How do cells organize into hexagonal packing in the fruit fly wing?​

The images on the left show how cells are packed in the developing Drosophila wing at 17 and 32 hours After Pupal Formation (APF). Each cell is color-coded by its hexatic magnitude, a measure of local hexagonal order: the brighter the color, the closer the arrangement is to a perfect hexagonal crystal.

What biophysical mechanism drives this striking transition from disordered to highly ordered cell packing?
Fig. 1: Cell hexatic in fly wing tissue at early (17 hAPF) and late (32 hAPF) development. hAPF = hours After Pupal Formation

Could critical exponents of phase transitions depend on history?​​

Why it matters: Critical exponents are usually expected to be universal, independent of how a system starts. We show when this can fail in nonequilibrium systems.

Description: We identify general criteria for absorbing-state phase transitions that predict when critical exponents can depend on the initial condition. The Fig. 2 illustrates the basic idea: depending on how the state space is connected, the long-time dynamics may either forget its starting point or retain a memory of it.
Fig. 2. To explain the idea simply, consider two Markov dynamics on the state space S={0,1,…,9}, shown in (a) and (c). The corresponding time evolution, measured by the average state index, is shown in (b) and (d). Depending on how the states are connected, the system can show either (i) unique long-time behavior, with no memory of the initial state, as in (a,b), or (ii) non-unique behavior, where the long-time dynamics depends on the initial condition, as in (c,d).

​For more details, see our paper below. 👇

From Ising model to Kitaev Chain: An introduction to topological phase transition.

A unifying, pedagogical bridge between three archetypes of phase transitions—2D thermal Ising, the 1D transverse-field Ising chain, and the 1D Kitaev chain—showing how one Ising critical point underlies symmetry-breaking and topological transitions.
Majorana chain in two limits: for g≫1, Majoranas pair on the same site; for g→0, they pair on neighboring sites, leaving unpaired zero modes at the chain ends.

How can we extract Ising critical exponents analytically?

In this video, I walk through a standard route to the critical exponents of the 2D Ising model on a torus using conformal field theory (CFT) and the operator product expansion (OPE). The idea is to map the 2D classical Ising model to a 1D quantum Ising chain on a ring, transform spins into free fermions via the Jordan–Wigner mapping, take the continuum limit, and then use CFT to identify the critical exponents.​

Do self‑organized systems share a universal class?
​Manna = Oslo ≠ DP.

Why this matters: Many systems produce cascades of all sizes—small ones often, big ones rarely. Sandpile models are a clean way to study this.

Question: In 2D, do different sandpile rules lead to the same large-scale behavior, and does the fixed-energy Manna model belong to the directed percolation (DP) universality class?

Result:
  1. Unlike in 1D, local rule details do not change the universality class in 2D: the driven Manna and Oslo models scale in the same way.
  2. The fixed-energy (closed) Manna model does not belong to the DP universality class.

For more details, see my dissertation below. 👇
Here, I explain what self‑organized criticality is.