# Effects of lattice geometry on thermopower properties of the repulsive
Hubbard model

Willdauany C. de Freitas Silva Instituto de Física, Universidade Federal do
Rio de Janeiro, Rio de Janeiro, RJ 21941-972, Brazil Maykon V. Araujo
Departamento de Física, Universidade Federal do Piauí, 64049-550 Teresina PI,
Brazil Sayantan Roy Department of Physics, The Ohio State University, Columbus
OH 43210, USA Abhisek Samanta Department of Physics, The Ohio State
University, Columbus OH 43210, USA Natanael de C. Costa Instituto de Física,
Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ 21941-972, Brazil
Nandini Trivedi Department of Physics, The Ohio State University, Columbus OH
43210, USA Thereza Paiva Instituto de Física, Universidade Federal do Rio de
Janeiro, Rio de Janeiro, RJ 21941-972, Brazil

###### Abstract

We obtain the Seebeck coefficient or thermopower $S$, which determines the
conversion efficiency from thermal to electrical energy, for the two-
dimensional Hubbard model on different geometries (square, triangular, and
honeycomb lattices) for different electronic densities and interaction
strengths. Using Determinantal Quantum Monte Carlo (DQMC) we find the
following key results: (a) the bi-partiteness of the lattice affects the
doping dependence of $S$; (b) strong electronic correlations can greatly
enhance $S$ and produce non-trivial sign changes as a function of doping
especially in the vicinity of the Mott insulating phase; (c) $S(T)$ near half
filling can show non-monotonic behavior as a function of temperature. We
emphasize the role of strong interaction effects in engineering better devices
for energy storage and applications, as captured by our calculations of the
power factor $PF=S^{2}\sigma$ where $\sigma$ is the dc conductivity.

###### pacs:

71.10.Fd, 71.30.+h, 02.70.Uu

##  I Introduction

Over the past decades, a great deal of interest has been given to increasing
the efficiency of electrical devices. As a possible route to this end, the
thermoelectric materials may play a crucial role, once they exhibit induced
voltage in presence of a temperature gradient, whose magnitude is quantified
by the Seebeck coefficient, the thermoelectric Power Factor, and, eventually,
the Figure of Merit Hébert  _et al._ (2015). However, there are many technical
issues that make the development of efficient thermoelectric materials a
challenge, e.g., the toxicity of the compounds, or their thermal instability.
There are few ways to overcome these problems: (i) optimizing the already
known compounds through band-structure engineering and nanostructuration
techniques Ding and Li (2015); Heremans  _et al._ (2008); Hicks and
Dresselhaus (1993), or (ii) seeking new classes of compounds which exhibit
unconventional properties usually related to strong electron-electron
interactions Wissgott  _et al._ (2010). In view of the increasing number of
novel correlated compounds, controlling and manipulating geometry and
correlations to enhance the thermopower properties is an open issue.

Despite intense experimental efforts, further theoretical investigations are
required, in particular to understand interacting electronic compounds, such
as NaxCoO2 or FeSb2, which exhibit unusual large thermopower response Wissgott
_et al._ (2010, 2011); Tomczak (2018). In the former, the combination of quasi
two-dimensional character with band topology and strong electronic
correlations make this material an interesting playground to examine
thermoelectricity. Once the charge carriers are confined to the hexagonal
layers of Co atoms, a disordered distribution of Na ions above and below it
can induce a charge imbalance. Using local-density approximation and dynamical
mean-field theory, Held et al Wissgott  _et al._ (2010) showed that disorder
combined with the pudding-mold band structure and strong correlations enforce
the electron-hole imbalance and enhances the thermopower by $200\%$ with
respect to the non-interacting case.

The thermopower of superconducting cuprates has been experimentally studied
Obertelli  _et al._ (1992); Tallon  _et al._ (1995); Honma and Hor (2008);
Benseman  _et al._ (2011), with different compounds displaying very similar
behavior with a sign change of the Seebeck coefficient near the maximum
critical temperature Obertelli  _et al._ (1992). This nearly universal
behavior has been the subject of theoretical interest, being ascribed to a
possible underlying critical point Garg  _et al._ (2011), to the presence of a
van Hove singularity McIntosh and Kaiser (1996), and has been recently
observed for the Hubbard Model with next-nearest neighbor hopping Wen O. Wang
and Devereaux (2023).

In view of these stimulating results, we explore how electron-electron
interactions and geometry affect the Seebeck coefficient, and the
thermoelectric Power Factor. To this end, we use unbiased Quantum Monte Carlo
simulations to study the single band repulsive Hubbard model. We analyze the
thermoelectric and electrical transport properties in the long wavelength DC
limit in two-dimensions, in the square, triangular and honeycomb lattices. Our
study finds a strong influence of particle hole symmetry of the many body
spectrum and DOS on the behavior of the Seebeck coefficient with respect to
doping. We show non trivial sign changes of the Seebeck coefficient that
signal a deviation of the Fermi surface from the Luttinger count and a
subsequent anomalous change of the type of carriers below and above half
filling. The sign change is also accompanied by a significant increase of the
Seebeck coefficient near half filling with respect to the non-interacting and
weakly interacting case. We show that despite using a simplified thermodynamic
formula for the Seebeck coefficient that is independent of dynamical
quantities, we are able to capture the effects of strong correlation.

The paper is organized as follows: In Section II we discuss the Hubbard model,
an introduction to the Seebeck coefficient, Kelvin formula and the auxiliary
field QMC method used to solve it. In Section III we discuss how to calculate
the entropy and present our results for this quantity. Section IV shows the
Local Density of States and conductivity results. In Section V we discuss the
Seebeck coefficient and in Section VI the Power Factor. Finally, in section
VII we summarize our findings.

##  II Model and methods

The repulsive Fermi Hubbard model describes electrons on a lattice with an
onsite repulsive interaction, with the Hamiltonian

$\displaystyle\mathcal{H}=$
$\displaystyle-t\sum_{\begin{subarray}{c}\langle\textbf{i},\textbf{j}\rangle\end{subarray},\sigma}\big{(}c_{\textbf{i},\sigma}^{\dagger}c_{\textbf{j},\sigma}+{\rm
H.c.}\big{)}-\mu\sum_{\begin{subarray}{c}\textbf{i}\end{subarray},\sigma}n_{\textbf{i},\sigma}$
$\displaystyle+U\sum_{\begin{subarray}{c}\textbf{i}\end{subarray}}(n_{\textbf{i},\uparrow}-1/2)(n_{\textbf{i},\downarrow}-1/2),$
(1)

where the sums run over sites of a given two-dimensional lattice, with
$\langle\mathbf{i},\mathbf{j}\rangle$ denoting nearest-neighbor sites. Here,
we use the second quantization formalism, with
$c^{\dagger}_{\mathbf{i},\sigma}$
($c^{\phantom{\dagger}}_{\mathbf{i},\sigma}$) being creation (annihilation)
operators of electrons on a given site $\mathbf{i}$, and spin $\sigma$, while
$n_{\mathbf{i},\sigma}\equiv
c^{\dagger}_{\mathbf{i},\sigma}c_{\mathbf{i},\sigma}$ are number operators.
The first two terms on the right-hand side of Eq. (II) correspond to the
hopping of electrons, and the chemical potential $\mu$, respectively, with the
latter determining the filling of the lattice. The third term describes the
local repulsive interaction between electrons, with interaction strength U;
the factor of $1/2$ is introduced to ensure invariance of the hamiltonian
under particle-hole transformations on bipartite lattices. This implies that
for the bipartite lattices we consider here (square and honeycomb lattices),
$\mu=0$ sets half-filling for all temperatures.

Our central quantities of interest are the transport coefficients, and their
behavior with respect to doping and strength of interactions. The transport
coefficients are defined through the following relations,

$\displaystyle\vec{j}$
$\displaystyle=\tensor{L}^{11}\vec{E}+\tensor{L}^{12}(-\vec{\nabla}T)$
$\displaystyle\vec{j}^{q}$
$\displaystyle=\tensor{L}^{21}\vec{E}+\tensor{L}^{22}(-\vec{\nabla}T)$ (2)

where $\vec{j}$ and $\vec{j}^{q}$ are the electrical and thermal currents, and
the $\tensor{L}$s are rank 2 tensors defining conductivities of the system.
The tensors $\tensor{L}^{11}$ and $\tensor{L}^{22}$ are the electrical and
thermal conductivities, and $\tensor{L}^{12}$($\tensor{L}^{21}$) are the
thermoelectrical(electrothermal) conductivities. The thermopower or Seebeck
coefficient is defined as

$\displaystyle S$
$\displaystyle=\frac{\big{(}\tensor{L}^{12}\big{)}_{xx}}{\big{(}\tensor{L}^{11}\big{)}_{xx}}=\frac{1}{T}\frac{\big{(}\tensor{L}^{21}\big{)}_{xx}}{\big{(}\tensor{L}^{11}\big{)}_{xx}},$
(3)

where the second equality is due to Onsager’s reciprocity relationsOnsager
(1931). Using linear response theory with respect to electrical field and
temperature, the Seebeck coefficient in the Kubo formalism can also be written
as

$\displaystyle
S(q_{x},\omega)=\frac{1}{T}\frac{\chi_{\hat{\rho}(q_{x}),\hat{K}(-q_{x})}(\omega)}{\chi_{\hat{\rho}(q_{x})\hat{\rho}(-q_{x})}(\omega)}$
(4)

where

$\displaystyle\chi_{\hat{\rho}(q)\hat{K}(-q)}(\omega)$
$\displaystyle\\!=\\!\lim_{\eta\rightarrow
0}\sum_{n,m}(f_{n}\\!-\\!f_{m})\frac{\langle n|\hat{\rho}(q)|m\rangle\langle
m|\hat{K}(-q)|n\rangle}{\omega+i\eta+\epsilon_{n}-\epsilon_{m}}$
$\displaystyle\chi_{\hat{\rho}(q)\hat{\rho}(-q)}(\omega)$
$\displaystyle=\lim_{\eta\rightarrow
0}\sum_{n,m}(f_{n}\\!-\\!f_{m})\frac{\langle n|\hat{\rho}(q)|m\rangle\langle
m|\hat{\rho}(-q)|n\rangle}{\omega+i\eta+\epsilon_{n}-\epsilon_{m}}$

define the electrothermal and electrical conductivities, respectively.
Evaluation of a Kubo-like formula (Eq (4)) is not easy for interacting systems
in the thermodynamic limit, although it is the most insightful. Alternate
formulas like the Mott formulaJonson and Mahan (1980), Heikes-MottHeikes  _et
al._ (1961), high frequency SeebeckShastry (2006, 2008); Zdroj (2007) and
Kelvin formulaPeterson and Shastry (2010a); Shastry (2008) exist but are
limited by their applicability to specific scenarios (weakly correlated metal
at low temperatures for the Mott formula, high-temperature limit for the
Heikes-Mott formula, and measurement of transport at high frequencies compared
to characteristic energy scale for high-frequency Seebeck formula). The Kelvin
formula was proposed by Lord Kelvin to provide reciprocity between Seebeck and
Peltier coefficients, and is calculated in the slow limit ($q_{x}\rightarrow
0,\omega\rightarrow 0$); it can be derived by taking the slow limit of Eq (4)
Peterson and Shastry (2010a). The Kelvin formula for the Seebeck coefficient
is

$\displaystyle S_{\rm Kelvin}$
$\displaystyle=\left.-\frac{1}{e}{\frac{\partial\mu}{\partial
T}}\right|_{V,n}=\left.\frac{1}{e}{\frac{\partial s}{\partial
n}}\right|_{T,V}\leavevmode\nobreak\ ,$

where the second equality follows from Maxwell’s relations. Although expressed
in terms of thermodynamic quantities, which are sufficient to capture the
effects from the many-body density of states, it misses kinematic factors like
contributions from velocities at the Fermi surface and relaxation times.
Nonetheless, the effects of strong correlations on the low-frequency transport
behavior are taken into account, as it retains the $\omega<U$ approximation,
which the high frequency formula, $S^{*}$ misses. The justification and
benchmark of using the Kelvin formula for strongly interacting systems like
the Hubbard model and fractionalized systems like $\nu=5/2$ FQHE states have
already been establishedArsenault  _et al._ (2013); Peterson and Shastry
(2010a).

We investigate the thermodynamics and transport properties of Eq. (II) on
three different lattices: square, triangular and honeycomb. In particular, we
examine the behavior of the entropy, conductivity, Local Density of States
(LDOS), Seebeck coefficient, and Power Factor as functions of lattice filling,
for different values of interaction strengths. To this end, we perform
unbiased determinant quantum Monte Carlo (DQMC) simulations Blankenbecler  _et
al._ (1981); Hirsch (1983, 1985); White  _et al._ (1989), a state-of-the-art
numerical method that maps a many-particle interacting fermionic system into a
single-particle (quadratic form) one, with the aid of bosonic auxiliary
fields. More details about the methodology may be found in, e.g., Refs.
Gubernatis  _et al._ , 2016; Becca and Sorella, 2017; dos Santos, 2003, and
references therein. Our DQMC simulations are performed for finite-sized
systems (with 100, 144 and 162 sites for the square, triangular, and honeycomb
lattices, respectively), and for interaction strengths $U/t=0$, 2, 4, 6, 8,
and 10; i.e., from non-interacting to strong coupling regimes. Throughout this
work, unless otherwise indicated, we consider $T/t=0.5$, which corresponds to
an energy scale low enough to observe the crossover towards an insulating
phase at half-filling Kim  _et al._ (2020); Simkovic  _et al._ (2020); Lenihan
_et al._ (2021). Hereafter, we define the lattice constant as unity, and the
hopping integral $t$ as the energy scale.

##  III Entropy

Figure 1: (Color online) Density $n$ as a function of chemical potential $\mu$
for the (a) square, (b) triangular, and (c) honeycomb lattices, for fixed
$T/t=0.5$, and different interaction strengths $U/t$. Here, and in all
subsequent figures, when not shown, error bars are smaller than symbol size. A
Mott plateau is formed above a critical interaction strength which depends on
lattice geometry. Note the width of this plateau is a measure of the charge
gap or “Mottness”. Figure 2: (Color online) Entropy $s$ as a function of the
electronic density $n$ for the (a) square, (b) triangular, and (c) honeycomb
lattices, for fixed $T/t=0.5$, and different interaction strengths $U/t$. Note
that the entropy is very different among the three lattices in the
noninteracting limit. In presence of sufficiently large interactions, a local
minimum appears at the half-filling for all three lattices; However, it
becomes qualitatively similar for the square and honeycomb lattices, while it
is different for the non-bipartite triangular lattice.

We start our analysis with the electronic density as a function of chemical
potential shown in Figure 1 for $T/t=0.5$ and the different lattices studied:
(a) square, (b) triangular, and (c) honeycomb. Particle-hole symmetry is
evident for the square and honeycomb lattices, where $n(\mu)=2-n(-\mu)$. In
all geometries, correlations lead to the formation of a Mott plateau
(insulating phase) around half-filling, although the critical interaction
strength for the onset of “Mottness” is strongly influenced by the lattice
geometry.

As the Seebeck coefficient is obtained from the entropy, we now turn to
discuss its behavior for these geometries. The entropy can be obtained from
the electronic density, $n(\mu)$, by integrating it over the chemical
potential $\mu$,

$s(\mu,T)=\int_{-\infty}^{\mu}d\mu\left.\frac{\partial n}{\partial
T}\right|_{\mu}$ (6)

Figure 2 (a) displays $s(\mu,T)$ as a function of $n$ for the square lattice.
One may notice that, at $T/t=0.5$, the results for $U/t=2$ (red circles
symbols) and $U/t=4$ (blue diamond symbols) exhibit very similar behavior to
the non-interacting one (solid black curve), once the temperature is high
enough to destroy the correlation effects for such small interaction
strengths. However as $U/t$ increases, e.g. $U/t=6$, 8, and 10 the entropy
presents a local minimum at half-filling, being drastically reduced as $T/t\to
0$, due to the Mott gap formation in the ground state Bonča and Prelovšek
(2003); Khatami and Rigol (2011); Simkovic  _et al._ (2020); Mikelsons  _et
al._ (2009); Paiva  _et al._ (2001). The increased entropy for the metallic
state in the vicinity of half-filling in the presence of interactions has been
observed for the cubic lattice and is relevant for cold fermionic atoms
trapped in optical lattices, where the metallic region of the atomic cloud is
used to absorb entropy and allow a central Mott region at a higher entropy per
particle Paiva  _et al._ (2011).

Unlike what is observed for the square lattice, the entropy behavior on the
triangular geometry only exhibits such a local minimum at $U/t=10$, the
largest interaction strength considered, as presented in Fig. 2 (b). This
suggests an absence of a Mott gap for weak and intermediate interaction
strengths. In fact, at $U/t\gtrsim 8$, a small dip starts to form around
$n=1$, in line with the expectation for a metal-to-insulator transition
occurring for $U/t\approx 7-8$ Yoshioka  _et al._ (2009); Shirakawa  _et al._
(2017); Garwood  _et al._ (2022). The lack of particle-hole symmetry in the
triangular lattice shown in Fig. 1 (c) is clearly also present in the entropy.

The entropy for the honeycomb lattice is more subtle. Similar to the square
and triangular lattices, a local minimum appears at half-filling in presence
of strong interaction; however, different from what is seen in the two
previous cases, here $s(n,T)$ displays a suppression at half-filling even for
the non-interacting case. This behavior is understood by recalling that the
honeycomb lattice has a vanishing DOS at half-filling, with van Hove
singularities below and above it. That is, the results of Fig. 2 confirm our
expectations that the entropy provides hints about the DOS, irrespective of
the interaction strengths. These odd features of the honeycomb lattice lead to
strong changes for the Seebeck coefficient at $U=0$, as we shall see in
Section V, and may obscure some of the the electronic correlation effects.

##  IV Local Density of states and conductivity

As discussed before, Fig. 1 signals the occurrence of a Mott insulating state
due to the presence of a plateau in the density as a function of chemical
potential, driven by correlations. The subtleties in the entropy require an
analysis of the suppression of spectral weight for different densities. In
order to avoid numerical analytical continuations, we examine the LDOS only at
the Fermi level ($\omega=0$), which is obtained through Trivedi and Randeria
(1995)

$N(\omega=0)\approx\frac{\beta}{\pi}G(|\mathbf{i}-\mathbf{j}|=0,\;\tau=\beta/2),$
(7)

where $G(\mathbf{r},\tau)$ is the real space and imaginary time Green’s
function calculated at $\mathbf{r}\equiv\mathbf{i}-\mathbf{j}=0$, and
imaginary time $\tau=\beta/2$ (hereafter, $\beta\equiv 1/(k_{B}T)$ is the
inverse of temperature, and $k_{B}$ the Boltzmann constant).

Figure 3: (Color online) Local density of states (LDOS) at the Fermi level at
$T/t=0.5$ as a function of density for the (a) square, (b) triangular, and (c)
honeycomb lattices. Note the similarity of the doping dependence of LDOS to
that of the entropy (Fig. 2) in the non-interacting limit. When correlations
increase, LDOS develops a dip near half-filling, leading to maxima at
intermediate densities which depend on the lattice geometry. The LDOS also
becomes qualitatively similar to the square and honeycomb lattices in the
strongly interacting limit, but is quite different from the triangular lattice
which, in contrast, is not bipartite and has an asymmetric particle-hole many-
body spectrum.

Fig. 3 shows $N(\omega=0)$ for the (a) square, (b) triangular, and (c)
honeycomb lattices. A common feature for all geometries is that the non-
interacting case is an upper bound for the LDOS, with data close to the empty
and completely filled systems showing a negligible dependence on the
interaction strength. For the temperature shown, $T/t=0.5$, we can see that
for the square lattice and $U/t=2$ the dip at half-filling has not developed
yet. As correlations increase, the dip starts to form, leading to maxima at
$n\approx 0.6$ and $n\approx 1.4$. The non-interacting ground state for the
honeycomb lattice is known to be a semi-metal, with a vanishing LDOS at half-
filling at $T/t\rightarrow 0$. Figure 3(c) shows a dip in the DOS at $T/t=0.5$
already for $U/t=0$, in line with entropy data. Similar to what is seen for
the square lattice, maxima develop as correlations increase, but the positions
are moved to $n=0.5$ and $n=1.5$. The triangular lattice LDOS shows a distinct
behavior, as can be seen in figure 3(b), with a broad peak for $U/t=0$ located
at $n\approx 1.6$. The effects of correlations are only relevant for $n\gtrsim
0.6$. For the larger values of $U/t$ considered, a small broad peak is present
at $n\simeq 0.75$ and a higher one at $n\simeq 1.6$, with a dip at half-
filling signaling the Mott state for large $U$.

To further investigate the transport properties, we now turn to the
longitudinal dc conductivity,

$\sigma_{dc}=\frac{\beta^{2}}{\pi}\Lambda_{xx}(\mathbf{q=0},\tau=\beta/2),$
(8)

in which

$\Lambda_{xx}(\mathbf{q},\tau)=\langle
j_{x}(\mathbf{q},\tau)j_{x}(-\mathbf{q},0)\rangle$ (9)

is the current-current correlation function with $j_{x}(\mathbf{q},\tau)$
being the Fourier transform of the unequal-time current operator

$j_{x}(\mathbf{i},\tau)=\mathrm{e}^{\tau\mathcal{H}}\left[it\sum_{\sigma}\left(c_{\mathbf{i}+\mathbf{x},\sigma}^{\dagger}c_{\mathbf{i},\sigma}^{\phantom{\dagger}}-c_{\mathbf{i},\sigma}^{\dagger}c_{\mathbf{i}+\mathbf{x},\sigma}^{\phantom{\dagger}}\right)\right]\mathrm{e}^{-\tau\mathcal{H}}.$
(10)

Here we also avoid analytical continuation, see, e.g., Refs. Trivedi  _et al._
, 1996; Denteneer  _et al._ , 1999; Mondaini  _et al._ , 2012.

Figure 4: (Color online) Longitudinal DC conductivity at $T/t=0.5$ as a
function of density for the (a) square, (b) triangular and (c) honeycomb
lattices. The dip at half-filling signifies the opening of the Mott gap, and
the presence (absence) of bipartiteness is reflected in the particle-hole
symmetric (asymmetric) behavior of the conductivity. The locations of the
peaks of $N(\omega=0)$ are very different from that of $\sigma_{DC}$ in the
strongly interacting regime. Note that the dip in LDOS in Fig. 3 is not
accompanied by a dip in $\sigma_{dc}$ in the honeycomb lattice.

Similar to what is seen for the LDOS, correlations reduce the conductivity,
with the non-interacting conductivity as an upper limit for all the geometries
studied as shown in Fig. 4. Once again, correlations are shown to be
irrelevant to transport properties for densities near the completely empty or
filled bands, while its effects increase as half-filling is approached, with
$\sigma_{dc}\to 0$ as $U/t$ increases. It is interesting to note that, for the
honeycomb lattice, the dip in the non-interacting DOS is not accompanied by a
dip in the conductivity [shown in Figs. 3(c) and 4(c), respectively]. For
large values of $U/t$, the maxima for the conductivity are not at the same
densities as the ones for the LDOS; for the square lattice, the conductivity
has maxima at $n=0.5$ and $n=1.5$. For the honeycomb lattice, the maxima are
at $n=0.6$ and $1.4$ and for the triangular lattice, the maxima are at
$n=0.45$ and $n=1.45$. For the triangular lattice, one can see that both the
$U/t=0$ peak and the higher intensity peak for the LDOS, which are above half-
filling, move to densities below half-filling for the conductivity.

##  V Seebeck coefficient

We now turn to the Seebeck coefficient, which is obtained from the entropy by
using the Kelvin formula Peterson and Shastry (2010b); Arsenault  _et al._
(2013),

$\displaystyle S_{\rm Kelvin}=\left.\frac{1}{e}{\frac{\partial s}{\partial
n}}\right|_{T,V}\leavevmode\nobreak\ .$ (11)

Within this approach, Fig. 5 displays the behavior of $S_{\rm Kelvin}$ in
units of $k_{B}/e^{2}$ as a function of the electronic density for the (a)
square, (b) triangular, and (c) honeycomb lattices.

At this point, it is worth recalling that the sign of the Seebeck coefficient
is directly related to the type of carrier, being negative for holes and
positive for electrons. As a consequence of particle-hole symmetry for the
square and honeycomb lattices, one has $S_{\rm Kelvin}(n)=-S_{\rm
Kelvin}(2-n)$, leading to $S_{\rm Kelvin}(n=1)=0$. For all geometries
examined, the effect of correlations is strongly dependent on density, being
negligible for $n\lesssim 0.3$ and $n\gtrsim 1.7$, as previously observed for
entropy, DOS and conductivity.

Figure 5: (Color online) Seebeck coefficient for different values of the
interaction strength at $T/t=0.5$ as a function of density for the (a) square,
(b) triangular, and (c) honeycomb lattices. For clarity, we have reduced the
set of $U/t$ values compared to previous plots. In the non-interacting limit,
the Seebeck coefficient changes sign at half-filling for bipartite lattices
(square and honeycomb), but at a finite doping for triangular lattice. With
strong interactions, there is an enhancement of the Seebeck coefficient near
half-filling, as well as an anomalous sign changes away from half-filling,
signaling a change in carrier type. Note that for the honeycomb lattice, there
is a sign change in Seebeck coefficient away from half filling even at
$U/t=0$, due to presence of Van Hove singularities from the Bloch bands.

As expected, the Seebeck coefficient for the non-interacting square lattice
presents only one sign change, at half-filling, as shown in Fig. 5 (a).
However, notice that in the presence of strong correlations ($U/t\gtrsim 6$)
there is an anomalous behavior, characterized by a change of sign for
densities away from half-filling, also displayed in Fig. 6 (a) for $n\approx
0.96$. In addition, there is a notable increase in the absolute value of
Seebeck coefficient, compared to the non-interacting case. For instance, at
$n=0.99$, $S_{\rm Kelvin}\approx 2.37\leavevmode\nobreak\ k_{B}/e^{2}$, at
$U/t=10$, while for $U/t=0$, $S_{\rm Kelvin}\approx-0.01\leavevmode\nobreak\
k_{B}/e^{2}$. This steep increase can be explained by noting that the Seebeck
coefficient, as defined in Eq. (3), is the ratio between the longitudinal
thermoelectric and electrical conductivities. As we approach the Mott
insulator at half-filling, electrons form local moments and electrical
transport is strongly reduced, as shown by both conductivity and LDOS in the
previous section. The fast-decreasing electrical conductivity in the vicinity
of half-filling must be accompanied by a non-vanishing thermoelectric current
for the peaks to form. We can understand this as follows: at half-filling and
strong correlations, each site is singly occupied and local moments are
completely formed, there is no electric or thermoelectric transport and the
Seebeck coefficient is zero. As we move slightly away from half-filling, there
is a background of local moments over which $p=1-n$ carriers lead to
thermoelectric and electric currents. This reduced number of carriers is in
line with the breaking of Luttinger count Osborne  _et al._ (2021); Gröber
_et al._ (2000); Sakai  _et al._ (2009) that has been established for the
Fermi Hubbard model and is in agreement with the change in carrier density in
Hall experiments for YBCO Badoux  _et al._ (2016).

Due to absence of particle-hole symmetry in the triangular lattice, the sign
change of $S_{\text{Kelvin}}$ in the non-interacting limit occurs away from
half-filling, at $n=1.42$, as shown in Fig. 5 (b), as opposed to the square
lattice, which has particle-hole symmetry, and exhibits the sign change
symmetrically around half filling. There is a range of densities, around
$0.5\leq n\leq 0.9$ where correlations lead to a small increase in modulus of
the Seebeck coefficient. In contrast to the square lattice, the peak in the
Seebeck coefficient for the triangular lattice at $n=0.99$ is only present for
very strong interactions. For $U/t=8$, $S_{\rm
Kelvin}\approx-0.18\leavevmode\nobreak\ k_{B}/e^{2}$, while for $U/t=0$, for
this electronic density $S_{\rm Kelvin}\approx-0.55\leavevmode\nobreak\
k_{B}/e^{2}$. Strong correlations, around $U/t=10$, are needed to change the
sign of the Seebeck coefficient at half-filling, as presented in Fig. 6 (b).
Interestingly, above half-filling and below the sign-change point for $U/t=0$,
the correlations are detrimental to the thermopower up to $U/t=8$, bringing
the Seebeck coefficient closer to zero.

Figure 6: (Color online) Seebeck coefficient as a function of $U/t$, at a
fixed $T/t=0.5$ for different densities in (a) square, (b) triangular and (c)
honeycomb lattices. For densities close to half-filling, the Seebeck
coefficient changes sign and enhances as $U/t$ is increased.

Fig. 5 (c) shows the Seebeck coefficient for the honeycomb lattice. For the
non-interacting system, $S_{\rm Kelvin}$ changes sign at $n=0.6$, 1.0, and
1.4, therefore the charge of the carriers is negative for $n<0.6$ and
$1.0<n<1.4$, while it is positive for $0.6<n<1.0$ and $n>1.4$. Interestingly,
these densities correspond to the peaks and dip positions for the
conductivity, as seen in Fig. 4 (c). We recall that such changes in carriers
come from the fact that the honeycomb lattice has two bands for the
noninteracting case, so one may expect changes from electron to hole
properties of the transport coefficients depending on the filling of each
band. However, correlations push the sign change to values closer to half-
filling, going from $n=0.6$ and $1.4$ at $U/t=0$ to $n=0.85$ and $1.15$ at
$U/t=8$. One thing to note is that although the behavior of the Seebeck
coefficient is very different between the square and the honeycomb lattice in
the noninteracting limit, adding interactions and transitioning to the Mott
insulating state seems to wash these differences away. This can be understood
by noting that in the noninteracting picture, the transport is determined by
the Bloch bands which are different for square and honeycomb lattices.
However, with strong interactions, Mott physics destroys the Bloch bands,
opens up a Mott gap, and forms upper and lower Hubbard bands. While details of
the single particle bands are erased, the information about the particle-hole
symmetry imprinted in the many body spectrum is still retained. Hence the
behavior of the Seebeck coefficient is qualitatively similar between the
square and honeycomb lattice but different from the triangular lattice.

Similar to the square lattice, there is a significant increase in the Seebeck
coefficient for the honeycomb lattice close to half-filling at $U/t=6$, 8 and
10, as shown for $n=0.96$ and $n=0.99$ in Fig. 6 (c). For $U/t=10$, for
example, we have that for $n=0.99$, $S_{\rm Kelvin}\approx
2.30\leavevmode\nobreak\ k_{B}/e^{2}$, while in the non-interacting case
($U/t=0$) for this density value, $S_{\rm Kelvin}\approx 0.03k_{B}/e^{2}$, an
enhancement of two orders of magnitude.

We also analyze the effects of temperature on the thermopower for the square
lattice at $U/t=10$. Figure 7 (a) displays $S_{\rm Kelvin}$ as a function of
$n$ for different $T/t$, while Fig. 7 (b) shows $S_{\rm Kelvin}$ as a function
of $T/t$ for different densities. Curves for different temperatures nearly
cross around $n=0.5$ and $n=1.5$ [Fig. 7(a)], leading to almost horizontal
lines for $n=0.53$ and $n=1.47$ in Fig. 7(b). For $n\lesssim 0.5$ and
$n\gtrsim 1.5$ reducing the temperature is detrimental to the thermopower, as
seen by a reduction of the modulus of the Seebeck coefficient for $n=0.2$. For
densities in the range $0.5\lesssim n\lesssim 0.9$ and $1.1\lesssim n\lesssim
1.5$, the behavior of $S_{\rm Kelvin}$ with temperature is non-monotonic, and
can change sign with $T/t$, as shown for $n=0.75$ in figure 7(b). Finally, the
anomalous behavior in the vicinity of half-filling has a marked dependence on
temperature, increasing the modulus of the Seebeck coefficient as $T/t$ is
reduced. This fast increase of the thermopower with decreasing temperature
close to half-filling has also been observed in the $t-J$ modelPeterson and
Shastry (2010a), Hubbard model on an FCC latticeArsenault  _et al._ (2013) and
$t-t^{\prime}-U$ Hubbard Model Wen O. Wang and Devereaux (2023).

Figure 7: (Color online) Seebeck coefficient for the square lattice at
$U/t=10$ as (a) a function of density for the square lattice at different
temperatures (b) a function of temperature for different densities. In panel
(a), the Seebeck coefficient shows anomalous behavior at low temperatures, and
approaches the free particle limit as $T/t$ is increased. In panel (b), at low
densities ($n=0.2$), the Seebeck coefficient has the expected sign, and
monotonically decreases with temperature. At a critical density $n\approx 0.5$
(also shown for $n\approx 1.5$ in the electron doped side), the Seebeck
coefficient is almost temperature independent. In the anomalous region
($0.75\lessapprox n\lessapprox 1.0$), peaks in the Seebeck coefficient start
to move towards the smallest temperature considered here, as one moves towards
the half-filling limit, where it becomes 0 for all temperatures.

##  VI Thermoelectric Power factor

Proceeding, we now discuss the effects of correlations to the thermoelectric
Power Factor (PF), defined as

$PF=S^{2}\sigma,$ (12)

where the dc conductivity ($\sigma$) and the Seebeck coefficient (S) were
obtained in Sections IV and V, respectively. Simultaneously increasing both
the modulus of the Seebeck coefficient and the conductivity maximizes the
Power Factor. Strategies to determine the Seebeck coefficient that leads to an
optimum thermoelectric Power Factor have been sought theoretically for systems
that can be described by the Boltzmann transport equation Pichanusakorn and
Bandaru (2009) and experimentally for CZTS thin films Ashfaq  _et al._ (2020)
and La-doped SrTiO3 Jalana and Stemmerb (2010) thin films.

Figure 8: (Color online) Thermoelectric Power factor as a function of density
for (a) square, (b) triangular and (c) honeycomb lattices. In the free
particle limit (almost empty and almost filled lattice), the Power factor
exhibits a peak for all values of $U/t$. Interaction causes additional peaks
to develop very close to the Mott insulating limit for large interaction
strengths. Additional features for intermediate doping also appear that are
strongly influenced by lattice geometry.

Fig. 8 shows the thermoelectric Power Factor as a function of density for
fixed $T/t=0.5$, and for the three analyzed geometries. A common feature to
all data is the presence of peaks close to the empty ($n\lesssim 0.1$) and
completely filled lattices ($n\gtrsim 1.9$). Starting with completely empty
(filled) lattices, as density increases (decreases) the conductivity
increases, and the Seebeck coefficient modulus decreases, leading to peaks
that are independent of interaction strength. As the conductivity goes to zero
for Mott insulators, $PF\to 0$ at half-filling for a geometry-dependent value
of $U/t$. For the square lattice, the hump in $PF$ for $U/t=0$ around
$n\approx 0.6$ (1.4) turns into a peak at $n\approx 0.5$ (1.5) with the
increase of correlations, the dominant contribution coming from the
conductivity.

The effect of correlations for the intermediate densities peak of the
honeycomb lattice is more subtle, as can be seen in figure 8(c). The peaks for
the non-interacting system at $n=0.85$ ($n=1.15$) decrease in intensity with
correlations for $U/t=2$ and 4, and then a shoulder develops at lower (higher)
densities for larger $U/t$. Close to half-filling once again correlations
drive the Seebeck coefficient peak up which in turn leads to the $PF$ peaks.

For the triangular lattice one can see that the non-interacting system has a
peak at $n=0.95$. Correlations move the peak to lower densities and increase
its intensity, as clearly seen in figure 8(b), where the peak for $U/t=10$ is
at $n\simeq 0.7$. As for the other geometries, a peak develops in the vicinity
of the Mott insulating state, here only seen for the larger values of $U/t$
studied.

Comparing the overall effects of correlations in the different geometries
analyzed, we see that there are clear correlation-induced peaks near half-
filling in all cases. For the square and honeycomb lattices, the non-
interacting thermoelectric power factor is zero at half-filling and is driven
to $PF\approx 0.1\ k_{B}^{2}/e^{2}h$. Around quarter and three-quarter
fillings as well, correlations also play a relevant role in increasing the
power factor for the square and honeycomb lattices. Correlations are very
effective in increasing the Power Factor in a triangular lattice, where
$PF\simeq 0.27\ k_{B}^{2}/e^{2}h$, the largest value obtained, for $U/t=10$ at
$n\sim 0.7$. For this geometry, we observe that density can be used to tune
the thermoelectric Power Factor.

##  VII Conclusions

We have studied the thermoelectric properties of strongly interacting two
dimensional systems with different geometries. Our results clearly show an
anomaly in the Seebeck coefficient in the vicinity of half-filling,
characterized by an enhanced response depending on both geometry and
interaction strength. The anomaly is characterized by a change in the sign of
the carriers which is accompanied by an interaction-induced increase. The
anomaly is also intensified with the reduction of temperature.

The thermoelectric Power Factor displays a competition between the Seebeck
coefficient and the conductivity. The anomaly in the Seebeck coefficient is
reflected in the $PF$, with correlation-driven peaks immediately below and
above half-filling at geometry-dependent values of $U/t$. The decreasing
conductivity near half-filling is the limiting factor in the intensity of $PF$
in this region of densities. Away from half-filling, at intermediate densities
(around $n=0.4-0.6$ and $n=1.4-1.6$) the peaks in $PF$ have a strong
contribution from the conductivity with positions strongly dependent on
geometry. For this range of densities, peak position and intensity can be
tuned by correlations. Although the Seebeck coefficient is smaller for the
triangular lattice, the Power Factor for this geometry shows the higher peak
values and the stronger tunability with density and correlations, making it a
strong candidate for enhanced thermoelectric properties.

## ACKNOWLEDGMENTS

Financial support from Fundação Carlos Chagas Filho de Amparo à Pesquisa do
Estado do Rio de Janeiro, grant numbers E-26/204.308/2021 (W.C.F.S.),
E-26/200.258/2023 - SEI-260003/000623/2023 (N.C.C.), E-26/200.959/2022 (T.P.),
and E-26/210.100/2023 (T. P.); CNPq grant numbers 313065/2021-7 (N.C.C.),
403130/2021-2 (T.P.), and 308335/2019-8 (T.P.) is gratefully acknowledged. We
also acknowledge support from INCT-IQ. We acknowledge support from NSF
Materials Research Science and Engineering Center (MRSEC) Grant No.
DMR-2011876 and NSF-DMR 2138905 (SR,AS,NT). We also acknowledge computational
resources from the Unity cluster at the Ohio State University.

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