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Quantum Time Crystals Quantum Time Crystals The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Wilczek, Frank. “Quantum Time Crystals.” Physical Review Letters 109.16 (2012). © 2012 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevLett.109.160401 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/76209 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. Selected for a Viewpoint in Physics week ending PRL 109, 160401 (2012) PHYSICAL REVIEW LETTERS 19 OCTOBER 2012 Quantum Time Crystals Frank Wilczek Center for Theoretical Physics Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 29 March 2012; published 15 October 2012) Some subtleties and apparent difficulties associated with the notion of spontaneous breaking of time- translation symmetry in quantum mechanics are identified and resolved. A model exhibiting that phenomenon is displayed. The possibility and significance of breaking of imaginary time-translation symmetry is discussed. DOI: 10.1103/PhysRevLett.109.160401 PACS numbers: 11.30.Àj, 03.75.Lm, 05.45.Xt Symmetry and its spontaneous breaking is a central not broken, but clearly there is a sense in which something theme in modern physics. Perhaps no symmetry is more is moving. fundamental than time-translation symmetry, since time- We can display the essence of this situation in a simple translation symmetry underlies both the reproducibility of model, that displays its formal structure clearly. Consider experience and, within the standard dynamical frame- a particle with charge q and unit mass, confined to a works, the conservation of energy. So it is natural to con- ring of unit radius that is threaded by flux 2=q. sider the question, whether time-translation symmetry The Lagrangian, canonical (angular) momentum, and might be spontaneously broken in a closed quantum- Hamiltonian for this system are, respectively, mechanical system. That is the question we will consider, L ¼ 1_ 2 þ ;_ ¼ _ þ ; and answer affirmatively, here. 2 1 2 (2) Here we are considering the possibility of time crystals, H ¼ 2ð À Þ : analogous to ordinary crystals in space. They represent spontaneous emergence of a clock within a time-invariant , through its role as generator of (angular) translations, dynamical system. Classical time crystals are considered in and in view of the Heisenberg commutation relations, is @ a companion Letter [1]; here the primary emphasis is on realized as Ài @ . Its eigenvalues are integers l, associated quantum theory. with the states jli¼eil. For these states we have Several considerations might seem to make the possi- hlj_ jli¼l À ; hljHjli¼1ðl À Þ2: bility of quantum time crystals implausible. The 2 (3) Heisenberg equation of motion for an operator with no The lowest energy state will occur for the integer l0 that intrinsic time dependence reads makes l À smallest. If is not an integer, we will have _ _ hÉjOjÉi¼ihÉj½H;OjÉi!0; (1) hl0jjl0i¼l0 À Þ 0: (4) ɼÉE The case when is half an odd integer requires special where the last step applies to any eigenstate ÉE of H. This consideration. In that case we will have two distinct states 1 seems to preclude the possibility of an order parameter that j Æ 2i with the minimum energy. We can clarify the could indicate the spontaneous breaking of infinitesimal meaning of that degeneracy by combining two simple time-translation symmetry. Also, the very concept of observations. First, that the combined operation Gk of ‘‘ground state’’ implies the state of lowest energy, but in multiplying wave functions by eik and changing ! any state of definite energy (it seems) the Hamiltonian þ k, for integer k, in the Lagrangian leaves the dynamics must act trivially. Finally, a system with spontaneous invariant. Indeed, if we interpret in L as embodying a breaking of time-translation symmetry in its ground state constant gauge potential, Gk is a topologically nontrivial must have some sort of motion in its ground state, and is gauge transformation on the ring, corresponding to therefore perilously close to fitting the definition of a the multiply valued gauge function A ! A þrÃ, perpetual motion machine. à ¼ k=q. Note that the total flux is not invariant under Ring particle model.—And yet there is a familiar this topologically nontrivial gauge transformation, which physical phenomenon that almost does the job. A super- cannot be extended smoothly off the ring, so L is modi- conductor, in the right circumstances, can support a stable fied. Second, that the operation of time-reversal T, imple- current-carrying ground state. Specifically, this occurs if mented by complex conjugation of wave functions, takes we have a superconducting ring threaded by a flux that is a jli!jÀli and leaves the dynamics invariant if simulta- fraction of the flux quantum. If the current is constant then neously ! . Putting these observations together, we nothing changes in time, so time-translation symmetry is see that the combined operation 0031-9007=12=109(16)=160401(5) 160401-1 Ó 2012 American Physical Society week ending PRL 109, 160401 (2012) PHYSICAL REVIEW LETTERS 19 OCTOBER 2012 ~ T ¼ G2 T (5) h jÈj i¼0: (9) leaves the Lagrangian invariant; it is a symmetry of [Normalization of j i depends on how the limit is taken. If the dynamics and maps jli!j2 À li. T~ interchanges we arrange hj0i!ð À 0Þ, then the proportionality jl Æ i!jl Ç i. Thus the degeneracy between those constant is ð2ÞÀ1.] states is a consequence of a modified time-reversal sym- Why then do we prefer one of the states ji as a 1 1 metry. We can choose combinations j þ 2ij À 2i that description of the physical situation? The reason is closely simultaneously diagonalize H and T~; for these combina- related to the emergent orthogonality of the different ji tions the expectation value of _ vanishes. states, as we now recall. We envisage that our system Returning to the generic case: For that are not half- extends over a large number N of identical subsystems integral time-reversal symmetry is not merely modified, having correlated values of the long-range order parameter but simply broken, and there is no degeneracy. How do we , but otherwise essentially uncorrelated. Then we can _ _ reconcile hl0jjl0i Þ 0 with Eq. (1)? The point is that , express the total wave function in the form despite appearances, is neither the time-derivative of YN a legitimate operator nor the commutator of the Éðx1; ...;xNÞ c ðxjÞ: (10) Hamiltonian with one, since , acting on wave functions j¼1 in Hilbert space, is multivalued. By way of contrast, op- For different values ; 0 we have therefore erators corresponding to single-valued functions of , ik YN spanned by trigonometric functions Ok ¼ e , do satisfy N hÉ 0 jÉ i hc 0 ðx Þjc ðx Þi ¼ ðf 0 Þ ! 0 jÉi¼jli j j (11) Eq. (1) for the eigenstates . j¼1 Wave functions of the quantized ring particle model 0 correspond to the (classical) wave functions that appear for Þ and large N, since jf0j < 1. Similarly, for any in the Landau-Ginzburg theory of superconductivity. finite set of local observables (that is, observables whose Those wave functions, in turn, heuristically describe arguments include only upon a finite subset of the xj), we the wave function for macroscopic occupation of the have single-particle quantum state appropriate to a Cooper NÀfinite hÉ0 jO1ðxaÞO2ðxbÞ...jÉi/ðf0Þ ! 0 (12) pair, regarded as a particle. Under this correspondence, 0 the nonvanishing expectation value of _ for the ground for Þ . Since the off-diagonal matrix elements vanish, state of the ring particle subject to fractional flux maps onto any world of local observations (including ‘‘observations’’ the persistent current in a superconducting ring. by the environment) can be described using a single ji Symmetry breaking and observability.—The choice of a state. Averaging over them, to produce j i, is a purely ground state that violates time-translation symmetry formal operation. Measurement of a nonsinglet observable must be based on some criterion other than energy mini- will project onto a ji state. mization. But what might seem to be a special difficulty This analysis, which elaborates [2], brings out several with breaking , because of its connection to the relevant points. The physical criterion that identifies useful Hamiltonian, actually arises in only a slightly different ‘‘ground states’’ is not simply energy, but also robust form for all cases of spontaneous symmetry breaking. observability—that is, relevance to the description of ob- Consider, for example, the breaking of number (or dually, servations in a world of mutually communicating observ- phase) symmetry. We characterize such breaking through a ers. Mathematically, that requirement is reflected in the complex order parameter, È, that acquires a nonzero orthogonality of the Hilbert spaces built upon ji states by expectation value, which we can take to be real: the action of physical observables. The large N limit is crucial for spontaneous symmetry breaking. It is only in h0jÈj0i¼v Þ 0: (6) that infinite degree of freedom, or (as it is usually called) infinite volume, limit, that the ji states and their Fourier We also have states ji related to j0i by the symmetry R transforms jji/ deijji, with definite charge j, be- operation. These are all energetically degenerate and mu- come degenerate, and the former are preferred. Important tually orthogonal in the appropriate ‘‘infinite volume’’ for present purposes: The preceding discussion applies, limit (see immediately below), and satisfy with only symbolic changes, when we consider possible hjÈji¼vei: (7) breaking of time-translation in place of phase symmetry.
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