_prob}, exponential localization is never reached. Even in the asymptotic limit of $s \gg 1$, using equation \ref{eqn:bdi_prob} one finds:
\begin{equation}
	-\expval{\ln(g)} \approx 2\sqrt{\frac{2s}{\pi}}.
	\label{eqn:lng_bdi}
\end{equation}
The typical localization length scales as $\xi_{\text{typ}} \sim  \sqrt{Ll}$ in this case, and becomes infinite in the limit of large system size.  Similarly $\xi_{avg} \sim L/\log s$, and diverges with $L$. It must be noted that this does not correspond to a perfectly conducting system, but one with a broad transport distribution. In fact, based on equation \ref{eqn:bdi_prob}, one finds that $\ln(g)$ is not self-averaging in the BDI symmetry class, even for arbitrarily long system sizes. This is the disorder induced critical point in 1D \cite{LocalizationDelocalizationDirtySC,MotrunichDamleHuse,Kamenev2016}. 



The existence of the critical point at zero energy is intimately connected to topology in non-interacting systems. In symmetry class BDI, it is possible to have topologically non-trivial gapped phases in 1D. These are classified by an integer-valued topological invariant of the band structure \cite{RyuTenFoldWay}, the winding-number. A disorder-free 1D metal in class BDI can be viewed as the critical point separating the topologically trivial phase with zero winding from one of the non-trivial phases with winding $\pm 1$. One can then imagine that adding disorder creates regions in each of these phases. On the domain walls between these disorder-induced topologically distinct regions, there exist low-energy states. It is these low-energy states which lead to the divergent density of states and localization length noted previously \cite{DysonDOS, TitovDOS, MotrunichDamleHuse}.



As shown above, the FP equation describes a 1-parameter scaling that changes abruptly from one symmetry class to another.  This poses a conundrum for scattering at non-zero energy in class BDI, where chiral symmetry does not impose any constraints on the S-matrix, and the relevant FP equation is technically in symmetry class AI. It follows that at sufficiently low energies, eqn. \ref{eq:FP} fails to describe transport distributions in such systems, which must cross over smoothly from that of class BDI at zero energy to that of class AI for sufficiently large energies (compared to the inverse relaxation-time, say). In this regard, energy can be thought of as a parameter that moves us away from criticality.  (As noted previously, zero-energy is not always a disorder-induced critical point, as symmetry preserving terms such as a staggered hopping can be added to tune the system away from criticality \cite{MotrunichDamleHuse, GruzbergSuperuniversality}; such terms are not considered here.)  A minimal description of the transport statistics in this finite energy cross-over from class BDI to AI thus requires one additional parameter, which clearly must be related to the energy; here we argue that the appropriate parameter is $r = \epsilon \tau$, where $\tau$ is the scattering time.  Note that a breakdown of one-parameter scaling also occurs in the standard Anderson localization problem near zero-energy and the band edges \cite{TitovShortRangeDisorder, TessieriBandCenterAnomaly, KandBandCenterAnomaly}; this is distinct from the phenomenon we discuss here.



Ref. \cite{RyuCrossover} suggested that the FP equation fails to describe  the finite enregy cross-over regime between symmetry class BDI and AI because the magnitude and phase of the reflection coefficient do not decouple. Specifically, writing the reflection coefficient as
\begin{equation}
	\mathcal{R} = \sqrt{R} e^{i \phi},
\end{equation}
where $\sqrt{R} = \tanh(x)$ is the amplitude and $\phi$ is the phase. The usual FP equation assumes that the probability distribution for the phase $Q(\phi)$ is stationary with respect to $s$, independent of $x$, and follows:
\begin{equation}
	Q(\phi) =
	\begin{cases}
		\frac{1}{2\pi}, & \text{AI} \\
		\frac{1}{2}[\delta(0) + \delta(\pi) ], & \text{BDI}
	\end{cases}
\end{equation}
Ref. \cite{RyuCrossover} derived FP equations which incorporate the flow of $\phi$ with $s$, and proposed approximate solutions to these in order to describe the scaling regime.   The drawback of this approach is that the resulting differential equations cannot be solved exactly; we return to this point in Sec. \ref{Sec:Results}, where the results of this work are compared to those of ref. \cite{RyuCrossover}.  



\section{Model and Methods} \label{Sec:Models} 



In this section, the models used to study the cross-over of transport from symmetry class BDI to AI are introduced. These are the 1D chain and metallic arm-chair graphene ribbon. The symmetries of the disordered models used to classify the symmetry class of these models is then discussed. Finally, a brief description of the methods used to compute transport numerically is given.



To study the cross-over and establish its properties, two tight binding systems are considered. The first is a 1D chain and the second is a graphene nanoribbon with arm-chair boundaries. These are shown in figure \ref{fig:lat}. The Hamiltonian of the 1D chain in the absence of disorder is given by:
\begin{equation}
	H = \sum_i t (c^{\dagger}_{i, B} c_{i, A} + c^\dagger_{i+1, A} c_{i, B}) + \text{h.c.},
	\label{eqn:ham_1d}
\end{equation}
where $c_{i, A}$ ($c_{i, B}$) are annihilation operators for electrons on the unit cell labeled by the index $i$ at the sub-lattice $A$ ($B$). See figure \ref{fig:lat} for the sub-lattice labeling convention used here. Similarly, the Hamiltonian for the arm-chair ribbon of width $W$ is given by:
\begin{multline}
	H = \sum_i\sum_{j\in\text{odd}}^{W} t\left( c^\dagger_{i+1, j, A} c_{i, j, B} \right) + \text{h.c} \\ + \sum_i\sum_{j\in\text{even}}^{W} t\left( c^\dagger_{i, j, A} c_{i, j, B} \right) + \text{h.c} \\ + \sum_i\sum_{j=1}^{W-1} t\left( c^\dagger_{i, j+1, B} c_{i, j, A} + c^\dagger_{i, j+1, A} c_{i, j, B} \right) + \text{h.c}.
	\label{eqn:ham_ac}
\end{multline}
In this case, the extra index $j$ labels the vertical position of the site within the unit cell of the arm-chair graphene ribbon. The convention used for labeling the sites within a unit cell is shown in figure \ref{fig:lat}. For both the 1D chain and the arm-chair ribbon, since there are no terms breaking the degeneracy between the spins, only one spin species is considered for simplicity. 



The 1D chain consists of a pair of left and right moving modes, with a linear dispersion close to zero energy. The spectrum of the arm-chair ribbon can be thought of as a 1D projection of the dispersion of 2D graphene at quantized values of momentum along the transverse direction of the arm-chair ribbon \cite{WakabayashiGrapheneStates}. The dispersion generally has a gap that is inversely proportional to the width $W$. However, for widths $W = 3 n -1$, where $n$ is an integer, the dispersion contains the Dirac point of graphene and is metallic with a linear band crossing at zero energy. The spectrum of a metallic arm-chair ribbon with $W=5$ is shown in figure \ref{fig:ac_spectrum}. 


Adding a random component to the hopping introduces disorder without violating the chiral symmetry that is integral to the problem, as will be shown.  In the rest of this work, a disorder strength of $V$ denotes that a random number chosen from the uniform distribution spanning $[-V, V]$ is added to every non-vanishing hopping matrix element in the Hamiltonian.



\begin{figure}
	\centering
	\begin{subfigure}{0.48\linewidth}
		\centering
		\includegraphics[width=\linewidth]{1d_chain.pdf}
		\caption{}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.48\linewidth}
		\centering
		\includegraphics[width=\