# Enhanced Condensation Through Rotation

Maxim Chernodub, Frank Wilczek

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Maxim Chernodub, Frank Wilczek. Enhanced Condensation Through Rotation. 2025. hal-04903167

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Distributed under a Creative Commons Attribution 4.0 International License
Enhanced Condensation Through Rotation

Maxim Chernodub¹,² and Frank Wilczek³,⁴,⁵,⁶,⁷,⁸

¹Institut Denis Poisson UMR 7013, Université de Tours, 37200 France

²Department of Physics, West University of Timişoara,
Bd. Vasile Pârvan 4, Timişoara 300223, Romania

³Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

⁴T. D. Lee Institute, Shanghai 201210, China

⁵Wilczek Quantum Center, Department of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai 200240, China

⁶Department of Physics and Origins Project, Arizona State University, Tempe, Arizona 25287, USA

⁷Department of Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden

⁸Nordita, Stockholm University and KTH Royal Institute of Technology,
Hannes Alfvén's väg 12, SE-106 91 Stockholm, Sweden

(Dated: January 6, 2025)

We argue that the rotation of a thin superconducting cylinder can increase the critical temperature of the superconducting phase transition substantially. The phenomenon can be interpreted as an effective negative moment of inertia associated with condensation of Cooper pairs. We give quantitative estimates for a thin cylinder of aluminum.

**Introduction.** The core observation we make here is very simple, when stated naively. Motivated by a “two-fluid” picture, one might expect that a superfluid or superconducting condensate decouples from the rotational motion of the normal component. This decreases the moment of inertia, and thus the energy that would otherwise be associated with rotational motion at an imposed velocity. Thus, it becomes energetically advantageous to put more substance into the condensate, and this effect increases the critical temperature. As we shall discuss, that naive motivating thought, after major revision and qualification, retains an important element of truth.

Below we review and extend the theory of the rotating superconductors, and show that the rotation of a thin superconducting cylindrical shell can catalyze the emergence of superconductivity. The effect is facilitated by the mismatch of the normal and supercurrent velocities and can be attributed to an effective negative moment of inertia for the condensate of Cooper pairs.

**Magnetic Coupling: General** Consider a solid superconducting cylinder rotating uniformly with a constant angular velocity $\Omega$ about its symmetry axis. At zero temperature, all electrons form Cooper pairs and condense into a charged superfluid, which interacts with a rotating, positively charged ionic lattice. In the absence of mechanical friction between the ionic lattice and the charged superfluid condensate, one might naively argue that the superfluid component would remain in a static, non-rotating state to minimize its kinetic energy. Such behavior would be analogous to the lack of rotational response expected of a neutral superfluid confined within a very slowly rotating vessel. Here, however, the rotation of the crystal lattice induces a circular electric current of positively charged ions. This current produces a magnetic field along the rotation axis, perceived by the charged superfluid as an external background field. The

magnetic field generated by the rotating crystal arises intrinsically from within the bulk at every point of the superconductor.

To mitigate the effect of this energetically costly bulk magnetic field, which frustrates superconductivity, the condensate produces a Meissner supercurrent. In this way, the negatively charged superfluid fraction synchronizes its velocity with the velocity of the positively charged crystal lattice, ensuring that in the bulk of the superconductor, the total electric current vanishes. Thus, even in the absence of a phonon-mediated coupling between the rotating ionic lattice and the condensate, the rotation rigidly drags the charged superfluid via a photon-mediated interaction in bulk.

Still, a rotating superconductor, regardless of its chemical composition, develops the bulk magnetic field (which is also called the “London magnetic field”) [1, 2]:

$$B_L = \frac{2mc}{e} \Omega. \qquad (1)$$

This field is generated by a surface layer of the cylinder, where the velocities of the normal and condensed electronic fractions differ from each other [3]. With this, the bulk vector potential relieves the potential for frustration associated with non-zero vorticity of the superflow.

**Magnetic Coupling: Thin Cylinder Geometry and Free Energy** Consider now, specifically, a hollow cylinder made of a thin superconducting film as shown in the inset of Fig. 1. Following the Little-Parks setup [4], we consider a thin superconducting film of a thickness $d$ deposited on a cylindrical insulator of a radius $R \gg d$. Contrary to the Little-Parks experiment, we do not apply an external magnetic field. If the thickness of the film $d$ is smaller than the London penetration length, $\lambda_L$, then the rotation of the ionic lattice produces a negligible Meissner current, and the kinetic energy of the condensate can
be neglected. In this case, at a finite temperature $T$ below the superconducting phase transition, $T < T_c$, the electrons are shared between the superconducting condensate and the normal electron component. In addition to requiring $d \lesssim \lambda_L$, we take the thickness of the film to be smaller than the coherence length, $d \lesssim \xi$. In this case, the absolute value of the superconducting order parameter $|\psi|$ is a spatially homogeneous quantity. Spatial dependence of the condensate appears only in its phase: $\psi(\mathbf{x}) = |\psi|e^{i\theta(\mathbf{x})}$ [5].

The Ginzburg-Landau approach to rotating superconductors has an extensive literature [3, 6-8]. Below, we reexamine the energy balance for a thin superconducting cylinder, bringing out the importance of the fact that normal and superconducting electrons share the same reservoir of charge carriers.

The total free energy of a rotating superconductor,

$$F = F_{\text{supr}} + F_{\text{mech}} + F_{\text{magn}}, \quad (2)$$

is a sum of the contributions coming from the superconducting condensate, $F_{\text{supr}}$, the classical mechanical motion of the non-superconducting electronic component $F_{\text{mech}}$, and the magnetic field generated by the circular motion of the electrically charged normal constituent, $F_{\text{magn}}$, respectively.

The Ginzburg-Landau (GL) free energy of the superconducting condensate $\psi = \psi(\mathbf{x})$ is [5]:

$$F_{\text{supr}} = \int_{V_s} d^3 x \left[ \frac{1}{4m} \left| \left( \frac{\hbar}{i} \nabla + \frac{2e}{c} \mathbf{A} \right) \psi \right|^2 + \alpha |\psi|^2 + \frac{\beta}{2} |\psi|^4 \right], \quad (3)$$

where the gauge field $\mathbf{A}$ corresponds to an electromagnetic background generated by the rotating environment, and $\alpha$ and $\beta > 0$ are the GL parameters. [9] The superconductivity is supported by the finite density of the Cooper pairs, $n_s = |\psi|^2$. Each pair has a mass of $2m$ and an electric charge of $-2e$ (twice that of an electron, with $m = m_e$ and $e = |e| > 0$). The integral in Eq. (3) is taken over the whole volume $V_s$ of the superconductor.

The mechanical energy of the normal component, the second term in Eq. (2), corresponds to the sum of the rotational kinetic energies of electrons in the normal state and the ions in the crystal lattice, respectively:

$$F_{\text{mech}} = \int d^3 x \left[ \frac{1}{2} \rho_n(\mathbf{x}) v_n^2(\mathbf{x}) + \frac{1}{2} \rho_I(\mathbf{x}) v_I^2(\mathbf{x}) \right]. \quad (4)$$

Here $\rho_n$ and $\rho_I$ are the mass densities of the normal fraction of electrons and the ionic lattice. The local velocity $v_n$ of the normal fraction of electrons and the velocity of the ionic lattice $v_I$ coincide,

$$v_n(\mathbf{x}) = v_I(\mathbf{x}) = v(\mathbf{x}) = \Omega \times \mathbf{x}, \quad (5)$$

because the phonon-mediated interaction synchronizes their rotational motion in thermal equilibrium.

The mechanical rotational energy of the normal-state electrons can be inferred by noticing that both normal and superconducting electrons share a common reservoir. Consequently, the number density of normal electrons $n_n$ is directly related to the number density of the superconducting Cooper pairs $|\psi|^2$: a stronger superconducting condensate leads to a reduced population of normal electrons and vice versa. To calculate the density of normal electrons, we notice that in thermal equilibrium, the superconductor is electrically neutral at every point. Therefore, the electric charge density of the superconducting component, $-2e|\psi|^2$, is compensated by the sum of the charge densities of the normal component, $-en_n$, and the ionic lattice, $+eZ_I n_I$. The latter is expressed via the number density of ions, $n_I$, and the electric charge of each ion, $+eZ_I$. Then, the local neutrality condition, $-2e|\psi(\mathbf{x})|^2 - en_n + eZ_I n_I = 0$, gives us the number density of the normal electrons, $n_n(\mathbf{x}) = Z_I n_I - 2|\psi(\mathbf{x})|^2$, as well as their mass density:

$$\rho_n(\mathbf{x}) \equiv mn_n(\mathbf{x}) = m(Z_I n_I - 2|\psi(\mathbf{x})|^2). \quad (6)$$

The local number density of ions, $n_I$, does not depend on the angular velocity $\Omega$ since the non-relativistic rotation does not deform the ionic lattice. Thus, the mass density of the ions is a constant quantity, $\rho_I(\mathbf{x}) = M_I n_I$, where $M_I$ is an effective mass of an ion in the crystal.

Equations (4), (5) and (6) provide us with the rotational energy of the normal part of the system, which incorporates the ions and the normal fraction of electrons:

$$F_{\text{mech}} = F_{\text{mech}}^{(0)} + \frac{I_s}{2} \Omega^2. \quad (7)$$

Here, the first term

$$\begin{aligned} F_{\text{mech}}^{(0)} &= \int_{V_s} d^3 x (M_I + mZ_I) n_I \frac{r_{\perp}^2 \Omega^2}{2} \\ &= \pi (M_I + mZ_I) n_I L_z d R^3 \Omega^2, \end{aligned} \quad (8)$$

corresponds to the rotational energy of the system in the absence of the superconducting condensate, $\psi = 0$, if all electrons were in the normal state. [10] Contribution (8) does not depend on the superconducting order parameter $\psi$ and, therefore, it will be disregarded below. We use cylindrical coordinates $\mathbf{x} = (r_{\perp}, \varphi, z)$ with the symmetry axis of the cylinder pointing out along the $z$ direction.

The last term in Eq. (7) has an appearance of the rotational kinetic energy of a classical body possessing the moment of inertia $I_s$. Remarkable properties of this term are that the emergent effective classical moment of inertia $I_s$ (i) depends explicitly on the quantum superconducting condensate $\psi$ and (ii) has a negative value in the superconducting state with $\psi \neq 0$:

$$I_s = -2m \int_{V_s} d^3 x r_{\perp}^2 |\psi(\mathbf{x})|^2 \le 0. \quad (9)$$
This effect can alternatively be interpreted as a negative moment of inertia associated with the condensate of Cooper pairs: the larger the density of the superconducting pairs, the lower the total rotational energy carried by electrons in the normal fraction. In other words, an increase in the superconducting density diminishes the rotational energy of the normal electronic component. [11]

Note that the negative moment of inertia (9) saturates when all the available electrons have condensed.

The energy of the magnetic field enters as the last term in Eq. (2). In the presence of the superconducting condensate, the normal component —that comprises both the normal electrons and the ionic lattice— has a non-vanishing charge density. The circular motion of electric charges leads to a circular electric current density $J = J_I + J_n$, associated with the rotation of the ionic lattice, $J_I = eZ_I n_I v_I$, and the normal fraction of electrons, $J_n = -e n_n v_n$. Using Eq. (5) together with the condition of the local charge neutrality, we obtain that the electric current density of the normal component is proportional to the superconducting density $|\psi|^2$:

$$J(\mathbf{x}) = 2e|\psi(\mathbf{x})|^2(\Omega \times \mathbf{x}). \quad (10)$$

The electric current (10) generates the magnetic field $B_\Omega = \nabla \times A$ according to the Ampère law:

$$\nabla \times B_{\Omega}(\mathbf{x}) = \frac{4\pi}{c} J(\mathbf{x}). \quad (11)$$

This field enters the free energy (2) via the gauge potential $A$ in the Ginzburg-Landau functional (3) and also contributes directly to the energy of the system:

$$F_{\text{magn}} = \frac{1}{8\pi} \int d^3 x \, B_{\Omega}^2(\mathbf{x}). \quad (12)$$

Here, the integral is evaluated over the entire space because the generated magnetic field extends beyond the spatial boundaries of the superconductor.

For a thin film, the current density (10) can be written in the form: $J(\mathbf{x}) = 2e|\psi|^2 \Omega R d\theta \cdot \delta(r_\perp - R)e_\varphi$, where $e_\varphi$ is a polar vector. Then, the Ampère equation (11) gives us the magnetic field parallel to the cylinder axis $\mathbf{e}_z$,

$$B_{\Omega} = \frac{8\pi e}{c} |\psi|^2 R d\Omega \Theta(R - r_{\perp}) \mathbf{e}_z, \quad (13)$$

where $\Theta(x)$ is the Heaviside function with $\Theta(x) = 1$ for $x > 0$ and $\Theta(x) = 0$ otherwise.

The classical magnetic field (13) is proportional to the superconducting density $n_s \equiv |\psi|^2$ that controls the electric charge density of the normal component (the normal fraction of electrons and the ionic lattice). In the normal phase, the condensate vanishes $\psi = 0$, the rotating system becomes electrically neutral, and no magnetic field should be produced in agreement with Eq. (13).

It will be convenient to work with the condensate $\psi$ normalized to its zero-temperature value $\psi_0$ in a non-rotating superconductor [12]:

$$|\bar{\psi}|^2 = \frac{|\psi|^2}{|\psi_0|^2}, \quad |\psi_0|^2 = \frac{|\alpha_0|}{\beta_0} = \frac{mc^2}{8\pi e^2} \frac{1}{\lambda_0^2}, \quad (14)$$

where $\alpha_0 = \alpha(T=0)$ and $\beta_0 = \beta(T=0)$ are the parameters of the GL model (3) at zero temperature. The penetration depth $\lambda_0$ and the coherence length $\xi_0$ at $T=0$, expressed via the parameters of the GL model (3), are:

$$\lambda_0^2 = \frac{mc^2\beta_0}{8\pi e^2|\alpha_0|}, \quad \xi_0^2 = \frac{\hbar^2}{4m|\alpha_0|}. \quad (15)$$

We also use the angular frequency $\Omega$ in the dimensionless units and introduce the geometrical factor $\gamma$:

$$\bar{\Omega} = \frac{\Omega}{\Omega_0}, \quad \Omega_0 = \frac{\hbar}{2m\xi_0 R}, \quad \gamma = \frac{Rd}{\lambda_0^2}. \quad (16)$$

The magnetic field (13) appears only in the interior of the cylinder and vanishes outside it. Its contribution (12) to the total free energy (2) is:

$$F_{\text{magn}} = \frac{L_z}{2} \left( \frac{\phi_0}{4\pi\xi_0} \right)^2 \gamma^2 \bar{\Omega}^2 |\bar{\psi}|^4, \quad (17)$$

where $L_z$ is the length of the cylinder and $\phi_0 = 2\pi\hbar/(2e)$ is the magnetic flux quantum.

The free energy (3) for the superconducting condensate,

$$F_{\text{supr}} = 2\pi R d L_z \left( m v_s^2 |\psi|^2 + \alpha |\psi|^2 + \frac{\beta}{2} |\psi|^4 \right), \quad (18)$$

where the superfluid velocity of Cooper pairs

$$v_s = \frac{1}{2m} \left( \hbar \nabla \theta + \frac{2e}{c} A \right) = \frac{\hbar}{2mR} \left( n + \bar{\phi}_{\Omega} \right) e_{\varphi}, \quad (19)$$

is defined by the winding number $n \in \mathbb{Z}$ of the phase of the condensate, $\theta \equiv \arg \psi = n\varphi$, and the total magnetic flux $\phi_\Omega = \pi R^2 B_\Omega$ of the magnetic field (13) produced by the rotating cylinder:

$$\bar{\phi}_{\Omega} = \frac{\phi_{\Omega}}{\phi_0} = \frac{\gamma R}{2\xi_0} \bar{\Omega} |\bar{\psi}|^2, \quad (20)$$

given in units of the elementary flux quantum $\phi_0$. Inserting these values, we have

$$F_{\text{supr}} = \gamma L_z \left( \frac{\phi_0}{4\pi\xi_0} \right)^2 \left[ \left( \frac{\xi_0}{R} \right)^2 (n + \bar{\phi}_{\Omega})^2 |\bar{\psi}|^2 + \frac{\alpha}{|\alpha_0|} |\bar{\psi}|^2 + \frac{1}{2} \frac{\beta}{\beta_0} |\bar{\psi}|^4 \right]. \quad (21)$$

Finally, the contribution of the negative moment of inertia, Eqs. (7) and (9), associated with the superconducting electrons can conveniently be written as follows:

$$F_{\text{mech}} = -\gamma L_z \left( \frac{\phi_0}{4\pi\xi_0} \right)^2 |\bar{\psi}|^2 \bar{\Omega}^2. \quad (22)$$
Combining magnetic (17), superconducting (21) and mechanical (22) terms, we get the total free energy (2):

$$F = F_0 f(\bar{\psi}, \bar{\Omega}, n), \quad F_0 = \gamma L_z \left( \frac{\phi_0}{4\pi\xi_0} \right)^2, \quad (23)$$

where

$$f(\bar{\psi}, \bar{\Omega}, n) = \frac{1}{2} \left( \frac{\beta}{\beta_0} + \gamma \bar{\Omega}^2 \right) |\bar{\psi}|^4 + \left[ \frac{\alpha}{|\alpha_0|} - \bar{\Omega}^2 + \left( \frac{\xi_0}{R} \right)^2 \left( n + \frac{\gamma \bar{\Omega} R}{\xi_0} |\bar{\psi}|^2 \right)^2 \right] |\bar{\psi}|^2. \quad (24)$$

The final term (i.e., the term proportional to $(\xi_0/R)^2$) in Eq. (24) can be neglected. Indeed, in the thermodynamically favored ground state, the winding number $n \in \mathbb{Z}$ adjusts itself in such a way that $n + \gamma R \bar{\Omega} |\bar{\psi}|^2 / \xi_0 \sim O(1)$. On the other hand, the prefactor in this term is small ($\xi_0 \ll R$), while the other contributions entering the second line of Eq. (24) are of the order of unity.

Upon omitting that term, $f(\bar{\psi}, \bar{\Omega}, n) = f(\bar{\psi}, \bar{\Omega}) + \dots$, we arrive at

$$f(\bar{\psi}, \bar{\Omega}) = a(T, \bar{\Omega}) |\bar{\psi}|^2 + \frac{1}{2} b(\bar{\Omega}) |\bar{\psi}|^4, \quad (25)$$

where $a = \alpha / |\alpha_0| - \bar{\Omega}^2$ and $b = \beta / \beta_0 + \gamma \bar{\Omega}^2$.

Following the original GL prescription, we assume a linear temperature dependence of $\alpha = \alpha_0(T/T_c - 1)$ and neglect temperature dependence in the self-interaction of the condensate, $\beta = \beta_0$. Then the coefficients in Eq. (25) become

$$a(T, \Omega) = \frac{T}{T_c^{(0)}} - 1 - \frac{\Omega^2}{\Omega_0^2}, \quad b(\Omega) = 1 + \gamma \frac{\Omega^2}{\Omega_0^2}, \quad (26)$$

where $T_c^{(0)} \equiv T_c(\Omega = 0)$ is the critical temperature of the superconducting transition in a non-rotating cylinder.

The total free energy of the thin rotating cylinder, Eqs. (23) and (25), has a form of the standard Ginzburg-Landau potential with the coefficients (26) modified by rotation. The effect of rotation has a straightforward physical interpretation: (i) the negative moment of inertia associated with the condensate of Cooper pairs renders the condensation energetically more favorable by decreasing the coefficient $a$; (ii) the rotating environment generates a magnetic field, which translates into the enhanced coupling $b$ of the interaction of Cooper pairs.

The onset of superconductivity is determined by the requirement $a(T, \Omega) = 0$. Using Eq. (26), we obtain the influence of rotation on the critical temperature:

$$T = T_c^{(0)} \left( 1 + \frac{\Omega^2}{\Omega_0^2} \right), \quad (27)$$

where the characteristic angular velocity $\Omega_0$ is given in Eq. (16). Rotation increases the critical superconducting temperature (27), as shown in Fig. 1.

<figure><img src="image_2.png" /><figcaption>FIG. 1. The critical temperature $T_c(\Omega)$, Eq. (27), of the superconducting phase transition of a thin cylinder (shown in the inset) rotating with the angular frequency $\Omega$, given in units of the characteristic frequency (16).</figcaption></figure>

**Estimating the strength of the effect.** The rotational catalysis of superconductivity is more pronounced at a larger radius of the cylinder. Indeed, an increase in the radius $R$ at a fixed angular frequency $\Omega$ also increases the rotational kinetic energy of a non-condensed electron, $\epsilon_{\text{kin}} = m\Omega^2 R^2 / 2$, thus promoting the formation of static Cooper pairs and reducing the overall energy of the system. To estimate the strength of this rotational effect, we choose a macroscopically large radius of the cylinder, $R = 1$ cm, which is substantially larger than the one ($\simeq 0.7$ µm) used in the Little-Parks experiment [4].

The characteristic angular velocity (16) is inversely proportional to the coherence length $\xi_0$, implying that a lower frequency of rotation can be achieved in superconducting materials with a larger $\xi_0$. Therefore, we choose a superconducting film made of pure aluminum (Al) since this material has an exceptionally long coherence length in bulk, $\xi_{Al} \simeq 1.6$ µm [12]. Notice that elemental tin (Sn) used in the original Little-Parks experiment has a much shorter coherence length, $\xi_{Sn} \simeq 0.23$ µm [12]. Also, for sufficiently small thicknesses, $d \sim 50$ nm, the penetration length $\lambda$ is larger than the width $d$ [13], implying that the film satisfies the required conditions $d \lesssim \lambda_0$ and $d \lesssim \xi_0$.

Taking $R = 1$ cm and $\xi_0 = 1$ µm, we get from Eq. (16) the characteristic rotation rate $\nu_0 = \Omega_0 / (2\pi) \simeq 0.9$ kHz, which does not seem outlandish. A cylinder, rotating at $\Omega = \Omega_0$, will have a twice higher critical temperature of superconducting transition (27) compared to a static case: $T_c(\Omega = \Omega_0) = 2T_c(\Omega = 0)$.

An experiment can be performed in a rotating cylindrical jar made of a thermally conducting dielectric material, covered by an aluminum film at the lateral surface, and filled with superfluid helium. The superconducting critical temperature of a 50 nm-thick aluminum film is about 1.25 K [14, 15], while helium loses superfluidity at 2.17 K. This nearly two-fold temperature margin allows us to test the increase of the superconducting tempera-
ture due to rotation (27). The critical superconducting
temperature can be found by illuminating the rotating
cylinder with microwave photons and measuring their
absorption coefficient, which serves as a reliable tool for
detection of the superconducting energy gap [12, 16].

In the Supplemental Material, we assess several factors
that could potentially challenge the experimental imple-
mentation of the proposed mechanism and demonstrate
that their impact is negligible.

**Conclusions.** We show that at finite temperatures, the condensate of Cooper pairs can possess a negative moment of inertia due to sharing a common reservoir of electrons with the normal, non-condensed fraction. This property becomes apparent in a thin rotating superconducting cylinder, where the superconducting condensate decouples from the rotational motion. We argue that the rotation can lead to a significant enhancement of the critical temperature of the superconducting transition. We estimated the effect in a cylinder made of a thin aluminum film and pointed out its experimental feasibility.

For superfluid helium 4 we must consider the possi-
bility of vortex creation, which ruins an analysis based
on rigid motion, unless the rotation is exceedingly slow.
Analogous effects in other, more complex superfluids are
under study.

MC is partially supported by the EU's NextGen-
erationEU instrument through the National Recover-
ery and Resilience Plan of Romania - Pillar III-C9-
I8, managed by the Ministry of Research, Innovation
and Digitization, within the project FORQ, contract
no. 760079/23.05.2023 code CF 103/15.11.2022. FW is
supported by the U.S. Department of Energy under grant
Contract Number DE-SC0012567 and by the Swedish Re-
search Council under Contract No. 335-2014-7424.

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[9] For a rotating lattice, the GL energy (3) can be extended by an additional term accounting for the energy associated with the relative motion between the ionic lattice and the superconducting condensate [3]. We omit this term as it leads only to a renormalization of the effective electron mass ($m \rightarrow m^*$), which does not play a significant role in our analysis.

[10] Hereafter, we ignore geometrical $O(d/R)$ subleading corrections that arise due to a finite film thickness $d \ll R$.

[11] A negative value of the rotational energy associated with the superconducting fraction of electrons, Eqs. (7) and (9), does not in any way imply that the electrons in the superconducting state possess a negative mass. On the contrary, the mass of a Cooper pair is a positive quantity. A negative moment of inertia has also been found in numerical simulation of a completely different physical system, a hot gluon plasma [17].

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Supplemental Material

Here, we discuss several factors associated with an experimental realization of the proposed mechanism.

(i) Thin superconducting films generally exhibit reduced coherence lengths $\xi$ compared to the same materials in bulk. This effect originates primarily due to the surface scattering and reduced electron mean free paths since decreasing the film thickness $d$ eventually reduces the grain size in the material [15]. However, elemental aluminum is a very clean metal for which the coherence length $\xi$ remains relatively long even in thin films, being close to its bulk value [18]. Even higher values of $\xi \simeq 8.86 \mu$m were reported in cleaner aluminum films grown by molecular beam epitaxy [19].

(ii) An aluminum film of the thickness $d \simeq 50$ nm has the penetration depth $\lambda_0 \simeq 120$ nm [13], implying that in our setup, the geometrical factor (16) is a very large number, $\gamma \sim 3.5 \times 10^4$. Therefore, one can suspect that the effective magnetic field generated by the cylinder (13),

$$B_{\Omega} = \frac{\gamma \phi_0 \bar{\Omega}}{2\pi R \xi_0} |\bar{\psi}|^2, \qquad (A.1)$$

could reach rather high values that might potentially destroy the superconductivity in thin film.

The maximal strength of the generated magnetic field (A.1) is reached at zero temperature, when all electrons are condensed in the form of Cooper pairs, implying $|\bar{\psi}| = 1$. For our set of parameters, the cylinder rotating with the typical angular frequency $\Omega = \Omega_0$ at zero temperature produces the magnetic field (A.1) of the order of 10 G, which is substantially weaker than the critical value [12] $B_c \simeq 100$ G for the bulk aluminum at $T = 0$. Moreover, the critical value $B_{c\parallel}$ of the magnetic field parallel to the film is strongly enhanced compared to the bulk critical field $B_c$ [20]. For example, for an aluminum film of the thickness $d \sim 100$ nm, the critical magnetic field is $B_{c\parallel} \sim 10^4$ G [15]. These estimations indicate that the magnetic field produced by the rotating cylinder cannot substantially affect the emerging superconductivity.

For reference, the London magnetic field (1) at the characteristic rotation rate $\nu_0 = \Omega_0/(2\pi) \simeq 0.9$ kHz corresponding to the chosen characteristic frequency (16) has a much smaller value: $B_L \simeq 6.6 \times 10^{-4}$ G. This field would have been produced in a solid cylinder that generates large screening Meissner currents in bulk (as opposed to a hollow cylinder made of a thin superconducting film that we discuss in our article).

(iii) A large value of the geometrical factor $\gamma$ decreases the superconducting condensate $\psi$ as a result of the enhancement of self-interaction $b$ of Cooper pairs (26) in the GL potential (25). For our parameters at $\Omega = \Omega_0$, we get the suppression by two orders of magnitude, since $|\psi|/\psi_0 = \sqrt{|a|/b}$ and $b \simeq \gamma\Omega^2/\Omega_0^2$ for $\gamma \gg 1$. However, the condensate is still formed, albeit with a smaller density of condensed Cooper pairs.

Notice that for clean, weakly disordered thin aluminum films, the superconducting transition is a second-order transition, similar to bulk aluminum [18]. Within our simple approach based on the GL formalism, the rotation does not change the order of the phase transition.

(iv) The centrifugal force acting on an electron of the normal, non-condensed fraction,

$$F_{\text{cf}} = -m\Omega \times (\Omega \times R) \equiv m\Omega^2 R e_{\rho}, \qquad (A.2)$$

leads to a voltage drop

$$\Delta V = \frac{F_{\text{cf}}d}{e} = \frac{\hbar^2 \bar{\Omega}^2}{4me\xi_0^2 R}, \qquad (A.3)$$

along the radial direction $\rho$ across the film. For our set of parameters, this effect is also negligible, $\Delta V \sim 10^{-13}$ V.

(v) One should also make sure that the fast rotation of the cylindrical jar containing superfluid helium does not destroy the superfluidity itself. While the jar spinning at the angular frequency of the order of $(1-10) \text{s}^{-1}$ should definitely lead to the formation of a lattice of quantized vortices with normal-fluid cores, superfluid helium will only experience a transition to a normal, non-superfluid state when these vortex cores start to overlap. This transition appears at the practically unachievable angular frequencies of $\Omega_{c2} \sim 10^{11} \text{s}^{-1}$ [21]. Therefore, the helium superfluidity will not be destroyed at the suggested experimental parameters.