Abstract
Crystals arise as the result of the breaking of a spatial translation symmetry. Similarly, translation symmetries can also be broken in time so that discrete time crystals appear. Here, we introduce a method to describe, characterize, and explore the physical phenomena related to this phase of matter using tools from graph theory. The analysis of the graphs allows to visualizing time-crystalline order and to analyze features of the quantum system. For example, we explore in detail the melting process of a minimal model of a period-2 discrete time crystal and describe it in terms of the evolution of the associated graph structure. We show that during the melting process, the network evolution exhibits an emergent preferential attachment mechanism, directly associated with the existence of scale-free networks. Thus, our strategy allows us to propose a previously unexplored far-reaching application of time crystals as a quantum simulator of complex quantum networks.
INTRODUCTION
Symmetries are of utmost importance in condensed matter physics and statistical mechanics owing to the strong relation between quantum phases of matter and symmetry breaking (1–4). Among the zoo of symmetry-broken phases, discrete time crystals (DTCs) play a fundamental role on their own due to the type of symmetry involved (5). The time-crystalline phase, analogous to “space” crystals, arises when time (instead of space) translation symmetry is broken. The existence of quantum time crystals was originally proposed by Wilczek (6), and a discrete version of time crystals can be realized by periodically driven quantum systems (7–9). It is in such cases that the system dynamics exhibits a subharmonic response with respect to the characteristic period of the drive caused by the synchronization in time of the particles of a many-body system (5).
The exploration of time crystals is a very active field of research and several experimental realizations with trapped ions (10), dipolar spin impurities in diamond (11), ordered dipolar many-body systems (12), ultracold atoms (13), and nuclear spin-1/2 moments in molecules (14) have been achieved. Yet, an intuitive and complete insight into the nature of time crystals and its characterization, as well as a set of proposed applications, is lacking. We here provide tools based on graph theory and statistical mechanics to fill this gap. We propose a new strategy for the study and understanding of time symmetry-broken phases and their related phenomena (15, 16). As an example, we characterize the time-crystalline order and its melting.
Our approach allows us to identify an instance of a perturbed time crystal as a novel physical platform where to simulate complex networks whose evolution is governed by preferential attachment mechanisms (17, 18). This type of networks, far from being regular or random, contains nontrivial topological structures present in many biological, social, and technological systems. Small-world and scale-free networks are two of the most popular examples, the latter being commonly characterized through power law degree distributions that can be explained from the presence of a preferential attachment mechanism. The simulation of such networks has wide applicability, ranging from the study and understanding of behaviors present in communication or internet networks (19), the development of new algorithms in deep learning (14), or the analysis of genetic and neural structures in biological systems (20).
RESULTS
Floquet theory in a nutshell
The fundamentals of this work rely on the exploration of the dynamics of driven many-body quantum systems described by time-periodic Hamiltonians
DTCs of period 2T
We now move to a well-known period-2 DTC (2T-DTC) model (10, 15) consisting of a one-dimensional chain with n spin-1/2 particles and governed by a time-dependent Hamiltonian with period T = T1 + T2
Here,
Crucially, the Floquet operator depends on the error ϵ, which determines how the time crystal will melt. In our work, we choose α = 1.51, J0T2 = 0.06 with a disorder strength WT2 = π, which are similar values used in the recent experiment (10). The key feature of this unitary evolution is that at each period T, as shown in Fig. 1A, the system evolves between the states ∣↑↑⋯↑〉 and ∣ ↓↓⋯ ↓ 〉, when the state is initialized at one of these two. This periodic evolution is robust against moderate errors ϵ in
(A) On the left, diagram of the 2T-DTC dynamics with no rotation error, ϵ. The initial state (∣Ψ(0)〉 = ∣ ↑ ↑ ↑ … ↑ 〉), represented with the green arrows pointing up, is recovered after two periods of the driving protocol. From the first period, we obtain the unitary, U(T), that will be used as the Floquet operator,
Floquet graphs
Let us now introduce the concept of a Floquet graph as the graphical representation of the effective Hamiltonian,
The effective Hamiltonian,
To clarify the percolation on the graph, we define clusters as subsets of nodes, which are connected with a path. The graph is percolated when all nodes belong to a single cluster (28). We want to stress that the nodes represent many-body states ∣i〉 and not physical spin sites l, so the graph spans through the full Hilbert—configuration—space (see Fig. 1C). This percolation rule was proven to be useful to detect many-body localization to thermal phase transitions in undriven spin systems (27).
Time crystal graphs
Under the framework of Floquet graphs, the dynamics of the described 2T-DTC can now be investigated. The effective Hamiltonian
Melting of the 2T-DTC
Consider now the case where our system is initialized to ∣ψ(0)〉 = ∣2n − 1〉 = ∣ ↑ ↑ ⋯ ↑ 〉 (10). This state is a superposition of Floquet states,
(A) Graph representation obtained from
Conserved quantities
Let us now analyze what factors are determinant in the robustness of each pair of configurations or, alternatively, how they cluster as the crystal melts. With our system represented as a graph, we can now focus on how to interpret its structural properties in terms of conserved quantities. To do so, it is of fundamental importance to note that the system is switching stroboscopically between two configurations, ∣i〉 and ∣2n − 1 − i〉. If we prepare the system in a symmetry-broken state in terms of its total magnetization and observe the system every two periods (2T), the system will effectively remain in a manifold of states with a fixed quasienergy. For no error, the graph will be formed by decoupled single nodes. In that sense, if we sample the dynamics every two periods, the system is in an effective equilibrium state determined by conserved quantities that are defined stroboscopically. Now, let us construct the conserved quantities in the absence of error and show how they are destroyed when clusters in the graph are formed (Fig. 3).
(A) Graph representation obtained from
To calculate the conserved quantities, let us consider the square of the Floquet operator of Eq. 2, where
In the absence of error, ϵ = 0, the disorder cancels out exactly and the effective Hamiltonian reads
Breaking symmetries
For zero error, the 2n quasienergies are local integrals of motion and the Hilbert space can be classified by using the symmetries that preserve the aforementioned conserved quantities. In turn, the effective Hamiltonian
This allows us to understand how ϵ destroys the symmetries by coupling different symmetry multiplets. This can be observed from Fig. 3 (B to E) as off-diagonal entries start populating the effective Hamiltonian matrix due to the error. This situation resembles the Kolmogorov-Arnold-Moser (KAM) theory in classical mechanics, which describes how invariant tori are broken in phase space under the effect of perturbations (29), with our perturbation being ϵ. First of all, the transverse field term
Preferential attachment
Let us now put the focus on the network topology for moderate levels of rotation error (ϵ ∼ 0.012). In the previous section, we have seen that the quantum terms that appear in Eq. 7 break the conservation of number of domain walls. The error lifts the degeneracies present in the spectrum, and new transitions between close states appear. In terms of the graph, the nodes with the same or similar number of domain walls connect following a preferential attachment mechanism. As most nodes in the network have n/2 or n/2 − 1 (n: even) number of domain walls, and they have very close quasienergies, these nodes easily connect to each other with a small value of error. This leads to the appearance of the large degree hub nodes as well as the heavy-tailed degree distributions shown in Fig. 2C. The tail of these distributions can be fit to a power law distribution, which is a characteristic of scale-free networks. Following the recommendations of Clauset et al. (30), we further test the goodness of fit of the power law against the lognormal distribution for each of these distributions and observe that the power law favors over the lognormal distribution (see the “Power law fitting of the degree distributions” section in Methods for further details).
In addition, exploring the behavior of the average degree of the graph obtained from
The nodes are distinguished by their number of domain walls and averaged individually for each value of domain walls, as well as averaging over 100 realizations of disorder. Notice that some of the curves exhibit a local maximum approximately between 0.01 < ϵ < 0.03. This is exactly the range in which we observe the scale-free behavior emerging in our system. Comparing each curve also confirms that the states with a lower number of domain walls are more robust against the error in a sense that they have lower degrees.
This behavior not only has consequences for the dynamics of the time crystals but also can be used to perform quantum simulation of networks with exotic properties such as scale-free–like ones (18). Note that the scale-free–like networks shown in Fig. 2 are only observed in the effective Hamiltonian obtained from a single period
DISCUSSION
Our approach of representing DTCs in terms of graphs is key to understand the structure of such an exotic phase of matter and to envisage prospective applications. We explicitly show that by translating a 2T-DTC into a graph theory language, we can study, in detail, how the crystal order disappears as an increasing error melts the time crystal. Among others, it allowed us to identify the crucial role symmetries and conserved quantities play in the resilience of the crystal, by observing the preferential attachment mechanism present in the formation of clusters for this specific model of time crystal. Crucially, the nature of the obtained graphs for moderate levels of error suggests that such systems could be used as scale-free–like network simulators, giving rise to a previously unknown application of such devices in the field of complex networks (32, 33). Because our networks span the configuration space, such simulation could be done with moderate numbers of qubits and thus available noisy intermediate-scale quantum (NISQ) platforms (ranging from ion traps to superconducting qubit chips) could be used for its implementation. Structural information about the network (i.e., the degree distribution) using DTCs in NISQ devices could be experimentally obtained by exploiting a quantum walk in the configuration space. One could track down the number of configurations visited dynamically in a given time by measuring a set of l-point spin correlation functions, a measure that is experimentally accessible with current technology. The number of configurations visited in a given time for different initial conditions is related to the degree of the different nodes of the graph, unveiling the complexity of the network (see section SV for more details on our proposed experimental protocol). Future work will include further investigation and characterization on the nature of these networks as well as the study of additional phenomena present in time crystals in terms of graphs. We believe that the use of this formalism will not only lead to a more complete and deep understanding of DTCs and their related phenomena but also be very advantageous in the study of periodically driven quantum systems.
METHODS
Floquet theory: Percolation rule and graph structure
One of the main tools used in our work is the percolation rule, which establishes when there is a link between two nodes of a graph representing the effective Hamiltonian. In this section, we explain in detail the motivation of the percolation rule and its interpretation in terms of Floquet theory. To have an intuitive picture of the percolation rule, let us consider only two configurations ∣i〉 and ∣j〉 with energies Ei and Ej in the absence of drive, respectively. For simplicity, let us assume that the drive induces transitions only between the aforementioned configurations. Within the two-state approximation, the Hamiltonian from Eq. 1 reads
In general, the transitions occur when the driving is resonant with the energy level spacing between two configurations, i.e., when Ej − Ei = ℏkω, where k is an integer and ω = 2π/T is the frequency of the drive. Therefore, when the drive is switched on, the quasienergies λs1 and λs2, corresponding to the energies Ei and Ej, respectively, exhibit an anticrossing (21). In the language of Floquet theory, the latter is referred to as an m-photon resonance because the quasienergies are defined modulo ω.
If we restrict ourselves to the configurations ∣i〉 and ∣j〉, the effective Hamiltonian can be written as
From this, we can see that the energy gap for the quasienergies scales as
Now, let us consider a more detailed derivation of the percolation rule by using a locator-like expansion for driven systems. Here, we explicitly derive the percolation rule and motivate its origin in terms of resonances. Let us first shortly discuss the most relevant aspects of the locator expansion. In the Anderson model, one can expand the resolvent in terms of the hopping strength. For strong disorder, this can be understood as a perturbative expansion of the localized wave functions in the hopping term of the Hamiltonian, which is referred to as the locator expansion. The latter usually diverges due to resonances (34).
In driven quantum systems, the Hamiltonian is periodic in time and all the important information is contained in the Floquet operator
The operator
Although Eq. 11 looks different from the resolvent for undriven systems (34), they share the same structure if we consider the expansion
The next step is to use the perturbative expansion of Eq. 12 in the hopping term
This allows us to calculate the transition probability between two configurations ∣i〉 and ∣j〉, as follows
Power law fitting of the degree distributions
Let us first define the concept of degree within the context of graph theory. The degree of a vertex v belonging to a graph G is the number of edges connected to that vertex v. The degree distribution is therefore a probability distribution of vertex degrees over the whole graph G. Here, we explain how the degree distributions of the graphs obtained from the effective Hamiltonian of the 2T-DTC can be fit to a power law function p(k) ∝ k−β. We have followed the method by Clauset et al. (30). First, the degree distributions P(k) are obtained by measuring log-binned histograms of the degrees k of the graph. We have obtained the distributions from 100 realizations of the graph with different disorder values for statistical convergence. Generally, the power law behavior is observed in the large degree tail of the distribution rather than the entire domain. Therefore, we estimate the lower bound k = kmin where the best power law fit can be obtained. This is done by estimating the power law exponent β by applying the maximum likelihood method on a certain portion k ≥ kcutoff of the distribution. Once the exponent is estimated, the Kolmogorov-Smirnov distance between the distribution and the fit is computed. We compute this with various kcutoff > 0, and the kmin is chosen at the kcutoff where the Kolmogorov-Smirnov distance is minimized. Once the kmin and β are estimated, we examine the goodness of fit of the power law using a comparative test. We consider an alternative probability distribution function that may be a good fit for our data and apply a likelihood ratio test between the power law and the alternative function. Here, we chose the lognormal function as the alternative, which is another heavy-tailed function. From this comparative test, we conclude that the power law function well describes the degree distributions of our graphs.
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/42/eaay8892/DC1
This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.
REFERENCES AND NOTES
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