The Green's-function Monte Carlo (GFMC) method is used to calculate very accurate ground-state energies of the two-dimensional, spin-1/2 Heisenberg antiferromagnet. The computations are performed on L\ifmmode\times\else\texttimes\fi{}L square lattices up to L=16 with varying uniform magnetization, which allows the extraction of the perpendicular susceptibility (\ensuremath{\chi}) and spin-wave velocity (c). These two quantities are the lattice- or cutoff-dependent parameters that allow one to map the long-wavelength properties of the antiferromagnet onto the nonlinear \ensuremath{\sigma} model and so are of general interest. Systematic errors present in previous GFMC calculations are addressed and corrected to yield results in excellent agreement with other numerical methods. I find, for the ground-state energy per site, -0.669 34(3); the susceptibility renormalization factor, ${\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}$=0.535(5); and the spin-wave velocity renormalization factor,
Zc=1.10(3). Finite-size effects in the extraction of
Zc and ${\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}$ are discussed. The value of ${\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}$ computed here is in agreement with the series-expansion results of Singh and of Zheng, Oitmaa, and Hamer, thereby clearing up a previous inconsistency between the series-expansion and quantum Monte Carlo predictions.
The ground-state parameters of the two-dimensional
S=1/2 antiferromagnetic Heisenberg model are calculated using the stochastic series expansion quantum Monte Carlo method for $L\ifmmode\times\else\texttimes\fi{}L$ lattices with
L up to
16. The finite-size results for the energy
E, the sublattice magnetization
M, the long-wavelength susceptibility ${\ensuremath{\chi}}_{\ensuremath{\perp}}(q=2\ensuremath{\pi}/L)$, and the spin stiffness ${\ensuremath{\rho}}_{s},$ are extrapolated to the thermodynamic limit using fits to polynomials in
1/L, constrained by scaling forms previously obtained from renormalization-group calculations for the nonlinear $\ensuremath{\sigma}$ model and chiral perturbation theory. The results are fully consistent with the predicted leading finite-size corrections, and are of sufficient accuracy for extracting also subleading terms. The subleading energy correction $(\ensuremath{\sim}{1/L}^{4})$ agrees with chiral perturbation theory to within a statistical error of a few percent, thus providing numerical confirmation of the finite-size scaling forms to this order. The extrapolated ground- state energy per spin is $E=\ensuremath{-}0.669437(5)$. The result from previous Green's function Monte Carlo (GFMC) calculations is slightly higher than this value, most likely due to a small systematic error originating from ``population control'' bias in GFMC. The other extrapolated parameters are
M=0.3070(3), ${\ensuremath{\rho}}_{s}=0.175(2)$, ${\ensuremath{\chi}}_{\ensuremath{\perp}}=0.0625(9)$, and the spin-wave velocity
c=1.673(7). The statistical errors are comparable with those of previous estimates obtained by fitting loop algorithm quantum Monte Carlo data to finite-temperature scaling forms. Both
M and ${\ensuremath{\rho}}_{s}$ obtained from the finite-
T data are, however, a few error bars higher than the present estimates. It is argued that the
T=0 extrapolations performed here are less sensitive to effects of neglected higher-order corrections, and therefore should be more reliable.
We combine the density-matrix renormalization-group (DMRG) technique with Green function Monte Carlo (GFMC) simulations using a special representation of the DMRG wave function. As a test case we apply the method to the two-dimensional frustrated Heisenberg antiferromagnet. By supplementing the branching in GFMC simulations with stochastic reconfiguration we get a stable simulation with a small variance also in the region where the fluctuations due to the minus sign problem are maximal. The sensitivity of the results to the choice of the guiding wave function is extensively investigated. In agreement with earlier calculations it is found from the DMRG wave function that for small ratios of the next-nearest-to-nearest neighbor coupling strength the system orders as a N\'eel-type antiferromagnet and for large ratios as a columnar antiferromagnet. The spin stiffness suggests an intermediate regime without magnetic long-range order. The energy curve indicates that the columnar phase is separated from the intermediate phase by a first-order transition. The combination of the DMRG and GFMC techniques allows us to substantiate this picture by calculating also the spin correlations in the system. We observe a pattern of spin correlations in the intermediate regime which is in between dimerlike and plaquette-type ordering, states that have recently been suggested. It is a state with strong dimerization in one direction and weaker dimerization in the perpendicular direction and thus it lacks the square symmetry of the plaquette state.
The ground-state energy, staggered magnetization, and correlation function are computed for the two-dimensional spin-(1/2 quantum antiferromagnetic Heisenberg model (believed to be relevant to the high-temperature superconducting materials without doping) on lattices up to 24\ifmmode\times\else\texttimes\fi{}24 by Monte Carlo methods extrapolated to zero temperature. A more accurate value for the ground-state energy is determined by the method of Neumann and Ulam by extrapolating from lattices up to 12\ifmmode\times\else\texttimes\fi{}12.
In this thesis the quantum Monte Carlo loop algorithm is applied to study the properties of correlation functions in low-dimensional quantum antiferromagnets. Two different types of lattice models are investigated. The first system is the quantum critical XXZ chain. The second class of models emerges when coupling the square lattice Heisenberg model to phonons. In case of the quantum critical XXZ chain, the loop algorithm is used to compute the two-point spin and dimer correlation functions at low temperatures. Starting from the well-known asymptotic expressions from conformal field theory for the decay of the correlations, the crossover from the finite-temperature to the ground state behavior is investigated. In addtion, the Monte Carlo data are used to extract numerical estimates for the correlation amplitudes. The deviations from asymptotic scaling at low finite temperatures and intermediate distances are examined. The results are also used to study the algebraic and logarithmic corrections in the ground state. The method is further applied to compute the low-temperature spin correlations of the square lattice Heisenberg model coupled to dispersionless Einstein phonons. Also, statically dimerized Heisenberg models in two dimensions are investigated. Here the quantum Monte Carlo technique is used to determine the specific pattern of dimerized bonds which has the largest gain in the magnetic energy. A direct comparison of ground state energies is only possible for a small number of dimerized systems. For this reason an alternative approach is developed, which allows to study a significantly larger class of models. Starting from a Landau expansion of the free energy, this amounts to a computation of specific structure factors of the dimer correlations of the uniform Heisenberg model. As a consequence, a Landau theory for the two-dimensional counterpart of the spin-Peierls transition is established. This adiabatic approach also applies to the Heisenberg model coupled to dispersive phonons. In this case the data for the dimer correlations are used to study the impact of the spin-phonon coupling on the phonon dispersion relations. In particular, the softening of phonon modes is investigated, which is relevant for experimental studies on substances with a quasi two-dimensional magnetic structure.
The tensor network algorithm, a family of prevalent numerical methods for quantum many-body problems, aptly captures the entanglement properties intrinsic to quantum systems, enabling precise representation of quantum states. However, its computational cost is notably high, particularly in calculating physical observables like correlation functions. To surmount the computational challenge and enhance efficiency, we propose integrating the Green's function Monte Carlo (GFMC) method with the projected entangled pair state (PEPS) ansatz. This approach combines the high-efficiency characteristics of Monte Carlo with the sign-free nature of tensor network states and proves effective in addressing the computational bottleneck. To showcase its prowess, we apply this hybrid approach to investigate the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice, a model notorious for its sign problem in quantum Monte Carlo simulations. Our results reveal a substantial improvement in the accuracy of ground-state energy when utilizing a preliminary PEPS as the guiding wave function for GFMC. By calculating the structure factor and spin-spin correlation functions, we further characterize the phase diagram, identifying a possible columnar valence-bond state phase within the intermediate parameter range of
0.52<J2/J1<0.58. This comprehensive study underscores the efficacy of our combined approach, demonstrating its ability to accurately simulate frustrated quantum spin systems while ensuring computational efficiency.