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Absence of Quantum Time Crystals week ending PRL 114, 251603 (2015) PHYSICAL REVIEW LETTERS 26 JUNE 2015 Absence of Quantum Time Crystals † Haruki Watanabe1,* and Masaki Oshikawa2, 1Department of Physics, University of California, Berkeley, California 94720, USA 2Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan (Received 28 March 2015; published 24 June 2015) In analogy with crystalline solids around us, Wilczek recently proposed the idea of “time crystals” as phases that spontaneously break the continuous time translation into a discrete subgroup. The proposal stimulated further studies and vigorous debates whether it can be realized in a physical system. However, a precise definition of the time crystal is needed to resolve the issue. Here we first present a definition of time crystals based on the time-dependent correlation functions of the order parameter. We then prove a no-go theorem that rules out the possibility of time crystals defined as such, in the ground state or in the canonical ensemble of a general Hamiltonian, which consists of not-too-long-range interactions. DOI: 10.1103/PhysRevLett.114.251603 PACS numbers: 11.30.-j, 05.70.Ln, 73.22.Gk Recently, Wilczek proposed a fascinating new concept of independent. To see this, recall that the expectation value time crystals, which spontaneously break the continuous ˆ ˆ ≡ 0 ˆ 0 0 hXi is defined as hXi h jXPj i at zero temperature T ¼ −β ˆ −β time translation symmetry, in analogy with ordinary crys- ˆ ≡ ˆ H ˆ En and hXi trðXe Þ=Z ¼ nhnjXjnie =Z at a finite tals that break the continuous spatial translation symmetry −1 temperature T ¼ β > 0, where j0i is the ground state and [1–3].Liet al. soon followed with a concrete proposal for ˆ ≡ −βH ˆ an experimental realization and observation of a (space-) Z trðe Þ is the partition function. Clearly, hnjOðtÞjni ÆiE t time crystal, using trapped ions in a ring threaded by an is time independent since two factors of e n cancel ˆ Aharonov-Bohm flux [4–6]. While the proposal of time against each other and hence hOðtÞi is time independent. crystals was quite bold, it is, on the other hand, rather Yet it is too early to reject the idea of time crystals just natural from the viewpoint of relativity: since we live in the from this observation, since a similar argument would Lorentz invariant space-time, why don’t we have time preclude ordinary (spatial) crystals. One might naively crystals if there are ordinary crystals with a long-range define crystals from a spatially modulating expectation ˆ ˆ order in spatial directions? value of the density operator ρˆðx~Þ¼e−iP~·x~ρˆð0~ÞeiP~·x~. However, the very existence, even as a matter of principle, The unique ground state of the Hamiltonian in a finite of time crystals is rather controversial. For example, Bruno ˆ ~ 0 0 [7] and Nozières [8] discussed some difficulties in realizing box is nevertheless symmetric and hence Pj i¼ , imply- ρˆ 0 time crystals. However, since their arguments were not fully ing that h ðx~Þi is constant over space at T ¼ . Likewise, ˆ ˆ ˆ general, several new realizations of time crystals, which at a finite temperature, hρˆðx~Þi≡tr½e−iP~·x~ρˆð0~ÞeiP~·x~e−βH=Z avoid these no-go arguments, were proposed [9,10]. ˆ cannot depend on position since P~ and Hˆ commute. More In fact, a part of the confusion can be attributed to the generally, the equilibrium expectation value of any order lack of a precise mathematical definition of time crystals. parameter vanishes in a finite-size system, since the Here, we first propose a definition of time crystals in the Gibbs ensemble is always symmetric. This, of course, equilibrium, which is a natural generalization of that of does not rule out the possibility of spontaneous symmetry ordinary crystals and can be formulated precisely also for breaking. time crystals. We then prove generally the absence of time A convenient and frequently used prescription to detect a crystals defined as such, in the equilibrium with respect to spontaneous symmetry breaking is to apply a symmetry- an arbitrary Hamiltonian which consists of not-too- breaking field. For example, in the case of antiferromagnets long-range interactions. We present two theorems: one on a cubic lattice, we apply a staggered magnetic field applies only to the ground state, and the other applies to the ~ ~ ~ ~ ≡ π 1 … 1 equilibrium with an arbitrary temperature. hsðRÞ¼Ph cosðQ · RÞ [Q ð =aÞð ; ; Þ] by adding a Naively, time crystals would be defined in terms of the term − h ðR~Þsˆz to the Hamiltonian, where R~’s are ˆ ˆ ˆ R~ s R~ expectation value hOðtÞi of an observable OðtÞ.IfhOðtÞi lattice sites and sˆz is the spin on the site R~. One computes exhibits a periodic time dependence, the system may be R~ regarded as a time crystal. However, the very definition of the expectation value of the macroscopic order parameter, ˆ which is the staggered magnetization in the case of an eigenstates Hjni¼Enjni immediately implies that the ˆ antiferromagnet, under the symmetry breaking field and expectation value of any Heisenberg operator OðtÞ ≡ then take the limit V → ∞ and h → 0 in this order. The ˆ ˆ eiHtOˆ ð0Þe−iHt in the Gibbs equilibrium ensemble is time nonvanishing expectation value of the macroscopic order 0031-9007=15=114(25)=251603(5) 251603-1 © 2015 American Physical Society week ending PRL 114, 251603 (2015) PHYSICAL REVIEW LETTERS 26 JUNE 2015 R parameter, in this order of limits, is often regarded as a Φˆ d ϕˆ 0 −iG~ ·x~ ϕˆ 0 where G~ ¼ V d x ðx;~ Þe . Note again that h ðx;~ Þi definition of spontaneous symmetry breaking (SSB). itself is a constant over space in the Gibbs ensemble, which is In the case of crystals, we apply a potential vðx~Þ¼ ˆ P symmetric.Forinstance,wesetϕ ¼ ρˆ forordinarycrystalsand ~ ϕˆ ˆ h G~ vG~ cosðG · x~Þ with a periodic position dependence. ¼ sα for spin-density waves. In terms of the LRO, one can Here, G~’s are the reciprocal lattice of the postulated therefore characterize crystals using only the symmetric crystalline order. ground state or ensemble, which itself does not have a finite This prescription is quite useful but unfortunately is not expectation value of the order parameter [14]. straightforwardly applicable to time crystals. The sym- Let us now define time crystals, in an analogous manner metry-breaking field for time crystals has to have a periodic to the characterization of ordinary crystals in terms of the time dependence. In the presence of such a field, the spatial LRO. Generalizing Eqs. (1) and (2), we could say “energy” becomes ambiguous and is defined only modulo the system is a time crystal if the correlation function ϕˆ ϕˆ 0 0 → the frequency of the periodic field, making it difficult to limV→∞h ðx;~ tÞ ð ; Þi fðtÞ is nonvanishing for large select states or to take statistical ensembles based on energy enough jx~j and exhibits a nontrivial periodic oscillation in eigenvalues. Therefore, an alternative definition of time time t (i.e., is not just a constant over time). In terms of the crystals is called for, and indeed we will propose a integrated order parameter defined above, the condition definition of time crystals which is applicable to very reads general Hamiltonians. ˆ ˆ − ˆ ˆ 2 Time-dependent long-range order.— lim heiHtΦe iHtΦi=V ¼ fðtÞ: ð3Þ In order to circum- V→∞ vent the problem in defining time crystals using a time- dependent symmetry-breaking field, here we define time When f is a periodic function of both space and time, we crystals based on the long-range behavior of correlation call it a space-time crystal, in which case we have functions. In fact, all conventional symmetry breakings can be defined in terms of correlation functions, without iHtˆ Φˆ −iHtˆ Φˆ 2 lim he G~ e −G~ i=V ¼ fG~ ðtÞ; ð4Þ introducing any symmetry-breaking field. That is, we V→∞ say the system has a long-range order (LRO) if the where f ðtÞ is the Fourier component of fðt; x~Þ.For equal-time correlation function of the local order parameter G~ example, Li et al. [4] investigated a Wigner crystal in a ϕˆ ðx;~ tÞ satisfies ring threaded by a Aharonov-Bohm flux and predicted its spontaneous rotation, which would be a realization of a lim hϕˆ ðx;~ 0Þϕˆ ðx~0; 0Þi → σ2 ≠ 0 ð1Þ V→∞ space-time crystal. If this were indeed the case, the density at (x1;t1) and (x2;t2) would be correlated as illustrated for jx~ − x~0j much greater than any microscopic scales. in Fig. 1. OneR can equivalently use the integrated order parameter One might think that we could define time crystals based Φˆ ≡ d ϕˆ 0 on the time dependence of equal-position correlation V d x ðx;~ Þ, for which the long-range order is ˆ 2 2 2 functions. Should we adopt this definition, however, rather defined as lim →∞hΦ i=V ¼ σ ≠ 0. For example, in V trivial systems would qualify as time crystals. For example, the case of the quantum transverse Ising model ˆ P P consider a two-level system H −Ω0σ =2 at T 0 and ˆ − σzσz − Γ σx ¼ z ¼ H ¼ h~r;~r0i ~r ~r0 ~r ~r, the local order parameter ϕˆ σz ð~r; tÞ is identified with ~r or its coarse graining. It has been proven quite generally that the LRO σ ≠ 0 guarantees the corresponding SSB, namely, a nonvanishing expect- ation value of the order parameter in the limit of the zero symmetry-breaking field taken after the limit V → ∞ [11,12].
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