We study the mechanism of decay of a topological (winding-number) excitation due to finite-size effects in a two-dimensional valence-bond solid state, realized in an
S=1/2 spin model $(J\ensuremath{-}Q$ model) with six-spin interactions and studied using projector Monte Carlo simulations in the valence bond basis. A topological excitation with winding number
∣W∣>0 contains domain walls, which are unstable due to the emergence of long valence bonds in the wave function, unlike in effective descriptions with the quantum dimer model (which by construction includes only short bonds). We find that the lifetime of the winding number in imaginary time, which is directly accessible in the simulations, diverges as a power of the system length
L. The energy can be computed within this time (i.e., it converges toward a ``quasieigenvalue'' before the winding number decays) and agrees for large
L with the domain-wall energy computed in an open lattice with boundary modifications enforcing a domain wall. Constructing a simplified two-state model which can be solved in real and imaginary time, and using the imaginary-time behavior from the simulations as input, we find that the real-time decay rate out of the initial winding sector is exponentially small in
L. Thus, the winding number rapidly becomes a well-defined conserved quantum number for large systems, supporting the conclusions reached by computing the energy quasieigenvalues. Including Heisenberg exchange interactions which bring the system to a quantum-critical point separating the valence-bond solid from an antiferromagnetic ground state (the putative ``deconfined'' quantum-critical point), we can also converge the domain-wall energy here and find that it decays as a power law of the system size. Thus, the winding number is an emergent quantum number also at the critical point, with all winding number sectors becoming degenerate in the thermodynamic limit. This supports the description of the critical point in terms of a U(1) gauge-field theory.
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 perform highly accurate density matrix renormalization group (DMRG) simulations to investigate the ground-state properties of the spin-
1/2 antiferromagnetic square lattice Heisenberg
J1-
J2 model. Based on studies of numerous long cylinders with circumferences of up to 14 lattice spacings, we obtain strong evidence for a topological quantum spin liquid state in the region $0.41\ensuremath{\le}{J}_{2}/{J}_{1}\ensuremath{\le}0.62$, separating conventional N\'eel and striped antiferromagnetic states for smaller and larger
J2/J1, respectively. The quantum spin liquid is characterized numerically by the absence of magnetic or valence bond solid order, and nonzero singlet and triplet energy gaps. Furthermore, we positively identify its topological nature by measuring a nonzero topological entanglement entropy $\ensuremath{\gamma}=0.70\ifmmode\pm\else\textpm\fi{}0.02$, extremely close to $\ensuremath{\gamma}=\mathrm{ln}(2)\ensuremath{\approx}0.69$ (expected for a
Z2 quantum spin liquid) and a nontrivial finite size dimerization effect depending upon the parity of the circumference of the cylinder. We also point out that a valence bond solid, and indeed any discrete symmetry breaking state, would be expected to show a constant correction to the entanglement entropy of opposite sign to the topological entanglement entropy.
Dirac fermions in (2 + 1) dimensions with dynamically generated anticommuting SO(3) antiferromagnetic and
Z2 Kekul\'e valence-bond solid (KVBS) masses map onto a field theory with a topological $\ensuremath{\theta}$ term. This term provides a mechanism for continuous phase transitions between different symmetry-broken states: topological defects of one phase carry the charge of the other and proliferate at the transition. The $\ensuremath{\theta}$ term implies that a domain wall of the
Z2 KVBS order parameter harbors a spin-
1/2 Heisenberg chain, as described by a (1 + 1)-dimensional SO(3) nonlinear sigma model with $\ensuremath{\theta}$ term at $\ensuremath{\theta}=\ensuremath{\pi}$. Using pinning fields to stabilize the domain wall, we show that our auxiliary-field quantum Monte Carlo simulations indeed support the emergence of a spin-
1/2 chain at the
Z2 topological defect. Surprisingly, the consequences of the topological term are seen far from the critical point such that the physics of apparently unrelated model systems are naturally understood by invoking them. This concept can be generalized to higher dimensions where (2 + 1)-dimensional SO(4) or SO(5) theories with topological terms are realized at a domain wall.
The ground-state and thermodynamic properties of quantum mixed-spin chains of1/2-1/2-1-1and 3/2-3/2-1-1are investigated by a quantum Monte Carlo simulation with the loop-cluster algorithm. For 1/2-1/2-1-1 chain, we find it has two phases separated by an energy-gap vanishing point in the ground-state. For 3/2-3/2-1-1 chain, the numerical results show two energy-gap vanishing points isolated by different phases in its ground-state. Our calculations indicate that all these ground state phases can be understood by means of valence-bond-solid picture, and the thermodynamic behavior at finite temperatures is continuous as a function of parameterα=J2/J1.
The large-N limit of a nearest-neighbor SU(N) antiferromagnet on a bipartite lattice exhibits in dimensions d\ensuremath{\ge}2 a zero-temperature phase transition between a N\'eel ordered state and a resonating-valence-bond state. Here it is shown in d=1,2 that topological effects produce spin-Peierls or valence-bond-solid order in the non-N\'eel phase with a ground-state degeneracy which varies periodically with ``spin'' for fixed N with periodicity given by the coordination number of the lattice. Thus a non-N\'eel phase of the spin-(1/2 Heisenberg model on a square lattice would be a spin-Peierls state with a fourfold degeneracy due to broken lattice rotational symmetry.