Valence Bond State Research Articles

Topological order in Gutzwiller-projected wave functions for quantum antiferromagnets

We study the topological order in a resonating valence bond (RVB) state derived from the Gutzwiller projection of the BCS-like mean-field state. We propose the construction of the topological excitation on the projected RVB state through the Gutzwiller projection of the mean-field state with inserted ${Z}_{2}$ flux tube. We prove that all projected RVB states derived from bipartite effective theories, no matter what the gauge structure in the mean-field ansatz is, are positive definite in the sense of the Marshall sign rule, which provides a universal origin for the absence of topological order in such an RVB state.

Quasi-adiabatic continuation in gapped spin and fermion systems: Goldstone’s theorem andflux periodicity

We apply the technique of quasi-adiabatic continuation to study systems withcontinuous symmetries. We first derive a general form of Goldstone’s theoremapplicable to gapped nonrelativistic systems with continuous symmetries. Wethen show that, for a fermionic system with a spin gap, it is possible to insertπ flux into a cylinder with only exponentially small change in the energy of the system, ascenario which covers several physically interesting cases such as an s-wave superconductoror a resonating valence bond state.

A phenomenological theory of the pseudogap state

An ansatz is proposed for the coherent part of the single particle Green’s function in a doped resonant valence bond (RVB) state by analogy with the form derived by Konik and coworkers for an array of 2-leg Hubbard ladders near half-filling. The parameters of the RVB state are taken from the renormalized mean field theory of Zhang and coworkers for underdoped cuprates. The ansatz shows good agreement with recent angle resolved photoemission on underdoped cuprates and resolves an apparent disagreement with the Luttinger sum rule.

Role of spatial distortions on the quadratic nonlinear optical properties of octupolar organic and metallo-organic molecules

Following on the recent experimental demonstration of a discrepancy between the nonlinear optical (NLO) behavior of several pi-conjugated chromophores and their assumed octupolar symmetry, the authors investigate how geometrical distortions influence the NLO response of multipolar push-pull molecules. Their analytical model is set on a basis of valence-bond and charge-transfer states to estimate the hyperpolarizability of organic and metallo-organic chromophores using the lowest possible number of variables. Since symmetry breakdown changes the definition of the molecular Cartesian framework, tensorial spherical coordinates are implemented. The evolution of the nonlinear molecular anisotropy with possible rotational deviations is then evaluated for two recently studied chromophores. Zero-frequency calculations show that, outside optical resonance, weak geometrical distortions lead to strong anisotropy variations in agreement with experimental data. Their goal is to underscore which molecular engineering strategies should be applied when designing a photoisomerizable nonlinear octupole.

Valence bond theory for chemical dynamics

This essay provides a perspective on several issues in valence bond theory: the physical significance of semilocal bonding orbitals, the capability of valence bond concepts to explain systems with multireferences character, the use of valence bond theory to provide analytical representations of potential energy surfaces for chemical dynamics by the method of semiempirical valence bond potential energy surfaces (an early example of specific reaction parameters), by multiconfiguration molecular mechanics, by the combined valence bond-molecular mechanics method, and by the use of valence bond states as coupled diabatic states for describing electronically nonadiabatic processes (photochemistry). The essay includes both ab initio and semiempirical approaches.

Entanglement in Finitely Correlated Spin States

We consider entanglement properties of pure finitely correlated states (FCS). We derive bounds for the entanglement of a spin with an interval of spins in an arbitrary pure FCS. Finitely correlated states are also known as matrix product states or generalized valence-bond states. The bounds become exact in the case where one considers the entanglement of a single spin with a half-infinite chain to the right of it. Our bounds provide a proof of the recent conjecture by Benatti, Hiesmayr, and Narnhofer that their necessary condition for nonvanishing entanglement in terms of a single spin and the memory of the FCS is also sufficient. We also generalize the study of entanglement in the Affleck-Kennedy-Lieb-Tasaki model by Fan, Korepin, and Roychowdhury. Our result permits a more efficient calculation, numerically and in some cases analytically, of the entanglement of arbitrary finitely correlated quantum spin chains.

Identity SN2 Reactions X−+CH3X→XCH3+X− (X=F, Cl, Br, and I) in Vacuum and in Aqueous Solution: A Valence Bond Study

The recently developed (L. Song, W. Wu, Q. Zhang, S. Shaik, J. Phys. Chem. A 2004, 108, 6017) valence bond method coupled with a polarized continuum model (VBPCM) has been applied to the identity SN2 reaction of halides in the gas phase and in aqueous solution. The barriers computed at the level of the breathing orbital VB method (P. C. Hiberty, J. P. Flament, E. Noizet, Chem. Phys. Lett. 1992, 189, 259), BOVB and VBPCM//BOVB, are comparable to CCSD(T) and CCSD(T)//PCM results and to experimentally derived barriers in solution (W. J. Albery, M. M. Kreevoy, Adv. Phys. Org. Chem. 1978, 16, 85). The reactivity parameters needed to apply the valence bond state correlation diagram (VBSCD) method (S. Shaik, J. Am. Chem. Soc. 1984, 106, 1227), were also determined by VB calculations. It has been shown that the reactivity parameters along with their semiempirical derivations provide a satisfactory qualitative and quantitative account of the barriers.

Identification of an RVB liquid phase in a quantum dimer model with competing kinetic terms

Starting from the mean-field solution of a spin-orbital model of $\mathrm{Li}\mathrm{Ni}{\mathrm{O}}_{2}$, we derive an effective quantum dimer model (QDM) that lives on the triangular lattice and contains kinetic terms acting on four-site plaquettes and six-site loops. Using exact diagonalizations and Green's function Monte Carlo simulations, we show that the competition between these kinetic terms leads to a resonating valence bond (RVB) state for a finite range of parameters. We also show that this RVB phase is connected to the RVB phase identified in the Rokhsar-Kivelson model on the same lattice in the context of a generalized model that contains both the six-site loops and a nearest-neighbor dimer repulsion. These results suggest that the occurrence of an RVB phase is a generic feature of QDM with competing interactions.

Topological Structure of Phase Vortex in Resonating Valence Bond Superconductivity

In this paper, based on the Schrödinger equation and theΨ-mapping theory, the accurate expression for the gradient ofresonating valence bond superconducting phase Θs is found.The expression of ∇⃗Θs is just the velocity flow V⃗without considering the coefficient. The curl of ∇⃗Θs is wherethe vortex lies, and has important relation to δ2(Ψ) and animportant relation to the zero points of resonating valence bondsuperconducting order parameter Ψ. The topological structure of thevortex is characterized by the Ψ-mapping topological numbers Hopf-indexand Brouwer degrees. The Ginzberg–Landau equation in resonating valencebond state also is discussed in this theory. The magnetic property isdiscussed also.

D-Wave Resonating Valence Bond States of Fermionic Atoms in Optical Lattices

We study controlled generation and measurement of superfluid d-wave resonating valence bond (RVB) states of fermionic atoms in 2D optical lattices. Starting from loading spatial and spin patterns of atoms in optical superlattices as pure quantum states from a Fermi gas, we adiabatically transform this state to an RVB state by a change of the lattice parameters. Results of exact time-dependent numerical studies for ladders systems are presented, suggesting generation of RVB states on a time scale smaller than typical experimental decoherence times.

Some formal results for the valence bond basis

In a system with an even number of SU ( 2 ) spins, there is an overcomplete set of states—consisting of all possible pairings of the spins into valence bonds—that spans the S = 0 Hilbert subspace. Operator expectation values in this basis are related to the properties of the closed loops that are formed by the overlap of valence bond states. We construct a generating function for spin correlation functions of arbitrary order and show that all nonvanishing contributions arise from configurations that are topologically irreducible. We derive explicit formulas for the correlation functions at second, fourth, and sixth order. We then extend the valence bond basis to include triplet bonds and discuss how to compute properties that are related to operators acting outside the singlet sector. These results are relevant to analytical calculations and to numerical valence bond simulations using quantum Monte Carlo, variational wavefunctions, or exact diagonalization.

Entanglement and quantum phase transitions in quantum mixed spin chains

The ground entanglement and thermal entanglement in quantum mixed spin chains consisting of two integer spins 1 and two half integer spins 1/2 arrayed as ${1/2}-{1/2}-1-1$ in a unit cell with antiferromagnetic nearest-neighbor couplings $J_1$($J_2$) between the spins of equal (different) magnitudes, are investigated by adopting the log-negativity. The ground entanglement transition found here is closely related with the valence bond state transition, and the thermal entanglement near the critical point is calculated and shown that two distinct behaviors exist in the nearest neighbor same kind of spins and different kind of spins, respectively. The potential application of our results on the quantum information processing is also discussed.

Phenomenological theory of the pseudogap state

An ansatz is proposed for the coherent part of the single particle Green's function in a doped resonant valence bond (RVB) state by analogy with the form derived by Konik and co-workers for a doped spin liquid formed by an array of two-leg Hubbard ladders near half-filling. The parameters of the RVB state are taken from the renormalized mean field theory of Zhang and co-workers for underdoped cuprates. The ansatz shows good agreement with recent angle resolved photoemission on underdoped cuprates and resolves an apparent disagreement with the Luttinger sum rule. The transition in the normal state from a doped RVB spin liquid to a standard Landau Fermi liquid, that occurs in the renormalized mean field theory, appears as a quantum critical point characterized by a change in the analytic form of the Green's function. A $d$-wave superconducting dome surrounding this quantum critical point is introduced phenomenologically. Results are also presented for the Drude weight and tunneling density of states as functions of the hole density.

Schwinger Boson Approach to the Fully Screened Kondo Model

We apply the Schwinger boson scheme to the fully screened Kondo model and generalize the method to include antiferromagnetic interactions between ions. Our approach captures the Kondo crossover from local moment behavior to a Fermi liquid with a nontrivial Wilson ratio. When applied to the two-impurity model, the mean-field theory describes the "Varma-Jones" quantum phase transition between a valence bond state and a heavy Fermi liquid.

Physics of the Resonating Valence Bond (Pseudogap) State of the Doped Mott Insulator: Spin-Charge Locking

The properties of the pseudogap phase above T(C) of the high-T(C) cuprate superconductors are described by showing that the Anderson-Nambu SU(2) spinors of a resonating valence bond spin gap "lock" to those of the electron charge system because of the resulting improvement of kinetic energy. This enormously extends the range of the vortex liquid state in these materials. A heuristic description of the nonlocal electrodynamics of this pseudogap-vortex liquid state is proposed.

Quantum antiferromagnetism and highTcsuperconductivity: A close connection between thet−Jmodel and the projected BCS Hamiltonian

A connection between quantum antiferromagnetism and high ${T}_{c}$ superconductivity is theoretically investigated by analyzing the $t\text{\ensuremath{-}}J$ model and its relationships to the Gutzwiller-projected BCS Hamiltonian. After numerical corroboration via exact diagonalization, it is analytically shown that the ground state of the $t\text{\ensuremath{-}}J$ model at half filling [i.e., the two-dimensional (2D) antiferromagnetic Heisenberg model] is entirely equivalent to the ground state of the Gutzwiller-projected BCS Hamiltonian with strong pairing. Combined with the high wave function overlap between the ground states of the $t\text{\ensuremath{-}}J$ model and the projected BCS Hamiltonian at moderate doping, this equivalence provides strong support for the existence of superconductivity in the $t\text{\ensuremath{-}}J$ model. The relationship between the ground state of the projected BCS Hamiltonian and Anderson's resonating valence bond state (i.e., the projected BCS ground state) is discussed.

Effective Hamiltonian for Fermions in an Optical Lattice across a Feshbach Resonance

We derive the Hamiltonian for cold fermionic atoms in an optical lattice across a broad Feshbach resonance, taking into account both multiband occupations and neighboring-site collisions. Under typical configurations, the resulting Hamiltonian can be dramatically simplified to an effective single-band model, which describes a new type of resonance between the local dressed molecules and the valence bond states of fermionic atoms at neighboring sites. On different sides of such a resonance, the effective Hamiltonian is reduced to either a t-J model for the fermionic atoms or an XXZ model for the dressed molecules. The parameters in these models are experimentally tunable in the full range, which allows for observation of various phase transitions.

Ground State Projection of Quantum Spin Systems in the Valence-Bond Basis

A Monte Carlo method for quantum spin systems is formulated in the valence-bond basis. The nonorthogonality allows for an efficient importance-sampled projection of the ground state out of an arbitrary state. The method provides access to resonating valence-bond physics, enables a direct estimator for the singlet-triplet gap, and extends the class of models that can be studied without negative-sign problems. As a demonstration, the valence-bond distribution in the ground state of the 2D Heisenberg antiferromagnet is calculated. Generalizations of the method to fermion systems are also discussed.

Algebraic spin liquid as the mother of many competing orders

We study the properties of a class of two-dimensional interacting critical states -- dubbed algebraic spin liquids -- that can arise in two-dimensional quantum magnets. A particular example that we focus on is the staggered flux spin liquid, which plays a key role in some theories of underdoped cuprate superconductors. We show that the low-energy theory of such states has much higher symmetry than the underlying microscopic spin system. This symmetry has remarkable consequences, leading in particular to the unification of a number of seemingly unrelated competing orders. The correlations of these orders -- including, in the staggered flux state, the Neel vector and the order parameter for the columnar and box valence-bond solid states -- all exhibit the SAME slow power-law decay. Implications for experiments in the pseudogap regime of the cuprates and for numerical calculations on model systems are discussed.

SU(2)-invariant spin-12Hamiltonians with resonating and other valence bond phases

We construct a family of rotationally invariant, local, $S=1∕2$ Klein Hamiltonians on various lattices that exhibit ground-state manifolds spanned by nearest-neighbor valence bond states. We show that with selected perturbations such models can be driven into phases modeled by well-understood quantum dimer models on the corresponding lattices. Specifically, we show that the perturbation procedure is arbitrarily well controlled by a new parameter which is the extent of decoration of the reference lattice. This strategy leads to Hamiltonians that exhibit (i) ${Z}_{2}$ resonating valence bond (RVB) phases in two dimensions, (ii) $U(1)$ RVB phases with a gapless ``photon'' in three dimensions, and (iii) a Cantor deconfined region in two dimensions. We also construct two models on the pyrochlore lattice, one model exhibiting a ${Z}_{2}$ RVB phase and the other a $U(1)$ RVB phase.