Thermodynamics of a Two-Band Hubbard Model

W.-H. STEEB and E. MARSCR: Thermodynamics of a Two-Band Hubbard Model 403

phys. stat. sol. (b) 66, 403 (1974)

Subject classification: 13 and 18

Institut fur Theoretische Physik der Universitat Kiel

Thermodynamics of a Two-Band Hubbard Model

BY W.-H. STEEB and E. MARSCH

Using a variational principle the approximate free energy of a simple two-band Hub- bard model is calculated. Numerical results are presented for the simple cubic lattice and these are given for the free energy, chemical potential, and energy gap as function of temperature. The values are compared with those of Dichtel et al. By introduction of a second band a lower transition temperature TN is obtained. The system shows a second-order phase transition.

Mit Hilfe eines Variationsprinzips wird die approximative freie Energie von einem ein- fachen Zwei-Band-Hubbard-Model1 berechnet. Die numerischen Rechnungen werden fur das kubisch-einfache Gitter ausgefuhrt. Resultate werden angegeben fur die freie Energie, das chemische Potential und die Energielucke als Funktion der Temperatur. Die Werte werden mit Auswertungen von Dichtel et al. verglichen. Durch die Einfuhrung des zweiten Bandes wird eine niedrigere ubergangstemperatur erreicht. Der Phasenubergang ist von zweiter Ordnung.

1. Introduction Dichtel et al. [l] have calculated by use of a variational principle thermo-

dynamic quantities of the Hubbard model. They started with the well-known Hamiltonian

H , = Tnm~,+,drno + u 2 dnftdn&$dnJ (1) nmo 12

and an inequality for the free energy. The calculations are performed for nearest neighbour hopping (hopping strength J ) , neutral model, and AB lattice. They found for all values of the coupling constant U two phases cor- responding to two different values for the free energy of which the smaller one has to be taken as the physical free energy. It is shown that the condensed phase, as long as it exists, always leads to the lower value of the free energy. The condensed phase is purely antiferromagnetic and the Hartree phase is paramagnetic. The phase transition is of second order. In the half-filled case (neutral model) the transition temperature k T N (NBel temperature) tends to U / 4 for U > J. This is in consistence with results in the strong correlated limit [2]. Numerical calculation must be restricted to the range U ZJ ( z number of nearest neighbours). Further, Dichtel et al. found that the chemical potential is independent of temperature.

The aim of this paper is to investigate the influence of a free band on the thermodynamic quantities where we use the same approximation (for the Hubbard term) like Dichtel et al. So we start with the Hamiltonian

26'

404 W.-H. STEEB and E. MARSCH

The first tern1 of (2) is the Hubbard Hamiltonian. In Section 2 a short deriva- tion of the variational principle (inequality for the grand thermodynamic potential) is given. In Section 3 the ansatz for the trial density matrix is made. A remark is made about the choice of the trial density matrix in the strongly correlated limit. In Section 4 the grand thermodynamic potential is calculated and the self -consistent equations are determined. Further we consider the chemical potential and the other quantities which we get from the self-consistent equations. Numerical calculations are performed in Section 5 and a discussion of the results is given. Our calculations are restricted to the simple cubic lattice and nearest-neighbour hopping for the Hubbard term.

2. The Variational Principle The grand canonical density matrix, W,, for a system with Hamiltonian H

and number operator Ne a t temperature l /B and chemical potential p can be derived by specifying it to be that which minimizes the grand thermodynamic potential

l2 = Tr { W(H - p N e ) } + - Tr { W In W} (3) 1 B

over all W . W is a linear, hermitian Hilbert-Schmidt operator with unit trace. The condition that the first functional derivative vanishes leads uniquely to

The fact that W , gives a minimum follows from the form of the second functional derivative a t W = W,. This operator is positive definite, since W , is positive definite. So we get an inequality for the grand potential [3] :

1 l2 5 Tr { Wt(H - p N J } + -Tr { W , In W,} , B

where the trial density matrix Wt is a linear, hermitian positive definite operator with unit trace.

3. Choice of the Trial Density Matrix In the following we consider only the Hubbard Haniiltonian because the

second term in (2) can be calculated exactly. One chooses a t,rial density matrix exp (...)

W - - Tr {exp (...)}

for which the trace 2, = Tr {exp (. . .)}

can be evaluated exactly. One possibility is (in Bloch (7)

representation)

exp (-B t: [ E ~ ( K ) & d Z r y + EAK) &+&+I)

~r exp -B ,Z [E,(K)Z..~&~ + E,(K) &+zK+])} ' (8)

K W , =--

{ ( K

The quasi-particle operators d+ and 2 obey also the ACR. The summation over K extends throughout the first Brillouin zone. E,(K) and E,(K) represent

Thermodynamics of a Two-Band Hubbard Model 405

the energy spectrum of the quasi-particle (chemical potential included). They play the role of real variation parameters. E,(K) and E,(K) are determined by

From (8) we obtain

2, = Tr {exp (...)} = 17 (1 + e-fiE1(K)) (1 + e--BEs(K)) (10) I(

and by use of an operator identity

(12) 1 - -

(’kraK&) = 1 + e,5Ea(K) = - f 2 m I

where f,(K) and f2(Zi) are the Fermi distributions. The operators 8, d a n d a+, d are connected by a Bogolyubov-Valentin transformation [4]. Since the Hub- bard operator commutes with the number operator N,, we require that also N , is invariant under this transformation (in contrast to the BCS theory).

We make the ansatz -

(13) A = eiSA e-i8 . , S = S f , where

with 1

Q = (n, n, n) - We obtain from (13) and (14)

a is the lattice constant for the simple cubic lattice. OLK plays also the role of a real variational parameter and is determined by

SO we have three equations for the unknown quantities E,(K), E,(K), and aK. A motivation of this ansatz is given in the next section.

A remark should be made about the strongly correlated limit. In this range holds U > J and our approximation is unrealistic. Here we start with

-+ a _ - _ ~ X P (-11 2 d&dn+d&d,+ - 2 2 ,Z dnt n t - i l3 ,Z xz+&+)

n n n 9 (18) ~~ -. ._ -_ ~ w,= - - Tr exp -ill ,Z +a a t Z+d n$ n+ - 1, L’ Z$d,+ -. 1 3 2 &Z!J”o>

{ ( n n n


where n runs over all lattice sites. d f , d and df, d a r e connected by -

(19) A = ,is A .-is

where S is the Fourier transform of (14),

The variational parameters are &, A2&, and qm-n.

4. Calculation of the Free Energy Now we want to calculate the grand thermodynamic potential and the self-

consistent equations of our extend model. We start with the Hamiltonian in Bloch representation

U H = r: T ( K ) dkadKa f - a(Ki - K, - K4) d&,:1,dK,tdk8+dK,& f

KO K A K&

and

we obtain with


and for the grand thermodynamic potential

Qapp = ---In l I 2cosh(BEK) + 2cosh B K I

U 4 2 U 2 U 4N - p N + - (N, - Nfr) + - N - __ [(N, - Nfr)2 - - Sz,)2] f Qfr 7 (29)

where

Purther we obtain the self-consistent equations

(32) where we use the abbreviations

(--)wt is the expectation value. Analogous to Dichtel et al. [ l ] the gap equation (equation (30)) has two

solutions where the trivial one A = 0 is for the paramagnetic state how it can be found by calculation of the transverse susceptibility. A lower energy one, if it exists for A > 0, corresponds to an antiferromagnetic state.


Now we want to give a motivation about the test Hamiltonian. TO approximate the Coulomb interaction by a one-particle term one can introduce magnetic fields fixed a t each lattice site R,. The most general form for such an operator is

where CJ means the vector built up of the Pauli-spin matrices and $2 the row vector (a$+, a$+).

Calculating the free energy of the Hubbard model using the test operator for the Hubbard Hamiltonian

H,= - J Z (dk+ffdnt + dn+j+dnl) - Z $ $ ( H a 0) gn - p 2 d k J m ) (37) n,i n n, 0

the Hn should be variational parameters to minimize the free-energy functional. But this program is too difficult to perform, therefore we choose

After Fourier transformation we obtain

A K-dependent rotation (see equation (23)) makes H, diagonal:

(40) aK aK d K + Q $ - sin- cos- 2 2

We obtain the form Hi = 2 [&(K) Zkf&+ f Ez(K) & ~ K $ ] * (41)

K

5. Numerical Results and Discussion

For numerical calculation we transform sunis over the first Brillouin zone as

and use the energy density of states established by Jelitto [5]. For our parameters U and B we choose

and N = N,. Further the energy spectrum of the s-band is

(43)

(44) K2h2

&(K) = __ 2m* ’

where m* is the band niass.


t

-2 - ' I I 1 1 0 0.2 0.4 0.6 08 1.0 1.2 7.4

kBT13 - - 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Fig. 1 Fig. 2 Fig. 1. Free energy as a function of temperature. (1) B/J = 0; (2) B/J = - 1; (3) without

free band Fig. 2. Chemical potential as a function of temperature. (1) B/J = 0; ( 2 ) B/J = - 1

With the relation J = A2/2rna2 we obtain m

E(K) = (Ka)' J -. 172. * (45)

As a first approximation we use m/m* = 1. Further we set [ = 0. In case of the neutral model without free band the free energy of the condensed phase (also in the Hartree phase) has a single minimum a t X,, = 0. In our model we take also S, = 0. We calculate the free energy, the chemical potential, and the energy gap A as function of temperature.

From Fig. 1 we observe that the free energy is a continuous and concave function of the temperature T . The system for il > 0 (antiferromagnetic case) has a lower free energy up to the NBel temperature TN where the energy gap vanishes. In contrast to the result of Dichtel et al. the free energy is lower. The chemical potential (Fig. 2) is (for A > 0 and A = 0) a continuous and concave function of T .

In the neutral model without free band the chemical potential is independent of temperature. The energy gap (Fig. 3) as a function of X looks like the curve of Dichtel et al. By introduction of the second band we obtain a lower NBel temperature. The phase transition is of second order. For increasing ratios BIJ the curves approach to the curve of Dichtel et al., although the transition temperature is lower. The results are only in the range U j ZJ good for com- parison with practical values.

Fig. 3. Energy gap as a function of temperature. (1) B/J = 0; ( 2 ) B/J = - 1;

(3) without free band

410 W.-H. STEEB and E. MARSCH: Thermodynamics of a Two-Band Hubbard Model

For calculations in the range U > J we must take the trial density matrix from (17). This leads to a transition temperature of P / U . I n our approximation we can extend the results. Are there solutions of the free energy for Sz+ 0 which are lower than for S, = 0 ?: This can lead to ferromagnetism. Further we have the requirement that Q is also a variation parameter. This yields new self-consistent equations. In this case we cannot use the density of states of Jelitto [5]. We also take no notice of s-d hybridization. These questions will be discussed in a following paper.

References [l] K. DICHTEL, R. J. JELITTO, and H. KOPPE, 2. Phys. 261,173 (1972). [2] T . A. KAPLAN and R. A. BARI, J. appl. Phys. 41,875 (1970). [3] N. D. MERMIN, Ann. Phys. (U.S.A.) 21, 99 (1963). [4] H. G. BECHER, Z. Naturf. 28a, 332 (1973). [51 R. J. JELITTO, J. Phys. Chem. Solids 30, 609 (1969).

(Received May 30, 1974)

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Thermodynamics of a Two-Band Hubbard Model