A number of many-body problems that are highly complex in three dimensions can be solved in a low-dimensional system. Simple one-dimensional Hamiltonians that describe such systems have yielded exact, closed-form solutions for their dynamic and static properties. Low-dimensional magnets have gained significant attention for their distinct characteristics that facilitate the analysis of quantum model ground and excited states, the examination of potential new phases of matter, and exploration of the interplay between thermal and quantum fluctuations. The chapter’s main topic is theoretical ideas and outcomes concerning one-dimensional magnetism, and their relevance to real materials and experiments.
A number of many-body problems that are highly complex in three dimensions can be solved in a low-dimensional system. Simple one-dimensional Hamiltonians that describe such systems have yielded exact, closed-form solutions for their dynamic and static properties. Low-dimensional magnets have gained significant attention for their distinct characteristics that facilitate the analysis of quantum model ground and excited states, the examination of potential new phases of matter, and exploration of the interplay between thermal and quantum fluctuations. The chapter’s main topic is theoretical ideas and outcomes concerning one-dimensional magnetism, and their relevance to real materials and experiments.
The description of solids at a microscopic level is complex, involving the interaction of a huge number of its constituents, such as ions or electrons. It is impossible to solve the corresponding many-body problems analytically or numerically, although much insight can be gained from the analysis of simplified models. An important example is the Hubbard model, which describes interacting electrons in narrow energy bands, and which has been applied to problems as diverse as high-Tc superconductivity, band magnetism, and the metal-insulator transition. This 2005 book presents a coherent, self-contained account of the exact solution of the Hubbard model in one dimension. The early chapters will be accessible to beginning graduate students with a basic knowledge of quantum mechanics and statistical mechanics. The later chapters address more advanced topics, and are intended as a guide for researchers to some of the more topical results in the field of integrable models.
The investigation of many-body quantum systems stands as a cornerstone in contemporary physics research. These systems exhibit a richness of phenomena, characterized by the emergence of collective behavior, which leads to the formation of quasi-particle excitations with occasionally unconventional statistics, alongside the occurrence of quantum phase transitions. Their significance extends further as they serve as models for understanding the behavior of real quantum materials, underlining the necessity for their comprehension.In the realm of theoretical physics, exact solutions are often elusive for many-body systems and are limited to a few select cases or toy models. However, real materials are often more complicated with additional dominant interactions beyond solvable limits. Consequently, the study and understanding of generic Hamiltonians necessitate resorting to computational simulations. Utilizing computational resources becomes indispensable, albeit challenging, as the Hilbert space expands exponentially with the number of spins, demanding substantial memory and computational bandwidth.However, recent advancements in quantum computing present a promising avenue for tackling these challenges. Quantum computers offer the potential to simulate many-body quantum systems with minimal overhead, overcoming the limitations of classical computational techniques. Although current quantum computing platforms operate within the confines of the NISQ era -noisy intermediate-scale quantum- they hold immense promise for the exact modelling of many-body systems.In this doctoral research, our objective is to investigate the predictive capabilities of quantum computers through simulating and benchmarking quantum many-body systems relevant to magnetic materials. We showcase three different projects on studying frustrated magnetism: 1) simulating non-equilibrium dynamics of frustrated Ising models on an analog quantum annealer, 2) studying frustrated Rydberg atom quench dynamics on a neutral atom platform, and 3) simulating the Kitaev honeycomb model and probing its topological order on a digital quantum computer. By leveraging the potential of quantum computing in this domain, this research aims to advance both theoretical understanding and practical applications in quantum physics.
We have a strong interest in the interplay between geometrical frustrations and quantum fluctuations in the strongly correlated matter because it causes the emergence of new exotic quantum phases. For example, in the past decade, we have found a number of new Mott insulator materials on a triangular layer: Cs2CuCl4 [1] and organic charge transfer salts [2], such as κ-(BEDT-TTF)2 X and β-Z[Pd(dmit)2]2. There are geometrical frustrations and quantum fluctuations in these materials. Probably, the interplay has an important role for the disordered behaviors at very low temperatures in some materials. In order to study the low temperature property of strongly correlated matter in low-dimensional systems, we need a theoretical tool beyond the mean-field theory. The numerical approach is a candidate. Unfortunately, there are serious drawbacks in the conventional numerical methods. For example, quantum Monte Carlo (QMC) methods are powerful tools because they are unbiased. However, the weight of QMC samples can be negative in quantum frustrated magnets, leading to a cancellation in sign, and the accuracy of simulations fatally decreases (this is the so-called negative sign problem.) Exact diagonalization gives us the detail of a ground state, but it can only be applied to small systems. On the other hand, the variational method is flexible, and it is powerful if we choose the appropriate variational wave function. In the past decade, based on the study of entanglement, we discussed the new types of wave functions for quantum many-body systems. We define the probability amplitudes of these wave functions by the tensor contractions of small tensors. Since the tensor contractions can be drawn as a network, it is called a tensor network state. In this paper, we will report the performance of tensor network state for the variational study of quantum frustrated magnets. There are two major tensor network states: projected entangled pair state(PEPS) [3] and multiscale entanglement renormalization ansatz(MERA) [4]. In the following, we will focus on the MERA, and we will report the performance for S = 1/2 spatially anisotropic triangular antiferromagnets [5]. We organize this paper as follows: In Sec. 2, we will briefly introduce the tensor network. In addition, we will explain a MERA tensor network for triangular lattice models in detail. In Sec. 3, we will show the performance of MERA for the S = 1/2 antiferromagnetic Heisenberg model on a
When sizable quantum fluctuations and strong frustration mechanisms act in concert to repel the formation of conventional long-range order in quantum magnets, they can make way for massively entangled spin liquid phases which may imbue the material with extraordinary properties. \nThe search for such curious phases of matter has proceeded for several decades, gaining extra momentum some fifteen years ago when Kitaev proposed an analytically solvable model for a quantum spin liquid with anyonic excitations on the basis of realistic microscopic spin exchange terms [Kitaev, Annals of Physics 321, 2 (2006)]. Attempts to identify different models or materials which harbor quantum spin liquid ground states have kept researchers -- experimentalists and theorists alike -- in suspense ever since. However, the simulation of quantum many-body systems poses a serious challenge even to modern numerical techniques, particularly in the case of frustrated quantum magnetism in three spatial dimensions. Such models evade tractability by many established approaches, leaving a methodological void. \n \nIn this thesis, we report on recent progress in cutting-edge implementations of the pseudo-fermion functional renormalization group (pf-FRG), which has originally been proposed by Reuther and Wölfle in the context of two-dimensional frustrated quantum magnetism with highly symmetric spin interactions [Reuther and Wölfle, Phys. Rev. B 81, 144410 (2010)]. Reflecting the growing interest in models with SU(2) symmetry-breaking spin exchange terms that has followed the unearthing of Kitaev's honeycomb model, we present a generalized implementation of the pf-FRG which is suited to numerically simulate arbitrary microscopic models with diagonal or off-diagonal two-spin interactions, even in three-dimensional frustrated quantum magnets. We provide insight into the inner workings of the method which has emerged over the course of the last couple of years, arguing that the pf-FRG formalism simultaneously combines aspects of a large-S expansion as well as a large-N expansion on equal footing, thus being able to resolve the subtle interplay between magnetic ordering tendencies and disruptive quantum fluctuations. \n \nMoreover, on a case by case basis we explore the stability of quantum spin liquids in paradigmatic models of frustrated quantum magnetism and elucidate the joint action of geometric frustration, exchange frustration, and quantum fluctuations to inhibit the formation of magnetic long-range order. Examples include: (i) a Heisenberg spin model on the three-dimensional diamond lattice where geometric frustration arises from antiferromagnetic next-nearest neighbor interactions. The additional competition with nearest neighbor interactions leads to an unusually large ground state degeneracy already on classical level. The quantum-to-classical transition is studied by systematically varying the spin length and the different roles of quantum fluctuations and thermal fluctuations are discussed. The theory is applied to interpret experiments on the spin-liquid candidate NiRh2O4; (ii) a Heisenberg antiferromagnet on the face centered cubic lattice which is augmented by competing bond-directional interactions, evolving around the concurrent manifestation of geometric frustration and exchange frustration. A situation is identified where both mechanisms collude to give rise to an unusually large degree of frustration over a wide parameter regime. We discuss related experimental findings on the highly frustrated iridate compound Ba2CeIrO6; (iii) a Heisenberg antiferromagnet on the kagome lattice with additional Dzyaloshinskii-Moriya interactions. The robustness of the unperturbed kagome antiferromagnet's spin liquid ground state against low-symmetry Dzyaloshinskii-Moriya interactions is investigated. Implications for the spin-liquid candidate herbertsmithite are discussed.
This thesis investigates quantum magnetism in two-dimensional synthetic systems, where strong correlations, geometric frustration, and enhanced quantum fluctuations produce unconventional magnetic phases. I focus on two major classes of engineered platforms: moiré materials and quantum simulators, which offer precise control over lattice geometry, interaction strengths, and dynamics, enabling access to regimes beyond conventional condensed-matter approaches. The first project studies moiré magnetism in strained multilayers of CrBr₃, where strain generates a spatial modulation of the interlayer exchange alternating between ferromagnetic and antiferromagnetic. With S=3/2 moments, the system admits a classical bilayer description with competing interactions. I develop a theoretical framework capturing the resulting non-collinear orders and reproducing experimental signatures, demonstrating how moiré engineering stabilizes non-trivial magnetic textures. The second project examines quantum magnetism in a WSe₂/WS₂ heterobilayer at three-quarters filling, where electrons form an effective kagome lattice. From the underlying Hubbard model, I derive a frustrated Heisenberg Hamiltonian with competing exchanges. Using Schwinger-boson mean-field theory, I map a phase diagram of non-collinear orders and spin-liquid phases. For realistic parameters, the ground state is a chiral spin liquid, highlighting transition-metal dichalcogenide moiré materials as platforms for topological and strongly correlated magnetism. The third project investigates quantum phase-transition dynamics with superconducting circuit quantum simulators, focusing on Hamiltonian ramps in the 2D XY model. An analogue-digital protocol drives the system from a gapped staggered state to the gapless XY phase. Analysis of correlations reveals scaling of the correlation length with ramp rate, broadly consistent with Kibble–Zurek predictions, with deviations reflecting finite-size, dimensionality, and experimental constraints. The fourth project develops a Holstein–Primakoff spin-wave description for the same XY ramps. Adapting the quantization axis to the instantaneous magnetization, this solvable quadratic theory provides magnon dispersions, time- and space-dependent correlations, and single-magnon dynamics, including decay and scattering. This framework offers a detailed picture of non-equilibrium quasiparticle behavior in a 2D magnet under time-dependent fields. Together, these projects show how moiré materials and quantum simulators enable controlled studies of frustrated and strongly correlated magnetism in two dimensions. By combining analytical modeling with experimentally relevant conditions, this thesis advances understanding of how frustration, topology, and non-equilibrium dynamics shape quantum magnetic phenomena in low-dimensional systems.
Nonequilibrium properties of quantum materials present many intriguing properties, among them athermal behavior, which violates the eigenstate thermalization hypothesis. Such behavior has primarily been observed in disordered systems. More recently, experimental and theoretical evidence for athermal eigenstates, known as ``quantum scars,'' has emerged in nonintegrable disorder-free models in one dimension with constrained dynamics. In this Rapid Communication, we show the existence of quantum scar eigenstates and investigate their dynamical properties in many simple two-body Hamiltonians with ``staggered'' interactions, involving ferromagnetic and antiferromagnetic motifs, in arbitrary dimensions. These magnetic models include simple modifications of widely studied ones (e.g., the XXZ model) on a variety of frustrated and unfrustrated lattices. We demonstrate our ideas by focusing on the two-dimensional frustrated spin-1/2 kagome antiferromagnet, which was previously shown to harbor a special exactly solvable point with ``three-coloring'' ground states in its phase diagram. For appropriately chosen initial product states---for example, those which correspond to any state of valid three-colors---we show the presence of robust quantum revivals, which survive the addition of anisotropic terms. We also suggest avenues for future experiments which may see this effect in real materials.