Valence Bond State Research Articles

Study of two-spin entanglement in singlet states

We study the entanglement properties of two-spin subsystems in spin-singlet states. The average entanglement between two spins is maximized in a single valence-bond (VB) state. On the other hand, Ev2 (the average entanglement between a subsystem of two spins and the rest of the system) can be maximized through a homogenized superposition of the VB states. The maximal Ev2 rapidly increases with system size and saturates at its maximum allowed value. We adopt two ways of obtaining maximal Ev2 states: (1) imposing homogeneity on singlet states; and (2) generating isotropy in a general homogeneous state. By using these two approaches, we construct explicitly four-spin and six-spin highly entangled states that are both isotropic and homogeneous. Our maximal E2v states represent a new class of resonating-valence-bond states which we show to be the ground states of the infinite-range Heisenberg model.

Hybrid valence-bond states for universal quantum computation

The spin-3/2 Affleck-Kennedy-Lieb-Tasaki (AKLT) valence-bond state on a hexagonal lattice was shown to be a universal resource state for measurement-based quantum computation (MBQC). Can AKLT states of higher spin magnitude support universal MBQC? We demonstrate that several hybrid two-dimensional AKLT states involving a mixture of spin-2 and other lower-spin entities, such as spin-3/2 and spin-1, are also universal for MBQC. This significantly expands universal resource states in the AKLT family. Even though frustration may be a hindrance to quantum computational universality, lattices can be modified to yield AKLT states that are universal. The family of AKLT states thus provides a versatile playground for quantum computation.

Entanglement Hamiltonian of the quantum Néel state

2D projected entangled pair states (PEPS) provide a unique framework giving access to detailed entanglement features of correlated (spin or electronic) systems. For a bi-partitioned quantum system, it has been argued that the entanglement spectrum (ES) is in a one-to-one correspondence with the physical edge spectrum on the cut and that the structure of the corresponding entanglement Hamiltonian (EH) reflects closely bulk properties (finite correlation length, criticality, topological order, etc). However, entanglement properties of systems with spontaneously broken continuous symmetry are still not fully understood. The spin-1/2 square lattice Heisenberg antiferromagnet provides a simple example showing spontaneous breaking of SU(2) symmetry down to U(1). The ground state can be viewed as a ‘quantum Néel state’ where the classical (Néel) staggered magnetization is reduced by quantum fluctuations. Here I consider the (critical) resonating valence bond (RVB) state doped with spinons to describe such a state; this enables the use of the associated PEPS representation (with virtual bond dimension D = 3) to compute the EH and the ES for a partition of an (infinite) cylinder. In particular, I find that the EH is (almost exactly) a chain of a dilute mixture of heavy (↓ spins) and light (↑ spins) hardcore bosons, where light particles are subject to long-range hoppings. The corresponding ES shows drastic differences with the typical ES obtained previously for ground states with restored SU(2)-symmetry (on finite systems).

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.

End states in a one-dimensional topological Kondo insulator in large-Nlimit

To gain further insight into the properties of interacting topological insulators, we study a one-dimensional model of topological Kondo insulators which can be regarded as the strongly interacting limit of the Tamm-Shockley model. Treating the model in a large-$N$ expansion, we find a number of competing ground-state solutions, including topological insulating and valence bond ground states. One of the effects to emerge in our treatment is a reconstruction of the Kondo screening process near the boundary of the material (``Kondo band-bending''). Near the boundary for localization into a valence bond state, we find that the conduction character of the edge state grows substantially, leading to states that extend deeply into the bulk. We speculate that such states are the one-dimensional analog of the light $f$-electron surface states which appear to develop in the putative topological Kondo insulator, ${\mathrm{SmB}}_{6}$.

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.

Magnetic and nonmagnetic phases in dopedAB2 t−JHubbard chains

We discuss the rich phase diagram of doped AB2 $t-J$ chains by using data from density matrix renormalization group and exact diagonalization techniques. The $J$ vs $\delta$ (hole doping) phase diagram exhibits regions of itinerant ferrimagnetism, incommensurate, resonating valence bond and Nagaoka states, phase separation, and Luttinger liquid (LL) physics. Several features are highlighted, such as the modulated ferrimagnetic structure, the occurrence of Nagaoka spin polarons in the underdoped regime and small values of $J=4t^2/U$, where $t$ is the first-neighbor hopping amplitude and $U$ is the on-site repulsive Coulomb interaction, incommensurate structures with nonzero magnetization, and strong-coupling LL physics in the high-doped regime. We also verify that relevant findings are in agreement with the corresponding findings in square and n-leg ladder lattices. In particular, we mention the instability of Nagaoka ferromagnetism against $J$ and $\delta$.

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.

Mechanistic investigations on E-N bond-breaking and ring expansion for N-heterocyclic carbene analogues containing the group 14 elements (E).

The potential-energy surfaces for the ring-expansion reactions (i)Pr N(CH)2N((i)Pr)E :(Rea-E) + SiH2Ph2 → six-membered ring heterocyclcic product (E = C, Si, Ge, Sn, and Pb) and (i)Pr N(CH)2N((i)Pr)C :(Rea-C) + EH2Ph2 → six-membered ring heterocyclcic product are studied at the M06-2X/Def2-TZVP level of theory. These theoretical investigations suggest that for a given SiH2Ph2, the relative reactivity of Rea-E toward the ring-ring expansion reaction decreases as the atomic weight of the central atom E increases, that is, in the order Rea-C ≫ Rea-Si > Rea-Ge > Rea-Sn > Rea-Pb. However, for a given Rea-C, these theoretical observations demonstrate that the reactivity of the EH2Ph2 molecule that undergoes the ring-expansion reaction decreases in the order SiH2Ph2 ≈ GeH2Ph2 ≈ SnH2Ph2 > PbH2Ph2 ≫ CH2Ph2. This theoretical study indicates that both the electronic structure and steric effects play a crucial role in determining their reactivities. The model conclusions are consistent with available experimental findings. Furthermore, a valence bond state correlation diagram model can be used to rationalize the computational results. These results allow a number of predictions to be made.

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.

Chiral phases of two-dimensional hard-core bosons with frustrated ring exchange

We study the zero-temperature phase diagram of two-dimensional hard-core bosons on a square lattice with nearest-neighbor and plaquette (ring-exchange) hoppings, at arbitrary densities, by means of a hierarchical mean-field theory. In the frustrated regime, where quantum Monte Carlo suffers from a sign problem, we find a rich phase diagram where exotic states with nonzero chirality emerge. Among them, novel insulating phases, characterized by nonzero bond-chirality and plaquette order, are found over a large region of the parameter space. In the unfrustrated regime, the hierarchical mean-field approach improves over the standard mean-field treatment as it is able to capture the transition from a superfluid to a valence bond state upon increasing the strength of the ring-exchange term, in qualitative agreement with quantum Monte Carlo results.

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.

Local magnetism and spin correlations in the geometrically frustrated cluster magnetLiZn2Mo3O8

LiZn$_2$Mo$_3$O$_8$ has been proposed to contain $S~=~1/2$ Mo$_3$O$_{13}$ magnetic clusters arranged on a triangular lattice with antiferromagnetic nearest-neighbor interactions. Here, microwave and terahertz electron spin resonance (ESR), $^7$Li nuclear magnetic resonance (NMR), and muon spin rotation ($\mu \textrm{SR}$) spectroscopies are used to characterize the local magnetic properties of LiZn$_2$Mo$_3$O$_8$. These results show the magnetism in LiZn$_2$Mo$_3$O$_8$ arises from a single isotropic $S~=~1/2$ electron per cluster and that there is no static long-range magnetic ordering down to $T~=~0.07\,\textrm{K}$. Further, there is evidence of gapless spin excitations with spin fluctuations slowing down as the temperature is lowered. These data indicate strong spin correlations which, together with previous data, suggest a low-temperature resonating valence-bond state in LiZn$_2$Mo$_3$O$_8$.

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.

Deconfined quantum criticality and conformal phase transition in two-dimensional antiferromagnets

Deconfined quantum criticality of two-dimensional SU(2) quantum antiferromagnets featuring a transition from an antiferromagnetically ordered ground state to a so-called valence-bond solid state, is governed by a non-compact CP1 model with a Maxwell term in spacetime dimensions. We introduce a new perspective on deconfined quantum criticality within a field-theoretic framework based on an expansion in powers of for fixed number N of complex matter fields. We show that in the allegedly weak first-order transition regime, a so-called conformal phase transition leads to a genuine deconfined quantum critical point. In such a transition, the gap vanishes when the critical point is approached from above and diverges when it is approached from below. We also find that the spin stiffness has a universal jump at the critical point.

Simulation of non-Abelian gauge theories with optical lattices

Many phenomena occurring in strongly correlated quantum systems still await conclusive explanations. The absence of isolated free quarks in nature is an example. It is attributed to quark confinement, whose origin is not yet understood. The phase diagram for nuclear matter at general temperatures and densities, studied in heavy-ion collisions, is not settled. Finally, we have no definitive theory of high-temperature superconductivity. Though we have theories that could underlie such physics, we lack the tools to determine the experimental consequences of these theories. Quantum simulators may provide such tools. Here we show how to engineer quantum simulators of non-Abelian lattice gauge theories. The systems we consider have several applications: they can be used to mimic quark confinement or to study dimer and valence-bond states (which may be relevant for high-temperature superconductors).

Preparing topological projected entangled pair states on a quantum computer

Simulating exotic phases of matter that are not amenable to classical techniques is one of the most important potential applications of quantum information processing. We present an efficient algorithm for preparing a large class of topological quantum states, the $G$-injective projected entangled pair states (PEPS), on a quantum computer. Important examples include the resonant valence bond states, conjectured to be topological spin liquids. The runtime of the algorithm scales polynomially with the condition number of the PEPS projectors and inverse polynomially in the spectral gap of the PEPS parent Hamiltonian.

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.

Confinement and Deconfinement of Spinons in Two Dimensions

We use MonteCarlo methods to study spinons in two-dimensional quantum spin systems, characterizing their intrinsic size λ and confinement length Λ. We confirm that spinons are deconfined, Λ→∞ and λ finite, in a resonating valence-bond spin-liquid state. In a valence-bond solid, we find finite λ and Λ, with λ of a single spinon significantly larger than the bound state-the spinon is soft and shrinks as the bound state is formed. Both λ and Λ diverge upon approaching the critical point separating valence-bond solid and Néel ground states. We conclude that the spinon deconfinement is marginal in the lowest-energy state in the spin-1 sector, due to weak attractive spinon interactions. Deconfinement in the vicinity of the critical point should occur at higher energies.