Solid State Quantum Chemistry with ΘΦ (ThetaPhi): Spin-Liquids, Superconductors and Magnetic Superstructures Made Computationally Available

Abstract

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 ThetaPhi package.

Full text

Solid State Quantum Chemistry with ΘΦ (ThetaPhi):
Spin-Liquids, Superconductors and Magnetic
Superstructures Made Computationally Available
A. Tchougr´eeff∗
, E. Plekhanov†
, R. Dronskowski‡
January 28, 2021
Abstract
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.
Keywords: Superconducting phases, Spin-liquid phases, Spirally ordered magnetic
phases, RVB, BCS, electronic structure of crystals.
∗A.N. Frumkin Institute of Physical Chemistry and Electrochemistry of Russian Academy of Science,
Moscow, Russia
†King’s College London, Theory and Simulation of Condensed Matter (TSCM), The Strand, London
WC2R 2LS, United Kingdom and A.N. Frumkin Institute of Physical Chemistry and Electrochemistry of
Russian Academy of Science, Moscow, Russia
‡Institut f¨ur anorganische Chemie, RWTH Aachen University, Landoltweg 1, D-52056, Aachen, Germany;
J¨ulich-Aachen Research Alliance, JARA-HPC, RWTH Aachen University, 52056 Aachen, Germany and
Hoffmann Institute of Advanced Materials, Shenzhen Polytechnic, 7098 Liuxian Blvd, Nanshan District,
Shenzhen, China
1

The available solid-state electronic-structure codes are devoid of hot topics of physics: incom-
mensurate magnetic, superconducting and spin-liquid electronic states/phases. The temper-
ature dependence of the solutions of electronic problem is not accessible either. These gaps
are closed by the proposed ΘΦ package.
2

INTRODUCTION
Somewhat simplified, the presently available solid-state quantum-mechanical packages avail-
able are all based either on density-functional theory (DFT) for the electron-electron inter-
actions or on the Hartree-Fock (HF) approximation for the electronic wave function.1Given
that the majority of materials is potentially metallic and taking into account the notorious
problems of HF theory to deal with that, the overwhelming success of DFT for solid-state
questions2is easily understandable, in particular when DFT is combined with plane waves
and the pseudopotential/PAW approximations allowing to structurally optimize whatever
kind of material based on exact Hellmann-Feynman forces. For certain nonmetallic materi-
als, however, post-HF approaches are a powerful alternative, 3as are semi-empirical approx-
imations to the HF approach for large systems.4By this, the repertory of the types of the
ground states accessible to the available software is significantly restricted. Practically, some
of the important types of the electronic states of solids cannot be reproduced since they
simply have not been programmed. The most striking (and scandalous) examples of inac-
cessible states are the Bardeen-Cooper-Schrieffer (BCS) state5,6 necessary for description of
superconductors (SC) and the related resonating valence bonds (RVB) state introduced in
the solid state context7as an option for systems with frustrated antiferromagnetic interac-
tions ultimately leading to the formation of spin-liquid phases.8,9 The RVB states have been
hypothesized to be those of high-Tccuprate superconductors. 10 Even more familiar electronic
states of solids - the magnetically ordered ones with an arbitrary pitch (superstructure) vec-
tor - are not easily accessible by the available numerical tools. The magnetically ordered
states can be obtained by extending the chemical unit cells to super-cells and setting the
primeval magnetic moments with broken symmetry (BS) in the input. Within this technol-
ogy only the simplest ordered magnetic states, ferromagnetic or antiferromagnetic with very
simple magnetic super-lattices, can be accessed. Thus, the presence of either BCS or RVB
or complex magnetic states of solids is incurred indirectly. Namely, the presence of occupied
anti-bonding one-electron states in the vicinity of the Fermi level and its intimate relation-
ship with itinerant ferromagnetism was apparent when the bonding proclivities of metallic
materials with strong exchange splitting were accessible from density-functional theory for
3

the first time11,12 In addition, the presence of occupied non-bonding one-electron states as
found in, say, metallic bcc-Cr 12,13 indicated a possibility of another kind of instability of the
primary symmetric structure, namely the formation of an antiferromagnetic ground state.
Although positioning the Fermi level in anti-bonding or non-bonding one-electron levels is a
rational way to prepare ferromagnetic or antiferromagnetic metallic materials by design,14
it is not possible to predict what specific BS phase is going to appear. Moreover, the BS
solutions of non-programmed types never come to surface in the calculations performed by
standard tools and thus cannot be conjectured on the basis of numerical experiments. An-
other feature so far missing in the available software is the temperature dependence of the
solutions of the electronic problems. This feature is, however, important due to characteristic
physics: transitions among the high-temperature symmetric and various low-temperature BS
phases are observed experimentally, the magnetic and superconducting phases being most
spectacular. Thus, we recently undertook some developments with a goal to heal 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 arbi-
trary superstructure vectors15 and (ii) the energy optimization with respect to the vector of
magnetic superstructure16 in the ΘΦ package. By the present paper we round up the devel-
opment of ΘΦ by adding the option of having an RVB state (for more detailed explanation of
these see below) as a result of the solution of an electronic problem. The paper is organized
as follows: in the next Section we present the necessary theoretical concepts in a form more
suitable for computational chemistry community in variance with papers15,16 more oriented
towards the computational physics one; next we describe implementation details; furtheron
we give test examples of applying developed software. Finally, we discuss the results of test
calculations and give some perspectives.
THEORY ACCOUNT
Like in the standard approach, the ΘΦ (Theta-Phi) program approximates the solution for
the many-particle electronic Hamiltonian of a crystal. Its most general form admitted by
ΘΦ involves one-electron terms (the core attraction and hopping) and two-electron terms:
4

that is one- and two-center Coulomb repulsion operators as well as the one- and two-center
spin-spin exchange operators. Their explicit forms are available. 15
Central objects in any mean-field theory are the mean-field version of the Hamiltonian,
the Fockian, and the density matrix. In that or another form, they are present in all existing
software. The electronic self-consistency problem is to be solved by requiring that a density
matrix ρand the Hamiltonian define a Fockian F[ρ], and the density stemming from its
ground state has to be equal to that defining the Fockian:
ρ⇔ρ(F[ρ]) (1)
This is depicted in Figure 1. Hence, the derived density matrix approximates the ground
state of electrons in a crystal, that is, its zero temperature phase. Figure 1
- here
In variance with the standard setting we want to be able to access different electronic
phases (magnetic, superconducting [alias BCS], spin-liquid [alias RVB]) which are tempera-
ture dependent. In order to address the temperature dependence one needs to minimize not
the ground state energy, rather the Helmholtz free energy instead. Similarly, the additional
phases (BCS, RVB, etc.) need to be included in the consideration. Thus, the general proce-
dure of solving the electronic self-consistency problem Eq. (1) and Fig. 1 for a crystal needs
to be modified at several points as described in the subsequent Subsections.
Decoupling
The first step of any mean-field theory is the decoupling serving to reduce the original
many particle Hamiltonian to the mean-field Hamiltonian (Fockian) F. In the original
Hamiltonian the electron-electron interactions (e.g. the Coulomb ones) enter through the
products of four electronic creation and destruction operators c†
1c2c†
3c4. Based on Wick’s
theorem17 the products of four operators are replaced by an expression keeping products
of only two operators complemented by the averages h...iof the product of two remaining
operators, by so called normal (Dc†
1c2E) and anomalous (Dc†
1c†
3Eand hc4c2i) elements of the
5

density matrix:
c†
1c2c†
3c4⇒Dc†
1c2Ec†
3c4+c†
1c2Dc†
3c4E−Dc†
1c2EDc†
3c4E
−Dc†
1c4Ec†
3c2−c†
1c4Dc†
3c2E+Dc†
1c4EDc†
3c2E(2)
+Dc†
1c†
3Ec4c2+c†
1c†
3hc4c2i − Dc†
1c†
3Ehc4c2i.
Note that the products of the two averages (the last terms in each row) are serving to
make the mean value of the Fockian equal to the mean value of the original Hamiltonian
calculated in the same approximation. In the standard software only the first two rows of Eq.
(2) (habitually called direct or Hartree and exchange terms, respectively) are implemented,
whereas the anomalous averages are forcedly set to zero. These averages are characteristic
for the BCS and RVB solutions of the electronic structure problem, thus we retain them to
be able to access the latter. The Fockian obtained through such decoupling is also known as
the Fock-Bogolyubov operator.18 The non-vanishing anomalous averages allow us to break
the symmetry tacitly implied in all quantum chemistry software: the conservation of the
number of particles (electrons).
As usual we assume that each unit cell rhomes Lspatial states (atomic orbitals, AOs)
numbered by subscript l= 1, . . . , L. For the sake of compactness of the subsequent presen-
tation we assemble the Fermi operators corresponding to the spin projection s=↓,↑in each
of the AOs, pertaining to the unit cell rof the solid, in a so called Nambu vector:19
Ψ†
r=c†
1r↓, c†
1r↑, ..., c†
Lr↓, c†
Lr↑, c1r↓, c1r↑, ..., cLr↓, cLr↑.(3)
It contains electron c†and hole ccreation (electron destruction) operators and allows for
uniform treatment of the normal, superconducting, spin-liquid and magnetically ordered
states.
Since the anomalous averages are retained in Eq. (2) non-vanishing matrix elements
between the electron (e) and hole (h) states appear in the Fockian operator as well, so that
the latter acquires the block form:
F=

Fee Feh
Fhe Fhh 
(4)
where, Feh =F†
he;Fhh =−Fee . The detailed forms of decoupled operators are available.15
6

In terms of the Fockian operator, the problem of minimizing the Helmholtz free energy
reads:
A=hF−µNi − θς →min.(5)
Here, h...iis the mean value of the enclosed operator calculated with the density matrix ρ:
h...i= Tr (...ρ)/Trρ;ςis the Shannon entropy to appear here since the temperature θ=kBT
appears in the energy units. Since the Fockian contains the off-diagonal blocks mixing the
electron and the hole states the quasi-particle states do not refer to any definite number
of electrons. As a result the number of electrons Nis not conserved in the target BCS
and RVB states and thus it is included in the optimization procedure accompanied by the
chemical potential µof electrons taking care about the conservation of their correct number
on average.
The Helmholtz energy minimization then performs iteratively (as in the standard soft-
ware), each iteration, as well as usual, divided in two stages: (i) the diagonalization of the
operator F−µN producing the spectrum of its quasi-particles; and (ii) the filling producing
the new density ρ0which assures the minimum of the Helmholtz free energy for the spec-
trum of quasi-particles, obtained by the diagonalization. The obtained density ρ0is then
reused for the decoupling of the original Hamiltonian and the iterations are repeated until
the convergence in the density matrix and (free) energy is achieved.
Diagonalization
In terms of Nambu vectors Eq. (3) the Fockian operator rewrites:
F=X
r,τ
Ψ†
rF(τ)Ψr+τ+ const,(6)
with the numerical constant so selected that hFi=hHi15, and where Ψris a column, formed
by the Fermi operators Hermitian conjugated to those in the row Ψ†
r. The summation extends
to all unit cells rand to lattice vectors τof the relative shifts of the unit cells containing
the states coupled through the matrices F(τ). It is translation invariant as it should be in
a crystal.
7

Diagonalization of the translationally invariant operators is habitually done as well in
two steps: first, the transition from the basis of local atomic states to that of the Bloch
sums (e.g. see1) is performed. The translation invariant operator is block-diagonalized at
this step: that is, reduces it to a (direct) sum of matrices numbered by the wave vectors k.
Next, the k-numbered matrix blocks are diagonalized one-by one.
At this point the (block) diagonalization procedure is so modified that it opens access
to electronic phases with arbitrary magnetic superstructure. In the latter the electronic
magnetic momenta (if any) appearing in the unit cells shifted by a lattice vector τare
rotated relative to each other by an angle (τ , Q)around a unit vector n, where Qis a
superstructure (pitch) wave vector. Since either nor Qmay be arbitrarily directed and even
more Qmay be arbitrarily small, the momenta in the neighbor unit cells may be relatively
rotated by an arbitrarily small angle giving rise to arbitrary (magnetic) spiral states. This
is reached by rotating the spin-quantization axes in each unit cell by the angles ϑ= (r, Q)
around nand builing the (generalized) Bloch sums from the local states quantized relative
these, unit cell specific, axes.20–22
Formally, we replace the usual numerical phase factors for the local states in the unit
cell r:exp (ikr)by 2×2spin-rotation matrices exp (ikr) Ωn,ϑ transforming the one-electron
states for the l-th AO with “up” and “down” spin projections relative to the laboratory z-axis
to the similar ones, but with the spin projections relative to the unit cell specific z-axes. The
matrices are given by:
Ωn,ϑ =σ0cos ϑ
2−i(n,σ) sin ϑ
2,(7)
where σ= (σx, σy, σz)is the vector of Pauli matrices, while σ0is the 2×2identity matrix.
The matrices exp (ikr) Ωn,ϑ are applied to each of the pairs of operators clr↓, clr↑in the
Nambu vector (and their Hermitian conjugates to the respective c†
lr↓, c†
lr↑pairs). Then, as
usual, summation over ris performed:
Ψk=1
√KX
r
exp (ikr) Ωn,(r,Q)Ψr,(8)
where Kis the number of the k-points involved in the calculation. This makes the Fockian
8

operator block-diagonal with the blocks numbered by the wave vectors k:
F=X
k
Ψ†
kF(k)Ψk,(9)
F(k)=X
τ
eikτ Ωn,(τ,Q)F(τ)
(here we omitted the immaterial const which can be restored in the end to maintain the
overall energy balance). Even in this case the dimension of the Fockian blocks F(k)remains
the same: 4L×4Lso that arbitrary magnetic superstructures can be addressed without
increasing the size of the unit cell.
Filling
Lastly, the filling generating the new density matrix from the eigenvectors of the Fockian
blocks F(k)is somewhat different from the usual filling of the particle states by electrons
implemented in existing solid state packages. Namely, at this point the desired temperature
dependence of electronic phases enters.
Indeed, diagonalization of F(k)’s yields its eigenvalues εα,k α= 1 ÷4Land eigenvectors.
The latter are assembled in unitary matrices Ξk. The matrix Ξkwhen applied to the corre-
sponding generalized Bloch sum of the Nambu vectors Eq. (8) produces the Nambu vector
of canonical quasi-particles:
Bk= (β1k, β2k, . . . , β4Lk) = ΞkΨk.(10)
As we already mentioned, the quasi-particles solving the energy minimum problem for the
F−µN operator are superpositions (linear combinations) of the electron and hole states as
suggested by Bogolyubov, Tolmachev and Tyablikov yet more than 60 years ago.23–26 Thus
they are referred to as bogolons in the BCS context. In their basis the sought density matrix
is diagonal. The equilibrium populations of the quasi-particle states βα,k assuring the free
energy minimum for a given spectrum εα,k of quasi-particles18,27,28 is given by :
ρ(k)
αα0=δα,α0fεα,k (µ)
θ(11)
where f(x) = (1 + ex)−1- the Fermi-Dirac distribution.
9

Since the number of electrons nα,k in the quasi-particle states is not mandatorily equal to
unity, in order to keep the average number of electrons in the unit cell equal to N0prescribed
by the chemical composition:
1
KX
α,k
fεα,k (µ)
θnα,k =hNi=N0,(12)
the chemical potential µof electrons is adjusted. As θ→0the Fermi distribution flows to
the standard step-like one, whereas the chemical potential µof electrons flows to the Fermi
energy εF.
At the same time, it can be shown18 that the spectrum of bogolons εα,k obtained by
diagonalization of the F(k)−µN blocks of the structure Eq. (4) is always symmetric with
respect to zero; namely: for each quasi-particle state with positive energy εα,k there exists
a quasi-particle state with the opposite energy −εα,k and vice versa, so that the chemical
potential of quasiparticles equals to zero (as it is in Eq. (11)).
The density Eq. (11) is to be used in the decoupling of the interaction terms. For this
end it must be back transformed to the representation of the original AOs, which is done
easily with use of the matrices Ξk:
hΨrΨ†
r+τi=X
k
exp (−ikτ ) Ω†
n,(τ,Q)Ξ†
kρ(k)Ξk.
The result does not depend on rbecause of the translational invariance (the expressions
ΨrΨ†
r+τare themselves matrices filled by all possible products of the Fermi operators from
the respective Nambu vectors). Because of the translational invariance, one can introduce
equivalent 4L×4Lmatrix blocks:
ρ(τ) = 1δτ,0− hΨrΨ†
r+τi.(13)
each accumulating matrix elements of the density between the states pertinent to the unit
cells shifted by the lattice vectors τ, while 1means a 4L×4Lidentity matrix; ρ(τ= 0) is
Hermitian, while at a finite shift τ6= 0 the following relation holds:
ρ(τ) = ρ†(−τ),
i.e. Hermitian conjugate of a density matrix at the lattice shift τis the density matrix at
the lattice shift −τassuring the hermiticity of the density matrix as a whole.
10

Physical properties
Once self-consistency is reached, the observables can be calculated, that is correlation func-
tions as well as band structure and density of electronic states (DOS). Below we summarize
the formulas for some of them,15 but many others can be defined in addition. At the self-
consistency, the internal energy Uis just an average of the Fockian operator F:
U=hFi=Tr (F ρ)
Trρ=1
KX
α,k
fεα,k
θεα,k
(hereinafter we omit the µ-dependence of the spectrum for the sake of brevity).
The (Helmholtz) free energy Aand the entropy ςEq. (5) are defined by the spectrum of
the canonical quasi-particles:
A=−θX
α,k
ln 1 + e−εα,k/θ ,(14)
ς=−∂A
∂θ =U−A
θ=−X
α,k nfεα,k
θln fεα,k
θ+1−fεα,k
θln 1−fεα,k
θo.
The expression Eq.(14) is used in practical calculations as anticipated above. The (electronic)
specific heat can be easily obtained:
CV/kB=∂U
∂θ =−θ∂2A
∂θ2.
The uniform Pauli magnetic susceptibility is given by:
χP=∂Sz
tot
∂Hz
,
where Sz
tot is the total magnetic moment of the unit cell:
Sz
tot =X
l
Sz
l,0=1
2X
lnDc†
0l↑c0l↑E−Dc†
0l↓c0l↓Eo,
expressed through the density matrix elements between the local states.
The band structure along the path connecting the high symmetry points in the Brillouin
zone is obtained by plotting εα,k along it. Finally, the density of states is obtained from εα,k
as usual:
D() = 1
KX
α,k
δ(−εα,k).
11

Closing the discussion of the observables, we note that the solutions of self-consistency
equations have their domains of existence as functions of Hamiltonian parameters, filling,
temperature and any other external condition. These domains form the system’s phase
diagrams, which can be directly compared with the experiment and/or other theories. As
well, there might exist (and this is the standard situation) several solutions or phases for the
same values of the Hamiltonian parameters, and/or temperature. In this case, owing to the
variational principle, the phase with the lowest free energy must be chosen, which paves the
way to description of the phase transitions at finite temperature.
IMPLEMENTATION DETAILS
Self-consistency cycle
The general (free) energy optimization procedure as applied to the electronic phases of a
crystal can be solved in two ways:
1. either as a self-consistency procedure (as in details described above): starting from a
trial ρand iteratively calculating new ρ0at each step, taking care about the chemical
potential. In order to stabilize such a procedure usually some forms of mixing are
used. We implemented the simplest linear mixing, such that ρ⇐αρ0+ (1 −α)ρwith
06α61;
or
2. by solving the self-consistency equation Eq. (1) supplemented with the constraint
fixing the number of particles as an equation for roots using the globally convergent
Newton-Raphson method29 with numerically estimated derivatives of the right hand
side of Eq. (1). It can be shown that the self-consistency condition is equivalent to the
evanescence of the derivatives of the energy with respect to relevant matrix elements
of the density. The energy is traced at each step to ensure that the chosen trial step
actually leads to a minimum.
Both approaches are implemented in ΘΦ.
12

Of course, a self-consistency procedure can find an extreme of the free energy which is
not necessary the (global) minimum. Therefore, we also track the values of the free energy
as self-consistency proceeds and control that it always goes down.
In case of multi-orbital systems and complicated lattices the inter-relations between den-
sity matrix elements at various distances (see e.g. Ref.30 ) may become very involved. It
is advisable to analyze them before using ΘΦ. We shall discuss this in more details in the
Examples Section below.
Matrix elements and porting with existing programs
It is a common place to contrast model and quantum chemical (ab initio or semi-empirical)
codes. Meanwhile, the difference is merely quantitative: codes of either type operate with
matrices formed according to prescriptions of the Theory Section with the only difference
that the model approaches use a small number of states in the unit cell (small number of
bands) and with respective (Fockian) matrix elements dependent on relatively small num-
ber of model parameters, whereas quantum chemistry packages employ realistic number of
atomic states and calculate the required matrix elements from the parameters of these latter
or pseudopotentials. The ΘΦ package bridges the model approach and quantum chem-
istry one by implementing the true multi-band scheme. As for the Hamiltonian/Fockian
matrix elements, the user can either specify them all manually, assuming an abstract the-
oretical model, or alternatively use an additional functionality allowing the import of the
hopping matrix elements ts,s0
l,l0(τ)from existing ab initio codes. This is achieved by reading the
tight-binding Hamiltonians, generated by either of the two well-known codes: wannier9031
and lobster32–34 serving to post-process the outputs of the DFT/PAW calculations. By
doing this, the precise, material-specific analysis of electronic phases of various crystals
can be made. ΘΦ reads the files wannier90_hr.dat (for wannier90) or RealSpaceHamil-
tonians.lobster (for lobster) and populates the hopping matrix. Both wannier90 and
lobster have in turn interfaces to the major codes (like vasp,abinit,castep,Quan-
tumEspresso etc.), which allows the ΘΦ user to profit from the results obtained by them.
Additionally, we can recommend using the MagAˆ
ıxTic package 35 to obtain the estimates
of the Jαβ
l,l0(τ)- diatomic exchange - parameters. In the future releases we plan to provide a
13

functionality of directly evaluting the Hamiltonian/Fockian matrix elements from chemical
composition/crystal structure data as in all quantum chemistry codes.
EXAMPLES
The features of ΘΦ described in the previous Sections has been implemented and tested15,16
against simple analytic models, the 1D Hubbard superconductor, 2D doped graphene su-
perconductor and square lattice doped antiferromagnet as well as against realistic models of
magnetic states of α- and γ-Fe. Below we shall briefly review these results before specially
concentrating upon the newest feature implemented in ΘΦ: spin-liquid (alias RVB) states
as compared to the magnetically ordered states of CuNCN.
Superconducting and magnetic phases of Hubbard model
The 1D single band Hubbard model is the simplest one for, say, electrons in polyene molecules
(CH)n,36 where each C atom holds one π-orbital or of individual chains of a metallic carbon
allotrope hypothetized by Hoffmann et al.37 as depicted in Figure 2. In the model setting Figure 2
- here
it is characterized by three parameters: inter-site ("π-π") one-electron hopping t, on-site
electron-electron interaction Uand the average electron population per site 1−η, where η
is referred to as a doping level. Accepting hereinafter t= 1 to be the energy scale, the other
model parameters as well as temperature express in its terms. Depending on the sign of U
either a magnetic (U > 0) or superconducting (U < 0) states come out as solutions of the
mean-field electronic problem. The latter corresponds to the density matrix of the form:
ρ=








n/2 0 0 −∆?
0n/2 ∆?0
0 ∆ 1 −n/2 0
−∆ 0 0 1 −n/2








.
The dimensionality of four is that of the subspace of quasipatricles which within the ΘΦ
setting stems from a single AO per unit cell by including two respective spin-projection
states and two corresponding hole states. The components of the density matrix, thus, have
14

the following meaning: n=c†
scs,∆ = hc†
↑c†
↓iboth independent on the site index which is
thus omitted for brevity.
The ΘΦ results for this simple model are shown in Fig. 3. As expected from the BCS Figure 3
- here
theory, the temperature dependence of the anomalous matrix element of the density ∆(the
superconducting order parameter) is described with a very good accuracy by:
∆(θ)=∆0s1−θ
θc3
.(15)
The temperature dependence of the solution of the electronic problem is a very remarkable
feature provided by ΘΦ . Incidentally, the U-dependence of ∆0at lowest temperature is very
well described by another BCS formula:
∆0(U) = −8e2π
U
U,(16)
for small absolute values of negative i.e. attractive Uas well nicely numerically reproduced
by ΘΦ.
Extending similar treatment to a square lattice of sites with the hopping tbetween the
nearest neighbors (a kind of model for one layer of simple cubic α-Po with only π-orbitals
included in the consideration) provides additional set of tests went through by ΘΦ. For
the square lattice 2D Hubbard model we studied tentative magnetic states of this model
(U > 0) under finite doping which may serve as a model of the corresponding simple cubic
layer deposited over some support. The single band Hubbard model with magnetic order
can be characterized by the density matrix of the form:
ρ=








n−m
20 0 0
0n+m
20 0
0 0 1 −n−m
20
0 0 0 1 −n+m
2








.(17)
Here, as well, the dimensionality of four is that of the space of quasiparticles stemming from
one AO per unit cell; in variance with 1D superconducting phase the anomalous averages
occupying the (northeast) antidiagonal are vanishing; n= 1 −ηis the average site occu-
pation, while mis the average site magnetization. Although the matrix refers to the single
15

unit cell (site) and describes the imbalance of the occupations of the states with different
spin projection - seemingly of a ferromagnetic state - this imbalance takes place with respect
to the adjustable local spin quantization axes as implemented in ΘΦ. Thus, depending on
the pitch wave vector Q, Eq. (17) describes an arbitrary magnetically ordered state (e.g.
Q= (π, π)results in the antiferromagnetic state) since the momenta of the magnitude mare
rotated by the angle (Q, τ)relative to each other in the unit cells shifted by the translation
vector τ. The results of the ΘΦ study of this model are depicted in Figure 4. Figure 4
- here
One can see that at the vanishing doping the Hubbard model treated by ΘΦ consistently
yields a checkerboard antiferromagnetic state with ∆Q= 0 provided the pitch vector is set
to be
aQ= (π, π) + (∆Q, ∆Q)
(ais the lattice constant) in a full agreement with the general theory and the numerical
findings.21 At a moderate doping level of η= 0.15,ΘΦ produces an energy profile shown on
the left of Figure 4 identical to that of Ref.21 .
Superconducting states of graphene
Superconductivity in pristine graphene does not occur naturally, although recently the super-
conductivity has been reported in graphene bilayers rotated at a series of “magic” angles. 38,39
Several approaches predicted superconductivity with various order parameter symmetries on
2D honeycomb lattice, among which auxiliary-boson theory (ABF)30,40 or even self-consistent
Bogoliubov-de Gennes approach to spatially inhomogeneous superconductivity.41,42 It is ex-
cessive to reiterate that neither of mentioned solutions of the electronic structure problem is
accessible to the available solid state quantum chemistry codes.
We benchmarked ΘΦ by reproducing several superconducting phases of various symme-
tries in graphene. We remind that on 2D honeycomb (graphene) lattice each lattice site
connects to three nearest neighbor (NN) sites and the unit cell of graphene contains two
atoms (see Fig. 5). With only NN hopping active, each unit cell is connected to its four
nearest neighbors at the lattice shifts τ=±a1,±a2.Figure 5
- here
Superconducting phases on the honeycomb lattice with the interactions between the NN
16

sites shifted by vectors δifeature the anomalous averages ∆AB(δi) = hc†
Ar↑c†
Br+δi↓i. The local
C3symmetry axes at each site stipulate relations between the NN averages corresponding to
the irreducible representations of this group: one-dimensional a1and two-dimensional e(the
former dubbed as (extended) s- and the latter d-phase in physical context although both
refer to the states of πelectrons residing in the graphene pzorbitals) the latter represented
by two components: x2−y2- and xy-waves. In the case of the sphase, all ∆AB (δi)are equal,
while in the x2−y2case: ∆AB(δ1) = 0 and ∆AB (δ2) = −∆AB(δ3); finally in the xy case:
∆AB(δ1) = −2∆AB (δ2) = −2∆AB(δ3).
The model Hamiltonian used for testing the ΘΦ capacity to reproduce the superconduct-
ing phases of graphene π-electrons was the t-J model Hamiltonian, featuring the one-electron
hopping between NN sites of the honeycomb lattice as well as a sizable exchange interaction
Jcoming from on-site Hubbard repulsion U:J=2t2
U.40 The value of tin graphene is known
for a long time: t= 2.4eV, 43,44 and reference therein, while the estimated U= 3.3t.30,40
Within the mean-field approach in ΘΦ according to the formalism outlined in the Theory
Section we could achieve perfect agreement with the reference data,30 as can be seen from
the right panel of Fig. 5, where we compare the critical superconducting temperature Tc
as a function of doping η(per C atom) at J= 1 in the scale30 precisely corresponding to
the ΘΦ value of J=2
3(for more detailed explanation see15). This agreement holds for
both s- and x2−y2-wave phases (we could not find literature data for Tcin the xy-wave
phase). We would like to emphasize that each point in the right panel of Fig. 5 is a result
of a calculation similar to that represented on the left panel of Fig. 3 with many points
performed to identify the critical temperature for a given level of doping. This proves the
solidity of our benchmark.
To have an insight into the electronic structure of these two solutions, we compare the Figure 6
- here
density of states (DOS) and the band structure in the extended-sand x2−y2-wave supercon-
ducting phases, as shown in Fig.6. A characteristic feature of the broken symmetry states is
the opening of the gap in the spectrum of (quasi)-particle energies. We observed this in the
cases of Hubbard chain and square lattice. Similarly, in the case of graphene the degeneracy
of the upper and lower electronic bands in the Dirac points of the Brillouin zone (Kpoints)
characteristic for graphene is lifted. One can see that in the s-wave superconducting phase,
17

there is a full gap in the DOS, while in the x2−y2-wave superconducting phase the DOS
goes to zero linearly in the vicinity of zero energy. Moreover, for the s-wave phase the true
gap opens at K(and K0) points in the Brillouin zone, while for the x2−y2-wave case two
nodal points appear on the line M→K→Γ0remarkably shifted from the K(and K0)
points - that is no true gap opens rather a DOS depletion whose exact position depends on
doping ηand J. Notice that in the latter case, a difference appears between Kand K0as
well as between M=π(1
2,1
2)and M0=π(1
2,0). Energetically, the extended-sphase is the
lowest one at J > 0.75t(in the notations of Ref.40 ), while for J < 0.75tthe d-wave phases
(almost indistinguishable between each other in free energy) are most favorable.
Magnetic phases of iron
After testing ΘΦ against toy models like cyclic or two-dimensional Hubbard or t-J graphene
ones we switched to true multiband models exemplified by metallic iron. The magnetism
of iron is of vital fundamental and technological importance and was extensively studied
(see e.g. review45 and references therein as well as46,47). Iron features versatile mag-
netic/structure phase diagram of which we have chosen α- and γ-Fe to benchmark ΘΦ.
In α-Fe we compared the FM phase as obtained in ΘΦ and DFT+Ucalculations in vasp,
while in γ-Fe after comparing the NM and FM phase with vasp DFT+U, with use of ΘΦ
we addressed the AFM phase as well.
The DFT simulations werer performed by us with use of vasp 48 in the PBE/GGA 49
version (for details see15 ). From them we extracted the effective tight-binding Hamiltonian
used as input for ΘΦ with use of wannier90.31
For the sake of comparability with vasp results we used isotropic Ujj0=Uand Jj j0=J,
although ΘΦ in principle can handle arbitrary interaction matrices.
For α-Fe we used the bcc lattice with a= 2.858˚
A and a 8×8×8Monkhorst-Pack
k-point mesh50 (both in ΘΦ and vasp). In this case we considered a ferromagnetic solution
(FM) and a non-magnetic solution (NM) and used the energy of the latter as a reference
point for comparing the relative stability of the former. It can be seen from Table 1 that Table 1 -
here
18

the agreement between ΘΦ and vasp is in general good:1the FM state gains the energy
with respect to the NM one and the d-shell magnetic moment is very close. The energy
gain in ΘΦ is approximately 20% greater, due to more adequate of the intra-atomic orbital
hopping terms as explained in more details.15 The comparison of the band structures is Figure 7
- here
presented in Figure 7. The overall agreement is very good. The minor discrepancy can be
attributed to the different local basis sets used in vasp and in ΘΦ. In vasp, the DFT+Uis
formulated for the charges inside PAW spheres, while in ΘΦ for the charges occupying the
Maximally Localized Wannier Orbitals (MLWO). This yields a somewhat smaller splitting
between lower and upper Hubbard bands in ΘΦ. In addition, at larger U, the margins of
the disentanglement window, where the wannier90 bands start to deviate from the vasp
ones, may come closer to the Fermi level, thus inducing further bias.
For γ-Fe we considered the fcc lattice with a= 3.583˚
A and a 8×8×8Monkhorst-Pack
k-point mesh.50 In addition to the FM and NM states mentioned above, in this case we
also considered an antiferromagnetic state (AFM) with the pitch vector Q=π(0,1,1) in
terms of the vectors of the reciprocal lattice given relative to the primitive cell: the state
with an alternation of oppositely magnetized ferromagnetic planes along the z-axis of the
conventional cubic unit cell.
We found that in γ-Fe, the NM state has the highest energy among all the states consid-
ered. It can be seen from Table 2 that, at U= 6 eV and J= 1 eV, the FM and AFM spiral Table 2 -
here
states with the above value of Qare lower than the NM one and are close in energy, being
the FM state slightly lower (by 0.04 eV in ΘΦ and 0.08 eV in vasp respectively). We have
checked that by increasing U, the AFM state eventually wins over the FM one, although the
band structure agreement becomes somewhat worse because the edges of the wannier90
disentanglement window are pushed closer to the Fermi level. We point out that both the
FM and AFM states energy gains coming from vasp are well reproduced in ΘΦ. Finally,
the d-orbital magnetic moment in ΘΦ appears to be identical to the vasp one up to the
second digit. The very good correspondence of the band structure of FM γ-Fe modeled by
two methods is presented in Figure 8. Figure 8
- here
1The values of the magnetic moment in this and subsequent Tables serve only to show the closeness
between the results stemming from vasp and ΘΦ calculations.
19

Spin-liquid phases of CuNCN
The most important feature added to ΘΦ at the present stage is the capacity to have the
spin liquid states/phases, alias resonating valence bond (RVB) states, as solutions of the
electronic problem: the capacity so far not accessible to any of the publicly available codes.
The existence of such phases is conjectured for high-Tccuprate superconductors,10,51 her-
bertsmithite family of compounds 9to mention a few. Among them we recollect an enigmatic
copper compound a nitrogen-containing analog of copper oxide synthesized recently.52 It can
be derived from an oxide of the rock salt structure by replacing the formal O2- anions by the
carbodiimide dianions NCN2- which together with the Jahn-Teller distortion characteristic
for hexacoordinated Cu2+ produces relatively simple layered structure represented in Figure
9 (left). The physics of this material is rather peculiar: 53 being an insulator, it shows temper- Figure 9
- here
ature independent (Pauli-like) paramagnetism from the room temperature down to approx-
imately 100 K when the latter switches to the gapped (Arrhenius-like) regime, although no
magnetically ordered state could be detected down to the lowest investigated temperature.
At the same time, some characteristic structural effects could be detected along with the
variations of the magnetic properties. The entirety of the experimental findings in relation
to CuNCN with a high probability allows for a consistent interpretation of its physics as a
sequence of transitions between the quasi-one-dimensional and two-dimensional spin-liquid
phases. For the purpose of the present paper we show that the ΘΦ package reproduced these
features consistently and numerically.
The physical properties of CuNCN fairly map onto the c-a-ca Heisenberg model on a
rectangular lattice54 in the ac crystallographic plane with the exchange interactions between
the nearest neighbors: Jcalong the c-axis, Jaalong the a-axis (each Cu atom has two
neighbors along each of the two directions), and Jac along the diagonals of the rectangles
whose vertices are occupied by the Cu atoms (each atom has four neighbors along the ac
diagonals which otherwise would be considered as next nearest neighbors whose presence
leads to frustration considered to be a prerequisite for the formation of a spin-liquid phase8).
The scheme of the relevant Heisenberg model is shown in Figure 9 (right).
The spin-liquid phases are considered to be a physical manifestation of the RVB solutions
20

of electronic problems for respective materials (Hamiltonians). The electronic states of the
RVB type are superpositions of the individual configurations where all the electron spins
residing in all nodes of a lattice form singlet pairs:55
c†
r↑c†
r0↓−c†
r↓c†
r0↑.
The most known example of such a state is that of benzene depicted in Figure 10. The Figure
10 - here
corresponding pairings are shown in Figure 10 by colored ellipsoids which indicate formation
of spin-pairings between the (electrons residing in the orbitals attached to the) vertices, they
join. Contrary to the widespread belief, the RVB state of benzene has very little to do with
the H¨uckel state, that is, the Slater determinant of the occupied molecular (crystal) orbitals
or the symmetry adapted linear combinations of the orbitals attached to the vertices. This is
the only option available in the standard solid state quantum chemistry codes. As explained
in Refs.55,56 the weight of the RVB state in the single determinant wave function (for benzene)
is not more than 12.5% and even more - the corresponding amount - the contribution of the
RVB states to it vanishes when it comes to an infinite system, a crystal.56 Thus, in order
to be able to access either the single determinant (ultimately, the only available in the
standard codes) and an RVB solution within a single procedure special precautions need
to be made. It can be shown,57 that the RVB state can be covered by an independent
(quasi)particle density Eq. (11) if off-diagonal anomalous averages are considered together
with their normal off-diagonal density counterparts. This manifests in appearance of non-
vanishing order parameters ζτ=qξ2
τ+|∆τ|2;ξτ=c†
rscr+τ scombining the normal averages
ξτand the anomalous ones ∆τ- the characteristic of a spin-liquid phase.58 When it comes
to CuNCN as described by the c-a-ca model (Figure 9 - right) the RVB states can be
expected (Figure 11). In each of the RVB states we show pairings related to the respective Figure
11 - here
exchange interactions which are indicated by the colors of ellipses. In the respective states
characterized by the presence of non-vanishing order parameters ζτfor the allowed values
of the neighbor vectors τall indicated pairings are present as indicated by the ⊕signs.
The analysis54 of the c-a-ca model in the high temperature approximation reveals a rich
diagram of the RVB/spin-liquids phases. Specifically, depending on the relation between
the exchange constants Jc,Ja, and Jac and temperature one can expect formation of eight
21

electronic phases in which the corresponding anomalous averages along respective directions
in the crystal are non-vanishing in various combinations as depicted in Figure 11.
The ΘΦ package as applied to this model exactly reproduces the critical temperatures at
which the respective anomalous averages/order parameters bounce from zero, coming from
the high temperature expansion of the free energy. Important acquisitions for the lowest
temperature region (T < 0.02J) available only with help of ΘΦ as compared to the high-
temperature estimates are (i) the temperature dependence of the order parameters ζτin the
vicinity of zero temperature, following the same law Eq. (15) as the anomalous averages in
the Hubbard chain model with on-site attraction and (ii) the parametric phase diagram at
the temperature flowing to zero. As for the latter, the numeric study with ΘΦ only partially
confirms the result of our previous study. 59 Indeed, as one can conjecture from inspection
of the sequence of phase diagrams in Figure 12 (right) with decrease of the temperature the Figure
12 - here
richness of the spin-liquid phases degenerates, so that all phases except two 2D spin-liquids
depicted yellow (three order parameters are non-vanishing) and orange (two non-vanishing
order parameters ζaand ζc) are squeezed out to the respective lines where either Jaor Jc
vanish. The stability analysis of the “orange” spin-liquid with respect to emergence of a
non-vanishing ζac the “yellow” spin-liquid59 shows the coefficient at ζ2
ac in the free energy
expansion performed in the “orange” phase to be always positive. This phase is stable with
respect to the transition into the “yellow” phase. This had been interpreted59 as an indication
of the global “orange” spin-liquid ground state of the c-a-ca model. The numerical results of
the present work, however, show that the stability of the “orange” phase was interpreted too
widely. The stability of the “orange” phase59 means only that the transition to the “yellow”
one cannot be of the second order, but may fairly be a first-order transition implying some
non-homogeneity of the material previously suspected from experiment.53 We return to this
possibility below. Thus, the zero temperature parametric phase diagram of spin-liquids
contains two regions occupied, respectively, by the “yellow” and “orange” spin-liquids, the
finding established numerically with help of ΘΦ.
22

Spiral magnetic states in CuNCN
The spin-liquid (RVB) phases are not the only possible ones in the c-a-ca model of CuNCN.
More traditional, spin spiral states can be easily simulated within ΘΦ, too. These states can
be stabilized by building the periodic spin patterns with the pitch vector Q. For the above
mentioned “experimental” parameter values showing the temperature dependence of the spin-
liquid order parameters depicted in Figure 12 and fitting the experimental data for CuNCN
at intermediate temperatures we, with use of ΘΦ, performed a scan through the Qvector in
the first Brillouin zone and plotted the energy of a magnetic state of the c-a-ca model as a
function of Q(Figure 13). The energy landscape presents two minima at (π, 0) and (0, π),Figure
13 - here
and one saddle point around (π/2, π/2) together with two fairly flat regions around (0,0) and
(π, π). The latter correspond to an absence of any magnetic solution at the corresponding
values of the Q-vector. The minimum at (π, 0) corresponds to a spin density wave with spins
arranged parallel and alternating “up”-”down” orientation at nearest neighbor distance along
a-axis, the one at (0, π)alternate along c-axis. Finally the energetically unstable saddle
point corresponds to an incommensurate spin-density wave with the spin quantization axis
making a complete rotation at approximately every two steps along the ca direction. The
minima depths are: −0.3187 at (π, 0) and −0.4601 at (0, π), respectively (in units of J).
The depth of the saddle point is −0.06J. The minimum at (0, π)is lower because in the
present example Jc> Ja. Therefore, the globally stable spin spiral state in CuNCN at the
given values of Jparameters is realized at Q= (0, π)according to ΘΦ calculation with the
minimum energy value of −0.4601J. Remarkably, the spin-liquid phase has the energy of
−0.125J. Interestingly, the spiral magnetic solution in the Heisenberg system like CuNCN at
the experimental values of Jtends to stabilize commensurate spin waves ((0, π)and (π, 0)) as
opposed to e.g. the 2D square lattice with doping as described by the Hubbard Hamiltomian
considered above. This is a consequence of several factors: i) comparable values of all three
antiferromagnetic Jvalues; ii) inter-site connectivity of the Heisenberg couplings which tends
to favor the (commensurate) order at the corresponding distances; and iii) absence of the
kinetic energy term loosing from the spiral order and the on-site Coulomb repulsion gaining
from the spiral order, which creates an interplay leading to the incommensurate spin spirals.
23

These issues will be considered elsewhere.
Thus, according to numerical analysis of the present paper the ground state (lowest tem-
perature phase) of the c-a-ca model of CuNCN is a state with frozen momenta, as established
by the μSR measurements.60 Due to significant difference between the spin-ordered and spin-
liquid states (completely different set of non-vanishing order parameters) one can conjecture
that the respective solutions represent different local minima of the free energy so that the
transition between them is a first order one. As a consequence, a temperature range where
the phases coexist may occur; in this range, the material is heterogeneous with spatial areas
occupied by either one or another (respectively, spin frozen and spin liquid ones) phase.
CONCLUSIONS, AND PERSPECTIVES
As of today, the available solid-state electronic-structure codes suffer from a limited func-
tionality with respect to the possible magnetic superstructures. The generally implemented
technology of multiplying the chemical unit cells allows to mimic only commensurate mag-
netic structures and even the simplest ones, since increasing the unit cell size by a factor of n
leads to the n3increase of the required computational resources due to employed diagonaliza-
tion procedures. Moreover, alternative types of electronic states like BCS and/or RVB corre-
sponding to physically interesting superconducting and spin-liquid electronic states/phases of
materials are not available at all. The temperature dependence of the solutions of electronic
problem (an important component of the physical behavior of superconducting, spin-liquid
and magnetic materials) is not accessible either. These gaps are partially closed by the new
ΘΦ package which will be further developed.
On examples of two model Hamiltonians (the Hubbard one for one-dimensional chain
and for two-dimensional square lattice and the t-J one for graphene) we demonstrate the
capacity of ΘΦ to reproduce the temperature dependent BCS solutions in these models. The
existence of RVB (spin-liquid) states/phases in CuNCN has been confirmed as well as the
existence of characteristic dependency of the RVB order parameters on temperature, known
from analytic theory. At the same time, it is shown that in CuNCN, the spin-ordered state
has a lower energy for the ratio of the exchange parameters fitting the experimental data.
24

Since the tight-binding Hamiltonians are the inputs for the proposed procedure, the ports
to the sources of such have been implemented. With use of the tight-binding parameters
as extracted from the vasp calculation of the ferromagnetic bcc and antiferromagnetic fcc
iron projected on the 3d4s4plocal basis the band structures and the respective magnetic
momenta together with their relative energies are fairly reproduced. The magnetically or-
dered states known from experiment are reproduced through optimization of energy with
respect to the superstructure wave vector, without extending the chemical unit cell. By this
the applicability of the code to the multi-band models of interacting electrons in solids is
demonstrated. The ΘΦ package is available at the the NetLaboratory portal 61 for registered
users.
ACKNOWLEDGMENTS
This work is partially supported by the German Research Foundation through the Collab-
oration Research Grant 761 (Steel ab initio) and by the Russian Science Foundation (grant
№ 19-73-10206).
25

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R. Dronskowski, S. Kikkawa, and A. Stein (Wiley, 2017), vol. 5, ISBN 9783527691036.
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0020-7608; 1097-461X.
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0034-4885.
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in the anisotropic triangular lattice. mean-field rvb analytical results. (2010), URL
https://arxiv.org/abs/1008.0182.
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29

62. K. Momma and F. Izumi, J Appl Crystallogr 44, 1272 (2011).
30

Figure 1: ΘΦ program flowchart diagram. Basically, it goes about the same entities (en-
closed in rectangles), operations (enclosed in ellipses), checks (enclosed in a rhombus) and
transitions shown by arrows, as appear in all solid-state quantum-chemistry packages. They
all are modified within ΘΦ in order to accommodate temperature dependence, superconduct-
ing and spin-liquid states as well as magnetic superstructure as explained in the subsequent
Subsections.
Figure 2: The structure of a hypothetized37 metallic carbon allotrope featuring the sp2
hybridization of each carbon atom forming quasi-isolated chains of the π-AOs further char-
acterized by the nearest neighbor hopping parameter tand the on-site electron-electron
interaction parameter U. Picture is produced with use of the VESTA 3 software.62
Figure 3: Left: temperature at U=−1and right Udependence at θ= 10−5of the super-
conducting paring ∆for local superconducting phase of 1D Hubbard model. The calculation
is performed for n= 1 (half-filling) of 50000 sites. The inset shows the details of the (∆)
approach to zero while U→0. The blue lines on the left and the inset are the respective
analytic formulae Eqs. (15) and (16). Adapted from. 15
Figure 4: Left: A benchmark of ΘΦ incommensurate spin state phase diagram. The axes
represent inverse interaction 1/U , doping ηand pitch vector offset from (π, π)normalized
to π. Right: Spin rotation pattern in a 20 ×20 2D square lattice chunk with a pitch vector
of Q= (π, π) + (∆Q, ∆Q)with ∆Q=π
10 , that is for η≈0.02 and U≈10. Looking
attentively one can notice that the NN spins are approximately antiparallel and become
exactly antiparralel after ten steps for either combination of unit cell vectors. For twenty such
steps spins become parallel suggesting the the magnetic supercell required for reproducing
such a spiral state to be 20 ×20 as well. We stress once again, that within ΘΦ this result is
achieved by diagonalizing Fockian matrices for one unit cell containing one pzAO. Adapted
from.16
31

Figure 5: Left panel: graphene unit cell showing two types of sites (A and B), vectors
{δi}connecting NN sites, and the unit cell vectors {a1(2)}. The dotted rhombus shows a
"chemical" unit cell as comprising two atoms of A and B sublattices, respectively. Right
panel: Comparison of the doping dependence θc(η)for s-wave and x2−y2-wave phases of
graphene with the data.30 The points were extracted from the graphs30, while the lines are
calculated by ΘΦ. For the details of the calculations and the difference in the exchange
parameter definitions see the text. Adapted from.15
Figure 6: Left panel: band structure along the path Γ→M→K→Γ0→M0→Γ
and DOS for the s-wave superconducting phase of graphene. Right panel: Band structure
along the path Γ→M→K→Γ0→M0→Γand DOS for x2−y2-wave superconducting
phase of graphene. Notice that in this case, a difference appears between K=π(1
3,2
3)and
K0=π(2
3,1
3)as well as between M=π(1
2,1
2)and M0=π(1
2,0). In either case Γ0point is
the center of the adjacent BZ with the k-space coordinates (1,0);η= 0.1/atom, θ= 10−5.
In either case it goes about the states of π-electrons of a graphene sheet. Adapted from. 15
Figure 7: Band structure comparison between ΘΦ (left panel) and vasp (right panel) for
ferromagnetic α-Fe. Adapted from.15
Figure 8: Band structure comparison between ΘΦ (left panel) and vasp (right panel) for
antiferromagnetic calculations in γ-Fe. Adapted from.15
Figure 9: Structure of CuNCN material (left) and the respective antiferromagnetic interac-
tions within an ac-layer according to c-a-ca model (right). Adapted from.15
Figure 10: Simplest and most widespread example of an RVB state in a molecule: benzene
with antiferromagnetic coupling between adjacent sites bearing one electron (spin) each.
Its wave function is a superposition (as shown by ⊕symbols) of five linearly independent
pairings known as Rumer basis.
32

Figure 11: Exemplary RVB states of the c-a-ca model of CuNCN. Three order parameters
ζc, ζa, ζca are possible. They may be either vanishing or non-vanishing producing altogether
eight possible combinations of which we represent five nontrivial ones. One more with all
three ζτ’s vanishing is a high-temperature disordered phase (shown in gray in the parametric
phase diagrams in Figure 12) and two more (1D and Q1D with spin-pairings along the
different crystallographic directions).
Figure 12: Various CuNCN properties calculated by ΘΦ. Left: spin-liquid order parameters
ζa,ζcand ζac as functions of temperature for Ja= 0.2346, Jc= 0.376, Jac = 0.3894 cor-
responding to the values of exchange parameters fitting an entirety of experimental data.54
Right: parametric phase diagrams for the c-a-ca model for different temperatures.
Figure 13: Energy as a function of pitch vector in CuNCN at Ja= 0.2346, Jc= 0.376, Jac =
0.3894. Energy is expressed in units of J≡Ja+Jc+Jac.Jin CuNCN is estimated to be
3270 K.54 Contour lines are added for better visualization. Calculations are performed at
θ= 10−6J. The red dot on the energy axis represents the RVB state solution.
33

H, ρ
decoupling
F[ρ]
diagonalization
ε, Ξ
filling
ρ’
update
ρ = (1 – α)ρ + αρ’ |ρ – ρ’| < η
ρ
solution,
properties
NO
YES
Figure 1
A. Tchougr´eeff, E. Plekhanov,
R. Dronskowski
J. Comput. Chem.
34

Figure 2
A. Tchougr´eeff, E. Plekhanov,
R. Dronskowski
J. Comput. Chem.
35

0.000
0.004
0.008
0.012
0.016
0 0.002 0.004 0.006 0.008 0.01
∆(θ)
θ
0.00
0.10
0.20
0.30
0.40
0.50
−10 −8−6−4−2 0
Δ(U)
U
0.000
0.004
0.008
0.012
0.016
−0.9 −0.6 −0.3 0
Figure 3
A. Tchougr´eeff, E. Plekhanov,
R. Dronskowski
J. Comput. Chem.
36

0
0.1
0.2 0
0.1
0.2
0.3
0.2
0.4
0.6
0.8
1
1/U
η
a
∆
Q/
π
Figure 4
A. Tchougr´eeff, E. Plekhanov,
R. Dronskowski
J. Comput. Chem.
37

δ
1
δ
2
δ
3
a1
a2
A B
0.00
0.02
0.04
0.06
0.08
0.10
0 0.05 0.1 0.15 0.2
s
d
θ
c
η
(per C atom)
Figure 5
A. Tchougr´eeff, E. Plekhanov,
R. Dronskowski
J. Comput. Chem.
38

−10
−8
−6
−4
−2
0
2
4
6
8
10
ΓM K Γ′ M′ Γ
Energy, (
t
)
0.05 0.1 0.15
−10
−8
−6
−4
−2
0
2
4
6
8
10
ΓM K Γ′ M′ Γ
Energy, (
t
)
0.05 0.1 0.15
Figure 6
A. Tchougr´eeff, E. Plekhanov,
R. Dronskowski
J. Comput. Chem.
39

ΓH N ΓP H
–10
–5
0
5
10
15
Energy (eV)
–10
–5
0
5
10
15
,H N ,P H
Energy (eV)
Figure 7
A. Tchougr´eeff, E. Plekhanov,
R. Dronskowski
J. Comput. Chem.
40

ΓX W K ΓL U W L K
–10
–5
0
5
10
15
Energy (eV)
–10
–5
0
5
10
15
,X W K ,L U W L K
Energy (eV)
Figure 8
A. Tchougr´eeff, E. Plekhanov,
R. Dronskowski
J. Comput. Chem.
41

Figure 9
A. Tchougr´eeff, E. Plekhanov,
R. Dronskowski
J. Comput. Chem.
42

Figure 10
A. Tchougr´eeff, E. Plekhanov,
R. Dronskowski
J. Comput. Chem.
43

2D RVB
⊕
⊕
Q1D RVB
⊕
⊕
2D RVB
⊕
⊕
ζτ= 0, τ =c, a;
ζca 6= 0
ζc6= 0;
ζτ= 0, τ =a, ca
ζτ6= 0, τ =c, ca;
ζa= 0
ζτ6= 0, τ =c, a;
ζca = 0
ζτ6= 0,
τ=c, a, ca
Figure 11
A. Tchougr´eeff, E. Plekhanov,
R. Dronskowski
J. Comput. Chem.
44

0.00
0.03
0.06
0.09
0.12
ζ
a
0.00
0.06
0.12
0.18
0.24
0.30
ζ
c
0.00
0.03
0.06
0.09
0 0.04 0.08 0.12 0.16
ζ
ac
θ
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
θ
=0.20
Jc
Ja
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
θ
=0.04
Jc
Ja
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
θ
=0.02
Jc
Ja
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
θ
=0.00001
Jc
Ja
Figure 12
A. Tchougr´eeff, E. Plekhanov,
R. Dronskowski
J. Comput. Chem.
45

0
π/2
π
π/2
π
−0.50
−0.40
−0.30
−0.20
−0.10
0.00
−0.1 −0.2 −0.3
−0.4
Qa
Qc
Figure 13
A. Tchougr´eeff, E. Plekhanov,
R. Dronskowski
J. Comput. Chem.
46

NM FM (ΘΦ) FM(vasp)
∆Etot (eV) 0−2.7−2.1
µ(µB)0 3.4 3.1
Table 1: Total energy gain Etot and Fe magnetic moment comparison between ΘΦ and VASP
for ferromagnetic calculations in α-Fe.
47

NM FM (ΘΦ) AFM (ΘΦ) FM(VASP) AFM (VASP)
∆Etot (eV) 0−2.64 −2.60 −2.32 −2.24
µ(d-shell, µB)0 3.24 3.10 3.24 3.10
Table 2: Total energy gain ∆Etot and Fe d-shell magnetic moment comparison between ΘΦ
and VASP for ferromagnetic and anti-ferromagnetic [Q=π(0,1,1)] calculations in γ-Fe.
48