How electrons break into pieces inside one-dimensional matter

Hari Padma
5 min readJul 30, 2022

What makes a cow a cow? A cow has four limbs, a pair of horns, a tail, and eats grass (among other things). What makes an electron an electron? An electron is arguably defined by two traits — its charge (the stuff that makes up electricity), and its spin (the stuff that makes up magnetism). However, this basic identity of an electron quite literally falls apart when it’s confined to live inside certain ‘one-dimensional’ materials.

Before diving into how this happens, or what a one-dimensional material even is, let’s look at something more familiar — electrons inside regular old three-dimensional materials. Since electrons are electrically charged, they strongly interact with each other via the Coulomb interaction. Pauli’s exclusion principle further demands that no two electrons can be at the same place. As a result, the motion of each electron is correlated with every other electron in the material. Describing the behavior of such a system of interacting electrons within the framework of quantum mechanics is an incredibly difficult mathematical problem. Luckily, as it turns out, this complex behavior of interacting electrons can be modeled remarkably well by assuming that the electrons are non-interacting, or free, albeit with some dynamical traits such as the effective mass appropriately modified. This is the essence of Fermi liquid theory developed by Lev Landau. Its disarming simplicity notwithstanding, it accurately describes the behavior of electrons in almost all the metals we encounter in everyday life.

Free electrons (left) and interacting electrons (right) moving in a three-dimensional material. Here, interacting electrons behave just like free electrons, albeit ‘dressed’ by interactions (yellow clouds).

If a free electron is a cow, you might say an interacting electron in a three-dimensional material is a cow in a dress. It looks a bit different, and it cannot move quite as fast — but it’s still a cow.

The simple expectation that an electron should fundamentally behave like an electron falls apart inside certain ‘one-dimensional’ materials. These can come in different shapes and forms, for example chains of copper and oxygen atoms, organic hydrocarbon complexes, and even ultra-cold atomic gases. Loosely speaking, one-dimensional materials are those in which atoms are strongly bonded along one dimension, and very weakly bonded along the other two dimensions. This means that the electrons live in what is essentially a one-dimensional world.

The life of an electron is quite different in one dimension. In general, strong Coulomb repulsion and Pauli’s exclusion principle mean that when electrons move around, they like to avoid each other. However, this is not easy to do in one dimension, since the electrons literally have no way to get around each other. Indeed, if you naively try to model these electrons as individual entities, you start running into infinities in the math, a sign that something is seriously wrong with the theory. This is because in one dimension, for one electron to move, all the electrons must move; the motion is collective.

The collective motion of interacting electrons in a one-dimensional material.

This collective behavior of electrons in one dimension was described mathematically by Shin’ichiro Tomonaga, Joaquin Mazdak Luttinger, and subsequently Duncan F. Haldane, in a model known as the ‘Tomonaga-Luttinger liquid’. Here, instead of treating each electron individually, you describe them by their collective motion. The technique is mathematically analogous to how collective atomic vibrations in materials, or phonons, the quanta of sound, are described. For the technically inclined, this method is called ‘bosonization’, the name referring to the fact that electrons, which are fermions, are mathematically transformed into objects that resemble collective bosonic excitations.

This is where things get bizarre — in the resulting solutions for the collective electronic motion, charge and spin, the two fundamental traits that define an electron, move independent of each other, at different speeds! Once again drawing a parallel to cows, it’s as if you lined up a herd of cows, and instead of moving collectively, one after another, the heads and bodies of the cows moved at different speeds and separated from each other. This remarkable, counter-intuitive phenomenon, a consequence of Coulomb interaction in concert with quantum mechanics, is called ‘spin-charge separation’.

A schematic of the dispersion of electrons in three-dimensional materials (left), compared to experimentally measured dispersion of spinons and holons in a one-dimensional material (right). The dispersion (Kim, et al. Nature Physics (2006)) was measured using angle-resolved photoemission spectroscopy, on the compound SrCuO2.

Spin-charge separation manifests in different ways. A characteristic, defining property of any type of wave (or equivalently by wave-particle duality, any type of particle) is its dispersion, i. e. the relationship between its energy and its wavelength. For example, it is the dispersion of light waves (or equivalently, photons) in air that gives rise to rainbows. The dispersion of electrons in solids is routinely measured using spectroscopic tools. However, when you measure the dispersion of electrons in such one-dimensional materials, instead of a single dispersion curve, what you detect is two separate dispersion curves — one for spin, and one for charge, just as predicted by the theory. If you carry out a simulation of interacting electrons in one dimension, you observe that the spin and charge collective excitations move at different speeds. And indeed, when you write down the equations describing such a system, the spin and charge components neatly separate into two independent sets of equations.

Numerical simulation of charge (upper curves) and spin (lower curves) densities in a model of 1D interacting electrons called the Hubbard model. The results show that the spin and charge propagate at different speeds, and hence separate from each other. The results are from work by Jagla, et al. Physical Review B (1993).

It’s as if the electron we knew died, and from it emerged two new particles, each with its own set of unique physical properties — one with only spin and no charge, (referred to as a spinon) and another with only charge and no spin, (referred to as a chargon).

This is an example of emergence; when interactions between a multitude of individual objects give rise to new phenomena requiring a fundamentally new, higher level physical description. Materials are bursting at their seams with such emergent phenomena. Emergence is indeed at the heart of the field of condensed matter physics, the study of materials. For what it’s worth, emergence is also why cows exist. Hijacking the oft-used idiom, there are no cows in the fundamental laws of physics! While almost all emergent phenomena (phonons, clouds, plant cells, cows) in some way meaningfully transcend the sum of their constituent parts, spin-charge separation is a somewhat rare instance of emergence that fractionalizes its constituent parts.

This is but a brief glimpse into the fascinating world of interacting electrons. Recent experiments have shown that in addition to spin-charge separation, electrons in one-dimensional materials can further fractionalize, giving rise to emergent orbitons, which are particles that carry the orbital angular momentum quantum number. The Nobel Prize winning discovery of the fractional quantum Hall effect, one of the most stunning instances of emergence in physics, is a phenomenon where the electronic charge itself fractionalizes. Condensed matter physics is the gift that keeps on giving.



Hari Padma

Postdoctoral researcher at Harvard, studying the physics of materials.