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Condensed-matter physics

Transitions on triangles

An exquisitely sensitive technique shows that a magnetic field only a few hundred times greater than Earth's can bring an exotic phase of matter known as a quantum spin liquid to an ordered magnetic state. See Letter p.612

Contrary to expectation and experience, cooling a material will not always freeze it, no matter how low the temperature falls. Two factors can act to preserve the liquid state even down to absolute zero: fluctuations away from order and towards disorder, which are especially important when atoms are confined to sheets and chains; and geometrical frustration, a failure to simultaneously satisfy all the constraints imposed by interactions between neighbouring atoms. On page 612 of this issue, Pratt et al.1 probe a model two-dimensional frustrated system that has magnetic spins as stand-ins for atoms and a quantum spin-liquid phase that is stable down to the lowest temperatures accessible. This is the layered molecular system denoted κ-(BEDT-TTF)2Cu2(CN)3.

When a system is precariously balanced between solid and liquid, or more generally between order and disorder, and the temperature hovers just above absolute zero, the range and impact of quantum fluctuations grow and the fundamental nature of the system's ground (minimal-energy) state becomes apparent. Moreover, low-energy excitations, collective and widespread, become accessible in new ways. Pratt and colleagues find1 that small magnetic fields applied to the model spin-liquid system tip the balance from disorder to order by exciting and condensing collective modes of spins. Ramping up the magnetic field disrupts the collective magnetic order and induces the possibility of shorter-range spin pairings.

Quantum spin liquids touch on some of the central themes of condensed-matter physics: correlations, competition between ground states, frustration, fluctuations, and protected degrees of freedom. The model system discussed here1 allows these themes to be explored quantitatively. Its magnetic properties are determined by the fact that its lattice consists of linked, corner-sharing triangles with electronic (S = 1/2) spins at each of the corners2. Think of the spins as compass needles that want to line up antiparallel to each other: an antiferromagnetic ground state. If the needle at the apex of the triangle points north, then each of the other two needles should point south to satisfy the constraint provided by interacting with the apex. But the two needles at the base of the triangle also interact with each other and impose the incompatible constraint of one needle pointing north and one pointing south. Each needle switches from north to south with little cost in energy. The spins can continue to fluctuate even at extremely low temperatures, refusing to order into the antiferromagnet3, just as atoms in a liquid fluctuate away from the lattice positions that they would assume in an ordered solid.

Significantly, the effects of these spin fluctuations are amplified because of the effective two-dimensional nature of the material. Consider the following analogy: a few spectators queuing to enter a football stadium can step out of line and easily dissolve that (one-dimensional) line, and a small number of band members stepping out of position at the half-time show will disrupt the (two-dimensional) formation, but a good portion of the total stadium attendance must leave their seats for (three-dimensional) order to be disturbed. Combined with the well-defined geometry of a spin-1/2, frustrated triangular antiferromagnet, the reduced dimensionality leads to an interesting and complex phase diagram4.

Pratt et al.1 show that the native spin-liquid state can be frozen and become ordered by the application of a magnetic field of 14 millitesla, which is only a few hundred times stronger than Earth's magnetic field. Given that the external magnetic field points in one direction and the spins at the corners of the triangles want to point in opposite directions, this ordering is not as simple as just aligning the spins with the applied field. Rather, the magnetic field provides the means to excite spins to energies above their ground state. The excitations then collectively condense into the long-range, ordered, antiparallel configuration of antiferromagnetism, following a scenario analogous to Bose–Einstein condensation in superfluid helium.

How are Pratt and colleagues able to deduce all this information? They use an exquisitely sensitive technique called muon spin rotation — which, as its name implies, is performed with subatomic particles known as muons — and their samples are cooled to a few hundredths of a degree above absolute zero. The muons are injected into the material with their own spins aligned along one direction, and are then used to sample the local magnetic environment and to communicate the presence of order or disorder by decaying within microseconds. Not only does the quantum phase transition between spin liquid and weak antiferromagnet — 'weak' because the ordered state still feels the disruption of fluctuations — show up clearly, but there is also evidence for a transition to another ordered state at much higher applied magnetic fields. Unfortunately, the local nature of the information obtained from the muon-spin-rotation technique makes it difficult to identify definitively the character of the high-field order.

Pratt and colleagues' work1 provides a compelling opportunity for serious confrontation between theory and experiment, including determination of the critical exponents that define the fundamental and generalizable character of the quantum spin liquid. More generally, it illuminates the ramifications of frustration, a concept useful in fields as diverse as economics, computer design and physics. In three-dimensional magnetic solids, order and disorder can mix, and geometrically frustrated states can exist in which only a fraction of the magnetic spins can change direction with little cost in energy. These blocks of spins are surrounded by frozen neighbours, but protected from them5. They not only offer the opportunity to probe the general structure of the state, but also, if they can be preferentially accessed, may serve to encode and manipulate information6.


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Correspondence to Thomas F. Rosenbaum.

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Rosenbaum, T. Transitions on triangles. Nature 471, 587–588 (2011).

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