Fluctuations of Random Matrix Products and 1D Dirac Equation with Random Mass

Abstract

We study the fluctuations of certain random matrix products $\Pi _N=M_N\cdots M_2M_1$ of $\mathrm {SL}(2,\mathbb {R})$, describing localisation properties of the one-dimensional Dirac equation with random mass. In the continuum limit, i.e. when matrices $M_n$’s are close to the identity matrix, we obtain convenient integral representations for the variance $\Gamma _2=\lim _{N\rightarrow \infty }\mathrm {Var}(\ln ||\Pi _N||)/N$. The case studied exhibits a saturation of the variance at low energy $\varepsilon$ along with a vanishing Lyapunov exponent $\Gamma _1=\lim _{N\rightarrow \infty }\ln ||\Pi _N||/N$, leading to the behaviour $\Gamma _2/\Gamma _1\sim \ln (1/|\varepsilon |)\rightarrow \infty$ as $\varepsilon \rightarrow 0$. Our continuum description sheds new light on the Kappus–Wegner (band center) anomaly.

Appendices

Appendix 1: Details of the Derivation of Eq. (4.4)

1.1 $\varepsilon ^2>0$

For real energy, we see from the SDE (2.1) that the process $z(x)$ flows towards $\mathbb {R}$. We have introduced the change of variable $\zeta =\mp \ln (\pm z/|\varepsilon |)/2$ for $z\in \mathbb {R}_\pm$, implying that the new process $\zeta (x)$ crosses $\mathbb {R}$ twice when $z(x)$ does once. Hence the change of variable maps the SDE (2.1) onto the couple of SDEs

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}x}\zeta = |\varepsilon |\,\cosh 2\zeta \pm m(x) = -\mathcal {U}_\pm '(\zeta ) + \sqrt{g}\,\eta (x). \end{aligned}$$

(10.1)

In the main text we used $\left\langle m(x)\right\rangle =0$ and the local nature of the mass correlation to disregard the sign. Here we consider for the moment the general case $\left\langle m(x)\right\rangle =\mu \,g$ and introduce a couple of potentials $\mathcal {U}_\pm (\zeta )=-(|\varepsilon |/2)\sinh 2\zeta \mp \mu \zeta$ related to the cases $z(x)>0$ and $z(x)<0$, respectively. $\eta (x)$ is a normalised Gaussian white noise with zero mean. The process is characterised by two stationary distributions $\mathcal {P}^\pm (\zeta )$, each normalized, related to $f(z)$ for $z\in \mathbb {R}_\pm$. For example, the Lyapunov exponent is given by [16]

$$\begin{aligned} \gamma _1 = \left\langle z+m(x)\right\rangle = \mu g+ \frac{|\varepsilon |}{2} \left[ \int \mathrm{d}\zeta \,\mathcal {P}^+(\zeta )\,e^{-2\zeta } -\int \mathrm{d}\zeta \,\mathcal {P}^-(\zeta )\,e^{+2\zeta } \right] . \end{aligned}$$

(10.2)

Now considering the case $\mu =0$ for which $\mathcal {P}^+=\mathcal {P}^-$, leads to $\gamma _1 = -|\varepsilon |\,\left\langle \sinh 2\zeta \right\rangle$, i.e.

$$\begin{aligned} \gamma _1 = 2\left\langle \mathcal {U}(\zeta )\right\rangle . \end{aligned}$$

(10.3)

Fluctuations may be discussed in a similar way. A crucial observation is that, in the original SDE (2.1), the diffusion effectively vanishes at $z=0$, implying the absence of correlations between the process at coordinates $x$ and $x'$ associated to $z(x)>0$ and $z(x')<0$. It follows that the contributions of the fluctuations related to the two intervals $z\in \mathbb {R}_+$ and $z\in \mathbb {R}_-$ simply add. The second term of (2.5) takes the form

$$\begin{aligned}&\frac{1}{2}\bigg ( \int \mathrm{d}\zeta \mathrm{d}\zeta '\, |\varepsilon |e^{-2\zeta } \mathcal {G}^+(\zeta |\zeta ')\, (-2|\varepsilon |)\sinh 2\zeta '\, \mathcal {P}^+(\zeta ')\\&\quad +\, \int \mathrm{d}\zeta \mathrm{d}\zeta '\, (-|\varepsilon |)e^{+2\zeta } \mathcal {G}^-(\zeta |\zeta ')\, (-2|\varepsilon |)\sinh 2\zeta '\, \mathcal {P}^-(\zeta ') \bigg ). \end{aligned}$$

For $\mu =0$ we have $\mathcal {G}^+=\mathcal {G}^-$ leading to Eq. (4.4).

1.2 $\varepsilon ^2<0$

For imaginary energy the analysis is slightly different: the process $z(x)$ is trapped on $\mathbb {R}_+$ and $\zeta (x)$ does not flow across $\mathbb {R}$. The change of variable is simply $\zeta =-(1/2)\ln (z/|\varepsilon |)$. The new process is trapped by the potential well $\mathcal {U}(\zeta )=(|\varepsilon |/2)\cosh 2\zeta -\mu \zeta$. The equilibrium distribution is $\mathcal {P}(\zeta )\propto \exp \big [-(2/g)\mathcal {U}(\zeta )\big ]$. When $\mu =0$ the potential is symmetric. We can symmetrize the expression $\gamma _1=|\varepsilon |\left\langle e^{-2\zeta }\right\rangle$, leading to $\gamma _1=|\varepsilon |\left\langle \cosh 2\zeta \right\rangle$, i.e. again to (10.3). Eq. (2.5) leads to

$$\begin{aligned} \gamma _2 = g&+ 2|\varepsilon |\left\langle \zeta \,e^{-2\zeta }\right\rangle + 2|\varepsilon |^2 \int \mathrm{d}\zeta \mathrm{d}\zeta '\, e^{-2\zeta }\, \mathcal {G}(\zeta |\zeta ')\, \cosh 2\zeta ' \,\mathcal {P}(\zeta ') . \end{aligned}$$

(10.4)

The second term can be obviously symmetrized, which gives the second term of (4.4). Symmetrization of the third integral term works as follows: the propagator may be decomposed over the left/right eigenvectors of the forward generator $\fancyscript{G}^\dagger$ as

$$\begin{aligned} \mathcal {G}(\zeta |\zeta ') = \sum _{n>0} \frac{\Phi _n^\mathrm {R}(\zeta )\Phi _n^\mathrm {L}(\zeta ')}{\fancyscript{E}_n} \end{aligned}$$

(10.5)

where $\fancyscript{G}^\dagger \Phi _n^\mathrm {R}(\zeta ) = -\fancyscript{E}_n\Phi _n^\mathrm {R}(\zeta )$ and $\fancyscript{G} \Phi _n^\mathrm {L}(\zeta ) = -\fancyscript{E}_n\Phi _n^\mathrm {L}(\zeta )$. Because the potential $\mathcal {U}(\zeta )$ is symmetric, the eigenvectors have a symmetry property $\Phi _n^\mathrm {L/R}(-\zeta )=(-1)^n\Phi _n^\mathrm {L/R}(\zeta )$. Integration over $\zeta '$ in (10.4) selects only the contributions of even eigenvectors which allows one to symmetrize the integrand with respect to $\zeta \rightarrow -\zeta$, leading to Eq. (4.4).

It is remarkable that despite the dynamics of the process $\zeta (x)$ being quite different for real and imaginary $\varepsilon$, we have found a unique representation for both $\gamma _1$, Eq. (4.9), and $\gamma _2$, Eq. (4.4), expressed in terms of the potential $\mathcal {U}(\zeta )$.

Appendix 2: Direct Calculation of $\gamma _2$ for $\varepsilon =0$

The study of the case $\varepsilon =0$ shows some subtlety related to the choice of the norm of the matrix. In the usual case, the statistical properties of the RMP are independent of the precise definition of the norm [2, 4]. Bougerol and other authors propose

$$\begin{aligned} ||M|| = \mathrm {Sup}\{ |Mx|,\;\ x\in \mathbb {R}^2,\;\ |x|=1\} \end{aligned}$$

(10.6)

where $|x|$ is the norm on the vector space.

In the numerical calculation, we have parametrized the spinor as $\Psi =e^{\xi }(\sin \Theta ,-\cos \Theta )$, in the spirit of the phase formalism [26], and study the statistical properties of $\xi (x)=(1/2)\ln \big [\Psi (x)^\dagger \Psi (x)\big ]$, usually setting $\Theta (0)=\Theta _0=0$. Let us discuss the general case where $\Theta _0$ may differ from $0$. Since $\Psi (x_{N+1}^-)=\Pi _N\Psi (x_1^-)$, the numerical procedure corresponds to considering the norm

$$\begin{aligned} ||\Pi _N||_{\Psi _0} = |\Pi _N\Psi _0| \ \text{ with } \ \Psi _0= \left( \begin{array}{c} \sin \Theta _0 \\ -\cos \Theta _0 \end{array} \right) , \end{aligned}$$

(10.7)

i.e. $\xi (x_{N+1}^-)=\ln ||\Pi _N||_{\Psi _0}$. We also introduce another possible definition of the norm

$$\begin{aligned} |||\Pi _N||| = \int _{|\Psi _0|=1}\mathrm{d}\Psi _0\, ||\Pi _N||_{\Psi _0} , \end{aligned}$$

(10.8)

closer to the spirit of (10.6).

For $\varepsilon =0$, the matrix product $\Pi _N$ can be studied rather directly: the angles vanish $\theta _n=0$ and the matrices $M_n$ commute. Hence we can write

$$\begin{aligned} \Pi _N = \left( \begin{array}{l@{\quad }l} e^{\Lambda } &{} 0 \\ 0 &{} e^{-\Lambda } \end{array} \right) \ \text{ with } \ \Lambda = \sum _{n=1}^N\eta _n . \end{aligned}$$

(10.9)

The distribution of the random variable $\Lambda$ is given by the central limit theorem: $\left\langle \Lambda \right\rangle =\rho x\left\langle \eta _n\right\rangle =0$ and $\mathrm {Var}(\Lambda )=\rho x\left\langle \eta _n^2\right\rangle =gx$ (we consider that $x$ is fixed and $N$ fluctuates with $\left\langle N\right\rangle =\rho x$). We have

$$\begin{aligned} \ln ||\Pi _N||_{\Psi _0} = \frac{1}{2} \ln \left[ \cosh 2\Lambda -\cos 2\Theta _0\,\sinh 2\Lambda \right] . \end{aligned}$$

(10.10)

We examine first the particular case $\Theta _0=0$, leading to $\ln ||\Pi _N||_{\Psi _0} = -\Lambda$. We immediatly deduce that $\left\langle \ln ||\Pi _N||_{\Psi _0}\right\rangle =0$ and $\mathrm {Var}(\ln ||\Pi _N||_{\Psi _0})=gx$, which would lead to $\gamma _1=0$ and, incorrectly, to $\gamma _2=g$. The choice $\Theta _0=\pi /2$ leads to a similar conclusion. This reflects the statistical properties of the two particular zero energy solutions

$$\begin{aligned} \left( \begin{array}{c} 1 \\ 0 \end{array}\right) \,e^{\int ^x\mathrm{d}x'\,m(x')} \ \text{ and } \ \left( \begin{array}{c} 0 \\ 1 \end{array}\right) \,e^{-\int ^x\mathrm{d}x'\,m(x')} \end{aligned}$$

(10.11)

selected by the choices $\Theta _0=\pi /2$ and $\Theta _0=0$, respectively.

We now consider the case of an arbitrary initial vector, with $\Theta _0\notin \{0,\,\pi /2\}$. In the $N\rightarrow \infty$ limit, the large $\Lambda$ behaviour of the norm is selected: $\ln ||\Pi _N||_{\Psi _0} \simeq |\Lambda | +\mathop {\theta _\mathrm {H}}\nolimits (\Lambda )\,\ln |\sin \Theta _0| +\mathop {\theta _\mathrm {H}}\nolimits (-\Lambda )\,\ln |\cos \Theta _0|$. Some algebra gives, for $gx\gg 1$,

$$\begin{aligned} \left\langle \ln ||\Pi _N||_{\Psi _0} \right\rangle \simeq \sqrt{\frac{2gx}{\pi }} + \frac{1}{2}\ln \left| \frac{1}{2}\sin 2\Theta _0\right| \end{aligned}$$

(10.12)

and

$$\begin{aligned} \mathrm {Var}(\ln ||\Pi _N||_{\Psi _0}) \simeq gx\left( 1-\frac{2}{\pi }\right) + \frac{1}{4}\ln ^2|\tan \Theta _0|. \end{aligned}$$

(10.13)

Note that the average value is reminiscent of the average of the logarithm of the transmission probability [27] (this calculation was first performed in Ref. [34] in another context). Interestingly, the behaviours (10.12, 10.13) were shown to persist in a quasi-1D situation with an odd number of channels (see the review [35] and references therein). We obtain

$$\begin{aligned} \gamma _1&= 0 \end{aligned}$$

(10.14)

$$\begin{aligned} \gamma _2&= g\left( 1-\frac{2}{\pi }\right) = g\times 0.363380... \end{aligned}$$

(10.15)

We can easily repeat this calculation with the second norm. Averaging of (10.10) over the angle $\Theta _0$ gives

$$\begin{aligned} |||\Pi _N||| = \frac{2e^{|\Lambda |}}{\pi }\, \mathbf {E}\left( \sqrt{1-e^{-4|\Lambda |}}\right) , \end{aligned}$$

(10.16)

where $\mathbf {E}(k)$ is the elliptic integral [36]. We deduce the asymptotic behaviours $\ln |||\Pi _N|||\simeq (3/4)\Lambda ^2$ for $|\Lambda |\ll 1$ and $\ln |||\Pi _N|||\simeq |\Lambda |-\ln (\pi /2)$ for $|\Lambda |\gg 1$, leading again to (10.14, 10.15).

In conclusion: for $\varepsilon \ne 0$, the calculation of the cumulants $\gamma _n$ is insensitive to the precise definition of the norm, i.e. to the precise choice of the initial spinor. In the Monte Carlo simulation, we have chosen $\Theta _0=0$ in order to set a Dirichlet boundary condition for the first component of the spinor. On the other hand, setting $\varepsilon =0$, the behaviour of $\gamma _2$ as a function of $\Theta _0$ presents two discontinuities precisely at $0$ and $\pi /2$. We understand these singular values as resulting from a lack of ergodicity in the matrix space when considering the Abelian subgroup describing the case $\varepsilon =0$. Hence, the value $g$ found for $\Theta _0=0$ or $\pi /2$ should not be taken as the correct result.

Appendix 3: Two Other Subgroups of Random Matrices of $\mathrm {SL}(2,\mathbb {R})$

It is well-known that the random Kronig–Penney model $\big [-\partial _x^2 + \sum _n v_n \,\delta (x-x_n) \big ]\psi (x) = E\,\psi (x)$ for energy $E=k^2$ is controlled by transfer matrices of the form

$$\begin{aligned} M_n = \left( \begin{array}{c@{\quad }c} \cos \theta _n &{} -\sin \theta _n \\ \sin \theta _n &{} \cos \theta _n \end{array} \right) \left( \begin{array}{c@{\quad }c} 1 &{} u_n \\ 0 &{} 1 \end{array} \right) \end{aligned}$$

(10.17)

where $\theta _n=k\,(x_{n+1}-x_{n})>0$ and $u_n=v_n/k$. The Schrödinger equation with negative energy $E=-k^2$ involves matrices of the form [16]

$$\begin{aligned} M_n = \left( \begin{array}{c@{\quad }c} \cosh \theta _n &{} \sinh \theta _n \\ \sinh \theta _n &{} \cosh \theta _n \end{array} \right) \left( \begin{array}{c@{\quad }c} 1 &{} u_n \\ 0 &{} 1 \end{array} \right) \end{aligned}$$

(10.18)

with the same definitions for $\theta _n$ and $u_n$.

The study of the continuum limit, $\ell _n\rightarrow 0$ and $v_n\rightarrow 0$ with $\left\langle v_n\right\rangle =0$ and $\sigma =\left\langle v_n^2\right\rangle /\left\langle \ell _n\right\rangle$ fixed can be done along the same lines as in the paper. In this more simple case, the Riccati variable $z(x)=\psi '(x)/\psi (x)$ obeys the SDE $z'(x)=-E-z(x)^2+V(x)$. In the continuum limit $V(x)$ is a Gaussian white noise of variance $\sigma$ and the process is characterised by the (backward) generator $\fancyscript{G} = ({\sigma }/{2})\partial _z^2-(E+z^2)\partial _z$. We arrive at

$$\begin{aligned} \gamma _2 = 2 \,\int \mathrm{d}z\mathrm{d}z'\, z\, G(z|z') \, z'\, f(z') \end{aligned}$$

(10.19)

where

$$\begin{aligned} f(z) = \frac{2N}{\sigma } f_0(z) \int _{-\infty }^z\frac{\mathrm{d}t}{f_0(t)} \ \ \text{ with } \ f_0(z)=e^{-\frac{2}{\sigma }\mathcal {U}(z)} \end{aligned}$$

(10.20)

is the stationary distribution, involving the potential $\mathcal {U}(z) = Ez+(1/3)z^3$ and the integrated density of states $N(E)$, given in Ref. [37] for instance (also recalled in Ref. [25]). The equation

$$\begin{aligned} \fancyscript{G}^\dagger G(z|z') = f(z) - \delta (z-z') \end{aligned}$$

(10.21)

for the propagator can be solved:

$$\begin{aligned} G(z|z')= \frac{1}{N(E)} \left\{ f(z) \left[ c(z')+\int _{-\infty }^z\mathrm{d}t\,f(t)\right] - f_0(z) \int _{-\infty }^z\mathrm{d}t\,\frac{f(t)^2}{f_0(t)} + \frac{f_0(z_>)f(z_<)}{f_0(z')} \right\} \end{aligned}$$

(10.22)

where

$$\begin{aligned} c(z') +\frac{1}{2} = \frac{\sigma }{2N(E)} \left[ \int _{-\infty }^{+\infty }\mathrm{d}z'' \, f(z'')^2 f(-z'') -f(-z')\,f(z') \right] - \int _{-\infty }^{z'} \mathrm{d}z''\, f(z''). \end{aligned}$$

(10.23)

We can analyse the limiting behaviours of the variance (10.19). In the high energy regime, $k=\sqrt{E}\gg \sigma ^{1/3}$ we obtain the expansions

$$\begin{aligned} f(z) = \frac{k/\pi }{z^2+k^2} + \frac{\sigma k}{\pi }\frac{z}{(z^2+k^2)^3} + \mathcal {O}(\sigma ^2) \end{aligned}$$

(10.24)

(recall that $N(E)={k}/{\pi }+\mathcal {O}(\sigma ^2)$) and

$$\begin{aligned} G(z|z')&=\left[ \frac{1}{z^2+k^2} +\sigma \frac{z}{(z^2+k^2)^3} \right] \Omega (z,z')\nonumber \\&\quad \quad +\,\frac{3\sigma }{16\pi k^3} \left( \frac{1}{z^2+k^2}-\frac{4k^4}{(z^2+k^2)^3}\right) + \frac{\mathop {\theta _\mathrm {H}}\nolimits (z-z')}{z'^2+k^2}\frac{f_0(z)}{f_0(z')} +\mathcal {O}(\sigma ^2) \end{aligned}$$

(10.25)

where

$$\begin{aligned} \Omega (z,z') = \frac{1}{2}\mathop {\mathrm {sign}}\nolimits (z'-z)+\frac{1}{\pi } \big [\arctan (z/k)-\arctan (z'/k)\big ] . \end{aligned}$$

(10.26)

When introducing these expressions in (10.19), the term $\mathcal {O}(\sigma ^0)$ seems at first sight logarithmically divergent but is eliminated by symmetry (i.e. integrals must be understood as principal parts). We get

$$\begin{aligned} \gamma _2 = \frac{k\sigma }{\pi } \int \mathrm{d}z\, \frac{z^2}{[\mathcal {U}'(z')]^3} +\mathcal {O}(\sigma ^2) = \frac{\sigma }{8E}+\mathcal {O}(\sigma ^2) \end{aligned}$$

(10.27)

i.e. we have recovered the asymptotic relation $\gamma _2\simeq \gamma _1$ for $E\rightarrow \infty$ (SPS).

For $E=0$, the fluctuations are finite $\gamma _2=\tilde{c}\,\sigma ^{1/3}$ where $\tilde{c}$ is a dimensionless constant of order unity (calculated explicitly in Ref. [33]). $\gamma _2$ is maximum for a negative value of the energy, however the numerics shows that the ratio $\gamma _2/\gamma _1$ reaches its maximum at $E=0$ (Fig. 5).

The limit $k=\sqrt{-E}\gg \sigma ^{1/3}$ is more easy to handle. In this case the potential $\mathcal {U}(z)$ develops a deep well at $z=k$, where the process is most of the “time” trapped. This dominates the fluctuations, which are those of the Ornstein–Uhlenbeck process,

$$\begin{aligned} \gamma _2 \underset{E\rightarrow -\infty }{\simeq } \frac{\sigma }{4(-E)} . \end{aligned}$$

(10.28)

The fluctuations thus decay faster as energy decreases than in the Dirac case studied in the paper, since the relation between the two models involves the mapping $E\leftrightarrow \varepsilon ^2$. Recalling that $\gamma _1\simeq \sqrt{-E}$ in this case shows that $\gamma _2\ll \gamma _1$ (no SPS).

Monte Carlo simulations are in perfect agreement with these behaviours (see Fig. 5).

Fig. 5

(Color online) Left Plot of the Lyapunov exponent (red circles) and the variance (blue squares) for $\sigma =1$ obtained by Monte Carlo simulations. Comparison with limiting behaviours (10.27) and (10.28) (dashed black lines). Right SPS, $\gamma _2/\gamma _1\simeq 1$, holds for $E\gg \sigma ^{2/3}$

The problem considered in this appendix was studied earlier in Refs. [33, 38] in another context and with a different method: the generalised Lyapunov exponent (2.2) is obtained as the largest eigenvalue of the operator $\fancyscript{G}^\dagger +qz$ [7]. The perturbative treatment [33] gives an integral representation

$$\begin{aligned} \gamma _2 = 2 \int \mathrm{d}z \, (z-\gamma _1)\, \varphi _1(z) \end{aligned}$$

(10.29)

where

$$\begin{aligned} \varphi _1(z) = N\left( \frac{2}{\sigma }\right) ^2f_0(z) \int _{-\infty }^z\frac{\mathrm{d}z'}{f_0(z')} \int _{-\infty }^{z'}\mathrm{d}z''\,(\gamma _1-z'')\,f_0(z'') \int _{-\infty }^{z''}\frac{\mathrm{d}z'''}{f_0(z''')} . \end{aligned}$$

(10.30)

Although it is not straightforward to prove the equivalence between (10.19) and (10.29), they seem to give similar results (see Fig. 1 of Ref. [33]).