Neutron Scattering and Magnetism
Laboratory for Solid State Physics · ETH Zurich

Frustrated Quantum Magnets

Geometric frustration of interactions implies that there are multiple possible quantum ground states with very close energies. It takes a very small push, such as the application of a magnetic field or perhaps some small anisotropy terms in the Hamiltonian, to favor one such state over another. The most likely mechanism is the so-called order-from-disorder. Among competing states it selects the one with the softest excitation spectrum, and can be of either classical (thermal) or quantum origin. Today, frustrated quantum magnets are by far the most active area of our research.

First observation of a ℤ2 vortex crystal

Z2 vortex crystal spin texture mnrc phase diagram

Left: calculated spin texture of a ℤ2 vortex crystal on a triangular lattice. Arrows show the local spin directions; color encodes the out-of-plane spin component. The vortex cores (red) themselves crystallize into a regular triangular superlattice, woven by the three symmetry-equivalent wave vectors q1, q2, q3. Right: magnetic phase diagram of (CD3ND3)2NaRuCl6 for the H || c field direction. Color indicates the measured magnetostriction coefficient.

At first glance, the organic antiferromagnet (CD3ND3)2NaRuCl6 seems to be yet another realization of the S = 1/2 triangular lattice Heisenberg model, perhaps the most studied problem in all of frustrated magnetism. In truth, it is a rather special breed. Its magnetic ions are heavy: in Ru3+, strong spin-orbit coupling entangles the spin and orbital degrees of freedom into jeff = 1/2 pseudospins and generates "Kitaev" terms, exchange interactions whose anisotropy axes differ from bond to bond. For such triangular Heisenberg-Kitaev magnets, theory makes a spectacular prediction: even a tiny Kitaev term condenses topological point defects, the so-called ℤ2 vortices, directly into the ground state, where they arrange into a periodic superlattice, a "vortex crystal". This is a direct analog of skyrmion lattices in chiral helimagnets and of Abrikosov vortex lattices in type-II superconductors. The catch: no one has ever observed this state in a real material.

We may have just changed that. We have grown the first large single crystals of this compound and combined thermodynamic, magneto-elastic and neutron scattering experiments at millikelvin temperatures to map out its magnetism [1]. The material turns out to be amazingly rich: below TN ≈ 0.23 K it orders in two steps, and a magnetic field applied along the c axis induces a cascade of no fewer than seven distinct ordered phases, both commensurate and incommensurate. The best part is the zero-field ground state itself. Neutron diffraction reveals a complex multi-q structure with three incommensurate Bragg peaks, precisely the fingerprint expected of the ℤ2 vortex crystal.

Is it the real thing? The "smoking gun", a polarized-neutron look directly at the vortex cores, is still ahead of us. What is certain is that (CD3ND3)2NaRuCl6 is only the first member of an entirely new family of 4d and 5d triangular lattice magnets that we can now grow as large single crystals. The hunt is on.

Spin supersolid on a triangular lattice

The two continuous degeneracies of the Y state Excitation spectrum of the spin supersolid

The two continuous degeneracies of the "Y" state, illustrated on its three sublattice spins, and the measured spectrum. Left: a global rotation of all transverse spin components about the easy axis. This is the true U(1) symmetry of the XXZ Hamiltonian; its breaking produces a gapless Goldstone mode. Center: the "umbrella" deformation connects classical ground states of exactly equal energy, yet it is not a symmetry of the Hamiltonian. The degeneracy is accidental, and the corresponding mode survives only as a gapped pseudo-Goldstone mode. Right: magnetic excitation spectrum of K2Co(SeO3)2 measured by inelastic neutron scattering: a broad continuum with a roton-like minimum at the M point in place of conventional spin waves. Inset: a zoom on the K point, where the pseudo-Goldstone mode appears.

A supersolid sounds like a contradiction in terms: a state of matter that is simultaneously a rigid, ordered "solid" and a coherent, freely flowing "superfluid". Proposed for solid helium more than half a century ago, it has evaded unambiguous observation ever since. Quantum magnets offer another route. Consider the S = 1/2 easy-axis XXZ antiferromagnet on a triangular lattice. Its Ising part is famously frustrated: as Wannier showed back in 1950, it never orders and retains a macroscopic ground-state entropy. The small transverse exchange picks a spectacular compromise out of this enormous manifold: the "Y" state, in which a collinear spin density wave along the easy axis (the "solid") coexists with transverse XY order, the magnetic incarnation of a Bose-Einstein condensate (the "superfluid"). The two orders set in at separate Berezinskii–Kosterlitz–Thouless transitions: a spin supersolid. All that remains is to find the right material. K2Co(SeO3)2 is precisely that: a rare realization of the XXZ model on an ideal triangular lattice, lying very close to the Ising limit, and ordering exactly as prescribed [1]. We even recover Wannier's entropy experimentally just above the ordered phase [2].

Symmetry-wise, the supersolid is a rather peculiar object, as the two animations illustrate. Rotating all transverse spin components about the easy axis costs no energy and never will: it is a true U(1) symmetry of the Hamiltonian, and its breaking produces a gapless Goldstone mode. The "umbrella" deformation also connects classical ground states of equal energy, but it is protected by no symmetry whatsoever. Quantum fluctuations lift this accidental degeneracy, order-from-disorder at work, and the would-be Goldstone mode acquires a gap. Our ultra-high-resolution neutron spectroscopy finds precisely this pseudo-Goldstone mode at the K point, with a gap of a mere 60 μeV [2].

The dynamics is equally striking. Instead of sharp magnons, the zero-field spectrum is dominated by a broad continuum of excitations, the kind of response one would sooner expect of a quantum spin liquid. In applied fields the continuum gradually coalesces into coherent spin waves, and the two symmetry sectors respond in completely different ways: the Goldstone excitations remain gapless all the way up to the quantum critical point, where the supersolid gives way to the m = 1/3 up-up-down magnetization plateau, while the pseudo-Goldstone gap steadily grows. All of this is in excellent quantitative agreement with quantum Monte Carlo simulations of the underlying model [2], making K2Co(SeO3)2 a unique platform for studying supersolidity in a real, gram-sized crystal.

Braided Ising spin tubes: the kagome magnet that wasn't

Field-induced magnetic structures of the braided Ising spin tube

Magnetic structures of a single braided spin tube in Nd3BWO9, calculated from the two-parameter Ising model; the competing couplings J1 and J′1 are indicated. Left to right: the antiferromagnetic ground state in zero field; the ↑↑↓ "spiral" phase behind the m = 1/3 magnetization plateau in an axial field; the spin-flop and double spin-flop states that produce the m = 1/4 and m = 1/2 plateaus in transverse fields. The small circles show the two flopped structures viewed along the tube axis.

No lattice in frustrated magnetism is more celebrated than kagome, the pattern of corner-sharing triangles. The kagome antiferromagnet is one of the oldest models in the field and a prime suspect for hosting a quantum spin liquid, yet good material realizations are famously scarce: the few known ones are plagued by structural disorder or by unwanted extra interactions. The recent discovery of the rare-earth family R3BWO9, with perfectly ordered "breathing" kagome planes of magnetic ions, therefore caused a great deal of excitement. The Nd member did everything an exotic frustrated magnet is supposed to do: it orders only below 0.3 K, at a temperature ten times smaller than the interaction scale, produces a whole collection of fractional magnetization plateaus in applied fields, and at elevated temperatures its magnetic structure even turns incommensurate [1].

Our single-crystal neutron scattering experiments tell a rather different story [2]. The culprit is single-ion physics. The ground state doublet of the Nd3+ ion is composed almost entirely of the maximal angular momentum states, making the moments extremely Ising-like, and the local easy axes of neighboring ions in the kagome plane are nearly orthogonal, at 89° to one another. For such spins, symmetry all but forbids the in-plane exchange: the celebrated kagome network is magnetically mute. What remains are the shortest bonds in the structure, which connect ions in adjacent planes. They weave the spins into triple "braids" of Ising chains, twisted triangular spin tubes running along the crystal axis. Note that each chain on its own is completely unfrustrated. It is the braiding, with competing ferro- and antiferromagnetic couplings intertwining at every third site, that generates frustration of an entirely different, one-dimensional kind.

The payoff is a model that is classical, one-dimensional, and contains all of two exchange constants. It quantitatively accounts for essentially everything we measure: the zero-field ground state, the field-induced structures shown in the figure and the fractional plateaus they produce (the first such plateau cascade ever reported in a one-dimensional magnet), the specific heat across the entire field-temperature plane, and the local spin-flip excitations seen by neutron spectroscopy. Even the incommensurate order at elevated temperatures comes out naturally: it is driven by thermally proliferating domain walls, and the calculated temperature dependence of the propagation vector matches experiment with no free parameters. You cannot always get what you want. We went looking for kagome physics and found none, but what we found instead is rarer still: a frustrated magnet whose spectacularly rich behavior can be understood and simulated to astounding accuracy.

Confinement of fractional excitations: the Zeeman ladder

Zeeman ladder spectrum of Cs2CoBr4

Magnetic excitation spectrum of Cs2CoBr4 along the chain direction, measured by high-resolution neutron spectroscopy at 40 mK with two incident neutron energies. What was once mistaken for a broad scattering continuum resolves into a hierarchy of sharp dispersive bound states of kink pairs: a "Zeeman ladder".

Quarks cannot be pulled apart. The force binding them does not weaken with separation, so an isolated quark would cost an infinite energy: quarks are permanently confined into hadrons. Remarkably, the same physics can be studied in a quantum magnet, at millikelvin temperatures and milli-electronvolt energies. The elementary excitations of anisotropic spin chains are not spin waves but "kinks", mobile domain walls that each carry only a fraction of a spin flip and normally appear in scattering experiments as broad continua. Weak couplings between the chains change everything: pulling two kinks apart leaves behind a string of overturned spins whose energy grows linearly with its length. The pair is confined. Quantum mechanics in a linear potential is a textbook problem: the continuum collapses into a discrete hierarchy of two-kink bound states whose energies are spaced, in supreme mathematical elegance, according to the negative zeros of the Airy function. The resulting spectrum is known as a "Zeeman ladder".

This is exactly what we found in Cs2CoBr4 [1]. Our earlier experiments on this material had revealed what looked like a gapped excitation continuum. When we returned to it with much sharper energy resolution, the "continuum" turned out to be nothing of the sort: it resolved into a sequence of at least nine sharp bound states, confirmed independently by THz absorption spectroscopy, with energies that follow the Airy sequence almost perfectly. Note that all previously known Zeeman ladders live in strongly Ising-like spin chains. Cs2CoBr4 is nearly planar, about as far from the Ising limit as one can get, and its bound states are confined to individual chains only at special wave vectors; elsewhere in the Brillouin zone they propagate as genuinely two-dimensional objects.

The material itself had one more surprise in store. Cs2CoBr4 was long believed to be a distorted triangular lattice antiferromagnet, a cousin of the famous spin liquid candidate Cs2CuCl4. It is not. Neutron spectroscopy in the fully field-polarized state, the only regime where exchange constants can be measured reliably, revealed a rather different architecture: a network of frustrated zigzag spin ladders with XYZ anisotropy [2]. This is what makes the problem tractable: quasi-one-dimensional systems are exactly where matrix product state numerics excel. Together with our theory colleagues in Geneva we showed that a single frustrated XYZ ladder in a self-consistent Weiss field reproduces the measured spectrum, and that confinement survives where it seemingly has no right to: in this model no spin component is conserved, and nothing remains of the strong Ising anisotropy of the classic examples [3]. Polarized neutron measurements even separate two distinct sequences of bound states, longitudinal and transverse to the ordered moment, just as the theory prescribes.