# Fermionic and bosonic partition functions at imaginary chemical potential as
Bloch functions

Evangelos G. Filothodoros1

###### Abstract

We point out that the phase transitions of the $d+1$ Gross-Neveu and
$CP^{N-1}$ models at finite temperature and imaginary chemical potential can
be mapped to transformations of Hubbard-like regular hexagonal to square
lattice with the intermediate steps to be specific surfaces (irregular
hexagonal kind) with an ordered construction based on the even indexed Bloch-
Wigner-Ramakrishnan polylogarithm function. The zeros and extrema of the
Clausen $Cl_{d}(\theta)$ function play an important role to the analysis since
they allow us not only to study the fermionic and bosonic theories and their
phase transitions but also the possibility to explore the existence of
conductors arising from the correspondence between the partition functions of
the two models and the Bloch and Wannier functions that play a crucial role in
the tight-binding approximation in solid state physics.

Keywords: Fermion-boson map; Hubbard lattice; conductor; Bloch function;
Wannier function

## 1\. Introduction

In the realm of condensed matter physics the Hubbard model stands as a
cornerstone in the study of electronic correlations within quantum materials
[1, 2]. It describes electrons in a solid that interact with each other
through short-range repulsive interactions. Specifically, the Hubbard model
introduces a contact interaction between particles of opposite spin on each
site of a lattice. When applied to electron systems, these interactions are
typically expected to be repulsive, arising from the screened Coulomb
interaction. In this work, we believe that if we examine in detail the setup
of the higher dimensional thermal windows for fermions and bosons at imaginary
chemical potential, we may connect them to the transformation of Hubbard-like
lattices to square lattice and the appearance of insulator’s identities. An
Appendix contains some technical details and useful formulae for the Bloch-
Wigner function.

## 2\. The Hubbard model and a brief review of the charges of the $U(N)$
fermionic Gross-Neveu and the bosonic CPN-1 models at imaginary chemical
potential in odd dimensions

In the absence of a chemical potential, the fermionic Gross-Neveu and bosonic
CPN-1 models exhibit distinct symmetry-breaking patterns at finite temperature
$T$. The Gross-Neveu model displays a parity-broken phase at low temperatures,
which vanishes beyond a critical temperature. Conversely, the CPN-1 model
follows the usual continuous symmetry-breaking pattern at zero temperature,
but this broken phase ceases to exist for $T>0$. Instead, when the coupling
reaches its critical value at $T=0$, a finite-temperature scaling regime
emerges, characterized by a non-zero thermal mass for the scalars. From a
previous work, detailed in [3], reveals that introducing an imaginary chemical
potential alters this situation. The phase structures of both models can be
mapped onto each other. Additionally, we have explored the significance of the
celebrated Bloch-Wigner function on the unit circle, leveraging it in our
calculations of phase transition-thermal windows [4]. In a pursuit to
illuminate bosonization physics, we have delved into the fermion-boson map in
higher dimensions, comparing our continuum models’ phase transformations with
those of specific Hubbard-like lattices. By utilizing the zeros and
maximizations of Bloch-Wigner functions introduced by Zagier [3, 4, 5, 6], we
have uncovered an alternative perspective that draws parallels to the
emergence of conductors. We consider that the charge arising as the eigenvalue
of the charge $N$-normalized fermion number density operator can separate the
insulating from the conducting state of a Hubbard-like model of fermions [7].
These investigations extend across arbitrary odd dimensions.

### 2.1. The Hubbard model at imaginary chemical potential

The Hubbard model is an approximate model that offers an insight view into how
interacted electrons give as a result insulating or conducting phenomena in a
crystal. Let’s see the Hubbard Hamiltonian.

$H=-t\sum_{[i,j]_{s}}c^{\dagger}_{is}c_{js}+U\sum_{i}n_{iu}n_{id}-i\mu\sum_{i}(n_{iu}+n_{id})$
(1)

where the notations $iu$ and $id$ imply spin up and spin down respectively,
$n$ is the number operator and $c^{\dagger},c$ are the creation and
destruction operators that create an electron in a state $\Phi_{R}$ and
destroy an electron in a state $\Phi_{R}$, where $\Phi_{R}$ is the Wannier
function. Wannier functions are orthogonal functions, used in solid state
physics, that refer to localized molecular orbitals with periodic boundary
conditions. Since they are orthogonal

$\int_{crystal}\Phi(x)_{1}^{*}\Phi(x)_{2}dx=0$ (2)

or

$e^{\int_{crystal}\Phi(x)_{1}^{*}\Phi_{2}(x)dx}=1$ (3)

The first term is the kinetic term which shows how electrons move from one
site to a neighbour site of the lattice. The second term adds an energy $U$ if
the site already has two electrons. At last we have the imaginary chemical
potential term. So, what does the chemical potential actually do? Firstly, we
use the imaginary chemical potential to avoid the sign problem that arises in
Monte Carlo simulations (configurations in the partition function that may be
negative or complex). Mainly, the imaginary chemical potential places the
fermions and bosons in such positions, creating a complex structure of
hexagonal lattices, sometimes creating an accumulation of charge and sometimes
its nullification [8]. So we have an image of a conductor for the first case
and an insulator for the second case.

### 2.2. The charge $Q_{df}$ of fermions

The GN model in $d$ Euclidean dimensions is described by the action [4]

$\displaystyle
S_{1f}=-\int_{0}^{\frac{1}{T}}\\!\\!\\!dx^{0}\int\\!\\!d^{d-1}\bar{x}\left[\bar{\psi}^{a}(/\\!\\!\\!\partial-i\gamma_{0}\alpha)\psi^{a}+\frac{G_{d}}{2({\rm
Tr}\mathbb{I}_{d-1})N}\left(\bar{\psi}^{a}\psi^{a}\right)^{2}+i\alpha
NQ_{df}\right]\,,$ (4)

with $Q_{df}$ the $N$-normalized $d$-dimensional fermionic number density and
$a=1,2,..N$. For odd $d$ we take the dimension of the gamma matrices to be
${\rm Tr}{\mathbb{I}}_{d-1}=2^{\frac{d-1}{2}}$. We see that the model also has
$3$ terms. The first one is the kinetic term, the second is the interaction
term that contributes to the energy of the fermions and the the third one is
the imaginary chemical potential term with $Q$ the number eigenvalue of the
fermion number operator. The canonical partition function for an auxiliary
scalar field $\sigma$ is expressed as follows:

$\displaystyle Z_{1f}(\frac{1}{T},Q_{df})$ $\displaystyle=\int({\cal
D}\alpha)({\cal D}\sigma)e^{-NS_{1f,eff}}\,,$ (5) $\displaystyle S_{1f,eff}$
$\displaystyle=iQ_{df}\int_{0}^{\frac{1}{T}}\\!\\!\\!dx^{0}\\!\\!\int\\!\\!d^{d-1}\bar{x}\,\alpha-\frac{{\rm
Tr}\mathbb{I}_{d}}{2G_{d}}\int_{0}^{\frac{1}{T}}\\!\\!\\!dx^{0}\\!\\!\int
d^{d-1}\bar{x}\,\sigma^{2}+\rm
Tr\ln\left(/\\!\\!\\!\partial-i\gamma_{0}\alpha+\sigma\right)_{\frac{1}{T}}\,.$
(6)

where $\sigma$ plays the role of the mass of the fermion condensate.

The $d$-dimensional charge gap equation becomes:

$iQ_{df}=\lim_{\epsilon\rightarrow 0}{\rm
Tr}\mathbb{I}_{d-1}T\int^{\Lambda}\\!\\!\frac{d^{d-1}\bar{p}}{(2\pi)^{d-1}}\sum_{n=-\infty}^{\infty}\frac{e^{i\omega_{n}\epsilon}(\omega_{n}-\alpha_{*})}{\bar{p}^{2}+(\omega_{n}-\alpha_{*})^{2}+\sigma_{*}^{2}},.$
(7)

We will discuss below in some detail the cases $d=5$ and $d=7$ in order to
exhibit some of the general features of the higher dimensional models.
Starting with $d=5$ and using the results of [3, 4], we have the form of the
charge gap equation as

$\frac{\pi^{2}Q_{5f}}{T^{4}}-3i\left[D_{4}(-z_{*})+\frac{1}{6}\ln^{2}\\!|z_{*}|D_{2}(-z_{*})\right]=0$
(8)

where $z_{*}=e^{-\frac{\sigma_{*}}{T}-\frac{i\alpha_{*}}{T}}$. When an
imaginary chemical potential is present the situation becomes more intriguing.
We encounter nontrivial zeros of $D_{3}(-z_{*})$ on the unit circle. A brief
exploration in Mathematica revealed two zeroes for $D_{3}(-z)$ on the unit
circle. Remarkably their positions were approximated to high accuracy by
rational multiples of $\pi$ as

$\displaystyle
D_{3}(-e^{\frac{-i\alpha_{*}}{T}})=Cl_{3}(\frac{\alpha_{*}}{T}\pm\pi)=0\Rightarrow$
$\displaystyle\frac{\alpha_{*}}{T}\approx\frac{7\pi}{13}\,{\rm
or}\,\frac{\alpha_{*}}{T}=\frac{19\pi}{13}\,\,\,({\rm mod}\,2\pi)\,.$ (9)

When we use the periodic properties of Clausen functions, we encounter the
following relevant results:

$Cl_{3}\left(\frac{6\pi}{13}\right)=Cl_{3}\left(\frac{20\pi}{13}\right)=0.000362159\,.$
(10)

and

$Q_{5f,extr}=\pm i\frac{3T^{4}}{\pi^{2}}Cl_{4}\left(\frac{6\pi}{13}\right)$
(11)

since $Cl_{4}(\pm 6\pi/13)\approx\pm 0.995777$ are the maximum (minimum)
values of $D_{4}(-z)$ on the unit circle. We saw that this pattern generalizes
to all dimensions. Finally, when $\frac{\alpha_{*}}{T}=\pi$ the gap equation
coincides - apart the overall $\sigma_{*}$ factor - with the corresponding one
of the $CP^{N-1}$ that will be given below. The charge is $Q_{5f}=0$ and the
system has been bosonized (insulator mode) [9].

The seven-dimensional case shows how our results are generalized to higher
dimensions. The gap equation is

$\frac{4\pi^{3}}{3T^{6}}Q_{7f}+15i\left[D_{6}(-z_{*})+\frac{1}{10}\ln^{2}\\!|z_{*}|D_{4}(-z_{*})+\frac{1}{120}\ln^{4}\\!|z_{*}|D_{2}(-z_{*})\right]=0$
(12)

Moving on the non zero chemical potential we can look for zeros of the
critical gap equation on the unit circle. Again, their positions are
remarkably well approximated, better than in $d=5$, by rational multiples of
$\pi$ as

$\displaystyle
D_{5}(e^{\frac{-i\alpha_{*}}{T}})=Cl_{5}(\frac{\alpha_{*}}{T}\pm\pi)=0\Rightarrow$
$\displaystyle\frac{\alpha_{*}}{T}\approx\frac{26\pi}{51}\,{\rm
or}\,\frac{76\pi}{51}\,\,\,({\rm mod}\,2\pi)\,.$ (13)

The relevant result is

$Cl_{5}\left(\frac{25\pi}{51}\right)=Cl_{5}\left(\frac{77\pi}{51}\right)=0.000129657\,.$
(14)

and we found at these points that

$Q_{7f,extr}=\mp i\frac{45T^{6}}{4\pi^{3}}Cl_{6}\left(\frac{25\pi}{51}\right)$
(15)

since $Cl_{6}(\pm 25\pi/51)\approx\pm 0.999151$ are the maximum (minimum)
values of $D_{6}(-z)$ on the unit circle.

### 2.3. The charge $Q_{db}$ of bosons

The action of the bosonic theory for general $d$ is from [4]

$\displaystyle
S_{2b}=\int_{0}^{\frac{1}{T}}\\!\\!\\!dx^{0}\\!\\!\int\\!\\!d^{d-1}x\left[|(\partial_{0}-i\alpha)\phi^{a}|^{2}+|\partial_{i}\phi|^{2}+i\lambda(\bar{\phi}^{a}\phi^{a}-\frac{N}{g_{d}})+iNQ_{db}\alpha\right]\,,\,\,\,a=1,2,..,N\,,$
(16)

where the auxiliary scalar field $\lambda$ enforces the constraint
$|\phi|^{2}=N/g_{d}$ and $Q_{db}$ is the eigenvalue density of the
$N$-normalized $U(1)$ charge density operator
$\hat{Q}_{db}=-ig\bar{\phi}^{a}\overset{\leftrightarrow}{\partial}_{0}\phi^{a}/N$.
In the given model, we have both a global $SU(N)$ symmetry and a global $U(1)$
symmetry. The latter can be easily gauged by introducing a non-propagating
abelian gauge field. Upon integrating out the scalar fields, we arrive at the
canonical partition function as

$\displaystyle Z_{2b}(\frac{1}{T},Q_{db})$ $\displaystyle=\int({\cal
D}\alpha)({\cal D}\lambda)e^{-NS_{2b,eff}}\,,$ (17) $\displaystyle S_{2b,eff}$
$\displaystyle=iQ_{db}\int_{0}^{\frac{1}{T}}\\!\\!\\!dx^{0}\\!\\!\int\\!\\!d^{d-1}\bar{x}\,\alpha+i\frac{1}{g_{d}}\int_{0}^{\frac{1}{T}}\\!\\!\\!dx^{0}\\!\\!\int\\!\\!d^{d-1}\bar{x}\,\lambda-\rm
Tr\ln\left(-(\partial_{0}-i\alpha)^{2}-\partial^{2}+i\lambda\right)_{\frac{1}{T}}\,.$
(18)

The $d$-dimensional charge gap equation is

$iQ_{db}=-\lim_{\epsilon\rightarrow
0}2T\int\frac{d^{d-1}\bar{p}}{(2\pi)^{d-1}}\sum_{-\infty}^{\infty}\frac{e^{i\omega_{n}\epsilon}(\omega_{n}-\alpha_{*})}{\bar{p}^{2}+(\omega_{n}-\alpha_{*})^{2}+m_{*}^{2}}$
(19)

where the bosonic frequencies are $\omega_{n}=2\pi nT$ and we have set
$i\lambda_{*}\equiv m_{*}^{2}$ in order to facilitate the comparison with the
fermionic gap equation (7) as $\sigma_{*}$ and $m_{*}$ have the same
dimensions.

The edges of the bosonic thermal windows represents a switch in the statistics
where the bosonic becomes fermionic. These are given by roots on the unit
circle of $D_{d-2}(z_{*})$ and we find for example for $d=5$ and $d=7$

$D_{3}(e^{-i\frac{\alpha_{*}}{T}})=Cl_{3}(\frac{\alpha_{*}}{T})=0\,\,\,\Rightarrow\,\,\,\frac{\alpha_{*}}{T}=\frac{6\pi}{13}\,{\rm
or}\,\frac{\alpha_{*}}{T}=\frac{20\pi}{13}\,,\,\,\,({\rm mod}\,2\pi)\,.$ (20)

At the edges of the bosonic thermal windows, the charge becomes extremized.
Specifically, it is given by the following expression:

$Q_{5b,extr}=\mp\frac{3iT^{4}}{4\pi^{2}}Cl_{4}\left(\frac{6\pi}{13}\right)=\frac{Q_{5f,extr}}{4}$
(21)

for $\frac{\alpha_{*}}{T}=6\pi/3$ and $20\pi/13$ respectively or

$Q_{7b,extr}=\mp\frac{45iT^{6}}{32\pi^{3}}Cl_{6}\left(\frac{25\pi}{51}\right)=\frac{Q_{7f,extr}}{8}$
(22)

for $\frac{\alpha_{*}}{T}=25\pi/51$ and $77\pi/51$.

So if one wants to use a fermionic-bosonic model with equal charges then he
has to consider for every boson with charge $Q_{b}$ a fermion with charge
$Q_{f}=2^{\frac{d-1}{2}}Q_{b}$, or for every fermion a number of
$\frac{d-1}{2}$ bosons, where $d=3,5,7,...$.

## 3\. Large $N$ fermions and bosons on the lattice

Consider a system in three dimensions and at finite temperature $T$ with a
global $U(1)$ charge operator $\hat{Q}$. Its canonical partition function can
be formally calculated as the thermal average over states with fixed $\hat{Q}$
as

$Z_{c}(\frac{1}{T},B)=Tr\left[\delta(\hat{Q}-B)e^{-\frac{\hat{H}}{T}}\right]$
(23)

If the eigenvalues $B$ of $\hat{Q}$ are integers, namely if the system
contains elementary excitations, an explicit representation of (23) can be
written as

$Z_{c}(\frac{1}{T},B)=\int_{0}^{2\pi}\\!\frac{d\theta}{2\pi}\,e^{i\theta
B}\,\rm
Tr\left[e^{\frac{\hat{H}}{T}-i\theta\hat{Q}}\right]=\int_{0}^{2\pi}\\!\frac{d\theta}{2\pi}\,e^{i\theta
B}\,Z_{gc}(\frac{1}{T},i\mu=i\theta T),$ (24)

where $Z_{gc}(T,i\mu)$ is the grand canonical partition function with
imaginary chemical potential $i\mu$ [10, 11, 12, 13].

In the simple systems we are interested in one expects that

$Z_{gc}(\frac{1}{T},i(\mu+2\pi
kT))=Z_{gc}(\frac{1}{T},i\mu)\,,\,\,\,k\in\mathbb{Z}$ (25)

One then notices with Bloch’s theorem as follows [14]. In quantum system,
taken here to be $1d$ for clarity, in a periodic potential with period $a$ the
energy eigenstates are the Bloch waves

$\psi_{k}(x)=e^{ikx}u(x)\,,\,\,\,u(x+a)=u(x)$ (26)

where $k$ is the lattice momentum vector [15]. The transition amplitude
between two Bloch waves with different lattice momenta is

$\langle\psi_{k_{1}}|\psi_{k_{2}}\rangle=\int_{0}^{a}dxe^{i(k_{2}-k_{1})x}|u(x)|^{2}$
(27)

In spherical coordinates we may write

$\langle\psi_{k_{1}}|\psi_{k_{2}}\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta
e^{i(k_{2}-k_{1})\theta}|u(\theta)|^{2}$ (28)

Notice that $B$ may be thought of as a charge transfer ’momentum’. According
to the theoretical background of the Hubbard model, this charge transfer
’momentum’ is a potential term that allows the fermions to travel on lattice
sites. This is like a term that describes the transition between conducting
and insulating systems. When $B$ is $0$ the Hubbard-like lattice is an
insulator and when $B$ is max the lattice is a conductor.

A suitable expression of a $1d$ Bloch-wave is of the form:

$\psi_{k}(\theta)=e^{ik\theta}u(\theta)$ (29)

where $u$ function has the periodicity of the lattice $2\pi$ since
$u(\theta)=u(\theta+2\pi)$, like $Z_{gc}(\beta,i\mu=i\theta T)$ has the
periodicity of the chemical potential. The fermionic states in a Hubbard model
is described by the Wannier functions that defined by

$\Phi_{X}(x)=\frac{1}{\sqrt{N}}\sum_{k}e^{-ikX}\psi_{k}(x)$ (30)

where $X$ is a Bravais lattice vector in cartesian coordinates for a $1d$
quantum system [16, 17, 18, 19]. According to the replacement rule the sum can
be written as an integral, since $N$ is very large, like

$\sum_{k}=\frac{N}{V}\int_{BZ}dk$ (31)

where $V$ is the volume of the $BZ$ Brillouin zone.

Interestingly the Bloch functions can be written in terms of Wannier functions
like

$\psi_{k}(x)=\frac{1}{\sqrt{N}}\sum_{X}e^{ikX}\Phi_{X}(x)$ (32)

So, in spherical coordinates the $\psi$ Bloch function may be written as

$\psi_{k}(\theta)=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta
e^{ik\Theta}\Phi_{\Theta}(\theta)$ (33)

where $\Theta$ is a Bravais lattice vector in spherical coordinates. So the
picture is that the canonical partition function of the Gross Neveu model is
like a Bloch state that varies continuously with $k$ (or $B$) and the grand
canonical partition function is like a Wannier state at imaginary chemical
potential $i\Theta$. The grand canonical partition function is a localized
system with periodic boundary conditions arising from the imaginary chemical
potential. This periodicity is an interesting quasi-periodicity since we have
the appearance of the Bloch-Wigner-Ramakrishnan polylogarithm function
$D_{m}(z)$. It is interesting that this function exhibits a very interesting
behaviour regarding periodicity. For $m=even$, the function is not periodic
since its sum involves alternating terms, and the series does not repeat in a
regular pattern. For $m=odd$, the function has a more intricate structure.
While the individual terms in the series are not periodic, the presence of the
logarithmic term $\log^{m}|x|$ introduces a quasi-periodicity. Specifically
the function satisfies the functional

$D_{m}(1/x)=(-1)^{m-1}D_{m}(x)$ (34)

Then we have two parts in partition function $Z_{c}$. The charge part with
$even$ index of $D_{m}(z)$ function and the $\rm Tr\log$ and the grand
canonical part with $odd$ index of $D_{m}(z)$. The first part has no
periodicity but the second part has a quasi-periodicity of the hypothetical
quasi-crystal of the band theory. When we take the points of $D_{m}(z)$ on the
unit circle we have the Clausen function which is periodic with period $2\pi$
and the matching between Wannier function and the grand canonical partition
function is obvious. The grand canonical parts of $Z_{1f}$ and $Z_{2b}$ are an
orthogonal set of Wannier functions where

$\int_{0}^{2\pi}Z_{1f_{gc}}Z_{2b_{gc}}=0$ (35)

So the transition amplitude on the unit circle between the two Bloch states of
the Gross-Neveu and $CP^{N-1}$ models is from [4]:

$Z^{d}_{fb}(\frac{\alpha_{*}}{T})\Bigl{|}_{\sigma_{*}=0}\equiv\left[Z_{1f}^{(d)}(\frac{\alpha_{*}}{T}+\pi)Z_{2b}^{(d)}(\frac{\alpha_{*}}{T})\right]\Bigl{|}_{\sigma_{*}=0}=e^{4\pi
N\frac{V_{d-1}}{S_{d}T^{1-d}}Cl_{d-1}(\frac{\alpha_{*}}{T})}$ (36)

The advantage here is that the divergent terms from the bosonic partition
function’s exponential come with the opposite sign compared to the fermionic
partition function so their contribution is zero. At zero temperature
$T\rightarrow 0$ the ratio $V_{d-1}/S_{d}T^{1-d}\rightarrow 1$ (i.e. we can
think of $V_{d-1}$ as the surface of a very large sphere). Then the Clausen’s
functions with even index at $d\rightarrow\infty$ limits are
$\sin(\frac{\alpha_{*}}{T})$. Therefore we write

$\lim_{d\rightarrow\infty}Z^{d}_{fb}(\frac{\alpha_{*}}{T})\Bigl{|}_{\sigma_{*}=0}=e^{4\pi
N\sin(\frac{\alpha_{*}}{T})}\,.$ (37)

Suppose that we have a large number of $N$ fermions (blue) and $N$ bosons
(red) on Euclidean space, where the charge of every fermion is
$2^{\frac{d-1}{2}}$ times the bosonic charge. Here $d$ plays the role of the
order of the family of lattices covering the Euclidean space. We also suppose
that the large Euclidean surface is covered by regular hexagons with the help
of the imaginary chemical potential mentioned before and let us assume that
the fermions and bosons are on the vertices of these lattices.

Figure 1: Fermions and bosons on Euclidean surface

Let’s see for example the case of the regular hexagon and its "perturbed"
relatives of the hexagons conjecture. Based on the hypothesis we had
previously, we can, in line with the generalization of the thermodynamic study
of Gross-Neveu and CPN-1 theories, give a more general hypothesis on the
observation that the inner angles of a regular hexagon are exactly $2\pi/3$
and there is a lattice point at $2\pi/3$ on the unit circle so the
maximization of Clausen function which is equal to the imaginary part of
$Li_{d}(-z)$, where $d=2,4,6,...$, on the unit circle for specific angles
$7\pi/13$,$26\pi/51$,$103\pi/205$ and so on. These values come from an
analytic formula for the approximate positions of the zeros of all
$D_{2n-1}(z)$, $n=1,2,..$ functions on the unit circle. We obtain:

$D_{2n-1}(e^{-i\frac{\alpha_{*}}{T}})\equiv
Cl_{2n-1}(\frac{\alpha_{*}}{T})=0\Leftrightarrow\frac{\alpha_{*}}{T}\approx\theta_{n},2\pi-\theta_{n}\,({\rm
mod}\,2\pi)\ $ (38)

where

$\theta_{n}=\frac{\pi}{2}\left(1-\frac{5}{4^{n+1}-(-1)^{n+1}}\right)\ $ (39)

for $n=1,2,3,..$. There is an interesting approximation of these zeros in
[20].

The particles are the lattice points on Euclidean space of irregular hexagons.
At a large $d$ limit the conjecture ends up to the square lattice
construction. Somehow as dimension increases the 6th and 5th sides disappear
and the particles are getting closer to make some pairs. We may say that these
pairs are similar to Cooper pairs. The conjecture turns to be as follows:

Table 1. Positions of the lattice points on the unit circle for hexagonal
model
---
Order $d$ | Lattice point $1$ | Lattice point $2$ | Lattice point $3$ | Lattice point $4$ | Lattice point $5$ | Lattice point $6$
3 | $\pi/3$ | $2\pi/3$ | $\pi$ | $4\pi/3$ | $5\pi/3$ | $2\pi$
5 | $6\pi/13$ | $7\pi/13$ | $\pi$ | $19\pi/26$ | $20\pi/26$ | $2\pi$
7 | $25\pi/51$ | $26\pi/51$ | $\pi$ | $76\pi/51$ | $77\pi/51$ | $2\pi$
9 | $102\pi/205$ | $103\pi/205$ | $\pi$ | $307\pi/205$ | $308\pi/205$ | $2\pi$
… | … | … | … | … | … | …
$\infty$ | $\pi/2$ | $\pi/2$ | $\pi$ | $3\pi/2$ | $3\pi/2$ | $2\pi$

One may see the transformation of a hexagonal Hubbard lattice to a square one
and compare with the phase transitions of the Gross-Neveu model and $CP^{N-1}$
model at imaginary chemical potential. The picture is like the one that
follows:

”Studying the correspondence of thermal windows with the transformation of a
hexagonal Hubbard lattice to square lattice we have fermions at high
temperature (edges of a fermionic thermal window) and bosons at the beginning
of their fermionization area (edges of a bosonic thermal window). The thermal
windows close for both of them and we have fermions with the same charge that
getting closer to each other, with a strong interaction (clockwise for
fermions to $\pi/2$ and anti-clockwise for fermions to $3\pi/2$, anti-
clockwise for bosons to $\pi/2$ and clockwise for bosons to $3\pi/2$). The
picture is like the Hubbard model of electrons in a periodic potential which
predicts the theory of conductivity. Fermions from hexagons (order $3$
lattice) are on lattice points $2\pi/3$, $4\pi/3$ and $0$ and the fermionized
bosons are on $\pi/3$, $5\pi/3$ and $\pi$ (the bosons are in the beginning of
their "fermionization" area for the $CP^{N-1}$ model) and moving to make a
pair with fermions. We see that the chemical potential is moving them closer
and closer until they coincide on angle $\pi/2$ on the unit circle where
$D_{even}$ function has its maximum value for the Gross-Neveu model
($D_{\infty}(\pi/2)=Cl_{\infty}(\pi/2)=sin(\pi/2)=1$). At order $\infty$ of
the model we have a bound of two fermions with total charge $2Q_{f}$, at
$\pi/2$ and $3\pi/2$. We must not forget that while in the Hubbard model the
fermions have the same charge, in the two models of fermions and bosons the
fermions and bosons have charges connected by the relationship
$Q_{f}=2^{\frac{d-1}{2}}Q_{b}$. This picture could correspond to the creation
of Cooper pairs of conductivity.”

### 3.1. Generalized Bragg Law

Imagine a beam of particles with a wavelength similar to the spacing between
atoms in a crystal. When this beam encounters a lattice plane within the
crystal, it gets scattered. The incident and reflected waves from the lattice
plane remain in phase if the difference in their path lengths which determines
the constructive interference, is given by the formula:
$n\lambda=2d\sin\theta$, where $\theta$ is the scattering angle and $d$ is the
interplanar distance.

Suppose now that a particle travels in a crystal lattice with lattice spacing
$d$, with momentum $q_{i}$ and collides with a lattice point with an exchange
of momentum $q=q_{i}-q_{f}$, where $q_{f}$ is the the momentum of the
reflected particle. Since a wave has momentum $q=\hbar k$ ($k$ is the wave
number) in reciprocal space momentum transfer is given by $Q=k_{i}-k_{f}$ ($k$
is a vectorial quantity). Then we are able to calculate a reciprocal lattice
vector $G=Q$ with the relation to the lattice spacing as $G=2\pi/d$.
Consequently the total transfer momentum from N particles of a beam scattered
by a crystal lattice is:

$Q_{total}=4\pi Nsin\theta$ (40)

For a perturbed version of Bragg Law we may define a new function $\Delta$
where

$\Delta(\theta)\rightarrow\left(sin\theta+\sum_{k=2}^{\infty}\frac{sink\theta}{k^{m}}\right)$
(41)

for arbitrary even $m=2,4,...,\infty$. This $\Delta$ function is Clausen
function $Cl_{m}(\theta)$. So the Bragg Law of diffraction of a lattice with
lattice space $d$ may have a generalized version:

$n\lambda=2d\Delta(\theta)\rightarrow
n\lambda=2d\left(sin\theta+\sum_{k=2}^{\infty}\frac{sink\theta}{k^{m}}\right).$
(42)

The ratio $sin\theta/\Delta(\theta)$ for $m\rightarrow\infty$ is $1$, follows
from the observation that this limit is actually well defined

$\lim_{m\rightarrow\infty}Cl_{m}(\theta)=\sin\theta\,$ (43)

where $m=2n,n=1,2,3,4...$..

Figure 2: The ratio $sin\theta/\Delta(\theta)$ for $\theta\in[-2\pi,2\pi]$

The sum in the parenthesis is the Clausen function so:

$n\lambda=2dCl_{m}(\theta)$ (44)

The generalized transferred momentum is

$Q_{gen}=G$ (45)

where $G$ is the reciprocal lattice vector with $G=2\pi/d$. For $n=1$,
$\lambda=1$ and $N$ beam particles we have:

$Q_{gen}=4\pi NCl_{m}(\theta)$ (46)

So, (36) may be equivalent to the calculation of the overlap between two Bloch
wavefunctions that differ by lattice momentum $Q_{gen}$ when one is the
incident and the other the reflected wave from a lattice point.

## 4\. Generalized thermal windows and lattice points

Let’s see in detail Figure 3. Interestingly there is an equivalent picture of
the thermal windows in the Gross-Neveu and $CP^{N-1}$ models at finite T and
imaginary chemical potential. In the Gross-Neveu model and its change in the
statistics we have temperature (the chemical potential) that "creates" the
thermal windows, where inside them we have a chiral symmetry breaking in order
to make some fermion condensates. Outside these windows we have Fermi
statistics. On the other hand the imaginary chemical potential "creates" the
fermionization thermal windows for bosons where it is always possible to tune
the coupling to a special value, such that the thermal mass vanishes. Outside
these windows we have Bose statistics. This picture is reminiscent to the
behaviour of the Gross-Neveu model where for a nonzero temperature it was
possible to tune the coupling in order to restore parity. With the conjecture
of the thermal windows we have the equivalent picture of the region inside the
hexagon. Later, we will observe that the boundaries of the thermal windows
align with the positions of lattice points on the hexagonal lattice. At these
points, $D_{2}(-z*)$ has its maximum value.

From [4] the generalized thermal windows are:

Table 2. Generalized Thermal Windows for the $GN$ model
---
Dimensions | Closing T | Opening T
3 | $\frac{3\alpha_{*}}{4\pi}$ | $\frac{3\alpha_{*}}{2\pi}$
5 | $\frac{13\alpha_{*}}{19\pi}$ | $\frac{13\alpha_{*}}{7\pi}$
7 | $\frac{51\alpha_{*}}{76\pi}$ | $\frac{51\alpha_{*}}{26\pi}$
9 | $\frac{205\alpha_{*}}{307\pi}$ | $\frac{205\alpha_{*}}{103\pi}$
11 | $\frac{819\alpha_{*}}{1228\pi}$ | $\frac{819\alpha_{*}}{410\pi}$

Let’s focus on the case of the $3d$ theory thermal window. The borders for
$\alpha_{*}$ are $2\pi T/3$ and $4\pi T/3$. These are the points where the
$D_{2}(-z*)$ takes its maximum value (imaginary part) on the unit circle. On
the unit circle $D_{2}(-z*)=Cl_{2}(\pi-\frac{\alpha_{*}}{T})$. The generalized
thermal windows for the $CP^{N-1}$ model are:

Table 3. Generalized Thermal Windows for the $CP^{N-1}$ model
---
Dimensions | Closing T | Opening T
3 | $\frac{3\alpha_{*}}{5\pi}$ | $\frac{3\alpha_{*}}{\pi}$
5 | $\frac{13\alpha_{*}}{20\pi}$ | $\frac{13\alpha_{*}}{6\pi}$
7 | $\frac{51\alpha_{*}}{77\pi}$ | $\frac{51\alpha_{*}}{25\pi}$
9 | $\frac{205\alpha_{*}}{308\pi}$ | $\frac{205\alpha_{*}}{102\pi}$
11 | $\frac{819\alpha_{*}}{1229\pi}$ | $\frac{819\alpha_{*}}{409\pi}$

Let’s focus again on the case of the $3d$ theory thermal window. The borders
for $\alpha_{*}$ are $\pi T/3$ and $5\pi T/3$. These are the points where the
$D_{2}(z*)$ takes its maximum value (imaginary part) on the unit circle. On
the unit circle $D_{2}(z*)=Cl_{2}(\frac{\alpha_{*}}{T})$. The interesting
aspect involves placing all these points on the unit circle which is
circumscribed about a regular hexagon as we have explained before. So the
lattice points of the "supersymmetric" model are lying on the unit circle at
$\alpha=0,\pi/3,2\pi/3,\pi,4\pi/3,5\pi/3$. The "supersymmetric" model when the
regular hexagon turns to square has a superconducting picture arising from the
identities of the a bosonic and fermionic models at imaginary chemical
potential.

Figure 3: GN and CP thermal windows on lattice Figure 4: GN and CP thermal
windows on perturbed lattice Figure 5: GN and CP thermal windows on square
lattice

## 5\. Summary and discussion

The main message of this work is that a fermionic and a bosonic theory at
imaginary chemical potential for arbitrary odd dimensions, have a phase
structure which is connected to a hexagonal Hubbard model of fermions.
Particularly, the fermion-boson duality at imaginary chemical potential
enables us to relate the thermal windows of the fermionic $U(N)$ Gross-Neveu
model and the bosonic CPN-1 model at finite temperature to lattice
transformations of hexagonal Hubbard-like lattices to square lattice.
Theoretical studies have predicted that certain systems can undergo a
transition from a Hubbard hexagonal lattice to a square lattice. Since our
calculations have unveiled the relevance of the canonical partition functions
of the models to Bloch states, we think that our results offer a new window
into the utility of the Gross-Neveu and $CP^{N-1}$ models as a tool with which
the creation of a Cooper-pair can be studied. The overlap between two Bloch
wavefunctions that differ by lattice momentum $Q$ may be equivalent to the
calculation of the total charge of a Cooper-pair of fermions or the momentum
transfer that one Bloch wave gives to a lattice when reflected by a lattice
point.

## Statement of Competing Interests

The author report that he does not have any competing interests associated
with this research work.

##  Appendix A The Bloch-Wigner-Ramakrishnan functions $D_{m}(z)$

From the usual analytic continuation of the polylogarithms

$Li_{m}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{m}}\,,\,\,\,z\in{\mathbb{C}}\setminus[1,\infty)\,,\,\,\,m=1,2,3,..\,.$
(47)

one can define the following Bloch-Wigner-Ramakrishnan functions [5, 6] as

$D_{m}(z)=\Re\left(i^{m+1}\left[\sum_{k=1}^{m}\frac{(-\ln|z|)^{m-k}}{(m-k)!}Li_{k}(z)-\frac{(-\ln|z|)^{m}}{2m!}\right]\right)$
(48)

These are real functions of complex variable, analytic in
${\mathbb{C}}\setminus\\{0,1\\}$. In the text we utilized the following
properties of $D_{m}(z)$’s.

$\displaystyle D_{m}(1/z)$ $\displaystyle=(-1)^{m-1}D_{m}(z)$ (49)
$\displaystyle\frac{\partial}{\partial z}D_{m}(z)$
$\displaystyle=\frac{i}{2z}\left(D_{m-1}(z)+\frac{i}{2}\frac{(-i\ln|z|)^{m-1}}{(m-1)!}\frac{1+z}{1-z}\right)$
(50)

On the unit circle we have

$\displaystyle
D_{2n-1}(e^{-i\theta})=(-1)^{n}\Re[Li_{2n-1}(e^{-i\theta})]=(-1)^{n}Cl_{2n-1}(\theta)\,,$
(51) $\displaystyle
D_{2n}(e^{-i\theta})=(-1)^{n+1}\Im[Li_{2n}(e^{-i\theta})]=(-1)^{n}Cl_{2n}(\theta)\,$
(52)

for $n=1,2,3,..$. The Clausen functions $Cl_{m}(\theta)$ are defined as

$Cl_{2n-1}(\theta)\equiv\sum_{k=1}^{\infty}\frac{\cos
k\theta}{k^{2n-1}}\,,\,\,\,Cl_{2n}(\theta)\equiv\sum_{k=1}^{\infty}\frac{\sin
k\theta}{k^{2n}}\,,\,\,\,n=1,2,..$ (53)

## References

  * [1] K. Veschgini and M. Salmhofer, Low-energy effective theory at a quantum critical point of the two-dimensional Hubbard model: Mean-field analysis, Physical Review B, 98 (2108) 235131, [https://link.aps.org/doi/10.1103/PhysRevB.98.235131].
  * [2] Dong, X., Gull, E. and Millis, A.J., Quantifying the role of antiferromagnetic fluctuations in the superconductivity of the doped Hubbard model, Nat. Phys., 18, (2022) 1293–1296, [https://doi.org/10.1038/s41567-022-01710-z].
  * [3] E. G. Filothodoros, A. C. Petkou, and N. D. Vlachos, $3d$ fermion-boson map with imaginary chemical potential, Phys. Rev. D95 (2017), no. 6 065029, [http://arxiv.org/abs/1608.07795].
  * [4] E. G. Filothodoros, Anastasios C. Petkou, Nicholas D. Vlachos, The fermion–boson map for large d, Nuclear Physics B 941 (2019) Pages 195-224,[http://arxiv.org/abs/1803.05950].
  * [5] D. Zagier, The dilogarithm function, In Frontiers in Number Theory, Physics and Geometry II, (2006, P. Cartier, B. Julia, P. Moussa, P. Vanhove (eds.), Springer-Verlag, Berlin-Heidelberg-New York) 3–65.
  * [6] D. Zagier, The Bloch-Wigner-Ramakrishnan polylogarithm function, Math. Ann. 286 (1990) (1990), no. 1-3 613–624.
  * [7] J. Murugan and H. Nastase, Particle-vortex duality in topological insulators and superconductors, JHEP 05 (2017) 159, [http://arxiv.org/abs/1606.01912].
  * [8] M. Körner, K. Langfeld, D. Smith, and L. von Smekal, Density of states approach to the hexagonal Hubbard model at finite density, Physical Review D, 102, (2020) 054502,[https://journals.aps.org/prd/abstract/10.1103/PhysRevD.102.054502].
  * [9] E. G. Filothodoros, Strongly Coupled Fermions in Odd Dimensions and the Running Cut-off $\Lambda_{d}$, American Journal of Modern Physics, 13(1) (2024) 1-11, [http://ajmp.org/article/10.11648/j.ajmp.20241301.11].
  * [10] M. D’ Elia, M. P. Lombardo, Finite density QCD via an imaginary chemical potential, Physical Review D, 67 (2003) 014505, [https://journals.aps.org/prd/abstract/10.1103/PhysRevD.67.014505].
  * [11] M. P. Lombardo, Lattice QCD at finite density: imaginary chemical potential, Proceedings of Science, CPOD2006, (2006) 003, [https://arxiv.org/pdf/hep-lat/0612017].
  * [12] M. Alford, New possibilities for QCD at finite density, Nuclear Physics B - Proceedings Supplements, 73, (1999) 161-166, [https://doi.org/10.1016/S0920-5632(99)85015-4].
  * [13] M. Alford, A. Kapustin, and F. Wilczek, Imaginary chemical potential and finite fermion density on the lattice, Physical Review D, 59, (1999) 054502, [https://journals.aps.org/prd/abstract/10.1103/PhysRevD.59.054502].
  * [14] E. G. Filothodoros, Fermions Coupled to Chern-Simons Gauge Field or Imaginary Chemical Potential and the Bloch Theorem, International Journal of High Energy Physics, 10(2) (2023) 12-19, [https://sciencepublishinggroup.com/article/10.11648/j.ijhep.20231002.11].
  * [15] T. Li, L. Duca, M. Reitter, F. Grusdt, E. Demler, M. Endres, M. Schleier-Smith, I. Bloch, and U. Schneider, Bloch state tomography using wilson lines, Science 352 (2016) 1094.
  * [16] A. Ambrosetti, and P. L. Silvestrelli, Introduction to Maximally Localized Wannier Functions, Reviews in Computational Chemistry, 29 (2016), [https://doi.org/10.1002/9781119148739.ch6].
  * [17] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized Wannier functions: Theory and applications, Reviews of Modern Physics, 84 (2012) 1419, [https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.84.1419].
  * [18] A. Bermudez, E. Tirrito, M. Rizzi, M. Lewenstein, and S. Hands, Gross–Neveu–Wilson model and correlated symmetry-protected topological phases, Annals of Physics, 399 (2018) 149-180, [https://doi.org/10.1016/j.aop.2018.10.007].
  * [19] Yuan-Da Liao, Xiao-Yan Xu, Zi-Yang Meng, and Jian Kang, Correlated insulating phases in the twisted bilayer graphene, Chinese Physics B, 30 (2021) 017305, [https://dx.doi.org/10.1088/1674-1056/abcfa3].
  * [20] R. J. Etienne, On the Zeros and Extrema of Generalised Clausen Functions , Lecture Notes of TICMI vol. 22 (2021), 91-113 , [http://www.viam.science.tsu.ge/others/ticmi/lnt/vol22/6-R. J. Etienne].