We discuss a projector Monte Carlo method for quantum spin models formulated in the valence bond basis, using the S=1/2 Heisenberg antiferromagnet as an example. Its singlet ground state can be projected out of an arbitrary basis state as the trial state, but a more rapid convergence can be obtained using a good variational state. As an alternative to first carrying out a time consuming variational Monte Carlo calculation, we show that a very good trial state can be generated in an iterative fashion in the course of the simulation itself. We also show how the properties of the valence bond basis enable calculations of quantities that are difficult to obtain with the standard basis of Sz eigenstates. In particular, we discuss quantities involving finite-momentum states in the triplet sector, such as the dispersion relation and the spectral weight of the lowest triplet.
We show how efficient loop updates, originally developed for Monte Carlo simulations of quantum spin systems at finite temperature, can be combined with a ground-state projector scheme and variational calculations in the valence-bond basis. The methods are formulated in a combined space of spin
z components and valence bonds. Compared to schemes formulated purely in the valence-bond basis, the computational effort is reduced from up to
O(N2) to
O(N) for variational calculations, where
N is the system size, and from
O(m2) to
O(m) for projector simulations, where
m⪢N is the projection power. These improvements enable access to ground states of significantly larger lattices than previously. We demonstrate the efficiency of the approach by calculating the sublattice magnetization
Ms of the two-dimensional Heisenberg model to high precision, using systems with up to $256\ifmmode\times\else\texttimes\fi{}256$ spins. Extrapolating the results to the thermodynamic limit gives
Ms=0.30743(1). We also discuss optimized variational amplitude-product states, which were used as trial states in the projector simulations, and compare results of projecting different types of trial states.
We develop a generalization of the singlet sector valence-bond basis projection algorithm of Sandvik, Beach, and Evertz (Sandvik 2005 Phys. Rev. Lett. 95 207203; Beach and Sandvik 2006 Nucl. Phys. B 750 142; Sandvik and Evertz 2010 Phys. Rev. B 82 024407) to cases in which the ground state of an antiferromagnetic Hamiltonian has total spin S tot = 1/2 in a finite-size system. We explain how various ground state expectation values may be calculated by using generalizations of the estimators developed in the singlet case, and illustrate the power of the method by calculating the ground state spin texture and bond energies in an L × L Heisenberg antiferromagnet with L odd and free boundaries.
A Quantum Monte Carlo method of calculating operator expectation values for the ground state of the nearest-neighbor Heisenberg model on large square lattices is presented, along with some comparisons of the results obtained from this method and alternative methods for 2D lattices. In the simulations the ground state is projected out from an arbitrary state and sampled in a valence\nbond basis spanning the spin singlet subspace. The projection method is then extended to be used on a simplified Hubbard model at half-filling, which is\naptly called the T0-less Hubbard model, for arbitrary relative strengths of the electron hopping and electron-electron repulsion energy parameters. Results for calculated ground state energies in 1D are presented. These calculations are performed using a spin-charge separated basis, in which the valence bonds\nare generalized to also carry charge.
The set of valence-bond states --- states in which localized spin-1/2 particles are correlated in singlet pairs said to be connected by valence bonds --- provides a useful basis for visualizing singlet ground states of quantum spin systems. For example, the ground state of the uniform one-dimensional nearest-neighbor spin-1/2 antiferromagnetic (AFM) Heisenberg model (the prototypical spin-liquid state) can be viewed as a strongly fluctuating liquid of valence bonds with a power-law length distribution. This intuitive picture directly reflects the long-range spin correlations in this state, as well as the existence of gapless excitations created by breaking long bonds. Valence-bond states also play a key role in describing the physics of random spin-1/2 AFM Heisenberg chains. For these systems, it was shown by Fisher, using a real space renormalization group analysis, that on long-length scales the ground state is described by a single valence-bond state known as a random singlet state. This single valence-bond state should be viewed as a caricature of the true ground state, which will certainly exhibit bond fluctuations on short-length scales. In valence-bond Monte Carlo (VBMC) simulations valence-bond states are used to stochastically sample singlet ground states of quantum spin systems. One of the appealing features of VBMC is that if one imagines viewing the sampled valence-bond states over many Monte Carlo time steps the resulting ''movie" would correspond closely to the intuitive resonating valence bond picture described above. For random Heisenberg chains (and related models) VBMC should therefore provide a useful method for directly studying the phenomenon of random singlet formation on long-length scales, while at the same time capturing the short-range fluctuations which will always be present. In this dissertation I present results of VBMC studies for a class of models which include the uniform and random spin-1/2 AFM Heisenberg chains, as well as models describing chains of interacting non-Abelian quasiparticles --- exotic quasiparticles conjectured to exist in certain fractional quantum Hall states. In addition to numerically computing and analyzing the so-called valence-bond entanglement scaling in these models, I introduce a new quantity which I refer to as the valence-bond fluctuation (the central new result and the main contribution of this dissertation). It is shown that this quantity, which is easy to compute in valence-bond Monte Carlo, provides a direct signature of random singlet phase formation by essentially allowing one to directly ''see" the ''locking" of the ground state into a particular valence-bond state on long-length scales. A detailed scaling analysis of this new quantity is then used to extract the dependence of the fluctuation length scale on disorder strength. Where possible, the results are compared to previous numerical and analytic work on the relevant models.
We discuss designer Hamiltonians—lattice models tailored to be free from sign problems (“de-signed”) when simulated with quantum Monte Carlo (QMC) methods but which still host complex many-body states and quantum phase transitions of interest in condensed matter physics. We focus on quantum spin systems in which competing interactions lead to nonmagnetic ground states. These states and the associated quantum phase transitions can be studied in great detail, enabling direct access to universal properties and connections with low-energy effective quantum field theories. As specific examples, we discuss the transition from a Néel antiferromagnet to either a uniform quantum paramagnet or a spontaneously symmetry-broken valence-bond solid (VBS) in SU(2) and SU(N) invariant spin models. We also discuss anisotropic (XXZ) systems harboring topological Z2 spin liquids and the XY* transition. We briefly review recent progress on QMC algorithms, including ground-state projection in the valence-bond basis and direct computation of the Renyi variants of the entanglement entropy.