Resonating Valence Bond State Research Articles

Novel constructive method for the quantum dimer model in spin-1/2 Heisenberg antiferromagnets with frustration on a diamond-like-decorated square lattice

We study spin-1/2 Heisenberg antiferromagnets on a diamond-like-decorated square lattice. The diamond-like-decorated square lattice is a lattice in which the bonds in a square lattice are replaced with diamond units. The diamond unit has two types of antiferromagnetic exchange interactions, and the ratio λ of the diagonal bond strength to that of the other four edges controls the frustration strength. For 0.974 < λ < 2, the present system has a nontrivial macroscopic degeneracy, which is called the macroscopically degenerated tetramer-dimer (MDTD) states. The MDTD states are identical to the Hilbert space of the Rokhsar–Kivelson (RK) quantum dimer model (QDM). By introducing further neighbor couplings in the MDTD states, we calculate the second-order effective Hamiltonian, which is exactly the same as the square-lattice QDM with a finite hopping amplitude t and dimer-dimer interaction v. Furthermore, we calculate v/|t| as a function of the ratio λ in the Heisenberg model and examine which phases of the square-lattice QDM appear in our obtained states. Our obtained QDM has a region where λ exhibits a finite hopping amplitude (|t| > 0) and repulsive interaction between dimers (v > 0). This suggests the possibility of realizing the resonating valence bond (RVB) state because the RVB state is obtained at v = |t|, which is known as the RK point.

Interplay of SU(2), point group, and translational symmetry for projected entangled pair states: Application to a chiral spin liquid

Projected entangled pair states (PEPS) provide exact representations for many non-chiral topologically ordered states whereas their range of applicability to interacting chiral topological phases remains largely unsettled. In this context, the symmetries of the local PEPS tensors are crucial for determining the characteristic topological features of the state. In this article we examine the constraints that arise when different symmetries are imposed simultaneously on the local tensor such as internal $SU(2)$, point group and translation symmetry. We show how the interplay of these symmetries manifests in the entanglement spectrum which is the main diagnostic tool for chiral topological order. We apply our results to a spin liquid PEPS introduced previously as a chiral generalization of the resonating valence bond state. Our findings explain the discrepancies observed between the entanglement spectrum of this state and the expected edge spectrum described by a chiral conformal field theory. Finally, in a certain parameter region where this PEPS possesses an additional $U(1)$ symmetry we are able to resolve these discrepancies and obtain an entanglement spectrum with the expected state countings and conformal weight.

Resonating valence bonds and spinon pairing in the Dicke model

Resonating valence bond (RVB) states are a class of entangled quantum many body wavefunctions with great significance in condensed matter physics. We propose a scheme to synthesize a family of RVB states using a cavity QED setup with two-level atoms coupled to a common photon mode. In the lossy cavity limit, starting with an initial state with MM atoms excited and NN atoms in the ground state, we show that this setup can be configured as a Stern Gerlach experiment. A measurement of photon emission collapses the wavefunction of atoms onto an RVB state composed of resonating long-ranged singlets. Each emitted photon reduces the number of singlets by unity, replacing it with a pair of lone spins or ‘spinons’. As spinons are formed coherently in pairs, they are analogous to Cooper pairs in a superconductor. To simulate pair fluctuations, we propose a protocol in which photons are allowed to escape the cavity undetected. This leads to an inchoate superconductor – mixed quantum state with a fluctuating number of spinon pairs. Remarkably, in the limit of large system sizes, this protocol reveals an underlying quantum phase transition. Upon tuning the initial spin polarization, the emission exhibits a continuous transition from a dark state to a bright state. This opens an exciting route to simulate RVB states and superconductivity.

Topological Z2 resonating-valence-bond spin liquid on the square lattice

A one-parameter family of long-range resonating valence bond (RVB) state on the square lattice was previously proposed to describe a critical spin liquid (SL) phase of the spin-$1/2$ frustrated Heisenberg model. We provide evidence that this RVB state in fact also realises a topological (long-range entangled) $\mathbb{Z}_2$ SL, limited by two transitions to critical SL phases. The topological phase is naturally connected to the $\mathbb{Z}_2$ gauge symmetry of the local tensor. This work shows that, on one hand, spin-$1/2$ topological SL with $C_{4v}$ point group symmetry and $SU(2)$ spin rotation symmetry exists on the square lattice and, on the other hand, criticality and nonbipartiteness are compatible. We also point out that, strong similarities between our phase diagram and the ones of classical interacting dimer models suggest both can be described by similar Kosterlitz-Thouless transitions. This scenario is further supported by the analysis of the one-dimensional boundary state.

Fate of the open-shell singlet ground state in the experimentally accessible acenes: A quantum Monte Carlo study.

By means of the Jastrow correlated antisymmetrized geminal power (JAGP) wave function and quantum Monte Carlo (QMC) methods, we study the ground state properties of the oligoacene series, up to the nonacene. The JAGP is the accurate variational realization of the resonating-valence-bond (RVB) ansatz proposed by Pauling and Wheland to describe aromatic compounds. We show that the long-ranged RVB correlations built in the acenes' ground state are detrimental for the occurrence of open-shell diradical or polyradical instabilities, previously found by lower-level theories. We substantiate our outcome by a direct comparison with another wave function, tailored to be an open-shell singlet (OSS) for long-enough acenes. By comparing on the same footing the RVB and OSS wave functions, both optimized at a variational QMC level and further projected by the lattice regularized diffusion Monte Carlo method, we prove that the RVB wave function has always a lower variational energy and better nodes than the OSS, for all molecular species considered in this work. The entangled multi-reference RVB state acts against the electron edge localization implied by the OSS wave function and weakens the diradical tendency for higher oligoacenes. These properties are reflected by several descriptors, including wave function parameters, bond length alternation, aromatic indices, and spin-spin correlation functions. In this context, we propose a new aromatic index estimator suitable for geminal wave functions. For the largest acenes taken into account, the long-range decay of the charge-charge correlation functions is compatible with a quasi-metallic behavior.

Complementary views on electron spectra: From fluctuation diagnostics to real-space correlations

We study the relation between the microscopic properties of a many-body system and the electron spectra, experimentally accessible by photoemission. In a recent paper [Phys. Rev. Lett. 114, 236402 (2015)], we introduced the "fluctuation diagnostics" approach, to extract the dominant wave vector dependent bosonic fluctuations from the electronic self-energy. Here, we first reformulate the theory in terms of fermionic modes, to render its connection with resonance valence bond (RVB) fluctuations more transparent. Secondly, by using a large-U expansion, where U is the Coulomb interaction, we relate the fluctuations to real space correlations. Therefore, it becomes possible to study how electron spectra are related to charge, spin, superconductivity and RVB-like real space correlations, broadening the analysis of an earlier work [Phys. Rev. B 89, 245130 (2014)]. This formalism is applied to the pseudogap physics of the two-dimensional Hubbard model, studied in the dynamical cluster approximation. We perform calculations for embedded clusters with up to 32 sites, having three inequivalent K-points at the Fermi surface. We find that as U is increased, correlation functions gradually attain values consistent with an RVB state. This first happens for correlation functions involving the antinodal point and gradually spreads to the nodal point along the Fermi surface. Simultaneously a pseudogap opens up along the Fermi surface. We relate this to a crossover from a Kondo-like state to an RVB-like localized cluster state and to the presence of RVB and spin fluctuations. These changes are caused by a strong momentum dependence in the cluster bath-couplings along the Fermi surface. We also show, from a more algorithmic perspective, how the time-consuming calculations in fluctuation diagnostics can be drastically simplified.

Unifying neural-network quantum states and correlator product states via tensor networks

Correlator product states (CPS) are a powerful and very broad class of states for quantum lattice systems whose (unnormalised) amplitudes in a fixed basis can be sampled exactly and efficiently. They work by gluing together states of overlapping clusters of sites on the lattice, called correlators. Recently Carleo and Troyer (2017 Science 355 602) introduced a new type sampleable ansatz called neural-network quantum states (NQS) that are inspired by the restricted Boltzmann model used in machine learning. By employing the formalism of tensor networks we show that NQS are a special form of CPS with novel properties. Diagramatically a number of simple observations become transparent. Namely, that NQS are CPS built from extensively sized GHZ-form correlators making them uniquely unbiased geometrically. The appearance of GHZ correlators also relates NQS to canonical polyadic decompositions of tensors. Another immediate implication of the NQS equivalence to CPS is that we are able to formulate exact NQS representations for a wide range of paradigmatic states, including superpositions of weighed-graph states, the Laughlin state, toric code states, and the resonating valence bond state. These examples reveal the potential of using higher dimensional hidden units and a second hidden layer in NQS. The major outlook of this study is the elevation of NQS to correlator operators allowing them to enhance conventional well-established variational Monte Carlo approaches for strongly correlated fermions.

Randomness-induced quantum spin liquid on honeycomb lattice

Quantum entanglement in magnetic materials is expected to yield a quantum spin liquid (QSL), in which strong quantum fluctuations prevent magnetic ordering even at zero temperature. This topic has been one of the primary focuses of condensed-matter science since Anderson first proposed the resonating valence bond state in a certain spin-1/2 frustrated magnet in 1973. Since then, several candidate materials featuring frustration, such as triangular and kagome lattices, have been reported to exhibit liquid-like behavior. However, the mechanisms that stabilize the liquid-like states have remained elusive. Here, we present a QSL state in a spin-1/2 honeycomb lattice with randomness in the exchange interaction. That is, we successfully introduce randomness into the organic radial-based complex and realize a random-singlet (RS) state (or valence bond glass). All magnetic and thermodynamic experimental results indicate the liquid-like behaviors, which are consistent with those expected in the RS state. Our results suggest that the randomness or inhomogeneity in the actual systems stabilize the RS state and yield liquid-like behavior.

Generating resonating-valence-bond states through Dicke subradiance

Dicke's original thought experiment with two spins coupled to a photon mode has recently been experimentally realized. We propose extending this experiment to N spins and show that it naturally gives rise to highly entangled states. In particular, it gives rise to dark states which have resonating valence bond (RVB) character. We first consider a system of N two level spins in a cavity with only one spin in the excited state. This initial state is a linear combination of a dark state and a bright state. We point out the dark state is a coherent superposition of singlets with resonating valence bond character. We show that the coupling to the photon mode takes the spin system into a mixed state with an entangled density matrix. We next consider an initial state with half of the spins in the excited state. We show that there is a non-zero probability for this to collapse into a dark state with RVB character. In the lossy cavity limit, if no photon is detected within several decay time periods, we may deduce that the spin system has collapsed onto the dark RVB state. We show that the probability for this scales as 2/N, making it possible to generate RVB states of 20 spins or more.

Competition between spin liquids and valence-bond order in the frustrated spin- 12 Heisenberg model on the honeycomb lattice

Using variational wave functions and Monte Carlo techniques, we study the antiferromagnetic Heisenberg model with first-neighbor $J_1$ and second-neighbor $J_2$ antiferromagnetic couplings on the honeycomb lattice. We perform a systematic comparison of magnetically ordered and nonmagnetic states (spin liquids and valence-bond solids) to obtain the ground-state phase diagram. N\'eel order is stabilized for small values of the frustrating second-neighbor coupling. Increasing the ratio $J_2/J_1$, we find strong evidence for a continuous transition to a nonmagnetic phase at $J_2/J_1 \approx 0.23$. Close to the transition point, the Gutzwiller-projected uniform resonating valence bond state gives an excellent approximation to the exact ground-state energy. For $0.23 \lesssim J_2/J_1 \lesssim 0.4$, a gapless $Z_2$ spin liquid with Dirac nodes competes with a plaquette valence-bond solid. In contrast, the gapped spin liquid considered in previous works has significantly higher variational energy. Although the plaquette valence-bond order is expected to be present as soon as the N\'eel order melts, this ordered state becomes clearly favored only for $J_2/J_1 \gtrsim 0.3$. Finally, for $0.36 \lesssim J_2/J_1 \le 0.5$, a valence-bond solid with columnar order takes over as the ground state, being also lower in energy than the magnetic state with collinear order. We perform a detailed finite-size scaling and standard data collapse analysis, and we discuss the possibility of a deconfined quantum critical point separating the N\'eel antiferromagnet from the plaquette valence-bond solid.

Analytical recursive method to ascertain multisite entanglement in doped quantum spin ladders

We formulate an analytical recursive method to generate the wave function of doped short-range resonating valence bond (RVB) states as a tool to efficiently estimate multisite entanglement as well as other physical quantities in doped quantum spin ladders. We prove that doped RVB ladder states are always genuine multipartite entangled. Importantly, our results show that within specific doping concentration and model parameter regimes, the doped RVB state essentially characterizes the trends of genuine multiparty entanglement in the exact ground states of the Hubbard model with large onsite interactions, in the limit which yields the $t$-$J$ Hamiltonian.

RVB states in doped band insulators from Coulomb forces: theory and a case study of superconductivity in BiS2 layers

Doped band insulators, HfNCl, WO3, diamond, Bi2Se3, BiS2 families, STO/LAO interface, gate doped SrTiO3, MoS2 and so on are unusual superconductors. With an aim to build a general theory for superconductivity in doped band insulators, we focus on the BiS2 family which was discovered by Mizuguchi et al in 2012. While maximum Tc is only ∼11 K in , a number of experimental results are puzzling and anomalous in the sense that they resemble high Tc and unconventional superconductors. Using a two orbital model of Usui, Suzuki and Kuroki, we show that the uniform low density free Fermi sea in is unstable towards formation of the next nearest neighbor Bi-S-Bi diagonal valence bond (charged −2e Cooper pair) and their Wigner crystallization. Instability to this novel state of matter is caused by unscreened nearest neighbor coulomb repulsions (V ∼ 1 eV) and a hopping pattern with sulfur mediated diagonal next nearest neighbor Bi-S-Bi hopping t’ ∼ 0.88 eV, as well as larger than nearest neighbor Bi-Bi hopping, t ∼ 0.16 eV. Wigner crystals of Cooper pairs quantum melt for doping around x = 0.5 and stabilize certain resonating valence bond states and superconductivity. We study a few variational RVB states and suggest that BiS2 family members are latent high Tc superconductors, but challenged by competing orders and the fragile nature of many body states sustained by unscreened Coulomb forces. One of our superconducting states has dXY symmetry and a gap. We also predict a 2d Bose metal or vortex liquid normal state, as charged −2e valence bonds survive in the normal state.

SU(2)1chiral edge modes of a critical spin liquid

Protected chiral edge modes are a well-known signature of topologically ordered phases like the Fractional Quantum Hall States. Recently, using the framework of projected entangled pair states (PEPS) on the square lattice, we constructed a family of chiral Resonating Valence Bond states with $\mathbb{Z}_2$ gauge symmetry. Here we revisit and analyze in full details the properties of the edge modes as given by their Entanglement Spectra on a cylinder. Surprisingly, we show that the latter can be well described by a chiral SU(2)$_1$ Conformal Field Theory (CFT), as for the $\nu=1/2$ (bosonic) gapped Laughlin state, although our numerical data suggest a critical bulk compatible with an emergent $U(1)$ gauge symmetry. We propose that our family of PEPS may physically describe a boundary between a chiral topological phase and a trivial phase.

Double-semion topological order from exactly solvable quantum dimer models

We construct a generalized quantum dimer model on two-dimensional nonbipartite lattices including the triangular lattice, the star lattice and the kagome lattice. At the Rokhsar-Kivelson (RK) point, we obtain its exact ground states that are shown to be a fully gapped quantum spin liquid with the double-semion topological order. The ground-state wave function of such a model at the RK point is a superposition of dimer configurations with a nonlocal sign structure determined by counting the number of loops in the transition graph. We explicitly demonstrate the double-semion topological order in the ground states by showing the semionic statistics of monomer excitations. We also discuss possible implications of such double-semion resonating valence bond states to candidate quantum spin-liquid systems discovered experimentally and numerically in the past few years.

Chiral topological spin liquids with projected entangled pair states

Topological chiral phases are ubiquitous in the physics of the Fractional Quantum Hall Effect. Non-chiral topological spin liquids are also well known. Here, using the framework of projected entangled pair states (PEPS), we construct a family of chiral spin liquids on the square lattice which are generalized spin-1/2 Resonating Valence Bond (RVB) states obtained from deformed local tensors with $d+i\, d$ symmetry. On a cylinder, we construct four topological sectors with even or odd number of spinons on the boundary and even or odd number of ($\mathbb{Z}_2$) fluxes penetrating the cylinder which, we argue, remain orthogonal in the limit of infinite perimeter. The analysis of the transfer matrix provides evidence of short-range (long-range) triplet (singlet) correlations as for the critical (non-chiral) RVB state. The Entanglement Spectrum exhibits chiral edge modes, which we confront to predictions of Conformal Field Theory, and the corresponding Entanglement Hamiltonian is shown to be long ranged.

Striped critical spin liquid in a spin-orbital entangled RVB state in a projected entangled-pair state representation

We introduce a spin-orbital entangled (SOE) resonating valence bond (RVB) state on a square lattice of spins $\frac{1}{2}$ and orbitals represented by pseudospins $\frac{1}{2}$. Like the standard RVB state, it is a superposition of nearest-neighbor hard-core coverings of the lattice by spin singlets, but adjacent singlets are favored to have perpendicular orientations and, more importantly, an orientation of each singlet is entangled with orbitals state on its two lattice sites. The SOE-RVB state can be represented by a projected entangled pair state (PEPS) with a bond dimension $D=4$. This representation helps to reveal that the state is a superposition of striped coverings conserving a topological quantum number. The stripes are a critical quantum spin liquid. We propose a spin-orbital Hamiltonian supporting a SOE-RVB ground state.

Spinon excitation spectra of the J1-J2 chain from analytical calculations in the dimer basis and exact diagonalization

The excitation spectrum of the frustrated spin-$1/2$ Heisenberg chain is reexamined using variational and exact diagonalization calculations. We show that the overlap matrix of the short-range resonating valence bond states basis can be inverted which yields tractable equations for single and two spinons excitations. Older results are recovered and new ones, such as the bond-state dispersion relation and its size with momentum at the Majumdar-Ghosh point are found. In particular, this approach yields a gap opening at $J_2=0.25J_1$ and an onset of incommensurability in the dispersion relation at $J_2=9/17J_1$ [as in S. Brehmer \emph{et al.}, J. Phys.: Condens. Matter \textbf{10}, 1103 (1998)]. These analytical results provide a good support for the understanding of exact diagonalization spectra, assuming an independent spinons picture.

Semionic resonating valence-bond states

The nature of the kagome Heisenberg antiferromagnet (HAFM) is under ongoing debate. While recent evidence points towards a Z_2 topological spin liquid, the exact nature of the topological phase is still unclear. In this paper, we introduce semionic Resonating Valence Bond (RVB) states, this is, Resonating Valence Bond states which are in the Z_2 ordered double-semion phase, and study them using Projected Entangled Pair States (PEPS). We investigate their physics and study their suitability as an ansatz for the HAFM, as compared to a conventional RVB state which is in the Toric Code Z_2 topological phase. In particular, we find that a suitably optimized "semionic simplex RVB" outperforms the equally optimized conventional "simplex RVB" state, and that the entanglement spectrum (ES) of the semionic RVB behaves very differently from the ES of the conventional RVB, which suggests to use the ES to discriminate the two phases. Finally, we also discuss the possible relevance of space group symmetry breaking in valence bond wavefunctions with double-semion topological order.

Detection of symmetry-enriched topological phases

Topologically ordered systems in the presence of symmetries can exhibit new structures which are referred to as symmetry enriched topological (SET) phases. We introduce simple methods to detect the SET order directly from a complete set of topologically degenerate ground state wave functions. In particular, we first show how to directly determine the characteristic symmetry fractionalization of the quasiparticles from the reduced density matrix of the minimally entangled states. Second, we show how a simple generalization of a non-local order parameter can be measured to detect SETs. The usefulness of the proposed approached is demonstrated by examining two concrete model states which exhibit SET: (i) a spin-1 model on the honeycomb lattice and (ii) the resonating valence bond state on a kagome lattice. We conclude that the spin-1 model and the RVB state are in the same SET phases.

Topology and criticality in the resonating Affleck-Kennedy-Lieb-Tasaki loop spin liquid states

We exploit a natural projected entangled-pair state (PEPS) representation for the resonating Affleck-Kennedy-Lieb-Tasaki loop (RAL) state. By taking advantage of PEPS-based analytical and numerical methods, we characterize the RAL states on various two-dimensional lattices. On square and honeycomb lattices, these states are critical since the dimer-dimer correlations decay as a power law. On the kagome lattice, the RAL state has exponentially decaying correlation functions, supporting the scenario of a gapped spin liquid. We provide further evidence that the RAL state on the kagome lattice is a ${\mathbb{Z}}_{2}$ spin liquid, by identifying the four topological sectors and computing the topological entropy. Furthermore, we construct a one-parameter family of PEPS states interpolating between the RAL state and a short-range resonating valence bond state and find a critical point, consistent with the fact that the two states belong to two different phases. We also perform a variational study of the spin-1 kagome Heisenberg model using this one-parameter PEPS.