# 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, 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 FY2020 running from **April 1**^{st}** 2020** until **March 31**^{st}** 2021**.

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

### Semi-classical simulation of spin-1 magnets

Spin-1 is special. Indeed, beside displaying dipolar degrees of freedom, a spin-1 can also exhibit on-site quadrupolar degrees of freedom, while retaining its quantum characteristics. Therefore, spin-1 systems have often been use to refer to spin-nematic order in magnetic insulators, Fe-based superconductors or cold atoms. Unlike for spin-1/2, which in the classical limit can be represented by an O(3) vector, for spin-1, an O(3) vector does not completely describe all of what a spin-1 can do. This is due to the fact that a spin-1 intrinsically has quadrupoles, which cannot be describe by an O(3) vector.

In this work, we develop a method based on the extension of the usual su(3) algebra describing a quantum spin-1 into the u(3) algebra. This allows a united framework which enables us to treat dipolar and quadrupolar degrees of freedom on an equal footing. Within our u(3) formalism, we derive equations of motion (EoM) for the objects living in the u(3) algebra, which take a simple form suitable for numerical implementation. Moreover our u(3) approach enables the appropriate formulation to both classical and quantum derivations.

We illustrate our method by applying it to the well-known Bilinear-Biquadratic model on the triangular lattice for the ferroquadupolar state. We support our study through classical low-temperature expansion and classical Monte Carlo (MC) in order to probe the thermodynamical properties as well as quantum multi-bosons theory and molecular dynamics (MD) simulations that allow us to access dynamics. We show that at sufficiently low temperature we can correct for the classical statistics in numerical simulations and retrieve the fully quantum zero-temperature analytical results. Finally, we confirm that our method is also applicable to anisotropic models, which is of experimental relevance.

These results open the door to the efficient simulation of spin-1 magnets, and of spin-1 models arising in the context of cold atoms, and Fe-based supercondcutors.

This work is described in the preprint: "*Semi-classical simulation of spin-1 magnets*" Kimberly Remund, Rico Pohle, Yutaka Akagi, Judit Romhànyi and Nic Shannon, arXiv:2203.09819.

### Spinning on the edge...

Just as people are said to be "frustrated" when they experience conflicting desires, so magnets are said to be frustrated when it proves impossible to satisfy all of the interactions within them. In magnets, this can lead to a situation where radically different forms of magnet order lie very close in energy, and compete even at the scale of a few atoms. This has lead to a paradigm of "multiple-phase competition" explored in earlier work by the Theory of Quantum Matter (TQM) Unit.

Yb2Ti2O7 is one of the best studied of frustrated pyrochlore magnets, and provides one of the most compelling examples of multiple--phase competition. None the less, many of its properties, particularly with respect to dynamics, remain ambiguous. And exploring the consequences of multiple phase competition for dynamics has proved difficult, since it requires the ability to measure on the scale of micro electron volts (\mu eV). This is approximately one millionth of the energy of the chemical bonds within Yb2Ti2O7, and over a thousand times smaller than the energies usually measured in inelastic neutron scattering, the technique of choice for atomic--scale magnetic dynamics.

In a recent study, the TQM Unit have collaborated with experimental scientists based at Oak Ridge National Laboratory (ORNL) and John Hopkins University, as well as former TQM Unit members at MPI-PKS Dresden, CNRS and ETH Zurich, to shed new light on the dynamics of Yb2Ti2O7, at the scale of \mu eV. By making novel use of a neutron-scattering instrument originally designed to study liquids, our collaborators were able to resolve for the first time the energy dependence of "rods" of scattering, characteristic of Yb2Ti2O7.

Remarkably, when these data are plotted as a function of energy over temperature, experimental measurements taken at different energies and temperatures collapse onto a single curve, as shown in Figure 1. Dynamical scaling of this form is extremely unusual, and is found to be perfectly described by the theory of multiple phase competition.

These results, described in the preprint "*Dynamical scaling as a signature of multiple phase competition in Yb2Ti2O7*", arXiv:2202.11085, open a new perspective for the mechanism of multiple-phase competition, which may be relevant to many different frustrated materials.

### Making the most of the time (and qu-bits) available...

Quantum computers of the present, "noisy intermediate-scale quantum " (NISQ) era are severely limited in both the time for which they can run calculations, and the number of qu-bits on which they can run them. For this reason, every nanosecond of computational time, and every available qu-bit is precious. It is therefore of vital interest to identify approaches which can be used to maximise the accuracy of results obtained from finite resources, and in finite time.

In this work, a collaboration of the TQM Unit with researchers from The Hong Kong University of Science and Technology, Tsinghua University, and the University of Wisconsion-Madison, we show that simple extrapolations of results as a function of time, or variance (statistical error), can dramatically increase the accuracy of results obtained with limited resources. We do this through a mixture of numerical simulation, and explicit experimental demonstration, making use of "Jakarta", an IBM quantum computer based on superconducting qu-bits.

We find that extrapolations schemes can lead to (multiple) order-of-magnitude improvements in the accuracy obtained in three of the most popular approaches to quantum computing; quantum annealing; variational quantum eigensolvers (VQE); and quantum imaginary time evolution, at fixed evolution time or circuit depth. These results suggest that simple extrapolation techniques may prove an valuable resource for the NISQ era.

These results are described in the preprint: "*Energy Extrapolation in Quantum Optimization Algorithms*", Chenfeng Cao, Yunlong Yu, Zipeng Wu, Nic Shannon, Bei Zeng, Robert Joynt, arXiv:2109.08132.

### What would a magnetic monopole look like ?

The possibility of finding a magnetic monopole - a particle which acts a point--like source of magnetic field - has fascinated physicists for over a century. None the less, the existence of magnetic monopoles as elementary particles remains an enigma. And while there is compelling evidence for the existence of emergent excitations which behave like magnetic monopoles in spin ice, the direct observation of a point-like source of magnetic field in these materials remains an open challenge.

In this work, we explore what can be learnt about magnetic monopoles from holographic images taken with an electron microscope. This approach shows great promise since it combines the atomic-scale resolution needed to resolve an individual monopole, with extreme sensitivity to magnetic charge, through the Aharonov-Bohm effect. Through experiments on artificial spin ice, we show that holograms can be used to directly image magnetic monopoles, and measure their local magnetic charge. And through numerical simulations, we demonstrate that holographic experiments are capable of resolving magnetic monopoles in pyrochlore spin ice, and their dynamics.

These results suggest that the observation of magnetic monopoles in spin ice is a realistic possibility in an electron microscope with sufficiently-high phase resolution.

These results is described in the preprint *"Observing Magnetic Monopoles in Spin Ice using Electron Holography"*, Ankur Dhar, L. D. C. Jaubert, Cathal Cassidy, Tsumoru Shintake, and Nic Shannon, arXiv:2112.01362.

### Power-laws in 3-d hard-plate lattice gases.

Fully-packed dimer models on bipartite lattice models are of interest in condensed matter physics as a host to highly-correlated Coulomb liquid phases via an effective field theory in which each fully-packed dimer configuration is mapped to a divergence-free vector field on the links of the links of the lattice. This effective field theory correctly describes the power-law columnar phase in two dimensions and the power-law dipolar correlations seen in three dimensions.

This naturally piques the curiosity on what happens when dimers are generalised to plates, where one plate is a composite object consisting of two parallel dimers with infinite attractive potential. The key question then is whether such a lattice gas displays correlated liquid phases that could be understood by thinking in terms of tensor-valued analogs of the vector fields.

To this end we numerically explore the anisotropic fully-packed model using pocket cluster Monte Carlo algorithm and find that it hosts a crystalline sublattice phase, a columnar phase and a layered phase, wherein the the system separates into bilayers with interlayer and intralayer power-law correlations between plates.

We also explore this model in the presence of vacancies using a transfer matrix based line update Monte Carlo algorithm. In the presence of vacancies, the interlayer power-law correlations in the layered phase decay exponentially. Further increasing the density of vacancies disorders the system.

This work is described in the preprints, "*Spontaneous layering and power-law order in the three-dimensional fully-packed hard-plate lattice gas*" G. Rakala, D. Mandal, S. Biswas, K. Damle, D. Dhar, R. Rajesh, arXiv:2109.02619 and "*Phases of the hard-plate lattice gas on a three-dimensional cubic lattice*" D. Mandal, G. Rakala, K. Damle, D. Dhar, R. Rajesh", arXiv:2109.02611

**Parafermionic multicriticality in Ising models**

The critical properties and associated universality of phase transitions in magnets can be well understood under Wilson’s renormalisation group (RG) framework in terms of the RG flows of the effective Hamiltonian where the nature of the transition is determined by the codimension of the ‘relevant’ coupling constants near the fixed points of the Hamiltonian. Apart from the translational, rotational and global scale invariance present at critical fixed points, a conformal invariance (local scale invariance) was first hypothesised by Polyakov in systems with local interactions.

The fixed points of two-dimensional lattice models with discrete symmetries are described by a class of minimal conformal models with known central charge. Moreover some lattice models show extended superconformal symmetry at criticality between bosons and associated Majorana fermions of the underlying conformal model, making them the first models to confirm the predicted but elusive realization of supersymmetry. In two dimensional planar models with discrete symmetries, a dual parafermionic conformal symmetry was shown to exist. Parafermions are spinful generalisations of Majorana fermions which obey non-Abelian statistics and have been proposed as a robust route to topological quantum computing.

Examples of conformal, superconformal and parafermionic symmetries in condensed matter systems abound and have been of sustained interest. However experimental access to these symmetries is extremely hard due to the presence of two or more ‘relevant’ coupling constants in the vicinity of the associated fixed points of the underlying effective Hamiltonian and thus requiring a fine-tuned control over the parameter space of the physical Hamiltonian.

In this work we study the frustrated antiferromagnetic Ising model on a triangular lattice with further neighbor couplings using worm algorithms. We numerically establish the presence of Z6 parafermionic multicriticality at the pinch-off point of the two-step melting transition. Moreover we show that this multicritical point controls the melting physics in a large experimentally accessible neighbourhood of parameter space.

This work is described in the preprint, "*Melting of three-sublattice order in triangular lattice Ising antiferromagnets: Power-law order, Z6 parafermionic multicriticality, and weakly first order transitions*" G. Rakala, N. Desai, S. Shivam, K. Damle, arXiv:2109.03178

### Quantum spin solver near saturation: QS^{3}

Exact diagonalization (ED) method is a traditional and one of the most powerful numerical methods to understand the nature of quantum many-body systems from finite-size cluster calculations. This method enables us to obtain the eigenvectors and eigenvalues of our target Hamiltonian and evaluate all static, dynamical, and thermal quantities without any bias. However, the only disadvantage of the ED method is that the accessible system size is severely limited to small clusters even if we use the modern largest supercomputers. We develop new open-source software -quantum spin solver near saturation: QS^{3}- based on the thick-restart/conventional Lanczos algorithm. This software is powerful for spin-1/2 XXZ-type quantum spin models near fully polarized states, which can be mapped to dilute hardcore Bose systems. All calculations in QS^{3} are performed in the U(1) and translational symmetry-adapted bases without using the bit representation, and this software can support large-scale finite-size calculations of more than 1000 sites (see Fig. 1). The users can also compute the dynamical spin structure factor by the continuous fractional expansion on the large-scale finite-size cluster (Fig. 2) . The open-source code is available from https://github.com/QS-Cube/ED.

This work is described in the preprint, "*Quantum spin solver near saturation: QS ^{3}*" Hiroshi Ueda, Seiji Yunoki and Tokuro Shimokawa, arXiv:2107.00872

### Quantum supremacy, sooner ?

The current generation of quantum computers are exploratory machines, capable of manipulating the information stored in tens of qu-bits, but subject to noise which severely limits the number of operations they can perform, and the accuracy to which they can work. One of the great challenges of this “noisy, intermediate-scale quantum” (NISQ) era, is to identify algorithms which can exploit the advantages of quantum calculation, while running on a technologically achievable device.

One of the leading contenders is the “quantum approximate optimization algorithm” (QAOA), a recipe for transforming a simple many-qubit wavefunction into one which encodes the solution to a classical optimization problem. The QAOA works by optimizing the schedule according to which unitary operators are applied to qubits, and is capable of solving problems which would be difficult on a classical computer. However the accuracy achieved by the QAOA depends heavily on the number of operations performed, with high levels of accuracy requiring more operations than can sensibly be performed on a NISQ device. Any algorithm which requires less operatorions to achieve the desired accuracy is therefore significant.

In this work, a collaboration of the TQM Unit with researchers from Tsinghua University, The Hong Kong University of Science and Technology, and the University of Wisconsion, Madison, we show how a dramatic speedup in the QAOA can be achieved through the use of an “adaptive bias” (ab-). In simulations of the solution of the canonical “Maxcut” problem, the ab-QAOA proved to be up to 18 times as fast as the standard QAOA, without any increase in the complexity of the hardware needed to run it. These results will facilitate the solution of classically hard optimisation problems using existing quantum computers, and could hasten the day when quantum computers show meaningful advantage over classical computers in real-world problems.

This work is described in the preprint “*Quantum Approximate Optimization Algorithm with Adaptive Bias Fields*”, Yunlong Yu, Chenfeng Cao, Carter Dewey, Xiang-Bin Wang, Nic Shannon, Robert Joynt, arXiv:2105.11946