🔬 mesoscale physics

Strain-induced Berry phase in chiral superconductors

This paper demonstrates that adiabatically varying in-plane strain in a $p_x+ip_y$ chiral superconductor induces a $\pi$ Berry phase in the order parameter due to a half-rotation of the gap function, a topological effect that manifests as observable signatures in the superfluid stiffness tensor, vortex geometry, and upper critical field.

Original authors: Canon Sun, Marcel Franz, Joseph Maciejko

Published 2026-08-06

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CC BY 4.0

Original authors: Canon Sun, Marcel Franz, Joseph Maciejko

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where materials don't just sit there; they dance. In the realm of condensed matter physics, scientists study how electrons move through solids, looking for the most exotic dance moves of all: superconductivity. This is a state where electricity flows with zero resistance, like a frictionless slide. But some superconductors are more complicated than others. While most are like a simple, synchronized line dance, "unconventional" superconductors are more like a complex ballet with multiple partners. In these materials, the electrons pair up in ways that can break fundamental rules of symmetry, such as time-reversal symmetry (the idea that a movie of the process would look the same played backward).

The key to understanding these materials lies in their "order parameter," a fancy term for the collective state of the electron pairs. Think of this order parameter as a compass needle that points in a specific direction, defining the material's internal magnetic and structural personality. Sometimes, these compass needles can get stuck in a state of confusion, or "degeneracy," where two different directions are equally good. Scientists have long wondered: what happens if you gently push or pull on these materials? Does the material simply bend, or does it reveal a hidden, topological secret? This is the question that drives the search for new quantum states, because finding them could lead to ultra-fast, energy-efficient electronics and even quantum computers that don't lose information.

In this study, the authors explore what happens when you stretch a specific type of superconductor, known as a $p_x + ip_y$ chiral superconductor, using a precise amount of pressure. They found that when you apply this "strain," the material doesn't just change its shape; it undergoes a fascinating transformation in its internal structure. Imagine the material's phase diagram (a map showing its state at different temperatures and pressures) as a landscape. Usually, the transition from a normal metal to a superconductor is a single, smooth hill. However, the authors discovered that under strain, this single hill splits into two distinct peaks, creating a valley between them. This valley is shaped like a "Dirac cone," a sharp, cone-like feature that acts as a gateway to a new kind of physics.

The most exciting discovery is what happens when you walk around this cone. The authors showed that if you slowly (adiabatically) change the strain in a circle around this cone, the material's internal "compass needle" (the order parameter) doesn't just spin around and return to where it started. Instead, it performs a "half-rotation." It's as if you were holding a ribbon, and after walking in a circle around a pole, the ribbon had twisted by 180 degrees, swapping its front and back sides. This twist is called a "Berry phase" of $\pi$ (pi). It's a topological signature, a permanent mark left on the material that proves it has a hidden, knotted structure in its quantum state.

This half-rotation isn't just a theoretical curiosity; it leaves a fingerprint on the material's physical properties. The authors explain that this twist changes how the material resists the flow of supercurrents, a property called "superfluid stiffness." You can think of this stiffness as the material's "muscle tone." Because of the half-rotation, the material's muscle tone becomes anisotropic, meaning it's stronger in one direction and weaker in another, and this direction rotates as you change the strain. This rotation can be detected by looking at the shape of tiny whirlpools, called Abrikosov vortices, that form inside the superconductor. Just like a spinning top that wobbles in a specific way, these vortices will change their shape and orientation in a way that reveals the hidden half-rotation. Similarly, the magnetic field required to destroy the superconductivity (the upper critical field) will point in different directions depending on the strain, acting like a compass that spins as you walk around the cone.

The paper suggests that this phenomenon is a robust way to distinguish between different types of superconductors. If a material shows this specific "winding" behavior in its vortex shapes and magnetic limits when you stretch it, it confirms that the material has a symmetry-protected, multi-component order parameter. The authors didn't just guess this; they used mathematical models and simulations to show that the Berry phase of $\pi$ is a direct consequence of the cone-shaped phase diagram. While they focused on a specific theoretical model of a $p_x + ip_y$ superconductor, their findings offer a general recipe for how to spot these exotic quantum states in real-world materials, potentially helping scientists identify similar behaviors in complex compounds like $Sr_2RuO_4$ or $UTe_2$ . The study doesn't claim to have built a new device yet, but it provides a clear, topological "smoking gun" for identifying these elusive quantum states in the lab.

Technical Summary: Strain-induced Berry phase in chiral superconductors

Problem Statement
The paper addresses the characterization of the topological properties of multicomponent superconducting order parameters, specifically focusing on the intermediate phase between the superconducting transition and the time-reversal symmetry (TRS) breaking transition in a $p_x + ip_y$ superconductor under applied strain. While symmetry-protected multicomponent superconductors (such as $UPt_3$ , $UTe_2$ , and potentially $Sr_2RuO_4$ ) are known to exhibit split transitions under symmetry-breaking perturbations, the topological nature of the order parameter in the region between these split transitions remains to be fully elucidated. The authors investigate whether the adiabatic variation of strain around the critical point in the phase diagram induces a non-trivial Berry phase and how this topology manifests in observable physical quantities.

Methodology
The authors employ a combination of microscopic tight-binding modeling and Ginzburg-Landau (GL) theory to analyze a quasi-two-dimensional $p_x + ip_y$ superconductor on a tetragonal lattice ( $D_{4h}$ point group).

Microscopic Model: Electrons are treated as spinless with a band Hamiltonian $\hat{H}_0$ and a pairing interaction $\hat{H}_{int}$ favoring $p$ -wave pairing. In-plane strain ( $\epsilon$ ) is introduced, which reduces the crystal symmetry from $D_{4h}$ to $C_{2h}$ . This strain modifies the Fermi surface geometry and the electronic hopping amplitudes, leading to a strain-dependent reciprocal effective mass tensor $M^{-1}(\epsilon)$ .
Ginzburg-Landau Expansion: The free energy is expanded to fourth order in the order parameter fields $\Delta_\alpha$ ( $\alpha = x, y$ ). The quadratic coefficients $A_{\alpha\beta}$ are derived to linear order in strain, revealing that the degeneracy between the $p_x$ and $p_y$ channels is lifted.
Topological Analysis: The authors analyze the eigenvectors of the matrix $A_{\alpha\beta}$ in the region between the two split transitions. They parameterize the strain space using polar coordinates $(\epsilon_3, \epsilon_1)$ and track the evolution of the order parameter vector as the strain is varied adiabatically along closed loops enclosing the Dirac point (the multicritical point where the transitions merge).
Observable Probes: The study connects the abstract Berry phase to measurable quantities by calculating the symmetry-adapted components of the superfluid stiffness tensor ( $K_3, K_1$ ), the geometry of Abrikosov vortices, and the angular dependence of the upper critical field ( $B_{c2}$ ). Numerical minimization of the GL free energy is used to verify the topological winding of these quantities.

Key Contributions and Results

Dirac Cone Structure: The application of strain splits the single superconducting transition into two: a transition to a non-chiral superconducting state followed by a TRS-breaking transition to the chiral $p_x + ip_y$ state. In the temperature-strain phase diagram, these transitions form a Dirac cone structure.
$\pi$ Berry Phase: When the strain is varied adiabatically along a closed path enclosing the Dirac cone, the order parameter acquires a Berry phase of $\pi$ . This arises because the normalized eigenvector of the quadratic coefficient matrix, which dictates the relative phase of the $p_x$ and $p_y$ components, rotates by only half a turn ( $\pi$ ) when the strain angle completes a full $2\pi$ rotation. Consequently, the superconducting gap function performs a half-rotation in momentum space, swapping the positions of the gap lobes and nodes.
Topological Signature in Stiffness Tensor: The authors demonstrate that this $\pi$ Berry phase manifests as a non-trivial winding in the symmetry-adapted components of the superfluid stiffness tensor, $K_3 = (K_{xx} - K_{yy})/2$ and $K_1 = K_{xy}$ . Specifically, the complex combination $K_3 + iK_1$ winds by $2\pi$ around the origin of the strain plane when the strain loop encloses the Dirac point, whereas it remains contractible for loops that do not enclose the point.
Experimental Manifestations:
- Vortex Geometry: The topology of the stiffness tensor dictates the shape of Abrikosov vortices. For a strain loop enclosing the Dirac point, the semimajor axis of the elliptical vortex profile reverses its orientation after one cycle, whereas it returns to its original configuration for a contractible loop.
- Upper Critical Field: The direction of the maximum upper critical field $B_{c2}$ tracks the orientation of the gap maxima. The authors show that the angle of maximum $B_{c2}$ undergoes an inversion (exchange of positions) for a non-contractible strain loop, providing a direct probe of the gap function's half-rotation.
- Thermal Conductivity: The paper notes that thermal conductivity components exhibit similar winding behavior, though it is complicated by Fermi surface distortions in the normal state.

Significance
The paper claims to provide a protocol for distinguishing between symmetry-protected and accidental degeneracies in the phase diagrams of unconventional superconductors. By identifying the $\pi$ Berry phase as a topological invariant of the order parameter in the intermediate phase, the work offers a theoretical basis for detecting the multicomponent nature of the superconducting state through non-phase-sensitive measurements. Specifically, the ability to infer the presence of a Dirac point and the associated topological winding through the evolution of vortex shapes and the upper critical field under strain provides a concrete experimental pathway to characterize the topology of the order parameter in materials like $Sr_2RuO_4$ and other candidate multicomponent superconductors. The results generalize to other multicomponent systems beyond the specific $p_x + ip_y$ model studied.

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