Quantum Time Crystals from Hamiltonians with Long-Range Interactions

TL;DR

This paper proposes a new Hamiltonian model with long-range interactions that enables the realization of quantum time crystals in closed systems, breaking continuous time-translational symmetry and connecting it to discrete cases.

Contribution

It introduces a Hamiltonian with long-range multispin interactions that allows for stable quantum time crystals in closed systems, bypassing previous no-go theorems.

Findings

Quantum time crystals are stable at zero temperature.

The model breaks continuous time-translational symmetry.

It links continuous and discrete TTS in quantum systems.

Abstract

Time crystals correspond to a phase of matter where time-translational symmetry (TTS) is broken. Up to date, they are well studied in open quantum systems, where external drive allows to break discrete TTS, ultimately leading to Floquet time crystals. At the same time, genuine time crystals for closed quantum systems are believed to be impossible. In this study we propose a form of a Hamiltonian for which the unitary dynamics exhibits the time crystalline behavior and breaks continuous TTS. This is based on spin-1/2 many-body Hamiltonian which has long-range multispin interactions in the form of spin strings, thus bypassing previously known no-go theorems. We show that quantum time crystals are stable to local perturbations at zero temperature. Finally, we reveal the intrinsic connection between continuous and discrete TTS, thus linking the two realms.

Equations

$$\underset{N\to \infty }{\lim }⟨\hat{\Phi }(t)\hat{\Phi }(0)⟩=f(t),$$

$$\hat{H}={ϵ}_0|E_0⟩⟨E_0|+{ϵ}_1|E_1⟩⟨E_1|+\sum _{j>1}{ϵ}_j|E_j⟩⟨E_j|,$$

$$f_N(t)=⟨E_0|{\hat{M}}_z(t){\hat{M}}_z(0)|E_0⟩=⟨E_0|e^{i\hat{H}t}{\hat{M}}_z{e}^{-i\hat{H}t}{\hat{M}}_z|E_0⟩$$

$$=e^{i{ϵ}_{0}t}⟨E_0|{\hat{M}}_z^2|E_0⟩⟨E_1|e^{-i\hat{H}t}|E_1⟩=\mathcal{O}{e}^{-i({ϵ}_1-{ϵ}_0)t},$$

$$|G_{\mp }⟩=\frac{1}{\sqrt{2}}(|\uparrow \uparrow ...\uparrow ⟩\mp |\downarrow \downarrow ...\downarrow ⟩),$$

$$\hat{H}=-|G_{+}⟩⟨G_{+}|,$$

$$\hat{\rho }=\frac{1}{m}\sum _{i=1}^m|E_0^{\{i\}}⟩⟨E_0^{\{i\}}|.$$

$$f_N(t)=$$

$$+Z^{-1}\sum _{j>0}{e}^{-\beta {ϵ}_j}{e}^{i{ϵ}_{j}t}⟨E_j|{\hat{M}}_z{e}^{-i\hat{H}t}{\hat{M}}_z|E_j⟩,$$

$$f_N(t)=$$

$$+Z^{-1}\sum _{j>0}\sum _{k\ge 0}{e}^{-\beta {ϵ}_j}{e}^{i({ϵ}_j-{ϵ}_k)t}|⟨E_j|{\hat{M}}_z|E_k⟩{|}^2.$$

$$\hat{H}=-\frac{J}{N(N-1)}\sum _{1\le i<j\le N}{\sigma }_x^{(1)}{\sigma }_x^{(2)}...{\sigma }_y^{(i)}...{\sigma }_y^{(j)}...{\sigma }_x^{(N)},$$

$${\hat{H}}_0=-\sum _{j=1}^N{\sigma }_z^{(j)}{\sigma }_z^{(j+1)},$$

$${\hat{H}}_1={\sigma }_x^{(1)}{\sigma }_x^{(2)}...{\sigma }_x^{([N/2])}-{\sigma }_x^{([N/2]+1)}...{\sigma }_x^{(N)},$$

$$\underset{N\to \infty }{\lim }⟨\mathrm{GS}|e^{i\hat{H}(J)t}{\hat{M}}_z{e}^{-i\hat{H}(J)t}{\hat{M}}_z|\mathrm{GS}⟩$$

$$=\underset{N\to \infty }{\lim }⟨\mathrm{GS}|M_z^2|\mathrm{GS}⟩e^{-i2(\sqrt{1+J^2}-1)t}$$

$$={\sin }^2(\frac{3\pi }{8})e^{-i2(\sqrt{1+J^2}-1)t},$$

$$\displaystyle\log\hat{\mathcal{U}}_{\mathrm{step}}=\lim_{\rho\rightarrow 0}\frac{1}{\rho}\Big{(}\hat{\mathcal{U}}_{\mathrm{step}}^{\rho}-1\Big{)},$$

$${\hat{H}}_{\mathrm{DTC}}=-\frac{1}{N}\sum _{j=1}^N{\sigma }_z^{(j)}{\sigma }_z^{(j+1)}+(-1{)}^{(N-1)/2}\frac{\pi }{2}\prod _j{\sigma }_x^{(j)},$$

Full text

*Introduction.—*Crystalline structures are ubiquitous in nature, and surround us in everyday life. Formally, a crystal is defined as a structure of atoms which is periodically ordered in space, and is stable to external perturbations AshcroftMermin . Generalized to various other constituents, the concept was found applicable in many areas of science, including photonic crystals in optics Lodahl2015 ; Istrate2006 , synthetic cold atom lattices Bloch2008 , granular materials Beloborodov2007 etc. While thought to be inherent to the spacial degrees of freedom, the contemporary understanding of physics in space-time posed the question of possibility to have a crystalline structure in the time dimension. This would correspond to a time-periodic behavior, which persists perpetually. For the classical case, the time crystal (TC) was described by Shapere and Wilczek Shapere2012 . Finally, the quantum case was postulated by Frank Wilczek Wilczek2012 , theorizing the breaking of time-translational symmetry. This was suggested to be observed in superconducting rings Wilczek2013 or trapped ion Li2012 systems. However, despite the basic time crystal formulation remains an intriguing concept, the subsequent studies have shown that ground state TC cannot be realized in the simple setting Bruno2013C ; Bruno2013 , and are ruled out by the no-go theorem.

The formalized definition of a quantum time crystal was set by Watanabe and Oshikawa (W-O) Oshikawa , who proposed to closely follow the definition for spatial crystals. It should be noted, however, that there is a difference between breaking the space and time-translational symmetries. In the case of spontaneous breaking of the space translational symmetry the ground state is highly degenerate in the thermodynamic limit. This results in the fact that the ground state of a crystal does not have a definite momentum even though the Hamiltonian commutes with the momentum operator. The continuous time translational symmetry is present in isolated systems where the Hamiltonian does not depend on time (the Hamiltonian commutes with the evolution operator) but the ground state must have a certain energy by definition. Instead, W-O defined the condition for the correlation function to be a periodic function of time $t$ as

[TABLE]

where the operator $\hat{\Phi}(t)$ corresponds to an integrated order parameter (e.g. average density or magnetization) at time $t$, and the averaging over the ground state of the system is considered in the thermodynamic limit of infinite number of lattice sites $N$. At the same time, W-O have proven that non-vanishing periodic oscillations are impossible unless two conditions are violated: 1) the system is out-of-equilibrium; or 2) the interaction is of long-range type, decaying slower than power law $\sim r^{-\alpha}$ for $\alpha>0$. This left the question of possibility for quantum time crystals to remain in the air.

Analyzing the first constraint, the consideration of the driven quantum system previously has allowed breaking TTS in discrete time Sacha2015 . The idea relies on the successive application of driver and interaction unitaries, and ultimately led to the appearance of Floquet or discrete time crystals (DTC) Else2016 ; Khemani2016 ; Sacha2015 . For the chain of spin-1/2 particles, the protocol reads as simultaneous spin flip application to the entire chain, which leads to the oscillations of magnetization, followed by Ising type interaction unitary. The major feature of the described system is a subharmonic response which is stable to drive perturbations Else2016 ; Yao2017 . As the driven system is prone to heating in high frequency limit, it also involves many-body localized unitary evolution, although later this condition was relaxed if the finite time scale is considered Else2017 ; Huang2018 ; YuAngelakis2019 . The successful theoretical prediction of time-periodic structures in driven systems was followed by its experimental observation in trapped ions ZhangMonroe2017 , silicon vacancy-based quantum simulators ChoiLukin2017 , superfluid helium-3 AuttiVolovik2018 , NMR systems Pune2018 ; Rovny2018 . Finally, many works targeting other platforms and generalizations were presented, including TC in fermionic lattices Huang2018 and Majorana fermions NTU2019 , dissipative DTCs OSullivan2018 , topological DTCs Giergiel2019 , discrete time quasicrystals Giergiel2019b , cosmological space-time crystals DasGhosh2018 ; DasGhosh2019 , and described the dynamical phase transition for DTCs Kosior2018 . The emergent field up to date was reviewed in several in-depth studies Sacha2018 ; Else2019 . Further directions represent exploration of quantum scar physics in nonequilibrium setting Turner2018 ; WWHo2019 ; Choi2019 ; Iadecola2019 or modification of TC definition to nonlocal order parameters Medenjak2019 . Finally, to study the breaking of continuous TTS, several open system realizations were considered up to date, including fermionic cold atom lattices Buca2019 and Rydberg atoms Gambetta2019 .

Although the number of systems demonstrating discrete time crystalline behavior grows rapidly for driven-dissipative and Floquet systems, the original issue of possibility to break continuous time-translational symmetry and existence of genuine time crystals for closed quantum system remains an open challenge. In this Letter we use the original time crystal definition and exploit the second loophole in W-O no-go theorem to show that there is a class of static Hamiltonians which host TC phase for the unitary dynamics. In particular, using W-O definition for spin-1/2 many-body system, we find Hamiltonian with spin string-type interactions which corresponds to a system with a ground state being a nondegenerate maximally entangled state. The minimal size of the spin-string to support TC phase is $N/2$ for a system of $N$ particles, thus formally bypassing the applicability range of W-O theorem Oshikawa . The stability of the system is studied, showing resilience to local perturbations.

*Continuous time crystal Hamiltonian.—*We start by postulating that certain many-body Hamiltonians which contain terms involving all particles can support time crystalline phase and exhibits spontaneously broken continuous TTS. Here, we present a method of constructing a Hamiltonian satisfying the W-O definition of a quantum time crystal. To be concrete, we focus on an $N$-qubit system, and as an operator $\hat{\Phi}$ in the definition (1) we consider the total magnetization $\hat{M}_{z}=\sum_{i=1}^{N}\sigma^{(i)}_{z}/N$ along the z-axis, where $\sigma^{(i)}_{x,y,z}$ denote Pauli matrices acting on qubits at site $i$. In this case, the Watanabe-Oshikawa TC definition in Eq. (1) recasts as $\lim\limits_{N\rightarrow\infty}\langle\hat{M}_{z}(t)\hat{M}_{z}(0)\rangle=f(t)$ where $f(t)$ must be periodic. To construct a Hamiltonian satisfying this definition we start with a state $\lvert E_{0}\rangle$ from the Hilbert space that we chose as a ground state at energy $\epsilon_{0}$. Acting on $\lvert E_{0}\rangle$ by a non-unitary operator $\hat{M}_{z}$ and renormalizing the vector gives the partner state $\lvert E_{1}\rangle=\hat{M}_{z}\lvert E_{0}\rangle/\sqrt{\langle E_{0}\rvert\hat{M}_{z}^{2}\lvert E_{0}\rangle}$. The crucial point here is to demand that $\lvert E_{0}\rangle$ is chosen such that $\lvert E_{1}\rangle$ is orthogonal to $\lvert E_{0}\rangle$, i.e. $\langle E_{0}\lvert\hat{M}_{z}\rvert E_{0}\rangle=0$. Having done that, we assign an energy $\epsilon_{1}>\epsilon_{0}$ to the state $\lvert E_{1}\rangle$, and construct the Hamiltonian using projectors for the many-body states as

[TABLE]

where arbitrary real numbers $\epsilon_{1},\epsilon_{2},...,\epsilon_{2^{N}-1}$ must all be greater than $\epsilon_{0}$, with $\epsilon_{0}$ being a ground state energy, and arbitrary state vectors are chosen such that $\langle E_{0,1}\rvert E_{j}\rangle=0$ for $j>1$. We claim that Eq. (2) describes the quantum time crystal Hamiltonian if certain restrictions on $\lvert E_{0}\rangle$ are respected. To find these restrictions we calculate the correlation function at finite size $N$ of the system given in Eq. (1) for the Hamiltonian (2), which yields

[TABLE]

where $\mathcal{O}=\langle E_{0}\rvert\hat{M}_{z}^{2}\lvert E_{0}\rangle$ denotes the order parameter. If it remains non-zero $\mathcal{O}\neq 0$ at $N\rightarrow\infty$ (as for the case of long-range order in crystals Oshikawa ) and the energy difference $\omega=\epsilon_{1}-\epsilon_{0}\neq 0$, then $\lim_{N\rightarrow\infty}f_{N}(t)=f(t)$ is a periodic function of time in the thermodynamic limit, oscillating at frequency $\omega$. The latter condition on $\omega$ can be fulfilled easily by construction, whereas the former one needs an additional consideration.

We remind that our goal is to find such a state $\lvert E_{0}\rangle$ that: (a) $\langle E_{0}\rvert\hat{M}_{z}\lvert E_{0}\rangle=0$ for any $N$, and (b) $\mathcal{O}$ remains non-zero at $N\rightarrow\infty$. In order to analyze the conditions (a) and (b) it is convenient to expand $\lvert E_{0}\rangle$ in the eigenbasis of the Hermitian operator $\hat{M}_{z}$. The operator $\hat{M}_{z}$ has $N+1$ distinct eigenvalues $m_{k}=(N-2k)/N$, where $k=0,1,...,N$ each having degeneracy $C^{k}_{N}$. Denoting the corresponding eigenstates as $\lvert m_{k},d\rangle$, where $d=1,2,...,C^{k}_{N}$ is accounting for degeneracy, we can express $\lvert E_{0}\rangle=\sum_{k,d}c_{k,d}\lvert m_{k},d\rangle$, where $c_{k,d}$ are normalized to unity, giving $\langle E_{0}\lvert\hat{M}_{z}\rvert E_{0}\rangle=\sum_{k,d}|c_{k,d}|^{2}m_{k}$ and $\langle E_{0}\lvert\hat{M}_{z}^{2}\rvert E_{0}\rangle=\sum_{k}|c_{k,d}|^{2}m_{k}^{2}$. The eigenbasis $\lvert m_{k},d\rangle$ corresponds to states having $k$ spins pointing down and $N-k$ pointing up, and vice versa if we replace $k$ with $N-k$ ($m_{k}=-m_{N-k}$). Thus, in order to fulfill condition (a) we can choose any superposition such that $|c_{k,d}|=|c_{N-k,d}|$.

One of the examples of the states satisfying the requirements discussed above, is the pair of maximally entangled Greenbergen-Horne-Zeilinger (GHZ) states Toth2005

[TABLE]

as they fulfill $\hat{M}_{z}\rvert G_{-,+}\rangle=\rvert G_{+,-}\rangle$ and $\langle G_{-,+}\lvert G_{+,-}\rangle=0$. Thus, the simplest possible TC Hamiltonian can be constructed as

[TABLE]

where, in accordance with the notation introduced earlier, $\epsilon_{0}=-1$, $\epsilon_{1}=\epsilon_{2}=...=\epsilon_{2^{N}-1}=0$.

If the ground state is $m$-degenerate, $m$ remains finite at $N\rightarrow\infty$, and for each $i=1,...,m$ ground state $\lvert E_{0}^{\{i\}}\rangle$ the quantity $\langle E_{0}^{\{i\}}\rvert\hat{M}_{z}^{2}\lvert E_{0}^{\{i\}}\rangle$ does not vanish in the thermodynamic limit, then the system remains a quantum time crystal for the averaging in Eq. (1) performed over the zero temperature density matrix

[TABLE]

Moreover, the same conclusion holds for non-zero temperature once the contribution of excited states $~{}|\langle E_{j}|\hat{M}_{z}|E_{k}\rangle|^{2}$ remain finite in $N\rightarrow\infty$ limit.

Generalizing the zero temperature case considered above, we proceed to the case of non-zero temperature $T=(k\beta)^{-1}$, when the system is described by the density matrix $\hat{\rho}=Z^{-1}e^{-\beta\hat{H}}$, where $Z=\sum_{i}e^{-\beta\epsilon_{i}}$. Without loss of generality, we focus on the system described by the Hamiltonian (2) having the non-degenerate ground state. Then, the correlation function (1) for finite $N$ reads

[TABLE]

and inserting the unity decomposition in the last line right after $e^{-i\hat{H}t}$ we obtain

[TABLE]

From the analysis of Eq. (8) we conclude that if there is a finite number of non-zero terms $\langle E_{j}\rvert\hat{M}_{z}\rvert E_{k}\rangle$ in the thermodynamic limit and $Z$ does not diverge in the same limit, then the system represents a quantum time crystal for any temperature.

*Locality of TC Hamiltonian.—*Above we have presented the quantum time crystal Hamiltonian by construction, formulated in terms of the many-body projector and coupling operators. However, when it is rewritten in terms of physical interaction between qubits [e.g. taking the toy TC Hamiltonian from (5)], it reveals highly nonlocal character. One can easily check that resulting expression contains $N$-length spin strings of $\sigma_{x}$ operators with pairwise substitution of $\sigma_{y}^{(i)}\sigma_{y}^{(j)}$ couples, and even-length strings of $\sigma_{z}$ operators NoteBehemoths . This corresponds to the sum of GHZ-state stabilizer products Toth2005 , which must satisfy the conditions specified for Hamiltonian (2). While there are various ways the spin-strings can be combined, one can consider a particular example and verify that W-O conditions for TC are satisfied. For instance, taking the Hamiltonian of an $N$-qubit system in the form

[TABLE]

where there are exactly two $\sigma_{y}$ operators in each term filled with $\sigma_{x}$’s, we observe that its spectrum is bounded between $-J/2\leq\epsilon_{n}\leq J/2$. The non-degenerate ground state of the Hamiltonian (9) is given by $\lvert G_{-}\rangle$ with energy $\epsilon_{0}=-J/2$, and the highest exited state corresponding to $\epsilon_{2^{N}-1}=J/2$ energy (which is also non-degenerate) reads $\lvert G_{+}\rangle$. Thus, the Hamiltonian (9) fulfills W-O TC definition Oshikawa in the thermodynamic limit. It is worth noting that one could take a correlation function of the form $\langle E_{0}\rvert\sigma^{(i)}_{z}(t)\sigma^{(j)}_{z}(0)\rvert E_{0}\rangle$ and get absolutely the same result for the correlation function (1), namely $e^{-iJt}$ (the two correlation functions coincide for an arbitrary $N$). Thus, the provided Hamiltonian (9) implements a genuine time crystal formally bypassing W-O no-go theorem by exploiting nonlocality. We also note that long-range multispin interaction Hamiltonians may be potentially studied in the systems with slow but strong driving, where heating is precluded Haldar2018 ; Lazarides2014 .

Finally, we can ask the question: what is the most local Hamiltonian which satisfies TC conditions described after Eq. (2)? To answer it, one needs to find the Hamiltonian containing in the spectrum $\lvert G_{+}\rangle$ and the shortest interaction range. The ultimate bound on such interaction was given in Ref. FlorioPRL2011 , where the most local Hamiltonian comprises of at least $[N/2]$-length strings. It can be written as $\hat{H}(J)=\hat{H}_{0}+J\hat{H}_{1}$, and contains the nearest-neighbour Ising Hamiltonian

[TABLE]

for the periodic boundary conditions $(\sigma_{z}^{(N+1)}\equiv\sigma_{z}^{(1)})$, and spin-string interaction part

[TABLE]

which involves the half of the qubits. The ground state $\lvert\mathrm{GS}\rangle$ of the Hamiltonian $\hat{H}(J)$ satisfies $M_{z}\lvert\mathrm{GS}\rangle\sim\lvert G_{+}\rangle$ and $\langle\mathrm{GS}\lvert M_{z}^{2}\rvert\mathrm{GS}\rangle$ is finite at $N\rightarrow\infty$. These states have the energies $\hat{H}(J)\lvert G_{+}\rangle=-N\lvert G_{+}\rangle$ and $\hat{H}(J)\lvert\mathrm{GS}\rangle=[-N-2(\sqrt{1+J^{2}}-1)]\lvert\mathrm{GS}\rangle$. The correlation function (1) for the Hamiltonian $\hat{H}(J)$ is given by

[TABLE]

thus corresponding to the time crystalline behavior for the most local TC Hamiltonian $\hat{H}(J)$. In the absence of $\hat{H}_{1}$ the Hamiltonian $\hat{H}(J)$ reduces to the Ising Hamiltonian having $|G_{+,-}\rangle$ as degenerate ground states.

*Stability of time crystal.—*The important feature of time crystal phase is its stability towards perturbations in the Hamiltonian (extra magnetic field), which in the case of DTC has allowed to distinguish from otherwise trivial spin flip operation Else2016 ; Yao2017 . Examining the behavior of the Hamiltonian $\hat{H}(J)$ under magnetic field perturbations, $\hat{H}^{\prime}=\hat{H}(J)+\delta\hat{H}$ with $\delta\hat{H}=\sum_{j}(h_{x}^{j}\sigma_{x}^{(j)}+h_{y}^{j}\sigma_{y}^{(j)}+h_{z}^{j}\sigma_{z}^{(j)})$, we note that $\langle\mathrm{GS}\rvert\delta\hat{H}\lvert\mathrm{GS}\rangle=0$ and $\langle G_{+}\rvert\delta\hat{H}\lvert G_{+}\rangle=0$. Moreover, the perturbation of the form $\delta\hat{H}=\sum_{j}\sigma_{x,y}^{(j)}\sigma_{x,y}^{(j+1)}$ analogously does not overlap with $\lvert\mathrm{GS}\rangle$ or $\lvert G_{+}\rangle$. Thus, these types of perturbations do not affect the part of the Hamiltonian that is responsible for TC behavior, $\hat{H}_{01}=\epsilon_{0}\lvert E_{0}\rangle\langle E_{0}\rvert+\epsilon_{1}\lvert E_{1}\rangle\langle E_{1}\rvert$ (for $\hat{H}(J)$ the ground state is $\lvert E_{0}\rangle\equiv\lvert\mathrm{GS}\rangle$ and $\lvert E_{1}\rangle\sim\lvert G_{+}\rangle$).

If the temperature is non-zero, then the analysis of stability becomes more subtle as exited states come into play and contribute to the correlation function, leading to the generation of higher harmonics. Therefore, it is important to understand the scaling of the number of additional distinct harmonics with the size of the system. At non-zero temperature, our investigation of the Hamiltonian $\hat{H}(J)$ is limited to numerical analysis based on exact diagonalization which showed that the number of additional distinct harmonics grows with the size of the system. Thus, we expect the melting of the time crystal at non-zero temperature, which is shown in Fig. 1. However, the analysis of the TC phase at non-zero temperature in the thermodynamic limit remains an open question.

*Relation to discrete time crystals.—*To better understand the origin of TC behavior for continuous time, we can study its relation to the discrete counterpart Else2016 . We recall that standard DTC protocol is based on the unitary operation comprising a sequence of gates for an $N$-qubit chain. It reads as: 1) start with all-down or all-up spin string and apply spin flip operator $\hat{U}_{\mathrm{X}}=\exp(-i\pi/2\sum_{j}\sigma_{x}^{(j)})=\prod_{j}(-i\sigma_{x}^{(j)})$; 2) evolve the state with Ising Hamiltonian $\hat{U}_{\mathrm{Ising}}=\exp(-i\tau\sum_{j=1}^{N}J_{j,j+1}\sigma_{z}^{(j)}\sigma_{z}^{(j+1)})$, thus completing the single step unitary $\hat{\mathcal{U}}_{\mathrm{step}}=\hat{U}_{\mathrm{Ising}}\cdot\hat{U}_{\mathrm{X}}$; 3) continue for many stroboscopic periods $\hat{\mathcal{U}}_{\mathrm{DTC}}=\hat{\mathcal{U}}_{\mathrm{step}}\cdot\hat{\mathcal{U}}_{\mathrm{step}}\cdot...$ . The algorithm is sketched in Fig. 2 (a). The resulting stroboscopic dynamics then shows persistent oscillations of $M_{z}$ magnetization with period twice bigger than the drive $\prod_{j}(-i\sigma_{x}^{(j)})$.

Given the unitary operation, we can formally define the static Hamiltonian which governs the dynamics as $\hat{H}_{\mathrm{DTC}}=i\log(\hat{\mathcal{U}}_{\mathrm{step}})$. Generally, the logarithm can be computed the using Dynkin’s formula, but the infinite order resummation needed to obtain the closed form expression seems infeasible. Instead, we can analyze the structure of $\hat{H}_{\mathrm{DTC}}$ by studying the unitary operator in Fig. 2 (a). First, exploiting the Euler’s formula we can combine the $\sigma_{x}$-string and the layer of Ising terms into a single block. Denoting phases as $J_{j,j+1}\tau=\phi_{j}$, the odd $j$ sublattice unitary reads $\prod_{j}(-1)\{\cos(\phi_{j}/2)\sigma^{(j)}_{x}\sigma^{(j+1)}_{x}+\sin(\phi_{j}/2)\sigma^{(j)}_{y}\sigma^{(j+1)}_{y}\}$, which we denote as blocks “XX + YY” in Fig. 2 (b). These blocks multiplied altogether contain various $\sigma_{x}$ and $\sigma_{y}$ strings, from length $2$ to length $N$. Finally, the multiplication of the second sublattice Ising interaction either leaves string unaltered, or changes $\sigma_{x}\leftrightarrow\sigma_{y}$ (with some phase factor). Once the matrix $\hat{\mathcal{U}}_{\mathrm{step}}$ is obtained, the effective Hamiltonian can be inferred employing the replica trick Vajna2017 . This relies on taking formally the limit

[TABLE]

and corresponds to taking the powers of previously derived unitary matrix. In this case, the odd powers contribute to the same stings we observe in the Hamiltonian (9) [see Fig. 2(c)], albeit with different prefactors, and cross-terms from even powers contribute to the $\sigma_{z}$-strings of even length.

Finally, let us relate the physics of discrete time crystals through the properties of $\hat{H}_{\mathrm{DTC}}$ Hamiltonian and continuous time crystal Hamiltonian described in the previous sections. In particular, if the ground state for both Hamiltonians is all-down or all-up spin string or their superposition, than one may expect that breaking of continuous TTS is inherently related to breaking of the discrete TTS for the corresponding Floquet TC. To yield direct comparison, we can consider the odd size $N$ system with Ising interaction fixed $\phi_{j,j+1}=J_{j,j+1}\tau=-1/N$. Then, the matrix exponentials and logarithm can be calculated numerically, and their analysis provides a concise structure of the effective DTC Hamiltonian

[TABLE]

which is analogous to the time crystal Hamiltonian $\hat{H}(J)$ introduced earlier in a sense that it has a form of the Ising Hamiltonian plus a non-local part lifting the degeneracy of the ground state. For $N=4k+3$ the non-degenerate ground state of the Hamiltonian (14) is $\lvert G_{+}\rangle$ and $\lvert G_{-}\rangle$ is an exited state, whereas for $N=4k+1$ it is the other way round. The energy difference between $\lvert G_{\pm}\rangle$ states is equal to $\pi$ for any odd $N$. Therefore, the Hamiltonian (14) is nothing but a quantum time crystal and the corresponding correlation function for any odd $N$ is $f(t)=e^{-i\pi t}$.

*Conclusions and outlook.—*While previously it was believed to be impossible in the case of non-driven system, we have shown that time crystals can emerge for Hamiltonians with long-range interactions in the form of spin strings. We described the minimal continuous TC Hamiltonian as a sum of strings encompassing halves of a spin system, and find that at zero temperature TC is stable. Finally, we note that nonlocal many-body operators which induce time crystalline behavior represent a specific example of hermitian quantum behemoths Khaymovich2019 , possibly relating the two notions.