Spin revolution breaks time reversal symmetry of rolling magnets

The classical laws of physics are usually invariant under time reversal. Here, we reveal a novel class of magnetomechanical effects rigorously breaking time-reversal symmetry. These effects are based on the mechanical rotation of a hard magnet around its magnetization axis in the presence of friction and an external magnetic field, which we call spin revolution. The spin revolution leads to a variety of symmetry breaking phenomena including upward propulsion on vertical surfaces defying gravity as well as magnetic gyroscopic motion that is perpendicular to the applied force. The angular momentum of spin revolution differs from those of the magnetic field, the magnetic torque, the rolling axis, and the net torque about the rolling axis. The spin revolution emerges spontaneously, without external rotations, and offers various applications in areas such as magnetism, robotics and energy harvesting.

To describe the lifting effect, we start with a small initial rotation of the tubes by an angle  ( = /10   in Fig. 4b). The tubes will not be moved anymore, but the spheres may roll with velocity ( ) / d t dt  . In the next step we calculate e 1,2 ( ) q M t  by solving two coupled equations (Eq. (2), one for each sphere) for an instantaneous ( ) t  . The resulting

Code availability
The codes used for this study are available from the corresponding authors on reasonable request.

A: Rolling Torque
In the coordinate system of Fig. 1 (c) of the main manuscript, the rotational part of the motion of a rolling sphere is described by the rotational equation of motion

B: Competition between rolling and magnetic torques
A net mechanical torque producing a change in angular momentum c. .

D: Local and global time-reversal symmetry and spatial reflection symmetry.
In classical mechanics the operation of time reversal T is defined as , with t being the time and s the trajectory of an object of mass m . As a consequence the velocities and momenta (first derivatives in time) change the sign under T , while the forces and accelerations (second derivatives in time) do not ( ). A prototypical example for time-reversal symmetry breaking is the motion of an electron in a magnetic field due to the Lorentz force = . For the initial velocity v  and the magnetic field B  given in Fig. 1S the forward-intime electron's trajectory (left) differs from its backward-in-time trajectory (top-right), if the field orientation remains unchanged. Hence, the mapping of trajectory s in Eq. 13 does not hold or, in other words, the time-reversal symmetry is locally broken. Globally, however, the time-reversal symmetry of this motion is preserved, because the magnetic field is odd ( ) under the time transformation and, hence, if the orientation of field is reversed, the forward-in-time and backward-in-time trajectories coincide (bottom-right in Fig. 1S a) and no symmetry breaking occurs.
The spatial reflection symmetry of embodiment showing lifting force is shown in Fig. 1S b. For the xy reflection plane z z   , this symmetry is preserved only if the gravitational force is also reflected. That is, from the point of view of a person in the northern hemisphere, the downwards motion of the spheres is possible only in its southern counterpart.

E: Magnetic field and magnetic attraction force
The magnetic field from a sphere 1 acting on a sphere 2 can be calculated as 0 1 12 1 12 1 12 12 5 3 12 where 1 M  is the magnetic moment of the sphere 1, while 12 r  is the distance vector between the two spheres. The attractive force acting on sphere 1 can be determined as

F: Microscopic description of the lifting force
Initially, the spheres' magnetization is oriented along the radius-vectors of the tubes (dashed black lines in Fig. 3(e) of the main manuscript). Instantaneous fields of each sphere acting on the other sphere 12 ( ) B t  and 21 ( ) B t  can be calculated exactly (9) and have small mismatch with these radius-vectors (blue arrows in Fig. 3(e), please note that all angles are exaggerated for the sake of clarity). The magnetic torque tries to rotate 1,2 ( ) M t  towards the instantaneous fields. The magnetic attraction force m ( ) F t  (see Fig. 3 Fig. 3e)) and the tube's wall. This drifting force leads to an upward rolling of the magnets. The numerically calculated trajectory of such a motion is shown in Fig. 2S and described in the Methods section. It is a formidable task to find an analytical description of this complicated movement. The upper limit of the lifting force, however, can be approximated by l 12 m 12 m 12 with k being the friction coefficient. As the rolling friction is tiny (0.05-0.1 for metal/plastic interfaces), the lifting force can reach significant values.

G: Other tube's geometries
In the case with two vertical tubes, the gravitational force leads to an inclination of the magnetization axis with respect to the surface's normal within the zx-plane (see Fig. 3S(a)). This deviation alone would allow the sphere to roll left or right along the tube's circumference rotating around the magnetization axis. The tube's rotation (external force appl F  ) in the xy-plane leads to the inclination of the magnetization axis with respect to the surface's normal within the xy-plane (see Fig. 3S(b)). This global inclination of the magnetization axis with respect to the surfaces' normal allows for two possible scenarios: the spheres roll up towards one another or down away from one another independent on the direction of the tube's rotation. Two other possibilities (away-upwards and down-towards are excluded by the orientation of the equilibrium magnetization axis). The away-downwards scenario leads to increasing distances between the spheres. Hence, if the magnetic attraction is stronger than the gravitation, the spheres will roll upwards decreasing magnetic potential energy and breaking the time reversal symmetry. Summarizing, the deviation of mg  from to the  plane defined by appl F  and N  is necessary to achive the SR.
If mg    , e.g. the tubes lie on a horizontal surface, the SR does not appear, because the deviation of the magnetization axis with respect to the surface normal due to gravitation and due to tube's rotation lie in the same plane (zy-plane in Fig. 3S(c-d)). Hence, there two rolling scenarios returning the spheres back to the smallest separation distance (rolling right or left along the x-axis and towards one another) are energetically degenerate. Therefore, on average the sphere will go neither right nor left, just performing statistical movements at the origin. Already tiniest deviation from the tube's horizontality, however, is enough to achive the SR.   Movie S1: The system to observe the spin revolution The system consists of a glass or plexiglass test tube fixed on a tripod. The inclination and spatial orientation of the glass tube can be smoothly changed.

Movie S2: Spin revolution
This video shows a magnetic NiFeB sphere ( 2 s 0.5A M m   , = 0.0003 m kg, = 0.003 R m) located within a test tube. Initially, its magnetization vector coincides with that of the Earth's magnetic field. The south magnetic pole of the sphere is marked by a dark cross. Inclination of the tube leads to a depart of the magnetization from its initial orientation and to a new eauilibrium magnetization orientation corresponding to the minimal net torque about the rolling axis. When the rolling torque becomes comparable with the magnetic torque, the sphere revolves up about the new equilibrium magnetization axis and rolls down the incline.

Movie S3: Change of the revolution direction and drift
If one puts a non-magnetic sphere into a test tube and lets the tube roll in the direction v  coinciding with appl F  , the sphere will roll in the same direction and remain at initial position with respect to the tube due to the friction. If one replaces a non-magnetic sphere by a magnetic one (for example,  Fig. 2(f) of the main manuscript) the sphere changes the direction of drift to the opposite.

Movie S4: Flexible change of the revolution's angular velocity
Various sources of external magnetic field other than the Earth's magnetic field can be used to realize the magnetic revolution. This video gives an example of such an implementation. Here, a permanent or electro-magnet producing a field is situated underneath a non-magnetic plane, on which a magnetic sphere is positioned. Initially, the magnetic sphere (or an object of another shape) is attracted by the magnetic field and rests. In the next step, the magnetic field starts moves/changes. As a consequence, the magnet revolves-up and moves in the same direction. The strength and orientation of angular momentum depends on the strength and orientation of the field and the driving velocity. If the orientation of equilibrium magnetization reverses, the angular momentum reverses as well.

Movie S5: Lifting force
This video shows an embodiment of spin revolution, comprising two non-magnetic tubes with magnetic spheres inside them. Independently of the angular velocity of the tubes the magnetic spheres lift up the tubes.