# 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.

### 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 is described in the preprint: "Quantum paramagnetic states in the spin-1/2 distorted honeycomb-lattice Heisenberg antiferromagnet -- application to Cu_{2}(pymca)_{3}(ClO_{4})" Tokuro Shimokawa, Ken'ichi Takano, Zentaro Honda, Akira Okutani and Masayuki Hagiwara, arXiv:2206.13064

### 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 accepted for publication in Modern Physics Letters A: "*A Note on Shape Invariant Potentials for Discretized Hamiltonians*" Jonas Sonnenschein, Mirian Tsulaia, arXiv:2205.10100