The spin-1=2 Heisenberg model on the pyrochlore lattice is an iconic frustrated three-dimensional spin system with a rich phase diagram.Besides hosting several ordered phases, the model is debated to possess a spin-liquid ground state when only nearest-neighbor antiferromagnetic interactions are present.Here, we contest this hypothesis with an extensive numerical investigation using both exact diagonalization and complementary variational techniques.Specifically, we employ a resonating-valence-bond-like, manyvariable, Monte Carlo ansatz and convolutional neural network quantum states for (variational) calculations with up to 4 × 4 3 and 4 × 3 3 spins, respectively.We demonstrate that these techniques yield consistent results, allowing for reliable extrapolations to the thermodynamic limit.We consider the ðλ; j 2 =j 1 Þ parameter space, with j 2 , j 1 being nearest and next-to-nearest neighbor interactions and λ the XXZ interaction anisotropy.Our main results are (1) the determination of the phase transition between the putative spin-liquid phase and the neighboring magnetically ordered phase and (2) a careful characterization of the ground state in terms of symmetry-breaking tendencies.We find clear indications of a dimer order with spontaneously broken inversion and rotational symmetry, calling the scenario of a featureless quantum spin liquid into question.Our work showcases how many-variable variational techniques can be used to make progress in answering challenging questions about three-dimensional frustrated quantum magnets.
The spin-1=2 Heisenberg model on the pyrochlore lattice is an iconic frustrated three-dimensional spin system with a rich phase diagram.Besides hosting several ordered phases, the model is debated to possess a spin-liquid ground state when only nearest-neighbor antiferromagnetic interactions are present.Here, we contest this hypothesis with an extensive numerical investigation using both exact diagonalization and complementary variational techniques.Specifically, we employ a resonating-valence-bond-like, manyvariable, Monte Carlo ansatz and convolutional neural network quantum states for (variational) calculations with up to 4 × 4 3 and 4 × 3 3 spins, respectively.We demonstrate that these techniques yield consistent results, allowing for reliable extrapolations to the thermodynamic limit.We consider the ðλ; j 2 =j 1 Þ parameter space, with j 2 , j 1 being nearest and next-to-nearest neighbor interactions and λ the XXZ interaction anisotropy.Our main results are (1) the determination of the phase transition between the putative spin-liquid phase and the neighboring magnetically ordered phase and (2) a careful characterization of the ground state in terms of symmetry-breaking tendencies.We find clear indications of a dimer order with spontaneously broken inversion and rotational symmetry, calling the scenario of a featureless quantum spin liquid into question.Our work showcases how many-variable variational techniques can be used to make progress in answering challenging questions about three-dimensional frustrated quantum magnets.
We investigate the ground-state properties of the spin-1/2 pyrochlore Heisenberg antiferromagnet using pseudofermion functional renormalization group techniques. The first part of our analysis is based on an enhanced parton mean-field approach, which takes into account fluctuation effects from renormalized vertex functions. Our implementation of this technique extends earlier approaches and resolves technical difficulties associated with a diagrammatic overcounting. Using various parton ans\"atze for quantum spin liquids, dimerized and nematic states our results indicate a tendency for lattice symmetry breaking in the ground state. While overall quantum spin liquids seem unfavorable in this system, the recently proposed monopole state still shows the strongest support among all spin liquid ans\"atze that we have tested, which is further confirmed by our complementary variational Monte Carlo calculations. In the second part of our investigation, we probe lattice symmetry breaking more directly by applying the pseudofermion functional renormalization group to perturbed systems. Our results from this technique confirm that the system's ground state either exhibits broken C3 rotation symmetry, or a combination of inversion and C3 symmetry breaking.
The nature of quantum spin liquids is studied for the spin-1/2 antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites, J1, and those between next-nearest-neighbor sites, J2. We perform variational Monte Carlo simulations together with the quantum-number-projection technique and clarify the phase diagram in the ground state together with its excitation spectra. We obtain the nonmagnetic phase in the region 0.4<J2/J1≤0.6 sandwiched by the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations of the spin gap support the notion that the triplet excitation from the singlet ground state is gapless in the range of 0.4<J2/J1≤0.5, while the gapped valence-bond-crystal (VBC) phase is stabilized for 0.5<J2/J1≤0.6. The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the C4v symmetry. The power-law behaviors of the spin-spin and dimer-dimer correlation functions in the gapless region are consistent with the emergence of the algebraic quantum-spin-liquid phase (critical phase). The exponent of the spin correlation ⟨S(0)S(r)⟩∝1/rz+η at a long distance r appears to increase from z+η∼1 at J2/J1∼0.4 toward the continuous transition to the VBC phase at J1/J1∼0.5. Our results, however, do not fully exclude the possibility of a direct quantum transition between the staggered AF and VBC phases with a wide critical region and deconfined criticality.
Closely competing valence bond crystal orders in the ground state of the spin-21 antiferromagnetic Heisenberg model on the pyrochlore lattice: a large scale unrestricted variational study
The spin-21 antiferromagnetic Heisenberg model on the pyrochlore lattice(PAFH) is arguably the most well known strongly frustrated quantum magnet in three spatial dimension. However, due to the rapid scaling of Hilbert space with the linear size of such a three dimensional system, the nature of its ground state in the thermodynamic limit remains elusive after about 30 years' intense debate. Here we apply a recently developed powerful algorithm to perform large scale unrestricted variational optimization of the ground state of the spin-21 PAFH from the resonating valence bond(RVB) theory perspective. We find a highly competitive candidate ground state of the system. This novel state features a maximally resonating valence bond crystal(VBC) pattern with 2a1×2a2×2a3 periodicity. There are at least four levels of hierarchical structure in such a VBC state, with the first and the second level of hierarchy related to the breaking of the inversion and the translational symmetry. Intriguingly, we find that within the RVB framework such a maximally resonating VBC state is almost degenerate with a recently proposed VBC state that is obtained from dressing hard hexagon covering of the pyrochlore lattice, although they have very different structures. We also find that further symmetry breaking will occur in the dressed hard hexagon VBC state under unrestricted optimization, which results in strong disparity in ⟨S^u2⟩ for up and down tetrahedrons as we observe in the maximally resonating VBC state. We show that the maximally resonating VBC state found here will be favored by a tiny next-neighboring exchange coupling over the dressed hard hexagon covering VBC state.
Ground state of the S=1/2 pyrochlore Heisenberg antiferromagnet: A quantum spin liquid emergent from dimensional reduction
The quantum antiferromagnet on the pyrochlore lattice offers an archetypal frustrated system, which potentially realizes a quantum spin liquid characterized by the absence of standard spontaneous symmetry breaking even at zero temperature, unusually as an isotropic 3D system. Despite tremendous progress in the literature, however, the nature of the ground state of the fully quantum-mechanical spin Hamiltonian on the pyrochlore lattice still remains elusive. Here, we show that an unconventional type of quantum spin liquid is born out from the pyrochlore system after the self-organized dimensional reduction leading to confined states in 2D layers. This conclusion is obtained from state-of-the-art variational Monte Carlo (VMC) simulations at zero temperature. Quantum spin liquids triggered by the emergent dimensional reduction is an unexplored route of the spin-liquid formation. The dimensional reduction from 3D to 2D is a consequence of a conventional spontaneous symmetry breaking, while the resultant decoupling of layers enables the emergence of a 2D quantum spin liquid that is adiabatically disconnected from trivial product states and exhibits strong quantum entanglement. The stabilized quantum spin liquid exhibits an algebraic decay of correlations and vanishing excitation gap in the thermodynamic limit. The wave-function structure supports the fractionalization of the spin into spinons. This spin-liquid ground state persists in the presence of spin-orbit interactions, which expands the possibilities of realizing quantum spin liquids in real pyrochlore-structured materials.
We present a comprehensive computational study of the phase diagram of the frustrated S=1/2 Heisenberg antiferromagnet on the honeycomb lattice, with second-nearest (J2) and third-neighbor (J3) couplings. Using a combination of exact diagonalizations (EDs) of the original spin model, of the Hamiltonian projected into the nearest-neighbor short-range valence-bond basis, and of an effective quantum dimer model, as well as a self-consistent cluster mean-field theory, we determine the boundaries of several magnetically ordered phases in the region ${J}_{2},{J}_{3}\ensuremath{\in}[0,1]$, and find a sizable magnetically disordered region in between. We characterize part of this magnetically disordered phase as a plaquette valence-bond crystal phase. At larger J2, we locate a sizable region in which staggered valence-bond crystal correlations are found to be important, either due to genuine valence-bond crystal (VBC) ordering or as a consequence of magnetically ordered phases, which break lattice rotational symmetry. Furthermore, we find that a particular parameter-free Gutzwiller projected tight-binding wave function has remarkably accurate energies compared to finite-size extrapolated ED energies along the transition line from conventional N\'eel to plaquette VBC phases, a fact that points to possibly interesting critical behavior---such as a deconfined critical point---across this transition. We also comment on the relevance of this spin model to model the spin liquid region found in the half filled Hubbard model on the honeycomb lattice.
We address the ground-state properties of the long-standing and much-studied three-dimensional quantum spin liquid candidate, the S=1/2 pyrochlore Heisenberg antiferromagnet. By using SU(2) density-matrix renormalization group (DMRG), we are able to access cluster sizes of up to 128 spins. Our most striking finding is a robust spontaneous inversion symmetry breaking, reflected in an energy density difference between the two sublattices of tetrahedra, familiar as a starting point of earlier perturbative treatments. We also determine the ground-state energy, E_{0}/N_{sites}=-0.490(6)J, by combining extrapolations of DMRG with those of a numerical linked cluster expansion. These findings suggest a scenario in which a finite-temperature spin liquid regime gives way to a symmetry-broken state at low temperatures.