Vacancy-induced local moments in quantum paramagnetic phases: An SU($N$) designer Hamiltonian study

Abstract

We explore the effects of non-magnetic impurities (vacancy disorder) on the quantum paramagnetic phases stabilized by SU($N$) designer Hamiltonians on bipartite lattices. Using the results of our quantum Monte Carlo simulations, we demonstrate that isolated vacancies seed emergent spin $S=1/2$ moments in their vicinity when the low-temperature state has valence bond solid order. Indeed, our quantum Monte Carlo results for the low-temperature susceptibility in such regimes shows clear evidence of the vacancy-induced Curie tails associated with these emergent moments, and our zero-temperature projector Monte Carlo results on the ground-state wavefunction in the valence bond basis provide additional evidence in support of this picture. Further, for such designer Hamiltonians on the Lieb lattice with two additional sites on each bond of a square lattice, we identify a low-temperature spin liquid-like regime with no sign of spin or valence bond order. This liquid-like regime serves as a test bed for validating a recently-developed argument concerning the effects of vacancy disorder in such low temperature regimes. Consistent with this argument, we find that isolated vacancies do not seed emergent local moments in such spin liquids. Instead, in the presence of vacancy disorder, emergent local moments are associated with the presence of monomers in maximum-density dimer packings of the corresponding diluted lattice.

Figures (9)

• Figure 1: Impurity susceptibilities associated with the response to uniform static fields that couple to $\mathcal{Q}_{\alpha\alpha}^{\rm tot}$ and $n^{\rm tot}_{\alpha\alpha}$ respectively, for two well-separated vacancies in the columnar VBS phase of the honeycomb-lattice $SU(2)$$J\hbox{-}Q_3$ model. The data are obtained for $J_m=1.0$ and $Q_3=2.0$ on an $L \times L$ honeycomb lattice (with $L^2$ unit cells) with periodic boundary conditions, containing two isolated vacancies placed on opposite sublattices and separated by a distance $L/2$. Left panel: Data for $\chi_{\rm imp}^{\mathcal{Q}}$ reveals the presence of a clear low-temperature Curie tail, $\chi_{\rm imp}^{\mathcal{Q}} \propto 1/T \equiv \beta$, consistent with the presence of two effectively free SU($2$) moments induced by the vacancies. The solid line is a linear fit whose slope matches the Curie constant of free spin-$\frac{1}{2}$ moments. Right panel: Impurity susceptibility $\chi_{\rm imp}^{n}$ also displays Curie-like behavior over the same temperature range, indicating that the vacancy-induced moments contribute to both uniform and staggered channels in the VBS phase.
• Figure 2: Right panel: Schematic representation of a columnar valence bond solid ordered state (valence bonds are represented by thick red bonds) on the honeycomb lattice, with two isolated vacancies (represented by gray circles). Each vacancy nucleates a localized free spin-$1/2$ moment (indicated by yellow arrows) on neighboring sites. Left panel: In the presence of the two vacancies, a singlet state that tries to minimizes local distortions to the columnar VBS pattern unavoidably features a domain wall (color coded green). The domain wall separates two symmetry-related columnar VBS patterns on either side and incurs an energy cost proportional to its length in the columnar VBS phase.
• Figure 3: Susceptibility $\chi_{\Psi}$ associated with the VBS order parameter $\Psi$, defined in Eq. \ref{['eq:chi_vbs']}, plotted as a function of $1/L$ for the $SU(N)$ nearest neighbor singlet projector Hamiltonian on the $K=2$ Lieb lattice at $N=8$ and several low-temperature (large $\beta$) values. The susceptibility decreases systematically with increasing system size and extrapolates to zero in the thermodynamic limit, indicating the absence of long-range columnar VBS order at all temperatures studied.
• Figure 4: A schematic illustration of two $\mathcal{R}$-type regions, which are created by deleting sites shown as crossed. Red denotes the $A$ sublattice, while black denotes the $B$ sublattice. Each $\mathcal{R}$-type region traps a monomer, whose possible locations are shown as the green sites in the illustration. Note that one $\mathcal{R}$-type region is seeded by vacancies on the $A$ sublattice, whereas the other such region is created by deleting some $B$ sublattice sites.
• Figure 5: Impurity susceptibilities associated with the response to uniform static fields that couple to $\mathcal{Q}_{\alpha\alpha}^{\rm tot}$ and $n^{\rm tot}_{\alpha\alpha}$ respectively. The data is displayed for two isolated vacancies separated by $L/2$ in an otherwise pure $L \times L$, $K = 2$ Lieb lattice with $L^2$ unit cells and periodic boundary conditions imposed on both directions. Left panel: The impurity susceptibility $\rm \chi_{imp}^{\mathcal{Q}}$ (defined in the text) of the $K = 2$ Lieb lattice SU($N$) projector model at $N = 8$ shows absolutely no sign of a Curie tail at low temperature. Right panel: The impurity susceptibility $\rm \chi_{imp}^{n}$ (defined in the text) of the $K = 2$ Lieb lattice SU($N$) projector model at $N = 8$ also shows absolutely no sign of a Curie tail at low temperature.