Resonating Valence Bond State Research Articles

Quantifying entanglement with scattering experiments

We show how the entanglement contained in states of spins arranged on a lattice may be quantified with observables arising in scattering experiments. We focus on the partial differential cross-section obtained in neutron scattering from magnetic materials but our results are sufficiently general such that they may also be applied to, e.g., optical Bragg scattering from ultracold atoms in optical lattices or from ion chains. We discuss resonating valence bond states and ground and thermal states of experimentally relevant models--such as Heisenberg, Majumdar-Ghosh, and XY model--in different geometries and with different spin numbers. As a by-product, we find that for the one-dimensional XY model in a transverse field such measurements reveal factorization and the quantum phase transition at zero temperature.

Valence bond liquid phase in the honeycomb lattice materialLi2RuO3

The honeycomb lattice material Li2RuO3 undergoes a dimerization of Ru4+ cations on cooling below 270C, where the magnetic susceptibility vanishes. We use density functional theory calculations to show that this reflects the formation of a 'valence bond crystal', with a strong bond disproportionation. On warming, x-ray diffraction shows that discrete three-fold symmetry is regained on average, and the dimerization apparently disappears. In contrast, local structural measurements using high-energy x-rays, show that disordered dimers survive at the nanoscale up to at least 650C. The high temperature phase of Li2RuO3 is thus an example of a valence bond liquid, where thermal fluctuations drive resonance between different dimer coverages, a classic analogue of the resonating valence bond state often discussed in connection with high T$_c$ cuprates.

Mean-field RVB ground states of lattice models of CuNCN

Recently we proposed to describe the fascinating physics of copper carbodiimide, CuNCN, with help of the anisotropic triangular antiferromagnetic Heisenberg model with the parameters Ja and Jab extending along the a, and a ± b lattice directions and a new frustrated Heisenberg antiferromagnetic model with exchange parameters Jc, Ja, and Jac, extending along the c, a, and a ± c (c-a-ca model) directions assuming the resonating valence bond (RVB) type of the corresponding phases. Here we discuss possible RVB ground states of these models in the mean-field approximation and show that in either case it is a two-dimensional RVB state. The difference between the models is that in the ground state of the triangular model the quasiparticle spectrum features a finite (although exponentially small) energy gap for arbitrary weak Jab whereas that of the c-a-ca model shows two pseudogaps and a linear dependence of the quasiparticle density of states in the low-energy range.

High-resolution neutron diffraction study of CuNCN: New evidence of structure anomalies at low temperature

Copper carbodiimide (CuNCN) is the nitrogen-containing analogue of cupric oxide. Based on high-resolution neutron-diffraction data, CuNCN's lattice parameters are derived as a function of the temperature. In accordance with a recent synchrotron study, a clear trend in the cell parameter a is observed accompanying the changing magnetic behavior. With decreasing temperature, a slowly decreases to a minimum at ~100 K after which it rises again. The same trend-albeit more pronounced-is observed for the c lattice parameter at ~35 K. The herein presented neutron powder-diffraction data also support the conjectured sequence of transitions from the high-temperature one-dimensional resonating valence-bond (RVB) state to a transient two-dimensional RVB state and eventually, at lowest temperatures, into another two-dimensional RVB state, presumably the ground state.

Field-induced superfluids and Bose liquids in projected entangled pair states

In two-dimensional incompressible quantum spin liquids, a large enough magnetic field generically induces "doping" of polarized S=1 triplons or S=1/2 spinons. We review a number of cases such as spin-3/2 AKLT or spin-1/2 Resonating Valence Bond (RVB) liquids where the Projected Entangled Pair States (PEPS) framework provides very simple and comprehensive pictures. On the bipartite honeycomb lattice, simple PEPS can describe Bose condensed triplons (AKLT) or spinons (RVB) superfluids with transverse staggered (N\'eel) magnetic order. On the Kagome lattice, doping the RVB state with deconfined spinons or triplons (i.e. spinon bound pairs) yields uncondensed Bose liquids preserving U(1) spin-rotation symmetry. We find that spinon (triplon) doping destroys (preserves) the topological Z_2 symmetry of the underlying RVB state. We also find that spinon doping induces longer range interactions in the entanglement Hamiltonian, suggesting the emergence of (additive) log-corrections to the entanglement entropy.

Antiferromagnetic long-range order in the uniform resonating valence bond state on the square lattice

With an extensive variational Monte Carlo simulation, we show that the uniform resonating valence bond (U-RVB) state on the square lattice is antiferromagnetic long-range ordered. The ordered moment is estimated to be . Such a behavior is quite unexpected from the slave boson mean-field treatment or the Gutzwiller approximation of the uniform RVB state.

Evidence for dimer crystal melting in the frustrated spin-ladder system BiCu2PO6.

In the spin ladder compound BiCu(2)PO(6), there exists a decisive dynamics of spin excitations that we classify and characterize using inelastic light scattering. We observe an interladder singlet bound mode at 24 cm(-1) and two intraladder bound states at 62 and 108 cm(-1) in the leg (bb) and the rung (cc) polarization as well as a broad triplon continuum extending from 36 cm(-1) to 700 cm(-1). Though isolated spin ladder physics can roughly account for the observed excitations at high energies, frustration and interladder interactions need to be considered to fully describe the spectral distribution and scattering selection rules at low and intermediate energies. In addition, we attribute the rich spectrum of singlet bound modes to a melting of a dimer crystal. Our study provides evidence for a Z(2) quantum phase transition from a dimer to a resonating valence bond state driven by singlet fluctuations.

Plaquette resonating valence bond state in a frustrated honeycomb antiferromagnet

We study the proposed plaquette-RVB (pRVB) state in the honeycomb lattice $J_1-J_2$ model with frustration arising from next-nearest neighbour interactions. Starting with the limit of decoupled hexagons, we develop a plaquette operator approach to describe the pRVB state and its low energy excitations. Our calculation clarifies that the putative pRVB state necessarily has f-wave symmetry - the plaquette wavefunction is an antisymmetric combination of the Kekul\'e structures. We estimate the plaquette ordering amplitude, ground state energy and spin gap as a function of $J_2/J_1$. The pRVB state is most stable around $J_2/J_1 \sim 0.25$. We identify the wavevectors of the lowest triplet excitations, which can be verified using exact diagonalization or DMRG studies. When $J_2$ is reduced, we can have either a deconfined Quantum Phase Transition (QPT) or a first-order transition into a N\'eel state. When $J_2$ is increased, we surmise that the system undergoes a first order phase transition into a state which breaks lattice rotational symmetry.

Entanglement in gapless resonating-valence-bond states

We study resonating-valence-bond (RVB) states on the square lattice of spins and of dimers, as well as SU(N)-invariant states that interpolate between the two. These states are ground states of gapless models, although the SU(2)-invariant spin RVB state is also believed to be a gapped liquid in its spinful sector. We show that the gapless behavior in spin and dimer RVB states is qualitatively similar by studying the Rényi entropy for splitting a torus into two cylinders. We compute this exactly for dimers, showing it behaves similarly to the familiar one-dimensional log term, although not identically. We extend the exact computation to an effective theory believed to interpolate among these states. By numerical calculations for the SU(2) RVB state and its SU(N)-invariant generalizations, we provide further support for this belief. We also show how the entanglement entropy behaves qualitatively differently for different values of the Rényi index n, with large values of n proving a more sensitive probe here, by virtue of exhibiting a striking even/odd effect.

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.

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.

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.

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.

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.

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.

Resonating Valence Bond State with Antiferromagnetic Order in Ce Compounds

The Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction model, which takes account of one-dimensional (1D) interactions responsible for the formation of the resonating valence bond (RVB) state and weak 3D interactions responsible for antiferromagnetic (AFM) order, is investigated on the basis of a mean field theory with the two-sublattice model. It is shown that the results of the present calculation are compatible with the properties of CeRu 2 Al 10 and CeOs 2 Al 10 , whose ordered states are assigned to a uniform RVB state accompanied by AFM order.