View from the arXiv: Feb 07 - Feb 11 2022

Welcome to ‘View from the arXiv’, where each week I’ll put together a short list of new preprints which have appeared on arxiv.org during the week which I’ve found interesting. I’ll focus on the categories ‘Disordered Systems and Neural Networks’, ‘Quantum Gases’ and ‘Strongly Correlated Electrons’, and in particular the first two as these are the main areas of the arXiv which I follow. This is an entirely subjective list of things which appeal to me, and of course there are far too many interesting papers to be able cover all of them so I just choose one or two from each day to highlight here. As these are all preprints which have not yet been peer-reviewed, remember to take any claims and conclusions with a grain of salt and be sure to cast a critical eye over the work if you’re interested in more details. (And let’s face it, this caveat should be applied to any published work too…!)

Feb 7th

Maximal quantum chaos of the critical Fermi surface, by Maria Tikhanovskaya, Subir Sachdev, and Aavishkar A. Patel: The rather terse abstract and very technical nature of this paper might make this look as though it’s a niche observation, but hiding in this paper is an extremely novel result. Few quantum systems are known which are ‘maximally chaotic’, which means they thermalize as quickly as is allowed by the laws of physics (or in other terms, they are fast scramblers of information which saturate the Lyanpunov bounds). The poster child for this fast-scrambling behaviour is the Sachdev-Ye-Kitaev model (SYK), which although theoretically rich and extremely interesting – keep an eye out for a soon-to-appear work on this by my collaborators and I! – is quite an unusual model that doesn’t resemble more typical microscopic models of solids. Here, Sachdev (the very same ‘S’ as in ‘SYK’) and co-authors show that in fact a two-dimensional critical system can exhibit the same properties. This is a very interesting development, perhaps opening up a wider world of maximally chaotic SYK-type physics in easier-to-realise systems, although the authors do point out that higher-order contributions could in principle suppress the Lyyapunov exponent slightly from its maximal value, and that future work is needed to say for sure.

Bounds in Nonequilibrium Quantum Dynamics, by Zongping Gong and Ryusuke Hamazaki: The title may not inspire much excitement, but this huge review covers a wide range of fascinating topics relating to how quantum systems far from equilibrium behave. From ‘speed limits’ of unitary dynamics, to a detailed look at thermalisation processes in quantum systems, to bounds on the propagation of quantum information (known as Lieb-Robinson bounds) and much, much more, this is well worth putting on your reading list if you’re interested in the deep mathematical properties and mysteries of quantum systems far from equilibrium. I can’t write much more about it because I’ve barely scratched the surface, but from what I’ve read so far, this looks to be a very insightful and comprehensive work!

A Practical Guide to the Numerical Implementation of Tensor Networks I: Contractions, Decompositions and Gauge Freedom, by Glen Evenbly: Anyone who’s ever searched for information on tensor networks and how to understand or program them will probably at some point have visited tensors.net, the excellent website run by Glen Evenbly, the author of this new article. This work fills an interesting gap in the literature, aiming to be a guide to the numerical implementation of tensor network codes, rather than a purely theoretical overview or a detailed description of some particular algorithm. As with his website, this is a clearly written, very pedagogical guide that I can see myself coming back to refer to in the very near future, as several of my projects involve tensor networks (or simply tensor operations, in some cases) and we’re in the early stages of coding some things that don’t exist in any current library, so this guide looks like an invaluable resource for the efficient numerical implementation of a variety of tensor network algorithms.

Feb 8th

On the relationship between the ground-state wave function of a magnet and its static structure factor by Jorge Quintanilla: This technical-looking paper puts forward two theorems that, while dry-sounding, could have quite far-reaching consequences. Working with a Heisenberg spin model, the theorems relate the Hamiltonian and ground state wavefunction respectively to the two-point correlation functions. Loosely, the idea is that not only does the Hamiltonian determine the (ground state) wavefunction, which determines the form of correlation functions, but that this relation holds in both directions. If you know the form of the correlation function, in other words, you can work backwards and determine the coefficients of the Hamiltonian and the ground state wavefunction. Now, I haven’t gone through the details and I’m not certain what the limitations of the theorems are in practical terms, but this idea holds quite some promise. In the paper, the claim is made that this could be useful for future machine-learning approaches – and in particular the idea that wavefunctions can be inferred directly from experimental measurements without requiring the input of a model Hamiltonian – but I wonder if these theorems might be even more general. Might similar theorems also hold for excited states, and might they be applicable to studies of many-body localisation or other disordered systems? Could we reconstruct MBL eigenstates from an understanding of the l-bits and their correlations/interactions? I think this may be an interesting and exciting question.

Introduction to the dynamics of disordered systems: equilibrium and gradient descent, by Giampaolo Folena, Alessandro Manacorda, Francesco Zamponi: Anyone who’s read more than a few of these posts will know that the dynamics of disordered systems is one of my main research interests. This manuscript is based on lecture notes given by one of the authors (Francesco Zamponi, a highly respected name in the field) at several summer schools and it’s a review of both the equilibrium and non-equilibrium properties of glassy systems. The review starts by motivating the study of glasses in terms of a variety of applications including optimisation problems, machine learning and simulated annealing (for quantum computers) before moving on to the dynamics and phase diagrams of some glassy models. The main model looked at here is a prototypical model of a spin glass known as the p-spin model (which I’ve worked on in the quantum case!). This is a really nice review, coming mostly from a statistical mechanics perspective (something I’m familiar with, but isn’t really my ’native tongue’ when it comes to studies of disordered systems) and it contains a lot of nice insights presented in a readable way that’s accessible to non-experts in the field. Check it out if you’re interested in glassy or disordered systems!

Feb 9th

Exceptional dynamics of interacting spin liquids, by Kang Yang, Daniel Varjas, Emil J. Bergholtz, Sid Morampudi, and Frank Wilczek: The first of two papers in today’s post on quantum spin liquids, this work examines non-Hermitian effects in spin liquid phases. A spin liquid is a phase of matter that exists only at low temperatures, and occurs when quantum fluctuations are strong enough to prevent a spin model from magnetically ordering. Because of these strong fluctuations, the system also exhibits long-range entanglement, in contrast to other non-magnetic phases such as paramagnets, and consequently spin liquids exhibit some unusual prpoerties. In this paper, the authors show that the system can be described in terms of an effective non-Hermitian Hamiltonian (resulting from studying the Dyson equation and expanding the non-Hermitian self-energy to leading order). This model exhibits so-called ’exceptional’ degeneracies which lead to unusual properties, detailed in the paper, and suggest that certain experimental probes may be able to detect these degeneracies in the single-particle spectrum. The authors also suggest that the symmetries of interactions and disorder, far from destroying or suppressing spin liquid signatures, may instead be the cause of exceptional degeneracies, and end with a suggestion that studies of these effects in 3D may lead to further interesting phenomena.

Scars from protected zero modes and beyond in U(1) quantum link and quantum dimer models, by Saptarshi Biswas, Debasish Banerjee, and Arnab Sen: Typical many-body quantum systems, left to their own devices, are expected to eventually reach a thermal equilibrium under their own unitary evolution. There are several exceptions to this rule, such as many-body localized systems and quantum glasses, where the majority of eigenstates are non-thermal, however in recent years a new class of systems have exhibited in which only a small sub-set of states fail to thermalise. These states are known as quantum scars, and when prepared in an initial state with a large overlap with a scar state, these systems will also fail to thermalise. This work examines scar states in an unusual pair of systems, and connects them to so-called zero modes. It’s an interesting study of quantum scars and their thermalisation properties from what seems to me to be a rather unusual point of view, and it sheds some interesting light on the phenomenon of weak ergodicity breaking.

Hole Spectral Function of a Chiral Spin Liquid in the Triangular Lattice Hubbard Model, by Wilhelm Kadow, Laurens Vanderstraeten, Michael Knap: The second quantum spin liquid paper of the day – and almost picking up where the previous one left off – this work examines the spin liquid phase in the Hubbard model on a triangular lattice, and concentrates on computing a quantity known as the hole spectral function capable of revealing several important and distinctive properties. The authors suggest that this function could be readily measured in experiments using angle-resolved photoemission spectroscopy (ARPES), providing a way in which spin liquids could be unambiguously observed in real materials. This raises the question in my mind of whether quench spectroscopy (which I’ve worked on in the past) could also be generalized to an ARPES measurement, as in ultracold atomic gases this is typically easier to measure than conventional spectral functions while containing the same physical information, although I’m not sure the required spatially resolved measurements are possible in solid state systems.

Feb 10th

Symmetry-induced many-body quantum interference in chaotic bosonic systems: an augmented Truncated Wigner method, by Quirin Hummel, and Peter Schlagheck: I’ve mentioned a few times in these posts that I’m very curious to learn more about the truncated Wigner method (TWA), as it seems like a very interesting method but its limitations have always been unclear to me. This work is a lengthy technical treatise on precisely the limitations of the TWA, ways in which the method can be augmented to overcome some of these limitations, and what we can learn about a system when the TWA fails to agree with other exact methods. In a nutshell, the key point is that interference between different trajectories (in the path integral sense) is not fully taken into account in the TWA approach, rendering it quasi-classical in nature and unable to fully capture purely quantum phenomena such as dynamical localisation and quantum scars, both of which are topics of great interest to me. This work shows in detail exactly which contributions the TWA does and does not correctly account for, and provides a lot of insight into how the failure of the TWA can imply the existence of quantum phenomena which have their origins in interference effects. The authors put this idea on a rigorous footing, and while this work is not for the faint-hearted, it makes for a very interesting read.

Entanglement estimation in tensor network states via sampling, by Noa Feldman, Augustine Kshetrimayum, Jens Eisert, and Moshe Goldstein: Tensor network methods – as described above in an earlier summary from this week’s new preprints – have become the de facto numerical standard for investigating quantum systems in one dimension, as they provide a natural way to calculate properties that characterise the ‘quantumness’ of a particular system. One particularly key example is the (von Neumann or Rènyi) entanglement entropy across a bipartition in the middle of the system, where the entanglement spectrum can simply be read off from the singular value decomposition (a step routinely performed in any tensor network calculation). In higher dimensions, however, computing entanglement measures using tensor networks is much harder and can be computationally prohibitive without additional tricks or approximations. Even in one-dimension, computing the entanglement between two arbitrary subsystems can still be a significant challenge using existing mmethods. This work tackles the problem of obtaining entanglement measures in systems of arbitrary dimension based on a stochastic sampling technique, bypassing some limitations of previous approaches and allowing numerical simulations to compute challenging entanglement measures far more efficiently. This opens the door to a fuller characterisation of quantum mechanical effects in systems in greater than one dimension, as well as potentially being a useful tool for exotic entanglement measures even in low-dimensional systems. (Full disclosure, I work in the same research group as authors Augustine Kshetrimayum and Jens Eisert, though I was not in any way involved with this work: it’s only featured here because I find it interesting!)

Entanglement Hamiltonian during a domain wall melting in the free Fermi chain, by Federico Rottoli, Stefano Scopa, and Pasquale Calabrese: I first encounterted the Bisognano-Wichmann theorem following a conversation with PhD student Jan Schneider when I worked in Paris, and ever since then I’ve been fascinated by it. The idea is that the entanglement spectrum of a given Hamiltonian can be mapped onto the energy spectrum of a different Hamiltonian: by solving the energy spectrum of this effective model, one gains information about the entanglement spectrum of the original model. This could be tremendously useful both for experiments (which can access the otherwise hard to measure entanglement properties of a model by directly simulating the effective entanglement Hamiltonian) and theory, opening up the possibility of using standard techniques to obtain complex entanglement measures. The sticking point is that determining the entanglement Hamiltonian is hard, and few exact results exist. Those that do focus on homogeneous and/or equilibrium scenarios. This work goes beyond these prior works in considering a setting which is inhomogeneous (a domain wall state) and non-equilibrium, and by making use of prior work on quantum hydrodynamics the authors are able to provide predictions for the form of the entanglement Hamiltonian even in this highly non-trivial situation. That said, the particular model they considered does have known solutions from various other method, which is why the authors used it as a testbed to develop their framework. It will be very interesting to see whether this method has further success in more complex models, and I look forward to seeing the results of future work on this topic. (Incidentally, I also learned from this paper that the domain-wall scenario considered here even allows for a non-equilibrium conformal field theory, which I did not know and is a very neat property.)

Feb 11th

Entanglement Hamiltonians: from field theory, to lattice models and experiments, by M. Dalmonte, V. Eisler, M. Falconi, and B. Vermersch: Continuing the theme from yesterday of looking at entanglement Hamiltonians, this review article appeared today surveying the state of the art in the field of entanglement Hamiltonians. It covers exactly solvable models, various important theorems for entanglement Hamiltonians, as well as directions beyond exact results and even experimental ways to implement and measure entanglement Hamiltonians. A very good complement to the entanglement Hamiltonian paper mentioned above in yesterday’s list!

Many-body localization in a tilted potential in two dimensions, by Elmer V. H. Doggen, Igor V. Gornyi, and Dmitry G. Polyakov: Many-body localisation (MBL) is an interesting effect that typically occurs when the interplay of disorder and interparticle interactions in a quantum system can lead to the breakdown of thermalisation, preventing the movement of particles and only allowing extremely slow growth of entanglement. This phenomenon is in some aspects well-understood, but two particular outstanding questions are whether MBL can ever be realised in systems without random disorder, and whether it is stable in two dimensional systems. This work tackles both questions at once, using what I suspect must have been extremely challenging matrix product state simulations to study a two dimensional system with a so-called Wannier-Stark disorder-free potential. Rather than study a square lattice, the authors instead study a rectangular lattice that is longer in one dimension than the other, as a two-dimensional square system would likely be extremely difficult to simulate. Various quantities are considered, including standard measures like the imbalance and entanglement entropy, and the authors conclude that stable localised states do exist in two dimensions, but only at very large field strengths, and only for particular states. This sounds more reminiscent of quantum scars or Hilbert space fragmentation than conventional MBL, and it will be interesting to see how this picture evolves in the near future. (Not least because it’s something that I’m working on as well, albeit from a slightly different point of view…!)