# 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.

This work is described 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 2017 running from April 1^{st} 2017 until March 31^{st} 2018.

Some of the more recent projects are described below.

### Recent Results

#### 1. Multipolar edge modes in the breathing kagome model

Since the discovery of topological insulators, a new phase of matter, the concepts of topology have quickly spread, revolutionizing today's physics. Interestingly, the discoveries made in weakly on non-interacting electron systems can be conveyed to the non-interacting excitations of correlated magnetic insulators. The physical realization of Haldane's model in the magnon spectrum of iron-based honeycomb insulators and the Weyl magnons emerging in breathing pyrochlore lattice illustrate well the progress of such magnetic analogues. Advantageously, the topological properties of magnon modes are easily controlled by magnetic field.

Here we take a relevant spin model of the trimerized kagome lattice treating the entangled trimers as the magnetic building blocks. The larger local Hilbert space characterizing the trimers naturally allows for multiplet excitations. In the chiral magnetic state, an effective spin-1/2 and spin-3/2 band is formed beside the gapless magnon mode. A small magnetic field removes the degeneracy of the multiplets and the excitations become topologically nontrivial with Chern numbers \(C_m = 2m\).

Increasing the magnetic field, the bands of the quartet undergo a topological transition when a spin-3/2 Dirac cone is formed by the touching of four bands. The spin-1/2 doublet carries only dipolar degrees of freedom, providing an analogue to the electronic systems. The spin-3/2 quartet, on the other hand, encompass higher multipolar characters which in the topologically nontrivial regime is manifested in novel multipolar edge states.

Topologically robust edge states traveling unimpeded at the boundaries are appealing for low- energy consuming fast spintronical devices. Multipolar edge states, put forward here, may be the stepping stone for new directions in these endeavors. A quadrupolar edge mode, for example, can couple to electric field and open a new route towards electric access and control of edge states emerging in the excitation spectrum of magnetic insulators.

These results are dicribed in the preprint : "*Multipolar edge states in the breathing kagome model*" J. Romhányi arXiv:1801.07950

(a) Spin-3/2 Dirac cone formed by the quartet excitations and the surface mapped out by the d vectors. The origin of the vectors touches the surface corresponds to the degeneracy of the bands at the Dirac point.

(b) In the topologically nontrivial regime the fully gapped bands are characterized by large Chern numbers and the origin is encompassed by the surface of the d vectors which form a skyrmion in the Brillouin zone.

(c) Opening the boundaries in the y-direction we find in-gap chiral edge modes.

(d) Novel quadrupolar edge states appearing as a consequence of larger spin-3/2 multiplet.

#### 2. Magnetoelectric memory function with optical readout

The formation of an ordered stated is conventionally associated with the breaking of a symmetry. For example, magnetic order breaks time reversal symmetry while the ordering of electric dipole moments arises with the breaking of spatial inversion. In multiferroic materials more than one primary order coexist. Magnetoelectric multiferroics break both inversion and time-reversal symmetries. This collective symmetry breaking can lead to a coupling between the order parameters, so that one order can be manipulated with the conjugate field of the other. The prospect of electric control of magnetism makes magnetoelectric materials desirable for applications. In this research we demonstrate another functional property of magnetoelectric compounds, the so called optical directional anisotropy. On the account of the coupling between the magnetic and electric degrees of freedoms, the reciprocity of light propagation can be violated. In other words, the absorption in a material becomes different for light propagating in the opposite directions.

In the new generation magnetoelectric memory devices the different magnetoelectric domains serve as the bits of information. Here we reveal the optical read-out of such a magnetoelectric memory state, realized in antiferromagnetic and antiferroelectric LiCoPO_{4}. LiCoPO_{4} exhibits linear magnetoelectric effect marked by finite components of the magnetoelectric susceptibility, \(\chi^{em}_{yx}\). The different domains are characterized by positive and negative \(\chi^{em}_{yx}\), enhancing or a reducing the refractive index \(N\). For example, in case of linearly polarized light with\( (E^\omega\| y, H^\omega\| x) \)propagating along the \(+z\) direction the refractive index is \(N_{+z} = \sqrt{\epsilon_{yy}\mu_{xx}}\pm\chi^{em}_{yx}\), where \(\epsilon_{yy}\) and \(\mu_{xx}\) are elements of the dielectric permittivity and magnetic permeability tensors, and the sign \(\pm\) distinguishes the two domains. Therefore, when the optical magnetoelectric effect is sufficiently strong, the domain characterized by \(\chi^{em}_{yx}(\omega)<0\) may become transparent, while the other domain absorbs photons more intensely. Reversing the light propagation is equivalent to interchanging the domains: \(N_{-z} = \sqrt{\epsilon_{yy}\mu_{xx}}\mp\chi^{em}_{yx}\). The direction of one-way-transparency depends on the crossed electric and magnetic fields. Such optical magnetoelectric effect may be used in hybrid optical architectures making LiCoPO_{4} an archetype of future optical memory devices and logic gates.

These results are described in the preprint : "*Magnetoelectric memory function with optical readout*" V. Kocsis, Y. Tokunaga, Y. Taguchi, Y. Tokura, J. Vit, T. Rõõm, U. Nagel, J. Romhanyi, K. Penc, I. Kézsmárki, and S. Bordács, arXiv:1711.08124

**Figure 2.1**:Magnetoelectric domains and optical magnetoelectric effect in LiCoPO_{4}.

(a) Local electric polarization and magnetization inside the unit cell for the two magnetoelectric domains $\alpha$ and $\beta$.

(b) Domains $\alpha$ and $\beta$ selected by the poling fields \(H_x>0,E_y>0\) (red) and \(H_x>0,E_y<0\) (blue).

(c) Absorption spectra measured after the four possible poling. The sign of the crossed fields selects the domains.

(d) Absorption after poling in the \(H_x>0,E_y>0\) (red) and\( H_x>0,E_y<0\) (blue) configurations for light propagation along the +z direction (full symbols) and the −z direction (open symbols).

#### 3. Frustrating quantum spin ice

The search for quantum spin liquids (QSLs) is one of the defining problems of the current era of condensed matter physics. Many of the most promising candidates are found among rare-earth pyrochlore oxides R_{2}M_{2}O_{7}, a family of materials which includes the celebrated "spin ices" Ho_{2}Ti_{2}O_{7} and Dy_{2}Ti_{2}O_{7}. At present our understanding of these systems is based on perturbative theories around the limit of classical spin ice, and on Quantum Monte Carlo simulations in the case where the transverse exchange has unfrustrated sign. This leaves open the question of what happens perturbative limit, particularly where transverse exchange is frustrated and Quantum Monte Carlo simulations are unfeasible. This question is of particular relevance in understanding Pr-based pyrochlores, since microscopic calculations predict them to exhibit frustrated transverse exchange.

In this work, we address the problem of quantum spin ice with frustrated transverse exchange, considering a very simple model: the XXZ model on a pyrochlore lattice. We show that upon increasing the strength of frustrated transverse exchange the U(1) QSL known from quantum spin ice transforms into a new phase: the nematic QSL. The nematic QSL possesses strong quantum entanglement and gapless emergent photons, just like U(1) spin liquid. In addition, it acts as a quantum spin nematic - a quantum analogue of a classical liquid crystal. These results show that even the simplest models of pyrochlore magnets can support a range of different QSL ground states. They also introduce a new method, the cluster-variational calculation (cVAR), which may be helpful in a wide range of frustrated quantum systems.

**Figure 3.1**: Phase Diagram for frustrated quantum spin ice : (a) Quantum ground state found in cluster-variational (cVAR) calculations. (b) Finite-temperature phase diagram found in classical Monte Carlo simulations. The model considered is the XXZ model on a pyrochlore lattice, with * *\(J_{zz} = J\cos\theta, J_\pm = -J/2\sin\theta\).

These results are described in the preprint : "*Quantum spin ice with frustrated transverse exchange: from pi-flux phase to nematic quantum spin liquid*" Owen Benton, L. D. C. Jaubert, Rajiv Singh, Jaan Oitmaa, Nic Shannon arXiv:1802.09198

#### 4. 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 4.1**: 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.

This work was published as O. Coquand *et. al* "*A glassy phase in quenched disordered graphene and crystalline membranes*, Phys. Rev. E **97**, 030102 (2018).

#### 5. 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 5.1:** Spin configurations for 0- and π-coplanar states and umbrella state, respectively.

**Figure 5.2:** Phase boundaries among the two coplanar and umbrella states just below the saturation field.

This work was published 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).

#### 6. 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 6.1**: 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.

This work was published as K. Essafi, O. Benton & L.D.C. Jaubert "*Generic Nearest-Neighbour Kagome Model: XYZ & Dzyaloshinskii-Moriya, and Comparison to Pyrochlore*", Phys. Rev. B **96**, 205126 (2017).

#### 7. 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 7.1**: 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 was published in Nature Physics, "*Experimental signatures of emergent electrodynamics in Pr _{2}Hf_{2}O_{7}*"