# Catching up with the quantum-thermo crowd

You have four hours to tour Oxford University.

What will you visit? The Ashmolean Museum, home to da Vinci drawings, samurai armor, and Egyptian mummies? The Bodleian, one of Europe’s oldest libraries? Turf Tavern, where former president Bill Clinton reportedly “didn’t inhale” marijuana?

Felix Binder showed us a cemetery.

Of course he showed us a cemetery. We were at a thermodynamics conference.

The Fifth Quantum Thermodynamics Conference took place in the City of Dreaming Spires.Participants enthused about energy, information, engines, and the flow of time. About 160 scientists attended—roughly 60 more than attended the first conference, co-organizer Janet Anders estimated.

Weak measurements and quasiprobability distributions were trending. The news delighted me, Quantum Frontiers regulars won’t be surprised to hear.

Measurements disturb quantum systems, as early-20th-century physicist Werner Heisenberg intuited. Measure a system’s position strongly, and you forfeit your ability to predict the outcomes of future momentum measurements. Weak measurements don’t disturb the system much. In exchange, weak measurements provide little information about the system. But you can recoup information by performing a weak measurement in each of many trials, then processing the outcomes.

Strong measurements lead to probability distributions: Imagine preparing a particle in some quantum state, then measuring its position strongly, in each of many trials. From the outcomes, you can infer a probability distribution $\{ p(x) \}$, wherein $p(x)$ denotes the probability that the next trial will yield position $x$.

Weak measurements lead analogously to quasiprobability distributions. Quasiprobabilities resemble probabilities but can misbehave: Probabilities are real numbers no less than zero. Quasiprobabilities can dip below zero and can assume nonreal values.

What relevance have weak measurements and quasiprobabilities to quantum thermodynamics? Thermodynamics involves work and heat. Work is energy harnessed to perform useful tasks, like propelling a train from London to Oxford. Heat is energy that jiggles systems randomly.

Quantum properties obscure the line between work and heat. (Here’s an illustration for experts: Consider an isolated quantum, such as a spin chain. Let $H(t)$ denote the Hamiltonian that evolves with the time $t \in [0, t_f]$. Consider preparing the system in an energy eigenstate $| E_i(0) \rangle$. This state has zero diagonal entropy: Measuring the energy yields $E_i(0)$ deterministically. Considering tuning $H(t)$, as by changing a magnetic field. This change constitutes work, we learn in electrodynamics class. But if $H(t)$ changes quickly, the state can acquire weight on multiple energy eigenstates. The diagonal entropy rises. The system’s energetics have gained an unreliability characteristic of heat absorption. But the system has remained isolated from any heat bath. Work mimics heat.)

Quantum thermodynamicists have defined work in terms of a two-point measurement scheme: Initialize the quantum system, such as by letting heat flow between the system and a giant, fixed-temperature heat reservoir until the system equilibrates. Measure the system’s energy strongly, and call the outcome $E_i$. Isolate the system from the reservoir. Tune the Hamiltonian, performing the quantum equivalent of propelling the London train up a hill. Measure the energy, and call the outcome $E_f$.

Any change $\Delta E$ in a system’s energy comes from heat $Q$ and/or from work $W$, by the First Law of Thermodynamics, $\Delta E = Q + W.$  Our system hasn’t exchanged energy with any heat reservoir between the measurements. So the energy change consists of work: $E_f - E_i =: W$.

Imagine performing this protocol in each of many trials. Different trials will require different amounts $W$ of work. Upon recording the amounts, you can infer a distribution $\{ p(W) \}$. $p(W)$ denotes the probability that the next trial will require an amount $W$ of work.

Measuring the system’s energy disturbs the system, squashing some of its quantum properties. (The measurement eliminates coherences, relative to the energy eigenbasis, from the state.) Quantum properties star in quantum thermodynamics. So the two-point measurement scheme doesn’t satisfy everyone.

Enter weak measurements. They can provide information about the system’s energy without disturbing the system much. Work probability distributions $\{ p(W) \}$ give way to quasiprobability distributions $\{ \tilde{p}(W) \}$.

So propose Solinas and Gasparinetti, in these papers. Other quantum thermodynamicists apply weak measurements and quasiprobabilities differently.2 I proposed applying them to characterize chaos, and the scrambling of quantum information in many-body systems, at the conference.3 Feel free to add your favorite applications to the “comments” section.

All the quantum ladies: The conference’s female participants gathered for dinner one conference night.

Wednesday afforded an afternoon for touring. Participants congregated at the college of conference co-organizer Felix Binder.3 His tour evoked, for me, the ghosts of thermo conferences past: One conference, at the University of Cambridge, had brought me to the grave of thermodynamicist Arthur Eddington. Another conference, about entropies in information theory, had convened near Canada’s Banff Cemetery. Felix’s tour began with St. Edmund Hall’s cemetery. Thermodynamics highlights equilibrium, a state in which large-scale properties—like temperature and pressure—remain constant. Some things never change.

With thanks to Felix, Janet, and the other coordinators for organizing the conference.

1Oxford derives its nickname from an elegy by Matthew Arnold. Happy National Poetry Month!

3Michele Campisi joined me in introducing out-of-time-ordered correlators (OTOCs) into the quantum-thermo conference: He, with coauthor John Goold, combined OTOCs with the two-point measurement scheme.

3Oxford University contains 38 colleges, the epicenters of undergraduates’ social, dining, and housing experiences. Graduate students and postdoctoral scholars affiliate with colleges, and senior fellows—faculty members—govern the colleges.

# The math of multiboundary wormholes

Xi Dong, Alex Maloney, Henry Maxfield and I recently posted a paper to the arXiv with the title: Phase Transitions in 3D Gravity and Fractal Dimension. In other words, we’ll get about ten readers per year for the next few decades. Despite the heady title, there’s deep geometrical beauty underlying this work. In this post I want to outline the origin story and motivation behind this paper.

There are two different branches to the origin story. The first was my personal motivation and the second is related to how I came into contact with my collaborators (who began working on the same project but with different motivation, namely to explain a phase transition described in this paper by Belin, Keller and Zadeh.)

During the first year of my PhD at Caltech I was working in the mathematics department and I had a few brief but highly influential interactions with Nikolai Makarov while I was trying to find a PhD advisor. His previous student, Stanislav Smirnov, had recently won a Fields Medal for his work studying Schramm-Loewner evolution (SLE) and I was captivated by the beauty of these objects.

SLE example from Scott Sheffield’s webpage. SLEs are the fractal curves that form at the interface of many models undergoing phase transitions in 2D, such as the boundary between up and down spins in a 2D magnet (Ising model.)

One afternoon, I went to Professor Makarov’s office for a meeting and while he took a brief phone call I noticed a book on his shelf called Indra’s Pearls, which had a mesmerizing image on its cover. I asked Professor Makarov about it and he spent 30 minutes explaining some of the key results (which I didn’t understand at the time.) When we finished that part of our conversation Professor Makarov described this area of math as “math for the future, ahead of the tools we have right now” and he offered for me to borrow his copy. With a description like that I was hooked. I spent the next six months devouring this book which provided a small toehold as I tried to grok the relevant mathematics literature. This year or so of being obsessed with Kleinian groups (the underlying objects in Indra’s Pearls) comes back into the story soon. I also want to mention that during that meeting with Professor Makarov I was exposed to two other ideas that have driven my research as I moved from mathematics to physics: quasiconformal mappings and the simultaneous uniformization theorem, both of which will play heavy roles in the next paper I release.  In other words, it was a pretty important 90 minutes of my life.

Google image search for “Indra’s Pearls”. The math underlying Indra’s Pearls sits at the intersection of hyperbolic geometry, complex analysis and dynamical systems. Mathematicians oftentimes call this field the study of “Kleinian groups”. Most of these figures were obtained by starting with a small number of Mobius transformations (usually two or three) and then finding the fixed points for all possible combinations of the initial transformations and their inverses. Indra’s Pearls was written by David Mumford, Caroline Series and David Wright. I couldn’t recommend it more highly.

My life path then hit a discontinuity when I was recruited to work on a DARPA project, which led to taking an 18 month leave of absence from Caltech. It’s an understatement to say that being deployed in Afghanistan led to extreme introspection. While “down range” I had moments of clarity where I knew life was too short to work on anything other than ones’ deepest passions. Before math, the thing that got me into science was a childhood obsession with space and black holes. I knew that when I returned to Caltech I wanted to work on quantum gravity with John Preskill. I sent him an e-mail from Afghanistan and luckily he was willing to take me on as a student. But as a student in the mathematics department, I knew it would be tricky to find a project that involved all of: black holes (my interest), quantum information (John’s primary interest at the time) and mathematics (so I could get the degree.)

I returned to Caltech in May of 2012 which was only two months before the Firewall Paradox was introduced by Almheiri, Marolf, Polchinski and Sully. It was obvious that this was where most of the action would be for the next few years so I spent a great deal of time (years) trying to get sharp enough in the underlying concepts to be able to make comments of my own on the matter. Black holes are probably the closest things we have in Nature to the proverbial bottomless pit, which is an apt metaphor for thinking about the Firewall Paradox. After two years I was stuck. I still wasn’t close to confident enough with AdS/CFT to understand a majority of the promising developments. And then at exactly the right moment, in the summer of 2014, Preskill tipped my hat to a paper titled Multiboundary Wormholes and Holographic Entanglement by Balasubramanian, Hayden, Maloney, Marolf and Ross. It was immediately obvious to me that the tools of Indra’s Pearls (Kleinian groups) provided exactly the right language to study these “multiboundary wormholes.” But despite knowing a bridge could be built between these fields, I still didn’t have the requisite physics mastery (AdS/CFT) to build it confidently.

Before mentioning how I met my collaborators and describing the work we did together, let me first describe the worlds that we bridged together.

3D Gravity and Universality

As the media has sensationalized to death, one of the most outstanding questions in modern physics is to discover and then understand a theory of quantum gravity.  As a quick aside, Quantum gravity is just a placeholder name for such a theory. I used italics because physicists have already discovered candidate theories, such as string theory and loop quantum gravity (I’m not trying to get into politics, just trying to demonstrate that there are multiple candidate theories). But understanding these theories — carrying out all of the relevant computations to confirm that they are consistent with Nature and then doing experiments to verify their novel predictions — is still beyond our ability. Surprisingly, without knowing the specific theory of quantum gravity that guides Nature’s hand, we’re still able to say a number of universal things that must be true for any theory of quantum gravity. The most prominent example being the holographic principle which comes from the entropy of black holes being proportional to the surface area encapsulated by the black hole’s horizon (a naive guess says the entropy should be proportional to the volume of the black hole; such as the entropy of a glass of water.) Universal statements such as this serve as guideposts and consistency checks as we try to understand quantum gravity.

It’s exceedingly rare to find universal statements that are true in physically realistic models of quantum gravity. The holographic principle is one such example but it pretty much stands alone in its power and applicability. By physically realistic I mean: 3+1-dimensional and with the curvature of the universe being either flat or very mildly positively curved.  However, we can make additional simplifying assumptions where it’s easier to find universal properties. For example, we can reduce the number of spatial dimensions so that we’re considering 2+1-dimensional quantum gravity (3D gravity). Or we can investigate spacetimes that are negatively curved (anti-de Sitter space) as in the AdS/CFT correspondence. Or we can do BOTH! As in the paper that we just posted. The hope is that what’s learned in these limited situations will back-propagate insights towards reality.

The motivation for going to 2+1-dimensions is that gravity (general relativity) is much simpler here. This is explained eloquently in section II of Steve Carlip’s notes here. In 2+1-dimensions, there are no “local”/”gauge” degrees of freedom. This makes thinking about quantum aspects of these spacetimes much simpler.

The standard motivation for considering negatively curved spacetimes is that it puts us in the domain of AdS/CFT, which is the best understood model of quantum gravity. However, it’s worth pointing out that our results don’t rely on AdS/CFT. We consider negatively curved spacetimes (negatively curved Lorentzian manifolds) because they’re related to what mathematicians call hyperbolic manifolds (negatively curved Euclidean manifolds), and mathematicians know a great deal about these objects. It’s just a helpful coincidence that because we’re working with negatively curved manifolds we then get to unpack our statements in AdS/CFT.

Multiboundary wormholes

Finding solutions to Einstein’s equations of general relativity is a notoriously hard problem. Some of the more famous examples include: Minkowski space, de-Sitter space, anti-de Sitter space and Schwarzschild’s solution (which describes perfectly symmetrical and static black holes.) However, there’s a trick! Einstein’s equations only depend on the local curvature of spacetime while being insensitive to global topology (the number of boundaries and holes and such.) If M is a solution of Einstein’s equations and $\Gamma$ is a discrete subgroup of the isometry group of $M$, then the quotient space $M/\Gamma$ will also be a spacetime that solves Einstein’s equations! Here’s an example for intuition. Start with 2+1-dimensional Minkowski space, which is just a stack of flat planes indexed by time. One example of a “discrete subgroup of the isometry group” is the cyclic group generated by a single translation, say the translation along the x-axis by ten meters. Minkowski space quotiented by this group will also be a solution of Einstein’s equations, given as a stack of 10m diameter cylinders indexed by time.

Start with 2+1-dimensional Minkowski space which is just a stack of flat planes index by time. Think of the planes on the left hand side as being infinite. To “quotient” by a translation means to glue the green lines together which leaves a cylinder for every time slice. The figure on the right shows this cylinder universe which is also a solution to Einstein’s equations.

D+1-dimensional Anti-de Sitter space ($AdS_{d+1}$) is the maximally symmetric d+1-dimensional Lorentzian manifold with negative curvature. Our paper is about 3D gravity in negatively curved spacetimes so our starting point is $AdS_3$ which can be thought of as a stack of Poincare disks (or hyperbolic sheets) with the time dimension telling you which disk (sheet) you’re on. The isometry group of $AdS_3$ is a group called $SO(2,2)$ which in turn is isomorphic to the group $SL(2, R) \times SL(2, R)$. The group $SL(2,R) \times SL(2,R)$ isn’t a very common group but a single copy of $SL(2,R)$ is a very well-studied group. Discrete subgroups of it are called Fuchsian groups. Every element in the group should be thought of as a 2×2 matrix which corresponds to a Mobius transformation of the complex plane. The quotients that we obtain from these Fuchsian groups, or the larger isometry group yield a rich infinite family of new spacetimes, which are called multiboundary wormholes. Multiboundary wormholes have risen in importance over the last few years as powerful toy models when trying to understand how entanglement is dispersed near black holes (Ryu-Takayanagi conjecture) and for how the holographic dictionary works in terms of mapping operators in the boundary CFT to fields in the bulk (entanglement wedge reconstruction.)

Three dimensional AdS can be thought of as a stack of hyperboloids indexed by time. It’s convenient to use the Poincare disk model for the hyperboloids so that the entire spacetime can be pictured in a compact way. Despite how things appear, all of the triangles have the same “area”.

I now want to work through a few examples.

BTZ black hole: this is the simplest possible example. It’s obtained by quotienting $AdS_3$ by a cyclic group $\langle A \rangle$, generated by a single matrix $A \in SL(2,R)$ which for example could take the form $A = \begin{pmatrix} e^{\lambda} & 0 \\ 0 & e^{-\lambda} \end{pmatrix}$. The matrix A acts by fractional linear transformation on the complex plane, so in this case the point $z \in \mathbb{C}$ gets mapped to $z\mapsto (e^{\lambda}z + 0)/(0z + e^{-\lambda}) = e^{2\lambda} z$. In this case

Start with $AdS_3$ as a stack of hyperbolic half planes indexed by time. A quotient by A means that each hyperbolic half plane gets quotiented. Quotienting a constant time slice by the map $z \mapsto e^{2\lambda}z$ gives a surface that’s topologically a cylinder. Using the picture above this means you glue together the solid black curves. The green and red segments become two boundary regions. We call it the BTZ black hole because when you add “time” it becomes impossible to send a signal from the green boundary to the red boundary, or vica versa. The dotted line acts as an event horizon.

Three boundary wormhole:

There are many parameterizations that we can choose to obtain the three boundary wormhole. I’ll only show schematically how the gluings go. A nice reference with the details is this paper by Henry Maxfield.

This is a picture of a constant time slice of $AdS_3$ quotiented by the A and B above. Each time slice is given as a copy of the hyperbolic half plane with the black arcs and green arcs glued together (by the maps A and B). These gluings yield a pair of pants surface. Each of the boundary regions are causally disconnected from the others. The dotted lines are black hole horizons that illustrate where the causal disconnection happens.

Torus wormhole:

It’s simpler to write down generators for the torus wormhole; but following along with the gluings is more complicated. To obtain the three boundary wormhole we quotient $AdS_3$ by the free group $\langle A, B \rangle$ where $A = \begin{pmatrix} e^{\lambda} & 0 \\ 0 & e^{-\lambda} \end{pmatrix}$ and $B = \begin{pmatrix} \cosh \lambda & \sinh \lambda \\ \sinh \lambda & \cosh \lambda \end{pmatrix}$. (Note that this is only one choice of generators, and a highly symmetrical one at that.)

This is a picture of a constant time slice of $AdS_3$ quotiented by the A and B above. Each time slice is given as a copy of the hyperbolic half plane with the black arcs and green arcs glued together (by the maps A and B). These gluings yield what’s called the “torus wormhole”. Topologically it’s just a donut with a hole cut out. However, there’s a causal structure when you add time to the mix where the dotted lines act as a black hole horizon, so that a message sent from behind the horizon will never reach the boundary.

Lorentzian to Euclidean spacetimes

So far we have been talking about negatively curved Lorentzian manifolds. These are manifolds that have a notion of both “time” and “space.” The technical definition involves differential geometry and it is related to the signature of the metric. On the other hand, mathematicians know a great deal about negatively curved Euclidean manifolds. Euclidean manifolds only have a notion of “space” (so no time-like directions.) Given a multiboundary wormhole, which by definition, is a quotient of $AdS_3/\Gamma$ where $\Gamma$ is a discrete subgroup of Isom($AdS_3$), there’s a procedure to analytically continue this to a Euclidean hyperbolic manifold of the form $H^3/ \Gamma_E$ where $H^3$ is three dimensional hyperbolic space and $\Gamma_E$ is a discrete subgroup of the isometry group of $H^3$, which is $PSL(2, \mathbb{C})$. This analytic continuation procedure is well understood for time-symmetric spacetimes but it’s subtle for spacetimes that don’t have time-reversal symmetry. A discussion of this subtlety will be the topic of my next paper. To keep this blog post at a reasonable level of technical detail I’m going to need you to take it on a leap of faith that to every Lorentzian 3-manifold multiboundary wormhole there’s an associated Euclidean hyperbolic 3-manifold. Basically you need to believe that given a discrete subgroup $\Gamma$ of $SL(2, R) \times SL(2, R)$ there’s a procedure to obtain a discrete subgroup $\Gamma_E$ of $PSL(2, \mathbb{C})$. Discrete subgroups of $PSL(2, \mathbb{C})$ are called Kleinian groups and quotients of $H^3$ by groups of this form yield hyperbolic 3-manifolds. These Euclidean manifolds obtained by analytic continuation arise when studying the thermodynamics of these spacetimes or also when studying correlation functions; there’s a sense in which they’re physical.

TLDR: you start with a 2+1-d Lorentzian 3-manifold obtained as a quotient $AdS_3/\Gamma$ and analytic continuation gives a Euclidean 3-manifold obtained as a quotient $H^3/\Gamma_E$ where $H^3$ is 3-dimensional hyperbolic space and $\Gamma_E$ is a discrete subgroup of $PSL(2,\mathbb{C})$ (Kleinian group.)

Limit sets:

Every Kleinian group $\Gamma_E = \langle A_1, \dots, A_g \rangle \subset PSL(2, \mathbb{C})$ has a fractal that’s naturally associated with it. The fractal is obtained by finding the fixed points of every possible combination of generators and their inverses. Moreover, there’s a beautiful theorem of Patterson, Sullivan, Bishop and Jones that says the smallest eigenvalue $\lambda_0$ of the spectrum of the Laplacian on the quotient Euclidean spacetime $H^3 / \Gamma_E$ is related to the Hausdorff dimension of this fractal (call it $D$) by the formula $\lambda_0 = D(2-D)$. This smallest eigenvalue controls a number of the quantities of interest for this spacetime but calculating it directly is usually intractable. However, McMullen proposed an algorithm to calculate the Hausdorff dimension of the relevant fractals so we can get at the spectrum efficiently, albeit indirectly.

This is a screen grab of Figure 2 from our paper. These are two examples of fractals that emerge when studying these spacetimes. Both of these particular fractals have a 3-fold symmetry. They have this symmetry because these particular spacetimes came from looking at something called “n=3 Renyi entropies”. The number q indexes a one complex dimensional family of spacetimes that have this 3-fold symmetry. These Kleinian groups each have two generators that are described in section 2.3 of our paper.

What we did

Our primary result is a generalization of the Hawking-Page phase transition for multiboundary wormholes. To understand the thermodynamics (from a 3d quantum gravity perspective) one starts with a fixed boundary Riemann surface and then looks at the contributions to the partition function from each of the ways to fill in the boundary (each of which is a hyperbolic 3-manifold). We showed that the expected dominant contributions, which are given by handlebodies, are unstable when the kinetic operator $(\nabla^2 - m^2)$ is negative, which happens whenever the Hausdorff dimension of the limit set of $\Gamma_E$ is greater than the lightest scalar field living in the bulk. One has to go pretty far down the quantum gravity rabbit hole (black hole) to understand why this is an interesting research direction to pursue, but at least anyone can appreciate the pretty pictures!

# Rock-paper-scissors, granite-clock-idea

I have a soft spot for lamassu. Ten-foot-tall statues of these winged bull-men guarded the entrances to ancient Assyrian palaces. Show me lamassu, or apkallu—human-shaped winged deities—or other reliefs from the Neo-Assyrian capital of Nineveh, and you’ll have trouble showing me the door.

Assyrian art fills a gallery in London’s British Museum. Lamassu flank the gallery’s entrance. Carvings fill the interior: depictions of soldiers attacking, captives trudging, and kings hunting lions. The artwork’s vastness, its endurance, and the contact with a three-thousand-year-old civilization floor me. I tore myself away as the museum closed one Sunday night.

I visited the British Museum the night before visiting Jonathan Oppenheim’s research group at University College London (UCL). Jonathan combines quantum information theory with thermodynamics. He and others co-invented thermodynamic resource theories (TRTs), which Quantum Frontiers regulars will know of. TRTs are quantum-information-theoretic models for systems that exchange energy with their environments.

Energy is conjugate to time: Hamiltonians, mathematical objects that represent energy, represent also translations through time. We measure time with clocks. Little wonder that one can study quantum clocks using a model for energy exchanges.

Mischa Woods, Ralph Silva, and Jonathan used a resource theory to design an autonomous quantum clock. “Autonomous” means that the clock contains all the parts it needs to operate, needs no periodic winding-up, etc. When might we want an autonomous clock? When building quantum devices that operate independently of classical engineers. Or when performing a quantum computation: Computers must perform logical gates at specific times.

Wolfgang Pauli and others studied quantum clocks, the authors recall. How, Pauli asked, would an ideal clock look? Its Hamiltonian, $\hat{H}_{\rm C}$, would have eigenstates $| E \rangle$. The labels $E$ denote possible amounts of energy.

The Hamiltonian would be conjugate to a “time operator” $\hat{t}$. Let $| \theta \rangle$ denote an eigenstate of $\hat{t}$. This “time state” would equal an even superposition over the $| E \rangle$’s. The clock would occupy the state $| \theta \rangle$ at time $t_\theta$.

Imagine measuring the clock, to learn the time, or controlling another system with the clock. The interaction would disturb the clock, changing the clock’s state. The disturbance wouldn’t mar the clock’s timekeeping, if the clock were ideal. What would enable an ideal clock to withstand the disturbances? The ability to have any amount of energy: $E$ must stretch from $- \infty$ to $\infty$. Such clocks can’t exist.

Approximations to them can. Mischa, Ralph, and Jonathan designed a finite-size clock, then characterized how accurately the clock mimics the ideal. (Experts: The clock corresponds to a Hilbert space of finite dimensionality $d$. The clock begins in a Gaussian state that peaks at one time state $| \theta \rangle$. The finite-width Gaussian offers more stability than a clock state.)

Disturbances degrade our ability to distinguish instants by measuring the clock. Imagine gazing at a kitchen clock through blurry lenses: You couldn’t distinguish 6:00 from 5:59 or 6:01. Disturbances also hinder the clock’s ability to implement processes, such as gates in a computation, at desired instants.

Mischa & co. quantified these degradations. The errors made by the clock, they found, decay inverse-exponentially with the clock’s size: Grow the clock a little, and the errors shrink a lot.

Time has degraded the lamassu, but only a little. You can distinguish feathers in their wings and strands in their beards. People portray such artifacts as having “withstood the flow of time,” or “evaded,” or “resisted.” Such portrayals have never appealed to me. I prefer to think of the lamassu as surviving not because they clash with time, but because they harmonize with it. The prospect of harmonizing with time—whatever that means—has enticed me throughout my life. The prospect partially underlies my research into time—perhaps childishly, foolishly—I recognize if I remove my blurry lenses before gazing in the mirror.

The creation of lasting works, like lamassu, has enticed me throughout my life. I’ve scrapbooked, archived, and recorded, and tended memories as though they were Great-Grandma’s cookbook. Ancient civilizations began alluring me at age six, partially due to artifacts’ longevity. No wonder I study the second law of thermodynamics.

Yet doing theoretical physics makes no sense from another perspective. The ancient Egyptians sculpted granite, when they could afford it. Gudea, king of the ancient city-state of Lagash, immortalized himself in diorite. I fashion ideas, which lack substance. Imagine playing, rather than rock-paper-scissors, granite-diorite-idea. The idea wouldn’t stand a chance.

Would it? Because an idea lacks substance, it can manifest in many forms. Plato’s cave allegory has manifested as a story, as classroom lectures, on handwritten pages, on word processors and websites, in cartloads of novels, in the film The Matrix, in one of the four most memorable advertisements I received from colleges as a high-school junior, and elsewhere. Plato’s allegory has survived since about the fourth century BCE. King Ashurbanipal’s lion-hunt reliefs have survived for only about 200 years longer.

The lion-hunt reliefs—and lamassu—exude a grandness, a majesty that’s attracted me as their longevity has. The nature of time and the perfect clock have as much grandness. Leaving the British Museum’s Assyrian gallery at 6 PM one Sunday, I couldn’t have asked for a more fitting location, 24 hours later, than in a theoretical-physics conversation.

With thanks to Jonathan, to Álvaro Martín-Alhambra, and to Mischa for their hospitality at UCL; to Ada Cohen for the “Art history of ancient Egypt and the ancient Near East” course for which I’d been hankering for years; to my brother, for transmitting the ancient-civilizations bug; and to my parents, who fed the infection with museum visits.

Click here for a follow-up to the quantum-clock paper.

# A poem for Stephen Hawking

Everyone is talking
My good friend
Explained how time can end.
And clued us in
On how time can begin.

Always droll,
“Now, wait a minute, Jack,
A black hole ain’t so black!”

Those immortal words he said,
Which millions now have duly read,
Hit physics like a ton of bricks.
Well, that’s how Stephen got his kicks.

Always grinning through his glasses,
He brought science to the masses,
Displayed a rare capacity
For humor and audacity.

And that’s why, on this somber day,
With relish we can gladly say:
“Thanks, Stephen, for the things you’ve done.
And most of all, thanks for the fun!”

And though there’s more to say, my friend,
This poem, too, must, sadly, end.

# Techs in flux & Rock & Roll

Each year, 10000 physicists descend on one of America’s finest inner cities in a ritual known as the American Physical Society’s March Meeting. If you are thinking that this is going to be one big nerd fest, you’re about right. From my experience, the backpacks, poster tubes, non-brand clothing, and distracted looks will be very easy to distinguish among the inhabitants of downtown LA (this year’s location) come next week.

However, with that many physicists, you will find a few trying to make science cool, or at least having fun while they try. One relatively untapped market in my opinion is montages. Take the Imagine Dragons song Believer, whose music video has lead signer Dan Reynolds mostly getting his ass kicked by veteran brawler Dolph Lundgren. Who says that training montages can’t also be for mental training? Sub out Dan for a young graduate student, replace Dolph with an imposing physicist, and substitute boxing with drama about writing equations on paper or a blackboard. Don’t believe it can be cool? I don’t blame you, but science montages have been done before, playing to science’s mystical side. And with sufficient experience, creativity, and money, I believe the sky is the limit.

But back to more concrete things. Having fun while trying to promote science is the main goal of the March Meeting Rock ‘n Roll Physics Sing-Along — a social and outreach event where a band of musicians, mostly scientists attending the meeting, plays well-known songs whose lyrics are substituted for science-themed prose. The audience then sings the new technically oriented lyrics along with the performers. Below is an example with the Smashmouth song I’m a Believer, but we play all kinds of genres, from power ballads to Britney Spears.

This year, we have an especially exciting line-up as we are joined by professional science entertainer, Einstein’s girl Gia Mora! Some of you may remember Gia from her performance with John Preskill at One Entangled Evening. She will join us to perform, among other hits, the funky E=mc^2:

The sing-along is run by the curator of all things related to physics songs, singer and songwriter Prof. Walter F. Smith of Haverford College. Adept at using songs to help teach physics, Walter has carefully collected a database of such songs dating back to the early 20th century; he believes that James Clerk Maxwell may have been the first song parody-er with his version of the lyrics to the Scotch Air Comin’ Thro’ the Rye. You can see James jamming alongside Emmy Noether, Paul Dirac, and Satyendra Bose below to questionable lyrics. The most well-known US physics song pioneer is Harvard grad Tom Lehrer, who recorded his first album in the 50s. Contrary to the general nature of scientists to be constantly worried about preserving their neutral academic self-image, Lehrer tackled edgy topics with creativity and humor.

The sing-along started in 2006, where the only accompaniment was a guitar and bongo, growing into a full rock band later on. The drums were first played by a Soviet-born physicist named Victor, and that has yet to change today despite it being a different person. The rest of the band this year consists of Walter, his wife Marian McKenzie on the flute, Lev Krayzman from Yale on the guitar, Prof. Esa Räsänen from Tampere University of Technology on the bass, Lenny Campanello from the University of Maryland on the keyboard, and of course the talented Gia Mora on voice. We hope that you can join us next week, as this year’s sing-along is sure to be one for the books!

March Meeting Rock-n-Roll Physics Sing-along
Wednesday, March 7, 2018
9:00 PM–10:30 PM
J.W. Marriott Room: Platinum D

See you there!

# The Ground Space of Babel

Librarians are committing suicide.

So relates the narrator of the short story “The Library of Babel.” The Argentine magical realist Jorge Luis Borges wrote the story in 1941.

Librarians are committing suicide partially because they can’t find the books they seek. The librarians are born in, and curate, a library called “infinite” by the narrator. The library consists of hexagonal cells, of staircases, of air shafts, and of closets for answering nature’s call. The narrator has never heard of anyone’s finding an edge of the library. Each hexagon houses 20 shelves, each of which houses 32 books, each of which contains 410 pages, each of which contains 40 lines, each of which consists of about 80 symbols. Every symbol comes from a set of 25: 22 letters, the period, the comma, and the space.

The library, a sage posited, contains every combination of the 25 symbols that satisfy the 410-40-and-80-ish requirement. His compatriots rejoiced:

All men felt themselves to be the masters of an intact and secret treasure. There was no personal or world problem whose eloquent solution did not exist in some hexagon. [ . . . ] a great deal was said about the Vindications: books of apology and prophecy which vindicated for all time the acts of every man in the universe and retained prodigious arcana for his future. Thousands of the greedy abandoned their sweet native hexagons and rushed up the stairways, urged on by the vain intention of finding their Vindication.

Probability punctured their joy: “the possibility of a man’s finding his Vindication, or some treacherous variation thereof, can be computed as zero.”

Many-body quantum physicists can empathize with Borges’s librarian.

A handful of us will huddle over a table or cluster in front of a chalkboard.

“Has anyone found this Hamiltonian’s ground space?” someone will ask.1

A Hamiltonian is an observable, a measurable property. Consider a quantum system S, such as a set of particles hopping between atoms. We denote the system’s Hamiltonian by H. H determines how the system’s state changes in time. A musical about H swept Broadway last year.

A quantum system’s energy, E, might assume any of many possible values. H encodes the possible values. The least possible value, E0, we call the ground-state energy.

Under what condition does S have an amount E0 of energy? S must occupy a ground state. Consider Olympic snowboarder Shaun White in a half-pipe. He has kinetic energy, or energy of motion, when sliding along the pipe. He gains gravitational energy upon leaving the ground. He has little energy when sitting still on the snow. A quantum analog of that sitting constitutes a ground state.2

Consider, for example, electrons in a magnetic field. Each electron has a property called spin, illustrated with an arrow. The arrow’s direction represents the spin’s state. The system occupies a ground state when every arrow points in the same direction as the magnetic field.

Shaun White has as much energy, sitting on the ground in the half-pipe’s center, as he has sitting at the bottom of an edge of the half-pipe. Similarly, a quantum system might have multiple ground states. These states form the ground space.

“Has anyone found this Hamiltonian’s ground space?”

“Find” means, here,“identify the form of.” We want to derive a mathematical expression for the quantum analog of “sitting still, at the bottom of the half-pipe.”

“Find” often means “locate.” How do we locate an object such as a library? By identifying its spatial coordinates. We specify coordinates relative to directions, such as north, east, and up. We specify coordinates also when “finding” ground states.

Libraries occupy the physical space we live in. Ground states occupy an abstract mathematical space, a Hilbert space. The Hilbert space consists of the (pure) quantum states accessible to the system—loosely speaking, how the spins can orient themselves.

Libraries occupy a three-dimensional space. An N-spin system corresponds to a 2N-dimensional Hilbert space. Finding a ground state amounts to identifying 2N coordinates. The problem’s size grows exponentially with the number of particles.

An exponential quantifies also the size of the librarian’s problem. Imagine trying to locate some book in the Library of Babel. How many books should you expect to have to check? How many books does the library hold? Would you have more hope of finding the book, wandering the Library of Babel, or finding a ground state, wandering the Hilbert space? (Please take this question with a grain of whimsy, not as instructions for calculating ground states.)

A book’s first symbol has one of 25 possible values. So does the second symbol. The pair of symbols has one of $25 \times 25 = 25^2$ possible values. A trio has one of $25^3$ possible values, and so on.

How many symbols does a book contain? About $\frac{ 410 \text{ pages} }{ 1 \text{ book} } \: \frac{ 40 \text{ lines} }{ 1 \text{ page} } \: \frac{ 80 \text{ characters} }{ 1 \text{ line} } \approx 10^6 \, ,$ or a million. The number of books grows exponentially with the number of symbols per book: The library contains about $25^{ 10^6 }$ books. You contain only about $10^{24}$ atoms. No wonder librarians are committing suicide.

Do quantum physicists deserve more hope? Physicists want to find ground states of chemical systems. Example systems are discussed here and here. The second paper refers to 65 electrons distributed across 57 orbitals (spatial regions). How large a Hilbert space does this system have? Each electron has a spin that, loosely speaking, can point upward or downward (that corresponds to a two-dimensional Hilbert space). One might expect each electron to correspond to a Hilbert space of dimensionality $(57 \text{ orbitals}) \frac{ 2 \text{ spin states} }{ 1 \text{ orbital} } = 114$. The 65 electrons would correspond to a Hilbert space $\mathcal{H}_{\rm tot}$ of dimensionality $114^{65}$.

But no two electrons can occupy the same one-electron state, due to Pauli’s exclusion principle. Hence $\mathcal{H}_{\rm tot}$ has dimensionality ${114 \choose 65}$ (“114 choose 65″), the number of ways in which you can select 65 states from a set of 114 states.

${114 \choose 65}$ equals approximately $10^{34}$. Mathematica (a fancy calculator) can print a one followed by 34 zeroes. Mathematica refuses to print the number $25^{ 10^6 }$ of Babel’s books. Pity the librarians more than the physicists.

Pity us less when we have quantum computers (QCs). They could find ground states far more quickly than today’s supercomputers. But building QCs is taking about as long as Borges’s narrator wandered the library, searching for “the catalogue of catalogues.”

What would Borges and his librarians make of QCs? QCs will be able to search unstructured databases quickly, via Grover’s algorithm. Babel’s library lacks structure. Grover’s algorithm outperforms classical algorithms just when fed large databases. $25^{ 10^6 }$ books constitute a large database. Researchers seek a “killer app” for QCs. Maybe Babel’s librarians could vindicate quantum computing and quantum computing could rescue the librarians. If taken with a grain of magical realism.

1Such questions remind me of an Uncle Alfred who’s misplaced his glasses. I half-expect an Auntie Muriel to march up to us physicists. She, sensible in plaid, will cross her arms.

“Where did you last see your ground space?” she’ll ask. “Did you put it on your dresser before going to bed last night? Did you use it at breakfast, to read the newspaper?”

We’ll bow our heads and shuffle off to double-check the kitchen.

2More accurately, a ground state parallels Shaun White’s lying on the ground, stone-cold.

# Fractons, for real?

“Fractons” is my favorite new toy (short for quantum many-body toy models). It has amazing functions that my old toys do not have; it is so new that there are tons of questions waiting to be addressed; it is perfectly situated at the interface between quantum information and condensed matter and has attracted a lot of interest and efforts from both sides; and it gives me excuses and new incentives to learn more math. I have been having a lot of fun playing with it in the last couple years and in the process, I had the great opportunity to work with some amazing collaborators: Han Ma and Mike Hermele at Boulder, Ethan Lake at MIT, Wilbur Shirley at Caltech, Kevin Slagle at U Toronto and Zhenghan Wang at Station Q. Together we have written a few papers on this subject, but I always felt there are more interesting stories and more excitement in me than what can be properly contained in scientific papers. Hence this blog post.

How I first learned about Fractons

Back in the early 2000s, a question that kept attracting and frustrating people in quantum information is how to build a quantum hard drive to store quantum information. This is of course a natural question to ask as quantum computing has been demonstrated to be possible, at least in theory, and experimental progress has shown great potential. It turned out, however, that the question is one of those deceptively enticing ones which are super easy to state, but extremely hard to answer. In a classical hard drive, information is stored using magnetism. Quantum information, instead of being just 0 and 1, is represented using superpositions of 0 and 1, and can be probed in non-commutative ways (that is, measuring along different directions can alter previous answers). To store quantum information, we need “magnetism” in all such non-commutative channels. But how can we do that?

At that time, some proposals had been made, but they either involve actively looking for and correcting errors throughout the time during which information is stored (which is something we never have to do with our classical hard drives) or going into four spatial dimensions. Reliable passive storage of quantum information seemed out of reach in the three-dimensional world we live in, even at the level of a proof of principle toy model!

Given all the previously failed attempts and without a clue about where else to look, this problem probably looked like a dead-end to many. But not to Jeongwan Haah, a fearless graduate student in Preskill’s group at IQIM at that time, who turned the problem from guesswork into a systematic computer search (over a constrained set of models). The result of the search surprised everyone. Jeongwan found a three-dimensional quantum spin model with physical properties that had never been seen before, making it a better quantum hard drive than any other model that we know of!

The model looks surprising not only to the quantum information community, but even more so to the condensed matter community. It is a strongly interacting quantum many-body model, a subject that has been under intense study in condensed matter physics. Yet it exhibits some very strange behaviors whose existence had not even been suspected. It is a condensed matter discovery made not from real materials in real experiments, but through computer search!

Excitations (stars) in Haah’s code live at the corner of a fractal.

In condensed matter systems, what we know can happen is that elementary excitations can come in the form of point particles – usually called quasi-particles – which can then move around and interact with other excitations. In Jeongwan’s model, now commonly referred to as Haah’s code, elementary excitations still come in the form of point particles, but they cannot freely move around. Instead, if they want to move, four of them have to coordinate with each other to move together, so that they stay at the vertices of a fractal shaped structure! The restricted motion of the quasi-particles leads to slower dynamics at low energy, making the model much better suited for the purpose of storing quantum information.

But how can something like this happen? This is the question that I want to yell out loud every time I read Jeongwan’s papers or listen to his talks. Leaving aside the motivation of building a quantum hard drive, this model presents a grand challenge to the theoretical framework we now have in condensed matter. All of our intuitions break down in predicting the behavior of this model; even some of the most basic assumptions and definitions do not apply.

The interactions in Haah’s code involve eight spins at a time (the eight Z’s and eight X’s in each cube).

I felt so uncomfortable and so excited at the same time because there was something out there that should be related to things I know, yet I totally did not understand how. And there was an even bigger problem. I was like a sick person going to a doctor but unable to pinpoint what was wrong. Something must have been wrong, but I didn’t know what that was and I didn’t know how to even begin to look for it. The model looked so weird. Interaction involved eight spins at a time; there was no obvious symmetry other than translation. Jeongwan, with his magic math power, worked out explicitly many of the amazing properties of the model, but that to me only added to the mystery. Where did all these strange properties coming from?

From the unfathomable to the seemingly approachable

I remained in this superposition of excited state and powerless state for several years, until Jeongwan moved to MIT and posted some papers with Sagar Vijay and Liang Fu in 2015 and 2016.

Interaction terms in a nicer looking fracton model.

In these papers, they listed several other models, which, similar to Haah’s code, contain quasi-particle excitations whose motion is constrained. The constraints are weaker and these models do not make good quantum hard drives, but they still represent new condensed matter phenomena. What’s nice about these models is that the form of interaction is more symmetric, takes a simpler form, or is similar to some other models we are familiar with. The quasi-particles do not need a fractal-shaped structure to move around, instead they move along a line, in a plane, or at the corner of a rectangle. In fact, as early as 2005 – six years before Haah’s code, Claudio Chamon at Boston University already proposed a model of this kind. Together with the previous fractal examples, these models are what’s now being referred to as the fracton models. If the original Haah’s code looks like an ET from beyond the milky way, these models at least seem to live somewhere in the solar system. So there must be something that we can do to understand them better!

Obviously, I was not the only one who felt this way. A flurry of papers appeared on these “fracton” models. People came at these models armed with their favorite tools in condensed matter, looking for an entry point to crack them open. The two approaches that I found most attractive was the coupled layer construction and the higher rank gauge theory, and I worked on these ideas together with Han Ma, Ethan Lake and Michael Hermele. Each approach comes from a different perspective and establishes a connection between fractons and physical models that we are familiar with. In the coupled layer construction, the connection is to the 2D discrete gauge theories, while in the higher rank approach it is to the 3D gauge theory of electromagnetism.

I was excited about these results. They each point to simple physical mechanisms underlying the existence of fractons in some particular models. By relating these models to things I already know, I feel a bit relieved. But deep down, I know that this is far from the complete story. Our understanding barely goes beyond the particular models discussed in the paper. In condensed matter, we spend a lot of time studying toy models; but toy models are not the end goal. Toy models are only meaningful if they represent some generic feature in a whole class of models. It is not clear at all to what extent this is the case for fractons.

Step zero: define “order”, define “topological order”

I gave a talk about these results at KITP last fall under the title “Fracton Topological Order”. It was actually too big a title because all we did was to study specific realizations of individual models and analyze their properties. To claim topological order, one needs to show much more. The word “order” refers to the common properties of a continuous range of models within the same phase. For example, crystalline order refers to the regular lattice organization of atoms in the solid phase within a continuous range of temperature and pressure. When the word “topological” is added in front of “order”, it signifies that such properties are usually directly related to the topology of the system. A prototypical example is the fractional quantum Hall system, whose ground state degeneracy is directly determined by the topology of the manifold the system lives in. For fractons, we are far from an understanding at this level. We cannot answer basic questions like what range of models form a phase, what is the order (the common properties of this whole range of models) characterizing each phase, and in what sense is the order topological. So, the title was more about what I hope will happen than what has already happened.

But it did lead to discussions that could make things happen. After my talk, Zhenghan Wang, a mathematician at Microsoft Station Q, said to me, “I would agree these fracton models are topological if you can show me how to define them on different three manifolds”. Of course! How can I claim anything related to topology if all I know is one model on a cubic lattice with periodic boundary condition? It is like claiming a linear relation between two quantities with only one data point.

But how to get more data points? Well, from the paper by Haah, Vijay and Fu, we knew how to define the model on cubic lattices. With periodic boundary conditions, the underlying manifold is a three torus. Is it possible to have a cubic lattice, or something similar, in other three manifolds as well? Usually, this kind of request would be too much to ask. But it turns out that if you whisper your wish to the right mathematician, even the craziest ones can come true. With insightful suggestions from Michael Freedman (the Fields medalist leading Station Q) and Zhenghan, and through the amazing work of Kevin Slagle (U Toronto) and Wilbur Shirley (Caltech), we found that if we make use of a structure called Total Foliation, one of the fracton models can be generalized to different kinds of three manifolds and we can see how the properties of the model are related to certain topological features of the manifold!

Foliation.

Foliation is the process of dividing a manifold into parallel planes. Total foliation is a set of three foliations which intersect each other in a transversal way. The xy, yz, and zx planes in a cubic lattice form a total foliation and similar constructions can be made for other three manifolds as well.

Things start to get technical from here, but the basic lesson we learned about some of the fracton models is that structural-wise, they pretty much look like an onion. Even though onions look like a three-dimensional object from the outside, they actually grow in a layered structure. Some of the properties of the fracton models are simply determined by the layers, and related

to the topology of the layers. Once we peel off all the layers, we find that for some, there is nothing left while for others, there is a nontrivial core. This observation allows us to better address the previous questions: we defined a fracton phase (one type of it) as models smoothly related to each other after adding or removing layers; the topological nature of the order is manifested in how the properties of the model are determined by the topology of the layers.

The onion structure is nice, because it allows us to reduce much of the story from 3D to 2D, where we understand things much better. It clarifies many of the weirdnesses of the fracton model we studied, and there is indication that it may apply to a much wider range of fracton models, so we have an exciting road ahead of us. On the other hand, it is also clear that the onion structure does not cover everything. In particular, it does not cover Haah’s code! Haah’s code cannot be built in a layered way and its properties are in a sense intrinsically three dimensional. So, after finishing this whole journey through the onion field, I will be back to staring at Haah’s code again and wondering what to do with it, like what I have been doing in the eight years since Jeongwan’s paper first came out. But maybe this time I will have some better ideas.