Many-Body Localisation: A Flow Equation Approach

When asked what are the big outstanding challenges in modern physics, a number of things may spring to mind. Perhaps the search for a theory of quantum gravity, that elusive unification of relativity with quantum mechanics, or maybe the LHC’s continued search for physics beyond the Standard Model. By comparison, the question of how impurities in quantum materials can affect their properties seems quaintly pedestrian, but it turns out to be every bit as fundamental a question as its ostensibly deeper cosmological cousins.

With the recent publication of our new paper in Physical Review B as a Rapid Communication (open-access arXiv link here), now seems like an opportune moment to talk a little about the significance of this problem, and our latest contribution towards solving it.

Many-Body Localisation

Since the 1950s, we’ve known that in theoretical models of non-interacting particles in one-dimension (e.g. wires) and two dimensions (e.g. flat sheets), the particles are localised by any concentration of disorder. This means that they can’t move: they’re effectively stuck in place, and the material is an insulator. This is known as Anderson localisation, and it’s perhaps the simplest type of localisation transition, where we can tune a material between metallic and insulating states by switching on and off the disorder. (In three dimensions, the particles will be Anderson-localised only once the disorder is strong enough – for weak disorder, the particles can still move and the material would remain metallic.)

You can visualise Anderson localisation in one dimension by picturing water flowing through a pipe. If you added disorder to the pipe in the form of a blockage, suddenly the water wouldn’t be able to flow any more. It would get stuck, or ‘localised’ in one region. This is pretty much what happens to non-interacting quantum particles in disordered wires, except instead of a physical barrier holding them in place, it’s an electromagnetic field: this disorder could come from atomic impurities, or perhaps defects in the crystal structure of a solid, or a whole host of other scenarios.

It’s easy to ‘localise’ a system in one dimension, as a single blockage is enough to stop the flow of particles along a wire, or liquid along a pipe.

It’s easy to ‘localise’ a system in one dimension, as a single blockage is enough to stop the flow of particles along a wire, or liquid along a pipe.

For a long time, it was tacitly assumed that this result would not remain true in the presence of the inter-particle interactions present in many real materials: these interactions, it was thought, would lead to quantum tunnelling events which would allow the particles to burrow through the electromagnetic barriers holding them in place, rendering the material metallic once more. And in any case, it was thought for sure that this result would not hold for a system able to exchange heat with its surroundings, so the result was effectively shelved for decades, until experimentalists achieved such a degree of control that extremely good isolation of experimental setups from their environments was indeed possible, and the question became relevant once more: do interactions indeed delocalise an isolated, disordered quantum material?

In 2006, it was shown that, in fact, they do not: if the disorder is strong enough, the material remains localised (insulating), but this is no longer simple non-interacting Anderson localisation. It’s a new type of quantum many-body effect, a collective phenomena produced by the interactions between all the particles in the system, and this leads to some surprising consequences. We call it many-body localisation, and understanding it is one of the biggest open problems in modern condensed matter physics.

Paradigm Shift

Left to their own devices, most materials will reach a thermal equilibrium, or ground state. Imagine a Newton’s cradle, for example: you can give it energy and set it going, but eventually it will dissipate its energy to its surroundings in the form of heat and sound, and will reach a stationary ground state. The same is true for almost everything else we encounter in daily life: a cup of hot coffee left alone in a room will give up its energy to the room until it has cooled down, or an ice cube will absorb energy from the room until it melts into a pool of room-temperature water.

Many-body localised (MBL) materials turn out to be near-perfect insulators of both heat and electricity, with an extremely slow internal dissipation of energy. What this means is that they don’t reach a thermal equilibrium even within themselves, unlike the vast majority of other phases of matter. MBL matter becomes ‘fragmented’ into small regions known as localised bits (l-bits) or Local Integrals of Motion (LIOMS) depending on your preference. MBL matter effectively cannot distribute energy within itself: it never reaches a true thermal equilibrium state. Imagine if a Newton’s cradle suddenly got stuck halfway through a swing, or if your cup of coffee suddenly fragmented into hot and cold regions that don’t exchange heat with each other: that’s what many-body localisation does to quantum matter. In fact, even the idea of temperature stops being meaningful in an MBL phase – does a single particle have a temperature? Do two? We can only start talking about things like temperature when we have enough particles to start treating them statistically, but MBL matter becomes fragmented into small regions that are essentially cut off from each other, and such a statistical description becomes difficult.

Any image of how quantum particles behave will be wrong, but for what it’s worth this is kind of how I picture l-bits: small regions of electrons that bunch together into composite particles known as l-bits, which have only a small ‘overlap’ with the other l-bits nearby to them. YMMV.

Any image of how quantum particles behave will be wrong, but for what it’s worth this is kind of how I picture l-bits: small regions of electrons that bunch together into composite particles known as l-bits, which have only a small ‘overlap’ with the other l-bits nearby to them. YMMV.

All of this is important because most quantum mechanical phenomena are very fragile and easily destroyed outside of ideal lab conditions by, for example, thermal effects. To study interesting quantum mechanical effects in real materials, researchers often have to cool their samples down to cryogenic temperatures just a few degrees above absolute zero. But, because MBL matter is largely insensitive to temperature and seems very robust, it could offer us an escape route: it could be a way to protect certain quantum mechanical properties from thermal effects and make them robust enough to be used in practical quantum technologies of the future.

Before we get there, however, we need to further develop our understanding of what MBL is. Particular open questions include understanding how and when disordered materials fail to relax towards thermal equilibrium, whether MBL can be realised in two- and three-dimensional systems, and just how robust it really is, both in more realistic theoretical models and in real experiments.

The big challenge here is that because of the peculiarities of this phase of matter, equilibrium quantum statistical mechanics - the cornerstone of condensed matter physics - is no longer valid.

We’re faced with the problem of simultaneously studying strong quantum mechanical interactions (a hard problem), the effects of random disorder (a hard problem), and non-equilibrium physics (you guessed it - a hard problem). MBL isn’t just a niche phenomenon in and of itself, therefore: it’s the gateway to a deeper level of emergent physics that our current tools just can’t capture. The theoretical methods we have are no longer adequate: to study these materials and to stand any hope of understanding them, we need new tools and a fundamentally new understanding of emergent many-body quantum effects.

Most of the existing work on MBL falls into one of two categories. On the pen-and-paper side, there are (relatively) simplified phenomenological models and perturbative calculations, which capture some of the main physics, but involve various approximations. On the computational side, there are powerful exact numerical methods, but because of the sheer complexity of the calculations involved, most simulations can only handle a few tens of particles. Real materials can contain tens of billions of particles and even synthetic quantum matter may contain several hundred, so these simulations are often too small to give us more than a glimpse of the underlying many-body physics.

What we really need, then, is a technique that bridges the gap between these two avenues of investigation, one that can use the insights from the pen-and-paper work to simplify the calculations we ultimately have to feed to the computer. And on that note…

Our Work

That’s where we come in. In our new paper, Marco Schiró and I have developed a new way to investigate many-body localisation, based around a pre-existing technique known as the flow equation method. In a nutshell, this method takes a complicated problem and transforms it into a simpler-looking one that’s easier for a computer to solve. We use a continuous unitary transform to take the equation studied by exact computational methods and morph it into a simpler so-called ‘diagonal’ form, which turns out to be the same as the phenomenological model used in many of the pen-and-paper studies. One of the main advantages of the flow equation method, therefore, is that it bridges the gap between these two existing avenues of investigation (in a precise mathematical sense).

In doing so, we have to make an approximation called a ‘truncation’ which involves the neglect of some terms in the final equations that we believe to be extremely small: we tested this by benchmarking against the gold-standard numerical method (known as exact diagonalisation) and find that the flow equation results are extremely accurate in the many-body localised phase. Flow equations have been used before to tackle this problem in their exact form, but the truncation we employ makes the problem much less computationally complex, and therein lies the key advantage to this approach.

With that done, the truncated flow equation method then allows us to:

  • Simulate over 10x as many particles as exact numerical implementations : In our paper, we show results for up to 144 particles, but this is by no means the limit. (Many of the simulations were run on my Macbook Air, with its relatively paltry processing power.) For comparison, exact implementations of flow equations are currently limited to around 12 particles.
  • Simulate MBL matter in two dimensions : Because we can reach such large numbers of particles, we can do meaningful simulations of MBL-type-physics in two dimensions. We find strong indications that MBL persists in two dimensions. Our work was the first to find this, followed very shortly afterwards by another work studying a different model with different methods which nonetheless reached the same conclusion, and provided further evidence for the existence of MBL in two dimensions.
  • Compute the properties of l-bits directly : The ability to connect the microscopic models of interacting particles to the phenomenological models of MBL allows us to directly calculate many quantities which were out of the reach of previous pen-and-paper calculations.
  • Study the dynamics of these materials : The relatively simple ‘diagonal’ form of our equations allows us to compute the non-equilibrium dynamics of MBL matter. In the paper, we demonstrate this by showing how local observables behave after a sudden parameter change known as a quench, however the method also works for correlation functions and composite non-local objects such as out-of-time-ordered correlators.

Such advantages come at a price, however, and the trade-off is that the truncated flow equation method only works well in the many-body localised phase: it’s no longer valid when the disorder strength is decreased and the material begins to become metallic once more. This unfortunately means that our method cannot, in its current incarnation, be used to study the delocalisation transition, i.e. when MBL breaks down, which is one of the other major open questions in the field. Further development is needed, and this is just one of many directions we’re considering for the future.


In the meantime though, the truncated flow equation method provides a powerful, transparent framework for calculating properties of the many-body localised phase. It allows us to start from microscopically exact models of interacting particles, transform the problem into a computationally simpler one, and calculate the dynamical properties of physical observables. With this, we can reproduce the results of a variety of other calculations, but also begin to do new things that (so far), no other methods can. While there’s definite room for improvement in our implementation, there are already a lot of really neat things that can be done with this technique even in its current incarnation, and I’m excited to see what other researchers do with it in the near future.

We may not be on the verge of solving the mysteries of many-body localisation any time soon, but in studying it, condensed matter and statistical physics researchers across the world are developing the tools and laying the foundations for a new understanding of emergent behaviour in quantum matter, and in doing so, perhaps even laying the foundations for the materials and technologies of the future. Some things really are more than the sum of their parts, and when it comes to many-body quantum matter, it looks like we’ve still only just begun to understand that.

Usual disclaimer: this post was written by me, not any of my colleagues or co-workers, and represents my views only. If you don’t like this post, blame me, not them!