Exact Thermodynamics of Pairing and Charge-spin Separation Crossovers in Small Hubbard Nanoclusters

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by  A. N. Kocharian

Exact Thermodynamics of Pairing and Charge-spin Separation
Crossovers in Small Hubbard Nanoclusters
Armen N. Kocharian
7 Department of Physics and Astronomy,
0 California State University, Northridge,
0 CA 91330-8268
2

n Gayanath W. Fernando
a Department of Physics, University of Connecticut,
J
Storrs, CT 06269 and IFS, Hantana Rd., Kandy, Sri Lanka

1

Tun Wang and Kalum Palandage
]
l Department of Physics, University of Connecticut,
e
Storrs, CT 06269
-
r
t
s James W. Davenport
t. Computational Science Center,
a Brookhaven National Laboratory,
m
Upton, NY 11973
-
d Theexactnumericaldiagonalization andthermodynamicsinanensembleofsmallHubbardclus-
n ters in the ground state and finite temperatures reveal intriguing insights into the nascent charge
o and spin pairings, Bose condensation and ferromagnetism in nanoclusters. The phase diagram off
c
half filling strongly suggests the existence of subsequent transitions from electron pairing into un-
[
saturated and saturated ferromagnetic Mott-Hubbard like insulators, driven by electron repulsion.

1 Rigorouscriteriafortheexistenceofquantumcriticalpointsinthegroundstateandcorresponding
v crossovers at finite temperatures are formulated. The phase diagram for 2×4-site clusters illus-
2 trates how these features are scaled with cluster size. The phase separation and electron pairing,
2 monitored by a magnetic field and electron doping, surprisingly resemble phase diagrams in the
0 family of doped high Tc cuprates.
1
0 PACSnumbers: 65.80.+n,73.22.-f,71.10.Fd,71.27.+a,71.30.+h,74.20.Mn
7
0
/ I. INTRODUCTION tencharacterizedbycrossovertemperaturesbelowwhich
t
a excitation (pseudo)gaps in the normal-state are seen to
m
develop [17]. The detailed manner in which T and
c
Despite tremendous experimental and theoretical ef-
- crossover temperature changes under variation of elec-
d forts, there is still no microscopic theory that can yield
tron concentration, magnetic field or pressure (Coulomb
n comprehensive support for the bare Coulomb interac-
interaction)isalsooffundamentalinterestfortheformu-
o tion originated pairing correlations, phase separation
c lation of the microscopic models responsible for nascent
and pseudogap phenomena in clusters, small nanopar-
: superconductivity[18]. Inthe optimally dopedcuprates,
v ticles, transition metal oxides and high T cuprates
c the correlation length of dynamical spin fluctuations is
i [1,2,3,4,5,6,7,8,9,10]. Therecentdiscoveryofthefer-
X very small [19] and hence short-range fluctuations are
romagnetic insulators at room-temperature has further
r dominantoverlong-rangeones. Therefore,amicroscopic
a stimulated a great interest related to the role of on-site
theory,withshort-rangedynamicalcorrelations,cangive
Coulombinteractionintheoriginofferromagnetism[11].
useful insight into nascent superconductivity in clusters
ElectronsinafiniteHubbardlattice,subjectedtostrong
and the rich physics observed in the high-T cuprates.
c
on-site electron repulsion near half filling, can lead to
In our opinion, thermodynamic properties of small Hub-
spontaneousferromagnetism[12]andthe finite tempera-
bard clusters under variation of composition, size, struc-
ture phasediagramis expectedto be applicable to disul-
ture,temperatureandmagneticfieldhavenotbeenfully
fides [13]. Moderate Coulomb interaction can also lead
explored, although there have been numerous exact cal-
to phase separation and formation of mesoscale struc-
culations [20, 21]. From this perspective the exact diag-
tures (such as ”stripes”) under doping of Mott insu-
onalization of small Hubbard nanoclusters can give in-
lating ”parent” materials with highly correlated elec-
sights related to the origin of superconductivity and fer-
trons, including high T cuprates [14, 15, 16, 17]. Al-
c romagnetism in an ensemble of clusters, nanoparticles,
though the experimental determination of various inho-
and eventually, nanomaterials [22, 23].
mogeneous phases in the cuprates is still somewhat con-
troversial[15,16],the underdopedhighT superconduc- The following questions are central to our study: (i)
c
tors (HTSCs) have many common features and are of- Using exact cluster studies, is it possible to obtain a mi-
2
croscopic understanding of charge and spin separation ters with periodic boundary conditions [23]. In addi-
and electron charge pairing and identify various possible tion,here wepresentthe resultsofexactnumericaldiag-
incipient phases at finite temperatures? (ii) Is it pos- onalization of 2 4 square 2D clusters, using numerical
×
sible for ferromagnetism to occur in Mott-Hubbard like eigenvaluesinthe aboveensemblesto study thermaland
insulatorsawayfromhalffilling? (iii)Dothesenanoclus- quantum fluctuations.
ter phase diagrams retain important features, such as
quantum critical points and crossovers that are known
for mesoscopic structures and large thermodynamic sys-
tems? (iv) Whentreatedexactly,whatessentialfeatures In nanoparticles electrons and holes are limited to
canthesimpleHubbardclusterscapturethatareincom- small regions and an effect known as quantum confine-
mon with the transition metal oxides, cuprates, etc.? ment yields discrete spectrum and energy-level spacings
In addition, 4-site (square) cluster is the basic build- betweenfilledandemptystatesthatcanmodifythether-
ingblockoftheCuO2 planesintheHTSCsanditcanbe modynamics. Because the small clusters are far from
used as a block reference to build up larger superblocks thermodynamic limit, one can naively think that the
in 2D of desirable L L sizes [24, 25, 26, 27, 28]. Short standard tools for the description of phase transitions
range electronic corr×elations provide unique insight into are not applicable and other concepts are needed. How-
the Nagaoka ferromagnetism in small 2D and 3D fer- ever,itisimportanttorealizethatphasetransitionsand
romagnetic particles [12, 13], exact thermodynamics of correspondingtemperaturedrivencrossoversinthegrand
many-body physics, which are difficult to obtain from canonicalensemblecanverywellbedefinedandclassified
approximate methods [22]. Here we present also strong for finite systems without the use of the thermodynamic
evidenceunderliningtheoccurrenceofsaturatedandun- limit. It is further shown also how spatially inhomoge-
saturatedferromagnetism[29],particle-particle,particle- neous configurations like phase separations can be ob-
holepairingsandcorrespondingtemperaturedrivenBose tained using the canonical ensemble. We illustrate how
condensation(BC)crossoversforspinandchargedegrees phasetransitionsonthevergeofaninstabilityandphase
in mesoscale structures. separation(segregation)canbedefinedandclassifiedun-
The paper is organized as follows. In the following ambiguously in finite Hubbard clusters [23]. It is thus
section we present the model and formulate exact ther- possible to define phase transitions and crossovers even
modynamics in grand canonical ensemble approach. In insmallsystemswithlocalCoulombforces,whicharenot
thethirdsection,weintroducethechargeandspinpseu- thermodynamically stable. There is also strong support
dogaps and define corresponding new order parameters. withregardtotheeffectivenessofthegrandcanonicalap-
In section four, we present the results of numerical cal- proach for studies of magnetism in small clusters due to
culations for electron density and magnetization versus recent experimental findings that average magnetization
chemical potential and illustrate how to calculate vari- of a ferromagnetic cluster is a property of the ensemble
ous phase boundaries for particle-particle/hole pairing, ofisolatedclustersbutnotoftheindividualcluster[1,2].
phase separation instabilities, quantum critical points in
the ground state and crossovers at finite temperatures.
The results for the 2 4 cluster are used to illustrate
× One can eliminate next nearest neighbor couplings by
how these features are scaled with the cluster size. The
replacing the planar square lattice with independent 4-
concluding summary is presented in the closing section.
site clusters and treat them as an ensemble immersed
in a particle heat reservoir, where electrons can trans-
fer from cluster to cluster due to the thermal fluctua-
II. MODEL AND FORMALISM
tions [23]. Earlier we formulated a new scheme in which
thermodynamical/statisticalnotions,concepts and theo-
The quantum and thermal fluctuations of electrons in
retical methodologies are tailored for the calculations of
finite clusters can be described by the Hubbard Hamil-
exact thermodynamics in a grand canonical ensemble of
tonian, placed in a magnetic field h as,
4-site clusters. Degrees of freedom for charge and spin,
electronandspinpairings,temperaturecrossovers,quan-
H =−t c+iσ cjσ −µ niσ+ Uni↑ni↓− tumcriticalpoints,etc. wereextracteddirectly fromthe
<iXj,σ>σ Xi,σ Xi thermodynamics of these clusters [22, 23]. The grand
hγ (n n ), (1) partition function Z (where the number of particles N
i↑ i↓
Xiσ − and the projection of spin sz are allowed to fluctuate)
and its derivatives are calculated exactly without taking
with the hopping amplitude t (energies are measured in the thermodynamic limit [22]. The charge and spin fluc-
the units of t) and on-site Coulomb interaction U 0. tuation responses to electron or hole doping level (i.e.
Here γ = 1 is the magnetic moment of an electron≥and chemical potential µ) and an applied magnetic field (h)
2
µ is the chemical potential for the ensemble of clusters. resulting in weak saddle point singularities (crossovers),
Thisworkutilizes statisticalcanonicalandgrandcanoni- whichdisplayclearlyidentifiable,prominentpeaksincor-
calensembles using analyticaleigenvaluesfor 4-site clus- responding charge and spin density of states [23].
3
III. CHARGE AND SPIN PSEUDOGAPS ∆e−h(T) 0[22]. Accordingly,the condition∆c(T)<0
≥
withµ−(T)>µ+(T)giveselectron-electronpairingwith
We apply the grand canonical ensemble of decoupled positive energy, ∆P(T)>0 [23]. Using the above defini-
clusters in contact with a bath reservoirallowing for the tions for µ± we can combine them and write:
pcualratticelethneumabboevrettohflerumctoudaytne.amIticispsrtorapiegrhttiefosrawnadrdsotmoecaol-f ∆c(T)= ∆e−h(T), for µ+ >µ− (5)
(cid:26)∆P(T), for µ− >µ+,
these results for the 2- and 4-site clusters were reported
earlier [22]. Using these eigenvalues, we have evaluated where ∆e−h(T) 0 serves as a natural order parameter
the exact grand partition function and thermal averages and will be calle≥d a MH-like pseudogap at nonzero tem-
suchasmagnetizationandsusceptibilitiesnumericallyas perature, since χ has a small, but nonzero weight inside
a function of the set of parameters T,h,µ,U . Using the gap at infinitesimal temperature. At T =0, this gap
{ }
peaksinspinandchargesusceptibilities,phasediagrams will be labeled a true gap since χ is exactly zero inside.
c
in a T vs µ plane for arbitrary U and h can be con- For a given U, we define the crossover critical tempera-
structed. This approach also allows us to obtain quan- ture T (U) at which the electron-hole pairing pseudogap
c
tum critical points (QCPs) and rigorous criteria for var- vanishes and a Fermi liquid state, with µ− = µ+, be-
ious sharp transitions, such as the Mott-Hubbard (MH), comes stable.
antiferromagnetic (AF) or ferromagnetic (F) transitions At zero temperature, the expression for electron bind-
in the ground state [22] and charge (particle-particle) or ing energy ∆P(T) is identical to true gap introduced in
spincondensationatfinitetemperatures[23]usingpeaks Ref. [35] and at nonzero temperature it will be called
in charge or spin susceptibilities (see below). a pairing pseudogap. We can easily trace the peaks of
The difference in energies for a given temperature be- χ (µ) at finite temperature, and find T (µ) at which a
c c
tweenconfigurationswithvariousnumbersofelectronsis possiblemaximumoccursforagivenµ. Atfinitetemper-
obtained by adding or subtracting one electron (charge) ature,thechargesusceptibilityisadifferentiablefunction
and defined as, of µ and the peaks, which may exist in a limited range
of temperature, are identified easily from the conditions,
∆c(T)=[E(M 1,M′;U,T) E(M +1,M′;U,T)] ′ ′′
2[−E(M,M′;U,T−)−E(M +1,M′;U,T−)],(2) χhac(lfµ±fil)li=ng0µw0i=thµχ+c(=µ±µ)−<=0.U2TvhaencishhaersgeatpsTeMudHogwahpicaht
′ ′′
where E(M,M′;U,T)is the lowestcanonical(ensemble) canbeidentifiedclearlywithχc(µ0)=0andχc(µ0)=0;
energy with a given number of electrons N = M +M′ i.e. as the temperature corresponding to a point of in-
determined by the number of up (M) and down (M′) flexioninχc(µ). Thisproceduredescribesrigorouscondi-
spins. The quantity ∆c(T) is related to the discretized tions for identifying the MH transition temperature and
a similar one can be carriedout for the spin gap, ∆s(T),
secondderivative of the energy with respect to the num-
byfollowingtheevolutionofspinsusceptibilitiesχ (µ)as
ber of particles, i.e. charge susceptibility χ . We define s
c
afunctionofµ. Possiblepeaksinthe zeromagneticfield
the chemical potential energies at finite temperature as
spin susceptibility χ (µ), when monitored as a function
µ , s
±
of µ, can be used to define an associated temperature,
µ+ =E(M +1,M′;U :T) E(M,M′;U,T) (3) Ts(µ), as the temperature at which such a peak exists
−
µ =E(M,M′;U :T) E(M 1,M′;U :T). (4) andaspinpseudogapastheseparationbetweentwosuch
−
− − peaks.
Eqs. (3) and (4) at T = 0 are identical to those in- Similar to the charge plateaus seen in N versus µ,
troduced in [30], near half filling. One usually refers to we can trace the variation of magnetizatiohn isz versus
h i
MH transitions, plateaus or gaps in N vs µ, as those anappliedmagnetic fieldh andidentify the spinplateau
h i
occurring only at half filling N = 4 in the 4-site clus- features which can be associated with staggered mag-
h i
ter with lower and upper Hubbard subbands separated netization or short range AF. We calculate the critical
by the energy gap. At half filling, one can recall MH magnetic field h for the onset of magnetization which
c±
and AF critical temperatures, at which corresponding depends on U, T and N, by flipping a down spin, or an
MH and AF gaps disappear. Similar steps (gaps) at upspin[23]. The differencein energiesforgiventemper-
other N in charge and spin degrees will be referred ature between configurations with various spins defines
h i
to as MH-like and AF-like plateaus respectively in or- the spin pseudogap
der to distinguish these. This terminology would also
apply to all the labelings in Fig. 1 and the rest of the ∆s(T)= h+−h−, (6)
2
paper. Usingthe definitions (3) and(4), the correspond-
ing charge energy gap, ∆c(T), at finite temperature can where h+ and h− identify the peak positions in the
be written as a difference, ∆c(T) = µ+ µ−. No- h space for the spin susceptibility χs. The condition
tice that µ+(T) and µ−(T) identify peak −positions in ∆s(T) > 0 yields spin (AF-like) paring, ∆AF(T) > 0,
a T µ space for charge susceptibility χ at finite tem- with a positive (spin) pseudogap [23]. Accordingly, the
c
pera−ture. The condition ∆c(T) > 0 provides electron- condition∆s(T)<0 with h−(T)>h+(T)yields a ferro-
hole (excitonic) excitations, with a positive pseudogap, magneticpairingpseudogap,∆F(T)>0,forspin(triplet
4
4-site cluster at temperature T = 0.02 U U (0) (a critical value; see Ref. [23]) and relatively
c
≥
low temperatures. Thus at low temperature, N (ex-
4 4 pressedasa function ofµin Fig.1) evolvessmohothily for
U U (0),andshowsfiniteleapsacrosstheMHplateaus
) U = 4 ≤ c
> U = 20 at N = 2,4. In Fig. 1, in the vicinity N = 3, one
N U → ∞ canhnoitice two phases. At U U (T), a nehgaitive charge
< c
r (3 U = -4 3 gapwithmidgapstatesisasig≤natureofelectron-electron
be similar behavior near <N>=3 pairing at low temperatures. For U ≥ Uc(T), the MH-
m like charge gap in (5) is positive µ+ > µ−, which favors
u electron-hole pairing similar to MH gap at half filling.
n
n 2 2 As U increases, sz versus µ reveals islands of stability;
ro 15 the minimal spinh stiate sz = 0 at U Uc(0); unsatu-
elect ptibility10 U = 4 sraattuedrafteerdroNmaagganoektaisfmerrhoshmziaig=ne21tisamt,Uwc(it0≤h) m<aUxim<umUFs(p0i)n;
rage 1 ge susce 5 1 htshzeic=ha23rgaetaUnd>sUpFin(0g)a(pnsoattshNownin3Fgigra.d1)u.aAllysUsat→ura∞te,
ave char 0-3 -2 -1µ 0 1 2 ttioveitlsy.mAaxtimNum →2,2(w2e−h√av2e)htfualnild≃ch2a−rg√e-3spvianlureescornescpileiac--
h i ≃
0 0 tion when the spin gap at quarter filling approaches the
-4 -3 -2 -1 0 1 2 charge gap, 4(√2 1)t, as U . The chemical po-
chemical potential (µ) − → ∞
tential gets pinned upon doping in the midgap states at
N 3 and U =4. Such a density profile of N versus
h i≃ h i
µ near N = 3, closely resembles the one calculated at
FIG. 1: Variation of average electron concentration versus µ h i
U = 4fortheattractive4-siteHubbardclusterinFig.1
inensembleof4-siteclustersforvariousU values. Theresult −
and, is indicative of possible particle or hole pairing.
for U = −4 is given for comparison with U = 4. The inset
shows the variation of charge susceptibility χc versus µ for
U =4.
B. Quantum critical points
or quadruplet) coupling. Using the above definitions for The exactexpressionfor the chargegap, ∆c(U :0), at
3
h± we can combine both as, N 3 has been derived earlier in Ref. [22]. At zero
h i ≃
temperature, the sharp transition at critical parameter
∆s(T)=(cid:26)∆∆FAF(T(T)), ffoorr hh−+ >>hh−+. (7) yUice(l0d)s,isUdce(0fi)ne=d4fr.5o8m39t9h9e3c8o.nTdihteionva∆nic3s(hUincg:0g)ap≡a0twUhc(ic0h)
inthegroundstatecanbedirectlylinkedtothequantum
This natural order parameter in a multidimensional pa- critical point [31] (QCP) for the onset of pair formation.
rameter space µ,U,T at nonzero temperature will be Indeed the QCP, U = U (0), separates electron-electron
c
called a spin pairing pseudogap. We define the crossover pairing from MH-like electron-hole pairing regime. The
temperatures TsP and TF as the temperatures at which QCP turns out to be a useful point for the analysis of
the corresponding spin pseudogaps vanish and a spin the phase diagram at zero and non-zero temperatures.
paramagnetic state, with h− = h+, is stable. Similarly, The QCP and the corresponding singular doping depen-
following the peaks in zero magnetic field spin suscepti- denciesonthechemicalpotentialandthedeparturefrom
bility, a spin pseudogap ∆s(T), and an associated Ts(µ) that point at nonzero crossover temperatures on T µ
−
can be defined [23]. phase diagrams for various U values are given in sec-
tion IVF. Exactly at U = U (0) there is no charge-spin
c
separationand,aspinparamagneticstatecoexistswitha
IV. RESULTS Fermi liquid similar to non-interacting electrons, where
U = 0. At zero temperature the analytical expression
A. hNi versus µ for the charge gap first was derived in Ref. [22]. Using
definition introduced in (5), the corresponding electron-
electron pairing gap, ∆P(0),
In Fig. 1 for the 4-site cluster, we explicitly show the
variation of N versus µ below half filling for various 2 γ U
h i ∆P(0)= (16+U2)cos +
U values in order to track the variation of charge gaps −√3 3 − 3
p
with U. The opening of the gap is a local correlation
2 α
effect, andclearly does not follow fromlong range order, (48+U2)cos 32+U2+4 64+3U2, (8)
3 3 −q
as exemplified here. For infinitesimal U > 0, true gaps p p
at N = 1,2 develop and they increase monotonically exists only at at U < U (0). In contrast, the electron-
c
h i
with U. In contrast,the charge gap at N =3 opens at holepairingiszeroatU <U (0)andexistsintheground
c
h i
5
state only for all U > U (0). The electron-hole pairing
c
gap, ∆e−h(0), within the range U <U <U (0) is T vs U phase diagram near
c F 0.30
P
2 γ U TC - Electron-electron pairing
∆e−h(0)= (16+U2)cos + + 0.25 TC - Onset of MH like
√3p 2 3 3 α Fermi liquid TC T PS - pSaprianm paaginrientgic insulator
32+U2+4 64+3U2 (48+U2)cos , (9) 0e.20 TF - Saturated ferromagnetic
q − 3 3 ur insulator
p p at TUF - Unsaturated ferromagnetic
AbovethecriticalpointU ≥UF(0)theelectron-holegap 0mper.15 M o t t P-Haruabmbaargdn eitnisculato r insulator
is e P
0T.10 (TC)
2 γ
∆e−h(0)= (16+U2)cos TUF
√3p 3 − 0.05 Electron-electron (TPS) TFU
2 α U pairing TF
(48+U2)cos + +4. (10) Spin pairing Unsaturated ferromagnetic insulator Ferromagnetic insulator
3 3 3 0.00
p 0.0 UC 5.0 10.0 15.0 UF 20.0 25.0 30.0
where corresponding parameters in Eqs. (8), (9) and U
(10) α = arccos (4U U3)/(16 + U2)32 and γ =
a4.r5c8co4s,{t(h4eUe)n/(er13g6y+g{Ua3p23)∆32}−c3.(U27A:t0)3reblaetcio9vmeleys}lpaorgsietivUe fo≥r FgthrIaeGmc.r2no:sesCaorrvooesprstotivemmearpltleyermadtpouperreeadtfuohrrNetsihv=eeor3snursseegtUiomffoet.rh4He-esMirteHeT-plcihkdaeseepnaodrtiaaes--
N = 3. With increasing temperature, this pseudogap magnetic insulator with electron-hole pairing gap and spin
h∆c(iU : T) increases. Due to (ground state) level cross- (gapless) liquid, onset of electron-electron pairing below TP,
3
ings, the spin degeneracy in an infinitesimal magnetic Bose condensation of electron charge and coupled spin pairs
fieldisliftedatQCP,UF(0)=8+4√7 18.583(10)and below TcP, unsaturated (TUF) and saturated ferromagnetic
the groundstate becomesa ferromagne≃ticinsulatorwith (TF)crossoversinMH-likeinsulatorswithelectron-holepair-
the maximum spin sz = 3 [23, 32]. The critical value, ing. Note that electron-electron pairing is unlikely to occur
U (T) at which ∆c(hU :iT)2= 0, depends on the temper- aboveT >0.08inaspin(gapless) liquidphase. Theenergies
c 3 andtemperaturesaremeasuredinunitsoft=1,thehopping
ature. At U < U (0), ∆c(T) in a limited range of U is
c 3 parameter.
negative [23]. Thus accordingto Eq. (2), the states with
N = 3 become energetically less favorable when com-
h i
pared with N =2 and N =4 states. This is explicit
it is not a priori obvious that such a mechanism in an
h i h i
evidencefortheelectronphaseseparationinstabilityand
ensemble of small clusters can survive at finite tempera-
the existence (at finite temperature) of particle-particle
tures. Our exact solution demonstrates that it does sur-
orhole-holebindingdespiteabareelectronrepulsion. At
vive. In Fig. 2, we show condensation with bound, dou-
zero temperature, U > U (0) and N = 3, the calcu-
F bleelectronchargeanddecoupledspinatU U (0)and
hlastzeid=sp23i,ngap∆s3(U :T),fortransithionifromhszi= 21 to aTrsPe(aUls)o≤conTde≤nsTecdP(wUit)h. ’BboesloownizTasPti(oUn’),ofthspeins≤painndcdcehgarregees
degrees and possible ’superconductivity’. As an impor-
32+U2+4√64+3U2 U tant observation we found is that the variation of pair-
∆s3(U :T =0)= p 2 −2− 3(,11) ing gaps, ∆P(U) and ∆e−h(U) versus U, in the ground
state closely follows the variation of TP(U), and T (U)
c c
isalsonegative,whichmanifestsathermodynamicinsta- respectively. For U U (0) there is correlation between
c
bility (χs < 0) and the phase separation into ferromag- the spinpairinggap≤∆P(U)andcorrespondingcrossover
ennestiecm”bdleomofaicnlus”stewrist.hThhszuis =we23caanndclahsssziify=an−d32seienvtahre- tfoeumnpdersaimtuirlaerTcsPor(rUel)avtieosrnssusinUU. AsptalcaergbeertwUee≥n fUecr(r0o)m,awge-
ious phases, phase transitions and phase separations in netic spin pairing gaps and corresponding ferromagnetic
finitesystems. Therefore,phasetransitionsininteracting crossovertemperatures.
many-body system is not a characteristicof an infinitely Our calculations for the U U range may also be
c
large thermodynamic system, but can well be defined in used to reproduce the behavior≤of T (p) versus pressure
c
canonical and grand canonical ensembles for finite sys- p in the HTSCs, if we assume that the parameter U de-
tems without the use of the thermodynamic limit. creases with increasing pressure. In Fig. 2, the crossover
temperaturesforelectron-electron,TP,electron-hole,T ,
c c
and,coupled spin, TP, pairingsareplotted as a function
s
C. Electron charge and spin pairings of U. The shown condensation of both doubled electron
charge, and zero spin sz = 0 (singlet) degrees below TP
s
The fact that electron pairing can arise directly from is indicative of a possible mechanism of superconductiv-
the on-site electron repulsion between electrons in small ity in the HTSCs near optimal doping. In the HTSCs,
clusters was found quite early [33, 34, 35, 36]. However, superconducting transition temperature T generally in-
c
6
creaseswithpressurefirst,reachesthemaximumvalueat 4
some critical pressure p and then decreases with pres-
c
sure [39] in agreement with our result for the variation <N>
ofTP(U)versusU. Thesepressureeffectsinthecuprate 3
s
family reflect the changes in electron pairing due to the
moderateCoulombinteractionU. Notice,thatatenough >
slahrogwenUi,nTFc(ipg). 2d.ecrTehaissesmwigithht Uexpulnadinerwphryesstuhree,praesssituries z> or <s2 µµµ === µµµPP ++- 000...000000505 µP = 0.4746, U = 2.5, T = 0.003
Tc(p)decreasesacrosssomeoforganicandfamiliesalkali <N P
1
doped fullerene superconductors [40]. <sz>
Our results for N 3 in Fig. 2 suggest that the en-
≈
hancementofT inthe parentalMHcuprates,withrela-
c 0
tivelylargeU intheoptimallydopedHTSCs,canbedue
to an increase of pairing by decreasing U under pressure
-0.2 -0.1 0 0.1 0.2
ratherthananincreaseofthe pressure-inducedholecon- magnetic field
centration. Thus it appears that the 4-site cluster near
N 3 indeed captures the essential physics of the pres-
sur≈e effect on electron pairing and BC in HTSCs near FIG. 3: The average number of electrons and magnetization
optimal doping. Similarly, our calculations in Fig. 2 dependencies versus an applied magnetic field at U = 2.5,
for U U range can also be useful in predicting the T = 0.003 and µP = 0.4746. Note how a small change in
≥ c the chemical potential (from µP ±0.005) forces the (paired)
variation of T (p) and T (p) versus pressure p in fer-
UF F systemintoamagneticspinliquidstatewithanunpairedspin
romagnetic insulators and Co-doped anatase TiO2 [11]. and N ≈3.
D. Charge-spin separation tions can be monitored by small variation of a chemical
potential or magnetic field. The coupling between the
spin and charge degrees of freedom is manifested due
Until recently, electrons were thought to carry their
to the composite nature of the electron, having charge
charge and spin degrees simultaneously. It is certainly
(holon)and spin(spinon). Thus,the spin singletpairing
true for non-interacting electrons, where U = 0. At
in the spin sector and the electron-electron or hole-hole
zerotemperature,aparamagneticFermiliquidwithzero
pairings in the charge sector are not independent but
charge and spin gaps in Fig. 2 exists only at U = 0 and
demonstrate coherency and a strong coupling between
U = U (0). At finite temperatures the corresponding
c
them, which is necessary for possible ’superconductiv-
chargepseudogaps vanish atparticle-partricle/holepair-
ity’. Below the critical temperature of crossoverTP, the
ing crossover temperatures and there is also no sign of c
charge and spin separation above T and TP. However, charge degrees are coupled and simultaneous condensa-
c c tion of spin degrees below TP results in reconciliationof
astemperaturedecreasesbelowthesecrossovertempera- s
charge and spin and possible full ’bosonization’ of elec-
tures only the charge pseudogaps are formed, while spin
trons. Thus the electron fragments into the charge and
excitations remain gapless and these paramagnetic spin
spinexcitationstatesandsuperconductivityarisesdueto
liquidstatescoexistingwithelectron-electronorMH-like
the charge-spin separation and subsequent condensation
electron-holepairingswillbelabeledasacharge-spinsep-
ofthesechargeandspindegreesthatarebothbosonicin
aration. Gaplessspin responsedue to smallvariationsof
nature [5].
magneticfieldorelectronconcentration(chemicalpoten-
tial)isindependentofthechargedegreesoffreedom. By
decreasing temperature, a partial charge-spin reconcilia-
tion takes place in Fig. 2 below ferromagnetic crossover E. Phase separation
temperaturesduetotheformedspinpseudogapwithcou-
pled charge and spin degrees. Accordingly, electron-hole Themechanismofphaseseparation(i.esegregation)in
pairs and ferromagnetic coupled spins are bounded in small clusters near N =3 is also quite straightforward
h i
charge and spin sectors below ferromagnetic crossover and simply depends on the pairing conditions in charge
temperature. The effect of charge-spin separation be- and spin channels respectively [23]. At U > U (0) in
c
comes stronger with increasing U > Uc(0), since the Fig. 2, we notice a stable ferromagnetic state with un-
distinction between the corresponding charge and spin saturated magnetization and a ferromagnetic state with
crossovertemperatures increases with U (see Fig. 2). saturated Nagaoka magnetization at larger U > U (0)
F
In Fig. 3, we examine the behavior of charge and spin values. As temperature increases the system undergoes
degrees of freedom below the spin condensation temper- a smooth crossover from a ferromagnetic MH-like insu-
ature TP. Fig. 3 shows how the spin degrees, at an lator with ∆e−h = 0 and ∆ = 0 into a paramagnetic
infinitessimal magnetic field, follow the charge. Notice MH insulator wit6h sz 0.FT6he ferromagnetic critical
h i ≈
how strong inter-configuration charge and spin fluctua- temperature T (U) increases monotonically with U and
F
7
as U the limiting T (U ) = 0.0125 value ap-
F
proac→hes∞to the maximum spin→pa∞iring temperature TP.
s (T vs ) 4-site phase diagram for U=4
A ferromagnetic phase with broken symmetry was ob- 0.08 close up nea r P
tained here in the presence of an infinitesimal magnetic P
spin susceptibility TC for onset of
field and increasing temperature leads to a F-PM tran- 0.07 peak positions charge pairing
from canonical
sition and restoration of the symmetry in paramagnetic 0.06 ensemble
from canonical
phase. The level crossing at Uc(0) (UF(0)) in the pres- e ensemble
enceofanappropriateinfinitesimalmagneticfieldbrings 0atur.05 (TPS (spin pairing T)
aboutphaseseparationofthethermodynamicallyunsta- 0er.04
p
ble ensemble of clusters with h− > h+ into spin up and 0em.03
spin down ferromagnetic ”domains” (see section IVA). t
0.02
It appears that the canonical approach yields also an
adequate estimation of possible pair binding instability 0.01
near N = 3 in an ensemble of small clusters at rela-
h i 0.00
tively low temperature and moderate U U (0). New 0.640 0.645 0.650 0.655 0.660 0.665 0.670
c
≤ chemical potential (
important features appear if the number of electrons
N = 3 is kept fixed for the whole system of decou-
h i
pled clusters, placed in the (particle) bath, by allowing
FIG.4: ThephaseT-µdiagramincanonicalensembleoffour-
theparticlenumberoneachseparateclustertofluctuate.
One is tempted to think that due to symmetry, there is psihteascelubsteelorws cTlocPsestuoghgNesits≈t3heanedxisµtPen=ce0o.6f5e8leacttrUon=-el4e.ctTrohne
a single hole on each cluster within the N = 3 set in
h i pairing at finite temperature with unpaired spin states. The
the ensemble. However, due to thermal and quantum
zero (magnetic) spin susceptibility peak in spin (melted) liq-
fluctuations in the density of holes between the clusters uid state terminates at finite TsP and as temperature is fur-
U < Uc(0)), it is energetically more favorable to form ther lowered the spin pairs begin to form. This picture with
pairs of holes. In this case, snapshots of the system at µ−(T) > µ+(T) supports the idea that there is inhomoge-
relativelylowtemperaturesandatacriticalvalue(µ in neous,electronicphaseseparationhere. AsU increasesabove
P
Fig.4)ofthechemicalpotentialwouldrevealequalprob- Uc(0) = 4.584, these inhomogeneities disappear and a stable
abilitiesoffinding(only)clustersthatareeitherholerich unsaturatedferromagneticMH-likeinsulatingregionemerges
around optimal doping shown in Fig. 5.
( N = 2) or hole poor ( N = 4). Thus ensemble of 4-
h i h i
site clusters at N = 3 is thermodynamically unstable,
h i
µ+ µ−, and can lead to macroscopic phase separation
≤
onto hole-richand hole-poorregionsrecently detected in
a nonzero magnetic field or a finite temperature is re-
super-oxygenatedLa2−xSrxCuO4+y,withvariousSrcon- quired to break the pairs. From a detailed analysis, it
tents [10, 23].
becomesevidentthatthesystemisonthevergeofanin-
The level crossing under slight doping (infinitesimal stability; the paired phase competing with a phase that
variation of chemical potential) brings about phase sep- suppresses pairing which has a high, zero-field magnetic
aration of ensemble of clusters and this can be linked to susceptibility. As the temperature is lowered, the num-
the formation of inhomogeneity, consisting of hole-poor ber of N 3 (unpaired) clusters begins to decrease
(superconducting)andhole-rich(antiferromagnetic)”do- while ahmiixt≈ure of (paired) N 2 and N 4 clus-
mains”atU <Uc(0). Thuswecanconcludethatensem- ters appears. In Fig. 4 theh spii≈n pairinghphia≈se below
ble of 4-site clusters with N =3 at relatively low tem- TP competes with a phase (having a high magnetic sus-
h i s
peratures is unstable for all U >0 values with regard to ceptibility) that suppresses pairing at ‘moderate’ tem-
phase separation due to spontaneous symmetry braking peratures. Surprisingly, the critical doping µ (which
P
into ferromagnetic or superconducting ”domains” either corresponds to a filling factor of 1/8 hole-doping away
in spin or charge sectors respectively. from half filling), where the above pairing fluctuations
The T-µ phase diagramfor the 4-site cluster near µ , take place when U < U (0), is close to the doping level
P c
is shown in Fig. 4. This exact phase diagram at U = 4 nearwhichnumerousintriguingpropertieshavebeenob-
in the vicinity of the optimally doped (N 3) regime served in the hole-doped HTSCs. For example, the spin
has been constructed based on the condition≈∆c(T) < 0 pseudogapcan be drivento zero also by applying a suit-
with µ−(T) > µ+(T), reported earlier in Ref. [23]. The able magnetic field. This factor leads to the stability of
electron pairing temperature, TP, identifies the onset of electron ”dormant” magnetic configuration in a narrow,
c
charge pairing. As temperature is further lowered, spin critical doping region close to N 3, competing with
pairs begin to form at TP. At this temperature (with N = 2 and N = 4 stateshasih≈ave seen in a recent
s h i h i
zero magnetic field), spin susceptibilities become very experiment [10]. Thus our results, at electron concen-
weakindicatingthe disappearanceofthe N 3states. tration N 3, clearly demonstrate phase separation
h i≈ h i ≈
Below this spin pairing temperature, only paired states andbreakdownofFermi liquid behavior due to electron-
in ”metallic” liquid (∆e−h = 0) in the midgap region electron/hole pairings and Nagaoka ferromagnetism in
are observed to exist having a certain rigidity, so that the absence of long-range order.
8
F. Phase diagrams

4-site phase diagram: U=6
0.6
In Figs. 4 and 5, the phase diagrams for the ensem- charge max.
ble of 4-site clusters are shown, where we define the 0.5 spin max.
charge peak (i.e maxima) T (µ) to be the temperature re II TAF
with maximumχc(µ) ata givcenµ. Possiblepeaksin the 0ratu.4 TAF( TUF
zero magnetic field spin susceptibility χ (µ), when mon- e
s p
itored as a function of µ, can also be used to define an m
associated temperature, Ts(µ). In addition, for a given 0r te.3 TC(
µ, TAF(µ) defines the temperature at which AF (spin) ve
gap disappears, i.e. ∆s(TAF) = 0. These have been 0sso.2 III TS(
cTocn(µst)r,uTcste(µd)aalmndosTtAeFx(cµlu)s,ivdeelfiynuedsinpgretvhioeutselym.peWraetuhraevse, 0cro.1 TUF( II I III
identifiedthefollowingphasesinthesediagrams: (I)and
(II) are charge pseudogap phases separated by a phase 0.0
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
boundary where the spin susceptibility reaches a maxi-
chemical potential (
mum,with∆e−h(T)>0,∆AF(T)=0;atfinitetempera-
ture,phaseIhasahigher N andcoupledspincompared
h i FIG. 5: Temperature T vs chemical potential µ phase dia-
to phase II spin liquid phase; Phase (III) is a MH-like
gram for the grand canonical ensemble of four-site clusters
antiferromagneticinsulator with bound charge and spin,
at U = 6 and h = 0. Regions I, II and III are quite simi-
when ∆e−h(T)>0, ∆AF(T)>0; phase separation (PS) lar to the ones found in enlarged Fig. 4 for U =4 [23], again
in T µ plane for U = 4 with a vanished charge gap at showingstrongcharge-spinseparation. However,achargegap
−
N = 3, now corresponding to the opening of a pairing opens as a new bifurcation (I and II phases) which consists
hgapi(∆P(T) > 0) in the electron-electron channel with ofchargeandspinpseudogaps,(replacingthelinephasePin
∆c(U :T)<0. Wehavealsoverifiedthewellknownfact the previous figure: see text). In the the vicinity of hNi≈3
3
that the low temperature behavior in the vicinity of half below TUF(µ) unsaturated ferromagnetism coexists with the
MH-likeinsulator.
filling,withchargeandspinpseudogapphasescoexisting,
represents an AF insulator [22]. However, away from
half filling, we find very intriguing behavior in thermo-
dynamical charge and spin degrees of freedom. In both (mainly)onthe QCPatµs,atwhichthe spinpseudogap
phasediagrams,wefindsimilarMH-like(I),(II)andAF- disappears. The regions (I) and (II), with the prevail-
like (III) charge-spinseparated phases in the hole-doped ing charge gap ∆c(T)> 0, are separated by a boundary
regime. In Fig. 5, spin-charge separation in phases (I) where the spin gap vanishes. At critical doping µs with
and (II) originates for relatively large U in the under- Ts 0, the zero spin susceptibility χs exhibits a sharp
→
dopedregime. Incontrast,Fig.4showstheexistence(at maximum. The critical temperature Ts(µ), which falls
U = 3) of a line phase (with pairing) similar to U < 0 abruptly to zero at critical doping µs, implies [15] that
case with electron pairing (∆P(T)>0), when the chem- thepseudogapcanexistindependently ofpossiblesuper-
ical potential is pinned up on doping within the highly conductingpairinginFig.5. Incontrast,Fig.4showsthe
degenerate midgap states near (underdoped) 1/8 filling. existence at U = 3 of a line phase (with pairing) similar
to U < 0 case with a spin pseudogap (∆s(T) > 0) and
Among other interesting results, rich in variety for
electronpairingpseudogap(∆P(T)>0),whenthechem-
U > 0, sharp transitions and quantum critical points
ical potential is pinned up on doping within the highly
(QCPs) are found between phases with true charge and
degenerate midgap states near (underdoped) 1/8 filling.
spin gaps in the ground state; for infinitesimal T > 0,
these gaps are transformed into ‘pseudogaps’ with some We have also seen that a reasonably strong magnetic
nonzero weight between peaks (or maxima) in suscepti- fieldhasadramaticeffect(mainly)ontheQCPatµs,at
bilities monitoredas a function of doping (i.e. µ) as well which the spin pseudogap disappears. It is evident from
as h. We have also verified the well known fact that the our exact results that the presence of QCP at zero tem-
low temperature behavior in the vicinity of half filling, perature andcriticalcrossovertemperatures, givestrong
withchargeandspinpseudogapphasescoexisting,repre- support for cooperative characterof existing phase tran-
sentsanAFinsulatorintheHubbardclusters[22]. How- sitions and crossovers in finite size clusters as in large
ever,awayfromhalffilling,wefindveryintriguingbehav- thermodynamic systems [37, 38].
ior in thermodynamical charge and spin degrees of free- As an important remark, in the noninteracting case,
dom. In both phase diagrams, we find similar MH-like U =0, the charge and spin peaks follow one another (in
(I), (II) and AF-like (III) charge-spin separated phases sharpcontrastto the U =4 and6 cases,inregionsI and
in the hole-doped regime. In Fig. 5, spin-charge separa- II where charge (as well as spin) maxima and minima
tion in phases (I) and (II) originates for relatively large are well separated) indicating that there is no charge-
U in the underdoped regime. We have also seen that spin separation,evenin the presence of a magnetic field.
a reasonably strong magnetic field has a dramatic effect Inthe entire rangeofµ, the chargeandspinfluctuations
9
exact cluster simulations of the Hubbard model displays
incipient pairing, ferromagnetism, phase separation and

2 4-site cluster at temperature T = 0.02
8 8 other phenomena are remarkably similar to those found
in small nanoparticles, transition metal oxides and high
er 7 7 Tc cuprates [14].
b
m6 6
u
n
n
o5 5 V. CONCLUSION
r
ct
ele4 4
e In summary, we have illustrated how to obtain phase
rag3 U = 2 3 diagramsandidentifythepresenceoftemperaturedriven
e U = 5
av U = 25 particle-particle/hole and spin pairing crossovers below
2 2
U U <Uc(0), quantum critical points (µs, µc) and charge-
1 1 spinseparationregionsforanyU >Uc(0)intheensemble
-1 0 1 2 3 of 4-site Hubbard clusters as doping (or chemical poten-
chemical potential (
tial) is varied. Specifically, our exact solution pointed
out an important difference between the U = 4 and
U = 6 phase diagrams near half filling (i.e. one hole off
FIG. 6: Variation of average electron concentration versus µ
half filling), which can be tied to electron-electron pair-
inensembleof2×4clusterswiththecouplingsc=1between
ing and possible superconductivity and ferromagnetism
the squares for various U values, h = 0 and T = 0.02. Here
in doped HTSCs and transition metal oxides or disul-
we can easily identify regions quite similar to theones found
in Fig. 1, showing strongcharge-spin separation with various fides respectively. Ourresultsshowthe pairingcrossover
crossover temperatures. This phase suggests the existence of near N 3 and strongly suggest that particle-particle
h i≈
electron-electron pairing at low temperatures, T ≤0.028. pairing and spin coupling can exist at U < U (0), while
c
particle-holebinding andferromagnetismis presumedto
occur for U > U (0). Exactly at U = U , paramagnetic
c c
directly follow one another,i.e. without charge-spinsep- Fermi liquid is stable in the ground state and all finite
aration. For interacting electrons, we notice similar re- temperatures. At U > U (0), there is another subse-
F
spond of charge and spin degrees at N = 3 only at quenttransitionintoasaturatedferromagneticinsulator
h i
singlepoint, U =Uc(0). Inthe atomiclimit, t=0,a full with maximum spin hszi ≈ 23. It is also apparent that
charge-spinseparation,exactlyathalffilling,takesplace short-range correlations alone are sufficient for pseudo-
in the groundstate and the correspondingMH crossover gaps to form in small 4-site and larger 2 4 clusters,
×
temperature of the metal-insulator transition occurs at which can be linked to the generic features of phase di-
TMH =U/(2ln2). agrams in temperature and doping effects seen in the
HTSCs. Theexactclustersolutionshowshowchargeand
spin gaps are formed at the microscopic level and their
G. Coupled 4-site clusters behaviorasafunctionofdoping(i.e. chemicalpotential),
magnetic field and temperature. The spin and charge
We have also carried out exact numerical diagonaliza- crossover temperatures can also be associated with the
tion and calculations of the charge gap and pairing in formationof pairinggaps belowTP(U). As temperature
c
8-site, 2 4 planar clusters with periodic boundary con- decreases further, a simultaneous BC of spin degrees of
ditions in×both directions, to illustrate similar effects on freedomtakesplacebelowTP(U). TheincreaseofTP(U)
s s
thepropertiesdescribedaboveforthe4-siteclusters. The withdecreaseofU belowUcinFig.2reproducesthevari-
pairingfluctuationsthatareseenforthe4-siteclusterare ationofTcversuspintheoptimallyandnearlyoptimally
foundtoexistfor2 4laddersnearhalffilling( N 7) doped HTSC materials [39], which suggests a significant
× h i≈
as well. Most of the trends observed for the 4-site clus- increase of pairing temperature Tc under pressure or U
ters, such as the MH-like charge (pseudo) gap, AF-like due to the enhancement of the zero temperature charge
(pseudo)gap,(spin)pseudogap,electronandspinpairing and spin pairing gaps. In addition, our calculations pro-
(pseudo) gaps andwith correspondingcrossovertemper- vide important benchmarks for comparison with Monte
atures are also observed here. Moreover, we find corre- Carlo, RSRG, PCT and other approximations.
sponding critical U (0) and U (0) values for the pairing Finally, we have developed novel theoretical concepts
c F
instabilities and vanishing of corresponding pseudogaps and techniques, which are especially suitable and effi-
at finite temperatures T and T similar to the 4-site cientforunderstandingofnascentsuperconductivityand
UF F
cluster. Thefluctuationsthatoccurhereatoptimaldop- magnetism using the canonical and grand canonical en-
ingareamongthestateswith N 6,7and8electrons. semble for small clusters. Our results show the cooper-
h i≈
At N 7, we clearly observe the dormant magnetic ative nature of phase transition phenomena in finite-size
h i ≈
state as noticed in the 4-site cluster with a slight varia- clusters similar to large thermodynamic systems. The
tionofthechemicalpotentialormagneticfield. Thusour developed approach allows an exact and unbiased study
10
of the Fermi liquid instabilities in small clusters without to continuous media, and applied for understanding of
the assumption of a broken symmetry. The small nan- phaseseparationandincipientspontaneoussuperconduc-
oclusters exhibit particle-particle,spin-spin pairings in a tivityandferromagnetisminsmallnanometer-scaleclus-
limited range of U, µ and T and share very important ters and Nb, Co nanoparticles [1, 2, 3, 4]. This research
intrinsic characteristics with the HTSCs and magnetic wassupportedinpartbytheU.S.DepartmentofEnergy
oxides. Theseideascouldbeusefulindifferentareasout- under Contract No. DE-AC02-98CH10886.
side the cluster field, to systems ranging from molecules
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