Scattering expansion for localization in one dimension: From disordered wires to quantum walks

We consider a one-dimensional “sample” consisting of $N$ sites labeled by $n=1,\dots,N$ (Fig. 1).

The sample is attached to infinite “leads” and is a scattering region for incoming and outgoing waves in the leads. Each site in the sample is associated with an array $\mathbf{D}_{n}$ of disorder variables. For instance, in the Anderson model with diagonal disorder, $\mathbf{D}_{n}$ would be the onsite potential, i.e., an array of length $1$. The disorder is assumed to be i.i.d. with some normalized probability distribution $\rho(\mathbf{D}_{n})$, and we write disorder averages of any quantity $X$ as $\langle X\rangle_{n}\equiv\int d\mathbf{D}_{n}\ \rho(\mathbf{D}_{n})X$ (any site $n$) and $\langle X\rangle_{1\dots N}\equiv\int\left[\prod_{n=1}^{N}d\mathbf{D}_{n}\ \rho(\mathbf{D}_{n})\right]X$ (the average over all $N$ sites).

We assume that each site has a $2\times 2$ disordered $S$ matrix $\mathcal{S}_{n}$ which we parametrize by

	$\displaystyle\mathcal{S}_{n}$	$\displaystyle=\begin{pmatrix}t_{n}&r_{n}^{\prime}\\ r_{n}&t_{n}^{\prime}\end{pmatrix}$		(1a)
		$\displaystyle=\begin{pmatrix}\sqrt{T_{n}}\exp[i\phi_{t_{n}}]&\sqrt{R_{n}}\exp[i\phi_{r_{n}^{\prime}}]\\ \sqrt{R_{n}}\exp[i\phi_{r_{n}}]&\sqrt{T_{n}}\exp[i\phi_{t_{n}^{\prime}}]\end{pmatrix},$		(1b)

where $t_{n}$ and $t_{n}^{\prime}$ ($r_{n}$ and $r_{n}^{\prime}$) are the local transmission (reflection) amplitudes, $T_{n}=|t_{n}|^{2}=|t_{n}^{\prime}|^{2}$ ($R_{n}=|r_{n}|^{2}=|r_{n}^{\prime}|^{2}$) are the local transmission (reflection) coefficients, and $\phi_{t_{n}}$ and $\phi_{t_{n}^{\prime}}$ ($\phi_{r_{n}}$ and $\phi_{r_{n}^{\prime}}$) are the transmission (reflection) phases, which unitarity constrains by $R_{n}+T_{n}=1$ and $\phi_{t_{n}}+\phi_{t_{n}^{\prime}}-\phi_{r_{n}}-\phi_{r_{n}^{\prime}}=\pi$ (modulo $2\pi$). Here and throughout, the subscript $n$ indicates dependence on the disorder variables $\mathbf{D}_{n}$ of site $n$. We consider only the single channel case, i.e., the amplitudes are complex numbers and not matrices.

We consider the scattering problem of the $N$-site sample. The $S$ matrix for the sample, denoted $\mathcal{S}_{1\dots N}$, relates incoming and outgoing scattering amplitudes in the leads by

$$\begin{pmatrix}\Psi_{R}^{+}\\ \Psi_{L}^{-}\end{pmatrix}=\mathcal{S}_{1\dots N}\begin{pmatrix}\Psi_{L}^{+}\\ \Psi_{R}^{-}\end{pmatrix}.$$

(2)

The $S$ matrix is obtained in the usual way by multiplying the scattering transfer matrices of the $N$ sites and converting the resulting scattering transfer matrix back to an $S$ matrix (see Appendix A). We write

$$\mathcal{S}_{1\dots N}=\begin{pmatrix}\sqrt{T_{1\dots N}}\exp[i\phi_{t_{1\dots N}}]&\sqrt{R_{1\dots N}}\exp[i\phi_{r_{1\dots N}^{\prime}}]\\ \sqrt{R_{1\dots N}}\exp[i\phi_{r_{1\dots N}}]&\sqrt{T_{1\dots N}}\exp[i\phi_{t_{1\dots N}^{\prime}}]\end{pmatrix},$$

(3)

in which the parameters are constrained by unitarity to satisfy the same equations mentioned below Eq. (1b). No further discussion of the leads is necessary at this point; the problem is to study the sample $S$ matrix (3) given the local $S$ matrix (1a), which in any particular problem depends on the properties of the sample and leads.

The central object of our study is the joint probability distribution of the minus logarithm of the transmission coefficient (denoted $s_{1\dots N}\equiv-\ln T_{1\dots N}$) and of either one of the reflection phases (we choose the right-to-left phase, $\phi_{r_{1\dots N}^{\prime}}$). We focus at first on the localization length and the marginal distributions of $s$ and $\phi_{r^{\prime}}$ (which indeed are determined by the joint distribution). We write the joint and marginal distributions as

	$\displaystyle P_{N}(s^{\prime},\phi_{r^{\prime}}^{\prime})$	$\displaystyle\equiv\langle\delta(s_{1\dots N}-s^{\prime})\delta(\phi_{r_{1\dots N}^{\prime}}-\phi_{r^{\prime}}^{\prime})\rangle_{1\dots N},$		(4a)
	$\displaystyle P_{N}(s)$	$\displaystyle\equiv\int_{-\pi}^{\pi}d\phi_{r^{\prime}}\ P_{N}(s,\phi_{r^{\prime}}),$		(4b)
	$\displaystyle p_{N}(\phi_{r^{\prime}})$	$\displaystyle\equiv\int_{0}^{\infty}ds\ P_{N}(s,\phi_{r^{\prime}}),$		(4c)

where the subscript $N$ on the left-hand side, in departure from our usual convention, indicates dependence on the number $N$ rather than on $\mathbf{D}_{N}$ ⁴⁴4Explicitly, we have (e.g.) $p_{N}(\phi_{r^{\prime}}^{\prime})=\int d\mathbf{D}_{1}\dots d\mathbf{D}_{N}\ \rho(\mathbf{D}_{1})\dots\rho(\mathbf{D}_{N})\delta[\phi_{r^{\prime}}(\mathbf{D}_{1},\dots,\mathbf{D}_{N})-\phi_{r^{\prime}}^{\prime}]$, where $\phi_{r^{\prime}}(\mathbf{D}_{1},\dots,\mathbf{D}_{N})\equiv\phi_{r_{1\dots N}^{\prime}}$. Due to the disorder being i.i.d., the probability distributions depend only on the number of consecutive sites $N$ and not on their location. Also, to make contact with the notation used in the Introduction, we take the spacing between sites to be the unit of distance; then the length of the sample is $L=N$.. As discussed in the Introduction, for a large sample ($N\gg 1$) the probability distribution of $s$ is Gaussian with the mean growing linearly with $N$ (although the Gaussian form may not accurately calculate moments of $T$ and $T^{-1}$ [42]). The localization length $L_{\text{loc}}$ is then determined by

$$\langle s_{1\dots N}\rangle_{1\dots N}=\frac{2}{L_{\text{loc}}}N+O(N^{0}).$$

(5)

The variance [denoted $\sigma(N)^{2}$] of $P_{N}(s)$ also grows linearly with $N$, and SPS holds when the mean and variance are related by an equation (see below).

In this discussion, we present results from the literature as they apply to our setup, using our notation throughout. We focus on prior work most relevant to our work and do not attempt a comprehensive review. We first review explicit, model-independent results for the localization length and for SPS from the literature, and then we discuss the relation of our work to the literature on the joint distribution.

In Ref. [27], Anderson et al. proposed that there would be a length scale $\ell_{p}$ beyond which complete phase randomization would occur. According to this uniform phase hypothesis, for $N\gg\ell_{p}$ the phase of $r_{1\dots N}^{\prime}r_{N+1}$, which we write as $\nu_{1\dots N}\equiv\phi_{r_{1\dots N}^{\prime}}+\phi_{r_{N+1}}$, is distributed uniformly in $(-\pi,\pi)$ and independently of $T_{N+1}$. [We will use “uniform” to refer to a phase distribution being uniform over $(-\pi,\pi)$, unless otherwise noted.] This hypothesis yields an explicit formula for the localization length in terms of a local average [27]:

$$\frac{2}{L_{\text{loc}}}=\langle-\ln T_{n}\rangle_{n}\qquad(\text{any site }n).$$

(6)

Furthermore, the same hypothesis yields SPS by relating the mean and variance through [27]

$$\lim_{N\to\infty}\frac{\sigma(N)^{2}}{2N}=\frac{2}{L_{\text{loc}}}.$$

(7)

However, the uniform phase hypothesis was shown to fail in a number of cases including strong disorder (see [43] and references therein, as well as [44, 45, 46, 47]), multiple scattering channels [48], and anomalies in the Anderson model [49, 50]. In Ref. [34], though, Lambert and Thorpe obtained a more general formula for the localization length that accounts for deviations from phase uniformity:

$$\frac{2}{L_{\text{loc}}}=\langle-\ln T_{n}\rangle_{n}\\ +\langle\ln\left[1+R_{N+1}+2\sqrt{R_{N+1}}\cos(\nu_{1\dots N})\right]\rangle_{1\dots N+1},$$

(8)

in which $N\gg 1$ and $n$ is any site. Equation (8) recovers the uniform phase result (6) when $\nu_{1\dots N}$ is uniformly distributed independently of $R_{N+1}$ [since $\int_{-\pi}^{\pi}d\nu\ \ln(1+R+2\sqrt{R}\cos\nu)=0$]. In Refs. [34, 35], Lambert and Thorpe furthermore showed that, in a model consisting of randomly spaced delta function scatterers of equal strength, the probability distribution of $\nu_{1\dots N}$ is generically non-uniform even for an arbitrarily long chain. The second term in (8) makes a substantial correction to the first, particularly for weak disorder. In a more general setting, an expression for the probability distribution of $\nu_{1\dots N}$ at the zeroth order in disorder strength (with the thermodynamic limit taken before the limit of zero disorder) was derived by Lambert et al. in Ref. [51]; the authors then used this probability distribution in Eq. (8) to find that the localization length is finite in band gaps and infinite outside of band gaps, as expected [51].

The significance of the delta-function model considered by Refs. [34, 35] was challenged by Stone et al. in Ref. [52], in which the authors showed numerically that in the Anderson model with weak, diagonal disorder with vanishing mean onsite energy, the probability distribution of $\nu_{1\dots N}$ is indeed uniform for large $N$. Stone et al. obtained a formula analogous to (8) for the variance $\sigma(N)^{2}$; using this formula, Eq. (8), and the uniformity of the distribution of $\nu_{1\dots N}$, they obtained both the known weak disorder expansion for the inverse localization length and the SPS relation (7). The authors further argued that the delta-function model considered by Lambert and Thorpe is a special case and that any model with the potential being positive as often as negative would yield results similar to those in the Anderson model.

A number of works (including some that we have cited above) have studied the distribution of the reflection phase $\phi_{r_{1\dots N}^{\prime}}$ rather than of $\nu_{1\dots N}$. We can understand this as follows. By construction, $\nu_{1\dots N}$ is distributed uniformly provided that either or both of $\phi_{r_{1\dots N}^{\prime}}$ and $\phi_{r_{N+1}}$ are distributed uniformly. The challenging case, then, is what happens when the local phase $\phi_{r_{N+1}}$ is non-uniformly distributed: Does the distribution of the sample phase $\phi_{r_{1\dots N}^{\prime}}$ become uniform for large $N$, or not? From here on, we mean by the “uniform phase hypothesis” the assertion that either or both of the sample reflection phases, $\phi_{r_{1\dots N}^{\prime}}$ and $\phi_{r_{1\dots N}}$, are distributed uniformly for large $N$.

Another approach to the SPS relation was provided by Deych et al. in Refs. [30, 31]. Deych et al. argue that the SPS relation holds in the regime $L_{\text{loc}}\gg\ell_{s}$, where $\ell_{s}$ is a new length scale that they identify, without appeal to any uniform phase hypothesis. In a setup encompassing both the Anderson model with diagonal disorder and the case of equally spaced delta functions with random strengths [52, 30, 31], Deych et al. provide analytical results in the Lloyd model (i.e., disorder following a Cauchy distribution). They obtain the SPS relation, albeit with an extra prefactor of $2$ that they attribute to the peculiarity of the Cauchy distribution; they also present numerical evidence for the SPS relation following from the condition $L_{\text{loc}}\gg\ell_{s}$ in a more generic model (an array of square well potentials with random widths) [30, 31].

The work with most overlap with ours is that of Schrader et al. in Ref. [32] (see also Ref. [53]), which presents an explicit formula for the inverse localization length and variance and a proof of the SPS relation, all at the leading order in a small parameter. In particular, Schrader et al. proved $2/L_{\text{loc}}=D\lambda^{\prime 2}+O(\lambda^{\prime 3})$ and $\lim_{N\to\infty}\sigma(N)^{2}/(2N)=2/L_{\text{loc}}+O(\lambda^{\prime 3})$, where $D$ is given in terms of averages over local quantities [unlike Eq. (8), which includes an average over the whole sample] and where $\lambda^{\prime}$ is a small parameter that Schrader et al. identified as “the effective size of the randomness” [32]. The quantity $D$ consists of a sum of two terms, one that recovers the uniform phase result (6) at leading order in $\lambda^{\prime}$ and a second that contributes due to departures from the uniform phase hypothesis [32]. Schrader et al. point out that although the second term vanishes in the Anderson model (with diagonal disorder and vanishing mean onsite energy), it is in general not a small correction to the first term (for instance, in the random polymer model [54]). Schrader et al. also calculate $2/L_{\text{loc}}$ and $\lim_{N\to\infty}\sigma(N)^{2}/(2N)$ to the next non-vanishing order in the Anderson model ($\lambda^{\prime 4}$), showing that both the uniform phase formula (6) and the SPS relation (7) break down at this order [32].

Let us next review prior work on the joint distribution of $s$ and $\phi_{r^{\prime}}$. Given a maximum entropy ansatz, Ref. [55] found that the probability distribution of $s$ satisfies a Fokker-Planck equation (which in turn yields a Gaussian distribution for $s$) ⁵⁵5Reference [55] obtains a Fokker-Planck equation for the probability distribution of $\rho\equiv R/T$. In the localized regime that is our focus, Ref. [55] uses this Fokker-Planck equation to obtain a Gaussian distribution for $s$ (there denoted $z$), as we have stated in the main text. The phase variables $\mu,\nu$ in Ref. [55], which are distributed uniformly [c.f. Eq. (2.6) there)], are related to the reflection phases by $\phi_{r^{\prime}}=-2\mu$ and $\phi_{r}=-2\nu-\pi$ [c.f. Eqs. (2.3a) and (2.3b) there].. The maximum entropy ansatz yields a factorization of the joint distribution of $s$ and $\phi_{r^{\prime}}$ into a product of the two marginal distributions, with the marginal distribution of $\phi_{r^{\prime}}$ being uniform. This calculation applies when either or both of the sample reflection phase and local reflection phase are distributed uniformly.

While our work focuses on the case of a single scattering channel (i.e., $2\times 2$ S matrices), it is useful for further comparison with the literature to consider the case of an arbitrary number $N_{c}$ of channels ($2N_{c}\times 2N_{c}$ S matrices). In this more general case, the $S$ matrix may be parametrized by $N_{c}$ transmission coefficients and $4N_{c}^{2}-N_{c}$ phases [57, 58]. The generalization of the uniform phase hypothesis to this case is known as the isotropy assumption and is believed to hold when there are many channels ($N_{c}\gg 1$) with sufficient coupling between them (see Ref. [59] and references therein). Under this assumption, the joint probability distribution of the $N_{c}$ transmission coefficients satisfies a version of the Fokker-Planck equation known as the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation [48, 57]. For the case $N_{c}=2$, it has been shown in particular models that the isotropy assumption breaks down and that the transmission coefficients are entangled with the phases in the Fokker-Planck equation [48, 59].

For the case $N_{c}=1$, which is our focus, general and analytical results for the joint distribution of $(s,\phi_{r^{\prime}},\phi_{t})$ were obtained by Roberts in Ref. [60]; see also Ref. [43]. These works obtained factorization of the joint distribution of $(s,\phi_{r^{\prime}})$ into the product of separate distributions for $s$ and $\phi_{r^{\prime}}$, apparently in conflict with our result. In Sec. IV, we support our result with analytical calculations and numerical checks, and we provide an explanation for the discrepancy with Refs. [60, 43].

Our first general result in this paper is a series expansion of the inverse localization length with the local reflection strength $|r_{n}|$ as a small parameter. In the course of this calculation, we also present results on SPS and on the marginal distribution of the reflection phase.

Our scattering expansion is defined by rescaling $r_{n}\to\lambda r_{n}$ and $r_{n}^{\prime}\to\lambda r_{n}^{\prime}$ in the local $S$ matrix (1a) (with $t_{n}$ and $t_{n}^{\prime}$ also rescaled to maintain unitarity), and, roughly speaking, expanding in the real and positive parameter $\lambda$. (As we discuss below, it is necessary in the calculation to consider the order of limits $N\to\infty$ and $\lambda\to 0^{+}$ in a particular way to maintain the system always in the localized regime. However, this subtlety does not appear in the final results, which may be understood as straightforward expansions in $\lambda$.) For convenience, we let the dependence on $\lambda$ be implicit in our notation and refer informally to an expansion in $|r_{n}|$. Thus, the reflection amplitudes $r_{n}$ and $r_{n}^{\prime}$ are both first order and the reflection coefficient $R_{n}$ is second order. While we assume for now that $|r_{n}|$ is linear in the small parameter $\lambda$, we explain below the simple modifications to our results in the more general case of $|r_{n}|$ starting at linear order but also having higher-order corrections.

Before presenting our calculation, we first note that the result of Schrader et al. [32] mentioned in the previous section has the following corollary: our parameter $\lambda$ is a valid choice for their parameter $\lambda^{\prime}$, and their formula for the localization length and variance may be written in terms of local averages at any site $n$ as (see Appendix B for the calculation) ⁶⁶6In the case of $t_{n}=t_{n}^{\prime}$ and $r_{n}=r_{n}^{\prime}$, a result related to Eq. (9) has also been obtained in Refs. [93, 94, 95] using the WSA approach developed there. See Appendix C for details.

$$\frac{2}{L_{\text{loc}}}=\langle R_{n}\rangle_{n}-2\text{Re}\left[\frac{\langle r_{n}\rangle_{n}\langle r_{n}^{\prime}\rangle_{n}}{1+\langle r_{n}r_{n}^{\prime}/R_{n}\rangle_{n}}\right]+O(|r_{n}|^{4}),$$

(9)

and

$$\lim_{N\to\infty}\frac{\sigma(N)^{2}}{2N}=\frac{2}{L_{\text{loc}}}+O(|r_{n}|^{4}),$$

(10)

where we have also improved their results from $O(|r_{n}|^{3})$ error to $O(|r_{n}|^{4})$ error (see next paragraph). The first term on the right-hand side of Eq. (9) is the small $|r_{n}|$ expansion of the uniform phase result (6), while the second term is the contribution from deviations from the uniform phase hypothesis. We emphasize again that the second term is not, in general, a small correction to the first; indeed, both terms are of the same order in the small parameter. We also note that unlike the Lambert and Thorpe formula (8), Eq. (9) involves only local averages and no averaging over the whole chain.

In writing Eqs. (9) and (10), we have in fact gone beyond the immediate corollary of the result of Schrader et al., which would yield the leading order contributions as given above but with $O(|r_{n}|^{3})$ error instead of $O(|r_{n}|^{4})$ ⁷⁷7In the particular case of the Anderson model with onsite disorder with vanishing mean, Ref. [32] calculates the third and fourth order terms and finds that the third order terms indeed vanish.. The third order terms, and indeed all odd orders, vanish due to the following symmetry argument. The localization length and variance must be invariant under phase redefinitions of the incoming and outgoing scattering amplitudes; in particular, we can redefine $\Psi_{\alpha}^{\pm}\to e^{\pm i\phi/2}\Psi_{\alpha}^{\pm}$ ($\alpha=R,L$), which sends $r_{n}\to e^{-i\phi}r_{n}$ and $r_{n}^{\prime}\to e^{i\phi}r_{n}^{\prime}$. It follows that $r_{n}$ and $r_{n}^{\prime}$ must appear in equal numbers in each term (hence there are no terms of the order of an odd power of $|r_{n}|$). This argument relies on the existence of series expressions for $2/L_{\text{loc}}$ and $\lim_{N\to\infty}\sigma(N)^{2}/(2N)$ with coefficients involving only local averages of integer powers of $r_{n}$, $r_{n}^{\prime}$, $v_{n}$, and $R_{n}$ (note that the latter two are invariant under the phase re-definition). In this section we show that such a series exists for $2/L_{\text{loc}}$, and in Sec. IV we show it for $\lim_{N\to\infty}\sigma(N)^{2}/(2N)$.

We have thus explicitly connected SPS to the weakness of the local reflection strength, and we have extended the SPS relation to one more order [Eq. (10)]. Our next task is to apply the scattering expansion to the inverse localization length, recovering Eq. (9) and providing a recursive calculation of higher orders. We show that all orders of the series depend, as in Eq. (9), only on averages involving the local reflection amplitudes $r_{n}$ and $r_{n}^{\prime}$. We explicitly verify the vanishing of the third order and present the next non-vanishing order ($|r_{n}|^{4}$).

A key ingredient in our calculation is the limiting form as $N\to\infty$ of the probability distribution of the reflection phase, which we write as

$$p_{\infty}(\phi_{r^{\prime}})\equiv\lim_{N\to\infty}p_{N}(\phi_{r^{\prime}}).$$

(11)

(In Appendix D, we show that this limit exists given the assumption that localization occurs.) Our approach is based on the following series formula, which we derive below, that expresses the inverse localization length in terms of the Fourier coefficients $p_{\infty,\ell}\equiv\int_{-\pi}^{\pi}\frac{d\phi_{r^{\prime}}}{2\pi}e^{-i\ell\phi_{r^{\prime}}}p_{\infty}(\phi_{r^{\prime}})$ and the moments of $r_{n}$:

$$\frac{2}{L_{\text{loc}}}=\langle-\ln T_{n}\rangle_{n}-4\pi\text{Re}\left[\sum_{\ell=1}^{\infty}\frac{1}{\ell}p_{\infty,-\ell}\langle r_{n}^{\ell}\rangle_{n}\right].$$

(12)

Equation (12) is related to the Lambert and Thorpe formula (8), with the essential difference being that we focus on the distribution of $\phi_{r^{\prime}}$ rather than $\nu$ and work in frequency space. Note that the uniform phase formula (6) is recovered from Eq. (12) in two non-exclusive special cases: (i) the local reflection phase is uniformly distributed independently of the local reflection coefficient (then $\langle r_{n}^{\ell}\rangle_{n}=0$ for $\ell>0$), or (ii) the reflection phase distribution of the sample is uniform. Case (i) is an example of strong phase disorder. The difficulty of applying Eq. (12), in the case that (i) does not hold, is that it has been shown in many examples that the reflection phase distribution $p_{\infty}(\phi_{r^{\prime}})$ can be non-uniform (see references cited in Sec. II.2) and in general the distribution is only known numerically (although Schrader et al. [32] calculated $p_{\infty,\pm 1}$ in an equivalent form).

In order to use Eq. (12), we calculate the probability distribution of the reflection phase order by order in the scattering expansion. We show that the zeroth-order contribution is (generically) the uniform distribution, and that at any fixed order of the expansion, only finitely many of the Fourier coefficients $p_{\infty,\ell}$ are non-vanishing. We further show that the expansion coefficients are local averages that can be calculated recursively. Together with Eq. (12), this recursive expansion of the Fourier coefficients $p_{\infty,\ell}$ yields our scattering expansion of the inverse localization length.

In the remainder of this section, we derive the above results (Sec. III.2), then apply them to disordered wires (Sec. III.3) and to quantum walks (Sec. III.4). We note here that our calculation in this section only concerns the localization length and not the variance; however, we do obtain Eq. (10) by a different approach in Sec. IV, and it is convenient to apply this result in this section to comment on SPS in a broad class of PARS.

We compare a sample of size $N$ with a sample of size $N+1$ using the following exact relations between the transmission coefficients and reflection phases (for completeness we derive them in Appendix A):

	$\displaystyle T_{1\dots N+1}$	$\displaystyle=\frac{T_{1\dots N}T_{N+1}}{\left|1-\sqrt{R_{1\dots N}}e^{i\phi_{r_{1\dots N}^{\prime}}}r_{N+1}\right|^{2}},$		(13a)
	$\displaystyle\phi_{r_{1\dots N+1}^{\prime}}$	$\displaystyle=\phi_{r_{1\dots N}^{\prime}}+\pi+\phi_{r_{N+1}}+\phi_{r_{N+1}^{\prime}}$
		$\displaystyle\ -\operatorname{Arg}\left[1-\sqrt{R_{1\dots N}}r_{N+1}e^{i\phi_{r_{1\dots N}^{\prime}}}\right]$
		$\displaystyle\ -\operatorname{Arg}\left[1-r_{N+1}e^{i\phi_{r_{1\dots N}^{\prime}}}/\sqrt{R_{1\dots N}}\right]$
		$\displaystyle\qquad\qquad\qquad\qquad\qquad(\text{modulo }2\pi).$		(13b)

A basic assumption of our calculation is that localization occurs: that is, for large $N$ the sample reflection coefficient $R_{1\dots N}\approx 1$ in all disorder realizations ⁸⁸8Whenever we require a property to hold for all disorder realizations, we expect that it would also suffice for that property to hold for almost all realizations (i.e., all except a set of measure zero).. We can then replace $R_{1\dots N}\to 1$ in the logarithm of Eq. (13a) and in Eq. (13b); in other words, we keep $\ln T_{1\dots N}$ and constant terms, but neglect terms linear or higher in $T_{1\dots N}=1-R_{1\dots N}$. We thus obtain

	$\displaystyle s_{1\dots N+1}$	$\displaystyle=s_{1\dots N}+g_{N+1}(\phi_{r_{1\dots N}^{\prime}}),$		(14a)
	$\displaystyle\phi_{r_{1\dots N+1}^{\prime}}$	$\displaystyle=\phi_{r_{1\dots N}^{\prime}}+h_{N+1}(\phi_{r_{1\dots N}^{\prime}}),$		(14b)

where

	$\displaystyle g_{n}(\phi)$	$\displaystyle=-\ln T_{n}+\ln(1-r_{n}e^{i\phi}-r_{n}^{*}e^{-i\phi}+R_{n}),$		(15a)
	$\displaystyle h_{n}(\phi)$	$\displaystyle=\pi+\phi_{r_{n}}+\phi_{r_{n}^{\prime}}+i\ln\frac{1-r_{n}e^{i\phi}}{1-r_{n}^{*}e^{-i\phi}}.$		(15b)

Let us pause to present an intuitive picture, in which we temporarily think of the increasing sample size $N$ as representing time. Then, Eq. (14b) indicates that the reflection phase performs a random walk on the circle; furthermore, the probability distribution of the step size depends on the current position that the walk has reached. From this point of view, it is clear that the long-time distribution of the phase should generally be non-uniform, since the walk will spend more time in regions where the step size is smaller. From Eq. (14a), we see that the variable $s$ performs a random walk on the half-line with a step size that depends on the current position of the phase walk, though not on the current position of $s$. While the average position reached by $s$ after a long time may be obtained by treating the phase as simply another disorder variable in the step size (i.e., sampling the phase from its long-time distribution), the variance of $s$ and the correlations between $s$ and the phase seem to require a more careful treatment (Sec. IV).

To clarify a technical point about our expansion, we temporarily restore the true small parameter $\lambda$. While we are interested in the regime of small $\lambda$, we cannot simply expand in $\lambda$ for fixed $N$, since the point $\lambda=0$ corresponds to a sample transmission coefficient of unity, inconsistent with our replacing $R_{1\dots N}\to 1$. Instead, we must suppose that for any fixed $\lambda>0$, there is some sufficiently large $N_{0}(\lambda)$ for which the sample transmission coefficient is negligible for any $N\geq N_{0}(\lambda)$ in every disorder realization [63]. [$N_{0}(\lambda)$ is of order of the localization length, so in view of our results below, it will turn out that $N_{0}(\lambda)\sim 1/\lambda^{2}$.] All expansions in powers of $\lambda$ in our calculations below must be understood as occurring in the regime of $\lambda>0$ with $N\geq N_{0}(\lambda)$ ⁹⁹9A similar point is mentioned in Ref. [26] (footnote 24).. This subtlety need not be considered in our final answer for $2/L_{\text{loc}}$, and we have there a simple expansion in $\lambda$.

We make a few comments now about our conventions. Here and throughout we use a superscript to indicate the order in $|r_{n}|$ (i.e., $X^{(j)}\sim|r_{n}|^{j}$ for any quantity $X$). Also, since the disorder is i.i.d., we frequently replace the disorder average over site $N+1$ by an average over any site $n$. Finally, all phases are constrained to be in $[-\pi,\pi)$, and any equations between non-exponentiated phases are to be understood as holding modulo $2\pi$.

Next, we develop the scattering expansion for the Fourier coefficients $p_{\infty,\ell}$. A recursion relation for $p_{N}(\phi_{r^{\prime}})$ is determined by $p_{N+1}(\phi_{r^{\prime}})=\langle\delta(\phi_{r_{1\dots N+1}^{\prime}}-\phi_{r^{\prime}})\rangle_{1\dots N+1}$, from which Eq. (14b) then yields

$$p_{N+1}(\phi_{r^{\prime}})=\int_{-\pi}^{\pi}d\phi_{r^{\prime}}^{\prime}\ p_{N}(\phi_{r^{\prime}}^{\prime})\\ \times\langle\delta\bm{(}\phi_{r^{\prime}}^{\prime}+h_{n}(\phi_{r^{\prime}}^{\prime})-\phi_{r^{\prime}}\bm{)}\rangle_{n}.$$

(16)

We then send $N\to\infty$ and take the Fourier transform of both sides to obtain [noting Eq. (15b)]

$$p_{\infty,\ell}=(-1)^{\ell}\langle v_{n}^{-\ell}\int_{-\pi}^{\pi}\frac{d\phi}{2\pi}\ e^{-i\ell\phi}p_{\infty}(\phi)A_{n,\ell}(\phi)\rangle_{n},$$

(17)

where we have defined

	$\displaystyle v_{n}$	$\displaystyle=\frac{r_{n}r_{n}^{\prime}}{R_{n}},$		(18a)
	$\displaystyle A_{n,\ell}(\phi)$	$\displaystyle=\left(\frac{1-r_{n}e^{i\phi}}{1-r_{n}^{*}e^{-i\phi}}\right)^{\ell}.$		(18b)

Eq. (17) can be solved order by order in the scattering expansion, as we now show.

We write the expansions of $p_{\infty}(\phi)$ and $A_{n,\ell}(\phi)$ as $p_{\infty}(\phi)=\sum_{j=0}^{\infty}p_{\infty}^{(j)}(\phi)$ and $A_{n,\ell}(\phi)=\sum_{j=0}^{\infty}A_{n,\ell}^{(j)}(\phi)$, with corresponding expansions for the Fourier coefficients. Note that by the normalization condition, $\int_{-\pi}^{\pi}d\phi\ p_{\infty}(\phi)=1$, we have $p_{\infty,0}^{(j)}=0$ for all $j\geq 1$.

We now equate terms of the same order on both sides of Eq. (17). Since $A_{n,\ell}^{(0)}=1$, the quantity $p_{\infty,\ell}^{(q)}$ appears on both sides of the order $q$ equation (for any $q\geq 0$). This has two immediate consequences. First, the zeroth-order part of Eq. (17) is

$$p_{\infty,\ell}^{(0)}=(-1)^{\ell}\langle v_{n}^{-\ell}\rangle_{n}p_{\infty,\ell}^{(0)},$$

(19)

and second, for any $q\geq 1$, the order $q$ part of Eq. (17) for $\ell\neq 0$ can be re-arranged to

$$p_{\infty,\ell}^{(q)}=\frac{(-1)^{\ell}}{1-(-1)^{\ell}\langle v_{n}^{-\ell}\rangle_{n}}\langle v_{n}^{-\ell}\sum_{j=0}^{q-1}\sum_{\ell^{\prime}=-\infty}^{\infty}p_{\infty,-\ell^{\prime}}^{(j)}\\ \times A_{n,\ell;\ell+\ell^{\prime}}^{(q-j)}\rangle_{n},$$

(20)

where $A_{n,\ell;\ell^{\prime}}=\int_{-\pi}^{\pi}\frac{d\phi}{2\pi}\ e^{-i\ell^{\prime}\phi}A_{n,\ell}(\phi)$ [and where we have written $\int_{-\pi}^{\pi}\frac{d\phi}{2\pi}\ e^{-i\ell\phi}p_{\infty}^{(j)}(\phi)A_{n,\ell}^{(q-j)}(\phi)$ as a sum over Fourier coefficients].

Equation (19) implies (see next paragraph) that $p_{\infty,\ell}^{(0)}=0$ for all $\ell\neq 0$. The $\ell=0$ Fourier coefficient is fixed by the normalization of the probability distribution, so we find that the zeroth-order distribution is uniform:

$$p_{\infty}^{(0)}(\phi_{r^{\prime}})=\frac{1}{2\pi}.$$

(21)

Equations (20) and (21) rely on assumptions of “reasonable” disorder and “generic” model parameters, as we now explain. For instance, the limit of no disorder at all technically meets our definitions so far, but must be excluded. Also, in applying our approach to any particular model, we must consider the model as part of some family of models (e.g., with parameters taking a continuous range of values), since our results are expected to hold generically but may break down at some special values of parameters (the anomalous momenta of the Anderson model are an example). A precise statement of these assumptions is (a) that localization occurs (an assumption that we have used already), and (b) that the following inequality holds for all integers $\ell\neq 0$:

$$\langle e^{i\ell(\phi_{r_{n}}+\phi_{r_{n}^{\prime}}+\pi)}\rangle_{n}\neq 1,$$

(22)

where $n$ is any site. We expect that any choice of disorder distribution and model parameters that violates (22) is “non-generic” (since a particular alignment of reflection phases across disorder realizations is required). [In Appendix F, we consider the case that (22) is only assumed for $0<|\ell|\leq\ell_{\text{max}}$ for some finite $\ell_{\text{max}}$.] Noting that $v_{n}=e^{i(\phi_{r_{n}}+\phi_{r_{n}^{\prime}})}$, we see then that for $\ell\neq 0$, we have $(-1)^{\ell}\langle v_{n}^{-\ell}\rangle_{n}\neq 1$, so indeed Eq. (19) implies $p_{\infty,\ell}^{(0)}=0$ and the denominator in Eq. (20) is non-vanishing.

Equation (20) is almost in a form in which we can calculate the reflection phase distribution to many orders. The point is that the right-hand side involves only the reflection phase distribution at lower orders than the order $q$ on the left-hand side; thus, in principle we can start with Eq. (21) and iterate Eq. (20) to calculate higher orders. To make this process more efficient, we show next that only finitely many Fourier coefficients are non-vanishing at any given order. In particular, we show that for any $q\geq 0$,

$$p_{\infty,\ell}^{(q)}=0\qquad(\text{for }|\ell|>q),$$

(23)

and we show that the sum over $\ell^{\prime}$ on the right-hand side of Eq.  (20) can be truncated to a finite sum:

$$p_{\infty,\ell}^{(q)}=\frac{(-1)^{\ell}}{1-(-1)^{\ell}\langle v_{n}^{-\ell}\rangle_{n}}\langle v_{n}^{-\ell}\sum_{j=0}^{q-1}\sum_{\ell^{\prime}=0}^{j}p_{\infty,-\ell^{\prime}}^{(j)}A_{n,\ell;\ell+\ell^{\prime}}^{(q-j)}\rangle_{n}\qquad(-q\leq\ell<0).$$

(24)

The Fourier coefficients $A_{n,\ell;\ell^{\prime}}$ that we need are easily calculated to many orders from the definition (18b), and the positive frequency coefficients ($\ell>0$) can be obtained from $p_{\infty,-\ell}=p_{\infty,\ell}^{*}$.

To derive Eqs. (23) and (24), we use some simple properties of the function $A$. We start by noting that $A_{n,\ell;\ell^{\prime}}^{(j)}=0$ for $|\ell^{\prime}|>j$. Suppose now that for some $q\geq 1$, $p_{\infty,\ell^{\prime}}^{(j)}=0$ when $0\leq j\leq q-1$ and $|\ell^{\prime}|>j$ [this holds for $q=1$ by Eq. (21)]. Then the only terms on the right-hand side of Eq. (20) that can be non-zero are those with $|\ell^{\prime}|\leq j$ and $|\ell+\ell^{\prime}|\leq q-j$, which implies $|\ell|\leq q$ for these terms. Thus, Eq. (23) holds by induction. The sum over $\ell^{\prime}$ in Eq. (20) can therefore be truncated to $\ell^{\prime}=-j$ to $j$. Fixing $\ell<0$, we can see that the negative $\ell^{\prime}$ terms can be dropped since $A_{n,\ell;\ell^{\prime}}=0$ for $\ell^{\prime}<\ell$; thus we obtain Eq. (24). Further improvement (dropping terms in the sum that must be zero) is possible, but Eq. (24) suffices for our purposes.

We can now calculate higher orders of the reflection phase distribution in a completely mechanical, recursive fashion using Eq. (24), starting from $p_{\infty,\ell}^{(0)}=\delta_{\ell,0}/(2\pi)$. To present compactly the first few terms of these expansions, we use the following notation:

	$\displaystyle\alpha_{j}$	$\displaystyle=\frac{1}{1-(-1)^{j}\langle v_{n}^{j}\rangle_{n}},$		(25a)
	$\displaystyle\gamma^{(1)}$	$\displaystyle=\alpha_{1}\langle r_{n}^{\prime}\rangle_{n},$		(25b)
	$\displaystyle\gamma^{(2)}$	$\displaystyle=\alpha_{2}\langle r_{n}^{\prime}(r_{n}^{\prime}-2\gamma^{(1)}v_{n})\rangle_{n},$		(25c)
	$\displaystyle\gamma_{1}^{(3)}$	$\displaystyle=\alpha_{1}\langle r_{n}(\gamma^{(1)}r_{n}^{\prime}-\gamma^{(2)}v_{n})\rangle_{n},$		(25d)
	$\displaystyle\gamma_{3}^{(3)}$	$\displaystyle=\alpha_{3}\langle r_{n}^{\prime}({r_{n}^{\prime}}^{2}-3\gamma^{(1)}r_{n}^{\prime}v_{n}+3\gamma^{(2)}v_{n}^{2})\rangle_{n}.$		(25e)

Then we obtain, transforming back from frequency space (see Appendix F for further detail)

$$2\pi p_{\infty}(\phi_{r^{\prime}})=1+2\text{Re}\biggr{[}\left(\gamma^{(1)}+\gamma_{1}^{(3)}\right)e^{-i\phi_{r^{\prime}}}\\ +\gamma^{(2)}e^{-2i\phi_{r^{\prime}}}+\gamma_{3}^{(3)}e^{-3i\phi_{r^{\prime}}}\biggr{]}+O(|r_{n}|^{4}),$$

(26)

which shows explicitly the corrections to the uniform distribution that is obtained at zeroth order. Let us also note that interchanging $r_{n}\leftrightarrow r_{n}^{\prime}$ in Eqs. (25b)–(25e) yields the corresponding expression for the limiting distribution of the left-to-right reflection phase ($\phi_{r_{1\dots N}}$).

We proceed to use the reflection phase distribution to calculate the inverse localization length in the scattering expansion. We start by deriving the relation (12) between the localization length and the Fourier coefficients $p_{\infty,\ell}$. From the recursion relation (14a) for $s$, it follows that for sufficiently large $N$, $\langle s_{1\dots N}\rangle_{1\dots N}$ increases by the same constant amount (which by definition is $2/L_{\text{loc}}$) each time $N$ is increased by one:

$$\langle s_{1\dots N+1}\rangle_{1\dots N+1}-\langle s_{1\dots N}\rangle_{1\dots N}\\ =\int_{-\pi}^{\pi}d\phi_{r^{\prime}}\ p_{\infty}(\phi_{r^{\prime}})\langle g_{N+1}(\phi_{r^{\prime}})\rangle_{N+1}.$$

(27)

The site $N+1$ on the right-hand side can be replaced by any site $n$, since the disorder is by assumption distributed identically across the sites. We thus obtain an expression for the inverse localization length in terms of the reflection phase distribution:

	$\displaystyle\frac{2}{L_{\text{loc}}}$	$\displaystyle=\langle\int_{-\pi}^{\pi}d\phi\ p_{\infty}(\phi_{r^{\prime}})g_{n}(\phi_{r^{\prime}})\rangle_{n}$		(28a)
		$\displaystyle=\langle-\ln T_{n}\rangle_{n}+\int_{-\pi}^{\pi}d\phi_{r^{\prime}}\ p_{\infty}(\phi_{r^{\prime}})$
		$\displaystyle\qquad\times\langle\ln(1-r_{n}e^{i\phi_{r^{\prime}}}-r_{n}^{*}e^{-i\phi_{r^{\prime}}}+R_{n})\rangle_{n}.$		(28b)

Equation (28b) can also be obtained directly from the Lambert and Thorpe expression [Eq. (8)], or (using the correspondence discussed in Appendix B) from the Furstenberg formula quoted in Ref. [32]. Expressing the second term in frequency space is straightforward [65] (see Appendix E) and yields Eq. (12).

Since the odd orders vanish due to the symmetry argument presented below Eq. (10), we may write

$$\frac{2}{L_{\text{loc}}}=\sum_{j=1}^{\infty}\left(\frac{2}{L_{\text{loc}}}\right)^{(2j)},$$

(29)

where the superscript indicates the order in the expansion. Furthermore, for any $j\geq 1$, Eq. (12) yields

$$\left(\frac{2}{L_{\text{loc}}}\right)^{(2j)}=\frac{1}{j}\langle R_{n}^{j}\rangle_{n}-4\pi\text{Re}\left[\sum_{\ell=1}^{j}\frac{1}{\ell}p_{\infty,-\ell}^{(2j-\ell)}\langle r_{n}^{\ell}\rangle_{n}\right],$$

(30)

where we noted that Eq. (23) allows us to drop the terms $\ell=j+1$ to $2j$ in the sum. Reading off the Fourier coefficients from Eq. (26), we then calculate the first two non-vanishing orders of the scattering expansion for the inverse localization length (the main result of this section):

$$\frac{2}{L_{\text{loc}}}=\langle R_{n}\rangle_{n}-2\text{Re}\left[\frac{\langle r_{n}\rangle_{n}\langle r_{n}^{\prime}\rangle_{n}}{1+\langle r_{n}r_{n}^{\prime}/R_{n}\rangle_{n}}\right]+\frac{1}{2}\langle R_{n}^{2}\rangle_{n}\\ -\text{Re}\left[\alpha_{2}(\langle r_{n}^{2}\rangle_{n}-2\alpha_{1}\langle r_{n}\rangle_{n}\langle r_{n}v_{n}\rangle_{n})(\langle r_{n}^{\prime 2}\rangle_{n}-2\alpha_{1}\langle r_{n}^{\prime}\rangle_{n}\langle r_{n}^{\prime}v_{n}\rangle_{n})+2\alpha_{1}^{2}\langle r_{n}\rangle_{n}\langle r_{n}^{\prime}\rangle_{n}\langle r_{n}r_{n}^{\prime}\rangle_{n}\right]+O(|r_{n}|^{6}).$$

(31)

On the right-hand side, the first two terms are second order and recover Eq. (9). The remaining terms are fourth order. We have explicitly checked [using Eq. (12)] that the third- and fifth-order contributions vanish, consistent with the general symmetry argument. Our recursive formula (24) can straightforwardly generate still higher orders of Fourier coefficients $p_{\infty,\ell}$, and thus of $2/L_{\text{loc}}$.

A final technical point is that we have so far assumed that $|r_{n}|$ is strictly linear in the small parameter $\lambda$. However, it may be the case in a particular problem that $|r_{n}|$ starts at linear order in some parameter $\lambda$ but also has higher-order corrections in $\lambda$ (for instance, this occurs in the Anderson model, where $\lambda$ is essentially the disorder strength). There are two straightforward steps to adapt the results of this section to this more general case. First, the condition (22) is required to hold at the zeroth order in $\lambda$ as $\lambda\to 0$. Second, the series for $2/L_{\text{loc}}$ is to be truncated to a fixed order in $\lambda$, which implies in particular that each term at a given order $j$ in the $|r_{n}|$ expansion contributes generally at all orders $j^{\prime}\geq j$ in $\lambda$.

In this section, we apply our general results to the Anderson model and to a broad class of PARS. In the Anderson model, we consider the case of diagonal disorder (i.e., disorder in the onsite energies but not the tunneling) and show that our next-to-leading order result (31) agrees with and extends a known formula for the weak disorder expansion of the inverse localization length. We also verify our result for the probability distribution of the reflection phase with the literature, and we discuss the “anomalous” values of the momentum. In the PARS case, we use the Born series to show that Eqs. (9) and (10) demonstrate the SPS relation to the first two orders in the potential strength. As a special case, we obtain the SPS relation in the same square well array model considered by Deych et al. (see Sec. II). We also verify our higher-order formula (31) for the inverse localization length by comparing to another special case in the literature.