Long-Range Prethermal Phases of Nonequilibrium Matter

We prove the existence of nonequilibrium phases of matter in the prethermal regime of periodically driven, long-range interacting systems, with power-law exponent $\alpha >d$, where $d$ is the dimensionality of the system. In this context, we predict the existence of a disorder-free, prethermal discrete time crystal in one dimension—a phase strictly forbidden in the absence of long-range interactions. Finally, using a combination of analytic and numerical methods, we highlight key experimentally observable differences between such a prethermal time crystal and its many-body localized counterpart.

Popular Summary

Nonequilibrium systems can exhibit phenomena fundamentally richer than their static counterparts. In particular, certain phases of matter that are forbidden in equilibrium, such as discrete time crystals, have found new life in periodically driven, out-of-equilibrium systems. However, the age-old question remains: How can one avoid heating in such a driven, strongly interacting, many-body system? The conventional answer is to introduce disorder to the system, but this significantly limits the scope of the experiments and models that one can consider. Alternatively, one can consider the high-frequency driving regime, where prethermalization occurs, and the heating timescale is known to be very long. Here, we extend this prethermal approach to long-range, power-law interacting systems, proving the existence of disorder-free prethermal phases of matter in such systems. A particular highlight is the prediction of a prethermal discrete time crystal in one dimension—a phase of matter that requires long-range interactions to stabilize it.

Through a careful analysis of the dynamics of long-range interacting systems, we rigorously prove two important results: first, that long-range interacting systems possess an effective Hamiltonian in the high-frequency regime and second, that such an effective Hamiltonian can host emergent symmetries. Such symmetries can be used to define new phases of matter with no static analog. The prototypical example is the prethermal discrete time crystal—a phase that exhibits collective subharmonic oscillations that remain synchronized for long periods of time.

These results bring to the table a broader class of models and interactions with which to explore out-of-equilibrium phases and phenomena. This is of particular relevance to a broad class of quantum simulation platforms characterized by long-range interactions such as trapped ions, Rydberg atom arrays, ultracold polar molecules, and solid-state spin defects.

Figure 1

(a) Schematic phase diagram for a one-dimensional prethermal time crystal as a function of interaction power law and energy density. The 1D PDTC can only exist for long-range interactions (i.e., $J_{ij}\propto |i-j{|}^{-\alpha }$) with power law $1<\alpha <2$ and an energy density that lies in the symmetry-broken phase of the prethermal Hamiltonian $D^{*}$. (b) PDTC Floquet dynamics depicting the magnetization $M(t)$ for a system size $L=28$. The robust period doubling behavior, which survives for exponentially long times in the frequency of the drive $\omega$, signals prethermal time crystalline order. (c) Table summarizing our analytical results. The star indicates that for this case prethermal phases exist provided that we assume that local observables relax to the Gibbs state of $D^{*}$, which we expect since this is the state that maximizes the entropy subject to the constraint of conservation of energy.

Figure 2

(a) [(b)] Illustration of operator spread via the action of a short- [long-]range Hamiltonian, Eq. (17) [Eq. (19)]. In the short-range case (a), the operator remains close to its original location. For the operator to spread to a far away location, it requires many actions of $H^{\mathrm{sr}}$, which leads to a correspondingly large increase in its support; the range and support are closely related notions of size. In the long-range case (b), this need not be the case. The operator can very quickly spread across the system without a significant increase to its support; the range and the support of the operator capture very different notions of size. (c) An $R$-ranged set is a set where any two elements can be connected via a sequence of “jumps” (within the set) of size no greater than $R$. We illustrate this concept with the gray, green, and orange sets, each representing a different $R$-ranged set. Crucially, this definition is closed: when two $R$-ranged sets have a nonempty intersection, their union is also an $R$-ranged set (e.g., the gray and green sets). If they do not intersect, the union of two $R$-ranged sets need not form an $R$-ranged set (e.g., the green and orange sets).

Figure 3

Evolution of an $L=22$ spin chain under the short-range model (left-hand column) and the long-range model. For the latter, we consider a “cold” initial state near the top of the spectrum of $D^{*}$ (center column) and another “hot” state near the center of the spectrum (right-hand column). (a)–(c) Evolution of the energy density $⟨D⟩/L$. Regardless of the model or initial state, the heating timescale ${\tau }^{*}$ (which measures the approach to infinite temperature) scales exponentially in the frequency of the drive. (d)–(f) Evolution of the entanglement entropy $S_{L/2}$. At intermediate times and large frequencies we observe a plateau corresponding to the entanglement entropy of the prethermal state, as independently corroborated by the evolution under the $\omega \to \infty$ limit of our Floquet evolution (captured on even periods by the evolution under $D$). Analogous to the energy density, at late times ($t>{\tau }^{*}$), the entanglement entropy approaches its infinite temperature value of $[L\log (2)-1]/2$ [94]. (g)–(i) Evolution of $M(t)$ for even (full line) and odd periods (dashed line). In both the short-range model (g) and the “hot” long-range initial state (i), any period doubling behavior of the magnetization quickly decays as the system approaches, independently of frequency, the prethermal state at${\tau }_{\mathrm{pre}}$. By contrast, in the “cold” long-range initial state (h), the magnetization exhibits a robust period doubling behavior for as long as the energy density remains conserved; the decay of both quantities occurs at ${\tau }^{*}=\mathcal{O}(e^{\omega /J_{\text{local}}})$ and the prethermal time crystal is robust. This distinction is even clearer when considering the $\omega \to \infty$ limit of our Floquet evolution, where the magnetization shows no signs of decay.

Figure 4

(a) [(b)] Decay timescale of the time crystalline order parameter ${\tau }_{\mathrm{TC}}$ as a function of the energy density of the initial state for the short- [long-]range model. In the short-range model (a), ${\tau }_{\mathrm{TC}}$ is fast, independent of frequency, and in agreement with the decay timescale of the magnetization $M(t)$ if the system were evolved according to $D$ alone (red squares). In the long-range model (b), an analogous behavior occurs near the center of the spectrum. However, as one moves to higher energies across the paramagnetic to ferromagnetic phase transition (red shaded region), ${\tau }_{\mathrm{TC}}$ becomes exponentially dependent on the frequency of the drive and ${\tau }_{\mathrm{TC}}$ approaches ${\tau }^{*}$. In this regime, ${\tau }_{\mathrm{TC}}$ is set by the exponentially slow heating rather than the prethermal dynamics for all frequencies—the prethermal time crystal is stable.

Figure 5

Schematic explanation of the behavior near the transition of the long-range model (Fig. 4). There are two competing timescales: the heating time ${\tau }^{*}$ and the magnetization decay time ${\tau }_{\mathrm{mag}}$ of the prethermal Hamiltonian $D^{*}$ [captured by the red squares in Figs. 4 and 4]. As the system approaches the phase transition into the ferromagnetic phase (shaded region) from the paramagnetic side, ${\tau }_{\mathrm{mag}}$ diverges (red dashed line). The relaxation time ${\tau }_{\mathrm{TC}}$ is given by the smaller of these two timescales. In (most of) the paramagnetic phase, ${\tau }_{\mathrm{mag}}$ is smaller and approximately frequency independent; while in the ferromagnetic phase, ${\tau }^{*}$ is smaller; ${\tau }_{\mathrm{TC}}$ shares its strong frequency dependence.

Figure 6

Analysis of the evolution of different single spin operators—${\sigma }_4^z$, ${\sigma }_{10}^z$, ${\sigma }_4^x$ and ${\sigma }_{10}^x$—for the different conditions considered in Fig. 3: the short-range model (a),(d),(g),(j), a “cold” initial state in the long-range model (b),(e),(h),(k), and a “hot” initial state (e),(f),(i),(l). On the different single spin observables, we observe the approach to a position-independent constant within the prethermal regime, consistent with the plateau observed in the $\omega \to \infty$ limit Floquet evolution, further suggesting that the system has approached a thermal state of the prethermal Hamiltonian $D^{*}$. By increasing the frequency of the driven system, we observe this agreement extending to later time, highlighting that the disagreement occurs due to the late time heating which becomes meaningful ${\tau }^{*}\sim e^{\omega /J_{\text{local}}}$. We also note that this simple picture is more complex in the case of ${\sigma }^x$. In this case, one needs to account for the small frame rotation $\mathcal{U}$ which can induce a finite overlap between $\mathcal{U}{\sigma }^x{\mathcal{U}}^{†}$ and an observable that fails to commute with $X$.

Figure 7

Example of the fitting procedure for extracting the decay times for a particular initial state evolved with the long-range Floquet evolution. We apply the same procedure to all initial states in both the short- and the long-range model. We observe that a simple exponential decay captures the approach of different observables to their thermal values: (a) energy density $⟨D(t)⟩/L$, (b) time crystalline order parameter $\mathrm{\Delta }M(t)$, (c) $\widehat{x}$ magnetization $M_x(t)$ (here plotted with a moving average over five points for clarity), and (d) half-chain entanglement entropy $S_{L/2}(t)$. (e) The entanglement entropy provides an extra timescale ${\tau }_{\mathrm{pre}}$ which captures the approach to the prethermal state. The $x$ axis in the shaded region is linear with time to emphasize the early time entanglement entropy behavior. (f) Comparison of the different decay times. The decay time of the energy density ${\tau }_{D^{*}}^{*}$, entropy ${\tau }_{S_{L/2}}^{*}$, and $\widehat{x}$ magnetization ${\tau }_{S_x}^{*}$ provide different estimates of the true thermalization timescale of the system ${\tau }^{*}$. Because this particular initial state is a “cold” state of the long-range model, it hosts a prethermal time crystal; the decay of the time crystalline order parameter also occurs at ${\tau }^{*}$. The agreement of all these timescales further corroborates the existence of a prethermal time crystal and the existence of a single thermalization timescale. Finally, we observe that ${\tau }_{\mathrm{pre}}$ occurs at a much earlier, frequency-independent timescale.

Figure 8

Analogous to Fig. 7, but considering an initial state time evolved with the short-range Floquet evolution. As in Fig. 7, we observe that a simple exponential decay captures the broad features of the approach of the different quantities to their thermal values. Moreover, we also observe a good agreement between the ${\tau }_{D^{*}}^{*}$, ${\tau }_{S_{L/2}}^{*}$, and ${\tau }_{S_x}^{*}$ as measures of the thermalization time ${\tau }^{*}$. However, unlike the long-range case, the time crystalline order parameter (b) decays at a much faster, frequency-independent, timescale. This time is on the same order of ${\tau }_{\mathrm{pre}}$, further corroborating that, in this case, the decay of the time crystalline order arises from the dynamics of the prethermal Hamiltonian.

Figure 9

Evolution of the half-chain entanglement entropy $S_{L/2}(t)$ for different initial states evolved under the long-range $D$. We observe that, for states away from the phase transition (blue lines), the evolution is characterized by a fast approach to a well-defined constant plateau. However, for initial states near the phase transition (red lines), the approach takes a very long time, displaying a slowly growing entropy for very long times and displaying large fluctuations. The initial states, marked in red, correspond to the state lying in the transition region in Fig. 4 of the main text.

Figure 10

Quantum Monte Carlo calculation exhibiting a peak in the heat capacity. We identify the position of the peak as the position of the finite-size transition or crossover between paramagnetic and ferromagnetic phases. This occurs at $T_c^{L=22}/J=3.97\pm 0.44$. Using the measured value of the energy density $⟨D⟩/L$ as a function of temperature, we can directly obtain the critical energy density ${\epsilon }_C^{L=22}=-(1.83\pm 0.37)$.