from the high- to the low binding energy part
[58]. Consequently, a well-defined quasiparticle
(QP) peak develops near the optimal doping. In con-
trast, the intensity near the S / /� ( , )� �2 2 point re-
mains suppressed for the entire underdoped regime so
that a QP peak is observed only for x � 015. .
Another peculiar feature of the ARPES band dis-
persion is extensively discussed in the literature saddle
point at the X point, the so-called flat band [59]. As
hole doping increases, the flat band moves mono-
tonically upwards and crosses the Fermi level EF at
x � 0 2. . This is reflected in the enhancement of the
DOS at the chemical potential N( )� observed by
AIPES [60].
The experimental distribution of the photoemission
spectral weight near the X and S points in doped
LSCO has been nicely reproduced using the DMFT
approach for vertical SC stripes obtained within the
Hubbard model [20]. As a consequence of the stripe
order, the obtained spectra along the �–X–M path
were not equivalent to those along the �–Y–M one,
with � � ( , )0 0 and Y � ( , )0 � . Moreover, as in the ex-
periment, the spectral weight along the �–X direction
was suppressed close to the � point and simulta-
neously enhanced at the X point. Furthermore, in the
framework of stripes, the flat QP band near the X
point with a large intensity at the maximum below the
chemical potential � follows from a superposition of
the dispersionless 1D metallic band along the x direc-
tion, formed by holes propagating along the vertical
domain walls, and an insulating band that stems from
the AF domains. In contrast, an AF band at the Y
point is characterized by a high binding energy well be-
low � and consequently the spectral weight at �� al-
414 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
Marcin Raczkowski et al.
most vanishes. Finally, a distinct gap for charge exci-
tations should open at � near the S point. This gap
follows indeed from the stripe structure — while the
system may be metallic along the stripes, i.e., in the
antinodal directions �–X or �–Y, the low-energy exci-
tations should be noticeably suppressed along the
nodal direction �–S crossing all the stripes. This con-
jecture is also supported either by the ED studies [18]
or by the analytical approach based on variational
trial wave function within the string picture [19],
both applied to the t–t
–t
–J model, or by the CPT for
the t–J model [22].
In fact, the low-energy spectral weight of
Nd-LSCO at x � 012. , a model compound for which
the evidence of spin and charge stripe order is the
strongest, is also mostly concentrated in flat regions
along the �–X and �–Y directions, while there is only
little spectral weight along the �–S direction [61]. On
the other hand, ARPES spectra of both LSCO and
Nd-LSCO at x � 015. have revealed not only the pres-
ence of flat bands around the X andY points, but also
the existence of appreciable spectral weight at EF in
the nodal region [62]. While the observation of flat
segments might be directly ascribed to 1D domain
walls [63], detection of nodal spectral weight poses a
formidable task to develop a theory that would de-
scribe the electronic structure resembling the FS of a
fully 2D system because, as it was already stressed
out, the nodal spectral weight is expected to be sup-
pressed in a static SC stripe picture [18–20,22]. In-
deed, the experimentally established FS looks rather
like the one arising from disorder or from dynamically
fluctuating stripes [63].
Alternatively, guided by the CPT results showing
that while the SC stripes yield little spectral weight
near the nodal region, the BC ones reproduce quite
well the nodal segments [22], Zhou et al. [62] have
conjectured that the experimental FS may result from
the coexistence of the SC and BC stripes. Within this
framework, upon increasing doping the BC stripes are
formed at the expense of the SC ones. This scenario is
particularly interesting because it has been shown that
the BC stripe, in contrast to its SC counterpart, en-
hances superconducting pairing correlations [64]. The
relevance of a bond order at the doping level x � 015.
is supported by recent studies of the ARPES spectra in
a system with the BC stripes [65]. These studies have
yielded pronounced spectral weight both in the nodal
and antinodal directions, reproducing quite well the
experimental results in Nd-LSCO and LSCO [62].
Furthermore, the stripe scenario would also explain
the origin of the already discussed two components
seen in the ARPES spectra at the X point near x � 0 05.
[58]. Indeed, the response from the AF insulating re-
gions would be pushed to the high binding energies
due to the Mott gap, whereas the charge stripes would
be responsible for the other component near EF .
Existence of DWs should also give rise to the ap-
pearance of new states inside the charge-transfer gap
that would suppress the shift of the chemical potential
� in the underdoped regime x /� 1 8 where � increases
linearly. Such pinning of � in LSCO was indeed de-
duced from XPS experiments [66]. In contrast, in the
overdoped region with a lock-in effect of �, the num-
ber of stripes per unit cell saturates, doped holes pene-
trate into the AF domains, and consequently � would
move fast with doping in agreement with the experi-
mental data. The picture of broadened stripes and
holes spreading out all over the AF domains above
x /� 1 8 is also indicated by the doping dependence of
the resistivity and the Hall coefficient RH in
Nd-LSCO. Namely, a rapid decrease in the magnitude
of RH for doping level x /� 1 8 at low temperature
provides evidence for the 1D charge transport,
whereas for x /� 1 8, relatively large RH suggests a
crossover from the 1D to 2D charge transport [67]. Al-
together, it appears that the metallic stripe picture
does capture the essence of the low-lying physics for
Nd-LSCO and LSCO systems.
Conversely, it is important to note that so far
no evidence of IC peaks has been detected in any
electron-doped cuprates superconductors. Instead,
the neutron scattering experiments have estab-
lished only commensurate spin fluctuations as in
Nd2�xCe xCuO4 (NCCO), both in the superconduct-
ing and in normal state [68]. Moreover, observation of
such peaks is consistent with the XPS measurements
in NCCO showing that the chemical potential in-
creases monotonously with electron doping [69].
3. Numerical results
In this Section we attempt a systematic investiga-
tion of the properties and relative stability of filled
vertical and diagonal stripes. We shall see that in
spite of the difficulty to stabilize the ground state
with half-filled stripes (one hole per every two atoms
in a DW), the mean-field framework is useful as pro-
viding a generic microscopic description of filled
inhomogeneous reference structures with the filling of
one doped hole per stripe unit cell. Their special sta-
bility rests on a gap that opens in the symmetry bro-
ken state between the highest occupied state of the
lower Hubbard band and the bottom of the so-called
mid-gap bands, i.e., some additional unoccupied
bands lying within the Mott-Hubbard gap that are
formed due to holes propagating along DWs [26].
Here, we extend early HF studies of the filled DWs
[4–7] and determine a phase diagram of the Hubbard
Stripe phases-possible ground state of the high-Tc superconductors
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 415
model with an anisotropic nearest-neighbor hopping t
by varying the on-site Coulomb repulsionU and inves-
tigating locally stable structures for representative
hole doping levels x /� 1 8 and x /� 1 6. We also re-
port the changes in stability of the stripe structures in
the extended Hubbard model due to the next-neighbor
hopping t
and to the nearest neighbor Coulomb inter-
action V. Finally, in order to gain a comprehensive
understanding of the competition between different
types of stripes in a realistic model, we include lattice
degrees of freedom induced by a static Peierls elec-
tron-lattice coupling.
3.1. Extended single-band Hubbard model
The starting point for the analysis of stripe struc-
tures is the extended single-band Hubbard model,
which is widely accepted as the generic model for a
microscopic description of the cuprate superconduc-
tors [70],
H t c c U n n V n nij
ij
i j i
i
i i j
ij
�
� �� � �� �
� �
† , (1)
where the operator ci
† ( )cj creates (annihilates)
an electron with spin � on lattice site i (j), and
n c c c ci i i i i� �
� � � �
† † stands for the electron density.
The hopping tij is t on the bonds connecting nearest
neighbors sites � �i j, and t
for second-neighbor sites,
while the on-site and nearest neighbor Coulomb inter-
actions are, respectively,U and V.
The model can be solved self-consistently in real
space within the HF, where the interactions are de-
coupled into products of one-particle terms becoming
effective mean fields that act on each electron with the
same strength. This approximation basically involves
solving an eigenvalue problem. The obtained wave-
functions form a new potential and hence the Hamil-
tonian for a new eigenvalue problem. Typically, the
new potential is chosen as some linear combination of
the current and preceding potential. The iterations are
continued until the input and output charge density
and energy do not change within some prescribed ac-
curacy. The most significant drawback of this method
is that it neglects correlations. Electron correlation
changes the system properties and manifests itself in
the decrease of the ground state energy. The difference
between the energy of the exact ground state and the
energy obtained within the HF is thus called the
correlation energy. It arises from the fact that an
electron’s movement is correlated with the electrons
around it, and accounting for this effect lowers further
the energy, beyond the independent electron approxi-
mation.
We do not consider noncollinear spin configura-
tions, and use the most straightforward version of the
HF with a product of two separate Slater determi-
nants for up and down spins, whence,
n n n n n n n ni i i i i i i i� � � � � � � �� � � � �
� �� �� . (2)
A similar decoupling is performed for the nearest
neighbor Coulomb interaction. Calculations were
performed on 12 12� (16 16� ) clusters for x /� 1 6
( )x /� 1 8 with periodic boundary conditions, and we
obtain stable stripe structures with AF domains of
width five atoms for x /� 1 6 and seven atoms for
x /� 1 8. Typical solutions at x /� 1 8 are shown in
Fig. 2 with the local hole density,
� � �
� � �� �n n nhi i i1 , (3)
scaled by the diameter of the black circles and the
length of the arrows being proportional to the ampli-
tude of local magnetization density,
� � � �
�� �S n ni
z
i i
1
2
| | . (4)
These structures possess nonmagnetic DWs with
enhanced hole density which separate AF domains
having hole density almost unchanged with respect to
the undoped case. Note that the AF sites on each side
of the DWs have a phase shift of �.
In order to appreciate better the microscopic rea-
sons of such arrangement let us consider a small clus-
ter consisting of three atoms filled by two electrons
and one hole (with respect to half-filling with the
416 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
Marcin Raczkowski et al.
lx
ly
Fig. 2. Vertical site-centered (VSC) and diagonal site-cen-
tered (DSC) stripe phases as found for U/t � 5 at hole
doping x /� 1 8. The length of arrows is proportional to
the magnetization � �Si
z and the hole density � �nhi is scaled
by the diameter of black circles.
electron density n � 1 per site). For simplicity we as-
sume that the electrons are confined to the considered
cluster owing to large Coulomb interaction U t�� ,
and we do not take into account any interactions with
the AF background. There are two possible candidates
for the ground state. The first one corresponds to a
hole added to three atoms of a single AF domain in
which, if we suppose that a �-spin electron is replaced
by a hole, the two remaining �-spin electrons can be
found in one of three allowed configurations: { , , }� �0 ,
{ , , }� � 0 , and { , , }0� � (the other configurations are ex-
cluded by the Pauli principle). Hence, this polaronic
state gives the total energy,
E tP �
2 , (5)
and the Coulomb interactionU does not contribute.
A different situation is obtained when a hole oc-
cupies instead a DW separating two AF domains.
Delocalization leads then to similar three configu-
rations to those obtained above with opposite spins:
{ , , }� �0 , {� �, ,0}, and { , , }0� � , but in addition, three con-
figurations with one doubly occupied site {��, ,0 0},
{ , , }0 0�� , and { , , }0 0�� , can be reached as excited states
which cost Coulomb energyU. Moreover, three other
configurations with interchanged �- and �-spins are
then also accessible via the decay of double occupan-
cies: {� �, ,0 }, {� �, ,0}, and {0, ,� �}. In the regime of large
U, the total energy in the ground state can be found in
a perturbative way, and as a result one obtains,
E t
t
US �
2
4 2
. (6)
Therefore, the Hilbert space for the latter solitonic
solution is larger and one finds that this solution is al-
ways more stable than the polaronic one [26]. The ar-
gument applies also to 2D systems, where the DWs
are more stable than the lines of polarons in an AF
background.
We compare the stability of such nonmagnetic SC
domain walls with the BC stripe phases in which DWs
are formed by pairs of magnetic atoms, as obtained
by White and Scalapino [12] (cf. Fig. 3). In the
three-band model, SC (BC) stripes correspond to
DWs centered at metal (oxygen) sites, respectively
[71–74].
3.2. Effect of hopping anisotropy
We begin by setting t
� 0 and V � 0 with the goal
of elucidating the effects of hopping anisotropy on the
stripes. This is motivated by the fact that the first de-
tection of static stripes in both charge and spin sectors
was accomplished in Nd-LSCO [29] indicating that
rare-earth elements doping is in some way helpful for
pinning the stripe structure. Indeed, it produces a
structural transition in the system from the LTO to
LTT phase [75]. Both phases involve a distortion of
the CuO2 plane by rotation of the CuO6 octahedra.
In the LTO phase the tilt axis runs diagonally within
the copper plane, such that all the oxygen atoms are
displaced out of the plane. Conversely, in the LTT
phase this rotation takes place around an axis oriented
along the planar Cu
O bonds. Therefore, oxygen at-
oms on the tilt axis remain in the plane, while the ones
in the perpendicular direction are displaced out of the
plane. This provides a microscopic origin for in-plane
anisotropies — the Cu–Cu hopping amplitude t de-
pends on the Cu
O bond and it is isotropic in the LTO
phase and anisotropic in the LTT one. For a physical
tilt angle of order 5°, the relative anisotropy taking
t ty x� ,
�t
x y
y
t t
t
�
| |
, (7)
is weak and amounts to �t � 0 015. [76,77]. The direc-
tion with a larger hopping amplitude coincides with
the direction of a stronger superexchange coupling J.
The possible relationship between this anisotropy
and the onset of stripe phases has been intensively
studied within anisotropic Hubbard (t tx y� ) or t-J
(t tx y� , J Jx y� ) models by means of various tech-
niques: unrestricted HF approach [76], DMRG [77],
and QMC method [23]. The in-plane anisotropies
might also be represented theoretically by on-site po-
Stripe phases-possible ground state of the high-Tc superconductors
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 417
lx
ly
Fig. 3. Vertical bond-centered (VBC) and diagonal
bond-centered (DBC) stripe phases as found for U/t � 5
at hole doping x /� 1 8. The meaning of the arrows and
black circles as in Fig. 2.
tentials as in the QMC study by Riera [24]. All these
investigations have shown a pronounced tendency to
forming stripe phases, which manifests itself by the re-
duction of their energy [76,77], accompanied by the
appearance of IC peaks in the spin and charge struc-
ture factor [23,24]. It appears that a finite anisotropy
of the next-nearest neighbor hopping term t
might
play a role in stabilizing diagonal incommensu-
rate peaks observed in the spinglass phase of LSCO
( . . )0 02 0 06� �x [41–43,45–47]. Indeed, although the
LTO phase is usually considered as isotropic, which
is the case for nearest neighbor hopping and inter-
action, a different length of the orthorhombic axes im-
plies the need for an anisotropic t
parameter. Exact
diagonalization studies incorporating such anisotropy
have shown that it strongly amplifies hole correlations
along one direction and suppresses them along the
other, resulting in a 1D pattern of holes [78].
It turns out, however, that the variation of the hop-
ping anisotropy �t (7) has only a little visible effect
on the local hole density,
n l n nh x l lx x
( ) ,( , ), ( , ),�
� � �� �1 0 0 (8)
shown in Fig. 4 as a function of the x-direction coor-
dinate lx for a given y-direction coordinate ly � 0,
even at the unrealistically large anisotropy level
�t � 0 22. , corresponding to t /tx � 11. and t /ty � 0 9. .
Similarly, the anisotropy does not modify the modu-
lated magnetization density,
S l n nx
l
l l
x
x x
( ) ( ) ,( , ), ( , ),�
�
�� �1
1
2 0 0
with a site dependent factor ( )
1 lx compensating
modulation of the staggered magnetization density
within a single AF domain.
In contrast, the strong effect of finite anisotropy �t
(7) is clearly demonstrated by variation of the expec-
tation values of the bond hopping terms along the x-
and y-directions,
E l t c ct
x
x x l lx x
( ) ,
( , ),
†
( , ),
�
� � �� �
0 10
h. c. (10)
E l t c ct
y
x y l lx x
( ) .
( , ),
†
( , ),
�
� � ��
0 1
h. c. (11)
These features are seen in Fig. 4. For the VSC stripes
one finds a large anisotropy in the values of the ki-
netic energies (10) and (11), which becomes espe-
cially pronounced beside the stripes, and is strongly
reinforced by the hopping anisotropy. Therefore, tak-
ing into account that the hopping between two differ-
ent charge densities is favored over motion between
equal densities, one should expect that transverse
charge fluctuations will always tune the direction of
DWs along the weaker hopping direction in the
anisotropic model. Analogous conclusion based on
Fig. 5 might be drawn concerning the orientation of
the VBC stripes.
Regarding diagonal stripes, although a finite aniso-
tropy in hopping is also reflected in the kinetic energy
anisotropy, a system with either the DSC or DBC
stripe pattern becomes topologically frustrated and
consequently may gain less kinetic energy compared to
a system with vertical stripes, taking a full advantage
of the hopping anisotropy (cf. Tables 1 and 2).
418 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
Marcin Raczkowski et al.
0 4 8 12 16
0.1
0.2
0.3
0.4
0 4 8 12 16
0.1
0.2
0.3
0.4
0 4 8 12 16
–0.4
–0.2
0
0.2
0.4
0 4 8 12 16
–0.4
–0.2
0
0.2
0.4
0 4 8 12 16
–1.2
–1.0
–0.8
–0.6
–0.4
0 4 8 12 16
–1.2
–1.0
–0.8
–0.6
–0.4
0 4 8 12 16
lx lx
–0.8
–0.7
–0.6
–0.5
–0.4
0 4 8 12 16
–0.8
–0.7
–0.6
–0.5
–0.4
n
h
S
E
/t tx
E
/t ty
Fig. 4. Local hole n lh x( ) (top) and magnetization S lx
( )
(second row) density; kinetic energy E lt
x
x( ) (third row)
and E lt
y
x( ) (bottom) projected on the bonds in the
x-(y)-directions, respectively, of the VSC (left) and DSC
(right) stripe phases shown in Fig. 2 (open circles) as
well as of the ones obtained in the anisotropic model with
t /tx y � 122. (filled circles). For clarity, the latter are
shifted by one lattice constant from the origin of the coor-
dinate system.
Table 1. Site-normalized ground-state energy Etot, kinetic
energy ( , )E Et
x
t
y , and potential energy EU in the isotropic
Hubbard model with U/t � 5 and x /� 1 8 as obtained for
different stripe phases: vertical site-centered (VSC), diago-
nal site-centered (DSC), vertical bond-centered (VBC) and
diagonal bond-centered (DBC). In the HF, both types of
vertical stripes are degenerate.
Et
x/t Et
y/t EU/t E /ttot
VB(S)C –0.6753 –0.6147 0.4900 –0.8000
DBC –0.6375 –0.6375 0.4726 –0.8024
DSC –0.6368 –0.6368 0.4696 –0.8040
Table 2. The same as in Table 1 but with the hopping ani-
sotropy �t � 022. .
Et
x/t Et
y/t EU/t E /ttot
DBC –0.8143 –0.4807 0.4815 –0.8135
DSC –0.8098 –0.4836 0.4793 –0.8141
VB(S)C –0.8304 –0.4776 0.4938 –0.8142
The effect of an increasing anisotropy illustrates
the phase diagram shown in Fig. 6 determined by va-
rying U and the ratio t /tx y of the nearest-neighbor
hoppings in the x- and y-directions, while maintaining
constant t t t /x y� �( ) 2. We observe the generic
crossover from vertical to diagonal stripes with in-
creasing Coulomb interaction reported in early HF
studies [4–7]. The transition from the VSC to DSC
stripes appears in the isotropic case at U/t � 41. for
x /� 1 8, and at a higher value U/t � 4 6. for x /� 1 6
(cf. Fig. 6,a). The corresponding phase boundary be-
tween the VBC and DBC stripes is shifted towards
stronger Coulomb interaction and occurs atU/t � 4 4.
(5.0) for x /� 1 8 (x /� 1 6), respectively (cf.
Fig. 6,b).
The results shown in Fig. 6 have a simple physical
interpretation. Stripe phases occur as a compromise
between, on the one hand, the AF interactions be-
tween magnetic ions and the local Coulomb interac-
tions responsible for charge localization, and the ki-
netic energy of doped holes which on the contrary
favors charge delocalization. The kinetic energies in
Table 1 show further that the vertical stripes are more
favorable for charge dynamics. This result, which is
not immediately obvious, has however a straightfor-
ward origin. Namely, the HF always leads to a large
spin polarization since it is the only way to minimize
the on-site Coulomb repulsion. Indeed, removal of a
�-spin electron at site i leads to relaxation of the
�-spin electron energy level at this site. As a conse-
quence, an alternating on-site level shift develops
yielding an energetical motivation for the symmetry
breaking and forming the AF order.
However, the renormalization of the double occu-
pancy energy involves a strong reduction of the ki-
netic energy in the �-spin channel between site i and
its neighboring sites, as an electron incoming into this
site encounters a high energy potentialU ni� �� . There-
fore, in the HF approximation we shall be able to
identify dynamically favorable stripe patterns only by
Stripe phases-possible ground state of the high-Tc superconductors
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 419
0 4 8 12 16
0.1
0.2
0.3
0.4
0 4 8 12 16
0.1
0.2
0.3
0.4
0 4 8 12 16
–0.4
–0.2
0
0.2
0.4
0 4 8 12 16
–0.4
–0.2
0
0.2
0.4
0 4 8 12 16
–1.2
–1.0
–0.8
–0.6
–0.4
0 4 8 12 16
–1.2
–1.0
–0.8
–0.6
–0.4
0 4 8 12 16
lx lx
–0.8
–0.7
–0.6
–0.5
–0.4
0 4 8 12 16
–0.8
–0.7
–0.6
–0.5
–0.4
n
h
S
E
/t tx
E
/t ty
Fig. 5. The same as in Fig. 4 but for the BC stripe phases
shown in Fig. 3.
3.5 4.0 4.5 5.0 5.5 6.0
U/t
0.90
0.95
1.00
1.05
1.10
4.0 4.5 5.0 5.5 6.0
U/t
0.90
0.95
1.00
1.05
1.10
a b
DSC
VSC
DBC
HBCHSC
VBC
t
/t x
Fig. 6. Phase diagrams for stable: (a) site-centered (SC),
and (b) bond-centered (BC) stripe structures obtained in
the anisotropic Hubbard model on a 16 16� cluster for
doping x /� 1 8 (solid lines) and on a 12 12� cluster for
x /� 1 6 (dashed lines). Parameters: t� � 0, V � 0.
comparing appropriate local magnetization densities.
For example, charge fluctuations occur more readily
in the VSC stripe geometry presumably due to their
greater overall width indicating weaker correlation ef-
fects (cf. Fig. 4). This explains their stability at small
U where the consequent cost in potential energy EU
becomes insignificant. By contrast, the DSC stripes
are narrower having larger hole density along non-
magnetic DWs. Moreover, magnetization density of
their nearest neighbor sites is markedly enhanced as
compared to the corresponding VSC stripe magnetiza-
tion, as shown in Fig. 4 and in Table 3. The former
also illustrates that the bonds connecting DWs with
their nearest neighboring sites perpendicularly to the
walls, have the main contribution to the kinetic en-
ergy gain, in fact suppressed here by larger spin polar-
ization. Taken together, the above features are re-
flected in a more localized character of the DSC
stripes, with a lower net double occupancy and hence
a more favorable on-site energy EU (cf. Table 1). This
clarifies the mechanism of the transition from the VSC
to DSC stripes with increasingU.
Table 3. Local hole � �nhi and magnetization � �Si
z density
of the site-centered stripes shown in Fig. 2, all labeled by
decreasing hole density in the x-direction. In parentheses
the values for the extended hopping model with t t� � / .015
are given.
i 1 2 3 4 5
VSC
� �nhi 0.364 0.234 0.067 0.014 0.006
(0.378) (0.234) (0.060) (0.013) (0.006)
� �Si
z 0.000 0.222 0.348 0.381 0.384
(0.000) (0.234) (0.357) (0.382) (0.384)
DSC
� �nhi 0.388 0.193 0.070 0.032 0.020
(0.405) (0.195) (0.066) (0.028) (0.017)
� �Si
z 0.000 0.262 0.352 0.373 0.380
(0.000) (0.272) (0.360) (0.377) (0.382)
Turning now to the analogous crossover between
the BC stripes, we shall again compare local hole and
magnetization densities on and around their DWs. In
contrast to the SC case, a VBC stripe phase possesses
larger hole density along DWs, as illustrated in Fig. 3
and Table 4, suggesting that it is more localized than
the DBC one. Nevertheless, a better renormalization
of the double occupancy energy EU by the latter (cf.
Table 1) follows from a stronger spin polarization not
only of the DW atoms but also their nearest neighbors
(cf. Fig. 3 and Table 4). This enhancement is directly
responsible for a substantial reduction of the kinetic
energy along bonds joining these atoms. Correspond-
ingly, it accounts for a crossover from the DBC to
VBC stripes in the smallU regime when the larger ki-
netic energy gain becomes crucial.
Table 4. The same as in Table 3 but for the bond-centered
stripes. VBC stripe is unstable in the extended hopping
model with t /t� � 015. — data in parentheses.
i 1 2 3 4
VBC
� �nhi 0.326 0.136 0.030 0.007
� �Si
z 0.118 0.301 0.371 0.384
DBC
� �nhi 0.314 0.115 0.047 0.023
(0.323) (0.110) (0.046) (0.021)
� �Si
z 0.145 0.322 0.365 0.378
(0.155) (0.333) (0.368) (0.380)
We would like to emphasize that the above transi-
tion between different types of stripe phases is not an
artefact of the HF and occurs also between filled
stripes obtained within more realistic approaches in-
cluding local electron correlations. Indeed,
slave-boson studies of the Hubbard model at the dop-
ing x /� 1 9 have established that the transition from
the filled VSC to DSC stripe phase appears at the
value U/t � 5 7. , being much higher than that pre-
dicted by the HF, which yieldsU/t � 3 8. [14]. In this
method, enhanced stability of the VSC stripes follows
from an additional variational parameter per each site
di , reducing the on-site energy without a strong sup-
pression of the kinetic energy. Remarkably, the total
energy difference between the vertical SC and BC
stripes at both doping levels is comparable to the accu-
racy of the present calculation. Such degeneracy was
also reported in the HF studies of the charge-transfer
model [71]. However, when electron correlations are
explicitly included the BC stripes are more stable at
and above x /� 1 8 doping [21,79].
3.3. Effect of the next-neighbor hopping t
We now turn to the effect of a next-neighbor hop-
ping t
on the relative stability of the stripes. There are
numerous experimental and theoretical results which
support the presence of finite t
in the cuprates. For ex-
ample, recent slave-boson studies have revealed that
the phenomena of the half-filled vertical stripes in
LSCO requires a finite next-neighbor hopping
t t
/ .� 0 2 [16].
Let us pause now for a moment to clarify the influ-
ence of t
on the DOS as well as on the FS using the
electronic band which follows from a simple tight-
binding model [80],
E t k k t k kx y x y( ) (cos cos ) cos cos .k �
�
2 4 (12)
420 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
Marcin Raczkowski et al.
By the reduction from the CuO2 multiband model
to an effective single-band model it has been found
that t � 0 and t
� 0 for hole doped system, and t � 0
and t
� 0 in electron doped system [70]. Although an
accidental cancellation of the various contributions re-
sults in almost perfect electron-hole symmetry of the
nearest neighbor hopping t, the next-neighbor hop-
ping t
asymmetry appears owing to the fact that the
dominant contribution to the latter comes from a di-
rect O–O hopping tpp in the case of a hole hopping.
On the contrary, an electron hopping follows from a
third order Cu � O � O � Cu process, being there-
fore dominated by the Cu–O hopping element tpd .
In the noninteracting limit the role of t
is to shift
the van Hove singularity away from the middle of the
band, either to higher or to lower energy depending on
its sign [80]. Figure 7 shows the tight-binding DOS,
centered at � 0 with the condition N d( )� � 0,
and the occupied states at the doping x /� 1 4. In the
hole-doped case, with the vacuum as the zero electron
state, the van Hove singularity lies in the lower part
of the band. Conversely, in the case of electron dop-
ing, with the vacuum as the zero hole state, the van
Hove singularity is shifted towards higher energy part
of the band, unoccupied by holes.
Apart from breaking the electron-hole symmetry,
the extra parameter t
modifies the shape of the FS of
the free electrons and indeed it becomes more consis-
tent with the FS topology seen by ARPES [59,81,82].
In the electron-doped system NCCO, the low-energy
spectral weight at the doping x � 0 04. is concentrated
in small electron pockets around the (��,0) and
(0,� �) points. Upon increasing doping, one observes
both the modification of the hole pockets and the
emergence of new low-lying spectral weights around
( , )� �� �/ /2 2 . Finally, at x � 015. the FS pieces
evolve into a large holelike curve centered at
M � ( , )� � . In contrast, it has been observed that in
the lightly doped regime( . )x � 0 03 doped holes in
LSCO enter into the hole pockets around
(� �� �/ /2 2, ) points [83], implying that the FS is
holelike and centered at the M point. However, in the
heavy overdoped regime x � 0 3. it converts into the
electronlike FS around the � � ( , )0 0 point.
Figure 8,a shows that the model (12) with t
� 0 has
a nested square FS at half-filling which becomes
electronlike and shrinks around the � point upon hole
doping. However, negative t
�
0 3. removes the FS
nesting at half filling, and the FS expands in the
( , )�k 0 and ( , )0 � k directions, while contracts along
the nodal ( , )k k� and ( , )�k k directions due to a large
gradient dE/dk along the latter. Indeed, the
eigenenergy map, illustrated in Fig. 9,a, has in this
case a valleylike character with a minimum at the �
point. Therefore the FS turns into a holelike one with
experimentally observed arcs (cf. Fig. 8,a). In con-
trast, the nearest neighbor hopping t
with the same
sign as t interchanges the expansion- and contraction
directions which results in the electronlike FS.
Stripe phases-possible ground state of the high-Tc superconductors
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 421
0.80.8
0.60.6
0.40.4
0.20.2
00
N
(
)(
1
/t
)
�
–4 –4–2 –20 02 24 46 6
�/t �/t
a b
Fig. 7. Effect of the next-neighbor hopping t t� � / .03 on
the noninteracting 2D DOS at the doping x /� 1 4: (a)
hole doping (t � 1); (b) electron doping (t � 1). Dotted
line shows the Fermi energy in the undoped case, whereas
the gray area shows the states occupied by either electrons
(a) or holes (b).
k
/ y
�
1.01.0
00
–1.0–1.0
k /x �
–1.0 0 1.0
k /x �
–1.0 0 1.0
a b
Fig. 8. FS obtained in the tight binding model at the dop-
ing x /� 1 4: (a) hole doping with t � 1 and: t� � 03.
(black solid line), t� � 03. (gray solid line), and t� � 0
(dashed line); (b) electron doping with t � 1 and: t� � 03.
(black solid line), t� � 03. (gray solid line), and t� � 0
(dashed line). The long-dashed line in both panels corre-
sponds to the undoped case with t� � 0. The excessively
large value of | | .t� � 03 as compared to LSCO was chosen
only for more clarity of the figure.
a b
3
0
–3
–6
6
3
0
–3
–� –�
–� –�
0 0
0 0
� �
� �
kx kx
ky
ky
E(k)/tE(k)/t
Fig. 9. Eigenenergy maps of the tight-binding model (12)
with t t� � / .03 as obtained for: (a) hole doping (t � 1);
(b) electron doping (t � 1).
Regarding the electron doped case with t �
1,
shown in Fig. 8,b, positive t
� 0 3. (dark solid line)
also leads to the appearance of arc segments of the FS
and makes it closer to experimental observations. In
this case, however, the minimum energy is found at
the M point, as illustrated in Fig. 9,b. It should be
noted in passing that this FS describes the same situa-
tion as the one obtained with t � 1 and t
� 0 3. , indi-
cated by the gray solid line in Fig. 8,a. In fact, the
sign of t is less important and turns out to be equiva-
lent to the ( , )� � shift of the momentum without
changing the corresponding eigenvalues. Conse-
quently, in order to imitate the effect of hole and elec-
tron doping it is sufficient to study the Hamiltonian
(1) only below half-filling and the alternation bet-
ween two regimes is possible by the particle-hole
transform,
c ci
i
i
† ( ) ,�
1 (13)
mapping the model (12) with t
� 0 onto the one with
t
� 0. Therefore, in order to avoid any further confu-
sion concerning the signs of t and t
in Eq. (12), we
set hereafter t to be positive; then a negative t
( / )t t
� 0 corresponds to hole doping, whereas a posi-
tive one (t t
�/ 0) indicates electron doping.
The remarkable differences of the electronic struc-
ture due to the broken hole-electron symmetry by t
,
result in different phase diagrams of LSCO and
NCCO. In the former the long-range AF order is al-
ready suppressed in the lightly doped regime x � 0 03. ,
while in the latter the antiferromagnetism is known to
be quite robust at increasing electron doping, hence
only commensurate spin fluctuations are observed at
x � 015. [68]. The robustness of the commensurate spin
fluctuations in the electron doped regime is consistent
with the ED studies of the t–t
–J [84,85] and t–t
–t
–J
[86,87] models. It is also supported by the conclusion
that a negative t
promotes incommensuration at a
lower doping level than a positive one, reached using
the QMC technique applied to the extended Hubbard
model [88]. Finally, the XPS measurements in NCCO
show that the chemical potential monotonously in-
creases with electron doping [69], whereas its shift is
suppressed in the underdoped region of LSCO [66].
These data have been nicely reproduced in Ref. 86 for
both compounds, except for the low doping regime of
LSCO where stripes are expected. All these numerical
and experimental results indicate that doped electrons
might selforganize in a different way than holes do —
in the latter case DWs are formed. Nevertheless, sta-
ble diagonal stripes with one doped electron per site in
a DW have been obtained in the slave-boson studies of
a more realistic extended three-band model [73], so
the problem is still open.
Turning back to the competition between stripes in
a doped system, Fig. 10,a shows that negative t
stabi-
lizes the DSC stripes, whereas positive t
favors the
VSC ones, within the parameter range where t
does
not drive a stripe melting. Analogous crossover from
vertical stripes at small | |t
to more complex in shape
diagonal ones at t t
�
/ .01 and t t
�
/ .0 2 has been
found in other HF studies [89]. The explanation is
contained in Table 5: negative t
gives a positive ki-
netic energy contribution, which is much more readily
minimized by the diagonal charge configuration. In-
deed, despite the solitonic mechanism yielding a no-
ticeable kinetic energy loss due to the transverse hop-
ping t t
�
/ .015, the overall kinetic energy loss in the
case of DSC stripes along the diagonal ( )11 and
antidiagonal ( )11 directions is smaller than the corre-
sponding one for the VSC stripe. A more careful anal-
ysis shows that hole propagation along the DSC stripe
results in a contribution having the same sign as t
.
However, it is entirely canceled by the ones coming
from diagonal bonds of the AF domains so that
Et
x y
�
� � 0.
One observes further that positive t
reduces the
anisotropy between the kinetic energy gains in the x-
and y-directions for the VSC stripes, and makes their
sum more favorable, while negative t
has the opposite
effect. For the DSC stripes the total kinetic energy
also follows the same trend. The explanation of these
results follows from the reinforcement of stripe order
by a negative t
(cf. values in parenthesis in Table 3),
which suppresses the hopping contributions, and its
smearing out by positive t
where hopping is enhanced.
These trends agree with the earlier finding within the
DMFT that the VSC stripe phase is destabilized by
kink fluctuations [21]. However, this stripe (dis)or-
dering tendency also leads to a considerably greater
change in the Coulomb energy EU , listed in Table 5,
for the DSC than for VSC stripes, which contributes
significantly to the predominance of the former struc-
ture for negative t
. In fact, it follows from the in-
422 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
Marcin Raczkowski et al.
3.5 4.0 4.5 5.0 5.5 6.0 6.0
U/t
–0.2
–0.1
0
0.1
0.2
3.5 4.0 4.5 5.0 5.5
U/t
–0.2
–0.1
0
0.1
0.2
a b
DSC
V(H)SC
DBC
V(H)BC
t
/t’
Fig. 10. Phase boundaries for: (a) site-centered, and (b)
bond-centered stripes as obtained in the extended Hubbard
model with the next-neighbor hopping t� for doping
x /� 1 8 (solid line) and x /� 1 6 (dashed line).
crease of hole density within the nonmagnetic stripes
and the magnetization density enhancement within
the AF domains (cf. Table 3).
Like their SC counterparts, DBC stripes are also
stabilized by negative t
resulting in a phase diagram
shown in Fig. 10,b. In this case, expelling holes from
the AF domains enhances not only magnetization of
their atoms but also increases magnetic moment of the
hole rich DWs, as illustrated in Table 4. This enhance-
ment must, however, strongly suppress the dominant
transverse kinetic energy gain of the VBC stripes.
Therefore, the latter are already unstable at
t t
�
/ .015.
It is worth noting that a finite diagonal hopping t
should directly affect the competition between the
d-wave pairing correlations and stripes. Indeed, a sys-
tematic comparison of stripe and pairing instabilities
within the DMRG framework has shown that when
the stripes are weakened by positive t
, the latter are
strongly enhanced due to increasing pair mobility
[13]. This effect is accompanied by a simultaneous en-
hancement of the AF correlations [87]. Conversely,
negative t
reinforcing a static stripe order results in
the suppression of pair formation in the underdoped
region, as found both in the DMRG technique and
Variational Monte Carlo (VMC) [90]. However, the
enhanced pairing correlation, attributed to the change
of the FS topology in LSCO, has been obtained in the
optimally doped and overdoped regimes [91].
3.4. Effect of the nearest-neighbor Coulomb
interaction V
We now investigate the changes in the stripe stabil-
ity due to either repulsive (V � 0) or attractive
(V � 0) nearest-neighbor Coulomb interaction, which
give the phase boundaries between the VSC and DSC
stripe phases shown in Fig. 1,a. We have found that
realistic repulsive V favors the latter. The tendency
towards the DSC stripe formation at V � 0 is primar-
ily due to a large difference between charge densities
at the atoms of the DW itself and at all their near-
est-neighbor sites, a situation which is avoided in the
case of VSC stripe phases (cf. Fig. 2). Consequently,
the former optimize better the repulsive potential en-
ergy component EV , as shown by the data reported in
Table 4. Similarly, the fact that the nearest-neighbor
interaction V is well minimized only by inhomo-
geneous charge densities makes the DBC stripe phase
more favorable than the VBC one, as shown in
Fig. 11,b. While this is also the leading mechanism for
both diagonal stripe suppression atV � 0, the asymme-
try of the curve in Fig. 11 arises from the fact that the
lower U values at the transition favor the higher ki-
netic energy contributions available for the vertical
stripes.
Stripe phases-possible ground state of the high-Tc superconductors
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 423
3.5 4.0 4.5 5.0 5.5 6.0 6.0
U/t
–0.4
–0.2
0.0
0.2
0.4
V
/t
3.5 4.0 4.5 5.0 5.5
U/t
–0.4
–0.2
0.0
0.2
0.4
a b
DSC
V(H)SC
DBC
V(H)BC
Fig. 11. Phase diagrams for the site-centered (a) and
bond-centered (b) stripes obtained in the extended Hub-
bard model with the nearest-neighbor Coulomb interaction
V for doping x /� 1 8 (solid line) and x /� 1 6 (dashed
line).
Table 5. Energies per site: ground-state energy Etot, kinetic energy contributions for the bonds along (10) Et
x, (01) Et
y,
(11) Et
x y
�
� and (11) Et
x y
�
� directions, as well as the potential energy EU, all normalized per one site, in the extended hop-
ping Hubbard model with U/t � 5 and x /� 1 8. VBC stripe is unstable at t t� � / .015.
t t� � �/ .015 Et
x/t Et
y/t Et
x y/t�
� Et
x y/t�
� EU/t E /ttot
VSC –0.15 –0.6876 –0.5886 0.0140 0.0140 0.4778 –0.7704
DBC –0.15 –0.6279 –0.6279 0.0000 0.0183 0.4562 –0.7813
DSC –0.15 –0.6275 –0.6275 0.0000 0.0188 0.4533 –0.7829
DBC 0.15 –0.6442 –0.6442 0.0000 –0.0282 0.4883 –0.8283
DSC 0.15 –0.6437 –0.6437 0.0000 –0.0279 0.4855 –0.8298
VB(S)C 0.15 –0.6612 –0.6372 –0.0169 –0.0169 0.4997 –0.8325
Table 6. Energies per site: ground-state energy Etot, ki-
netic energy ( , )E Et
x
t
y and potential energy ( , )E EU V com-
ponents in the extended Hubbard model with the nea-
rest-neighbor Coulomb interaction V for U/t � 5 and
x /� 1 8.
V/t Et
x/t Et
y/t EU/t EV/t E /ttot
DBC –0.4 –0.6322 –0.6322 0.4626 –0.6194 –1.4212
DSC –0.4 –0.6319 –0.6319 0.4602 –0.6193 –1.4229
VB(S)C –0.4 –0.6655 –0.6083 0.4749 –0.6251 –1.4240
VB(S)C 0.4 –0.6838 –0.6214 0.5063 0.6207 –0.1782
DBC 0.4 –0.6424 –0.6424 0.4829 0.6176 –0.1843
DSC 0.4 –0.6412 –0.6412 0.4789 0.6171 –0.1864
However, it has been argued based on the results
obtained using the SBA that an increasing repulsive
interaction V favors half-filled vertical stripes, hence
the latter take over at V/t � 01. in the parameter re-
gime of x /� 1 8 andU/t � 10 [15]. This finding could
naturally explain the appearance of filled diagonal
stripes in the nickelates, provided that they were char-
acterized by a small V term, and the stability of the
half-filled vertical ones in the Nd-codoped cuprates
due to possibly larger value ofV. It is also worth men-
tioning other HF [92] and variational [93] studies in
which a variety of intriguing stripe phases, coexisting
at V/t � 15. with charge order, has been found in a
broad doping region.
3.5. Effect of the lattice deformations
So far, we have demonstrated that a finite aniso-
tropy of the transfer integral t can tip the balance be-
tween vertical and diagonal stripes. Here we will
show that such anisotropy naturally emerges in a
doped system with DWs, described by a single-band
Peierls-Hubbard Hamiltonian,
H t u c c U n n K uij
ij
ij i j i
i
i ij
ij
�
� �� � �� �
� �
( ) .† 1
2
2
(14)
In this model we keep only the leading term and as-
sume a linear dependence of the nearest neighbor hop-
ping element tij on the lattice displacements uij ,
t u t uij ij ij( ) ( )� �0 1 � . (15)
Furthermore, we include the elastic energy � K
which allows to investigate the stability of the system
with respect to a given lattice deformation and to de-
termine the equilibrium configuration. For conve-
nience, we parametrize the electron-lattice coupling
with a single quantity, � �� 2
0t /K, with the parame-
ter values K/t0 18� and � � 3 assumed following the
earlier HF studies [26]. As previously, we focus on
the doping x /� 1 8 (x /� 1 6) and present the results
of calculations performed on 16 16� ( )12 12� clusters,
respectively, with periodic boundary conditions.
These calculations have shown that such clusters give
the most stable filled stripe solutions for the selected
doping levels. The model (14) was solved self-consis-
tently in real space within the HF (2). Thereby, we
used an approximate saddle-point formula for the
equilibrium relation between the actual deformation
424 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
Marcin Raczkowski et al.
0 4 8 12 16
0.1
0.2
0.3
0.4
0 4 8 12 16
0.1
0.2
0.3
0.4
0 4 8 12 16
–0.4
–0.2
0
0.2
0.4
0 4 8 12 16
–0.4
–0.2
0
0.2
0.4
0 4 8 12 16
0.08
0.10
0.12
0.14
0.16
0 4 8 12 16
0.08
0.10
0.12
0.14
0.16
0 4 8 12 16
lx
–1.4
–1.2
–1.0
–0.8
–0.6
–0.4
0 4 8 12 16
lx
–1.4
–1.2
–1.0
–0.8
–0.6
–0.4
n
h
E
/t t
0
x
S
u
;D
x(0
) ,
Å
Fig. 12. Local hole n lh x( ) (top) and magnetization S lx
( )
(second row) density; fractional change of the length for
the bonds to the right nearest-neighbor along the x-direc-
tion ux
( )0 (circles) and double occupancy D lx( ) (squares)
(third row), as well as the kinetic energy E lt
x
x( ) projected
on the bonds in the x-direction (bottom) of the VSC
(left) and DSC (right) stripe phases, as obtained in the
Peierls-Hubbard model (14) with U/t � 5,
� 05. and
x /� 1 8 (filled symbols). For comparison the results ob-
tained with
� 0 are shown by open symbols.
uij of a given bond and the bond-charge density
� �c ci j
† ,
u
t
K
c cij i j
( ) † ,0 0�
�
� � �� h.c. (16)
being a consequence of the linearity assumption in
Eq. (15).
Quite generally, it is a widely spread out belief
that inhomogeneous states at finite doping are very
sensitive to small changes of �, supported both by the
HF [94,95] and ED studies [96]. Further, it has been
shown that the electron-lattice interaction favors DW
solutions over other possible phases, such as isolated
polarons or bipolarons [26]. Therefore, a complete dis-
cussion of the stripe phase stability in correlated ox-
ides has to include the coupling to the lattice.
We turn now to the most important aspect of this
Section. Figures 12 and 13 illustrate the effect of the
finite electron-lattice coupling � � 0 5. on the SC and
BC stripes, respectively. Both figures give a clear
demonstration that, in contrast to the hopping aniso-
tropy �t (7) discussed above, finite � markedly modi-
fies both the local hole density (8) and modulated
magnetization (9) [cf. also Table 3 with 7 (SC stripes)
and Table 4 with 8 (BC stripes)]. Basically, the influ-
ence of � resembles the effect of positive t
, smearing
out the stripe order by ejecting holes from the DWs,
being however much stronger. In fact, hole deloca-
lization not only suppresses the magnetization within
the AF domains, but also noticeably quenches mag-
netic moments of the BC domain walls. These trends
can be understood by considering energy increments:
the kinetic Et , on-site EU , and elastic energy EK , as
explained below.
Table 7. Local hole � �n ih and magnetization � �Si
z density
at nonequivalent atoms of the SC stripe phases, all labeled
by decreasing hole density in the x-direction, in the
Peierls-Hubbard model on a 16 16� cluster with U/t � 5,
� 05. and x /� 1 8.
i 1 2 3 4 5
VSC � �nhi 0.270 0.212 0.103 0.038 0.022
� �Si
z 0.000 0.146 0.259 0.310 0.321
DSC � �nhi 0.292 0.179 0.094 0.058 0.046
� �Si
z 0.000 0.193 0.277 0.305 0.314
Table 8. The same as in Table 7 but for the BC stripe
phases.
i 1 2 3 4
VBC � �nhi 0.255 0.156 0.063 0.026
� �Si
z 0.074 0.209 0.291 0.319
DBC � �nhi 0.248 0.130 0.073 0.049
� �Si
z 0.103 0.243 0.294 0.312
One should realize that a system described by the
Hamiltonian (14) might be unstable towards lattice
deformations only if the covalency increase is large
enough to compensate both the EU and EK energy
cost. Without the electron-lattice coupling, a compro-
mise solution is mainly reached by developing a strong
magnetic order in the AF domains, where a possible
kinetic energy gain is irrelevant, and by forming non-
magnetic or weakly magnetic DWs with large hole
density. As we have already shown, transverse charge
fluctuations around the DWs yield the leading kinetic
energy contribution. However, enhanced covalency
and mixing of the lower � �d and higher � ��d U
energy states between a DW and the surrounding sites
partly delocalize these states and increase double oc-
cupancy,
Stripe phases-possible ground state of the high-Tc superconductors
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 425
0 4 8 12 16
0.1
0.2
0.3
0.4
0 4 8 12 16
0.1
0.2
0.3
0.4
0 4 8 12 16
–0.4
–0.2
0
0.2
0.4
0 4 8 12 16
–0.4
–0.2
0
0.2
0.4
0 4 8 12 16
0.08
0.10
0.12
0.14
0.16
0 4 8 12 16
0.08
0.10
0.12
0.14
0.16
0 4 8 12 16
lx
–1.4
–1.2
–1.0
–0.8
–0.6
–0.4
0 4 8 12 16
lx
–1.4
–1.2
–1.0
–0.8
–0.6
–0.4
n
h
E
/t t
0
x
S
u
;D
x(0
) ,
Å
Fig. 13. The same as in Fig. 12 but for the bond-centered
stripes.
D l n nx l lx x
( ) .( , ), ( , ),� � �� �0 0 (17)
Indeed, in the � � 0 case, double occupancy D lx( )
reaches its maximum at the DWs, as illustrated in
Figs. 12 and 13. The only exception is the DSC stripe
phase (right panels of Fig. 12) with the largest D lx( )
in the AF domains. As a consequence, the latter is the
most localized one with the smallest kinetic energy
gain (cf. Table 1).
The situation changes when turning on the elec-
tron-lattice coupling. When the electrons couple to
the lattice (� � 0), the bonds contract, and the saddle
point values of the distortions (16): u uij ij
( )0 � � � along
(10) and (01) direction, respectively, are finite. How-
ever, a nonuniform charge distribution results in a dif-
ferent bondlength in the cluster. This is illustrated in
Figs. 12 and 13 showing a fractional change of the
length for the bonds to the right nearest-neighbor
along the x-direction ux
( )0 (third row). Although the
values of uij
( )0 in the AF domains are also substantial,
the largest lattice deformations � � �c ci j
† appear ei-
ther on the bonds connecting atoms of the DWs with
their nearest neighbors (cf. Fig. 12), or on the bonds
which join two atoms of the bond-centered DWs (cf.
Fig. 13). Accordingly, a strengthening nearest neigh-
bor hopping (15) enables a larger kinetic energy gain
on these bonds (cf. bottom of Figs. 12 and 13).
Table 9. Ground-state energy Etot per site, kinetic energy
( , )E Et
x
t
y and potential energy ( , )E EU K components, as ob-
tained in the Peierls-Hubbard model. Parameters: U/t � 5,
� 05. , and x /� 1 8.
Et
x/t Et
y/t EU/t EK/t E /ttot
DBC –0.9679 –0.9679 0.6478 0.2548 –1.0332
DSC –0.9670 –0.9670 0.6450 0.2544 –1.0346
VB(S)C –1.0496 –0.9248 0.6719 0.2638 –1.0387
As expected, the increasing covalency is accompa-
nied by partial quenching of magnetic moments. In or-
der to appreciate this tendency, let us consider a site
in the AF domain with larger density of �-spin elec-
trons (at A sublattice). Once the magnetization is re-
duced, the corresponding �-spin energy level which
belongs to the lower Hubbard band is pushed up-
wards, and the �-spin of the upper Hubbard band goes
down. As a result, the locally raised �-spin state be-
comes stronger mixed with �-spin states at the sur-
rounding sites of B sublattice, and simultaneously
bond-charge density increases. At the same time, elec-
trons, jumping forth and back between the central site
with the �-spin polarization and its nearest neighbors
with the �-spin one, enhance considerably double oc-
cupancy D lx( ), as shown in Figs. 12 and 13. This
weakens the stripe order and results in a more uniform
distribution of D lx( ). Of course, the increase of the
elastic energy and concomitant enhancement of the
on-site energy, both owing to finite bond contractions
(16), is compensated by the kinetic energy gain and
the total energy is lowered (cf. Tables 1 and 9).
We close this Section with the phase diagrams
shown in Fig. 14. They were obtained by varying U
and the coefficient �, while maintaining constant
K/t0 18� . The increased stability of vertical stripes
follows from the relative stronger enhancement of the
local hopping elements (15) (and consequently larger
gain of the kinetic energy), especially on the bonds in
the direction perpendicular to the DWs itself.
4. Summary
In summary, we have shown that a competition be-
tween magnetic energy of interacting almost localized
electrons and the kinetic energy of holes created by
doping leads to the formation of new type of coexist-
ing charge and spin order — the stripe phases. We
have shown that vertical (horizontal) and diagonal
stripes dominate the behavior of the charge structures
formed by doping the antiferromagnet away from half
filling, using the solutions obtained for the Hubbard
model within the HF approximation in the physically
interesting regime of the Coulomb interaction. The de-
tailed charge distribution and the type of stripe order
depend on the ratio U/t, on the value of the
next-neighbor hopping t
, and on the nearest-neighbor
Coulomb interaction V. We have also shown that a
strong electron-lattice coupling might be responsible
for the appearance of the vertical stripes observed in
the superconducting cuprates at x /� 1 8.
Altogether, although some experimentally ob-
served trends could be reproduced already in the HF
approach, the presented results indicate that strong
electron correlations play a crucial role in the stripe
426 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
Marcin Raczkowski et al.
4.0 5.0 6.0 7.0 7.0
U/t
0
0.1
0.2
0.3
0.4
4.0 5.0 6.0
U/t
0
0.1
0.2
0.3
0.4
a b
DSC
VSC
DBC
VBC
� �
Fig. 14. Phase diagrams for site-centered (a) and bond-
centered (b) stripe structures as calculated from the
Peierls-Hubbard model for doping x /� 1 8 (solid line) and
x /� 1 6 (dashed line).
phases and have to be included for a more quantitative
analysis. Further progress both in the experiment and
in the theory is necessary to establish the possible role
of stripes in the phenomenon of high temperature su-
perconductivity.
M. Raczkowski was supported by a Marie Curie
fellowship of the European Community program un-
der number HPMT2000-141. This work was supported
by the the Polish Ministry of Scientific Research and
Information Technology, Project No. 1 P03B 068 26,
and by the Minist�re Fran�ais des Affaires Etrang�res
under POLONIUM 09294VH.
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| id | nasplib_isofts_kiev_ua-123456789-120185 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-01T20:34:54Z |
| publishDate | 2006 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Raczkowski, M. Oles, A.M. Fresard, R. 2017-06-11T12:05:42Z 2017-06-11T12:05:42Z 2006 Stripe phases — possible ground state of the high-Tc superconductors / M. Raczkowski, A.M. Oles, R. Fresard // Физика низких температур. — 2006. — Т. 32, № 4-5. — С. 411-429. — Бібліогр.: 96 назв. — англ. 0132-6414 Pacs: 71.10.Fd, 71.27.+a, 74.25.–q, 74.72.–h https://nasplib.isofts.kiev.ua/handle/123456789/120185 Based on the mean-field method applied either to the extended single-band Hubbard model or to the single-band Peierls-Hubbard Hamiltonian we study the stability of both site-centered and bond-centered charge domain walls. The difference in energy between these phases is found to be small. Therefore, moderate perturbations to the pure Hubbard model, such as next nearest neighbor hopping, lattice anisotropy, or coupling to the lattice, induce phase transitions, shown in the corresponding phase diagrams. In addition, we determine for stable phases charge and magnetization densities, double occupancy, kinetic and magnetic energies, and investigate the role of a finite electron-lattice coupling. We also review experimental signatures of stripes in the superconducting copper oxides. M. Raczkowski was supported by a Marie Curie fellowship of the European Community program under number HPMT2000-141. This work was supported by the the Polish Ministry of Scientific Research and Information Technology, Project No. 1 P03B 068 26, and by the Ministre Franais des Affaires Etrangres under POLONIUM 09294VH. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур General Aspects Stripe phases — possible ground state of the high-Tc superconductors Article published earlier |
| spellingShingle | Stripe phases — possible ground state of the high-Tc superconductors Raczkowski, M. Oles, A.M. Fresard, R. General Aspects |
| title | Stripe phases — possible ground state of the high-Tc superconductors |
| title_full | Stripe phases — possible ground state of the high-Tc superconductors |
| title_fullStr | Stripe phases — possible ground state of the high-Tc superconductors |
| title_full_unstemmed | Stripe phases — possible ground state of the high-Tc superconductors |
| title_short | Stripe phases — possible ground state of the high-Tc superconductors |
| title_sort | stripe phases — possible ground state of the high-tc superconductors |
| topic | General Aspects |
| topic_facet | General Aspects |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120185 |
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