What are some publications which continue Schrödinger's "What is life?" discussion? I'm looking for modern publications on the physical nature of life in which the primary reference is to the discussion started by Schrödinger in 1944 in his book "What is life". 
For example, publications that suggest amendments (which have been generally accepted) to his ideas, similar books (not text books) that take into consideration new discoveries and offer a modern follow up, compiled works, reviews, and the like. 
 A: Schrödinger's work is known for two distinct ideas relating to the nature of living systems.
The first is what he called "order from disorder," meaning the way in which organisms can maintain a low entropy (or high free energy) state by increasing the entropy of their environment. (I.e. by eating low-entropy food and excreting high-entropy waste). Although Schrödinger often gets the credit for this idea it is actually due to Boltzmann, around 50 years earlier. The most well-known work extending this idea is Prigogine's concept of "dissipative structures". An awful lot has been written about this idea, some of it good, some of it bad, and most of it kind of obvious when you get down to it. Searching for "dissipative structures" would be a good place to start.
The second of Schrödinger's ideas is what he called "order from order", which is essentially the idea that life can store information using molecules. This idea really is due to Schrödinger as far as I know, although I'm happy to be corrected. This idea came to fruition in Francis and Crick's discovery of DNA, and so you will find that amendments and extensions to this idea take up a large proportion of any modern biochemistry text book, although they won't necessarily be expressed in the same way.
A: The article  Broken symmetry, emergent properties, dissipative structures, life: are they related? co-authored by Phil Anderson discusses the emergence of life as emergence of long-range order in non-equilibrium systems. Anderson extensively refers to Prigogine and co-workers, but I am not sure whether one can recommend a specific work: perhaps starting with Kondepudi & Prigogine's Modern Thermodynamics is a good idea. This book takes a non-traditional view of thermodynamics, treating it as a theory of non-equilibrium processes, and discussing the relevant laws (or absence/equivalents thereof out of equilibroum).
I am not aware of more recent works of the same caliber, but the activity in this area continues and one could find something by following references, e.g., in this work, which builds heavily on the theory of dynamic kinetic stability. This article has a rather lucid discussion of several key points related to our understanding of the emergence of life: e.g., emergence of self-replicating molecules could not be a result of an equilibrium first-order phase transition.
