We present numerical evidence for the existence of an extended valence bond solid (VBS) phase at
T=0 in the kagome
S=1/2 Heisenberg antiferromagnet with ferromagnetic further-neighbor interactions. The VBS is located at the boundary between two magnetically ordered regions and extends close to the nearest-neighbor Heisenberg point. It exhibits a diamondlike singlet covering pattern with a 12-site unit cell. Our results suggest the possibility of a direct, possibly continuous, quantum phase transition from the neighboring
q=0 coplanar magnetically ordered phase into the VBS phase. Moreover, a second phase which breaks lattice symmetries, and is of likely spin-nematic type, is found close to the transition to the ferromagnetic phase. The results have been obtained using large-scale numerical exact diagonalization. We discuss implications of our results on the nature of the nearest-neighbor Heisenberg antiferromagnet.
Frustrated one-dimensional (1D) magnets are known as ideal playgrounds for new exotic quantum phenomena to emerge. We consider an elementary frustrated 1D system: the spin- \frac{1}{2} 1 2 ferromagnetic ( J_1 J 1 ) Heisenberg chain with next-nearest-neighbor antiferromagnetic ( J_2 J 2 ) interactions. On the basis of density-matrix renormalization group calculations we show the existence of a finite spin gap at J_2/|J_1|>1/4 J 2 / | J 1 | > 1 / 4 and we find the ground state in this region to be a valence bond solid (VBS) with spin-singlet dimerization between third-neighbor sites. The VBS is the consequence of spontaneous symmetry breaking through order by disorder. Quite interestingly, this VBS state has a Affleck-Kennedy-Lieb-Tasaki-type topological order. This is the first example of a frustrated spin chain in which quantum fluctuations induce gapped topological order.
We consider the
S=1/2 Heisenberg model with nearest-neighbor interaction
J and an additional multispin interaction
Q3 on the square lattice. The
Q3 term consists of three bond-singlet projectors and is chosen to favor the formation of a valence-bond solid (VBS) where the valence bonds (singlet pairs) form a staggered pattern. The model exhibits a quantum phase transition from the N\'eel state to the VBS as a function of
Q3/J. We study the model using quantum Monte Carlo (stochastic series expansion) simulations. The N\'eel-VBS transition in this case is strongly first order in nature, in contrast to similar previously studied models with continuous transitions into columnar VBS states. The qualitatively different transitions illustrate the important role of an emerging
U(1) symmetry in the latter case, which is not possible in the present model due to the staggered VBS pattern (which does not allow local fluctuations necessary to rotate the local VBS order parameter).
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.
Various lattice geometries and boundary conditions are used to investigate valence-bond-solid (VBS) ordering in the ground state of an
S=1/2 square-lattice quantum spin model---the
J-
Q model, in which four- or six-spin interactions
Q are added to the standard Heisenberg exchange
J. Ground-state results for finite systems (with up to thousands of spins) are obtained using an unbiased projector quantum Monte Carlo method. It is found that great care has to be taken when extrapolating the order parameter to infinite lattice size, in particular, in cylinder geometry. Even though strong VBS order exists in two dimensions, and is established clearly with increasing system size on $L\ifmmode\times\else\texttimes\fi{}L$ lattices (or ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ lattices with a fixed aspect ratio
Lx/Ly of order 1), only short-range VBS correlations are observed on long cylinders (when ${L}_{x}\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed
Ly). The correlation length increases with the cylinder width, until long-range order sets in at a ``critical'' width. This width is very large even when the 2D order is relatively strong. For example, for a system in which the order parameter is
70 of the largest possible value,
Ly=8 is required for ordering. Extrapolations of the VBS order parameter based on correlation functions (the square of the order parameter) for small $L\ifmmode\times\else\texttimes\fi{}L$ lattices can also be misleading. For a
20-ordered system, results for
L up to $\ensuremath{\approx}$20 appear to extrapolate clearly to a vanishing order parameter, while for larger lattices the scaling behavior crosses over and extrapolates to a nonzero value (with exponentially small finite-size corrections). The VBS order parameter also exhibits interesting edge effects related the known emergent U(1) symmetry close to a ``deconfined'' critical point, which, if not considered properly, can lead to wrong conclusions for the thermodynamic limit. The observed finite-size behavior for small $L\ifmmode\times\else\texttimes\fi{}L$ lattices and long cylinders is very similar to that predicted for a
Z2 spin liquid. The results therefore raise concerns about recent numerical work claiming
Z2 spin-liquid ground states in 2D frustrated quantum spin systems, in particular, the Heisenberg model with nearest and next-nearest-neighbor couplings. Based on the results presented so far, a VBS state in this system cannot be ruled out.
We investigate the ground state phase diagram of the spin-
1/2 antiferromagnetic Heisenberg model on the star lattice using infinite projected entangled pair states (iPEPS) and high-order series expansions. The model includes two distinct couplings:
Jd on the dimer bonds and
Jt on the trimer bonds. While it is established that the system hosts a valence bond solid (VBS) phase for ${J}_{d}\ensuremath{\ge}{J}_{t}$, the ground state phase diagram for
Jd<Jt has remained unsettled. Our iPEPS simulations uncover a first-order phase transition at ${J}_{d}/{J}_{t}\ensuremath{\approx}0.18$, significantly lower than previously reported estimates. Beyond this transition, we identify a close competition between two valence bond crystal (VBC) states; a columnar VBC and a $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC, with the latter consistently exhibiting lower energy across all finite bond dimensions. The high-order series expansion supports this by finding that the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC state indeed becomes energetically favorable, but only at sixth order in perturbation theory, revealing the subtle nature of the competition between candidate states.