# Nobel Prize: Haldane, neutron science and the ILL

**The Institut Laue Langevin (ILL) was privileged to learn of the 2016 Nobel Prize for Physics, awarded for the theoretical studies into exotic quantum states in hard condensed matter conducted by Thouless, Kosterlitz and Haldane.**

**Duncan Haldane worked as a post-doctoral researcher in the ILL's Theory group from 1977 to 1981. It is during this period that he started to develop his seminal work on one-dimensional quantum liquids and spin chains. Since then, neutron science has played a major role in the experimental investigation of these systems.**

We have known ever since the 1920s that elementary particles and, in particular, electrons possess a so-called spin. The spin of the electron can be shown experimentally by passing the particles through a magnetic field with a gradient. This causes the originally uniform beam to split into two clearly distinct families, which are referred to as the spin states of the electron. The mathematical operator describing this property of the electron takes the form of a vector in space acting on the so-called spinors describing the spin states themselves. The magnitude of this vector is given by the spin quantum number, which for a free electron is 1/2. In complex systems the spins can combine. However, irrespective of the combination, they can only take on multiples of 1/2. The spinors are not directly observable physical quantities. Although not intuitively obvious, we can show mathematically that when “rotating” an electron its spinors will only come back to their original value after 720°.

The question of what happens when we put interacting spins on a lattice has intrigued physicists for the last 80 years. When you place classical arrows equidistantly along a straight line and make them interact in such a way that any two of them tend to align parallel but with opposite direction then it takes only a bit of thinking to find out that the most favorable state is an ordered chain of arrows alternatively pointing up and down along the same arbitrary direction. When you ask the same question for quantum mechanical systems, so-called antiferromagnetic Heisenberg chains, the problem becomes extremely intriguing. An exact solution was developed very early on for spin-1/2 chains. The fundamental state of the chain is at zero temperature just at the critical point to long-range order, and can be excited with arbitrarily small energy input. However, the generalizations of this solution to higher dimensions and higher spin values turned out an impossible enterprise. Nevertheless everybody assumed that nothing dramatic should be expected when going from spin-½ to spin 1, spin 3/2 and so forth. Therefore, when Duncan Haldane presented his calculations, which predicted, that integer spin chains would behave completely different, his assertions were met with a lot of skepticism. Haldane stated that integer anti-ferromagnetic Heisenberg chains possess very short-range order in the ground state and in order to excite them you have to overcome a barrier, i.e. you need a finite amount of energy. The reason for this fundamental difference is rooted in the topological fact outlined at the very beginning that half-integer spinors need 720o rotations instead of 360o rotations for integer spinors to be mapped onto themselves. Haldane’s result was so unexpected that he had real difficulties publishing it, as explained in his Nobel lecture where he shows the original print in the form of an Internal ILL report. Duncan Haldane was at the time member of the ILL’s theory group.

It took only a few years to verify Haldane’s statement experimentally. The main difficulty resided in the distinction of the “Haldane gap” from more trivial sources of gapped excitations in real materials. ILL experiments provided the first direct proof of the triplet character of the gapped excitation (in CsNiCl3, [2]), as well as the proof of the triplet excitation in the first non-ordering spin-1 chain NENP [3] and the first direct comparison between a spin-1 and a spin-3/2 compound of otherwise identical structure (AgVP2S6 and AgCrP2S6), with gapped excitations for the spin-1 and gapless excitations of the spin 3/2 compound [4]. In the case of CsNiCl3 the use of polarized neutrons was crucial.

As Haldane pointed out [1], the one-dimensional quantum spin-1 antiferromagnet at zero temperature can be related to the classical two-dimensional planar ferromagnet at finite temperature. According to Kosterlitz and Thouless, the latter disposes of vortex-like topological excitations which condense into the ground-state upon lowering temperature. The quantum spin-1 chain’s ground state is characterized by a similar type of topological non-local order, analogous to the topological order responsible for the fractional quantum Hall effect [5].

Since then, over the last three decades, many other neutron scattering experiments have confirmed and elaborated in detail Haldane’s work, at the ILL and elsewhere.

[1] FDM Haldane, ILL preprint SP-81/95, arxiv1612.00076v1 (pdf - 439 Ki).

[2] M Steiner, K Kakurai, JK Kjems, D Petitgrand, and R Pynn, J Appl Phys 61, 3959 (1987).

[3] LP Regnault, WAC Erkelens, J Rossat-Mignod, JP Renard, M Verdaguer, WG Stirling, C Vettier, conference paper, DOI: 10.1007/978-3-642-73107-5_28 (Springer, 1987).

[4] H Mutka, JL Soubeyroux, G Bourleaux, P Colombet, PRB 39, 4820 (R) (1989).

[5] SM Girvin and DP Arovas, Physica Scripta T27, 156 (1988).