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Toda lattice invariants and the Harper's Hamiltonian thermodynamics

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J. Phys. A: Math. Gen.27 (1994) 5687-5692. Printed in the U K

Toda lattice invariants and the Harper's Hamiltonian thermodynamics S A Ktitorovt and L Jastrabikx TA F Ioffe Physical and Technical Institute, Russian Academy of Sciences, Polytechnicheskaya Street 26, 194021 SI Petersburg, Russia # Institute of Physics, Czech Academy of Sciences, Na Slovance 2, 18040 Prague 8, Czech Republic Received 17 December 1994, in final form 29 April 1994 Abstract. A connection between the Toda lattice additive integrals of motion and the Harper's Hamiltonian thermodynamics is established. The Harper's Hamiltonian is shown to be a special case of the Lax operator for the Toda lattice.

1. Inboduction The study of the Harper's Hamiltonian has a rich history, and began with the early work o f [ 1,2]. In the numerical work by Hofstadter, a fractal nature of the spectrum was discovered [ 3 ] . Apart from numerical computations, plenty o f different mathematical methods have been applied: quasiclassical WKB [4,5] and instanton [6] approaches, renormalization group [7] and C* algebra [SI methods, etc. Recent interest in the model stems from a possible application to a study of superconducting networks [9], wire networks [IO], the lattice Ginzburg-Landau theory of superconductivity near the upper critical magnetic field [ I 11, anyon superconductivity [ 121, organic conductors [13], the quantum Hall effect [14], etc. Most of the work dedicated to a study of the Harper's model is directed to a calculation of all eigenstates of the operator. But for a physicist this is too detailed for the great majority of cases. More than that, sometimes it is not easy to use effectively such a large amount o f information. All physicists need is a calculation of traces, determinants and other invariants which can be used for a computation of observables. The aim of this work is to develop an approach to the Harper's Hamiltonian giving the possibility of a direct calculation o f observables without obtaining eigenvalues and eigenvectors.

2. Spectral characteristics of an operator and observables

The Epstein zeta function CH(s)of an operator H is determined as follows [ 151: (1)

CH(S)=Ck-' n

0305-4470/94/165687 t 061El9.50

0 1994 IOP Publishing Ltd

5687

5688

S A Ktiforov and L Jastrabik

where A,, are eigenvalues of the operator H, H$" = U". The one-loop fluctuation correction to a classical free energy functional with the kinetic term b*H$ [IO] 6F=-(1/2) tr log(H+r)

(3)

can be expressed as [I51 6F=-rB(O).

(4)

On the other hand, the quantum partition function or the theta function of the operator H i s determined as W p ) = t r p = C exp(-pL)

"

where the density matrix p satisfies the Bloch equation ap/ap= -HP.

In accordance with Minakshisundaram's theorem [ 161,

Cds) = r-'(S)

dP B-' tr p(P)

(7)

where r(s)is the gamma function. Thus we see that the fluctuation correction to the classical free energy (3) and the quantum partition function (5) are linked by the common Epstein zeta function. Comparing (4), (5) and (7) we can write Fco

SF= -1im d/ds r-'(s)J dt tr-' tr(p(x, x, t)-po(x, x, 1)). S-0

(8)

0

Here p is the density matrix or the thermal kernel of the operator satisfying ( 6 ) ;po is a regulator. The theta function of the operator has the asymptotic high-temperature expansion [ 151

+ p + . ..)

= ( 4 7 ~ p ) - ~ ' ~ (aju ~

(9)

where ut are Seeley coefficients [15]; d is the space dimension. In the important and rather generic case we are interested by ultraviolet divergences (small 1 in (IO)); therefore, in this case only a few first Seeley coefficients are necessary. Apart from the thermodynamics written above, formulae can be applied to a calculation of spectral characteristics of the operator. For instance, the density of states

can be written in the following form: g(l)=(27~)-' d / d l tr log[(H- A+iO)/(H-A-iO)].

(1 1)

Our goal now is to express these values in terms of the Toda lattice invariants in the case of the Harper's operator.

Toda lattice invariants

5689

3. Connection between the Toda lattice problem and the Harper's Hamiltonian (i) The equation of motion for the Toda lattice can be written as [16] dQ./dr=P.

e.- 111-exp[ - (a+- Q.)l

dP.ldt=exp[-(&-

I

(12)

or, introducing new variables, I

I

b.= iP,

an's exp[-(Q.+~-&)I

(13)

we have da./dt=a,(b,-b,+t)

db./dt=2(a:+

I

-4).

(14)

Equations (14) can be written in the matrix form as dL/dt = EL - L 5

(15)

or, equivalently, as L(t)= u(l)L(o)u-l(t)

(16)

where the unitary matrix U(t) evolves according to the equation dU/dt = EU(f ) .

(17)

Therefore, L(t) is unitary equivalent to L ( 0 ) [ 161. For a periodic Toda lattice with the boundary conditions

b. = bn+N

a, = a.+ N

(18)

matrices L and 5 can be written as

L=

bl ai 0 ai bz a2 0 az b3

...

...

0

0

...

I

0 0

...

aN 0

B=

0

0 a3

0 al 0 0

-a1

0 02

0 -a2 0

bN-1

ON-i

aN-1

bx

... 0

UN

...

-a3

0 UN-1

-UN

0

One of the most important properties of the model is that the eigenvalues of the Lax operator are integrals of motion: di/dt=O.

(20)

This means that the determinant det(L1-L) is invariant too. It can be expanded as 1161

+ . . .+ uN-+

det(L - u)= AN+ LN-II1

I

I ~

(21)

5690

S A Klitorov and L Jastrabik

where the coefficients I, are polynomials in a. and b. or of the dynamical variables P. and Q.. The roots of the equation (22)

det(L-W)=O

are eigenvalues of the Lax operator L.One can see from (21) that all I, are constants: dIi/dt=O.

(23)

They are simply the integrals of motion first found by H6non [17]. Now we have the necessary minimum of information concerning the Toda lattice to consider our problem. (ii) The Harper's Hamiltonian appeared for the first time in the study of Landau level splitting and widening due to the lattice symmetry of a crystal potential [I, 21. It can be written in the form

N y = exp(iyl (m,n)) y(m + 1, n ) + expf-iyl (m - 1, n ) ) y ( m - 1, n ) +exp(iyz(m, n ) ) y ( m , n +l)+exp(-iyl(m,n-l))yv(m,n-

1)

(24)

where

jm,. Ai m+ I?

n ( m gn)= (elbc)

",n=l

dxi

y ~ ( mn) , = (eibc)

Jm,n

Ai

&.

(25)

Uniformity of the magnetic field transforms the gauge invariance into the following simple constraint [IS]:

y l ( m , n ) + y ~ ( m + l , n ) - y , ( m , n + l ) - y 2 ( m , n ) = 2 n f 4 i 4 ~ o ) = a (26) where +,,=(hc)/e is the quantum of flux and 4 is a flux through a plaquette. In the Landau gauge y(m, n) =0, y(in, n)=am. Then the lattice translation operator T. corre sponding to the non-magnetic translation in the y-direction on the one plaquette commutes with the Hamiltonian. Thus the solution of the Schrodinger equation factorizes: y(m, n ) =exp(ikin)$(n)

(27)

where k=pa withp beinga quasi-wavevector component and a being the latticespacing. The function $(n) satisfies the well known Harper's ('almost Mathieu') equation

4 (n+ 1) + $ (n - 1) + 2 cos(an + k)4(n) = E 4 (n).

(28)

(iii) The central idea of this work is the following observation: the Harper's Hamiltonian written in the matrix form is a special case of the Lax operator L for the Toda lattice with a.=

1

b,= 2 cos(an+ k).

(29)

If 4/40=p/q with p and q being whole numbers, L is a q x q matrix.

4. Expression of the Harper's Hamiltonian determinant in terms of HBnon's invariants

The matrix elements of the Harper Hamiltonian determined by (29) have to be considered as the initial conditions for the Toda evolution equations.

5691

Toda lattice invariants

Notice that the ‘time’ parameter t is connected with the evolution equations (14) and (15) and has nothing to do with the non-stalionary Schrodinger equation H y = i ayr/at, where H = L. The determinant of the matrix (2L-2AI) can be written as [16]

(

J=det(2L-221)=exp -

:r:

1 Aka2

i

(a&a&+,)

n )/I1

(a-22)

(30)

where I is the identity matrix, N = q ,

AI= (2~~)’. (31) Isospectral deformations due to the evolution equation do not change the spectral invariants and, therefore, we may calculate the determinant (30) with initial values of Bj=2bj

Bh, Ak.

The wanted log det(L- AI) differs from the value log J by an unimportant additive constant. On the other hand, J can be expanded into a series which stops at the Nth step ifp and q are whole numbers: I ~. ~ = ( 2 2 ) ~ + 1 ~ ( 2 2 ) ~. .- +~ +

(32)

where I,, is the nth HLnon’s integral of motion,

Substituting into (32) and (33) the formulae for A , and B. (see (31)) and taking into

account (29), we have our problem formally solved. Connections between J and other values of interest are given in section 2. We conclude that the established connection between the Harper’s Hamiltonian and the Toda lattice can be used for calculation of spectral and thermodynamic values relevant to this operator. As an example of a physical problem which can be effectively solved with the method suggested here, we can mention the problem of the fluctuation corrections to the free energy of the lattice Landau-Ginzburg model with a background magnetic field, considered using rather rough approximations in [I I]. An investigation of the case of irrational a seems to be of particular interest. This study is now in progress.

Acknowledgments The authors acknowledge useful discussions with B N Shalaev, L Trlifaj, M Bednar and V N Popov. References [IJ Harper P G 1955 Proc. Phys. Soc. London A 68 874 [Z] Zilberman G E 1956 Zh. Ekrp. Teor.. F i z 30 1092 [3] Hofstadler D R 1976 Phys. Rev. B 14 2239 (41 Ostlund S and Pandit R 1984 Phys. Rev. B 29 1394 [ 5 ] Ra”a1 R 1985 1.Phys. Frome 46 1345 [6] Freed D and Harvey J A 1990 Phys. Rev. B 41 I1328 (71 Wilkinson M 1987 J. Phys. A: Math. Gen. M 4337

5692

S A Ktitorov and L Jastrabik

181 Rammal R and Bellisard J 1990 3. Phjir. France 51 1803 [9] Doucot B and Ramma1 R 1986 J . Phys. France 47 913 [IO] R a m " R 1986 Physics and Fobrimtion of Microsfruelures ed M Kelly and C Weisbuch (Berlin: Springer) p 303 [II ] Ktitorov S A , Petrov Yu V, Shalaev B N and Sherstinov V G 1992 Inl. J . Mod. Pliys. B 6 1209 [ I Z ] Chen Y H, Wilczek F, Witten E and Halprin B I 1989 L f . 3. Mod. Phys. B 3 1001 1131 Rammal R and Bellisard J 1990 Europhys. Lelf. 13 205

1141 Thouless D I, Kohmoto M.Nightingale M P and den Nus M 1982 Phys. Rev. M I . 49 405 [IS] Hurt N 1983 Ceomefric Quantirarion in Acrion (Dordrecht: Reidel) [I61 Toda M 1981 Tlteory ofNonlinear Loltiees (Berlin: Springer) [I71 Hinon M 1974Phs.Rev. 8 9 1921 [I81 Rammal R and Bellisard J 1990 3. Pliys, Frunce 51 1803