Resonating Valence Bond Research Articles

Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet

The experimental realization of quantum spin liquids is a long-sought goal in physics, as they represent new states of matter. Quantum spin liquids cannot be described by the broken symmetries associated with conventional ground states. In fact, the interacting magnetic moments in these systems do not order, but are highly entangled with one another over long ranges. Spin liquids have a prominent role in theories describing high-transition-temperature superconductors, and the topological properties of these states may have applications in quantum information. A key feature of spin liquids is that they support exotic spin excitations carrying fractional quantum numbers. However, detailed measurements of these 'fractionalized excitations' have been lacking. Here we report neutron scattering measurements on single-crystal samples of the spin-1/2 kagome-lattice antiferromagnet ZnCu(3)(OD)(6)Cl(2) (also called herbertsmithite), which provide striking evidence for this characteristic feature of spin liquids. At low temperatures, we find that the spin excitations form a continuum, in contrast to the conventional spin waves expected in ordered antiferromagnets. The observation of such a continuum is noteworthy because, so far, this signature of fractional spin excitations has been observed only in one-dimensional systems. The results also serve as a hallmark of the quantum spin-liquid state in herbertsmithite.

Frustrated Resonating Valence Bond States in Two Dimensions: Classification and Short-Range Correlations

Resonating valence bond (RVB) states are of crucial importance in our intuitive understanding of quantum spin liquids in 2D. We systematically classify short-range bosonic RVB states into symmetric or nematic spin liquids by examining their flux patterns. We further map short-range bosonic RVB states into projected BCS wave functions, on which we perform large-scale Monte Carlo simulations without the minus sign problem. Our results clearly show that both spin and dimer correlations decay exponentially in all the short-range frustrated (nonbipartite or Z2) bosonic RVB states we studied, indicating that they are gapped Z2 quantum spin liquids. Generically, we conjecture that all short-range frustrated bosonic RVB states in 2D have only short-range correlations.

Correlation Functions in SU(2)-Invariant Resonating-Valence-Bond Spin Liquids on Nonbipartite Lattices

We introduce a MonteCarlo scheme based on sampling of Pfaffians to investigate Anderson's resonating-valence-bond (RVB) spin liquid wave function on the kagome and the triangular lattice. This eliminates a sign problem that prevents utilization of the valence bond basis in MonteCarlo studies for nonbipartite lattices. Studying lattice sizes of up to 600 sites, we calculate singlet-singlet and spin-spin correlations and demonstrate how the lattice symmetry is restored within each topological sector as the system size is increased. Our findings are consistent with the expectation that the nearest-neighbor RVB states describe a topological spin liquid on these nonbipartite lattices.

Mott insulators in plaquettes

We study small systems of Mott insulating ultracold atoms under the influence of gauge potentials and spin-orbit couplings. We use second order perturbation theory in tunneling, derive an effective theory for the Mott insulators with one atom per site, and solve it exactly. We find dramatic changes in the level structure and in the amount of degeneracies expected. We also demonstrate the dynamical behavior as the barriers between plaquettes are gradually removed and find potentially high overlap with the resonating valence bond (RVB) state of the larger system.

Magnetic response in the underdoped cuprates

We examine the dynamical magnetic response of the underdoped cuprates by employing a phenomenological theory of a doped resonant valence bond state where the Fermi surface is truncated into four pockets. This theory predicts a resonant spin response which with increasing energy (0 to 100meV) appears as an hourglass. The very low energy spin response is found at (pi,pi +- delta) and (pi +- delta,pi) and is determined by scattering from the pockets' frontside to the tips of opposite pockets where a van Hove singularity resides. At energies beyond 100 meV, strong scattering is seen from (pi,0) to (pi,pi). This theory thus provides a semi-quantitative description of the spin response seen in both INS and RIXS experiments at all relevant energy scales.

Resonating valence bond states in the PEPS formalism

We study resonating valence bond (RVB) states in the Projected Entangled Pair States (PEPS) formalism. Based on symmetries in the PEPS description, we establish relations between the toric code state, the orthogonal dimer state, and the SU(2) singlet RVB state on the kagome lattice: We prove the equivalence of toric code and dimer state, and devise an interpolation between the dimer state and the RVB state. This interpolation corresponds to a continuous path in Hamiltonian space, proving that the RVB state is the four-fold degenerate ground state of a local Hamiltonian on the (finite) kagome lattice. We investigate this interpolation using numerical PEPS methods, studying the decay of correlation functions, the change of overlap, and the entanglement spectrum, none of which exhibits signs of a phase transition.

Spin excitations in a single La2CuO4 layer

Cuprates and other high-temperature superconductors consist of two-dimensional layers that are crucial to their properties. The dynamics of the quantum spins in these layers lie at the heart of the mystery of the cuprates. In bulk cuprates such as La(2)CuO(4), the presence of a weak coupling between the two-dimensional layers stabilizes a three-dimensional magnetic order up to high temperatures. In a truly two-dimensional system however, thermal spin fluctuations melt long-range order at any finite temperature. Here, we measure the spin response of isolated layers of La(2)CuO(4) that are only one-unit-cell-thick. We show that coherent magnetic excitations, magnons, known from the bulk order, persist even in a single layer of La(2)CuO(4), with no evidence for more complex correlations such as resonating valence bond correlations. These magnons are, therefore, well described by spin-wave theory (SWT). On the other hand, we also observe a high-energy magnetic continuum in the isotropic magnetic response that is not well described by two-magnon SWT, or indeed any existing theories.

Gapped spin-liquid phase in theJ1−J2Heisenberg model by a bosonic resonating valence-bond ansatz

We study the ground-state phase diagram of the spin-$1/2$${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice with an accurate bosonic resonating-valence-bond (RVB) wave function. In contrast to the RVB ansatz based on Schwinger fermions, the representation based on Schwinger bosons, supplemented by a variational Monte Carlo technique enforcing the exact projection onto the physical subspace, is able to describe a fully gapped spin liquid in the strongly frustrated regime. In particular, a fully symmetric ${Z}_{2}$ spin liquid is stable between two antiferromagnetic phases; a continuous transition at ${J}_{2}=0.4{J}_{1}$, when the Marshall sign rule begins to be essentially violated, and a first-order transition around ${J}_{2}=0.6{J}_{1}$ are present. Most importantly, the triplet gap is found to have a nonmonotonic behavior, reaching a maximum around ${J}_{2}=0.51{J}_{1}$, when the lowest spinon excitation moves from the $\ensuremath{\Gamma}$ to the $M$ point, i.e., $k=(\ensuremath{\pi},0)$.

RVB gauge theory and the topological degeneracy in the honeycomb Kitaev model

We relate the Z2 gauge theory formalism of the Kitaev model to the SU(2) gauge theory of the resonating valence bond physics. Furthermore, we reformulate a known (Feng et al (2007 Phys. Rev. Lett. 98 087204), Chen and Hu (2007 Phys. Rev. B 76 193101), Chen and Nussinov (2008 J. Phys. A: Math. Theor. 41 075001) and Mandal et al (2006 Int. Conf. on Physics Near the Mott Transition)) Jordan–Wigner transformation of the Kitaev model on a torus in a general way that shows that it can be thought of as a Z2 gauge-fixing procedure. We give an explicit construction of the generators of large gauge transformations on a torus in terms of the spin operators. Using these and the non-trivial loop operators, we construct four mutually anti-commuting operators which commute with the Hamiltonian enabling us to prove that all eigenstates of this model, for the time-reversal symmetric case, are fourfold degenerate in the thermodynamic limit.

Topological and entanglement properties of resonating valence bond wave functions

We examine in details the connections between topological and entanglement properties of short-range resonating valence bond (RVB) wave functions using Projected Entangled Pair States (PEPS) on kagome and square lattices on (quasi-)infinite cylinders with generalized boundary conditions (and perimeters with up to 20 lattice spacings). Making use of disconnected topological sectors in the space of dimer lattice coverings, we explicitly derive (orthogonal) "minimally entangled" PEPS RVB states. For the kagome lattice, we obtain, using the quantum Heisenberg antiferromagnet as a reference model, the finite size scaling of the energy separations between these states. In particular, we extract two separate (vanishing) energy scales corresponding (i) to insert a vison line between the two ends of the cylinder and (ii) to pull out and freeze a spin at either end. We also investigate the relations between bulk and boundary properties and show that, for a bipartition of the cylinder, the boundary Hamiltonian defined on the edge can be written as a product of a highly non-local projector with an emergent (local) su(2)-invariant one-dimensional (superfluid) t--J Hamiltonian, which arises due to the symmetry properties of the auxiliary spins at the edge. This multiplicative structure, a consequence of the disconnected topological sectors in the space of dimer lattice coverings, is characteristic of the topological nature of the states. For minimally entangled RVB states, it is shown that the entanglement spectrum, which reflects the properties of the edge modes, is a subset (half for kagome RVB) of the spectrum of the local Hamiltonian, providing e.g. a simple argument on the origin of the topological entanglement entropy S0=-ln 2 of Z2 spin liquids. We propose to use these features to probe topological phases in microscopic Hamiltonians and some results are compared to existing DMRG data.

Unsynchronized resonance of covalent bonds in the superconducting state

DFT calculations performed on different cluster models of cuprates (LaBa2Cu3O6.7, La1.85Sr0.15CuO4, YBa2Cu3O7, TlBa2Ca2Cu3O8.78, HgBa2Ca2Cu3O8.27), metallic systems (Nb3Ge, MgB2) and the pnictide LaO0.92F0.08FeAs made evident the occurrence of unsynchronized resonance of covalent bonds in the superconducting state, as predicted by Pauling´s resonating valence bond (RVB) theory. For cuprates, the unsynchronized resonance involves electron transfer between Cu atoms accompanied by a decrease in the charge of the La, Sr, Y and Ca atoms. For MgB2, electron transfer occurs in the Mg layer, while the B layer behaves as charge reservoir. For Nb3Ge, unsynchronized resonance occurs among the Ge atoms, which should be responsible for charge transfer. For LaO0.92F0.08FeAs, the results suggest that both La-O and Fe-As layers are involved in the mechanism of superconductivity. The identification of unsynchronized resonances in these systems provides evidence which supports RVB as a suitable theory for high-temperature superconductivity (high-T C).

Resonating Valence Bond Wave Functions and Classical Interacting Dimer Models

We relate properties of nearest-neighbor resonating valence-bond (NNRVB) wave functions for SU(g) spin systems on two-dimensional bipartite lattices to those of fully packed interacting classical dimer models on the same lattice. The interaction energy can be expressed as a sum of n-body potentials V(n), which are recursively determined from the NNRVB wave function on finite subgraphs of the original lattice. The magnitude of the n-body interaction V(n) (n>1) is of order O(g(-(n-1))) for small g(-1). The leading term is a two-body nearest-neighbor interaction V2(g) favoring two parallel dimers on elementary plaquettes. For SU(2) spins, using our calculated value of V2(g=2), we find that the long-distance behavior of the bond-energy correlation function is dominated by an oscillatory term that decays as 1/|r|α with α≈1.22. This result is in remarkable quantitative agreement with earlier direct numerical studies of the corresponding wave function, which give α≈1.20.

Non-Abelian fractional quantum Hall states for hard-core bosons in one dimension

I present a family of one-dimensional bosonic liquids analogous to non-Abelian fractional quantum Hall states. A new quantum number is introduced to characterize these liquids, the chiral momentum, which differs from the usual angular or linear momentum in one dimension. As their two-dimensional counterparts, these liquids minimize a $k$-body hard-core interaction with the minimum total chiral momentum. They exhibit global order, with a hidden organization of the particles in $k$ identical copies of a one-dimensional Laughlin state. For $k=2$ the state is a $p$-wave paired phase corresponding to the Pfaffian quantum Hall state. By imposing conservation of the total chiral momentum, an exact parent Hamiltonian is derived which involves long-range tunneling and interaction processes with an amplitude decaying with the chord distance. This family of non-Abelian liquids is shown to be in formal correspondence with a family of spin-$\frac{k}{2}$ liquids which are total singlets made out of $k$ indistinguishable resonating valence bond states. The corresponding spin Hamiltonians are obtained.

Experimental Realization of Plaquette Resonating Valence-Bond States with Ultracold Atoms in Optical Superlattices

The concept of valence-bond resonance plays a fundamental role in the theory of the chemical bond and is believed to lie at the heart of many-body quantum physical phenomena. Here we show direct experimental evidence of a time-resolved valence-bond quantum resonance with ultracold bosonic atoms in an optical lattice. By means of a superlattice structure we create a three-dimensional array of independent four-site plaquettes, which we can fully control and manipulate in parallel. Moreover, we show how small-scale plaquette resonating valence-bond (RVB) states with s- and d-wave symmetry can be created and characterized. We anticipate our findings to open the path towards the creation and analysis of many-body RVB states in ultracold atomic gases.

Possible valence-bond condensation in the frustrated cluster magnet LiZn2Mo3O8

The emergence of complex electronic behaviour from simple ingredients has resulted in the discovery of numerous states of matter. Many examples are found in systems exhibiting geometric magnetic frustration, which prevents simultaneous satisfaction of all magnetic interactions. This frustration gives rise to complex magnetic properties such as chiral spin structures orbitally-driven magnetism, spin-ice behavior exhibiting Dirac strings with magnetic monopoles, valence bond solids, and spin liquids. Here we report the synthesis and characterization of LiZn2Mo3O8, a geometrically frustrated antiferromagnet in which the magnetic moments are localized on small transition metal clusters rather than individual ions. By doing so, first order Jahn-Teller instabilities and orbital ordering are prevented, allowing the strongly interacting magnetic clusters in LiZn2Mo3O8 to probably give rise to an exotic condensed valence-bond ground state reminiscent of the proposed resonating valence bond state. Our results also link magnetism on clusters to geometric magnetic frustration in extended solids, demonstrating a new approach for unparalleled chemical control and tunability in the search for collective, emergent electronic states of matter.

Superatom representation of high-TC superconductivity

A “super-atom” conceptual interface between chemistry and physics is proposed in order to assist in the search for higher TC superconductors. The plaquettes generating the checkerboard superstructure in the cuprates, the C60 molecules in K3C60, and the Mo6S82- clusters in Chevrel phase materials offer such candidate super-atoms. Thus, in the present study high-TC superconductivity HTSC is articulated as the entanglement of two disjoint electronic manifolds in the vicinity of a common Fermi energy. The resulting HTSC ground state couples near-degenerate protected local super-atom states to virtual magnons in an antiferromagnetic AFM embedding. The composite Cooper pairs emerge as the interaction particles for virtual magnons mediated “self-coherent entanglement” of super-atom states. A Hückel type resonating valence bond RVB formalism is employed in order to illustrate the real-space Cooper pairs as well as their delocalization and Bose Einstein condensation BEC on a ring of super-atoms. The chemical potential μBEC for Cooper pairs joining the condensate is formulated in terms of the super-exchange interaction, and consequently the TC in terms of the Neél temperature. A rationale for the robustness of the HTSC ground state is proposed: achieving local maximum “electron correlation entropy” at the expense of non-local phase rigidity.

Comments on the d-wave pairing mechanism for cuprate high T c superconductors: Higher is different?

The question of pairing glue for the cuprate superconductors (SC)is revisited and its determination through the angle resolved photo-emission spectroscopy (ARPES) is discussed in detail. There are two schools of thoughts about the pairing glue question: One argues that superconductivity in the cuprates emerges out of doping the spin singlet resonating valence bond (RVB) state. Since singlet pairs are already formed in the RVB state there is no need for additional boson glue to pair the electrons. The other instead suggests that the d-wave pairs are mediated by the collective bosons like the conventional low $T_c$ SC with the alteration that the phonons are replaced by another kind of bosons ranging from the antiferromagnetic (AF) to loop current fluctuations. An approach to resolve this dispute is to determine the frequency and momentum dependences of the diagonal and off-diagonal self-energies directly from experiments like the McMillan-Rowell procedure for the conventional SC. In that a simple d-wave BCS theory describes superconducting properties of the cuprates well, the Eliashberg analysis of well designed high resolution experimental data will yield the crucial frequency and momentum dependences of the self-energies. This line of approach using ARPES are discussed in more detail in this review, and some remaining problems are commented.

Structural Optimization by Quantum Monte Carlo: Investigating the Low-Lying Excited States of Ethylene.

We present full structural optimizations of the ground state and of the low lying triplet state of the ethylene molecule by means of Quantum Monte Carlo methods. Using the efficient structural optimization method based on renormalization techniques and on adjoint differentiation algorithms recently proposed [Sorella, S.; Capriotti, L. J. Chem. Phys.2010, 133, 234111], we present the variational convergence of both wave function parameters and atomic positions. All of the calculations were done using an accurate and compact wave function based on Pauling's resonating valence bond representation: the Jastrow Antisymmetrized Geminal Power (JAGP). All structural and wave function parameters are optimized, including coefficients and exponents of the Gaussian primitives of the AGP and the Jastrow atomic orbitals. Bond lengths and bond angles are calculated with a statistical error of about 0.1% and are in good agreement with the available experimental data. The Variational and Diffusion Monte Carlo calculations estimate vertical and adiabatic excitation energies in the ranges 4.623(10)-4.688(5) eV and 3.001(5)-3.091(5) eV, respectively. The adiabatic gap, which is in line with other correlated quantum chemistry methods, is slightly higher than the value estimated by recent photodissociation experiments. Our results demonstrate how Quantum Monte Carlo calculations have become a promising and computationally affordable tool for the structural optimization of correlated molecular systems.

Grand canonical variational approach for thet−Jmodel

Gutzwiller-projected BCS wave function or the resonating-valence-bond (RVB) state in the 2D extended t-J model is investigated by using the variational Monte Carlo technique. We show that the results of ground-state energy and excitation spectra calculated in the grand-canonical scheme allowing particle number to fluctuate are essentially the same as previous results obtained by fixing the number of particle in the canonical scheme if the grand thermodynamic potential is used for minimization. To account for the effect of Gutzwiller projection, a fugacity factor proposed by Laughlin and Anderson few years ago has to be inserted into the coherence factor of the BCS state. Chemical potential, particle number fluctuation, and phase fluctuation of the RVB state, difficult or even impossible to be calculated in the canonical ensemble, have been directly measured in the grand-canonical picture. We find that except for La-214 materials, the doping dependence of chemical potential is consistent with experimental findings on several cuprates. Similar to what has been reported by scanning tunneling spectroscopy experiments, the tunneling asymmetry becomes much stronger as doping decreases. We found a very large enhancement of phase fluctuation in the underdoped regime.

A quantum spin‐liquid in correlated relativistic electrons

AbstractIn recent years, an increasing number of systems displaying exotic quantum states like unconventional superconductivity, quantum spin‐liquids, or topological states were experimentally found. Here we summarize findings in quantum Monte Carlo simulations of correlated electrons on a honeycomb lattice, the structure of graphene, that revealed an unexpected spin‐liquid emerging between a state described by massless Dirac fermions and an antiferromagnetically ordered Mott insulator. Moreover, we found that this quantum‐disordered state is a resonating valence‐bond (RVB) liquid, akin to the one proposed for high temperature superconductors. This was the first unbiased determination of a RVB‐liquid in an electronic system.