Valence Bond State Research Articles

Magnetic ground state of La2LiMoO6 : A comparison with other Mo5+ ( S=1/2 ) double perovskites

${\mathrm{La}}_{2}\mathrm{Li}\mathrm{Mo}{\mathrm{O}}_{6}$ is a double perovskite (DP) with $P{2}_{1}/n$ symmetry based on the ${\mathrm{Mo}}^{5+}$ ion, $4{d}^{1}$, ${{t}_{2\mathrm{g}}}^{1}$, $S=1/2$. It is isostructural with ${\mathrm{Sr}}_{2}\mathrm{Y}\mathrm{Mo}{\mathrm{O}}_{6}$, the magnetic ground state of which is apparently a very unusual collective spin singlet or valence-bond glass state as is the case for cubic ($Fm\text{\ensuremath{-}}3m$) ${\mathrm{Ba}}_{2}\mathrm{Y}\mathrm{Mo}{\mathrm{O}}_{6}$. Initial studies of ${\mathrm{La}}_{2}\mathrm{Li}\mathrm{Mo}{\mathrm{O}}_{6}$ suggested a different ground state from the other DPs but no clear conclusions could be drawn. A more detailed study is presented here including magnetic susceptibility, heat capacity, and elastic neutron-scattering results. This DP is now well characterized as an antiferromagnet, ${T}_{\mathrm{N}}=18\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, via observation of magnetic Bragg peaks in neutron scattering and an anomaly in the magnetic heat capacity. The ordering wave vector is $\mathbit{k}=(1/2\phantom{\rule{4pt}{0ex}}1/2\phantom{\rule{0.16em}{0ex}}0)$, consistent with a type I face-centered-cubic magnetic structure, and the ordered moment on ${\mathrm{Mo}}^{5+}$ is $0.32(11)\phantom{\rule{0.16em}{0ex}}{\ensuremath{\mu}}_{\mathrm{B}}$, much reduced from the spin-only value of $1\phantom{\rule{0.16em}{0ex}}{\ensuremath{\mu}}_{\mathrm{B}}$. The index, $f=|{\ensuremath{\vartheta}}_{c}|/{T}_{\mathrm{N}}\ensuremath{\sim}3$, indicates a low level of frustration. The heat-capacity data above ${T}_{\mathrm{N}}$ can be interpreted in terms of a one-dimensional spin-correlation model, as can the low-temperature data which follow a ${T}^{1}$ power law. This is consistent with an earlier suggestion. The difference with isostructural ${\mathrm{Sr}}_{2}\mathrm{Y}\mathrm{Mo}{\mathrm{O}}_{6}$ is attributed to differences in the local distortion of the Mo--O octahedron and the resulting orbital ordering.

Efficient variational contraction of two-dimensional tensor networks with a non-trivial unit cell

Tensor network states provide an efficient class of states that faithfully capture strongly correlated quantum models and systems in classical statistical mechanics. While tensor networks can now be seen as becoming standard tools in the description of such complex many-body systems, close to optimal variational principles based on such states are less obvious to come by. In this work, we generalize a recently proposed variational uniform matrix product state algorithm for capturing one-dimensional quantum lattices in the thermodynamic limit, to the study of regular two-dimensional tensor networks with a non-trivial unit cell. A key property of the algorithm is a computational effort that scales linearly rather than exponentially in the size of the unit cell. We demonstrate the performance of our approach on the computation of the classical partition functions of the antiferromagnetic Ising model and interacting dimers on the square lattice, as well as of a quantum doped resonating valence bond state. Tensor network states provide an efficient class of states that faithfully capture strongly correlated quantum models and systems in classical statistical mechanics. While tensor networks can now be seen as becoming standard tools in the description of such complex many-body systems, close to optimal variational principles based on such states are less obvious to come by. In this work, we generalize a recently proposed variational uniform matrix product state algorithm for capturing one-dimensional quantum lattices in the thermodynamic limit, to the study of regular two-dimensional tensor networks with a non-trivial unit cell. A key property of the algorithm is a computational effort that scales linearly rather than exponentially in the size of the unit cell. We demonstrate the performance of our approach on the computation of the classical partition functions of the antiferromagnetic Ising model and interacting dimers on the square lattice, as well as of a quantum doped resonating valence bond state.

Triplet Resonating Valence Bond State and Superconductivity in Hund's Metals.

A central idea in strongly correlated systems is that doping a Mott insulator leads to a superconductor by transforming the resonating valence bonds (RVBs) into spin-singlet Cooper pairs. Here, we argue that a spin-triplet RVB (tRVB) state, driven by spatially, or orbitally anisotropic ferromagnetic interactions can provide the parent state for triplet superconductivity. We apply this idea to the iron-based superconductors, arguing that strong on site Hund's interactions develop intra-atomic tRVBs between the t_{2g} orbitals. On doping, the presence of two iron atoms per unit cell allows these interorbital triplets to coherently delocalize onto the Fermi surface, forming a fully gapped triplet superconductor. This mechanism gives rise to a unique staggered structure of on site pair correlations, detectable as an alternating π phase shift in a scanning Josephson tunneling microscope.

Quantum-classical crossover in the spin- 12 Heisenberg-Kitaev kagome magnet

The spin-1/2 Heisenberg kagome antiferromagnet is one of the paradigmatic playgrounds for frustrated quantum magnetism, with an extensive number of competing resonating valence bond (RVB) states emerging at low energies, including gapped and gapless spin liquids and valence bond crystals. Here we revisit the crossover from this quantum RVB phase to a semiclassical regime brought about by anisotropic Kitaev interactions, and focus on the precise mechanisms underpinning this crossover. To this end, we introduce a simple parametrization of the classical ground states (GSs) in terms of emergent Ising-like variables, and use this parametrizaton: i) to construct an effective low-energy description of the order-by-disorder mechanism operating in a large part of the phase diagram, and ii) to contrast, side by side, exact diagonalization data obtained from the full basis with that obtained from the restricted (orthonormalized) basis of classical GSs. The results reveal that fluctuation corrections from states outside the restricted basis are strongly quenched inside the semiclassical regime (due to the large anisotropy spin gaps), and that the RVB phase survives up to a relatively large value of Kitaev anisotropy $K$. We further find that the pure Kitaev model admits a subextensive number of one-dimensional symmetries, which explains naturally the absence of classical and quantum order by disorder reported previously.

Minimal Matrix Product States and Generalizations of Mean-Field and Geminal Wave Functions.

Simple wave functions of low computational cost but which can achieve qualitative accuracy across the whole potential energy surface (PES) are of relevance to many areas of electronic structure theory as well as to applications to dynamics. Here, we explore a class of simple wave functions, the minimal matrix product state (MMPS), that generalizes many simple wave functions in common use, such as projected mean-field wave functions, geminal wave functions, and generalized valence bond states. By examining the performance of MMPSs for PESs of some prototypical systems, we find that they yield good qualitative behavior across the whole PES, often significantly improving on the aforementioned ansätze.

Quantum dimer model containing Rokhsar-Kivelson point expressed by spin- 12 Heisenberg antiferromagnets

We obtain a quantum dimer model (QDM) containing a Rokhsar-Kivelson (RK) point expressed by spin-$\frac{1}{2}$ Heisenberg antiferromagnets on a diamond-like decorated square lattice. This lattice has macroscopically degenerated nonmagnetic ground states, which are equivalent to the Hilbert space of a square-lattice QDM. Then, a square-lattice QDM containing the RK point as a second-order effective Hamiltonian is obtained by introducing further neighbor couplings as perturbation. Our model can provide a new method for the experimental realization of a resonating valence bond state in the QDM.

Piercing the rainbow state: Entanglement on an inhomogeneous spin chain with a defect

The {\em rainbow state} denotes a set of valence bond states organized concentrically around the center of a spin 1/2 chain. It is the ground state of an inhomogeneous XX Hamiltonian and presents maximal violation of the area law of entanglement entropy. Here, we add a tunable exchange coupling constant at the center, $\gamma$, and show that it induces entanglement transitions of the ground state. At very strong inhomogeneity, the rainbow state survives for $0 \leq \gamma \leq 1$, while outside that region the ground state is a product of dimers. In the weak inhomogeneity regime the entanglement entropy satisfies a volume law, derived from CFT in curved spacetime, with an effective central charge that depends on the inhomogeneity parameter and $\gamma$. In all regimes we have found that the entanglement properties are invariant under the transformation $\gamma \longleftrightarrow 1 - \gamma$, whose fixed point $\gamma = \frac{1}{2}$ corresponds to the usual rainbow model. Finally, we study the robustness of non trivial topological phases in the presence of the defect.

Emergence of superconductivity in doped multiorbital Hubbard chains

We introduce a variational state for one-dimensional two-orbital Hubbard models that intuitively explains the recent computational discovery of pairing in these systems when hole doped. Our ansatz is an optimized linear superposition of Affleck–Kennedy–Lieb–Tasaki valence-bond states, rendering the combination a valence-bond liquid dubbed orbital resonant valence bond. We show that the undoped (one-electron/orbital) quantum state of two sites coupled into a global spin singlet is exactly written employing only spin-1/2 singlets linking orbitals at nearest-neighbor sites. Generalizing to longer chains defines our variational state visualized geometrically expressing our chain as a two-leg ladder, with one orbital per leg. As in Anderson’s resonating valence-bond state, our undoped variational state contains preformed singlet pairs that via doping become mobile, leading to superconductivity. Doped real materials with one-dimensional substructures, two near-degenerate orbitals, and intermediate Hubbard U/W strengths—W the carrier’s bandwidth—could realize spin-singlet pairing if on-site anisotropies are small. If these anisotropies are robust, spin-triplet pairing emerges.

Consistent Scaling Exponents at the Deconfined Quantum-Critical Point**Supported by the National Science Foundation (USA) under Grant No. DMR-1710170 and by the Simons Foundation under a Simons Investigator Award.

We report a quantum Monte Carlo study of the phase transition between antiferromagnetic and valence-bond solid ground states in the square-lattice S = 1/2 J–Q model. The critical correlation function of the Q terms gives a scaling dimension corresponding to the value ν = 0.455 ± 0.002 of the correlation-length exponent. This value agrees with previous (less precise) results from conventional methods, e.g., finite-size scaling of the near-critical order parameters. We also study the Q-derivatives of the Binder cumulants of the order parameters for L2 lattices with L up to 448. The slope grows as L1/ν with a value of ν consistent with the scaling dimension of the Q term. There are no indications of runaway flow to a first-order phase transition. The mutually consistent estimates of ν provide compelling support for a continuous deconfined quantum-critical point.

Bose-Einstein condensation of deconfined spinons in two dimensions

The transition between the N\'{e}el antiferromagnet and the valence-bond solid state in two dimensions has become a paradigmatic example of deconfined quantum criticality, a non-Landau transition characterized by fractionalized excitations (spinons). We consider an extension of this scenario whereby the deconfined spinons are subject to a magnetic field. The primary purpose is to identify the exotic scenario of a Bose-Einstein condensate of spinons. We employ quantum Monte Carlo simulations of the \mbox{$J$-$Q$} model with a magnetic field and perform a quantum field theoretic analysis of the magnetic field and temperature dependence of thermodynamic quantities. The combined analysis provides compelling evidence for the Bose-Einstein condensation of spinons and also demonstrates an extended temperature regime in which the system is best described as a gas of spinons interacting with an emergent gauge field.

Topological Z2 resonating-valence-bond quantum spin liquid on the ruby lattice

We construct a short-range resonating valence-bond state (RVB) on the ruby lattice, using projected entangled-pair states (PEPS) with bond dimension $D=3$. By introducing non-local moves to the dimer patterns on the torus, we distinguish four distinct sectors in the space of dimer coverings, which is a signature of the topological nature of the RVB wave function. Furthermore, by calculating the reduced density matrix of a bipartition of the RVB state on an infinite cylinder and exploring its entanglement entropy, we confirm the topological nature of the RVB wave function by obtaining non-zero topological contribution, $\gamma=-\rm{ln}\ 2$, consistent with that of a $\mathbb{Z}_2$ topological quantum spin liquid. We also calculate the ground-state energy of the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on the ruby lattice and compare it with the RVB energy. Finally, we construct a quantum-dimer model for the ruby lattice and discuss it as a possible parent Hamiltonian for the RVB wave function.

Oriented External Electric Fields and Ionic Additives Elicit Catalysis and Mechanistic Crossover in Oxidative Addition Reactions.

Judiciously applied oriented external electric fields (OEEFs) exert catalytic effects on the kinetics and improve the thermodynamics of chemical reactions. Herein, we examine the ability of OEEFs to assist catalysts and show that the rate of oxidative addition between palladium catalysts and alkyl/aryl electrophiles can be controlled by an OEEF applied along the direction of electron reorganization (the "reaction axis"). The concerted mechanism of oxidative addition proceeds through a transition state with moderate charge transfer character. We demonstrate that OEEFs along the reaction axis can control this charge transfer and impart electrostatic catalysis. When the applied field exceeds a certain critical value (∼0.15 V/Å), we observed a mechanistic crossover from the concerted to a dissociative CSNAr type of reactivity for aryl electrophiles. To our surprise, alkyl electrophiles follow a hitherto unexplored SN2 pathway for the reaction with large transition state stabilization at relatively low OEEFs. A valence-bond state correlation diagram (VBSCD) is employed to comprehend the results. Finally, although the catalytic effect of salt additives in oxidative addition is known, its mechanism is still under debate. Our findings further show evidence that salt additives exert electric-field effects on the rate of cross-coupling reactions, and their cocatalytic effects can be judiciously reproduced by applied external electric fields. As such, we propose that the use of additives (anionic or cationic) is an experimentally viable strategy to implement external electric-field effects in routinely used oxidative addition catalysis.

A valence bond perspective of the reaction force formalism

The reaction force formalism represents a convenient approach to analyze the course of a reaction step. From this analysis, the reaction path can be separated in a number of regions that are associated to either structural changes or electronic reorganization. This empirical observation is rationalized in this work on the basis of a simple two-state valence bond correlation diagram. We demonstrate that the ratio between the integrated reaction force and the region of interest ($$w_{\text{ii}}/w_{\text{i}}$$ for the forward reaction and $$w_{\text{iii}}/w_{\text{iv}}$$ for the backward reaction) increases with the ratio between the quantum mechanical resonance energy and the energy required to reach the crossing point at the transition state, we call to this ratio the strength of the resonance. This observation means that the size of the transition region (region ii and iii), that includes the transition state, depends on the strength of the resonance, and the structural zones (region i and iv), that are regions associated with the pure valence bond state curves (no resonance). We propose a simple analytical relationship for $$w_{\text{ii}}/w_{\text{i}}$$ and $$w_{\text{iii}}/w_{\text{iv}}$$ based on three parameters: (i) the quantum mechanical resonance energy, (ii) the energy of the reaction and (iii) the overlap between the VB structures at the transition state. The previous conclusions were supported by a reaction force analysis of a $${\text{S}}_{N}2$$ reactions, $${\text{X}}^{-} + {\text{CH}}_{3}{-}{\text{Y}} \rightarrow {{\text{X}}{-}{\text{CH}}}_{3} + {\text{Y}}^{-} ({\text{X}} = {\text{F}}, {\text{Cl}}, {\text{Br}})$$. The valence bond parameters for these reactions are estimated from empirical considerations. A very good agreement is found between the computed reaction force ratios and the predicted one.

Charge-ordered states in the t−J−U model

We study two charge-ordered states in a $t\ensuremath{-}J\ensuremath{-}U$ model with the method of renormalized mean-field theory. Compared with the $t\ensuremath{-}J$ model, which excludes the existence of one site occupied by two electrons, finite Coulomb interaction $U$ will allow the appearance of doublons and produce more charge fluctuations. The charge density wave states accompanied by Cooper pair density modulations have similar features with the states found in the $t\ensuremath{-}J$ model. The difference is that the modulation of doublons exists, and it has the same wavelength and phase as holes. Charge-ordered states disappear at the $U/t\ensuremath{\le}9$ region when the system does not correspond to the strongly correlated resonating valence-bond state. This feature indicates that local Coulomb interactions are the origin of charge-ordered states observed in cuprates. In addition, we calculate the strength of superconducting coherence and the gap depth of the nodal pair-density-wave state based on the local density of states. These two quantities also vary with lattice sites periodically. The oscillations have same wave vector and phase as the modulation of holes. These results are consistent with recent experiments.

Critical colored-RVB states in the frustrated quantum Heisenberg model on the square lattice

We consider a family of SU(2)-symmetric Projected Entangled Paired States (PEPS) on the square lattice, defining colored-Resonating Valence Bond (RVB) states, to describe the quantum disordered phase of the J_1-J_2J1−J2 frustrated Heisenberg model. For J_2/J_1\sim 0.55J2/J1∼0.55 we show the emergence of critical (algebraic) dimer-dimer correlations – typical of Rokhsar-Kivelson (RK) points of quantum dimer models on bipartite lattices – while, simultaneously, the spin-spin correlation length remains short. Our findings are consistent with a spin liquid or a weak Valence Bond Crystal in the neighborhood of an RK point.

Sequential quantum phase transitions in J1−J2 Heisenberg chains with integer spins (S>1) : Quantized Berry phase and valence-bond solids

On the basis of a Berry-phase analysis, we study the ground state of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chain for $S=2,3,4$. We find that changes in the Berry phase occur $S$ times for spin-$S$ systems, indicating sequential phase transitions. The sequential phase transitions are associated with the change in valence-bond configurations of the ground states. To demonstrate this numerically, we connect the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg Hamiltonian with the generalized Affleck-Kennedy-Lieb-Tasaki model and show that the different phases are connected to the different valence-bond solid states.

SU(4) topological resonating valence bond spin liquid on the square lattice

We generalize the construction of the spin-1/2 SU(2) Resonating Valence Bond (RVB) state to the case of the self-conjugate $\bf 6$-representation of SU(4). As for the case of SU(2) [J-Y. Chen and D. Poilblanc, Phys. Rev. B $\textbf{97}$, 161107(R) (2018)], we use the Projected Entangled Pair State (PEPS) formalism to derive a simple (two-dimensional) family of generalized SU(4) RVB states on the square lattice. We show that, when longer-range SU(4)-singlet bonds are included, a local gauge symmetry is broken down from U(1) to $\mathbb{Z}_2$, leading to the emergence of a short-range spin liquid. Evidence for the topological nature of this spin liquid is provided by the investigation of the Renyi entanglement entropy of infinitely-long cylinders and of the modular matrices. Relevance to microscopic models and experiments of ultracold atoms is discussed.

Solving the quantum dimer and six-vertex models one electric field line at a time

The nature and the very existence of the resonant plaquette valence bond state that separates the classical columnar phase and the Rokhsar and Kivelson point in the quantum dimer model remains unsettled. Here we take a different line of attack on this model, and on the closely related six vertex model, by exploiting the global conservation law of the number of electric field lines. This allows us to study a single fluctuating electric field line which we show maps exactly onto a one dimensional spin chain. In the case of the six vertex model, the electric field line maps onto the celebrated spin 1/2 XXZ model which can be solved exactly. In the quantum dimer model, the electric field line is mapped onto a two-leg spin 1/2 ladder, which we study using numerical exact diagonalization. Our findings are consistent with the existence of three distinct phases including a Luttinger liquid phase, the one-dimensional precursor to the two-dimensional plaquette valence bond solid. The uncanny resemblance of our quasi-one-dimensional electric field line problem to the full two-dimensional problem suggests that much of the behavior of the latter might be understood by thinking of it as a closely packed array of field lines which themselves are undergoing non-trivial phase transitions.

Noise-tolerant signature of ZN topological order in quantum many-body states

Topologically ordered states are fundamentally important in theoretical physics, which are also suggested as promising candidates to build fault-tolerant quantum devices. However, it is still elusive how topological orders can be affected or detected under noises. In this work, we find a quantity, termed as the ring degeneracy $\mathcal{D}$, which is robust under pure noise to detect both trivial and intrinsic topological orders. The ring degeneracy is defined as the degeneracy of the solutions of the self-consistent equations that encode the contraction of the corresponding tensor network(TN). For the $Z_N$ orders, we find that the ring degeneracy satisfies a simple relation $\mathcal{D} = (N + 1)/2 + d$, with $d = 0$ for odd $N$ and $d = 1/2$ for even $N$. Simulations on several non-trivial states (two-dimensional Ising model, $Z_N$ topological states, and resonating valence bond states) show that the ring degeneracy can tolerate noises up to a strength associated to the gap of the TN boundary theory.

Symmetry-Projected Jastrow Mean-Field Wave Function in Variational Monte Carlo.

We extend our low-scaling variational Monte Carlo (VMC) algorithm to optimize the symmetry-projected Jastrow mean-field (SJMF) wave functions. These wave functions consist of a symmetry-projected product of a Jastrow and a general broken-symmetry mean-field reference. Examples include Jastrow antisymmetrized geminal power, Jastrow-Pfaffians, and resonating valence bond states, among others, all of which can be treated with our algorithm. We will demonstrate using benchmark systems including the nitrogen molecule, a chain of hydrogen atoms, and 2-D Hubbard model that a significant amount of correlation can be obtained by optimizing the energy of the SJMF wave function. This can be achieved at a relatively small cost when multiple symmetries including spin, particle number, and complex conjugation are simultaneously broken and projected. We also show that reduced density matrices can be calculated using the optimized wave functions, which allows us to calculate other observables such as correlation functions and will enable us to embed the VMC algorithm in a complete active-space self-consistent field calculation.