5 appear with opposite signs. This 1+1 dimensional sine- Gordon equation appears as the Euler-Lagrange equa- tion of motion of a system with a Lagrangian with a kinetic-energy term proportional to ρ$_{s}$ ( ∂$_{t}$φ ( x, t )) 2 and a potential energy term proportional Δ$_{t}$ n cos( φ ( x, t ). Since total energy (integrated over position x ) is conserved by this dynamics it follows that the variation of the typical value of ρ$_{s}$ ( ∂$_{t}$φ ) 2 along the space-time boundary cannot be larger than ∼ Δ$_{t}$ n . When this energy conservation condition is mapped from the regular sine-Gordon equa- tion to the (imaginary time) elliptic sine-Gordon equa- tion we can conclude that the typical value of ⃗ ∇ φ · ˆ n along the boundary of ˜ A near the source contact cannot differ from the typical value of ⃗ ∇ φ · ˆ n along the boundary near the drain contact by more than ∼ Δ$_{t}$ n . It follows that the current flowing through the system from source to drain should not be much larger than where W is the length of the contract region, or the width of a Hall bar assumed to be contacted at its edges. We will refer to I c E as the edge critical current, since it is limited by the length of one edge. For stronger inter- layer coupling then critical current is expected to vary as (Δ$_{t}$ n ) 1 / 2 once λ is smaller than the Hall bar length. Finally we note that because of hot-spot effects in trans- port with large Hall angles the pseudospin transfer torque will act at the sample corners. Since the supercurrent is converted into coherent pseudospin precession over the length scale λ , the effective size of the contact region will be ∼ λ when the Hall angle is large and λ is smaller than W . We therefore estimate that the critical current is close to independent of Δ$_{t}$ n , under these circumstances. We refer to this last critical current as the corner critical current. In the following section we compare numerical LLS crit- ical currents with these rough estimates. The numerical calculations we describe below are sim- ilar to those carried out in micromagnetic descriptions of ferromagnetic metal spin-transfer torque physics, but are applied here to pseudospin transfer physics in condensed bilayers. We divide the system area into pixels within which the pseudospin magnetization is assumed to be constant. Except where noted we used square 10 l × 10 l pixels where l is the magnetic length. We rewrite the spin-transfer torque term in Eq.( 6) in the discretized form: As mentioned in the previous section, we identify the critical current as the circuit current value above which a time independent solution no longer exists. When the external current is small, the Gilbert damping term in the LLS equation relaxes the pseudospin magnetization into time-independent configurations. The behavior of the pseudospin as the current increases is partly analo- gous to the behavior of a damped pendulum driven by an increasingly strong torque. The current which flows into a pixel is assumed to have the same pseudospin polarization as the pixel from which it is incident. Currents entering or exiting from the con- tacts are assumed to have m$_{z}$ = 1 for top layer contacts and m$_{z}$ = − 1 for bottom layer contacts. In this way the pseudospin transfer torque and the Landau-Lifshitz precessional torque acting on each pixel’s pseudospin can be evaluated. Note that the pseudospin torque depends not only on current paths, but also on the pseudospin magnetization configuration. where ⃗ m$_{L,R}$ is the magnetization orientation to the left and right of a boundary separating two-pixels. The Hall angle used in these calculations was therefore tan − $^{1}$(20) over the largest part of the sample in which the pseu- dospin magnetization is close to collinear and planar. The results we report on are not sensitive to the Hall angle, provided that it is large. where [ G ] is the conductance matrix that describes the local conductivity including Hall components, { V } is a vector of local voltages in each pixel, and { I } is a vec- tor composed of currents that flow between pixels. Us- ing current conservation conditions the dimension of the matrix can be reduced to the pixel number N . Given the source and drain contact voltages, we can solve for the internal voltage and current distributions and for the variation of source and drain currents across the contacts. Note that quantum Hall physics comes into play through the Hall contributions to the conductivity matrix [ G ]. The local Hall conductivity was set to σ$_{xy}$ = e $^{2}$/h , which is close to the appropriate value for ν = 1 whether or not the Hall plateau is fully formed, and the longitudinal conductivity was set to where k labels the four neighboring sites, A is the pixel area, and α labels components of the magnetization ori- entation in the pixel of interest. I$_{n}$ is the quasiparticle number current flow from neighbor site n into this pixel. In this article we estimate values of I$_{n}$ by solving the resistor network model obtained by discretizing a contin- uum model with a local conductivity that includes a Hall component. { I } = [ G ] { V } (15) I c E ∼ eW ρ$_{s}$ ℏ λ (12) σ$_{xx}$ = 0 . 05 exp( − m 2 $_{z}$/W ) ( ⃗ m · ⃗ m$_{L}$ )( ⃗ m · ⃗ m$_{R}$ ) e $^{2}$/h (16) I c C ∼ eρ$_{s}$ ℏ , (13) IV. MODEL CALCULATIONS A. Critical Current Identification ˙ m$_{α}$ |$_{ST}$ = ∑ k I$_{n}$ nA$_{pixel}$ [ m$_{k,α}$ − ( ⃗ m$_{k}$ · ⃗ m ) m$_{α}$ ] (14)