# Multiple Current Reversals Using Superimposed Driven Lattices
## 2 Setup, Equations of Motion and Symmetries  
We consider \(N\) non-interacting classical particles of mass \(m\) in a two dimensional (2D) periodic potential \(V(\mathbf{r})=V_{B}(\mathbf{r})+V_{G}(\mathbf{r})\). The separable potential due to the 'background lattice' is represented by \(V_{B}(\mathbf{r})=\tilde{V}_{B}(\cos^{2}\frac{\pi\chi}{l}+\cos^{2}\frac{\pi \chi y}{l})\) with potential height \(\tilde{V}_{B}\) and spatial period \(l\) in both \(x\) and \(y\) directions. On top of the lattice \(V_{B}\), we superimpose two finite lattices of 2D Gaussian barriers \(V_{G1}\) and \(V_{G2}\) localized in different regions which can be described by the potential \(V_{G}(\mathbf{r})=\sum_{m,n=-\infty}^{+\infty}\tilde{U}_{G}(\mathbf{r}_{nm})e ^{-a(\mathbf{r}-\mathbf{r}_{nm})^{2}}\) with the barriers centered at positions \(\mathbf{r}_{nm}=(ml,nl)\) where \(m,n\in\mathbb{Z}\) (see Figure 1a). These two lattices also have spatial period \(l\) along both \(x\) and \(y\) directions. \(\tilde{U}_{G}(\mathbf{r}_{nm})\) denote the potential height of the barrier located at \(\mathbf{r}_{nm}\) and \(\alpha\) is a measure of the widths of the barriers. In addition, the lattices are driven by an external bi-harmonic periodic driving force \(\mathbf{f}_{\mathbf{D}}(t)=a(\cos\omega t+0.5\cos 2\omega t,0)\) along the \(x\)-direction with driving amplitude \(a\) and frequency \(\omega\). This force is spatially independent. Introducing dimensionless variables \(x^{\prime}=\frac{\chi}{l},y^{\prime}=\frac{y}{l}\) and \(t^{\prime}=\omega t\) and dropping the primes for simplicity, the equation of motion for a single particle at position \(\mathbf{r}=(x,y)\) with velocity \(\dot{\mathbf{r}}=(\dot{x},\dot{y})\) reads  
\[\begin{array}{rcl}\dot{\mathbf{r}}&=&\mathbf{F}_{\mathbf{B}}(\mathbf{r})+ \mathbf{F}_{\mathbf{G}}(t)+\mathbf{F}_{\mathbf{D}}(t)\\ &=&U_{B}\pi(\sin 2\pi x,\sin 2\pi y)+2\beta\sum_{m,n=-\infty}^{+\infty}U_{G}( \mathbf{R}_{mn})\left(\mathbf{r}-\mathbf{R}_{mn}\right)e^{-\beta(\mathbf{r}- \mathbf{R}_{mn})^{2}}+d(\cos t+0.5\cos 2t,0)\end{array} \tag{1}\]  
where \(\mathbf{F}_{\mathbf{B}}(\mathbf{r}),\mathbf{F}_{\mathbf{G}}(\mathbf{r})\) and \(\mathbf{F}_{\mathbf{D}}(t)\) denote the forces due to the background lattice, superimposed lattices of Gaussian barriers and external driving respectively. The system is described by the four dimensionless parameters: \(U_{B}=\frac{\tilde{V}_{B}}{m!\omega^{2}}\) denoting the effective potential height of the lattice \(V_{B}\), \(U_{G}(\mathbf{R}_{mn})=\frac{\tilde{U}_{G}(\mathbf{r}_{nm})}{m!\omega^{2}}\) denoting the effective potential heights of the Gaussian barriers, \(\beta=\alpha l^{2}\) and the effective driving amplitude \(d=\frac{a}{ml\omega^{2}}\). \(\mathbf{R}_{mn}=(m,n)\) denote the positions of the maxima of the Gaussian barriers which coincides with the positions of the potential maxima of the background lattice \(V_{B}\). In this dimensionless form, the system has a spatial period \(L=1\) in both \(x\) and \(y\) directions and a temporal period \(T=2\pi\).  
Our setup breaks the generalized time reversal symmetry \(S_{t}\): \(t\longrightarrow-t+\tau\), \(\mathbf{r}\longrightarrow\mathbf{r}+\boldsymbol{\delta}\) (for arbitrary constant translations \(\boldsymbol{\delta}\) and \(\tau\) of space and time respectively) and the generalized parity symmetry \(P_{x}\): \(x\longrightarrow-x+\delta\), \(t\longrightarrow t+\tau\) in the \(x\)-direction. As a result, directed transport of a particle ensemble is expected along the \(x\)-direction [7]. Since the setup preserves the generalized parity symmetry along the \(y\)-direction: \(P_{y}\): \(y\longrightarrow-y+\delta\), \(t\longrightarrow t+\tau\), directed transport is not possible along this direction. Throughout the following discussions, by 'transport' we would always refer to the directed transport along the \(x\)-direction.