Valence Bond Solid Ground State Research Articles

Structural and magnetic studies of the frustrated S = 1 kagome magnet NH4Ni2Mo2O10H3

The strong geometric frustration of the kagome antiferromagnets (KAFMs) can destabilise conventional magnetic order and lead to exotic electronic states, such as the quantum spin-liquid state observed in some S=12 KAFM materials. However, the ground state of S = 1 KAFM systems are less well understood. Spin nematic phases and valence bond solid ground states have been predicted to form but a paucity of experimental realisations restricts understanding. Here, the S = 1 KAFM NH4Ni2Mo2O10H3 is presented, which has the 3-fold symmetry of the kagome lattice but significant site depletion, with ∼64% site occupancy. Frustration and a competition between exchange interactions are evidenced through the suppression of order below the Weiss temperature |θW| and observation of ferromagnetic and antiferromagnetic characteristics in the magnetisation data. A semi spin glass ground state is predicted based on the ac-field frequency dependence of the magnetic transition and ferromagnetic signal.

Gapped magnetic ground state in quantum spin liquid candidate κ-(BEDT-TTF)2Cu2(CN)3.

Geometrical frustration, quantum entanglement, and disorder may prevent long-range ordering of localized spins with strong exchange interactions, resulting in an exotic state of matter. κ-(BEDT-TTF)2Cu2(CN)3 is considered the prime candidate for this elusive quantum spin liquid state, but its ground-state properties remain puzzling. We present a multifrequency electron spin resonance (ESR) study down to millikelvin temperatures, revealing a rapid drop of the spin susceptibility at 6 kelvin. This opening of a spin gap, accompanied by structural modifications, is consistent with the formation of a valence bond solid ground state. We identify an impurity contribution to the ESR response that becomes dominant when the intrinsic spins form singlets. Probing the electrons directly manifests the pivotal role of defects for the low-energy properties of quantum spin systems without magnetic order.

Magnon Bose–Einstein condensation and superconductivity in a frustrated Kondo lattice

Motivated by recent experiments on magnetically frustrated heavy fermion metals, we theoretically study the phase diagram of the Kondo lattice model with a nonmagnetic valence bond solid ground state on a ladder. A similar physical setting may be naturally occurring in [Formula: see text], [Formula: see text], and [Formula: see text] compounds. In the insulating limit, the application of a magnetic field drives a quantum phase transition to an easy-plane antiferromagnet, which is described by a Bose-Einstein condensation of magnons. Using a combination of field theoretical techniques and density matrix renormalization group calculations we demonstrate that in one dimension this transition is stable in the presence of a metallic Fermi sea, and its universality class in the local magnetic response is unaffected by the itinerant gapless fermions. Moreover, we find that fluctuations about the valence bond solid ground state can mediate an attractive interaction that drives unconventional superconducting correlations. We discuss the extensions of our findings to higher dimensions and argue that depending on the filling of conduction electrons, the magnon Bose-Einstein condensation transition can remain stable in a metal also in dimensions two and three.

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.

Entanglement entropy for the valence bond solid phases of two-dimensional dimerized Heisenberg antiferromagnets

We calculate the bipartite von Neumann and second R\'enyi entanglement entropies of the ground states of spin-1/2 dimerized Heisenberg antiferromagnets on a square lattice. Two distinct dimerization patterns are considered: columnar and staggered. In both cases, we concentrate on the valence bond solid (VBS) phase and describe such a phase with the bond-operator representation. Within this formalism, the original spin Hamiltonian is mapped into an effective interacting boson model for the triplet excitations. We study the effective Hamiltonian at the harmonic approximation and determine the spectrum of the elementary triplet excitations. We then follow an analytical procedure, which is based on a modified spin-wave theory for finite systems and was originally employed to calculate the entanglement entropies of magnetic ordered phases, and calculate the entanglement entropies of the VBS ground states. In particular, we consider one-dimensional (line) subsystems within the square lattice, a choice that allows us to consider line subsystems with sizes up to $L' = 1000$. We combine such a procedure with the results of the bond-operator formalism at the harmonic level and show that, for both dimerized Heisenberg models, the entanglement entropies of the corresponding VBS ground states obey an area law as expected for gapped phases. For both columnar-dimer and staggered-dimer models, we also show that the entanglement entropies increase but do not diverge as the dimerization decreases and the system approaches the N\'eel--VBS quantum phase transition. Finally, the entanglement spectra associated with the VBS ground states are presented.

Continuous Easy-Plane Deconfined Phase Transition on the Kagome Lattice.

We use large scale quantum MonteCarlo simulations to study an extended Hubbard model of hard core bosons on the kagome lattice. In the limit of strong nearest-neighbor interactions at 1/3 filling, the interplay between frustration and quantum fluctuations leads to a valence bond solid ground state. The system undergoes a quantum phase transition to a superfluid phase as the interaction strength is decreased. It is still under debate whether the transition is weakly first order or represents an unconventional continuous phase transition. We present a theory in terms of an easy plane noncompact CP^{1} gauge theory describing the phase transition at 1/3 filling. Utilizing large scale quantum MonteCarlo simulations with parallel tempering in the canonical ensemble up to 15552 spins, we provide evidence that the phase transition is continuous at exactly 1/3 filling. A careful finite size scaling analysis reveals an unconventional scaling behavior hinting at deconfined quantum criticality.

Field theory of symmetry-protected valence bond solid states in (2+1) dimensions

This paper describes a semiclassical field-theory approach to the topological properties of spatially featureless Affleck-Kennedy-Lieb-Tasaki type valence bond solid ground states of antiferromagnets in spatial dimensions one to three. Using nonlinear sigma models set in the appropriate target manifold and augmented with topological terms, we argue that the path integral representation of the ground-state wave functional can correctly distinguish symmetry-protected topological ground states from topologically trivial ones. The symmetry-protection feature is demonstrated explicitly in terms of a dual field theory, where we take into account the nontrivial spatial structure of topological excitations, which are caused by competition among the relevant ordering tendencies. A temporal surface contribution to the action originating from the bulk topological term plays a central role in our study. We discuss how the same term governs the behavior of the so-called strange correlator. In particular, we find that the path integral expression for the strange correlator in two dimensions reduces to the well-known Haldane expression for the two point spin correlator of antiferromagnetic spin chains.

Theory of triplon dynamics in the quantum magnetBiCu2PO6

We provide a theory of triplon dynamics in the valence bond solid ground state of the coupled spin-ladders modelled for BiCu$_2$PO$_6$. Utilizing the recent high quality neutron scattering data [Nature Physics (2015), DOI: 10.1038/NPHYS3566] as guides and a theory of interacting triplons via the bond operator formulation, we determine a minimal spin Hamiltonian for this system. It is shown that the splitting of the low energy triplon modes and the peculiar magnetic field dependence of the triplon dispersions can be explained by including substantial Dzyaloshinskii-Moriya and symmetric anisotropic spin interactions. Taking into account the interactions between triplons and the decay of the triplons to the two-triplon continuum via anisotropic spin interactions, we provide a theoretical picture that can be used to understand the main features of the recent neutron scattering experimental data.

Bulk entanglement spectrum in gapped spin ladders

We study the bulk entanglement of a series of gapped ground states of spin ladders, representative of the Haldane phase. These ground states of spin $S/2$ ladders generalize the valence bond solid ground state. In the case of spin 1/2 ladders, we study a generalization of the Affleck-Kennedy-Lieb-Tasaki and Nersesyan-Tsvelik states and fully characterize the bulk entanglement Hamiltonian. In the case of general spin $S$ we argue that in the Haldane phase the bulk entanglement spectrum of a half integer ladder is either gapless or possess a degenerate ground state. For ladders with integer valued spin particles, the generic bulk entanglement spectrum should have an entanglement gap. Finally, we give an example of a series of trivial states of higher spin $S>1$ in which the bulk entanglement Hamiltonian is critical, signaling that the relation between topological states and a critical bulk entanglement Hamiltonian is not unique to topological systems.

Numerical linked-cluster expansion for the distorted kagome lattice Heisenberg model

Motivated by experimental results for the thermodynamic properties of the Rb2Cu3SnF12 material and the discovery of its valence-bond solid ground state, we utilize the numerical linked-cluster expansions (NLCEs) and devise an expansion tailored to solve the Heisenberg model on a pinwheel-distorted kagome lattice. Using the exchange interactions that are relevant to Rb2Cu3SnF12, we calculate its uniform spin susceptibility and find a very good agreement with experiment. Next, we focus on the ground state of a simplified model of the distorted kagome lattice and take advantage of a zero-temperature Lanczos-based NLCE to study the approach to the ground state of the uniform kagome lattice Heisenberg model using a tuning parameter. Together with results from exact diagonalization of finite clusters, we find evidence that a phase transition occurs before the uniform limit is reached.

Influence of Dzyaloshinskii-Moriya interactions on magnetic structure of a spin-12deformed kagome lattice antiferromagnet

Motivated by the recent neutron-scattering experiment on Rb${}_{2}$Cu${}_{3}$SnF${}_{12}$ [Nature Phys. 6, 865 (2010)], we investigate the effect of Dzyaloshinskii-Moriya interactions in a theoretical model for the magnetic structure of this material. Considering the valence bond solid ground state, which has a 12-site unit cell, we develop the bond operator mean-field theory. It is shown that the Dzyaloshinskii-Moriya interactions significantly modify the triplon dispersions around the $\ensuremath{\Gamma}$ point and cause a shift of the spin-gap (the minimum triplon gap) position from the $K$ to $\ensuremath{\Gamma}$ point in the first Brillouin zone. The spin gap is also evaluated in exact diagonalization studies on a 24-site cluster. We discuss a magnetic transition induced by the Dzyaloshinskii-Moriya interactions in the bond operator framework. Moreover, the magnetization process under external magnetic fields is studied within the exact diagonalization approach. We find that the results of both approaches are consistent with the experimental findings.

Topological liquids and valence cluster states in two-dimensional SU(N)magnets

We study the zero-temperature phase diagram of a class of two-dimensional SU$(N)$ antiferromagnets. These models are characterized by having the same type of SU$(N)$ spin placed at each site of the lattice, and share the property that, in general, more than two spins must be combined to form a singlet. An important motivation to study these systems is that they may be realized naturally in Mott insulators of alkaline-earth atoms placed on optical lattices; indeed, such Mott insulators have already been obtained experimentally, although the temperatures are still high compared to the magnetic exchange energy. We study these antiferromagnets in a large-$N$ limit, finding a variety of ground states. Some of the models studied here have a valence-bond solid ground state, as was found in prior studies, yet we find that many others have a richer variety of ground states. Focusing on the two-dimensional square lattice, in addition to valence cluster states (which are analogous to valence-bond states), we find both Abelian and non-Abelian chiral spin liquid ground states, which are magnetic counterparts of the fractional quantum Hall effect. We also find a ``doubled'' chiral spin liquid ground state that preserves time-reversal symmetry. These results are based on a combination of rigorous lower bounds on the large-$N$ ground-state energy and a systematic numerical ground-state search. We conclude by discussing whether experimentally relevant SU$(N)$ antiferromagnets, away from the large-$N$ limit, may be chiral spin liquids.

Topological term in the non-linear σ model of the [formula omitted] spin chains

We show that there is a topological (Berry phase) term in the non-linear σ model description of the SO ( 5 ) spin chain. It distinguishes the linear and projective representations of the SO ( 5 ) symmetry group, in exact analogy to the well-known θ-term of the SO ( 3 ) spin chain. The presence of the topological term is due to the fact that π 2 ( SO ( 5 ) SO ( 3 ) × SO ( 2 ) ) = Z . We discuss the implication of our results on the spectra of the SO ( 5 ) spin chain, and connect it with a recent solvable SO ( 5 ) spin model which exhibits valence bond solid ground state and edge degeneracy.

Pinwheel valence-bond solid and triplet excitations in the two-dimensional deformed kagome lattice

Determining ground states of correlated electron systems is fundamental to understanding novel phenomena in condensed matter physics. A difficulty, however, arises in a geometrically frustrated system in which the incompatibility between the global topology of an underlying lattice and local spin interactions gives rise to macroscopically degenerate ground states, potentially prompting the emergence of quantum spin states, such as resonating valence bond (RVB) and valence bond solid (VBS). Although theoretically proposed to exist in a kagome lattice -- one of the most highly frustrated lattices in two dimensions (2D) being comprised of corner-sharing triangles -- such quantum-fluctuation-induced states have not been observed experimentally. Here we report the first realization of the "pinwheel" VBS ground state in the S=1/2 deformed kagome lattice antiferromagnet Rb2Cu3SnF12. In this system, a lattice distortion breaks the translational symmetry of the ideal kagome lattice and stabilizes the VBS state.

Optical spin-1 chain and its use as a quantum-computational wire

Measurement-based quantum computing, a powerful alternative to the standard circuit model, proceeds using only local adaptive measurements on a highly entangled resource state of many spins on a graph or lattice. Along with the canonical cluster state, the valence-bond solid ground state on a chain of spin-1 particles, studied by Affleck, Kennedy, Lieb, and Tasaki (AKLT), is such a resource state. We propose a simulation of this AKLT state using linear optics, wherein we can make use of the high-fidelity projective measurements that are commonplace in quantum-optical experiments, and describe how quantum logic gates can be performed on this chain. In our proposed implementation, the spin-1 particles comprising the AKLT state are encoded on polarization biphotons: three-level systems consisting of pairs of polarized photons in the same spatio-temporal mode. A logical qubit encoded on the photonic AKLT state can be initialized, read out, and have an arbitrary single-qubit unitary applied to it by performing projective measurements on the constituent biphotons. For MBQC, biphoton measurements are required which cannot be deterministically performed using only linear optics and photodetection.

AKLT models with quantum spin glass ground states

We study AKLT models on locally tree-like lattices of fixed connectivity and find that they exhibit a variety of ground states depending upon the spin, coordination and global (graph) topology. We find a) quantum paramagnetic or valence bond solid ground states, b) critical and ordered N\'eel states on bipartite infinite Cayley trees and c) critical and ordered quantum vector spin glass states on random graphs of fixed connectivity. We argue, in consonance with a previous analysis, that all phases are characterized by gaps to local excitations. The spin glass states we report arise from random long ranged loops which frustrate N\'eel ordering despite the lack of randomness in the coupling strengths.

ENTANGLEMENT IN VALENCE-BOND-SOLID STATES

This article reviews the quantum entanglement in Valence-Bond-Solid (VBS) states defined on a lattice or a graph. The subject is presented in a self-contained and pedagogical way. The VBS state was first introduced in the celebrated paper by I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki (abbreviation AKLT is widely used). It became essential in condensed matter physics and quantum information (measurement-based quantum computation). Many publications have been devoted to the subject. Recently entanglement was studied in the VBS state. In this review, we start with the definition of a general AKLT spin chain and the construction of VBS ground state. In order to study entanglement, a block subsystem is introduced and described by the density matrix. Density matrices of 1-dimensional models are diagonalized and the entanglement entropies (the von Neumann entropy and Rényi entropy) are calculated. In the large block limit, the entropies also approach finite limits. Study of the spectrum of the density matrix led to the discovery that the density matrix is proportional to a projector.

Geometric entanglement in a one-dimensional valence-bond solid state

In this paper we provide the analytical derivation of the global geometric entanglement per block for the valence-bond solid ground state of the spin-1 Affleck-Kennedy-Lieb-Tasaki chain. In particular, we show that this quantity saturates exponentially fast to a constant when the sizes of the blocks are sufficiently large. Our result provides an example of an analytical calculation of the geometric entanglement for a gapped quantum many-body system in one dimension and far away from a quantum critical point.

Entanglement of the valence-bond-solid state on an arbitrary graph

The Affleck–Kennedy–Lieb–Tasaki (AKLT) spin interacting model can be defined on an arbitrary graph. We explain the construction of the AKLT Hamiltonian. Given certain conditions, the ground state is unique and known as the valence-bond-solid (VBS) state. It can be used in measurement-based quantum computation as a resource state instead of the cluster state. We study the VBS ground state on an arbitrary connected graph. The graph is cut into two disconnected parts: the block and the environment. We study the entanglement between these two parts and prove that many eigenvalues of the density matrix of the block are zero. We describe a subspace of eigenvectors of the density matrix corresponding to non-zero eigenvalues. The subspace is the degenerate ground states of some Hamiltonian which we call the block Hamiltonian.

Designing spin-1 lattice models using polar molecules

We describe how to design a large class of always on spin-1 interactions between polar molecules trapped in an optical lattice. The spin degrees of freedom correspond to the hyperfine levels of a ro-vibrational ground state molecule. Interactions are induced using a microwave field to mix ground states in one hyperfine manifold with the spin entangled dipole–dipole coupled excited states. Using multiple fields, anistropic models in one, two, or three dimensions can be built with tunable spatial range. An illustrative example in one-dimension is the generalized Haldane model, which at a specific parameter has a gapped valence bond solid ground state. The interaction strengths are large compared to decoherence rates and should allow for probing the rich phase structure of strongly correlated systems, including dimerized and gapped phases.