# Research & Annual Reports

The Theory of Quantum Matter 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 frustrated magnets. Recurring themes include quantum and classical spin liquids, especially those where topology plays an important role.

Recent projects are described below, with links to preprints.

Earlier work, along with journal references, can be found in the **Annual Reports** listed in the menu on the left side of this page. Reports are designated by the Japanese financial year, with FY 2016 running from April 1^{st} 2016 until March 31^{st} 2017.

### Activities and Findings

#### 1. Experimental signatures of emergent electrodynamics in Pr_{2}Hf_{2}O_{7}

Spin ice, a magnet which mimics the behaviour of protons in water ice, has become famous as a condensed-matter system which realises magnetic monopole excitations. Recently there has also been considerable excitement about the possibility of a "quantum spin ice", in which spin fluctuate between different ice-like configurations. Following extensive numerical simulations, there is now a theoretical consensus that such a system would realise an emergent electrodynamics, in which the magnetic monopoles of spin ice become fractional spin excitations, accompanied by quantized excitations of a U(1) gauge field: photons. Identifying a quantum spin ice in experiment, however, has proved challenging.

In an earlier work [O. Benton et al., Phys. Rev B 86, 075154 (2012)], members of the TQM Unit identified a key experimental feature of the photons in a quantum spin ice; the suppression of the "pinch points" seen in neutron scattering experiments on classical spin ice. Now, in collaboration with a team of experimental physicists based at the Paul Scherer Institute (Switzerland), members of the TQM have probed the magnetic excitations of a promising candidate quantum spin ice, Pr_{2}Hf_{2}O_{7}. This analysis confirms the existence of suppressed pinch--point features in Pr_{2}Hf_{2}O_{7}, consistent with the emergent electrodynamics sought in a quantum spin ice.

**Figure 1.1**: Evidence of emergent electrodynamics in Pr_{2}Hf_{2}O_{7} from inelastic neutron scattering experiments. a) Inelastic neutron scattering at low energy. b) Equivalent prediction of the simplest model of a quantum spin ice. c) Quantitative fit of a quantum spin ice model to a cross section of the data, showing suppression of a pinch point.

This work is described in the preprint "*Experimental signatures of emergent electrodynamics in Pr _{2}Hf_{2}O_{7}*", R. Sibile

*et al.,*arXiv:1706.03604

#### 2. Generic Nearest-Neighbour Kagome Model: XYZ & Dzyaloshinskii-Moriya, and Comparison with Pyrochlore

The kagome lattice, made of corner-sharing triangles, is an archetype of geometrical frustration. Motivated by recent rare-earth materials which may be expected to exhibit highly anisotropic exchange interactions, Dr Karim Essafi, Dr Owen Benton and Dr Ludovic Jaubert studied the most general nearest-neighbour Hamiltonian allowed by symmetry (group C_{3v}). This Hamiltonian includes XYZ and Dzyaloshinskii-Moriya (DM) interactions.

Their group theoretical approach allows for a systematic description of the ground-state for a broad region of parameter space. The resulting picture is a connected map of ordered phases and spin liquids (see Fig.2.1). In particular it sheds new light on the mapping between three different spin liquids (along the XXZ, XXZ^{+} and XXZ^{-} lines).

**Figure 2.1**: Zero-temperature phase diagram of the XXZ model with DM interactions (XXZDM). The phase diagram can be divided into five continuous regions when represented on a sphere. The corresponding spin configuration to each region are shown in panel (d).

The XYZDM model is “asymmetric” with respect to spin chirality. This chiral asymmetry mixes states with positive chirality (kappaz = +1) with in-plane ferromagnetic states. This mixing leads the lattice to for stripe orders, with or without crossings.

The preprint of this work appeared as K. Essafi, O. Benton & L.D.C. Jaubert "*Generic Nearest-Neighbour Kagome Model: XYZ & Dzyaloshinskii-Moriya, and Comparison to Pyrochlore*", arXiv:1706.09101 (2017).

#### 3. A Glassy Phase in Quenched Disordered Graphene and Crystalline Membranes

Graphene is one of the most studied materials due to its unique mechanical, optical, thermal, chemical and electronic properties: high mechanical strength, optical transmittance, thermal conductivity, carrier mobility. These properties make graphene a very prommising compound for a broad range of technological applications: energy storage, ultrafilltration, gaz/electrochemical sensors, drug/gene delivery, bioimaging, chemo/bio sensing, *etc*. Properties of materials in general can be affected by the presence of defects, lattice imperfections (dislocations, grain boundaries, *etc.*) or impurities/vacancies. These defects are either generated during the chemical production process or explicitly added via dilution.

Motivated by the effect of defects on graphene, graphene-like materials and more generally crystalline membranes, Dr Karim Essafi from the TQM unit in collaboration with Oliver Coquand and Prof. Dominique Mouhanna (UPMC, France) and Dr Jean-Philippe Kownacki (Univ. of Cergy, France) investigate the flat phase of crystalline membranes submitted to both metric and curvature quenched disorders using a nonperturbative renormalization group approach. They identify a second order phase transition controlled by a finite-temperature/finite-disorder fixed point unreachable within the leading order of epsilon=4-D and 1/d-expansions. This critical point divides the flow diagram into two basins of attraction: that associated with the finite-temperature fixed point controlling the long distance behaviour of disorder-free membranes and that associated with the zero-temperature/finite-disorder fixed point (see Fig.3.1). Their work thus strongly suggests the existence of a whole low-temperature glassy phase for quenched disordered graphene, graphene-like compounds and crystalline membranes.

**Figure 3.1**: The Renormalization Group flow for quenched disordered membranes. P_{1} is the Gaussian, unstable, fixed point. P_{4} is fully attractive and associated with disorder-free membranes. P_{5} is also fully attractive and controls the low-temperature phase of disordered membranes. Finally P_{c}, unstable w.r.t. temperature governs the phase transition between the two phases.

The existence of this new glassy phase has potential of numerous applications in 2D condensed matter and biophysics.

The preprint of this work appeared as O. Coquand *et. al* "*A glassy phase in quenched disordered graphene and crystalline membranes*, arXiv:1708.08364 (2017).

#### 4. Exact diagonalization and cluster mean-field study of triangular-lattice XXZ antiferromagnets near saturation

Macroscopic degeneracy in frustrated magnetism could be a source of nontrivial magnetic states which are often lifted by fluctuation effects, either thermal or quantum. Tokuro Shimokawa from the TQM unit and co-workers have investigated the effect of the quantum fluctuation on the ordering process of the triangular antiferromagnets with XXZ-type anisotropy near saturation.

In this model, it has been suggested the possible realization of a nontrivial coplanar phase, called π-coplanar (or π) phase, in a certain range of anisotropy parameter J/J_{z} besides the well-known 0-coplanar (or V) and umbrella phases. For the most quantum case of S=1/2, a careful identification analysis with the large-scale exact diagonalization calculations shows that the lowest eigenstate is a chirally antisymmetric combination of finite-size umbrella states for J/J_{z} 2.218 while it corresponds to a coplanar phase for J/J_{z} 2.218. However, we demonstrate that the distinction between 0-coplanar and π-coplanar phases in the latter region is fundamentally impossible from the symmetry-preserving finite-size calculations with fixed magnon number.

We also perform a cluster mean-field plus scaling analysis which enable us to distinguish between the 0- and π-coplanar states since the symmetry of the system is broken by self-consistent mean-fields even on finite-size systems. The obtained results, together with the previous large-S analysis, indicate that the π-coplanar phase exists for any S except for the classical limit (S → ∞). The existence range in J/J_{z} is largest in the most quantum case of S=1/2.

**Figure 4.1:** Spin configurations for 0- and π-coplanar states and umbrella state, respectively.

**Figure 4.2:** Phase boundaries among the two coplanar and umbrella states just below the saturation field.

The paper of this work appeared as Daisuke Yamamoto *et al.*, "*Exact diagonalization and cluster mean-field study of triangular-lattice XXZ antiferromagnets near saturation*", Phys. Rev. B, 96, 014431 (2017).