# 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 FY2019 running from April 1st 2019 until March 31st 2020.

Results which appeared since April 1st, 2021, are described below:

### 1. Hidden phases born of a quantum spin liquid

Quantum spin liquids (QSL) have generated considerable excitement as phases of matter with emergent gauge structures and fractionalized excitations. In this context, phase transitions out of QSLs have been widely discussed as Higgs transitions from deconfined to confined phases of a lattice gauge theory.  However the possibility of a wider range of novel phases, occuring between these two limits, has yet to be systematically explored.

In this work, we develop a formalism which allows for interactions between fractionalised quasiparticles coming from the constraint on the physical Hilbert space, and can be used to search for exotic, hidden phases. Taking pyrochlore spin ice as a starting point, we show how a U(1) QSL can give birth to abundant daughter phases, without need for fine–tuning of parameters. These include a (charged–) Z2 QSL, and a supersolid.  We discuss implications for experiment, and numerical results which support our analysis.

These results are of broad relevance to QSL subject to a parton description, and offer a new perspective for searching exotic hidden phases in pyochlore magnets.

This work is described in the preprint: "Hidden phases born of a quantum spin liquid: Application to pyrochlore spin ice" Hyeok-Jun Yang, Nic Shannon and SungBin Lee, arXix:2102.06189

### 2. Generic Field-Driven Phenomena in Kitaev Spin Liquids: Canted Magnetism and Proximate Spin Liquid Physics

Quantum spin liquids are a fascinating example of the “More is Different” philosophy of modern condensed matter physics, featuring fractionalized excitations and emergent gauge structures that can only exist within the confines of a many-body system. For a Quantum spin liquid, a natural question to ask is what happens when an external magnetic field is applied? Here, we will address this question in the context of the famous two-dimensional Kitaev model considering lattices beyond the honeycomb.

Depending on the lattice, the Kitaev model hosts different quantum spin liquids with Abelian or non-Abelian topological order in the low-field limit. Both spin liquids have in common that they are protected by the excitation gap of its flux excitations in the presence of a small magnetic field, but may give way to field-induced phenomena at intermediate field strengths. Sandwiched between the low-field spin liquid physics and the high-field spin-polarized phase, the exploration of magnetic phenomena in this intermediate regime, however, often remains elusive to controlled analytical approaches. We numerically study such intermediate-field magnetic phenomena for two representative Kitaev models on the square-octagon and decorated honeycomb lattice. Using a combination of exact diagonalization and density matrix renormalization group techniques, as well as linear spin-wave theory, we establish the generic features of Kitaev spin liquids in an external magnetic field. While ferromagnetic models typically exhibit a direct transition to the polarized state at a relatively low field strength, antiferromagnetic couplings not only substantially stabilizes the topological spin liquid phase, but generically lead to the emergence of a distinct field-induced intermediate regime, separated by a crossover from the high-field polarized regime.

Our results suggest that, for most lattice geometries, this regime generically exhibits significant spin canting, antiferromagnetic spin-spin correlations, and an extended proximate spin liquid regime at finite temperatures. Notably, we identify a symmetry obstruction in the original honeycomb Kitaev model that prevents, at least for certain field directions, the formation of such canted magnetism without breaking symmetries.

This work was published as: "Generic Field-Driven Phenomena in Kitaev Spin Liquids: Canted Magnetism and Proximate Spin Liquid Physics" Ciarán Hickey, Matthias Gohlke, Christoph Berke and Simon Trebst, Phys. Rev. B 103, 064417 (2021).

### 3. Speedup of the quantum adiabatic algorithm using delocalization catalysis

The great hope of quantum computers is that they will ultimately be able to solve problems intractable through classical computation.    One approach to quantum computing is the "adiabatic algorithm", in which a wave function encoding the answer to simple problem, is smoothly evolved into a wave function which encodes the solution of another, more complex, problem.   Unfortunately, the time it takes for this approach to converge grows rapidly with the number of qubits involved.  So any approach which can speed it up, without sacrificing accuracy, is highly desirable.

In this work, we revist an approach to accelerating the quantum adiabatic alogrithm (QAA) known as "catalysis", in which the evolution of the wave function is governed by a Hamiltonian with additional terms, intended to speed its convergence on the final answer.   Considering random field Ising model, catalysed by an XY interaction between spins, we find that the greatest speedup occurs where the catalysis leads to a many-body delocalization of eigenstates, allowing a quantum computer to more quickly explore the space of possible solutions.

These results offer the first evidence that many-body delocalisation could play an important role in quantum computers, and have the potential to enhance the success of the QAA in real-world applications.

This work was published as: "Speedup of the Quantum Adiabatic Algorithm using Delocalization Catalysis" Chenfeng Cao, Jian Xue, Nic Shannon and Robert Joynt, Phys. Rev. Research 3, 013092 (2021).

### 4. Quantum typicality study on the finite-temperature mirror symmetry breaking of the S=1/2 Shastry-Sutherland model

Quantum frustrated magnets have been known as a source of the exotic phenomenon because of the competing interactions and quantum fluctuations, but we have known difficulties in estimating the thermal properties of them except for very few examples.

The quantum typicality method is one of state of art numerical methods to investigate the finite-temperature properties of the quantum frustrated magnetism without any bias in our quantum Hamiltonian although the treated system sizes are still small.  A problem we may have is if the typicality method works well to reveal the presence of the finite-temperature phase transition because we may feel that the smallness of the system sizes is crucial, especially near the transition temperature.

We applied this method to a simple two-dimensional quantum frustrated model, S=1/2 Shastry-Sutherland Heisenberg model. This model is known to have a plaquette-singlet ground state which breaks spontaneously the mirror symmetry of the Hamiltonian in a parameter region 0.68<J/JD<0.76, where J and JD are inter and intra-dimer interactions (a). It is also known the mirror symmetry breaking pattern in this state can not be observed in the two-point correlation function level in finite-size systems because the expected two-fold degeneracy does not happen in the ground state of the finite-size systems. In contrast to the ground state properties in the finite-size clusters, in our paper, our quantum typicality calculations succeed in revealing clear signatures of the finite-temperature phase transition associated with the mirror symmetry breaking within the two-point correlation function level even in finite-size clusters.

We also find that the origin of these surprising signatures is from the presence of several two-fold degenerated excited states by means of the thick-restarted Lanczos method, and this fact can also mean that the typicality method has the ability to detect the presence of the degenerated excited states. Indeed, the following figures, the real-space nearest-neighbor two-point correlation function at moderate temperature, can show the two plaquette-singlet patterns (c) or (d) depending on the different initial random states in our quantum typicality method. We hope that our findings have a future potential to understand the thermal properties of the other quantum frustrated magnets via the typicality method.

This work is post as a preprint on arXiv: “Signatures of the finite-temperature mirror symmetry breaking in the S=1/2 Shastry-Sutherland model”, Tokuro Shimokawa, arXiv:2012.15546

### 5. Novel features of Spin Hall and Chern insulator phases realized by triplet excitations

Triplet excitations found in disordered magnetic insulators are a promising new avenue for the study of topological band-theory. These quasi-particles carry momentum, energy and spin and may realize diverse topological phases. Unlike magnons, which exist by virtue of broken time-reversal symmetry, triplons do not require magnetic order to appear. Therefore, although a TR-breaking field may cause them to exhibit quantum Hall like physics as evidenced by the thermal transport coefficients, under TR-symmetry other phases such as $$\mathcal{Z}_2$$ can also be realized as shown by [D. G. Joshi and A. P. Schnyder, Phys. Rev. B 100, 020407 (2019)].

However, triplets are bosonic in nature and therefore do not enjoy the same protection as electrons in the Kane-Mele model. This leaves open the question of which symmetry could protect this phase and whether there is a faithful analogue to the Kane-Mele model realizable in a magnetic insulator. In this work we study the bilayer kagome lattice model. We derive a bond-wave Hamiltonian with all possible symmetry allowed nearest neighbor spin-exchange interactions and include several next-nearest neighbor terms.

In analogy to [H. Kondo, Y. Akagi, H. Katsura, Physical Review B. 99, 041110(R) (2019)] we define a pseudo-time reversal operator, which we show is realized by $$TR \times U (1)$$symmetry. While this is a valid symmetry of the Hamiltonian the $$\mathcal{Z}_2$$ topology is preserved, however we find that the allowed symmetric exchange anisotropies break this symmetry and thereby destroy the topology. We support this claim by calculating the triplet bands and the topological invariants associated with them in the continuum limit and verify the existence and absence of topological edge-modes in the finite lattice. Additionally we calculate the Thermal Hall and spin Nernst signals in the TR preserving case and the applied magnetic field case.

This work is post as a preprint on arXiv: “Novel features of Spin Hall and Chern insulator phases realized by triplet excitations”, Andreas Thomasen, Karlo Penc, Nic Shannon and Judit Romhányi, arXiv:2012.11765

### 6. Fracton excitations in classical frustrated kagome spin models

Fractons are exotic topological excitations in many-body systems. They are immobile due to intrinsic dynamical constraints from gauge symmetries, dipole conservations, or subsystem symmetries. Fractons are at the center of a wide range of interesting theoretical physics problems, from quantum information, quantum gravity, to topological order and non-conventional critical behaviors.

The experimental realization of fractons, however, is not a trivial issue due to the complicated interactions required. Here we explore fractons may exist in one of the most well-known frustrated magnet model: the Kagome magnetic with dominant antiferromagnetic interactions for the nearest neighbors.

We found that with properly-tune further neighbor interactions, the model host fracton excitations that are immobile, with unconventional mobility for the fracton bound states. We then further investigate several variations of the model and the fate of fractons.

Given the relative simpleness of these models, it is possible to realize them in insulators or cold atom experiments. This work also sets the foundation for future experimental pursuit of fractons.

This work is post as a preprint on arXiv: "Fracton excitations in classical frustrated kagome spin models", Max Hering, Han Yan, and Johannes Reuther, arXiv:2012.08167

### 7. Emergence of a nematic paramagnet via quantum order-by-disorder and pseudo-Goldstone modes in Kitaev magnets

The interplay of competing interactions and quantum fluctuations in spin systems can give rise to new and exciting physics. A prominent realisation with these competing interaction are Kitaev materials, which exhibit strong spin-orbit coupling leading to bond-dependent interactions between spin-1/2 constituents. If the Kitaev interaction is dominant, these materials have the potential to realize a quantum spin liquid phase. However, many such materials have additional interactions that stabilize magnetically ordered states. A paradigmatic example is \alpha-RuCl3, which is known to stabilize zigzag magnetic order. Upon applying a magnetic field, however, the zigzag order vanishes, while recent thermal Hall conductivity measurements indicate the existence of the much-desired Kitaev spin liquid. Consequently, Kitaev materials exposed to an external magnetic field have recently been a subject of intensive studies.

In this work, we investigate the Kitaev-Gamma-Gamma' model which has been suggested as a minimal model for \alpha-RuCl3. By using matrix product state techniques and linear spin wave theory, we show that a nematic paramagnet emerges in the quantum model in a magnetic field along the [111] direction. The nematic paramagnet is characterized by a spontaneous breaking of a lattice-rotational symmetry. We trace its origin to the frustrated ferromagnetic phase of the corresponding classical model. A homogeneous canting of the magnetic moments away from the field axis occurs as a result of a competition between the magnetic field and the anisotropic spin-exchange couplings. Classically, no preferred canting direction exists resulting in a continuous, U(1)-symmetric, manifold of ground states. A mechanism known as quantum order-by-disorder selects a discrete set of states, the nematic paramagnetic states, out of an emergent continuous manifold of ground states in the classical model. The continuous symmetry implies a gapless Nambu-Goldstone mode, however, quantum fluctuations introduce a small gap. Such a phenomenology has become known as a pseudo-Goldstone mode.

The nematic paramagnet exists in a wide range of parameters. Thus, this phase is likely relevant to \alpha-RuCl3 and possibly other Kitaev materials. Consequently, we complement our work by presenting dynamical signatures, i.e. the dynamical spin structure factor, of the nematic paramagnet and the adjacent high-field paramagnet. Although we find the Kitaev spin liquid to be stabilized only in the vicinity of pure Kitaev coupling, remnants of the fractional excitations of the KSL continue to exist in wider range of parameters illustrating a proximate spin liquid.

In summary, this work elucidates the origin of a nematic paramagnetic phase that is stabilized in a wide range of parameters relevant to Kitaev materials, and presents dynamical signatures that are potentially relevant to understand recent experiments on Kitaev materials.

This work was published in the article: "Emergence of a nematic paramagnet via quantum order-by-disorder and pseudo-Goldstone modes in Kitaev magnets", Matthias Gohlke, Li Ern Chern, Hae-Young Kee and Yong Baek Kim, Phys. Rev. Research 2, 043023 (2020).