Dynamics of the kicked Bose-Hubbard model

(1)

Universit`

a di Pisa

DIPARTIMENTO DI FISICA

Laurea Magistrale in Fisica

Dynamics of the kicked Bose-Hubbard model

Candidato:

Michele Fava

Matricola 508722

Relatori:

Prof. Rosario Fazio

Dott. Angelo Russomanno

(2)

(3)

Introduction

The description of many-body systems can be naturally formulated in the language of statistical physics. In this context, the microscopical underpinning of statistical mechanics is a crucial point. In fact, while statistical physics is an extremely powerful description, in the end this description must emerge from the underlying microscopic dynamics. While in classical systems the emergence of statistical mechanics is well understood in terms of chaotic behaviour in phase space and the ergodic hypothesis [1], in quantum mechanics many questions are still open.

In order to have a really fundamental answer to the question “How can a quantum system thermalize?”, one cannot study the dynamics of the system coupled to an external thermal bath. Indeed, even if this may grant some insight on the problem, any answer that may emerge would be unsatisfactory in the end, since one would be unable to explain why the external bath was in a thermal state to begin with. Thus, any attempt to answer this question should consider an isolated system and try to explain the emergence of a thermal behaviour from its own unitary dynamics.

The problem of the thermalization of isolated quantum systems was first tackled in [2] by Von Neumann himself a few years after the theoretical foundations of quantum physics were laid. In more recent years, the interest in the topic has reborn, mainly prompted by the new experimental platforms, such as ultra-cold atoms [3–7] or trapped ions [8, 9], which allow experimentalists to control and measure many-body systems that can be considered isolated for quite long times. Further interest has also arisen from the exciting discovery of a new out-of-equilibrium phase of matter, which cannot be captured by the statistical physics paradigm, i.e. many-body localization. Many-body localization (see [10] for a review), in fact, represents a new quantum phase of matter where thermalization is prevented by the combined effect of interaction and static spatial disorder.

More recently, considerable interest has focused on thermalization of local observables in periodically-driven quantum (also termed Floquet) systems. This problem is, in fact, both theoretically interesting and practically useful, as we will explain in short.

From a foundational perspective, Floquet thermalization is peculiar already in 1

(6)

classical systems and even more in the quantum case. In fact, being energy not conserved, the thermal ensemble in this case must yield every state with the same probability independently from its energy, as a T = ∞ canonical ensemble. It is then easy to see that thermalization is strictly connected to energy absorption from the driving. Another important aspect, in order to have thermalization, is the structure of the eigenstates of the dynamics (Floquet states) which must be strongly entangled and locally equivalent to the T = ∞ thermal ensemble (Eigenstate thermalization). Furthermore, quantum Floquet systems can support new exciting phases of matter, such as time crystals [11], which do not have a counterpart in static systems. Therefore, it is practically important to understand when thermalization occurs and when it is suppressed, in order to stabilize these peculiar phases.

Floquet systems are also acquiring an increasing relevance for quantum simulators. A lot of interesting Hamiltonians are indeed difficult to be practically simulated in such settings. A widely applied solution to this difficulty is to engineer a periodic driving protocol which, at least for short times, is able to mimic a desired effective Hamiltonian. Exploiting this approach, sometimes called Floquet engineering, a many interesting phenomena have been observed [12]. However, if at long times the system thermalizes to a T = ∞ ensemble, all the properties the states may have at short times (in the so called prethermal regime) will be lost at long enough times. It would be extremely useful, then, to find non-thermalizing Floquet systems, where the properties of the state in the prethermal regime are not destroyed in the long-time dynamics.

Moreover, one may wonder whether the thermal or non-thermal behaviour is dom-inated by the underlying semi-classical dynamics, or new quantum phenomena may emerge, completely altering the classical picture. On the one hand, one may expect that quantum fluctuations somehow foster thermalization and thus may bring classically non-ergodic systems to thermalization anyway. On the other hand, it is known that the phenomenon of many-body localization heavily relies on quantum coherence and does not have a classical counterpart.

Along this line, an interesting single-particle phenomenon, known as dynamical localization, was discovered in the 80s [13] in the kicked rotor model, a Floquet system where the driving is provided by a kick, i.e. a perturbation that is periodically switched on as a δ in time. In the classical kicked rotor model, it was known that there is a transition from a regular behaviour to a chaotic one, associated with a constant energy absorption from the driving. Instead, the corresponding quantum model is never thermal since the energy absorbed saturates after a certain time. It was in fact pointed out [14, 15] that this phenomenon can be understood as a form of Anderson localization in energy space, via a mapping to a hopping problem on a pseudo-disordered lattice.

In more recent years, a debate has been opened regarding whether dynamical localization may be present also in many-body systems or if, increasing the system size, dynamical localization will eventually be broken and ergodicity recovered. Thus, various works have focused on variants of coupled kicked rotors [16–19]. However, the issue

(7)

3 is far from being settled for two main reasons. First of all, with the exception of [19] in which a new mapping is proposed, the connection between dynamical localization and Anderson localization was not investigated in the case of coupled rotors. Second, numerical studies of coupled kicked rotors are limited (so far a maximum of3 coupled rotors has been considered), due to the huge Hilbert space of the model.

In the present Thesis, we try to overcome the second difficulty by considering, instead, the kicked Bose-Hubbard model with number conservation, which in fact allows for numerical studies up to quite large system sizes. In fact, we will show both analytically and numerically that dynamical localization can appear also in a Bose-Hubbard dimer (a chain with only L= 2 sites) with a finite number N of bosons in it.

We will then move to analyse how localization properties change when the number of interacting bosons increases. One way of going towards this many-body scenario is to keep the number of sites L fixed and to increase N . In this respect, we will discuss in great detail the dimer and we will show, both analytically and numerically, that as N increases the model gradually recovers ergodicity.

Afterwards, we will study localization properties as a function of L aiming to understand the behaviour of the model as both L and N increase, while N/L is kept fixed. For this purpose, analytical methods do not provide us with a useful insight any more. Our study then heavily relies on the numerics. We identify a few useful indicators of dynamical localization, apart from the energy absorption suppression. We then employ finite-size numerical simulations (exact diagonalization, Krylov technique and time-evolving block decimation) to evaluate the considered indicators and understand if dynamical localization can survive as L increases. We will show in this way that there seems to be a parameter region where dynamical localization can persist up to the maximum L which we are able to access numerically, suggesting the possibility of dynamical localization also in the thermodynamic limit. We furthermore stress that dynamical localization in the model under study is completely due to quantum effects. Indeed, we also simulate the semi-classical dynamics of the model and find a very different behaviour showing diffusion and T = ∞ thermalization. This analysis further confirms that the effects observed are a genuine quantum phenomenon of dynamical localization.

Finally, we also study the kicked Bose-Hubbard model in the presence of static disorder. We argue that the system may undergo a transition to a many-body localized phase. Indeed we numerically show that, at a finite system size and for strong enough disorder, the entanglement entropy in the chain logarithmically increases in time.

The Thesis is organised as follows. After introducing the phenomenon of dynamical localization in Chapter 2 we quickly review the state of the art concerning thermalization of many-body systems in Chapter 3, emphasising in particular Floquet systems and the debate on dynamical localization in many body systems (Sec. 3.4). We then move to discuss the original results, dealing with the kicked Bose-Hubbard model. In Chapter 4 we study dynamical localization in the dimer, first for a fixed value of N and then we characterize how localization properties change as N increases. Afterwards in Chapter 5,

(8)

we move to discuss the other many-body scenario where L is large. Finally we conclude in Chapter 6 by summarizing the main results and by discussing perspective of future work.

(9)

Chapter 2

Dynamical localization

In this chapter we will introduce the phenomenon of dynamical localization by considering a paradigmatic model displaying this phenomenon: the kicked rotor model. This model has been extensively studied in the literature and in this chapter we are going to review the existing results. In the process, we will also use the aforementioned model as an easy ground to introduce the key concepts related to thermalization.

The chapter is organised as follows. In Sec. 2.1 we introduce the classical kicked rotor model and, in the meanwhile, we also introduce thermalization of classical systems and its relationship with chaos. In Sec. 2.2, we discuss the quantization of the model and explain how its phenomenology differs from the classical one. In this occasion we will also explain the key role of Floquet states for thermalization, by introducing the diagonal ensemble. Finally, we discuss the connection between dynamical localization and Anderson localization in Sec. 2.3. In the end (Sec. 2.4), we will review some experimental realizations of the model exploiting cold atoms in an optical trap.

We furthermore report in Appendices 2.A and 2.B a brief introduction, respectively, to Floquet theory and Anderson localization.

2.1 Classical kicked rotor

The kicked rotor is a prototypical periodically driven system. The model describes a particle constrained to move on a circle lying on a horizontal plane. The particle moves freely on the circle, but periodically with period τ it receives a kick in a fixed direction.

We will denote with ϕ the angle of the particle on the circle, and p its conjugated momentum. Taking the momentum of inertia of the particle to be1, the Hamiltonian describing the system evolution is then

H(t) = p 2 2 + k cos(ϕ) +∞ X n=−∞ δ(t − nτ ) (2.1)

Such a system will not have any conserved quantity, not even the energy since the 5

(10)

Hamiltonian is time-dependent. Thus if we were to study the statistical mechanics of the kicked rotor, we should assume, having no a priori information, that for computing the long-time average of some quantity O(ϕ, p) we should use a distribution function uniform in all phase space U(ϕ, p)1_{, i.e.}

lim T →∞ 1 T Z T 0 dtO(ϕ(t), p(t)) = Z dϕdpO(ϕ, p)U (ϕ, p) (2.2)

This is what is usually called ergodic hypothesis and heavily relies on the assumption that the motion is chaotic. In the following, we will discuss the validity of this statement in the kicked rotor model.

Being interested in the long-time dynamics, we can restrict ourselves to studying the stroboscopic evolution of the model in order to simplify the problem, i.e. substitute

lim T →∞ 1 T Z T 0 dtO(ϕ(t), p(t)) −→ lim N →∞ 1 N N X n=0 O(ϕ(nτ ), p(nτ )) (2.3)

By denoting with pn and ϕn the momentum right after t= nτ , we have the recurrence relations

ϕn+1 = ϕn+ pnT pn+1= pn+ k sin(ϕn+1)

(2.4)

which can be rewritten in term of only one control parameter K = kT , with a momentum rescaling I_n= pnT

ϕn+1= ϕn+ In In+1= In+ K sin(ϕn+1)

(2.5)

This is the so called standard or Chirikov map, which has been extensively studied both analytically and numerically (see for example [20, 21]). It is well known that there exists a critical value Kc' 1, such that for K < Kcall orbits in phase space are regular. Instead for K > K_c chaotic regions start to appear (Fig. 2.1) and regular orbits remain stable only in certain islands. As K increases the regular islands shrink and eventually for K_{& 8 almost all the phase space is chaotic.}

When the dynamics is chaotic, we may expect the ergodic hypothesis to hold. So, for example, p will perform a random walk and on average p2 will increase without bounds. So that in the infinite time limit hp2i = ∞, in agreement with the statistical mechanics description. Being the system extremely simple, it is possible to study this behaviour explicitly. By squaring the second of the equations 2.4 and averaging on a set of initial conditions, we obtain

hp2_n+1i = hp2_ni + 2khpnsin(ϕn)i + k2hsin2(ϕn)i (2.6)

1_{Since, the uniform distribution over an unbounded set is ill-defined, we should think of it as a limit}

(11)

2.1. CLASSICAL KICKED ROTOR 7

(a) (b)

(c) (d)

Figure 2.1: The standard map for various values of K. Since Eq. (2.5) is invariant under a shift I 7→ I+ 2π, it is useful to fold I in an interval which we have chosen to be [0, 2π].

(12)

If K is large enough we may neglect correlations between momentum and position thus obtaining

hp2_n+1i = hp2_ni + k2hsin2(ϕn)i = hp2ni + k2

2 (2.7)

So the momentum will increase diffusively, i.e. as hp2(t)i = Dt with D ∼ k_2τ2. In spite of the brutal approximation, it turns out that the result is quite accurate. The diffusion can indeed be observed in numerical studies. The approximation, instead, fails in predicting the exact value of D, which is affected by corrections due to momentum-position correlation that we neglected.

Apart from the specific details of the kicked rotor model, the previous observations can be generalized to all ergodic Floquet systems. Indeed, if a Floquet system is ergodic, long time averages are reproduced by statistical expectation values such as Eq. (2.2) where the distribution function is uniform throughout all phase space since there are no integrals of motion. Such distribution is formally equivalent to the one of a canonical ensemble with infinite temperature2. This analogy has given rise to the terminology “infinite-temperature ensemble” widely used in the literature to describe thermal states

of Floquet systems.

The spreading over all phase space has deep consequences for what concerns the dynamics. Imagine we perform the following protocol. We initialize the system in a state with a low energy H₀, either meant as the static part of the Hamiltonian for a kicked system or the average Hamiltonian for a generic Floquet system. Then we let the system evolve. If the system is ergodic as we are assuming, on average H(t) must increase, until it saturates (on average) to the value determined by the infinite temperature ensemble. It is intended that if this value is infinite, as for the kicked rotor, this saturation is never reached and the energy will keep on increasing at all times. This phenomenon, usually termed energy absorption or heating3 dynamics is often used as a key signature of ergodicity, or its absence, in both experiments and numerical simulations.

2.2 Quantum kicked rotor

The Hamiltonian of the quantum model, can be obtained from the classical one, promoting the variables ϕ and p to operators satisfying the usual commutation relation[ ˆϕ,pˆ] = i~. The fundamental object in the quantum model is then the Floquet operator, which gives the evolution over a period (see Appendix 2.A for a quick introduction to Floquet theory). Choosing as initial time the instant right before a kick, then we have

ˆ UF = exp −iτ 2~pˆ 2 exp −i ~kcos ˆϕ (2.8) 2

Since the temperature is infinite, it is not even necessary to specify the Hamiltonian of the canonical ensemble. In fact, any Hamiltonian will yield the same distribution function.

3_{We would like to stress however that heat is a quite inappropriate word in this context. In fact, the}

(13)

2.2. QUANTUM KICKED ROTOR 9 A basis, which we will often use is the one of the momentum eigenstates |mi, which satisfyp |miˆ = ~m |mi with m ∈ Z. The discreteness of the eigenvalue arises from the 2π periodicity of ϕ.

Again, if we were to study the statistical mechanics of the model, we would translate the uniform distribution of the classical system into a density matrixρˆth proportional to the identity4. So, as for the classical system, statistical mechanics predicts hpˆ2_{i = ∞. If} the system were ergodic, then we should observe the same energy absorption dynamics characterized by an unbounded growth of hψ(t)|ˆp2|ψ(t)i.

The dynamics can be easily studied numerically, applying alternatively on the state the two time evolution operators: the free propagation part exp −_2~iτpˆ2 and the kicked evolution exp −i

~kcos ˆϕ. The first one is diagonal in the momentum basis, while the second one is diagonal in the position basis and will thus allow hopping between different momentum eigenstates. Its matrix elements in the momentum basis can in fact be readily calculated to be m0 exp −i ~kcos ˆϕ m = (−i)m−m0Jm−m0(k/~) (2.9)

where Jν denotes the cylindrical Bessel function of ν-th order. Furthermore, to highlight the relevant parameter it is customary to choose the units of time such that τ = 1. We can thus recognize that the dynamics is determined only by the rescaled Planck constant ~ and by k/~.

Via a numerical simulation of the dynamics starting from momentum eigenstates one can easily recognize that the system is never ergodic for any choice of parameters [13]. In fact, for every value of k and ~, hψ(n)| ˆp2_{|ψ(n)i starts to increase linearly. This is} in agreement with the classical dynamics, which is retrieved substituting K = k and taking the limit ~ → 0. However, after a certain time, usually called localization time tloc, the energy growth of the quantum model saturates, as shown in Fig. 2.2. Instead, the corresponding classical model will continue to absorb energy from the driving for an infinite time, as already discussed in the previous section.

This peculiar phenomenon where, in spite of the underlying classical dynamics being ergodic, the quantum model does not thermalize is called dynamical localization. A further feature of dynamical localization is the localization of Floquet states |φ_α(t)i (the eigenstates of the stroboscopic dynamics, see Appendix 2.A) in the momentum basis |mi. For definiteness, from now on we will work on the Floquet states at time t= 0−, which we will only denote as |φαi. These states can be numerically calculated by diagonalizing the operator ˆUF expressed in the momentum basis. Then, for a fixed eigenstate |φαi, the coefficients hm|φαi are peaked near a given m0(α) and decay exponentially as |m−m0(α)| is increased, i.e. | hm|φ_αi | ∼ exp−|m−m0(α)|

ξloc

, as shown in Fig. 2.3.

The suppression of energy absorption and the localization of Floquet states are in

4_{As before, being the Hilbert space infinite dimensional, we must define ˆ}_ρ

thvia a regularization such

(14)

Figure 2.2: Figure from [22]. The expectation value for hˆp2_{(t)i for the quantum dynamics with}

parameters k= 11.6 and ~ = 1. The black curve is the one for the initial state |m = 0i, the red curve is averaged over5 initial states with m ∼ 1. Instead the blue curve is the classical diffusion of the classical kicked rotor with K= 11.6. We clarify the meaning of the green curve in Sec. 2.4.

fact deeply connected, as we are going to explain. In particular, we will now see that the localization of the eigenstates implies the suppression of energy absorption. First of all, notice that the long time average of an observable ˆO can be easily calculated once Floquet states are known. To understand why, we will again work under the assumption that time averages can be replaced by stroboscopic time averages, which can be expressed as ¯ O= lim N →∞ 1 N N X n=0 D ψ(nτ ) Oˆ ψ(nτ ) E = Trρˆ∞Oˆ (2.10) where ˆ ρ∞= lim N →∞ 1 N N X n=0 ˆ U_Fn|ψ(0)i hψ(0)| ˆU_F†n (2.11) is the average density matrix. Once at this point, it is useful to decompose the Floquet operator in terms of the Floquet states |φ_αi

ˆ UF = X α e−iµατ_|φ αi hφα| (2.12)

Substituting the ˆUF decomposition into the expression for the average density matrix, we immediately recognize that, among all the terms |φαi hφβ|, only those with α = β yield a non-zero result5, while all the other average to zero. Thus the average density matrix is diagonal in the Floquet basis, and in particular

ˆ ρ∞=

X

α

|hφα| ψ(0)i|2|φαi hφα| (2.13)

5_{Here we are assuming that there are no degeneracies in the quasi-energy spectrum, i.e. µ}

α6= µβ if

(15)

2.3. DYNAMICAL AND ANDERSON LOCALIZATION RELATIONSHIP 11

Figure 2.3: Figure from [22]. |φα,m|2 for three typical eigenstates. The numerical parameters

are the same as Fig. 2.2

This form of the average density matrix is referred to as diagonal ensemble, being ρ∞ diagonal in the Floquet basis. At time t= 0 the density matrix |ψ(0)i hψ(0)| may have a lot of matrix elements outside the diagonal, but as time passes they are suppressed by the average in time, eventually leaving only those along the diagonal.

At this point we have an equilibrium density matrix, i.e. a density matrix which does not evolve in time. Under which conditions and to what degree such a density matrix can be reproduced by the thermal density matrix ρth is a hard question. Indeed the problem dates back to the dawn of quantum mechanics and was first tackled by Von Neumann himself in [2]. And only in very recent years it seems that physicists are getting closer to answer this question for many-body systems. We postpone this discussion to the next Chapter, and we show now that localization of Floquet states is incompatible with thermalization.

Indeed, if we use the diagonal ensemble in Eq. (2.13) to calculate for example the long time average ofpˆ2, we obtain

p2 ₌X α |hφα| ψ(0)i|2φα pˆ2 φ_α (2.14)

Suppose now we take as initial state a momentum eigenstate |mi (or anyway a state˜ localized around a given momentum ~ ˜m). Then |hφα|ψ(0)i|2will be negligible for all those Floquet states whose localization centre m₀(α) is far away from ˜m, i.e. |m0(α) − ˜m| ξloc. However, for these α, the expectation value hφα|ˆp2|φαi will be finite, since the eigenstates are localized around a finite m0(α).6

We have thus seen that localization of Floquet states implies that T = ∞ thermal-ization is not possible and that the energy absorbed by the system is suppressed.

2.3 Dynamical localization as Anderson localization in

en-ergy space

It was shown in [14, 15] that the equation for Floquet states of the kicked rotor model ˆ_U_F_e+iµα/~_|φ

αi = |φαi (2.15)

6_{For a better estimation of the absorbed energy, we refer to the argument in [19], which is partly}

(16)

is strictly connected to the eigenstate equation for a hopping problem with pseudo-disorder on the 1d lattice of momentum eigenstates |mi. This second problem is well studied in the literature and it is known that the one-particle eigenstates of a disordered 1d system are always localized. This phenomenon takes the name of Anderson localization and we refer to Appendix 2.B for a brief introduction to Anderson localization. Here we show the connection between dynamical localization and Anderson localization.

To see this, we can rewrite

exp −i ~kcos ˆϕ = 1 + iW ( ˆϕ) 1 − iW ( ˆϕ) (2.16)

where W( ˆϕ) is equal to − tan (k cos ˆϕ/~) and also

exp −i ~(ˆp 2_{/2 − µ} α) = 1 + iV (ˆp) 1 − iV (ˆp) (2.17)

with, as before, V(ˆp) = − tanpˆ2/2−µα ~

. Once at this point, with a few manipulations one can show that Eq. (2.15) is equivalent to

ˆ_V _{+ ˆ}_W 1

1 − i ˆW |φαi

| {z }

=|χi

= 0 (2.18)

Expanding in the momentum eigenstates, introducing where needed the identity decom-position1 =P

m|mi hm|, and defining χm= hm|χi, we obtain

mχm+ X

r

Wrχm−r = −W0χm (2.19)

where m= tan µα/2~ − m2~/4 and Wr are the coefficients of the Fourier expansion of W W(ϕ) = +∞ X r=−∞ Wreirϕ (2.20)

We can interpret Eq. (2.19) as the eigenstate equation for an Hamiltonian which describes hopping on a 1d lattice whose sites are labelled by m. Unlike the Hamiltonians we discus in Appendix 2.B, here we do not have random on-site energies m, still the tangent with m2 _{in its argument can be regarded as pseudo-random. Indeed it has been} checked that such a pseudo-random sequence can induce Anderson localization [23].

Another crucial point in Eq. (2.19) is that the hopping range is finite. Typically, for simplicity, when discussing Anderson localization on a lattice, a tight-binding Hamiltonian is considered, as we do in Appendix 2.B. Here, however, the hopping term Wris non-zero also if the distance between the two sites |r| is larger than1. Still, in order to apply Anderson theory of localization, what matters is that |W_r| decreases sufficiently fast with |r|, as in fact happens here since |Wr| decays exponentially on a length scale of k/~.

(17)

2.4. EXPERIMENTAL OBSERVATIONS 13

Hence, being in a 1d lattice, χ_m will always be localized, i.e. χ_m ∼ exp−|m−m0| ξloc

for some m0, as we know from Anderson theory. What information does this gives us on φ_α,m= hm|φαi? It turns out that the localization properties of |φαi and |χi are the same, in fact one can re-express

|χi = 1 + exp − i

~kcos ˆϕ

2 |φαi (2.21)

But since the matrix elements m

exp −i

~kcos ˆϕ

m0 decay rapidly when |m − m0| increases, the φα,m are related to the χm through a transformation which is local in m-space. Thus if χm are localized, also φα,m will be. The localized structure of φα,m with an exponentially decaying tail has indeed been observed in various numerical simulations [13–15]. An example of the Floquet state structure is reported in Fig. 2.3 for the reader’s convenience.

We have thus shown, reporting the argument from [14, 15], that the Floquet states of the kicked rotor model are localized in momentum space. Furthermore Anderson localization in momentum space hinders energy absorption and prevents the system from being ergodic.

2.4 Experimental observations

Apart from the theoretical analysis, the kicked rotor model has also attracted some experimental attention. This is mainly due to the fact that the model allows for indirect characterization of Anderson localization. In fact, while in 1d Anderson localization can be observed directly [24], in dimension2 and 3 a direct observation is much harder, at least without exploiting classical-waves systems which mimic Schrödinger equation [25]. Instead it was shown [26] that the kicked rotor driven with n incommensurate frequencies can be mapped on a localization problem in n dimensions. This has allowed, for example, the experimental characterization of Anderson localization in 3d [27, 28].

Given that the effect we want to observe is due to quantum coherence, it is essential to prevent the system from interacting with any external environment, with which the system would become entangled, thus losing coherence. In this respect, one of the main experimental platforms for studying isolated system are ultra-cold atoms experiments, where the effective Hamiltonian determining the time evolution of the atoms is engineered through laser beams [29]. This makes it particularly hard to realize the circular geometry of the kicked rotor model.

Looking back at the mapping, we can recognize however that the only point where we used spatial periodicity was to determine the discreteness of the eigenvalues of p.ˆ The same effect can thus be achieved in a periodic lattice with lattice spacing a. Indeed we know from Bloch theory that, as long as we do not alter the spatial periodicity of the Hamiltonian during time, the quasi-momentum β, −π/a < β < π/a, of the particle will be a conserved quantity throughout the whole time evolution. We can

(18)

thus restrict ourselves to studying one β sector at the time. Fixing a value of β 6= 0, just amounts to a change in the boundary condition of the rotor problem. Indeed, instead of requiring the monodromy of the wave-function ψ(ϕ + 2π), we would now require ψ(ϕ + 2π) = eiβa_{ψ(ϕ). This condition, in turn implies that the eigenstates of} ˆ

p are of the formp |m, βiˆ = ~(m + aβ) |m, βi with m ∈ Z. So the eigenvalues of ˆp are again a discrete set. Furthermore the eigenvalue of the free hamiltonianpˆ2 will again depend quadratically on m, which is an essential feature in order to have pseudo-random potentials in the mapping.

To sum up, to see dynamical localization, we can exploit a 1d optical lattice, where the static part of the Hamiltonian is simplypˆ2/2M , which means that for most of the time the optical potential is actually turned off. To realize the kick, instead, a potential with spatial periodicity a must be turned on. In order for the kick to be most resembling of the kicked rotor model, one can choose a potential proportional tocos(kLx) with kL= 2π/a. This potential can be achieved for example by means of an optical standing wave obtained by two counter-propagating laser beams with wave-vector kL.

With a proper units rescaling [28], and provided that the potential is switched on for a sufficiently short time, the atomic motion can be described by the kicked rotor model, with the only difference in the β-dependent boundary conditions mentioned above. Practically there is no way to fix a value of β, but each particle will participate to various sectors. In this way, the observations are actually averaged over β. Such average does not pose any problem, it is in fact very much resemblant of an average over the specific disorder configuration m, which is typically performed on disordered systems to reduce fluctuations. Indeed this is the only relevant effect of the average, which instead does not change localization properties, as it is also clear from Fig. 2.2.

The observations reported above allowed for the experimental realization of the kicked rotor model. Indeed dynamical localization has been observed in [30–32]. This platform, apart from allowing to measure hˆp2i, also allows for the characterization of the momentum distribution, which, at long enough times, turns out to be in perfect agreement with an exponential decay, as reported in Fig. 2.4.

Appendix

2.A

Floquet theory

In this section we review some basic notions on the Floquet formalism [33–35] which are useful to the connection between Anderson localization and dynamical localization.

Formally the Hamiltonian of a periodically driven system is invariant by time translations of nτ for n ∈ Z. This is very resemblant of the discrete space translation symmetry in a crystal. One can thus expect that something similar to Block theorem [36] rephrased in the time domain may apply here. Indeed Floquet theorem states that there exists a set of states {|φα(t)i} with the following properties:

(19)

2.B. BRIEF REVIEW OF ANDERSON LOCALIZATION 15

Figure 2.4: Figure from [32]. Time steps shown are 0 kicks (light solid line), 17 kicks (dash-dotted), 34 kicks (dashed), 51 kicks ((dash-dotted), and 68 kicks (heavy solid) The vertical scale is logarithmic and in arbitrary units. The corresponding numerical parameters are ~ = 2.08 and k '12.8.

(i) each |φα(t)i solves the Schrödinger equation, i.e.

∂t+ i ˆH(t)

|φα(t)i = 0 (ii) at all times the set {|φα(t)i} is a complete orthonormal set

(iii) |φα(t + τ )i = e−iµατ /~|φα(t)i.

The states |φα(t)i are called Floquet states and µα are their quasi-energies, which, very much as quasi-momentum in a crystal, are defined up to translations of multiples of ~ω = ~2π/τ . Thus, quasi-energies are typically folded in the equivalent of the first Brillouin zone, i.e. −~ω/2 < µα < ~ω/2.

Finally note that property (iii) identifies Floquet states as the eigenstates of the Floquet operators defined as

ˆ UF(t0, t0+ τ ) = T exp −i Z t0+τ t0 dtH(t) (2.22)

thus allowing for their numerical calculation.

2.B

Brief review of Anderson localization

Suppose we want to describe the motion of a particle in one dimension under the influence of an external potential V(x). The time evolution is then generated by the Hamiltonian

ˆ H= pˆ

2

2m+ V (ˆx) (2.23)

If V is flat or even periodic in space, we know from Bloch theory that the eigenstates of ˆ

(20)

it is known that the particle will spread ballisticallly in the whole space. The same would be true also in classical mechanics if the energy of the particle is high enough.

Anderson localization is the suppression of transport due to the interplay of (quantum) coherence and spatial disorder. We would like to stress that spatial disorder alone does not produce this phenomenon. Imagine in fact the dynamics of a classical particle in a random V(x). If the particle energy E is smaller than the fluctuations amplitude W of V(x), the particle will travel in the same direction, until it reaches a point x at which V(x) = E. At this point the particle will turn back and will go on oscillating between two of such points. However we can prevent this from happening if we give to the particle an initial energy E_{& W : in this case the particle will always travel in the} same direction.

On the opposite, quantum mechanical motion has completely different properties. At short time the wave-function ψ(x) starts to spread diffusively. But later the average dynamics becomes frozen with a wave-function |ψ(x)| ∼ exp−|x−x0|

ξloc

. Furthermore, the localization length ξ_loc will be quite independent of the initial state and on the specific details of disorder. This phenomenon is well known in solid-state physics and has also been observed experimentally, for example in [24].

To understand its physical origin, and also to make contact with the kicked rotor model, it is quite useful to rephrase the problem (2.23) as a hopping problem on a lattice. This can be done, for example, if we consider a potential V(x) with quite deep holes. Thus the low-energy physics can be described in terms of states |ii, i.e. the lowest energy state in the i-th hole. Such a state will have a random energy _i and will also have some non-zero tunnelling amplitude t_j,j±1 with the states centred on neighbouring sites. Thus the low-energy physics can be captured by a tight-binding Hamiltonian

ˆ H=X j j|ji hj| + X j (tj,j+1|ji hj + 1| + h.c.) (2.24)

with j distributed randomly. Here, the width W of the distribution of j reflects the disorder level of the potential V(x)

To understand why localization takes place, we may imagine to put a particle on site 0 and study the transport over the lattice. A first hint can be already achieved at the first order in perturbation theory, where the hopping termP

j(tj,j+1|ji hj + 1| + h.c.) is the perturbation. In fact, the Fermi golden rule would predict the decay rate of the state |0i to be Γ(1)= 2π ~ h |t0,1|2δ(0− 1) + |t0,−1|2δ(0− −1) i (2.25) The crucial point is that, being the on-site energies disordered the δ-function will yield a non-zero result only with0 probability. Thus the state |0i is stable at first order in perturbation theory, suggesting that there can be no transport along the chain. It is also clear, once at this point that any finite order calculation ofΓ will contain a finite number of δ-functions, thus will be always zero. Thus, the crucial point to check is if

(21)

2.B. BRIEF REVIEW OF ANDERSON LOCALIZATION 17 the re-summation of the whole perturbative series may yield aΓ 6= 0. This calculation was performed by Anderson in [37] and showed that indeed, where the series converges, h0|ψ(t)i tends to a finite vale as t → ∞, which implies that the particle remains localized in the region surrounding the initial site.

With more powerful tools, such as the scaling theory for localization [38], it was shown that 1d and 2d systems are always localized, no matter how weak the disorder W or how strong the hopping. We refer to [39] for an extensive review on the subject.

(22)

(23)

Chapter 3

Many-body Floquet systems

In the latest decades there has been a renewed interest in the study of thermalization of many-body quantum systems, prompted by the new experimental platforms of ultra-cold atoms in optical lattices [3–7] and trapped ions [8, 9]. Indeed, these platforms allow for simulations of closed many-body system for quite long times. In particular, due to these experimental developments, it became possible to study thermalization also in many-body Floquet systems. Thermalization of Floquet systems is in fact quite peculiar, given the absence of energy conservation. So, if thermalization occurs, the system must thermalize to a T = ∞ ensemble, which corresponds to a thermal density matrix proportional to the identity.

In the present chapter we aim at briefly reviewing the basic concepts regarding thermalization of many-body quantum systems, with a particular focus on Floquet systems. Unless otherwise noted we will be referring to periodically driven models. However, it will be sometimes useful to also consider non-driven systems, where some concepts may arise more naturally.

The chapter is then organised as follows. First of all, in Sec. 3.1, we introduce the notion of thermalization in quantum mechanics. Its meaning is in fact much less obvious than in classical physics. In Sec. 3.2 we introduce the Eigenstate Thermalization Hy-pothesis (ETH): the mechanism which nowadays is believed to be behind thermalization of isolated many-body systems. Afterwards, in Sec. 3.3 we review the main known classes of non-thermalizing systems. Finally, in Sec. 3.4 we discuss the recent literature inquiring the possibility for dynamical localization to persist also in many-body systems.

3.1 Thermalization in quantum mechanics

While in classical physics ergodic behaviour is related to chaos, a naive counterpart of chaos cannot exist in quantum mechanics. In fact, the time evolution is described by the Schrödinger equation, which is linear in the state |ψ(t)i. Thus the dynamics of a quantum state cannot be chaotic. Then, it seems impossible for a quantum system to “forget” its initial conditions and relax to an ensemble described by few parameters.

(24)

To better explain why, consider two initial states |ψi and |φi. The evolved states after an arbitrarily long time t can be obtained from the initial ones by applying the unitary time-evolution operator Û , |ψ(t)i = Û |ψi and |φ(t)i = Û |φi. It is then apparent that the trace distance [40] between the two states

D(|ψ(t)i , |φ(t)i) =p1 − | hφ(t)|ψ(t)i |2 _(3.1) will be conserved throughout the time evolution, i.e. D(|ψ(t)i , |φ(t)i) = D(|ψi , |φi). We recall that the trace distance between two states (poorly speaking) quantifies how much easily they can be distinguished via a measurement (more rigorously a POVM [41]). Thus we can imagine taking as |ψi and |φi two states with different properties. For example, in a spin system, we may take |ψi to be a state with a non-zero magnetization along a given direction, while |φi has a zero magnetization. Since the two states can be distinguished easily at the initial time through their magnetization, we can expect their distance to be very close to1. After time t, since the trace distance is conserved, it will exist a new measurement which will allow to distinguish |ψ(t)i and |φ(t)i as easily as before. On the other hand, if the system thermalizes to a T = ∞ ensemble1 _{there must} be a sufficiently long time ˜t, after which |ψ(˜t)i and |φ(˜t)i will yield the same thermal expectation value for every measurement. This argument then seems to suggest that thermalization is impossible and that the system will always retain memory of its initial condition.

The previous observation however already suggests the subtle point which may invalidate this argument in many-body systems. Indeed we are implicitly assuming, by employing the trace distance, that we can arbitrarily choose any Hermitian operator acting in the extremely highly dimensional Hilbert space of the many-body system and use it to discriminate between |ψ(t)i and |φ(t)i. However, it is quite unrealistic that the measurement of an arbitrary many-body operator can actually be performed in a laboratory. So, in principle, if we had access to the whole set of Hermitian operators acting on the whole Hilbert space of the system, we would be able to recover the information of the initial state even at long times. Still, it could happen that with physically accessible measurements, we cannot distinguish the two states at long times.

So, while there will be for sure operators that do not thermalize, we can imagine that it could exist a subset of observables whose expectation value is correctly predicted by statistical ensembles. Then, in order to discuss thermalization, one must specify which measurements can be performed and must be considered physically accessible and by exclusion which measurements cannot. While this may not be so clear for a single particle, in the context of many body physics this distinction arises quite naturally.

To fix ideas, suppose we are dealing with a lattice model, such as a spin model or a (Bose-)Hubbard model. The simplest observables which can be measured in this

1_{The same argument holds also for non-driven systems, with the only caveats that in this case |ψi}

(25)

3.2. THE EIGENSTATE THERMALIZATION HYPOTHESIS 21 context are operators ˆOj which act non-trivially only on site j, or spatial averages of such quantities. An observable of this type is for example the average magnetization in a spin model, or a site occupation for a Hubbard model. Other quantities which are physically measurable and which are of great relevance in statistical mechanics are n-point correlation functions. Anyway, in the idealization of such measurements, the operator related to them will always act non-trivially only on a finite number of sites. At the same time the system, to be in the thermodynamic limit, will always be indefinitely vast outside the measurement region. In this respect, it is natural to state that physical observables are the ones which act non-trivially only on a finite and spatially localized portion of the system.

To formalize this notion, we can imagine taking the system S and bipartite it in two subsystems A and B, where A is a finite subsystem, which occupies a finite region in real space, while B is infinitely extended outside of A. Then we say the system to be thermal if the reduced density matrix on the subsystem A,ρˆA(t) = TrB(|ψ(t)i hψ(t)|) reproduces at long times a reduced thermal density matrix ρˆA,th = TrB(ˆρth). Where ˆ

ρth is the thermal density of S as a whole which, in a Floquet system, is proportional to the identity.

This is the viewpoint usually taken in all the literature concerning the problem of thermalization and physically it corresponds to requiring that, even if the system is not coupled to any thermal bath, it is able to act itself as a thermal bath for any of its subsystems A which have a finite extension. In the next section we are going to discuss how this physical requirement reflects in the structure of the eigenstate of the dynamics.

3.2 The eigenstate thermalization hypothesis

In this section we are going to discuss the role of the eigenstates of the dynamics in the thermalization process. In particular we are going to introduce the Eigenstate Thermal-ization Hypothesis (ETH) for Floquet systems, which is nowadays widely accepted as the mechanism behind thermalization of many-body driven systems. Afterwards we are going to mention the ETH for static systems, where the system keeps memory of the initial energy. Starting from the two version of the ETH, we are going to expand the discussion with the aim at understanding the role of integrals of motion in many-body systems. Finally we are going to discuss the possibility of a description of Floquet systems by means of a local effective Hamiltonian, which would amount to an absence of T = ∞ thermalization.

In order to introduce the ETH and its context, it may be useful to look back at our discussion of the kicked rotor model. When we discussed the statistical mechanics of the classical kicked rotor, we argued that, there being no integrals of motion whatsoever, the statistical description would require to assume a uniform distribution throughout all phase space. Instead, had there been an integral of motion I(ϕ, p), we should have used a different distribution, vanishing outside of the manifold with I(ϕ, p) = I(ϕ(t = 0), p(t =

(26)

0)) and uniform inside this manifold. This is because the orbits in phase space would have been constrained on the manifold with fixed I, and they would have uniformly explored every region in the manifold, being the dynamics ergodic.

When we discussed the quantum kicked rotor, we introduced the thermal density matrix ρˆth ∝ 1 as the quantum version of the classical uniform distribution. There is nevertheless an important difference between the classical and quantum dynamics. Indeed, the stroboscopic time evolution generated by ˆUF has infinitely many integrals of motion, since ˆUF commutes with each projector on a Floquet state |φαi hφα|, while an ergodic classical system has no integrals of motion.

This feature is also apparent in the form of the average density matrixρˆ∞, which can always be expressed via a diagonal ensemble as in Eq. (2.13). The diagonal ensemble, in fact, keeps memory of the initial condition with respect to the observables |φαi hφα| in the coefficients Cα= | hφα|ψ(0)i |2.

On the other hand, if the system is ergodic, it must be that at long times ˆ

ρA,∞= TrBρˆ∞= X

α

CαTrB|φαi hφα| (3.2)

is equal toρˆA,th which does not depend on the initial state |ψ(0)i, and indeed can be obtained from Eq. (3.2) by putting all Cα equal among them. It then seems paradoxical that the system can both retain memory of all the C_α and, at the same time, relax to a thermal ensemble independent from the initial condition.

Once at this point, however, the solution to this apparent paradox is not difficult. In fact, we already noticed in the previous section that globally the information on the initial state cannot be lost. However the projectors |φ_αi hφ_α| are highly non local operators, so, according to the previous classification, they must be regarded as physically inaccessible. So, even if their expectation value, C_α, keeps memory of the initial state, thermalization is not hindered. What matters, instead, is that the expectation values of local operators should not retain memory of the initial state. This means that any local measurement (performed at long times) should not allow to reconstruct the distribution of the C_α.

The mechanism2 by which the Cα do not affect the expectations of local observable is a variant of the Eigenstate Thermalization Hypothesis (ETH), which states that

∀α : Tr_B|φαi hφα| = ˆρA,th (Eigenstate Thermalization Hypothesis) (3.3) It is thus apparent that, on a local level (the finite subsystem A), set by the Cα.

ETH is also strictly connected to the self-generated randomness of the quantum ergodic dynamics. Indeed it was shown that, when the ETH holds, the Floquet operator

2

We would like to add that there are in the literature also different approaches to explain thermal-ization, with which we will not deal in the present text. We instead refer to [42] for an extensive review on the subject, which however is mainly focused on non-driven systems.

(27)

3.2. THE EIGENSTATE THERMALIZATION HYPOTHESIS 23 ˆ

UF can be treated as a random matrix [43], once expressed in a physical basis. Here with physical basis we mean a basis identified as the eigenstates of a set of local operators (see examples below). As a result, the eigenstates of ˆUF have a random structure, which implies that they are highly entangled states and delocalized in every physical basis. Thanks to the entanglement and correlations, the partial trace of the eigenstates can give rise locally to a highly mixed density matrix as in Eq. (3.3). This is impossible if the eigenstates have small entanglement, thus the ETH is generally incompatible with localization of eigenstates in a physical basis (whose elements have no entanglement). Examples of set of operators that define a physical basis are:

• for a spin system, the set { ˆS_jz: ∀ sites j}; • for a system of coupled rotors {ˆpj : ∀ rotors j};

• for a bosonic or fermionic system {ˆnj : ∀ sites j}, which identifies the Fock basis. The ETH was first formulated for time-translational-invariant systems [44–47], context in which it has been confirmed numerically in various papers (the most famous being [48]). In these circumstances, Floquet states are replaced by the eigenstates |ψαi of the hamiltonian ˆH, satisfying ˆH |ψαi = Eα|ψαi. The ETH then asserts that

∀α : TrB|ψαi hψα| = TrBρˆmicro(Eα) (3.4) whereρˆmicro(Eα) denotes the micro-canonical density matrix with energy Eα. Later on, the variant of the ETH reported above was numerically found to hold in various Floquet systems [43, 49].

From the above discussion of the ETH, we can still extract some more information. In fact, the ETH can also provides us insight in what is the role of the integrals of motion in quantum many-body physics. One particular instance of integral of motion is a projector on an eigenstate |φ_αi hφ_α|. We have seen that this operator is highly non-local, since it acts non-trivially on the whole many-body Hilbert space. About its role in the long-time dynamics, we have seen above that the expectation value of this operator, i.e. C_α, does not carry any information at all on the local structure of the state, i.e. the reduced density matrix on a local subsystem A. Very different is instead the role of local3 integrals of motion, like the Hamiltonian of a time-independent system. Operators of this type have a local structure, i.e. they are sums of terms acting only on subsystems of finite size (typically only two neighbouring sites). About these integrals of motion, we know that their expectation value at the initial time strongly influences the long-time dynamics. For example, in energy-conserving systems, the micro-canonical density matrices will have very different properties, also at a local level, depending on

3

We are using the terminology commonly employed in this context, even it could be misleading. In fact, unlike in the previous discussion about local observables, when referred to integrals of motion locality is usually intended in a different way. As it is explained in the following discussion, an integrals of motion is said to be local if it has a local structure.

(28)

the energy E. The physical idea above is that the expectation value of an integral of motion with a local structure bears a piece of information on the initial state, which constrains the local structure of the state also at long times. On the opposite, conserved operators without a local structure do not impose any constraint on the local structure of the state at long times. Hence thermalization of local observables is possible [50]. This picture finds also further confirmation in the studies of integrable systems (see next section).

This fact has a relevant application in Floquet systems. It is, in fact, much debated in the literature whether or not Floquet systems can have a local integral of motion which plays the role of an effective Hamiltonian. Indeed one could define a Floquet Hamiltonian ˆHF as ˆUF = exp

−i ˆHFτ

. The Magnus expansion [51], a perturbative expansion of ˆHF for high driving frequencies, suggests that where the perturbative series converges the Floquet Hamiltonian is local [52]. However it was shown more recently that the Magnus expansion is valid only at finite times [53, 54], thus being able to describe only the so-called prethermal regime [55], where in fact the system behaves as if the dynamics were generated by an effective local Hamiltonian. In some systems there is also a finite-size crossover from a regime with a local ˆHF to an ergodic behaviour, but in the end it is mainly believed that the dynamics is always ergodic in the thermodynamic limit [43].

3.3 Non-thermalizing systems

In the previous section we have discussed the mechanism behind thermalization of isolated quantum systems, which is valid in a great number of models displaying thermal behaviour, as we have seen. However not all systems do thermalize. It is extremely interesting then to understand which features can hinder thermalization. Furthermore, non-thermal systems could in principle support totally new phases of matter, since they are not subject to the laws of statistical physics used to classify thermal phases [56].

Apart from the obvious theoretical interest, it is also practically important to find non-thermal systems, in particular Floquet ones. For example, in the context of ultra-cold atoms in optical lattices, a variety of effects such as topological band structures can be simulated using a periodically driven Hamiltonian [12]. However, in order to observe the desired phenomena, the dynamics must be stable and non-thermalizing for long times.

In this section, we aim to give a very brief introduction to the main classes of non-thermalizing systems and their respective features. Initially, we will discuss integrable systems, to focus afterwards on many-body localized systems.

First of all, however, we shall mention that in the literature there have been presented cases [57–59] of Floquet systems which are neither integrable nor many-body localized, and that are believed to be non-thermalizing. It is furthermore interesting to notice that,

(29)

3.3. NON-THERMALIZING SYSTEMS 25 at least for what concerns [57, 58], the physical origin of the non-thermal behaviour seems to be the underlying classical dynamics, whose non-ergodic behaviour is somehow robust against quantum fluctuations: a phenomenon completely different from dynamical localization, which is a genuine quantum effect absent in the classical dynamics.

3.3.1 Integrable models

In classical mechanics, a model is said to be integrable if it has an extensive number of integrals of motion. For definiteness, assume that we are dealing with a N -particle system with N independent integrals of motion. It is clear from the previous discussion that such a model cannot satisfy the ergodic hypothesis, since the motion will not explore all the phase space, being constrained on a N -dimensional torus [60, 61] immersed in the2N -dimensional phase space4 _.

In quantum mechanics instead the situation is not that clear-cut. As we already mentioned, all systems have a super-extensive number (i.e. exponential in the system size) of integrals of motion: the projectors on the Floquet states. However, as discussed in the previous section, typically systems are unable to locally keep memory of their initial expectation value. So it is not so clear what may be a good definition of quantum integrability. The question is far from being settled (see [42] for an extensive discussion) and we do not aim at reviewing the possible answers here.

It is known, however, that quantum systems that have an extensive number of local integrals of motion cannot relax to a thermal ensemble. Such systems, which we will just refer to as integrable from now on, fall into two classes, up to the present knowledge. The simplest type is given by models that, via an appropriate transformation, can be reduced to free particles (called quasi-particles) Hamiltonians, such as the Ising chain [62], or its periodically driven version [63, 64] (see also 5.2). The Hamiltonian or the Floquet operator thus commutes with the occupation number of every Fourier modes, whose number scales like L. Furthermore from the number operators, local integrals of motion can be constructed as in [65]. As a consequence, the long-time dynamics keeps memory of their initial expectation value, thus hindering thermalization. The other class is composed by systems solvable through Bethe Ansatz [66], which we will not deal with in this Thesis.

Still one may wonder if the steady state of an integrable model can be described by an ensemble which is simpler than the diagonal one in Eq. (2.13). In fact, the dynamics has only O(L) constraints fixed by the integrals of motion, while the the diagonal ensemble is described by O(dL_{) O(L) parameters where d is the local Hilbert space} dimension of the model5 _{. It was supposed, and verified in a variety of models that,}

4

In autonomous (i.e. non-driven) systems, the dynamics to be ergodic should uniformly explore a 2N − 1 manifold, because of the constraint arising from energy conservation. So, also autonomous integrable systems cannot be ergodic.

5

For a lattice model, the global Hilbert space is, in fact, a tensor product of L copies of the Hilbert space on each site: Hglobal=N_jHj, with j denoting the spatial index of the site in the lattice. The

(30)

indeed, integrable systems relax to a so-called Generalized Gibbs Ensemble (GGE) both in driven [64, 67] and static systems [65, 68]. The density matrix associated to the GGE ensemble is the one that maximizes the entropy of the thermal state, while satisfying the constraints related to the initial values of the local integrals of motion [68]. We would like to point out, however, that this statement is far from obvious. For example localized (non-interacting) systems cannot be described by any ensemble simpler than the diagonal one [69].

The presence of local integrals of motion, and hence integrability, in all the known cases requires a fine tuning of the parameters in the Hamiltonian. Considering for example the simplest class of integrable models, where the Hamiltonian or the Floquet operator can be diagonalized in terms of free quasi-particles, it is clear that an arbitrary perturbation in the Hamiltonian, even if weak, will produce an interaction between quasi-particles, thus breaking integrability.

It is much debated, instead, how an integrability-breaking perturbation will affect the dynamics.6 Surely for a strong enough perturbation the system should recover a thermal behaviour. However, it is not clear how the transition from integrability to thermalization develops. At finite system size a crossover regime between ergodic and integrable behaviour emerges [70–72] at a finite perturbation strength, but this, in fact, may be due to finite size fluctuations [73]. It is mainly believed, instead, that in the thermodynamic limit any arbitrarily small perturbation will result in the recovery of thermalization [74, 75] at infinite times. However, the weaker the perturbation, the longer the thermalization time will be. Being instead the relaxation to the GGE much faster, the system will remain for a long time in a prethermal regime, where the state of the system can be effectively described by a GGE ensemble [76].

Finally, we would like to point out the differences between integrable models and dynamically-localized many-body system, if they exist. First of all, dynamical localiza-tion is a genuine quantum phenomenon, which does not have a classical counterpart, unlike integrability. Furthermore, for the long-time dynamics to be non-thermal in the thermodynamic limit, the Hamiltonian parameters of an integrable model seem to require a fine-tuning. Instead, one may expect dynamical localization to be robust against small perturbations, since at least in the single-particle case, it does not rely on specific details of the Hamiltonian.

3.3.2 Many-Body Localized systems

Many-Body Localization (MBL) emerged as an answer to the following question. Is Anderson localization robust to interactions between different electrons? The question was settled in the seminal paper [77], which showed that indeed transport can be suppressed at large distances if the energy of the system is below a certain many-body

6_{In this respect, Floquet systems have been less studied so far, hence we content ourselves with}

(31)

3.4. DYNAMICAL LOCALIZATION IN MANY-BODY SYSTEMS 27 mobility edge. It is furthermore clear that the absence of transport hinders any possibility of thermalization. In fact, if we imagine to initialize the system in a state with large-scale inhomogeneities, the absence of transport prevents the system to restore homogeneity even at long times. Transport suppression is related to the fact that MBL systems exhibit an extensive number of local operators7 (i.e. acting non-trivially only on a finite region), usually called l-bits (local-bits), which commute with the Floquet operator or the Hamiltonian [78, 79], if the system is non-driven. Notice that, as a consequence, MBL systems have a super-extensive number of local integrals of motion, since not only the l-bits operators are local, but also an arbitrary product of these is still local. This is in sharp contrast with integrable systems, which only have an extensive number of local integrals of motion.

This class of systems has been studied extensively in time-translational invariant settings. In this context, in fact it is the only class of non-thermalizing systems which does not require fine tuning of the Hamiltonian parameters. A complete review of the subject is beyond the scope of the present text, we instead refer the interested reader to the excellent review [10], which mainly deals with non-driven systems.

Also for Floquet systems there are analytical [80], numerical [81–83] and experimen-tal [84] evidences that MBL can survive in presence of driving, if the driving frequency is high enough. In all studies it has been highlighted that gradually lowering the frequency, a critical point is typically found, beyond which localization is destroyed and ergodicity restored.

As we reported, MBL heavily relies on spatial disorder. At the present time, the possibility of non-thermal behaviour in a clean (translationally invariant) and not fine-tuned many-body system is an open question. In the next section we are going to review the existing studies concerning this possibility in the context of dynamical localization.

3.4 Dynamical localization in many-body systems

After the first studies on dynamical localization concerning single-particle systems, the interest in the phenomenon died out for many years until it rose again recently. In more recent years, physicists renewed their interest in dynamical localization. The main question, both interesting and non-trivial, aim at understanding if dynamical localization can survive in a many-body context, or if interactions among particles will eventually destroy dynamical localization, recovering ergodicity.

First of all, the classical dynamics of a chain of L coupled kicked rotors with Hamiltonian H(t) =X j p2_j 2 + K X j cos(ϕj− ϕj+1) ∞ X n=−∞ δ(t − nτ ) (3.5) 7

(32)

has been studied [16, 85]. It emerged that in the thermodynamic limit the classical system is always diffusive and thermalizing for any value of K, even if the diffusion coefficient is exponentially suppressed with1/K.

The quantum behaviour of coupled kicked rotors is instead much debated. In fact, also numerical studies of the system are greatly limited by the computational complexity. While classically one can study chain lengths up to L= 2000 of rotors [16], quantum simulations have been performed at maximum for3 rotors in [18, 19].

To overcome this problem, it was proposed in [17] the study of an integrable version of the kicked rotor model,

ˆ H(t) =X j αjpˆj+ X j6=j0 Jij( ˆϕj− ˆϕj+1) + X j K( ˆϕj) ∞ X n=−∞ δ(t − nτ ) (3.6)

It was analytically shown that, as long as αj/αj0 and α_j/2π are irrational for every j

and j0, the momentum growth of each rotor is bounded and Floquet states are localized in the L-dimensional momentum space. It was pointed out however that, as soon as αj = αj0 for some j and j0, hp_ji is not bounded any more during the time evolution, so

spatial disorder is playing a crucial role, as for an MBL system.

Furthermore, it remains unclear to what extent the observation reported above is due to the integrability of the model and to what extent it can be considered general. In an attempt to clarify this aspect, a model of relativistic rotors with static Hamiltonian H0 =P_j

q

Cjp2+ M_j2had been considered in [18]. This model had been chosen because it interpolates between the usual rotor and the linear one, and in doing so it breaks the integrability of the linear model. In the paper numerical evidence of persistence of localization up to3 rotors is presented. It is then argued that dynamical localization may persist even in the thermodynamic limit for this specific model. In fact, the authors suppose that if the model were to diffuse at short times, after a certain time it would enter the ultra-relativistic regime. In this regime, the Hamiltonian can become arbitrarily close to that of the integrable model. So, in the end, energy absorption would become suppressed.

Still, it remains unclear how much of these studies can be general and how much is strictly related to the specific model at hand. Furthermore it is unclear if localization really derives from quantum mechanics. For one relativistic rotor it was shown in [18] that also the classical system is never chaotic in the ultra-relativistic regime. And one may further suspect that the same behaviour could hold for arbitrary many linear rotors, being the model integrable.

A new approach, which does not seem to rely on very specific details of the Hamilto-nian, is instead taken in the more recent work [19]. In the article, the authors introduce a new mapping useful to study dynamical localization. Since this method is one of the main tools of the present work, we will discuss it in some detail.

(33)

3.4. DYNAMICAL LOCALIZATION IN MANY-BODY SYSTEMS 29 The paper deals with a chain of coupled kicked rotors, described by the Hamiltonian

ˆ H(t) =X j ˆ p2_j 2 | {z } ˆ H0 + K   X j cos ˆϕˆj+ α cos( ˆϕˆj− ˆϕˆj+1)   | {z } ˆ H1 ∞ X n=−∞ δ(t − nτ ) (3.7)

To study dynamical localization in this system a new mapping, which we will refer to as U+ U†, is constructed. First of all, we write the Floquet operator which evolve the system from time −τ /2 to τ /2

ˆ U_FS = exp −iτHˆ0 2 ! exp−i ˆH1 exp −iτHˆ0 2 ! (3.8)

Notice that we have put ~ = 1 and we will use such units from now on.

To determine if the system is in a dynamically localized state, one needs to study the coefficients ˜φα(m) = hm| ˜φαi, where | ˜φαi is an eigenstate of ˆU_FS and |mi is defined as the eigenstate of ˆH0 which satisfies pˆj|mi = mj|mi for each j. In order to do so, we recall from Floquet theory that ˆU_FS can be written as P

αe−iτ µα| ˜φαi h ˜φα|. So, each Floquet state | ˜φαi will satisfy

ˆ U_FS| ˜φαi = e−iτ µα| ˜φαi ˆ_US F † | ˜φαi = e+iτ µα| ˜φαi (3.9)

To obtain the mapping we project both these equations on hm| and insert the identity decomposition_{1 =}P

m|mi hm| after every exponential of Eq. (3.8). Finally we sum the two equations to obtain the U + U† mapping

X

m0_6=m

Wm,m0φ˜_α(m0) + _mφ˜_α(m) = 2 cos(µ_ατ) ˜φ_α(m) (3.10)

where, denoting with H0(m) =P_jm2j/2,

Wm,m0 = e−iτ H0(m)+H0(m0) 2 D m e −i ˆH1 m 0E_{+ e}+iτH0(m)+H0(m 0₎ 2 D m e +i ˆH1 m 0E (3.11) m= 2< e−iτ H0(m) D m e −i ˆH1 m E (3.12) For a single kicked rotor, this mapping yields the same results as the standard one, with the only advantage that this new mapping does not display the unphysical divergences of the standard one. The great advantage of this mapping is that it can be extended to an arbitrary number of rotors, while the standard one cannot, as also pointed out in [17]. A further merit of the U + U† mapping is that, in principle, it can also be applied to other systems, thus opening up the possibility of investigating dynamical localization in new models. One of such possibilities is dynamical localization

(34)

Figure 3.1: Image adapted from [19]. Phase diagram of the kicked rotor model according to [19]. For L >2 rotors a mobility edge appears, and the mobility edge scales exponentially fast to 0 as L grows.

in the kicked Bose-Hubbard model, which is the topic of this Thesis.

In [19], the U + U† mapping allows the authors to make predictions about the localization properties of an L rotor chain. Indeed they notice that the Hilbert space of the L-rotors chain is given by the points in the lattice ZL, where each point m in the lattice corresponds to the state |mi previously defined. In this lattice they furthermore show that the hopping terms Wm,m0 decay exponentially as the distance |m − m0| is

increased. Hence, for L= 2, the Hilbert space is 2d and from Anderson scaling theory it is known that the system will always be localized. Instead for L ≥3, a mobility edge appears as a function of K. Indeed, also the numerics shows that for L= 3 there is a transition from a localized phase at low K to a delocalized one at high K.

Furthermore, the authors argue that in the thermodynamic limit the system will always be delocalized. In fact, applying Anderson scaling theory, they find that the critical value of K, above which the system is delocalized, should tend to0 exponentially fast as L → ∞. This claim, however is no longer supported by the numerics, since L= 3 is the maximum system size achievable. To overcome this obstacle, they numerically study the dynamics in a mean field approximation, which is argued to be exact in the thermodynamic limit if the rotors have infinite-range interactions instead of being in a 1d geometry. The mean field analysis, indeed, displays unbounded energy absorption as claimed before. The emerging picture is summarized in Fig. 3.1.

In the rest of the Thesis, we will present the original work done. We will study dynamical localization in the kicked Bose-Hubbard model, with a particular focus on how many-body effects alter dynamical localization.

Dynamics of the kicked Bose-Hubbard model

Universit`

a di Pisa

Dynamics of the kicked Bose-Hubbard model

Contents

Chapter 1

Introduction

Chapter 2

Dynamical localization

2.1

Classical kicked rotor

2.2

Quantum kicked rotor

2.3

Dynamical localization as Anderson localization in

en-ergy space

2.4

Experimental observations

Appendix

2.A

Floquet theory

2.B

Brief review of Anderson localization

Chapter 3

Many-body Floquet systems

3.1

Thermalization in quantum mechanics

3.2

The eigenstate thermalization hypothesis

3.3

Non-thermalizing systems

3.4

Dynamical localization in many-body systems