News from the Front, XV: Nicely Entangled

This is one of my more technical posts about research activity. It is not written with wide readability in mind, but you may still get a lot out of it since the first part especially talks about about research life.

Some years ago (you’ll perhaps recall), I came up with an interesting construction that I called a “Holographic Heat Engine”. Basically, it arises as a natural concept when you work in what I call “extended” gravitational thermodynamics where you allow the spacetime cosmological constant to be dynamical. It is natural to associate the cosmological constant with a dynamical pressure (in the usual way it appears as a pressure in Einstein’s equations) and if you work it though it turns out that there’s a natural conjugate quantity playing the role of volume, etc. Black hole thermodynamics (that you get when you turn on quantum effects, giving entropy and temperature) then get enhanced to include pressure and volume, something that was not present for most of the history of the subject. It was all worked out nicely in a paper by Kastor et. al. in 2009. So…anyway, once you have black holes in that setup it seemed to me (when I encountered this extended framework in 2014) that it would be wilful neglect to not define heat engines: closed cycles in the p-V plane that take in heat, output heat, and do mechanical work. So I defined them. See three old posts of mine, here, here, and here, and there are others if you search.

Well, one of the things that has been a major quest of mine since 2014 is to see if I can make sense of the extended thermodynamics for quantum field theory, and then go further and translate the heat engines and their properties into field theory terms. This seemed possible to me all the way back then since for positive pressure, the cosmological constant is negative, and when you have gravity with negative cosmological constant you’ve got duality to strongly coupled field theories. So those heat engines must be some kind of special tour in the field theories. the efficiency of an engine must mean something about the tour. Moreover, since the efficiency of the engine is bounded by the Carnot efficiency, it seems we have a naturally defined dimensionless number that has a fundamental bound… Alarm bells ringing! – Opportunity knocking to learn something new and powerful! Maybe even important!

So I chipped away at this for some time, over years, doing various projects that improved my understanding of the engines over the years, even finding a novel application or two, but still looking for that direct relationship to field theory. Then about this time last year I realized what the key to making progress could be: quantum information theoretic quantities. Here was the main thought:

My core idea (see the thoughts at the end of my 2014 paper) was always that thermodynamic processes that change the pressure ought to have something to do with moving in the space of theories, visiting a whole series of theories connected to each other in some way. Deformation by an operator and following the evolution under renormaliztion group (RG) flow seemed to me a nice example of this, but there was a big obstruction to making this work in the examples I was thinking of: It all seemed to require detailed understanding of the flows at finite temperature on the field theory side, since it was thermodynamics I was trying to match to on the gravity side. This is very poorly explored territory (compared to zero temperature) and so there was not much to match to in making a new connection.

Sidestepping that was the key to my new approach: in the beautiful work of Casini, Huerta, and Myers from 2011, they’d shown that quantum information quantities for a conformal field theory (CFT) at zero temperature can be related to thermal properties of a specially sliced anti-de Sitter (AdS) spacetime. In particular, the entanglement entropy computation can be converted into a thermal entropy computation! They used this observation as a way of proving the Ryu-Takganayaki formula for spherical entanglement boundaries. That got me thinking. What if there is a way of taking the extended thermodynamics of some spacetimes (so, with finite temperature), and mapping it to the zero temperature physics of quantum information quantities of a field theory? The fact that the specially sliced AdS spacetime in the work of Casini et. al. is a special case of a family of black holes (ones with hyperbolic horizons as opposed to the usual spherical ones) was tantalizing to me. It seemed a straightforward guess that allowing the full dynamics of those black holes to come into play would be key and could lead to something interesting. Casini et. al would then prove to be a very special slice of a much larger story.

With that I began filling up notebooks with ideas and partial computations. Even scribblings of the opening paragraphs of a paper bringing together these diverse fields under one roof – I was that sure I was on the right track. Although, I had a long way to go still – I was looking for just the right thing to calculate that would crystallise it all.

But then things intervened. I had to focus on the final production stages of the book (The dialogues), working on many nearly-final proofs, going back and forth with a copy-editor at the publisher who (although well-meaning) clearly just did not understand the graphic novel form (and so a ton of time was wasted there), and then dealing with a million things both positive and negative about the final production stages… then the Fall came and went, seeing me taken up with so many things including serving on a big job search committee, teaching, the book launch, with all its endless headaches and hiccups, joys and pleasures, and months of doing publicity for the book (far more than I’d expected I’d need to). And all the usual teaching and other duties. And of course making sure to spend quality time with my family. And so I kept meaning to get back to that exciting project and those ideas I was sure were just so nice and sweet for tying it all together… but many things kept me from re-immersing.

But then some things changed. The first is that I got a bit less busy. The second is that a new student, Felipe Rosso, came by and said he wanted to talk to me about research. He was interested in some of the things I’d said (at one of our group meetings where we do a brief “what I’m thinking about” around the room once a semester) about connecting extended gravitational thermodynamics, heat engines and quantum field theory via quantum information quantities. But get this… he came to me saying he was interested, and while he was waiting for us meet to talk, the Winter break intervened, he (with no prompting from me!) taught himself about heat engines, did some lovely calculations about a certain aspect (finding a more general formula for the efficiency of heat engines made from static black holes), and wrote them up in a paper! So he was clearly ready to jump right in to my project on heat engines – I would not have to spend time motivating him or giving him background reading. He also began to take my quantum field theory class after the break, and so we had that reason to keep running into each other and talking about field theory… We started to meet from time to time, and I’d show him some of my older ideas, and we tried out ideas together and it turned into a fantastic collaboration… testing out ideas on each other, sharpening them through argument, making progress on the main thrust of the project, getting stuck, more progress, getting frustrated, killing the whole idea, resurrecting it, and so on and so forth, accelerating in our efforts as I got more time to focus on it (the semester eventually ended, and book promotion events became less frequent, etc).

By mid-May we began to write and refine our computations and our understanding until, last week, we put this paper out. I’m delighted with it.

In our paper, entitled “Holographic Heat Engines, Entanglement Entropy, and Renormalization Group Flow”, we do several significant things. Here’s the abstract:

We explore a fruitful connection between the physics of conformal field theories (CFTs) in d-dimensional Minkowski spacetime and the extended gravitational thermodynamics of hyperbolic black holes in (d+1)-dimensional anti–de Sitter spacetime. The CFTs are reduced on a region bounded by a sphere. We show that Renormalization Group (RG) flows between CFTs are specific thermodynamic processes in the (p,V) plane, where the irreversibility of coarse–graining flows from the UV to the IR corresponds to the Second Law of thermodynamics, preventing heat from flowing from low temperature to high. We observe that holographic heat engines using the black holes as a working substance correspond to specific combinations of CFT flows and deformations. We construct three special engines whose net heat and work can be described in terms of changes of entanglement entropy across the sphere. Engine efficiencies emerge as simple functions of the ratio of the number of degrees of freedom of two CFTs.

I’ll let you read the paper for more details, but one of the things I’m most pleased with is the sentence: “We show that Renormalization Group (RG) flows between CFTs are specific thermodynamic processes in the (p,V) plane, where the irreversibility of coarse–graining flows from the UV to the IR corresponds to the Second Law of thermodynamics, preventing heat from flowing from low temperature to high.” This is a big deal to me. Usually there’s usually an analogy drawn between properties of RG flow and the Second Law of thermodynamics… There’s a notion that the non-increase of the number of degrees of freedom when you course-grain a theory, going from the UV to the IR, is very much like the non-decreasing of entropy, and the accompanying ideas of irreversibility and so on. Well, one of the consequences of connecting the extended gravitational thermodynamics to the CFTs is that it’s no longer an analogy: irreversibility of an RG flow is the same as irreversibility of heat flow from high temps to low temps! You’ll see when you read.

One way to think about what we’ve done is that in the (p,V) plane (see figure), every single point means something in a field theory, or family of field theories. They are reduced on a ball (bounded by a sphere). The work of Casini et. al., in a single field theory, represents one point on that green line, which has an associated temperature, and entropy, and a pressure and volume in the extended thermodynamics. The entropy translates into entanglement entropy of the theory across the sphere, and the pressure counts the number of degrees of freedom in the theory. By going to the extended thermodynamics we see that it is a typical point on that whole green line. Any of them is a CFT (reduced on a sphere). From the point of view of gravity those points are massless black holes. Moving form one point to another is a thermodynamic process, which wee interpret in field theory. For example, going up the line (increasing pressure), irreversibly giving up heat (reducing volume and entropy) corresponds exactly to RG flow in field theory, where the number of degrees of freedom decreases irreversibility.

Moving off the line, you have the full family of hyperbolic black holes with non-zero mass. Such moves (we suggest) correspond to deforming the conformal field theory (still reduced on the sphere), for example by deforming the state (horizontal) or adding deformations that change the degrees of freedom (vertical). We’d love to fill out more of that dictionary with some examples.

Making heat engines requires using combinations of moves in the p,V plane. The three heat engines we consider (and there are others you can make too) are very special in that the operations they are built out of have heat and work that only depend upon quantities defined where we understand the transition most (the green line). That’s really great since we can characterise a whole class of engines by computations we can make on these. Crucially, the efficiency of the engines can be written entirely in terms of the ratio of degrees of freedom of the CFTs at points A and C. Anyway, why don’t I let you read the paper to find out the details?

Enjoy!

-cvj