Profile of David J. Thouless, J. Michael Kosterlitz, and F. Duncan M. Haldane, 2016 Nobel Laureates in Physics

One of the most profound and mysterious facts in science is that abstract mathematics can beautifully describe various phenomena realized in the real world. The deep connection between geometry and physics, which dates back to ancient Greece, continues to surprise us and most remarkably has borne fruit as general relativity. Topology is a branch of geometry, where two objects are congruent if one can transform one to the other with continuous deformation. Topology deals with qualitative aspects of objects but provides extremely accurate and astonishing predictions in theoretical physics.

The Nobel Prize in Physics has been awarded to three theoretical physicists, all of whom were born in Great Britain but are now US citizens: see Fig. 1. Most senior is David J. Thouless, born in September 1934 and now retired from the University of Washington in Seattle, who was awarded one-half of the prize. Next is J. Michael Kosterlitz, born in Aberdeen in June 1943, where his father, the noted biochemist Hans Walter Kosterlitz, was working since 1934 as a refugee from Nazi Germany. Mike, as he is called by friends, is at Brown University in Physics. He shares one-half of the Prize with F. Duncan M. Haldane, born in September 1951, and educated in St. Paul's School, London. Haldane, now at Princeton University, spent 4 years in France, where he met his wife Odile. The Nobel Prize citation reads, “for theoretical discoveries of topological phase transitions and topological phases of matter.” The breakthrough these theorists made explores the deep connection between topology and physical phenomena in quantum physics.

Fig. 1.

(Left to Right) David J. Thouless, photo by Kiloran Howard and image courtesy of © Trinity Hall, Cambridge University; J. Michael Kosterlitz, image courtesy of Brown University; and F. Duncan M. Haldane, photo by D. Applewhite and image courtesy of Princeton University.

Quantum physics is formulated in terms of the wavefunction $\mathrm{\Psi }\left(q\right)$, which describes the properties of a quantum state in terms of the coordinates q. In superfluids and superconductors, the most crucial wavefunction can be expressed as $\mathrm{\Psi }\left(r\right)=R\left(r\right)e^{i\mathrm{\Theta }\left(r\right)}$, where the phase field $\mathrm{\Theta }\left(r\right)$ gives rise to quantum coherence. This quantum coherence results in fluid flow without friction and current flow without resistance.

After obtaining his BA degree at Trinity Hall, Cambridge University, David Thouless went to Cornell University to obtain his PhD by studying the theory of nuclear matter under Hans Bethe, who was later awarded a Nobel Prize for his own work. After postdoctoral research at University of California, Berkeley, he moved to Cambridge University where, from 1961 to 1965, he held a position at Churchill College. Then he moved to Birmingham University becoming Professor of Mathematical Physics in Rudolph Peierls' department. Eventually, he returned to the United States and, in 1980, took up his position in Seattle from where he retired in 2003 and where his wife, Margaret, held a faculty position in pathobiology from 1980 to 2004.

During his period in Birmingham, he answered a basic question regarding the Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity, namely, the following: How is it that a manifestly Landau–mean-field theory works so well in many superconductors, in practice, whereas in a superfluid the same theory fails badly in the critical region? Indeed, Thouless most helpfully identified clearly the crucial, and exceptionally long, length scale that distinguishes most directly the (traditional) superconductors from liquid ⁴He.

While at Birmingham, Thouless supervised Michael Kosterlitz as a talented postdoctoral associate. Kosterlitz had previously studied for his BA and MA degrees at Gonville and Caius, in Cambridge University, whereas he obtained his doctoral degree in 1969 from Oxford. After working with Thouless in Birmingham, he spent 2 years at Cornell. By then he was already married to his wife Berit. There, with Nelson and Fisher, he mastered the then-new renormalization-group approach.

Shortly afterward, he established, with David Thouless, the striking Kosterlitz–Thouless transition in a superfluid. Quite unexpectedly, that predicted a universal discontinuity in the superfluid density (1). Additionally, experiments soon verified this totally new and unanticipated result. On return to England, he became a faculty member at Birmingham, rising to the rank of “reader” until, in 1982, he became a Professor of Physics at Brown University in Providence, Rhode Island.

As regards the Nobel Prize, Kosterlitz working with Thouless (2) in 1973 considered the classical statistical mechanics of a phase field $\mathrm{\Theta }\left(r\right)$ for a 2D superfluid or superconductor, to reveal the mechanism of the phase transition to the normal state.* They found that a vortex and an antivortex as shown as in Fig. 2, characterized by the topological winding numbers ±1, respectively, play essential roles for the destruction of the rigidity of $\mathrm{\Theta }\left(r\right)$ rather than its slowly varying fluctuation. One can regard the vortex and antivortex as positive and negative charges, respectively, with an effective Coulomb interaction between them. Therefore, at low temperature, a vortex and an antivortex form a bound pair and become inert. As the temperature rises, other vortices and antivortices intervene between the vortex and antivortex, screening the attractive interaction between them, causing the binding to become weaker and weaker. Eventually, at a critical temperature$T_{KT}$, vortices and antivortices become unbound and the medium behaves like a neutral plasma. Once these free vortices and antivortices exist, the rigidity of $\mathrm{\Theta }\left(r\right)$ is lost and the system is in the normal state.

Fig. 2.

The vortex (Left) and antivortex (Right) of the phase field $\mathrm{\Theta }\left(r\right)$ of the wavefunction. The directions of the arrows indicate the phase $\mathrm{\Theta }\left(r\right)$ at each position $r$ as the angle measured from the horizontal axis.

Later, Thouless et al. (4) also considered the wavevector $k$ as the coordinate $q$, which characterizes the electronic states in solids. In two dimensions, the boundary of the region of the wavevector $k$ forms a closed loop. Thouless et al. discovered that the winding number for the mapping from this closed loop in $k$-space to the wavefunction defines a topological number, that is, a Chern number, which is directly related to the quantized Hall conductance. This was the first example of a topological band structure, which is now widely generalized to the concept of topological insulators.

Meanwhile, Haldane had gained his undergraduate degrees and a PhD at Cambridge University—the latter, in 1978, under the tutelage of P. W. Anderson. After his period at the Institut Laue-Langevin, Haldane joined the University of Southern California in 1981. He contributed notably to the theory of Luttinger liquids, to 1D Heisenberg spin chains, to exclusion statistics, and, more recently, to entanglement spectra.

However, in 1988 Haldane published an important paper (5) proposing an explicit model on the honeycomb lattice that embodied the topological bands introduced by Thouless et al. (4). The hallmark of such “topological insulators” is the existence of a robust edge or “surface metallic state” while the bulk remains an insulator. Earlier, in 1983, Haldane had also considered, from a topological viewpoint, a 1D antiferromagnetic quantal spin chain, in the case of large spins (6). Basically, this is a standard model in condensed matter physics, but, so it turns out, highly nontrivial because the quantal zero-point fluctuations mean that the standard antiferromagnetically ordered Néel state is not the ground state. As a result of the destruction of the antiferromagnetic order, the system becomes, in fact, a quantal spin liquid.

Haldane predicted the essential behavior that depends upon the value of the spin quantum number S. What he discovered is that the contributions to the quantum amplitude can be classified into topological sectors characterized by an integer $Q$, in which the quantum amplitude is multiplied by a phase factor $e^{i2\pi SQ}$. This factor is always 1 for integer S, whereas it alternates between +1 and −1 for half-integer S. This results in an astonishing prediction that S = 1 behaves more quantum mechanically compared with S = 1/2 in stark contrast to the usual common sense that larger S means more classical. This arises because the negative interference of quantum amplitude between the even and odd $Q$ sectors reduces the quantum fluctuation for half-odd integer S. The large quantum fluctuation for S = 1 results in a gap in the excitation spectrum, now called the “Haldane gap.” This Haldane gap was later confirmed experimentally. Quantum spin liquids are still a hot topic in current condensed matter physics, attracting intensive interest, and now being extended to higher dimensions as symmetry-protected topological phases include interaction effects. One might say that Haldane was the first to launch this deep issue in condensed matter physics.

These works are a breakthrough in many respects. First, they revealed the novel mechanisms of phase transitions in which the topological defects are essential. Second, they discovered the nontrivial topological phases of matter characterized by integers and sometimes combined with symmetries. More generally speaking, these studies have revealed the role of topological objects in nonperturbative effects. Namely, most of the theoretical analysis had been based on perturbative expansion with respect to the interactions or field configurations. This means that only states that continue smoothly from the unperturbed, usually trivial, states were considered. Nonperturbative effects come from different topological sectors, that is, with finite topological indices. This concept is ubiquitous all through modern theoretical physics.