IrRep Package Troubleshooting For Topological Compounds

Hello everyone, I'm Cauã FS, a BSc student from Brazil, and I'm excited to share some insights and questions I've encountered while working with the IrRep package for analyzing topological compounds. In this article, I will discuss some doubts about running the IrRep package, focusing on specific issues I faced while analyzing topological compounds, such as ZrTe2 and ZrTe5, and their variations. I'll walk you through my process, the challenges I encountered, and the solutions I'm exploring. I hope this helps other students and researchers facing similar issues.

Background

My initial work involved running code for topological compounds, and I was able to achieve reasonable agreement with the Topological Materials Database, especially for enforced semimetals like ZrTe2 and topological insulators like ZrTe5. However, when I started working with variations of ZrTe5 that included intercalated atoms while maintaining the same space group, I ran into some challenges.

Initial Successes and New Challenges

Initially, I achieved good agreement with the Topological Materials Database for compounds like ZrTe2 and ZrTe5. However, when dealing with variations of ZrTe5 (with intercalated atoms but the same space group), issues arose. This article delves into these challenges and seeks potential solutions.

1. Resolving Errors in IrRep Calculations for Intercalated ZrTe5

Understanding the Error Message: When analyzing a variation of ZrTe5 with intercalated atoms, I encountered the following error message: "The traces for some symmetry operations at some maximal k-vector do not correspond to any direct sum of irreps in the space group given. Sometimes the problem can be solved by increasing the resolution in the calculations." This error suggests that the traces for certain symmetry operations at specific k-vectors do not align with any direct sum of irreducible representations (irreps) within the given space group. Essentially, the symmetry properties calculated by IrRep don't match the expected symmetry behavior for the material's crystal structure. The error message hints that increasing the resolution of the calculation might resolve the issue. For my IrRep input, I used:

irrep -kpnames=GM,T,Y,Z,R,S -IBend=106 -Ecut=110 -EF=auto -code=espresso -prefix=tmp/zrte5y2 > output

The Role of Ecut: The error message suggested increasing the resolution, which led me to consider the Ecut parameter. In IrRep, Ecut represents the energy cutoff, and I wondered if increasing its value would indeed address the problem. My question was: Is increasing Ecut the right approach to resolve this specific error? This links with my 5th question about the appropriate Ecut value in IrRep calculations, which I will discuss later. The output files related to this issue can be found here:

• check_topo_yzrte5.txt
• output_irreps_2y.txt

Possible Causes and Solutions: This type of error typically arises when the computational parameters used in the calculation are not sufficient to accurately capture the electronic structure and symmetry properties of the material. The symmetry operations and k-vectors are critical in determining the topological properties, and any inaccuracies in their representation can lead to such errors. Increasing the resolution, as suggested, generally means using a higher energy cutoff (Ecut in this case) and a denser k-point mesh. A higher Ecut ensures that more plane waves are included in the basis set, providing a more accurate representation of the electronic wavefunctions. A denser k-point mesh, on the other hand, better samples the Brillouin zone, improving the accuracy of the symmetry operations and band structure calculations.

When dealing with complex materials or those with heavy elements, such as the variations of ZrTe5, higher resolution is often necessary. The intercalated atoms can introduce additional electronic states and interactions that require a more refined calculation. The error message indicates that the symmetry operations are not being accurately represented, which is a strong indicator that the current resolution is insufficient. By increasing Ecut and potentially the k-point density, the calculation can better capture the electronic structure, resolving the mismatch between the calculated traces and the expected irreps. Furthermore, it's crucial to ensure that the crystal structure used in the calculation is accurately relaxed and that all atomic positions are correctly represented. Errors in the crystal structure can also lead to discrepancies in the symmetry operations and, consequently, the irrep analysis.

2. Clarifying the Use of the -spinor Flag in SOC Calculations

Understanding the -spinor Flag: My second question was a confirmation regarding the -spinor flag. I wanted to verify whether it is necessary to use this flag when performing calculations with Spin-Orbit Coupling (SOC). In other words, if SOC is included in the calculations, is the -spinor flag automatically implied? Reviewing examples, particularly those on GitHub, showed that the -spinor flag was not used in calculations with SOC. This observation led me to believe that the flag is redundant when SOC is already enabled.

SOC and the -spinor Flag: Spin-Orbit Coupling (SOC) is a relativistic effect that becomes significant in materials containing heavy elements. It couples the electron's spin and its orbital motion, leading to splitting of electronic bands and altering the electronic structure significantly. When SOC is included in a calculation, the wavefunctions are represented as spinors, which are two-component functions that account for the spin degree of freedom. The question arises: does the IrRep package automatically handle the spinor nature of wavefunctions when SOC is enabled, or is the -spinor flag necessary to explicitly instruct the code to treat the wavefunctions as spinors?

Based on the observation that the -spinor flag is not used in the GitHub examples with SOC, it seems that IrRep may automatically detect and handle spinor wavefunctions when SOC is included in the calculation setup. This would mean that the flag is indeed redundant. However, it is always good practice to confirm this behavior with the code's documentation or the developers to avoid any potential misinterpretations. If the code automatically handles spinors with SOC, explicitly using the -spinor flag might not cause an error, but it would be unnecessary. On the other hand, if the flag is crucial and not automatically implied, omitting it could lead to incorrect results. For clarity, referring to the IrRep package documentation or seeking clarification from the developers would provide a definitive answer. Additionally, comparing results obtained with and without the -spinor flag (while keeping SOC enabled) could offer empirical evidence of whether the flag has any impact on the calculations.

3. Time-Reversal Symmetry in IrRep Calculations

Time-Reversal Symmetry Considerations: My third question pertains to Time-Reversal Symmetry (TRS). Is TRS assumed to be conserved in IrRep calculations by default, or is it necessary to explicitly use the --time-reversal flag? This is a crucial consideration because TRS plays a significant role in the topological properties of materials. Time-reversal symmetry is a fundamental symmetry in physics, stating that the laws of physics are invariant under the transformation t → -t. In the context of electronic structure calculations, TRS implies that for every electronic state with momentum k and spin s, there exists a corresponding state with momentum -k and spin -s that has the same energy. This symmetry has profound implications for the band structure and the presence of topological states.

Implications of Time-Reversal Symmetry: In materials where TRS is conserved, band degeneracies can occur at time-reversal invariant momenta (TRIM points) in the Brillouin zone, leading to phenomena such as Dirac and Weyl semimetals. However, if TRS is broken (e.g., by the presence of magnetic ordering), these degeneracies can be lifted, and the topological properties of the material can change significantly. Therefore, correctly accounting for TRS is essential when analyzing topological materials. The question is whether IrRep, by default, assumes that TRS is conserved, or whether the user needs to explicitly specify the conservation of TRS using the --time-reversal flag. If TRS is assumed by default, the code might not correctly handle systems where TRS is broken, leading to inaccurate results.

On the other hand, if the flag is mandatory for both TRS-conserved and TRS-broken systems, the user needs to be aware of this requirement to avoid errors. To address this, one should consult the IrRep package documentation to understand the default behavior and the proper usage of the --time-reversal flag. It is also useful to consider the physical system being studied. If the material is known to be non-magnetic and TRS is expected to be conserved, the flag might not be necessary if the code assumes TRS by default. However, for magnetic materials or systems with complex magnetic order, explicit specification using the flag may be crucial. Testing the calculations with and without the --time-reversal flag (if applicable) can provide additional insights into the code's behavior and the importance of this setting for the specific material under study.

4. Addressing Orthogonality and Symmetry Matrix Issues When Separating by Inversion Eigenvalues

Inversion Eigenvalues and Orthogonality: When separating my system by inversion eigenvalues (specifically, number 5), I encountered a couple of concerning messages. First, I received an orthogonality warning: "Orthogonality (largest of diag. <psi_nk|psi_mk>): 1.00000 > 1e-5 this appeared when considering isymsep=N for inversion." This message indicates that the orthogonality between certain wavefunctions is not being maintained within the expected tolerance. Wavefunctions, which describe the quantum mechanical state of electrons in a material, must be orthogonal to ensure that they represent distinct, independent states. Orthogonality is a fundamental requirement in quantum mechanics, ensuring that different electronic states are independent and do not interfere with each other. In numerical calculations, deviations from perfect orthogonality can arise due to numerical approximations and computational limitations.

Understanding the Orthogonality Warning: The warning message "Orthogonality (largest of diag. <psi_nk|psi_mk>): 1.00000 > 1e-5" indicates that the overlap integral between some wavefunctions is larger than the tolerance threshold of 1e-5. This means that some wavefunctions, which should be orthogonal, have a non-negligible overlap, suggesting a potential issue with the calculation. This issue often arises when the basis set used to represent the wavefunctions is incomplete, or when numerical errors accumulate during the iterative self-consistent field (SCF) procedure. The fact that this message appeared when considering isymsep=N for inversion suggests that the issue is related to the way the code is handling inversion symmetry. The isymsep flag is used to separate states based on their symmetry eigenvalues, and an orthogonality issue when using this flag indicates that states with different inversion eigenvalues might not be properly orthogonalized.

Non-Zero Elements in Symmetry Matrix: Second, I received a warning about non-zero elements in the symmetry matrix: "WARNING: matrix of symmetry has non-zero elements between states of different energy: 0.02034485110352721 Printing matrix of symmetry at k=[0.5 0.5 0.5]." This warning suggests that the symmetry operations are mixing states with different energies, which should ideally not happen. A symmetry matrix represents how the symmetry operators of the crystal's space group transform the electronic states. Ideally, the symmetry matrix should be block-diagonal, with each block corresponding to a different energy level. Non-zero elements between states of different energies indicate that the symmetry operations are not properly preserving the energy separation, which can lead to incorrect band structure and topological property calculations. This is particularly concerning because, in principle, symmetry operations should only mix states within the same energy level. The non-zero elements suggest that there might be numerical inaccuracies or issues with the symmetry handling in the code. This could be due to insufficient convergence, an inadequate basis set, or errors in the implementation of the symmetry operations. The output files related to this issue can be found here:

• output_inversion.txt

Possible Solutions: Here are some steps to troubleshoot and potentially resolve these issues:

1. Increase Convergence Criteria: Tighten the convergence criteria for the self-consistent field (SCF) calculations. This means reducing the tolerance for energy and charge density changes between iterations. Inadequate convergence can lead to wavefunctions that are not fully optimized and may have orthogonality issues. Specifically, look for parameters controlling the convergence of the charge density, energy, and forces in your DFT code and reduce their tolerances. This ensures a more accurate solution of the Kohn-Sham equations. Sometimes, the default convergence criteria are not sufficient for complex systems, leading to inaccuracies in the wavefunctions and the symmetry operations.

2. Increase Basis Set Size: As mentioned in the context of Ecut, an incomplete basis set can lead to orthogonality issues. In plane-wave codes, this means increasing the energy cutoff (Ecut). A larger basis set provides a more complete representation of the wavefunctions, reducing the likelihood of overlap between supposedly orthogonal states. When the basis set is too small, the wavefunctions are not represented with sufficient detail, which can lead to numerical errors and loss of orthogonality. Increasing the basis set size allows for a more accurate representation of the electronic states, leading to improved orthogonality and more reliable symmetry analysis.

3. Check k-Point Density: Ensure that the k-point mesh is sufficiently dense. A sparse k-point mesh can lead to inaccurate Brillouin zone integrations and affect the orthogonality of wavefunctions. A denser k-point mesh provides a more accurate sampling of the Brillouin zone, which is crucial for obtaining reliable electronic structure results. This is particularly important for systems with complex band structures or small energy gaps. Insufficient k-point sampling can lead to inaccuracies in the band energies and wavefunctions, contributing to orthogonality and symmetry issues. The choice of k-point mesh should be based on the system's size and complexity, and convergence tests should be performed to ensure that the results are not sensitive to the k-point density.

4. Verify Symmetry Operations: Double-check the symmetry operations defined for the system. Incorrect symmetry definitions can lead to the mixing of states with different energies. Ensure that the space group and the corresponding symmetry operations are correctly defined in the input files. Errors in the symmetry definitions can lead to the incorrect application of symmetry operators, resulting in the mixing of states with different energies and incorrect symmetry eigenvalues. This can also lead to the non-zero elements in the symmetry matrix between states of different energies.

5. Examine the Crystal Structure: Ensure that the crystal structure is accurately relaxed and that all atomic positions are correct. Errors in the crystal structure can affect the symmetry operations and lead to the issues observed. Inaccurate atomic positions can break or distort the crystal's symmetry, leading to the incorrect application of symmetry operations and the mixing of states with different energies. Therefore, it is crucial to start with a well-relaxed crystal structure and to carefully verify the atomic positions used in the calculations.

6. Consult the Code Documentation and Developers: Review the IrRep package documentation for specific recommendations on handling orthogonality and symmetry issues. If the problem persists, consider reaching out to the developers for support. The documentation might provide specific guidelines or troubleshooting steps for these types of issues. The developers might also have insights into the code's behavior and be able to offer specific solutions or bug fixes. Consulting the documentation and the developers can save significant time and effort in resolving these complex issues.

5. Determining the Appropriate Ecut Value in IrRep Calculations

Understanding the Ecut Flag: My final question centers on the Ecut flag. As the IrRep paper explains, Ecut is in units of eV, and I'm using Quantum ESPRESSO (QE), which uses Rydberg (Ry) as its natural unit for energy. My calculations employ a cutoff of 110 Ry. Some examples used surprisingly low ecutwfc values (10–20 Ry) for TQC/EBRs analysis, which made me less concerned initially. However, I now question whether I should use a much larger Ecut value in IrRep, specifically converting my QE cutoff to eV: -Ecut = 110 Ry × 13.605 ≈ 1500 eV instead of the 110 eV I used based on examples. I've noticed that the Ecut value in IrRep doesn't perfectly align with the ecutwfc value in QE, leading to my confusion. This question is crucial because the energy cutoff directly affects the accuracy and computational cost of the calculations.

The Role of Ecut in IrRep: The Ecut parameter in IrRep determines the energy cutoff for the plane-wave basis set used in the calculations. In the context of density functional theory (DFT), the electronic wavefunctions are expanded in a basis set of plane waves. The energy cutoff defines the maximum kinetic energy of the plane waves included in this expansion. A higher Ecut means more plane waves are included, leading to a more accurate representation of the wavefunctions and the electronic structure. However, it also increases the computational cost. The ecutwfc parameter in Quantum ESPRESSO serves a similar purpose, defining the energy cutoff for the wavefunctions.

Conversion and Consistency: The initial examples using low ecutwfc values in QE might have been sufficient for specific cases where the electronic structure was relatively simple, or the focus was on qualitative analysis rather than high accuracy. However, for more complex systems, or when quantitative accuracy is needed, a higher Ecut is generally necessary. The conversion from Rydberg to eV is indeed crucial when transferring parameters between QE and IrRep. The question of whether to use 1500 eV instead of 110 eV in IrRep highlights the importance of this conversion. While 110 Ry in QE is equivalent to approximately 1500 eV, the context in which these values are used in IrRep might differ from their usage in QE. The fact that Ecut in IrRep doesn't perfectly align with ecutwfc in QE suggests that there might be different considerations for the optimal value in each code. This could be due to the specific algorithms and approximations used in IrRep, or the way symmetry operations and irrep calculations are performed.

Suggestions and Tips for Setting Ecut:

1. Convergence Testing: The most reliable way to determine the appropriate Ecut value is through convergence testing. Perform a series of IrRep calculations with different Ecut values, monitoring relevant properties such as band energies, irrep traces, or topological invariants. The Ecut value should be increased until the properties of interest converge, meaning they no longer change significantly with further increases in Ecut. This ensures that the results are not sensitive to the choice of energy cutoff. Convergence testing helps to balance the need for accuracy with computational efficiency. Starting with a lower Ecut value and gradually increasing it allows you to observe the impact on the results and identify the point at which further increases provide diminishing returns.

2. Consider the System: The complexity of the electronic structure of the material should also influence the choice of Ecut. Materials with heavy elements, strong electronic correlations, or complex band structures generally require higher Ecut values. The presence of core electrons, for example, necessitates a larger basis set to accurately represent the wavefunctions near the nuclei. Strong electronic correlations, such as those found in transition metal oxides, require a more detailed description of the electronic interactions, which can be achieved with a higher Ecut. Similarly, complex band structures with narrow bands or band crossings often require a higher resolution in reciprocal space, which is facilitated by a larger basis set.

3. Consult Literature and Examples: Review the literature for similar materials or calculations to see what Ecut values have been used. This can provide a starting point for your own calculations. However, always perform convergence testing to ensure that the chosen value is appropriate for your specific system and computational setup. Examples and published results can offer valuable guidance, but it's essential to adapt the parameters to the specific needs of your system. Different codes and computational methods might have different requirements for the energy cutoff, so it's important to consider the context of the published results.

4. Check the IrRep Documentation: Consult the IrRep package documentation for specific recommendations or guidelines on setting Ecut. The documentation might provide insights into the code's internal workings and the factors that influence the optimal Ecut value. The documentation might also include specific examples or tutorials that demonstrate how to choose Ecut for different types of materials or calculations. This can be a valuable resource for understanding the nuances of the code and avoiding common pitfalls.

Conclusion

I hope this detailed exploration of my challenges with the IrRep package has been helpful. Through addressing these questions, I've gained a deeper understanding of the software and its intricacies. I want to express my gratitude to Professor Stepan Tsirkin for his invaluable assistance and for developing such an amazing tool. The IrRep package is truly a game-changer for the field of topological materials! I encourage anyone facing similar issues to explore these solutions and to share their experiences. Let's continue to learn and grow together in this exciting field.

For further information on topological materials and related topics, you can explore resources like the Topological Materials Database.