Z. Phys. B - Condensed Matter 82, 369-374 (1991) Condensed

Zeitsehrift M a t t e r for Physik B �9 Springer-Verlag 1991

Dynamic response functions of Hubbard model in Gutzwiller approximation T. Li 1, Y.S. Sun 1, and P. WSlfle 2'*

1 Department of Physics, University of Florida, Gainesville, FL 32611, USA 2 Institut fiir Theorie der Kondensierten Materie, Universit/it Karlsruhe, W-7500 Karlsruhe, Federal Republic of Germany

Received October 24, 1990

We calculate the two-point dynamic response functions of the Hubbard model at any filling from a slave-boson representation in the paramagnetic saddle point approxi- mation. We use a spin-rotation invariant form of the representation due to Kotliar and Ruckenstein, for which the saddle point corresponds to Gutzwiller's varia- tional approximation. In addition to the spin and charge susceptibilities, which have the familiar RPA form in the limit of small wavevector q, the double and single- occupancy response functions are here calculated for the first time.

Variational methods are a powerful tool in uncover- ing the subtle correlation effects present in systems of strongly interacting fermions on a lattice. A number of new ground states of such systems have been proposed recently as candidate states for high-T~ superconductivity [1, 2, 3] and their energies have been evaluated approxi- mately. Most of these states are of the so-called Gutz- willer type, i.e. consist of a weakly correlated state with or without long range order and a projection operator removing configurations with doubly occupied sites (or else reducing their statistical weight). These theories are necessarily confined to a description of static properties in the ground state. The calculation of dynamical proper- ties requires one to go beyond a variational ansatz.

It is particularly useful to develop a dynamical theory which has a static limit of the form of a Gutzwiller- correlated ground state. As shown by Kotliar and Ruck- enstein (KR) [4], the Gutzwiller result [5] may be ob- tained as a saddle point of a suitably constructed slave boson representation. The KR formulation is not mani- festly spin-rotation invariant, and therefore not suitable for a calculation of e.g. the spin dynamics. However, an invariant form of the slave-boson representation has recently been proposed [6, 7]. Using this extension of

* Work partially performed at Department of Physics, University of Florida, Gainesville, FL 32611, USA

the KR theory we present here a calculation of the com- plete set of dynamical correlation functions for the Hub- bard model at the saddle point corresponding to the paramagnetic Gutzwiller solution.

There are three tyes of two-point correlation func- tions, which can be formed out of the occupation number operator ni~('c ) for electrons of spin a and the double occupancy operator D/(z) = n~t (z) ni+ (z):

B ~ o, (i - i', ~ - r') = < f [n~o (r) n~, ~, ( z ' ) ] > Tr ( i - i', z - - z') = < T [D i(z) ni,, (z ' ) ] > C (i - i', ~ - "c') = < f [D i (z) D i, ( { ) ] >, (1)

where T is the time ordering operator for imaginary times z. Only four of these seven correlation functions are independent for a system with space-time inversion symmetry.

In this paper we present results of an approximate calculation of the above correlation functions for the Hubbard model of fermions on a lattice

H = ~ [ i j c i ; c j a J ~ - U 2 1 ~ i t t'~i] " . (2) i , j , a i

We are in particular interested in the parameter regimes of medium to large on site repulsion U and particle den- sities close to the half-filled band. In this limit the motion of particles on the lattice is severely restricted, since most hopping processes take place onto already occupied sites and doubly occupied lattice sites are very costly in ener- gy. The physics of motion of fermions in constrained Hilbert space is most conveniently described in a slave boson representation [4].

The Hubbard model can be expressed in terms of slave-boson operators ei, Pi, di for empty, singly occupied and doubly occupied sites and fermion operators f~ as [6, 4]

H = Z t,~ Z f . +, z?~o, ~ . , ,J . , , + e 2 a?- d,, i , j aq" i

(3)

370

where

_Z i = [ ( 1 - - di+ d i ) ~-o - P + P i ] - 1/2 (e + Pi if- P+ eli) [(1

--e? el)Z_o-p) p,] t/z

and the underbar denotes a 2 x 2 matrix in spin space, _% is the unit matrix and p~ is the time reverse of operator _Pi"

The number of slave bosons per lattice site is confined to one, implying the local constraint

e~ e~ + d~ + d~ + tr(p~ P0 = 1 (4)

and the number of fermions is equal to the sum of the number of p-bosons and twice the number of d-bosons:

(p? p,)~p + d/d~ ~ = f~ f~ (5)

With the help of the local constraints (4) and (5), the correlation functions defined in (1) may be expressed as correlation functions of the slave boson fields. Thus one obtains for the double occupancy operator D~ =di + d,, the empty site occupation operator E i - ( 1 - n i ~)(1- ni , )= e + ei and the single occupancy operator of state with spin projection a along a quantization axis S: Pi~ =- ni~(1 -- n i_ ~) = t r [(-~o + a g. $) p+ Pi], where z is the vector of Pauli matrices. The spin correlation function in Fourier space is then given by

)i~(k) = ~, aa' ( 5n~( - k ) g)n~,(k)}

= ~ aa' (5P~(-k) 6P~,(k)) (6)

where, e.g. OP~(k)= P~(k)--6k, o(P~) and (P~) is the mean field value.

The charge correlation function takes the form

z~(k)= ~ (6n~( - k ) 5n~(k)}

= ( [ 6 D ( - - k ) - b E ( - - k ) ] [6D(k)--6E(k)]} (7)

For the remaining two independent correlation functions we choose

gee= ~ (6 [n~(1-n_~)] 6 [n~,(1-n_~,)]}

= ~ (SPo(-k) bP~,(k)) (8) ffff,

and

;goo= ( S D(-- k) 5 D(k) } (9)

describing the density response at singly occupied sites and doubly occupied sites.

The correlation functions may be expressed as func- tional integrals over coherent states, e.g.

( Ei('c) E j(Z') ) = 1 ~ D [boson] Ei('c ) Ej(z') exp

- ~ d~ ~.ef(~c) (10) 0

where Z is the partition function. The effective Lagran- gian has been derived in [-6, 7] as

Sole + ~ f f (11) ~e~ff(z) = B F

with

,J)fff ('i;) = 2 [0~i e{ + (a~ -- 2 fi, o + U) d? + 2 (~, - fl~o) ~', P~u i #

--4([1,. p,) P,o]

and

~fffr(Z) = -- trln [(0~-- #0 + flio) ~ , bi3 + fli" %~, ~i2 + t,2 ~ zj~r ~,~ <].

a r

Here, ei, di, Pi, (#= 0, 1, 2, 3) and ~i, fliu are real fields [8], and fli = (ill, flz, fla), etc.

The functional integral in (10) cannot be done in gen- eral form. A useful class of approximations is introduced by considering the saddle point approximation. As usual, it may be shown that the saddle point solution becomes exact in the limit N ~ oo of a suitably defined N-orbital model, consisting of N flavors of fermions (i.e. replacing f ~ by f~, , in (3), where m= 1, 2, . . .N, summing over m in the hopping term and in the r.h.s, of the constraint (5) and putting the r.h.s, of (4) equal to N). The simplest saddle point solution is translation invariant and spin rotation invariant and corresponds to a paramagnetic, metallic phase. This mean field solution is discussed in appendix A (see also refers 9, 10, 11).

In order to calculate the correlation functions one must go beyond mean field theory. The first useful ap- proximation is obtained by calculating the effect of Gaussian fluctuations about the mean field, which pro- vide the leading correction in a 1IN expansion. To this end we expand the effective action in second order in the fluctuations about the mean field:

r + ~ 0T( - k) ~(k) 0(k) (12) ~d t ~ e f f = ~ k

where

O(k) = (5 e, 6d, 6Po, 6flo, 6a; 6 P l , 5 i l l ; ~P2 ; ~ f 1 2 ; 6P3,6fl3)

is the eleven-component fluctuation vector of bose fields about the mean field values and S(k) is the fluctuation matrix with elements Sij(k)=Sj~(-k). We have intro- duced the notation (k=(k, con), where oJn=2~nT and

. . . . fi-1 ~ L-1 ~ with L the number of lattice sites.

The correlation functions of the fluctuation fields 4J~,

D~j(k)- (0~(-- k) tpj(k)} (13)

are given in the approximation for ~ e f f defined by (12) in terms of the components of the inverse fluctuation matrix S- 1 (k) as

- - 1 --1 D,j(k)-~S, j (k). (14)

The correlation functions defined above may be ex- pressed in mean field approximation in terms of the S~j by using 6 0 = 6 ( d + d)=2d6d, etc., with the result

)~(k) = 2p 2 $66 ~ (k), (15)

zc(k)=2[eZS;a~(k)+d2Sd(k)-2edS~d(k)] , (16)

)~ep(k) = 2p~ S3d (k) (17)

and

Zoo(k) = 2 d z S;11 (k). (18)

The matrix S has been calculated for the half-filled band case in [7] for the spin-rotation invariant representation (and in [8] for the original K R formulation). For general band filling the elements of the fluctuation matrix are given in appendix B.

It follows from the transformation properties of the fluctuation fields under spin rotations that S decomposes into a direct sum of a 5 x 5 matrix A associated with the scalar components (fie, 6d, 6po, 6~, 6flo) of r and 3 identical 2 x 2 matrices B associated with the vector components (6pl, 6/31), (3p2, 6fl2) and (6pa, aft3).

The spin susceptibility X, is given in terms of the matrix elements of B as

z~(k)= 2p~ Sv7 S66 S77 _ $27 . (19)

Substituting the expression for Sij into (19) we find

z~(k) Zo(k)

where

1 + Ak ~(o (k) + A1 )~1 (k) + A2 [Z 2 ( k ) - Z0 (k) Z2 (k)] (20)

I- 32z~ /Ozt\Zl

cg z r A 1 = p o l z c3pl

A2=(4p2o)- ~z2 (C~Z~ lZ \Opd

and c~, rio, z, ek and ;~,(k) have been defined in Appendix A and B. The spin susceptibility as given by (20) does not have the standard RPA form, since zl(k) and )~2(k) in general differ in their co, k dependence from g0(k). However, in the longwavelength limit Ikl ~ kv, using the expansion

)c,(k)=(2te)";~o(k)+O(~F), n = l , 2

e k = e o + O (k~). (21)

One obtains from (20)

Zo(k) Zs(k)-- 1 + F~ Zo (k)/NF" (22)

371

Here NF is the renormalized density of states, Nv = z - 2 N ~ (for both spin projections) at the Fermi level, t F is the value of tk at the Fermi level and the Landau parameter F~ is given by

/ ,~2z / g z \ Z \ ~?z F~ = "N~v-2 f ~ - fi~ + e~ ~z ~-~p2 + ~ ~pl po [ ) + 4 '~ tv Z -~Plpl ] " ( 2 3 )

(~Z ~2Z Substituting the expressions for :~-flo, ~ and Op~'

one obtains

Fg = - 1 + (1 - 82)(x 4 - 82)/(2 x 2 - x 4 - 32) 2 (24)

in accordance with (19) of Ref. 10, except for a misprint there (i - x 2 --+ 1 - 6e). Here x = e + d, and is obtained as solution of (A.2).

In the weak-coupling and strong-coupling regimes at any filling one finds F~ explicitly as

Fg= - 2 u u ~ l (25a)

1+[0[ g = - l + 4 ~ u > l (25b)

where u = U/U~ and U~ is defined in Appendix A. In the strong coupling limit u ~ 0% F~ ~ - 1, i.e. the

system becomes unstable with respect to ferromagnetism for any band filling.

In the limit of small 6 (near half-filling) one finds

= - , u < l (26a)

and

1+161~ F ~ = - l q 4u ' u > l (26b)

with ~=(1--t/U) 1/2. The limiting value of Fg as 3-~0 has been calculated first by Lavagna [11].

Calculation of the charge susceptibility involves in- verting the 5 x 5 submatrix A of the fluctuation matrix S. For general band-filling the result is

zc(k) = 2 Po eS44 D-- (3edF22-dPo F,2 + d2 F21). (27)

The remaining response functions are given by

Xvv (k) = 2 p2 e ~ A (F11 d - F~ 2 e) (28)

and

Zoo(k) = 2d2 po e $44 F21 D (29)

with ~j and D defined as

Fll = $44(P0 dStl - Po eS12-edS13 + e2 S23) - (dSl~-eS24)(po $14-eS34)

372

/'22 =$44(P02 $I2--P0 eS23--Po dS13q-edS33)

- (Po S14-eS34)(Po $ 2 4 - d S 3 4 )

F12 = $44 (Po dS12 - Po eS22 - d2 $1 a + e dS23 )

- - (Po $24. -- dS34)(dSl~. -- eS24)

F2 t = $44(P0 z S ~ - 2 p o eSxa + e2Sa3)

- (Po S~4- eS34) 2

and

(30a)

D = F11F22- F12 F2,. (30b)

In the limit k/kF ~ 1, the correlation functions simplify considerably. For example, )~(k) is then given by the RPA form

zo(k) z~(k) = 1 + F~ Zo(k)/Ne

, (31)

where the spin-symmetric Landau parameter Fg may be expressed in terms of the hole density c5 and the quantity x = e + d as

N~ ~ E ~ e 6 ~?z 2 1 0flo (32) O = ~ z 2 Oa N ~ 2 Oa

where N ~ is the average density of occupied states, i.e. ~ o = N ~ for fiat DOS, and

OZ 2 46p 2 x 2 2 [ 2 ~X2 2 ~p2] 86 - ( 1 - 6 ) 2 t - - i - z~[P~ ~ - + x 0~- ] '

Off 0 Ucx2 [ p2 1_}_a2 ] aa 2 (x2 +6) 2 + (1 -62) 2 p2

ax2r;o < x pg 1 + ~ 6 - [ ~ 2 (x 2 -- 6)2j

apo ~ G ~F 1 a q + T a - ~- x [ x 2 ~ 7 + ] ~ 7 ~ ] , (33)

with

U ~ 2[ Pg - ,5 2 flo-?-~ L~+~po-�89 The derivatives of x 2 and p2 are obtained as

(34)

Op 2 a l / a 2 ) 0 x 2 (35) 69c~ - x2 ~ - ~ - 1 ~

and

8 x 2 2u6 06 - x z (2 u - 2 + 3 X2) " (36)

In the weak-coupling and strong-coupling regimes at any filling one finds

F~ = 2 u, u .~ l

1 1 + 1 6 1 u > > l . F~= 16[ 4u ' (37)

In the limit of small 3 the expression (32) for Fg takes the form

u ( 2 - u ) . . . . 2, F ~ = ~ + u t o ;, u<l

1 2 - 1 / u

Fg= 2181 ~ , u > l . (38)

In the limit of 6 ~ 0 and in the strong coupling regime u > l , F~ is seen to diverge as 1/161 (see Refs. 10 and 11). Since the effective mass also diverges as 1/1~[, the static compressibility )~(0) remains essentially unrenormalized in this limit.

For general wavevector k, the correlation functions deviate from the RPA form. In the following we give the results for the case of a half-filled band. One finds for the charge correlation functions zfik)

zc(k) = 4d 2 $44 Dz-D~ (39) Do

where

Do = D1 D2 + ($1 ~ - $1~) D~ = - D/(po eS4,,)

and

(40 a)

Dx = 2d2 -(S11 - $12) $44

D2 =p~(S11 +S12)+2d2S33-4dpo S13

03 = ~22 ($14 + S24)- ]//2dS34. (40b)

Substituting the expression (B.1) for Sij , one finds

~o(k) (41) zc(k) = 1 +fd )~o(k)

where

;~o (k) = ,~o (k) + 2 Z~ (k)//~2 (k) (42)

and

/~2 (k) = D2 (k)/[8 d 2 (1 -- 2 d 2)(1 -- 4 d 2)2]

G ek--eo--�89 + (43)

- 2 ( 1 - 4 d 2 ) 2 2 d ~ ( 1 - 2 d 2)

In the limit k--+0, Z1 vanishes and )~o(k)~Xo(k), so that zc(k) reduces to the usual RPA form 1-12]. Hence FJ =fd N(Ev) may be identified with the Landau parameter.

Finally, we give the limiting forms for Z~(k) and Zee(k) at half-filling and for Ikl ~ ke,

1 Zep(k) - Uc" (44)

The correlation function of the density of singly occupied sites, Zpp(k), is seen to be given by a constant, whereas the dynamics of the density of doubly occupied sites

is given by that of the charge operator. In the limit U U~, when )~c(k) -+ 0, the dynamical part of ZoD(k) disap-

pears. The static parts of both functions are independent of U.

In summary, we have calculated all two point dynam- ical density correlation functions of the Hubbard model within a slave boson scheme corresponding to Gutz- willer's variational method, using the paramagnetic mean field solution as our starting point. There are four independent correlation functions in this case, for which we chose the spin and charge susceptibilities )~,, Z~ and the correlation functions for the densities of doubly occu- pied and singly occupied sites, )~DD, ZFe respectively. We give the general result, valid for all band fillings and arbitrary interaction strength. In the long wavelength limit Z~, Z~ take the form obtained in Fermi liquid theory. We calculate the Landau parameters Fg, Fg for all fi, U, thereby extending previous calculations [10, 11, 12]. At short wavelengths (k ~ kv) the result deviates signifi- cantly from the simple RPA structure. The correlation functions ZoD, Zel, are evaluated for the half-filled band case. It is found that in the long wavelength limit )~e*' is given by a constant, whereas Z~D is proportional to (Z~ + const).

We thank Dr. R. Fresard for useful discussions. This work was supported in part by ESPRIT project MESH 3041 (PW) and by DARPA grant MDA 972-88-J-1006.

Appendix A

At the saddle point (for N = 1), the values of the bose fields may be expressed in terms of the variable x = e + d as

d = (x 2 - 6)/2 x

e =(x2+a) /2x

p~ = 1 - - (X 4 -F 62)/2 X 2

G = c ~ - I G 1+

p =/~=o.

Here x is a solution of

(A.1)

(1 - - X 2) X4/(X 4 - (~2)= U/Uc ' (A.2)

with

~geff

U ~ = - 8 I dssN(s) --oO

(A.3)

and N(e) is the bare density of states for both spins (#~ff =(#o-flo)/Z 2 is the quasiparticle chemical potential). The hole concentration ~ is defined in terms of the number of particles per site n as fi = 1 - n, and is positive (negative) for less (more) than half-filled band. The saddle point solution is identical to the Gutzwiller variational approximation [5, 9, 10].

373

For weak to medium coupling strength, U < U~, the solution of (A.2) in the limit of small ~5 is given by

X 2 = 1 - - U -}- '52/(1 - - U) 2 (A.4)

with u = U/U~ and fi<(1 - u ) 3/2. It is well known [9] that the effective mass of the fermions is enhanced for small 6 and U --+ U~,

m, 1 l_ ' s z [ u 2 ]-1 ~-~ Z2 - - m 2p~X 2~ 1--Ue+'sa(1--U)2(l+2u--uZ)

(a.5)

In the strong coupling limit U > Uc and for small fi the solution of (A.2) is given by [10]

x2 = 161/g_ 1 6z 2 ~-(1-~' )+0(,53) (A.6)

with ~ = (1 - UdU) 1/2. Note that X 2 is a nonanalytic func- tion of fi in this case. For given U > G, the ground state energy has a cusp as a function of fi at '5 = 0 and the chemical potential has a discontinuity [ 11]. The effective mass in this case is also enhanced

_ ( _ l - ja i l m* l+[a] 1 (A.7) m 2[6[ 2u ]"

Appendix B

The elements of the fluctuation matrix S o are obtained by expanding the effective action given by (11) in terms of the fluctuating parts of the Bose fields up to second order:

S 1 l (k) ~- 0( -}- S11 (k)

$22 (k) ~-- 0{ - - 2 flo + U + 322 (k)

$33 (k) = o~ --/~0 -~ 333 (k)

So(k) = So(k); i =#j; i, j < 3,

1 (k) z S14(k)= - ~ Zl

Sl5(k)=e

S24(k) = 1 Oz - 2 d - ~ ~ (k) z ad

$2 s (k) = d 1 ~z

s3~(k) = - p o - ~ z~ (k) z O o

S35 (k) = !%

1 S44 (k) = - - ~ Zo (k)

$66 (k) ~-- S 88 (k) : S 10,10 (k) = o~ - flo -~- 866 (k)

1 877 (k) = 899 (k) = S 11,11 (k) = - - ~ ZO (k)

1 Oz, $67 (k) = 889 (k) = $1O.ll (k) = - Po - ~- X l (k) Z p-~-.

374

Ttze matr ix S is l~ermitian g~j= Sj~( -k) and the remain- ing elemenls of S are zero. In the above we have intro- duced the abbrevia t ions

8Zz [ 1 2 1 8z 8z 63 ~c30 + ek - -gZ Zz(k) (B.2)

with ~9~=e, d, Po for i = 1 , 2, 3, ~ i = p l for i = 6 , and

~ =X t._~ G(p) (B.3) p~

with the single f-part ic le Green 's funct ion

1 GO))- ico,-E~' co, = ( 2 n + 1) ~ T (B.4)

and the fermion spect rum

Ek = z 2 tk-- #0 +/~o. (B.5)

Fo r a simple cubic lattice in d-dimension

d

t k = - - 2 t ~. COS kia. (B.6) i= ! .

The quantil ies )~,(k) are dynamic response functions of the fermion system :

Z, (k) = - X (t~ + t~ + 0" O~ (p) O~ (p + k) ptr

= Z , ( - k), n = 0 , 1, 2. (B.7)

The last equali ty implies that S~j(k)=S~j(-k)=Ss~(k). The derivatives of e at the saddle point are given by

/ 2 x e \ 63 Z2 - e r12 Pg x k l + ~'~- 3 e

':? zz63 d ( 2 x d \_

63z 2 8po =4qZP~ xz(1 + 2t/ZP~

8z~ = 4172p ~ 5(1 --2t/2p~ x 2) 8p~

8pl 63Px

and

632Z

63e 2

632z

63 e 63 d 632z

632z 63 e 63 p o 632z 63p~

= 2 ~ /~ .3 Po x (3 + 3 p~ ,12 (1 + 6 2) + p~)

1 = ~/2t/[1 + 2tlZp~ + ~ 6 + 2qaxepZ+6 (1 _ 3)z ]

-- 2]/~.3 po {x [1 + 2qepg(1 +202) ] - 2 ( e - d ) 5 }

(B.9)

where q 2 = 1 / ( 1 - 8 ~ ) . The derivates OZz/Od z and 632Z/c3d 63P0 are obta ined from 632z/63e2 and 632z/63e 63Po by interchanging e and d, and 6 ---, - 6 respectively.

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