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A trial wavefunction approach to the frustrated square-lattice Heisenberg model

Xiaoming Zhang

Published Jan 1, 2011• 0 citations
Publisher

Abstract

Models of interacting quantum spins have contributed significantly to our understanding of magnetism. The Heisenberg model on square lattice, which exhibits semiclassical N´eel order, is one of the canonical models. However, with frustration introduced by competing interactions, the system becomes computationally intractable. Exotic quantum phases that have no magnetic long range order may be present due to the frustration. The quantum phases do not have any classical analogue, and a valence bond crystal state with translational and/or rotational symmetry breaking as well as a quantum spin liquid state have been proposed as potential candidates. We construct several trial wavefunctions in the resonating valence bond basis and apply large-scale unbiased calculations to examine the possible descriptions of the ground state in the strongly frustrated region. An analytical master equation is also proposed to provide an approximate solution for the valence bond states. Our numerical and analytical studies suggest that the frustrated ground state exhibits a one-dimensional-like behaviour.

DOI
10.7939/r3r614

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Models of interacting quantum spins have contributed significantly to our understanding of magnetism. The Heisenberg model on square lattice, which exhibits semiclassical N´eel order, is one of the canonical models. However, with frustration introduced by competing interactions, the system becomes computationally intractable. Exotic quantum phases that have no magnetic long range order may be present due to the frustration. The quantum phases do not have any classical analogue, and a valence bond crystal state with translational and/or rotational symmetry breaking as well as a quantum spin liquid state have been proposed as potential candidates. We construct several trial wavefunctions in the resonating valence bond basis and apply large-scale unbiased calculations to examine the possible descriptions of the ground state in the strongly frustrated region. An analytical master equation is also proposed to provide an approximate solution for the valence bond states. Our numerical and analytical studies suggest that the frustrated ground state exhibits a one-dimensional-like behaviour.
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 present variational results for the ground state of the antiferromagnetic quantum Heisenberg model with frustrating next-nearest-neighbour interactions. The trial wave functions employed are of resonating-valence-bond type, elaborated to account for various geometric motifs of adjacent bond pairs. The calculation is specialized to a square-lattice cluster consisting of just sixteen sites, large enough that the system can accommodate nontrivial singlet dimer correlations but small enough that exhaustive enumeration of states in the total spin zero sector is still feasible. A symbolic computation approach allows us to generate an algebraic expression for the expectation value of any observable and hence to carry out the energy optimization exactly. While we have no measurements that could unambiguously identify a spin liquid state in the controversial region at intermediate frustration, we can say that the bond-bond correlation factors that emerge do not appear to be consistent with the existence of a columnar valence bond crystal. Furthermore, our results suggest that the magnetically disordered region may accommodate two distinct phases.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
We consider a resonating-valence-bond state as a trial wave function for the ground state of the antiferromagnetic Heisenberg model on the square lattice. The energy and the staggered spin-spin correlation functions are calculated on a small lattice and are compared with the results of the exact diagonalization of Oitmaa and Betts. The trial energy is only about 4.7% different from the exact ground-state energy on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The importance of the singlet pairs between the next-nearest antiferromagnetic sites is inferred. An argument against the previous interpretation of the numerical data as being evidence of long-range order is presented.
We investigate the ground-state phase diagram of an anisotropic Heisenberg model on the honeycomb lattice with competing interactions. We use quantum Monte Carlo simulations, as well as linear spin-wave and Ising series expansions, to determine the phase boundaries of the ordered magnetic phases. We find a region without any classical order in the vicinity of a highly frustrated point. Higher-order correlation functions in this region give no signal for long-range valence-bond order. The low-energy spectrum is derived via exact diagonalization to check for topological order on small-size periodic lattices.
We investigate the phase diagram of hard-core bosons on a square lattice with competing interactions. The hard-core bosons can also be represented by spin-1/21/21/2 operators and the model can therefore be mapped onto an anisotropic J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg model. We find the N\'eel state and a collinear antiferromagnetic state as classical ordered phases to be suppressed by the introduction of ferromagnetic exchange terms in the xxx-yyy plane that result in a ferromagnetic phase for large interactions. For an intermediate regime, the emergence of new quantum states like valence bond crystals or supersolids is predicted for similar models. We do not observe any signal for long-range order in terms of conventional order or dimer correlations in our model and find an exponential decay in the spin correlations. Hence, all evidence is pointing toward a quantum-disordered ground state for a small region in the phase diagram.