Frustrated magnets with highly degenerate ground states are at the heart of hunting exotic states of matter. Recent studies in spin-orbit coupled honeycomb magnets have generated immense interest in bond-dependent interactions, appreciating a symmetric off-diagonal $\mathrm{\ensuremath{\Gamma}}$ interaction which exhibits a macroscopic degeneracy in the classical limit. Here, we study a generic spin model and discover a chiral-spin (CS) ordering with spontaneously broken time-reversal symmetry near the dominant $\mathrm{\ensuremath{\Gamma}}$ region. The CS phase is demonstrated to possess a staggered chirality relation in different sublattices, and it exhibits gapless excitations, as revealed by the vanishing energy gap and the finite central charge on cylinders. Although there is a vestige of a tiny peak in the corner of the second Brillouin zone, the magnetic order is likely to vanish as the system size increases. Finally, we also attempt to gain insight into the possible topological signature of the CS phase by calculating the dynamic structure factor and the modular $\mathcal{S}$ matrix.
Frustrated magnets with highly degenerate ground states are at the heart of hunting exotic states of matter. Recent studies in spin-orbit coupled honeycomb magnets have generated immense interest in bond-dependent interactions, appreciating a symmetric off-diagonal \ensuremathΓ interaction which exhibits a macroscopic degeneracy in the classical limit. Here, we study a generic spin model and discover a chiral-spin (CS) ordering with spontaneously broken time-reversal symmetry near the dominant \ensuremathΓ region. The CS phase is demonstrated to possess a staggered chirality relation in different sublattices, and it exhibits gapless excitations, as revealed by the vanishing energy gap and the finite central charge on cylinders. Although there is a vestige of a tiny peak in the corner of the second Brillouin zone, the magnetic order is likely to vanish as the system size increases. Finally, we also attempt to gain insight into the possible topological signature of the CS phase by calculating the dynamic structure factor and the modular S matrix.
Abstract A family of spin–orbit coupled honeycomb Mott insulators offers a playground to search for quantum spin liquids (QSLs) via bond-dependent interactions. In candidate materials, a symmetric off-diagonal Γ term, close cousin of Kitaev interaction, has emerged as another source of frustration that is essential for complete understanding of these systems. However, the ground state of honeycomb Γ model remains elusive, with a suggested zigzag magnetic order. Here we attempt to resolve the puzzle by perturbing the Γ region with a staggered Heisenberg interaction which favours the zigzag ordering. Despite such favour, we find a wide disordered region inclusive of the Γ limit in the phase diagram. Further, this phase exhibits a vanishing energy gap, a collapse of excitation spectrum, and a logarithmic entanglement entropy scaling on long cylinders, indicating a gapless QSL. Other quantities such as plaquette-plaquette correlation are also discussed.
The higher-spin Kitaev magnets, in which the Kitaev interaction and off-diagonal exchange couplings are overwhelmingly large, have emerged as a fertile avenue to explore exotic phases and unusual excitations. In this paper, we study the quantum phase diagram of the spin-1 Kitaev-$\mathrm{\ensuremath{\Gamma}}$ model on the honeycomb lattice using density-matrix renormalization group. It harbors six distinct phases and the intriguing findings are three magnetically ordered phases in which both time-reversal symmetry and lattice symmetry, albeit of different sorts, are broken spontaneously. The chiral spin state originates from the order-by-disorder effect and exhibits an almost saturated scalar spin chirality at the quantum level. Depending on the relative strength of the two interactions, it also features a columnarlike or plaquettelike dimer pattern as a consequence of the translational symmetry breaking. In parallel, the nematic ferromagnets are situated at the ferromagnetic Kitaev side and possess small but finite ferromagnetic ordering. The lattice-rotational symmetry breaking enforces nonequivalent bond energy along one of the three bonds. Although the intrinsic difference between the two nematic ferromagnets remains elusive, the discontinuities in the von Neumann entropy, hexagonal plaquette operator, and Wilson loop operator convincingly suggest that they are separated via a first-order phase transition.
The intricate interplay between frustration and spin chirality, induced by the Dzyaloshinskii-Moriya interaction, holds promise for unveiling novel phases in frustrated quantum magnets. This study investigates the ground state phase diagram of the spin-21 square lattice ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model upon incorporation of the chiral interaction ${J}_{\ensuremath{\chi}}$. Employing exact diagonalization (ED) with full lattice symmetries, we analyze the phase evolution as a function of J2 at fixed ${J}_{\ensuremath{\chi}}$, utilizing highly symmetric energy levels. Critical level crossings and ground state fidelity susceptibility (FS) techniques aid in pinpointing phase boundaries: Magnetic to nonmagnetic phase transitions are identified through critical level crossings between gapless magnetic excitations and quasidegenerate ground states of nonmagnetic phases. Direct transitions between two nonmagnetic phases are characterized by FS peaks due to avoided ground state level crossing. Based on these observations, we identify an anticipated chiral spin liquid (CSL) state and an adjacent nematic spin liquid (NSL) phase within a significant range of ${J}_{\ensuremath{\chi}}$, demarcated by a nearly vertical boundary line at ${J}_{2}\ensuremath{\approx}0.65$. This critical line terminates at the lower boundary in ${J}_{\ensuremath{\chi}}$ of a magnetic ordered chiral spin solid phase, which gains prominence with increasing ${J}_{\ensuremath{\chi}}$ from both CSL and NSL phases. The topological nature of the CSL is confirmed using the modular S matrix of minimally entangled states on a torus and the entanglement spectra of even and odd sectors on a cylinder, employing ED and SU(2)-symmetric density matrix renormalization group method, respectively. Furthermore, a comprehensive discussion on the nature of the NSL is provided, exploring aspects such as ground state degeneracy, local bond energy landscape, and singlet and triplet gaps on various tori, offering substantial evidence supporting the nematic nature of the NSL.
Motivated by two-dimensional bilayer systems we numerically study an anti-ferromagnetic spin-1/2 Heisenberg model on the honeycomb lattice with a nearest-neighbour exchange coupling J1, and a next-nearest-neighbour exchange coupling J2′ for the B sublattice sites only. Using infinite Projected Entangled-Pair States (iPEPS), variational uniform matrix-product states on cylinders, and exact diagonalization, we find an extended symmetric regime 0.4≲J2′/J1≲0.6, in which local observables show no magnetic, valence-bond, or chiral spin order. At J2′/J1=0.5, the PEPS correlation length grows systematically with bond dimension, and an inverse-correlation-length extrapolation favors a vanishing limit. We also find that various observables scale algebraically with the finite bond-dimension-induced correlation length, which points to a gapless spin liquid ground state. We propose a Z2 Dirac spin liquid parton state, with Dirac points that are protected by translation, time-reversal and three-fold rotation symmetry, as a promising candidate state to explain the numerical results. We also discuss the possibility that the symmetric ground state is a featureless, short-range entangled state with a small gap.
Frustrated magnetism provides a rich platform for realizing exotic quantum phases in condensed matter systems. Among these, quantum spin liquids (QSLs) are a particularly intriguing class of states that exhibit no magnetic ordering and preserve all symmetries down to zero temperature. Just as conventional magnets can be thought of as spin crystals,QSLs represent their liquid analogs. This thesis explores two broad types of QSLs in two-dimensional magnets, with a focus on their dynamic properties and the nature of transitions from conventional magnetic states into QSL phases. QSLs are topological phases that host fractionalized excitations known as spinons. In these states, mobile spinons are coupled to an emergent gauge field, forming a highly entangled phase of matter. The first part of this thesis focuses on Dirac spin liquids (DSLs), a class of QSLs where spinons exhibit a gapless Dirac dispersion. To study the dynamicsof these spin liquids beyond leading-order mean-field theory, we develop a novel large-N fermionic limit of gapless spinons. In this limit, all gauge excitations are suppressed, and the leading-order corrections instead arise from order parameter fluctuations, which may condense and drive transitions into symmetry-broken phases from the parent DSL. Incorporating random phase approximation (RPA) corrections to the spin susceptibility, we find that sharp, spin-wave–like modes are recovered in the excitation spectrum. This methodology successfully captures characteristic features of the dynamics near phase transitions and in weakly symmetry-broken phases proximate to a DSL. We apply this approach to microscopic models, including the J1-J2 XXZ model on the triangular lattice and the J1-J3 XXZ model on the honeycomb lattice. These models closely represent a wide range of frustrated materials, such as triangular and honeycomb cobaltates and triangular ytterbium compounds. Our theoretical predictions show excellent agreement with experimentally measured inelastic neutron scattering spectra from these materials. Gapless spinons can also sometimes become gapped through chiral symmetry breaking, leading to the formation of a chiral spin liquid (CSL). The CSL is an archetypal bosonic realization of a fractional quantum Hall (FQH) state at half-filling, and has been proposed as a time-reversal–breaking ground state in a variety of frustrated spin models. The second part of this thesis addresses the intriguing possibility of realizing a CSL through the "melting" of a chiral spin crystal—a conventional magnet with non-coplanar ordering. Such a transition offers a physically appealing analogue to the melting of a Wigner crystal into a fermionic FQH liquid. Focusing on the kagome lattice, where chiral interactions are knownto stabilize a CSL ground state, we study the effects of additional long-range Heisenberg interactions and demonstrate a phase transition into a non-coplanar octahedral magnetic order. To describe the universal class of transition between a CSL and a non-coplanar anti-ferromagnetic order, we then develop a novel Chern-Simons–matter field theory framework, constrained by symmetry principles and symmetry fractionalization patterns. This thesis thus provides new insights into the dynamical signatures and phase transitions of quantum spin liquids, offering a bridge between theoretical models and experimental observations in frustrated magnetic systems.
A number of lattices exhibit moat-like band structures, i.e. a band with infinitely degenerate energy minima attained along a closed line in the Brillouin zone. If such a lattice is populated with hard-core bosons, the degeneracy prevents their condensation. At half-filling, the system is equivalent to s=1/2 XY model at zero magnetic field, while absence of condensation translates into the absence of magnetic order in the XY plane. Here we show that the ground state breaks the time-reversal as well as inversion symmetries. This state, which may be identified with the chiral spin liquid, has a bulk gap and chiral gapless edge excitations. The applications of the developed analytical theory include an explanation of recent numerical findings and a suggestion for the chiral spin liquid realizations in experiments with cold atoms in optical lattices.