On Thouless bandwidth formula in the Hofstadter model
aa r X i v : . [ m a t h - ph ] M a r ON THOULESS BANDWIDTH FORMULA IN THE HOFSTADTER MODEL
ST´EPHANE OUVRY (*) AND SHUANG WU (*)
Abstract.
We generalize Thouless bandwidth formula to its n -th moment. We obtain a closedexpression in terms of polygamma, zeta and Euler numbers. (*) LPTMS, CNRS-Facult´e des Sciences d’Orsay, Universit´e Paris Sud, 91405 Orsay Cedex,France 1. Introduction
In a series of stunning papers stretching over almost a decade [1] Thouless obtained a closedexpression for the bandwidth of the Hofstadter spectrum [2] in the q → ∞ limit. Here the integer q stands for the denominator of the rational flux γ = 2 πp/q of the magnetic field piercing a unitcell of the square lattice; the numerator p is taken to be 1 (or equivalently q − .Let us recall that in the commensurate case where the lattice eigenstates ψ m,n = e ink y Φ m are q -periodic Φ m + q = e iqk x Φ m , with k x , k y ∈ [ − π, π ], the Schrodinger equationΦ m +1 + Φ m − + 2 cos( k y + γm )Φ m = e Φ m (1)reduces to the q × q secular matrix m p/q ( e, k x , k y ) = k y ) − e · · · e − iqk x k y + πpq ) − e · · · · · · · · · () 1 e iqk x · · · k y + ( q − πpq ) − e acting as m p/q ( e, k x , k y ) . Φ = 0 (2)on the q -dimensional eigenvector Φ = { Φ , Φ , . . . , Φ q − } . Thanks to the identitydet( m p/q ( e, k x , k y )) = det( m p/q ( e, , − − q (cos( qk x ) − qk y ) − , the Schrodinger equation det( m p/q ( e, k x , k y )) = 0 rewrites [3] asdet( m p/q ( e, , − q (cos( qk x ) − qk y ) −
1) (3)The polynomial b p/q ( e ) = − [ q ] X j =0 a p/q (2 j ) e j materializes in det( m p/q ( e, , m p/q ( e, , − q = ( − q e q b p/q (1 /e ) (4)so that eq. (3) becomes e q b p/q (1 /e ) = 2(cos( qk x ) + cos( qk y )) . Date : October 15, 2018. In the sequel p is understood to be equal to 1. + - - - - e - - e - e Figure 1. p = 1 , q = 3 , e q b p/q (1 /e ) = e − e : the 3 horizontal red segments arethe energy bands; the 3 red dots are the mid-band energies; the 6 black dots arethe ± a p/q (2 j )’s (with a p/q (0) = −
1) are related to the Kreft coefficients [4]: how to get an explicitexpression for these coefficients is explained in Kreft’s paper.We focus on the Hofstadter spectrum bandwidth defined in terms of the 2 q edge-band energies e r (4) and e r ( − r = 1 , , . . . , q, solutions of e q b p/q (1 /e ) = 4 and e q b p/q (1 /e ) = − e r (4)’s and the e r ( − e (4) ≤ e (4) ≤ . . . ≤ e q (4) and e ( − ≤ e ( − ≤ . . . ≤ e q ( − − q +1 q X r =1 ( − r (cid:0) e r ( − − e r (4) (cid:1) . (5)The Thouless formula is obtained in the q → ∞ limit aslim q →∞ ( − q +1 q q X r =1 ( − r (cid:0) e r ( − − e r (4) (cid:1) = 32 π ∞ X k =0 ( − k k + 1) (6)(see also [5]). We aim to extend this result to the n -th moment defined as( − q +1 q X r =1 ( − r (cid:0) e nr ( − − e nr (4) (cid:1) . (7)which is a natural generalization of (5): one can think of it as n Z − ˜ ρ p/q ( e ) e n − de Computing also the bandwidth n -th moment( − q +1 q X r =1 ( − r (cid:0) e r ( − − e r (4) (cid:1) n , (8)here defined for n odd, would be of particular interest. We will come back to this question in the conclusion. N THOULESS BANDWIDTH FORMULA IN THE HOFSTADTER MODEL 3 + - - - e - - e - e + Figure 2. p = 1 , q = 4 , e q b p/q (1 /e ) = e − e + 4 : the 4 horizontal red segmentsare the energy bands; the 4 red dots are the mid-band energies; the 8 black dotsare the ± ρ p/q ( e ) is the indicator function with value 1 when | e q b p/q (1 /e ) | ≤ n is even –we will see later how to give a non trivial meaning to the n -th moment in this case. Therefore we focus on (7) when n is odd and, additionnally, when q isodd, in which case it simplifies further to − q X r =1 ( − r e nr (4) = 2 q X r =1 ( − r e nr ( −
4) (9)thanks to the symmetry e r ( −
4) = − e q +1 − r (4). As said above, the e r (4)’s are the roots of e q b p/q (1 /e ) = 4 that is, by the virtue of (4), those ofdet( m p/q ( e, , . The first moment : Thouless formula
The key point in the observation of Thouless [1] is that if evaluating the first moment rewrittenin (9) as 2 P qr =1 ( − r e r (4) when q is odd seems at first sight untractable, still, • thanks to det( m p/q ( e, , m p/q ( e, , − det( m ++ p/q ( e )) det( m −− p/q ( e )) ST´EPHANE OUVRY (*) AND SHUANG WU (*) where m ++ p/q ( e ) = e − · · · e − πpq ) 1 · · · · · · · · · () 11 0 0 · · · e − q −
12 2 πpq ) − m −− p/q ( e ) = e − πpq ) 1 0 · · · e − πpq ) 1 · · · · · · · · · () 11 0 0 · · · e − q −
12 2 πpq ) + 1 are matrices of size ( q + 1) / q − / e r (4)’s split intwo packets e ++ r , r = 1 , , . . . , ( q + 1) / , the roots of det( m ++ p/q ( e )) = 0 and e −− r , r =1 , , . . . , ( q − / , those of det( m −− p/q ( e )) = 0 • and thanks to P qr =1 ( − r e r (4) happening to rewrite as − q X r =1 ( − r e r (4) = q +12 X r =1 | e ++ r | − q − X r =1 | e −− r | (9) becomes tractable since it reduces to the sum of the absolute values of the roots of twopolynomial equations.Indeed using [1] 2 iπ Z ix − ix (cid:18) zz − a − (cid:19) dz = 4 aπ arctan( xa ) , lim x →∞ aπ arctan( xa ) = 2 | a | and lim x →∞ iπ Z ix − ix (cid:18) zz − a − (cid:19) dz = lim x →∞ iπ Z ix − ix (cid:18) − log z − az (cid:19) dz one gets 2 q +12 X r =1 | e ++ r | − q − X r =1 | e −− r | = 2 iπ lim x →∞ Z ix − ix log z det( m −− p/q ( z ))det( m ++ p/q ( z )) ! . (10)Making [1] further algebraic manipulations on the ratio of determinants in (10) in particular interms of particular solutions { Φ , Φ , . . . , Φ q − } of (2) –on the one hand Φ = 0 and on the otherhand Φ ( q − / = Φ ( q +1) / – and then for large q taking in (1) the continuous limit lead to, via thechange of variable y = qz/ (8 πi ),lim q →∞ q q +12 X r =1 | e ++ r | − q − X r =1 | e −− r | = 32 Z ∞ log (cid:18) Γ(3 / y ) y Γ(1 / y ) (cid:19) dy This last integral gives the first momentlim q →∞ q q X r =1 ( − r (cid:0) e r ( − − e r (4) (cid:1)! = 4 π (cid:18) ψ (1) (cid:18) (cid:19) − π (cid:19) (11)which is a rewriting of (6) ( ψ (1) is the polygamma function of order 1). N THOULESS BANDWIDTH FORMULA IN THE HOFSTADTER MODEL 5 The n -th moment n odd: to evaluate the n -th moment one follows the steps above by first noticing that − q X r =1 ( − r e nr (4) = q +12 X r =1 | e ++ r | n − q − X r =1 | e −− r | n holds. Then using 2 iπ Z ix − ix z n z − a − n − X k =0 a k z n − − k ! dz = 4 a n π arctan( xa ) , lim x →∞ a n π arctan( xa ) = 2 | a n | andlim x →∞ iπ Z ix − ix z n z − a − n − X k =0 a k z n − − k ! dz = lim x →∞ iπ Z ix − ix − nz n − log z − az + n − X k =1 a k kz k ! dz (12)one gets2 q +12 X r =1 | e ++ r | n − q − X r =1 | e −− r | n = 2 iπ lim x →∞ Z ix − ix nz n − log z det( m −− p/q ( z ))det( m ++ p/q ( z )) ! − n − X k =1 P q +12 r =1 ( e ++ r ) k − P q − r =1 ( e −− r ) k kz k dz. (13)In the RHS of (12) the polynomial z n − P n − k =1 a k kz k cancels the positive or nul exponents in the ex-pansion around z = ∞ of the logarithm term z n − log z − az . Likewise, in (13), the same mechanismtakes place for − z n − P n − k =1 P q +12 r =1 ( e ++ r ) k − P q − r =1 ( e −− r ) k kz k ! with respect to z n − log (cid:18) z det( m −− p/q ( z ))det( m ++ p/q ( z )) (cid:19) .Additionally, the polynomials can be reduced to their k even components. Further algebraic ma-nipulations in (13) and, when q is large, taking the continous limit, lead to, via the change ofvariable y = qz/ (8 πi ),lim q →∞ q n q +12 X r =1 | e ++ r | n − q − X r =1 | e −− r | n = (8 πi ) n − Z ∞ ny n − log (cid:18) Γ(3 / y ) y Γ(1 / y ) (cid:19) + n − X k =2 ,k even E k k k y k dy. (14)To go from (13) to (14) one has used that for k even, necessarily lim q →∞ q k q +12 X r =1 ( e ++ r ) k − q − X r =1 ( e −− r ) k = (2 π ) k | E k | (15)where the E k ’s are the Euler numbers. Indeed in (14), as it was the case in (12,13), the polynomial P n − k =2 ,k even E k / ( k k ) y n − − k cancels the positive or nul exponents in the expansion around y = ∞ of the logarithm term y n − log (cid:16) Γ(3 / y ) y Γ(1 / y ) (cid:17) [6]. It amounts to a fine tuning at the infinite upper (15) is also strongly supported by numerical simulations. More generally the k -th moments P q +12 r =1 ( e ++ r ) k and P q − r =1 ( e −− r ) k can be directly retrieved from the coefficients of det( m ++ p/q ( e )) and det( m −− p/q ( e )) respectively.In particular one finds P q +12 r =1 e ++ r = 2 and P q − r =1 e −− r = −
2; for k odd lim q →∞ P q +12 r =1 ( e ++ r ) k = 4 k / q →∞ P q − r =1 ( e −− r ) k = − k /
2; for k even lim q →∞ /q (cid:18)P q − r =1 ( e ++ r ) k (cid:19) = lim q →∞ /q (cid:18)P q − r =1 ( e −− r ) k (cid:19) = (cid:0) kk/ (cid:1) /
2. This last result can easily be understood in terms of the number (cid:0) kk/ (cid:1) of closed lattice walks with k steps [7]. ST´EPHANE OUVRY (*) AND SHUANG WU (*) integration limit so that after integration the end result is finite. Performing this last integralgives the n -th momentlim q →∞ q n q X r =1 ( − r ( e nr ( − − e nr (4)) ! = 4 π (cid:18) ( − n − ψ ( n ) (cid:18) (cid:19) − n (cid:0) n +1 − (cid:1) ζ ( n + 1) n ! (cid:19) (16)which generalizes the Thouless formula (11) to n odd ( ψ ( n ) is the polygamma function of order n ).3.2. n even: as said above the n -th moment trivially vanishes when n is even. In this case, weshould rather consider a n -th moment restricted to the positive –or equivalently by symmetrynegative– half of the spectrum . In the q odd case it is − ( q − / X r =1 ( − r ( e nr ( − − e nr (4))+ e n ( q +1) / (cid:16) ( − q +12 (cid:17) = q X ( q +3) / ( − r ( e nr ( − − e nr (4))+ e n ( q +1) / (cid:16) ( − q − (cid:17) . (17)It is still true that q +12 X r =1 ( e ++ r ) n − q − X r =1 ( e −− r ) n = q X ( q +3) / ( − r ( e nr ( − − e nr (4)) + e n ( q +1) / (cid:16) ( − q − (cid:17) where, since n is even, absolute values are not needed anymore, a simpler situation. It follows thatthe right hand side of (16) also gives, when n is even, twice the q → ∞ limit of the half spectrum n -th moment as defined in (17), up to a factor q n .3.3. Any n : one reaches the conclusion that4 π (cid:18) ( − n − ψ ( n ) (cid:18) (cid:19) − n (cid:0) n +1 − (cid:1) ζ ( n + 1) n ! (cid:19) (18)= 2 π n +1 n ! ∞ X k =0 ( − k (2 k + 1) n +1 = 2 π n ! (cid:0) ζ ( n + 1 ,
14 ) − ζ ( n + 1 ,
34 ) (cid:1) yields q n times the n -th moment when n is odd and twice the half spectrum n -th moment when n is even. Numerical simulations do confirm convincingly this result (eventhough the convergenceis slow). In the n even case one already knows from (15) that (18) simplifies further to2 | E n | (cid:0) π (cid:1) n from which one gets for the n → ∞ -moment scaling µ n !2 n where µ = lim n →∞ ,n even − n π n | E n | n ! = 2 . . . . . Instead of n R − ˜ ρ p/q ( e ) e n − de one considers n Z − ˜ ρ p/q ( e ) e n − de = n Z ˜ ρ p/q ( e ) e n − de. . When n is odd it is also twice the half spectrum n -th moment n Z − ˜ ρ p/q ( e ) e n − de = 2 n Z ˜ ρ p/q ( e ) e n − de. . N THOULESS BANDWIDTH FORMULA IN THE HOFSTADTER MODEL 7 Conclusion and opened issues (18) is certainly a simple and convincing n -th moment generalization of the Thouless band-width formula (6). It remains to be proven on more solid grounds for example in the spirit of[5]. In the definition of the n -th moment (7) one can view the exponent n as a magnifying loop ofthe Thouless first moment. (18) was obtained for p = 1 (or q − p = 1 where numerical simulations indicate a strong p dependencewhen n increases, an effect of the n -zooming inherent to the n -th moment definition (7).In the n even case, twice the half spectrum n -th moment ends up being equal to 2 | E n | (2 π/q ) n ,a result that can be interpretated as if, at the n -zooming level, they were 2 | E n | bands each oflength 2 π/q . It would be interesting to see if this Euler counting has a meaning in the context oflattice walks [7] (twice the Euler number 2 | E n | counts the number of alternating permutations in S n ). Finally, returning to the bandwidth n -th moment defined in (8) for n odd, and focusing againon q odd, one can expand q X r =1 ( − r (cid:0) e r ( − − e r (4) (cid:1) n = − ( n − / X k =0 (cid:18) nk (cid:19) ( − k q X r =1 ( − r e r ( − k e r (4) n − k (19)where the symmetry e r ( −
4) = − e q +1 − r (4) has again been used. The k = 0 term − P qr =1 ( − r e r (4) n is the n -th moment discussed above and one knows that multiplying it by q n ensures in the q → ∞ limit a finite scaling. Let us also multiply in (19) the k = 1 , . . . , ( n − / q n : one checksnumerically thatlim q →∞ − q n q X r =1 ( − r e r ( − k e r (4) n − k = n − kn lim q →∞ − q n q X r =1 ( − r e r (4) n = n − kn π (cid:18) ( − n − ψ ( n ) (cid:18) (cid:19) − n (cid:0) n +1 − (cid:1) ζ ( n + 1) n ! (cid:19) . Using ( n − / X k =0 (cid:18) nk (cid:19) ( − k n − kn = 0one concludes that in the q → ∞ limit the bandwidth n -th moment is such thatlim q →∞ q n q X r =1 ( − r (cid:0) e r ( − − e r (4) (cid:1) n = 0when n is odd, a fact which is also supported by numerical simulations . Clearly, multiplying thesum in (19) by q n is insufficient, a possible manifestation of the fractal structure [5] of the bandspectrum. We leave to further studies the question of finding a right scaling for the bandwidth n -th moment. 5. Acknowledgements
S. O. acknowlegdes interesting discussions with Eug`ene Bogomolny and Stephan Wagner andthank Alain Comtet for a careful reading of the manuscript. Discussions with Vincent Pasquierare also acknowledged. Similarly, when n is even, the bandwidth n -th moment now defined as q X r =1 (cid:0) e r ( − − e r (4) (cid:1) n is such that lim q →∞ q n P qr =1 (cid:0) e r ( − − e r (4) (cid:1) n = 0. ST´EPHANE OUVRY (*) AND SHUANG WU (*)
References [1] D.J. Thouless, ”Bandwidths for a quasiperiodic tight-binding model”, Phys. Rev. B 28 (1983) 4272-4276;”Scaling for the Discrete Mathieu Equation”, Commun. Math. Phys. 127 (1990) 187-193; D.J. Thouless andY. Tan, ”Total bandwidth for the Harper equation. III. Corrections to scaling ”, J. Phys. A Math. Gen. 24(1991) 4055-4066.[2] D.R. Hofstadter, ”Energy levels and wave functions of Bloch electrons in rational and irrational magneticfields”, Phys. Rev. B (1976) 2239.[3] W. Chambers, Phys. Rev A140 (1965), 135143.[4] C. Kreft, ”Explicit Computation of the Discriminant for the Harper Equation with Rational Flux”, SFB 288Preprint No. 89 (1993).[5] B. Helffer and P. Kerdelhu´e, ”On the total bandwidth for the rational Harper’s equation”, Comm. Math. Phys.173 (1995), no. 2, 335-356 ; Y. Last, ”Zero Measure for the Almost Mathieu Operator”, Commun. Math. Phys.164 (1994) 421-432; ” Spectral Theory of Sturm-Liouville Operators on Infinite Intervals: A Review of RecentDevelopments ”, in W.O. Amrein, A.M. Hinz, D.B. Pearson ”Sturm-Liouville Theory: Past and Present”,99120 (2005) Birkhauser Verlag Basel/Switzerland.[6] It can be shown explicitly that − P ∞ k =2 , k even E k k k y k is the series expansion of log (cid:16) Γ(3 / y ) y Γ(1 / y ) (cid:17) as y → ∞→ ∞