# Oxford physicists solve 10-year-old puzzles

28 September 2020

Three physicists at the University of Oxford’s Department of Physics have finally solved puzzles unearthed by one of the trio ten years ago.

Professor Radu Coldea’s paper titled Quantum criticality in an Ising chain: experimental evidence for emergent E8 symmetry appeared in *Science* in 2010. It documented a spectacular series of neutron scattering measurements of a material, Cobalt Niobate (chemical formula CoNb_{2}O_{6}). Cobalt niobate is a three-dimensional crystal where the magnetic ions form many one-dimensional chains described by the Ising model (see below). These experiments found strong evidence for 1D Ising physics – but there were some unexplained anomalies. One was that normally immobile domain walls were in fact moving -- which indicated additional energetics allowing them to ‘hop’ from site to site. And secondly, there was unexplained decay in the dispersion; this was surprising because decay is not allowed for the Ising model for symmetry reasons, and had to be attributed to minor details, such as a slight misalignment in the orientation of the external magnetic field in the experiments. Despite attempts by various groups to reconcile these observations with the robust signatures of Ising criticality, no satisfactory explanation was ever given for these puzzles.

Ten years on, the puzzles have been resolved in a paper that appears in the most recent *Proceedings of the National Academy of Sciences* (PNAS). The research was led by Michele Fava, a second year DPhil student in Oxford’s Rudolph Peierls Centre for Theoretical Physics, working with his DPhil supervisor Professor Siddharth Parameswaran, and building on insights from Professor Coldea.

*Image caption (main panel): Dynamical structure factor of a single Ising chain in CoNb _{2}O_{6}. This quantity, which can be probed by inelastic neutron scattering experiments, indicates how the system responds to a magnetic probe of the given energy and wavevector. It gives a quantitative picture of the low-energy 'quasiparticle' excitations of the system. The colormap shows numerical simulations of the microscopic model introduced in the new paper; the position of the peaks predicted by the theoretical model agrees well with the experimentally-determined values indicated by the white dots. A key result of the new work was providing a microscopic explanation for the curvature or 'dispersion' of these peaks away from their minima.*

Image caption (inset): A single Ising chain in CoNb_{2}O_{6} consists of a series of magnetic Co^{2}+ ions (blue dots) arranged in a zig-zag pattern, each surrounded by an octahedron of oxygen ions. The zig-zag arrangement provided an important clue to the microscopic model constructed by the Oxford trio.

### A new perspective

Fava decided to revisit the original symmetry model tapping into Professor Coldea’s expertise and identified a subtlety missed by previous theoretical models proposed for this material: the 'real' symmetry of the model was not just linked to the spins, but also to the way in which they were spatially arranged in a zigzag pattern along the chain.

With this new insight, the trio were able to build microscopic models that allowed for these zigzag arrangements. The models differ from the usual Ising model in that the interactions are slightly different on the zigs and the zags: while the Ising transformation of rotating spins by 180 degrees around their y-axis changes the energy, as does swapping 'zigs' for 'zags', combining both transformations leaves the energy invariant: the model has a hidden symmetry!

This sort of spin-rotation combined with a zig-to-zag transformation of space is closely related to something called a 'glide' symmetry, a particular area of expertise of Parameswaran's. Together, Fava and Parameswaran realised that this insight changed the picture: it allows domain walls to hop, explaining their dispersion; and excitations could decay, but only if they also picked up a big shift in their momentum – which precisely matched the experimentally observed quasiparticle decay.

### Explaining the remaining puzzle

One question remained unanswered: why was the experiment so close to the idealised Ising problem? Fava and Parameswaran realised that there is a precise mapping that makes it clear why this material is linked to an Ising-chain system where the spins align anti ferromagnetically – thus explaining why the physics near the critical point is captured by the idealised problem.

'There is an enormous sense of satisfaction in solving these particular scientific puzzles,' explains Fava who led the work. 'It is a privilege to be able to build on the work of colleagues and our microscopic model will serve others in their work on domain wall dynamics and quasiparticle decay.'

Glide symmetry breaking and Ising criticality in the quasi-1D magnet CoNb2O6, M Fava, R Coldea and SA Parameswaran, Proceedings of the National Academy of Sciences, 25 September 2020.

For more information on the research from the two groups, see their websites:

Professor Parameswaran: https://www.sites.google.com/site/sidparameswaran

Professor Coldea: https://www2.physics.ox.ac.uk/research/quantum-magnetism-and-quantum-phase-transitions

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### Ising model

Theorists like simple models that capture the key features of a system and arguably one of the most famous models in theoretical physics is the Ising model: a cartoon picture for magnetism and phase transitions, where systems go from one macroscopic state to another. In the Ising model, atoms spins are viewed as placed in a regular array or grid, each in one of two states: up or down.

In the **classical Ising model**, the competition is between the desire of spins to align parallel to each other to minimise energy and the fact that at any temperature above absolute zero, they also like to maximise entropy or their “disorder”; low-energy configurations have low entropy, and vice versa. So, raising the temperature tips the balance between the energy and entropy leading to a transition between a ferromagnet and a thermal paramagnet.

In the **quantum Ising model**, one thinks of physics at zero temperature ( – zero temperature is technically impossible, but experiments can get close enough that it’s the right starting point). Here, the energy-entropy consideration is not important, but it’s instead the competition between spins that want to align all-up or all-down (the energetics don’t care which) or parallel to an external field that points “sideways”. When the field is low, the energetics want the spins to be all-up (↑↑↑↑) or all-down (↓↓↓↓), but don’t have a preference for one over the other. The system is then said to be in a ferromagnetic ground state which has to choose one of these options: it must spontaneously break the symmetry of the interactions. When the field is high, the system’s lowest-energy state or ground state is a field-aligned paramagnet: (→→→→), which preserves the up-down symmetry of the model. The transition between these limits occurs at a quantum critical point, near which various physical quantitities show “scaling” behaviour: their dependence on external parameters like the applied field and the temperature are captured by a universal set of power laws.

The 1D quantum Ising model is an especially rich testbed for ideas of quantum criticality, quantum field theory and conformal field theory. It also has a long history at Oxford: among one of the earliest studies of the quantum Ising model was performed by the late Sir Roger Elliot (1928-2018; Wykeham Professor of Physics 1974-88) and his collaborators at the Rudolf Peierls Centre for Theoretical Physics [R J Elliott, P Pfeuty, and C Wood *Phys. Rev. Lett.* 25, 443 (1970)]. In the same year, one of those collaborators, Pierre Pfeuty, presented an exact solution of the 1D model relevant to this work [Annals of Physics, 57, 79 (1970)]. Incidentally, Sir Rudolf Peierls himself made an influential contribution to the physics of the Ising model, providing a clever argument that clarified when the model would have a sharp phase transition. [RE Peierls, *Proc. Cambridge Phil. Soc.* 32, 477 (1936).]

Top image from left to right: Michele Fava, Radu Coldea and Siddharth Parameswaran (© John Cairns)