In order to explore some properties of fermionic systems on graphs numerically
as well as display the graphs alongside their fermionic counterparts, we
created a small Python script that is part of the public domain and can be
found at https://mage.uber.space/dokuwiki/material/fermion-graph .

###### Acknowledgements.

M. P. acknowledges helpful discussions with Alexander Steinicke, the
hospitality provided by Aisha during the final stage of this work, and
financial support by the Erwin Schrödinger Fellowship J 4107-N27 of the FWF
(Austrian Science Fund). R. v. L. acknowledges the Academy of Finland for
support under project no. 317139.

##  Appendix A Expression for the constrained-search functional in the
triangle example

Here we give the full derivation of the closed, analytical expression for the
constrained-search density functional $\tilde{F}$ for a non-interacting, two-
particle system on the triangle graph with Hamiltonian (42) that is referenced
in Section VI.3. A general trial wave function $\Psi$ given in the
$\\{e_{I}\\}_{I}$ basis of $\mathcal{H}_{2}$ is

$\Psi=c_{1}\,e_{1}\wedge e_{2}+c_{2}\,e_{1}\wedge e_{3}+c_{3}\,e_{2}\wedge
e_{3}$ (133)

and we want to minimize (80) under the constraint
$|c_{1}|^{2}+|c_{2}|^{2}+|c_{3}|^{2}=1$. Now since

$\rho_{1}=|c_{1}|^{2}+|c_{2}|^{2},\quad\rho_{2}=|c_{1}|^{2}+|c_{3}|^{2},\quad\rho_{3}=|c_{2}|^{2}+|c_{3}|^{2},$
(134)

and $\rho_{1}+\rho_{2}+\rho_{3}=2$, we can parametrize

$c_{1}=\sqrt{1-\rho_{3}}\,\mathrm{e}^{\mathrm{i}\varphi_{1}},\quad
c_{2}=\sqrt{1-\rho_{2}}\,\mathrm{e}^{\mathrm{i}\varphi_{2}},\quad
c_{3}=\sqrt{1-\rho_{1}},$ (135)

where a global phase factor has been split off. Because $\rho$ is fixed, the
search space is just $(\varphi_{1},\varphi_{2})\in[0,2\pi)^{2}$. In the next
step we calculate the expectation value with respect to $H(0)$ from (42) and
get

$\langle\Psi,H(0)\Psi\rangle=4+2\left(\alpha_{1}\cos\varphi_{1}-\alpha_{2}\cos\varphi_{2}-\alpha_{3}\cos(\varphi_{1}-\varphi_{2})\right),$
(136)

where we defined

$\alpha_{1}=\sqrt{(1-\rho_{1})(1-\rho_{3})},\quad\alpha_{2}=\sqrt{(1-\rho_{1})(1-\rho_{2})},\quad\alpha_{3}=\sqrt{(1-\rho_{2})(1-\rho_{3})}.$
(137)

Since we limit ourselves to $\mathcal{P}_{3,2}^{+}$ all $\alpha_{i}>0$, while
at the border of the triangular density domain an explicit solution is easily
possible. This is because then always one $\rho_{i}=1$ and consequently two
$\alpha_{i}$ are zero. At the edge where $\rho_{A}$ is located this leads to
$\tilde{F}(\rho)=4-2\alpha_{1}$ and similarly to an expression with
$\alpha_{2},\alpha_{3}$ at the other edges. In order to find the minimum of
(136) for the interior of $\mathcal{P}_{M,N}$ over all
$\varphi_{1},\varphi_{2}$ that gives the value of $\tilde{F}(\rho)$ we
consider the expression

$T(\varphi_{1},\varphi_{2})=\alpha_{1}\cos\varphi_{1}-\alpha_{2}\cos\varphi_{2}-\alpha_{3}\cos(\varphi_{1}-\varphi_{2})$
(138)

for fixed values of $\alpha_{i}$ and solve

$\displaystyle 0=\frac{\partial
T}{\partial\varphi_{1}}=-\alpha_{1}\sin\varphi_{1}+\alpha_{3}\sin(\varphi_{1}-\varphi_{2}),$
(139) $\displaystyle 0=\frac{\partial
T}{\partial\varphi_{2}}=\alpha_{2}\sin\varphi_{2}-\alpha_{3}\sin(\varphi_{1}-\varphi_{2})$
(140)

in order to find a minimum for $T(\varphi_{1},\varphi_{2})$. Adding both
equations gives

$-\alpha_{1}\sin\varphi_{1}+\alpha_{2}\sin\varphi_{2}=0\quad\Longrightarrow\quad\sin\varphi_{2}=\frac{\alpha_{1}}{\alpha_{2}}\sin\varphi_{1}.$
(141)

Rewriting (139) as

$0=-\alpha_{1}\sin\varphi_{1}+\alpha_{3}(\sin\varphi_{1}\cos\varphi_{2}-\cos\varphi_{1}\sin\varphi_{2})$
(142)

and using the result (141) from before then gives

$\sin\varphi_{1}\left(-\alpha_{1}+\alpha_{3}\left(\cos\varphi_{2}-\frac{\alpha_{1}}{\alpha_{2}}\cos\varphi_{1}\right)\right)=0.$
(143)

One option that immediately follows is that $\sin\varphi_{1}=0$ which then
immediately yields $\sin\varphi_{2}=0$, which gives
$\varphi_{1},\varphi_{2}\in\\{0,\pi\\}$. We obtain

$\displaystyle(\varphi_{1},\varphi_{2})=(0,\pi)\quad$
$\displaystyle\mapsto\quad T(0,\pi)=\alpha_{1}+\alpha_{2}+\alpha_{3},$ (144)
$\displaystyle(\varphi_{1},\varphi_{2})=(0,0)\quad$ $\displaystyle\mapsto\quad
T(0,0)=\alpha_{1}-\alpha_{2}-\alpha_{3},$ (145)
$\displaystyle(\varphi_{1},\varphi_{2})=(\pi,0)\quad$
$\displaystyle\mapsto\quad T(\pi,0)=-\alpha_{1}-\alpha_{2}+\alpha_{3},$ (146)
$\displaystyle(\varphi_{1},\varphi_{2})=(\pi,\pi)\quad$
$\displaystyle\mapsto\quad T(\pi,\pi)=-\alpha_{1}+\alpha_{2}-\alpha_{3}.$
(147)

The case $(\varphi_{1},\varphi_{2})=(0,\pi)$ giving
$\alpha_{1}+\alpha_{2}+\alpha_{3}>0$ is always larger than the other ones and
can thus be disregarded. In order to make $T$ minimal by choosing a density in
$\mathcal{P}_{3,2}$ we have to, let us say for (145), minimize $\alpha_{1}$
and maximize the other two, which is clearly achieved by $\rho_{2}=1$ and
$\rho_{1}=\rho_{3}=\tfrac{1}{2}$, exactly the exceptional point $\rho_{A}$
again. The lower bound for (145)-(147) is thus $-\tfrac{1}{2}$ which means
$\tilde{F}=4+2T\geq 3$, the correct ground-state eigenvalue of $H(0)$. The
remaining case of (143) is when

$-\alpha_{1}+\alpha_{3}\left(\cos\varphi_{2}-\frac{\alpha_{1}}{\alpha_{2}}\cos\varphi_{1}\right)=0\quad\Longrightarrow\quad\cos\varphi_{2}=\frac{\alpha_{1}}{\alpha_{2}}\left(\cos\varphi_{1}+\frac{\alpha_{2}}{\alpha_{3}}\right).$
(148)

We then find that

$\displaystyle
1=\sin^{2}\varphi_{2}+\cos^{2}\varphi_{2}=\left(\frac{\alpha_{1}}{\alpha_{2}}\right)^{2}\sin^{2}\varphi_{1}+\left(\frac{\alpha_{1}}{\alpha_{2}}\right)^{2}\left(\cos\varphi_{1}+\frac{\alpha_{2}}{\alpha_{3}}\right)^{2},$
(149)

which gives

$\displaystyle\left(\frac{\alpha_{2}}{\alpha_{1}}\right)^{2}$
$\displaystyle=\sin^{2}\varphi_{1}+\left(\cos\varphi_{1}+\frac{\alpha_{2}}{\alpha_{3}}\right)^{2}=1+2\frac{\alpha_{2}}{\alpha_{3}}\cos\varphi_{1}+\left(\frac{\alpha_{2}}{\alpha_{3}}\right)^{2},$
(150)

from which we deduce

$\cos\varphi_{1}=\frac{\alpha_{2}\alpha_{3}}{2\alpha_{1}^{2}}-\frac{\alpha_{2}}{2\alpha_{3}}-\frac{\alpha_{3}}{2\alpha_{2}}\quad\text{and}\quad\cos\varphi_{2}=\frac{\alpha_{3}}{2\alpha_{1}}+\frac{\alpha_{1}}{2\alpha_{3}}-\frac{\alpha_{1}\alpha_{3}}{2\alpha_{2}^{2}},$
(151)

where for the second equation we used (148). We have therefore found an
explicit expression for the angles of the stationary point

$(\varphi_{1},\varphi_{2})=\left(\arccos\left(\frac{\alpha_{2}\alpha_{3}}{2\alpha_{1}^{2}}-\frac{\alpha_{2}}{2\alpha_{3}}-\frac{\alpha_{3}}{2\alpha_{2}}\right),\arccos\left(\frac{\alpha_{3}}{2\alpha_{1}}+\frac{\alpha_{1}}{2\alpha_{3}}-\frac{\alpha_{1}\alpha_{3}}{2\alpha_{2}^{2}}\right)\right).$
(152)

It now remains to evaluate $T$ in this optimal point. From (151) we find

$\displaystyle\alpha_{1}\cos\varphi_{1}-\alpha_{2}\cos\varphi_{2}=-\frac{\alpha_{1}\alpha_{2}}{\alpha_{3}}$
(153)

and we proceed to evaluate the term

$\displaystyle\cos(\varphi_{1}-\varphi_{2})$
$\displaystyle=\cos\varphi_{1}\cos\varphi_{2}+\sin\varphi_{1}\sin\varphi_{2}=\cos\varphi_{1}\cos\varphi_{2}+\frac{\alpha_{1}}{\alpha_{2}}\sin^{2}\varphi_{1}=\cos\varphi_{1}\cos\varphi_{2}+\frac{\alpha_{1}}{\alpha_{2}}\left(1-\cos^{2}\varphi_{1}\right)$
(154)
$\displaystyle=\frac{\alpha_{1}}{\alpha_{2}}-\frac{1}{\alpha_{2}}(\alpha_{1}\cos\varphi_{1}-\alpha_{2}\cos\varphi_{2})\cos\varphi_{1}=\frac{\alpha_{1}}{2\alpha_{2}}+\frac{\alpha_{2}}{2\alpha_{1}}-\frac{\alpha_{1}\alpha_{2}}{2\alpha_{3}^{2}},$

where we used (141), (151), and (153). Therefore we obtain the expression

$\alpha_{3}\cos(\varphi_{1}-\varphi_{2})=\frac{\alpha_{1}\alpha_{3}}{2\alpha_{2}}+\frac{\alpha_{2}\alpha_{3}}{2\alpha_{1}}-\frac{\alpha_{1}\alpha_{2}}{2\alpha_{3}}$
(155)

and together with (153) this gives

$T=\alpha_{1}\cos\varphi_{1}-\alpha_{2}\cos\varphi_{2}-\alpha_{3}\cos(\varphi_{1}-\varphi_{2})=-\frac{\alpha_{1}\alpha_{3}}{2\alpha_{2}}-\frac{\alpha_{2}\alpha_{3}}{2\alpha_{1}}-\frac{\alpha_{1}\alpha_{2}}{2\alpha_{3}}$
(156)

at the stationary point. Now since

$\frac{\alpha_{1}\alpha_{2}}{\alpha_{3}}=1-\rho_{1},\quad\frac{\alpha_{2}\alpha_{3}}{\alpha_{1}}=1-\rho_{2},\quad\frac{\alpha_{1}\alpha_{3}}{\alpha_{2}}=1-\rho_{3}$
(157)

this gives a value at the minimum of

$T=-\frac{1}{2}(3-\rho_{1}-\rho_{2}-\rho_{3})=-\frac{1}{2}\quad\Longrightarrow\quad\tilde{F}=4+2T=3.$
(158)

This is also the lower bound for (145)-(147) as already demonstrated, so we
have definitely found a minimum here, but it remains to find out for which
densities the derivation we performed was even valid. After all, we need to
check that

$-1\leq\cos\varphi_{1}\leq 1\quad\text{and
thus}\quad-1\leq\frac{\alpha_{2}\alpha_{3}}{2\alpha_{1}^{2}}-\frac{\alpha_{2}}{2\alpha_{3}}-\frac{\alpha_{3}}{2\alpha_{2}}\leq
1,$ (159)

which equivalently gives

$-\alpha_{1}\leq\frac{\alpha_{2}\alpha_{3}}{2\alpha_{1}}-\frac{\alpha_{2}\alpha_{1}}{2\alpha_{3}}-\frac{\alpha_{1}\alpha_{3}}{2\alpha_{2}}\leq\alpha_{1}.$
(160)

From (157) we then have that

$\frac{\alpha_{2}\alpha_{3}}{2\alpha_{1}}-\frac{\alpha_{2}\alpha_{1}}{2\alpha_{3}}-\frac{\alpha_{1}\alpha_{3}}{2\alpha_{2}}=\frac{1}{2}(1-\rho_{2}-(1-\rho_{1})-(1-\rho_{3}))=\frac{1}{2}(\rho_{1}+\rho_{3}-\rho_{2}-1)=\frac{1}{2}(1-2\rho_{2})$
(161)

and therefore $-2\alpha_{1}\leq 1-2\rho_{2}\leq 2\alpha_{1}$. Squaring this
equation gives the condition

$(1-2\rho_{2})^{2}\leq
4\alpha_{1}^{2}=4(1-\rho_{1})(1-\rho_{3})=4(1-\rho_{1})(\rho_{1}+\rho_{2}-1),$
(162)

which can be checked to be equivalent to

$\left(\sum_{i}\left(\rho_{i}-\frac{2}{3}\right)^{\\!\\!2}\right)^{\\!\\!\frac{1}{2}}\leq\frac{1}{\sqrt{6}},$
(163)

precisely the condition for the incircle region $\mathcal{C}$ already derived
in (99). This means the result $\tilde{F}=3$ is valid inside $\mathcal{C}$
while the three other results from (145)-(147) are for the spike regions
$\mathcal{S}_{1},\mathcal{S}_{2},\mathcal{S}_{3}$. This allows us to collect
all results in the full expression for $\tilde{F}$ already given in (100),

$\tilde{F}(\rho)=\left\\{\begin{array}[]{ll}3&\rho\in\mathcal{C}\\\
4+2\left(\alpha_{1}-\alpha_{2}-\alpha_{3}\right)&\rho\in\mathcal{S}_{1}\\\
4+2\left(-\alpha_{1}-\alpha_{2}+\alpha_{3}\right)&\rho\in\mathcal{S}_{2}\\\
4+2\left(-\alpha_{1}+\alpha_{2}-\alpha_{3}\right)&\rho\in\mathcal{S}_{3}.\end{array}\right.$
(164)

Finally, let us remark that an equivalent approach to retrieve an expression
for the constrained-search density functional would be to use the Lagrange
multiplier method for finding the minimum of the Hermitian form
$\langle\Psi,H_{0}\Psi\rangle$ under the constraints (134).

## References

  * Hohenberg and Kohn [1964] P. Hohenberg and W. Kohn,  _Inhomogeneous electron gas,_Phys. Rev. 136, B864 (1964).
  * Levy [1979] M. Levy,  _Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem,_Proc. Natl. Acad. Sci. USA 76, 6062 (1979).
  * Lieb [1983] E. H. Lieb,  _Density functionals for Coulomb-systems,_Int. J. Quantum Chem. 24, 243 (1983).
  * Katriel  _et al._ [1981] J. Katriel, C. J. Appellof, and E. R. Davidson,  _Mapping between local potentials and ground state densities,_Int. J. Quantum Chem. 19, 293 (1981).
  * Englisch and Englisch [1984] H. Englisch and R. Englisch,  _Exact density functionals for ground state energies, II. Details and remarks,_Phys. Stat. Sol. (B) 124, 373 (1984).
  * Chayes  _et al._ [1985] J. T. Chayes, L. Chayes, and M. B. Ruskai,  _Density functional approach to quantum lattice systems,_J. Stat. Phys. 38, 497 (1985).
  * Schönhammer  _et al._ [1995] K. Schönhammer, O. Gunnarsson, and R. M. Noack,  _Density-functional theory on a lattice: Comparison with exact numerical results for a model with strongly correlated electrons,_Phys. Rev. B 52, 2504 (1995).
  * Xianlong  _et al._ [2006] G. Xianlong, M. Polini, M. P. Tosi, V. L. Campo Jr., K. Capelle, and M. Rigol,  _Bethe ansatz density-functional theory of ultracold repulsive fermions in one-dimensional optical lattices,_Phys. Rev. B 73, 165120 (2006).
  * Giesbertz and Ruggenthaler [2019] K. J. Giesbertz and M. Ruggenthaler,  _One-body reduced density-matrix functional theory in finite basis sets at elevated temperatures,_Phys. Rep. 806, 1 (2019).
  * Carrascal  _et al._ [2015] D. Carrascal, J. Ferrer, J. C. Smith, and K. Burke,  _The Hubbard dimer: a density functional case study of a many-body problem,_J. Phys.: Cond. Matt. 27, 393001 (2015).
  * Lima  _et al._ [2003] N. Lima, M. F. Silva, L. N. Oliveira, and K. Capelle,  _Density functionals not based on the electron gas: Local-density approximation for a Luttinger liquid,_Phys. Rev. Lett. 90, 146402 (2003).
  * Ijäs and Harju [2010] M. Ijäs and A. Harju,  _Lattice density-functional theory on graphene,_Phys. Rev. B 82, 235111 (2010).
  * Saubanere and Pastor [2014] M. Saubanere and G. Pastor,  _Lattice density-functional theory of the attractive Hubbard model,_Phys. Rev. B 90, 125128 (2014).
  * Garrigue [2018] L. Garrigue,  _Unique continuation for many-body Schrödinger operators and the Hohenberg–Kohn theorem,_Math. Phys. Anal. Geom. 21, 27 (2018).
  * Penz  _et al._ [2019] M. Penz, A. Laestadius, E. I. Tellgren, and M. Ruggenthaler,  _Guaranteed convergence of a regularized Kohn–Sham iteration in finite dimensions,_Phys. Rev. Lett. 123, 037401 (2019).
  * Penz  _et al._ [2020] M. Penz, A. Laestadius, E. I. Tellgren, M. Ruggenthaler, and P. E. Lammert,  _Erratum: Guaranteed convergence of a regularized Kohn–Sham iteration in finite dimensions,_Phys. Rev. Lett. 125, 249902 (2020).
  * Coe  _et al._ [2015] J. P. Coe, I. D’Amico, and V. V. França,  _Uniqueness of density-to-potential mapping for fermionic lattice systems,_EPL 110, 63001 (2015).
  * Dimitrov  _et al._ [2016] T. Dimitrov, H. Appel, J. I. Fuks, and A. Rubio,  _Exact maps in density functional theory for lattice models,_New J. Phys. 18, 083004 (2016).
  * Odlyzko [1981] A. M. Odlyzko,  _On the ranks of some $(0,1)$-matrices with constant row sums, _J. Austral. Math. Soc. (Series A) 31, 193 (1981).
  * Kohn [1983] W. Kohn,  _$v$ -representability and density functional theory, _Phys. Rev. Lett. 51, 1596 (1983).
  * Diestel [2005] R. Diestel,  _Graph theory_ , 3rd ed. (Springer, 2005).
  * Grünbaum [2003] B. Grünbaum,  _Convex polytopes_ , 2nd ed. (Springer, 2003).
  * Rispoli [2008] F. J. Rispoli,  _The graph of the hypersimplex,_ arXiv preprint (2008), arXiv:0811.2981 [math.CO] .
  * Harriman [1981] J. E. Harriman,  _Orthonormal orbitals for the representation of an arbitrary density,_Phys. Rev. A 24, 680 (1981).
  * Chung [1994] F. R. K. Chung,  _Spectral graph theory_ (AMS, 1994).
  * Biyikoglu  _et al._ [2007] T. Biyikoglu, J. Leydold, and P. F. Stadler,  _Laplacian eigenvectors of graphs_ (Springer, 2007).
  * ten Haaf  _et al._ [1995] D. F. B. ten Haaf, H. J. M. van Bemmel, J. M. J. van Leeuwen, W. van Saarloos, and D. M. Ceperley,  _Proof for an upper bound in fixed-node monte carlo for lattice fermions,_Physical Review B 51, 13039 (1995).
  * Pino  _et al._ [2007] R. Pino, O. Bokanowski, E. V. Ludeña, and R. L. Boada,  _A re-statement of the Hohenberg–Kohn theorem and its extension to finite subspaces,_Theor. Chem. Account 118, 557 (2007).
  * de Figueiredo and Gossez [1992] D. G. de Figueiredo and J.-P. Gossez, Comm. Par. Diff. Eq. 17, 339 (1992).
  * Regbaoui [2001] R. Regbaoui, in  _Carleman estimates and applications to uniqueness and control theory_ (Birkhäuser, 2001) pp. 179–190.
  * Isozaki and Morioka [2014] H. Isozaki and H. Morioka,  _A Rellich type theorem for discrete Schrödinger operators,_Inverse Probl. Imaging 8, 475 (2014).
  * Peyerimhoff  _et al._ [2017] N. Peyerimhoff, M. Täufer, and I. Veselić,  _Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space,_Nanosyst.: Phys. Chem. Math. 8, 216 (2017).
  * Longstaff [1977] W. E. Longstaff,  _Combinatorial solution of certain systems of linear equations involving $(0,1)$ matrices, _J. Austral. Math. Soc. (Series A) 23, 266 (1977).
  * Ullrich and Kohn [2002] C. A. Ullrich and W. Kohn,  _Degeneracy in density functional theory: Topology in the v and n spaces,_Phys. Rev. Lett. 89, 156401 (2002).
  * Prosalov [2000] V. Prosalov,  _Problems and theorems in linear algebra_ (AMS, 2000).
  * Rellich [1937] F. Rellich,  _Störungstheorie der Spektralzerlegung, I. Mitteilung,_Mathematische Annalen 113, 600 (1937).
  * Rellich [1969] F. Rellich,  _Perturbation theory of eigenvalue problems_ (Gordon and Breach Science Publishers, 1969).
  * Kato [1995] T. Kato,  _Perturbation theory for linear operators_ (Springer, 1995).
  * Baumgärtel [1985] H. Baumgärtel,  _Analytic perturbation theory for matrices and operators_ (Birkhäuser, 1985).
  * Lammert [2018] P. E. Lammert,  _In search of the Hohenberg–Kohn theorem,_J. Math. Phys. 59, 042110 (2018).
  * Minc [1988] H. Minc,  _Nonnegative matrices_ (Wiley, 1988).
  * Meyer [2000] C. D. Meyer,  _Matrix analysis and applied linear algebra_ (SIAM, 2000).
  * Englisch and Englisch [1983] H. Englisch and R. Englisch,  _V-representability in finite-dimensional spaces,_J. Phys. A 16, L693 (1983).
  * Rockafellar [1970] R. T. Rockafellar,  _Convex analysis_ (Princeton University Press, 1970).
  * Kvaal  _et al._ [2014] S. Kvaal, U. Ekström, A. M. Teale, and T. Helgaker, J. Chem. Phys. 140, 18A518 (2014).
  * Lammert [2007] P. E. Lammert,  _Differentiability of Lieb functional in electronic density functional theory,_Int. J. Quantum Chem. 107, 1943 (2007).
  * Lammert [2006] P. E. Lammert,  _Coarse-grained v representability,_J. Chem. Phys. 125, 074114 (2006).
  * Lammert [2010] P. E. Lammert,  _Well-behaved coarse-grained model of density-functional theory,_Phys. Rev. A 82, 012109 (2010).
  * Rössler  _et al._ [2018] T. Rössler, C. Verdozzi, and C.-O. Almbladh, Eur. Phys. J. B 91, 219 (2018).
  * Levy [1982] M. Levy,  _Electron densities in search of Hamiltonians,_Phys. Rev. A 26, 1200 (1982).
  * Schönhammer and Gunnarsson [1987] K. Schönhammer and O. Gunnarsson,  _Discontinuity of the exchange-correlation potential in density functional theory,_J. Phys. C 20, 3675 (1987).
  * Ceperley [1991] D. M. Ceperley,  _Fermion nodes,_J. Stat. Phys. 63, 1237 (1991).
  * Bajdich  _et al._ [2005] M. Bajdich, L. Mitas, G. Drobný, and L. K. Wagner,  _Approximate and exact nodes of fermionic wavefunctions: Coordinate transformations and topologies,_Phys. Rev. B 72, 075131 (2005).
  * Mitas [2006] L. Mitas,  _Structure of fermion nodes and nodal cells,_Phys. Rev. Lett. 96, 240402 (2006).
  * Ullrich [2005] C. A. Ullrich,  _Nonuniqueness in spin-density-functional theory on lattices,_Phys. Rev. B 72, 073102 (2005).
  * Zaslavsky [1982] T. Zaslavsky,  _Signed graphs,_Discr. Appl. Math. 4, 47 (1982).
  * Belardo  _et al._ [2019] F. Belardo, S. M. Cioabă, J. H. Koolen, and J. Wang,  _Open problems in the spectral theory of signed graphs,_ arXiv preprint (2019), arXiv:1907.04349 [math.CO] .
  * Li and Ullrich [2008] Y. Li and C. A. Ullrich,  _Time-dependent v-representability on lattice systems,_J. Chem. Phys. 129, 044105 (2008).
  * Tokatly [2011] I. Tokatly,  _Time-dependent current density functional theory on a lattice,_Phys. Rev. B 83, 035127 (2011).
  * Farzanehpour and Tokatly [2012] M. Farzanehpour and I. Tokatly,  _Time-dependent density functional theory on a lattice,_Phys. Rev. B 86, 125130 (2012).
  * Uimonen  _et al._ [2011] A.-M. Uimonen, E. Khosravi, A. Stan, G. Stefanucci, S. Kurth, R. van Leeuwen, and E. K. U. Gross,  _Comparative study of many-body perturbation theory and time-dependent density functional theory in the out-of-equilibrium Anderson model,_Phys. Rev. B 84, 115103 (2011).
  * Stefanucci and Kurth [2015] G. Stefanucci and S. Kurth,  _Steady-state density functional theory for finite bias conductances,_Nano Lett. 15, 8020 (2015).
  * Jacob  _et al._ [2020] D. Jacob, G. Stefanucci, and S. Kurth,  _Mott metal-insulator transition from steady-state density functional theory,_Phys. Rev. Lett. 125, 216401 (2020).
  * Karlsson  _et al._ [2011] D. Karlsson, A. Privitera, and C. Verdozzi,  _Time-dependent density-functional theory meets dynamical mean-field theory: Real-time dynamics for the 3D Hubbard model,_Phys. Rev. Lett. 106, 116401 (2011).
  * Kartsev  _et al._ [2013] A. Kartsev, D. Karlsson, A. Privitera, and C. Verdozzi,  _Three-dimensional dynamics of a fermionic Mott wedding-cake in clean and disordered optical lattices,_Sci. Rep. 3, 2570 (2013).