Pre-thermal Phases of Matter Protected by Time-Translation Symmetry
In a periodically driven (Floquet) system, there is the possibility for new phases of matter, not present in stationary systems, protected by discrete time-translation symmetry. This includes topological phases protected in part by time-translation symmetry, as well as phases distinguished by the spontaneous breaking of this symmetry, dubbed “Floquet time crystals”. We show that such phases of matter can exist in the pre-thermal regime of periodically-driven systems, which exists generically for sufficiently large drive frequency, thereby eliminating the need for integrability or strong quenched disorder that limited previous constructions. We prove a theorem that states that such a pre-thermal regime persists until times that are nearly exponentially-long in the ratio of certain couplings to the drive frequency. By similar techniques, we can also construct stationary systems which spontaneously break continuous time-translation symmetry. We argue furthermore that for driven systems coupled to a cold bath, the pre-thermal regime could potentially persist to infinite time.
Much of condensed matter physics revolves around determining which distinct phases of matter can exist as equilibrium states of physical systems. Within a phase, the properties of the system vary continuously as external parameters are varied, while different phases are separated by phase transitions, at which the properties change abruptly. An extremely rich set of observed phases can be characterized by symmetry. The best known example is spontaneous symmetry-breaking, as a result of which the equilibrium state of the system is less symmetrical than the Hamiltonian. More recently, a set of uniquely quantum phases—symmetry-protected topological (SPT) phases Gu and Wen (2009); Pollmann et al. (2010, 2012); Fidkowski and Kitaev (2010); Chen et al. (2010, 2011a); Schuch et al. (2011); Fidkowski and Kitaev (2011); Chen et al. (2011b, 2013); Levin and Gu (2012); Vishwanath and Senthil (2013); Wang et al. (2014); Kapustin (2014); Gu and Wen (2014); Else and Nayak (2014); Burnell et al. (2014); Wang et al. (2015); Cheng et al. (2015), including topological insulators Hasan and Kane (2010); Qi and Zhang (2011), and symmetry-enriched topological (SET) phases Maciejko et al. (2010); Essin and Hermele (2013); Lu and Vishwanath (2016); Mesaros and Ran (2013); Hung and Wen (2013); Barkeshli et al. (2014); Cheng et al. (2016)—has been discovered. These phases, while symmetric, manifest the symmetry in subtly anomalous ways, and are distinct only as long as the symmetry is preserved. We can collectively refer to these three classes of phases as symmetry-protected phases of matter.
Thus far, the concept of symmetry-protected phases of matter has not been as succesful in describing systems away from equilibrium. Recently, however, it was realized that certain periodically-driven “Floquet” systems can exhibit distinct phases, akin to those of equilibrium systems Khemani et al. (2016). In this paper, we show that there is, in fact, a very general set of non-equilibrium conditions under which such phases can arise, due to a remarkable phenomenon called ”pre-thermalization”. In Floquet systems, pre-thermalization occurs when a time-dependent change of basis removes all but a small residual time-dependence from the Hamiltonian, and thus allows the properties of the system to be mapped approximately onto those of a system in thermal equilibrium. The residual time-dependence is nearly exponentially-small in a large parameter of the original Hamiltonian of the system. One can then talk about a “pre-thermal regime” in which the system reaches a thermal equilibrium state with respect to the approximate effective time-independent Hamiltonian that results from neglecting the small residual time dependence. In this regime, the system can exhibit phases and phase transitions analogous to those seen in thermal equilibrium, such as symmetry-protected phases. Nevertheless, in the original non-rotating frame, the system remains very far from thermal equilibrium with respect to the instantaneous Hamiltonian at any given time. After the characteristic time , which is nearly exponentially-long in the large parameter , other physics (related the residual time-dependence) takes over.
In this paper, we show that pre-thermal systems can also exhibit phases of matter that cannot exist in thermal equilibrium. These novel phases can also be understood as symmetry-protected phases but of a variety that cannot occur in thermal equilibrium: these phases are protected by discrete time-translation symmetry. While these include topological phases protected by time-translation symmetry von Keyserlingk and Sondhi (2016); Else and Nayak (2016); Potter et al. (); Roy and Harper (), perhaps the most dramatic of these are “time crystals” that spontaneously break time-translation symmetry. The idea of time crystals that spontaneously break continuous time-translation symmetry was first proposed by Wilczek and Shapere Wilczek (2012); Shapere and Wilczek (2012), but finding a satisfactory equilibrium model has proven difficult and some no-go theorems exist Li et al. (2012); Bruno (2013a, b, c); Noziéres (2013); Volovik (2013); Watanabe and Oshikawa (2015). In this paper, we construct pre-thermal “Floquet time crystals”, which spontaneously break the discrete time-translation symmetry of periodically-driven systems Else et al. (2016) 111For an alternative view of such systems that focuses on other symmetries of the discrete time-translation operator, see Refs. Khemani et al., 2016; von Keyserlingk and Sondhi, ; von Keyserlingk et al., 2016.. Floquet time crystals are the focus of this paper, but as a by-product of our analysis, we also find pre-thermal – i.e. non-equilibrium – time crystals that spontaneously break continuous time-translation symmetry. We also construct SPT and SET phases protected by discrete time-translation symmetry.
Periodically-driven systems have long been considered an unlikely place to find interesting phases of matter and phase transitions since generic driven closed systems will heat up to infinite temperature D’Alessio and Rigol (2014); Lazarides et al. (2014); Ponte et al. (2015). It has been known that the heating problem can be avoided Abanin et al. (2016); Ponte et al. (2015a, b); Lazarides et al. (2015); Iadecola et al. (2015) if the system is integrable or if the system has sufficiently strong quenched disorder that it undergoes many-body localization (MBL) Basko et al., 2006, ; Oganesyan and Huse, 2007; Z̆nidaric̆ et al., 2008; Pal and Huse, 2010; Bardarson et al., 2012; Bauer and Nayak, 2013; Serbyn et al., 2013, 2013; Huse et al., 2014. However, integrability relies on fine-tuning, and MBL requires the system to be completely decoupled from the environment Levi et al. (2016); Fischer et al. (2016); Nandkishore et al. (2014); Gopalakrishnan and Nandkishore (2014); Johri et al. (2015); Nandkishore (2015); Li et al. (2015); Nandkishore and Gopalakrishnan (); Hyatt et al. (). Furthermore, the disorder must be sufficiently strong, which may be difficult to realize in an experiment but does not constitute fine-tuning.
The central result of this paper is therefore to show that pre-thermalization makes it possible for non-equilibrium phases protected by time-translation symmetry to occur in more generic non-equilibrium systems without the need for fine-tuning, strong disorder, or complete decoupling from the environment. Remarkably, these non-equilibrium phases and phase transitions, which have have no direct analogues in thermal equilbrium, have a mathematical formulation that is identical to that of equilibrium phases, though with a different physical interpretation. Since MBL is not a requirement, it is conceivable that pre-thermal time-translation protected phases could survive the presence of coupling to an environment. In fact, we will discuss a plausible scenario by which these phases can actually be stabilized by coupling to a sufficiently cold thermal bath, such that the system remains in the pre-thermal regime even at infinite time.
The structure of the paper will be as follows. In Section II, we state our main technical result. In Section III, we apply this to construct prethermal Floquet time crystals which spontaneously break discrete time-translation symmetry. In Section IV, we show that a continuous time-translation symmetry can also also be spontaneously broken in the pre-thermal regime for a system with a time-independent Hamiltonian. In Section V, we outline how our methods can also be applied to construct SPT and SET phases protected by time-translation symmetry. In Section VI, we discuss what we expect to happen for non-isolated systems coupled to a cold thermal bath. Finally, we discuss implications and interpretations in Section VII.
Ii Pre-Thermalization Results
The simplest incarnation of pre-thermalization occurs in periodically-driven systems when the driving frequency is much larger than all of the local energy scales of the instantaneous Hamiltonian Abanin et al. (2015a, 2017, b); Kuwahara et al. (2016); Mori et al. (2016) (see also Refs. Bukov et al., 2015; Canovi et al., 2016; Bukov et al., 2016 for numerical results). The key technical result of our paper will be a theorem generalizing these results to other regimes in which the driving frequency is not greater than all the local scales of the Hamiltonian, but there is nevertheless some separation of energy scales. This will allow us to show that time-translation protected phases can exist in the pre-thermal regime. More precisely, in the models that we construct, one local coupling strength is large and the others are small; the drive frequency is large compared to the small couplings, and the parameter is the ratio of the drive frequency to the largest of the small local couplings. The term in the Hamiltonian with large coupling must take a special form, essentially that of a symmetry generator, that allows it to avoid heating the system.
Accordingly, we will consider a time-dependent Hamiltonian of the form , where and are periodic with period . We assume that , where is the local energy scale of . We further assume that has the property that it generates a trivial time evolution over time cycles: , where
We claim that such a time evolution will exhibit pre-thermalizing behavior for even if the local energy scale of is comparable to . In other words, such a system exhibits pre-thermalizing behavior when the frequency is large compared some of the couplings (those in ) but not others (those in ), as promised in the introduction.
An easy way to see that this claim is true is to work in the interaction picture (treating as the “interaction”). Then we see that the time evolution of the total Hamiltonian over time cycles is given by
where is the representation of in the interaction picture, and ensures that the time evolution operator Eq. (2) is the same in the interaction and Schrödinger pictures. If we rescale time as , then Eq. (2) describes a system being driven at the large frequency by a drive of local strength 1, which by the results of Refs. Abanin et al., 2015a, 2017, b; Kuwahara et al., 2016; Mori et al., 2016 will exhibit pre-thermalizing behavior for .
On the other hand, since the above argument for pre-thermalization required coarse-graining the time period from to , it prevents us from identifying phases of matter, such as time crystals or Floquet SPT phases, that are protected by time translation symmetry. The problem is that the time-translation symmetry by is what allows different phases of matter to be sharply distinguished. This symmetry is still present, of course (because the coarse-graining is a feature of our description of the system, not the system itself), but it is no longer manifest. Therefore, it is not at all transparent how to understand the different phases of matter in this picture.
In order to proceed further, we will need a new approach. In this paper, we develop a new formalism that analyzes itself rather than , allowing the effects of time-translation symmetry to be seen in a transparent way. Our central tool is a theorem that we will prove, substantially generalizing those of Abanin et al.Abanin et al. (2017). A more precise version of our theorem will be given momentarily, and the proof will be given in Appendix A; the theorem essentially states that there exists a time-independent local unitary rotation such that , where is the time evolution of over one time cycle, and is a quasi-local Hamiltonian that commutes with . The dynamics at stroboscopic times are well-approximated by for times , where . This result combines ideas in Ref. Abanin et al., 2017 about (1) the high-frequency limit of driven systems and (2) approximate symmetries in systems with a large separation of scales. Recall that, in the high-frequency limit of a driven system, the Floquet operator can be approximated by the evolution (at stroboscopic times) due a time-independent Hamiltonian, . Meanwhile, in a static system with a large separation of scales, , where is much larger than the couplings in but , Ref. Abanin et al., 2017 shows that there is a unitary transformation such that where , i.e. the system has an approximate symmetry generated by . Our theorem states that, after a time-independent local unitary change of basis, a periodic Hamiltonian , with satisfying the condition given above, can be approximated, as far as the evolution at stroboscopic times is concerned, by a binary drive that is composed of two components: (1) the action of over one cycle, namely and (2) a static Hamiltonian that is invariant under the symmetry generated by .
These results might seem surprising, because they imply that the evolution over one time period commutes with a symmetry [or in the original basis], despite the fact that the microscopic time-dependent Hamiltonian had no such symmetry. We interpret this “hidden” symmetry as a shadow of the discrete time-translation symmetry. (For example, the evolution over time periods also commutes with , but if we add weak -periodic perturbations to break the discrete time-translation symmetry then this is no longer the case.) Thus, our theorem is precisely allowing us to get a handle on the implications of discrete time-translation symmetry. Compare Ref. von Keyserlingk et al. (2016), where a similar “hidden” symmetry was constructed for many-body-localized Floquet time crystals.
The preceding paragraphs summarize the physical meaning of our theorem. A more precise statement of the theorem, although it is a bit more opaque physically, is useful because it makes the underlying assumptions manifest. The statement of the theorem makes use of an operator norm that measures the average over one Floquet cycle of the size of the local terms whose sum makes up a Hamiltonian; the subscript parametrizes the extent to which the norm suppresses the weight of operators with larger spatial support. An explicit definition of the norm is given in Appendix A. The theorem states the following.
Consider a periodically-driven system with Floquet operator:
where , and satisfies for some integer . We assume that can be written as a sum of terms acting only on single sites . Define . Assume that
Then there exists a (time-independent) unitary such that
where is local and ; are independent of time; and
The exponent is given by
Unpacking the theorem a bit in order to make contact with the discussion above, we see that it states that there is a time-independent unitary operator that transforms the Floquet operator into the form with and local , up to corrections that are exponentially small in . These “error terms” fall into two categories: time-independent terms that do not commute with , which are grouped into ; and time-dependent terms, which are grouped into . Both types of corrections are exponentially-small in . Since they are exponentially-small , these terms do not affect the evolution of the system until exponentially-long times, (for some constant ). It is not possible to find a time-independent unitary transformation that exactly transforms the Floquet operator into the form because the system must, eventually, heat up to infinite temperature and the true Floquet eigenstates are infinite-temperature states, not the eigenstates of an operator of the form with local . In the interim, however, the approximate Floquet operator leads to Floquet time crystal behavior, as we will discuss in the next Section.
The proof of Theorem 1 constructs and through a recursive procedure, which combines elements of the proofs of pre-thermalization in driven and undriven systems given by Abanin et al. Abanin et al., 2017.
In the case of pre-thermal undriven systems, the theorem we need has essentially already been given in Ref. Abanin et al., 2017, but we will restate the result in a form analogous with Theorem 1, which entails some slightly different bounds (however, they are easily derivable using the techniques of Ref. Abanin et al., 2017).
Consider a time-independent Hamiltonian of the form
where . We assume that can be written as a sum of terms acting only on single sites . Define , and assume that
Then there exists a local unitary transformation such that
where and satisfies
Here, we have defined, following Ref. Abanin et al., 2017, the symmetrized operator according to
which, by construction, satisfies .
Iii Pre-thermalized Floquet time crystals
iii.1 Basic Picture
The results of the previous section give us the tools that we need to construct a model which is a Floquet time crystal in the pre-thermalized regime. Our approach is reminiscent of Ref. von Keyserlingk et al., 2016, where the Floquet-MBL time crystals of Ref. Else et al., 2016 were reinterpreted in terms of a spontaneously broken “emergent” symmetry. Here, “emergent” refers to the fact that the symmetry is in some sense hidden – its form depends on the parameters on the Hamiltonian in a manner that is not a priori known. Furthermore, it is not a symmetry of the Hamiltonian, but is a symmetry of the Floquet operator.
In particular, suppose that we have a model where we can set . (Thus ). We then have , where the quasi-local Hamiltonian by construction respects the Ising symmetry generated by . This Ising symmetry corresponds to an approximate “emergent” symmetry of (“emergent” for the reason stated above and approximate because it an exact symmetry of , not , and therefore is approximately conserved for times .) Suppose that spontaneously breaks the symmetry below some finite critical temperature . For example, working in two dimensions or higher, we could have plus additional smaller terms of strength which break integrability. We will be interested in the regime where the heating time , where is the thermalization time of .
Now consider the time evolution , starting from a given short-range correlated state . We also define the rotated states . At stroboscopic times , we find that . Since , we see that at even multiples of the period, , the time evolution of is described by the time-independent Hamiltonian . Thus, we expect that, after the time , the system appears to be in a thermal state of at temperature . Thus, , where is a thermal density matrix for at some temperature , and the approximate equality means that the expectation values of local observables are approximately the same. Note that for , the Ising symmetry of is spontaneously broken and must either select a nonzero value for the order parameter or have long-range correlations. The latter case is impossible given our initial state, as long-range correlations cannot be generated in finite time. Then, at odd times , we have
(since commutes with .) Therefore, at odd times, the order parameter
Thus, the state of the system at odd times is different from the state at even times, and time translation by is spontaneously broken to time translation by .
The above analysis took place in the frame rotated by . However, we can also consider the expectation values of operators in the original frame, for example . The rotation is close to the identity in the regime where the heating time is large222Specifically, it follows from the construction of that , and is the regime where the heating time is large., so has large overlap with and therefore will display fractional frequency oscillations. We recall that the condition for fractional frequency oscillations in the pre-thermalized regime is that (a) must spontaneously break the Ising symmetry up to a finite critical temperature ; and (b) the energy density with respect to of must correspond to a temperature . In Figure 1, we show the expected behavior at low temperatures and contrast it with the expected behavior in a system which is not a time crystal in the pre-thermal regime.
iii.2 Example: periodically-driven Ising spins
Let us now consider a concrete model which realizes the behavior descrived above. We consider an Ising ferromagnet, with a longitudinal field applied to break the Ising symmetry explicitly, and driven at high frequency by a very strong transverse field. Thus, we take
and we choose the driving profile such that
ensuring that the “unperturbed” Floquet operator implements a pulse, , and we can set . (If the driving does not exactly implement a pulse, this is not a significant problem since we can just incorporate the difference into .) This implies that , and we assume that .
Then by the results of Section II (with playing the role of here), we find a quasi-local Hamiltonian , where
In particular, in the case where the pulse acts instanteously, so that
we find that
(this Hamiltonian is integrable, but in general the higher order corrections to will destroy integrability.) More generally, if the delta function is smeared out so that the pulse acts over a time window , the corrections from Eq. (29) will be at most of order . Therefore, so long as , then in two dimensions or higher, the Hamiltonian will indeed spontaneously break the Ising symmetry up to some finite temperature , and we will observe the time-crystal behavior described above.
iii.3 Field Theory of the Pre-Thermal Floquet Time Crystal State
The universal behavior of a pre-thermal Floquet time crystal state can be encapsulated in a field theory. For the sake of concreteness, we derive this theory from the model analyzed in the previous section. The Floquet operator can be written, up to nearly exponential accuracy, as:
Consequently, the transition amplitude from an initial state at time to a final state at time can be written in the following form, provided :
where and ; recall that is or for, respectively, even or odd.
The second line of Eq. (31) is just the transition amplitude for the quantum transverse field Ising model in -dimensional spacetime, with . The model has nearest-neighbor interaction (29) together with higher-order terms that are present in the full expression for . Hence, it can be represented by the standard functional integral for the continuum limit of the Ising model:
where has minima at when the parameters in the Ising model place it in the ordered phase. This functional integral is only valid for wavevectors that are less that a wavevector cutoff: , where and is the spatial lattice spacing. Although the right-hand side of (33) has a continuous time variable, it is only equal to the original peridiodically-driven problem for stroboscopic times for . Note the left-hand side of (33) is also well-defined for arbitrary times, i.e. for continuous , although it, too, only corresponds to the original problem for integer . Thus the continuous-time effective field theory has a frequency cutoff that we are free to choose. Although the functional integral only corresponds to the original problem for stroboscopic times, the functional integral is well-defined for all times. As a result of the factor of in , the field is related to the Ising spin according to . In other words, the field in the functional integral has the intepretation of the temporally-staggered magnetization density, just as, in the corresponding description of an Ising anti-ferromagnet, this field would be the spatially-staggered magnetization. Discrete time-translation symmetry, has the following action: . Thus, the symmetry-breaking phase, in which , is a pre-thermal Floquet time crystal, in which TTSB occurs, as expected.
The rotated Floquet operator has an approximate symmetry generated by the operator since and . Hence, commutes with the (unrotated) Floquet operator . It is not a microscopic symmetry in the conventional sense, since does not commute with the time-dependent Hamiltonian , except for special fine-tuned points in the Floquet time crystal phase. However, since it commutes with the Floquet operator, it is a symmetry of the continuum-limit field theory (33). (See Ref. von Keyserlingk et al., 2016 for a discussion of Floquet time crystals in the MBL context that focuses on such symmetries, sometimes called “emergent symmetries”.) Within the field theory (33), this symmetry acts according to , i.e. it acts in precisely the same way as time-translation by a single period. Again, this is analogous to the case of an Ising anti-ferromagnet, but with the time-translation taking the place of spatial translation. Thus, it is possible to view the symmetry-breaking pattern as . The unbroken symmetry is generated by the combination of time-translation by one period and the action of .
However, there is an important difference between a Floquet time crystal and an Ising antiferromagnet. In the latter case, it is possible to explicitly break the the Ising symmetry without breaking translational symmetry (e.g. with a uniform longitudinal magnetic field) and vice versa (e.g. with a spatially-oscillating exchange coupling). In a Floquet time crystal, this is not possible because there is always a symmetry regardless of what small perturbation (compared to the drive frequency) is added to the Hamiltonian. The only way to explicitly prevent the system from having a symmetry is to explicitly break the time-translation symmetry. Suppose the Floquet operator is . When a weak perturbation with period is added, the Floquet operator can be written in the approximate form where is due to the doubled-period weak perturbation, but it is not possible to guarantee that . Thus there is a symmetry generated by an operator of the form only if time-translation symmetry is present – i.e. it is a consequence of time-translation symmetry and pre-thermalization.
This functional integral is computed with boundary conditions on at and . Time-ordered correlation functions can be computed by inserting operators between the factors of . However, if we are interested in equal-time correlation functions (at stroboscopic times ),
then we can make use of the fact that the system rapidly pre-thermalizes to replace by a thermal state:
where is determined by . The latter has an imaginary-time functional integral representation:
This equation expresses equal-time correlation functions in a pre-thermal Floquet time crystal in terms of the standard imaginary-time functional integral for the Ising model but with the understanding that the field in the functional integral is related to the Ising spins in the manner noted above.
In order to compute unequal-time correlation functions, it is convenient to use the Schwinger-Keldysh formalism Schwinger (1961); Keldysh (1964) (see Ref. Kamenev, for a modern review). This can be done by following the logic that led from the first line of Eq. (31) to the second and thence to Eq. (33). This will be presented in detail elsewhere Else et al. ().
We close this subsection by noting that the advantage of the field theory formulation of a pre-thermal Floquet time crystal is the salience of the similarity with the equilibrium Ising model; for instance, it is clear that the transition out of the Floquet time crystal (e.g. as a function of the energy of the initial state) in the pre-thermal regime is an ordinary Ising phase transition. The disadvantage is that it is difficult to connect it to measurable properties in a quantitative way because the field has a complicated relationship to the microscopic degrees of freedom.
iii.4 Relation to formal definitions of time crystals
In the above discussion, we have implicitly been adopting an “operational” definition of time-crystal: it is a system in which, for physically reasonable initial states, the system displays oscillations at a frequency other than the drive frequency forever (or at least, in the pre-thermal case, for a nearly exponentially long time.) This is a perfectly reasonable definition of time crystal, but it has the disadvantage of obscuring the analogies with spontaneous breaking of other symmetries, which tends not to be defined in this way. (Although in fact it could be; for example, an “operational” definition of spontaneously broken Ising symmetry, say, would be a system in which the symmetry-breaking order parameter does not decay with time for physically reasonable initial statesPekker et al. (2014).) It was for this reason that in Ref. Else et al., 2016 we introduced a formal definition of time-translation symmetry-breaking in MBL systems in terms of eigenstates (two equivalent formulations of which we called TTSB-1 and TTSB-2.)
The definitions TTSB-1 and TTSB-2 of Ref. Else et al., 2016 are natural generalizations of the notion of “eigenstate order” used to define spontaneous breaking of other symmetries in MBL Huse et al. (2013); Pekker et al. (2014). On the other hand they, like the notion of eigenstate order in general, are not really appropriate outside of the MBL context. In this subsection, we will review the usual formal definitions of spontaneous symmetry breaking in equilibrium. Then we will show how they can be extended in a natural way to time-translation symmetries, and that these extended versions are satisfied by the pre-thermal Floquet time crystals constructed above.
Let us first forget about time-translation symmetry, and consider a time-independent Hamiltonian with an Ising symmetry generated by . Let be a steady state of the Hamiltonian; that is, it is invariant under the time evolution generated by . (Here, we work in the thermodynamic limit, so by we really mean a function which maps local observables to their expectation values; that is, we define a state in the -algebra sense Haag (1996).) Generically, we expect to be essentially a thermal state. If the symmetry is spontaneously broken, then can obey the cluster decomposition (i.e. its correlations can be short-ranged), or it can be invariant under the symmetry , but not both. That is, any state invariant under the symmetry decomposes as , where and have opposite values of the Ising order parameter, and are mapped into each other under . Thus, a formal definition of spontaneously broken Ising symmetry can be given as follows. We call a symmetry-invariant steady state state an extremal symmetry-respecting state if there do not exist states and such that for some , where and are symmetry-invariant steady states. We say the Ising symmetry is spontaneously broken if extremal symmetry-invariant steady states do not satisfy the cluster decomposition. Similar statements can be made for Floquet systems, where by “steady state” we fnow mean a state that returns to itself after one time cycle.
We can now state the natural generalization to time-translation symmetry. For time-translation symmetry, “symmetry-invariant” and “steady state” actually mean the same thing. So we say that time-translation symmetry is spontaneously broken if extremal steady states do not satisfy the cluster decomposition. This is similar to our definition TTSB-2 from Ref. Else et al., 2016 (but not exactly the same, since TTSB-2 was expressed in terms of eigenstates, rather than extremal steady states in an infinite system), so we call it TTSB-2. We note that TTSB-2 implies that any short-range correlated state , i.e. a state which satisfies the cluster decomposition, must not be an extremal steady state. Non-extremal states never satisfy the cluster decomposition, so we conclude that short-range correlated states must not be steady states at all, so they cannot simply return to themselves after one time cycle. (This is similar to, but again not identical with, TTSB-1 in Ref. Else et al., 2016.)
We note that, for clean systems, the only steady state of the Floquet operator is believed to be the infinite temperature stateD’Alessio and Rigol (2014); Lazarides et al. (2014); Ponte et al. (2015) which always obeys the cluster property, and hence time translation symmetry is not broken spontaneously. This does not contradict our previous results, since we already saw that time translation symmetry is only spontaneously broken in the pre-thermal regime, not at infinitely long times. Instead, we should examine the steady states of the approximate Floquet operator which describes the dynamics in the pre-thermal regime. We recall that, after a unitary change of basis, , where commutes with and spontaneously breaks the Ising symmetry generated by (for temperatures ). Hence . Any steady state of must be a steady state of , which implies (if its energy density corresponds to a temperature ) that it must be of the form , where is an Ising symmetry-breaking state of temperature for the Hamiltonian . Hence, we see (since is invariant under ) that . So if is a steady state of and not just , we must have . But then the state clearly violates the cluster property. Hence, time translation is spontaneously broken.
Iv Spontaneously-broken continuous time-translation symmetry in the pre-thermal regime
iv.1 Basic Picture
The pre-thermalized Floquet time crystals discussed above have a natural analog in undriven systems with continuous time translation symmetry. Suppose we have a time-independent Hamiltonian
where the eigenvalues of are integers; in other words, for time , the condition holds for all . We also assume that is a sum of local terms of local strength ; and is a local Hamiltonian of local strength . Then by Theorem 3.1 of Ref. Abanin et al., 2017, restated in Theorem 2 in Section II), there exists a local unitary such that such that and the local strength of is . As noted in Theorem 2 in Section II), the first term in the explicit iterative construction of in Ref. Abanin et al., 2017 is , where
As a result of this theorem, such a system has an approximate U(1) symmetry generated by that is explicitly broken only by nearly exponentially-small terms. Consequently, is conserved by the dynamics of for times . We will call the Hamiltonian the “pre-thermal” Hamiltonian, since it governs the dynamics of the system for times short compared to . We will assume that we have added a constant to the Hamiltonian such that is positive-definite; this will allow us to abuse terminology a little by referring to the expectation value of as the “particle number”, in order to make analogies with well-known properties of Bose gases, in which the generator of the symmetry is the particle number operator. In this vein, we will call the electric potential, in analogy with (negatively) charged superfluids.
We will further suppose that is neither integrable nor many-body localized, so that the dynamics of will cause an arbitrary initial state with non-zero energy density and non-zero to rapidly thermalize on some short (compared to ) time scale . The resulting thermalized state can be characterized by the expectation values of and , both of which will be the same as in the initial state, since energy and particle number are conserved. Equivalently, the thermalized state can be characterized by its temperature (defined with respect to ) and effective chemical potential . In other words, all local correlation functions of local operators can be computed with respect to the density matrix . The chemical potential has been introduced to enforce the condition .
Now suppose that we choose such that spontaneously breaks the symmetry in some range of temperature and chemical potential . Suppose, further, that we prepare the system in a short-range correlated initial state such that the energy density (and hence, its temperature) is sufficiently low, and the number density sufficiently high, so that the corresponding thermalized state spontaneously breaks the symmetry generated by . Then, the preceding statement must be slightly revised: all local correlation functions of local operators can be computed with respect to the density matrix for some satisfying . The limit is taken after the thermodynamic limit is taken; the direction of the infinitesimal symmetry-breaking field is determined by the initial state. To avoid clutter, we will not explicitly write the in the next paragraph, but it is understood.
Consider an operator that satisfies . (For example, if we interpret as the particle number, we can take to be the particle creation operator.) Its expectation value at time is given by
According to the discussion in Appendix B, which makes use of the result of Watanabe and Oshikawa Watanabe and Oshikawa (2015), the trace on the right-hand-side of the second equality must be independent of time. Hence, so long as (which we assume to be true for some order parameter in the symmetry-breaking phase), we find that the expectation value of oscillates with frequency given by the “effective electrochemical potential” due to the winding of the phase of .
If the dynamics were exactly governed by , then the system would oscillate with period forever. As it is, these oscillations will be observed until the exponentially late time . At infinitely long times, the system approaches a thermal state of the full Hamiltonian . Since is small, this is approximately the same as a thermal state of . However, because is not exactly zero, the particle number is not conserved and in equilibrium the system chooses the particle number that minimizes its free energy, which corresponds to the “electrochemical potential” being zero, . Since this corresponds to zero frequency of oscillations, it follows that no oscillations are observed at infinite time.
The above discussion is essentially the logic that was discussed in Refs. Wilczek, 2013; Volovik, 2013; Watanabe and Oshikawa, 2015, where it was pointed out that a superfluid at non-zero chemical potential is a time crystal as a result of the well-known time-dependence of the order parameter Pethick and H. Smith (2008). However, there is an important difference: the U(1) symmetry is not a symmetry of the Hamiltonian of the problem and, therefore, does not require fine-tuning but, instead, emerges in the limit, thereby evading the criticism Nicolis and Piazza (2012); Castillo et al. (2014); Thies (2014); Volovik (2013); Watanabe and Oshikawa (2015) that the phase winds in the ground state only if the U(1) symmetry is exact.
iv.2 Example: XY Ferromagnet in a Large Perpendicular Field
Consider the concrete example of a spin-1/2 system in three spatial dimensions, with Hamiltonian
We take , and the longitudinal magnetic field plays the role of in the preceding section. We take and to vanish except for nearest neighbors, for which , , and . (We do not assume .) The local scale of is given by , so that the condition is satisfied if . In this case, is (to first order) the Hamiltonian of an ferromagnet:
Then, starting from a short-range correlated state with appropriate values of energy and , we expect that time evolution governed by causes the system to “pre-thermalize” into a symmetry-breaking state with some value of the order parameter . According to the preceding discussion, the order parameter will then rotate in time with angular frequency (where is determined by the initial value of ) for times short compared to the thermalization time .
iv.3 Field Theory of Pre-Thermal Continuous-TTSB Time Crystal
For simplicity we will give only the imaginary-time field theory for equal-time correlation functions deep within the pre-thermal regime; the Schwinger-Kelysh functional integral for unequal-time correlation functions, with nearly exponentially-small thermalization effects taken into account, will be discussed elsewhere Else et al. (). Introducing the field , we apply Eq. (39) to the XY ferromagnet of the previous section, thereby obtaining the effective action:
The represents higher-order terms. The U(1) symmetry generated by acts according . Time-translation symmetry acts according to for any . Thus, when develops an expectation value, both symmetries are broken and a combination of them is preserved according to the symmetry-breaking pattern , where the unbroken is generated by a gauge transformation by and a time-translation .
From the mathematical equivalence of Eq. (42) to the effective field theory of a neutral superfluid, we see that (1) in 2D, there is a quasi-long-range-ordered phase – an ‘algebraic time crystal’ – for initial state energies below a Kosterlitz-Thouless transition; (2) the TTSB phase transition in 3D is in the ordinary XY universality class in 3D; (3) the 3D time crystal phase has Goldstone boson excitations. If we write , and integrate out the gapped field , then the effective action for the gapless Goldstone boson is of the form discussed in Ref. Castillo et al., 2014.
V Pre-thermalized Floquet topological phases
We can also apply our general results of Section II to Floquet symmetry-protected (SPT) and symmetry-enriched (SET) topological phases, even those which don’t exist in stationary systems. (We will henceforth use the abbreviation SxT to refer to either SPT or SET phases.)
As was argued in Refs. Else and Nayak, 2016; Potter et al., , any such phase protected by symmetry is analogous to a topological phase of a stationary system protected by symmetry , where the extra corresponds to the time translation symmetry. Here the product is semi-direct for anti-unitary symmetries and direct for unitary symmetries. For simplicity, here we will consider only unitary symmetries. Similar arguments can be made for anti-unitary symmetries.
We will consider the class of phases which can still be realized when the is refined to . That is, the analogous stationary phase can be protected by a unitary representation of the group . Then, in applying the general result of Section II, we will choose such that its time evolution over one time cycle is equal to , where is the generator of . Then it follows that, for a generic perturbation of small enough local strength , there exists a local unitary rotation (commuting with all the symmetries of ) such that , where , is a quasi-local Hamiltonian which commutes with , and well describes the dynamics until the almost exponentially large heating time .
Now let us additionally assume (since we want to construct a Floquet-SxT protected by the symmetry , plus time-translation) that the Floquet operator is chosen such that it has the symmetry . Specifically, this means that it is generated by a periodic time evolution such that, for all , , By inspection of the explicit construction for and (see Appendix A), it is easy to see that in this case is a symmetry-respecting local unitary with respect to , and commutes with . That is, the rotation by preserves the existing symmetry as well as revealing a new symmetry generated by (which in the original frame was “hidden”).
Therefore, we can choose to be a Hamiltonian whose ground state is in the stationary SxT phase protected by . It follows (by the same arguments discussed in Ref. Else and Nayak, 2016 for the MBL case) that the ground state will display the desired Floquet-SxT order under the time evolution generated by . Furthermore, since Floquet-SxT order is invariant under symmetry-respecting local unitaries, the ground state of will display the desired Floquet-SxT order under .
We note, however, that topological order, in contrast to symmetry-breaking order, does not exist at nonzero temperature (in clean systems, for spatial dimensions ). Thus, for initial state mean energies that corresponds to temperatures satisfying , where is the bulk energy gap, the system will exhibit exponentiall-small corrections to the quantized values that would be observed in the ground state. This is no worse than the situation in thermal equilbirum where, for instance, the Hall conductance is not precisely quantized in experiments, but has small corrections . However, preparing such an initial state will be more involved than for a simple symmetry-breaking phase. For this reason it is more satisfactory to envision cooling the system by coupling to a thermal bath, as discussed in Section VI, which is analogous to how topological phases are observed in thermal equilibrium experiments – by refrigeration.
Vi Open systems
So far, we have considered only isolated systems. In practice, of course, some coupling to the environment will always be present. One can also consider the effect of classical noise, for example some time-dependent randomness in the parameters of the drive, so that successive time steps do not implement exactly the same time evolution. The Floquet-MBL time crystals of Ref. Else et al., 2016 are not expected to remain robust in such setups, since MBL will be destroyed. Since some amount of coupling to the environment is inevitable in realistic setups, this limits the timescales over which one could expect to observe Floquet-MBL time crystals experimentally.
However, the situation could be quite different for the pre-thermal time crystals of this work. A complete treatment is beyond the scope of the present work, so in this section we will confine ourselves to stating one very interesting hypothesis: Floquet case time-crystals can actually be stabilized in open systems so that the oscillations actually continue forever for any initial state (in contrast to the case of isolated systems, in which, as discussed previously, the oscillations continue only up to some very long time, and only for some initial states). We will not attempt to establish this more rigorously, but simply discuss a plausible scenario by which this would occur. The idea, as depicted in Figure 2, is that the heating due to the periodic driving, as well as classical noise sources and other stray couplings to an environment, can be counteracted by cooling from a coupling to a sufficiently cold thermal bath. Provided that the resulting steady-state has sufficiently low “energy”, we will argue that that oscillations at a fraction of the drive frequency will be observed in this steady state. Here “energy” means the expectation value of the effective Hamiltonian which describes the dynamics in the prethermal regime. We discuss this hypothesis further, and show that it indeed implies periodic oscillations, in Appendices C and D. We also note that this argument does not apply to the continuous-time time crystals of Section IV, since in that case low energy is not a sufficient condition to observe oscillations even in an isolated system; there is also a dependence on the chemical potential .
In this paper, we have described how phases protected by time-translation symmetry can be observed in the pre-thermal regime of driven and undriven quantum systems. This greatly increases the set of experimental systems in which such phases can be observed, since, as opposed to previous proposals, we do not require many-body localization to robustly prevent the system from heating to infinite temperature. While many-body localization has been observed in experiments Schreiber et al. (2015); Smith et al. (2016); Choi et al. (2016), the ideas put forward in this paper significantly reduce experimental requirements as strong disorder is not required.
Our Theorem 1 implies that the time-translation-protected behavior (for example, the fractional-frequency oscillations in the Floquet time crystal) can be observed to nearly exponentially-late times, provided that the drive frequency is sufficently high. However, the rigorous bound given in the theorem – which requires a drive frequency times larger than the local couplings in the time-dependent Hamiltonian – may not be tight. Therefore, it would be interesting to check numerically whether (in the Floquet time crystal case, say) long-lived oscillations are observed in systems with drive frequency only moderately larger than the local couplings. This may be challenging in small systems, in which there isn’t a large separation of energy scales between the local coupling strength and the width of the many-body spectrum (which the frequency should certainly not exceed). In one-dimensional systems, oscillations will not be observed to exponentially-long (in the drive frequency) times, but will have a finite correlation time for any non-zero energy density initial state. However, there will be a universal quantum critical regime in which the correlation time will be the inverse effective temperature.
Although naive application of Theorem 1 suggests that the ideal situation is the one in which the drive frequency becomes infinitely large, in practice very high-frequency driving will tend to excite high energy modes that were ignored in constructing the model lattice Hamiltonian. For example, if the model Hamiltonian describes electrons moving in a periodic potential in the tight-binding approximation, high frequency driving would excite higher orbitals that were excluded. Thus, the driving frequency needs to be much greater than the local energy scales of the degrees of freedom included in the model Hamiltonian (except for one particular coupling, as discussed in Section III), but also much less than the local energy scales of the degrees of freedom not included. (One cannot simply include all degrees of freedom in the model Hamiltonian, because then the norm of local terms would be unbounded, and Theorem 1 would not apply.)
In the case of undriven systems, we have shown that continuous time-translation symmetry breaking can similarly occur on nearly exponentially-long time intervals even without any fine-tuning of the Hamiltonian, provided that there is a large separation of scales in the Hamiltonian. We show how in certain cases this can be described in terms of approximate Goldstone bosons associated with the spontaneously-broken time-translation symmetry.
Our analysis relied on the construction of hidden approximate symmetries that are present in a pre-thermal regime. The analogous symmetries in MBL systems, where they are exact, were elucidated in the interesting work of von Keyserlingk et al. von Keyserlingk et al. (2016). In the time-translation protected phases discussed here, the symmetry generated by the operator is enslaved to time-translation symmetry since, in the absence of fine-tuning, such a symmetry exists exists only if time-translation symmetry is present. (That is, if we add fields to the Hamiltonian that are periodic with period and not period , then the hidden symmetry no longer exists.) Moreover, this symmetry is broken if and only if time-translation symmetry is broken. (Similar statements hold in the MBL casevon Keyserlingk et al. (2016).) In the Floquet time crystal case, the hidden symmetry generated by acts on the order parameter at stroboscopic times in the same way as time-translation by (a single period of the drive), and therefore it does not constrain correlation functions any more than they already are by time-translation symmetry. The same observation holds for the approximate symmetry generated by in the undriven case.
However, there are systems in which time crystal behavior actually does “piggyback” off another broken symmetry. This does require fine-tuning, since it is necessary to ensure that the system posseses the “primary” symmetry, but such tuning may be physically natural (e.g. helium atoms have a very long lifetime, leading to a U(1) symmetry). The broken symmetry allows a many-body system to effectively become a few-body system. Thus, time crystal behavior can occur in such systems for the same reason that oscillations can persist in few-body systems. Oscillating Bose condensates (e.g. the AC Josephson effect and the model of Ref. Sacha, 2015) can, thus, be viewed as fine-tuned time crystals. They are not stable to arbitrary time-translation symmetry-respecting perturbations; a perturbation that breaks the “primary” symmetry will cause the oscillations to decay. Indeed, most few-body systems are actually many-body systems in which a spontaneously-broken symmetry approximately decouples a few degrees of freedom. A pendulum is a system of atoms that can be treated as a single rigid body due to spontaneously-broken spatial translational symmetry: its oscillations owe their persistence to this broken symmetry, which decouples the center-of-mass position from the other degrees of freedom.
With the need for MBL obviated by pre-thermalization, we have opened up the possibility of time-translation protected phases in open systems, in which MBL is impossible Levi et al. (2016); Fischer et al. (2016); Nandkishore et al. (2014); Gopalakrishnan and Nandkishore (2014); Johri et al. (2015); Nandkishore (2015); Li et al. (2015); Nandkishore and Gopalakrishnan (); Hyatt et al. (). In fact, since the results of Appendix D show that TTSB can occur in non-thermal states, it is possible for the coupling to a cold bath to counteract the heating effect that would otherwise bring an end to the pre-thermal state at time . This raises the possibility of time-translation protected phases that survive to infinite times in non-equilibrium steady states; the construction of such states is an interesting avenue for future work.
Acknowledgements.We thank D.E. Liu and T. Karzig for helpful discussions. We thank W.W. Ho for pointing out a typo in a previous version of the manuscript. D.E. acknowledges support from the Microsoft Corporation. Part of this work was completed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293.
Note added: After the submission of this paper, two experimental papers (J. Zhang et al., arXiv:1609.08684 and S. Choi et al., arXiv:1610.08057) have appeared with evidence consistent with the observation of a Floquet time crystal. We note that the J. Zhang et al. paper implements disorder by addressing each ion sequentially. A pre-thermal version of this experiment would not need disorder, thereby sidestepping this bottleneck standing in the way of experiments on larger systems. The Choi et al. paper occurs in a system that is unlikely to be many-body localize, and therefore occurs during a slow approach to equilibrium. This is unlikely to correspond to a prethermal regime, but the approximate short-time form of the time evolution entailed in our Theorem 1 might still be relevant to understanding the results.
Appendix A Rigorous proof of pre-thermalization results
a.0.1 Definition of the norm
Let’s suppose, for the sake of concreteness, that we have a spin system with a local time-dependent Hamiltonian of the form:
Here are the components of the spins, and are lattice sites. In the first line, we have explicitly written the -site and -site terms; the represents terms up to -site terms, for some finite . It is assumed that these interactions have finite range such that all of the sites in a -site term are within distance . In the second line, we have re-expressed the Hamiltonian in a more generic form in terms of -site terms with . To avoid clutter, we have not explicitly denoted the -dependence of . We define the local instantaneous norm according to
where is the operator norm of at a given instant of time and
We make this choice of -dependence of , following Ref. Abanin et al., 2017 for reasons that will be clear later. We then average the instantaneous norm over one cycle of the drive:
It is only in this step that we differ from Abanin et al. Abanin et al. (2017), who consider the supremum over rather than the average. In analyzing the Floquet operator, i.e. the evolution due to at stroboscopic times, it is the total effect of , which is determined by its integral over a cycle, that concerns us. Error terms that act over a very short time, even if they are relatively strong, have little effect on the Floquet operator so long as their norm, as defined above, is small. Finally, we define the global time-averaged norm of the Hamiltonian :
The term in square braces restricts the sum to -tuples that contain the site .
a.0.2 More technical statement of Theorem 1
Theorem 1 stated above will follow from the following slightly more technical formulation. For notational simplicity we work in units with .
Consider a periodically-driven system with Floquet operator:
where satisfies for some integer , and we assume that can be written as a sum of terms acting on single sites . Define . Then there exists a sequence of quasi-local such that, defining , we have
where ; are independent of time; and
where we have defined , and