Triangular lattice and frustration

In plain words

A triangular lattice is a flat net of triangles, each atom with six neighbours. Put a tiny magnet that wants to point opposite to its neighbours on every corner, and a triangle cannot oblige: once two corners point up and down, the third has no good choice. That standoff is called frustration. It can stop a magnet from ordering at all, leaving room for unusual states such as .

Going deeper

On a triangle, three spins that each want to point opposite to their neighbours cannot all be satisfied. The best they can do is a 120° compromise – or, when quantum effects win, no order at all.

Three spins, no winner

Take three atomic magnets on the corners of a triangle and let each one want to point opposite to its neighbours. Two can manage it, one up and one down; the third is then next to one of each and can satisfy only one. A triangular lattice is built entirely of such triangles, so for limited to up and down there is an enormous number of arrangements with the same, equally unsatisfying, energy. When the spins can point in any direction, the best compromise is for the three to sit at 120° to each other – which spreads the dissatisfaction evenly and does produce order, though a fragile one.

The same frustration appears on other lattices built from triangles, such as the , and on the when each bond prefers a different spin direction, as in α-RuCl₃.

What frustration does

Frustration lowers the temperature at which a magnet orders, sometimes to zero. A useful measure compares two temperatures: the one at which the spins start to feel each other strongly, read from how the magnetism follows temperature well above any transition, and the one at which order sets in. In an ordinary magnet the two are similar; in a strongly frustrated one the first can be ten or a hundred times the second.

If order never arrives, quantum fluctuations can take over and produce a quantum spin liquid, in which the spins stay entangled and restless even at absolute zero. The triangular lattice is where this idea was first proposed, in 1973, and it remains one of the main places to look.

Triangles in 2D materials

Many layered compounds put their magnetic atoms on a triangular lattice: the metal atoms of NiI₂ and of other transition metal dihalides do, as do those of several . Nb₃Cl₈ goes a step further: its niobium atoms form clusters of three, and the clusters themselves sit on a triangular lattice, each carrying one unpaired electron. The same geometry shapes charge as well as spin, as in the star-of-David of -TaS₂, whose clusters also form a triangular lattice.

In a , frustration competes with the the layer needs to order at all, so the number of layers, and stacking can tip the balance – tuning knobs that bulk frustrated magnets do not have.

For specialists

A Bravais lattice with six nearest neighbours per site. With nearest- the three spins of a triangle cannot all be antiparallel: Ising spins have a macroscopically degenerate ground state, while Heisenberg spins settle into the 120° state. This geometric frustration suppresses ordering and enhances quantum fluctuations; frustration also arises from competing exchange paths and from bond-dependent Kitaev interactions on the honeycomb lattice. Its strength is gauged by the ratio |θ_CW|/T_N.

Where this comes from

Correlated flat bands and quantum spin liquid state in a cluster Mott insulator Hu et al. · Communications Physics 6, 172 (2023) cited by 47
Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet Banerjee et al. · Nature Materials 15, 733 (2016) cited by 839

Quantum spin liquid

A magnet whose atomic magnets never settle into a pattern, even at absolute zero. Quantum effects keep them fluctuating and entangled – linked so that none has a direction of its own – so instead of freezing into order the material stays restless, like a liquid that never turns solid. Disturbances in it can behave like fractions of an electron, carrying its magnetism but not its charge.

A ground state of a frustrated magnet with no symmetry breaking down to zero temperature, marked by long-range entanglement, an emergent gauge structure and fractionalised excitations such as spinons or Majorana fermions. Honeycomb materials with bond-dependent Kitaev exchange – α-RuCl₃ above all – are the leading layered candidates, but the evidence is indirect: a scattering continuum and thermal Hall signals, against residual magnetic order in zero field.

Full explanation

Spin

A built-in property of every electron that makes it a tiny magnet. Measure it along any direction and you only ever find one of two answers, ‘up’ or ‘down’. Despite the name, nothing is actually spinning – the word stuck from an early picture – but the magnetism is real: countless electron spins lined up are what make a fridge magnet stick. Spintronics tries to carry information in spin rather than in charge.

The intrinsic angular momentum of the electron, ħ/2, with a magnetic moment of almost exactly one Bohr magneton; a measurement along any axis yields one of two values. In solids spin couples to orbital motion through spin–orbit coupling and to other spins through exchange; in 2D materials it sets magnetic order, spin–valley locking and how long a spin survives as a carrier of information.

Full explanation

Honeycomb lattice

A flat net of hexagons, like a beehive, with an atom at every corner. Graphene is the famous example, but the same pattern gives boron nitride its band gap, gives graphene its massless electrons, and gives some magnets interactions that cannot all be satisfied at once.

A triangular Bravais lattice with a two-site basis, so every site has three nearest neighbours on the other sublattice. Equivalent sublattices with nearest-neighbour hopping give Dirac cones at K and K′ (graphene); inequivalent ones open a gap with valley-contrasting Berry curvature (hBN, TMDC monolayers seen from above); buckling mixes in spin–orbit coupling (the Xenes); and bond-dependent exchange on the same geometry gives the exactly solvable Kitaev model behind the α-RuCl₃ spin-liquid programme.

Full explanation

Exchange interaction

The force that lines up the tiny magnets of neighbouring atoms in a magnetic material – all the same way in a ferromagnet, alternating in an antiferromagnet. Despite what it does, it is not magnetic in origin: it comes from the electric repulsion between electrons combined with a quantum rule that keeps two electrons from sharing the same state, which makes their energy depend on whether their spins point the same way. It is far stronger than the magnetic pull between atomic magnets, which is why iron stays magnetic up to 770 °C.

The spin-dependent energy that arises from the Coulomb repulsion together with the antisymmetry of the many-electron wavefunction, modelled as a coupling J between neighbouring spins in Heisenberg, Ising or XY form. In insulators it is mostly mediated by ligands – superexchange, whose sign follows the Goodenough–Kanamori rules: antiferromagnetic for metal–ligand–metal angles near 180°, often ferromagnetic near 90°, as in the edge-sharing octahedra of CrI₃ – in metals by conduction electrons (RKKY, double exchange), and with strong spin–orbit coupling it becomes anisotropic or bond-dependent (Kitaev). In van der Waals magnets the intralayer exchange of a few meV sets the ordering scale, while a much weaker interlayer exchange decides ferro- or antiferromagnetic stacking and depends on the stacking itself; ordering a monolayer additionally requires anisotropy, by the Mermin–Wagner theorem.

Full explanation