# FY 2022 Research & Annual Reports

The Theory of Quantum Matter (TQM) Unit carries out research into a wide range of problems in condensed matter and statistical physics, with an emphasis on the novel phases of matter found in frustrated magnets. Other topics of current interest include the application of machine learning to problems in physics, and algorithmic aspects of quantum computing.

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 FY2021 running from **April 1**^{st}** 2021** until **March 31**^{st}** 2022**.

Recent research results, and preprints, are described below.

### Spin nematics meet spin liquids: Exotic quantum phases in the spin-1 bilinear-biquadratic model with Kitaev interactions

New discoveries are often made on the border between different disciplines. One major discipline in solid state physics is dedicated to quantum spin liquids, an unconventional state of matter accompanied by emergent gauge fields, topological order, and fractionalized excitations. Another concept is that of spin nematics, a magnetically ordered state dominated by quadrupole moments, which breaks spin-rotation symmetry by selecting an axis, while not choosing a particular direction. Usually seen as two separate areas of study, we are interested in combining those two disciplines, by asking the question: “What happens, when a spin nematic and a spin liquid meet?”

To answer this question, we showed that the *S*=1 Kitaev model under the influence of bilinear-biquadratic interactions hosts many unconventional ordered and disordered phases. We obtain a comprehensive phase diagram including chiral ordered and quadrupolar ordered phases, in addition to already known ferro, antiferro, zigzag and stripy phases. Intriguingly, we find that the competition between Kitaev and positive biquadratic interactions also promotes a noncoplanar finite-temperature spin liquid state, with macroscopic degeneracy and finite scalar chirality

Our results show that the competition between spin liquid and spin nematic phases is a promising way to explore new magnetic states of matter.

Figure: Ground state phase diagram of the *S*=1 Kitaev model under the influence of bilinear-biquadratic interactions hosts many ordered and disordered phases, such as ferromagnetic (FM), antiferromagnetic (AFM), zigzag, and stripy orders, and spin-nematic ferroquadrupolar (FQ) order. The competition between chiral magnetic order, the Kitaev spin liquid (SL), and quadrupolar semi-order (SO) gives rise to unconventional phases, such as the twisted conical (TC), quasi-one-dimensional (q1D) coplanar, and noncoplanar (NC) ordered phases.

This work was published as "*Spin nematics meet spin liquids: Exotic quantum phases in the spin-1 bilinear-biquadratic model with Kitaev interactions*", R. Pohle, N. Shannon, Y. Motome, Phys. Rev. B 107, L140403 (2023).

### Extended Quantum Spin Liquid with Spinon-like Excitations in an Anisotropic Kitaev-Gamma Model

Quantum spin liquids have become an important research subject in condensed matter physics due to their exotic emergent properties: fractional excitations, topological order, emergent gauge fields, anyonic exchange statistics, etc. Kitaev's honeycomb model is a paradigmatic spin-1/2 model in this context, due to being exactly solvable and featuring a quantum spin liquid ground state in terms of itinerant Majorana fermions in a static \(\mathbb Z_2\) gauge field. The bond-dependent Kitaev spin-exchange is realized in certain magnets with strong spin-orbit coupling. This mechanism, however, introduces additional spin exchanges that spoil the exact solvability of the Kitaev model. Many candidate Kitaev materials have been proposed among which alpha-\(RuCl_3\) has gained much attention due to a putative QSL phase in an in-plane magnetic field, and even more so, since the measurement of a half-quantized thermal hall effect was reported suggesting the existence of emergent Majorana fermions. As a minimal model for alpha-RuCl_3, the spin-1/2 Kitaev-\(\Gamma (K\Gamma)\) model on the honeycomb lattice with ferromagnetic Kitaev exchange and positive symmetric off-diagonal \(\Gamma\) exchange has been proposed. Many different methods have been applied, yet no clear understanding of its quantum ground state has emerged. Among the suggested ground state phases are not only magnetically ordered states, such as zigzag, ferromagnet, six-sublattice, or incommensurate spiral order, but also quantum paramagnetic phases such as a putative gapped QSL, a lattice-nematic paramagnet, or a gapless QSL with multiple Majorana-Dirac nodes.

Within this project we like to change the perspective and focus instead on an aspect rarely included in theoretical works on Kitaev materials: we additionally tune the strength of the spin exchange spatially, ranging from a limit of uncoupled chains to spatially equal, yet still strongly anisotropic, spin exchange. In fact, such spatial anisotropy may either be intrinsic due to a reduced symmetry of the underlying lattice, such as \(C2/m\) instead of a full \(C_3\) rotational symmetry, or spatial anisotropy can be induced by applying external pressure or strain, which possibly realizes various different QSL.

This result is described in the preprint "Extended Quantum Spin Liquid with Spinon-like Excitations in an Anisotropic Kitaev-Gamma Model" Matthias Gohlke, Jose Carlos Pelayo, Takafumi Suzuki, arXiv:2212.11000

The one-dimensional limit features an emergent Tomonaga-Luttinger liquid (TLL) with the same critical properties as the antiferromagnetic Heisenberg (AFH) chain. Since the TLL is a critical state, most TLLs are unstable under adding small inter-chain coupling. An exception is \emph{sliding Luttinger liquids that occur if the inter-chain coupling competes with the dominant correlations along the chains. As a consequence, long-range order is suppressed. Here, we find that a similar mechanism arises in the strongly anisotropic K\(\Gamma\) model: The TLL phase of the K\(\Gamma\) chain turns into an extended QSL phase. This QSL retains the dominant algebraic correlations along the chains and features spinon-like excitations that are characteristic for the AFH chain. This contrasts the emergent Majorana fermions and \(Z_2\) fluxes of the KSL.

**Quantum computers can, sometimes, get what they need**

While quantum computers hold great promise, the devices available in the near future will continue to be “noisy, intermediate-scale quantum” (NISQ) platforms, capable of performing only a limited number of quantum operations. For this reason, the development of the quantum computing rests in part on finding algorithms which can run within the limited resources of a NISQ machine.

In an earlier work, [Yunlong Yu et al., Phys. Rev. Research 4, 023249 (2022)] we introduced an alogrithm, the “adaptive-bias Quantum Approximate Optimisation Algorithm” (ab-QAOA), which is capable of solving difficult classical optimization problems with limited quantum resources. In a continuation of this project, we have now extended the ab-QAOA to a canonical hard classical optimization problem, "SAT" or the simultaneous satisfiability of logical clauses.

In collaboration with researchers from Tsinghua University, The Hong Kong University of Science and Technology, and the University of Wisconsion, Madison, we have now shown how the the ab-QAOA is able to solve a class of SAT problems using only resources achievable in a NISQ device. This confirms the ab-QAOA as a quantum algorithm which has the potential to bring a speedup in solving real-world problems.

Figure: Comparison of the resources required to solve a SAT optimisation problem on a quantum computer, using the established Quantum Approximate Optimization Algorithm (QAOA), and the adaptive-bias Quantum Approximate Optimisation Algorithm” (ab-QAOA). The problem considered is Max-1-3-SAT+, for which the QAOA shows a dramatic increase in the resources required ("level p") as the clause density ("α") is increased. In contrast the ab-QAOA is able to solve problems at all clause densities considered, with much lower overhead in resources.

This work is described in the preprint "Solution of SAT Problems with the Adaptive-Bias Quantum Approximate Optimization Algorithm", Yunlong Yu, Chenfeng Cao, Xiang-Bin Wang, Nic Shannon and Robert Joynt, arXiv:2210.02822

### Spurious Symmetry Enhancement and Interaction-Induced Topology in Magnons

From Néel order in the mid 20th century to skyrmion phases in the 21st, magnetically ordered materials have been a constant source of insights into the collective behavior of matter. The coherent spin wave excitations, or magnons, about these magnetic textures provide invaluable information about magnetic structures and couplings. They are also interesting in their own right: as a window into many-body interactions and quasiparticle breakdown, as a platform for investigating band topology, and as an essential ingredient in the functioning of many spintronics devices. One of the most useful theoretical tools at our disposal to understand magnons is an expansion in powers of inverse spin \(S\) based on the Holstein-Primakoff bosonization of quantum spins. The single particle spectrum arising from spin wave theory to lowest order---known as linear spin wave theory (LSWT)---is often used with great success to constrain magnetic couplings from experimental data. This theory is known to fail qualitatively in cases where coupling between single and multi-particle states becomes important such as in highly frustrated magnets and non-collinear spin textures, but also close to quantum phase transitions. Another, more subtle way, in which LSWT can fail qualitatively is called order-by-disorder where spurious ground states and symmetry enhancement exist at the semi-classical level that are lifted by fluctuations. In some instances of quantum order-by-disorder, a spurious continuous symmetry forces the presence of a gapless mode within LSWT where none should be present. In this project, we focus on a related instance of this physics where, instead of failing to capture degeneracy breaking in the ground state, the LSWT instead does not fully capture symmetries that affect degeneracies higher up in the excitation spectrum. The goals of this paper are to spell out ways in which spin wave theory can lead to spurious degeneracies in excitations across the Brillouin zone and to supply a simple, general way to resolve them.

The cases we consider fall into two classes: The first class is where the lattice symmetries are not manifest for exchange couplings between moments out to \(n\)th nearest neighbors but where the symmetries do manifest for longer-range couplings ({\it shell anomaly}). The second class is more subtle in that LSWT does not capture certain kinds of exchange anisotropy ({\it anisotropy blindness}). Then, LSWT fails to produce the correct magnon spectrum at a qualitative level and spurious symmetry-protected magnon degeneracies occur. We show that degeneracy breaking occurs by carrying out a matrix product state based time evolution (DMRG+tMPO) to resolve band splittings nonperturbatively. While the most straightforward LSWT does not capture the symmetries of the magnetic Hamiltonian, one may show that the symmetry breaking terms, treated perturbatively, lead to effective magnon hopping terms that do resolve spurious degeneracies. This fact leads us to propose a general solution to the problem by including all symmetry-allowed exchange couplings out to some shell.

These results are described in the preprint "Spurious Symmetry Enhancement and Interaction-Induced Topology in Magnons" Matthias Gohlke, Alberto Corticelli, Roderich Moessner, Paul A. McClarty, Alexander Mook, arXiv:2211.15157

### Quantum paramagnetic states in the spin-1/2 distorted honeycomb-lattice Heisenberg antiferromagnet - application to Cu_{2}(pymca)_{3}(ClO_{4})

Quantum magnetism has been an essential subject of many investigations through the cooperation of experiments and theory. A spin-1/2 honeycomb-lattice antiferromagnet, Cu_{2}(pymca)_{3}(ClO_{4}), recently joined a family of such subjects. The exciting feature is that the material exhibit gapped paramagnetic behavior down to 2 K, although the simple spin-1/2 honeycomb-lattice antiferromagnetic Heisenberg model exhibits the Neel ordered ground state. The recent X-ray diffraction experiments suggested at least three intralayer exchange interactions in the honeycomb-lattice plane. In our work, we numerically investigated the ground-state properties and dynamics of the extended Honeycomb-lattice antiferromagnetic Heisenberg model with three exchange interactions \(J_A, J_B, J_C\) as a minimal model for the Cu_{2}(pymca)_{3}(ClO_{4}). We remeasured the magnetic susceptibility of its polycrystalline sample with special care and determined the exchange interactions of this material through the comparison with numerical results based on a quantum Monte Carlo method. The obtained interactions revealed that a hexagonal-singlet-type (HS) state could be realized in this material. We also report the ground-state phase diagram and the evolution of the spin gap in the \( J_A/J_C-J_B/J_C\) plane (see the left-hand side figure), which are informative for future and other honeycomb-lattice materials. The characteristic four energy band structures in the HS state in the spin dynamics (see the right-hand side figure) are helpful in clarifying the ground and excited states of Cu_{2}(pymca)_{3}(ClO_{4}) by future neutron scattering measurements.

This work was published as: "Quantum paramagnetic states in the spin-1/2distorted honeycomb-lattice Heisenberg antiferromagnet: Application to Cu_{2} (pymca)_{3} (ClO_{4})" Tokuro Shimokawa, Ken'ichi Takano, Zentaro Honda, Akira Okutani and Masayuki Hagiwara, Phys. Rev. B 106, 134410 (2022)

### SUSY Quantum Mechanics on a lattice

The ultimate goal of quantum mechanics is to solve Schrödinger's equation. This equation is postulated to describe the dynamics of quantum mechanical objects. Focusing on a fixed moment in time one can study the time-independent form of this equation which consists of two parts:

One part contains information about a local potential surrounding the object in question the other part contains information about its kinetic energy. This second part usually contains a differential operator of the second order. Starting with Newton and Leibnitz many physicists and mathematicians have developed what is known as differential calculus to solve these problems. To unleash the full power of calculus one must have the underlying space continuous. However, there are some physical systems, for instance the interior of a crystal or nucleus, in which this assumption might not entirely be justified. In these discrete settings, obtaining a solution to Schrödinger's equation without the use of differential calculus becomes much harder.

On the other hand, even for the continuous case only a few potentials allow for an exact solution. Interestingly, all these known exact solutions can be obtained by the means of supersymmetric quantum mechanics. In this framework one uses a super potential which factorize the initial potential. Upon the super potential one can define operators that act similarly to the raising and lowering operators known from the algebraic solution of the quantum harmonic oscillator. The algebra of these operators allows for the construction of two partner Hamiltonians whose spectra and eigenfunctions are equivalent up to the ground state. Therefore, solving one of these partner Hamiltonians immediately yields the solution of the other. This procedure can be further generalized and thus one finds the solution of an entire Hierarchy of Hamiltonians [1].

In our work we combined the concept of "Exact discretization", which is a formulation of discrete operators that have many of the desired properties of differential operators recently developed in References [2,3], with the principles of supersymmetric quantum mechanics and suggest a computational procedure allowing to easily solve some lattice problems.

[1] F. Cooper, A. Khare and U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rept. 251, 267-385 (1995)

[2] V. Tarasov, “Exact discretization of Schrödinger equation,” Phys. Lett. A 380, 68-75 (2016)

[3] V. Tarasov, “Exact discretization by Fourier transforms,” Commun. Nonlinear Sci. Numer. Simul. 37, 31-61 (2016)

This work was published as "*A Note on Shape Invariant Potentials for Discretized Hamiltonians*" Jonas Sonnenschein, Mirian Tsulaia, Modern Physics Letters A Vol. **37**, No. 23, 2250153 (2022)