Valence-bond Phase Research Articles

Plaquette valence bond state in the spin- 12 J1−J2 XY model on the square lattice

We studied the ground-state phase diagram of spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model on the square lattice with first-neighbor ${J}_{1}$ and second-neighbor ${J}_{2}$ antiferromagnetic interactions using both infinite density matrix renormalization group (iDMRG) and DMRG approaches. We show that a plaquette valence bond phase is realized in an intermediate region $0.50\ensuremath{\le}{J}_{2}/{J}_{1}<0.54$ between a N\'eel magnetic ordered phase at ${J}_{2}/{J}_{1}<0.50$ and a stripy magnetic ordered phase at ${J}_{2}/{J}_{1}\ensuremath{\ge}0.54$. The plaquette valence bond phase is characterized by finite dimer orders in both the horizontal and vertical directions. Contrary to the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model, we do not find numerical evidence for a quantum spin liquid phase in the ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model.

Multicritical Deconfined Quantum Criticality and Lifshitz Point of a Helical Valence-Bond Phase.

The S=1/2 square-lattice J-Q model hosts a deconfined quantum phase transition between antiferromagnetic and dimerized (valence-bond solid) ground states. We here study two deformations of this model-a term projecting staggered singlets, as well as a modulation of the J terms forming alternating "staircases" of strong and weak couplings. The first deformation preserves all lattice symmetries. Using quantum MonteCarlo simulations, we show that it nevertheless introduces a second relevant field, likely by producing topological defects. The second deformation induces helical valence-bond order. Thus, we identify the deconfined quantum critical point as a multicritical Lifshitz point-the end point of the helical phase and also the end point of a line of first-order transitions. The helical-antiferromagnetic transitions form a line of generic deconfined quantum-critical points. These findings extend the scope of deconfined quantum criticality and resolve a previously inconsistent critical-exponent bound from the conformal-bootstrap method.

Dynamical properties of Néel and valence-bond phases in the J1–J2 model on the honeycomb lattice

By using a variational Monte Carlo technique based upon Gutzwiller-projected fermionic states, we investigate the dynamical structure factor of the antiferromagnetic S = 1/2 Heisenberg model on the honeycomb lattice, in presence of first-neighbor (J1) and second-neighbor (J2) couplings, for J2 < 0.5J1. The ground state of the system shows long-range antiferromagnetic order for J2/J1 ≲ 0.23 (Néel phase), plaquette valence-bond order for 0.23 ≲ J2/J1 ≲ 0.36, and columnar dimer order for J2/J1 ≳ 0.36. Within the Néel phase, a well-defined magnon mode is observed, whose dispersion is in relatively good agreement with linear spin-wave approximation for J2 = 0. When a nonzero second-neighbor super-exchange is included, a roton-like mode develops around the K point (i.e., the corner of the Brillouin zone). This mode softens when J2/J1 is increased and becomes gapless at the transition point, J2/J1 ≈ 0.23. Here, a broad continuum of states is clearly visible in the dynamical spectrum, suggesting that nearly-deconfined spinon excitations could exist, at least at relatively high energies. For larger values of J2/J1, valence-bond order is detected and the spectrum of the system becomes clearly gapped, with a triplon mode at low energies. This is particularly evident for the spectrum of the dimer valence-bond phase, in which the triplon mode is rather well separated from the continuum of excitations that appears at higher energies.

Valence bond phases of herbertsmithite and related copper kagome materials

Recent evidence from magnetic torque, electron spin resonance, and second harmonic generation indicate that the prototypical quantum spin liquid candidate, herbertsmithite, has a symmetry lower than its x-ray refined trigonal space group. Here, we consider known and possible distortions of this mineral class, along with related copper kagome oxides and fluorides, relate these to possible valence bond patterns, and comment on their relevance to the physics of these interesting materials.

Spontaneous topological transitions in a honeycomb lattice of exciton-polariton condensates due to spin bifurcations

We theoretically study the spontaneous formation of the quantum anomalous Hall effect in a graphene system of spin-bifurcated exciton-polariton condensates under nonresonant pumping. We demonstrate that, depending on the parameters of the structure, such as intensity of the pump and coupling strength between condensates, the system shows a rich variety of macroscopic magnetic ordering, including analogs of ferromagnetic, antiferromagnetic, and resonant valence bond phases. Transitions between these magnetic polarized phases are associated with dramatic reshaping of the spectrum of the system connected with spontaneous appearance of topological order.

Classification of trivial spin-1 tensor network states on a square lattice

Classification of possible quantum spin liquid (QSL) states of interacting spin-1/2's in two dimensions has been a fascinating topic of condensed matter for decades, resulting in enormous progress in our understanding of low-dimensional quantum matter. By contrast, relatively little work exists on the identification, let alone classification, of QSL phases for spin-1 systems in dimensions higher than one. Employing the powerful ideas of tensor network theory and its classification, we develop general methods for writing QSL wave functions of spin-1 respecting all the lattice symmetries, spin rotation, and time reversal with trivial gauge structure on the square lattice. We find $2^5$ distinct classes characterized by five binary quantum numbers. Several explicit constructions of such wave functions are given for bond dimensions $D$ ranging from two to four, along with thorough numerical analyses to identify their physical characters. Both gapless and gapped states are found. The topological entanglement entropy of the gapped states are close to zero, indicative of topologically trivial states. In $D=4$, several different tensors can be linearly combined to produce a family of states within the same symmetry class. A rich "phase diagram" can be worked out among the phases of these tensors, as well as the phase transitions among them. Among the states we identified in this putative phase diagram is the plaquette-ordered phase, gapped resonating valence bond phase, and a critical phase. A continuous transition separates the plaquette-ordered phase from the resonating valence bond phase.

Competing valence bond and symmetry-breaking Mott states of spin-32fermions on a honeycomb lattice

We investigate magnetic properties of strongly interacting four component spin-3/2 ultracold fermionic atoms in the Mott insulator limit with one particle per site in an optical lattice with honeycomb symmetry. In this limit, atomic tunneling is virtual, and only the atomic spins can exchange. We find a competition between symmetry breaking and liquid like disordered phases. Particularly interesting are valence bond states with bond centered magnetizations, situated between the ferromagnetic and conventional valence bond phases. In the framework of a mean-field theory, we calculate the phase diagram and identify an experimentally relevant parameter region where a homogeneous SU(4) symmetric Affleck-Kennedy-Lieb-Tasaki-like valence bond state is present.

Numerical studies of various Néel-VBS transitions in SU(N) anti-ferromagnets

In this manuscript we review recent developments in the numerical simulations of bipartite SU(N) spin models by quantum Monte Carlo (QMC) methods. We provide an account of a large family of newly discovered sign-problem free spin models which can be simulated in their ground states on large lattices, containing O(105) spins, using the stochastic series expansion method with efficient loop algorithms. One of the most important applications so far of these Hamiltonians are to unbiased studies of quantum criticality between Neel and valence bond phases in two dimensions - a summary of this body of work is provided. The article concludes with an overview of the current status of and outlook for future studies of the “designer” Hamiltonians.

Valence bond phases in S = 1/2 Kane–Mele–Heisenberg model

The phase diagram of the Kane–Mele–Heisenberg model in a classical limit [] contains disordered regions in the coupling space, as the result of competition between different terms in the Hamiltonian, leading to frustration in finding a unique ground state. In this work we explore the nature of these phases in the quantum limit, for a S = 1/2. Employing exact diagonalization in Sz and nearest neighbour valence bond bases, and bond and plaquette valence bond mean field theories, we show that the disordered regions are divided into ordered quantum states in the form of plaquette valence bond crystals and staggered dimerized phases.

Exotic spin orders driven by orbital fluctuations in the Kugel-Khomskii model

We study zero temperature phase diagram of the three-dimensional Kugel-Khomskii model on a cubic lattice using the cluster mean field theory and different perturbative expansions in the orbital sector. The phase diagram is rich, goes beyond the single-site mean field theory due to spin-orbital entanglement. In addition to the antiferromagnetic (AF) and ferromagnetic (FM) phases, one finds also a plaquette valence-bond phase with singlets ordered either on horizontal or vertical bonds. More importantly, for increasing Hund's exchange we identify three phases with exotic magnetic order stabilized by orbital fluctuations in between the AF and FM order: (i) an AF phase with two mutually orthogonal antiferromagnets on two sublattices in each $ab$ plane and AF order along the c axis (ortho-$G$-type phase), (ii) a canted-$A$-type AF phase with a non-trivial canting angle between nearest neighbor FM layers along the c axis, and (iii) a striped-AF phase with anisotropic AF order in the $ab$ planes. We elucidate the mechanism responsible for each of the above phases by deriving effective spin models which involve second and third neighbor Heisenberg interactions as well as four-site spin interactions going beyond Heisenberg physics, and explain how the entangled nearest neighbor spin-orbital superexchange generates spin interactions between more distant spins.

Entangled Spin-Orbital Phases in the d9Model

We investigate the phase diagrams of the spin-orbital $d^9$ Kugel-Khomskii model for a bilayer and a monolayer square lattice using Bethe-Peierls-Weiss method. For a bilayer we obtain valence bond phases with interlayer singlets, with alternating planar singlets, and two entangled spin-orbital (ESO) phases, in addition to the antiferromagnetic and ferromagnetic order. Possibility of such entangled phases in a monolayer is under investigation at present.

Entangled spin-orbital phases in the bilayer Kugel-Khomskii model

We derive the Kugel-Khomskii spin-orbital (SO) model for a bilayer and investigate its phase diagram depending on Hund's exchange $J_H$ and the $e_g$ orbital splitting $E_z$. In the (classical) mean-field approach with on-site spin $<S_i^z>$ and orbital $<\tau_i^z>$ order parameters and factorized spin-and-orbital degrees of freedom, we demonstrate a competition between the phases with either $G$-type or $A$-type antiferromagnetic (AF) or ferromagnetic long-range order. Next we develop a Bethe-Peierls-Weiss method with a Lanczos exact diagonalization of a cube coupled to its neighbors in $ab$ planes by the mean-field terms --- this approach captures quantum fluctuations on the bonds which decide about the nature of disordered phases in the highly frustrated regime near the orbital degeneracy. We show that the long-range spin order is unstable in a large part of the phase diagram which contains then six phases, including also the valence-bond phase with interlayer spin singlets stabilized by holes in $3z^2-r^2$ orbitals (VB$z$ phase), a disordered plaquette valence-bond (PVB) phase and a crossover phase between the VB$z$ and the $A$-type AF phase. When on-site SO coupling is also included by the $<S_i^z\tau_i^z>$ order parameter, we discover in addition two entangled phases which compete with $A$-type AF phase and another crossover phase in between the $G$-AF phase with occupied $x^2-y^2$ orbitals and the PVB phase. Thus, the present bilayer model provides an example of SO entanglement which generates novel disordered phases. We analyze the order parameters in all phases and identify situations where SO entanglement is crucial and factorization of the spins and orbitals leads to qualitatively incorrect results. We point out that SO entanglement may play a role in a bilayer fluoride K$_3$Cu$_2$F$_7$ which is a realization of the VB$z$ phase.

Quantum computational capability of a 2D valence bond solid phase

Quantum phases of naturally-occurring systems exhibit distinctive collective phenomena as manifestation of their many-body correlations, in contrast to our persistent technological challenge to engineer at will such strong correlations artificially. Here we show theoretically that quantum correlations exhibited in the 2D valence bond solid phase of a quantum antiferromagnet, modeled by Affleck, Kennedy, Lieb, and Tasaki (AKLT) as a precursor of spin liquids and topological orders, are sufficiently complex yet structured enough to simulate universal quantum computation when every single spin can be measured individually. This unveils that an intrinsic complexity of naturally-occurring 2D quantum systems—which has been a long-standing challenge for traditional computers—could be tamed as a computationally valuable resource, even if we are limited not to create newly entanglement during computation. Our constructive protocol leverages a novel way to herald the correlations suitable for deterministic quantum computation through a random sampling, and may be extensible to other ground states of various 2D valence bond phases beyond the AKLT state.

Extended quantum dimer model and novel valence-bond phases

We extend the quantum dimer model (QDM) introduced by Rokhsar and Kivelson so as toconstruct a concrete example of the model which exhibits the first-order phase transitionbetween different valence-bond solids suggested recently by Batista and Trugman and lookfor the possibility of other exotic dimer states. We show that our model contains threeexotic valence-bond phases (herringbone, checkerboard and dimer smectic) in theground-state phase diagram and that it realizes the phase transition from thestaggered valence-bond solid to the herringbone. The checkerboard phase hasfour-fold rotational symmetry, while the dimer smectic, in the absence of quantumfluctuations, has massive degeneracy originating from partial ordering only inone of the two spatial directions. A resonance process involving three dimersresolves this massive degeneracy and the dimer smectic becomes ordered (order fromdisorder).

Bose-Hubbard model on a star lattice

We analyze the Bose-Hubbard model of hardcore bosons with nearest-neighbor hopping and repulsive interactions on a star lattice using both quantum Monte Carlo simulation and dual vortex theory. We obtain the phase diagram of this model as a function of the chemical potential and the relative strength of hopping and interaction. In the strong-interaction regime, we find that the Mott phases of the model at 1/2 and 1/3 fillings, in contrast to their counterparts on square, triangular, and kagome lattices, are either translationally invariant resonant valence bond (RVB) phases with no density-wave order or have coexisting density-wave and RVB orders. We also find that upon increasing the relative strength of hopping and interaction, the translationally invariant Mott states undergo direct second-order superfluid-insulator quantum phase transitions. We compute the critical exponents for these transitions and argue using the dual vortex picture that the transitions, when approached through the tip of the Mott lobe, belong to the inverted XY universality class.

High-Order Coupled Cluster Method (CCM) Calculations for Quantum Magnets with Valence-Bond Ground States

In this article, we prove that exact representations of dimer and plaquette valence-bond ket ground states for quantum Heisenberg antiferromagnets may be formed via the usual coupled cluster method (CCM) from independent-spin product (e.g. Néel) model states. We show that we are able to provide good results for both the ground-state energy and the sublattice magnetization for dimer and plaquette valence-bond phases within the CCM. As a first example, we investigate the spin-half J 1–J 2 model for the linear chain, and we show that we are able to reproduce exactly the dimerized ground (ket) state at J 2/J 1=0.5. The dimerized phase is stable over a range of values for J 2/J 1 around 0.5, and results for the ground-state energies are in good agreement with the results of exact diagonalizations of finite-length chains in this regime. We present evidence of symmetry breaking by considering the ket- and bra-state correlation coefficients as a function of J 2/J 1. A radical change is also observed in the behavior of the CCM sublattice magnetization as we enter the dimerized phase. We then consider the Shastry-Sutherland model and demonstrate that the CCM can span the correct ground states in both the Néel and the dimerized phases. Once again, very good results for the ground-state energies are obtained. We find CCM critical points of the bra-state equations that are in agreement with the known phase transition point for this model. The results for the sublattice magnetization remain near to the “true” value of zero over much of the dimerized regime, although they diverge exactly at the critical point. Finally, we consider a spin-half system with nearest-neighbor bonds for an underlying lattice corresponding to the magnetic material CaV4O9 (CAVO). We show that we are able to provide excellent results for the ground-state energy in each of the plaquette-ordered, Néel-ordered, and dimerized regimes of this model. The exact plaquette and dimer ground states are reproduced by the CCM ket state in their relevant limits. Furthermore, we estimate the range over which the Néel order is stable, and we find the CCM result is in reasonable agreement with the results obtained by other methods. Our new approach has the dual advantages that it is simple to implement and that existing CCM codes for independent-spin product model states may be used from the outset. Furthermore, it also greatly extends the range of applicability to which the CCM may be applied. We believe that the CCM now provides an excellent choice of method for the study of systems with valence-bond quantum ground states.

Dzyaloshinskii-Moriya interactions in valence-bond systems

We investigate the effect of Dzyaloshinskii-Moriya interactions on the low temperature magnetic susceptibility for a system whose low energy physics is dominated by short-range valence bonds (singlets). Our general perturbative approach is applied to specific models expected to be in this class, including the Shastry-Sutherland model of the spin-dimer compound SrCu_2(BO_3)_2 and the antiferromagnetic Heisenberg model of the recently discovered S=1/2 kagome compound ZnCu_3(OH)_6Cl_2. The central result is that a short-ranged valence bond phase, when perturbed with Dzyaloshinskii-Moriya interactions, will remain time-reversal symmetric in the absence of a magnetic field but the susceptibility will be nonzero in the T\to 0 limit. Applied to ZnCu_3(OH)_6Cl_2, this model provides an avenue for reconciling experimental results, such as the lack of magnetic order and lack of any sign of a spin gap, with known theoretical facts about the kagome Heisenberg antiferromagnet.

Ground state phases of the spin-1/2J1–J2Heisenberg antiferromagnet on the square lattice: A high-order coupled cluster treatment

Using the coupled cluster method for high orders of approximation and complementary exact diagonalization studies we investigate the ground state properties of the spin-1/2 ${J}_{1}--{J}_{2}$ frustrated Heisenberg antiferromagnet on the square lattice. We have calculated the ground state energy, the magnetic order parameter, the spin stiffness, and several generalized susceptibilities to probe magnetically disordered quantum valence-bond phases. We have found that the quantum critical points for both the N\'eel and collinear orders are ${J}_{2}^{c1}\ensuremath{\approx}(0.44\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$ and ${J}_{2}^{c2}\ensuremath{\approx}(0.59\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$, respectively, which are in good agreement with the results obtained by other approximations. In contrast to the recent study by [Sirker et al. Phys. Rev. B 73, 184420 (2006)], our data do not provide evidence for the transition from the N\'eel to the valence-bond solid state to be first order. Moreover, our results are in favor of the deconfinement scenario for that phase transition. We also discuss the nature of the magnetically disordered quantum phase.

Unidirectionald-wave superconducting domains in the two-dimensionalt−Jmodel

Motivated by the recently observed pattern of unidirectional domains in high-${T}_{c}$ superconductors [Y. Kohsaka et al., Science 315, 1380 (2007)], we investigate the emergence of spontaneous modulations in the $d$-wave superconducting resonating valence bond phase using the $t$-$J$ model at $x=1∕8$ doping. Half-filled charge domains separated by four lattice spacings are found to form along one of the crystal axes, leading to modulated superconductivity with out-of-phase $d$-wave order parameters in neighboring domains. Both renormalized mean-field theory and variational Monte Carlo calculations reveal that the energies of modulated and uniform phases are very close to each other.

Resonating singlet valence plaquettes

We consider the simplest generalizations of the valence bond physics of SU(2) singlets to $\mathrm{SU}(N)$ singlets that comprise objects with $N$ sites---these are $\mathrm{SU}(N)$ singlet plaquettes with $N=3$ and $N=4$ in three spatial dimensions. Specifically, we search for a quantum mechanical liquid of such objects---a resonating singlet valence plaquette phase that generalizes the celebrated resonating valence bond phase for SU(2) spins. We extend the Rokhsar-Kivelson construction [Phys. Rev. Lett. 61, 2376 (1988)] of the quantum dimer model to the simplest SU(4) model for valence plaquette dynamics on a cubic lattice. The phase diagram of the resulting quantum plaquette model is analyzed both analytically and numerically. We find that the ground state is solid everywhere, including at the Rokhsar-Kivelson point where the ground state is an equal amplitude sum. By contrast, the equal amplitude sum of SU(3) singlet triangular plaquettes on the face-centered cubic lattice is liquid and is thus a candidate for describing a resonating single valence plaquette phase, given a suitably defined local Hamiltonian.