LOCAL p AIRING IN TRE ATTRACTIVERUBBARD MODEL

T .DI MATTEO, F .MANCINI, S.MARRA

1NFM e Dipartimento di Fisica Teorica e S.M.S.A.Universitd di Salerno , 1-84140 Baronissi(SA), 1taly

Received September 12, 1995

The Hubbard model with on-site attractive interaction (negative-U Hubbard model) is studied by means of the composite operatormethod in the static approximation for the case of half filling. At zerotemperature, when the strength of the attractive interaction equalsthe band width, the system exhibits a phase transition to a pair state,where alI the electrons are locally paired. The temperature Tp whichcontrols the cross over to the pair state is calculated as a functionof U. The double occupancy and the spin magnetic susceptibility Xoare calculated for various values of U and T. For strong attractiveinteraction Xo is strongly depressed, and tends to zero as T-+ Tp .

Introduction1.

An interesting problem in the physics of strongly correlated electron systemsis the study of systems w here the formation of electron pairs is possi ble in thereal space: local electron pairing. This problem has been extensively studiedboth from experimental and theoretical point of view; for an excellent andexhaustive review we refer the reader to the work of Ref. 1. The formationof local pairs requires an effective attraction which overcomes the repulsiveCoulomb potential. There are several mechanisms which can induce a shortrange attraction among the electrons ( coupling of the electrons with thelattice, coupling of the electrons with quasibosonic excitations, chemicalmechanisms, etc. ) and many physical systems exhibit properties relatedto the existence of local electrons pairs ( superconductors, transition metaloxides, alternating-valence compounds, etc. ). A simple model to inve~tigatethis problem is given by the extended Hubbard model

H = 2::: tijCt ( i) .C(]) + U 2::: nT( i)n! ( i) + 1/22::: Wij n( i)n(]) -Jl, 2::: n( i).ij i ij i

(1.1)The variable i stands for the lattice vector Ri. { C( i), c t ( i) } are annihilationand creation operators of c-electrons at site i, in the spinor notatioI1:

ct = (cr,cl), (1.2)

@ T .Di Matteo, F .Mancini, S.Marra, 1996 -ISSN 0452-9910. Condensed Matter Physics 1996 No 8 (109-117)

110 T.Di Matteo, F.Mancini, S.Marra

tij is the transfer integraI; the U- and W -terms describe on-site and intersite

interactions: respectively; n(i) = nT(i)+n!(i): with nO'(i) = c!(i)cO'('i) beingthe number operator of electrons with spin O"; J.l is the chemicaI potential.This modeI has been extensively studied for various lattice dimensjons bvmeans of different theoreticaI approaches and numericaI analysis: appropri-ate references can be found Ref.l. Many analytic and numerjcaI methodshave been developed for the study of strongly correlated electron systems.

.In the Iast :years we have been developing a method of calculatjon~ the com-posite operator method: based on the consjderation that in strongly corre-Iated systems it is not convenient to develop approximate solutions aroundthe originaI particles~ since in the observation their properties are stronglymodified by the interactions. Instead~ we develop approximate solutionsstarting from a convenient set of composite fields~ which correspond to theoriginaI fields dressed by interactions and compute their properties by a selfconsistent calculations. The method has been applied to different models:p-d model [2]: Heisenberg model [3]: Hubbard model [4]: Kondo-Heisenbergmodel [5]. In this article we shall consider the case of the Hubbard modelwith on-site attraction. In Section .2 we derive the explicit expression forthe Green :s function~ in the static approximation. In Section 3 we presentsome results for the double occupancy and the spin magnetic susceptibilityfor the case of half filling. By analytic method we can show that at zerotemperature the system exhibits a phase transition to a pair state: whereall the electrons are locally paired in a singlet state; the criticaI value U c forthe interaction potential at which the phase transition occurs is equal to Uc=-8t~ 8t being the band width.

The negative-U Hubbard model2.

Systems where the induced local attraction overcomes the on-site Coulombrepulsion can be conveniently described by the so-called negative- U Hubbardmodel; defined by the following Hamiltonian

(2.1)

with U < o. We shall study a squared two-dimensionallattice in the nearest.neighbour approximation; the hopping matrix has the expression

(2.2)tij

1Q(k) = 2[cos(k:1;a) + cos(kya)]: (2.3)

a being the lattice constant. In the framework of the composite operatormethod [4]: we introduce the field

~(i)

1}(i)(2.4)1/J(i)

where ~( i) and 1]( i) are the Hubbard operators

{2.5)~q(i) = cq(i)[l- n-q(i)]~

~~.OJ170"(i) = cO"(ì)n-0"(ì)..

By means of Hamiltonian (2.1) one obtains the equation of motion:

~~.I )

where the source term Jli) has the expresslon

-J.L{(i) -4tlcU(i) + 7r~'l.)J-(J.L -U)1J( i) + 4t7r( i)

j( i) = t;l.~)

with

CU(z) ~.l.~)-

7r(ì) lZ.IU}-

Let us introduce the retarded thermal <.;reen IS tunctlon

~:l.ll)

where the bracket indicates the thermal average; by decomposmg J~l) as:

~~.1L.)j( i) = f(-i V) + oj( i)

with the constrain~Z.I.j)({lijn(ì),'I/Jm(j)})E.T = O

the Fourier transform of S ( i, j) takes the expression [ 4]

~~.iq)f)'lk,w) =

where the kinetic f(k) a.nd the dyna.mica.l 2.:(k,W) pa.rts 01 the sel1 energya.re gi ven by

~~.l;))

(2.16)

f.lk)

~(k,w)

~

m(k)l-J.(k),

M(k,w)S-l(k,w).

~

witb

~~.l{)

(2.18~

(2.19~

llk,

m(k~

~~

M(k,wJ -

In order to explicitly calculate the Green lS tunctlon we must lntroauce someapproximation. In this article we compute S(k,w) by neglecting the dynam-ical part E(k,w). As shown in our previous works [4], this approximatioDseems quite reasonable, since we can reproduce with quite good accuracyrnost of the results obtained by means of numerical analysis. Then, weneed the knowledge of the normalization matrix given by (2.17) and of the

m-matrix, defined by (2.18). The attractive interaction will favour the for-mation of local singlet pairs and contrast the establishment of a magneticorder [6]j therefore, we restrict the analysis to the paramagnetic phase andwe have

(nj(i)) = ,

A straightforward calculation gives:

(n!(i)) n/2. (2.20)

~ IÒ1 I~2 )I(k)

n

1-2

O ~ ) (2.21)

f(k)

m12(k)I;1(k)

m22(k)I;1(k)(2.22)

mll(k) (2.23)

m12(k) (2.24)

m22(k) (2.25)

where

i:::J. (2.26)

(2.27)p

By means of tbese results, tbe matrix elements of tbe Green 's function (2.14 )bave tbe following expressions

(2.28)

2Q(k) L\.~(k)

E2(k) + iTJ 1,

S12(k,w) (2.29)1.] ,

E2(k) + iTJ:,I.}

(2.30)2Q(k) + ~E(k)JCA) -E2(k) + iTJ .

El(k).~

R(k) + Q(k),F;2(k)

.i

1

R(k) -Q(k). (2.31)

= 111 .[2Q(k) + ~}::;(k) +

4Q(k) w -El(k) + iTJ

(R[~( i)TJ t (j)])F.T = (R[T}( i)~t (j)])F.T = ,cc ~ -,

m12 r 1

Local pairing in the attractive Hubbard mode] 113--~

The definitions afe

{ L\ + a(k)[p + (l -n )122]}, (2.32)2t

111122

(2.33)

{1- n)m12(k)~E(k) = -U + 1 T T. (2.34)

n = (ct .c) = 2[1- (~(i)~t(i)) -(1](i)1]t(i))]. (2.35)

The amplitude ~ is expressed in terms of matrix elements of S( i, j), wherej is the first nearest neighbouring site to i, as

~ = (~!X(i)~t(i)) -(TJ!X(i)TJt(i)). (2.36)

The parameter p, defined by (2.27), is related to intersite charge, spin andpair fluctuations and is computed by means or the equation

(~(i)1]t(i)) = O, (2.37)

which recovers the Pauli principle. Equations (2.35)- (2.37) constitute a setof coupled equations which :fix in a self consistent way the three parameter/l., Ll and p. Once these quantities are evaluated, the Green 's functioncan be computed; then, the properties of the system can be computed.For example, one quantity which characterizes the system is the doubleoccupancy

(2.38)

which gives the average number of sites occupied by two electrons. Anotherimportant quantity which describes the magnetic properties is given by thespin magnetic susceptibility; in the present approximation this quantity isgiven by [7]

X(k,w) -

+ (2.39)

where Q~fJ-y6 is the retarded part or

(2.40)

Sa{3(k, w) being the eIements of the causaI thermaI Green '5 function.

.Lll.L22The expressions (2.28)-(2.30) for the matrix elements contain three param-eters, J.L,~ and p, which must be self consistently calculated. The chemicalpotential J.L is determined in terms of the particle density n by means of theequation

Qf212) + 1;1( Qf212 + 2Qf222 + Qf222)],

T. Di Matte~E.Yancini, S!!iarra114

Half filling and pair state3.

In a previous work [8] we have studied the system for various values of theparticle densityj the results are in good agreement with the data obtainedby numerical analysis [9] .In this article we concentrate on the case of halffilling. By analytic methods it is possible to show that at zero temper-

0.50

CI 0.40

0.30

Figure 1. The double occupancy D is reported as a function ofthe potential strength U /t for various values of the

reduced temperature KBT /t.

~E-o

Figure 2. The "critica!" temperature T p which characterizes the"phase" transition to the pair state is reported as afunction of the potentiaI strength U /t.

ature the system exhibits a phase transition: there is a critical value ofthe interaction Uc=-8t, such that for IUI > 8t the system is in a pair statecharacterized by the fact that all electrons are locally paired; the double OC-cupancy is equal to 1/2 and the uniform static spin magnetic susceptibility

Xo = X(o, O) vanished. Details of calcu)this note we limit ourselves to presentthe double occupancy is reported as aent values of temperature. In the lim

Figure 3. The uniform static spin magnetic susceptibility X(O)is reported as a function of the potential strength U /tfor various values of the reduced temperature I(BT /t.

Figure 4. The uniform static spin magnetic susceptibility X(D)is reported as a function of the reduced temperatureKBT /t for various values of the potential strength U /t.

upancy is equal to 1/4; by increasing IUllocal pajring is favoured andncreases; for zero temperature D reaches the maximum value D = 1/27 = -8t; for finite temperature the thermal fiuctuations tends to breaklocal pajrs and D tends asymptotically to the maximum value only inlimit oflUI -+ 00. To study the transition to the pajr state, for a fixed~ntial we have calculated the value of the temperature Tp such that-D < 10-4; the results are reported in figure 2 where Tp is given as

iations will be presented elsewhere, inthe results of calculations. In figure 1function of the interaction for differ-it of vanishing interaction the double

116 T. Di Matteo, F.Mancini, S.Marra

a function of IUI. The line in the plane (T ,U) indicates a crossover to theregime of pair state. Approximately, we might take Tp as the critical tem-perature for the breaking of electron pairs. From mean field results [1] weexpect that below Tp there is another critical temperature which separatesthe phase of uncorrelated pairs from a long-range ordered phase ( supercon-ductivity or/and charge density wave). The possible existence of an orderedphase in which the pairs are correlated is currently under investigation bymeans of COM.The uniform static spin magnetic susceptibility Xo is reported in figure 3as a function of U for different temperatures. At zero temperature Xo goesabruptly to zero when U reaches the value U = -8t, in correspondence tothe phase transition to the pair state. In this phase the electrons are allpaired in a singlet state and of course there is no magnetic answer. WhenT # O, Xo goes asymptotically to zero only in the limit of infinite U, butfor T < TpXo is practically zero, as shown in figure 3. Genera.lly, the spinmagnetic susceptibility is strongly depressed by the attractive interaction.This can be clearer seen in figure 4 where Xo is given as a function oftemperature for different values of the interaction. For weak attraction thesusceptibility has a Heisenberg-like dependence, as in the case of repulsiveattraction [7], though quite depressed. For strong attraction the dependenceof Xo on T is completely different : Xo ~ 0 as T ~ T p and increases with T .This is due to the fact that for strong attraction the electrons are practicallyall singlet paired and can give a non-zero contribution to the susceptibilityonly when the thermal fluctuations break the pair .

References

[1] Micnas R., Ranninger J ., Robaszkiewicz R. Superconductivity in nar-row-band systems with Iocal nonretarded attractive interactions. / / Rev.of Mod. Phys., 1990, voI. 62, p. 113.

[2] Ishihara I., Matsumoto H., Odashima S., Tachiki M., Mancini F. Mean-fieId analysis in the p-d modeI of oxide Superconductors. / / Phys. Rev.B, 1994, voI. 49, p. 1350;Mancini F ., Marra S., Villani D., Matsumoto H. Local magnetic mo-ment in the two-dimensional reduced p-d model. Preprint University ofSalerno (1995).

[3] Allega A.M., Odashima S., Matsumoto H., Mancini F. Static and dy-namical spin susceptibiIity in 2D antiferromagnetic Heisenberg model./ / Physica C, 1994, voI. 235-240, p. 2229.

[4] Mancini F ., Marra S., Matsumoto H. Doping dependence ofon-site quan-tities in the two-dimensional Hubbard modeI. / / Physica C, 1995, voI.244, p. 49;Mancini F ., Marra S., Matsumoto H. Energy and chemical potential inthe two-dimensional Hubbard model. / / Physica C, 1995, voI. 250, p.184.

[5] Matsumoto H., Allega A.M., Odashima S., Mancini F. Metal-insulatortransition in Kondo-Heisenberg model of oxide superconductors. / /Physica C, 1994, voI. 235-240, p. 2227.

[6] Lieb E.H. Two theorems on the Hubbard model. / / Phys. Rev. Lett.,1989, voI. 62, p. 1201.

~7] Mancini F ., Marra S., Matsumoto H. Spin magnetic susceptibility in thetwo- dimensional Hubbard model. / / Physica C (in press ).

117Loca} pairing in the attractive Hubbard mode}

[8] Di Matteo T ., Mancini F ., Marra S., Matsumoto H. AnaIysis of the two-dimensional negative- U Hubbard modeI by composite operator method.(Preprint JuIy 1995). Submitted to Physica B.

[9] Randeria M., Trivedi N ., Moreo A., ScaIettar R. T. Pairing and spin gapin the normaI state of short coherence Iength superconductors. / / Phys.Rev. Lett., 1992, voI. 69, p. 2001.

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