10/96/119

United Nations Educational Scientific and Cultural Organizationand

international Atomic Energy Agency

INTERNATIONAL CENTRE EOR THEORETICAL PHYSTCS

SPIN AND CHARGE ROTATION INVARIANT APPROACHTO THE HUBBARD MODEL

J.J. Vicenle Alvarez1, C.A. Balseiro1 and H.A. Ceccatlo2

In tern at,ion al Centre for Tlieorelica.l Physics, Triesle, Tla.ly.

ABSTRACT

We present a slave-boson formulation for the Iiubbard model that preserves the spin-

rotalion and pa.rlicle-liole HynnTietrieH of the pa.ramagnel.ic phase at liall filling. Fermion and

boson fields are treated GTI an equal footing in a I'miclioTial iiiLcgrai [orTrnilal.ion. W<; KIIOW

l.Iiat., wliilc; in 2 rind 3 diivKJiiKioiiK llicr<; is A Tnclril-iiiKiilalor Iransilion as OK; inUjraclion

I.J increases, in 1 dimension our approach always gives an insulating state. Short range

spin-spin correlations are included at the saddle point level for all values of // , giving a

good description of the insulating state. We show that the energy gap in the one-particle

Kjxjd.nnn increases [Venn zero as (U — Uc ) ' \ vvliere Uc IN OK; critical value of l.lie on-sik;

inl<;ra<:li()Ti and llie expoTuml. a is 1 and 1/2 for 2 and 3 diiTKmHioTiH r<;spe<:livdy.

MTRAMARE TRIESTE

July 1996

1 Permanent, address: Cenlro Almnico Bariloc.}i<; and TiiHJ.ilulo Ralseiro. 8400 Rariloelie,Argenlina.1 Permanent address: Instiliilo d< Fi'siea R.OHa.rio. Univcrsidad Nacional dc Rosario, "Bvd.27 dc Eebrc ro 210 "Bis. 2000 Rosario, Repiiblica Argcrilina.

I. INTRODUCTION

The iiubbard model, which has been studied for almost thirty years, is considered one

of the simplest models that contains the essential ingredients to understand the physics of

correlated metals [1]. Despite its simple; form, it provides a physical scenario lo describe I.IK;

melal-insulalor Iran si lion. j.}i<; formation of localized magnetic moments, and t.Tic: occurrence

of long-range magTi<;l.i<: order. T'}i<; discovery of higli-Tc snp<;r<:oTidiid.ivil.y [2] renewed the

interest in this model, since it provides a natural framework to describe many properties

of the cuprates. Moreover, it has been argued that superconductivity itself could be a

consequence of the local repulsion // [3].

The big effort, j.o obtain solutions l.o this model bot.Ti t.Tic: ground stale [4] [5] and the

llieriTiodynamic properties led lo only a lew controlled results. Probably the best, results

in the regime of interest, near half filling and intermediate coupling, correspond to those

obtained by numerical simulation [6]. However, the slave-boson approach [7] -applied to the

Iiubbard model by Kotliar and lUtckenstein (KH) [8]- seems to be an attractive starting

point, Cor a systematic study of t.Tic: model. Tlie KR approach is a functional inl.egral method

ill at. reproduees the Gulzwiller solulion [4] at llie saddle-point l<;vd. The advantage of lliis

melliod over the Gulzwiller variational wave function is that it, provides a systematic way to

improve the solution. As pointed out by Li, Wolfie and iiirschfeld [9], the KR theory does

not preserves the spin rotation invariance of the original iiamiltonian. They generalized

t.Tic: llieory and provided a way of formulating a funelional inlegral method vvliieli nalurally

preserves this invarianee. Tn any case. t.Tic: resulls at, th<; saddle-point level arc: equivalent

lo Gut.zwiller's solulion. and in th<; Mol.t. insulaling slate: spin-spin c:.orr<;lations ar<; not

included.

in the present work we introduce slave bosons in the Zou and Anderson [10] scheme to

describe the paramagnetic phase of the Iiubbard model. We devise a functional integral

melliod which presenls t.Tic: following advantages over previous l.Tieories:

/.— Tt. explicitly preserves t.Tic: full spin-rolation invariance and parlicle-liole symmetry

(charge- rotation invariance) of the original half-filled Iiubbard Iiamiltonian.

ii.— The insulating state includes short range spin-spin correlations already at the saddle

point, level.

One: disadvantage: of our approach is t.Tiat. t.Tic: U = 0 limit, is not exactly reproduced, since

we do not renormalize the hopping matrix elements as in the Kit formulation. However, this

renormalization is inconvenient when fluctuations are incorporated above the saddle-point

results [11].

The rest of t.Tic: paper is organized as follows: Tn Sec. 2 we: give: a. detailed description of

t.Tic: formalism, in Sec. 3 we: present. t.Tic: results for 1, 2 and 3 dimensions (D), and in Sec. 4

we: summarize our conclusions.

II. SLAVE-BOSON APPROACH

in this section we present the method starting from a functional integral theory that

makes apparent the spin- rotation and particle-hole symmetries of the original model. We

consider t.Tic: Hubbard HaniilIonian writl.en in the usual notation:

H = -l

Here cllT creates an electron at site i with spin a. nl!7 = cllTcirT and nt = n,.f + n.4. Note that

we: have included an on-site: energy U/2. Tn this notation t.Tic: chemical potential /./ is zero

for the hal [-filled case. Tn the following we will measure the energy in nnit.s of / = 1.

Tn order t.o make: apparent, the global symmetries of the model, we: introduce the following

operators:

/ . \ /

(2)

V « J

-4

ln this representation the iiamiltonian reads

U

<•'".?>

- 8 ) -/'

Global spin rot.ations arc: c:<.|nivalc:nt. t.o tli<; following transformations: ^V- = ySvTf,: , where <f

is a SU(2) matrix. Rot.ations in t.Tic: charge sect.or (part.ic:.l<;-hole t.ransformat.ions) c:.orr<;spond

3

to 4\; = t<jf, where g\ = (a-,)' g {<JZ)' with a,: a Pauli matrix and g a SI.J(2) matrix. Then, it

is straightforward to show that the iiamiltonian (3) is invariant under these transformations

for /./ = 0, i.e.. one particle per si la.

We introduce slave bosons in the Zou and Anderson [10] scheme, where the ferinioTi

operators are written in the following way:

t-iir — Sin (A)

H<;re .H^ is j.}i<; Kingle-occupancy ferinionic operator, and a] and d\ ar<; l.lie empty and double-

occupancy slave-boson creation operators respectively. The electron operator \t,; is then

given by

(5)

where the spin and charge fields are

* ? =•M ~-H

(6)

'' /

Under spin rotations and particle-hole transformations these operators change as

and .J = *l!j gf respectively, with gs and g\ as above.

T'}i<; Hamill.onian vvril.ten in terms of t}i<;se fi<;ld op<;ral.oi'K tal«;s l.lie form

(7)

and becomes equivalent to (1) provided t}i<; following constraint is satisfied:

= 1 . (8)

T'}i<; parl.ition fnncl.ion Z is giv<;n as a coher<;nt-Ktal.<; fnnclional integral of a Lagrangean

L(T) :

Z = / [D,i][Dc][Dd][DX]cxp - drL(r)Ju (9)

with

Mr) =

]Tr

Here A,; is a Lagrange multiplier introduced to enforce the constraint (8).

In order to go beyond the standard saddle-point approximation we introduce llubbard-

Stratonovich fields \ij (T): m terms of which the partition function reads

Z = [DX] [Ds] [De] [Dd] [DX] exp

The charge; an el spin Lagra.Tigeja.nK are; given by

- / ' dr f £ X\iXn + ^ (r) + 1, (r)° \

(11)

(12)

+ Ei(ai-/,-^(iTr[iPr*H + iJ ,

and

iTr [vpfflrj] - 2) - 2/i Ef 4 * •

Tn (9) I'emiioTiN and bosons Crin be <;xri.d.ly inU;gra.U;d out., lcjaving rin efl'ecl.ive pri.rl.iJ.ioTi

[Hnid.ioTi in UJI'TTIK of t.Tie HiibT)ar(l-Si.rri.J.oTiovi<:li fi<;ldK \-.;J a.Tid T igraTigc; nnill.ipli<;rK A.,;. Up i.o

this point everything is exact. To proceed further, we perform a saddle-point approximation

for the fields \ij and A,;. This procedure presents the following advantages over the standard

one: i) fermions and bosons are treated on an equal footing, ii) the saddle-point approxima-

tion preserves bot.Ti the global Npin-rot.al.ion invaria.nee and parl.iele-liole KyiviTVKJJ.rieK of OK;

paramagnetic phase at. half filliTig, regardless of l.lie ;\;.; valneK, and iii) j.}i<; Ka.d<ll<; point, in-

cludes KliorJ.-ra.nge a.nt.iferroTna.gnel.ic corrcjlal.ioTis (through t.he integrated bosoTi dyna.TnicK)

even in the large-/ •' insulating state. The disadvantage is that the I.J = 0 limit is not as well

reproduced as in KH's approach, where the hopping term includes normalization operators

introduced precisely to reproduce; the non-interacting case.

The Kaddle-point approximation corresponds to replacing t.he Hnbba.rd-St.ra.t.onovich

fi<;lds and La.gra.Tige; multipliers by c-Tnnnb<;rs and miniTnizing the effective action

-'••If \XjTi Xj7!. •• ^T) • -'-'ne saddle-point equations for the \; numbers are

and

The value of I.IK; Lagrange -multiplier A''f' is obtained by satisfying l.he constraint (8) (details

will be discussed in l.he next sect.ion).

Note that in the homogeneous state, where both the \ numbers and the Lagrange multi-

plier A are site-independent, the formalism is equivalent to a self-consistent diagonalization

of the following spin and charge Ilamiltonians:

kcr k

and

U>~ = Z^ Kck + A ) ekek + I ek + A ' I rfkdk - 2 ^ 5k ekrfk + e k 4 . (15)

H e r e t.Tie op<;ral.orK wil.li Kiibin<]<;x k a r e I.IK; F o u r i e r I r a n s t o r m of Kil.e o p e r a t o r s . (.£ = — 2/I ^ —

f / /2 — p. . <]%, = 2 /?7k for te rmiOTIS. r.£ = —2C-'7k, <t = 2f-'7k — 2/./ . r/£ = 2 / ?7k !'("• b o s o n s , a n d

7k = S . T cos ka . wil.li A:,v I.IK; r.v -cornpoTuml. of I.IK; Wri.v<;-ve<:l.or k . T I K ; prirarneUjrH A. B. C a n d

D are obtained from the self-consistency equations A = (e|e ;) — {d]dj) . 13 = (e-idj) + {ejd-i),

C = E^-sL'S^), and D = {.s.4.sjt)

TIK; energy is given by

while the nearest-neighbor isotropic spin-spin correlation function can be calculated as

/ c . c \ _ _ _ / ) 2

W'-a.?/ — o ^ *o

Tn I.IK; next , s ec t ion we pr<;senl. I.IK; reKiill.s for 1, 2 a n d 3 D .

III . RESULTS

We first discuss the results for the half-filling case, which corresponds to ft = 0. For

this case, in 2D and 31) two different phases -corresponding to the metallic phase for weak

coupling an t.Tic: insulating plia.se for strong coupling are obl.ained. TTie metallic plia.se is

characterized by a. condensation of the charge bosons. [12] As U increases t.Tic density of

condensed bosons decreases, and at a cril.ical value of t.Tie interaction Uc it. goes t.o zero.

The insulating phase is characterized by the absence of a boson condensate, i.e. the Mott

transition occurs when the effective chemical potential for the bosons ( — \S]J) is depinned

from the bottom of the bosonic excitation band, in 11) only the insulating phase occurs

since there is no Rose condensation.

Tn the niel.allic phase t.Tie number of empty and double-occupancy bosons which are

equal due to the electron-hole symmetry for j.i = 0- can be separated into two contributions:

the condensed and the uncondensed ones, in this way the constraint takes the form

(44)) + 5>L^> = i. fie)

TTie Lagrange mult.iply is pinned at t.Tie bott.om of t.Tie T>oson bands. The self-consisl.ent

solut.ion of (14) and (15) gives for t.liis case A = C = 0 and Asp = zD, where z is t.Tie lat.tice

coordination number. The density of the condensed bosons is then given by the constraint

f 16). Tor // > //,-, (e)2 = (d)2 = 0 and Xsp > zD, and its value is obtained by satisfying the

constraint.. Wit.Ti t.Tiis paramet.er and th<; K<;lf-coTisiKtent. solutions for B and D th<; energy

and different expectation valuers can be calculated.

Tn Fig. 1 we plot t.Tie energy as a. function of U for 1, 2 and 3D. TTie inset, sliows t.Tie ID

result and the exact energy obtained by the Bhete Ansatz for comparison. The following

points deserve a comment:

i) For I.J = 0 the exact results are not reproduced as anticipated above.

ii) Tn t.Tie large-/7 limit the energy is lower than zero due to t.Tie antiferromagnet.ic correla-

tions present for any finite value of the on-site interaction.

7

iii) While for 2D and 3D there is a critical value //,. at which the Mott transition occurs, in

"ID t.Tic: present approach gives an insulal.ing stale for all values of U.

iv) The energy and its //-derivative are continuous at the Mott transition. As we will

show below, the same occurs for the spin-spin, double occupation and other correlation

functions.

Tn our approximation the double occupation is always different, from zero, as shown in

Fig. 2. in particular, for // > I.L the double occupation is due to the fluctuations of the

vacuum, which are present because we have introduced the boson dynamic effects already at

saddle-point order. These fluctuations are responsible for the antiferromagnetic correlations

of t.Tie insulating stale. For U < Uc t.Tie condensate conlribul.es wit.b a macroscopic number

of empty and donT)le-occupaii<:y bosons.

The nearest-neighbor isotropic spin-spin correlation functions are shown in Fig. 3. For

small I.J they decrease as the dimension increases, as expected for uncorrelated electrons

in a lattice. This tendency is maintained for all values of //. For large // the correlations

satnrale and become U independent, like in the true antiferromagncM.ic ground stale. Notice

however t.Tiat. we are describing a paramagnetic (Mott.) insulaling phase, HO that in 2 and

3D these correlations do not. correspond t.o the st.andard Heisenberg correlations. Tn ID

our result is sensibly smaller than the Bethe ansatz result, since at saddle-point order our

approach introduces the short-range spin-spin correlations only partially. This quantitative

aspect can be correcl.ed by t.Tie inclusion of Gaussian fluct.uat.ions.

Away from half-filling the electron-hole symmet.ry must, be broken. Since according to

(15) t.Tie zero-point, excitalions of t.Tie c and d fields are paired, the number of empty and

double-occupancy condensed bosons must be different. The saddle-point equations force

their densities to be related by

W zC - ft + yJ{zC - {if + {zDf

U s i n g l l i is r e l a l i o n a n d a d j n s l i n g t h e c h e m i c a l p o l e n t i a l t o fix t.Tie n u m b e r of e l e c t r o n s , w e

c a n c a l c u l a l e all t.Tie se l f - cons i s t en t , p a r a m e t e r s A . B.C a n d D.

8

The one-particle gap l£fj of the Mott insulating phase is calculated as the jump in the

chemical potential when l.he tolal number of parlicle;s e;hange;s from 1 — c. l.o 1 -f r., find is

given by

This expression has to be evaluated numerically. The results are shown in Fig. 1, where

l.he gi-ip is plol.t.ed as a. I'mict.ioTi of U for 1,2 and 3D. Tn l.he 2D and 3D e;ase;s we; [bund

E,, <-- (U — Uc.)" '°r U —7- £/,-. l.he exponent a being 1 and 1/2 for 2D and 3D respectively.

Tn 3D il. is ea.K_y l.o oblain an aiia.lyl.ical expression for l.he gap when /7 —> Uc , and the

Kit dependence is reproduced, i.e., Kj 'CX A/(7; — Un). For the 2D case the expansion is

cumbersome, however the exponent a can be obtained numerically from Fig. 5, where In Ef!

vs In ((/ — //,-) is shown with the best numerical fit giving a- = 1.0 within our numerical

accuracy. Tn ID our approach gives a non zero gap for all values of U. including U = 0. This

is a conseqiKJiice of l.he fact, that our approximation does not reproduce l.he mi correlated case.

In order to overcome this difficulty fluctuations should be included. However, an interesting

point is that, contrary to what happens in the KH treatment, our approach makes apparent

the differences between 1. 2 and 3D due to the dynamics of the boson fields. Clearly, our

results are belter l.han KR'K for large U since; spin-spin correlations are present even for

U —> 'DC', and are less accurate for small U (nolice, }iowev<;r, that we do not. include; ad hoc.

r<;TiorinalizatioTi factors in this limit.).

We have also evaluated the kinetic energy in 2D as a function of doping for // = 6, that

is, above the critical value for the Mott transition at half filling (Fig. 6). There is a cusp at

S = 1 — rc = 0 and a steep drop as soon as we dope; l.he sysleni. Tn particular, l.he behavior

near half filling is given by EK(0) — EK{$) ^ $'^''\ with o.(U) ~ 0.6 for U = 6. Tn the inset,

l.o Fig. 6 we show l.he expecled linear behavior of /^-(O) a.s a fnnclion of \ jU.

IV. CONCLUSIONS

We presenled a fnnclional inl<;gral formulalion for l.he Hubbard model in terms of boson

and fermion fields. Bosons arc; associated wilh <;mply and double; occupane;y con figurations,

9

while fermions describe single occupancy states. The formalism preserves the spin rotation

in variance; for all occupations and J.IK; pa.rlicle;-liole HynnTiel.ry for one electron par silt. Fur-

t.Tiermore. bosons and fermioTiN are tre;ateel on a.n ee.|iia.l fooling. Tn l.TiiH work we sluelie;d J.IK;

saddle-point approximation of the corresponding functional integral theory.

We found that in 2D and 31) for one electron per site there is a metal-insulator transition

at a critical value //,. of the on-site interaction. The gap in the one particle spectrum behaves

as E,, oc (U — Uc)a for U —> Uc, wil.li an expone;nt a = 1 and 1/2 for 2D and 3D re;spee;lively.

Tn our approximation, for t.Tic: half-filling case t.Tic: double; occnpalion is always differeml. from

z.ero. For U < Uc the;re is a e:oTidc:nsat.ion of boNOTiN with a nia.croHcopic k = 0 conl.ribiij.ion j.o

the number of empty and double occupancy. Tor I.J > I.L the double occupancy occurs due to

the fluctuations of the vacuum, which are responsible for the antiferromagnetic correlations

of 1.1K; insulating stale. Tliese short-range; magnetic corrdalionH give; a. correction lo the;

ground stale; e;nergy which IN always le)vvcr than <AVXO. Fe>r la.rgc U the; spin-spin ce)rre;la.tion

Ha.tnrale;s and bce:oTncH U inele;pe;nele;nt; in lliis limit, t.Tic: e;nergy goe;s a.s '\/U in agreement,

with the canonical transformation that maps the llubbard llamiltonian onto the iieisenberg

model.

in order to treat fermions and bosons in an equivalent way, we did not introduce a

re;Tiorma.liza.l.ion of t.Tic: hopping matrix e;le;iTie;nl.H with an operator z as in t.Tic: KR formalism.

TTie price; we; paiel is thai t.Tic: TK)n-inle;rae;ling ca.se: is not reproduced e;_xa.ctly. Hovvc:ve;r, our

results give a better description in the intermediate and large // limit.

We expect the introduction of Gaussian fluctuations above the saddle-point values should

bring our results more in line with the exact ones. In particular, preliminary calculations in

"ID give; a remarkable; good agreement. wit.Ti t.Tic: exae;l e;nergy shown in the; inse;!. of Fig. 1,

especially for large; U. The;se re;siilts will be; prese;nteel in a. [brllie;oming publication.

10

REFERENCES

[I] J. Hnbbard, Proe. R. Soc. London, Ser A 276, 238 (1963).

[2] J. C. Bednorz and K. A. Muller. Z. Phys. B 61. 188 (1986).

[3] P. W. Anderson, Science 235 : 1196 (1987)

[1] M. Cutzwiller, Phys. Pev. Lett. 10. 159 (1963)

[5] W. 1<\ Brinkman and T. M. Pice, Phys. Pev. B 2, 4302 (1970)

[6] S.R. Whi le , D. J. Scalapino, R. L, Sugar, E. Y. Loh, J. E. Gnbernatis and R.T. Scaletar,

Phys. Rev. R 40, 506 (1989)

[7] S. E. Barnes. J. Phys. F 6, 6 (1976); P. Coleman, Phys. Rev. B 29. 3035 (1984)

[8] G. Kol.liar and A. E. Rnckensleiik Phys. Rev. Lell.. 57, 1362 (1986)

[9] T. Li, P. Wolfle and P. J. Hirschleld, Pliys. Rev. "B 40, 6817 (1989)

[10] Z. Zou and P. W. Anderson, Phys. Pev. B 37, 627 (1988)

[11] E. Arrigoni and G. C. Sl.rinai.i, Phys. Rev. Lell.. 7 1 , 3178 (1993)

[12] Notice tha t by using the constraint (8) the total charge can be writ ten Q = qc Y^i{Aei ~

d\d.j). tha t is. it can be assigned entirely to the bosons. Since the i iamil tonian (15) pairs

a and d pa.rlicles with opposite^ charges in its ground sta.le, only the charged condensate

ca.n respond lo an slatic <;l<;d.ric field. Consequently, the existence of a. boson condensate

is a necessary condition to have a metallic phase.

FIGURES

Fig. 1: Energy par silt as a function of t.Tic: on-site interaction U. for 1, 2 and 3D.

TTie arrows indicate t.Tic: point wliere t.Tic: metal-insulator transition occurs in 2D and 3D. Tn

the inset the energy of the ID case obtained with the saddle-point approximation (S-P) is

compared to the exact result.

Fig. 2: Double; occupation as a function of U for 1, 2 and 3D. Arrows indicate; the; points

where; iTie;tal-insulator LransitioiiK occur.

Fig. 3: Nearest-neighbor isotropic spin-spin correlation for 1, 2 and 3D as indicated.

Fig. 1: Clap Ea as a function of (/. In the ID case the gap obtained for small values of

U is ple>tt.e;d with a. dasheel line;.

Fig. b: hi(Ey) vs h\(U - Uc) for the; 2D case. The; line; is the; T>e;sl, fit, of the; pe>ints with

I.J close to //,- giving an exponent a = 1.0.

Fig. 6: Kinetic energy En as a function of the hole doping S in 2D for // = 6. Inset:

Kinetic energy at half filling as a function of 1/7/.

12

I!1 1

•

• n Vwv\

1

1

\

1

1 1

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i

\

\

a.CO

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• ^ •

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\

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act

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\

\

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o

\

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CM

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-

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18