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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,α0fεα,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

fεα,k (µ)

θnα,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

fεα,k

θεα,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 nfεα,k

θln fεα,k

θ+1−fεα,k

θln 1−fεα,k

θo.

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

θc3

.(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 scombining 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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(2013).

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R. Dronskowski, S. Kikkawa, and A. Stein (Wiley, 2017), vol. 5, ISBN 9783527691036.

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29

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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