Session B18: Focus Session: Low D/Frustrated Magnetism - Quantum Magnetism

Abelian and Non-Abelian Height Models

We present Monte Carlo simulations on a new class of lattice models in which the degrees of freedom are elements of an abelian or non-abelian finite group $G$, placed on directed edges of a two-dimensional lattice. The group product around any plaquette is constrained to be the group identity, as in a discrete gauge model, but in contrast a ``height model" only allows a certain subset of group elements to appear on edges. These models often realize a classical form of topological order, in that the ensemble breaks up into sectors labeled by loop products (group products taken around topologically non-trivial loops). Our implementation uses a non-local Monte Carlo update, whereby a pair of topological defects is created and later recombined after one diffuses; this allows the simulation to visit different topological sector. We measured two quantities as diagnostics of topological order (i) The relative probabilities of different sectors, which were found to converge to unity with increasing system size $L$. (ii) The probability distribution of the separation $R$ of a defect pair, which should approach a constant (be deconfined). Both results show exponential decay as a function of $L$ or $R$, as expected for a liquid-like phase having only topological order. As a check, we measured the same two quantities in a model equivalent to the 6-vertex model, known to be a critical state, and confirmed the algebraic decay in that case.   

Application of DFT+U for calculating magnetic parameters for manganese based molecular magnets

Single-molecule magnets are promising materials for molecular spintronics and quantum computing applications. Two methods feasible to predict the exchange coupling parameters of molecular magnets, broken symmetry Density Functional Theory and DFT with empirical Hubbard U parameter (DFT+U). In this contribution we apply DFT+U to study magnetic coupling for two Mn12-based molecular magnetic wheel using Vanderbilt Ultrasoft Pseudopotential plane wave DFT method implemented in Quantum ESPRESSO. Unlike most previous studies, we adjust U parameters for both metal and ligand atoms using five dineuclear organometallics as the benchmarks. Our study finds antiparallel spin alignment of the weakly interacting fragments of Mn$_{12}$, while the magnetic coupling inside the fragments are much stronger, both are in agreement with experimental observations.   

DMRG Study of Anisotropic Triangular Heisenberg Lattice

The anisotropic antiferromagnetic two-dimensional triangular Heisenberg lattice for spin $1/2$ describes certain classes of transition-metal oxides (TMOs) and chalcogenides (TMCs) supported by experimental data. The understanding of the ground state properties of this frustrated system from a theoretical point of view, however, has remained an extraordinary challenge. In the model under consideration, quasi-one-dimensional Heisenberg chains of uniform intrachain coupling strength $J$ interact with their neighboring chains via the triangular interchain coupling $J'$. By varying the anisotropy ratio $j=J'/J$ from $j=0$ (decoupled Heisenberg chains) to $j=1$ (uniform triangular lattice with finite Neel order like local magnetization), it was pointed out [1,2] that spin liquid properties up to remarkably high values of $j$ of about 0.85 exist. We present in detail our results on the incommensurable correlations using DMRG with special care given to finite size effects. We argue that incommensurable correlations persist throughout the entire range of $j\in[0,1]$.\\ $[1]$ S. Yunoki et al., PRB 74, 014408 (2006).\\ $[2]$ D. Heidarian et al., PRB 80, 012404 (2009).   

Interacting antikinks on a diamondback ladder I

Recently introduced ``antikinks'' are spin 1/2 excitations of the Heisenberg antiferromagnet on a sawtooth lattice. The idea is that they mimic spinons of the kagome antiferromagnet. Antikinks are triangles of spins which are not in their ground state. Treating antikinks as free non- interacting particles (a good approximation for the sawtooth chain), their energy was found to be substantially reduced by delocalization. We study antikinks on a ``diamondback'' ladder in which all spins are shared between two triangles. Consequently, in a uniform case the concentration of antikinks becomes 1/4 and they strongly interact, making such a model a much better approximation for the kagome case. We treat these effects perturbatively by allowing different Heisenberg couplings on the up- and downward oriented triangles, the two limiting cases being the sawtooth and uniform diamondback ladder. We find a non-monotonic, power-law decay of induced interactions between the antikinks with their separation. The consequences of these interactions will be discussed in this talk.   

Interacting antikinks on a diamondback ladder II

Recently introduced ``antikinks'' are spin 1/2 excitations of the Heisenberg antiferromagnet on a sawtooth lattice [1]. The idea is that they mimic spinons of the kagome antiferromagnet. Antikinks are triangles of spins which are not in their ground state. Treating antikinks as free non-interacting particles (a good approximation for the sawtooth chain), their energy was found to be substantially reduced by delocalization [1]. We study antikinks on a ``diamondback'' ladder in which all spins are shared between two triangles. Consequently, in a uniform case the concentration of antikinks becomes 1/4 and they strongly interact, making such a model a much better approximation for the kagome case. We treat these effects perturbatively by allowing different Heisenberg couplings on the up- and downward oriented triangles, the two limiting cases being the sawtooth and uniform diamondback ladder. We find a non-monotonic, power-law decay of induced interactions between the antikinks with their separation. The consequences of these interactions will be discussed in this talk. \\[4pt] [1] Z. Hao and O. Tchernyshyov, Phys. Rev. Lett. \textbf{103}, 187203 (2009)   

Partial Kondo screening in geometrically frustrated Kondo lattice systems

One of the most important concepts in Kondo lattice systems is competition between the Kondo coupling and the RKKY interaction. The competition leads to a quantum critical point between a magnetically-ordered state and a Fermi liquid state, and furthermore, it is the origin of novel phenomena around the quantum critical point, such as a non-Fermi liquid behavior and a superconductivity. To explore a new quantum phase resulting from the competition, we investigate the ground state of geometrically-frustrated Kondo lattice systems by employing a high-precision variational Monte Carlo simulation. We find that a partially-ordered state, in which a magnetic order and a Kondo spin singlet coexists, emerges between a magnetically-ordered state stabilized by the RKKY interaction and a Kondo spin liquid state stabilized by the Kondo coupling. We clarified that this new quantum phase is stabilized by quantum fluctuations as well as magnetic anisotropy, and that it is accompanied by a charge disproportionation. Ref. Y. Motome {\it et al.}, Phys. Rev. Lett. {\bf 105}, 036403 (2010).   

Study of SU(N) magnets on the cubic lattice

We consider a class of $SU(N)$ magnets that have the same spin on every lattice site, which is obtained as the completely antisymmetric tensor product of $m < N$ fundamental representations. These models, which can be realized in ultracold gases of alkaline earth atoms in optical lattice potentials, have the remarkable property that more than two spins must be combined to form a singlet. A recent study of this model on the square lattice in the large-$N$ limit found a chiral spin liquid ground state with topological order. Inspired by this result, we have studied the three-dimensional version of this model, solving it on the cubic lattice in the large-$N$ limit, which addresses the competition among a variety of non-magnetic states, including some with exotic order. We present results on the phase diagram as the fraction $m/N$ is varied.   

Trial wave function for the quantum Ising model at zero temperature

A trial wavefunction for the ground state of the transverse field Ising model is proposed. It is a product of pair wavefunctions, which is exact for up to three spins, and is amenable to Monte Carlo calculations. We study the phase transition that occurs at zero temperature as the transverse field varies. Results for the Ising ferromagnet and some spin-glass models will be given.   

Thermodynamics of deconfined bosonic spinons in two dimensions

We consider the quantum phase transition between a Neel antiferromagnet and a valence-bond solid (VBS) in a two-dimensional system of $S=1/2$ spins. Assuming that the excitations of the critical ground state are linearly dispersing deconfined spinons obeying Bose statistics, we derive expressions for the specific heat and the magnetic susceptibility at low temperature $T$. Comparing with quantum Monte Carlo results for the J-Q model, which is a candidate for a deconfined Neel--VBS transition, we find excellent agreement, including a logarithmic correction in the susceptibility. In our treatment, this is a direct consequence of a confinement length scale $\Lambda \propto \xi^{1+a} \propto 1/T^{1+a}$, where $\xi$ is the correlation length and $a>0$ (with $a\approx 0.2$ in the model). \newline Reference: A. W. Sandvik, V. N. Kotov, and O. P. Sushkov, arXiv:1010.2522 (2010).   

Properties of Resonating-Valence-Bond Spin Liquids and Critical Dimer Models

We use Monte Carlo simulations to study properties of resonating-valence-bond (RVB) spin liquid states for $s=1/2$ spins on 2D square lattices. It is well known that the spin-spin correlations decay exponentially in these states, but we find that the four-spin (valence-bond-solid, VBS, type) correlations are critical [1]. We compare various properties of the RVB with those of the classical dimer model (CDM), i.e., the exact ground state wavefunction of the critical Rokhsar-Kivelson quantum dimer model. It is well known that the CDM maps to a height model with a gradient-squared elasticity governed by a stiffness constant $K$. We show that also the RVB has such an effective classical field theory description, namely its (i) four-spin (dimer) correlations (ii) probabilities of different winding number sectors, and (iii) separation of monomer defect pairs, are all consistent with the same value of $K$ (which is higher than in the CDM, i.e., the RVB is closer to an ordered VBS state). In addition to the short-bond RVB we also consider systems with longer bonds, and again find consistency with the height-model description. We discuss implications of the critical fluctuations of the RVB states. \\[4pt] [1] Y. Tang, A. W. Sandvik, and C. L. Henley, arXiv:1010.6146.   

Quantum spin liquid in two-dimensional Kagome lattice spin-1/2 XY model with 4-site ring exchange

We have studied the 2D Kagome lattice spin-1/2 XY model with 4-site exchange. The ground state properties are investigated within the framework of the Stochastic Series Expansion quantum Monte Carlo (QMC) technique. We have found a featureless insulating phase in the regime of large 4-site exchange interaction. This novel phase is a potential candidate for a the $Z_2$ quantum spin liquid phase proposed by Balents, Girvin and Fisher [Phys. Rev. B, ${\bf 65}$, 224412 (2002)] in a related model. Our efforts to characterize this phase using large-scale QMC simulations are also discussed.   

Linear independence of nearest neighbor valence bond states on several 2D lattices

We show for several two-dimensional lattices that the spin-$1/2$ nearest neighbor valence bond states are linearly independent. To do so, we utilize and further develop a method recently introduced [1] for the kagome lattice. This method relies on the identification of an appropriate cell for the respective lattice, for which a certain local linear independence property can be demonstrated. Whenever this can be achieved, linear independence follows for arbitrarily large lattices that can be covered by such cells, for open or periodic boundary conditions. We report that this method is applicable to a number of 2D lattices including the kagome, honeycomb, square, pentagonal I and II, and the star lattice. Applications of general linear independence properties, such as the derivation of effective quantum dimer models, are discussed. Furthermore, motivated by a spin-$1/2$ Hamiltonian on the kagome lattice that has Anderson's resonating-valence-bond (RVB) spin liquid wave function(s) as ground state(s) [1], we mention possibilities to study the properties of this RVB wave function for the kagome and other frustrated lattices using Monte Carlo techniques. $[$1$]$ A. Seidel, Phys. Rev. B \textbf{80,} 165131 (2009).   

Symmetry Fractionalization in Two Dimensions

Topologically ordered states are often characterized by topological properties, such as braiding statistics and fusion rules, of their excitations. However, excitations also carry symmetry quantum number, namely a representation of a symmetry group, when a topologically ordered state respects the symmetry. If an excitation's symmetry quantum number cannot be obtained from a finite integer number of fundamental constituents of the system, we propose to call such phenomena ``symmetry fractionalization.'' We introduce a solvable SO(3) spin-rotational and time reversal invariant spin-1 model on the honeycomb and decorated honeycomb lattices. We show that the ground state is the equal-weight superposition of all valence loops, which we call ``resonating valence loop'' (RVL) state and which is a quantum spin liquid respecting all the symmetries of the model. Ends of broken loops are excitations with spin-1/2, which are deconfined spinons. Since spin-1/2 cannot be obtained from an integer numbers of spin-1, the system exhibits symmetry fractionalization (specifically the ``SO(3) symmetry fractionalization''). Moreover, for time-reversal $T$, a spinon has $T^{2}$ = -1, while integer spins have $T^{2}$ = +1. Consequently, the system also has ``time-reversal fractionalization.''   

Defect-driven phase transitions out of the Coulomb phase

Lattice models constrained by a local ``conservation law,'' such as close-packed dimer models on 3D bipartite lattices, exhibit an emergent ``Coulomb phase'' with characteristic power-law correlations. We have studied, by Monte Carlo simulations, phase transitions out of the Coulomb phase induced by introducing a finite fugacity of defect excitations in dimer models. We report two cases. (1) In the simple cubic dimer covering, we admit non- bipartite dimers (connecting sites in the same sublattice), which appear as effective charges with Coulomb-like interactions. Non- bipartite defects induce a transition immediately out of the Coulomb phase, exponentially damping the critical correlations via Debye screening. We characterize this transition by extracting the screening length from our numerical calculation of the dimer structure factor.(2) In the diamond lattice, we initially restrict the dimers to a 2D layer forming a (bipartite) honeycomb lattice, and then admit interlayer dimers. These bipartite dimers appear as dipoles and do not destroy the Coulomb phase, but induce an immediate transition from a 2D to 3D Coulomb phase.