Frustrated Lattices Research Articles

Nature of the Spin Liquid Ground State in a Breathing Kagome Compound Studied by NMR and Series Expansion.

In the vanadium oxyfluoride compound (NH_{4})_{2}[C_{7}H_{14}N][V_{7}O_{6}F_{18}] (DQVOF), the V^{4+} (3d^{1}, S=1/2) ions realize a unique, highly frustrated breathing kagome lattice composed of alternately sized, corner-sharing equilateral triangles. Here we present an ^{17}O NMR study of DQVOF, which isolates the local susceptibility of the breathing kagome network. By a fit to series expansion, we extract the ratio of the interactions within the breathing kagome plane, J_{∇}/J_{Δ}=0.55(4), and the mean antiferromagnetic interaction J[over ¯]=60(7) K. Spin lattice (T_{1}) measurements reveal an essentially gapless excitation spectrum with a maximum gap Δ/J[over ¯]=0.007(7). Our study provides new impetus for further theoretical investigations in order to establish whether the gapless spin liquid behavior displayed by DQVOF is intrinsic to its breathing kagome lattice or whether it is due to perturbations to this model, such as a residual coupling of the V^{4+} ions in the breathing kagome planes to the interlayer V^{3+} (S=1) spins.

Entropy of diluted antiferromagnetic Ising models on frustrated lattices using the Wang-Landau method.

We use a Monte Carlo simulation to study the diluted antiferromagnetic Ising model on frustrated lattices including the pyrochlore lattice to show the dilution effects. Using the Wang-Landau algorithm, which directly calculates the energy density of states, we accurately calculate the entropy of the system. We discuss the nonmonotonic dilution concentration dependence of residual entropy for the antiferromagnetic Ising model on the pyrochlore lattice, and compare it to the generalized Pauling approximation proposed by Ke etal. [Phys. Rev. Lett. 99, 137203 (2007)PRLTAO0031-900710.1103/PhysRevLett.99.137203]. We also investigate other frustrated systems, the antiferromagnetic Ising model on the triangular lattice and the kagome lattice, demonstrating the difference in the dilution effects between the system on the pyrochlore lattice and that on other frustrated lattices.

Orbital Dimer Model for the Spin-Glass State in Y_{2}Mo_{2}O_{7}.

The formation of a spin glass generally requires that magnetic exchange interactions are both frustrated and disordered. Consequently, the origin of spin-glass behavior in Y_{2}Mo_{2}O_{7}-in which magnetic Mo^{4+} ions occupy a frustrated pyrochlore lattice with minimal compositional disorder-has been a longstanding question. Here, we use neutron and x-ray pair-distribution function (PDF) analysis to develop a disorder model that resolves apparent incompatibilities between previously reported PDF, extended x-ray-absorption fine structure spectroscopy, and NMR studies, and provides a new and physical explanation of the exchange disorder responsible for spin-glass formation. We show that Mo^{4+} ions displace according to a local "two-in-two-out" rule on each Mo_{4} tetrahedron, driven by orbital dimerization of Jahn-Teller active Mo^{4+} ions. Long-range orbital order is prevented by the macroscopic degeneracy of dimer coverings permitted by the pyrochlore lattice. Cooperative O^{2-} displacements yield a distribution of Mo-O-Mo angles, which in turn introduces disorder into magnetic interactions. Our study demonstrates experimentally how frustration of atomic displacements can assume the role of compositional disorder in driving a spin-glass transition.

Emergent order in the kagome Ising magnet Dy3Mg2Sb3O14.

The Ising model—in which degrees of freedom (spins) are binary valued (up/down)—is a cornerstone of statistical physics that shows rich behaviour when spins occupy a highly frustrated lattice such as kagome. Here we show that the layered Ising magnet Dy3Mg2Sb3O14 hosts an emergent order predicted theoretically for individual kagome layers of in-plane Ising spins. Neutron-scattering and bulk thermomagnetic measurements reveal a phase transition at ∼0.3 K from a disordered spin-ice-like regime to an emergent charge ordered state, in which emergent magnetic charge degrees of freedom exhibit three-dimensional order while spins remain partially disordered. Monte Carlo simulations show that an interplay of inter-layer interactions, spin canting and chemical disorder stabilizes this state. Our results establish Dy3Mg2Sb3O14 as a tuneable system to study interacting emergent charges arising from kagome Ising frustration.

Extension of Lieb-Schupp theorem to Heisenberg models with higher-order interactions

We extend the Lieb-Schupp theorem to the Heisenberg models with higher order interactions on non-frustrated or frustrated finite lattices. These lattices are constructed by even numbered rings with or without crossing bonds and have reflection symmetry. The results show that all ground states have total spin zero in wide interaction parameters region which is not covered with the results of the Marshall-Lieb-Mattis type arguments.

Thermodynamics and magnetization reversal in artificial brickwork spin ice

The thermodynamics and magnetization reversal behavior of three artificial frustrated brickwork systems are investigated by means of the Monte Carlo method. Three frustrated systems have different array patterns of ferromagnetic nanoislands, and consequently different geometry symmetry and magnetic properties. The simulated results show that two brickwork systems which have only ‘three-spins’ vertex exhibit the long-range ordered magnetic ground state, and one brickwork system that contains mixed ‘two-spins’ and ‘four-spins’ vertex as well as ‘three-spins’ vertex has a high degeneracy of ground state and no long-range order. In all three frustrated systems, there occurs the phase transition from the magnetic ground order to disorder. Three frustrated brickwork lattices show significant differences in the reversal mechanism in the presence of magnetic field for different lattice spacings.

Nematic quantum liquid crystals of bosons in frustrated lattices

The problem of interacting bosons in frustrated lattices is an intricate one due to the absence of a unique minimum in the single-particle dispersion where macroscopic number of bosons can condense. Here we consider a family of tight-binding models with macroscopically degenerate lowest energy band, separated from other bands by a gap. We predict the formation of exotic states that spontaneously break rotational symmetry at relatively low filling. These states belong to three nematic phases: Wigner crystal, supersolid, and superfluid. The Wigner crystal phase is established exactly at low filling. Supersolid and superfluid phases, at larger filling, are obtained by making use of a projection onto the flat band, construction of an appropriate Wannier basis, and subsequent mean-field treatment. The nematic superfluid that we predict is uniform in real space but has an anisotropic momentum distribution, providing a novel scenario for Bose condensation with an additional nematic order. Our findings open up a promising direction of studying microscopic quantum liquid crystalline phases of bosons.

Heavy-Fermion Valence-Bond Liquids in Ultracold Atoms: Cooperation of the Kondo Effect and Geometric Frustration.

We analyze a microscopic mechanism behind the coexistence of a heavy Fermi liquid and geometric frustration in Kondo lattices. We consider a geometrically frustrated periodic Anderson model and demonstrate how orbital fluctuations lead to a Kondo-screened phase in the limit of extreme strong frustration when only local singlet states participate in the low-energy physics. We also propose a setup to realize and study this exotic state with SU(3)-symmetric alkaline-earth cold atoms.

ChemInform Abstract: Unusual Ordered Phases of Highly Frustrated Magnets: A Review

AbstractReview: 194 refs.

Chiral spin liquids in arrays of spin chains

We describe a coupled-chain construction for chiral spin liquids in two-dimensional spin systems. Starting from a one-dimensional zigzag spin chain and imposing SU(2) symmetry in the framework of non-Abelian bosonization, we first show that our approach faithfully describes the low-energy physics of an exactly solvable model with a three-spin interaction. Generalizing the construction to the two-dimensional case, we obtain a theory that incorporates the universal properties of the chiral spin liquid predicted by Kalmeyer and Laughlin: charge-neutral edge states, gapped spin-1/2 bulk excitations, and ground state degeneracy on the torus signalling the topological order of this quantum state. In addition, we show that the chiral spin liquid phase is more easily stabilized in frustrated lattices containing corner-sharing triangles, such as the extended kagome lattice, than in the triangular lattice. Our field theoretical approach invites generalizations to more exotic chiral spin liquids and may be used to assess the existence of the chiral spin liquid as the ground state of specific lattice systems.

Chiral spin density wave order on the frustrated honeycomb and bilayer triangle lattice hubbard model at half-filling.

We study the Hubbard model on the frustrated honeycomb lattice with nearest-neighbor hopping t_{1} and second nearest-neighbor hopping t_{2}, which is isomorphic to the bilayer triangle lattice, using the SU(2)-invariant slave boson theory. We show that the Coulomb interaction U induces antiferromagnetic (AF) chiral spin density wave (χSDW) order in a wide range of κ=t_{2}/t_{1} where both the two-sublattice AF order at small κ and the decoupled three-sublattice 120° order at large κ are strongly frustrated, leading to three distinct phases with different anomalous Hall responses. We find a continuous transition from a χSDW semimetal with the anomalous Hall effect to a topological chiral Chern insulator exhibiting the quantum anomalous Hall effect, followed by a discontinuous transition to a χSDW insulator with a zero total Chern number but an anomalous ac Hall effect. The χSDW is likely a generic phase of strongly correlated and highly frustrated hexagonal lattice electrons.

Unusual ordered phases of highly frustrated magnets: a review

We review ground states and excitations of a quantum antiferromagnet on triangular and other frustrated lattices. We pay special attention to the combined effects of magnetic field h, spatial anisotropy R and spin magnitude S. The focus of the review is on the novel collinear spin density wave and spin nematic states, which are characterized by fully gapped transverse spin excitations with Sz = ± 1. We discuss extensively the R − h phase diagram of the antiferromagnet, both in the large-S semiclassical limit and the quantum S = 1/2 limit. When possible, we point out connections with experimental findings.

Polar molecules in frustrated triangular ladders

Polar molecules in geometrically frustrated lattices may result in a very rich landscape of quantum phases, due to the non-trivial interplay between frustration, and two- and possibly three-body inter-site interactions. In this paper, we illustrate this intriguing physics for the case of hard-core polar molecules in frustrated triangular ladders. Whereas commensurate lattice fillings result in gapped phases with bond-order and/or density-wave order, at incommensurate fillings we find chiral-, two-component-, and pair-superfluids. We show as well that, remarkably, polar molecules in frustrated lattices allow, for the first time to our knowledge, for the observation of bond-ordered supersolids.

Magnetic frustration in lead pyrochlores

The rich phase diagrams of magnetically frustrated pyrochlores have maintained a high level of interest over the past 20 years. To experimentally explore these phase diagrams requires a means of tuning the relevant interactions. One approach to achieve this is chemical pressure, that is, varying the size of the non-magnetic cation. Here, we report on a new family of lead-based pyrochlores A$_2$Pb$_2$O$_7$ (A = Pr, Nd, Gd), which we have characterized with magnetic susceptibility and specific heat. Lead is the largest known possible B-site cation for the pyrochlore lattice. Thus, these materials significantly expand the phase space of the frustrated pyrochlores. Pr$_2$Pb$_2$O$_7$ has an absence of long-range magnetic order down to 400 mK and a spin ice-like heat capacity anomaly at 1.2 K. Thus, Pr$_2$Pb$_2$O$_7$ is a candidate for a quantum spin ice state, despite weaker exchange. Nd$_2$Pb$_2$O$_7$ transitions to a magnetically ordered state at 0.41 K. The Weiss temperature for Nd$_2$Pb$_2$O$_7$ is $\theta_{\text{CW}}$ = $-$0.06 K, indicating close competition between ferromagnetic and antiferromagnetic interactions. Gd$_2$Pb$_2$O$_7$ is a Heisenberg antiferromagnet that transitions to long-range magnetic order at 0.81 K, in spite of significant site mixing. Below its ordering transition, we find a $T^{3/2}$ heat capacity dependence in Gd$_2$Pb$_2$O$_7$, confirmation of a ground state that is distinct from other gadolinium pyrochlores. These lead-based pyrochlores provide insight into the effects of weakened exchange on highly frustrated lattices and represent further realizations of several exotic magnetic ground states which can test theoretical models.

Epitaxial growth of (1 1 1)-oriented spinel CoCr2O4/Al2O3 heterostructures

High quality (1 1 1)-oriented CoCr2O4/Al2O3 heterostructures were synthesized on the sapphire (0 0 0 1) single crystal substrates by pulsed laser deposition. The structural properties are demonstrated by in-situ reflection high energy electron diffraction, atomic force microscopy, X-ray reflectivity, and X-ray diffraction. X-ray photoemission spectroscopy confirms that the films possess the proper chemical stoichiometry. This work offers a pathway to fabricating spinel type artificial quasi-two-dimensional frustrated lattices by means of geometrical engineering.

Dynamical properties of interacting charge system on frustrated lattices

Dynamical properties in interacting fermion systems on geometrically frustrated lattices are introduced. As a minimal model for the electronic charge order phenomena, we adopt the so-called spin-less fermion V–t model where the hoppings and interactions of spin-less fermions are taken into account in frustrated lattices. By analyzing this model, we examine (1) dielectric and optical properties in a double triangular lattice, (2) a collective excitation in a dimer-molecular system on a triangular lattice, and (3) a photo-excited state in a triangular lattice. We demonstrate that frustration effects influence directly dynamical properties in these charge systems, and explain successfully recent some experimental results.

Bose–Einstein condensate in a bichromatic optical lattice: an exact analytical model

We provide an exact analytical model for the dynamics of a 1D Bose–Einstein condensate loaded in a bichromatic optical lattice. Although a host of exact solutions result from this novel method, we mainly concentrate on the solitonic excitations. The trapping potential and its depth of lattice frustration can be varied by tuning the power and the wavelengths of the two overlaying laser beams. Both attractive and repulsive regimes are thoroughly investigated. In the attractive domain, we obtain bright soliton, which reveals interesting variations with the depth of lattice frustration. Localization of the matter wave density is demonstrated as one of the applications in this regime. In the repulsive domain, dark soliton is obtained when the potential resembles an optical lattice. With appropriate tuning of the potential parameter, the dark soliton becomes modulated with an oscillatory background and gradually transforms to bright solitary trains.

Freezing and thawing of artificial ice by thermal switching of geometric frustration in magnetic flux lattices.

The problem of an ensemble of repulsive particles on a potential-energy landscape is common to many physical systems and has been studied in multiple artificial playgrounds. However, the latter usually involve fixed energy landscapes, thereby impeding in situ investigations of the particles' collective response to controlled changes in the landscape geometry. Here, we experimentally realize a system in which the geometry of the potential-energy landscape can be switched using temperature as the control knob. This realization is based on a high-temperature superconductor in which we engineer a nanoscale spatial modulation of the superconducting condensate. Depending on the temperature, the flux quanta induced by an applied magnetic field see either a geometrically frustrated energy landscape that favours an ice-like flux ordering, or an unfrustrated landscape that yields a periodic flux distribution. This effect is reflected in a dramatic change in the superconductor's magneto-transport. The thermal switching of the energy landscape geometry opens new opportunities for the study of ordering and reorganization in repulsive particle manifolds.

NaCaCo2F7 : A single-crystal high-temperature pyrochlore antiferromagnet

We report the magnetic characterization of the frustrated transition metal pyrochlore ${\text{NaCaCo}}_{2}{\mathrm{F}}_{7}$. This material has high spin Co${}^{2+}$ in CoF${}_{6}$ octahedra in a pyrochlore lattice and disordered nonmagnetic Na and Ca on the large-atom sites in the structure. Large crystals grown by the floating zone method were studied. The magnetic susceptibility is isotropic; the Co moment is larger than the spin-only value; and in spite of the large Curie Weiss theta (\ensuremath{-}140 K), freezing of the spin system, as characterized by peaks in the ac and dc susceptibility and specific heat, does not occur until around 2.4 K. This yields a frustration index of $f\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}\ensuremath{-}{\ensuremath{\theta}}_{\mathrm{CW}}/{T}_{f}\ensuremath{\approx}\phantom{\rule{0.16em}{0ex}}56$, an indication that the system is highly frustrated. The observed entropy loss at the freezing transition is low, indicating that magnetic entropy remains present in the system at 0.6 K. The compound may be the realization of a frustrated pyrochlore antiferromagnet with weak bond disorder. The high magnetic interaction strength, strong frustration, and the availability of large single crystals makes ${\text{NaCaCo}}_{2}{\mathrm{F}}_{7}$ an interesting alternative to rare earth oxide pyrochlores for the study of geometric magnetic frustration in pyrochlore lattices.

Proposal for Verification of the Haldane Phase Using Trapped Ions

A proposal to use trapped ions to implement spin-one XXZ antiferromagnetic chains as an experimental tool to explore the Haldane phase is presented. We explain how to reach the Haldane phase adiabatically, demonstrate the robustness of the ground states to noise in the magnetic field and Rabi frequencies, and propose a way to detect them using their characteristics: an excitation gap and exponentially decaying correlations, a nonvanishing nonlocal string order, and a double degenerate entanglement spectrum. Scaling up to higher dimensions and more frustrated lattices, we obtain richer phase diagrams, and we can reach spin liquid phase, which can be detected by its entanglement entropy which obeys the boundary law.