# The supercurrent diode effect and nonreciprocal paraconductivity due to the
chiral structure of nanotubes

James Jun He1,2,∗ Yukio Tanaka3 Naoto Nagaosa2,4 1International Center for
Quantum Design of Functional Materials (ICQD), Hefei National Laboratory for
Physical Sciences at Microscale, University of Science and Technology of
China, Hefei, Anhui 230026, China
2Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama, 351-0198,
Japan
3Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan
4Department of Applied Physics, University of Tokyo, Tokyo, 113-8656, Japan
*Corresponding author. Email<EMAIL_ADDRESS>
###### Abstract

The research interest in the supercurrent diode effect (SDE) has been growing.
It has been found in various kinds of systems, in a large part of which it may
be understood by combining spin-orbit coupling and Zeeman field. Here, we show
that there exists another mechanism of generating SDE in chiral nanotubes that
trap magnetic fluxes, without spin-orbit coupling or Zeeman field. We further
show that the same generalized Ginzburg-Landau theory leads to nonreciprocal
paraconductivity (NPC) near the transition temperature. The main features of
both the SDE and the NPC are revealed by their parameter dependence. Our study
suggests a new kind of platforms to explore nonreciprocal properties of
superconducting materials. It also provides a theoretical link between the SDE
and the NPC, which were often studied separately.

## INTRODUCTION

Nonreciprocal transport properties [1] near or inside the superconducting
phase of electronic systems have been attracting a lot of research attention
recently. It may manifest itself in nonreciprocal paraconductivity (NPC) [2,
4, 5, 8, 3, 7, 6] or in so-called supercurrent diode effect (SDE) [10, 9].

In superconductors (SCs) or Josephson junctions with broken inversion
($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries, the critical
currents along opposite directions, $J_{c\pm}$, may be unequal, leading to the
SDE. This effect has been found in various experimental systems [10, 11, 13,
16, 15, 14, 12, 17, 18], part of which may be understood by combining spin-
orbit coupling (SOC) and Zeeman field [19, 21, 20] which break $\mathcal{P}$
and $\mathcal{T}$, respectively. The SOC-Zeeman mechanism also works in one-
dimension [22, 23] and in systems with disorders [24]. There also exist
theories that consider symmetry breakings by internal magnetic [25, 27, 26,
29, 28], electric [30, 31] or valley [32] orders, finite momentum pairing [33,
34], unconventional superconductivity [36, 35], etc. However, systems with
magnetic orders may be understood in a way similar to those under Zeeman
fields, and superconductors with ferroelectric or valley orders, finite
momentum pairing, or $\mathcal{T}$-breaking are not conveniently found in
nature. Thus, it remains an open question whether there exist a new mechanism
to generate the SDE in state-of-the-art experimental systems. Finding such a
mechanism shall greatly enrich the choice of platforms to investigate the SDE
and promote the research in this direction.

While the SDE is a manifestation of a nonreciprocal SC below its transition
temperature $T_{c}$, the nonreciprocity can also be seen slightly above
$T_{c}$, where Cooper pairs start to form but coherent superconductivity is
not reached yet. In this regime, the trend of forming Cooper pairs makes a
large contribution to the conductivity, which is called the paraconductivity
[37, 38]. In systems where $\mathcal{P}$ and $\mathcal{T}$ are broken, the
paraconductivity in opposite directions may differ significantly, leading to
the NPC. Although nonreciprocal conductance may also exist in the normal state
at $T\gg T_{c}$, this effect can be enhanced by several orders of magnitude as
the temperature approaches $T_{c}$ [4]. Theories have shown that the NPC can
also originate from a combination of SOC and Zeeman field [2, 4]. Despite the
similarity in the conditions to realize SDE and the NPC, current theories have
not discussed the two in the same framework to the best of our knowledge.

Here, we show that both the SDE and the NPC exist in a chiral nanotube under a
magnetic field along its axial direction, and they can be obtained with the
same generalized Ginzburg-Landau theory. The inversion symmetry is broken by
the chiral structure of the nanotube without any SOC, and the magnetic field
plays its role through the orbital effect, i.e., Aharonov-Bohm effect, instead
of the Zeeman coupling. The resulting nonreciprocal signals strongly depend on
the magnetic flux, the nanotube radius, and the chiral angle. There exist a
periodicity in the magnetic flux through the tube, similar to the Little-Parks
oscillation [39], as well as a periodicity in the chiral angle. The interplay
of the magnetic flux and the chiral structure is the origin of both the SDE
and the NPC.

## RESULTS

## Chiral nanotubes near $T_{c}$

A nanotube near its superconductivity transition temperature $T_{c}$ may be
described by the following free energy,

$\displaystyle F=\int
d^{2}\bm{r}\psi^{*}(\bm{r})[\alpha+\xi(\hat{\bm{p}})+\frac{\beta}{2}|\psi(\bm{r})|^{2}]\psi(\bm{r}),$
(1)

where $\alpha\sim T-T_{c}$ and $\beta$ are the conventional Ginzburg-Landau
parameters. The displacement vector $\bm{r}=(x,y)$ is defined so that the
nanotube aligns alone the $x$-direction and the transverse coordinate $y$
circulates around the tube, as illustrated in FIG. 1. The term
$\xi(\hat{\bm{p}})=\sum_{ij}\xi_{ij}\hat{p}_{x}^{i}\hat{p}_{y}^{j}$ is the
kinetic energy of a Cooper pair. Apparently, a periodic boundary condition
should be applied along the $y$-direction. The momentum operator is
$\hat{\bm{p}}=-i\hbar\nabla_{\bm{r}}+2e\bm{A}(\bm{r})$. Considering a uniform
magnetic field applied along the $x$-direction, i.e
$\bm{H}=H_{x}\hat{\bm{x}}$, and assuming the nanotube wall thickness to be
negligible, the vector potential becomes $\bm{A}=\frac{\phi}{2\pi
R}\hat{\bm{y}}$, where $\phi=\pi R^{2}H_{x}$ is the magnetic flux through the
nanotube and $R$ is its radius. This is equivalent to a boundary condition
$\psi(\bm{r})=\psi(\bm{r}+2\pi R\hat{\bm{y}})\exp\\{-2\pi i\phi/\phi_{0}\\}$,
$\phi_{0}=h/2e$ being the magnetic flux quantum.

A Fourier transformation (taking into account the magnetic flux) leads to the
following equivalent form of Eq. (1),

$\displaystyle F=$ $\displaystyle 2\pi R\sum_{n}\int
dq[\alpha+\xi\left(\bm{p}\right)+\frac{\beta}{2}(2\pi
R)^{2}|\psi_{n}|^{2}]|\psi_{n}|^{2},$ (2)

where $q$ is the wavenumber along the tube and $\bm{p}=(\hbar
q,[n-\phi/\phi_{0}]\hbar/R)$. The integer $n$ labels the transverse Fourier
components. It is quantized due to the small circumference of the tube. We
have neglected the coupling between different $\bm{q}$-components in the
$|\psi|^{4}$ term, which does not affect the results of this study. It is
clear from Eq. (2) that $F$ is a periodic function of $\phi$, leading to the
Little-Parks oscillation, as will be seen later.

Figure 1: A schematic of a chiral nanotube formed by rolling a two-dimensional
sheet. The two coordinate systems, $(x_{0},y_{0})$ and $(x,y)$, are connected
by a rotation of the chiral angle $\theta$. A magnetic field $\bm{H}$ is
applied along the tube to generate nonreciprocal effects.

The chiral structure of the nanotube is reflected in the functional form of
$\xi(\bm{p})$. To see that, imagine a nano-ribbon obtained by cutting and
flattening the nanotube. When the local continuous rotational symmetry
($\mathcal{C}_{\infty}$) of this ribbon is reduced a discrete
$\mathcal{C}_{n}$, a chiral nanotube can be obtained if the rolling direction
mismatch all the high-symmetry directions. For simplicity, we consider here a
system with $\mathcal{C}_{2}$ and the kinetic term may be written as (up to
the 4-th order in the momentum)

$\displaystyle\xi(\bm{p}_{0})=$
$\displaystyle\frac{|\bm{p}_{0}|^{2}}{2m_{0}}+\frac{|\bm{p}_{0}|^{4}}{4m_{0}^{2}\zeta_{0}}+\frac{p_{x0}^{2}-p_{y0}^{2}}{2m_{1}}+\frac{(p_{x0}^{2}-p_{y0}^{2})^{2}}{4m_{1}^{2}\zeta_{1}}$
$\displaystyle+\frac{p_{x0}^{2}(p_{x0}^{2}-3p_{y0}^{2})+p_{y0}^{2}(p_{y0}^{2}-3p_{x0}^{2})}{4m_{2}^{2}\zeta_{2}}$
(3)

where $\bm{p}_{0}$ is defined in a coordinate system whose axes align with the
high-symmetry directions. It is generally different from that of $\bm{p}$
defined in the previous coordinate system whose $x$-axis is along the
nanotube. They are connected by a rotation of the chiral angle $\theta$, as
shown in FIG. 1. The first two terms in Eq. (3) preserves
$\mathcal{C}_{\infty}$ while the third term reduces it to $\mathcal{C}_{2}$.
Note that $m_{1}>m_{0}$ must hold for the mass along arbitrary direction to be
positive. The last two terms are $\mathcal{C}_{4}$ symmetric. The inclusion of
quartic terms is necessary to reveal the nonreciprocal properties, similar to
the case where such an effect is caused by magnetochiral anisotropy [2, 4, 21,
20].

Equation (3) can be rewritten as

$\displaystyle\xi(\bm{p})=\frac{p_{x}^{2}}{2m_{x}}+\frac{p_{y}^{2}}{2m_{y}}+\frac{p_{x}p_{y}}{m_{xy}}+\sum_{n=0}^{4}\kappa_{n}p_{x}^{n}p_{y}^{4-n}$
(4)

with $m_{x},m_{y},m_{xy}$ and $\kappa_{n}$ being functions (see Materials and
Methods) of the original parameters in Eq. (3). To see how a chiral nanotube
breaks $\mathcal{P}$, note that $p_{y}=(n_{y}-\phi/\phi_{0})\hbar/R$ is
defined along a circular coordinate and behaves as angular momentum (rather
than the usual momentum in a flat space). It remains unchanged under
$\mathcal{P}$ operation, consistent with the symmetry property of the magnetic
flux $\phi$ which should not change under spatial inversion. As a result, the
nanotube geometry leads to the symmetry operation
$(p_{x},p_{y})\xrightarrow{\mathcal{P}}(-p_{x},p_{y})$, and thus the
$p_{x}$-odd terms in Eq. (4) break $\mathcal{P}$.

The supercurrent is

$\displaystyle J_{x}$ $\displaystyle=-2e\int
dy\psi^{*}(\bm{r})\frac{d\xi}{d\hat{p}_{x}}\psi(\bm{r})$ (5)
$\displaystyle=-2e\sum_{n}\frac{2\pi R}{L}\int
dq\frac{\partial\xi(\bm{p})}{\partial p_{x}}|\psi_{n}(q)|^{2},$ (6)

where $L\rightarrow\infty$ is the length of the nanotube. With Eqs. (2), (4)
and (6), we study the SDE when $T<T_{c}$ and the NPC when $T>T_{c}$ in the
following.

## Supercurrent diode effect

When a supercurrent passes through the nanotube, the Cooper pairs acquire a
momentum $\bm{p}$ and a kinetic energy $\xi(\bm{p})$. The order parameter is
determined by the Ginzburg-Landau equation as

$\displaystyle|\psi_{n}(q)|^{2}=\frac{|\alpha|}{\beta(2\pi
R)^{2}}\left(1-\frac{\xi(\bm{p})}{|\alpha|}\right)$ (7)

and the supercurrent is

$\displaystyle J_{x}(n,q)$
$\displaystyle=\frac{-2eR}{L^{2}}\frac{|\alpha|}{\beta
R^{2}}\left(1-\frac{\xi(\bm{p})}{|\alpha|}\right)\frac{\partial\xi(\bm{p})}{\partial
p_{x}}.$ (8)

Note that $\alpha<0$ since $T<T_{c}$. The critical currents $J_{c\pm}$ are the
absolute values of the maximum and minimum, respectively, of $J_{x}(n,q)$ as
$n$ and $q$ are varied.

For general parameters, $J_{c\pm}$ can be determined numerically and the
resulting diode efficiency, $\eta\equiv\frac{I_{c+}-I_{c-}}{I_{c+}+I_{c-}}$,
is shown in FIG. 2 as functions of the magnetic flux $\phi$, the angle
$\theta$ and the temperature, respectively. FIG. 2(a) shows a periodicity in
$\phi$, similar to the Little-Parks oscillation. Different curves are for
various values of the ratio $r=R/l_{0}$, with $R$ being the radius of the
nanotube and $l_{0}=\hbar/\sqrt{2m_{0}T_{c}}$. When $r$ is small and
$\phi/\phi_{0}$ is close to a half-integer, the transverse momentum,
$p_{y}=(n-\phi/\phi_{0})\hbar/R\approx\hbar/(2rl_{0})$, costs so high a
kinetic energy $\xi(\bm{p})$ that it kills the superconductivity (i.e.,
$\psi_{n}\rightarrow 0$), leading to vanishing $J_{c\pm}$. We define $\eta$ in
this case to be zero, resulting in the curve with $r=1$ in FIG. 2 (a). As $r$
increases, $J_{c\pm}$ becomes nonzero for arbitrary magnetic flux and
discontinuities occur as $\phi/\phi_{0}$ changes across half-integers, which
originates from the quantization of the transverse index $n$ in Eq. (8). When
$r\gg 1$, discontinuities disappear while non-smooth kinks remain and $|\eta|$
decreases. From FIG. 2(b), one finds that $\eta$ vanishes whenever $\theta$
becomes a multiple of $\pi/2$. This is expected because the nanotubes in these
cases are not chiral and the inversion symmetry is preserved, forbidding the
SDE. As $\theta/\pi$ deviate from half-integers, $|\eta|$ increases sharply
and extreme values of $\eta$ are reached quickly. Note that the positions of
the extreme points depend on the ratio $m_{0}/m_{1}$, which measures the
strength (and the sign) of inversion symmetry breaking. The temperature
dependence has the usual feature $\eta\sim\sqrt{T_{c}-T}$, as shown in FIG. 2
(c).

Figure 2: The diode efficiency, $\eta=(J_{c+}-J_{c-})/(J_{c+}+J_{c-})$,
obtained by numerically solving for the critical currents $J_{c\pm}$ with Eq.
(8). (a) The dependence on the magnetic flux ($\phi_{0}=h/2e$ is the flux
quantum). The solid curves are for various values of the nanotube radius $R$,
normalized so that $r=R/l_{0}$, where $l_{0}=\hbar/\sqrt{2m_{0}T_{c}}$. The
dashed curve is the approximate result given by Eq. (9) with $r=30$. (b)
Dependence on the angle $\theta$ which corresponds to the chiral structure of
the nanotube. (c) The temperature dependence. The parameters are
$m_{0}=1,m_{1}=2,\zeta_{2}m_{2}\rightarrow\infty,\zeta_{0}/T_{c}=10,\zeta_{1}/T_{c}=20,r=2,\theta=0.6\pi,\phi/\phi_{0}=0.3$
and $T/T_{c}=0.9$ for all the results unless specified otherwise.

It is helpful to obtain the analytical form of $\eta$, which is possible when
$\zeta_{0,1,2}\gg T_{c}$ and thus the terms with $\kappa_{n}$ in Eq.(4) can be
treated as perturbations. We also assume $r$ to be small, and then varying the
transverse quantum number $n$ costs so much energy that $J_{c\pm}$ are
obtained with a fixed $n$ in Eq. (8). Under these conditions, the diode
efficiency is

$\displaystyle\eta=\frac{-4}{\sqrt{3}}$
$\displaystyle\left(4\kappa_{0}\frac{m_{x}^{2}}{m_{xy}}+\kappa_{1}m_{x}\right)m_{0}T_{c}$
$\displaystyle\times
b\sqrt{\frac{|\alpha|}{T_{c}}\frac{m_{x}}{m_{0}}-b^{2}\left(\frac{m_{x}}{m_{y}}-\frac{m_{x}^{2}}{m_{xy}^{2}}\right)},$
(9)

where $b=\phi/\phi_{0}-[\phi/\phi_{0}]$ ($[x]$ denotes the integer closest to
$x$). From Eq. (9) it becomes clear that either $m_{xy}^{-1}$ or $\kappa_{1}$
must be nonzero to achieve the SDE. The requirement, combined with Eqs. (14)
and (16), becomes $m_{1}^{-1}\neq 0$ and $\sin 2\theta\neq 0$, which is just
equal to requiring the nanotube to have a chiral structure. When the magnetic
filed $H_{x}$ is small, $\eta$ is linear in $H_{x}$ (note that $\phi=\pi
R^{2}H_{x}$). As the magnetic flux increases, the expression under the square
root becomes negative for small $|\alpha|$ since
$(\frac{m_{x}}{m_{y}}-\frac{m_{x}^{2}}{m_{xy}^{2}})$ is positive definite.
This results in a decrease of the transition temperature to $T_{c}^{\prime}$
with $\delta T_{c}=T_{c}-T_{c}^{\prime}\sim
b^{2}(\frac{m_{x}}{m_{y}}-\frac{m_{x}^{2}}{m_{xy}^{2}})\frac{m_{0}}{m_{x}}$.
And the temperature dependence of Eq. (9) may be written as
$\eta\sim\sqrt{T_{c}^{\prime}-T}$. A substitution of Eqs. (13)$-$(17) leads to
the dashed curves in FIG. 2 (a) and (c), which show great agreement with
previous numerical results except two situations, (i) $r\gg 1$ and
$\phi/\phi_{0}$ is close to a half integer and (ii) The temperature is far
below $T_{c}$. In both situations, the assumption that $J_{c\pm}$ can be
obtained with the same index $n$ in Eq. (8) no longer holds.

The differences in the SDE between chiral nanotube SCs and previously studied
spin-orbit coupled SCs [19, 21, 20] is clear now. The diode efficiency here is
controlled by the nanotube diameter and the chiral angle, while it is
determined by the SOC strength in spin-orbit coupled SCs. The sign change of
$\eta$ happens in both kinds of systems as the magnetic field is tuned.
However, the origins are rather different. In SOC SCs, $\eta$ changes sign due
to the higher-order (in momentum and in field strength) terms in the kinetic
energy of the Cooper pairs. Here, it is because the transverse index $n$
corresponding to the critical currents $J_{c\pm}$ is shifted. The sign of
$\eta$ changes exactly at $b=1/2$ here (i.e. when the number of flux quanta is
a half-integer) while the sign-flipping field-strength in SOC SCs depends on
multiple system parameters.

## Nonreciprocal paraconductivity

The nonreciprocity of superconducting materials manifests itself not only in
the SDE when $T-T_{c}<0$, but also in the NPC when $T_{c}\gg T-T_{c}>0$. In
the latter case, although the average order parameter vanishes, its quantum
fluctuations induce a significant contribution to the conductance, resulting
in a drop of resistance above $T_{c}$ before a finite order parameter is
established. The relation between the two phenomena has not been discussed
elsewhere although the symmetry requirements are very similar. In this
section, we calculate the paraconductivity of the chiral nanotubes described
by Eq. (2) and discuss it in the same framework as we discuss the SDE.

Figure 3: The linear and nonlinear paraconductivity of a chiral nanotube,
$\sigma_{1}$ and $\sigma_{2}$, normalized by
$\bar{\sigma}_{1}=\frac{k_{B}T}{T_{c}}\frac{e^{2}}{4\pi^{2}\hbar}\frac{\gamma
l_{0}}{R}$ and
$\bar{\sigma}_{2}=\frac{k_{B}T}{T_{c}^{2}}\frac{e^{3}}{6\pi^{2}\hbar}\frac{\gamma^{2}l_{0}^{2}}{R}$
respectively. (a) Magnetic-flux dependence, showing the Little-Parks
oscillation. (b) The evolution of the flux dependence as the normalized radius
$r$ is varied. (c) Dependence of $\sigma_{2}$ on the chiral angle $\theta$.
(d) The temperature dependence of the inverse of $\sigma_{1/2}$. The
parameters are the same as those in FIG. 2.

We calculate the paraconductivity using the time-dependent Ginzburg-Landau
theory [40] (see Materials and Methods). The resulting current density
$j_{x}=\sigma_{1}E+\sigma_{2}E^{2}+O(E^{3})$ where the linear conductivity

$\displaystyle\sigma_{1}=\gamma\frac{T}{T_{c}}\frac{e^{2}}{4\pi^{2}\hbar}\frac{l_{0}}{R}\sum_{n}\int
dx\frac{\partial^{2}_{x}f_{n}(x)}{[\alpha/T_{c}+f_{n}(x)]^{2}},$ (10)

and the nonreciprocal term

$\displaystyle\sigma_{2}=\gamma^{2}\frac{T}{T_{c}^{2}}\frac{e^{3}}{6\pi^{2}\hbar}\frac{l_{0}^{2}}{R}\sum_{n}\int
dx\frac{\partial^{3}_{x}f_{n}(x)}{[\alpha/T_{c}+f_{n}(x)]^{3}}.$ (11)

In the dimensionless function $f_{n}(x)=\xi(\bm{p})/T_{c}$, we made a change
of variables, $\bm{p}=[p_{x},p_{y}]\rightarrow[x\hbar q_{0},y_{n}\hbar
q_{0}]$, where $y_{n}=(n-\phi/\phi_{0})/Rq_{0}$ and $q_{0}=1/l_{0}$. The
substitution of Eq. (4) leads to

$\displaystyle f_{n}(x)$ $\displaystyle=\frac{1}{T_{c}}\xi\left(x\hbar
q_{0},y_{n}\hbar q_{0}\right)$
$\displaystyle=\frac{x^{2}}{2\tilde{m}_{x}}+\frac{y_{n}^{2}}{2\tilde{m}_{y}}+\frac{xy_{n}}{\tilde{m}_{xy}}+\sum_{i=0}^{4}\tilde{\kappa}_{n}x^{i}y_{n}^{4-i}$
(12)

where $\tilde{m}_{x/y/xy}=m_{0}^{-1}m_{x/y/xy}$ and
$\tilde{\kappa}_{i}=\kappa_{i}m_{0}(\hbar q_{0})^{2}$ are dimensionless
parameters.

The integrals in Eqs. (10) and (11) can be done numerically and the resulting
$\sigma_{1/2}$ are shown in FIG. 3 as functions of the magnetic flux $\phi$
and the chiral angle $\theta$. Little-Parks oscillations of both the linear
and nonlinear conductivities are found in FIG. 3(a). The maxima/minima of
$\sigma_{1}$ are at integer/half-integer values of $\phi/\phi_{0}$ since
$\sigma_{1}$ is an even function of $\phi$ and finite flux suppresses
superconductivity. On the hand, the nonreciprocal $\sigma_{2}$ is odd in
$\phi$ and it vanishes whenever $\phi/\phi_{0}$ becomes a integer. The flux
values for optimal $\sigma_{2}$ depend on the system parameters such as the
nanotube radius, as shown in FIG. 3(b). The curves resemble those in FIG. 2(a)
with the difference that they are smooth here because all the transverse
components $n\in(-\infty,\infty)$ of the order parameter contribute, unlike
the supercurrent which is given by a certain $n$. FIG. 3(c) shows the effect
of the chiral angle $\theta$. The angle dependence of $\sigma_{2}$ is of
similar amplitude to the flux dependence in FIG. 3(a). In FIG. 3(d), we find
that the temperature dependence of $\sigma_{1}$ is rather linear, which is
similar to higher-dimensional systems [40, 2, 4]. A difference here is a
shifted transition temperature $T_{c}^{\prime}$, so that
$\sigma_{1}^{-1}\sim(T-T_{c}^{\prime})$. The $T$-dependence of
$\sigma_{2}^{-1}$ is clearly of higher order and we do not find any single
power law.

## DISCUSSION

We have shown that superconducting chiral nanotubes with trapped magnetic flux
behave as supercurrent diodes, whose diode efficiency strongly depends on the
chiral angle. We also found, in the same theoretical framework, that the
paraconductivity of such chiral nanotubes near $T_{c}$ contains a
nonreciprocal part $\sigma_{2}$, whose dependence on the system parameters is
rather similar to that of the SDE efficiency $\eta$ and oscillates
periodically as the magnetic flux $\phi$ or the chiral angle $\theta$ is
varied. The results show that a combination of inversion symmetry breaking by
chiral structure and time-reversal symmetry breaking by magnetic flux can
induce nonreciprocal transport properties, including the SDE and the NPC, in
superconductors.

One may notice that actual nanotubes created in laboratories are mostly
related to honeycomb or triangular lattices while the nanotubes discussed here
are obtained by rolling a sheet of rectangular lattice. This choice is for
technical convenience. However, the main conclusions drew here shall generally
apply. To discuss a carbon nanotube (honeycomb) or a transition-metal-
dichalcogenide nanotube (triangular), terms up to the 6-th order in momentum
must be included when constructing their Ginzburg-Landau free energies, which
is not really meaningful considering the condition for the validity of the
Ginzburg-Landau theory itself. Thus, a study of realistic (carbon/NbSe2/…)
nanotubes may need to use the microscopic BCS theory, which can be done
numerically.

Although single superconductors are considered here, the nonreciprocal effects
discussed here shall apply to Josephson junctions where two conventional bulk
superconductors (Al, Pb, Nb, NbSe2, etc.) are connected by a chiral nanotube.
A study of such a system will be of great practical importance. In this
manuscript, we aim to clarify the physical principles and general features of
the nonreciprocal properties of superconducting chiral nanotubes, and leave
more detailed and realistic studies to future works.

Although one needs to break both $\mathcal{P}$ and $\mathcal{T}$ to obtain
unequal $J_{c\pm}$ [25, 41], it should be noted that there also exist
nonreciprocal properties in $\mathcal{T}$-preserving Josephson junctions. The
nonreciprocity may be observed in unequal retrapping currents $J_{r\pm}$ [42]
or in ac Josephson effects [9, 25]. The interaction between electrons plays an
important role in these cases. The design or improvement of supercurrent
diodes with strong electron interactions is a topic worth further
investigation.

## MATERIALS AND METHODS

## Parameters in the rotated coordinate system

By rotating the coordinate system by the chiral angle $\theta$, one obtains
the free energy form in Eq. (4) where the parameters are functions of those in
Eq. (3). The functional forms are

$\displaystyle\frac{1}{m_{x/y}}=\frac{1}{m_{0}}\pm\frac{\cos 2\theta}{m_{1}},$
(13) $\displaystyle\frac{1}{m_{xy}}=-\frac{\sin 2\theta}{m_{1}},$ (14)
$\displaystyle\kappa_{0}=\kappa_{4}=\frac{1}{4}\left(\frac{1}{\zeta_{0}m_{0}^{2}}+\frac{\cos^{2}2\theta}{\zeta_{1}m_{1}^{2}}+\frac{\cos
4\theta}{\zeta_{2}m_{2}^{2}}\right),$ (15)
$\displaystyle\kappa_{1}=-\kappa_{3}=\frac{\sin
4\theta}{2}\left(\frac{1}{\zeta_{1}m_{1}^{2}}+\frac{2}{\zeta_{2}m_{2}^{2}}\right),$
(16)
$\displaystyle\kappa_{2}=\frac{1}{4}\left(\frac{2}{\zeta_{0}m_{0}^{2}}+\frac{1-3\cos
4\theta}{\zeta_{1}m_{1}^{2}}-\frac{6\cos 4\theta}{\zeta_{2}m_{2}^{2}}\right).$
(17)

## Time-dependent Ginzburg-Landau theory

At a temperature slightly above $T_{c}$, the fluctuation of the order
parameter is determined by the following Langevin equation [40],

$\displaystyle\hbar\gamma\partial_{t}\psi(\bm{r},t)=-\left[\alpha+\xi(\hat{\bm{p}})\right]\psi(\bm{r},t)+\delta(\bm{r},t),$
(18)

where $\delta(\bm{r},t)$ is an uncorrelated random force and $\gamma$ is the
inverse of damping constant. Note that $\alpha>0$ and the static order
parameter vanishes, i.e. $\langle\psi_{n,q}(t)\rangle_{t}=0$. However, Eq.
(18) leads to a nonzero $\langle|\psi_{n,q}(t)|^{2}\rangle_{t}$, which is [40]

$\displaystyle\langle|\psi_{n,q}(t)|^{2}\rangle=\frac{2k_{B}T}{\hbar\gamma}\int_{-\infty}^{t}dt^{\prime}e^{-\frac{2}{\hbar\gamma}\int_{t^{\prime}}^{t}dt^{\prime\prime}[\alpha+\xi(t^{\prime\prime})]}.$
(19)

It is nonzero when an electric field $\bm{E}=E\hat{\bm{x}}$ is applied, making
$\xi(\bm{p}(t^{\prime\prime}))=\xi_{n}(q+2eEt^{\prime\prime})$. Combining Eqs.
(6) and (19), one obtains Eq. (10) and Eq. (11).

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###### Acknowledgements.

##
Acknowledgments

Funding: N.N. was supported by JST CREST Grant Number JPMJCR1874, Japan, and
JSPS KAKENHI Grant Number 18H03676. Y.T. was supported by Scientific Research
(A) (KAKENHI Grant No. JP20H00131), Scientific Research (B) (KAKENHI Grants
No. JP20H01857) and JSPS Core-to-Core program Oxide Superspin international
network (Grants No. JPJSCCA20170002). Author contributions: N.N. initiated and
supervised this work. Y.T. helped to analyze the problem. J.J.H. carried out
the calculations and wrote the manuscript with suggestions from all the
authors. Competing interests: The authors declare that they have no competing
interests. Data and materials availability:  All data needed to evaluate the
conclusions in the paper are present in the paper.


#  Fisher-information and the thermodynamics of scale-invariant systems

A. Hernando<EMAIL_ADDRESS>C. Vesperinas<EMAIL_ADDRESS>A. Plastino<EMAIL_ADDRESS>Departament ECM, Facultat de Física,
Universitat de Barcelona. Diagonal 647, 08028 Barcelona, Spain  Sogeti España,
WTCAP 2, Plaça de la Pau s/n, 08940 Cornellà, Spain  National University La
Plata, IFCP-CCT-CONICET, c.c. 727, 1900 La Plata, Argentina

###### Abstract

We present a thermodynamic formulation for scale-invariant systems based on
the minimization with constraints of Fisher’s information measure. In such a
way a clear analogy between these systems’s thermal properties and those of
gases and fluids is seen to emerge in natural fashion. We focus attention on
the non-interacting scenario, speaking thus of _scale-free ideal gases_
(SFIGs) and present some empirical evidences regarding such disparate systems
as electoral results, city populations and total citations in Physics
journals, that seem to indicate that SFIGs do exist. We also illustrate the
way in which Zipf’s law can be understood in a thermodynamical context as the
surface of a finite system. Finally, we derive an equivalent microscopic
description of our systems which totally agrees with previous numerical
simulations found in the literature.

###### keywords:

Fisher information , scale-invariance

††journal: Physica A

##  1 Introduction

Scale-invariant phenomena are rather abundant in Nature and display somewhat
unexpected features. Examples can be found that range from physical and
biological to technological and social sciences [1]. One may cite, among many
possibilities, that empirical data from percolation theory and nuclear
multifragmentation [2] reflect scale-invariant behaviour, as does the
abundance of genes in various organisms and tissues [3]. Additionally, we can
speak of the frequency of words in natural languages [4], scientific
collaboration networks [5], the Internet traffic [6], Linux packages links
[7], as well as of electoral results [8, 9], urban agglomerations [10, 11] and
firm sizes all over the world [12]. What characterizes these disparate systems
is the lack of a characteristic size, length or frequency for an observable
$k$ under scrutiny. This fact usually leads to a power law distribution
$p(k)$, valid in most of the domain of definition of $k$,

$p(k)\sim 1/k^{1+\gamma},$ (1)

with $\gamma\geq 0$. Special attention deserves the class of universality
defined by $\gamma=1$, which corresponds to the so-called Zipf’s law in the
cumulative distribution or the rank-size distribution [2, 3, 4, 6, 7, 10, 11,
12, 13]. Recently, Maillart et al. [7] have found that links’ distributions
follow Zipf’s law as a consequence of stochastic proportional growth. In its
simplest formulation such kind of growth assumes that an element of the system
becomes enlarged proportionally to its size $k$, being governed by a Wiener
process. The class $\gamma=1$ emerges from the condition of stationarity,
i.e., when the system reaches a dynamic equilibrium [13]. We will as well
propose to consider the case $\gamma=0$ as representative of a second class of
universality, since the ensuing behavior, empirically found by Costa Filho et
al. [8] with regards to the vote-distribution in Brazilian electoral results,
emerges as the result of multiplicative processes in complex networks [9].

In this paper we attempt to formulate a thermodynamic treatment common to
these systems. Our efforts are based on the minimization with appropriate
constraints of Fisher’s information measure (FIM), abbreviated as the MFI
approach. It is shown in [14] that MFI leads to a (real) Schreodinger-like
equation whose “potential” function is given by the constraints employed to
constrain the variational process. The interplay between constraints and
associated Lagrange multipliers turns our to be Legendre-invariant [14] and
leads to all known thermodynamic relations. Such result constitutes the
essential ingredient of our considerations here. We will first consider the
MFI treatment of the ideal gas (Seccion 3), not given elsewhere as far as we
are aware of, since it is indispensable to deal with it in order to fully
understand the methodology employed for scale-free systems, which is tackled
in Section 4. Applicatioms are discussed in Section 5 and some conclusiones
drawn in Section 6. We begin our considerations in Section 2 with a brief
Fisher’s sketch.

##  2 Minimum Fisher Information approach (MFI)

The Fisher information measure $I$ for a system described by a set of
coordinates $\mathbf{q}$ and physical parameters $\mathbf{\theta}$, has the
form [15]

$I(F)=\int_{\Omega}d\mathbf{q}F(\mathbf{q}|\mathbf{\theta})\sum_{ij}c_{ij}\frac{\partial}{\partial\theta_{i}}\ln
F(\mathbf{q}|\mathbf{\theta})\frac{\partial}{\partial\theta_{j}}\ln
F(\mathbf{q}|\mathbf{\theta}),$ (2)

where $F(\mathbf{q}|\mathbf{\theta})$ is the density distribution in a
configuration space ($\mathbf{q}$) of volume $\Omega$ conditioned by the
physical parameters collectively represented by the variable
$\mathbf{\theta}$. The constants $c_{ij}$ account for dimensionality, and take
the form $c_{ij}=c_{i}\delta_{ij}$ if $q_{i}$ and $q_{j}$ are uncorrelated,
where $\delta_{ij}$ is the Kronecker delta. As shown in [14], the thermal-
equilibrium state of the system can be determined by minimizing $I$ subject to
adequate prior conditions (MFI), like the normalization of $F$ or by any
constraint on the mean value of an observable $\langle A_{i}\rangle$ [14]. The
MFI is then cast as a variation problem of the form

$\delta\left\\{I(F)-\sum_{i}\mu_{i}\langle A_{i}\rangle\right\\}=0,$ (3)

where $\mu_{i}$ are appropriate Lagrange multipliers.

##  3 MFI treatment of the ideal gas

As a didactic introductory example, not discussed in [14], we will here
rederive, via MFI (something original as far as we know), the density
distribution, in configuration space, of the (translational invariant) ideal
gas (IG) [16], that describes non-interacting classical particles of mass $m$
with coordinates $\mathbf{q}=(\mathbf{r},\mathbf{p})$, where
$md\mathbf{r}/dt=\mathbf{p}$. The translational invariance is described by the
translational family of distributions
$F(\mathbf{r},\mathbf{p}|\mathbf{\theta}_{r},\mathbf{\theta}_{p})=F(\mathbf{r}^{\prime},\mathbf{p}^{\prime})$
whose form does not change under the transformations
$\mathbf{r}^{\prime}=\mathbf{r}-\mathbf{\theta}_{r}$ and
$\mathbf{p}^{\prime}=\mathbf{p}-\mathbf{\theta}_{p}$. We assume that these
coordinates are canonical [17] and uncorrelated. This assumption is introduced
into the information measure (2) setting $c_{ij}=c_{i}\delta_{ij}$, where
$c_{i}=c_{r}$ for space coordinates, $c_{i}=c_{p}$ for momentum coordinates.
The density can obviously be factorized in the fashion
$F(\mathbf{r},\mathbf{p})=\rho(\mathbf{r})\eta(\mathbf{p})$, and then [15] it
follows from the additivity of the information measure that $I=I_{r}+I_{p}$.
If $D$ is the dimensionality we have

$\begin{array}[]{rl}I_{r}=&\displaystyle c_{r}\int
d^{D}\mathbf{r}~{}\rho(\mathbf{r})\left|\mathbf{\nabla}_{r}\ln\rho(\mathbf{r})\right|^{2}\\\
I_{p}=&\displaystyle c_{p}\int
d^{D}\mathbf{p}~{}\eta(\mathbf{p})\left|\mathbf{\nabla}_{p}\ln\eta(\mathbf{p})\right|^{2}.\end{array}$
(4)

In extremizing FIM we constrain the normalization of $\rho(\mathbf{r})$ and
$\eta(\mathbf{p})$ to the total number of particles $N$ and to $1$,
respectively, i.e.,

$\int d^{D}\mathbf{r}~{}\rho(\mathbf{r})=N,\qquad\int
d^{D}\mathbf{p}~{}\eta(\mathbf{p})=1.$ (5)

In addition, we penalize infinite values for the particle momentum with a
constraint on the variance of $\eta(\mathbf{p})$ to a given empirically
obtained value, namely,

$\int
d^{D}\mathbf{p}~{}\eta(\mathbf{p})(\mathbf{p}-\overline{\mathbf{p}})^{2}=D\sigma_{p}^{2},$
(6)

where $\overline{\mathbf{p}}$ is the mean value of $\mathbf{p}$. For each
degree of freedom it is known from the Virial Theorem that the variance is
related to the temperature $T$ as $\sigma_{p}^{2}=mk_{B}T$, with $k_{B}$ the
Boltzmann constant. Variation thus yields

$\displaystyle\delta\left\\{c_{r}\int
d^{D}\mathbf{r}~{}\rho\left|\mathbf{\nabla}_{r}\ln\rho\right|^{2}+\mu\int
d^{D}\mathbf{r}~{}\rho\right\\}=0$ (7)

and

$\delta\left\\{c_{p}\int
d^{D}\mathbf{p}~{}\eta\left|\mathbf{\nabla}_{p}\ln\eta\right|^{2}+\lambda\int
d^{D}\mathbf{p}~{}\eta(\mathbf{p}-\overline{\mathbf{p}})^{2}+\nu\int
d^{D}\mathbf{p}~{}\eta\right\\}=0,$ (8)

where $\mu$, $\lambda$ and $\nu$ are Lagrange multipliers. Introducing now
$\rho(\mathbf{r})=\Psi^{2}(\mathbf{r})$ and varying (7) with respect to $\Psi$
leads to a Schroedinger-like equation [14, 18]

$\left[-4\nabla_{r}^{2}+\mu^{\prime}\right]\Psi(\mathbf{r})=0,$ (9)

where $\mu^{\prime}=\mu/c_{r}$. To fix the boundary conditions, we first
assume that the $N$ particles are confined in a box of volume $V$, and next we
take the thermodynamic limit $N,V\rightarrow\infty$ with $N/V$ finite. The
equilibrium state compatible with this limit corresponds to the ground state
solution ($\mu^{\prime}=0$), which is the uniform density
$\rho(\mathbf{r})=N/V$.

Introducing $\eta(\mathbf{p})=\Phi^{2}(\mathbf{p})$ and varying (8) with
respect to $\Phi$ leads to the quantum harmonic oscillator-like equation [18]

$\left[-4\nabla_{p}^{2}+\lambda^{\prime}(\mathbf{p}-\overline{\mathbf{p}})^{2}+\nu^{\prime}\right]\Phi(\mathbf{p})=0,$
(10)

where $\lambda^{\prime}=\lambda/c_{p}$ and $\nu^{\prime}=\nu/c_{p}$. The
equilibrium configuration corresponds to the ground state solution, which is
now a gaussian distribution. Using (6) to identify
$|\lambda^{\prime}|^{-1/2}=\sigma_{p}^{2}$ we get the Maxwell-Boltzmann
distribution, which leads to a density distribution in configuration space of
the form

$f(\mathbf{r},\mathbf{p})=\frac{N}{V}\frac{\exp\left[-(\mathbf{p}-\overline{\mathbf{p}})^{2}/2\sigma_{p}^{2}\right]}{(2\pi\sigma_{p}^{2})^{D/2}}.$
(11)

If $H$ is the elementary volume in phase space, the total number of
microstates is
$Z=N!H^{DN}\prod_{i=1}^{N}F_{1}(\mathbf{r}_{i},\mathbf{p}_{i})$, where
$F_{1}=F/N$ is the monoparticular distribution and $N!$ counts all possible
permutations for distinguishable particles. The entropy $S=-k_{B}\ln Z$ gets
then written in the form

$S=Nk_{B}\left\\{\ln\frac{V}{N}\left(\frac{2\pi\sigma_{p}^{2}}{H^{2}}\right)^{D/2}+\frac{2+D}{2}\right\\},$
(12)

where we have used the Stirling approximation for $N!$. This expression
agrees, of course with the known value entropic expression for the IG [16],
illustrating on the predictive power of the MFI formulation advanced in [14].

##  4 Scale invariant systems

We pass now to the leit-motif of the present communication and consider a one-
dimensional system with dynamical coordinates $\mathbf{q}=(k,v)$ where
$dk/d\tau=v$, with $\tau$ the time variable. We define $k$ as a discrete
coordinate, i.e. $k=k_{1},k_{2},\ldots,k_{M}$, where $k_{i}=i\Delta k$ and
$M\gg 1$, is the total number of bins of width $\Delta k$ in our system. In
order to address the scale-invariance behaviour of $k$ we change variables
passing to new coordinates $u=\ln k$ and $w=du/dt$. We work under the
hypothesis that $u$ and $w$ are canonically conjugated [17] and uncorrelated.
This assumption immediately leads to proportional growth since

$dk/dt=v=kw.$ (13)

For constant $w$ this equation yields an exponential growth $k=k_{0}e^{wt}$,
which represents uniform linear motion in $u$, that is, $u=wt+u_{0}$, with
$u_{0}=\ln k_{0}$ 111This exponential growth allows to identify the systems
that we study in this work in macroscopic fashion with those addressed in
[19].. It is easy to check that the scale transformation
$k^{\prime}=k/\theta_{k}$ leaves invariant the coordinate $w$, whereas the
coordinate $u$ transforms translationally as $u^{\prime}=u-\Theta_{k}$, where
$\Theta_{k}=\ln\theta_{k}$. Thus, the physics does not depend on scale and the
system is translationally invariant with respect to the coordinates $u$ and
$w$, entailing that the distribution of physical elements can be described by
the monoparametric translation families
$f(u,w|\Theta_{k},\Theta_{w})=f(u^{\prime},w^{\prime})$. By analogy with the
IG, we will call our system a “scale-free ideal gas” (SFIG), i.e., a system of
$N$ non-interacting elements. Taking into account that i) $u$ and $w$ are
canonical and uncorrelated ($c_{ii}=c_{i}\neq 0$ and $c_{uw}=c_{wu}=0$), so
the density distribution can be factorized as $f(u,w)=g(u)h(w)$, and ii) that
the Jacobian for our change of variables is $dkdv=e^{2u}dudw$, the information
measure $I=I_{u}+I_{w}$ can be obtained in the continuous limit as

$\begin{array}[]{rl}I_{u}=&\displaystyle
c_{u}\int_{\Omega}du~{}e^{2u}g(u)\left|\frac{\partial\ln g(u)}{\partial
u}\right|^{2}\\\ I_{w}=&\displaystyle
c_{w}\int_{-\infty}^{\infty}dw~{}h(w)\left|\frac{\partial\ln h(w)}{\partial
w}\right|^{2},\end{array}$ (14)

where $\Omega=\ln(k_{M}/k_{1})=\ln M$ is the volume defined in “$u$”-space.

###  4.1 MFI treatment of the scale-free ideal gas

The constraints to the given observables $\langle A_{i}\rangle$ in the
extremization problem determine the behaviour of the system. For the general
case, we constrain the normalization of $g(u)$ and $h(w)$ to the total number
of particles $N$ and to $1$, respectively

$\int_{\Omega}du~{}e^{2u}g(u)=N,\qquad\int_{-\infty}^{\infty}dw~{}h(w)=1.$
(15)

In addition, we penalize infinite values for $w$ with a constraint on the
variance of $h(w)$ to a given measured value

$\int_{-\infty}^{\infty}dw~{}h(w)(w-\overline{w})^{2}=\sigma_{w}^{2},$ (16)

where $\overline{w}$ is the average growth. The variation yields

$\delta\left\\{c_{u}\int_{\Omega}du~{}e^{2u}g\left|\frac{\partial\ln
g}{\partial u}\right|^{2}+\mu\int_{\Omega}du~{}e^{2u}g\right\\}=0$ (17)

and

$\delta\left\\{c_{w}\int_{-\infty}^{\infty}dw~{}h\left|\frac{\partial\ln
h}{\partial
w}\right|^{2}+\lambda\int_{-\infty}^{\infty}dw~{}h(w-\overline{w})^{2}+\nu\int_{-\infty}^{\infty}dw~{}h\right\\}=0,$
(18)

where $\mu$, $\lambda$ and $\nu$ are Lagrange multipliers. Introducing
$g(u)=e^{-2u}\Psi^{2}(u)$, and varying (17) with respect to $\Psi$ leads, as
is always the case with the MFI [14], to the Schroedinger-like equation

$\left[-4\frac{\partial^{2}}{\partial u^{2}}+4+\mu^{\prime}\right]\Psi(u)=0,$
(19)

where $\mu^{\prime}=\mu/c_{u}$. Analogously to the IG, we impose solutions
compatible with a finite normalization of $g$ in the thermodynamic limit
$N,\Omega\rightarrow\infty$ with $N/\Omega=\rho_{0}$ finite, where $\rho_{0}$
is defined as the _bulk density_. Solutions compatible with the normalization
of (15) are given by $\Psi(u)=A_{\alpha}e^{-\alpha u/2}$, where $A_{\alpha}$
is the normalization constant and $\alpha=\sqrt{4+\mu^{\prime}}$. In this
general case, the density distribution as a function of $k$ takes the form of
a power law: $g_{\alpha}(\ln k)=A^{2}/k^{2+\alpha}$. The equilibrium is always
defined for the MFI as the ground state solution [14], which corresponds to
the lowest allowed value $\alpha=0$.

Introducing now $h(w)=\Phi^{2}(w)$ and varying (18) with respect to $\Phi$
leads to the quantum harmonic oscillator-like equation [14, 18]

$\left[-4\frac{\partial^{2}}{\partial
w^{2}}+\lambda^{\prime}(w-\overline{w})^{2}+\nu^{\prime}\right]\Phi(w)=0,$
(20)

where $\lambda^{\prime}=\lambda/c_{w}$ and $\nu^{\prime}=\nu/c_{w}$. The
equilibrium configuration corresponds to the ground state solution, which is
now a Gaussian distribution. Using (16) to identify
$|\lambda^{\prime}|^{-1/2}=\sigma_{w}^{2}$ we get the Maxwell-Boltzmann
distribution

$h(w)=\frac{\exp\left[-(w-\overline{w})^{2}/2\sigma_{w}^{2}\right]}{\sqrt{2\pi}\sigma_{w}}.$
(21)

The density distribution in configuration space $F(k,v)dkdv=f(u,w)e^{2u}dudw$
is then

$F(k,v)=\frac{N}{\Omega
k^{2}}\frac{\exp\left[-(v/k-\overline{w})^{2}/2\sigma_{w}^{2}\right]}{\sqrt{2\pi}\sigma_{w}}.$
(22)

If we define $H=\Delta k^{2}/\Delta\tau$ as the elementary volume in phase
space, where $\Delta\tau$ is the time element, the total number of microstates
is $Z=N!H^{N}\prod_{i=1}^{N}F_{1}(k_{i},v_{i})$, where $F_{1}=F/N$ is the
monoparticular distribution function and $N!$ counts all possible permutations
for distinguishable elements. The entropy equation of state $S=-\kappa\ln Z$
reads

$S=N\kappa\left\\{\ln\frac{\Omega}{N}\frac{\sqrt{2\pi}\sigma_{w}}{H^{\prime}}+\frac{3}{2}\right\\},$
(23)

where $\kappa$ is a constant that accounts for dimensionality and
$H^{\prime}=H/(k_{M}k_{1})=H/(M\Delta k^{2})=1/(M\Delta\tau)$. Remarkably,
this expression has the same form as the one-dimensional IG ($D=1$ in (12));
instead of the thermodynamical variables $(N,V,T)$, here we deal with the
variables $(N,\Omega,\sigma_{w})$, which make the entropy scale-invariant as
well.

Figure 1: (colour on-line) a, rank-size distribution of the cities of the
province of Huelva, Spain (2008), sorted from largest to smallest, compared
with the result of a simulation with Brownian walkers (green squares). b,
rank-plot of the 2008 General Elections results in Spain. c, rank-plot of the
2005 General Elections results in the United Kingdom. (Red dots: empirical
data; blue lines: fit to (25)).

The total density distribution for $k$ is obtained integrating for all $v$ the
density distribution in configuration space. Accordingly, from (22) we get

$F(k)=\int dvF(k,v)=\frac{N}{\Omega}\frac{1}{k}=\frac{\rho_{0}}{k}.$ (24)

It can be shown that this it is just a uniform density-distribution in
$u$-space of the bulk density: $F(k)dk=f(u)e^{u}du=N/\Omega du=\rho_{0}du$.

##  5 Social examples of scale-free ideal gases

A common representation of empirical data is the so-called rank-plot or Zipf
plot [4, 11, 20], where the $j$th element of the system is represented by its
size, length or frequency $k_{j}$ against its rank, sorted from the largest to
the smallest one. This process just renders the inverse function of the
ensuing cumulative distribution, normalized to the number of elements. We call
$r$ the rank that ranges from 1 to $N$. For large $N$, the density
distribution (24) correspond to an exponential rank-size distribution

$k(r)=k_{M}\exp\left[-\frac{r-1}{\rho_{0}}\right].$ (25)

This behaviour, which corresponds to the class of universality $\gamma=0$ in
(1), is that empirically found by Costa Filho et al. [8] in the distribution
of votes in the Brazilian electoral results. We have found such a behaviour in
i) the city-size distribution of small regions (as in the province of Huelva
(Spain) [21]) and ii) electoral results (as in the 2008 Spanish General
Elections results [22]). We depict in figures 1a and 1b the pertinent rank-
sizes distributions in semi-logarithmic scale, where a straight line
corresponds to a distribution of type (25). A large portion of the
distributions can be fitted to (25), with a correlation coefficient of $0.994$
and $0.998$, respectively. From these fits we have obtain a bulk density of
$\rho_{0}=0.058$ for the General Elections results, and in the case of Huelva
of $\rho_{0}=17.1$ ($N=77$, $\Omega=4.5$). Using historical data for the later
[21], we have used the backward differentiation formula to calculate the
relative growth rate of the $i$-th city as

$w_{i}=\frac{\ln k_{i}^{(2008)}-\ln k_{i}^{(2007)}}{\Delta t}$ (26)

where $k_{i}^{(2007)}$ and $k_{i}^{(2008)}$ are the number of inhabitants of
the $i$-th city in $2007$ and $2008$, respectively while $\Delta t=1$ year. We
show in figure 2 the empirical rank-plot of the relative growth, where we have
obtained $\overline{w}=0.011$ years-1 and $\sigma_{w}=0.030$ years-1, compared
with the rank-plot of a Maxwell-Boltzmann distribution with the same mean
value and standard deviation.

Figure 2: (colour on-line) Rank-plot of the growth rate $w$ of the province of
Huelva between 2007 and 2008 (red dots) compared with a Boltzmann distribution
with the same mean value and standard deviation (blue line).

However, these regularities are not always obvious to the naked eye, as shown
for the case of the most voted parties in Spain’08 or for the whole
distribution of the 2005 General Elections results in the United Kingdom [23]
(figure 1c). In both cases, the competition between parties seems to play an
important role, and the assumption of non-interacting elements can be
unrealistic 222The effects of interaction are studied in [24], where we go
beyond the non-interacting system using a microscopic description based on
complex networks..

###  5.1 Bulk and Zipf regimes

The situation in which $N/\Omega\rightarrow\,{\rm constant}\,\neq 0$ as
$N,\Omega\rightarrow\infty$ will be referred to herefrom as the _bulk regime_.
Now, in a recent communication [25], we show that Zipf’s law ($\gamma=1$ in
(1) with a slope of $-1$ in the rank-plot) can be derived from the
extremization of Fisher’s information with no constraints. In the
thermodynamic context studied here, the absence of normalization can be
understood as the inability of the system to reach the thermodynamic limit,
i.e. $N/\Omega\rightarrow 0$ as $N,\Omega\rightarrow\infty$. In this case the
system can not follow (24). Zipf’s law emerges as this behaviour of the
density in what we will accordingly denominate the _Zipf regime_
($N/\Omega\rightarrow 0$). We digress on the conditions for both regimes in
the example discussed below.

We have studied the system formed by all Physics journals [26] ($N=310$) using
their total number of cites as coordinate $k$. If a journal receives more
cites due to its popularity, it becomes even more popular and therefore it
will receive more cites. Under such conditions proportional growth and scale
invariance are expected. Since we consider all sub-fields of Physics,
correlation effects are much lower than they would be should we only consider
journals pertaining to an specific sub-field. Accordingly, the non-interacting
approximation seems to be realistic in this instance. In figure 3 we depict
the rank-plot of the number of citations in Physic journals, and find a slope
approaching $-1$ for the most-cited journals in the logarithmic representation
(figure 3a) and an slope in the vicinity of r $+1$ for the less-cited journals
(figure 3b). For the central part of the distribution, the bulk density
reaches a value of $\rho_{0}\sim 57$ (figure 3c).

Figure 3: (colour on-line) a, rank-plot of the total number of cites of
physics journal, from most-cited to less-cited, in logarithm scale. b, sorted
from less-cited to most cited c, same as a, in semi-logarithm scale. (Red
dots: empirical data; blue line: fit to (25)).

This distribution shows a notably symmetric behaviour under the change
$k\rightarrow 1/k$ ($u\rightarrow-u$). We exhibit in figure 4 the raw
empirical data as compared with the distribution obtained from the
transformation $k^{\prime}=c/k$ ($u^{\prime}=-u+\ln c$), where $c=3.3\times
10^{6}$. The main part of the density distribution reaches the bulk density
obeying (24), whereas Zipf’s law emerges at the edges, which could be
understood as constituting the _surface_ of the system, since they explain how
the density (exponentially) falls from its bulk-value to zero in $u$-space
when the system is exposed to an infinitely empty volume. This effect is
clearly visible in figure 5, where the empirical density distribution $p(u)du$
in $u$-space is compared with the “fitted” density

$p(u)=\left\\{\begin{array}[]{ll}\rho_{Z}e^{u-u_{1}}&\mathrm{if~{}}u<u_{1}\\\
\rho_{0}&\mathrm{if~{}}u_{1}<u<u_{2}\\\
\rho_{Z}e^{u_{2}-u}&\mathrm{if~{}}u>2\end{array}\right.$ (27)

where $\rho_{Z}=18$, $\rho_{0}=57$, $u_{1}=5.2$ and $u_{2}=10$. These findings
lead us to conclude that the system consisting of Physics journals, when
sorted by total number of citations, is a perfect example of a finite scale-
free ideal gas at equilibrium.

Figure 4: (colour online) Rank-plot of the total number of cites of Physics
journal, from most-cited to less-cited, compared with the distribution
obtained from the inverse transformation $k^{\prime}=3.3\times 10^{6}/k$ where
$k$ is the number of cites. Figure 5: (colour online) Empirical density
distribution in $u$-space of the total number of cites of Physics journals,
compared with (27). The bulk regime and the Zipf regime at the edges is
clearly visible.

###  5.2 An accompanying microscopic description

The dynamics of the system under scrutiny here can be microscopically
described as a stochastic process using (13) together with the density
distribution (21). Treating $w$ as a random variable, the pertinent stochastic
equation of motion is written in the guise of a geometrical Brownian motion,
i.e., [19]

$dk=k\overline{w}dt+k\sigma_{w}dW,$ (28)

where $dW$ represents a Wiener process. In the $u$-space, this equation reads

$du=\overline{w}dt+\sigma_{w}dW,$ (29)

and is known to describe the celebrated Brownian motion (28), which exactly
describes the dynamical condition found empirically in [7] and also the
(stochastic) proportional growth model used in [13] to obtain Zipf’s law. We
thus dare to suggest that we are dealing here with a sort of “equivalent” of a
molecular dynamics’ simulation for gases/liquids [27].

Indeed, (29) describes $N$ Brownian walkers moving in a fixed volume $\Omega$
with uniform density in $u$-space, a model used in the literature to describe
the ideal gas [27]. This scenario can also be reproduced by our free-scale
ideal gas merely by choosing to represent the system with the coordinates
$(k,v)$. In figure 1a we show the rank-plot for $k$ of a system of $N=78$
geometrical Brownian walkers with $\overline{w}=0.011$ and $\sigma_{w}=0.030$
in a volume $\Omega=4.5$, with d $k_{1}=200$ in reduced units, which
approximately describes the distribution of the population of the province of
Huelva. We also show in figure 2 the rankplot of $w$ of the same random
walkers, compared with the empirical data.

##  6 Conclusions

Our present considerations derive from the fact that, as shown in [14],
thermodynamics can be reformulated in terms of the minimization with
appropriate constraints of Fisher’s information. We have applied such
reformulation in order to discuss the thermodynamics of scale-free systems and
derived the density distribution in configuration space and the entropic
expression for the equilibrium state of what we call SFIG: the scale-free
ideal gas (in the thermodynamic limit). We have encountered convincing
empirical evidences of the SFIG actual existence in sociological scenarios.
Thus, we have dealt with city populations, electoral results and citations in
Physics journals. In such a context it is seen that Zipf’s law emerges
naturally as the equilibrium density of the non-interacting system when the
volume grows without bounds, a situation that we call the Zipf regime. Using
empirical data we have revealed that this regime can be understood as a
density-decay at the “surface” separating the bulk from an empty and very
large volume. Finally, we have shown with a simulation of city-populations
that geometrical Brownian motion can describe such systems at a microscopic
level.

## Acknowledgments

We would like to thank D. Puigdomenech, D. Villuendas, M. Barranco, R.
Frieden, and B. H. Soffer for useful discussions. This work has been partially
performed under grant FIS2008-00421/FIS from DGI, Spain (FEDER).

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