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21-03-2013

A geometric representation of quantum mechanics developed by ESRF theorist Patrick Bruno tames the abstract world of magnetic systems.

Niels Bohr, a leading architect of quantum mechanics, famously remarked that anyone who was not shocked by quantum theory had not understood it. Today, quantum mechanics remains as weird and counterintuitive as ever, helped only by a strong command of mathematics. Sometimes even the best mathematical minds need help to work what it means.

Head of the ESRF’s theory group, Patrick Bruno, has found a way to represent complex quantum systems in a relatively simple geometric form. Based on a powerful but largely ignored concept invented by Italian physicist Ettore Majorana 80 years ago, Bruno’s formulation offers a deeper insight into the behaviour of magnetic “spin” systems.

Aged 26, Majorana devised an elegant representation that described a pure quantum state of a spin-*J* system, where *J* is the spin, as a constellation of 2*J* points on a sphere, analogous to stars on the celestial sphere. The result was contained in one of just 10 papers that Majorana published in his short but influential career; six years later he would board a boat on the island of Palermo bound for Naples, and mysteriously disappear.

Spin is a measure of the intrinsic angular momentum of a quantum system. A spin ½ system such as a single electron has two levels: ± ½, commonly called “up” and “down”. The standard way to represent this, using Schrödinger’s wavefunction or Heisenberg’s matrix formulation of quantum mechanics, is in terms of 2*J* + 1 complex numbers. This representation not only becomes complicated for higher spin systems, but there is no way to get a perception of what it means physically.

“You don’t care about lists of complex numbers moving on the screen on your computer,” explains Bruno. “In classical mechanics, motion is a point moving from one place to another in time, and the difficulty in quantum mechanics is to create that geometric picture.”

Being purely geometric, Majorana’s representation recovers the concept of a trajectory in quantum mechanics. The basic states of a two-level system correspond to a unique star at the “north” and “south” poles, and linear superpositions of theses states correspond to the star being located anywhere on the sphere. This is the familiar Bloch representation and is the essential ingredient of a qubit, the basic element of quantum information processing. “Manipulating qubits just means doing certain rotation operations on one point on a sphere,” Bruno explains. “More realistic systems have more than two levels, and the question is: how do we represent that in a geometric manner?”

This year, Bruno found a way to express the expectation value of physical observables in a spin-*J* system, for instance its dipole moment and energy, in terms of the Majorana stars. He used a novel diagramatic method by which he was able to map the quantum state of any spin-*J* system onto the thermodynamic partition of a fictitious classical gas on the sphere (*Phys. Rev. Lett*. 108 240402).

Having shown how to derive the observables of a spin system for a given configuration of stars, Bruno also worked out the “symplectic structure” of general spin-*J* systems. This structure, which in quantum mechanics is given by the geometric phase (a phase shift acquired by a system upon cyclic motion of the stars on the sphere), determines how a system evolves in response to forces, and thus determines its dynamics.

Essentially, Bruno has found the quantum mechanical counterpart of the well known Landau–Lifshitz equation, which describes the precession of an electron in a magnetic field and is the basis for understanding magnetic systems such as computer hard disks. He is now working on a second, longer, paper in which he will set out the applications of his new formulation. “Usually one is interested in systems that are more complex, such as infinite lattices of spins coupled together by exchange interactions, so it could be a ferromagnet for instance. I will be then able to describe systems that people investigate with neutrons or X-rays.”

The work will be particularly useful, he says, for making sense of experiments with nematic magnets, which the Landau–Lifshitz equation cannot solve because they contain states with zero magnetic moments. “This paper will presumably be much less cited than many other things that I have done, but it’s one with which I am particularly happy.”

Theorist John Hannay of the University of Bristol, UK, who has carried out similar work, told *ESRFnews *thathe was delighted when he learned of Bruno’s “very imaginative and neat re-interpretation” of the theory of Majorana stars. “I would not say my work, or his re-interpretation, impinges on any deep or philosophical problems of quantum mechanics, though I might hope that it indeed could possibly be useful in qubits or exotic magnets.”

*Matthew Chalmers*

**The ESRF in theory**

The ESRF’s theory group currently comprises four staff who conduct research into theoretical physics, mainly solid-state theory and magnetism, and assist users and beamline scientists. Calculations performed last year by a PhD student in the group, for instance, helped understand high-pressure studies of the magnetic behaviour of nickel at the ID24 beamline (*Phys. Rev. Lett*. 107 237202). The group also works alongside theorists from the Institut Laue-Langevin. “It is quite a unique environment,” says Bruno. “We are not expert in all experimental areas, but my door is always open and people are welcome to expose us to new problems.”

*This article originally appeared in ESRFnews, December 2012. *

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