scaling relation for the density of states of a disordered n-orbital model

Volume 99A, number 1 PHYSICS LETTERS 14 November 1983

SCALING RELATION FOR THE DENSITY OF STATES OF A DISORDERED n-ORBITAL MODEL *

Klaus ZIEGLER Institut fir Theoretische Physik, Universitiit Heidelberg, Philosophenweg 19, 6900 Heidelberg, Fed. Rep. Germany

Received 18 March 1983

A tight-binding model of a particle on a ddimensional lattice with n orbitals per lattice site is considered. It is shown that the density of states obeys a scaling relation at the n = - band edges with characteristic dimension d = 4.

Most properties of a single particle in a random po- tential seem to be clear as Thouless declared [I].

Believing in this, it is rather surprising that there are still contradictions in the perturbative evaluation of such a fundamental quantity as the density of states (DOS) of a disordered system. For example, Harris and Lubensky argue that the DOS vanishes at the mobility edge for spatial dimensions d > 4 but not for d < 4 [2]. In contrast, Wegner proved that the DOS is different from zero and finite for finite energy and arbitrary spatial dimensions [3].

We will study the n-orbital model with local unitary gauge invariance, also called the phase-invariant- ensemble (PIE) [4], to understand some more about the dimensional dependence of the DOS in a vicinity of the Anderson transition. This model is governed by the hamiltonian

H=n-Y* c I5 Ir,p>frp,r)p~ (r’,pI, r,r’ p,p’=l

(1)

where the tight-binding state Ir, p) is characterised by the lattice point r on a simple d-dimensional lattice and the orbital number p. The hermitean matrix f

f w, r’p’ =f;p', rp (2) has a gaussian distribution

* This work has been supported by the Deutsche Forschungs- gemeinschaft through Sonderforschungsbereich 123.

P[fJ =lv-l

’ exP(-$s ( w$)vl ,ccl l&p, r’p’l*), C3)

with a symmetric, positive definite matrix ~-1 and the normalisation factor N-1. Assuming translational invariance

(w-1)rr’ = (w- ‘)r_,’ (4)

the n = QD limit can be solved exactly [4]. The behavior of the DOS under scaling transforma-

tions has been studied for this model with large n in the continuum limit [5]. The result is a scaling relation in a vicinity of the n = = band edges %!Z,-,

p(E , rr) = n-Q+r*E(lEl- ED)) ,

with the exponent

t = 2/(6 - 6)

(5)

(6)

for spatial dimensions d < 2. The restriction to d < 2 is caused by the method, since the continuum limit of the model exists only below d = 2. The DOS is given by the average one-particle Green function Cl as

p(E,n)=Ta-lImG1(E+iO’,“) (7)

with

Gl(z,n)=JDKl P[fl (r,pl(z - W’lr,p). (8)

Cl is independent of r and p due to the translational invariance and the local unitary gauge invariance. A

0.031-9163/83/0000-0000/$03.00 0 1983 North-Holland 19


useful representation for a detailed examination of the l/n expansion of G, has been given in ref. [6] :

G&Y n) = -(4/J& (Ql,,),

( > = det(w/2rr) JD[Q;, Qi] . . . exp(-L),

with the lagrangian

(9)

and

Q,, r as a function of Q&, r is a solution of the non-lin- ear equation

Ql,r=n -Y2Q;,,-fA,,

A, = 2 C w&(Q;, r, - iQi, r,)-l rt

Xhl 2nY2.z + $1 t Q;, rg - iQ;, ,.#

2nY2.z + $1 - Q;, ,., + iQL, ?, (11)

S, = .y2(Q1, r + iQ2, ,I

= 2nY2Ql,,- Q;,,+iQ;,, which follows immediately from eqs. (8) and (12) of ref. [6]. Furthermore, we can also write

G,(z, n) = -(2n-y2/E;) (S ) r 2 (12)

because of

(CQl,, - iQz,,N= ((Q;,, - iQi,,P= 0. (13)

Eq. (8) of ref. [6] implies that S, is a solution of

S, = Q;, r t iQi, r - nY2Ar. (14)

We can solve this equation approximately by expand- ing the logarithmic factor of A, in powers of the vari-

ables s,, Qk, r and iterating the resulting equation:

s,=s,-SO) (15)

sr = ~~ [s,] = c u,,$Q;,,~ + iQ; rt> - 8 r'

m 1,

t c vr_r’ c c nl-%(l, 1’) r’ I=3 I’=1

X (s,I)""(Q;, r~ - iQi rf)‘-l t R,, m, (16)

with the matrices

v = (w - l/z’2)-1, u = WV,

and

z’ = z + ++S 0)

6 = (4/E;)(So t n y2E;/2z’)(4/E; - 1/z’2)-‘,

4 -1’1 v(& 1’) = 7 22’

( )O I’

(Z’ is the summation over odd numbers). The remain- der R, of the Taylor expansion is o(n(-m+l)/2). The choice of the translations So of S, is not com- pletely arbitrary, since we have to respect the cut of the logarithm [5,7] :

Rez’<@)O, ifRez<(>)O,

Imz’<@)O, ifImz<QO. (17)

Iteration of eq. (16) for sy with the starting value

so, r = s; WI (18) and

Sj+l,r = q~jJ1 0'2 01, (19)

yields an approximation for s, which can be used to determine GL(z, n) by (12). The approximations for

v-1 C(s,, r

(V = volume of the system) can be represented in Fourier space by Feynman diagrams. An example is given in fig. 1. The first contribution stands for

ahd(k),

the propagator is

4/E; - 1/zf2 t p2k2,

p2 = -(1/2d) Cr2 wr, r

20


1 YT

Fig. 1. Contributions to the average one-particle Green function up to two loops.

and a vertex of order 1 is o(r~~-~/~). We obtain in par- ticular the l/n expansion [8] with the choice 9 = 0. This expansion breaks down for z = fEO due to infra- red divergences [.5,8]. In this case we choose d # 0 with

z’=zu t f,

z. = Eo12, forRez>O,

= -E,/2, forRez<O, (20)

and determine E such that the sum of the leading diagrams is zero. Thus the leading contribution to G, is pulled into the constant So. Inspection of the diagrams up to two-loop order and power-counting for the higher order diagrams determines E as

e=n -21(6-a,(_~/6)1/(6-d)

X (5 t 6 ee2A t e-4A2)-1/6-a t o(nm4/@-d))

WJ),

f = *in-la-Y2(5 t 6em2A t es4A2)-v2

+o(np2) (d>4); (21)

with

A=z,(2z, -z), lA191e12.

a, b are real, positive numbers. The DOS reads now as a scaling relation:

p(E, n) = -z02 Im E t o(nm2s)

= n-~j$n2t(1El - Eo)) t o(n-a),

with

(22)

.$=2/(6-d) (d<‘Q,

= 1 (d>4),

according to (7). This agrees with the earlier result for d < 2 [5]. However, it shows that d = 4 is a characteristic dimension of the average one-particle Green function in contrast to the speculation d = 6 of ref. 161. Furthermore, the DOS does not vanish at E = So. It is important to remark that we have no one- loop contributions to G, for the PIE. Existence of such diagrams would decrease the characteristic dimension to d = 2 in the considered system.

References

[l] D.J. Thouless, in: Proc. Fourth Taniguchi Intern. Symp. 1981 (Springer, Berlin, 1982).

[2] A.B. Harris and T.C. Lubensky, Phys. Rev. B23 (1981) 2640.

[3] F. Wegner, Z. Phys. B44 (1981) 9. [4] F. Wegner, Phys. Rev. B19 (1979) 783. [S] K. Ziegler, Z. Phys. B48 (1982) 293. [6] K. Ziegler, Phys. Lett. 92A (1982) 339. [7] K. Ziegler, Thesis, Heidelberg (1982). [8] R. Oppermann and F. Wegner, Z. Phys. B34 (1979) 327.

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scaling relation for the density of states of a disordered n-orbital model

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