Floquet time crystals in clock models

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Universit`

a di Pisa

Dipartimento di Fisica

Laurea Magistrale in Fisica

Floquet time crystals in

clock models

Candidato

Federica Maria Surace

Relatore

Prof. Rosario Fazio

Sessione di Laurea del 20 Luglio 2017

Anno accademico 2016/2017

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Introduction

Spontaneous symmetry breaking is a concept of fundamental relevance in modern theo-retical physics. Realizations of a physical system that differ for their symmetry represent different states of matter: a symmetry breaking is related to a change in the phase of the system. One of the examples we are most familiar with is the phase transition from liquid to crystal that is associated to the spontaneous breaking of space translation symmetry. In analogy with crystals in space, Frank Wilczek recently proposed that spontaneous breaking of time translation symmetry can produce a “time crystal“. In his work, together with a first attempt to define a time crystal, Wilczek also suggested an example of a time-independent Hamiltonian that was claimed to have a time-periodic ground state. The idea was the following: a superconducting ring threaded by a magnetic flux can support a persistent current in its ground state. However, the current is perfectly homogeneous and stationary, and there are no physical observables that could reveal a time periodic dependence. Nevertheless, if the particles in the ring interact attractively, they localize in a specific position forming a lump and spontaneously breaking the rotational symmetry of the ring. A non-zero magnetic flux through the ring would induce a motion of the lump in the form of a persistent current. This periodic rotational motion can now be experimentally detected since the rotating lump is localized. Therefore, although the Hamiltonian is time-independent, the ground state exhibits a periodic motion that results in the oscillation of physical observables, thus spontaneously breaking the time translation symmetry of the Hamiltonian.

This idea immediately generated both enthusiasm and criticism. Experimental realiza-tions were proposed and other models were claimed to show spontaneous breaking of time translation symmetry. On the other hand, Patrick Bruno and Philippe Nozi`eres raised some objections against Wilczek’s conclusions and showed that a rotating motion was not possible in the model proposed. The problem was actively debated until 2015, when a very general no go theorem was proven by Watanabe and Oshikawa, showing that time crystals cannot exist either in the ground state or in thermal equilibrium. The theorem does not exclude time translation symmetry breaking in non-equilibrium settings. In fact, periodically driven many body systems can exhibit time-crystal behavior. Systems with a periodic drive have an underlying discrete time translation symmetry that can

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be spontaneously broken in a less symmetric phase. These are the so called Floquet time crystals introduced by Else, Bauer and Nayak and by von Keyserlingk, Khemani, Lazarides, Moessner and Sondhi. If the Hamiltonian is periodic with a period T , in the (discrete) time crystal phase the system will respond with a period nT . These subharmonic oscillations of physical observables associated to time translation symmetry breaking have some characteristic features that are not found in other similar phenomena occurring in non-correlated systems. In fact, time crystals originate from a many-body effect that crucially relies on the presence of interactions. As a consequence of this collective nature, oscillations with a multiple of the original period are remarkably stable to perturbations and persist for an infinitely long time in the thermodynamic limit.

Else, Bauer and Nayak proposed the first example of discrete time crystal. Their model consists of a disordered spin chain with Ising interactions, undergoing a periodic spin flip pulse. Disorder plays an essential role because it prevents the chain from indefinitely absorbing energy from the drive and from heating to an infinite temperature state. Absence of thermalization is a fundamental prerequisite for time crystals. The research for this novel phase of matter could benefit from the recent breakthroughs in the study of thermalization. In fact, a phenomenon known as many-body localization was discovered in 2005. Due to the presence of disorder, systems that are many-body localized cannot relax to a thermal state, hence they are the most promising candidates for time crystals. The model studied by Else, Bauer and Nayak was many body localized and exhibited the emergence of a phase with robust period doubling in the thermodynamic limit. Different experiments were performed confirming the predicted time-crystal oscillations: one has been realized using trapped atomic ions interacting via optical dipole forces; another one used an ensemble of nitrogen vacancy spin impurities in diamond.

Motivated by the results of Else, Bauer and Nayak, the main purpose of this thesis is to examine both analytically (where possible) and with numerical simulations the existence of a time crystal phase in a more general class of models. The paradigmatic example of a discrete time crystal with period doubling is based on an Ising chain, where a

Z2symmetry is spontaneously broken. Therefore, an analogous model with an underlying

Zn symmetry (the so called n-state clock model) and an appropriate drive of period T , is

expected to result in a time crystal with period nT .

The thesis is organized as follows: in chapters 2 and 3 the works related to time crystals are reviewed from the earliest proposals to the most recent experimental realizations. The main concepts associated with spontaneous symmetry breaking are introduced, highlighting those aspects that are needed to properly define time translation symmetry breaking. Wilczek’s model and its limitations are then reviewed; chapter 2 ends with the enunciation of the no-go theorem and with the discussion of the consequent interest for possible non-equilibrium time crystals. This topic is then resumed in chapter 3 with the specific focus on discrete (or Floquet) time crystals. After a general analysis of periodically driven system and their dynamical properties, a rigorous definition of Floquet time crystals is presented and illustrated through the example studied by Else, Bauer and Nayak. In conclusion, experimental realizations and other recent theoretical works are reviewed.

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The periodically driven 3-state clock model is considered in chapter 4. Using analytic calculations for an exactly solvable case, we show that two different situations are possible: if the model is non-chiral, i.e. if the parameters in the Hamiltonian are such that the interactions have an additional time reversal symmetry, a time crystal is not possible due to the degeneracies in the spectrum; on the other hand, if the model is chiral (with no time reversal symmetry) the system is expected to be a time crystal. This result can be generalized when a perturbation is added and the model is no longer exactly solvable. In fact, the time crystal features a remarkable stability to external perturbations, thanks to the existence of a ”local spectral gap“. Under very general assumptions, it is possible to deduce that the time crystal phase is not destroyed provided that the perturbation is sufficiently small. In this thesis, we specifically show that the subharmonic oscillations and the spectral properties characteristic of time crystal order are recovered in the thermodynamic limit for small perturbations from the solvable model. In order to do so, we identify some useful quantities that can be used as signatures of time translation symmetry breaking. The scaling of these quantities with the system size is predicted with analytic arguments and tested numerically. Numerical simulations are performed using exact diagonalization of finite size systems. The stability of the time crystal is studied and a phase diagram is derived. The non-chiral model exhibits a qualitatively different behavior from the chiral one. The phase diagram shows that the time crystal originating from a 3-state clock model is less and less stable as we approach the non-chiral case. In the non-chiral model, an infinitely small perturbation is able to destroy the time crystal order. This confirms our prediction about the absence of a time crystal phase for the non-chiral 3-state clock model.

In chapter 5 the general case of the n-state clock model is discussed. As in chapter 4, an exactly solvable case is first studied using analytic calculations. A time crystal with period nT is expected as long as the spectrum has no local degeneracies: the corresponding condition on the parameters of the Hamiltonian is derived. All the properties studied in the case n = 3 are shown to have a straightforward generalization to an arbitrary n. In conclusion, a slightly more general model is considered, showing that the n-state clock model can generate different time crystal phases depending on the drive.

In chapter 6 we review our results and discuss possible future directions for further research in this field.

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Chapter 2

Spontaneous time-translation

symmetry breaking

Matter is formed by atoms: they can organize in different ways which lead to many types of materials with a variety of distinctive properties. The emergence of distinct macroscopic phases of matter from the same microscopic constituents can be very elegantly accounted by means of Landau theory of symmetry breaking. Different organizations (or orders) of atoms in a material correspond to different symmetries. A system can change its symmetry, undergoing a phase transition. A change of the order that reduces the symmetry of the material is called symmetry breaking.

The concepts of symmetry and symmetry breaking have a fundamental importance in most areas of modern physics. A very general overview of the problem can be given, that applies to a number of different phenomena and different types of symmetries.

In this section we introduce spontaneous symmetry breaking with a particular attention to some aspects that will be necessary in the following chapters. We then explain the ideas that motivated the interest for time translation symmetry breaking and review the advances in the theory from the first proposals of time crystals to the search in a non-equilibrium context.

2.1 Spontaneous symmetry breaking

Natural phenomena involving an aggregate of a large number of particles cannot be understood from a simple extrapolation of the properties of the single particles in it [3]. A remarkable example is represented by systems that do not have all the symmetry of the physical laws which govern them: this can only happen for large systems or, in other words, in the “thermodynamic limit”. In a quantum system with an infinite number of degrees of freedom, spontaneous symmetry breaking (SSB) occurs when the Hamiltonian is invariant under a given symmetry group, but the physical realization of the system is not. According to Landau theory, spontaneous symmetry breaking is revealed by the so called “order parameter”. The order parameter is a local operator a(x) whose expectation

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value is zero for symmetry respecting states, and is non-zero for symmetry breaking ones. It should be noted that traditional statistical mechanics fails to detect spontaneous symmetry breaking in a finite size system. The density matrix ρ = e−βH _{for a system at}

temperature T = 1/kBβ necessarily has all the symmetry of H, hence it cannot describe

a symmetry breaking state. Calculating the average of a(x) with the usual formula

ha(x)i = T r[a(x)e

−βH_]

T r[e−βH_]

trivially yields ha(x)i = 0, because we are averaging on a symmetrical ensemble. A solution to this problem was formulated by Bogolyubov. According to his prescription, the correct averaged value can be obtained by introducing a symmetry breaking term in the Hamiltonian. Now, since the ensemble is not symmetric, the local order parameter can have a non-zero expectation value. To re-obtain the symmetric Hamiltonian we take the infinite volume limit and then send to zero the additional field. The symmetry is spontaneously broken if the average of the order parameter resulting from these limits is different from zero. The fundamental point is that we first have to take the thermodynamic limit and only then send to zero the symmetry breaking field: the limits have to be taken strictly in this order because they do not commute.

Bogolyubov’s prescription is often used to define spontaneous symmetry breaking. However, in many cases its application is not straightforward and an alternative definition is preferred. Such an equivalent definition can be formulated in terms of the long-range correlation of the order parameter

lim

|x−x0_|→∞ha(x)a(x

0_{)i = c,} _(2.1)

where the average can be calculated with the usual statistical mechanic approach. We can say that spontaneous symmetry breaking occurs when the correlations of the local (or microscopic) order parameter do not decay at long distance (i.e. c 6= 0) or, equivalently, when the integrated (or macroscopic) order parameter A =R

V dx d_{a(x) satisfies} lim V →∞ hA2_i V2 = C 6= 0. (2.2)

This special property featured by symmetry-breaking states is often known as long-range order. As an example of spontaneous symmetry breaking and long-range order we will now examine crystalline structures. This example is especially significant in our discussion because it was the analogy between space and time translation symmetries that inspired the search for time crystals.

Let us consider the example of (space) crystals. A basic assumption for our un-derstanding of the universe is that space is homogeneous and isotropic. Physical laws are translationally invariant, but many examples can be found of spontaneous break-ing of space translation symmetry [4]. Matter at sufficiently high temperature is in a homogeneous gaseous or liquid state and perfectly respects translation symmetry. The phase transition to a solid for lower temperatures takes the system to a much ordered

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2.1. SPONTANEOUS SYMMETRY BREAKING 11

phase where the atoms have a precise position in space and translation symmetry is spontaneously broken.

When the Coulomb interaction dominates over the kinetic energy, the ground state of a system of electrons has to minimize the potential energy

V =X

i6=j

V (|~ri− ~rj|).

In order to do so, the system of particles will assume a regular lattice structure with fixed relative distances between the atoms. This property of rigidity implies that different parts of the body do not act independently.

If the solid as a whole has a fixed momentum ~P , the Heisenberg uncertainty principle implies that the position of the center of mass is completely undefined. This fact does not contradict the rigidity of relative positions. In fact, the location of the center of mass may be completely undetermined, but relative positions are still perfectly defined.

The emergence of a long range order related to a spontaneous symmetry breaking is evidenced by the correlations in the density operator ˆρ(~r) =P

iδ(~r − ˆ~ri). The correlator

lim

V →∞hρ(~r)ρ(~r

0_{)i = f (~r − ~r}0₎ _(2.3)

reveals a periodic behavior as a function of ~r − ~r0_{. Let us consider a wavevector ~}_{G that}

matches the periodicity of the crystalline structure (i.e. that belongs to the reciprocal lattice). We can define the density operator modulated with the wavevector ~G

ρ_G~(~r) = ρ(~r)e−i ~G·~r,

and the integrated operator

Π_G~ =

Z

V

drd_ρ(~r)e−i ~G·~r_.

From Eq. (2.3) it follows that

lim

V →∞

Π_G~Π_{− ~}_G

V2 = fG~ 6= 0.

Based on our definition (see Eq. (2.2)), we can conclude that the long range order in Eq. (2.3) is the evidence of spontaneous symmetry breaking from a continuous space translation symmetry to a discrete one.

2.1.1 Cluster property

In view of what we are going to discuss in the rest of the paper it is useful to point out some aspects of SSB that, although not necessary for its definition, will be helpful to define time-crystals.

As an example we consider an Ising model at zero temperature and in the absence of an external field. This model has a global parity symmetry that is spontaneously broken

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in the two ground states |↑ . . . ↑i and |↓ . . . ↓i. These states have a non-zero expectation value of the order parameter, represented by the local magnetization. From the laws of quantum mechanics, we could argue that the system might exist in a linear combination of different ground states, for example

|±i = |↑ . . . ↑i ± |↓ . . . ↓i√

2 .

The states |±i (so called “cat states“) respect the parity symmetry of the Ising model, so, with this choice of states, the symmetry is not broken. Apparently, there is no reason to prefer |↑ . . . ↑i and |↓ . . . ↓i to give a physical description of the system.

While this is perfectly applicable to quantum mechanical systems with a finite number of degrees of freedom, a more careful analysis is required when the number of degrees of freedom is infinite [5], [6]. The crucial point is that, if the number of spins in our system is infinite, the states |±i are physically unrealizable. An infinitesimally small field in the longitudinal direction (explicitly breaking the symmetry) will result in a splitting between the two states |↑ . . . ↑i and |↓ . . . ↓i. The matrix element that connects one of these two states with the other (and hence also the probability of tunneling due to the perturbation) is exponentially small in the system size. This comes from the fact that the perturbation acts ”locally“1_{and cannot flips all the spins of the chain at the same}

time. Hence, these two eigenstates are stable in the thermodynamic limit, while the cat states |±i are strongly mixed by the perturbation and are unstable.

The cat states |±i are not physically realizable (in the limit of increasingly large volume) because they are destroyed by an arbitrarily small perturbation; moreover, no physical measurement can distinguish between them. In fact, if we take a superposition of |↑ . . . ↑i and |↓ . . . ↓i, in order to measure the relative phase we need a local observable having non-zero matrix element connecting them, but such an observable does not exist in the thermodynamic limit. Therefore, the two stable ground states describe different physical realizations of the system that are not connected by any operation. One can consider a single ground state (either |↑ . . . ↑i or |↓ . . . ↓i) to make equivalent predictions about measurable quantities.

The same argument applies to crystals and translation symmetry breaking. If the volume is finite, the ground state is unique and is realized for ~P = 0 (where ~P is the momentum of the solid as a whole), so the position of the center of mass is completely undetermined and translation symmetry is not broken. In the thermodynamic limit, the total mass diverges and states with different ~P become degenerate. Hence, sponta-neous symmetry breaking can occur2_{. The physical mechanism that explains the actual}

localization of the crystal in space is once again related to the presence of an external

1_{An Hamiltonian is local when it can be expressed as the sum of terms which are products of local}

observables: such local observables have support on a region with a finite extension in space. An operator that involves an extensive number of local observables is said to be global. Locality is a fundamental requirement in a physical theory.

2_{Note that, if the ground state of a system spontaneously breaks the symmetry, it must be degenerate:}

since the Hamiltonian is invariant under the given symmetry, if we apply the symmetry transformation to the ground state, we must obtain another ground state. This new ground state cannot coincide with the former, otherwise the symmetry would not be broken.

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2.2. A TIME CRYSTAL? 13

perturbation: this perturbation, however small, breaks the degeneracy of the ground states and selects a state with a defined position of the center of mass. Therefore, in order to describe our system we can choose a single ground state by fixing the position of the entire solid.

In general, if we have an infinite number of degrees of freedom, some states are “preferred” in the sense that they correspond to real physical configurations of the system of interest, while others do not. We will now give a more detailed analysis of the distinction between physical and unphysical states: some essential concepts introduced here will be used to define discrete time translation symmetry breaking in section 3.3.

The problem of physical realizability has a rigorous formulation and is connected to the requirement of locality of the theory [7]: our physical description of systems with an infinitely large number of degrees of freedom is constrained by the fact that we can only perform local measurement. We cannot act on the system over the whole space: any physically realizable operation is necessarily localized in space. Moreover, one of the fundamental principles of physics is that experiments that are located far apart in space have unrelated results. If this principle were not valid, it would be impossible to make any prediction about the result of an experiment, because this could require the knowledge of the entire universe.

The principle of unrelated distant experiment may be codified by the following property (known as cluster decomposition property or simply as cluster property)

lim

|~x− ~x0_|→∞hψ|a(x)b(x

0_{)|ψi − hψ|a(x)|ψi hψ|b(x}0_{)|ψi = 0.} _(2.4)

Cluster property has to be satisfied by any physically meaningful state |ψi for every pair of local observables a(x), b(x0_{). In other words, we cannot regard as realizable states}

the ones having long range (connected)3_{correlations.}

The states |↑ . . . ↑i and |↓ . . . ↓i satisfy cluster property and therefore represent physical states of our system. On the other hand, the cat states |±i violate cluster property and show long range connected correlations in local magnetization. From now on we will briefly refer to states satisfying cluster property as long-range correlated states, meaning that they have long range connected correlations.

We can now use Eq. 2.4 as a rigorous definition for realizable states. Therefore, we can say that spontaneous symmetry breaking occurs when the symmetry-preserving ground states violate cluster property. The connection of spontaneous symmetry breaking with the cluster property turns out to be very useful when defining non-equilibrium time crystals in chapter 3.3.

2.2 A time crystal?

Crystalline structures are examples of spontaneous breaking of space translation symmetry that can be found anywhere in nature. Can time translation symmetry be spontaneously

3_{We denote hABi}

C= hABi − hAi hBi as connected correlators, to distinguish them from ordinary

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broken as well? This question, posed by Frank Wilczek [8] in 2012, comes naturally from the observation that space and time are intrinsically related because of Lorentz invariance. From this perspective, it is plausible that, in analogy with ordinary crystals, time crystals could exist. While space crystals feature periodicity in space, time crystals should exhibit time periodicity.

However, this analogy is not straightforward. In fact, time translation symmetry is generated by the Hamiltonian itself, implying that the eigenstates |ψni must satisfy the

Heisenberg equation hψn| dO dt|ψni = i ~hψn|[H, O]|ψni = 0. (2.5) While this equation seems to preclude any time dependence (and thus any time crystal) in the eigenstates of the Hamiltonian, as suggested by Wilczek, an apparent violation to this condition can be realized in superconducting circuits with circulating currents.

Consider a superconducting ring threaded by a magnetic flux that is a fraction of the flux quantum. A particle of charge q and mass m is moving on the ring, that has radius a, and the magnetic flux is Φ. The ground state for the system can be found solving the Schr¨odinger equation for the Hamiltonian

H = 1 2m ~p −

q ~A c

!2

where ~A is the vector potential and is related to the magnetic flux as follows I

~

A · d~s = Φ.

From this equation we find that A =_2πaΦ on the ring and we can write

H = 1 2ma2 ~ i ∂ ∂φ− qΦ 2πc 2

where we denote by φ the coordinate indicating the angular position on the ring. Let us introduce the flux quantum Φ0= hcq and the fraction α =

Φ

Φ0. We get the following

expression for the Hamiltonian

H = ~ 2 2ma2 1 i∂φ− α 2 .

Its eigenstates are |li = eilφ _{with l integer because of the periodic condition on the ring.}

The energies of these states are El= ~

2

2ma2(l − α)2, so the ground state corresponds to

the integer value l0 that minimizes (l − α)2. If α is not an integer, l0− α is different from

0. Calculating h ˙φi on the eigenstates |li yields

hl| ˙φ|li = ~

ma2hl|(−i∂φ− α)|li =

~

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2.2. A TIME CRYSTAL? 15

Therefore, if α is not an integer, hl| ˙φ|li is non-zero even for the ground state |l0i.

The derivative of φ has non-zero expectation value, so apparently the system violates Eq. (2.5) and has some motion in the ground state. The problematic point comes from defining the operator φ: its expectation value is ambiguous because it is a multivalued function, defined only modulus 2π. All the single-valued functions of φ (such as eiφ_{) are}

not time-dependent in the ground state and their time derivatives have 0 expectation value.

The physical interpretation for this peculiar situation is the following: a magnetic flux threading the superconducting ring induces a persistent current in the ring. However, this current does not represent the actual measurable motion of a physical object. While we classically think of a current as something associated to the change in the position of a localized particle with time, the persistent currents in the superconductor are perfectly homogeneous and stationary in time. The particle is completely delocalized on the ring, its position is undetermined at any instant and hence we cannot say that it is actually changing with time.

The idea suggested by Wilczek is that if we induce a localization of the particle on the ring, then the current generated by the magnetic flux will represent the actual change in position of the particle itself. This effect can be accomplished by putting on the ring many particles that interact attractively.

The model is described by the N -particle Hamiltonian

H = ~ 2 2ma2 N X j=1 1 i ∂ ∂φj − α 2 − λ N − 1 N X j6=k,1 δ(φj− φk).

where the delta is intended as a well-defined periodic function on the ring. Wilczek formulated the product ansatz

Ψ(φ1, . . . , φN) = N

Y

j=1

ψ(φj)

Through this ansatz, that is equivalent to a mean field approximation, the problem reduces to a one-body non-linear Schr¨odinger equation

i~∂ψ ∂t =

~2

2ma2(−i∂φ− α)

2_{ψ − λ|ψ|}2_ψ _(2.6)

If α = 0 it is possible to find a solution of this equation in the form of a soliton. When the interactions are sufficiently strong this solution becomes energetically favorable. The soliton realizes a spontaneous breaking of ordinary translation symmetry because all the particles agglomerate to constitute a lump in a fixed position of the ring. Wilczek claims that a non-zero α induces a motion of the soliton, that starts rotating on the ring. Different wavefunctions parameterized by an integer l can in fact be found as solutions of Eq. (2.6). Each of them corresponds to a rotating soliton with a different angular momentum, in analogy with the single particle problem. Among them, the one with lowest energy minimizes (l − α)2_{. If α is not an integer, this solution corresponds to a}

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rotating lump in the ground state. Therefore, the minimal energy solution spontaneously breaks time translation symmetry realizing a time crystal.

Wilczek’s work stimulated a large number of reactions in the scientific community. An experimental proposal was formulated by Li, Gong, Yin, et al. [9]: their setting consisted of a cilindrically symmetric trapping potential threaded by a magnetic flux. When the magnetic flux was absent the atoms were expected to assume a crystalline order (Wigner crystal) due to Coulomb interactions. The claim was that with a non-zero magnetic flux the crystal would start to rotate, exhibiting a periodic motion. The motion could be observed through the Doppler shift of the ions or by exciting one ion to another hyperfine level and monitoring its position.

Another immediate reaction to Wilczek’s idea came from Bruno [10], [11]: he showed that both Wilczek and Li derived erroneous results. One of the arguments that he presented in support of his idea is that a periodic motion in the ground state leads to contradictory conclusions concerning energy conservation. In fact, rotating particles should radiate, but, being in the ground state, the system cannot reduce its energy. In Wilczek’s model, the problem came from the fact that the predicted ground state was not correct, and eigenstates with lower energy could be found. The true ground state of the system was stationary and had no time crystal order.

Later, other works by Bruno [12] and Nozi`eres [13] showed that an Aharanov Bohm magnetic flux cannot induce a rotation in the ground state and that the Heisenberg equation (Eq. 2.5) is indeed satisfied by any eigenstate for a certain class of systems, including Wilczek’s and Li’s models.

The application of the arguments given by Bruno and Nozi`eres is limited to the special case of a ring threaded by magnetic flux. It does not in general preclude the existence of time crystals. Other models were proposed as realizations of time crystals in different settings [14]. It was only in 2015 that a general no-go theorem was formulated by Watanabe and Oshikawa [15], stating the absence of quantum time crystals in equilibrium. Watanabe and Oshikawa formulated a precise definition of a time crystal, that was previously lacking. They defined a time crystal as a system where, for arbitrarily large |x|,

lim

V →∞hφ(x, t)φ(0, 0)i = f (t) (2.7)

with f (t) exhibiting nontrivial periodic oscillation in time. An essential requirement is that the correlation function in Eq. 2.7 is non-vanishing in the long distance limit. This definition has an evident analogy with Eq. (2.3) for ordinary space crystals. Note that the definition implies the long range order discussed in section 2.1.

An equivalent definition can be formulated in terms of the integrated order parameter

Φ(t) = Z V dxd_{φ(x, t)} lim V →∞ hΦ(t)Φ(0)i V2 = f (t)

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2.3. NON-EQUILIBRIUM TIME CRYSTALS 17 by proving that lim V →∞ hΦ(t)Φ(0)i V2 = c (2.8)

where c is constant in time, and Φ is a generic operator of the form Φ =R

V d

d_{x φ(x).}

More specifically, they proved the following statement: for any Hermitian operators A =R

V d

d_{xa(x) and B =}R

V d

d_{xb(x) (where a(x) and b(x) are local operators that act}

only near x) the following inequality is satisfied 1

V2| h0|Ae

−i(H−E0)t_{B|0i − h0|AB|0i | ≤ C} t

V

where E0 is the ground state energy and C is a constant that does not depend on t or V .

Setting A = B = Φ(0) and taking the limit V → ∞ yields

lim

V →∞

h0|eiHt/~_Φ(0)e−iHt/~_Φ(0)|0i

V2 = lim_{V →∞}

h0|Φ(0)Φ(0)|0i V2 .

The limit on the right does not depend on time, hence no time-dependent long range order is possible at T = 0. The authors also proved Eq. (2.8) in the case of a system at finite temperature T .

Thus, the existence of equilibrium time crystals was definitely ruled out.

2.3 Non-equilibrium time crystals

The no-go theorem only applies to equilibrium states, but non-equilibrium time crystals remained an open problem. Non-equilibrium physics is a vast and still largely unknown field [16]. Much progress has been done in recent years thanks to the extreme controllability of novel experimental techniques. However, while thermal equilibrium is a clear and defined concept, the term “non-equilibrium” refers to a wide range of situations.

For instance, an interesting example of non-equilibrium physics is represented by systems that persist for a long time in an intermediate state before reaching a complete thermal relaxation. Systems of this kind can have a time-crystal behavior that lasts for a long time and can be experimentally detected. Both Volovik [17] and Else, Bauer, and Nayak [2] have studied the time-crystal order that can emerge in an intermediate time before reaching equilibrium.

A system prepared in an excited state is another example of non-equilibrium setting. Time crystals in excited states have also been studied [18]: their existence was suggested by Wilczek’s model, once it was proven that the eigenstate he originally found in his work was not the true ground state.

Another scenario where the assumption of equilibrium does not apply is represented by periodically driven systems. If the Hamiltonian is time dependent, continuous time translation symmetry is obviously broken. But if the Hamiltonian is periodic in time, a discrete time translation symmetry remains. A time translation of a time t that is a multiple of the period T leaves the Hamiltonian unchanged. But the system can spontaneously break this symmetry, being invariant only under time translation of nT .

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This realization of spontaneous symmetry breaking goes under the name of Floquet or discrete time crystal [1]. Floquet time crystals will be the object of the next chapter and the general context of the work presented in this thesis.

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Chapter 3

Floquet Time crystals

As discussed in the previous chapter, in 2015 Watanabe and Oshikawa [15] proved that time crystals at equilibrium were impossible and finally put an end to the debate that followed Wilczek’s proposal [8]. However, the proof did not exclude the existence of time crystals out of equilibrium, leaving a vast set of possible directions still open to further investigation (see section 2.3). Among the possible non-equilibrium realizations, the most promising is probably represented by Floquet systems (i.e. systems with a time periodic Hamiltonian). Recent works have shown that this type of system can exhibit spontaneous time-translation symmetry breaking [1]. This phenomenon is defined in this way: given a time dependent Hamiltonian periodic in time with period T , there is some physical observable (an order parameter) oscillating with a period nT for a generic initial state. Hence the symmetry group generated by time translations of T is spontaneously broken to the subgroup generated by time translations of nT . Recently, Floquet time crystals have also been experimentally realized [19], [20].

This chapter is organized as follows: general properties of periodically driven systems are illustrated in sections 3.1 and 3.2; in section 3.3 we define Floquet time crystals and discuss their dynamical properties; an example of Floquet time crystal is presented in section 3.4; in conclusion, recent experimental and theoretical works about this topic are discussed in sections 3.5 and 3.6.

3.1 Floquet Theory

For Hamiltonians with a discrete space translation symmetry, the Bloch theorem holds: “eigenfunctions take the form of traveling plane waves modulated by an appropriate function with the lattice periodicity” [21]. An analogous theorem, known as Floquet theorem, holds for discrete time translation symmetry. We are now going to review the basic theory that describes systems with a time periodic Hamiltonian and explain the concepts that are essential for the definition of discrete time crystals.

We will be interested in quantum systems with Hamiltonians that are periodic functions of time,

H(t + T ) = H(t) 19

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where T is the period. In order to solve the time-dependent Schr¨odinger equation, we can exploit the symmetry of the Hamiltonian under discrete time translations. This symmetry enables the use of the Floquet formalism [22]–[25]. According to the Floquet theorem, the Schr¨odinger equation

H(t) − i~∂ ∂t

|Ψ(t)i = 0 has solutions of the form

|Ψα(t)i = exp(−iαt/~) |Φα(t)i

where |Φα(t)i is periodic in time with period T , i.e. |Φα(t)i = |Φα(t + T )i.

Floquet theorem is formally analogous to Bloch theorem in a periodic solid. Like the quasimomenta, the quasienergies α are defined up to translations of an integer

number of ~ω = ~2π/T ; in particular they can be folded in the first Brillouin zone −~ω/2 ≤ < ~ω/2.

As a consequence of the Floquet theorem, the states |Ψα(t)i satisfy

|Ψα(t0+ T )i = exp(−iαT /~) |Ψα(t0)i

which means that they are eigenstates of the time evolution operator

U (t0+ T, t0) = T exp − i ~ Z t0+T t0 H(t)dt !

with eigenvalues exp(−iαT /~).

Since we are interested in the breaking of the time translation symmetry of a period T , we will not study the evolution for a generic time t. Starting from t0= 0 we will only

examine the system at times kT with k integer (stroboscopic evolution). The Floquet states |Ψα(0)i = |Φα(0)i and the discrete time evolution operator defined as

Uf ≡ U (T, 0) =

X

α

e−iαT /~_|Φ

α(0)i hΦα(0)|

will therefore play a crucial role in the analysis of Floquet time crystals.

From the time evolution operator Uf, it is possible to define the so called Floquet

Hamiltonian Hf which satisfies

Uf = e−iHfT

Note that Hf, like the quasienergy, is not uniquely defined.

The existence of the Floquet Hamiltonian implies that the stroboscopic evolution is equivalent to the evolution with the time-independent Hamiltonian Hf. However, this

simplification is only apparent: finding the Floquet Hamiltonian in a closed form is not generally possible. Moreover, in general the Floquet Hamiltonian cannot be written as a sum of local terms: for a many-body system it might be highly non-local and unphysical. Therefore, the stroboscopic evolution of a periodically driven many-body system does not simply reduce to the undriven case but it can exhibit a much richer behaviour. For

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3.2. DYNAMICS OF PERIODICALLY DRIVEN MANY-BODY SYSTEMS 21

instance, the theorem proven by Watanabe and Oshikawa [15] and reported in section 2.2 only applies to local time independent Hamiltonian: it forbids time crystal order when the Floquet Hamiltonian is local, but makes no predictions about a non-local Hf. We

can then expect that Floquet time crystals could exist despite the fact that equilibrium time crystals are impossible.

3.2 Dynamics of periodically driven many-body

sys-tems

Periodically driven many body systems can exhibit dynamical properties that are not featured by the undriven ones. This fact enables the existence of time crystal order, therefore a discussion of this topic is needed in order to understand the indispensable prerequisites for time translation symmetry breaking.

The dynamics of (both driven and undriven) quantum many-body systems out of equilibrium has been intensely studied in recent years [16]. The experimental advances have made it possible to study non-equilibrium physics with unprecedented controllability and stimulated a new interest for the problems connected to quantum ergodicity and thermalization. We will now give a general introduction to these concepts, that are discussed in more detail in Appendix A.

A fundamental question that has been addressed concerns the evolution of closed quantum systems described by a time-independent or a periodic Hamiltonian. When a generic nonintegrable many-body system evolves under its own unitary evolution, it is expected to relax to a thermal equilibrium state. This thermal state is fully characterized by a small number of extensive conserved quantities (temperature, chemical potential, etc.) and all the other details about the local properties of the system’s initial state are spread over the entire system and are therefore inaccessible.

However, thermalization is not the only possible result of the evolution of a many-body system. Translationally invariant integrable systems have been studied for a long time and are known to be non thermalizing, because there exist infinitely many conserved quantities which can be written as sums of local or quasi-local operators. As a result, the dynamics of the system is constrained and the system cannot reach a thermal state: on the opposite, it can be described by what is known as a generalized Gibbs ensemble. The absence of thermalization in integrable systems relies on the specific type of Hamiltonian and can only happen for highly fine-tuned models: this property is not robust to small changes in the Hamiltonian.

On the other hand, when enough disorder is present, localization can take place. This phenomenon was first studied for non interacting electrons in a disordered potential and is known as Anderson localization [26]. The presence of disorder prevents the transport of heat (or electricity) and the system fails to thermalize (or conduct). It was shown by Basko, Aleiner, and Altshuler [27] that localization can also take place for interacting particles, where it goes under the name of many-body localization.

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preserve some information about the initial state [28]. This is related to the fact that in this regime there is a set of localized conserved charges which are constants of motion of the system (local integrals of motions or LIOMs) [65], [66]. While integrability is broken when the Hamiltonian is perturbed, MBL is highly robust and it can exist for interacting Hamiltonians as long as disorder is strong enough.

The behavior of a many-body system undergoing closed unitary evolution depends on the properties of its eigenstates. It was shown that thermalizing and MBL systems are characterized by different features in the statistics of the spectrum and in the entanglement properties of the eigenstates [54], [68], [69].

This picture can be applied to systems with a time-independent many-body Hamilto-nian as well as to Floquet many body systems. In contrast with the undriven case, in Floquet systems energy is not a conserved quantity: in the general ergodic case the system can absorb energy from the external drive indefinitely, so it is expected to thermalize to an infinite temperature thermal state. When this happens, Hf is non-local, otherwise the

system would be at finite temperature with respect to Hf.

Another possible result is the so called Floquet many-body localization: when Hf is

local and it is the Hamiltonian of a MBL system, thermalization is not possible. However, many body localization can occur even when the Floquet Hamiltonian Hf is non-local

and there is no complete set of local integrals of motions [1]. As will be discussed in the next section, this is the setting needed for spontaneous time translation symmetry breaking.

3.3 Definition of Floquet time crystals

While time-independent Hamiltonians have a continuous time translation symmetry, Floquet systems are only invariant under a discrete time symmetry and the generator is the unitary operator Uf = U (T, 0). This means that the symmetry respecting states

are the eigenstates of the operator Uf, i.e. Floquet eigenstates. This fact motivated the

following definition for Floquet time crystal given by Else, Bauer, and Nayak [1]: Definition 1 Time Translation Symmetry Breaking occurs if all the eigenstates of the Floquet operatorUf= U (T, 0) violate cluster property.

In other words, time translation symmetry is spontaneously broken in Floquet systems when symmetry respecting states are unphysical in the sense we defined in section 2.1.11_.

As a consequence, physical states will break the symmetry: they cannot show T -periodicity, but might be invariant only under time translations of 2T , 3T , etc.

The absence of T -periodicity for a generic physical state can be expressed as an equivalent definition of time translation symmetry breaking:

1_{Note that in a Floquet system quasienergy is only defined modulo 2π/T , so it is not possible to}

identify a ground state, i.e. a minimal quasienergy state. While for time independent Hamiltonians with spontaneous symmetry breaking cluster property is violated by symmetry respecting ground states (or finite temperature states), here we require that all the symmetry respecting states violate cluster property. A weaker formulation of this definition can be proposed, requiring that an extensive number of Floquet eigenstates (and not all of them) violates cluster property. This condition is sufficient to observe a time crystal behavior for a large number of initial states.

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3.3. DEFINITION OF FLOQUET TIME CRYSTALS 23

Definition 2 Time Translation Symmetry Breaking occurs if, for every time t and every physical state |ψ(t)i, there exists an observable Φ such that

hψ(t)|Φ|ψ(t)i 6= hψ(t + T )|Φ|ψ(t + T )i

where |ψ(t + T )i = U (t + T, t) |ψ(t)i.

This definition is especially useful for experimental observations of Floquet time crystals since it applies to measurable quantities. The operator Φ is an order parameter for this spontaneous symmetry breaking.

Another almost equivalent definition of time translation symmetry breaking was given by Khemani, Keyserlingk, and Sondhi [29]. It focused on the existence of a family of local order parameters Φi,α that transform under a non-trivial irreducible representation of

time translation symmetry. Since the group is abelian, it is equivalent to requiring that they transform with a non-zero phase θα

UfΦi,αUf†= e −iθα_Φ

i,α. (3.1)

Definition 3 Let G be the time translation symmetry group. Time translation symmetry is spontaneously broken to a subgroup H in an eigenstate |ψi if

lim

|i−j|→∞L→∞lim | hψ|Φi,αΦ †

j,α|ψi | = c06= 0

for every Φα which transforms non trivially under G, but trivially under H. The same

correlator disappears for everyΦα that transforms non trivially under H.

Note that, since |ψi is a Floquet eigenstate and Φi,αtransforms non trivially under

Uf, it follows that

hψ|Φı,α|ψi = 0.

This means that, when definition 3 is satisfied, the state |ψi violates cluster property and definition 1 is also satisfied.

We will now make some general considerations, based on the connection between the long-time dynamics of a many-body system and the existence of a time crystal phase. Ergodic Floquet systems: The Floquet eigenstates of an ergodic Floquet system are

infinite temperature states. As a consequence, time crystals are not expected to be possible in these systems, because infinite temperature states have trivial correlations. Moreover, it is expected that local operators spread under time evolution [30], [31]. A local order parameter Φi,α satisfying Eq. (3.1) does not spread under time

evolution and remains local. Therefore, no local order parameter can be found for thermalizing systems. The same argument precludes the existence of time crystals in non-localized integrable systems.

Systems with a MBL (Floquet) Hamiltonian: MBL Hamiltonian systems and Flo-quet systems with a MBL FloFlo-quet Hamiltonian have a set of local operators that do

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not spread under time evolution: these so called l-bits commute with each other and with Uf. However, these local operators transform with the trivial irreducible

repre-sentation because they commute with Uf and cannot be used as local parameters

for time translation symmetry breaking.

Based on these considerations, a time crystal phase was ruled out for systems with a MBL Floquet Hamiltonian and for thermalizing systems [1], [29]. The following example shows an alternative case: the Floquet evolution operator Uf has no local

Floquet Hamiltonian, but U2

f = e−2i ¯

HT _{where ¯}_{H is a (local) MBL Hamiltonian. This}

happens because there is a set of local operators (l-bits) that anticommute with Uf (and

hence commute with Uf2). Since these l-bits transform with a non trivial irreducible

representation (θ = π in Eq. (3.1)), they can serve as order parameters for time translation symmetry breaking.

3.4 Floquet time crystal in a disordered driven spin

chain

An example of Floquet time crystal was proposed by Else, Bauer, and Nayak [1] and consists of a disordered Ising chain to which a periodic drive is applied. The Hamiltonian is H(t) =    λP iσ x i 0 ≤ t < t1 P iJiσi+1z σzi + P ihxiσix+ P ihziσzi t1≤ t < t0+ t1= T (3.2)

where Ji, hzi and hxi are randomly chosen from uniform distributions in the intervals

[J/2, 3J/2], [0, hz_{], [0, h}x_{]. Disorder prevents thermalization, that would forbid the}

existence of a time crystal.

3.4.1 Solvable case

The problem is exactly solvable for hx= 0 and λt1= π/2. For this choice of parameters,

the Floquet evolution operator is

Uf = exp (−iHMBLt0) X (3.3) where HMBL= X i Jiσzi+1σiz+ X i hz iσ z i

and X is the operator that flips all the spins (up to an irrelevant phase)

X = e−iπ2 P iσ x i =Y i (iσx i).

Since HMBL commutes with every σiz, the eigenstates of HMBL are product states of

spins polarized in the z direction. They can be classified by the eigenvalues of σz

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3.4. FLOQUET TIME CRYSTAL IN A DISORDERED DRIVEN SPIN CHAIN 25

one has a set of quantum numbers {si= ±1}.

σz

i |{si}i = si|{si}i

The eigenvalues of HMBL are functions of {si} and can be separated in two parts: one

term (E+_{) is even in the σ}

z operators and is invariant under the spin flip X, the other

term (E−_{) is odd and changes sign when it transforms according to X.}

HMBL|{si}i = X i Jisi+1si | {z } E+_({s i}) +X i hz isi | {z } E−_({s i}) |{si}i = E+({si}) + E−({si}) |{si}i (3.4) HMBL|{−si}i = E+({si}) − E−({si}) |{−si}i (3.5) X |{±si}i = |{∓si}i (3.6)

Using Eq. (3.4), (3.5) and (3.6), it can be shown that the eigenstates of Uf are of the

form eit0E−({si})/2_|{s i}i ± e−it0E − ({si})/2_|{−s i}i (3.7)

with eigenvalues ± exp(it0E+({si})).

The Floquet states in Eq. (3.7) are cat-states with long range correlations: they are unphysical and violate cluster property, so the model is a Floquet time crystal according to the definition 1.

As a consequence, physical states cannot be periodic with period T . For instance, a state of the form |{si}i evolves acquiring a phase over a time 2T (it is an eigenstate

of U (2T, 0) = U2

f). The symmetry given by time translations of 2nT is not broken: the

system can respond with period 2T to an external drive of period T . This subharmonic oscillations with half of the original frequency are the evidence of the reduction of the symmetry group from G = Z to the subgroup H = 2Z.

3.4.2 General case and stability

We are now going to consider what happens when the constraints on the parameters hx

and λ enforced in section 3.4.1 are relaxed. It is not a priori clear if the time-crystal behavior is destroyed or not. An argument supporting the robustness of the time crystal was given based on the assumption that there exists a local spectral gap (see for example [1], [32], [33]).

The idea is the following: for a generic local operator Φ(x), the matrix element hi|Φ(x)|ji is different from zero only if the two states |ii and |ji are very similar and only differ locally. We expect that, being the system many-body-localized, two states |ii and |ji connected by Φ(x) have finite quasi-energy difference i− j > 0 (local spectral gap).

The existence of a local spectral gap implies that a local2_{perturbation in the operator}

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Uf can only perturb the eigenstates locally. Therefore, if the unperturbed eigenstates

violate cluster property, also the perturbed eigenstates do: long range correlations are preserved.

3.5 Experimental realizations

The evidence of a Floquet time crystal phase has been recently observed in two different experimental realizations.

A model very similar to Eq. (3.2) has been realized for an interacting spin chain of 10 trapped atomic ions [19]. Two hyperfine levels of171_Yb+ _{ions represented the effective}

spin-1/2 states. The following Floquet unitary operator was implemented:

Uf = e−iH3t3e−iH2t2e−iH1t1

         H1= Ω(1 − )Piσ y i H2=Pi,jJi,jσxiσjx H3=Pihiσxi

A Rabi rotation Ω = π/2t1 is first performed, such that for = 0 all the spins are

flipped. Then, the system evolves for a time t2 under long range interactions generated by

optical dipole forces. Interactions depend on distance with the power law Ji,j= J0/|i−j|α,

α = 1.51. Next, each spin was individually addressed by a tightly focused laser beam, generating an AC Stark shift. The Stark shift acts as a programmable disorder for the system: each hi is taken in the interval [0, W ].

The ions are initialized in the separable state of all spins pointing downwards in the x-direction. After many periods (up to 100), single-site magnetization is measured by collecting the spin-dependent fluorescence on a camera for site-resolved imaging. In this way, the time correlation function hσi(t)σi(0)i is obtained.

Thanks to the capability of individually tuning the parameters , J0 and W it was

possible to demonstrate the necessity of both interactions and disorder in order to get a stable time crystal phase.

For = 0 a 2T periodicity is expected even in the absence of disorder and interactions, because the Floquet unitary simply flips the spins after each period. On the other hand, a significant difference was observed for 6= 0. In the absence of disorder and interactions, when 6= 0 the correlation function shows beatings, due to the imperfect rotation of the spins over a period (Fig. 3.1a). The corresponding Fourier peak is symmetrically split in two peaks. When disorder is turned on, the different precessing frequencies produce a dephasing effect (Fig. 3.1b). Remarkably, if interactions are present the period of the oscillations is locked to 2T and time crystal order is restored(Fig. 3.1c). This can be clearly seen in the Fourier spectrum, that shows only one peak. The stability (or rigidity)

Hamiltonian is local when it can be expressed as the sum of terms which involve only a finite number of local observables. We will refer to a unitary operator as local if it can be written as the time evolution induced by a local Hamiltonian. We say that the perturbation of the Floquet unitary operator is local, if the perturbed Uf can be expressed in terms of the unperturbed Uf0as Uf= U0Uf0and U

0 _{is a local}

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3.5. EXPERIMENTAL REALIZATIONS 27

Figure 3.1: Time-evolved magnetizations of each spin and corresponding Fourier spectra. Each point is the average of 150 experimental repetitions. Figure from [19].

of the oscillations at 2T is evinced for not too large . Increasing the deviation from the perfect spin flip, the system enters the symmetry unbroken phase and time crystal order is lost(Fig. 3.1d).

A phase diagram was obtained showing the dependence on the perturbation and on the interaction strength (Fig. (3.2)). For a strong perturbation the system makes a transition to a thermal phase, where MBL is lost, or to a MBL phase with no time translation symmetry breaking.

Figure 3.2: Phase diagram of the discrete time crystal, obtained using the variance of the subharmonic peak amplitude as a signature of the transition. Figure from [19].

Another experiment [20] was performed using an ensemble of 106 nitrogen vacancy (NV) spin impurities in diamond. From each NV center with spin S = 1 an effective two level system was isolated by applying an external magnetic field. These isolated spin states were optically initialized/detected and manipulated via microwave radiation. The following Hamiltonian was realized:

H(t) =X i [Ωx(t)σix+ Ωy(t)σ y i + ∆iσzi] + X ij Jij r3 ij (σxiσ x j + σ y iσ y j − σ z iσ z j).

where ∆i are disordered on-site fields and Jij are the orientation-dependent coefficients

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procedure is the following: a continuous microwave driving along x with Rabi frequency Ωx is applied for a duration τ1; then, a driving along y with frequency Ωy is applied

for τ2. Approximate π-pulses are realized, with Ωyτ2 ' π, while during τ1 the system

effectively evolves due to the interactions. After repeating these operations for n periods T = τ1+ τ2, the spin polarization is read out.

Figure 3.3: Observed magnetization and its Fourier transform for different values of τ1

and Ωyτ2: (a) τ1 = 92 ns, Ωyτ2 = π, (b) τ1 = 92 ns, Ωyτ2 = 1.034π,(a) τ1 = 989 ns,

Ωyτ2= 1.034π. Figure from [20].

Figure 3.3a shows the oscillations in the magnetization and the Fourier peak for the exact π pulse. When Ωyτ2 is switched to 1.034π, the peak is split and the perfect 2T

oscillations are lost (Fig.3.3b). As the interactions are enhanced by increasing τ1, the 2T

periodicity and the central peak are restored (Fig. 3.3c).

The distinctive features of the time crystal order that have been observed in this system do not have a simple explanation in terms of many-body localization. In fact, even though many sources of disorder are present (lattice strain, paramagnetic impurities and the random positioning of NV centers), this 3D system is not expected to localize. A recent theoretical work [34] has shown that this robust time crystal order might be connected to a special case of slow thermalization.

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3.6. BRIEF REVIEW ON THE THEORETICAL ACTIVITY 29

3.6 Brief review on the theoretical activity

The introduction of Floquet time crystals stimulated an intense theoretical activity. Together with the already cited work by Else, Bauer, and Nayak [1], where Floquet time crystals were first defined, some works by Khemani, Lazarides, Moessner, et al. studied the properties of the Ising chain with periodic spin-flip [35]–[38]. They identified a phase denoted as π-spin glass that was later recognized as an example of Floquet time crystal.

The model consists of a binary drive

Uf = e−iH2t2e−iH1t1 (3.8)    H1= −Pihiσix+ Hint H2= −PiJiσziσ z i+1+ Hint

where Hint contains some weak interaction terms that are needed to prevent integrability

and Ji and hi are random variables uniformly distributed around the mean values J

and h. Four different phases can be found depending on ht1 and Jt2 (Fig. 3.4a):

these phases exhibit different spectral properties and different long distance correlations Ci,j= hΦα|σziσ

z

j|Φαi (with |i − j| → ∞) of the Floquet eigenstates |Φαi.

Figure 3.4: (a) Phase diagram for the binary drive of Eq.(3.8). (b) Floquet spectrum: the quasienergy axis running from 0 to 2π/T is shown as a circle, with location of Floquet eigenenergies shown, which are distributed randomly (paramagnets), in pairs (0SG) and in pairs diametrically separated by π/T (πSG). (c) Period doubling in the

πSG spatio-temporal order as seen in stroboscopic snapshots. Figure from [38].

Two phases (the so called trivial paramagnet and 0-π paramagnet) show Cij → 0 in

the long distance limit. On the contrary, the 0-spin glass (0-SG) and the π-spin glass (π-SG) have Cij6= 0. The 0-spin glass correspond to the ordinary spin glass phase that

exists also in the undriven case: as an effect of spontaneous symmetry breaking the eigenstates are found in pairs of cat states. The quasienergy splitting between the two states is exponentially small in the system size (Fig. 3.4b). On the other hand, the π-spin glass phase has no equivalent in the undriven case. The eigenstates are again classified in pairs of cat states, but the splitting is no more close to 0 (Fig. 3.4b). The quasienergies, in fact, have splitting π/T (with corrections exponentially small in the thermodynamic limit). The π-spin glass is an example of the period doubling time crystal (Fig. 3.4c) that is absolutely stable, i.e. it is stable to all small perturbations, including those explicitly

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breaking the Ising symmetry.

A model similar to the example discussed in section 3.4 was also studied by Yao, Potter, Potirniche, et al. [39]. These authors focused on the Hamiltonian

Hf(t) =    H1= (g − )Piσ x i 0 < t < T1 H2=PiJzσizσ z i+1+ Bizσ z i T1< t < T The coefficients Bz

i are random variables uniformly distributed in the interval [0, W ] and

g = π/(2T1), such that = 0 represents the deviation from an exact spin flip. The phase

diagram of this Hamiltonian is represented in Fig. (3.5).

Figure 3.5: Phase diagram of the discrete time crystal as a function of interaction strength Jz and pulse imperfections . Figure from [39].

In this model, the transitions from the discrete time crystal to the symmetry unbroken MBL and to the thermalizing phases are studied using various signatures. The authors claim that there is a phase transition from the time crystal to trivial paramagnet which is of the Ising type, with some key differences between this “hidden” Ising transition and the conventional transition. In conclusion, a realization of the discrete time crystal in a 1D array of long-range interacting trapped ions is proposed.

Some other works focus on clean systems and propose realizations of time crystals in the absence of disorder. In [40] a Floquet ladder model is studied. A binary drive alternates an Hubbard Hamiltonian that acts independently on the two chains with an inter-chain tunneling. Robust oscillations with a subharmonic frequency manifest with non-zero intra-chain interactions and are persistent in the limit L → ∞, where L is the length of the chains.

In [41] the Lipkin-Meshkov-Glick model is studied: it consists of N spin-1/2 interacting through an infinite range coupling. A clean time-crystal is possible due to the fact that all sites interact with each other, so the argument about the spreading of correlations that forbids the existence of possible order parameters does not apply here.

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Chapter 4

Floquet Time Crystal in a 3

state clock model

The example proposed by Else, Bauer, and Nayak [1] in their pioneering work about Floquet time crystals showed the emergence of period doubling for a driven disordered Ising chain. The Ising model is a special case of a more general class of models named n-state clock models. In this chapter we present a generalization of the Ising Floquet time crystal to a time crystal based on a clock model with 3 states. We study such a model through both analytic and numerical methods. We present the model in sections 4.1 and 4.2 and exactly solve it for a special case in section 4.2.1. In sections 4.3 and 4.4 we treat the general case discussing the stability for small perturbations of the parameters. To conclude, we examine the signatures of the time crystal phases: the oscillations of the order parameter (section 4.5) and the presence of multiplets in the spectrum (section 4.6).

4.1 A generalization of the Ising chain

The quantum Ising chain is a paradigmatic example of spontaneous breaking from the discrete symmetry group Z2 to the trivial identity subgroup. An analogous model can

be formulated where the spontaneously broken symmetry is Z3. This model is known

as 3-state quantum clock model. Its critical behavior has been studied with a density matrix renormalization group technique, highlighting a rich phase diagram [42].

To each site of the chain we associate an Hilbert space of dimension 3. We define the operators σi and τi that operate on the site i of the chain such that σ3i = 1, τi3= 1

and σiτi= ωτiσi, where ω = ei

2π

3 . We can give to these operators the following matrix

representation σ =    1 0 0 0 ω 0 0 0 ω2    τ =    0 0 1 1 0 0 0 1 0   

The three states of our chosen canonical basis are the three possible positions of the hand 31

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of the clock (the angles 0, 2π/3 and 4π/3) associated to the operator σ; the τ operator moves the hand of the clock a step forward. The matrices σi and τi can be seen as the

generalization of the Pauli matrices σz

i and σxi. Using this analogy, we can write the

equivalent of the disordered Ising model Hamiltonian H2=PiJiσizσi+1z +

P

ih x iσixas

the sum of nearest neighbors interactions and a flip term.

H3= X i Ji(eiϕσi†σi+1+ h.c.) + X i hx i(e iϕx_τ j+ h.c.) (4.1)

where ϕ, ϕx are real numbers (chiralities) and Ji, hxi, hzi are real numbers taken from a

random distribution.

4.1.1 Properties and symmetries of the 3-state clock model

The Hamiltonian (4.1) has a global Z3symmetry that can be represented by

Xτ=

Y

i

τi.

When ϕ = ϕx = 0 the Hamiltonian is also invariant under charge conjugation, that

induces the following transformation

σi→ σi† τi→ τi†

At this special point the combination of charge conjugation and Z3 symmetry realizes

the full S3 permutation symmetry and the model is known as the 3-state quantum Potts

model or the 3-state non chiral clock model.

In addition to the global symmetries, we now enumerate some useful properties of the model [42]. 1. The transformation ϕ0→ ϕ +2πn 3 ϕ 0 x→ ϕx+ 2πm 3 changes the Hamiltonian to

H3= X i Ji(ωneiϕσi†σi+1+ h.c.) + X i hx i(ωmeiϕxτj+ h.c.)

We can recover the original form of the Hamiltonian by transforming the operators

τ0 = ωmτ σ2j0 = ω n

σ2j σ02j+1= σ2j+1

2. The transformation

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4.2. THE PERIODICALLY DRIVEN MODEL 33

changes the Hamiltonian to

H3= X i Ji(e−iϕσi†σi+1+ h.c.) + X i hx i(e−iϕxτj+ h.c.)

We reobtain the original Hamiltonian after redefining the operators as follows

σ0 → σ† τ0 → τ†

3. Consider the duality transformation

µj+1 2 = j Y k=1 τk νj+1 2 = σ † jσj+1

the new operators satisfy the relations µ3 _{= 1, ν}3 _{= 1 and µν = ωνµ. The new}

Hamiltonian is H3= X i Jieiϕνi+1 2 + X i hxie iϕx_µ i+1 2µ † i−1 2 + h.c.

We can now exchange ϕ and ϕx and at the same time Ji and hxi and we get the

original Hamiltonian.

4.2 The periodically driven model

We can now generalize the example given in section 3.4 of the driven disordered Ising chain to the 3-state clock model. The first part of the evolution is represented by the unitary transformation Xτ =Qiτi. After this instantaneous flip operation, the system

evolves with the Hamiltonian HM BL for a time T . From now on we will set T = 1 or, in

other words, we will include the factor T in the coefficients Ji, hxi, h z i. HM BL = X i Ji(eiϕσ†iσi+1+ h.c.) + X i hx i(e iϕx_τ j+ h.c.) + X i hz i(e iϕz_σ i+ h.c.) (4.2) Uf = e−iHM BLXτ (4.3)

4.2.1 Solvable case

Following the procedure used by Else, Bauer, and Nayak, we first restrict to the special case hx= 0. The Hamiltonian in Eq. (4.2) reduces to

H0= X i Ji(eiϕσ†iσi+1+ h.c.) + X i hz i(e iϕz_σ i+ h.c.) (4.4)

and it commutes with all the operators σi.

The eigenstates and the quasienergies of the Floquet operator in Eq. (4.3) can be easily found using the fact that an eigenstate of Uf is also an eigenstate of Ufn for any

(34)

integer number n (note that the converse is not always true). The detailed calculations are reported in Appendix B.

Let us denote with |{si}i the eigenstates of the σ operators such that

σi|{si}i = si|{si}i

where si = 1, ω, ω2. The states |{si}i are eigenstates of H0, with

H0|{si}i = X i Ji(eiϕs†isi+1+ h.c.) | {z } E+_({s i}) +X i hz i(eiϕzsi+ h.c.) | {z } E−_({s i}) |{si}i where we denoted as E+_({s

i}) and E−({si}) the terms in the expression of the energy

that are respectively even and odd functions of the si. Most importantly, the states

|{si}i are also eigenstates of Uf3 (with eigenvalue exp (−3iE+({si}))) but not of Uf or

U2

f. Note that this implies that Uf|{si}i and Uf2|{si}i are also eigenstates of Uf3 with

the same eigenvalue. This triplet of degenerate states can be linearly combined to form three eigenstates of Uf: each of the corresponding eigenvalues must be a cube root of

exp (−3iE+_({s

i})). Using these observations, it is easy to derive that the eigenstates of Uf

can be expressed as superpositions of |{si}i, |{ωsi}i = Xτ|{si}i and |{ω2si}i = Xτ2|{si}i.

To simplify the expression, let us denote

|αi = |{si}i

|βi = e−iE−({ωsi})_|{ωs

i}i

|γi = e−i(E−({ωsi})+E−({ω2si}))_|{ω2_s

i}i

Then, the eigenstates of Uf are cat-states, superpositions of |αi, |βi and |γi with different

phases

|ψ({si}, 1)i = |αi + |βi + |γi (4.5)

|ψ({si}, ω)i = |αi + ω |βi + ω2|γi (4.6)

|ψ({si}, ω2)i = |αi + ω2|βi + ω |γi (4.7)

The three eigenvalues are respectively e−iE+_({s

i})_{, ωe}−iE+({si})_{, ω}2_e−iE+({si})_{. Hence, in}

our notation

Uf|ψ({si}, p)i = pe−iE

+_({s

i})_|ψ({s

i}, p)i

with p = 1, ω, ω2_{. As a result, these states form a triplet in the quasienergy spectrum}

(defined modulo 2π) with levels separated by 2π/3: E+_({s

i}), E+({si})−2π/3, E+({si})+

2π/3.

We can conclude that in our model all the spectrum can be classified in triplets with 2π/3 splitting. In general, multiplets with exact 2π/n splitting are crucial for the existence of a time crystal because they result in the absence of dephasing. Coherences between states in a multiplet are not destroyed by time evolution: they oscillate in stroboscopic

Floquet time crystals in clock models

Universit`

a di Pisa

Dipartimento di Fisica

Laurea Magistrale in Fisica

Floquet time crystals in

clock models

Federica Maria Surace

Prof. Rosario Fazio

Sessione di Laurea del 20 Luglio 2017

Anno accademico 2016/2017

Contents

Chapter 1

Introduction

Chapter 2

Spontaneous time-translation

symmetry breaking

2.1

Spontaneous symmetry breaking

2.1.1

Cluster property

2.2

A time crystal?

2.3

Non-equilibrium time crystals

Chapter 3

Floquet Time crystals

3.1

Floquet Theory

3.2

Dynamics of periodically driven many-body

sys-tems

3.3

Definition of Floquet time crystals

3.4

Floquet time crystal in a disordered driven spin

chain

3.4.1

Solvable case

3.4.2

General case and stability

3.5

Experimental realizations

3.6

Brief review on the theoretical activity

Chapter 4

Floquet Time Crystal in a 3

state clock model

4.1

A generalization of the Ising chain

4.1.1

Properties and symmetries of the 3-state clock model

4.2

The periodically driven model

4.2.1

Solvable case