In physics, a change from solid to liquid is called a phase transition, but in two-dimensional systems only a few atoms thick the usual theory fails. Kosterlitz and Thouless proposed that tiny vortices pair up at low temperature and unbind when heated, explaining how superfluid films or thin superconductors lose resistance. Haldane later showed that similar topological effects occur in one-dimensional spin chains and in thin films even without a magnetic field, predicting quantized electrical conduction. Topological phases are robust against impurities and fluctuations, so their properties survive small environmental changes and are attractive for devices. The theory revolutionized our understanding of the quantum Hall effect and topological insulators, leading to new standards for precision measurement and low-power electronics. Future applications include fault-tolerant quantum computing elements and advanced communication materials that could underpin tomorrow’s infrastructure.

The laureates’ work began with the Berezinskii–Kosterlitz–Thouless (BKT) transition, which describes phase changes in two-dimensional systems via the binding and unbinding of topological defects. In the KT framework, competition between the energy of vortex pairs and their entropy leads to a transition where pairs dissociate at a critical temperature, producing an exponential jump in the superfluid density. Thouless and Kosterlitz used renormalization-group analysis of the XY model to show that the transition is characterized by the universal exponent η = 1/4. Haldane, on the other hand, predicted that integer-spin one-dimensional Heisenberg chains possess a finite excitation gap (the Haldane gap) and introduced a honeycomb-lattice model in which the quantum Hall effect occurs without an external magnetic field. These systems are classified by Chern numbers and other topological invariants that guarantee phases cannot be smoothly connected without closing the gap. Edge states in topological materials transport spin or charge with negligible back-scattering, a property being explored for spintronics and quantum-information applications. Today the original ideas have expanded into studies of topological superconductors, symmetry-protected topological (SPT) phases, and higher-order topological insulators, making topology central to many-body physics. The concepts now permeate cold-atom platforms, photonic crystals, and mechanical metamaterials, fueling interdisciplinary research across condensed-matter and beyond.

The laureates formulated a non-perturbative description of the 2D U(1)-symmetric XY model, showing that superfluid density exhibits a universal jump at T_BKT due to vortex–antivortex unbinding. The Kosterlitz renormalization-group flow reduces to the Kosterlitz equations, yielding the hyperbolic flow line that separates bound and free vortex phases and giving critical exponents ν = 1/2 and η = 1/4. Thouless further explained the integer quantum Hall effect by identifying the Hall conductance σ_xy = (e^2/h)C with the Chern number C of the occupied Bloch bands, accounting for its exact quantization. Haldane mapped integer-spin SU(2) Heisenberg chains to a nonlinear sigma model with a topological θ-term, distinguishing gapped even-spin chains from gapless odd-spin chains via θ = 2πS. He also introduced the Haldane honeycomb model whose valence and conduction bands carry opposite Chern numbers, predicting a quantum Hall state without net magnetic flux, recently realized in cold-atom optical lattices. These breakthroughs imported Berry phase and Pontryagin invariants into solid-state physics, providing the conceptual foundation for bulk-edge correspondence, topological order, and symmetry-protected topological phases. The ideas now underpin K-theory classification schemes such as the Ryu–Schnyder table, guide research on strongly correlated and non-equilibrium driven systems, and inform modern renormalization-group approaches. Protection of quantum anomalies, emergence of Majorana zero modes, and prospects for topological quantum computation all rely on this framework, making it indispensable in both condensed-matter and field-theoretic contexts.