Local constraints govern the low-energy physics of frustrated matter, but familiar ice-type rules constrain flux-like quantities and are insensitive to handedness. Here we show that handedness itself can be imposed as an exact local quantum constraint without selecting an axis in spin space. We construct positive-semidefinite, SU(2)-invariant parent Hamiltonians whose complete zero-energy space on a tetrahedron has a prescribed chirality sign, rather than selecting a particular chiral wave function. For spin-1/2 the local term is a rank-one projector onto a chiral tetrahedral singlet, while for arbitrary spin it factorizes as $B^\dagger B$ through a singlet-annihilation operator, with a completely characterized kernel given by the span of the globally rotated chiral color-ice states. For coherent states, the same zero-energy condition becomes an $S$-independent nonlinear constraint in which three spin directions determine the fourth through a Möbius transformation; compositions of these maps define constraint holonomies on extended lattices. Connecting the same local constraint in different ways produces qualitatively different collective regimes: corner-sharing lattices retain exponentially large quantum ground-state kernels, with rigorous lower bounds exceeding conventional ice benchmarks; edge-sharing lattices support subdimensional plane or line zero modes; while triangular constructions suppress nonuniform coherent deformations and contain the complete Anderson tower of tetrahedral magnetic order at exactly zero energy. Two inequivalent triangular coverings further show that harmonic zero-mode counting does not determine the size of the quantum kernel. These results establish a tractable setting in which local handedness, nonlinear constraint geometry, and quantum degeneracy can be disentangled and related directly to the connectivity of the constraint network.
Local constraints govern the low-energy physics of frustrated matter, but familiar ice-type rules constrain flux-like quantities and are insensitive to handedness. Here we show that handedness itself can be imposed as an exact local quantum constraint without selecting an axis in spin space. We construct positive-semidefinite, SU(2)-invariant parent Hamiltonians whose complete zero-energy space on a tetrahedron has a prescribed chirality sign, rather than selecting a particular chiral wave function. For spin-1/2 the local term is a rank-one projector onto a chiral tetrahedral singlet, while for arbitrary spin it factorizes as B†B through a singlet-annihilation operator, with a completely characterized kernel given by the span of the globally rotated chiral color-ice states. For coherent states, the same zero-energy condition becomes an S-independent nonlinear constraint in which three spin directions determine the fourth through a Möbius transformation; compositions of these maps define constraint holonomies on extended lattices. Connecting the same local constraint in different ways produces qualitatively different collective regimes: corner-sharing lattices retain exponentially large quantum ground-state kernels, with rigorous lower bounds exceeding conventional ice benchmarks; edge-sharing lattices support subdimensional plane or line zero modes; while triangular constructions suppress nonuniform coherent deformations and contain the complete Anderson tower of tetrahedral magnetic order at exactly zero energy. Two inequivalent triangular coverings further show that harmonic zero-mode counting does not determine the size of the quantum kernel. These results establish a tractable setting in which local handedness, nonlinear constraint geometry, and quantum degeneracy can be disentangled and related directly to the connectivity of the constraint network.
In spin lattices, frustration happens when there exists a competition between different terms of the Hamiltonian, which cannot be simultaneously minimized in all local physical bases. Frustration can destabilize the bipartite Neel order, which is the conventional groundstate of antiferromagnetic Heisenberg-type models in two dimensions. Hence, the focus of two-dimensional quantum magnetism studies has been directed toward elucidating the zero-temperature properties of Heisenberg-type models on the Archimedean lattices that exhibit geometrical frustration. This type of frustration is of great interest since it can lead to exotic forms of the quantum matter, such as topological spin-liquid (SL) and many-sublattice long-range-ordered phases. The spin-1/2 triangular Heisenberg model (THM) is a prototypical model with geometrical frustration, which has the highest coordination number of all Archimedean lattices. However, there exist two major, distinct difficulties for numerical methods to efficiently determine the groundstate properties of the THM: the immense dimension of the Hilbert space and the geometrical frustration phenomenon. In this thesis, to deal with such difficulties, we employ the state-of-the- art non-Abelian matrix product states (MPS) and density matrix renormalization group (DMRG) methods. We exploit the symmetries of the Hamiltonian to reduce the working dimension of the Hilbert space and, to lessen the effects of the geometrical frustration, we depend on the local entanglement (area-law) nature of the MPS as it scales with the size of a lattice subspace with reduced spatial dimensions. In such a way, we thoroughly examine the phase diagram of the THM with nearest and next-nearest neighbor (NN and NNN) exchange couplings on finite- and infinite-length cylinders of widths up to 12 sites. In addition, we develop new numerical tools to analyze topological and symmetry-broken phases in the context of the SU(2)-symmetric translation-invariant MPS algorithm. We establish that on infinite cylinders, the anyonic sectors of a topological order can be classified through the fractionalization of global symmetries and degeneracy patterns of the entanglement spectrum (ES). On the other hand, we detect symmetry-breaking phase transitions by a combination of the correlation lengths and second and fourth cumulants of the magnetic order parameters, even though symmetry implies that the order parameter itself is strictly zero. Furthermore, the appearance of Nambu-Goldstone modes in the excitation spectrum can be detected using “tower of states” (TOS) levels in the momentum-resolved ES. By applying the new tools to the THM, we discover a variety of exotic groundstates such as a Z2-gauge toric-code-type topological order (with vison and spinon excitations), a NNN Majumdar-Gosh, and a long-range 120◦-ordered state.
Classical spin liquids are paramagnetic phases that feature nontrivial patterns of spin correlations within their ground-state manifold whose degeneracy scales with system size. Often they harbor fractionalized excitations, and their low-energy fluctuations are described by emergent gauge theories. In this work, we discuss a model composed of chiral three-body spin interactions on the pyrochlore lattice that realizes a novel classical chiral spin liquid whose excitations are fractonalized while also displaying a fracton-like behavior. We demonstrate that the ground-state manifold of this spin liquid is given by a subset of the so-called color-ice states. We show that the low-energy states are captured by an effective gauge theory which possesses a divergence-free condition and an additional chiral term that constrains the total flux of the fields through a single tetrahedron. The divergence-free constraint on the gauge fields results in two-fold pinch points in the spin structure factor and the identification of bionic charges as excitations of the system.
We present a comprehensive analytical theory of quantum spin liquids (QSLs) on the kagome lattice with Heisenberg exchange J1, next-nearest-neighbor exchange J2, Dzyaloshinskii-Moriya interaction D, and scalar spin chirality interaction λ. The work contains three principal contributions. First, we derive new mean-field analytic relations connecting the chiral interaction λ to the spinon gap, low-temperature thermodynamics, and thermal Hall response, with explicit spin-S dependence and generalization across kagome, triangular, and honeycomb lattices. The central result is Δspinon ∝ S³λ/J12 with lattice-dependent prefactors derived from self-consistent mean-field equations. Second, we derive a controlled effective model by projecting the kagome Hamiltonian onto the triangle-doublet subspace, obtaining an effective spin-chirality model on the triangular lattice with complex-valued interactions arising from the C3 symmetry of the chirality eigenstates. We compute all matrix elements of the effective spin operator in the chirality basis and characterize the inter-chirality coupling mediated by shared lattice sites. Third, we provide a rigorous map of obstructions to exact analytical solutions, systematically examining eight classes of proof techniques and identifying the precise mathematical reason each fails for the kagome Hamiltonian. The Lieb-Schultz-Mattis theorem provides the single rigorous exact result: any gapped ground state on the kagome lattice with spin-1/2 per site must exhibit topological order. All other results are classified by derivation category with explicit uncertainty quantification. AI-Generated Content Statement: This manuscript was generated primarily with the assistance of artificial intelligence language models, including ChatGPT. The work represents an exploratory demonstration of AI-assisted knowledge generation and prompt engineering methods rather than traditional experimental research.
Motivated by recent quantum Monte Carlo (QMC) simulations of the quantum Kagome ice model by Juan Carrasquilla, et al., [Nature Communications 6, 7421 (2015)], we study the ground state properties of this model on the triangular lattice. In the presence of a magnetic field h, the Hamiltonian possesses competing interactions between a Z2-invariant easy-axis ferromagnetic interaction J±± and a frustrated Ising term Jz. As in the U(1)-invariant model, we obtain four classical distinctive phases, however, the classical phases in the Z2-invariant model are different. They are as follows: a fully polarized (FP) ferromagnet for large h, an easy-axis canted ferromagnet (CFM) with broken Z2 symmetry for small h and dominant J±±, a {\it ferrosolid} phase with broken translational and Z2 symmetries for small h and dominant Jz, and two lobes with m=⟨Sz⟩=±1/6 for small h and dominant Jz. We show that quantum fluctuations are suppressed in this model, hence the large-S expansion gives an accurate picture of the ground state properties. When quantum fluctuations are introduced, we show that the {\it ferrosolid} state is the ground state in the dominant Ising limit at zero magnetic field. It remains robust for Jz→∞. With nonzero magnetic field the classical lobes acquire a finite magnetic susceptibility with no Sz-order. We present the trends of the ground state energy and the magnetizations. We also present a detail analysis of the CFM.
Motivated by recent quantum Monte Carlo (QMC) simulations of the quantum Kagome ice model by Juan Carrasquilla, et al., [Nature Communications 6, 7421 (2015)], we study the ground state properties of this model on the triangular lattice. In the presence of a magnetic field h, the Hamiltonian possesses competing interactions between a Z2-invariant easy-axis ferromagnetic interaction J±± and a frustrated Ising term Jz. As in the U(1)-invariant model, we obtain four classical distinctive phases, however, the classical phases in the Z2-invariant model are different. They are as follows: a fully polarized (FP) ferromagnet for large h, an easy-axis canted ferromagnet (CFM) with broken Z2 symmetry for small h and dominant J±±, a {\it ferrosolid} phase with broken translational and Z2 symmetries for small h and dominant Jz, and two lobes with m=⟨Sz⟩=±1/6 for small h and dominant Jz. We show that quantum fluctuations are suppressed in this model, hence the large-S expansion gives an accurate picture of the ground state properties. When quantum fluctuations are introduced, we show that the {\it ferrosolid} state is the ground state in the dominant Ising limit at zero magnetic field. It remains robust for Jz→∞. With nonzero magnetic field the classical lobes acquire a finite magnetic susceptibility with no Sz-order. We present the trends of the ground state energy and the magnetizations. We also present a detail analysis of the CFM.
We propose a chiral valence-bond solid (VBS) in which spin-singlet bond order itself spontaneously acquires handedness without magnetic order. Using a spin-lattice model on the diamond lattice, we demonstrate two enantiomorphic VBS states. In the presence of anisotropic spin interactions, this emergent handedness is inherited by chiral triplons with spin-split bands and spin textures that reverse between the two enantiomers. These textures generate a temperature-gradient-induced net spin response whose sign is set by the VBS handedness. The imaginary off-diagonal components of the dynamical spin structure factor encode the triplon polarization and change sign between the two enantiomers, enabling direct detection of chiral triplons and the selected handedness by polarized inelastic neutron scattering. Our results establish a direct connection between spontaneous chirality and magnetic excitations, opening avenues for exploring the interplay between quantum magnetism and chiral physics.