Wannier90 is an open-source computer program for calculating maximally-localised Wannier functions (MLWFs) from a set of Bloch states. It is interfaced to many widely used electronicstructure codes thanks to its independence from the basis sets representing these Bloch states. In the past few years the development of Wannier90 has transitioned to a community-driven model; this has resulted in a number of new developments that have been recently released in Wannier90 v3.0. In this article we describe these new functionalities, that include the implementation of new features for wannierisation and disentanglement (symmetry-adapted Wannier functions, selectivelylocalised Wannier functions, selected columns of the density matrix) and the ability to calculate new properties (shift currents and Berry-curvature dipole, and a new interface to many-body perturbation theory); performance improvements, including parallelisation of the core code; enhancements in functionality (support for spinor-valued Wannier functions, more accurate methods to interpolate quantities in the Brillouin zone); improved usability (improved plotting routines, integration with arXiv:1907.09788v1 [cond-mat.mtrl-sci]

The Allen-Heine-Cardona theory allows us to calculate phonon-induced electron self-energies from first principles without resorting to the adiabatic approximation. However, this theory has not been able to account for the change of the electron wavefunction, which is crucial if inter-band energy differences are comparable to the phonon-induced electron self-energy as in temperature-driven topological transitions. Furthermore, for materials without inversion symmetry, even the existence of such topological transitions cannot be investigated using the Allen-Heine-Cardona theory. Here, we generalize this theory to the renormalization of both the electron energies and wavefunctions. Our theory can describe both the diagonal and off-diagonal components of the Debye-Waller selfenergy in a simple, unified framework. For demonstration, we calculate the electron-phonon coupling contribution to the temperature-dependent band structure and hidden spin polarization of BiTlSe2 across a topological transition. These quantities can be directly measured. Our theory opens a new door for studying temperature-induced topological phase transitions in materials both with and without inversion symmetry.Interactions between electrons and phonons induce a temperature-dependent renormalization of electronic structures 1 . The Allen-Heine-Cardona (AHC) theory 2-4 is one of the current state-of-the-art methods to study the effect of electron-phonon coupling (EPC) on electronic structures from first-principles density functional theory (DFT) and density functional perturbation theory (DFPT). Zero-point renormalization and temperature dependence of the electronic band gap 5-13 , optical responses 14-16 , and topological properties 17 are being actively investigated with the AHC theory.In the AHC theory 2-4 , the matrix elements of the second-order derivatives of the electron potential is required to compute the Debye-Waller (DW) contribution to the electron self-energy. This matrix element is approximated using the rigid-ion approximation (RIA), which assumes that the potential is a sum of atomcentered contributions. Then, by invoking the translational invariance of the electron eigenvalues, one can calculate the second-order EPC matrix elements from the first-order EPC matrix elements. However, this method is applicable only to the band-diagonal part of the DW self-energy. An approximation scheme for the full DW self-energy matrix including off-diagonal components has been absent.The missing off-diagonal components of the self-energy are essential to describe the hybridization of energy eigenstates 1,[18][19][20][21][22] . Thus, the current scope of the firstprinciples AHC theory is limited in that the EPC-induced change of the electron eigenstate wavefunctions is neglected. This theoretical limitation even precluded a complete study of electronic structures across an EPCinduced topological transition. For example, in Ref. 17 , the renormalized electron energy was calculated only at Γ where the wavefunction hybridization is forbidden due to...

Autonomous quantum error correction utilizes the engineered coupling of a quantum system to a dissipative ancilla to protect quantum logical states from decoherence. We show that the Knill-Laflamme condition, stating that the environmental error operators should act trivially on a subspace, which then becomes the code subspace, is sufficient for logical qudits to be protected against Markovian noise. It is proven that the error caused by the total Lindbladian evolution in the code subspace can be suppressed up to very long times in the limit of large engineered dissipation, by explicitly deriving how the error scales with both time and engineered dissipation strength. To demonstrate the potential of our approach for applications, we implement our general theory with binomial codes, a class of bosonic error-correcting codes, and outline how they can be implemented in a fully autonomous manner to protect against photon loss in a microwave cavity.

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