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1 Max-Planck-Institut fur Mathematik in den Naturwissenschaften Leipzig Bloch electron in a magnetic eld and the Ising model by Igor V. Krasovsky Preprint no.:

3 Bloch electron in a magnetic eld and the Ising model I. V. Krasovsky Max-Planck-Institut fur Mathematik in den Naturwissenschaften Inselstr. 22{26, D-0403, Leipzig, Germany and B.I.Verkin Institute for Low Temperature Physics and Engineering 47 Lenina Ave., Kharkov 3064, Ukraine. Abstract. The spectral determinant det(h ; "I) of the Azbel-Hofstadter Hamiltonian H is related to Onsager's partition function of the 2D Ising model for any value of magnetic ux = 2P=Q through an elementary cell, where P and Q are coprime integers. The band edges of H correspond to the critical temperature of the Ising model the spectral determinant at these (and other points dened in a certain similar way) is independent of P. A connection of the mean of Lyapunov exponents to the asymptotic (large Q) bandwidth is indicated.

4 Although the problems of an electron in a constant magnetic eld and that of an electron on a periodic lattice were solved in the early days of quantum mechanics, the case where a magnetic eld and a lattice are present simultaneously still dees adequate understanding. The simplest model for this case is commonly referred to as the Hofstadter or Azbel-Hofstadter model [,2]. The corresponding Hamiltonian describes an electron on a two-dimensional (2D) N N square lattice with nearest-neighbour hopping subject to a perpendicular uniform magnetic eld: (H ) nx n y = nx; n y + nx+ n y + e ;inx n x n y; + e inx n x n y+ n x n y =0 ::: N ; () where = 2P=Q is the ux through an elementary cell (measured in units of the elementary ux), P and Q are coprime positive integers, = t =t 2 0 is a ratio of hopping amplitudes in x and y directions. We impose periodic boundary conditions ( N ny = 0 ny, ; ny = N; ny, similarly for n y ). This Hamiltonian and the related one-dimensional operator were studied in manyworks (see the reviews [3{5]). Usually, the Hamiltonian () is dened on an innite lattice at the outset in the present paper, however, we shall deal rst with a nite N and take the limit N!later on. It was noticed by Wiegmann and Zabrodin [6,3] that the model is related to integrable systems and the algebra U q (sl 2 ). In the present paper, we nd a dierent type of relation to integrable systems. We shall see that the spectral determinantdet(h ;"I) for any =2P=Q is related to Onsager's partition function of the 2D Ising model. It is well known that the spectrum of H for an innite lattice (on l 2 (Z)) consists of Q intervals (bands). When Q!, the total width W of the bands for 6= is known [7] to approach 4j ; j exponentially fast in Q for =, it is of order =Q [8]. Thouless formulated a conjecture [9,0] that for = the total width W 32G=Q as Q!, where G =;=3 2 +=5 2 ;=7 2 :::is Catalan's constant. (Notation A B means, henceforth, that A=B tends to in the limit.) This result was derived only for P =, P = 2 so far [0,,2] but is supposed to hold in the general case (even when P=Q tends to a nonzero limit) for which it is supported by extensive numerical data. In the present paper, we shall generalize this conjecture and recast it into a simpler type of statement by relating the quantity lim N! (=N 2 )lnjdet(h ; "I)j to the bandwidth. 2

5 Substitute nx ny = e ikyny nx, k y = 2k=N, k = 0 ::: N ; into () to get H = N; k=0 H k,theeigenvalue equation for H k being 2P (H k ) n = n; + n+ +2cos Q n + 2k! n = " n N (2) n = n x =0 ::: N ; : When n is allowed to range from ; to, the corresponding H k on l 2 (Z) is called the almost Mathieu operator. Let us assume that N is divisible by Q and substitute j+ql = e i!l j, j = 0 ::: Q ;,! = 2m, m = 0 ::: N=Q ; into (2). We get N=Q H k = N=Q; m=0 H km, (H km ) j = j; + j+ +2 cos 2P Q j + 2k! j N j =0 ::: Q; Q = e i! 0 ; = e ;i! Q; : Chamber's formula [3] gives the dependence of the spectral determinant of H km on k and m, namely: det(h km ; "I QQ )=(;) Q (") ; 2 Q cos 2k! 2m ; 2 cos (4) N=Q N=Q where (") is a polynomial of degree Q which depends on and, but not on k and m. It is easy to verify (4) by considering the matrix given by (3) and the one obtained after the substitution j = P Q; l=0 exp 2i(jlP=Q + jm=n + lk=n)l. 0 It is the expression (4) which implies that the spectrum of H in the limit N!consists exactly of Q bands: the image of the interval [;2( + Q ) 2( + Q )] under the inverse of the transform = ("). Thus, det(h ; "I) = N=Q; Y (;) N 2 k m=0 NY k=0 N=Q; Y m=0 det(h km ; "I QQ )= (") ; 2 Q cos 2k! Q 2m ; 2 cos : N=Q N=Q In the limit of innite lattice (N! ), let us replace 2k=N in (2) by a continuous parameter, denote H k by H, and consider the mean (") of 3 (3) (5)

6 Lyapunov exponents over all. By virtue of the 3-term recursion (2), any n is obtained from initial conditions 0,. The Lyapunov exponent (") corresponding to H describes the exponential rate of growth (or decay) of a solution to the equation (H ) n = " n, n =0 :::. More precisely, (") = lim n! n ; max = ln( 2 n + 2 n+) =2 : (6) Note that the dierence between H dened in the space of square-summable sequences with indices n =0 ::: and the almost Mathieu operator corresponding to n = ::: ; 0 ::: is that the spectrum of the latter is doubly degenerate. However, the normalized to unity density of states (x) is the same in both cases. According to the Thouless formula [4], (") = R ln j" ; xj (x)dx. Using this formula and (5), we get for the mean of (") over all (to ensure that the determinant is nonzero we assume that " has an imaginary part after taking the limit N!we can let ="! 0): (") = lim 2 Q N! Z Z 0 0 ln j det(h ; "I)j = 2 N ln j(") ; 2 Q cos x ; 2 cos yjdxdy: (7) On the other hand, Onsager's partition function Z for the 2D Ising model on a square N N lattice satises [5]: lim N! Z N ln Z ; 2 2 ln(2 sinh a0 )= Z ln a cosh a0 0 2cosh sinh a 0 ; 2 sinh a sinh a (8) cos x ; 2 cos y 0 dxdy: Here a =2J=T, a 0 =2J 0 =T, where T is the temperature J, J 0,interaction constants in x and y direction. We now set sinh a= sinh a 0 = Q (hence a 0 = arcsinh ( ;Q sinh a)) cosh a cosh a0 (") =2 sinh a 0 q =2 Q coth a + ;2Q sinh 2 a (9) to obtain the equivalence (as N!) for any " and coprime P and Q: j det ~t 2 (H ; "I)j Z 2=Q (0) 4

7 where ~t Q 2 =2sinha 0. A simple analysis of (9) shows that for any real T, j(")j 2( + Q ). The minimum of j(")j as a function of T, j(")j = 2( + Q ), corresponds to the critical temperature T c of the Ising model (at T c the argument of the logarithm in (7) and (8) vanishes at an integration limit). On the other hand, the energies " ei at which j(" ei )j = 2( + Q ) are the band edges of lim N! H. It is well known [6] that Z at T c equals the square of the partition function Z D of dimers on the lattice if r = p 2sinha and r 2 = p 2sinha 0 are interpreted as activities of dimers in x and y directions. Let us identify r 2 = t Q and r 2 2 = t Q 2 (recall that t and t 2 are hopping amplitudes in x and y directions, and = t =t 2 ). One result of Lieb and Loss [7] says that det H f ZD 2 as N!for the ux =, that is for P =,Q =2. Here fh = t 2 H a more symmetric form of the Azbel-Hofstadter Hamiltonian. Since it is known that " = 0 is a band edge for Q even, we have from (0) det H f Z 4=Q D, which is a generalization of the mentioned result to any P odd, Q even. Henceforth, we only consider the limit of innite N. Let us simplify expression (7) (note an important fact that depends on " and P only through (")!). Because of the obvious equality ( )= ln + (= Q =), it is sucient to consider only the case 0. In this case, taking the integral over y in (7) we get: () = 8 R >< Q 0 arccosh (jj=2+ Q cos x)dx jj 2( + Q ) R arccos(2;jj)=2 Q 0 arccosh (jj=2+ Q cos x)dx 2( + Q ) > jj > 2( ; Q ) Q >: 0 jj 2( ; Q ) () The fact that (") is zero on the image of the interval 2 [;2( ; Q ) 2( ; Q )] is in agreement with the general argument: this image is an intersection of the intervals of the spectra of the operators H. The generalized eigenfunction of any H is just a Bloch wave on a periodic lattice with Q atoms in an elementary cell. Therefore, the Lyapunov exponents (") are zero on this set. By denition, (") isamean of these exponents. Now we can note that in the limit Q!, xed, () tends to the following Lyapunov exponent on the spectrum of H: (")! 0 if, : 5

8 (")! ln if >, in accordance with a statement of Aubry and Andre [8]. (Naturally, we obtain the same asymptotic result for any individual (").) For T c (jj = 2(+ Q )), let us represent () in another form by reducing the integral to R =2 0 arcsinh ( Q=2 cos x)dx and then using dierentiation w.r.t. the parameter. We get for any >0: (" ei )= 4 Z Q=2 Q 0 arctan x dx =ln + 4 Z ;Q=2 x Q 0 arctan x dx: (2) x In what follows, we always consider (the most interesting from the point of view of bandwidth) case =. It is not, however, dicult to generalize the argument below to the case of any. For =, the expression (2) reads: (" ei )=4G=Q. In these terms, the Thouless conjecture says that the total bandwidth behaves as 8(" ei ) asymptotically for large Q. The following more general fact appears to hold and is supported by numerics. Let W (x) be the total length of the image of [0 x] under the inverse of the transform = (") (recall that the whole spectrum is the image of [;4 4], i.e., the total bandwidth is W = W (;4) + W (4)). Then for any P (including P changing with Q such that P=Q is not small) as Q!: W () 4() = 2 Q Z jj 0 K(x=4)dx 2 [;4 4]: (3) The last equation is just another form of () for = K(k) = R =2 0 ( ; k 2 sin 2 x) ;=2 dx is the complete elliptic integral of the rst kind. It is easy to derive (3) in the case P =,where we can use the known from semiclassics [2] asymptotic expression for ("). In this case, the main asymptotic contribution to W () comes from the bands in a small neighbourhood of " = 0. These bands have width of order =Q ln Q, while the others are exponentially narrow. For j"j, P =, andq!, (") 4cosh("Q=4) cos "Q 2 ln 4Q ; 2 arg ; 2 + i"q 4 ; Q 2 Hence, the asymptotic contribution of an individual band to W () is : (4) 2 Q ln Q arcsin jj=4 cosh("q=4) (5) 6

9 and the numberofsuch bands in an interval dt = d("q=4) is 2 dt ln Q. Therefore, the sum of all contributions W () 8 Z arcsin jj=4 Q 0 cosh t dt = 2 Z jj K(x=4)dx =4() (6) Q 0 which completes the derivation of (3) for P =. If we could show that for a general P, W () c(), where c is independent of, then the value of c could be found using a result of Last and Wilkinson [9] that for any coprime P and Q, P Q i= j 0 (" i )j ; = Q ;, where (" i )=0,i = ::: Q. This can be written as jw(0)j 0 ==Q (right or left derivative) because, obviously, jw()j 0 = P Q i= j 0 (" i )j ;, where (" i ) =, i = ::: Q. On the other hand, since K(0) = =2, we have j(0)j 0 ==4Q, which implies that c =4. Putting our results and conjectures together, we can roughly say that the logarithm of Onsager's partition function, the mean of Lyapunov exponents, and the asymptotic bandwidth are basically the same object which is universal in that for a xed value of, itdoesnot depend on P. I am grateful to P. B. Wiegmann for encouraging my interest in the problem. I also thank R. Seiler, J. Kellendonk, and D. J. Thouless for valuable suggestions which helped me to improve thispaper. References [] M. Ya. Azbel, Zh.Eksp.Teor.Fiz. 46, 929 (964) [ Sov.Phys.JETP 9, 634 (964) ] [2] D. R. Hofstadter, Phys.Rev. B4, 2239 (976) [3] P. B. Wiegmann, Prog.Theor.Phys. Suppl. 34, 7 (999) [4] Y. Last, XI th Intl. Congress Math. Phys. Proceedings. p.366 (Boston: Intl. Press, 995) S. Ya. Jitomirskaya ibid p.373 [5] Ch. Kreft and R. Seiler, J.Math.Phys. 37, 5207 (996) [6] P. B. Wiegmann and A. V. Zabrodin, Phys.Rev.Lett. 72, 890 (994) 7

10 [7] J. Avron, P. van Mouche, B. Simon, Commun.Math.Phys. 32, 03 (990) [8] Y. Last, Commun.Math.Phys. 64, 42 (994) [9] D. J. Thouless, Phys.Rev. B28, 4272 (983) [0] D. J. Thouless, Commun.Math.Phys. 27, 87 (990) [] D. J. Thouless and Yong Tan, J.Phys.A:Math.Gen. 24, 4055 (99) [2] G. I. Watson, J.Phys.A: Math.Gen. 24, 4999 (99) [3] W. Chambers, Phys.Rev. A40, 35 (965) [4] D. J. Thouless, J.Phys.C 5, 77 (972) J. Avron and B. Simon, Duke Math.J. 50, 369 (983) A. L. Figotin and L. A. Pastur, J.Math.Phys. 25, 774 (984) [5] L. Onsager, Phys.Rev. 65, 7 (944) [6] P. W. Kasteleyn, Physica 27, 209 (96) H. N. V. Temperley and M. E. Fisher, Philos. Mag. 6, 06 (96) M. E. Fisher, Phys.Rev. 24, 664 (96) [7] E. H. Lieb and M. Loss, Duke Math.J. 7, 337 (993) [8] S. Aubry, G. Andre, Ann. Israel Phys. Soc. 3, 33 (980) [9] Y. Last and M. Wilkinson, J.Phys.A: Math.Gen. 25, 623 (992) 8

THE DERIVATION OF THE FREE ENERGY OF THE ISING MODEL FROM THAT OF THE EIGHT-VERTEX MODEL D. A. LAVIS Department of Mathematics, King's College, Strand

THE DERIVATION OF THE FREE ENERGY OF THE ISING MODEL FROM THAT OF THE EIGHT-VERTEX MODEL D. A. LAVIS Department of Mathematics, King's College, Strand, London, WCR LS, England March 5, 99 Abstract The