Phase transitions are ubiquitous in nature. For equilibrium cases, the celebrated Landau theory has successfully described these phenomena on general grounds. Even for nonequilibrium transitions such as optical bistability, flocking transition, and directed percolation, one can often define Landau’s free energy in a phenomenological way to successfully describe the transition at a meanfield level. In such cases, the nonequilibrium effect is present only through the noise-activated spatial-temporal fluctuations that break the fluctuation-dissipation theorem.
In this lecture series, I will introduce a novel class of nonequilibrium phase transitions [1-2] and critical phenomena  that do not fall into this class by generalizing the Ginzburg-Landau theory to driven systems. Remarkably, the discovered phase transition is controlled by spectral singularity called the exceptional points that can only occur by breaking the detailed balance and therefore has no equilibrium counterparts. The emergent collective phenomena range from active time (quasi)crystals to exceptional point enforced pattern formation, hysteresis, to anomalous critical phenomena that exhibit anomalously large phase fluctuations (that diverge at d≤4) and enhanced many-body effects (that become relevant at d<8) . The inherent ingredient to these is the non-reciprocal coupling between the collective modes that arise due to the drive and dissipation. These phenomena occur in a broad class of many-body interacting nonequilibrium systems both in quantum and classical matter.
In the first (and a half?) part of the lecture, after formulating the general framework, I will give an example of such non-reciprocal phase transitions by considering models of Kuramoto model of synchronization, Vicsek model of flocking, and Swiss-Hohenberg equation of pattern formation, all generalized to have non-reciprocal interaction . I will show that the hydrodynamics of such model exhibit the above type of phase transitions. I will put some emphasis on the role of frustrations induced by non-reciprocity, which gives rise to a phenomenon analogous to the “order-by-disorder transition” known to occur in geometrically frustrated systems.
In the second part of the lecture, I will show how this theory offers a unified framework to describe two very different but similar states of matter, i.e., Bose-Einstein condensate (an equilibrium coherent state) and a photon laser (a nonequilibrium coherent state that requires population inversion) . Surprising enough, I will argue that these two classes of states can be either continuously tuned to crossover from one to the other, or may exhibit a discontinuous phase transition – just like in a liquid-gas phase diagram. I will show that the exceptional point marks the endpoint of the discontinuous phase transition boundary.
If time allows, I will also talk about the anomalous critical phenomena that arise at the exceptional point .
These frameworks lay the foundation of the general theory of critical phenomena in systems whose dynamics are not governed by an optimization principle.
 M. Fruchart*, R. Hanai*, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions. Nature 592, 363 (2021).
 R. Hanai, A. Edelman, Y. Ohashi, and P. B. Littlewood, Non-Hermitian phase transition from a polariton Bose-Einstein condensate to a photon laser. Phys. Rev. Lett. 122, 185301 (2019).
 R. Hanai and P. B. Littlewood, Critical fluctuations at a many-body exceptional point. Phys. Rev. Res. 2, 033018 (2020).
Zoom link: https://us02web.zoom.us/j/84135308219