Ian Affleck

Ian Affleck, a pioneer in the modern field-theory approach to condensed-matter physics, passed away on 4 October 2024. He received numerous honors, including the Canadian Association of Physicists Medal for Lifetime Achievement in Physics (2006), a fellow of the Royal Society (2010), the Onsager Prize (American Physical Society, 2012), and a foreign associate of the French Academy of Sciences (2015).

Ian started his career in high-energy physics, obtaining a PhD at Harvard University in 1979 under the guidance of Sidney Coleman. After being a junior fellow at Harvard, he was appointed as an assistant professor at Princeton University in 1981. Notable contributions to high-energy theory in his early career include the theory of baryogenesis (Affleck–Dine mechanism) and dynamical supersymmetry breaking (Affleck-Dine-Seiberg superpotential). These works have had lasting impacts even after he largely left the field of high-energy theory.

Between 1984 and 1985, Ian spent his sabbatical at CEA Saclay. Around that time, he shifted to condensed-matter physics. Capitalizing on his expertise, he became one of the leaders who transformed condensed-matter theory to that based on modern field-theory methods. He continued his studies of condensed-matter physics after moving to the University of British Columbia (UBC) as a professor in 1987, where he stayed until his retirement in 2021, except for two years at Boston University, between 2001 and 2003.

He was first interested in studying quantum Hall transitions with the nonlinear sigma model and found a connection to quantum spin chains. He made significant contributions to establishing the Haldane conjecture by generalizing the Lieb-Schultz-Mattis theorem to an arbitrary half-integer spin quantum number with Elliott Lieb, and by constructing the Affleck-Kennedy-Lieb-Tasaki (AKLT) model with an exact AKLT ground state for an arbitrary integer spin quantum number. The AKLT state was the first example of more general matrix product states in one dimension and a tensor network in higher dimensions, which are of increasing importance in contemporary numerical and analytical studies of many-body systems and quantum information. The AKLT state was also pivotal for later findings of the edge states and the non-local string order in the Haldane gap phase. Those features were identified as signatures of a symmetry-protected topological phase, more than 20 years after the construction of the AKLT state.

Ian also played a crucial role in applications of conformal field theory (CFT) to condensed-matter physics. First, he derived (independently of the similar work by Blote, Cardy, and Nightingale at the same time) the universal finite-size correction to the free-energy density of a 1D quantum critical system at finite temperatures, which is given in terms of the central charge of the corresponding CFT. This made CFT a practically important tool to study 1D quantum critical systems. Furthermore, with Andreas Ludwig, he developed the boundary CFT approach to multichannel Kondo effects, demonstrating the non-Fermi liquid property beautifully. Also with Ludwig, he introduced the universal boundary entropy (“non-integer ground-state degeneracy”) to characterize conformally invariant boundary conditions and conjectured “g-theorem,” which states that the boundary entropy monotonically decreases along the renormalization-group flow. The boundary entropy and g-theorem became essential not only in condensed-matter physics but also in high-energy physics, where Ian came from. While Affleck–Ludwig only provided a proof in the lowest-order perturbation theory, more general proofs have been given later, including the one based on the strong subadditivity in 2024.

Ian was strongly interested in understanding and predicting experimental results. In particular, he applied the bosonization method to an S = ½ Heisenberg antiferromagnetic chain, which is one of the most fundamental models in magnetism with many material realizations. He found the previously unexpected infinite slope of the specific heat near zero temperature with Eggert and Takahashi, and he developed a theory of field-induced gap and electron spin resonance with Oshikawa. These were quantitatively verified by experiments, demonstrating the power of Ian’s approach. With Halperin, he also collaborated with experimentalists (Hagiwara, Katsumata, and Renard) to establish the existence of the edge states in the Haldane gap phase, which was theoretically predicted based on the AKLT model.

Other notable contributions include Ian’s influential work with Marston on the large-N limit of the Heisenberg–Hubbard model in the context of high-T_c superconductors, studies with Sirker and Pereira on transport in 1D quantum systems and, most recently, with Rahmani, Pikulin, and Franz on a series of papers describing emergent supersymmetry in chains and ladders composed of strongly interacting Majorana zero modes.

Ian had a very high standard in science and tended to publish papers only after he understood the physics in detail. Even when he was working with students or postdocs, he often performed complicated analytical calculations by himself. With his pure passion for physics, his personal integrity, and his kindness, he was widely regarded as a role model among friends and in the wider community. He also did not hesitate to go out of his way to help others, especially young people.

Ian’s talks were always clear and instructive. Because of this, he was often invited to give lectures on advanced topics in summer schools and similar events. In particular, the lecture notes “Field Theory Methods and Quantum Critical Phenomena,” based on his lectures at the legendary 1988 Les Houches Summer School, has served as a bible for the field for many years after publication. At UBC, his students from first-year undergraduates to graduate students have praised his efforts toward creating a welcoming classroom, his deep knowledge of physics, and his passionate devotion to sharing this knowledge with them.

Ian loved nature and hiked many mountains around Vancouver, often with family, friends, or physics colleagues. He was swimming regularly, and he also enjoyed windsurfing on the sea. Until his retirement, he cycled daily between his home and office through the forest, rain or shine. Perhaps he was among those who took the most advantage of the beautiful nature around Vancouver. He spent his final days with his beloved wife, Glenda.

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