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Time crystals: Can diamagnetic currents drive a charge density wave into rotation?

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Time crystals: Can diamagnetic currents drive a charge density wave into rotation?

Philippe Nozieres`

Institut Laue Langevin - BP 156, F-38042 Grenoble Cedex 9, France, EU

received 18 July 2013; accepted in final form 26 August 2013 published online 23 September 2013

PACS 75.20.-g – Diamagnetism, paramagnetism, and superparamagnetism PACS 72.15.Nj – Collective modes (e.g., in one-dimensional conductors)

Abstract – It has been recently argued that an inhomogeneous system could rotate spontaneously in its ground state —hence a “time crystal” which is periodic in time. In this letter we present a very simple example: a superfluid ring threaded by a magnetic field which develops a charge density wave (CDW). A simple calculation shows that diamagnetic currents cannot drive rotation of the CDW, with a clear picture of the cancellation mechanism.

editor’s choice Copyright c EPLA, 2013

In a recent letter [1] Frank Wilczek introduced a revo- ν = 0, with a diamagnetic current J = −n0qA/mc.As- lutionary concept, “time crystals” which in their ground sume now that the ring presents a spontaneous charge den- state would be time dependent. A simple example is a ring sity wave with a density modulation n1. We do not need threaded by a magnetic field that breaks time-reversal in- a periodic n1, we only request a modulation, n1dx =0. variance. If a charge density wave (CDW) appears, will The current equation becomes diamagnetic currents put it in rotation? Such a challeng-    n0 + n1 qA ing proposal raised a vivid controversy [2], especially with J =  grad S − . my colleague Patrick Bruno who recently provided a gen- m c eral proof that it is impossible [3]. In order to clarify the underlying physics, I study here a very simple model in J is conserved and grad S cannot vanish. We must solve which elementary calculations can be done explicitly from the equation beginning to end. qA mJ Consider a circular ring with radius R, perimeter L =  S grad = c + n n 2πR, threaded by a magnetic flux Φ. The vector poten- 0 + 1 tial along the ring is A =Φ/L.AsamodeltakeaBose   with the two conditions grad Sdx =0and n1dx =0 condensate of charge q particles (for Cooper pairs q =2e) which will fix the modulation grad S and the unknown with a density n0 = N0/L. The superfluid phase is S,the current J. Integrating that equation over the ring we find current density on the ring and the energy are    2   n0 qA N0 qA qA 1 J =  grad S − ,E0 =  grad S − . + mJ =0. m c 2m c c n S πν The circulation of grad is quantized, equal to 2 ,where Note that regions of small n severely reduce the current J, ν p is an integer. The linear momentum per particle is = a feature pointed out long ago by Tony Leggett [4]. Here  grad S, without the gauge term. The angular momentum n1 is small and in lowest order we find per particle is   πν 2 2 n0qA n1 Lz = R grad S = R = ν. J − − . πR = 1 2 2 mc n0 We recover the usual quantization of angular momentum. Consider first a weak magnetic field whose flux is smaller The charge density wave reduces the diamagnetic current 2 than half a quantum: the ground state corresponds to by an amount of order n1. The ground-state energy may

57008-p1 Philippe Nozi`eres be written as unfolds as before. The energy in the laboratory frame may   be written as qA 1  2 E0 = J  grad S − = n n qA 2 c E 0 + 1  S −   0 = m grad c 2 2 2 2 qAJ N0q A n1 − − . n0 + n1 c = mc2 1 n2 2 2 0 = m  2    qA qA ×  S −  S − m2 2R2 Note that the charge density wave reduces the energy, a grad c grad c + Ω feature that should eventually be added to the usual Lan-     qA n 1  2 2 dau picture for the appearance of 1, = J  grad S − + n0mΩ R , 2 c α 2 4 2  ECDW = −αn1 + βn1 −→ n1 = . wherewehavesetA = A − mcΩR/q. The constant 2β current J  is   Diamagnetism enhances the charge density instability. n qA n2 J  − 0 − 1 . Finally, the angular momentum Lz, zero for a diamag- = 1 2 mc n0 netic current in a perfect ring, does not vanish any more when the density wave appears: We find the energy   2   2 N0 q A A n Lz = (n0 + n1) R grad S = E − 1 m2 2R2 . 0 = m c2 1 n2 + Ω   2 0 2 qA qA n1 R (n0 + n1) + mJ = N0R 2 . 2 c c n0 The correction due to rotation is of order Ω .Thereis no linear term that could generate spontaneous rotation. Hence a crucial question: could such an angular momen- Note that this second-order term vanishes if n1 =0:ro- tum induce a spontaneous rotation of the charge density tating something which does not exist cannot cost any wave? Standard wisdom says that rotating the frame at energy! In contrast, rotating the density wave costs an angular velocity Ω adds a term LzΩ to the energy: if true, energy, rotation of the charge density wave is unavoidable since mN n2 all other terms in the energy are quadratic in Ω. A wise 0 2 2 1 E1 = Ω R 2 . pedestrian approach is to stay in the laboratory frame 2 n0 when calculating energy, noting, however, that in that frame the current is no longer conserved. It is conserved The conclusion of this naive model is clear: a charge den- in the rotating frame of the charge density wave: we start sity wave is not driven to rotation by a diamagnetic cur- ν from that statement and we bring everything back to the rent in the ground state =0. k ν/ πR laboratory frame. Generalization to an excited state = 2 is straightforward. We still have Let J  be the constant current in the rotating frame. The previous calculation relates it to a vector potential A qA mJ  grad S = + unknown as of now, c n0 + n1     n0 + n1 qA whose circulation is J   S − ⇒   = m grad c = ν qA mJ n2   1 qA mJ = + 1+ 2 .  grad S = + . R c n0 n0 c n0 + n1 The constant current J and the energy become The current J in the laboratory frame is       n n2 ν qA qA    J 0 − 1 − ,E 1 J  S− . n0 + n1 qA = 1 2 = grad J J  n n R  S − . m n0 R c 2 c = +( 0 + 1)Ω = m grad c Since J is constant only the circulation of grad S matters, It follows that A = A + mcΩR/q: J  is related to A hence exactly as J was to A before. Rotating the charge density     N ν qA 2 n2 wave is tantamount to replacing the vector potential A by E 0 − − 1 .  = 1 2 A in the rotating frame. From there on the calculation 2m R c n0

57008-p2 Time crystals: Can diamagnetic currents drive a charge density wave into rotation?

The only difference is the replacement of eA/c by (eA/c− ∗∗∗ ν/R). From there on the calculation is unchanged. I wish to thank my colleagues Patrick Bruno and Our final conclusion is clear, but limited: a charge den- Andres Cano who introduced me to the challenge of time sity wave is not driven to rotation in a quantum coherent crystals. Numerous discussions with them were crucial in state, for instance by a diamagnetic current induced by a my search for simplicity. The reader is referred to the magnetic field, or by a persistent current in a coherent, paper of Patrick Bruno [3] which offers a much more phase locked, superfluid state. This is consistent with the general proof, necessarily more elaborate. His conclusions Ehrenfest theorem which states that the expectation value are fully consistent with my simple picture. of the time derivative of any observable A is identically zero in any eigenstate |ψn of the Hamiltonian: Additional remark: Nowadays basic concepts and sim- ple proofs may hardly find their way on highly reputed dA physics journals. That was the fate of the present letter, = i ψn| AH − HA|ψn =0. dt which I dedicate to young physicists who conceive original ideas and strive to get their work published. Any local motion of the charge density wave creates a local time dependence which is precluded. The ring is REFERENCES a finite system and CDW motion is a local issue. Such a conclusion holds for the ground state as well as for thermal [1] Wilczek F., Phys. Rev. Lett., 109 (2012) 160401. equilibrium where the density matrix is diagonal in the [2] Bruno P., Phys. Rev. Lett., 110 (2013) 118901; response |ψn basis. This is no longer true if a current is forced by Wilczek F., Phys. Rev. Lett., 110 (2013) 118902. in the ring, breaking thermal equilibrium. Then CDW [3] Bruno P., Phys. Rev. Lett., 111 (2013) 070402. dragging becomes possible. [4] Leggett A. J., Phys. Rev. Lett., 25 (1970) 1543.

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