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# Θ Φ : Solid state package allowing Bardeen-Cooper-Schrieffer and magnetic superstructure electronic states

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## Abstract

We propose the ΘΦ package which addresses two of the most important extensions of the essentially single-particle mean-field paradigm of the computational solid state physics: the admission of the Bardeen–Cooper–Schrieffer electronic ground state and allowance of the magnetically ordered states with an arbitrary superstructure (pitch) wave vector. Both features are implemented in the context of multi-band systems which paves the way to an interplay with the solid state quantum physics packages eventually providing access to the first-principles estimates of the relevant matrix elements of the model Hamiltonians derived from the standard DFT calculations. Several examples showing the workability of the proposed code are given.

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... Thus, we recently undertook some developments with a goal to heel the outlined deficiencies in the existing software which resulted in the package ΘΦ. 15,16 In previous works we implemented, respectively, (i) the BCS states and the magnetic phases with an arbitrary superstructure vectors 15 and (ii) the energy optimization with respect to the vector of magnetic superstructure 16 in the ΘΦ package. By the present paper we round up the development of ΘΦ by adding the option of having RVB state (for more detailed explanation of these see below) as a result of the solution of an electronic problem. ...
... Thus, we recently undertook some developments with a goal to heel the outlined deficiencies in the existing software which resulted in the package ΘΦ. 15,16 In previous works we implemented, respectively, (i) the BCS states and the magnetic phases with an arbitrary superstructure vectors 15 and (ii) the energy optimization with respect to the vector of magnetic superstructure 16 in the ΘΦ package. By the present paper we round up the development of ΘΦ by adding the option of having RVB state (for more detailed explanation of these see below) as a result of the solution of an electronic problem. ...
... Their explicit forms are available. 15 Central objects in any mean-field theory are the mean-field version of the Hamiltonianthe Fockian and the density matrix. In that or another form, they are present in all existing software. ...
Preprint
The available solid-state electronic-structure codes are devoid of hot topics of physics: incommensurate magnetic, superconducting and spin-liquid electronic states/phases. The temperature dependence of the solutions of electronic problem is not accessible either. These gaps are closed by the proposed ΘΦ (ThetaPhi) package.
... On the other hand, recently, our program ΘΦ 10,11 , allows for ISS, superconductivity of arbitrary order and Resonating Valence Bond (RVB) states in multi-orbital electronic systems at finite temperature. In addition, ΘΦ is capable to import the hopping parameters from major ab-initio codes by means of wannier90 12 and lobster 13-15 programs, which makes it possible to perform practically ab-initio strongly correlated magnetic or superconducting calculations. ...
... Secondly, in this paper, we present a general framework for the spin quantization axis rotation in a general case when no assumptions are made on the spin dependence of the Hamiltonian's hopping matrix elements. This general framework extends the one already presented in the Ref. 11 and allows to treat the cases when the hopping is spin-dependent and contains the spin-flip terms, thus permitting simulations of systems with spin-orbit coupling and explicit break of timereversal symmetry. ...
... • The interaction terms like Coulomb repulsion and Heisenberg exchange, which are typically fourth rang tensors, have to be transformed according to the formulas given in Ref. 11, and this task is facilitated by the fact that the most important contributions to them are local i.e. do not extend outside of the unit cell. ...
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Simulating the incommensurate spin density waves (ISDW) states is not a simple task within the standard \emph{ab-initio} methods. Moreover, in the context of new material discovery, there is a need for fast and reliable tool capable to scan and optimize the total energy as a function of the pitch vector, thus allowing to automatize the search for new materials. In this paper we show how the ISDW can be efficiently obtained within the recently released $\Theta\Phi$ program. We illustrate this on an example of the single orbital Hubbard model and of $\gamma$-Fe, where the ISDW emerge within the mean-field approximation and by using the twisted boundary conditions. We show the excellent agreement of the $\Theta\Phi$ with the previously published ones and discuss possible extensions. Finally, we generalize the previously given framework for spin quantization axis rotation to the most general case of spin-dependent hopping matrix elements.
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We present a software package GoGreenGo—an overlay aimed to model local perturbations of periodic systems due to either chemisorption or point defects. The electronic structure of an ideal crystal is obtained by worldwide‐distributed standard quantum physics/chemistry codes, and then processed by various tools performing projection to atomic orbital basis sets. Starting from this, the perturbation is addressed by GoGreenGo with use of the Green's functions formalism, which allows evaluating its effect on the electronic structure, density matrix, and energy of the system. In the present contribution, the main accent is made on processes of chemical nature, such as chemisorption or doping. We address a general theory and its computational implementation supported by a series of test calculations of the electronic structure perturbations for benchmark model solids: simple, face‐centered, and body‐centered cubium systems. In addition, more realistic problems of local perturbations in graphene lattice, such as lattice substitution, vacancy, and “on‐top” chemisorption, are considered. Point defects in crystals form a wide class of processes being of great importance in solid‐state chemistry. Only by considering surface chemistry one can propose a numerous examples ‐ from formation of isolated surface defects to single particle chemisorption and elementary reactions on catalysts' surfaces. Theoretical investigation of these processes, aiming to understand their mechanisms from the electronic structure perspective, presents one of many important branches of solid‐state chemistry deserving close attention. In this work we present a new software package GoGreenGo specifically designed to perform computationally effective quantum chemical calculations of local processes in solids and to provide results in “chemical” terms.
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We present a standalone ΘΦ (ThetaPhi) package capable to read the results of ab initio DFT/PAW quantum‐chemical solid‐state calculations processed through various tools projecting them to the atomic basis states as an input and to perform on top of this an analysis of so derived electronic structure which includes (among other options) the possibility to obtain a superconducting (Bardeen‐Cooper‐Schrieffer, BCS), spin‐liquid (resonating valence bond, RVB) states/phases as solutions of the electronic structure problem along with the magnetically ordered phases with an arbitrary pitch (magnetic superstructure) vector. Remarkably, different solutions of electronic‐structure problems come out as temperature‐dependent (exemplified by various superconducting and spin‐liquid phases) which feature is as well implemented. All that is exemplified by model calculations on 1D chain, 2D square lattice as well as on more realistic superconducting doped graphene, magnetic phases of iron, and spin‐liquid and magnetically ordered states of a simplest nitrogen‐based copper pseudo‐oxide, CuNCN, resembling socalled metal‐oxide framework (MOF) phases by the atomic interlinkage. The available solid‐state electronic‐structure codes are devoid of hot topics of physics: incommensurate magnetic, superconducting, and spin‐liquid electronic states/phases. The temperature dependence of the solutions of electronic problem is not accessible either. These gaps are closed by the proposed ΘΦ package.
Article
Simulating the incommensurate spin density waves (ISDW) states is not a simple task within the standard ab initio methods. Moreover, in the context of new material discovery, there is a need for fast and reliable tool capable to scan and optimize the total energy as a function of the pitch vector, thus allowing to automatize the search for new materials. In this paper we show how the ISDW can be efficiently obtained within the recently released ΘΦ program. We illustrate this on an example of the single orbital Hubbard model and of γ-Fe, where the ISDW emerge within the mean-field approximation and by using the twisted boundary conditions. We show the excellent agreement of the ΘΦ with the previously published ones and discuss possible extensions. Finally, we generalize the previously given framework for spin quantization axis rotation to the most general case of spin-dependent hopping matrix elements.
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Dyson's systematic approach to the reduction of the Heisenberg $S$-Matrix into a sum of "graph" terms can be simplified. A notation is introduced and an algebraic theorem is proved, which allow one to handle the reduction problem quite easily and in the same manner for any type of field.
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Simple, yet predictive bonding models are essential achievements of chemistry. In the solid state, in particular, they often appear in the form of visual bonding indicators. Because the latter require the crystal orbitals to be constructed from local basis sets, the application of the most popular density-functional theory codes (namely, those based on plane waves and pseudopotentials) appears as being ill-fitted to retrieve the chemical bonding information. In this paper, we describe a way to re-extract Hamilton-weighted populations from plane-wave electronic-structure calculations to develop a tool analogous to the familiar crystal orbital Hamilton population (COHP) method. We derive the new technique, dubbed "projected COHP" (pCOHP), and demonstrate its viability using examples of covalent, ionic, and metallic crystals (diamond, GaAs, CsCl, and Na). For the first time, this chemical bonding information is directly extracted from the results of plane-wave calculations.
Article
A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n + 1) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point. The simplex adapts itself to the local landscape, and contracts on to the final minimum. The method is shown to be effective and computationally compact. A procedure is given for the estimation of the Hessian matrix in the neighbourhood of the minimum, needed in statistical estimation problems.
Article
By combining the Dirac equation of relativistic quantum mechanics with the Bogoliubov-de Gennes equation of superconductivity we investigate the electron-hole conversion at a normal-metal-superconductor interface in graphene. We find that the Andreev reflection of Dirac fermions has several unusual features: (1) the electron and hole occupy different valleys of the band structure; (2) at normal incidence the electron-hole conversion happens with unit efficiency in spite of the large mismatch in Fermi wavelengths at the two sides of the interface; and, most fundamentally: (3) away from normal incidence the reflection angle may be the same as the angle of incidence (retroreflection) or it may be inverted (specular reflection). Specular Andreev reflection dominates in weakly doped graphene, when the Fermi wavelength in the normal region is large compared to the superconducting coherence length.
Article
The oxide superconductors, particularly those recently discovered that are based on La2CuO4, have a set of peculiarities that suggest a common, unique mechanism: they tend in every case to occur near a metal-insulator transition into an odd-electron insulator with peculiar magnetic properties. This insulating phase is proposed to be the long-sought "resonating-valence-bond" state or "quantum spin liquid" hypothesized in 1973. This insulating magnetic phase is favored by low spin, low dimensionality, and magnetic frustration. The preexisting magnetic singlet pairs of the insulating state become charged superconducting pairs when the insulator is doped sufficiently strongly. The mechanism for superconductivity is hence predominantly electronic and magnetic, although weak phonon interactions may favor the state. Many unusual properties are predicted, especially of the insulating state.
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