Valence Bond Crystal Research Articles

Renormalization group analysis of competing quantum phases in theJ1-J2Heisenberg model on the kagome lattice

Recent discoveries in neutron scattering experiments for Kapellasite and Herbertsmithite as well as theoretical calculations of possible spin liquid phases have revived interest in magnetic phenomena on the kagome lattice. We study the quantum phase diagram of the $S=1/2$ Heisenberg kagome model as a function of nearest neighbor coupling ${J}_{1}$ and second neighbor coupling ${J}_{2}$. Employing the pseudofermion functional renormalization group (PFFRG), we find four types of magnetic quantum order ($\mathbf{q}=0$ order, cuboc order,ferromagnetic order, and $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order) as well as extended magnetically disordered regions by which we specify the possible parameter regime for Kapellasite. In the disordered regime $\frac{{J}_{2}}{{J}_{1}}\ensuremath{\ll}1$, the flatness of the magnetic susceptibility at the zone boundary which is observed for Herbertsmithite can be reconciled with the presence of small ${J}_{2}>0$ coupling. In particular, we analyze the dimer susceptibilities related to different valence-bond crystal (VBC) patterns, which are strongly inhomogeneous, indicating the rejection of VBC order in the RG flow.

Quantum phases in the frustrated Heisenberg model on the bilayer honeycomb lattice

We use a combination of analytical and numerical techniques to study the phase diagram of the frustrated Heisenberg model on the bilayer honeycomb lattice. Using the Schwinger boson description of the spin operators followed by a mean field decoupling, the magnetic phase diagram is studied as a function of the frustration coupling $J_{2}$ and the interlayer coupling $J_{\bot}$. The presence of both magnetically ordered and disordered phases is investigated by means of the evaluation of ground-state energy, spin gap, local magnetization and spin-spin correlations. We observe a phase with a spin gap and short range N\'eel correlations that survives for non-zero next-nearest-neighbor interaction and interlayer coupling. Furthermore, we detect signatures of a reentrant behavior in the melting of N\'eel phase and symmetry restoring when the system undergoes a transition from an on-layer nematic valence bond crystal phase to an interlayer valence bond crystal phase. We complement our work with exact diagonalization on small clusters and dimer-series expansion calculations, together with a linear spin wave approach to study the phase diagram as a function of the spin $S$, the frustration and the interlayer couplings.

Geometry-induced pair condensation

We study a one-dimensional model of interacting bosons on a lattice with two flat bands. Regular condensation is suppressed due to the absence of a well defined minimum in the single particle spectrum. We find that interactions stabilize a number of non-trivial phases like a pair (quasi-) condensate, a supersolid at incommensurable fillings and valence bond crystals at commensurability. We support our analytical calculations with numerical simulations using the density matrix renormalization group technique. Implications for cold-atoms and extensions to higher dimensions are discussed.

Frustrated magnetism and resonating valence bond physics in two-dimensional kagome-like magnets

We explore the phase diagram and the low-energy physics of three Heisenberg antiferromagnets which, like the kagome lattice, are networks of corner-sharing triangles but contain two sets of inequivalent short-distance resonance loops. We use a combination of exact diagonalization, analytical strong-coupling theories, and resonating valence bond approaches, and scan through the ratio of the two inequivalent exchange couplings. In one limit, the lattices effectively become bipartite, while at the opposite limit heavily frustrated nets emerge. In between, competing tunneling processes result in short-ranged spin correlations, a manifold of low-lying singlets (which can be understood as localized bound states of magnetic excitations), and the stabilization of valence bond crystals with resonating building blocks.

Unveiling the Physics of the Doped Phase of thet−JModel on the Kagome Lattice

We investigate the ground state properties of the kagome lattice t - J model at low doping by variational Monte Carlo calculations. The resulting state possesses an interesting balance of spin exchange and kinetic exchange through the building blocks of stars which are linked by triangles and their internal hexagons. While the spin exchange is taking place mainly on the stars, hopping is favored on the hexagons. There is a density modulation, resulting in the holes having an effective static contribution. From this observation, how holes lead to dimerization in this model and why a particular valence bond crystal pattern is formed can be understood. Furthermore, we argue the optimal doping for this state. We discuss our result in connection with static impurities, and show the likely relevance to the diluted kagome lattice Heisenberg model, describing actual compounds.

Dimerized Solids and Resonating Plaquette Order inSU(N)-Dirac Fermions

We study the quantum phases of fermions with an explicit SU(N)-symmetric, Heisenberg-like nearest-neighbor flavor exchange interaction on the honeycomb lattice at half filling. Employing projective (zero temperature) quantum MonteCarlo simulations for even values of N, we explore the evolution from a weak-coupling semimetal into the strong-coupling, insulating regime. Furthermore, we compare our numerical results to a saddle-point approximation in the large-N limit. From the large-N regime down to the SU(6) case, the insulating state is found to be a columnar valence bond crystal, with a direct transition to the semimetal at weak, finite coupling, in agreement with the mean-field result in the large-N limit. At SU(4) however, the insulator exhibits a subtly different valence bond crystal structure, stabilized by resonating valence bond plaquettes. In the SU(2) limit, our results support a direct transition between the semimetal and an antiferromagnetic insulator.

Competing states in the SU(3) Heisenberg model on the honeycomb lattice: Plaquette valence-bond crystal versus dimerized color-ordered state

Conflicting predictions have been made for the ground state of the SU(3) Heisenberg model on the honeycomb lattice: Tensor network simulations found a plaquette order [Zhao et al, Phys. Rev. B 85, 134416 (2012)], where singlets are formed on hexagons, while linear flavor-wave theory (LFWT) suggested a dimerized, color ordered state [Lee and Yang, Phys. Rev. B 85, 100402 (2012)]. In this work we show that the former state is the true ground state by a systematic study with infinite projected-entangled pair states (iPEPS), for which the accuracy can be systematically controlled by the so-called bond dimension $D$. Both competing states can be reproduced with iPEPS by using different unit cell sizes. For small $D$ the dimer state has a lower variational energy than the plaquette state, however, for large $D$ it is the latter which becomes energetically favorable. The plaquette formation is also confirmed by exact diagonalizations and variational Monte Carlo studies, according to which both the dimerized and plaquette states are non-chiral flux states.

Weak plaquette valence bond order in the S = 1/2 honeycomb J1 - J2 Heisenberg model.

Using the density matrix renormalization group, we investigate the S = 1/2 Heisenberg model on the honeycomb lattice with first (J(1)) and second (J(2)) neighbor interactions. We are able to study long open cylinders with widths up to 12 lattice spacings. For J(2)/J(1) near 0.3, we find an apparently paramagnetic phase, bordered by an antiferromagnetic phase for J(2) ≲ 0.26 and by a valence bond crystal for J(2) ≳ 0.36. The longest correlation length that we find in this intermediate phase is for plaquette valence bond order. This correlation length grows strongly with cylinder circumference, indicating either quantum criticality or weak plaquette valence bond order.

Valence-bond crystals in the kagomé spin-1/2 Heisenberg antiferromagnet: a symmetry classification and projected wave function study

In this paper, we do a complete classification of valence-bond crystals (VBCs) on the kagomé lattice based on general arguments of symmetry only and thus identify many new VBCs for different unit cell sizes. For the spin-1/2 Heisenberg antiferromagnet, we study the relative energetics of competing gapless spin liquids (SLs) and VBC phases within the class of Gutzwiller-projected fermionic wave functions using variational Monte Carlo techniques, hence implementing exactly the constraint of one fermion per site. By using a state-of-the-art optimization method, we conclusively show that the U(1) Dirac SL is remarkably stable towards dimerizing into all 6-, 12- and 36-site unit cell VBCs. This stability is also preserved on addition of a next-nearest-neighbor super-exchange coupling of both antiferromagnetic and ferromagnetic (FM) type. However, we find that a 36-site unit cell VBC is stabilized on addition of a very small next-nearest-neighbor FM super-exchange coupling, i.e. |J2| ≈ 0.045, and this VBC is the same in terms of space-group symmetry as that obtained in an effective quantum dimer model study. It breaks reflection symmetry, has a nontrivial flux pattern and is a strong dimerization of the uniform RVB SL.

Absence of valence-bond supersolidity on the checkerboard lattice

The stability of the supersolid state of bosons reported recently on the checkerboard lattice in the vicinity of a valence-bond crystal phase is assessed. In contrast to previously identified supersolid phases of lattice bosons, this state would---if stable---provide superfluidity atop a solid with inherent quantum dynamics, restricted to the particular filling fraction of one quarter. Using large-scale quantum Monte Carlo simulations based on the stochastic series expansion method constrained to the canonical ensemble, the supersolid state is, however, found to be unstable in the thermodynamic limit. Converging crossing points in both the winding number fluctuations and the Binder ratio related to the order parameter of the valence-bond crystal instead exhibit a direct quantum melting transition of the crystal into the superfluid regime.

Valence bond crystal and possible orbital pinball liquid in at2gorbital model

We study a model for orbitally degenerate Mott insulators, where localized electrons possess t_2g degrees of freedom coupled by several, competing, exchange mechanisms. We provide evidence for two distinct strongly fluctuating regimes, depending on whether superexchange or direct exchange mechanism predominates. In the superexchange-dominated regime, the ground state is dimerized, with nearest neighbor orbital singlets covering the lattice. By deriving an effective quantum dimer model and analyzing it numerically, we characterize this dimerized phase as a valence bond crystal stabilized by singlet resonances within a large unit cell. In the opposite regime, with predominant direct exchange, the combined analysis of the original model and another effective model adapted to the local constraints, shows that subleading perturbations select a highly resonating ground state, with coexisting diagonal and off-diagonal long-range orbital orders.

Frustrated Heisenberg antiferromagnet on the checkerboard lattice:J1-J2model

We study the ground-state (gs) phases of the spin-half anisotropic planar pyrochlore (or crossed chain) model using the coupled cluster method (CCM). The model is a frustrated antiferromagnetic (AFM) $J_{1}$--$J_{2}$ system on the checkerboard lattice, with nearest-neighbor exchange bonds $J_{1}>0$ and next-nearest-neighbor bonds $J_{2} \equiv \kappa J_{1} > 0$. Using various AFM classical ground states as CCM model states we present results for their gs energy, average on-site magnetization, and susceptibilities to plaquette valence-bond crystal (PVBC) and crossed-dimer valence-bond crystal (CDVBC) ordering. We show that the state with Neel ordering is the gs phase for $\kappa < \kappa_{c_1} \approx 0.80 \pm 0.01$, but that none of the fourfold set of AFM states selected by quantum fluctuations at $O(1/s)$ in a large-$s$ analysis (where $s$ is the spin quantum number) from the infinitely degenerate set of AFM states that form the gs phase for the classical version of the model (for $\kappa>1$) survives the quantum fluctuations to form a stable magnetically-ordered gs phase for the spin-half case. The Neel state becomes susceptible to PVBC ordering at or very near to $\kappa = \kappa_{c_1}$, and the fourfold AFM states become infinitely susceptible to PVBC ordering at $\kappa = \kappa_{c_2} \approx 1.22 \pm 0.02$. In turn, we find that these states become infinitely susceptible to CDVBC ordering for all values of $\kappa$ above a certain critical value at or very near to $\kappa = \kappa_{c_2}$. We thus find a Neel-ordered gs phase for $\kappa<\kappa_{c_1}$, a PVBC-ordered phase for $\kappa_{c_1} < \kappa < \kappa_{c_2}$, and a CDVBC-ordered phase for $\kappa > \kappa_{c_2}$. Both transitions are probably direct ones, although we cannot exclude very narrow coexistence regions confined to $0.79 \lesssim \kappa \lesssim 0.81$ and $1.20 \lesssim \kappa \lesssim 1.22$ respectively.

The frustrated Heisenberg antiferromagnet on the honeycomb lattice: J1–J2 model

We study the ground-state phase diagram of the frustrated spin- antiferromagnet with J2 = xJ1 > 0 (J1 > 0) on the honeycomb lattice, using the coupled-cluster method. We present results for the ground-state energy, magnetic order parameter and plaquette valence-bond crystal (PVBC) susceptibility. We find a paramagnetic PVBC phase for xc1 < x < xc2, where xc1 ≈ 0.207 ± 0.003 and xc2 ≈ 0.385 ± 0.010. The transition at xc1 to the Néel phase seems to be a continuous deconfined transition (although we cannot exclude a very narrow intermediate phase in the range 0.21 ≲ x ≲ 0.24), while that at xc2 is of first-order type to another quasiclassical antiferromagnetic phase that occurs in the classical version of the model only at the isolated and highly degenerate critical point . The spiral phases that are present classically for all values are absent for all x ≲ 1.

Ground-state phases of the frustrated spin-12J1–J2–J3Heisenberg ferromagnet (J1&lt;0) on the honeycomb lattice withJ3=J2&gt;0

We study the ground-state (gs) properties of the frustrated spin-1/2 $J_{1}$--$J_{2}$--$J_{3}$ Heisenberg model on a honeycomb lattice with ferromagnetic (FM) nearest-neighbor ($J_{1}=-1$) exchange and frustrating antiferromagnetic (AFM) next-nearest-neighbor ($J_{2}>0$) and next-next-nearest-neighbor ($J_{3}>0$) exchanges, for the case $J_{3}=J_{2}$. We use the coupled cluster method in high orders of approximation, complemented by the exact diagonalization of a lattice with 32 sites, and calculate the gs energy, magnetic order parameter, and spin-spin correlation functions. We find a quantum phase transition between regions characterized by FM order and a form of AFM ("striped") collinear order at $J^{c}_{2} \approx 0.1095 \pm 0.0005$. We compare results for the FM case (with $J_{1}=-1$) to previous results for the corresponding AFM case (with $J_{1}=+1$). While the magnetic order parameters behave similarly for the FM and the AFM models for large values of the frustration parameter $J_{2}$, there are considerable differences between them for $J_{2}/|J_{1}| \lesssim 0.6$. For example, the quasiclassical collinear magnetic long-range order for the AFM model (with $J_{1}=+1$) breaks down at $J^{c_{2}}_{2} \approx 0.60$, whereas the "equivalent" point for the FM model (with $J_{1}=-1$) occurs at $J^{c}_{2} \approx 0.11$. Unlike in the AFM model (with $J_{1}=+1$), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical AFM phases (with N\'eel and striped order) for $J^{c_{1}}_{2} < J_{2} < J^{c_{2}}_{2}$, with $J^{c_{1}}_{2} \approx 0.47$, we find no clear indications in the FM model for an intermediate magnetically disordered phase between the phases exhibiting FM and striped order. Instead, the evidence points strongly to a direct first-order transition between the two ordered phases of the FM model.

Competing valence bond crystals in the kagome quantum dimer model

The singlet dynamics which plays a major role in the physics of the spin-1/2 Quantum Heisenberg Antiferromagnet (QHAF) on the Kagome lattice can be approximately described by projecting onto the nearest-neighbor valence bond (NNVB) singlet subspace. We re-visit here the effective Quantum Dimer Model which originates from the latter NNVB-projected Heisenberg model via a non-perturbative Rokhsar-Kivelson-like scheme. By using Lanczos exact diagonalisation on a 108-site cluster supplemented by a careful symmetry analysis, it is shown that a previously-found 36-site Valence Bond Crystal (VBC) in fact competes with a new type of 12-site "{\it resonating-columnar}" VBC. The exceptionally large degeneracy of the GS multiplets (144 on our 108-site cluster) might reflect the proximity of the Z_2 dimer liquid. Interestingly, these two VBC "emerge" in {\it different topological sectors}. Implications for the interpretation of numerical results on the QHAF are outlined.

Frustrated Heisenberg antiferromagnet on the honeycomb lattice: A candidate for deconfined quantum criticality

We study the ground-state (gs) phase diagram of the frustrated spin-1/2 $J_{1}$-$J_{2}$-$J_{3}$ antiferromagnet with $J_{2} = J_{3} =\kappa J_1$ on the honeycomb lattice, using coupled-cluster theory and exact diagonalization methods. We present results for the gs energy, magnetic order parameter, spin-spin correlation function, and plaquette valence-bond crystal (PVBC) susceptibility. We find a N\'eel antiferromagnetic (AFM) phase for $\kappa < \kappa_{c_{1}} \approx 0.47$, a collinear striped AFM phase for $\kappa > \kappa_{c_{2}} \approx 0.60$, and a paramagnetic PVBC phase for $\kappa_{c_{1}} \lesssim \kappa \lesssim \kappa_{c_{2}}$. The transition at $\kappa_{c_{2}}$ appears to be of first-order type, while that at $\kappa_{c_{1}}$ is continuous. Since the N\'eel and PVBC phases break different symmetries our results favor the deconfinement scenario for the transition at $\kappa_{c_{1}}$.

Phase diagram of a frustrated quantum antiferromagnet on the honeycomb lattice: Magnetic order versus valence-bond crystal formation

We present a comprehensive computational study of the phase diagram of the frustrated S=1/2 Heisenberg antiferromagnet on the honeycomb lattice, with second-nearest (J2) and third-neighbor (J3) couplings. Using a combination of exact diagonalizations of the original spin model, of the Hamiltonian projected into the nearest neighbor short range valence bond basis, and of an effective quantum dimer model, as well as a self-consistent cluster mean-field theory, we determine the boundaries of several magnetically ordered phases in the region J2,J3\in [0,1], and find a sizable magnetically disordered region in between. We characterize part of this magnetically disordered phase as a plaquette valence bond crystal phase. At larger J2, we locate a sizable region in which staggered valence bond crystal correlations are found to be important, either due to genuine valence bond crystal ordering or as a consequence of magnetically ordered phases which break lattice rotational symmetry. Furthermore we find that a particular parameter-free Gutzwiller projected tight-binding wave function has remarkably accurate energies compared to finite-size extrapolated ED energies along the transition line from conventional N\'eel to plaquette VBC phases, a fact that points to possibly interesting critical behavior - such as a deconfined critical point - across this transition. We also comment on the relevance of this spin model to model the spin liquid region found in the half-filled Hubbard model on the honeycomb lattice.

Quantum disordered ground state for hard-core bosons on the frustrated square lattice

We investigate the phase diagram of hard-core bosons on a square lattice with competing interactions. The hard-core bosons can be represented also by spin-1/2 operators and the model can therefore be mapped onto an anisotropic $J_1$-$J_2$-Heisenberg model. We find the N\'eel state and a collinear antiferromagnetic state as classical ordered phases to be suppressed for small ferromagnetic exchange terms $J_{1,2}^{x,y}$ and a ferromagnetic phase which orders in the x-y-plane for large $J_{1,2}^{x,y}$. For an intermediate regime the emergence of new quantum states like valence bond crystals or super-solids is predicted for similar models. We do not observe any signal for long-range order in terms of conventional order or dimer correlations in our model and find an exponential decay in the spin correlations. Hence, all evidence is pointing towards a quantum disordered ground state for a small region in the phase diagram.

Valence-bond crystal in the extended kagome spin-12quantum Heisenberg antiferromagnet: A variational Monte Carlo approach

The highly frustrated spin-$\frac{1}{2}$ quantum Heisenberg model with both nearest (${J}_{1}$) and next-nearest (${J}_{2}$) neighbor exchange interactions is revisited by using an extended variational space of projected wave functions that are optimized with state-of-the-art methods. Competition between modulated valence-bond crystals (VBCs) proposed in the literature and the Dirac spin liquid (DSL) is investigated. We find that the addition of a small ferromagnetic next-nearest-neighbor exchange coupling $|{J}_{2}|&gt;0.09{J}_{1}$ leads to stabilization of a 36-site unit cell VBC, although the DSL remains a local minimum of the variational parameter landscape. This implies that the VBC is not trivially connected to the DSL; instead it possesses a nontrivial flux pattern and large dimerization.

Symmetry Breaking on the Three-Dimensional Hyperkagome Lattice ofNa4Ir3O8

We study the antiferromagnetic spin-1/2 Heisenberg model on the highly frustrated, three-dimensional, hyperkagome lattice of Na(4)Ir(3)O(8) using a series expansion method. We propose a valence bond crystal with a 72 site unit cell as a ground state that supports many, very low lying, singlet excitations. Low energy spinons and triplons are confined to emergent lower-dimensional motifs. Here, and for analogous kagome and pyrochlore states, we suggest finite temperature signatures, including an Ising transition, in the magnetic specific heat due to a multistep breaking of discrete symmetries.