arXiv subjects

Farzan Vafa

Publications and source records attributed to Farzan Vafa.

15 recordsLinked to original sources

arXiv2026

Epithelia Realize Nematopolar Topological Defect Structures

We introduce a shape-based polar order parameter that captures the structural asymmetry of cells within epithelial monolayers. By combining bright-field imaging and traction force microscopy, we demonstrate that shape polarity serves as a unifying biomechanical metric, integrating the physical information encoded by nematic directors, principal stresses, and cellular motion. Furthermore, we show that the tissue organizes into a mixed polar-nematic phase, characterized by the coexistence of integer ($\pm 1$) and half-integer ($\pm 1/2$) defects. Through mechanical perturbations, we demonstrate that both substrate stiffness and cell-cell adhesion modulate the density of these excitations and the length of domain walls binding like-signed positive half-integer defects. Using a minimal continuum model of polar-nematic active matter, we establish that this mixed phase is fundamentally driven by the interplay of active stresses and polar-nematic elasticity. These findings provide a direct experimental evidence that epithelial monolayers behave as nematopolar matter, in which coupled polar and nematic elastic interactions jointly shape the active state

cond-mat.soft↗

arXiv2026

Non-equilibrium geometric forces steer spiral waves on folded surfaces

Spiral waves are ubiquitous signatures of non equilibrium dynamics, appearing across chemical, biological, and active systems. Yet, in many living systems these waves unfold on curved and folded surfaces whose geometry has rarely been treated as a dynamical factor. Here we show that surface curvature fundamentally shapes spiral wave behavior and can contribute to the organization of neural activity in the brain. Via analytical theory and simulations of the complex Ginzburg Landau equation (CGLE) on curved surfaces, we demonstrate that curvature enters through the Laplace Beltrami operator as a spatial modulation of effective diffusion. Gradients of this effective diffusion generate a geometric force on spiral defects, and the complex nature of the CGLE produces a complex mobility that leads to non central and non reciprocal responses. Applied to realistic cortical surfaces of the human brain, the model predicts that the pattern of cortical folding stabilizes and localizes spiral waves, while progressive smoothing of the surface erases these non equilibrium structures. This reveals that brain geometry is not a passive scaffold but an active physical constraint that shapes neural dynamics. More broadly, the same geometric mechanism provides a universal route by which curvature and topology control pattern formation across oscillatory, chemical, and active matter systems.

cond-mat.soft↗

arXiv2025

Phase diagram, confining strings, and a new universality class in nematopolar matter

We study a minimal model of a system with coexisting nematic and polar orientational orders, where one field tends to order and the other prefers isotropy. For strong coupling, the ordered field aligns the isotropic one, locking their orientations. The phase diagram reveals three distinct phases--nematopolar (aligned orders), nematic (independent orders), and isotropic (vanishing orders)--separated by continuous and discontinuous transitions, including a triple and a tricritical point. We find unique critical scaling for the nematopolar-nematic transition, distinct from standard nematic or polar universality classes. Additionally, in the locked nematopolar phase, we show nematic $+1/2$ topological defect pairs are connected and confined by strings with constant tension. These strings arise from frustration in locking the orientational orders and can be interpreted as elongated cores of $+1$ polar topological defects. When a sufficiently strong background field couples to the polar order, all topological defects are expelled from the region. Analytical predictions are quantitatively confirmed by numerical simulations.

cond-mat.soft↗

arXiv2024

An efficient solution to Hidden Markov Models on trees with coupled branches

Hidden Markov Models (HMMs) are powerful tools for modeling sequential data, where the underlying states evolve in a stochastic manner and are only indirectly observable. Traditional HMM approaches are well-established for linear sequences, and have been extended to other structures such as trees. In this paper, we extend the framework of HMMs on trees to address scenarios where the tree-like structure of the data includes coupled branches -- a common feature in biological systems where entities within the same lineage exhibit dependent characteristics. We develop a dynamic programming algorithm that efficiently solves the likelihood, decoding, and parameter learning problems for tree-based HMMs with coupled branches. Our approach scales polynomially with the number of states and nodes, making it computationally feasible for a wide range of applications and does not suffer from the underflow problem. We demonstrate our algorithm by applying it to simulated data and propose self-consistency checks for validating the assumptions of the model used for inference. This work not only advances the theoretical understanding of HMMs on trees but also provides a practical tool for analyzing complex biological data where dependencies between branches cannot be ignored.

stat.ML↗

arXiv2024

Defect ground states for liquid crystals on cones and hyperbolic cones

This contribution is intended for Journal of Physics A: Mathematical and Theoretical Special issue on Non-equilibrium Dynamics in Complex Systems: Celebrating the Contributions of Uwe Täuber on his 60th Birthday. Cones with orientational order in the local tangent plane provide a soft matter analog of the Aharonov-Bohm effect. We first review recent work on 2D liquid crystals with $p$-fold rotational symmetry on cones. We exploit an analogy with electrostatics to determine the ground state as a function of both the cone angle and the liquid crystal symmetry for both free and tangential boundary conditions applied at the cone base. There is an effective topological charge $-χ$ at the apex, where $2πχ$ is the deficit angle. The ground states are in general frustrated due to parallel transport along the azimuthal direction on the cone. In the case of tangential boundary conditions, the ground state changes as a function of $χ$, where the apex absorbs and emits quantized defect charges, with intricate dependence on both the deficit angle and $p$. We check our predictions numerically for a set of commensurate cone angles, whose surfaces can be polygonized as a perfect triangular or square mesh, and find excellent agreement. Cones with both free and tangential boundary conditions can also exhibit metastable states distinguished by quantized screening of the apex charge. We also present preliminary work on hyperbolic cones, where the Gaussian curvature singularity is negative at the apex. When free boundary conditions are applied at the base, the ground states are characterized by an effective topological charge at the apex, similarly to the case of conventional cones. However, when tangential boundary conditions are applied, decreasing the deficit angle (which is now negative) induces neutral defect pair nucleation at the apex followed by emission of a positive defect.

cond-mat.soft↗

arXiv2023

Periodic orbits, pair nucleation, and unbinding of active nematic defects on cones

Geometric confinement and topological constraints present promising means of controlling active materials. By combining analytical arguments derived from the Born-Oppenheimer approximation with numerical simulations, we investigate the simultaneous impact of confinement together with curvature singularity by characterizing the dynamics of an active nematic on a cone. Here, the Born-Oppenheimer approximation means that textures can follow defect positions rapidly on the time scales of interest. Upon imposing strong anchoring boundary conditions at the base of a cone, we find a a rich phase diagram of multi-defect dynamics including exotic periodic orbits of one or two $+1/2$ flank defects, depending on activity and non-quantized geometric charge at the cone apex. By characterizing the transitions between these ordered dynamical states, we can understand (i) defect unbinding, (ii) defect absorption and (iii) defect pair nucleation at the apex. Numerical simulations confirm theoretical predictions of not only the nature of the circular orbits but also defect unbinding from the apex.

cond-mat.soft↗

arXiv2023

Active topological defect absorption by a curvature singularity

Using the Born-Oppenheimer approximation, we present a general description of topological defects dynamics in $p$-atic materials on curved surfaces, and simplify it in the case of active nematics. We find that activity induces a geometric contribution to the motility of the $+1/2$ defect. Moreover, in the case of a cone, the simplest example of a geometry with curvature singularity, we find that the motility depends on the deficit angle of the cone and changes sign when the deficit angle is bigger than $π$, leading to the change in active behavior from contractile (extensile) to extensile (contractile) behavior. Using our analytical framework, we then identify for positively charged defects the basin of attraction to the cone apex and present closed-form predictions for defect trajectories near the apex. The analytical results are quantitatively corroborated against full numerical simulations. Provided the capture radius is small compared to the cone size, the agreement is excellent.

cond-mat.soft↗

arXiv2023

Statics and diffusive dynamics of surfaces driven by $p$-atic topological defects

Inspired by epithelial morphogenesis, we consider a minimal model for the shaping of a surface driven by $p$-atic topological defects. We show that a positive (negative) defect can dynamically generate a (hyperbolic) cone whose shape evolves diffusively, and predict that a defect of charge $+1/p$ leads to a final semi-cone angle $β$ which satisfies the inequality $\sinβ\ge 1 - \frac{1}{p} + \frac{1}{2p^2}$. By exploiting the fact that for axisymmetric surfaces, the extrinsic geometry is tightly coupled to the intrinsic geometry, we further show that the resulting stationary shape of a membrane with negligible bending modulus and embedded polar order is a deformed lemon with two defects at antipodal points. Finally, we close by pointing out that our results may be relevant beyond epithelial morphogenesis in such contexts as shape transitions in macroscopic closed spheroidal surfaces such as pollen grains.

cond-mat.soft↗

arXiv2022

Defect absorption and emission for $p$-atic liquid crystals on cones

We investigate the ground state configurations of $p$-atic liquid crystals on fixed curved surfaces. We focus on the intrinsic geometry and show that isothermal coordinates are particularly convenient as they explicitly encode a geometric contribution to the elastic potential. In the special case of a cone with half-angle $β$, the apex develops an effective topological charge of $-χ$, where $2πχ= 2π(1-\sinβ)$ is the deficit angle of the cone, and a topological defect of charge $σ$ behaves as if it had an effective topological charge $Q_\mathrm{eff} = (σ- σ^2/2)$ when interacting with the apex. The effective charge of the apex leads to defect absorption and emission at the cone apex as the deficit angle of the cone is varied. For total topological defect charge 1, e.g. imposed by tangential boundary conditions at the edge, we find that for a disk the ground state configuration consists of $p$ defects each of charge $+1/p$ lying equally spaced on a concentric ring of radius $d = (\frac{p-1}{3p-1})^{\frac{1}{2p}} R$, where $R$ is the radius of the disk. In the case of a cone with tangential boundary conditions at the base, we find three types of ground state configurations as a function of cone angle: (1) for sharp cones, all of the $+1/p$ defects are absorbed by the apex; (2) at intermediate cone angles, some of the $+1/p$ defects are absorbed by the apex and the rest lie equally spaced along a concentric ring on the flank; and (3) for nearly flat cones, all of the $+1/p$ defects lie equally spaced along a concentric ring on the flank. Here the defect positions and the absorption transitions depend intricately on $p$ and the deficit angle which we analytically compute. We check these results with numerical simulations for a set of commensurate cone angles and find excellent agreement.

cond-mat.soft↗

arXiv2021

Active nematic defects and epithelial morphogenesis

Inspired by recent experiments that highlight the role of nematic defects in the morphogenesis of epithelial tissues, we develop a minimal framework to study the dynamics of an active curved surface driven by its nematic texture. Allowing the surface to evolve via relaxational dynamics leads to a theory linking nematic defect dynamics, cellular division rates and Gaussian curvature. Regions of large positive (negative) curvature and positive (negative) growth are colocalized with the presence of positive (negative) defects. Applying this framework to the dynamics of cultured murine neural progenitor cells (NPCs) in an ex-vivo setting, we find that cells accumulate at positive defects and are depleted at negative defects. In contrast, applying this to the dynamics of a basal marine invertebrate \emph{Hydra} in an in-vivo setting, we show that activity stabilizes a bound $+1$ defect state by creating an incipient tentacle, while a bound $+1$ defect state surrounded by two $-1/2$ defects can create a stationary ring configuration of tentacles, consistent with observations.

cond-mat.soft↗

arXiv2021

Fluctuations can induce local nematic order and extensile stress in monolayers of motile cells

Recent experiments in various cell types have shown that two-dimensional tissues often display local nematic order, with evidence of extensile stresses manifest in the dynamics of topological defects. Using a mesoscopic model where tissue flow is generated by fluctuating traction forces coupled to the nematic order parameter, we show that the resulting tissue dynamics can spontaneously produce local nematic order and an extensile internal stress. A key element of the model is the assumption that in the presence of local nematic alignment, cells preferentially crawl along the nematic axis, resulting in anisotropy of fluctuations. Our work shows that activity can drive either extensile or contractile stresses in tissue, depending on the relative strength of the contractility of the cortical cytoskeleton and tractions by cells on the extracellular matrix.

cond-mat.soft↗

arXiv2020

Defect Dynamics in Active Polar Fluids vs. Active Nematics

Topological defects play a key role in two-dimensional active nematics, and a transient role in two-dimensional active polar fluids. In this paper, we study both the transient and long-time behavior of defects in two-dimensional active polar fluids in the limit of strong order and overdamped, compressible flow, and compare the defect dynamics with the corresponding active nematics model studied recently. One result is non-central interactions between defect pairs for active polar fluids, and by extending our analysis to allow orientation dynamics of defects, we find that the orientation of $+1$ defects, unlike that of $\pm 1/2$ defects in active nematics, is not locked to defect positions and relaxes to asters. Moreover, using a scaling argument, we explain the transient feature of active polar defects and show that in the steady state, active polar fluids are either devoid of defects or consist of a single aster. We argue that for contractile (extensile) active nematic systems, $+1$ vortices (asters) should emerge as bound states of a pair of $+1/2$ defects, which has been recently observed. Moreover, unlike the polar case, we show that for active nematics, a linear chain of equally spaced bound states of pairs of $+1/2$ defects can screen the activity term. A common feature in both models is the appearance of $+1$ defects (elementary in polar and composite in nematic) in the steady state.

cond-mat.soft↗

arXiv2020

Multi-defect Dynamics in Active Nematics

Recent experiments and numerical studies have drawn attention to the dynamics of active nematics. Two-dimensional active nematics flow spontaneously and exhibit spatiotemporal chaotic flows with proliferation of topological defects in the nematic texture. It has been proposed that the dynamics of active nematics can be understood in terms of the dynamics of interacting defects, propelled by active stress. Previous work has derived effective equations of motion for individual defects as quasi-particles moving in the mean field generated by other defects, but an effective theory governing multi-defect dynamics has remained out of reach. In this paper, we examine the dynamics of 2D active nematics in the limit of strong order and overdamped compressible flow. The activity-induced defect dynamics is formulated as a perturbation of the manifold of quasi-static nematic textures explicitly parameterized by defect positions. This makes it possible to derive a set of coupled ordinary differential equations governing defect (and therefore texture) dynamics. Interestingly, because of the non-orthogonality of textures associated with individual defects, their motion is coupled through a position dependent ``collective mobility" matrix. In addition to the familiar active self-propulsion of the $+1/2$ defect, we obtain new collective effects of activity that can be interpreted in terms of non-central and non-reciprocal interactions between defects.

cond-mat.soft↗

arXiv2017

Chern-Simons Theory and Wilson Loops in the Brillouin Zone

Berry connection is conventionally defined as a static gauge field in the Brillouin zone. Here we show that for three-dimensional (3d) time-reversal invariant superconductors, a generalized Berry gauge field behaves as a fluctuating field of a Chern-Simons gauge theory. The gapless nodal lines in the momentum space play the role of Wilson loop observables, while their linking and knot invariants modify the gravitational theta angle. This angle induces a topological gravitomagnetoelectric effect where a temperature gradient induces a rotational energy flow. We also show how topological strings may be realized in the 6 dimensional phase space, where the physical space defects play the role of topological D-branes.

cond-mat.mes-hall↗

arXiv2016

Symmetry Protected Topological Hopf Insulator and its Generalizations

We study a class of $3d$ and $4d$ topological insulators whose topological nature is characterized by the Hopf map and its generalizations. We identify the symmetry $\mathcal{C}^\prime$, a generalized particle-hole symmetry that gives the Hopf insulator a $\mathbb{Z}_2$ classification. The $4d$ analogue of the Hopf insulator with symmetry $\mathcal{C}^\prime$ has the same $\mathbb{Z}_2$ classification. The minimal models for the $3d$ and $4d$ Hopf insulator can be heuristically viewed as "Chern-insulator$ \rtimes S^1$" and "Chern-insulator$\rtimes T^2$" respectively. We also discuss the relation between the Hopf insulator and the Weyl semimetals, which points the direction for its possible experimental realization.

cond-mat.str-el↗