Critical behaviour of driven bilayer systems: A field-theoretic renormalisation group study

We investigate the static and dynamic critical behaviour of a uniformly driven bilayer Ising lattice gas at half filling. Depending on the strength of the interlayer coupling J , phase separation occurs across or within the two layers. The former transitions are controlled by the universality class of model A (corresponding to an Ising model with Glauber dynamics), with upper critical dimension dc = 4. The latter transitions are dominated by the universality class of the standard (single-layer) driven Ising lattice gas, with dc = 5 and a non-classical anisotropy exponent. These two distinct critical lines meet at a non-equilibrium bicritical point which also falls into the driven Ising class. At all transitions, novel couplings and dangerous irrelevant operators determine corrections to scaling. Center for Stochastic Processes in Science and Engineering, Physics Department Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0435, USA Fachbereich Physik, Universität Gesamthochschule Essen, D-45117 Essen, Federal Republic of Germany. Introduction. Driven diffusive lattice gases (DDLG), introduced by Katz et al. [1] to investigate far-from-equilibrium properties of interacting many-particle systems, are deceptively simple generalisations of the familiar equilibrium Ising lattice gas [2]. Particles diffuse on a lattice, controlled by not only the usual interparticle attraction and thermal bath (at temperature T ), but also a uniform external force. In conjunction with periodic boundary conditions, the latter drives the system into a non-equilibrium steady state with non-trivial current. A remarkable range of novel collective phenomena emerges, some aspects of which are now well understood while many others remain mysterious [3]. In particular, when further ‘slight’ modifications or generalisations are introduced, simulations often contradict equilibrium-based expectations. In this sense, an intuitive understanding of non-equilibrium steady states is still lacking. While most simulations of the original model were carried out on a single twodimensional (d = 2) square lattice, Monte Carlo studies have been reported recently for two coupled driven Ising lattices stacked to form a bilayer structure [4, 5, 6]. Upon tuning the (Ising) inter-layer coupling, the system makes a transition from a homogeneous high temperature phase into two distinct ordered states at low temperatures. A simple phase diagram was found [5], displaying two lines of continuous transitions which meet a line of ‘first-order’ transitions at a ‘bicritical’ point. The critical properties associated Critical behaviour of driven bilayer systems 2 with the two lines were conjectured. In the most recent study [6], anisotropic intralayer couplings were introduced, and certain critical properties were measured. In this letter, we report results from a field-theoretic renormalisation group (RG) study for the continuous transitions of this model. Even though we essentially confirm the original conjecture [5], some novel and curious features emerge near the bicritical point and along the line associated with repulsive inter-layer couplings. After a brief description of both the microscopic model and the field-theoretic description, we present our results. In all simulation studies, the ‘microscopic’ model consists of two fully periodic L1 × L2 square lattices, arranged in a bilayer structure (effectively, an L1 × L2 × 2 system). The sites, labeled by (j1, j2, j3), with j1,2 = 1, .., L1,2 and j3 = 1, 2, may be empty or occupied. Thus, the set of occupation numbers {n(j1, j2, j3)}, where n = 0 or 1, completely specifies a configuration. To access critical points, we use half-filled systems, i.e., ∑ n = L1L2. The particles interact, so that the Hamiltonian is given by H ≡ −J0 ∑ nn′ − Jnn′′, where n and n′ are nearest neighbours within a given layer, while n and n′′ differ only in the layer index. All studies focus on attractive intra-layer interactions, J0 > 0, while J can be of either sign. Since intra-layer anisotropies generate no qualitatively new features, J0 may be set to unity. The equilibrium phase diagram in the J-T plane is easily obtained, with some exactly known features. For example, a second-order transition occurs at TO ≃ 0.5673/kB and J = 0 [7]. To access the most interesting non-equilibrium properties, a conserved dynamics must be imposed. Typically, Kawasaki spin exchange with Metropolis rates is employed, i.e., particles hop to nearest neighbour holes with probability min{1, exp (−∆H/kBT )}, where ∆H is the energy change associated with the move. To model the effects of the drive, we add ±Eo to ∆H for hops against/along, say, the 1-axis [1], interpreting the particles as ‘charged’ in the presence of an external ‘electric’ field (Eo, 0, 0). Note that, with Metropolis rates, it is possible to study the ‘infinite’ Eo case: jumps against the field are simply never executed. When so driven, the phase diagram in the T -J plane can be found (schematically shown in Fig. 1 [4, 5, 6]). At high T , the system is disordered (D). At low T , the system phase separates: For sufficiently repulsive J , the fully ordered (T = 0) state displays densities 1 and 0 in the two layers, so that this phase is labeled ‘full-empty’ (FE). On the other hand, for J > 0 and low T , each layer phase separates individually, resulting in strips of particles (of width L2/2, at T = 0) ‘on top of each other’, and aligned with the drive. Thus, we label this state the ‘strip’ (S) phase. In the following, we invoke field-theoretic renormalisation group techniques, to investigate universal properties associated with the critical points. Model equations. This analysis has already been initiated in [3, 8]. We define the single-layer magnetisations φi(~x), i = 1, 2 as the coarse-grained versions of 2n(j1, j2, i) − 1, with the in-layer coordinate ~x generalized to d dimensions. To insure the proper Eo = 0 limit, we construct a Landau-Ginzburg-Wilson (LGW) Hamiltonian, Hc, which contains all terms, up to fourth order in φi and second order in ~ ∇φi, compatible with stability requirements and symmetries of the microscopic model. As usual, the explicit relationships [8] between the coarse-grained couplings and the Critical behaviour of driven bilayer systems 3