New math rules out some ‘ghost’ orders in superconductors

Grgur Palle and Rafael M. Fernandes of the University of Illinois Urbana-Champaign have identified the root cause of ambiguity in calculations modeling secondary phases, often called “ghost” orders, in superconductors. Their work demonstrates that redundancy relations, such as Fierz identities, are responsible for the inconsistencies arising from the standard large-N approach to modeling these complex materials.

The researchers propose an unbiased approach that accounts for how different vestigial channels interfere, yielding unique and reliable values for effective interactions, and revealing scenarios where vestigial orders are unexpectedly unstable. This analysis applies to materials ranging from iron pnictides to twisted bilayer graphene.

Large-N Approach Ambiguity Impacts Vestigial Order Predictions

Current methods for predicting partially ordered phases, known as vestigial phases, in materials like iron pnictides and twisted bilayer graphene may be flawed due to an inherent ambiguity in calculations. The standard large-N approach, used to model the Ginzburg-Landau action of primary order parameters, struggles with decoupling composite order parameters, potentially leading to inaccurate predictions of vestigial instabilities. This means researchers could be falsely enhancing or eliminating these secondary phases in their simulations, obscuring the true behavior of these complex materials.

The root of this issue lies in redundancy relations, such as Fierz identities, which demonstrate that different vestigial channels are not independent but instead interfere with each other. This interference isn’t accounted for in the traditional large-N approach, creating an ambiguity in how these channels are treated mathematically. The work highlights that this ambiguity isn’t a sign of new physics, but rather a consequence of how calculations are performed.

The approach applies only at weak coupling, while vestigial order can also be a strong-coupling effect, further complicating the analysis. To address this, a modified large-N approach has been proposed, treating all vestigial orders arising from the same primary order equally. This new method respects both the redundancy relations and the underlying symmetry-group structure, yielding unique values for the effective interactions of all vestigial channels.

Analysis of phase diagrams for X-point charge density waves on the tetragonal lattice reveals that, unlike the traditional approach, there is no stable vestigial order near the degeneracy point when λ₁ = λ₂ = 0, due to the competing channels. The revised approach offers a more reliable framework for understanding these subtle, yet significant, phases of matter.

Redundancy Relations and Symmetry Define Unbiased Large-N Method

These relationships prevent vestigial channels, subtle secondary phases, from being treated as independent entities in calculations, a practice that can lead to inaccurate predictions about material behavior. Researchers have now demonstrated that this interference explains why seemingly new phases can emerge as artifacts of calculation, rather than genuine physical phenomena. The revised approach expands the symmetry group of a multicomponent model, introducing an internal index to the primary order parameter.

This expansion, while structurally similar to traditional large-N methods, replaces bare vestigial coupling constants with unique, symmetrized values. The resulting saddle-point equations, accurate as the number of components approaches infinity, provide a consistent description of vestigial order. This ensures that the calculated interactions accurately reflect the underlying physics, rather than being distorted by the mathematical limitations of the model. The team argues that preserving these features, redundancy relations and vestigial channels, is important because they are specific to a given symmetry group and its irreducible representation.

Vestigial Orders Arise from Multicomponent Order Parameters

Regions exist within the parameter space of Landau coefficients where vestigial order remains unstable, a finding that contrasts with the standard large-N approach but aligns with both weak-coupling and variational methods. The analysis reveals that these instabilities are not necessarily novel phenomena, but rather consequences of how calculations are performed. The work highlights a competition between symmetry-distinct vestigial orders, a dynamic intensified by the coupling between composite order parameters facilitated by redundancy relations such as Fierz identities.

This competition manifests as a threshold behavior around points where all vestigial orders are degenerate, meaning they share the same energy. This suggests a revised methodology, starting with an infinite number of components and extrapolating to finite values, may be more effective than previous methods.

The agreement between the new large-N approach and Hartree-Fock calculations is not inherent to all large-N methods, but a specific outcome of this unbiased approach. Applying this modified large-N approach to three primary order types, two-component charge-density waves, two- and three-component superconductors and three-component spin-density waves, revealed exotic vestigial phases describing spin-quadrupolar, charge-4e superconducting and altermagnetic orders.

Competing Density-Wave and Superconducting Instabilities

The mathematical framework used to predict secondary phases in superconductors may have been overlooking an important factor: the inherent relationships between those phases themselves. Grgur Palle and Rafael M. This oversight stems from an ambiguity in decoupling composite order parameters within the standard large-N approach, a common technique for simplifying complex calculations. The team’s analysis reveals that seemingly distinct vestigial orders are not isolated, but rather interconnected through redundancy relations like Fierz identities.

These relations dictate how different vestigial channels interfere, meaning they cannot be analyzed in isolation as has been typical. Focusing on three-component superconductivity, a phenomenon possible in cubic systems but rarely explored, the researchers found parameter ranges where charge-4e superconductivity and ferromagnetic order emerge as leading vestigial instabilities. The authors note a gap in current understanding.

Exotic Vestigial Phases: Spin-Quadrupolar, Charge-4e, Altermagnetic

Analyses of charge-density waves, spin-density waves and multicomponent superconductors in tetragonal, hexagonal, and cubic systems revealed exotic vestigial phases describing spin-quadrupolar, charge-4e superconducting and altermagnetic orders. These phases emerge from the complex interplay of symmetry and competing instabilities within these materials. Spin-quadrupolar orders, while not typically leading vestigial instabilities, can become degenerate with other vestigial channels under specific conditions, occurring along fine-tuned lines within the parameter space defined by ${\lambda }_1$ and ${\lambda }_2$.

The symmetry groups governing these systems often combine crystalline space groups with internal continuous groups, global phase for superconductivity and spin-rotation for spin-density waves, further complicating possible vestigial phases. Competition between nearly degenerate vestigial instabilities can also enable composite orders, transforming nontrivially with respect to these internal groups, to become dominant.

Specifically, charge-4e order stabilizes in three-component superconductivity on the cubic lattice, while spin-current order emerges in M-point spin-density waves on the hexagonal lattice. Although spin-nematics represent one example of exotic spin-quadrupolar composite order, the analysis indicates it cannot be a leading vestigial instability, existing instead in degeneracy with other phases along fine-tuned parameter lines.

Intertwined Orders and Symmetry Breaking in Complex Systems

This means previously unexplained phases may have been misinterpreted, or even missed entirely, due to mathematical artifacts. The analysis reveals that multiple inequivalent large-N limits exist, with the limit best extrapolating to a finite system being one where coupling constants are fully symmetrized, as detailed in equation 42 of the study. The team extended this formalism to investigate more complex vestigial orders involving simultaneous broken symmetries in both real-space and spin-space, specifically examining spin-quadrupolar order.

These orders, described by a composite operator, can be interpreted as generalizations of spin-nematic orders, representing spontaneous breaking of rotational symmetry in spin space. This approach provides a framework for understanding how intertwined orders emerge and interact, offering a more accurate picture of complex material behavior than previously possible.

Parameter Space Defines Stability of Vestigial Orders

The work details how these vestigial instabilities, partially ordered states appearing at higher temperatures than the primary superconducting phase, are affected by the interplay of Landau coefficients. This suggests that their emergence is highly sensitive to specific conditions, rather than a robust feature of the system. While exotic spin-quadrupolar composite orders cannot be leading vestigial instabilities, they can achieve degeneracy with other vestigial phases along fine-tuned lines of the parameter space, further complicating possible ordered states.

Applications of Vestigial Order Across Material Systems

The symmetry-breaking patterns arising from vestigial order extend beyond nematic phases to encompass charge-density waves, ferromagnets and even charge-4e superconductors, demonstrating a richer diversity than previously understood. Experimental evidence for these vestigial phases has accumulated across a broad range of materials, including cuprates, iron pnictides and increasingly, in engineered systems like twisted bilayer graphene and van der Waals antiferromagnets, suggesting a widespread phenomenon.

Current calculations of these partially ordered states, often termed “ghost” orders, may be flawed due to an intrinsic ambiguity in how composite order parameters are decoupled within the standard large-N approach to modeling superconductivity. This ambiguity can lead to the false enhancement or complete elimination of vestigial instabilities, potentially misinterpreting the origins of these secondary phases.

In tetragonal lattices, the most negative symmetrized coupling constant dictates the dominant vestigial channel within the parameter space defined by (λ_1,λ_2). The analysis reveals the generic existence of regions in the parameter space of quartic Landau coefficients where no vestigial order is stable, in contrast to the standard large-N approach, but in agreement with weak-coupling and variational approaches.

Ginzburg-Landau Action and Fluctuations in Vestigial Phases

The theoretical description of vestigial order requires methods that account for fluctuations of the primary order parameter, with the large-N approach being the most common, though Gaussian variational, self-consistent Hartree-Fock, renormalization group, and Monte Carlo methods have also been employed. Analysis of a two-component real order parameter reveals three possible composite bilinears, with only two representing nontrivial channels associated with symmetry-distinct vestigial phases. While group theory can identify all possible composite order parameters and their relation to primary OP configurations, a microscopic calculation is necessary to determine which vestigial phase, if any, emerges.

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