#  Geometrical Perspective on Spin-Lattice Density-Functional Theory

Markus Penz  Max Planck Institute for the Structure and Dynamics of Matter and
Center for Free-Electron Laser Science, Hamburg, Germany  Department of
Computer Science, Oslo Metropolitan University, Oslo, Norway<EMAIL_ADDRESS>Robert van Leeuwen  Department of Physics, Nanoscience Center, University of
Jyväskylä, Finland

###### Abstract

A recently developed viewpoint on the fundamentals of density-functional
theory for finite interacting spin-lattice systems that centers around the
notion of degeneracy regions is presented. It allows for an entirely
geometrical description of the Hohenberg–Kohn theorem and
$v$-representability. The phenomena receive exemplification by an Anderson
impurity model and other small-lattice examples. The case of adiabatic change
and the time-dependent setting are examined as well.

###### Contents

  1. I Introduction
  2. II Spin-lattice systems
  3. III Density and potential spaces
  4. IV $v$-representability, non-uniqueness, and degeneracy regions
  5. V Sets of the potential-density mapping
  6. VI Anderson impurity model: mistaken non-$v$-representability and real convergence issues
  7. VII Adiabatic variation of the potential and geometric phase
  8. VIII Time-dependent setting
  9. IX Outlook
  10. A Unitary transformation of the Anderson impurity model

##  I Introduction

Within the field of density-functional theory (DFT), lattice models are
frequently considered to obtain rigorous theoretical statements in cases where
the continuum formulation proves too difficult [1; 2; 3; 4; 5], or to treat
physical situations in an approximate way [6; 7; 8; 9; 10; 11; 12; 13; 14].
When considering a finite lattice, this consists of $M$ lattice sites that
serve as the positional degrees-of-freedom for $N$ fermionic quantum
particles. Such settings arise directly from a space discretization or are
introduced as a model in themselves, often including spin and on-site
interaction in the form of a Hubbard model [15]. Since they are interpreted as
simplifications of the continuum case, it is commonly assumed that any
statement from continuum DFT equally holds in the lattice setting. This
adoption of known results applies especially to the Hohenberg–Kohn (HK)
theorem [16], which states that a ground-state ($v$-representable) density is
produced by a unique (up to a constant) potential. Yet Chayes  _et al._ [4],
after their proof that every density on an (even infinite) lattice with
$0<\rho_{i}<1$ can be produced by a well-defined potential
($v$-representability), carefully remarked that: “The HK theorem for fermions
at zero temperature remains an open problem.”

And indeed, in the study of ground-state solutions on finite lattice systems
without spin (at zero temperature), it was only recently discovered by the
authors that already in very simple lattice systems counterexamples to the HK
theorem exist [17]. For the spin-resolved version of density-functional theory
(SDFT), on the other hand, such counterexamples have already been discussed
before, both for the continuum [18; 19; 20] and on lattices [21]. So while
there is representability of ground-state densities on the lattice by a
corresponding choice of potentials, there is no _unique_ representability. The
phenomenon can be linked to the occurrence of a critical number of zero
components in the wave-function expansion with respect to the lattice basis
[17]. Interestingly, this again relates to degeneracy, since the potential can
then be varied without affecting the ground state until a level crossing with
an excited state is reached. This creates an intriguing geometrical picture
that explains the HK counterexamples as the intersection points of degeneracy
regions (themselves objects from algebraic geometry) in density space with
each other or with the boundary of the density domain [22]. A few examples for
degeneracy regions of simple lattice systems are given in Fig. 1.

1234 (a)

1234 (b)

1234 (c)

Figure 1: Three examples of degeneracy regions taken from Ref. [22] in their
respective tetrahedral and octahedral density domains, together with the
corresponding lattice.

A further surprising discovery is the existence of $v$-representable densities
even on the boundary of the density domain (i.e., with $\rho_{i}=0$ or $1$),
exactly where a density region touches the boundary. And even though
degeneracy occurs only for very special potentials, either being highly
symmetric or leading to an accidental degeneracy, the complex of all
degeneracy regions can occupy a large part of the total density domain. As
Ullrich and Kohn [23] expressed it in their pioneering work on topology in the
potential and density spaces: “Ground-state degeneracies in $v$ space are
‘rare’ [while] degeneracy in $\rho$ space is _not_ ‘rare’.” All this
highlights the special relevance of degeneracy when discussing DFT, not only
on lattices, and it offers exciting links to various fields of mathematics
like topology, algebraic geometry, linear algebra, group theory, and graph
theory.

In this perspective we aim at summarizing previous findings, especially those
from Refs. [17; 22], in a setting that explicitly includes the spin degree-of-
freedom. We do this by first introducing the basic setting (Sec. II) and the
density and potential spaces that form the basis of the further geometrical
inquiries (Sec. III). This sets the stage to explain the important notions of
$v$-representability, non-uniqueness, and degeneracy regions and to show how
they are all related (Sec. IV). We are such able to give a purely geometrical
formulation of the HK theorem. In order to assist the whole discussion, the
different involved sets in density and potential space are introduced and
explained separately (Sec. V). We continue with the very interesting example
of an Anderson impurity model, where several of the before theoretically
introduced phenomena can be found, and we describe their relation to the
Kohn–Sham method (Sec. VI). Next to the ground-state case, we also discuss
adiabatic variation of the potential and its connection to the geometric phase
(Sec. VII), as well as the time-dependent setting (Sec. VIII). Finally, we
summarize the remaining open questions and give an outlook towards possible
future research directions (Sec. IX).

##  II Spin-lattice systems

We consider putting $N$ spin-$\frac{1}{2}$ particles on $M>N/2$ sites. The
corresponding one-particle Hilbert space is thus $\mathbb{C}^{2M}$ which gets
promoted into the $N$-particle Hilbert space
$\mathcal{H}=(\mathbb{C}^{2M})^{\wedge N}$ by an $N$-fold antisymmetric tensor
product. Throughout this paper, we will remain in a _finite-dimensional_
Hilbert-space setting. The lattice indices will always be
$i,j,k\in\\{1,\ldots,M\\}$ and the spin indices
$\alpha,\beta\in\\{\uparrow,\downarrow\\}$ with respect to $\hat{\sigma}_{z}$.
We employ shorthand notation for index tuples, $i\alpha\equiv(i,\alpha)$. We
use the usual fermionic creation and annihilation operators
$\hat{a}^{\dagger}_{i\alpha},\hat{a}_{i\alpha}$ of the spin-lattice basis and
write $\hat{\rho}_{i\alpha}=\hat{a}^{\dagger}_{i\alpha}\hat{a}_{i\alpha}$ for
the number (density) operator. If one index is left out, this implies
summation over the other one, e.g.,
$\hat{\rho}_{i}=\hat{\rho}_{i\uparrow}+\hat{\rho}_{i\downarrow}$ or
$\hat{\rho}_{\uparrow}=\sum_{i}\hat{\rho}_{i\uparrow}$. The Hamiltonian of the
system will at first be allowed to be an entirely general self-adjoint
operator on $\mathcal{H}$ (next to spin-lattice hopping it may include
interactions between lattice sites, a Hubbard-$U$-type interaction between
spin components, etc.), but we already write the action of an external
potential and a magnetic field separately,

$\hat{H}=\hat{H}_{0}+\sum_{i}\left(v_{i}\hat{\rho}_{i}+\mathbf{B}_{i}\cdot\hat{\bm{m}}_{i}\right).$
(1)

Here, $\mathbf{B}_{i}\in\mathbb{R}^{3}$ and the magnetization operator at $i$
is given by a 3-vector of operators

$\hat{\bm{m}}_{i}=(\hat{a}^{\dagger}_{i\uparrow},\hat{a}^{\dagger}_{i\downarrow})\cdot\bm{\sigma}\cdot\begin{pmatrix}\hat{a}_{i\uparrow}\\\
\hat{a}_{i\downarrow}\end{pmatrix},$ (2)

with $\bm{\sigma}$ the 3-vector of Pauli matrices. This can equivalently be
rewritten with a complex 4-component potential at every lattice site,

$\hat{H}=\hat{H}_{0}+\sum_{i\alpha\beta}v_{i\alpha\beta}\hat{\rho}_{i\alpha\beta},$
(3)

where $v_{i\alpha\beta}=v_{i\beta\alpha}^{*}$ and we follow the standard
definition for the density-matrix operator
$\hat{\rho}_{i\alpha\beta}=\hat{a}^{\dagger}_{i\beta}\hat{a}_{i\alpha}$. Both
conventions, scalar potential and magnetic field, or 4-component potential,
are frequently employed. Yet, in the second formulation, the density-matrix
operator $\hat{\rho}_{i\alpha\beta}$ is not self-adjoint and yields complex
expectation values, and also the potential $v_{i\alpha\beta}$ is complex,
although their combined sum in Eq. (3) is again self-adjoint as it should.

In many cases one considers a _collinear_ setting, where the direction of the
magnetic field is fixed at every lattice point and only the magnetization in
direction of $\mathbf{B}_{i}$ matters. Let the magnitude of the magnetic field
be $B_{i}$ and the magnetization operator in the given direction
$\hat{m}_{i}$, then the Hamiltonian of Eq. (1) is reduced to

$\hat{H}=\hat{H}_{0}+\sum_{i}\left(v_{i}\hat{\rho}_{i}+B_{i}\cdot\hat{m}_{i}\right).$
(4)

With $\hat{\rho}_{i}=\hat{\rho}_{i\uparrow}+\hat{\rho}_{i\downarrow}$,
$\hat{m}_{i}=\hat{\rho}_{i\uparrow}-\hat{\rho}_{i\downarrow}$ and
$v_{i\uparrow}=v_{i}+B_{i}$, $v_{i\downarrow}=v_{i}-B_{i}$ we arrive at the
more usually applied formulation for the collinear setting, where two
potentials act individually on the spin components,

$\hat{H}=\hat{H}_{0}+\sum_{i\alpha}v_{i\alpha}\hat{\rho}_{i\alpha}.$ (5)

Finally, if the magnetic field is entirely removed or taken as fixed and
integrated into $\hat{H}_{0}$, one arrives at the usual DFT setting, where the
spin components cannot be addressed individually by the external potential,

$\hat{H}=\hat{H}_{0}+\sum_{i}v_{i}\hat{\rho}_{i}.$ (6)

In all those settings we can speak of a _graph_ instead of a lattice, since we
are concerned with how spin-lattice points are connected by $\hat{H}$ and
since this can happen in an irregular way. For $i\alpha\neq j\beta$, we say
that $i\alpha$ and $j\beta$ are connected (or _adjacent_), written
$i\alpha\sim j\beta$, if $\hat{H}$ includes any terms with
$\hat{a}^{\dagger}_{i\alpha}\hat{a}_{j\beta}$ (self-adjointness dictates that
it then also needs to include the adjoint terms). If the term is
$\hat{a}^{\dagger}_{i\alpha}\hat{a}_{j\alpha}$, this is called ‘hopping’,
while a Hubbard-$U$ interaction term connects different spin values,

$\displaystyle\hat{\rho}_{i\uparrow}\hat{\rho}_{i\downarrow}$
$\displaystyle=\hat{a}^{\dagger}_{i\uparrow}\hat{a}_{i\uparrow}\hat{a}^{\dagger}_{i\downarrow}\hat{a}_{i\downarrow}=(1-\hat{a}_{i\uparrow}\hat{a}^{\dagger}_{i\uparrow})\hat{a}^{\dagger}_{i\downarrow}\hat{a}_{i\downarrow}$
(7)
$\displaystyle=\hat{\rho}_{i\downarrow}-(\hat{a}^{\dagger}_{i\downarrow}\hat{a}_{i\uparrow})(\hat{a}^{\dagger}_{i\uparrow}\hat{a}_{i\downarrow}).$

If this graph of spin-lattice points decomposes into several connected
components then the Hamiltonian has a block-diagonal form and we actually
speak about totally separated systems that can be dealt with individually.
Such a situation automatically leads to degeneracy and the ensuing non-
uniqueness that will be discussed in Sec. IV. Remember that in some cases a
block-diagonal form can be achieved by unitary transformation of the
Hamiltonian, like in the example discussed in Sec. VI. This means that by
symmetry one is able to pass to a new basis and find new creation and
annihilation operators (like in a Bogoliubov transformation leading to a
_quasiparticle_ interpretation), such that the system decouples into
independent subsystems which will decrease computational complexity.
Systematically, this is performed by finding the _irreducible representations_
of the symmetry group. Yet, note that by allowing an independent potential at
every spin-lattice point, any form of symmetry will be broken.

As a trivial example of disconnected graphs, imagine that $\hat{H}_{0}$ in
Eqs. (5) or (6) does not connect the up/down spin components. Since the
$v_{i\alpha}\hat{\rho}_{i\alpha}$ cannot provide such a connection either, the
total Hamiltonian already decomposes into separate up- and down-spin systems.
If this situation is still considered as a single system then it directly
leads to (trivial) HK counterexamples [19; 21].

##  III Density and potential spaces

As soon as we move our attention to DFT, we need to choose certain observables
that serve as density variables and that couple linearly to the external
potentials. According to Sec. II, three choices offer themselves.

| density | potential
---|---|---
standard DFT | $\rho_{i}$ | $v_{i}$
collinear SDFT | $\rho_{i\alpha}$ | $v_{i\alpha}$
| $\rho_{i},m_{i}$ | $v_{i},B_{i}$
non-collinear SDFT | $\rho_{i\alpha\beta}$ | $v_{i\alpha\beta}$
| $\rho_{i},\mathbf{m}_{i}$ | $v_{i},\mathbf{B}_{i}$

Here, the density values themselves are always considered the expectation
values of the corresponding operators. This can either be with respect to a
pure state,
$\rho_{i}=\langle\hat{\rho}_{i}\rangle_{\Psi}=\langle\Psi,\hat{\rho}_{i}\Psi\rangle$,
or a mixed (ensemble) state,
$\rho_{i}=\langle\hat{\rho}_{i}\rangle_{\Gamma}=\operatorname{Tr}(\Gamma\hat{\rho}_{i})$,
and accordingly for the other density variables. Note that a totally general
and more abstract formulation employing a set of $M$ self-adjoint ‘density’
operators $\hat{o}_{i}$, $i\in\\{1,\ldots,M\\}$, that are linearly coupled to
‘potentials’ $v_{i}\in\mathbb{R}$,

$\hat{H}=\hat{H}_{0}+\sum_{i}v_{i}\hat{o}_{i},$ (8)

is possible as well [7; 24].

The usual density as site occupancy naturally allows $\rho_{i}\in[0,2]$ due to
the antisymmetry of the wave function and the possibility of filling two
different spin channels. In the spin-resolved formulation this changes to
$\rho_{i\alpha}\in[0,1]$. Additionally, we have the normalization to the
number of particles $N$,

$\sum_{i}\rho_{i}=\sum_{i\alpha}\rho_{i\alpha}=N.$ (9)

We thus define the density set

$\mathcal{D}^{\mathsf{DFT}}=\left\\{\rho\in\mathbb{R}^{M}\mid
0\leq\rho_{i}\leq 2,\sum\nolimits_{i}\rho_{i}=N\right\\}$ (10)

for standard lattice DFT with spin-$\frac{1}{2}$ particles. This results in a
doubly scaled $(M,N/2)$-hypersimplex [17; 25] for $\rho_{i}$ if $N$ even. If
$N$ odd the density space for $\rho_{i}$ is slightly more complex, but it is
always described by a $(M-1)$-dimensional convex polytope. In the case of
collinear SDFT we get

$\mathcal{D}^{\mathsf{cSDFT}}=\left\\{\rho\in\mathbb{R}^{2M}\mid
0\leq\rho_{i\alpha}\leq 1,\sum\nolimits_{i\alpha}\rho_{i\alpha}=N\right\\}.$
(11)

This results in a $(2M,N)$-hypersimplex for $\rho_{i\alpha}$ if $N$ even and a
more general $(2M-1)$-dimensional convex polytope if $N$ odd.

To determine the space of possible magnetizations
$\mathbf{m}_{i}=\langle\hat{\bm{m}}_{i}\rangle$ for non-collinear SDFT, let us
first consider just a single particle on a single vertex and the expectation
value with respect to a pure state $\Psi\in\mathbb{C}^{2}$. We have

$|\mathbf{m}|^{2}=|\langle\hat{\bm{\sigma}}\rangle_{\Psi}|^{2}=\langle\hat{\sigma}_{x}\rangle_{\Psi}^{2}+\langle\hat{\sigma}_{y}\rangle_{\Psi}^{2}+\langle\hat{\sigma}_{z}\rangle_{\Psi}^{2}=1,$
(12)

which means the magnetization vector $\mathbf{m}$ is always on a sphere. For
mixed states convex combinations of the magnetization vectors become possible,
so the value can also be inside the sphere and $|\mathbf{m}|\leq 1$. We
recognize exactly the Bloch sphere construction, where pure states are located
on the surface of the sphere and mixed states are always in the interior. The
space for magnetization is thus the unit ball, $\mathbf{m}\in\mathbb{B}$.
Adding another particle on the same site will force them to align antiparallel
with a total magnetization of zero, so this does not change the space. For
multiple vertices, the value of
$\mathbf{m}_{i}=\langle\hat{\bm{m}}_{i}\rangle_{\Psi}$ depends on the partial-
trace density matrix over all vertices $j\neq i$, so one already considers a
mixed state and thus, in general, has $|\mathbf{m}_{i}|\leq 1$. The total
space for magnetization is thus $\mathbb{B}^{M}$. In the collinear setting
this accordingly changes to $[-1,1]^{M}$. Whatever setting we use, the
complete density space for $(\rho_{i},\mathbf{m}_{i})$, $(\rho_{i},m_{i})$,
$\rho_{i\alpha}$, or just $\rho_{i}$ will always be denoted by $\mathcal{D}$.
It was previously noted in the context of standard lattice DFT that for any
$\rho\in\mathcal{D}$ there is indeed a $\Psi\in\mathcal{H}$ that gives this
density [17, Prop. 5]. This result can easily be extended to both variants of
SDFT.

For the following presentation we will mostly remain in the setting of
_collinear SDFT_ , while statements for the other DFT variants follow in a
similar manner. We write $\hat{H}_{v}$ for the Hamiltonian from Eq. (5) that
includes a potential $v_{i\alpha}$. This potential, according to Eq. (5), acts
in the form of an inner product on $\mathbb{R}^{2M}$ when the expectation
value is formed and we write

$\langle\hat{H}_{v}\rangle_{\Psi}=\langle\hat{H}_{0}\rangle_{\Psi}+\sum_{i\alpha}v_{i\alpha}\rho_{i\alpha}=\langle\hat{H}_{0}\rangle_{\Psi}+\langle
v,\rho\rangle.$ (13)

In this sense it is natural to see the potentials as dual to densities. But
$\mathcal{D}$ lies within an affine space of codimension 1, defined by the
normalization condition $\sum_{i\alpha}\rho_{i\alpha}=N$. Now, if two
potentials differ by just a constant, $v-v^{\prime}=c$, then their difference
acts as $\langle v-v^{\prime},\rho\rangle=cN$ on any $\rho\in\mathcal{D}$.
This means that $v,v^{\prime}$ cannot be distinguished as elements of the dual
space $\mathcal{V}=\mathcal{D}^{*}$. In other words, the elements
$[v]\in\mathcal{V}$ are equivalence classes where for all $v,v^{\prime}\in[v]$
it always holds that $v-v^{\prime}$ is constant. In the following, we will
denote potentials further simply as $v\in\mathcal{V}$ and understand them as
modulo a constant. This means that the usual statement in DFT that potentials
are defined “up to a constant”, since adding a constant does not change the
ground-state properties (it only shifts the ground-state energy), can actually
be suppressed. For each $v\in\mathcal{V}$ we now define the ground-state
energy functional

$E(v)=\inf_{\Psi}\langle\hat{H}_{v}\rangle_{\Psi},$ (14)

where the variation extends over all $\Psi\in\mathcal{H}$ normalized to 1.
This functional is concave in $v$ because of linearity in $v$ and the
properties of the infimum. Another immediate result is that this infimum is
realized by some (or rather many) $\Psi\in\mathcal{H}$ since the variation
domain is compact in the case of a finite lattice. One could thus equally
write ‘min’ instead of ‘inf’. The respective optimizer is then a ground-state
wave function for $\hat{H}_{v}$.

If for a given $\rho\in\mathcal{D}$ there is further a potential
$v\in\mathcal{V}$ such that the density is achieved by the corresponding
(possibly ensemble) ground state of $\hat{H}_{v}$ then one calls this density
_$v$ -representable_. This is a subtle notion, but it can be proven that all
densities from the interior of $\mathcal{D}$ are $v$-representable, while only
a certain few densities on the boundary of $\mathcal{D}$ have this property
[4; 17; 22]. Importantly, this notion depends on the choice of $\hat{H}_{0}$
that then also determines which boundary densities can be reached. We will
give a full characterization of $v$-representable densities in the spin-
lattice setting in Sec. IV below. One has to keep in mind that the situation
is rather different for the infinite-dimensional continuum setting of standard
DFT, where $v$-representability depends on the topology of the density space
and only limited results are available by now [26; 27; 28; 29].

##  IV $v$-representability, non-uniqueness, and degeneracy regions

One of the cornerstones in the presentation of DFT is usually the HK theorem
that establishes a mapping from $v$-representable densities to potentials. In
its usual formulation it states that for a fixed $\hat{H}_{0}$ and every
$\rho\in\mathcal{D}$ that is $v$-representable, a _unique_ $v\in\mathcal{V}$
can be found that retrieves this density from a ground state. This can be an
ensemble state (represented by a density matrix) in the case of ground-state
degeneracy. In Sec. VI we will discuss an example where the occurrence of
degeneracy was mistakenly taken as a failure of $v$-representability. But
there are counterexamples (discussed at the end of this section) that rule out
the full validity of this statement and show the _non-uniqueness_ of the
representing potential. Nevertheless, a characterization similar to that of
$v$-representability can also be found when such counterexamples arise and
this will lead us to an alternative and entirely geometrical formulation for
the HK theorem.

The theoretical basis for DFT is the _universal density functional_ that can
be defined on $\mathcal{D}$ as the density-matrix constrained-search
functional [30]

$F(\rho)=\inf_{\Gamma\mapsto\rho}\operatorname{Tr}\hat{H}_{0}\Gamma.$ (15)

Here, variation is over all density matrices that yield the given density
$\rho$. The ground-state energy for a given $v\in\mathcal{V}$ is then

$E(v)=\inf_{\Gamma}\operatorname{Tr}\hat{H}_{v}\Gamma=\inf_{\rho\in\mathcal{D}}\left\\{F(\rho)+\langle
v,\rho\rangle\right\\}.$ (16)

More on these functionals can be found in any mathematically oriented
introduction to DFT [31; 32].

We will now define the important notion of a _degeneracy region_
$D(v)\subseteq\mathcal{D}$ as the set of densities coming from all (ensemble)
ground states of the Hamiltonian $\hat{H}_{v}$ with a $v\in\mathcal{V}$ that
facilitates ground-state degeneracy. A density region consists of
$v$-representable densities (by definition), it is always convex and closed,
and it is further equal to the superdifferential of the concave ground-state
energy functional $E(v)$ (since $E(v)$ in Eq. (16) is the Legendre–Fenchel
transform of the universal density functional $F(\rho)$ and thus its
superdifferential yields all ground-state densities for a given potential [32,
Sec. VII]),

$\displaystyle D(v)$
$\displaystyle=\\{\rho\in\mathcal{D}\mid\Gamma\mapsto\rho,\operatorname{Tr}\hat{H}_{v}\Gamma=E(v)\\}$
(17) $\displaystyle=\overline{\partial}E(v):=\\{\rho\in\mathcal{D}\mid\forall
v^{\prime}\in\mathcal{V}:$ $\displaystyle\qquad\qquad\qquad
E(v)-E(v^{\prime})\geq\langle v-v^{\prime},\rho\rangle\\}.$

The maximal dimension of a degeneracy region when the degeneracy is $g$-fold
is $g^{2}-1$, reduced to $g(g+1)/2-1$ in the case of a real $\hat{H}_{0}$.
When the density variable does not contain any spin information, degeneracy
can be such that it only affects the internal spin degree-of-freedom and is
thus not expressed in the density alone. Then $D(v)$ remains a single point
and we call the degeneracy ‘internal’. This is resolved in SDFT with the
inclusion of the magnetization as an additional density variable. Then, full
spin degeneracy means that $D(v)$ extends over all of $\mathbb{B}^{M}$.

When degeneracy is due to symmetry and a variation of $v$ preserves this
symmetry, then this does not change the degeneracy of the ground state,
although the associated density region $D(v)$ of course changes. This way, a
family of potentials $v(\lambda)$, continuously parametrized by a vector
$\lambda\in\mathbb{R}^{\ell}$, can lead to a family of degeneracy regions
$D(v(\lambda))$ that we call a _degeneracy bundle_. It consists of degeneracy
regions that lie arbitrarily close together but never touch. Since the space
of densities has dimension $2M-1$ (in the collinear SDFT setting), we get a
maximum of $\ell\leq 2M-d-1$ parameters for a family of degeneracy regions
with dimension $d$ each. This reproduces an earlier result by Ullrich and Kohn
[23] that was later adapted to degeneracy regions [22]. But degeneracy
regions, if they are not part of the same bundle, can intersect and it is also
possible that they touch the boundary of the density domain itself. These two
situations are sketched in Fig. 2 and they have a special relevance in the
current discussion, since there are always infinitely many different
potentials that yield those density points as ground-state densities.

(a)

(b)

Figure 2: Degeneracy regions that intersect or touch the boundary of the
density domain.

They are thus counterexamples to the HK theorem and must be omitted in its
formulation (see below). How this situation arises can indeed be easily
recognized. If a density is produced by two different potentials $v$ and
$v^{\prime}$ then the linearity of the Hamiltonian means that any linear
combination of those potentials leads to the same density, _unless_ an energy-
level crossing is reached in either direction. This then gives rise to the two
possible situations, pictorially related to energy-level crossings in Fig. 3.

$\lambda$$E$$\lambda_{1}$$\lambda_{2}$ (a)

$\lambda$$E$$\infty$$v_{\rm A}$ (b)

Figure 3: The two situations of degeneracy regions intersecting and touching
the boundary related to energy level crossings.

Finally, we need to highlight that even though we presented these notions
within a specific spin-lattice DFT setting, they hold quite generally, even
for the infinite-dimensional continuum setting. Details about the proof and
the intricate shape of density regions can be found in our study on the
geometry of degeneracy [22]. We here summarize the results:

  1. (a)

If two density regions $D(v)$ and $D(v^{\prime})$ intersect then $D(v)\cap
D(v^{\prime})$ is either a single ground-state density point or a density
region itself. In both cases it results from all potentials that are a convex
combination of $v$ and $v^{\prime}$.

  2. (b)

If a density region $D(v)$ touches the boundary of $\mathcal{D}$ then this
density point results from all potentials that lie on a ray that extends from
$v$ to infinity.

  3. (c)

All densities that are not on the boundary of $\mathcal{D}$ are
$v$-representable. Densities on the boundary of $\mathcal{D}$ need to be part
of a degeneracy region in order to be $v$-representable. All $v$-representable
densities that are not described by a or b are even _uniquely_
$v$-representable.

We are thus in the position to formulate a purely geometrical HK theorem: _All
ground-state densities that are not on the boundary of the density domain and
that are not at the intersection of degeneracy regions are uniquely given by a
potential._

Examples for all mentioned situations a-c in the pure-lattice case can be
found in our previous works on the topic [17; 22] and seemed to have been
overlooked previously as counterexamples to the HK theorem.

We want to conclude this section with a brief discussion of the mentioned HK
counterexamples that specifically connect to spin in the collinear setting
[19; 21]. In those examples the up- and down-components of the spin lattice
remain unconnected by the Hamiltonian (in the sense given in Sec. II) and
consequently changing the potential $v_{i\alpha}$ by adding a constant for all
$i$ on only one component $\alpha\in\\{\uparrow,\downarrow\\}$ does not
influence the ground state, _unless_ this change leads to a level crossing.
But the occurrence of a level crossing means that the density is part of a
degeneracy region and a change of the potential in the other direction either
leads to another level crossing (a, intersecting degeneracy regions) or it
continues to infinity without a level crossing (b, touching the boundary of
the density domain at full magnetization $m_{i}\in\\{\pm 1\\}$). The non-
collinear setting for a single lattice vertex is instructive too. Here, at
$\mathbf{B}=0$, the whole Bloch sphere is a degeneracy region since no spin-
direction is preferred. Since the density domain is just the same Bloch
sphere, the degeneracy region touches the boundary at every $\mathbf{m}$ with
$|\mathbf{m}|=1$ (pure states) and indeed all fields $\mathbf{B}\neq 0$ lead
to non-uniqueness in the sense of b.

##  V Sets of the potential-density mapping

We start by defining the set $\mathcal{V}^{\mathsf{D}}$ of all potentials that
lead to degeneracy in the ground state and the complementary set
$\mathcal{V}^{\mathsf{\neg D}}$ where no degeneracy occurs. Obviously, the
complete space of potentials is the disjoint union of those sets,
$\mathcal{V}=\mathcal{V}^{\mathsf{\neg
D}}\;\dot{\cup}\;\mathcal{V}^{\mathsf{D}}$. The set $\mathcal{V}^{\mathsf{D}}$
is closed since degeneracy occurs only when (continuous) energy bands
intersect, which makes $\mathcal{V}^{\mathsf{\neg D}}$ open. Now define the
mapping from potentials to ground-state densities that we just call $\rho$ for
convenience.

$\rho:\mathcal{V}^{\mathsf{\neg D}}\longrightarrow\mathcal{D}$ (18)

It is well-defined on $\mathcal{V}^{\mathsf{\neg D}}$ since each such
potential leads to a _unique_ ground-state density due to the lack of
degeneracy. A beautiful result of Rellich [33, 34] then tells us that this
mapping is analytic, so in particular it is continuous. We also define the
image of this map, $\mathcal{D}^{\mathsf{\neg
D}}=\rho(\mathcal{V}^{\mathsf{\neg D}})$, which is found to be open [2; 17],
and further $\mathcal{D}^{\mathsf{D}}$ as the union of all degeneracy regions.
In many examples the occurrence of degeneracy in potential space is rare
($\mathcal{V}^{\mathsf{D}}$ has measure zero), while the corresponding
densities $\mathcal{D}^{\mathsf{D}}$ can fill a larger volume than their
complement $\mathcal{D}\setminus\mathcal{D}^{\mathsf{D}}$ [23; 22].

It is now important to note that the sets $\mathcal{D}^{\mathsf{D}}$ and
$\mathcal{D}^{\mathsf{\neg D}}$ are _not_ necessarily disjoint. Take the set
$\mathcal{D}^{\mathsf{\neg D}}\cap\mathcal{D}^{\mathsf{D}}$, then this
includes densities that come from different potentials, exactly
counterexamples to the HK from before. If we call $\mathcal{D}^{\mathsf{m}}$
the set of densities that arise from multiple different potentials (also
called ‘non-uv densities’ after ’non-uniquely $v$-representable’ in Ref. 17),
then clearly $\mathcal{D}^{\mathsf{\neg
D}}\cap\mathcal{D}^{\mathsf{D}}\subseteq\mathcal{D}^{\mathsf{m}}$. Sec. IV a-c
gives a complete classification of $\mathcal{D}^{\mathsf{m}}$ and also has
$\mathcal{D}^{\mathsf{m}}\subseteq\mathcal{D}^{\mathsf{D}}$, since all those
counterexamples are connected to degeneracy. It is known that
$\mathcal{D}^{\mathsf{m}}$ is closed and has measure zero in $\mathcal{D}$
[22, Cor. 10] and that densities in $\mathcal{D}^{\mathsf{m}}$ can only arise
from wave functions with a certain minimal number of zero coefficients in the
lattice basis [17, Sec. III.B]. Then there is the set
$\mathcal{D}^{\mathsf{\neg D}}\cup\mathcal{D}^{\mathsf{D}}$ that collects all
$v$-representable densities. In general it holds $\mathcal{D}^{\mathsf{\neg
D}}\cup\mathcal{D}^{\mathsf{D}}\subsetneqq\mathcal{D}$, since there can always
be densities on the boundary of $\mathcal{D}$ that are not $v$-representable.

All densities that come from multiple potentials are collected in
$\mathcal{D}^{\mathsf{m}}$, so we can invert the mapping Eq. (18) on
$\mathcal{D}^{\mathsf{\neg
D}}\setminus\mathcal{D}^{\mathsf{m}}=\mathcal{D}^{\mathsf{\neg
D}}\setminus\mathcal{D}^{\mathsf{D}}$. But what about continuity of the
inverse map $\rho^{-1}:\mathcal{D}^{\mathsf{\neg
D}}\setminus\mathcal{D}^{\mathsf{m}}\to\mathcal{V}$. This is always secured
only if we already limit ourselves to a _compact_ domain in Eq. (18), which
$\mathcal{V}^{\mathsf{\neg D}}$ is not since it is open. But using tools from
convex analysis we can still show that the inverse map is continuous almost
everywhere too and even extend it. First remember that if $\rho$ is a ground-
state density for potential $v$ then $-v$ is an element of the subdifferential
of the universal density functional $F$ at $\rho$,
$-v\in\underline{\partial}F(\rho)$. Since for
$\rho\in(\mathcal{D}^{\mathsf{\neg
D}}\cup\mathcal{D}^{\mathsf{D}})\setminus\mathcal{D}^{\mathsf{m}}$ this
subdifferential contains only a single element, $F$ is differentiable at
$\rho$ and we write $v=-\nabla F(\rho)$ [35, Th. 25.1]. Differentiability in
this context means Fréchet differentiability, which in turn means that the
inverse map

$\displaystyle\rho^{-1}:(\mathcal{D}^{\mathsf{\neg
D}}\cup\mathcal{D}^{\mathsf{D}})\setminus\mathcal{D}^{\mathsf{m}}$
$\displaystyle\to\mathcal{V}$ (19) $\displaystyle\rho$
$\displaystyle\mapsto-\nabla F(\rho),$

defined on the _uniquely $v$-representable_ densities, is continuous [35, Th.
25.5]. Actually this is more than an inverse to Eq. (18) because it now
includes degeneracy regions but it still does not include any boundary
densities. The domain of the inverse mapping above is exactly the one from the
geometrical formulation of the HK theorem from Sec. IV, “all ground-state
densities that are not on the boundary of the density domain and that are not
at the intersection of degeneracy regions”, because those densities arise from
multiple potentials. Yet we are not allowed to continue the inverse mapping to
all $v$-representable densities $\mathcal{D}^{\mathsf{\neg
D}}\cup\mathcal{D}^{\mathsf{D}}$ (this would require uniform continuity). This
means that a perfectly smooth density path can start to diverge if mapped back
to potential space as one gets close to $\mathcal{D}^{\mathsf{m}}$. Note that
this is different to the discontinuity of $\rho^{-1}$ in the usual continuum
setting [36], which depends on much more complicated topologies and where due
to the HK theorem no non-uniquely $v$-representable densities are expected to
exist.

Densities with zero components are only found on the boundary of
$\mathcal{D}$, but the only $v$-representable densities on the boundary are in
$\mathcal{D}^{\mathsf{m}}$ as stated in Sec. IV. Conversely, all ground-state
densities in $(\mathcal{D}^{\mathsf{\neg
D}}\cup\mathcal{D}^{\mathsf{D}})\setminus\mathcal{D}^{\mathsf{m}}$, which
removes the boundary densities, are strictly positive.

We exemplify these sets with the triangle-graph example ($M=3,N=2$) from Ref.
17, Sec. VI.C, depicted in Fig. 4.

$(1,1,0)$$(1,0,1)$$(0,1,1)$$\mathcal{D}^{\mathsf{m}}$$\mathcal{D}^{\mathsf{m}}$$\mathcal{D}^{\mathsf{m}}$$\mathcal{D}^{\mathsf{D}}$$\mathcal{D}^{\mathsf{\neg
D}}$$\mathcal{D}^{\mathsf{\neg D}}$$\mathcal{D}^{\mathsf{\neg D}}$123 Figure
4: Different density sets for the triangle graph. Here,
$\mathcal{D}^{\mathsf{D}}$ is a closed circular set, while the three
disconnected sets belonging to $\mathcal{D}^{\mathsf{\neg D}}$ are all open.
Three non-unique $v$-representable points belong to $\mathcal{D}^{\mathsf{m}}$
and lie exactly at the intersections of the incircle with the triangle.

##  VI Anderson impurity model: mistaken non-$v$-representability and real
convergence issues

We learnt in Sec. IV that every density (that is not on the boundary) in a
spin-lattice model is $v$-representable; still Rössler  _et al._ [37] reported
that they found a counterexample in exactly such a setting. Their claim rests
on a numerical procedure that aims at reaching the density of an interacting
system by adjusting the potential in a non-interacting system and that failed.
The statement of non-$v$-representability is _wrong_ , but analyzing the
situation that leads to this failure gives interesting insight into the
relevance of degeneracy, especially with respect to Kohn–Sham DFT.

Their example is the realization of an Anderson impurity model [38], depicted
in the top panel of Fig. 5, with $M=9,N=8$ and the Hamiltonian

$\hat{H}=t\sum_{i\sim
j,\alpha}\hat{a}^{\dagger}_{i\alpha}\hat{a}_{j\alpha}+\sum_{i\alpha}v_{i}\hat{a}^{\dagger}_{i\alpha}\hat{a}_{i\alpha}+U\hat{a}^{\dagger}_{1\uparrow}\hat{a}_{1\uparrow}\hat{a}^{\dagger}_{1\downarrow}\hat{a}_{1\downarrow}.$
(20)

The first two terms, without the interaction, form the one-particle
Hamiltonian $\hat{h}$. One always sets $v_{i\uparrow}=v_{i\downarrow}=v_{i}$.
Further, the potential is assumed to obey $v_{1}=v_{a}$,
$v_{2}=v_{3}=v_{4}=v_{5}=v_{b}$ and $v_{6}=v_{7}=v_{8}=v_{9}=v_{c}$ which
introduces a $\mathsf{C}_{4\mathsf{v}}$ symmetry to the system. Note that in
the case of degeneracy, a ground-state density of this system does not have to
necessarily share this symmetry, just the full degeneracy region does. This
also means that the density at the middle point of the degeneracy region, the
_equidensity_ that arises from an equally weighted statistical mixture of the
orthonormal basis states of the degenerate subspace (thus leading to
fractional occupation numbers), shares the symmetry. The same density is
achieved in thermally-assisted-occupation DFT [39], where a thermal ensemble
is chosen instead of the ground state, if the temperature approximates zero.

The benefit of such a symmetrical setting is that a fully
$\mathsf{C}_{4\mathsf{v}}$ symmetric (also taking spin-symmetry into account)
density, like the external potential, has only three different components
$(\rho_{a},\rho_{b},\rho_{c})$ which allows to plot them in 3D. With the
further normalization condition

$\rho_{a}+4\rho_{b}+4\rho_{c}=N/2$ (21)

such a density is even restricted to a 2D subplane. Further, a unitary
transform $\hat{U}$ of the one-particle Hamiltonian $\hat{h}$ in position
basis allows to reduce it into a form with five separate blocks, corresponding
to the irreducible representations of the point group
$\mathsf{C}_{4\mathsf{v}}$. The model and its unitary transformation are
displayed in Fig. 5, while the unitary matrix that was used can be found in
Appendix A.

123456789$t$$t$$t$$t$$t$$t$$t$$t$$t$$t$$t$$t$$U$$v_{a}$$v_{b}$$v_{b}$$v_{b}$$v_{b}$$v_{c}$$v_{c}$$v_{c}$$v_{c}$

123456789$2t$$2t$$\sqrt{2}t$$\sqrt{2}t$$U$$v_{a}$$v_{b}$$v_{c}$$v_{b}$$v_{c}$$v_{b}$$v_{c}$$v_{b}$$v_{c}$AB1B2C1C2

Figure 5: The Anderson impurity model under consideration. Top panel shows the
original lattice, bottom panel the unitarily transformed model where the
lattice separates into five independent blocks.

Ignoring the Hubbard-$U$ interaction for a moment, this yields a new spin-free
one-particle Hamiltonian $\hat{h}^{\prime}=\hat{U}\hat{h}\hat{U}^{\dagger}$
that equally acts on each spin-component.

$\hat{h}^{\prime}=\pNiceMatrix\Block[fill=blue!15,rounded-
corners]{3-3}{}v_{a}&2t0000000\\\ 2tv_{b}2t000000\\\ 02tv_{c}000000\\\
000\Block[fill=red!15,rounded-corners]{2-2}{}v_{b}\sqrt{2}t0000\\\
000\sqrt{2}tv_{c}0000\\\ 00000\Block[fill=red!15,rounded-
corners]{2-2}{}v_{b}\sqrt{2}t00\\\ 00000\sqrt{2}tv_{c}00\\\
0000000\Block[fill=yellow!30,rounded-corners]{1-1}{}v_{b}0\\\
00000000\Block[fill=yellow!30,rounded-corners]{1-1}{}v_{c}.$ (22)

Note not only the difference in the basis ordering compared to Rössler  _et
al._ [37], but also the hopping $\sqrt{2}t$ in the $2\times 2$-blocks (red
blocks in Eq. (22), B clusters) that was wrongly put $2t$ in the reference. We
will go on with this choice, assuming that it does not influence the effect
that was diagnosed as “non-$v$-representability”. (We also checked the ground
state for the hopping $2t$ but there occurred no qualitative difference.) The
important consequence of the transformation is that the Hamiltonian now blocks
into five submatrixes that correspond to smaller clusters that do not
interact. On the other hand, the on-site energies in the diagonal stay exactly
the same while the hopping inside each cluster just gets renormalized.
Finally, the interaction from $U$ stays exactly the same since the first
lattice site, the only place where it acts between particles of opposing spin,
is not changed. Only the first cluster (A cluster, marked in violet) is thus
interacting and one can solve for the ground state in each cluster separately,
as soon as the respective particle filling for each one is determined, which
amounts to a huge reduction in computational complexity.

Now Rössler  _et al._ [37] chose the potentials $v_{a}=-2.69,v_{b}=-1$ and
$v_{c}=0$ in order to deliberately create a degenerate ground state for the
interacting system with $U=1,t=1$ (yet this is not critical for the following
example). With these values and a total of $N=8$ particles, one finds the
ground states realized with the filling

$N_{\rm A}=3,N_{\rm B1}=2,N_{\rm B2}=2,N_{\rm C1}=1,N_{\rm C2}=0.$ (23)

Here, particles remain unpaired and thus a spin degeneracy occurs, but it is
only ‘internal’ and thus every ground state has the same spin-unresolved
density that defines our target density $\rho_{\odot}$.

Next, the interaction was turned off ($U=0$) and the potential parameter space
was scanned in order to find a representing potential for $\rho_{\odot}$. It
is found that with $v_{a}=2v_{b},v_{c}=0$ degeneracy occurs, but this time the
degeneracy regions extend in space and form a degeneracy bundle in the shape
of a deformed cylinder. At $v_{a}=v_{b}=v_{c}=0$ the degeneracy region is even
3-dimensional and forms a cone. Rössler  _et al._ [37] report that they cannot
reach the target density $\rho_{\odot}$ with any of these potentials, and
indeed it is not reached if only the equidensity of each degeneracy region is
considered, which seems to have been their choice. The difference between
$\rho_{\odot}$ and the equidensities is displayed in Fig. 6 which shows great
similarity with Fig. 2 from the reference. The further displayed energy
eigenvalues demonstrate how at $v_{b}=0$ three-fold degeneracy occurs together
with a jump of the density difference, since the computed density crosses the
whole degeneracy region. But $\rho_{\odot}$ is easily reached with another
density from the appropriate degeneracy region since it lies inside the
degeneracy bundle, so there is clearly no problem with $v$-representability.
The whole situation is visualized in Figs. 7 and 8. All displayed densities
are with respect to one spin channel only.

Figure 6: Difference between the target density and the equidensity of each
degeneracy region for $v_{a}=2v_{b}$, $v_{c}=0$. A minimum is attained close
to $v_{b}=-1$ but zero is not reached as can be seen in the zoomed-in inset.
In olive the eigenvalues (slightly displaced for better visibility) of the
spin-independent Hamiltonian are plotted to demonstrate the occurrence of
double (for $v_{b}<0$) and triple (at $v_{b}=0$) ground-state degeneracy.
Figure 7: The target density $\rho_{\odot}$ (red cross) is closely missed by
the equidensities (black dots; see inset) of the 2D degeneracy regions (blue
circles) that form a degeneracy bundle that attaches to the 3D degeneracy
region in the shape of a cone. The shaded plane is given by the condition Eq.
(21) for symmetric densities. Figure 8: Same as Fig. 7 with an in-plane view.

We want to study the appearance of the cone degeneracy region in more detail.
For $U=0$ and maximal symmetry at $v_{a}=v_{b}=v_{c}=0$ a three-fold
degeneracy is attained for the ground state in each spin channel. We call the
six lowest eigenstates of the (spin-independent) one-particle Hamiltonian Eq.
(22) $\phi_{1},\ldots,\phi_{6}$ where $\phi_{4},\phi_{5},\phi_{6}$ all have
the same eigenvalue. Then with $N/2=4$ particles in each spin channel the
degenerate ground-state subspace is spanned by the Slater-determinant states
$\Phi_{1}=\phi_{1}\wedge\phi_{2}\wedge\phi_{3}\wedge\phi_{4}$,
$\Phi_{2}=\phi_{1}\wedge\phi_{2}\wedge\phi_{3}\wedge\phi_{5}$,
$\Phi_{3}=\phi_{1}\wedge\phi_{2}\wedge\phi_{3}\wedge\phi_{6}$ with densities
$\rho_{\overline{1}},\rho_{\overline{2}},\rho_{\overline{3}}$. We further
define $\rho_{\overline{kl},i}=2\langle\Phi_{k},\hat{\rho}_{i}\Phi_{l}\rangle$
($k,l\in\\{1,2,3\\}$, $k\neq l$) which in this special case is zero for all
but one choice of indices (say $k=1$, $l=2$) where the result is also real-
valued. The degeneracy region is then formed as the convex hull of the set
[22]

$\left\\{x_{1}^{2}\rho_{\overline{1}}+x_{2}^{2}\rho_{\overline{2}}+x_{3}^{2}\rho_{\overline{3}}+x_{1}x_{2}\rho_{\overline{12}}\;\middle|\;\mathbf{x}\in\mathbb{R}^{3},|\mathbf{x}|=1\right\\}.$
(24)

Putting in spherical coordinates, $x_{1}=\sin\theta\cos\varphi$,
$x_{2}=\sin\theta\sin\varphi$, $x_{3}=\cos\theta$, we can transform the points
of the set into

$\displaystyle(\sin\theta)^{2}\left(\frac{\rho_{\overline{1}}+\rho_{\overline{2}}}{2}+\frac{\rho_{\overline{1}}-\rho_{\overline{2}}}{2}\cos(2\varphi)+\frac{\rho_{\overline{12}}}{2}\sin(2\varphi)\right)$
(25) $\displaystyle+$ $\displaystyle(\cos\theta)^{2}\rho_{\overline{3}}.$

The large bracket in the first line then describes an ellipse with center
$(\rho_{\overline{1}}+\rho_{\overline{2}})/2$ that gets double covered because
of the appearance of $2\varphi$. Since $(\sin\theta)^{2}+(\cos\theta)^{2}=1$
we further have convex combinations between all points of the ellipse and
$\rho_{\overline{3}}$ that thus forms the tip of the cone. Equation (24) can
be seen as the projection of the Veronese variety into projective 3-space
(called a Steiner surface), a topic studied by Degen [40] with the cone
appearing as (Ae) in his typology. Another famous shape, the Roman surface,
was already discovered in previous examples [22].

The issue faced by Rössler  _et al._ [37] was thus that $\rho_{\odot}$ lies
_inside_ a degeneracy region of the non-interacting system. The same effect
can manifest itself as a convergence issue for the Kohn–Sham method [41] if
the interacting and the auxiliary non-interacting systems have different
degeneracy structures. Just like in the example above, the target density
might lie within a degeneracy region $D(v_{\mathrm{s}})$ when the correct
exchange-correlation potential is applied to the Kohn–Sham system. This then
leads to redundancy when choosing the density. But already during the
iterative approach towards this density, when choosing a potential $v\approx
v_{\mathrm{s}}$, one will always remain _outside_ of $D(v_{\mathrm{s}})$ and
just approach the boundary of the degeneracy region as $v\to v_{\mathrm{s}}$
while never entering it. Densities inside $D(v_{\mathrm{s}})$ are thus
unattainable by an approximative algorithm if it is not specifically adjusted
such that it can enter the degeneracy region. The simplest modification is
_damping_ [42; 43], where an iteration step, e.g., between densities
$\rho_{i}\to\rho_{i+1}$ (or, analogously, between density matrices) is instead
taken as the convex combination

$\rho_{i}\to(1-\tau)\rho_{i}+\tau\rho_{i+1}.$ (26)

The step size is chosen such that the energy surely decreases with the finally
decided iteration step. Apart from leading to improved convergence of the
algorithm (only with such a modification it was possible to show that the
Kohn–Sham method in finite dimensions and with additional regularization does
in principle converge [44; 45]) the step Eq. (26) allows to enter inside a
degeneracy region also in cases where $\rho_{i+1}$ is always chosen on the
outside.

1234

Figure 9: Densities of randomly chosen potentials in a $M=4,N=2$ system with
tetrahedral symmetry. The lower panel also shows the degeneracy regions.

How degeneracy regions would be avoided by a usual Kohn–Sham iteration can
easily be visualized by just sampling potentials randomly and plotting their
respective densities. Those are almost always unique since the potentials
$\mathcal{V}^{\mathsf{D}}$ have measure zero in $\mathcal{V}$. Figure 9 shows
this situation with an example from Ref. 22 and how the intricate shape of the
degeneracy regions fits into the empty area.

##  VII Adiabatic variation of the potential and geometric phase

We will now allow for potentials $v(t)$ and $\mathbf{B}(t)$ that vary in time
and for all $t\in[0,T]$ lead to a non-degenerate ground state. The adiabatic
theorem [46] then states that if the system initially is in its ground state
and if further this variation is slow enough, it will also remain in the
instantaneous ground state of the Hamiltonian $\hat{H}(t)$. We determine this
ground state for the example of a graph with $M=4$ vertices where vertex 1 is
linked to all others but no other edges are present. Due to its shape this is
called the ‘claw graph’ and it clearly has a trigonal symmetry that can lead
to two-fold degeneracy.

If we put $N=2$ particles on the graph then for each spin component this leads
to the following two-particle Hamiltonian (following Ref. 17),

$\hat{H}_{0}=\begin{pmatrix}[r]4&0&0&1&1&0\\\ 0&4&0&-1&0&1\\\ 0&0&4&0&-1&-1\\\
1&-1&0&2&0&0\\\ 1&0&-1&0&2&0\\\ 0&1&-1&0&0&2\end{pmatrix}.$ (27)

We put $\mathbf{B}(t)=0$ and the (time-dependent) external potential
$v=(v_{1},v_{2},v_{3},v_{4})$ adds
$(v_{1}+v_{2},v_{1}+v_{3},v_{1}+v_{4},v_{2}+v_{3},v_{2}+v_{4},v_{3}+v_{4})$
along the diagonal of the Hamiltonian to yield $\hat{H}(t)$. We can easily
test that for $v=0$ the ground state is two-fold degenerate and that changing
just $v_{1}$ (that acts on the center vertex 1) does not change the symmetry
of the Hamiltonian, thus this degeneracy persists while the ground-state
density changes. The degeneracy region for any $v=(s,0,0,0)$,
$s\in\mathbb{R}$, is a precise circle (the two-fold degeneracy regions always
give ellipses as degeneracy regions [22, Sec. III]) and those circles are
stacked above each other to form a filled cylinder that runs diagonally
through the whole density domain in the shape of a octahedron (see Fig. 10).
Note that in this case the degeneracy regions come arbitrarily close to each
other and to the boundary of the density domain (for $|s|\to\infty$) but never
touch each other nor the boundary.

We now choose a closed path $v(t)$, $t\in[0,2\pi]$, in the potential space
that avoids the degeneracy at $v=(s,0,0,0)$,

$v(t)=\cos t\;(0,1,0,0)+\sin t\;(0,0,1,0).$ (28)

How this path looks in density space can be seen in Fig. 10, where we note
that it winds around the cylindrical degeneracy bundle such that it cannot be
contracted to a point without cutting through a degeneracy region.

1234 Figure 10: Degeneracy regions (collected into a single degeneracy bundle)
of the claw graph with a closed density path that winds them.

Let $\Psi(t)$ be the instantaneous ground state of $\hat{H}(t)$ that is unique
up to a complex global phase. We will choose it real in all its components
(which is always possible since the Hamiltonian is real) and Fig. 11 shows its
six components.

Figure 11: The six components of the instantaneous ground-state wave function
to $\hat{H}(t)$ with a potential winding the degeneracy bundle. Note the sign
jump when comparing the values at 0 and $2\pi$.

On first sight these look like nice continuous functions, but if one compares
closely the values at $t=0$ and $t=T=2\pi$, we see that a sign jump occurs.
This means that even though $\hat{H}(t)$ was varied continuously and has
$\hat{H}(0)=\hat{H}(T)$, the corresponding $\Psi(t)$ is _not_ continuous but
includes a $\pi$ phase jump. By lifting our condition of a real ground state
and including a complex global phase $\mathrm{e}^{\mathrm{i}\varphi/2}$ this
phenomenon can be cured, but we still include a phase change of $\pi$ while
going from $t=0$ to $t=2\pi$. This phase can be calculated as

$\gamma(T)=\mathrm{i}\int_{0}^{T}\langle\Psi(t),\partial_{t}\Psi(t)\rangle\,\mathrm{d}t$
(29)

and is exactly the geometric phase introduced by Berry [47]. The reason for
the phase accumulation is that the chosen path in density space cannot be
continuously contracted to a point (it is not homotopic to a point) without
intersecting a degeneracy region.

Note that if we choose a path that does not have this property, i.e., can be
continuously contracted to a point without intersecting a degeneracy region
(it is homotopic to a point), then no phase accumulation occurs and we can
always choose a real _and_ continuous $\Psi(t)$. This offers the possibility
to study the occurrence of degeneracy regions/bundles and the fundamental
group of $\mathcal{V}^{\mathsf{\neg D}}$ by calculating the accumulated phase
along different closed potential curves.

##  VIII Time-dependent setting

After having studied in detail the ground state theory we now turn our
attention to the time-dependent version of the theory. A particularly
interesting question in connection with the degeneracy structure is how a
potential that is used to slowly steer the density through a degeneracy region
has to look like. We address this question at the end of this section and
after first giving a general discussion of time-dependent density-functional
theory (TDDFT) [48; 49], but restricted to the lattice setting [50; 51].

Therein, we aim at solving the time-dependent Schrödinger equation

$\mathrm{i}\partial_{t}\Psi(t)=\hat{H}(t)\Psi(t)$ (30)

where the Hamiltonian is any one of the forms Eqs. (3)-(6) and in general we
allow for time-dependent potentials. However, for simplicity, let us restrict
to the spinless case in the following and consider a general time-evolution
with potential $v(t)=(v_{1}(t),\ldots,v_{M}(t))$ starting from a given initial
state $\Psi_{0}$ at time $t_{0}$. Suppose we are given a time-dependent
density $\rho(t)$ and an initial state $\Psi_{0}$ with density $\rho(t_{0})$.
Then we can ask whether there exists a potential $v(t)$ that generates
$\rho(t)$ from solving the time-dependent Schrödinger equation with initial
state $\Psi_{0}$ and using $v(t)$ as potential. This is the time-dependent
$v$-representability problem. In the continuum case, this problem has been
widely studied [52; 53; 54; 48] and most proofs rely on temporal analyticity
[55] although mathematically rigorous proofs are still lacking, see Ref. 56
for a review. In the lattice case the situation is more favorable and some
precise statements related to $v$-representability can be made. We will here
outline the main results. We take a time-dependent Hamiltonian of the form

$\hat{H}(t)=\sum_{i,j}h_{ij}\hat{a}^{\dagger}_{i}\hat{a}_{j}+\frac{1}{2}\sum_{i,j}w_{ij}\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{j}\hat{a}_{i}+\sum_{i}v_{i}(t)\hat{a}^{\dagger}_{i}\hat{a}_{i}.$
(31)

with $w_{ij}=w_{ji}$ real and symmetric. Applying the Heisenberg equation of
motion to the density operator and denoting
$\rho_{k}(t)=\langle\hat{\rho}_{k}\rangle_{\Psi(t)}$ we get the lattice-
equivalent of the continuity equation,

$\displaystyle\partial_{t}\rho_{k}$
$\displaystyle=-\mathrm{i}\langle[\hat{\rho}_{k},\hat{H}(t)]\rangle_{\Psi(t)}$
(32)
$\displaystyle=\sum_{j}2\mathop{\mathsf{missing}}{Im}h_{kj}\gamma_{jk}(t)=-\sum_{j}J_{kj}(t),$

where we defined the one-particle density matrix
$\gamma_{jk}(t)=\langle\hat{a}^{\dagger}_{k}\hat{a}_{j}\rangle_{\Psi(t)}$ and
the bond current
$J_{kj}(t)=2\mathop{\mathsf{missing}}{Im}h_{jk}\gamma_{kj}(t)$. Using the
Heisenberg equation of motion a second time and working out the commutators
gives

$\displaystyle\partial_{t}^{2}\rho_{k}(t)$
$\displaystyle=-\mathrm{i}\partial_{t}\langle[\hat{\rho}_{k},\hat{H}(t)]\rangle_{\Psi(t)}$
(33)
$\displaystyle=-\langle[[\hat{\rho}_{k},\hat{H}(t)],\hat{H}(t)]\rangle_{\Psi(t)}$
$\displaystyle=q_{k}(t)+\sum_{j}K_{kj}(t)v_{j}(t),$

where we defined

$q_{k}(t)=-\sum_{j}2\mathop{\mathsf{missing}}{Re}h_{kj}\langle[\hat{a}^{\dagger}_{k}\hat{a}_{j},\hat{H}_{0}]\rangle_{\Psi(t)}$
(34)

and

$K_{kj}(t)=2\mathop{\mathsf{missing}}{Re}\left(-h_{jk}\gamma_{kj}(t)+\delta_{kj}\sum_{i}h_{ji}\gamma_{ij}(t)\right)$
(35)

If we choose the gauge $v_{M}(t)=0$, we can rewrite Eq. (33) as

$\sum_{j=1}^{M-1}K_{kj}(t)v_{j}(t)=\partial_{t}^{2}\rho_{k}(t)-q_{k}(t).$ (36)

If we are given a density $\rho_{k}(t)$ over time we can prescribe
$\partial_{t}^{2}\rho_{k}$ on the right-hand side of this equation and try
solve for $v$ at any $t>t_{0}$ when the matrix $K$ and the vector $q$ are
known. The latter quantities are calculable from the wave function, which in
turn dependents on $v$ at all previous times. It was shown by Farzanehpour and
Tokatly [51] that solving Eq. (36) together with the time-dependent
Schrödinger equation allows for a $v$-representability solution for a
sufficiently short time interval $[t_{0},t_{0}+\Delta t]$, provided that
$K(t_{0})$ is invertible. This procedure then yields a potential that produces
the prescribed density in this time interval. The natural question to ask is
how the size of $\Delta t$ is determined. It is known that on lattices the
density becomes non-$v$-representable when it changes too fast from one
lattice site to the next [50; 51]. To see this, let us assume that
$\partial_{t}\rho_{k}$ in Eq. (32) becomes large. From the definition of the
bond current we see that its maximal value is reached when the imaginary parts
of $h_{jk}\gamma_{kj}$ for different $j$ are maximal, i.e., when
$\mathop{\mathsf{missing}}{Re}h_{jk}\gamma_{kj}=0$ for all $j$. But this
happens precisely when $K_{kj}=K_{jk}=0$ for all $j\neq k$ in which case it
follows from Eq. (35) that also $K_{kk}=0$; thereby the $k$th row and column
of the matrix vanishes and $K$ becomes non-invertible [51]. To specify a
$v$-representability domain we therefore have to restrict the density changes
on neighboring sites. One way would be to prove that under a condition of the
type

$\left|\frac{\partial_{t}\rho_{i}}{\rho_{i}}\cdot\frac{\partial_{t}\rho_{j}}{\rho_{j}}\right|\leq
C|h_{ij}|$ (37)

for some constant $C$ the matrix $K$ is invertible at all times. In such a
case $v$-representability can continue up to indefinite times. Such a
condition was explicitly derived for a simple dimer system [51], however for a
general lattice this remains an unproven conjecture.

We will next consider the linear-response regime. Let $v(t)$ and
$v_{\mathrm{s}}(t)$ be the potentials of an interacting and a non-interacting
system, respectively, that share the same density $\rho(t)$ and that both
evolve from initial states with the appropriate density. Also in the time-
dependent setting, the non-interacting system is called the Kohn–Sham system
[48; 49] and its external potential is typically split up as
$v_{\mathrm{s}}(t)=v(t)+v_{\textrm{Hxc}}(t)$ which defines the Hartree-
exchange-correlation potential $v_{\textrm{Hxc}}(t)$. Functional
differentiation of $v_{\mathrm{s}}(t)$ with respect to the density (assuming
that this can be appropriately defined) yields

$\frac{\delta v_{\mathrm{s},i}(t)}{\delta\rho_{j}(t^{\prime})}=\frac{\delta
v_{i}(t)}{\delta\rho_{j}(t^{\prime})}+\frac{\delta
v_{\textrm{Hxc},i}(t)}{\delta\rho_{j}(t^{\prime})}.$ (38)

If we define the functions

$\displaystyle\chi_{ij}(t,t^{\prime})$
$\displaystyle=\frac{\delta\rho_{i}(t)}{\delta v_{j}(t^{\prime})}$ (39)
$\displaystyle\chi_{\mathrm{s},ij}(t,t^{\prime})$
$\displaystyle=\frac{\delta\rho_{i}(t)}{\delta v_{\mathrm{s},j}(t^{\prime})}$
(40) $\displaystyle f_{\textrm{Hxc},ij}(t,t^{\prime})$
$\displaystyle=\frac{\delta
v_{\textrm{Hxc},i}(t)}{\delta\rho_{j}(t^{\prime})},$ (41)

where $\chi$ and $\chi_{\mathrm{s}}$ are the density response functions of the
interacting and the Kohn–Sham system and $f_{\textrm{Hxc}}$ is the Hartree-
exchange-correlation kernel. We can rewrite Eq. (38) in the form

$\chi=\chi_{\mathrm{s}}+\chi_{\mathrm{s}}\cdot f_{\textrm{Hxc}}\cdot\chi,$
(42)

where the product is defined as a matrix product in the lattice indices and
with an integration over time variables. Eq. (42) is commonly used to
calculate the excitation energies of the interacting system, which appear as
poles of $\chi$ after a Fourier transformation from time to frequency domain.
Similarly $\chi_{\mathrm{s}}$ contains the excitation energies of the
Kohn–Sham system. When an approximation for $f_{\textrm{Hxc}}$ is given, the
excitation energies of the interacting system can then be obtained from a
Kohn–Sham calculation.

We now want to make a connection between this formalism and the discussion of
degeneracy. When we perturb the ground state of a non-degenerate system with a
small time-dependent $\delta v(t)$, the response function $\chi$ is well-
defined. However, if the ground state is degenerate, poles in the response
function merge and make it ill-defined. If we assume that the changes are
adiabatic, this can be related to Rellich’s theorem [17; 34; 33]: when a
degenerate ground state is perturbed dependent on a small prefactor $\lambda$,
then Rellich’s theorem asserts that there exists a possible choice within the
set of degenerate states that depends analytically on $\lambda$, although most
choices will not. Physically this means that the response is only defined for
changes in very special directions in potential space. In practice, however,
almost all applications of linear-response TDDFT are carried out for systems
with non-degenerate ground states. For the calculation of excitation energies
of systems with degenerate ground states an alternative approach based on
choosing a non-degenerate excited state as a reference has been suggested
instead [57].

Let us finally come back to the question asked at the beginning of the
section. Suppose we consider two densities $\rho_{1}$ and $\rho_{2}$, just
outside a degeneracy region, and connect them with a smooth path $\rho(t)$
that crosses the degeneracy region. Let us further assume that the path is
traversed in a slow manner. We can then ask for the time-dependent potential
$v(t)$ that produces the given density path from a solution of the time-
dependent Schrödinger equation. Since the starting and end densities are just
outside a degeneracy region, the static potentials $v_{1}$ and $v_{2}$ that
have $\rho_{1}$ and $\rho_{2}$ as their respective ground states correspond to
nearly degenerate systems. The whole situation is depicted in Fig. 12. Since
the adiabatic theorem can only be applied when the change in the potential is
small compared to the energy gap $\Delta E$ between the ground and excited
states, we cannot reliably use the adiabatic theorem to derive an
approximation for $v(t)$. Such a study of driving the density slowly into a
degeneracy region was carried out by Rössler  _et al._ [37] and it was found
that the potential will exhibit fast oscillations to produce a nearly time-
constant density. For this reason we can expect that the presence of
degeneracy regions has a noticeable effect on the behavior of the potential in
TDDFT calculations. Apart from this pioneering work, we are unaware of studies
on this issue. The implications of the degeneracy structure in density space
for TDDFT is still largely unexplored and further research is warranted.

$E$$\Delta E$$\Delta E$$v_{1}$$v_{2}$$\rho_{1}$$\rho_{2}$ Figure 12: A time-
dependent density path through a degeneracy region on the top, where densities
just outside the degeneracy region correspond to potentials with a (small)
energy gap $\Delta E$.

##  IX Outlook

By offering a geometrical perspective on fundamental aspects of spin-lattice
DFT we hope to set the stage for further inquiries in this direction as well
as stimulate contributions from a wider mathematical field. Even though there
are clear links to graph theory and algebraic geometry, those areas remain
mostly untapped. We close with a list of open and unfinished topics that
continue the direction of research started here.

  * •

We showed how simple spin-lattice systems that are ubiquitous as model
systems, like the Anderson impurity model described in Sec. VI, can be studied
with respect to their degeneracy geometry. Yet, we did not develop tools to
efficiently and reliably find all degeneracy regions. As far as degeneracy due
to symmetry is concerned, the group-theoretic tools mentioned in Sec. II and
used in Sec. VI should do the trick. Additionally, the geometric phase from
closed integral contours in potential space provides an indicator for
degeneracy regions and relates to the fundamental group of
$\mathcal{V}^{\mathsf{\neg D}}$, as mentioned in Sec. VII.

  * •

In this and the preceding works on the subject [17; 22] already several
different degeneracy regions up to $\dim D=3$ have been discovered and
described. They range from ellipses to cones and the convex hull of the Roman
surface. In Ref. [22] we also suggested a first classification scheme for
degeneracy regions in terms of the degree of degeneracy and the nullity of the
map from the Veronese variety to density space. This scheme can definitely be
refined and many more examples of intricate degeneracy regions are possibly
still to be found.

  * •

The central objects of the Kohn–Sham method in DFT are the exchange-
correlation density functional $F_{\rm Hxc}(\rho)$ and its functional
derivative $v_{\rm Hxc}(\rho)$. The relative simplicity of the discussed spin-
lattice systems allows it to study these objects in-depth, also along the
‘adiabatic connection’ [58]. Degeneracy regions clearly play a role here,
since on them $F(\rho)$ has a constant derivative. A first simple yet non-
trivial trial system could be the Hubbard trimer at half filling, for which a
detailed study about the magnetization density functional is already available
[59].

  * •

Due to the availability of a rigorous HK result in the continuum [60], a
$v$-representable density is always _uniquely_ $v$-representable there. On the
other hand, and contrary to the lattice setting, no real criterion for
$v$-representability is available within the usual $L^{p}$-space formulation.
A possible remedy is to switch to the Sobolev space $H^{1}$ for densities,
where indeed $v$-representability holds for strictly positive densities on a
one-dimensional ring if also distributional potentials are considered [29].
Yet, this formulation again opens the general possibility of non-unique
$v$-representability, just like in the lattice case, but up to now no examples
are known. The precise connection between lattices and this setting, in the
form of a continuum limit and with respect to degeneracy regions and related
concepts, would be a further interesting object of study.

  * •

There have been considerable advances in the study of quantum-transport
phenomena using lattice DFT, as reviewed by Kurth and Stefanucci [61]. An
important application was the description of the Kondo effect in transport
through a single Anderson impurity, where it was found that the Kohn–Sham
potential becomes a discontinuous function of the site occupation number in
the zero-temperature limit [62]. For the case of larger lattices, such as the
Anderson impurity model discussed in this perspective, the Kohn–Sham potential
is likely to reveal structures related to the degeneracy regions which poses
certain challenges for the construction of future approximate DFT functionals
for quantum transport.

  * •

For the case of time-dependent DFT, the limitations of the currently available
$v$-representability proofs have been laid out in Sec. VIII together with a
possible strategy how to overcome them. The next necessary step would then be
a convergence proof for the corresponding time-dependent Kohn–Sham method. We
further conjectured a noticeable effect due to degeneracy on the (slow)
steering of a density through degeneracy regions that has yet to be
investigated.

  * •

A rigorous treatment of finite-temperature lattice DFT was given by Chayes
_et al._ [4] in which it was proven that any density in the interior of the
density domain is uniquely $v$-representable. In the low temperature limit the
density functionals are found to be strongly dependent the lattice occupation
number [63], which is intimately connected to the filling of levels according
to the magnitude of the chemical potential. Clearly, here both the degeneracy
structure and the particle number play a crucial role. While the dependence on
particle number has received considerable attention [63; 61; 64; 65], the
study of degeneracy in this context remains largely unexplored.

  * •

The states corresponding to the corners of the hypersimplices that form the
density domains for pure-lattice systems are naturally encoded as multi-qubit
basis states on a quantum computer [66]. For an arbitrary quantum state $\Psi$
the probability that qubit $i$ has value $1$ is given by the density
$\rho_{i}$, such that our geometrical representation may provide a useful
setting for quantum-computing applications. Given this natural setting, it is
also conceivable that the minimization needed for the constrained-search
functional $F(\rho)$ on our lattice systems can be efficiently implemented
using a quantum algorithm, as was recently done for the Hubbard dimer [67].
Since spin-lattices have special interest in the quantum computing community,
further connections to the field of quantum information [68] seem worthwhile
to explore.

## Data availability statement

The plots were created with the help of Python and Mathematica scripts that
can be downloaded at https://mage.uber.space/dokuwiki/material/fermion-graph.
All further data that support the findings of this study are available from
the corresponding author upon reasonable request.

###### Acknowledgements.

RvL acknowledges support from the Finnish Academy under project number 356906.

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##  Appendix A Unitary transformation of the Anderson impurity model

Here, we give the unitary transformation that leads to a block-diagonalization
of the model discussed in Sec. VI. Additionally to what is defined in Rössler
_et al._ [37, Eq. (2)] it also includes a permutation of the basis vectors and
a removal of imaginary off-diagonal contributions in the $2\times 2$-blocks.

$\hat{U}=\frac{1}{2}\begin{pmatrix}2&0&0&0&0&0&0&0&0\\\ 0&1&1&1&1&0&0&0&0\\\
0&0&0&0&0&1&1&1&1\\\ 0&-1&-\mathrm{i}&1&\mathrm{i}&0&0&0&0\\\
0&0&0&0&0&-(-1)^{1/4}&-(-1)^{3/4}&(-1)^{1/4}&(-1)^{3/4}\\\
0&-(-1)^{1/4}&(-1)^{3/4}&(-1)^{1/4}&-(-1)^{3/4}&0&0&0&0\\\
0&0&0&0&0&-1&\mathrm{i}&1&-\mathrm{i}\\\ 0&1&-1&1&-1&0&0&0&0\\\
0&0&0&0&0&1&-1&1&-1\end{pmatrix}.$ (43)