# Research & Annual Reports

The Theory of Quantum Matter (TQM) Unit carries out research into a wide range of problems in condensed matter theory, with a strong emphasis on the novel phases and excitations found in quantum matter.

This work is described in the **Annual Reports** listed in the menu on the left side of this page. These provide details of all of the research carried out by the TQM Unit, publications and presentations by Unit members, outreach activity, and seminars given by visitors to TQM in OIST. Reports are organised by the Japanese financial year, with FY2018 running from **April 1**^{st}** 2018** until **March 31**^{st}** 2019**.

More recent rsearch projects, completed after **April 1 ^{st} 2019**, are described below:

### 1. Interpreting frustrated magnets: man versus machine

Machine learning has already revolutionised the way we think about tasks as complex as driving a car. But can machines also do science ? In this project, we revisit a challenging problem in condensed-matter physics, the determination of the phase diagram of frustrated magnet, originally studied by Taillefumier et al., [Phys. Rev. X 7 041057 (2017)] (cf. FY2016 Annual Report). The model in question is the XXZ model on a pyrochlore lattice, and the particular challenge lies in the fact that this supports three different types of spin liquid phases, as well as both conventional, and unconventional forms of magnetic order (cf. upper panel of Fig.).

The machine learning method used was a Support Vector Machine (SVM) with a tensorial kernel (TK) and graphical analysis, previously introduced by J. Greitemann et al. [Phys. Rev. B 99, 060404(R) (2019)]. This method of machine learning does not require the machine to be "trained" on established examples, and can be "interpreted", i.e. it is possible to interrogate the machine about how it reached its final decision. Starting from spin configurations taken from classical Monte Carlo simulation, the TK-SVM correctly identified all of the phases found by Taillefumier et al. without any prior information about the type of phases present, completely reproducing the published phase diagram (cf. lower panel of Fig.) . Moreover, the machine also correctly identified the organisational principle behind each phase, determining the correct order parameters for each of the ordered phases, and the correct local constraint for each of the spin liquids.

The striking success of the TK-SVM in this case suggests that it will be possible to solve the phase diagrams of complex models without human input.

This work was a collaboration between members of the TQM Unit (OIST), members of the Theoretical Nanophysics Group (LMU Munich), and Dr Ludovic Jaubert of LOMA (CNRS Bordeaux). It is described in the preprint: "Identification of hidden order and emergent constraints in frustrated magnets using tensorial kernel methods", Jonas Greitemann, Ke Liu, Ludovic D.C. Jaubert, Han Yan, Nic Shannon, Lode Pollet, arXiv:1907.12322

### 2. Putative spin-nematic phase in BaCdVO(PO_{4})_{2}

Like Volborthite, described below, BaCdVO(PO_{4})_{2} is a quasi-two dimensional magnet, with competing ferromagnetic and antiferromagnetic interactions. In such systems, it has been proposed that a new state of matter, known as a "spin nematic", can occur in high magnetic fields. BaCdVO(PO_{4})_{2} has previously been discussed as a candidate for spin-nematic order [A. Smerald, H. T. Ueda, and N. Shannon, Phys. Rev. B 91, 174402 (2015)], but until now, relatively little has been known about the experimental phase diagram of this material.

In this paper, we identify for the first time the nature of the magnetic grounds state of BaCdVO(PO_{4})_{2} in the absence of magnetic field, and present evidence for the existence of a novel magnetic phase in applied magnetic field, consistent with predictions of spin-nematic order. Evidence in support of this conclusion comes from elastic neutron scattering and dynamical susceptibility measurements on powder samples of BaCdVO(PO_{4})_{2}. These results, in combination with recent thermodynamics measurements on single crystals [K. Y. Povarov, V. K. Bhartiya, Z. Yan, and A. Zheludev, Phys. Rev. B 99, 024413 (2019)], establish BaCdVO(PO_{4})_{2}, like Volborthite, as a strong candidate for spin-nematic order.

This work was published in the article: "Putative spin-nematic phase in BaCdVO(PO_{4})_{2}" M. Skoulatos, F. Rucker, G.J. Nilsen, A. Bertin, E. Pomjakushina, J. Ollivier, A. Schneidewind, R. Georgii, O. Zaharko, L. Keller, Ch. Rüegg, C. Pfleiderer, B. Schmidt, N. Shannon, A. Kriele, A. Senyshyn, and A. Smerald, Phys. Rev. B **100**, 014405 (2019)

### 3. Exotic states of matter mimicking the entanglement features of gravity

“Emergence” is a fascinating theme in condensed matter physics: many exotic theories that were invented to understand the fundamental constituents and interactions of the universe are found to exist in many-body systems --- not as the microscopic laws but as collective phenomena. It is thus intriguing to ask, are there any many-body systems that mimic gravity? In particular, can they mimic the informational properties of gravity like the entanglement structure, instead of the field-aspect ones like gravitons?

In this project, we give an affirmative answer, thanks to the recent development in understanding some exotic forms of matter called fracton states. Unlike conventional many-body systems where the global and local symmetries play the crucial role of determining their properties, the fracton states of matter have subsystem symmetries, which operate on a lower-dimensional space of the system, but not locally. Such systems were found to satisfy a few essential properties of gravity in the context of holography or AdS/CFT. That is, when embedded in the space of negative curvature, the fracton model shows very similar entanglement structure to gravity.

As a sequel to our first paper, making the initial discovery, the second paper provides more in-depth insight into the physics. We show that in a dual language of the system told in vertices and arrows, the fracton model is equivalent to a web of bit-threads — a well-known toy model construction widely used by people who study AdS/CFT. The dual description also helps us to quantify the holography breaking at finite temperature and the black hole microstate degrees of freedom.

This work leads to a universal picture of different toy models (tensor-networks, bit-threads, fracton states) with the advantage of known bulk theory. Hopefully, it will help us to make a more solid connection between toy models and genuine gravity.

This work is described in the preprint : "Hyperbolic Fracton Model, Subsystem Symmetry, and Holography II: The Dual Eight-Vertex Modelar" , Han Yan, arXiv:1906.02305

### 4. Possible observation of quantum spin-nematic phase in the frustrated magnet volborthite

Every child learns that water freezes in the cold of winter, and evaporates quickly in the heat of summer. Scientifically, these transformations between solid, liquid and gas are called phase transitions, and the fact that the same atoms can exhibit different phases lies at the heart of our understanding of the material world. A fourth phase of matter was discovered, by chance, late in 19th century, when Freiedrich Reinitzer tried to make crystals from molecules of cholesterol. Reinitzer noticed that as he cooled his solution of cholesterol towards its freezing point, it underwent a marked change in its physical properties, while remaining a liquid. This was the first observation of a "liquid crystal", a phase of matter in which rod- or disk-shaped molecules align like the atoms in a solid, while continuing to flow like a liquid.

Liquid crystals have since become an important part of every-day life, and are integral to the displays in most electronic devices. Meanwhile, the search for new phase of matter, including quantum analogies of liquid crystals, has become an increasingly important field of research. One long-sought example is the "quantum spin nematic", in which the quantum state of magnetic atoms mimic the rod-like molecules of a nematic liquid crystal. The possibility of this new phase of matter was first pointed half a century ago [M. Blume and Y. Y. Hsieh J Appl Phys 40, 1249 (1969); A. F. Andreev, I. A. Grishchuk, J Exp Theor Phys 97, 467 (1984)]. And it is now well understood how competing, or "frustrated", interactions between the ions in a magnet can give rise to a spin-nematic phase [A. V. Chubukov, Phys. Rev. B 44, 4693 (1991); N. Shannon et al., Phys. Rev. Lett., 96, 027213 (2006)]. None the less, despite their considerable interest, quantum spin nematics have proved very difficult to observe in experiment.

In this work, we report the possible observation of a quantum spin nematic in the naturally occuring mineral, volborthite. Volborthite contains copper atoms which are magnetic, with interactions of type needed to promote a spin-nematic state in high magnetic field [O. Janson et al., Phys. Rev. Lett. 117, 037206 (2016)]. We have probed the magnetic behaviour of the copper atoms by carrying out high-precision measurements of the way in which crystals of volborthite absorb heat, in fields of up to 33 Tesla. The results we find are consistent with the existence of a quantum spin-nematic phase, at temperatures below 1.5 Kelvin, for magnetic fields ranging from 25.5 to 27.5 Tesla.

This work was published in the article: "Possible observation of quantum spin-nematic phase in a frustrated magnet", Yoshimitsu Kohama, Hajime Ishikawa, Akira Matsuo, Koichi Kindo, Nic Shannon, and Zenji Hiroi, Proc. Natl. Acad. Sci. **116**, 10686 (2019).

### 5. Electromagnetism beyond Maxwell’s Equations

Condensed Matter is a beautiful playground for low-energy realisation of high-energy phenomena. This is especially true in strongly correlated magnetism, where spin-spin correlations can be described as effective gauge fields. Quantum spin ice is a canonical example of this physics, supporting an emergent electromagnetic gauge field. Searching for photon and magnetic monopoles in these systems is an exciting venue, but over the past years, a growing question has been stimulating the community: is it possible to go beyond electromagnetism ?

To join this quest, the approach of the TQM Unit has been to seek the realisation of higher-rank gauge fields, in collaboration with Dr. Owen Benton (RIKEN) and Dr. Ludovic Jaubert (CNRS, Univ. Bordeaux). Where electromagnetism is of rank 1, described by vector fields and scalar charges, higher-rank gauge fields require a higher dimension of description; e.g. the fields are tensors and the charges are, counter-intuitively, vectors ! As is often the case in physics, going to higher dimensions offer a plethora of new opportunities, but every rose has its thorn. To go from vector to tensor gauge fields, substantially more complex spin-spin correlations are needed. Finding a model stabilising such correlations is a challenge by itself, which is why most attempts so far have led to largely artificial Hamiltonians, away from materials.

The breakthrough of the TQM Unit has been to propose a realistic microscopic model supporting such rank-2 gauge field, on the breathing pyrochlore lattice made of alternating small and large tetrahedra. This proposal is based on a classical field theory, supported by Monte Carlo simulations, and shows the typical four-fold pinch point of a rank-2 gauge field, as would be observed in polarised neutron scattering experiments (cf. Fig.). The model is simply a Heisenberg antiferromagnet with Dzyaloshinskii–Moriya perturbations on the small tetrahedra. Such Hamiltonian is strongly reminiscent of the parametrisation of the Ba3Yb2Zn5O11 breathing pyrochlore, suggesting that future experimental realisations are only one step away!

This work is described in the preprint: “Rank-2 U(1) U(1) spin liquid on the breathing pyrochlore lattice”, Han Yan, Owen Benton, Ludovic D.C. Jaubert, Nic Shannon, arXiv:1902.10934.