A comprehensive theory for nematoelasticity

Theorists at the University of Illinois Urbana-Champaign address an experimental paradox by developing a general theory uniting a kind of order known as electronic nematicity with a crystal’s elasticity.

Electronic nematicity is a phase of some crystalline solids in which electrons’ collective properties, such as charge or spin densities, organize themselves into ordered patterns, lowering the crystal’s rotational symmetry. This phase is found across a wide range of diverse materials, making nematicity crucial to understanding emergent solid-state phenomena, such as unconventional superconductivity and magnetism. But lately, experimentalists have encountered a hurdle to understanding nematicity: despite exhibiting nematic order at macroscopic scales, at the microscopic level, many nematic materials seem to exhibit disorder instead.

To address this seeming paradox, theorists at the University of Illinois Urbana-Champaign have invented a new way of looking at the interactions between nematicity and elasticity, incorporating aspects of elasticity theory, whose impacts on nematicity have previously been overlooked. Their new model argues that a resolution to the paradox lies in how a crystal’s elasticity selectively couples to only certain types of nematic modes, while suppressing others associated with disorder.

This research was published in the journals Physical Review Letters and Physical Review B as Editors’ Suggestions on April 20th, 2026, and was featured in an APS Viewpoint.

Electronic nematicity

Symmetry is the property of an object that makes it look the same after some change is made: a square looks the same if you rotate it through 90, 180, 270, or 360 degrees—a four-fold rotational symmetry; so does a rectangle after a 180- or 360-degree rotation, a two-fold rotational symmetry. An object’s symmetries can also be changed, such as stretching a square into a rectangle, a reduction of symmetry known as symmetry breaking. Such symmetries and their breakings enable physicists to identify different states of matter and study them in spite of their complex interactions. In fact, symmetry breaking is a hallmark of interesting phenomenal including phase transitions.

One important symmetry-breaking phase in solids is electronic nematicity, in which the collective behavior of the electrons spontaneously breaks rotational symmetry. This process occurs when, for instance, a variable such as temperature is lowered below a critical value, inducing a phase transition into an ordered nematic state.

Illinois Physics Postdoctoral Research Associate Joe Meese led the current research, working with Illinois Physics Professor Rafael Fernandes.

Meese said, “The best examples of electronic nematicity are those in which electrons on a square atomic lattice collectively break the square’s four-fold symmetry down to the two-fold symmetry of a rectangle. The key idea is that the electrons themselves and their correlations break the lattice symmetry.”

This means that on a square lattice, the electronic degrees of freedom—charge, spin, the orbital states they occupy, and even the ways they superconduct—order themselves into highly correlated, rectangle-like configurations. These patterns can be directly observed by finding that the resistance measured along one direction, for instance, differs from the resistance measured along another direction, a clear sign of broken rotational symmetry.

First proposed as a correlated electronic state of matter in a 1998 paper co-authored by Illinois Physics Professor Eduardo Fradkin, electronic nematicity has been found across seemingly unrelated materials—two-dimensional electron gases, topological insulators, twisted bilayer graphene, the copper- and iron-based superconductors, and more—indicating that it plays an influential role in the emergence of correlated quantum phenomena, especially in some of the most interesting materials in condensed matter physics.

Recent experiments, however, have consistently raised a strange paradox.

“What’s been observed,” Meese noted, “is that if you look at the largest length scales, you can see nematic order. People observe the expected transition when lowering the temperature, and everything seems well-understood.

“But the problem is that when you look at microscopic scales, the nematicity is inhomogeneous.”

Inhomogeneity manifests as patches of order surrounded by large swaths of nematically disordered material. In other words, it appears that nematic order emerges at macroscopic levels despite a lot of disorder present at the microscopic level. How can order come from disorder?

Elasticity to the rescue

To solve this paradox, Meese and Fernandes turned to elasticity, which encodes the degree to which a material responds when bent, twisted, or stretched.

Meese described, “When electrons develop long-wavelength nematicity by symmetry, this has the same effect as an elastic strain, squishing, say, one axis and stretching the other. This symmetry perspective couples nematicity and elasticity to each other, an interaction we call ‘nematoelasticity.’”

In the last few years, elasticity has been implicated in some of nematicity’s most salient features, including direction-selective criticality, an effect where large fluctuations in nematicity appear in only certain momentum directions associated with lattice distortions.

Illinois Physics Professor Rafael Fernandes added, “People knew that a lattice’s properties can be directionally dependent. Since nematicity couples to the lattice, which responds by distorting, Joe realized we needed to understand elasticity more deeply.”

Like the contradictory nature of the supposed paradox, elasticity affects nematicity in seemingly opposing ways: Although it can promote nematicity by connecting different parts of a lattice, elasticity can also suppress nematicity through defects, which arise when the lattice is misaligned, missing atoms, or otherwise distorted, introducing strains that stamp out long-range nematic order. Such defects, no matter the skill of the crystal grower, exist in virtually every sample. Can a balance between elastic enhancement and suppression account for the paradox?

First off, a crystal’s strain comes in different types, and they’re not independent of one another. Indeed, they’re entangled through a set of equations called the compatibility relations (CRs), rules governing the different strains that ensure the crystal doesn’t break or crack. Other kinds of strain that don’t obey the CRs, called incompatible strain, originate from defects.

Despite being fundamental aspects of elasticity theory—derived as early as the 1860s—the CRs’ influence on nematicity has never been closely studied. Meese and Fernandes wondered if they could incorporate the CRs into a generalized theory of nematoelasticity.

A new basis from the old

Typically, nematic order is quantified by a set of five numbers called order parameters. Often expressed using a set of functions, a basis, similar to the atomic d-orbitals, these parameters describe all nematic configurations by specifying each as a point in an abstract five-dimensional space, just like a point in three-dimensional real space can be specified by three coordinates, x, y, and z.

“Unfortunately, it’s very cumbersome to incorporate the compatibility relations using the d-orbital basis,” Fernandes said. “This problem becomes very complicated.”

But as any mathematician knows, bases aren’t unique: a point expressed in one basis can be expressed just as well using a different one. The researchers needed a new basis, one that explicitly reveals the CRs’ effects on nematicity through the lattice.

Inspired by ideas from high-energy physics and guided by the fact that momentum directions of lattice distortions couple to nematicity in different ways, Meese built a basis that moves around with momentum, so that wherever it’s pointing, the basis follows. Like the d-orbital basis, this new one, called the helical basis, consists of five new order parameters, each corresponding to a different pattern of deformation, or mode, such as an expansion or compression.

“Those who study the dynamics of elasticity–or acoustics–would call this basis longitudinal and transverse phonons,” Meese explained. “This basis enables us to connect elasticity and nematicity in a tractable way, showing that they’re basically two sides of the same coin, and allows us to unify effects people have known about phonons to the disorder physics that people have studied in the past.”

Fernandes summarized, “Joe's insight was to find a basis where the compatibility relations are automatically satisfied without having to solve them. This new, helical basis naturally accounts for what’s compatible and what’s not, and everything becomes much simpler.”

The answer: selective coupling of elasticity to nematic order

Armed with their new basis, the researchers observed the behaviors of the basis’ order parameters in both defect-free and defect-containing crystals. They found that the CRs split up the parameters and the modes associated with them into two types, according to their energy cost: compatible and incompatible.

 

“The compatible modes satisfy the compatibility relations of the lattice,” Fernandes elaborated. “These are ‘good’ modes in that they don’t cost a lot of energy. On the other hand, incompatible modes that only appear in the presence of defects cost a lot of energy.”

Nature generally favors low-energy states—think of how a ball rolls down a hill to minimize its energy, rather than up. So splitting the modes up this way thus enabled Meese to show that the modes are not created equal.

Meese said, “I found that only specific momentum directions are impacted by disorder because they feel the effects of defects. But there exist other directions for which defects cannot make any strain appear. For these directions, defects cannot couple to nematicity.”

That is, elasticity doesn’t couple to nematicity in a straightforward way. Instead of generating a simple combination of enhancing and suppressive effects on nematicity, as was previously conjectured, Meese and Fernandes found that elasticity enhances only certain aspects of nematicity at the expense of others. Specifically, elasticity selectively enhances the low-energy compatible modes while suppressing the higher-energy incompatible modes, the ones feeling disorder.

How does this resolve the paradox? Because defects exist in virtually every real crystal sample, they’re bound to induce disorder, as seen at the microscopic level. However, through the CRs, elasticity “shields” the compatible modes from the disorder, enabling nematic order to emerge at macroscopic scales. The modes associated with disorder therefore don’t destroy nematicity at large scales, acting only on short-length, microscopic scales.

As a corollary, Meese noted, the suppression of the incompatible modes also means that in all crystals (even isotropic ones) that lack directionally dependent lattice properties, “direction selectivity–the coupling between momentum direction and nematic fluctuations–persists.” This is a surprising discovery given that physicists had previously attributed this effect to crystal anisotropy.

From nematoelasticity to nematoplasticity and beyond

These findings about the intricate coupling between nematicity and elasticity are far-reaching, as the CRs don’t just apply to specific crystals, but to all crystals, providing a general framework for nematoelasticity.

Fernandes summed up, “What Joe developed is a beautiful way of merging two theoretical frameworks—nematicity and elasticity—into a unified theory of nematoelasticity, showing how nematicity inherits the effects of the compatibility relations.”

Meese added, “What we did shows that we should always keep the elastic regime in the back of our minds, especially when we want to talk about macroscopic electronic nematicity. The compatibility relations show that once nematic states start deforming a crystal, elasticity’s complicated rulebook can alter electronic behaviors or lead to entirely new ones.”

With the resolution of the paradox, Meese and Fernandes are ready to take on new challenges: Recognizing the power of the helical basis invented by Meese, Fernandes is excited to begin applying it to both old and new problems.

“With this new tool,” Fernandes concluded, “we can revisit a lot of questions that condensed matter physicists working on electronic liquid crystalline phases, myself included, have studied in the past. For example, how does nematicity promote superconductivity? And how does nematicity affect quantum phase transitions?

“We can also go in completely new directions, such as asking how defects interact with each other by applying plastic, rather than elastic, deformations. If you stretch an elastic rubber band just a little bit, for example, it goes back to its original state. But if you strain it too much—a plastic deformation—it never goes back. This is because you’ve created new defects and moved them around. In this way, we can study how plastic deformations affect nematicity and vice versa, opening up an entirely new field called nematoplasticity.”

Meese looks forward to further investigations, noting, “I’m most excited by the prospect of nematic waves as sources of plastic deformation. It may be possible for these waves to push defects around, or even more dramatically, make new ones. If true, then instead of the lattice acting only as a vibrating background, it may be that the disorder within the crystal structure itself is important dynamical degrees of freedom, with properties that depend heavily on the electronic correlations in a material.”