In the early 1970s, David Thouless was thinking about vortices. Not the kind that form in a stirred cup of coffee, but the topological defects that can appear in thin films of superfluid helium—quantized swirls that carry a fixed amount of circulation. At the time, Thouless was a condensed-matter theorist at the University of Birmingham, working on phase transitions. But to describe how vortices could unbind and drive a transition, he reached for equations that had been written for ocean currents and atmospheric flows. That borrowing—taking the mathematical language of fluid dynamics and applying it to the electronic properties of solids—would eventually help reshape condensed-matter physics, leading to the discovery of topological insulators, Weyl semimetals, and a whole new classification of quantum phases.

A Borrowed Language from Fluids

The equations of fluid dynamics are built around a handful of core concepts: velocity, pressure, density, and—most crucially for topology—vorticity. Vorticity is a local measure of rotation in a flow, defined as the curl of the velocity field. In the late 19th century, Hermann von Helmholtz and Lord Kelvin showed that in an ideal fluid, vorticity is conserved along flow lines, and vortex lines cannot end inside the fluid—they form closed loops or extend to boundaries. This conservation law is topological in nature: it constrains the possible shapes and connections of vortex structures.

By the 1960s, physicists studying liquid crystals had begun to borrow these ideas. Liquid crystals are phases of matter that flow like liquids but retain some orientational order. Their molecules can align in patterns that contain topological defects—points or lines where the orientation is undefined. The mathematics used to classify these defects, developed by the French mathematician René Thom and applied by Pierre-Gilles de Gennes, drew heavily on the homotopy theory of maps from the material to an order-parameter space. But the intuitive picture came from fluids: defects were like vortices, and their stability came from topological conservation laws similar to Kelvin's theorem.

Thouless and his collaborator John Kosterlitz recognized that a similar mechanism could operate in two-dimensional superfluids. In a thin film of superfluid helium-4, the phase of the condensate can vary in space, and quantized vortices—points where the phase winds by a multiple of 2π—can exist. At low temperatures, vortices of opposite circulation bind together in pairs. As temperature increases, pairs unbind, and the free vortices destroy the superfluid order. The Kosterlitz-Thouless transition, for which they shared the 2016 Nobel Prize, was described using the language of vortex unbinding—a direct import from fluid dynamics.

This cross-pollination was not a one-way street. Fluid dynamicists had long used the concept of winding numbers—how many times the velocity field rotates around a point—to characterize vortices. In condensed matter, winding numbers became a way to classify topological defects in ordered systems, from magnets to superconductors. The bridge between continuum fluid models and lattice models of solids was built by mapping the phase of a superfluid wavefunction onto a velocity field, effectively treating the condensate as a fluid whose flow is irrotational except at singular points.

When Vortices Became Quasiparticles

Superfluid helium experiments in the 1960s and 1970s provided the first clear evidence that vortices could be treated as individual quantum objects—quasiparticles with well-defined properties. In rotating superfluid helium, arrays of quantized vortices form, each carrying exactly one quantum of circulation: h/m, where h is Planck's constant and m the mass of a helium atom. These vortices are topological excitations: they cannot be created or destroyed singly without changing the topology of the superfluid order parameter.

Alexei Abrikosov, working at the Kapitza Institute in Moscow, had earlier predicted a similar phenomenon in type-II superconductors. In a magnetic field, magnetic flux penetrates a superconductor in quantized tubes—Abrikosov vortices—each carrying one flux quantum. Abrikosov's 1957 paper, largely ignored at the time, described how these flux lines form a periodic lattice. The mathematics of vorticity and circulation applied directly: the supercurrent circulating around a flux line is exactly analogous to the flow around a fluid vortex, and the quantization condition comes from the requirement that the wavefunction be single-valued.

The connection deepened when theorists began to map vortex dynamics onto the Berry curvature of electronic bands. Berry curvature, introduced by Michael Berry in 1984, is a geometric property of quantum states that describes how the phase of a wavefunction changes as parameters are varied. In a solid, the Berry curvature of occupied electronic bands can act like a magnetic field in momentum space. Remarkably, the equations of motion for a wavepacket in a crystal, including the anomalous velocity due to Berry curvature, take the same form as the equations for a charged particle moving in a magnetic field—or, equivalently, for a fluid element with vorticity.

By the late 1980s, it was clear that the circulation of a superfluid vortex and the Berry curvature integral over a Brillouin zone were both manifestations of the same topological invariant: the Chern number. What had started as a description of fluid swirls had become a tool for classifying electronic band topology.

The Chern Number That Came from Circulation

The Chern number is a topological invariant that arises in the study of fiber bundles in mathematics, but its physical meaning in condensed matter is intimately tied to circulation. In the integer quantum Hall effect, discovered by Klaus von Klitzing in 1980, the Hall conductance of a two-dimensional electron gas in a strong magnetic field is quantized in integer multiples of e²/h, with extraordinary precision. The quantization is robust against disorder and sample details—a signature that it is topological in origin. In 1982, Thouless, together with Mahito Kohmoto, Peter Nightingale, and Marcel den Nijs (known as TKNN), derived a formula linking the quantized Hall conductance to the Chern number of the occupied electronic bands. The derivation used a mathematical object called the Chern-Simons form, originally developed in the context of gauge field theories. But the physical interpretation was fluid-like: the Hall current can be thought of as a circulation in momentum space, and the Chern number measures the net winding of the electronic wavefunction across the Brillouin zone. The fluid analogy is striking: in a fluid, the circulation around a closed loop is a topological invariant if the enclosed region contains no vorticity sources or sinks. In the quantum Hall system, the Hall conductance is proportional to the sum of Chern numbers over occupied bands—a kind of total circulation of the Berry curvature in momentum space. Just as Kelvin's theorem says that circulation is conserved in an ideal fluid, the Chern number is robust under continuous deformations of the Hamiltonian, as long as the energy gap remains open. This insight was transformative. It meant that the quantum Hall effect was not an isolated curiosity but the first example of a topological phase of matter—a state whose properties are determined not by local details but by global, topological features of its band structure. The fluid-dynamics-inspired language of circulation and vorticity provided the intuitive handle that made this abstraction concrete.

Turning Turbulence into Topology

Not all fluid concepts that migrated into condensed matter were about orderly vortices. Turbulence—the chaotic, multiscale motion of fluids—also contributed a topological invariant: helicity. Helicity measures the degree to which vortex lines are knotted or linked in a flow. It is conserved in ideal fluids, as shown by Keith Moffatt in 1969. Moffatt's helicity is defined as the integral of the dot product of velocity and vorticity over a volume, and it is related to the linking number of vortex filaments.

In magnetohydrodynamics, magnetic helicity plays a similar role: it measures the linkage of magnetic field lines and is conserved in ideal plasmas. This topological conservation law has been used to understand solar flares, dynamo action, and the stability of fusion plasmas. In condensed matter, helicity and its generalizations appear in the study of chiral anomalies in Weyl semimetals, where the interplay of electric and magnetic fields leads to a non-conservation of charge in individual Weyl nodes—an effect that mirrors the production of helicity in turbulent flows.

The modern twist is that topological protection—the immunity of certain surface states to disorder—can be understood in terms of fluid-like conservation laws. For example, the surface states of a topological insulator are robust because their existence is guaranteed by the bulk topology, much as a vortex line in a superfluid cannot end arbitrarily. The mathematics of homotopy, which classifies defects in ordered media, provides the rigorous foundation, but the fluid picture often guides intuition.

Experimental Signatures in Real Materials

The theoretical framework built from fluid analogies predicted a wealth of new materials. The first experimental confirmation of a topological insulator came in 2007, in bismuth-antimony alloys. Using angle-resolved photoemission spectroscopy (ARPES), researchers at Princeton and other institutions observed surface states with a Dirac cone—a linear dispersion of electrons that is topologically protected. The surface states were metallic while the bulk remained insulating, exactly as predicted by the topological band theory derived from Berry curvature and Chern numbers.

Subsequent experiments on materials such as Bi₂Se₃ and Bi₂Te₃ confirmed the presence of spin-momentum locking—a property where the electron's spin is tied to its direction of motion, analogous to the polarization of a wave in a chiral medium. In Weyl semimetals, predicted by theorists using a model that borrowed from fluid dynamics to describe the chiral anomaly, experiments observed negative magnetoresistance and Fermi arcs on the surface—direct signatures of the topological band structure.

ARPES has been the workhorse for these discoveries. The technique measures the energy and momentum of electrons ejected from a crystal by ultraviolet light, revealing the band structure directly. The topological surface states appear as sharp, linear bands crossing the Fermi level, and their spin texture can be mapped using spin-resolved ARPES. These measurements have confirmed the predictions of fluid-inspired models with remarkable fidelity.

What the Cross-Pollination Changed

The importation of fluid dynamics into condensed-matter topology changed the way physicists think about phases of matter. Before the 1980s, phases were classified by symmetry: solids break translational symmetry, magnets break rotational symmetry, and so on. The discovery of topological phases added a new dimension: phases that cannot be distinguished by any local order parameter but only by global, topological invariants. The fluid-dynamics-inspired concepts of circulation and vorticity provided the first concrete examples of such invariants in a condensed-matter context.

This new classification has been enormously predictive. It led to the prediction and discovery of topological insulators, topological superconductors, and Weyl semimetals—materials with exotic properties that could be useful for spintronics, quantum computing, and low-power electronics. The fluid analogies also inspired the search for higher-order topological states, where the protected modes appear at corners or hinges rather than surfaces. These states were predicted using the same mathematical tools of nested Wilson loops and Berry phases that trace back to the circulation integrals of fluid dynamics.

Yet the cross-pollination has limitations. Not all fluid concepts survive the transition to the quantum, lattice world. For example, the direct analogue of turbulent cascades in momentum space—a flow of energy from low to high momenta—has not been observed in electronic systems, though some theorists have proposed that similar mechanisms might operate in driven systems. The fluid picture is most powerful when it is backed by rigorous mathematics; without that, analogies can mislead.

Lessons for the Next Borrowed Tool

The story of fluid dynamics and condensed-matter topology offers lessons for future cross-disciplinary transfers. First, analogies require precise mathematical mapping: the success of the fluid-to-topology transfer depended on the fact that vorticity in fluids and Berry curvature in solids obey the same differential form equations. Second, not every fluid concept has a quantum analogue—turbulence, for instance, remains elusive in electronic systems. Third, experimental falsifiability is essential: the predictions of topological band theory were tested and confirmed by ARPES and transport measurements before the field gained broad acceptance.

Cross-disciplinary workshops and collaborations accelerated the transfer. The Aspen Center for Physics and the Kavli Institute for Theoretical Physics have hosted meetings where fluid dynamicists and condensed-matter theorists share methods. These exchanges have sparked new ideas, such as the application of machine learning to compute topological invariants from raw data—a technique that borrows from both fields.

One open question is how to extend topological classification to nonequilibrium and dissipative systems. Fluid dynamics has long dealt with dissipation through viscosity, and its methods for handling non-ideal flows may offer a path forward. Another is whether the fluid analogy can help understand the interplay of topology and interactions—a notoriously difficult problem in condensed matter. As the field moves beyond non-interacting electrons, the fluid-dynamics toolbox may prove even more valuable.

The borrowing that began with Thouless and superfluid vortices has become a permanent part of condensed-matter physics. A concrete example of its continued influence is the work of physicist B. Andrei Bernevig at Princeton, who used topological band theory to predict the quantum spin Hall effect in HgTe quantum wells—a prediction confirmed experimentally by Laurens Molenkamp's group at the University of Würzburg in 2007. This success demonstrated that the fluid-inspired topological invariants could guide the discovery of new materials with precision. The story shows that the boundaries between subfields are porous, and that a good idea—whether it comes from oceanography or gauge theory—can travel far if it arrives with the right mathematics. What remains to be seen is whether fluid dynamics will again provide the key to unlocking topological phases in driven or interacting systems, or whether the next borrowed tool will come from an entirely different field.