Unrestricted slave-boson mean-field approximation for the two-dimensional Hubbard model

14 Pages·0.16 MB·English

by  G. Seibold

Unrestricted slave-boson mean-field approximation for the two-dimensional Hubbard
model
G. Seibold and E. Sigmund
Institut f. Physik, Technische Universit¨at Cottbus,
PBox 101344, 03013 Cottbus, Germany
8 V. Hizhnyakov
9 Institute of Physics, Estonian Academy of Sciences, Riia 142, Tartu, Estonia
9
1 The Kotliar-Ruckenstein slave-boson scheme is used to allow for an unrestricted variation of
the bosonic and fermionic fields on the saddle-point level. Various inhomogeneous solutions, such
n
as spin polarons and domain walls, are discussed within the two-dimensional Hubbard model and
a
J compared with results of unrestricted Hartree-Fock (HF) calculations. We find that the present
approachdrasticallyreducesthepolarization ofthesestatesandleadstoincreaseddelocalized wave
4
1 functions as compared to the HF model. The interaction between two spin-polarons turns out to
beattractiveoverawiderangeof theon-siterepulsion U.Inaddition weobtain thecrossover from

] vertical todiagonal domain walls at a higher valueof U than predicted by HF.
l
e
- 71.27,71.10.Fd,75.10.Lp,75.60.CH
r
t
s
t. I. INTRODUCTION [13–15],innickeloxideanaloguesofthecopperoxides[16]
a
and in La Nd Sr CuO [17,18]. In the two latter
m 1.6−x 0.4 x 4
systemsthedensityfluctuationsarepinnedbytheunder-
- The unrestricted Hartree-Fock (HF) approach has
d lying lattice structure giving rise to a static charge- and
turned out to be a powerful tool in the calculationof in-
n spin-density wave. The low-temperature orthorhombic
homogeneousstatesintheHubbardmodel[1–6]. Among
o phase in the nickel oxides stabilizes the stripe structure
c themmagneticpolarons[1,8–11],domainwalls[2–5,7,29] alongthediagonalswhereasthelow-temperaturetetrago-
[ and vortex solutions [6] have been extensively studied
1 within the context of high Tc superconductors (HTSC). nofalashtrourciztuonretailnsLtrai1p.6e−pxhNads0e..4SArlxsCouinO4thfeavCoursOth-eplpainnensinogf
2
v Schrieffer, Wen and Zhang [8] have proposed the so- Bi2212 compounds the existence of charge stripe order
5 calledspin-bagmechanismaccordingtowhichaholecou- has been demonstrated with extended x-ray-absorption
3
1 plestothespindensityoftheantiferromagnetically(AF) fine structure (EXAFS) experiments [19].
ordered background of the CuO planes thus creating a
1 2
Thestripeinstabilitywaspredictedtheoreticallyin[4]
0 local reduction of the AF order parameter. Within this
withinaHFformalismappliedtothe extendedHubbard
8 modeltwoholesareattractedbysharingacommonbag.
9 In the simplest model the resulting superconducting or- model. Also most of the further investigations on the
/ striped phases [2,5,7,29] have been carried out within
t der parameter is of the order of the spin-density wave
a standard HF theory. However, the solutions obtained
gap which is large enough to lead to high-temperature
m
with HF are poor variational wave functions since they
superconductivity. The concept of spin polaron forma-
- aremuchtoo highinenergy. Within HF,the only mech-
d tion is also successfulin explaining the phase diagramin
anism avoiding double occupancy in order to reduce the
n the low and intermediate doping regime of the HTSC’s.
o Accordingtothe conceptofmicroscopicelectronicphase Hubbard repulsion is to renormalize the spin-dependent
c separation [9,10], the doping induced spin polarons (or on-site energy ǫi,σ = Uhni,−σi. Therefore the commen-
v: spin clusters) form a conducting subsystem in an back- surateantiferromagneticphasedisplaysasanalternating
Xi ground dominated by strong antiferromagnetic correla- shift of the spin-up and spin-down one-particle levels re-
spectively,whichoverestimatesbyfarthe polarizationof
tions. Due to the competition between short-range at-
r the AF order.
a traction and the long-range Coulomb repulsion, these
polaronsdo notform a homogeneoussea in anantiferro- Itiswellknownthatapartofthecorrelationsbetween
magnetic background. Instead, the holes are confined to electrons of opposite spins can be accounted for by us-
a percolative network which dimension D is lower than ing the Gutzwiller projector [21,22]. For the AF ordered
two (1 ≤ D ≤ 2). The resulting inhomogeneity in the system this approach leads to wave functions that are
electronic subsystem has also important implications on very close in energy to the solution obtained by Quan-
the pairing mechansim [12]. tum Monte Carlo simulations [23,24]. In addition, the
Besidesspin-polarons,domainwallsolutionspresently Gutzwiller wave function leads to a significantly lower
also attract a lot of interest in the field of HTSC. spin polarization in the intermediate U regime as com-
StripecorrelationshavebeenobservedinLa Sr CuO paredtotheHFresult[23]. TheGutzwillerapproachhas
2−x x 4
1
alsobeenusedtoimprovethesolutionsofinhomogeneous The rest of the paper is organized as follows: In Sec.
states such as spin polarons [11,25] and domain walls IIwegiveadetaileddescriptionofthe formalism,inSec.
[25,26]. However, due to computional limitations these III we present the results for spin polaronic and domain
calculations where done with an Ansatz for the charge- wallsolutionsrespectively, andin Sec. IV we summarize
andspin-densityprofile,i.ethenumberofvariationalpa- our conclusions.
rameters for charge and spin at the different lattice sites
has been strongly reduced.
In this paper we overcome the limitations discussed II. MODEL AND FORMALISM
above and present results for unrestriced spin-polaronic
and kink type Gutzwiller wave functions. Our approach We consider the two-dimensionalHubbardmodel ona
is based on the representation of the Hubbard model in square lattice, with hopping restricted to nearest neigh-
terms of fermions and slave bosons due to Kotliar and bors (indicated by the bracket <i,j >)
Ruckenstein (KR) [27]. The KR formulation is a func-
tional integral method that reproduces the Gutzwiller H =−t c† c +U n n (1)
X i,σ j,σ X i,↑ i,↓
solution at the saddle-point level. The advantage of this
<ij>,σ i
method is that it provides a systematic way to improve
the solution by expanding the fields around the saddle where c(†) destroys (creates) an electron with spin σ at
i,σ
point. The KR approachis a goodstarting point for our sitei,andn =c† c . Uistheon-siteHubbardrepul-
purposes, since at the saddle point level we can immedi- i,σ i,σ i,σ
sion and t the transfer parameter. For the calculations
atelyidentifythevariationalparameterswiththebosonic
in Sec. III we take t=1. Following KR we enlarge the
and fermionic fields. Since we don’t make any assump-
originalHilbert space by introducing four subsidiary bo-
tion on the spatial symmetry of these fields we therefore
(†) (†) (†) (†)
son fields e , s , s , and d for each site i. These
obtain various inhomogeneous textures which stability i i,↑ i,↓ i
operators stand for the annihilation (creation) of empty,
depends on doping and the value of the Hubbard repul-
singly occupied states with spin up or down, and dou-
sion U. In the present paper we will concentrate on the
blyoccupiedsites,respectively. Since thereareonlyfour
stabilityandshapeofspin-polaronicanddomainwallso-
possible states per site, these bosonprojectionoperators
lutions, respectively. Itturns outthat forcommensurate
must satisfy the completeness condition
filling (i.e. one hole along the wall) the parameter range
for the occurence of stripes is significantly enlarged in
e†e + s† s +d†d =1 (2)
comparison to the HF calculation. Within the investi- i i X i,σ i,σ i i
σ
gated range of U (4t ≤ U ≤ 10t) we don’t observe a
crossover to a polaronic Wigner crystal for commensu- Furthermore
rate doping, which means that the interaction between
the holes keeps attractive up to very large values of U. n =s† s +d†d (3)
i,σ i,σ i,σ i i
Then, in the physical subspace defined by Eqs. (2,3) the Hamiltonian (1) takes the form
H˜ =−t z† c† c z +U d†d (4)
X i,σ i,σ j,σ j,σ X i i
<ij>,σ i
1 1
z = (e†s +s† d ) (5)
i,σ i i,σ i,−σ i
e†e +s† s d†d +s† s
q i i i,−σ i,−σ q i i i,σ i,σ
and has the same matrix elements as those calculated for (1) in the original Hilbert space.
In the saddle-point approximation, all bosonic opera- GivenasystemwithN particleswefinallyobtainfor
el
tors are treated as numbers. The resulting effective one- the total energy
particle Hamiltonian describes the dynamics of particles
withmodulatedhoppingamplitudeandcanbediagonal- Nel
ized by the transformation Etot =−t X zi∗,σzj,σXΦ∗i,σ(k)Φj,σ(k)+UXd2i
<ij>,σ k=1 i
c = Φ (k)a (6)
i,σ X i,σ k (8)
k
where the orthogonality of the transformation requires
which has to be evaluated within the constraints (2,3,
Φ∗ (k)Φ (q)=δ . (7) 7). Thisisachievedbyaddingtheseconstraintsquadrat-
X i,σ i,σ kq
ically to Eq. (8)
i,σ
2
E =λ (e2+ s2 +d2−1)2 (9a) compare our calculations with the results of Yokoyama
C1 1X i X i,σ i
i σ and Shiba [23]. Since for this case their AF Gutzwiller
variationalapproach(AFGF)isequivalenttothesaddle-
point approximation of our slave boson method we find
E =λ ( Φ∗ (k)Φ (k)−s2 −d2)2 (9b)
C2 2X X i,σ i,σ i,σ i perfect agreement within the numerical error. In con-
i,σ k trast to the HF scheme, the AFGF leads to a signicfi-
cant reduction of the magnetization in the intermediate
U regime. On the other hand, both methods converge
E =λ ( Φ∗ (k)Φ (q)−δ )2 (9c)
C3 3X X i,σ i,σ kq rapidly to fully polarized Sz components in the large U-
k,q i,σ
limit. One should note that Quantum Monte Carlo sim-
ulations[24]stillleadtomuchsmallervaluesinthe large
E =λ ( Φ∗ (k)Φ (k)−N )2 (9d) U limit.
C4 4 X i,↑ i,↑ ↑
k,i
A. Spin Polarons
E =λ ( Φ∗ (k)Φ (k)−N )2 (9e)
C5 5 X i,↓ i,↓ ↓
The formation of spin polarons in the 2D Hubbard
k,i
model results from the competition between kinetic en-
We have added the last two conditions that turn out to ergygainandmagneticenergylosswhendopingthe sys-
be veryconvenientbecausethey allowto define the total tem away from half filling. Let us first consider the case
number of spin up and down particles N↑ +N↓ = Nel. where a particle with spin down has been removed from
The energy functional E{Φi,σ(k),ei,si,σ,di} = Etot + the half filled, antiferromagnetically (AF) ordered lat-
EC1+EC2+EC3+EC4+EC5 now hasto be minimized tice. Ifthevacancyisimmobile,thecostinthe magnetic
with respect to the fermionic and bosonic fields. Since energy is 4J∼ t2/U in the large U limit. However, this
the KR theory does not preserve the spin-rotation in- vacancy can gainkinetic energy via virtualhopping pro-
variance of the original hamiltonian all variational pa- cesses to the nearest neighbor sites, thus mixing some
rameters can be taken as real numbers [28]. For the probability of spin up occupation to the site where the
minimization procedure we have used a standard conju- particle has been removed from. Therefore, the result-
gate gradient algorithm. The gradients of the functional ing spin-density profile therefore has an inverted order
E{Φi,σ(k),ei,si,σ,di} can be calculated analytically and parameter at the site where the charge is located. Ac-
convergenceischeckedbyevaluatingthenormofthegra- cording to Nagaoka’s theorem [20] the size of this ferro-
dient. The accuracy of the solution can be controlled by magneticcoreisexpectedtobeverylargeforlargevalues
calculatingthevalueofEC1+EC2+EC3+EC4+EC5 at of U.
theendoftheiterationprocedure. Wegenerallyhaveset Fig.1 show the chargeandspin density profile of such
thevaluesoftheLagrangeparametersλ ...λ to104−105 a polaron obtained by unrestricted HF calculations and
1 5
which leads to an estimated Error at ≈0.0002. with the present method, respectively. The calculation
Inprincipleonecouldstartthe calculationwitha ran- was done on a 8×8 lattice with on-site repulsion U=6t.
dom configuration of the fermionic and bosonic fields. In both methods, the doped hole is mainly localized at
However, for a doped system there exist different self- site (4,5), But, whereasthe chargeatthis site is reduced
consistent solutions which are close in energy and deter- to hni = 0.53 within the HF approach, the unrestricted
miningthemostfavorablecanbedifficult. Therefore,we SB method gives a value of hni = 0.73 only. Moreover,
havegenerallystartedfromtheunrestrictedHFsolutions the AF order parameter ∆S = (−1)ix+iySz at the po-
i i
for spin polaronic or domain wall phases. The order of laron center is much less affected in the SB mean field
magnitude of the time needed to get convergence is half (MF) treatment (∆S = −0.07) than in the case of un-
i
an hour on a SGI Indy workstation. restricted HF (∆S = −0.21). This discrepancy can be
i
understood as follows. The HF theory only renormalizes
the spin dependent on-site energies. The removal of a
III. RESULTS spin down particle at site i leads to a relaxation of the
spin up on-site level at this site. As a consequence the
In a first step we consider a single spin-polaron and alternating on-site level shift, describing the AF order,
comparethespinandchargeprofilesobtainedwithinthe is changed at site i where 5 neighbored spin up states
HFandunrestrictedSBapproximation,respectively. We have now acquired nearly the same energy. Thus there
then extend the calculations to domain wall type solu- is a strong hybridization between the spin-up states and
tions and study the stability of these textures as a func- one obtaines a large value for the reversed spin order
tion of U. Before this, we evaluate the ground state en- parameter at the central site i. By contrast the kinetic
ergy and AF polarization for the half filled system and energy in the spin-down channel between site i and its
3
nearest neighbors is very much reduced since the cor- in eqs. (9) in order to keep the constraint induced error
responding spin-down on-site level is pushed to a high within the desired limits. This fact causes the difficulty
energy. Note that the HF method alwaysleads to a very in exploring the whole parameter range of attraction in
large spin-polarization, since this is the only way within Fig. 3.
this approximation to minimize the on-site repulsion.
Let us now consider the removal of a spin down par-
ticle at site i when calculated within the SB aproxima- B. Domain wall solutions
tion. Then the spin-up hopping channel allows for the
hybridization of neighboring spin up states with site i,
whichiscomparabletotheHFapproach. However,since For intermediate values of the on-site Hubbard repul-
wenowhaveanadditionalvariationalparameterpersite sion U Hartree-Fock theory predicts the existence of do-
(i.e. the boson field {d }) double occupancy can be min- mainwallsolutions,wherethedopedchargedcarriersare
i
imizedatsitei,whithoutverymuchreducingthekinetic localized within a stripe in horizontal or diagonal direc-
energy in the spin-down channel between site i and its tion. This stripe separates two AF ordered regions with
nearest neighbors. In fact the reason why the Gutziller opposite sign in the AF order parameter. Within the
wave function leads to a lower energy than HF theory HF approximation it was shown [5] that there is a tran-
is that there one can minimize double occupancy while sition from horizontal to diagonal stripes when the ratio
keepingthe kineticenergyatahighervalue ascompared between on-site repulsion U and transfer integral t ex-
to the HF approach. Inother words,the localizationbe- ceeds the criticalvalue ofU/t≈3.6. Froma constrained
havior is much more pronounced in the HF treatment Gutzwiller variation of hyperbolic-type domain walls it
than within the SB formalism. was concluded in [26], that this limit probably is shifted
Theenergydifferenceofthe solutionsinFig.1(U=6t) to much lower values.
is of the same order of magnitude than for the homoge- In the following we will compare energies for diagonal
neous AF solutions of Ref. [23]. The total energy per and horizontal stripes with the energy of isolated spin-
site calculated within the HF approximation is EHF = polaronsusing the unrestrictedSB scheme. The calcula-
−0.607t and for the SB method we get ESB = −0.645t. tionsaremadefordifferentlattice sizes(typically17×4,
Fig. 2 shows the order parameter ∆S as a function of 13×6, 9× 8) by applying appropriate boundary condi-
i
U at the perturbed sites and far from the spin polaron, tions for each domain wall type (see Fig. 4). The choice
evaluated with HF and SB approximations respectively. of the supercell is, of course, a delicate issue since peri-
It turns out that the SB method drastically reduces the odic boundary conditionsin principle requirean’uneven
polarization at the center of the polaron as compared to × uneven’ lattice for a diagonal wall and an ’uneven ×
HF. As can be further seen on Fig. 2, this reduction of even’latticeforaverticalstripe. Toavoidthecomparison
polarization is much stronger than in the residual AF between different lattice sizes for different domain wall
ordered plane. Within the SB scheme the spin polaron types we have choosen the ’shifted boundaries’ shown in
acquires a ferromagnetic core for values U > 5t whereas Fig. 4b. This means that along the y-direction the su-
this limit in HF is already achieved for U ≈3.3t. percellsareshiftedbytheextendofthe diagonaldomain
Finally we evaluate the static interaction between two wall in x-direction. Consider for example a diagonal do-
spin-polaronswhich is of special importance with regard main wall on a 13×6 lattice. Then the site (Nx,1) is
to the spin-bag model [8] and phase separation scenar- connected with site (Nx+6,6) for Nx < 8 and with site
ios [9] of the high-Tc superconductors. To calculate the (Nx+6-13,6)for Nx ≥8. Calculating the energy per site
bindingenergyweconsidertwoholeswithoppositespins forthehalf-filledAForderedsystem,wefindthatthetwo
placedattwoneighboredsitesandcomparetheenergyof kindsofboundaryconditionsinFig.4differintheresult
this configuration with the energy of two infinitely sep- by0.1%for U≈3t. This differencerapidlyvanishes with
arated spin-polarons. It should be noted that a positive increasing U.
bindingenergymeansattraction. Theresultsareplotted In Fig. 5 we show the charge- and spin density profile
inFig.3. Within the HF approachweobtainaninterac- ofaverticaldomainwallcalculatedwithunrestrictedHF
tion between the two polarons which is attractive up to and SB approximations respectively. The doping corre-
U ≈ 6.5t, in agreement with the results of Ref. [1]. The sponds to one hole per site in the domain wall. As for
binding energy displays a maximum at U ≈ 4.5t. How- the spin polarons studied in the previous section the SB
ever, within the SB approximation the parameter space result displays a charge- and spin profile that is consid-
of attraction is considerably enlarged. The maximum erably enlarged compared to the HF solution. This is in
of the binding energy, which is approximately twice the agreement with the calculations of Ref. [26]. The dip in
value of the HF calculation, now occurs at U ≈ 8.5t. the HF charge profile is nearly twice the value of the SB
Unfortunately, convergence of our variational approach approximation.
becomes very slow in the very large U-regime. In this Fig.6 showsthe energyper hole as a function ofU for
limit one has to increase very much the parameters λ varioustexturessuchasdomainwallsandspinpolarons.
i
4
Wealsohaveinvestigatedthehalf-filledwallwiththeon- to isolated polarons for U≤ 10t. This result supports
wallquadruplingoftheperiodwhichhasbeenintensively thedescriptionofdomainwallstructuresintheLa NiO
2 4
discussed by Zaanen and Ole´s in Ref. [29]. The energy compounds [35]where it is generallyarguedthat the oc-
hasbeencalculatedinastandardway[29]by comparing curence of diagonalwalls is a result ofmean field theory.
the energy of each texture with the energy of the ho- Within the present approachthe range of stability of di-
mogeneous AF ordered state with the same number of agonal walls is considerably enhanced in comparison to
holes (compared to half filling). In case of the diagonal the HF approximation, where for U≈ 8t the walls decay
wall we additionally have choosen the same shift of the into isolated polarons.
boundaries for the reference AF lattice.
Within the HF approximation we obtain a crossover
from vertical to diagonal domain walls for U≈ 3.8t and ACKNOWLEDGMENTS
a crossover to isolated spin-polarons at U≈ 8t. The en-
ergy for half-filled walls is always higher than for iso-
Valuable discussions with W. Essl are gratefully ac-
lated polarons. These results are in complete agreement
knowledged. We also thank A. Bill for a critical reading
with earlier HF studies of the two-dimensional Hubbard
of the manuscript.
model [5,29]. However,the rangeof stability for the ver-
tical stripe solutionis considerably enlargedin the unre-
strictedSBapproximationwhereweobtainthecrossover
at U≈ 5.7t This result is supported by Lancos diago-
nalization studies of the tJ-model [30] and Monte Carlo
methods of the one-bandHubbard model [31–33]. These
works report a shift of the static spin structure factor [1] W.P.SuandX.Y.Chen,Phys.Rev.B38,8879 (1988).
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(1989).
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[5] M. Inui and P. B. Littlewood, Phys. Rev. B44, 4415
we don’t observe a crossover to a spin-polaronic Wigner
(1991).
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[6] J.A.Verg´es,E.Louis,P.S.Lomdahl,F.Guinea,andA.
steadthe energy of half-filledwalls turns outto be lower
R. Bishop, Phys.Rev.B43, 6099 (1991).
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[7] H. J. Schulz, Phys.Rev.Lett. 64 1445 (1990).
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FIG.1. Charge- (hnii) and Spin- (∆Si) density profiles for a spin polaron on a 8×8 lattice. a) HF approximation; b) SB
approximation. The Hubbardon-site repulsion is U =6t.
FIG. 2. Spin-density order parameter ∆Si at the center of the spin polaron (two lower curves) and at maximum distance
from the polaron (two uppercurves). Solid line: SB approximation; Dashed line: HFapproximation.
FIG.3. Binding energy of a pair of polarons with opposite spins placed at neighbored sites on a 8×8 lattice. Dashed line:
HFapproximation; Solid line: SBapproximation
FIG.4. Sketchshowingtheboundaryconditionsiny-directionwhichhavebeenappliedtodescribevertical(a)anddiagonal
(b) domain walls respectively. In x-direction we have taken periodic conditions. The line crossings correspond to the lattice
sites and thedashed lattices mark thechoosen periodicity in the y-direction.
FIG.5. Charge-(a)andSpin-(b)densityprofilesinx-directionforaverticaldomainwallona17×4lattice. Thenumberof
holes is 4 since thebest energy is obtained when there is one hole per site in thewall. Solid lines: SB approximation; Dashed
lines: HF approximation; Short dashed: cosh- (for charge) and tanh- (for spin) functional fit to theSB solution.
FIG. 6. Binding energy per hole for vertical stripes (full lines), and diagonal stripes (dashed lines) with one hole per site
along the wall, vertical half-filled walls (dotted lines) and isolated spin polarons (dashed-dotted lines). The binding energy is
definedasthedifferenceinenergybetweenagiventextureandthehomogeneousAForderedlatticewiththesamedoping. The
domainwallsolutionshavebeencalculatedona9×8lattice,polaronsona8×8lattice. a)HFapproach;b)SBapproximation.
6
1 1
0.9 0.9
0.8 0.8
hnii 0.7 hnii 0.7
0.6 0.6
8 8
7 7
6 6
1 2 3 4 5 6 7 8 1 2 3 4 5 1 2 3 4 5 6 7 8 1 2 3 4 5
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
S 0.1 S 0.1
(cid:1)i 0 (cid:1)i 0
-0.1 -0.1
-0.2 -0.2
8 8
7 7
6 6
1 2 3 4 5 6 7 8 1 2 3 4 5 1 2 3 4 5 6 7 8 1 2 3 4 5
a) b)
0.5
0.4
0.3
0.2
S (cid:1)i
0.1
0
-0.1
-0.2
-0.3
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
U
0.07
0.06
e
l
o
h 0.05
r
e
p 0.04
y
g 0.03
r
e
n
e 0.02
g
n
di 0.01
n
i
b 0
-0.01
-0.02
4 6 8 10 12 14 16 18
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(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
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a) b)