Plaquette Order Research Articles

Rise and fall of plaquette order in the Shastry-Sutherland magnet revealed by pseudofermion functional renormalization group

The Shastry-Sutherland (SS) model as a canonical example of frustrated magnetism has been extensively studied. The conventional wisdom has been that the transition from the plaquette valence bond order to the Neel order is direct and potentially realizes a deconfined quantum critical point beyond the Ginzburg-Landau paradigm. This scenario, however, was challenged recently by improved numerics from density matrix renormalization group which offers evidence for a narrow gapless spin liquid between the two phases. Prompted by this controversy and to shed light on this intricate parameter regime from a fresh perspective, we report high-resolution functional renormalization group analysis of the generalized SS model. The flows of over 50 million running couplings provide a detailed picture for the evolution of spin correlations as the frequency/energy scale is dialed from the ultraviolet to the infrared to yield the zero-temperature phase diagram. The singlet dimer phase emerges as a fixed point, the Neel order is characterized by divergence in the vertex function, while the transition into and out of the plaquette order is accompanied by pronounced peaks in the plaquette susceptibility. The plaquette order is suppressed before the onset of the Neel order, lending evidence for a finite spin liquid region for ${J}_{1}/{J}_{2}\ensuremath{\in}(0.77,0.82)$, where the flow is continuous without any indication of divergence.

Higher-order topological quantum paramagnets

Quantum paramagnets are strongly-correlated phases of matter where competing interactions frustrate magnetic order down to zero temperature. In certain cases, quantum fluctuations induce instead topological order, supporting, in particular, fractionalized quasi-particle excitations. In this work, we investigate paradigmatic spin models and show how magnetic frustration can also give rise to higher-order topological properties. We first study the frustrated Heisenberg model in a square lattice, where a plaquette valence bond solid appears through the spontaneous breaking of translational invariance. Despite the amount of effort that has been devoted to study this phase, its topological nature has so far been overlooked. By means of tensor network simulations, we establish how such state belongs to a higher-order symmetry-protected topological phase, where long-range plaquette order and non-trivial topology coexist. This interplay allows the system to support excitations that would be absent otherwise, such as corner-like states in the bulk attached to dynamical topological defects. Finally, we demonstrate how this higher-order topological quantum paramagnet can also be induced by dipolar interactions, indicating the possibility to directly observe this phase using atomic quantum simulators.

Intrinsic transverse field in frustrated quantum Ising magnets: Physical origin and quantum effects

Transverse field Ising model is a common model in quantum magnetism and is often illustrated as an example for quantum phase transition. Its physical origin in quantum magnets, however, is actually not quite well-understood. The quantum mechanical properties of this model on frustrated systems are not well-understood either. We here clarify the physical origin, both extrinsic one and intrinsic one, for the transverse field of the quantum Ising model, and then explain the quantum effects in the Kagome system. We discuss the quantum plaquette order and the quantum phase transition out of this ordered state in the rare-earth Kagome magnets. Our specific results can find their relevance in the rare-earth tripod Kagome magnets.

SU(3) fermions on the honeycomb lattice at 13 filling

SU(N) symmetric fermions on a lattice, which can be realized in ultracold-atom-based quantum simulators, have very promising prospects for realizing exotic states of matter. Here we present the ground state phase diagram of the repulsive SU(3) Hubbard model on a honeycomb lattice at 1/3 filling obtained from infinite projected entangled pair states tensor network calculations. In the strongly interacting limit the ground state has plaquette order. Upon decreasing the interaction strength $U/t$ we find a first order transition at $U/t=7.2(2)$ into a dimerized, color-ordered state, which extends down to $U/t=4.5(5)$ at which the Mott transition occurs and the ground state becomes uniform. Our results may serve as a prediction and benchmark for future quantum simulators of SU(3) fermions.

Hidden Plaquette Order in a Classical Spin Liquid Stabilized by Strong Off-Diagonal Exchange.

We report a new classical spin liquid in which the collective flux degrees of freedom break the translation symmetry of the honeycomb lattice. This exotic phase exists in the frustrated spin-orbit magnets where a dominant off-diagonal exchange, the so-called Γ term, results in a macroscopic ground-state degeneracy at the classical level. We demonstrate that the system undergoes a phase transition driven by thermal order by disorder at a critical temperature T_{c}≈0.04|Γ|. This transition reduces the emergent spherical spin symmetry to a cubic one: spins point predominantly toward the cubic axes, yet seem to remain disordered at T<T_{c}. Importantly, we show that the phase transition corresponds to a hidden plaquette ordering of hexagonal fluxes, which explicitly breaks the cubic symmetry, a scenario that is confirmed by our extensive MonteCarlo simulations. We further compute the dynamical structure factors of the spin-liquid phase and reveal unusual dynamical properties of the hexagonal flux parameters.

Gapless spin liquid ground state of the spin- 12 J1−J2 Heisenberg model on square lattices

The spin-1/2 $J_1$-$J_2$ Heisenberg model on square lattices are investigated via the finite projected entangled pair states (PEPS) method. Using the recently developed gradient optimization method combining with Monte Carlo sampling techniques, we are able to obtain the ground states energies that are competitive to the best results. The calculations show that there is no N\'eel order, dimer order and plaquette order in the region of 0.42 $\lesssim J_2/J_1\lesssim$ 0.6, suggesting a single spin liquid phase in the intermediate region. Furthermore, the calculated staggered spin, dimer and plaquette correlation functions all have power law decay behaviours, which provide strong evidences that the intermediate nonmagnetic phase is a single gapless spin liquid state.

Dynamics and Instabilities of the Shastry-Sutherland Model.

We study the excitation spectrum in the dimer phase of the Shastry-Sutherland model by using an unbiased variational method that works in the thermodynamic limit. The method outputs dynamical correlation functions in all possible channels. This output is exploited to identify the order parameters with the highest susceptibility (single or multitriplon condensation in a specific channel) upon approaching a quantum phase transition in the magnetic field versus the J^{'}/J phase diagram. We find four different instabilities: antiferro spin nematic, plaquette spin nematic, stripe magnetic order, and plaquette order, two of which have been reported in previous studies.

Domain imaging across the magneto-structural phase transitions in Fe1+yTe

The investigation of the magnetic phase transitions in the parent compounds of Fe-based superconductors is regarded essential for an understanding of the pairing mechanism in the related superconducting compounds. Even though the chemical and electronic properties of these materials are often strongly inhomogeneous on a nanometer length scale, studies of the magnetic phase transitions using spatially resolved experimental techniques are still scarce. Here, we present a real space spin-resolved scanning tunneling microscopy investigation of the surface of Fe1+yTe single crystals with different excess Fe content, y, which are continuously driven through the magnetic phase transition. For Fe1.08Te, the transition into the low-temperature monoclinic phase is accompanied by the appearance of a chevron-patterned structural ordering due to the four 90° rotational domains of the monoclinic lattice. Each of the structural domains contains locally commensurate nanoscale diagonal double stripe antiferromagnetic spin order domains with π-phase slips accross domain boundaries. In the low-temperature phase of Fe1.12Te, on the other hand, the chevron pattern gets rather narrow and less well-defined, and an additional 90° rotated component of the spin-order with local plaquette order emerges. The simultaneous imaging of spin and structural order we show here gives valuable insights into the nature of the magneto-structural domains of Fe1+yTe near the tricritical point, which presumably add to the understanding of the mechanism of superconductivity in the related Fe1+yTexSe1−x material.

Plaquette order in the SU(6) Heisenberg model on the honeycomb lattice

We revisit the SU(6) Heisenberg model on the honeycomb lattice, which has been predicted to be a chiral spin liquid by mean-field theory [G. Szirmai et al., Phys. Rev. A 84, 011611 (2011)]. Using exact diagonalizations of finite clusters, infinite projected entangled pair states simulations, and variational Monte Carlo simulations based on Gutzwiller projected wave functions, we provide strong evidence in favour of the competing plaquette state, which was reported to be higher but close by in energy according to mean-field theory. This is further confirmed by the investigation of the model with a ring exchange term, which shows that there is a transition between the plaquette state and the chiral state at a finite value of the ring exchange term.

Generalized plaquette state in the anisotropic Shastry-Sutherland model

We use a generalized Schwinger boson approach to investigate the nature of the intermediate phase (separating the antiferromagnetic and dimer phases) in the Shastry-Sutherland model with exchange anisotropy. Our results confirm the plaquette nature of the intermediate phase and demonstrate that exchange anisotropy (both Ising-like and $XY$-like) weakens the plaquette order and ultimately suppresses it altogether for sufficiently strong anisotropy. Interestingly, we find that the low-energy excitation spectrum of the plaquette state changes dramatically near the boundary with the dimer phase---the location of the minimum in the (gapped) lowest excitation mode changes from $\mathbf{k}=(0,0)$ to four degenerate points.

Interfaces, strings, and a soft mode in the square lattice quantum dimer model

The quantum dimer model on the square lattice is equivalent to a $U(1)$ gauge theory. Quantum Monte Carlo calculations reveal that, for values of the Rokhsar-Kivelson (RK) coupling $\lambda < 1$, the theory exists in a confining columnar phase. The interfaces separating distinct columnar phases display plaquette order, which, however, is not realized as a bulk phase. Static "electric" charges are confined by flux tubes that consist of multiple strands, each carrying a fractionalized flux $\frac{1}{4}$. A soft pseudo-Goldstone mode emerges around $\lambda \approx 0$, long before one reaches the RK point at $\lambda = 1$.

Plaquette order in a dimerized frustrated spin ladder

We study the effect of spin-Peierls instability on the phase-diagram of a frustrated antiferromagnetic spin-1/2 ladder, with weak transverse and diagonal rung coupling. Our analysis focuses on a one-dimensional version of the model (i.e. a single two-leg ladder) where we consider two forms of spin-Peierls (SP) instabilities on the legs: columnar dimers (CD) and staggered dimers (SD). We particularly examine the regime of parameters (corresponding to an intermediate XXZ anisotropy) where the SP and rung coupling terms are equally relevant. In both the CD and SD cases we find that the effective field theory describing the system is a self-dual sine-Gordon model, which favors ordering and the opening of a gap to excitations. The order parameter, which reflects the interplay between the SP and rung interactions, represents a crystal of 4-spin plaquettes on which longitudinal and transverse dimers are in a superposition. Depending on the SP instability mode these plaquettes are closed or open, however both types spontaneously break reflection symmetry across the ladder. The closed plaquettes are stable, while the open plaquette-order is relatively fragile and the corresponding gap may be tuned to zero under extreme conditions. We further find that a first order transition occurs from the Plaquette order to a valence bond crystal (VBC) of dimers on the legs. It is suggestive that in a higher dimensional version of this system, this variety of distinct VBC states with comparable energies leads to the formation of domains. Effectively one-dimensional gapless spinon modes on domain boundaries can possibly account for the experimental observation of a spin-liquid behavior in a physical realization of the model.

Semiclassical evidence of columnar order in the fully frustrated transverse-field Ising model on the square lattice

We investigate the zero-temperature phase diagram of the fully frustrated transverse-field Ising model on the square lattice both in the classical limit and in the presence of quantum fluctuations. At the classical level (the limit of infinite spin $S$), we find that upon decreasing the transverse field $\ensuremath{\Gamma}$ this model exhibits a phase transition from the fully polarized state into an eightfold degenerate translational symmetry breaking state. This phase can be identified to correspond to plaquette order in the dimer language and remains the lowest-energy state in the entire range of fields below the critical one, ${\ensuremath{\Gamma}}_{c}$. The eightfold degenerate solution that corresponds to columnar order in the dimer language is a saddle point of the classical energy. It is degenerate with the plaquette solution at $\ensuremath{\Gamma}=0$ and is only slightly higher in energy in the whole interval $0&lt;\ensuremath{\Gamma}&lt;{\ensuremath{\Gamma}}_{c}$. The effect of quantum fluctuations is investigated in the context of a large S expansion both for the plaquette and columnar structures. For this purpose we employ an approximate method allowing to estimate from above the fluctuation-induced correction to the energy of a configuration which at the classical level is a saddle point of the energy, not a local minimum, and find that the harmonic quantum fluctuations show a clear tendency to overcome the energy difference between the two states. For relatively high fields, the transition from the plaquette to the columnar state takes place at values of $S$ large enough to be in the domain of validity of the harmonic approximation.

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.

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.

Evidence for Columnar Order in the Fully Frustrated Transverse Field Ising Model on the Square Lattice

Using extensive classical and quantum MonteCarlo simulations, we investigate the ground-state phase diagram of the fully frustrated transverse field Ising model on the square lattice. We show that pure columnar order develops in the low-field phase above a surprisingly large length scale, below which an effective U(1) symmetry is present. The same conclusion applies to the quantum dimer model with purely kinetic energy, to which the model reduces in the zero-field limit, as well as to the stacked classical version of the model. By contrast, the 2D classical version of the model is shown to develop plaquette order. Semiclassical arguments show that the transition from plaquette to columnar order is a consequence of quantum fluctuations.

Plaquette order and deconfined quantum critical point in the spin-1 bilinear-biquadratic Heisenberg model on the honeycomb lattice

We have precisely determined the ground state phase diagram of the quantum spin-1 bilinear-biquadratic Heisenberg model on the honeycomb lattice using the tensor renormalization group method. We find that the ferromagnetic, ferroquadrupolar, and a large part of the antiferromagnetic phases are stable against quantum fluctuations. However, around the phase where the ground state is antiferroquadrupolar ordered in the classical limit, quantum fluctuations suppress completely all magnetic orders, leading to a plaquette order phase which breaks the lattice symmetry but preserves the spin SU(2) symmetry. On the evidence of our numerical results, the quantum phase transition between the antiferromagnetic phase and the plaquette phase is found to be either a direct second order or a very weak first order transition.

Spin-12J1-J2Heisenberg antiferromagnet on a square lattice: A plaquette renormalized tensor network study

We apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg ${J}_{1}$-${J}_{2}$ model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with $L$ $=$ 8, 16, and 32. By treating tensor elements as variational parameters, we obtain the ground states for different ${J}_{2}/{J}_{1}$ values, and investigate staggered magnetizations, nearest-neighbor spin-spin correlations, and plaquette order parameters. In addition to the well-known N\'eel order and collinear order at low and high ${J}_{2}/{J}_{1}$, we observe a plaquettelike order at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.5$. A continuous transition between the N\'eel order and the plaquettelike order near ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.40{J}_{1}$ is observed. The collinear order emerges at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.62{J}_{1}$ through a first-order phase transition.

Metal-Insulator Transition of Fermions on a Kagome Lattice at 1/3 Filling

We discuss the metal-insulator transition of the spinless fermion model on a kagome lattice at 1/3 filling. The system is analyzed by using exact diagonalization, density-matrix renormalization group methods, and random-phase approximation. In the strong-coupling region, the charge-ordered ground state is consistent with the predictions of an effective model, i.e., plaquette order. We find that the qualitative properties of the metal-insulator transition are totally different depending on the sign of the hopping matrix elements, reflecting the difference in the band structure near the Fermi level.

Plaquette order in theJ1−J2−J3model: Series expansion analysis

Series expansion based on the flow equation method is employed to study the zero temperature properties of the spin-1/2 J1-J2-J3 antiferromagnet in two dimensions. Starting from the exact limit of decoupled plaquettes in a particular generalized J1-J2-J3 model we analyze the evolution of the ground state energy and the elementary triplet excitations in powers of all three inter-plaquette couplings up to fifth order. We find the plaquette phase to remain stable over a wide range of exchange couplings and to connect adiabatically up to the case of the plain J1-J2-J3 model, however not to the J1-J2 model at J3=0. Besides confirming the existence of such a phase, recently predicted by Mambrini, et al. [Phys. Rev. B 74, 144422 [2006)], we estimate its extent by Dlog-Pade analysis of the critical lines that result from closure of the triplet gap.