3 phase boundary 27 that separates ordered and disordered states. larization. In terms of these variables the LLS equations take the form The transport-current (Slonczewski $^{23}$) term in Eq.( 4) can be understood in several different ways. In the spin- tronics literature this term is normally motivated by an appeal to total spin conservation and referred to as the spin-transfer torque. The idea is that when a ferromag- net’s spin-polarized quasiparticles carry a transport cur- rent through a spatial region with a non-collinear mag- netization, they violate spin-conservation. The collective magnetization must therefore compensate by rotating at a constant rate which is proportional to the fermion drift velocity. The form we use for the pseudospin transfer torque assumes that each component of the pseudospin current is locally equal to the number current times the corresponding pseudospin direction cosine. This prop- erty does not hold locally in a microscopic theory, but is expected to be valid in the smooth pseudospin texture limit we address. (In metals an additional phenomeno- logical factor is required to account for the difference in drift velocity between majority and minority spin elec- trons.) Since pseudospin is not conserved in bilayer quantum Hall systems, as we can see from Eq.( 2) or Eq.( 3), this argument does not apply directly. If we appeal to a mean- field description of the pseudospin ferromagnet, however, we can obtain the same result by the following argument. Mean-field quasiparticles satisfy a single-particle Hamil- tonian for a particle in a magnetic field which experiences both a scalar potential, and a pseudospin-dependent po- tential that can be interpreted as a pseudospin effec- tive magnetic field. Following standard textbook deriva- tions it is possible 28 to derive an expression for the time- dependence of the contribution of a single quasiparticle to any component of the pseudospin density. The ex- pression contains a pseudospin precession term and an additional term which is the divergence of the current of that component of pseudospin. Summing over all quasi- particle states we obtain a precession term that depends on the configuration of the order parameter, and hence on the pseudospin-field to which it gives rise, even in the absence of a current. We obtain the additional current- driven term on the right-hand side of Eq.( 4) when the pseudospin currents are non-zero and space dependent. The additional term in the equation of motion can also be viewed 29 as a consequence of an altered relationship be- tween pseudospin-dependent effective magnetic field and pseudospin polarization direction for quasiparticles that carry a current. Eq.( 4) should in principle also include a current related damping term 30 which we ignore in the present paper. When quantum Hall bilayer pseudospin ferromagnets are tilted far from their easy plane, order tends 31 to be destroyed. For that reason we are often most interested in the limit in which m$_{z}$ is much less than 1. It is therefore convenient to express the pseudospin direction in terms of the azimuthal angle φ , which is the inter-layer phase difference, and m$_{z}$ which is proportional to the layer po- The three terms on the right-hand side can be identi- fied as contributions to the time-dependence of m$_{z}$ (or equivalently of the exciton density) due respectively to the divergence of the exciton supercurrent, coherent con- densate tunneling, and the divergence of the ˆ z (layer In this section we discuss approximate upper bounds on the critical current which are helpful in interpreting the numerical results described in the following section. We use a simplified version of the static limit of the ˙ m$_{z}$ LLS equation (Eq.( 6)) in which m$_{z}$ is assumed to be small: These equations do not on their own provide a closed description of pseudospin dynamics in the presence of electrical bias potentials and need to be supplemented by a theory which specifies the spatial dependence of the pseudospin current. In general this quantity depends 21 on the order parameter configuration as well as on the contact geometry and external current or voltage biases. The transport theory and the pseudospin dynamics the- ory are therefore not independent. In the present paper we study voltage biased Hall bars with source and drain leads at opposite ends, and with a variety of shapes and sizes. We assume, as a simplification, that the current distribution is defined by a local conductivity tensor with a large Hall angle. Given these simplifications, we are able to explicitly evaluate the maximum current at which time-independent order parameters are possible. Be- cause collective tunneling no longer contributes strongly to the dc interlayer current when the inter-layer phase is time-dependent, the interlayer conductance mechanism changes qualitatively when this maximum current is ex- ceeded. We therefore associate this current with the ex- perimental critical current. In Eq.( 6) we have written ⃗ j/n , which has units of veloc- ity, as the pseudospin velocity ⃗ v$_{ps}$ . ˙ m$_{z}$ = { − 2 n $_{ℏ}$ρ$_{s}$ m 2 ⊥ ⃗ ∇ $^{2}$φ + Δ$_{t}$ ℏ m$_{⊥}$ sin φ } − ( ⃗ v$_{ps}$ · ⃗ ∇ ) m$_{z}$ } + α$_{z}$ m 2 ⊥ ˙ φ ˙ φ = m$_{z}$ { 2 n ℏ ρ$_{s}$ ( ∣ ∣$^{⃗}$ ∇ φ ∣ ∣$^{2}$ + 2 m 4 ⊥ ∣ ∣$^{⃗}$ ∇ m$_{z}$ ∣ ∣$^{2}$ + 2 m$_{z}$ m 2 ⊥ ⃗ ∇ $^{2}$m$_{z}$ ) − 4 n $_{ℏ}$β − Δ$_{t}$ ℏ 1 m$_{⊥}$ cos φ } − ( ⃗ v$_{ps}$ · ⃗ ∇ ) φ } − α$_{φ}$ m 2 ⊥ ˙ m$_{z}$. (6) 0 = − ρ$_{s}$ ℏ ⃗ ∇ $^{2}$φ + 1 2 Δ$_{t}$ n ℏ sin φ − 1 2 ⃗ j · ⃗ ∇ m$_{z}$. (7) III. APPROXIMATE CRITICAL CURRENTS