Topological Twist for Phase Transitions

Robert-Jan Slager

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Topological Twist for Phase Transitions

Department of Physics and Astronomy, University of Manchester, Manchester, UK

June 23, 2025• Physics 18, 118

Contrary to conventional wisdom, so-called order parameters that distinguish symmetry-governed phases of matter can have topological structure.

Figure caption — **Figure 1:** Sun and Maciejko [3] considered a condensed-matter system whose phase diagram is parameterized by the temperature T and two additional quantities, $𝛼_{1}$ and $𝛼_{2}$ . The system has a high-temperature, symmetry-conserving phase (gray region) and a low-temperature, symmetry-breaking phase (blue region). These phases are distinguished by a quantity known as an order parameter that depends on T, $𝛼_{1}$ , and $𝛼_{2}$ . When the researchers tracked this parameter around a loop (purple) in the plane of $𝛼_{1}$ and $𝛼_{2}$ , they found that it could acquire a geometric phase. This phase is represented by the different direction of the purple arrow after it has traced a loop on the spherical surface. These findings suggest that order parameters can have hidden topological structure.

From materials developing magnetization patterns to metals becoming superconductors, a wide range of phase transitions can be qualitatively described by a single framework known as Ginzburg-Landau theory [1, 2]. This framework generally assumes that a key quantity in its descriptions, called an order parameter, has trivial topology. But now, Canon Sun and Joseph Maciejko at the University of Alberta, Canada, have shown that order parameters can have hidden topological structure [3]. The researchers have developed an extension to Ginzburg-Landau theory that incorporates such hidden topology, revealing features absent from the original framework.

Symmetry constitutes a fundamental concept in physics. It appears in many guises but is especially important when studying how interactions of countless microscopic constituents give rise to macroscopic order in condensed-matter systems. For example, below a critical temperature, an ordinary magnet has a net magnetization because its spins all align in the same direction, breaking rotational symmetry. If the magnet is heated above that temperature, it loses its magnetization as its spins point in random directions, restoring rotational symmetry.

Ginzburg-Landau theory is a universal, phenomenological tool for analyzing such phase transitions. In short, one considers the free energy (a quantity governing the system’s equilibrium state) to be a smooth function of an order parameter (a quantity distinguishing the different phases). Expanding this function in a power series and minimizing it yields an effective theory of the phase transition and the values of the order parameter. For the magnet example, the coefficients of the expansion depend on temperature and change sign at the critical temperature. This feature results in an order parameter that has a finite value below the critical temperature and is zero above it.

Given that these phases are characterized by their symmetry, there is a direct relationship between the parent symmetry of the system and the order parameter. This link is formalized by the mathematical description of reflections, rotations, and other symmetry operations. Any representation of a symmetry operation can be decomposed into its smallest building blocks, known as irreducible representations. For example, a matrix that describes how the corners of a triangle transform when the shape is rotated by 120° can be expressed as matrices that cannot be reduced further. Because the free energy, and thus the order parameter, must respect the parent symmetry, the order parameter has specific requirements from a representation standpoint. In particular, when a symmetry operation is applied to a system in its ordered phase, the order parameter must transform according to a single, precisely defined irreducible representation.

The success of symmetry perspectives in physics can hardly be overstated. But following the discovery of the quantum Hall effect in the 1980s [4], another, seemingly distinct, mathematical take on classifying matter gained traction: topological characterizations. These analyses can deliver quantized invariants—quantities, such as so-called winding numbers, that effectively take on the role of the order parameter. These quantities are insensitive to smooth changes in the system’s parameters, as long as an energy gap remains between the system’s ground and excited states.

Geometrically speaking, topological evaluations are closely related to global geometric phases, such as so-called Berry phases [5], that emerge when one follows an eigenstate slowly (adiabatically) around a loop in parameter space. As an example, consider a spin in a magnetic field that points radially outward and has a constant magnitude. One can track the adiabatic evolution of the spin’s eigenstate over a spherical surface of constant field strength back to the initial point to find that the state has acquired a global phase. Such phases can be quantized or directly linked to topological winding numbers, capturing the “winding” of the spin state along the closed path in parameter space.

Over the past two decades, it has become clear that symmetry and topology are not as distinct as initially thought. In fact, the presence of symmetries can act as a condition to guarantee the existence of generalized topological invariants. This concept underpinned the discovery of topological insulators that exhibit symmetry-induced invariants and of topological metals that host electronic-band touching points around which winding numbers can be defined [6]. Accordingly, in the past few years, uniform views on classifying such topological materials have emerged [7–9]. The idea is to precisely track the irreducible representations under which eigenstates transform as one moves between points of high symmetry in parameter space.

In their theoretical proposal, Sun and Maciejko show that even the hallmark of symmetry approaches—the order parameter—can have topological structure. Their key insight is that, although the order parameter in Ginzburg-Landau theory must transform according to a single irreducible representation, multiple order parameters associated with different orders could transform under the same irreducible representation and contribute simultaneously to form a composite order parameter. Returning to the magnet example, one could consider two different spin orderings that relate to the same irreducible representation. That scenario would give rise to a composite order parameter that depends not only on temperature but also on two additional parameters that describe the internal degrees of freedom of these simultaneous phases below the critical temperature. Crucially, such extra parameters allow one to make loops in the system’s phase diagram that can result in Berry phases and winding numbers—that is, topological structures (Fig. 1).

Sun and Maciejko specifically analyzed their topological Ginzburg-Landau theory for temperature-dependent superconducting phase transitions. Symmetry-broken superconducting orders are characterized by an order parameter that describes the system’s energy gap. Using a general model of interacting electrons, the researchers found that this region of phase space could contain two ordering patterns that relate to the same irreducible representation. By studying the resulting composite order parameter, the researchers identified different scenarios that give rise to nontrivial global phases and winding numbers.

In particular, Sun and Maciejko showed that, if time-reversal symmetry is preserved, the order parameter can acquire a Berry phase with a value of $𝜋$ after following a loop in parameter space. By contrast, if that symmetry is broken, one can obtain an analogue of a so-called topological Weyl semimetal in which winding numbers around band touching points can be defined. Moreover, the researchers note that the $𝜋$ -valued Berry phase qualitatively influences the Josephson effect [10]—the flow of current from one superconductor to another. Their theory predicts that, instead of changing sign, the current across the junction returns to its original value upon closing a loop in parameter space. This effect could be observed in future experiments.

This work by Sun and Maciejko opens many avenues for identifying new topological structures and observable effects in systems characterized by composite order parameters. Given its general applicability, the presented topological Ginzburg-Landau theory sets the stage for an exciting research agenda.

References

L. D. Landau, “On the theory of phase transitions,” Collected Papers of L. D. Landau (Pergamon, Oxford, 1965).
V. L. Ginzburg and L. D. Landau, “On the theory of superconductivity,” Collected Papers of L. D. Landau (Pergamon, Oxford, 1965).
C. Sun and J. Maciejko, “Topological Landau theory,” Phys. Rev. Lett. 134, 256001 (2025).
K. von Klitzing, “The quantized Hall effect,” Rev. Mod. Phys. 58, 519 (1986).
M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392, 45 (1984).
X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057 (2011); M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” 82, 3045 (2010); N. P. Armitage et al., “Weyl and Dirac semimetals in three-dimensional solids,” 90, 015001 (2018).
J. Kruthoff et al., “Topological classification of crystalline insulators through band structure combinatorics,” Phys. Rev. X 7, 041069 (2017).
H. C. Po et al., “Symmetry-based indicators of band topology in the 230 space groups,” Nat. Commun. 8, 50 (2017).
B. Bradlyn et al., “Topological quantum chemistry,” Nature 547, 298 (2017).
B. D. Josephson, “Possible new effects in superconductive tunnelling,” Phys. Lett. 1, 251 (1962).

About the Author

Robert-Jan Slager obtained his doctorate in 2016 at Leiden University in the Netherlands. He was then a postdoctoral researcher at the Max Planck Institute for the Physics of Complex Systems, Germany, and at Harvard University. He became a principal investigator at the University of Cambridge in 2019 and has been a professor of theoretical physics at the University of Manchester, UK, since October 2024.