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 iE 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]. While the reverse is not proved in general, it is expected to hold in many systems of interest. A crystalline order can also be defined by the correlation function. Namely, if the long-range correlation approaches FIG. 1 (color online). Time-dependent correlation. (a) Wigner to a periodic function crystal of ions in a ring threaded by an Aharonov-Bohm flux, as proposed in Ref. [4] as a possible realization of a time crystal. lim hϕˆ ðx;~ 0Þϕˆ ðx~0; 0Þi → fðx~ − x~0Þð2Þ V→∞ (b) Illustration of the time dependent correlation function, should the time crystal be indeed realized as a spontaneous rotation of for sufficientlylarge jx~−x~0j,thesystemexhibitsaspontaneous the density wave (crystal) in the ground state, as proposed. The Φˆ Φˆ 2 density-density correlation function between (x1;t1) and (x2;t2) crystalline order [13]. Equivalently, limV→∞h ~ ~ i=V ¼ G −G must exhibit an oscillatory behavior as a function of t1 − t2 for ≠ 0 ~ fG~ signals a density wave order with wave vector G, fixed x1 and x2.
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ˆ iHtˆ −iHtˆ ∥ ˆ ˆ ˆ ∥ 3−2 set ϕðtÞ ≡ σxðtÞ¼e σxe ¼ σx cos Ω0t þ σy sin Ω0t. ½A; ½H; A is at most of the order of V ¼ V ˆ ˆ ˆ The correlation function h0jϕˆ ðtÞϕˆ ð0Þj0i of the ground state [15,16]. The same is true for ∥½B; ½H; B ∥. Therefore, 0 1 0 T −iΩ0t combining Eqs. (6) and (7), we get the desired Eq. (5). j i¼ð ; Þ exhibits a periodic time dependence e . ˆ ˆ ˆ The same applies to the equal-position correlation function In this estimate of ∥½A; ½H; A ∥, we assumed the locality ˆ in independent two-level systems spread over the space. of the Hamiltonian; i.e., H is an integral of the Hamiltonian Clearly we do not want to classify such a trivial, uncorre- density hˆðx~Þ, which contain only local terms. It is easy to lated system as a time crystal. “Crystal” should be reserved see that the same conclusion holds even when there are for systems exhibit correlated, coherent behaviors, which interactions among distant points, provided that the inter- are captured by long-distance correlation functions, be it an action decays exponentially as a function of the distance. ordinary crystal or a time crystal. One can further relax this assumption to power-law Absence of long-range time order at T ¼ 0.—We now decaying interactions r−α (α > 0). When 0 < α
The integrand can be bounded by the Schwarz inequality as The Lieb-Robinson bound (8) tells us that jgABðtÞj ≤ d ½C3 þ C4ðvtÞ =V for some constants C3 4 that do not ˆ ; ˆ ˆ 1=2 −iðH−E0Þs ˆ 1=2 ˆ 1 jh0jAðH − E0Þ e ðH − E0Þ Bj0ij depend on t or V. Hence, as long as t ¼ oðV =dÞ, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi → 0 → ∞ ˆ ˆ ˆ ˆ ˆ ˆ jgABðtÞj as V . ≤ h0jAðH − E0ÞAj0ih0jBðH − E0ÞBj0i On the other hand, we have g ðtÞ¼f ðtÞ − f ð−tÞ, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AB AB BA 1 ≡ iHtˆ ˆ −iHtˆ ˆ 2 ˆ ˆ ˆ ˆ ˆ ˆ where fABðtÞ Phe Ae Bi=V . By inserting the com- ¼ h0j½A; ½H; A j0ih0j½B; ½H; B j0i: ð7Þ 1 2 plete set ¼ njnihnj, it can be readily shown that Z ˆ ˆ ˆ ∞ Each of H, A, and B involves a spatial integration and −iωt fABðtÞ¼ dωρABðωÞe ; ð10Þ introduces a factor of V, while each commutation relation −∞ reduces a factor of V, assuming that the equal-time Z ˆ ∞ ϕ1 x;~ t −βω − ω commutation relation of any two operators ð Þ and g ðtÞ¼ dωð1 − e Þρ ðωÞe i t; ð11Þ ˆ 0 0 AB AB ϕ2ðx~ ;tÞ can be nonzero only near x~ ¼ x~ . Hence, −∞
251603-3 week ending PRL 114, 251603 (2015) PHYSICAL REVIEW LETTERS 26 JUNE 2015 where ρABðωÞ is defined by This is nothing but the ac Josephson effect. In fact, in Ref. [9], a proposal of time crystal based on this effect X ˆ ˆ −β hmjAjnihnjBjmie Em was made. δðω − E þ E Þ: ð12Þ ZV2 n m However, in order to observe the ac Josephson effect, the n;m initial state simply must not be in the equilibrium. In order to see this, it is helpful to use the mapping of the ac Since limV→∞gABðtÞ¼0 for any given real value of t, 1 − −βω ρ ω 0 Josephson effect in two coupled condensates to a quantum Rð e ÞlimV→∞ ABð Þ¼ . Combined by the sum rule ωρ ω 0 ˆ ˆ 2 spin in a magnetic field. For simplicity, let us consider d ABð Þ¼fABð Þ¼hA Bi=V , we find condensates of bosons without any internal degree of ˆ ˆ 2 freedom, and suppose there is only one single-particle lim ρABðωÞ¼δðωÞ lim hA Bi=V : ð13Þ V→∞ V→∞ state in each condensate. Then the system can be described by the two set of bosonic annihilation-creation operators, † † Therefore, fABðtÞ, as a function of finite t in the thermo- a; a and b; b . The effective Hamiltonian of the system, in dynamic limit V → ∞, can be at most a finite constant that the limit of zero coupling between the two condensates, is † † does not depend on time. Thus (space-)time crystals do not † † a a−b b μ1þμ2 given as H ¼ μ1a a þ μ2b b ¼ðμ1 −μ2Þ 2 þ 2 N, exist at a finite temperature either. † † — where N ¼ a a þ b b is the total number of particles in Grand-canonical ensemble. Let us discuss systems the coupled system. Let us assume that the coupling to the with variable number of particles. The equilibrium of those outside environment is negligible in the time scale we are systems can be described by a grand-canonical ensemble. It interested in. N is then exactly conserved and can be is given by the Boltzmann-Gibbs distribution with respect Hˆ ˆ − μ ˆ μ regarded as a constant. As a consequence, the second term to ¼ H N, where is the chemical potential deter- in the Hamiltonian proportional to N can be ignored. mined by the property of the particle reservoir. Namely, the ˆ With N being exactly conserved, this system of coupled expectation value of an observable X is given by condensates can be mapped to a quantum spin model ˆ ˆ −βHˆ −βHˆ hXiμ ≡ trðXe Þ=Zμ, where Zμ ≡ tre . Although the by identifying the bosons as Schwinger bosons. The z statistical weight is given in terms of Hˆ , the time evolution Hamiltonian now reads H ¼ BS þ const, where B ¼ μ1 − μ z of the Heisenberg operator Ψˆ ðtÞ is still defined by Hˆ , i.e., 2 and S is the z component of the quantum spin with ˆ ˆ the spin quantum number S ¼ðN − 1Þ=2. Similarly, the Ψˆ t ≡ eiHtΨˆ 0 e−iHt. This mismatch can produce some ð Þ ð Þ current operator between the two condensates is given by trivial time dependence as we shall see now. If we define J ∝−iða†b − b†aÞ¼2Sy. The ac Josephson effect, in the Ψˆ ≡ iðHˆ −μNˆ ÞtΨˆ 0 −iðHˆ −μNˆ Þt ˆ Ψˆ 0 μðtÞ e ð Þe and assume ½N; ð Þ ¼ quantum spin language, is just a Larmor precession about ˆ −qΨð0Þ with q a real number that represents the U(1) the magnetic field. The oscillatory behavior of the current ˆ ˆ −iqμt charge of Ψ, then ΨðtÞ¼ΨμðtÞe . Therefore, even if in the ac Josephson effect just corresponds to the oscillation ˆ ˆ † 2 of the excitation value of Sy in the Larmor precession. fμ ¼ limV→∞½hΨμðtÞΨμð0Þiμ=V is time independent as ˆ ˆ † 2 In order to observe the Larmor precession, the initial we proved above, fðtÞ¼lim →∞½hΨðtÞΨ ð0Þiμ=V has a V state must have a nonvanishing expectation value of the −iqμt trivial time dependence fðtÞ¼fμe . This is consistent transverse component (Sx or Sy). This excludes the ground with the well-known fact that the order parameter of a state, in which the spin is fully polarized along the magnetic Bose-Einstein condensate has the trivial time dependence field in the z direction, as well as thermodynamic equilib- −iμt [18] as hψˆ ðx;~ tÞi ¼ ψ 0e . This type of time dependence rium at arbitrary temperature. In Ref. [9] it was argued that, has been discussed [19] also in the context of time crystals by taking the limit of weak coupling, the dissipation can be [20,21]. However, Volovik pointed out that this kind of time made arbitrarily small. While this is certainly true, the dependence cannot be measured as long as the particle lack of dissipation does not mean that the system is in an number is exactly conserved [22]. Indeed, the overall phase equilibrium, as it is clear by considering the spin Larmor of condensate cannot be measured unless one couples the precession in a magnetic field. Our result, which is valid for condensate to another one. We will discuss this phenome- equilibrium, of course does not exclude such spontaneous non in the following section. oscillations of nonequilibrium quantum states. The latter, Spontaneous oscillation of nonequilibrium states.—In however, are well known and should not be called time order to extract the time dependence of the condensate crystals without a further justification. order parameter, the system has to be attached to another Discussion.—In this Letter, we proposed a definition of system to allow change of the number of particles. As a time crystals and proved their absence in the equilibrium. simplest setup, we may prepare two condensates with The present result brings back the question: why there is no different chemical potentials μ1; μ2 and measure their time crystal, even though there surely exist crystals with a time-dependent interference pattern ∝ e−iðμ1−μ2Þt in terms spatial long-range order? We should recall that Lorentz of the current between the condensates, or, equivalently, invariance does not mean the complete equivalence the change of the number of particles in each condensate. between space and time: the time direction is still
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