On Thouless bandwidth formula in the Hofstadter model

TL;DR

This paper extends the Thouless bandwidth formula to higher moments, providing a closed-form expression involving special mathematical functions, thereby deepening the understanding of spectral properties in the Hofstadter model.

Contribution

It introduces a generalized formula for the n-th moment of the bandwidth in the Hofstadter model, expressed through polygamma, zeta, and Euler numbers.

Findings

Derived a closed-form expression for the n-th moment of the bandwidth.

Connected spectral properties to special functions like polygamma, zeta, and Euler numbers.

Abstract

We generalize Thouless bandwidth formula to its n-th moment. We obtain a closed expression in terms of polygamma, zeta and Euler numbers.

Equations

$${\Phi }_{m+1}+{\Phi }_{m-1}+2\cos (k_y+\gamma m){\Phi }_m=e{\Phi }_m$$

$$m_{p/q}(e,k_x,k_y)=\left(\begin{array}{cccccc}2\cos (k_y)-e& 1& 0& \cdots & 0& e^{-iq{k}_x}\\ 1& 2\cos (k_y+\frac{2\pi p}{q})-e& 1& \cdots & 0& 0\\ 0& 1& ()& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & ()& 1\\ e^{iq{k}_x}& 0& 0& \cdots & 1& 2\cos (k_y+(q-1)\frac{2\pi p}{q})-e\end{array}\right)$$

$$m_{p/q}(e,k_x,k_y).\Phi =0$$

$$\det (m_{p/q}(e,k_x,k_y))=\det (m_{p/q}(e,0,0))-2(-1{)}^q(\cos (q{k}_x)-1+\cos (q{k}_y)-1),$$

$$\det (m_{p/q}(e,0,0))=2(-1{)}^q(\cos (q{k}_x)-1+\cos (q{k}_y)-1)$$

$$b_{p/q}(e)=-\sum _{j=0}^{[\frac{q}{2}]}{a}_{p/q}(2j)e^{2j}$$

$$\det (m_{p/q}(e,0,0))+4(-1{)}^q=(-1{)}^q{e}^q{b}_{p/q}(1/e)$$

$$e^q{b}_{p/q}(1/e)=2(\cos (q{k}_x)+\cos (q{k}_y)).$$

$$e^q{b}_{p/q}(1/e)=4\mathrm{and}{e}^q{b}_{p/q}(1/e)=-4$$

$$e_1(4)\le e_2(4)\le \dots \le e_q(4)\mathrm{and}{e}_1(-4)\le e_2(-4)\le \dots \le e_q(-4)$$

$$(-1)^{q+1}\sum_{r=1}^{q}(-1)^{r}\big{(}e_{r}(-4)-e_{r}(4)\big{)}.$$

$$\lim_{q\to\infty}(-1)^{q+1}q\sum_{r=1}^{q}(-1)^{r}\big{(}e_{r}(-4)-e_{r}(4)\big{)}={32\over\pi}\sum_{k=0}^{\infty}(-1)^{k}{1\over(2k+1)^{2}}$$

$$(-1)^{q+1}\sum_{r=1}^{q}(-1)^{r}\big{(}e_{r}^{n}(-4)-e_{r}^{n}(4)\big{)}.$$

$$(-1)^{q+1}\sum_{r=1}^{q}(-1)^{r}\big{(}e_{r}(-4)-e_{r}(4)\big{)}^{n},$$

$$n{\int }_{-4}^4{\tilde{\rho }}_{p/q}(e)e^{n-1}de$$

$$-2\sum _{r=1}^q(-1{)}^r{e}_r^n(4)=2\sum _{r=1}^q(-1{)}^r{e}_r^n(-4)$$

$$\det (m_{p/q}(e,0,0))=0.$$

$$\det (m_{p/q}(e,0,0))=-\det (m_{p/q}^{++}(e))\det (m_{p/q}^{--}(e))$$

$$m_{p/q}^{++}(e)=\left(\begin{array}{cccccc}e-2& 2& 0& \cdots & 0& 1\\ 1& e-2\cos (\frac{2\pi p}{q})& 1& \cdots & 0& 0\\ 0& 1& ()& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & ()& 1\\ 1& 0& 0& \cdots & 1& e-2\cos (\frac{q-1}{2}\frac{2\pi p}{q})-1\end{array}\right)$$

$$m_{p/q}^{--}(e)=\left(\begin{array}{cccccc}e-2\cos (\frac{2\pi p}{q})& 1& 0& \cdots & 0& 1\\ 1& e-2\cos (\frac{4\pi p}{q})& 1& \cdots & 0& 0\\ 0& 1& ()& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & ()& 1\\ 1& 0& 0& \cdots & 1& e-2\cos (\frac{q-1}{2}\frac{2\pi p}{q})+1\end{array}\right)$$

$$-\sum _{r=1}^q(-1{)}^r{e}_r(4)=\sum _{r=1}^{\frac{q+1}{2}}|e_r^{++}|-\sum _{r=1}^{\frac{q-1}{2}}|e_r^{--}|$$

$$\frac{2i}{\pi }{\int }_{-ix}^{ix}\left(\frac{z}{z-a}-1\right)dz=\frac{4a}{\pi }\arctan (\frac{x}{a}),$$

$$\underset{x\to \infty }{\lim }\frac{4a}{\pi }\arctan (\frac{x}{a})=2|a|$$

$$\underset{x\to \infty }{\lim }\frac{2i}{\pi }{\int }_{-ix}^{ix}\left(\frac{z}{z-a}-1\right)dz=\underset{x\to \infty }{\lim }\frac{2i}{\pi }{\int }_{-ix}^{ix}\left(-\log \frac{z-a}{z}\right)dz$$

$$2\left(\sum _{r=1}^{\frac{q+1}{2}}|e_r^{++}|-\sum _{r=1}^{\frac{q-1}{2}}|e_r^{--}|\right)=\frac{2i}{\pi }\underset{x\to \infty }{\lim }{\int }_{-ix}^{ix}\log \left(\frac{z\det (m_{p/q}^{--}(z))}{\det (m_{p/q}^{++}(z))}\right).$$

$$\underset{q\to \infty }{\lim }2q\left(\sum _{r=1}^{\frac{q+1}{2}}|e_r^{++}|-\sum _{r=1}^{\frac{q-1}{2}}|e_r^{--}|\right)=32{\int }_0^{\infty }\log \left(\frac{\Gamma (3/4+y{)}^2}{y\Gamma (1/4+y{)}^2}\right)dy$$

$$\lim_{q\to\infty}q\left(\sum_{r=1}^{q}(-1)^{r}\big{(}e_{r}(-4)-e_{r}(4)\big{)}\right)={4\over\pi}\left(\psi^{(1)}\left(\frac{1}{4}\right)-\pi^{2}\right)$$

$$-\sum _{r=1}^q(-1{)}^r{e}_r^n(4)=\sum _{r=1}^{\frac{q+1}{2}}|e_r^{++}{|}^n-\sum _{r=1}^{\frac{q-1}{2}}|e_r^{--}{|}^n$$

$$\frac{2i}{\pi }{\int }_{-ix}^{ix}\left(\frac{z^n}{z-a}-\sum _{k=0}^{n-1}{a}^k{z}^{n-1-k}\right)dz=\frac{4a^n}{\pi }\arctan (\frac{x}{a}),$$

$$\underset{x\to \infty }{\lim }\frac{4a^n}{\pi }\arctan (\frac{x}{a})=2|a^n|$$

Full text

(*) LPTMS, CNRS-Faculté des Sciences d’Orsay, Université Paris Sud, 91405 Orsay Cedex, France

1. Introduction

In a series of stunning papers stretching over almost a decade [1] Thouless obtained a closed expression for the bandwidth of the Hofstadter spectrum [2] in the $q\to\infty$ limit. Here the integer $q$ stands for the denominator of the rational flux $\gamma=2\pi{p}/{q}$ of the magnetic field piercing a unit cell of the square lattice; the numerator $p$ is taken to be $1$ (or equivalently $q-1$)111In the sequel $p$ is understood to be equal to $1$..

Let us recall that in the commensurate case where the lattice eigenstates $\psi_{m,n}=e^{ink_{y}}\Phi_{m}$ are $q$-periodic $\Phi_{m+q}=e^{iqk_{x}}\Phi_{m}$, with $k_{x},k_{y}\in[-\pi,\pi]$, the Schrodinger equation

[TABLE]

reduces to the $q\times q$ secular matrix

[TABLE]

acting as

[TABLE]

on the $q$-dimensional eigenvector $\Phi=\{\Phi_{0},\Phi_{1},\ldots,\Phi_{q-1}\}$. Thanks to the identity

[TABLE]

the Schrodinger equation $\det(m_{p/q}(e,k_{x},k_{y}))=0$ rewrites [3] as

[TABLE]

The polynomial

[TABLE]

materializes in $\det(m_{p/q}(e,0,0))$

[TABLE]

so that eq. (3) becomes

[TABLE]

The $a_{p/q}(2j)$’s (with $a_{p/q}(0)=-1$) are related to the Kreft coefficients [4]: how to get an explicit expression for these coefficients is explained in Kreft’s paper.

We focus on the Hofstadter spectrum bandwidth defined in terms of the $2q$ edge-band energies $e_{r}(4)$ and $e_{r}(-4)$, $r=1,2,\ldots,q,$ solutions of

[TABLE]

respectively (see Figures 1 and 2).

If one specifies an ordering for the $e_{r}(4)$’s and the $e_{r}(-4)$’s

[TABLE]

the bandwidth is

[TABLE]

The Thouless formula is obtained in the $q\to\infty$ limit as

[TABLE]

(see also [5]). We aim to extend this result to the $n$-th moment defined as

[TABLE]

which is a natural generalization222Computing also the bandwidth $n$-th moment

(-1)^{q+1}\sum_{r=1}^{q}(-1)^{r}\big{(}e_{r}(-4)-e_{r}(4)\big{)}^{n},

(8) here defined for $n$ odd, would be of particular interest. We will come back to this question in the conclusion. of (5): one can think of it as

[TABLE]

where $\tilde{\rho}_{p/q}(e)$ is the indicator function with value 1 when $|e^{q}b_{p/q}(1/e)|\leq 4$ and 0 otherwise.

Trivially (7) vanish when $n$ 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 $q$ is odd, in which case it simplifies further to

[TABLE]

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 of

[TABLE]

2. The first moment : Thouless formula

The key point in the observation of Thouless [1] is that if evaluating the first moment rewritten in (9) as $2\sum_{r=1}^{q}(-1)^{r}e_{r}(4)$ when $q$ is odd seems at first sight untractable, still,

• •

thanks to $\det(m_{p/q}(e,0,0))$ factorizing as

[TABLE]

where

[TABLE]

[TABLE]

are matrices of size $(q+1)/2$ and $(q-1)/2$ respectively, so that the $e_{r}(4)$’s split in two packets $e_{r}^{++}$, $r=1,2,\ldots,(q+1)/2,$ the roots of $\det(m_{p/q}^{++}(e))=0$ and $e_{r}^{--}$, $r=1,2,\ldots,(q-1)/2,$ those of $\det(m_{p/q}^{--}(e))=0$

• •

and thanks to $\sum_{r=1}^{q}(-1)^{r}e_{r}(4)$ happening to rewrite as

[TABLE]

(9) becomes tractable since it reduces to the sum of the absolute values of the roots of two polynomial equations.

Indeed using [1]

[TABLE]

[TABLE]

and

[TABLE]

one gets

[TABLE]

Making [1] further algebraic manipulations on the ratio of determinants in (10) in particular in terms of particular solutions $\{\Phi_{0},\Phi_{1},\ldots,\Phi_{q-1}\}$ of (2) –on the one hand $\Phi_{0}=0$ and on the other hand $\Phi_{(q-1)/2}=\Phi_{(q+1)/2}$– and then for large $q$ taking in (1) the continuous limit lead to, via the change of variable $y=qz/(8\pi i)$,

[TABLE]

This last integral gives the first moment

[TABLE]

which is a rewriting of (6) ($\psi^{(1)}$ is the polygamma function of order 1).

3. The $n$-th moment

3.1. $n$ odd:

to evaluate the $n$-th moment one follows the steps above by first noticing that

[TABLE]

holds. Then using

[TABLE]

[TABLE]

and

[TABLE]

one gets

[TABLE]

In the RHS of (12) the polynomial $z^{n-1}\sum_{k=1}^{n-1}{a^{k}\over kz^{k}}$ cancels the positive or nul exponents in the expansion around $z=\infty$ of the logarithm term $z^{n-1}\log{z-a\over z}$. Likewise, in (13), the same mechanism takes place for $-z^{n-1}\left(\sum_{k=1}^{n-1}{\sum_{r=1}^{q+1\over 2}(e_{r}^{++})^{k}-\sum_{r=1}^{q-1\over 2}(e_{r}^{--})^{k}\over kz^{k}}\right)$ with respect to $z^{n-1}\log\left({z\det(m_{p/q}^{--}(z))\over\det(m_{p/q}^{++}(z))}\right)$. Additionally, the polynomials can be reduced to their $k$ even components. Further algebraic manipulations in (13) and, when $q$ is large, taking the continous limit, lead to, via the change of variable $y=qz/(8\pi i)$,

[TABLE]

To go from (13) to (14) one has used that for $k$ even, necessarily333(15) is also strongly supported by numerical simulations. More generally the $k$-th moments $\sum_{r=1}^{q+1\over 2}(e_{r}^{++})^{k}$ and $\sum_{r=1}^{q-1\over 2}(e_{r}^{--})^{k}$ can be directly retrieved from the coefficients of $\det(m_{p/q}^{++}(e))$ and $\det(m_{p/q}^{--}(e))$ respectively. In particular one finds $\sum_{r=1}^{q+1\over 2}e_{r}^{++}=2$ and $\sum_{r=1}^{q-1\over 2}e_{r}^{--}=-2$; for $k$ odd $\lim_{q\to\infty}\sum_{r=1}^{q+1\over 2}(e_{r}^{++})^{k}=4^{k}/2$ and $\lim_{q\to\infty}\sum_{r=1}^{q-1\over 2}(e_{r}^{--})^{k}=-4^{k}/2$; for $k$ even $\lim_{q\to\infty}1/q\left(\sum_{r=1}^{q-1\over 2}(e_{r}^{++})^{k}\right)=\lim_{q\to\infty}1/q\left(\sum_{r=1}^{q-1\over 2}(e_{r}^{--})^{k}\right)=\binom{k}{{k}/{2}}^{2}/2$. This last result can easily be understood in terms of the number $\binom{k}{{k}/{2}}^{2}$ of closed lattice walks with $k$ steps [7].

[TABLE]

where the $E_{k}$’s are the Euler numbers. Indeed in (14), as it was the case in (12,13), the polynomial $\sum_{k=2,k\;{\rm even}}^{n-1}E_{k}/(k4^{k})y^{n-1-k}$ cancels the positive or nul exponents in the expansion around $y=\infty$ of the logarithm term $y^{n-1}\log\left({\Gamma(3/4+y)^{2}\over y\Gamma(1/4+y)^{2}}\right)$ [6]. It amounts to a fine tuning at the infinite upper integration limit so that after integration the end result is finite. Performing this last integral gives the $n$-th moment

[TABLE]

which generalizes the Thouless formula (11) to $n$ odd ($\psi^{(n)}$ 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, we should rather consider a $n$-th moment restricted to the positive –or equivalently by symmetry negative– half of the spectrum444Instead of ${n}\int_{-4}^{4}\tilde{\rho}_{p/q}(e)e^{n-1}de$ one considers

${n}\int_{-4}^{0}\tilde{\rho}_{p/q}(e)e^{n-1}de={n}\int_{0}^{4}\tilde{\rho}_{p/q}(e)e^{n-1}de.$

.. In the $q$ odd case it is

[TABLE]

It is still true that

[TABLE]

where, since $n$ is even, absolute values are not needed anymore, a simpler situation. It follows that the right hand side of (16) also gives, when $n$ is even, twice the $q\to\infty$ 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 that

[TABLE]

yields $q^{n}$ times the $n$-th moment when $n$ is odd555When $n$ is odd it is also twice the half spectrum $n$-th moment

${n}\int_{-4}^{4}\tilde{\rho}_{p/q}(e)e^{n-1}de=2{n}\int_{0}^{4}\tilde{\rho}_{p/q}(e)e^{n-1}de.$

. and twice the half spectrum $n$-th moment when $n$ is even. Numerical simulations do confirm convincingly this result (eventhough the convergence is slow). In the $n$ even case one already knows from (15) that (3.3) simplifies further to

[TABLE]

from which one gets for the $n\to\infty$-moment scaling

[TABLE]

where

[TABLE]

4. Conclusion and opened issues

(3.3) is certainly a simple and convincing $n$-th moment generalization of the Thouless bandwidth 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 of the Thouless first moment. (3.3) was obtained for $p=1$ (or $q-1$): it would certainly be interesting to understand what happens for $p\neq 1$ where numerical simulations indicate a strong $p$ dependence when $n$ 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 $2|E_{n}|({2\pi/q})^{n}$, a result that can be interpretated as if, at the $n$-zooming level, they were $2|E_{n}|$ bands each of length ${2\pi/q}$. It would be interesting to see if this Euler counting has a meaning in the context of lattice 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 again on $q$ odd, one can expand

[TABLE]

where the symmetry $e_{r}(-4)=-e_{q+1-r}(4)$ has again been used. The $k=0$ term $-2\sum_{r=1}^{q}(-1)^{r}e_{r}(4)^{n}$ is the $n$-th moment discussed above and one knows that multiplying it by $q^{n}$ ensures in the $q\to\infty$ limit a finite scaling. Let us also multiply in (19) the $k=1,\ldots,(n-1)/2$ terms by $q^{n}$: one checks numerically that

[TABLE]

Using

[TABLE]

one concludes that in the $q\to\infty$ limit the bandwidth $n$-th moment is such that

[TABLE]

when $n$ is odd, a fact which is also supported by numerical simulations666Similarly, when $n$ is even, the bandwidth $n$-th moment now defined as

\sum_{r=1}^{q}\big{(}e_{r}(-4)-e_{r}(4)\big{)}^{n}

is such that \lim_{q\to\infty}q^{n}\sum_{r=1}^{q}\big{(}e_{r}(-4)-e_{r}(4)\big{)}^{n}=0.. Clearly, multiplying the sum in (19) by $q^{n}$ is insufficient, a possible manifestation of the fractal structure [5] of the band spectrum. 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 Eugène Bogomolny and Stephan Wagner and thank Alain Comtet for a careful reading of the manuscript. Discussions with Vincent Pasquier are also acknowledged.