A generalised Hubbard Hamiltonian : Influence of temperature and fractality

0038-1098/91$3.00+.00 PergamonPress plc

Solid State Communications, Vol.78,No. 8,pp.685-690, 1991. Printed in Great Britain.

A GENERALISED HUBBARD HAMILTONIAN : INFLUENCE OF TEMPERATURE AND FRACTALITY Sergio A. CANNAS *', Francisco A. TAMARIT *+ and Constantino TSALLIS* *

Centro Brasileiro de Pesquisas Fisicas/CNPq, Rua Xavier Sigaud 150, 22290-Rio de Janeiro-RJ, Brazil Facultad de Matematica, Astronom?a y Fisica/UNC, Laprida 854, 5000 Cordoba, Argentina

+

(Received January 4, 1991 by C.E.T. Go!&alves da Silva) We consider a generalised Hubbard Hamiltonian which is invariant under a Real Space Renonnalisation Group. Within the present approximation, the Bravais lattice is replaced by a diamond-like hierarchical lattice. We calculate the phase diagram associated with the half-filled band Hubbard Hamiltonian in d = 1, 2 and 3, in the full range of the dimensionless intra-atomic interaction U and hopping t parameters. The influence of the hierarchical lattice fractality is analysed as well.

Hamiltonian Hubbard The 111 (hereafter denoted by H) provides the the model- tb - study simple most effects of electrons in correlation metallic bands, like energy narrow magnetism [Z] and the metal-insulator transition). (Mott transition r3,41 Moreover, in the last years (see, for intance, [5,6]) there was an increasing interest in this model mainly because of its applications in the study of high Tc oxide superconductors [7-141. The dimensionless Hubbard Hamiltonian is defined as %

In order to study the thermodynamics of the Hubbard model, at least as far as criticality is concerned, we use a Real Space correlation-function-preserving Renormalisation Group (RG) method. Our approximation consists of replacing the Bravais lattices by diamond-like lattices, hierarchical namely those associated with the clusters shown in to Fig.1. In order obtain the RG recurrence equations, a decimation-like procedure is carried out over the degrees of freedom associated with the internal sites of the clusters. This method, first introduced by Suzuki et al [15], proved to be a useful one in a variety of quantum magnetic systems anisotropic spin l/2 Heisenberg ::,:i’ [16,17]). As it is well known, this approximation is valid at high temperatures as first dicussed by Takano et al [18] and revisited in Ref.[l6].

MM=-(3(H-p'N)= t

(c:~c,~+ c;&) x .U (1)

the (3= l/keT, cTcr(cIcr) is creation (annihilation) operator for an electron with spin u = up-arrow, downarrow in a Wannier state centere+d at the .c is site i of the lattice, n iu =c IU 1u the corresponding occupation number, and u' is the chemical potential. We shall band consider the half-filled case, implies p = U/2. We can then which rewrite the Hamiltonian (1) as where

The recurrence equations are obtained by imposing exp(M'+C) =

(c:OcJu + c;&) 1
(n,

-

n,

exp(H)

(3)

sites

where Ui denotes Hamiltonian the associated with the cluster, Ml denotes the renormalised Hamiltonian of the two-sites chain and the partial trace operation is carried out by summing over the set of occupation numbers associated with the internal sites of the cluster. In particular, Hamiltonian (2) does not satisfy relation (3). Therefore we have to search for a generalisation of the Hubbard Hamiltonian (2) which might satisfy this relation.

LH"= t

+ u/2

Tr Internal

12

i

685

686

Vol. 78, No. 8

A GENERALISED HUBBARD HAMILTONIAN (al

We now transformation

unitary (see as

introduce a defined U P' reference [19]). U p = ll ew( i

i7,Zi,.-b,)

(6)

unit arbitrary where the (;1,) are vectors and the (1,) are the parameters of (b) 1

-

i 0

-1

L

%2

the

transformation.

If

q,= "s for

every site i, and 3 lies in the x-y plane, this transformation generalises transparticle-hole exchange the formation. are symmetries of The (2) that preserved under (3) are the following ones : (a) rotational invariance (0-l"commutes with 2 = F gl);

2

(b) invariance under Up when G,= G is in the z direction and y,= 1 for all sites

(cl

1. *

In

this

trivial

case

phase

U

P

corresponds

change

of

the

to

a

Wannier

representation

I+?

=

i

1 %2

cell Renormalisation Fig. 1 : group transformation. L stands for the set of parameters (U,K,J,I,D,t). a) d = 1; b) d = 2; c; d = 3.

lines of Following along the Castellani et al [19] this generalised without Hamiltonian can be obtained explicitly carrying out the decimation procedure, by using the symmetries of the Hamiltonian (2) that are preserved by (3). As usual, we define the spin and charge operators as Sy=n

-n 1) Ik

S; = CT,%

pf = n,,+ n,,- 1 P;

= c,,c 1)

(4) = c;+c+ S1 P: $4 'b The fermionic character of the C's relations imposes the operators = c;c

(S.7) + (py’)

=

1 and Svpu~

0

(v,v’=x,Yrz)

where s: = s; + s;

P; = PI

SY

P;

i - -i(S: - S;)

+ P;

(5) =

-i(pT

-

P;)

v,(x)

(p,(x) + eiTi p,(x), being the Wannier orbital located

at site i). (cl invariance under U when ?I,=i lies P in the x-y plane provided that we choose with i,j being nearest7,= -V J , neighboring sites. Such a choice is not always possible frustrated (e.g. clusters), therefore the choice of the cluster must be carefully made in order to apply the present method. For simplicity let us first analyse the l-d case. In one dimension the smallest cluster which preserves the symmetry (c) is a four-site chain (see Fig. l-a). Indeed, if a three-site chain were used,then the Hamiltonian resulting from MH through (3), i.e. by taking the partial trace over the internal site, would not be invariant under (c); it would rather be invariant under U with 6 in the x-y plane and T,= 7

for P,ll i.

This Hamiltonian clearly will not contain the original Hubbard Hamiltonian as a particular case [19,20] . If the four-site chain is instead used, the partial trace in (2) over the two internal sites will leave the desired generalised Hamiltonian which satisfies more all our requirements. To be explicit, if we alternatively attribute +r and -r to each site of the chain, the four-site chain leaves, under decimation 7,= +r and Tz= -7 for the two-site chain, while the three-site chain leaves 7,= r2= +r . We are now able to generalise the half-filled band Hubbard Hamiltonian. We consider all the one- and two-site

which characterises a normal metallic stable There are two fully phase. namely points, fixed (U/t,l/t,I,J,K,D) = (k m,-,O,O,O,O). All

operators which are invariant under the above. A symmetries mentioned three linear combination of such terms yields the desired Hamiltonian, namely

04

c

=-J

-1

+

x2,.2,
- K 1 (S:)2(8;)2 + U/2 c i
[ P:Pf- (P:P;+ z

D 1 (C&c+ O

+;)I

C;&(r)

+ t Ix

U

(n,+-

We note the following properties : i)For J=K=I=D=O we recover the Hamiltonian band Hubbard half-filled (2), as expected. t=D=I=O the Hamiltonian ii) For (7) leads to a quantum analog of the BEG Hamiltonian (Blume-Emery-Griffiths) [211it can transformation (6) iii) Using grand partition be shown that the function associated with (7) satisfies Z(t,D) = H(-t,-D). Using transformation (6) it can be iv) Hamiltonian (7) preserves shown that the half-filled band character of (2). The above arguments not only apply to a linear chain, but to all two-terminal clusters whose topology enables them to satisfy the invariance property (c). The generalised Hamiltonian Mc being now constructed, the recurrence equations between the parameters of Hc can be

obtained

by

(sf)2 -

w;&+

c;&)

+

(7)

n,_o)2

This is the minimal Hamiltonian which satisfies (3) and contains (2) as a particular case.

and Mc'

687

A GENEBALISEDHUBBARDHAHILTONIAN

Vol. 78, No. 8

explicitly

computing the partial trace (3). These relations cannot be obtained analytically, since the calculation of exp(Mc) involves the diagonalization of very large matrices; consequently part of the calculations were done numerically. We analysed the RG flow of points in the parameter space (J,I,K,U,t,D) for d=l,2 and 3. Then we obtained the section of this complex phase diagram with the plane (i.e. (U,t) I = J = K = D = 0) for both signs of U. In d=l there is no phase transition for any value of U f 0 as expected [18]. For U/t > 0 the system is an insulator, for U/t = 0 it is a normal metal and for U/t < 0 it is in a metallic phase characterised by uncorrelated pairs of electrons. In d=2 we find the same structure as in d=l. All points in the U/t = 0 axis with l/t + 0 are attracted by the fixed point (U/t,l/t,I,J,K,D) = (O,m,O,O,O,O),

points with U/t 7 0 and l/t + 0 are point attracted fixed the by while points with (+m,@,O,O,O,O), U/t -z0 and l/t + 0 are attracted by (-m,m,O,O,O,O). In the limit U/t + 00 the Hubbard model is asymptotically equivalent to an antiferromagnetic Heisenberg therefore we can model, conclude that for U/t 7 0 and l/t * 0 the system is a paramagnetic insulator. There is no Mott transition for finite U/t in the ground state (i.e.,l/t = 0). These results are in agreement with previous ones obtained by a different RG technique [23], and by Monte Carlo calculations [ZO]. In the limit U/t + --m is asymptotically the Hubbard model equivalent to a gas of bosons with hard interactions. cores and long-ranged pairs of These bosons are bound electrons. However for d=2 there is neither superconductivity nor charge density waves for finite temperature. Therefore for U/t < 0 and l/t + 0 the system is in a metallic phase characterised by uncorrelated pairs of electrons without any magnetic order r251. For d=3 the calculated phase diagram is shown in Fig.2. For U/t > 0 the system is always an insulator and there is a second order phase transition from a to an paramagnetic state antiferromagnetic state. For u/t < 0 there is a second order phase transition from a characterised phase by uncorrelated pairs of electrons at high temperature to a mixed phase characterised by charge density waves (CDW) and singlet superconductivity (SS) without at low temperature, both magnetic order 1261. The two critical lines for U/t > 0 and U/t < 0 join at a point l/t=* 0 in the U/t = 0 axis (pure tight-binding limit). For U/t = 0 and 1/t > 1/t all the points are attracted by the i/t = m fixed point, that is, that region corresponds to a metallic phase. l/t < l/t we found an For the anomalous behavioui in renormalisation flow. All the points are

Vol. 78, No. 8

A GENEEULISED HUBBARD HAMILTONIAN

688 0.E

SInglet suQerConduclwlly and

charge

same physical state, since the grand partition function remains invariant under this transformation. Concerning the shape of the critical line, for say u/t > 0, one expects, for a Bravais lattice, the existence of a maximum for U/t + 0 [29,30]. This is not observed in the present calculation, possibly due to the fractality of the lattice. It is also worthy stressing that no Mott transition is observed at vanishing temperature and U/t* 0. In fact we localising effect think that the associated with the fractal character of the lattice is strong enough to destroy such possibility. In Fig.3 we show t as a function of

--I

poromaqnellc

onllferlomognellc

denslly

InsuIaIor

YOYIS

04. -20

1 -10

I 10

00

20

u/t

Fig 2 : Phase diagram of the filled Hubbard model in d = 3 hierarchical lattice dimensionless temperature).

halfin a (l/t =

the dimension d. We ibserve a critical to dimension dc I 2.41. In addition that, we find the same value for the lower critical dimension of the para-antiferromagnetic transition for u/t >>1. This suggests that the whole critical line disappears for d = d C'

attracted by limit cycles of order two rather than being attracted by normal fixed points. More precisely, the points are attracted by one or the other of two different cycles characterised as indicated in table I. The basins of attraction of these two cycles appear alternatively along the U/t = 0 axis for 0 < l/t < l/t . We believe that this behaviour is'owed to the fractality of the lattice that has been used. It is known that tight-binding systems present many peculiar properties when defined on fractal lattices: fractal spectrum, localised states, hierarchal states [27] and other anomalies (see Fig. 1 of Ref. [ZS]). Therefore any operation involving resealing (Eq. 3) will be strongly affected when Bravais lattices are replaced by fractal ones, as in our case. At this time, the physical meaning (if any) of these fixed cycles is not clear. --Note that the two- cycles are related by the transformation t + -t, D + -D, so it is clear that these two cycles in fact represent essentially the

1412-

lo081/tc

0604-

02 1

00+----iFig 3 : 1/t dimensionalit; lattice.

d

vs. of

the intrinsic the hierarchical

Table I : Limit cycles for U/t = 0 and 1/t < 1/t . The parameters (U,K,J,I,t) have fixed'values while the parameter D oscillates D

cycle

(2)

between

the

values

D"'and

.

J

t

(11

D’2’

U

K

I

A

1.406

0.703

0.07

0.07

7.513

0.607

-15.63

B

1.406

0.703

0.07

0.07

-7.513

-0.607

15.63

D

case, since (8) contains many of the been that have interactions basic proposed to explain this phenomenon [7,131. In the d=3 case the present approach does not exhibit the metal-insulator

Let us focus on the case of non case this half-filled band. In Hamiltonian (1) loses symmetry (c), and we obtain that the Hamiltonian which generalises (1) and remains invariant under (3) is the following one :

MG =

J

-

-

$2, <1,1'

I2

- K

P;P;

z


+

z ‘I.>>

(P:)*

Pf

+

(P;P;+

(Pf12

P;P;)

P;

]

nio 10

i

J>

I1

1,

(1%

n,+n,, + c1

(S:)2(ST)2 - U
El

+R

689

A GENERALISEDHUBBARDHAMILTONIAN

Vol. 78, No. 8

+

+

(8)

t

1'

(qcrcJcr+

c

+

D1

-

2D2 Z

(1,

c;&)

(“*_o+

nJ-&

1’0

tC;eC~(T+

‘;&)

n,_o

nJ-,,

U

The phase diagram associated with this Hamiltonian is under study and will be published elsewhere. approach the present In summary, shows to be a powerful one to study critical properties of a great variety of fermionic systems related to the Hubbard model, since the generalised explicitly contains Hamiltonian (8) which charge terms account for fluctuations, order and magnetic hopping. In particular we have seen that for d = 2 the fractality of the lattice does not affect, at least qualitatively, the phase diagram associated with the half-filled band Hubbard Model for lattices. In view of these Bravais results, we think that this approach can provide a new method to study high T superconductivity considering the

in CuO compounds b; band non-half-filled

transition which is expected in Bravais lattices. In spite of that, it provides an interesting method to analise the interplay between correlations and fractality in fermionic systems in the weak correlation limit u/t B:1. Moreover, in the strong correlation limit U/t > 1, the fractality seems to have a negligible influence on the phase our diagram. Consequently we expect model to be a useful one for studying superconductivity also in the same limit (u/t > 1). Work along these lines is in progress.

Acknowledgement - We wish to thank D. Prato, E.M.F.Curado, R.Maynard and L.M. Falicov for fruitful discussions. Two of us (S.A.C. and F.A.T) thank CNPq (Brazil) for financial support.

References [ll [21 [31 r41 r51 [61 [71 [f31 r91

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[lo] C. R. Proetto and L. M. Falicov, Phys. Rev. B 15, 1754 (1988). [ll] J.R. Schrieffer, X.G. Wen and S.C, Zhang, Phys. Rev. Lett. 60, 944 (1988). [12] Wei-ming Que and G. Kirczenow, 2. Phys.B 73, 425 (1989). [13] B. I. Shraiman and E. D. Siggia, Phys. Rev. Lett. 60, 740 (1988). [14] R. Micnas, J. Ranninger and S. Robaszkiewicz, J. Phys. C 21, L145 (1988). [15] M-Suzuki and H.Takano, Phys.Lett. 69A, 426 (1979) [16] A. 0. Caride, C. Tsallis and S. I. Zanette, Phys. Rev. Lett.51, 145 (1983); 51, 616 (1983).

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A GENERALISEDHUBBARDHAMILTONIAN

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J. Hirsch, Phys. Rev. Lett. 51, 1900 (1983). ~251 V.J. Emery, Phys. Rev. B 14,2989 (1976). S. Robaszkiewicz, R. Micnas and K. [261 A. Chao, Phys. Rev. B 23, 1447 (1981). Rammal, J. Physique 45, 191 ~271 R. (1984). C. Tsallis and R. Maynard, Phys. [281 Lett. A 129, 118 (1988). R.R. DeMarco, E.N. Economou and [291 C.T. White, Phys. Rev. B 18, 3968 (1978). J.E. Hirsh, Phys. Rev. B 35, 1851 [301 (1987). v41