Multi-criticality Systems Research Articles

Disordered graphene ribbons as topological multicritical systems

The low energy spectrum of a zigzag graphene ribbon contains two gapless bands with highly non-linear dispersion, $\epsilon(k)=\pm |\pi-k|^W$, where $W$ is the width of the ribbon. The corresponding states are located at the two opposite zigzag edges. Their presence reflects the fact that the clean ribbon is a quasi one dimensional system naturally fine-tuned to the topological {\em multicritical} point. This quantum critical point separates a topologically trivial phase from the topological one with the index $W$. Here we investigate the influence of the (chiral) symmetry-preserving disorder on such a multicritical point. We show that the system harbors delocalized states with the localization length diverging at zero energy in a manner consistent with the $W=1$ critical point. The same is true regarding the density of states (DOS), which exhibits the universal Dyson singularity, despite the clean DOS being substantially dependent on $W$. On the other hand, the zero-energy localization length critical exponent, associated with the lattice staggering, is not universal and depends on the topological index $W$.

Work-conserving dynamic time-division multiplexing for multi-criticality systems

Multi-core architectures pose many challenges in real-time systems, which arise from contention between concurrent accesses to shared memory. Among the available memory arbitration policies, time-division multiplexing (TDM) ensures a predictable behavior by bounding access latencies and guaranteeing bandwidth to tasks independently from the other tasks. To do so, TDM guarantees exclusive access to the shared memory in a fixed time window. TDM, however, provides a low resource utilization as it is non-work-conserving. Besides, it is very inefficient for resources having highly variable latencies, such as sharing the access to a DRAM memory. The constant length of a TDM slot is, hence, highly pessimistic and causes an underutilization of the memory. To address these limitations, we present dynamic arbitration schemes that are based on TDM. However, instead of arbitrating at the level of TDM slots, our approach operates at the granularity of clock cycles by exploiting slack time accumulated from preceding requests. This allows the arbiter to reorder memory requests, exploit the actual access latencies of requests, and thus improve memory utilization. We demonstrate that our policies are analyzable as they preserve the guarantees of TDM in the worst case, while our experiments show an improved memory utilization. We furthermore present and evaluate an efficient hardware implementation for a variant of our arbitration strategy.

Efficient Schedulability Test for Dynamic-Priority Scheduling of Mixed-Criticality Real-Time Systems

Systems in many safety-critical application domains are subject to certification requirements. In such a system, there are typically different applications providing functionalities that have varying degrees of criticality. Consequently, the certification requirements for functionalities at these different criticality levels are also varying, with very high levels of assurance required for a highly critical functionality, whereas relatively low levels of assurance are required for a less critical functionality. Considering the timing assurance given to various applications in the form of guaranteed budgets within deadlines, a theory of real-time scheduling for such multi-criticality systems has been recently under development. In particular, an algorithm called Earliest Deadline First with Virtual Deadlines (EDF-VD) has shown a lot of promise for systems with two criticality levels, especially in terms of practical performance demonstrated through experiment results. In this article, we design a new schedulability test for EDF-VD that extends these performance benefits to multi-criticality systems. We propose a new test based on demand bound functions and also present a novel virtual deadline assignment strategy. Through extensive experiments, we show that the proposed technique significantly outperforms existing strategies for a variety of generic real-time systems.

Dynamical stability of the quantum Lifshitz theory in 2+1 dimensions

The roles of magnetic and electric perturbations in the quantum Lifshitz model in $2+1$ dimensions are examined in this paper. The quantum Lifshitz model is an effective field theory for quantum multicritical systems, which include generalized two-dimensional (2D) quantum dimer models in bipartite lattices and their generalizations. It describes a class of quantum phase transitions between ordered and topological phases in $2+1$ dimensions. Magnetic perturbations break the dimer conservation law. Electric excitations, the condensation of which leads to ordered phases, have been studied extensively both in the classical three-dimensional model and in the quantum 2D model. The role of magnetic vortex excitations, the condensation of which drives these systems into a ${\mathbb{Z}}_{2}$ topological phase, has been largely ignored. Recent numerical studies claim that the quantum theory has a peculiar feature: the dynamical exponent $z$ flows continuously and the quantum theory is hence unstable to magnetic vortices. To study the interplay of both excitations, we perform a perturbative renormalization group study to one-loop order and study the stability of the theory away from quantum multicriticality. This is done by generalizing the operator-product expansion to anisotropic models. It is found that the dynamical exponent does not appear to flow, in contrast to the classical Monte Carlo study. Possible reasons for this difference are discussed at length.

Directed polymers with random interaction: An exactly solvable case.

We propose a model for two $(d+1)$-dimensional directed polymers subjected to a mutual $\delta$-function interaction with a random coupling constant, and present an exact renormalization group study for this system. The exact $\beta$-function, evaluated through an $\epsilon(=1-d)$ expansion for second and third moments of the partition function, exhibits the marginal relevance of the disorder at $d=1$, and the presence of a phase transition from a weak to strong disorder regime for $d>1$. The lengthscale exponent for the critical point is $\nu=1/2\mid\epsilon\mid$. We give details of the renormalization. We show that higher moments do not require any new interaction, and hence the $\beta$ function remains the same for all moments. The method is extended to multicritical systems involving an $m$ chain interaction. The corresponding disorder induced phase transition for $d>d_m=1/(m-1)$ has the critical exponent ${\nu}_m=[2d(m-1)-2]^{-1}$. For both the cases, an essential singularity appears for the lengthscale right at the upper critical dimension $d_m$. We also discuss the strange behavior of an annealed system with more than two chains with pairwise random interactions among each other.

Random-field mechanism in random-bond multicritical systems (abstract)

It is argued on general grounds that bond randomness drastically alters multicritical phase diagrams via a random-field mechanism. For example, tricritical points and critical endpoints are entirely eliminated (in spatial dimension d≤2) or depressed in temperature (d>2). These predictions are confirmed by a renormalization-group calculation.1 Another consequence of this phenomenon is that, under bond randomness, the phase transitions of q-state Potts models are second order for all q at dimensionality d≤2.