Superfluidity and Space-Time Translation Symmetry Breaking

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As Published http://dx.doi.org/10.1103/PhysRevLett.111.250402

Publisher American Physical Society

Version Final published version

Citable link http://hdl.handle.net/1721.1/85671

Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. week ending PRL 111, 250402 (2013) PHYSICAL REVIEW LETTERS 20 DECEMBER 2013

Superﬂuidity and Space-Time Translation Symmetry Breaking

Frank Wilczek Center for Theoretical Physics, MIT, Cambridge, Massachusetts 02139, USA (Received 2 October 2013; published 18 December 2013) I present a simple model that exhibits a temporal analogue of superconducting crystalline (Larkin- Ovchinnikov-Ferrell-Fulde) ordering, with a time-dependent order parameter. I sketch designs for minimally dissipative ac circuits, all based on time translation symmetry () invariant dynamics, exploiting weak links (Josephson effects). These systems violate spontaneously. I also discuss effective theories of that phenomenon, and space-time generalizations.

DOI: 10.1103/PhysRevLett.111.250402 PACS numbers: 11.30.j, 67.10.j, 74.20.Fg

X Recently there has been considerable interest in the ¼ y ¼ þ ¼ y Sþ bk ak S1 iS2 S possibility of spontaneous breaking of time translation k X X (5) symmetry [1–5]. Here I bring in ideas from superfluidity 1 y y S3 ¼ b b a a ; which both establish connections with central ideas of that 2 k k k k field and widen the experimental possibilities significantly. k k Microscopic model.—The energy functional define Hermitian pseudospin operators S1, S2, S3 that 4 2 satisfy the algebra of angular momentum and generate 1 2e 2e ELOFF / @~ A~ @~ A~ (1) isospinlike rotations between the a and b modes. 4 @ 2 @ 2 Since N, S~ , and S3 commute, we can construct the describes superconductors with wavelike or crystalline minimum energy states for H, given N, by maximizing S (Larkin-Ovchinnikov-Ferrell-Fulde [6,7], or ‘‘LOFF’’) (so S ¼ N=2) and choosing a state with definite S3.IfS3 is condensates. In the absence of a vector potential we will also allowed to vary, the minimum will occur for have, upon minimizing the energy in Eq. (1), "2 "1 N hS3i¼max ; : (6) 4g 2 ðx~Þ¼k~ x~ þ 0 (2) (The second alternative on the right-hand side, which ~ 2 ¼ for some wave vector k with k . Several possible saturates the population of the a modes, is essentially realizations of LOFF states have been identified [8–12]. trivial.) On the other hand it can be appropriate to hold It is natural to consider the possibility of a temporal the expectation value of S3 fixed. We can imagine, for analogue to the spatial behavior indicated in Eq. (2). What example, that the a and b modes correspond to states in we want, is that the pairing occurs between states separated distinct layers, whose total occupations can be fixed inde- by a characteristic frequency. For superconducting systems pendently. (Note that the assumed interaction term does not the absolute frequency dependence is rendered ambiguous require interlayer tunneling—this is just another way of by the possibility of time-dependent gauge transforma- saying that it commutes with S3.) As we can see by tions, or stated more simply by the lack of a natural zero rearranging of energy, so it is simplest to use the language of particle- y y ! y y hole pairing. ak bkbl al ak albl bk (7) For orientation purposes, let us begin by further special- izing to the transparent case of two flat bands with energies the assumed interaction corresponds, roughly, to an effective repulsion between density waves that does not depend "1 <"2, and the Hamiltonian on momentum transfer. "2 þ "1 Now we can follow the classic BCS procedure, postulat- H ¼ N þð"2 "1ÞS3 gðSSþ þ SþSÞ 2 ing a symmetry-breaking condensate. In this procedure, we

"2 þ "1 2 2 assume the ansatz ¼ N þð"2 "1ÞS3 2gðS~ S3Þ; (3) 2 i h; jSþj; i¼0e (8) where with 0 a number, ultimately ﬁxed self-consistently by the X X ¼ y þ y gap equation. Since N bk bk ak ak (4) k k ½H;Sþ¼ð"2 "1ÞSþ þ 2gðS3Sþ þ SþS3Þ (9) is the total occupation number and and

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d so that the spatial variation of the phases 1, 2 of the two hSþi¼ih½H;Sþi; (10) dt superconductors near the contact, and the vector potential across the link, can be neglected. Then the basic Josephson consistent classical evolution for requires relations are _ ¼ "2 "1 þ 4ghS3i: (11) 2 d ¼ eV @ ; (12) This vanishes if hS3i is fixed nontrivially by Eq. (6), but not dt if that expectation value is pinned at a different value. In the pseudospin formalism, this time dependence has a j ¼ gðÞ (13) simple interpretation: The condensate is an effective spin of fixed magnitude at a fixed angle to the z^ axis, and the where 2 1 is the relative phase, V is the voltage across the junction, gðÞ is a nontrivial 2-periodic func- "2 "1 term supplies an effective magnetic field in the z^ sin direction, which induces precession. tion often approximated as , and is a coupling The BCS condensation ansatz is overkill for this simple parameter introduced for later convenience. model, where all states with the total spin and expectation Now if V and are nonzero constants then according to Eqs. (12) and (13) we will have the time-dependent current value of S3 are degenerate eigenstates. Its virtue is its ability, at the price of more complicated algebra, to accom- 2 ð Þ¼ eV þ modate more complex, momentum-dependent energies j t g @ t 0 (14) and interactions than assumed in Eq. (3). One expects qualitative aspects of spontaneous symmetry breaking to where 0 is an integration constant. This presents a survive such generalizations. One can also consider manifestly time-dependent physical phenomenon, though bosonic systems along the same lines. Indeed, related nothing in the specification of the problem broke time techniques have been applied to discuss dynamic magnon translation symmetry. In that sense it is an example of condensation in liquid He3 [13] and density oscillations in spontaneous breaking. The occurrence of an undeter- two-component cold atomic gases [14,15]. In both those mined parameter (‘‘soft mode’’) 0 within a manifold of contexts, very long-lived oscillatory states have been solutions fits that interpretation. observed. On the other hand the movement of charge, in the In principle one can probe for LOFF behavior, or its presence of a potential difference, will involve dissipation temporal analogue, by comparing the phase relationships in a closed system with normal (nonsuperconducting) ele- among currents induced by weak-link contacts with a ments to close the circuit. If there is no external energy conventional superfluid at several points, an idea elabo- source of energy to sustain it, V will relax to zero. So if our rated below and in [16]. ambition is to exhibit highly persistent ‘‘ground state’’ breaking—separation of dynamics and measure- spontaneous time-translation symmetry, the standard ac ment.—The ac Josephson effect gives an example of Josephson effect does not quite serve. time-dependent, and thus time-translation symmetry That objection, however, is more formal than substan- breaking, behavior in a system specified by time- tial. We can cleanly separate the conceptually time- independent conditions. Since its standard realization dependent effect, Eq. (12), from its practical manifestation involves continual (alternating) current flow across a volt- Eq. (13). Specifically, by making and breaking contact we age, however, it describes behavior in a dissipative system. can arrange ! ðtÞ to vanish except at designated ‘‘mea- Here I will emphasize, and then build upon, the simple surement’’ times, and to be small even then. In other words, observation that by making the connection intermittent— we can choose to regard the separated superconductors as thus regarding it as a probe, rather than an intrinsic part, of the system of interest, and the junction as a measuring the dynamics—we can make the dissipation arbitrarily device. Then in the ground state we will have the time- small, while retaining the time dependence. This supplies dependent relation Eq. (14), which entails measurable us both with an example of a well understood system that physical consequences (and contains ðtÞ only as a multi- spontaneously breaks time-translation symmetry (‘‘time plicative factor), in a system with arbitrarily small dissi- crystal’’) and, when generalized, with a technique for pation. Practical implementation of a low-dissipation probing possible novel dynamical realizations. I will then switch in this context raises several challenging issues, as describe another, perhaps more elegant, way to avoid dis- discussed in the companion paper [16], where a concrete sipation, using two weak links. design is proposed. The design in [16] employs extended, Since they are basic to everything that follows, a quick as opposed to point, contacts, so more complicated equa- recollection of the Josephson phenomena is in order. (For a tions apply. simple conceptual introduction see [17]; for a more sophis- One can avoid normal components altogether, by clos- ticated introduction to the state of the art see [18].) We ing the circuit in an annular arrangement incorporating a consider two bulk superconductors connected by a weak second weak link. Ideally, if gðÞ¼sin, the junctions are link; for simplicity we suppose that the contact is localized, identical, and a magnetic flux of magnitude h=4e threads

250402-2 week ending PRL 111, 250402 (2013) PHYSICAL REVIEW LETTERS 20 DECEMBER 2013 the annulus, then the ac Josephson currents at the junctions remains nontrivial for V ¼ 0, giving a dissipationless time will be equal and opposite, due to a phase offset , thus crystal. Unfortunately practical identification is compli- allowing closure of the circuit. If those idealized conditions cated by the possibility of nontrivial internal potentials, are met approximately, then plausibly the system will settle which can also contribute to V. The bifurcation of frequen- into a mode of operation wherein some charge accumulates cies could be a more robust characteristic. (dc) at one or both junctions and an appropriate steady (dc) The condensate in a superconductor supporting both edge supercurrent supplies corrective flux, so that there is kinds of unconventional terms, Eq. (17) and (1), would no net charge accumulation per cycle. In this mode, we exhibit traveling waves in , realizing a space-time crystal. realize dissipation-free yet time-dependent current flow. A LOFF superconductor subject to a nonzero potential V (I am neglecting electromagnetic radiation, whose effect would also serve for that purpose. can be minimized in several ways [1]). Distinct but related Spatial Josephson effect—significance of vector poten- frequency locking phenomena in Josephson arrays have tial offset.—It is interesting, and falls naturally within our been discussed previously [19]. exploration of time $ space analogies, to consider the (Since the voltage difference can in principle relax, possibility of a spatial analog of Eq. (12), in the form unless special precautions are taken, one may choose not to regard this breaking as occurring in a ground state, in 2 d ¼ e ½ ð2Þ ð1Þ which case we could speak of a minimally dissipative @ Az Az : (19) symmetry breaking system, as distinct from a strict time dz crystal. To me, whether that is a useful distinction seems somewhat a matter of taste, hinging on to what extent one Just as jumps in A0 can be imprinted by parallel capacitor is willing to regard the special precautions as intrinsic to plates with opposite charge densities, jumps in Az can be defining ‘‘the system.’’) imprinted by parallel current sheets with opposite jz.Ifwe Effective theory.—Let us now adopt a broader perspec- imagine two superconductors on opposite sides of the tive, to consider the possible implications of less conven- x ¼ 0 plane, where such parallel current sheets are found, tional dynamics for superconductor 2. The effect of this then Eq. (19) will apply. (Of course, one will have to allow will be to modify Eq. (12). To set the stage for general- for small windows in the current sheets, where weak links izations, let us recall the default assumptions, which lead can form.) to Eq. (12), in a way suggestive for our purposes. The This effect exhibits a direct physical significance for energy functional of the superconductors contain terms vector potential offsets, similar in spirit to the Aharonov- of the form Bo¨hm effect. Indeed, if we draw a loop with short lines connecting the two superconductors at z ¼ z , z near 2e 2 a b / _ x ¼ 0, and joined up by lines inside the superconductors, Eminimal @ A0 : (15) the enclosed magnetic flux will be ½Azð2ÞAzð1Þðzb zaÞ, This form is consistent with the appropriate gauge and the change in specified by Eq. (19) reflects that flux symmetry directly. (Integral of Fxz in the xz plane.) The conventional ac Josephson effect can be interpreted in a similar way, but 2 0 ¼ þ e ð Þ 0 ¼ þ _ ð Þ now involving loops in the xt plane and enclosed electric @ t ;A0 A0 t : (16) flux. (Integral of Fxt in the xt plane.) One can, of course, If each superconductor minimizes an energy functional of combine the effects. this type, then Eq. (12) follows. Conclusions.—Temporal, and mixed spatiotemporal, On the other hand, suppose that the energy functional of analogues of LOFF ordering can arise in microscopic superconductor 2 is of a less conventional type, suggested models that plausibly might correspond to realizable sys- by an extension of the Landau-Ginzburg philosophy, in the tems. It appears to be possible to exhibit time-dependent form phenomena in time independent systems that are asymp- totically free of dissipation, in the limit of infrequent 3 _ 2e 4 _ 2e 2 measurement, or very nearly so. The preceding discussion Emotive / A0 A0 (17) 4 @ 2 @ of weak links is a concrete embodiment of the framing of the issue of observability of time-translation symmetry in 0 with > , while superconductor 1 is conventional. (The [1] and the related discussions in [3]. I have emphasized factor 3 is adopted for consistency with [2].) Then we will the language of superconductivity, but the central idea, that have, in place of Eq. (12), weak links can be used to probe unconventional order rffiffiffiffi d 2eV parameter dynamics, is more general. ¼ : (18) I am grateful to C. Nayak, A. Shapere, Z. Xiong, and dt @ 3 M. Hertzberg for helpful discussions. This work is sup- In principle, this behavior might be probed by use of ported by the U.S. Department of Energy under Contract Eq. (13), with a small intermittent . Note that Eq. (18) No. DE-FG02-05ER41360.

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