The fractional quantum Hall effect (FQHE) realized in two-dimensional electron systems under a magnetic field is one of the most remarkable discoveries in condensed matter physics. Interestingly, it has been proposed that FQHE can also emerge in time-reversal invariant spin systems, known as the chiral spin liquid (CSL) characterized by the topological order and the emerging of the fractionalized quasiparticles. A CSL can naturally lead to the exotic superconductivity originating from the condense of anyonic quasiparticles. Although CSL was highly sought after for more than twenty years, it had never been found in a spin isotropic Heisenberg model or related materials. By developing a density-matrix renormalization group based method for adiabatically inserting flux, we discover a FQHE in a spin-½ isotropic kagome Heisenberg model. We identify this FQHE state as the long-sought CSL with a uniform chiral order spontaneously breaking time reversal symmetry, which is uniquely characterized by the half-integer quantized topological Chern number protected by a robust excitation gap. The CSL is found to be at the neighbor of the previously identified Z2 spin liquid, which may lead to an exotic quantum phase transition between two gapped topological spin liquids.
The fractional quantum Hall effect (FQHE) realized in two-dimensional electron systems under a magnetic field is one of the most remarkable discoveries in condensed matter physics. Interestingly, it has been proposed that FQHE can also emerge in time-reversal invariant spin systems, known as the chiral spin liquid (CSL) characterized by the topological order and the emerging of the fractionalized quasiparticles. A CSL can naturally lead to the exotic superconductivity originating from the condense of anyonic quasiparticles. Although CSL was highly sought after for more than twenty years, it had never been found in a spin isotropic Heisenberg model or related materials. By developing a density-matrix renormalization group based method for adiabatically inserting flux, we discover a FQHE in a spin-½ isotropic kagome Heisenberg model. We identify this FQHE state as the long-sought CSL with a uniform chiral order spontaneously breaking time reversal symmetry, which is uniquely characterized by the half-integer quantized topological Chern number protected by a robust excitation gap. The CSL is found to be at the neighbor of the previously identified Z2 spin liquid, which may lead to an exotic quantum phase transition between two gapped topological spin liquids.
Kalmeyer-Laughlin (KL) chiral spin liquid (CSL) is a type of quantum spin liquid without time-reversal symmetry, and it is considered as the parent state of an exotic type of superconductor--anyon superconductor. Such an exotic state has been sought for more than twenty years; however, it remains unclear whether it can exist in a realistic system where time-reversal symmetry is breaking (T breaking) spontaneously. By using the density matrix renormalization group, we show that KL CSL exists in a frustrated anisotropic kagome Heisenberg model, which has spontaneous T breaking. We find that our model has two topological degenerate ground states, which exhibit nonvanishing scalar chirality order and are protected by finite excitation gap. Furthermore, we identify this state as KL CSL by the characteristic edge conformal field theory from the entanglement spectrum and the quasiparticles braiding statistics extracted from the modular matrix. We also study how this CSL phase evolves as the system approaches the nearest-neighbor kagome Heisenberg model.
The breaking of chiral and time-reversal symmetries provides a pathway to exotic quantum phenomena and topological phases. Recent work has extensively explored the resulting emergence of chiral charge orders and chiral spin liquids (CSLs) on the kagome lattice. Such CSLs are closely tied to bosonic fractional quantum Hall states with anyonic quasiparticles; however, their connection to nearby ordered states has remained a mystery. Here, we use spin-wave theory, parton Gutzwiller wave functions, and exact diagonalization to show that two distinct magnetic orders with uniform scalar chirality---the XYZ umbrella state and the octahedral spin crystal--emerge as competing orders in close proximity to the CSL. In this letter, we highlight the intimate link between a topologically ordered liquid and broken symmetry states with nontrivial real-space topology.
The kagome spin-21 systems have attracted intensive attention in recent years as the primary candidate for hosting different gapped spin liquids (SLs). To uncover the nature of the novel quantum phase transition between the SL states, we study a minimum XY model with the nearest-neighbor (NN) (Jxy), the second-NN, and the third-NN couplings (${J}_{2xy}={J}_{3xy}={J}_{xy}^{\ensuremath{'}}$). We identify the time-reversal-symmetry-broken chiral SL (CSL) with the turn on of a small perturbation ${J}_{xy}^{\ensuremath{'}}\ensuremath{\sim}0.06{J}_{xy}$, which is fully characterized by the fractionally quantized topological Chern number and the conformal edge spectrum as the $\ensuremath{\nu}=1/2$ fractional quantum Hall state. Interestingly, the NN XY model (${J}_{xy}^{\ensuremath{'}}=0$) is shown to be a critical SL state adjacent to the CSL, characterized by the gapless spin singlet and spin triplet excitations. The quantum phase transition from the CSL to the gapless critical SL is driven by the collapsing of the neutral (spin singlet) excitation gap. The effect of the NN spin-z coupling Jz is also studied, which leads to a quantum phase diagram with an extended regime for the gapless SL.
Topological phases in frustrated quantum magnetic systems have captivated researchers for decades, with the chiral spin liquid (CSL) standing out as one of the most compelling examples. Featuring long-range entanglement, topological order, and exotic fractional excitations, the CSL has inspired extensive exploration for practical realizations. In this Letter, we demonstrate that CSLs can emerge in a kagome lattice driven by vacuum quantum fluctuations over the noninteracting vacuum within a single-mode gyrotropic cavity. The gyrotropic cavity imprints quantum fluctuations with time-reversal symmetry breaking and stabilizes a robust CSL phase without external laser excitation. Moreover, we identify experimentally accessible observables-such as average photon number and transport properties-that reveal connections between photon dynamics and the emergent chiral order. Our findings establish a novel pathway for creating, controlling, and probing topological and symmetry-breaking quantum phases in strongly correlated systems.
Recent years have seen the discovery of a chiral spin liquid state - a bosonic analogue of a fractional Quantum Hall state first put forward by Kalmeyer and Laughlin in 1987 - in several deformations of the Heisenberg model on the Kagome lattice. Here, we apply state-of-the-art numerical techniques to one such model, where breaking of the time-reversal symmetry drives the system into the chiral phase. Our methods allow us to obtain explicit matrix-product state representations of the low-lying excitations of the chiral spin liquid state, including the topologically non-trivial semionic excitation. We characterize these excitations and study their energetics as the model is tuned towards a topological phase transition.
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.