Relaxation and (pre)thermalization in the one dimensional Bose-Hubbard model

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Università di Pisa

Dipartimento di Fisica “Enrico Fermi”

Corso di Laurea Magistrale in Fisica

Relaxation and (pre)thermalization in the

one dimensional Bose-Hubbard model

Tesi di Laurea Magistrale

Candidato

Emanuele Tirrito

Relatori

Prof. Pasquale Calabrese

Dott. Mario Collura

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Preface

The present work may be inscribed in the general framework of Quantum Statistical Mechanics and aims to investigate some aspects of the dynamics of the many-body quantum systems. More specifically, it focuses on the out of equilibrium dynamics induced by a quantum quench. Indeed, one of the most common procedures to bring a system out of equilibrium is the so-called

quan-tum quench that consists in preparing the system in an eigenstate of a given

initial Hamiltonian and bring it out of equilibrium by suddenly changing the parameters of the Hamiltonian. By studying the time evolution of the system, it is thus possible to investigate both ergodicity and (pre)thermalization which were introduced in Classical Statistical Mechanics and then extended by Von Neumann [97] to Quantum Statistical Mechanics. Both from a theoretical and experimental point of view, it has been shown that those principles are well defined for non-integrable quantum systems and that the stationary states are described by the Gibbs ensemble. As regards integrable systems, instead, the experiment carried out by Kinoshita et al. [54] is of paramount importance be-cause they showed that both ergodicity and thermalization fail in such systems. Moreover, the systems reach a state of equilibrium which is not described by the Gibbs ensemble and this is the reason why the so-called Generalised Gibbs ensemble (GGE) has been introduced [78] in order to describe such stationary states. The GGE density matrix thus assumes the following form:

ˆρ = 1

Ze −P

αλαIˆα

where ˆIα are the integrals of motion while λα are the Langrange

multipli-ers which are fixed by the initial condition. In the context of these studies, this work deals with the numerical analysis of a specific quantum quench in a quantum chain whose dynamics is governed by the Bose-Hubbard Hamilto-nian. More specifically, we carried out some simulations through the TEBD algorithm starting from an initial state |ψ0iwhich describes a "diluite" infinite

chain.

This thesis is divided into four chapters.

The first chapter provides a general overview of the the recent theoretical developments about the equilibration of the isolated quantum systems. In particular, the attention is focused on the role of the integrability. In fact,

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it is widely believed that one-dimensional integrable systems relax towards a non-thermal distribution that is described by a generalized Gibbs ensemble. Finally, we provide some experimental results which are at the basis of these theoretical studies.

In the second chapter, we deal with the TEBD algorithm [95] in depth. Firstly, we introduce some basic tools, i.e. the Singular value decomposition and the Schmidt decomposition. Then we show how we can decompose a state in a MPS language. Finally, we explain in detail the recipe in order to simulate the time evolution of a MPS.

In the third chapter, the Bose Hubbard model [41] is analysed. In par-ticular, we provide a general overview of the properties of this model and we analyse the superfluid and the Mott-Insulator phase. Finally, we introduce the concept of quantum phase transition and we study the Bose-Hubbard phase transitions through the mean field approach that leads to a qualitative phase diagram.

The fourth chapter is the core of this thesis. We present the results of the simulations which we carried out for different interactions and filling factors. In the non-interacting and hard-core limit the system is integrable and its dynamics can be solved analytically. In these cases, the exact time evolution and the stationary value for every observable were found. Moreover, we argue that the stationary state is described by the GGE which, for the particular quench we considered, coincides with the Gibbs ensemble. Whenever we turn on the interactions, one breaks integrability and, for this reason, we may expect that the long time steady state of the system is thermal. Nevertheless, our simulations show that for very long times the local observables relax to quasi-stationary values which can be described by a "deformed" GGE (DGGE) [34] characterized by quasi-conserved charges whose explicit expression is provided. Only for very long times, which cannot be reached by our simulations, they are expected to relax to the genuine thermal steady state.

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Chapter

1

Theoretical Framework

In the last two decades the study of the quantum many-body systems has been the focus of renewed scientific interest. The reasons for this interest are several. As far as the theoretical aspect is concerned, while a complete framework for the description of the equilibrium properties of such systems exists (although often explicit analytical calculations cannot be obtained), re-garding the non-equilibrium properties, a general understanding is much less satisfactory.

From the experimental point of view, instead, thanks to the technical progress in the field of cold atomic gases, a large number of experiments have been carried out in order to study the dynamics of closed quantum systems. In particular, it has been possible to construct quasi one-dimensional optical lat-tices that are so weakly interacting with the environment that their evolution can be considered unitary.

Moreover, this field may be also particularly important for engineering application. Indeed, the non-equilibrium quantum physics will be definitely crucial for future technologies. A quantum computer, for example, will require the capability of performing real time manipulation of interacting quantum systems.

In particular, in this thesis we will focus our attention on the out-equilibrium properties of the Bose-Hubbard model (see Chapter 3) by using numerical tech-niques in order to understand the behaviour of some quantities which cannot be treated analytically.

There are many ways to take a system out of equilibrium, such as applying a driving field or pumping energy or particles in the system through external reservoirs as in transport problems. With respect to closed quantum systems, the so called "quantum quench" is one of the most widely used protocol for studying the non-equilibrium dynamics. In practice, the system is prepared in a pure state, e.g. the ground state of a given Hamiltonian H(g0); then, at time t= 0, the parameter is suddenly quenched from g0 to a different value g. The

initial state is not an equilibrium state of the post-quench Hamiltonian and, therefore, it evolves non-trivially. Notice that, depending on the properties of the quenched parameter, we can identify two different types of quenches: global

quenches and local quenches. In a local quench, the change in the Hamiltonian

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is localized (there is no energy pumped into the system in the thermodynamic limit), instead, in a global quench, a change in the Hamiltonian affects the whole system (there is excess in energy which appears as quasi-particles that propagate in time [12–14,32]).

Through the study of unitary time evolution of a quantum many-body system we try to address the following issues:

• Will the system after the quench approach a well-defined steady state? • Is this state described by a thermal state?

• And if it is not, which is the statistical ensemble describing it? May it depend on the integrability of the post-quench Hamiltonian?

In order to find a satisfactory answer to these questions several experiments have been carried out. In particular, see for example Kinoshita et al. [54] and of Trotzky et al. [91]. (For further details on these experiments see paragraph 1.4)

The physical picture emerging from these experiments reveals that the in-tegrability causes both the violation of the ergodicity and the breakdown of the thermalization thus leading to a kind of generalized thermalization described by a statistical ensemble which maximizes entropy: the generalized Gibbs

En-semble (GGE) [18, 75–78]. Nonetheless, some aspects about the construction

of a proper GGE that is able to describe the stationary state of a generic in-tegrable model (after a quantum quench) are still lacking [36,46,74,100].

In the next section we will deal with the extension of the concepts of ergod-icity and thermalization in quantum statistical mechanics. In particular, we will study the effects of the integrability on the dynamics following a quantum quench.

1.1 Quantum Statistical Mechanics

A closed quantum system is completely described by a state vector |ψi be-longing in a complex Hilbert space. Its dynamics is governed by the Schroedinger

equation:

ı~∂

∂t|ψ(t)i = ˆH|ψ(t)i (1.1)

where ˆH is the Hamiltonian operator. The initial state |ψ0i can be expanded

in the basis {|αi} of the eigenstates of Hamiltonian as follows: |ψ0i=

X

α

cα|αi (1.2)

where cα = hα|ψ0i. The solution of equation (1.1), with the initial condition

given by equation (1.2), is therefore: |ψ(t)i =X

α

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1.1. QUANTUM STATISTICAL MECHANICS 9 where Eα are the eigenvalues of ˆH. An exact description of the initial state

requires the knowledge of all the cα whose number equals the Hilbert space

dimension, which usually grows exponentially with the number of primary constituents of the many-body system. Since generally we do not have access to the exact initial state, we cannot construct a wave function for the entire system. When we deal with a lack of information it is useful to use the statis-tical approach. Therefore, the single state is replaced by an ensemble of states |Ψi_i _{with an associated classical probability ω}

i such thatPiωi = 1. Given an

observable ˆA, the ensemble average is defined as:

h ˆAi=X

i

ωihΨi| ˆA|Ψii (1.4)

where hΨi_{| ˆ}_A|_Ψi_i_{is the expectation value of the observable ˆ}_A _{in the state |Ψ}i_i_.

Then we define the density matrix ˆρ =P

iωi|ΨiihΨi| in such a way that:

h ˆAi= T r[ˆρ ˆA]. (1.5)

We can define different density matrices associated to different ensembles. In fact, for an isolated quantum system the natural choice for ˆρ would be the

microcanonical ensemble [53]: ˆρmc = 1 Zmc X α |αihα| (1.6)

where the summation is over all the eigenstates {|αi} of ˆH such that Eα is

inside a window ∆ = [E0− δ, E0 + δ] centred around the initial state energy E0 = hψ0| ˆH|ψ0i and Zmc = Pα1 is the microcanonical partition function,

where δ can be arbitrary small.

The canonical ensemble [53], instead, is built by considering an open por-tion of a large isolated system. In this case the density matrix is:

ˆρc =

1

Zc

e−β ˆH (1.7)

where Zc is the partition function defined as Zc = T re−β ˆH. In the

thermo-dynamic limit, therefore, the microcanonical and canonical averages should coincide. Consequently, we might define the canonical temperature of our isolated system to be such that: E0 = T r[ˆρcHˆ] .

Finally, we can define the grancanonical ensemble [53]. It allows us to study open quantum systems which are able to exchange particles with the surround-ing environment. Therefore, in this ensemble the density operator acts in the Hilbert space with an arbitrary number of particles and it is defined as follows:

ˆρ = 1

ZGC

e−β ˆH−µ ˆN (1.8)

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Figure 1.1: Graphical representation of interference patterns for different hold times described by Grainer et al. [48]. In their experiment they showed that the matter wave field on the Bose-Einstein condensate undergoes a periodic series of collapses and revivals. They considered a Bose gas formed by R_b atoms at low temperature confined by a three-dimensional optical lattice. The system is initially prepared in a superfluid phase, i.e. the ground state of BH Hamiltonian with J U . The term

U is suddenly quenched U J . The system is left evolving for several intervals and

then is released from the trap. After τ = 550µs the system reaches again its initial configuration.

1.2 Ergodicity

In this section we will provide a general overview of the concept of er-godicity which is one of the main ingredients to understand both relaxation to a steady state and thermalization. We start introducing the definition of classical ergodicity.

Following Ref. [72], let us consider an isolated system described by a Hamil-tonian H(p, q). The volume Γ spanned by all the possible configurations is known as phase space. Due to the fact that we are considering a many-body system, we do not know neither the position nor the momentum of all par-ticles. Therefore, there is a lack of information. Consequently, we need to use a statistical approach to study this system. Taking into account the fact that the system is isolated and that the energy is constant, the usual choice to describe the system is the microcanonical ensemble, that is:

ρmc(p, q) = 1

Zmc

θ(∆ − |H(p, q) − E0|). (1.9)

In this framework, the dynamical observable A(p, q) will be caracterized by an average over the ensemble:

hAimc= Z Γ dpdqρmcA= 1 Zmc Z SE dpdqA(p, q) (1.10)

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1.2. ERGODICITY 11 where Zmc is a microcanonical partition function and SE is the shell in the

phase space with the constant energy. Now we can define the concept of er-godicity [97] in the following way: an isolated system is ergodic if for any

observable A(p,q) and almost any initial state (q0, p0) A= lim t→∞ 1 t Z t 0 dt0A(p(t0), q(t0)) = hAimc. (1.11)

Now we try to extend the concept of ergodicity to quantum many-body systems [97]. We consider a non-degenerate Hamiltonian with eigenvalues Eα. We

select a microcanonical shell SE and choose as the initial state a superposition

of eigenstates in this energy shell i.e. |ψ0i=

X

α

cα|αi (1.12)

where |αi is in SE. The time averaged density matrix is given by the diagonal

From the definition of ergodicity we have that ˆρmc = ˆρdiag which implies that

|cα|2 are equal. This condition can be satisfied only for a very special class of

states. Quantum ergodicity is, therefore, almost never realizable. The key to understand ergodicity is by focusing our attention on the observables rather than on the states themselves. The idea is the following: let us consider a set of macroscopic course grained observables { ˆMα}which are commuting operators

that define a macroscopic state. A natural definition of quantum ergodicity is: lim

t→∞hψ(t)| ˆMα|ψ(t)i = T r[ ˆMαˆρmc] = h ˆMαimc (1.14)

for every initial condition. This limit does not exist in finite systems because of quantum revivals which have been observed for the first time by Grainer

et al [48] (see Fig. 1.1). Furthermore, once this equivalence holds in the long

time limit, ergodicity can be redefined using the time everage i.e.

hψ(t)| ˆMα|ψ(t)i = T r[ ˆMαˆρdiag] = h ˆMαimc (1.15)

If h ˆMαit relaxes to a well-defined value, this one will coincide with the time

averaged state. Therefore, ˆρmc can be considered as equivalent to ˆρdiag. As

in classical mechanics, the ergodicity may fail due to many factors such as integrability [19,75–78].

Moreover, the concept of ergodicity is essential to understand the thermal-ization of systems with integrability breaking. We say that an observable ˆA thermalizes with respect to a given ensemble if A= h ˆAi. For a non-integrable

Hamiltonian, generically we expect that the quantum microcanonical ensemble can be applied:

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Thermalization has also been explained through the Eigenstate

Thermaliza-tion Hypothesis(ETH) [9, 31, 85, 88]. The objects of this hypothesis are the

expectation values of observables on the basis of the eigenstates of the Hamil-tonian hα| ˆA|αi, and the statement is that for natural, physically interacting

observables these expectation values are smooth and quite flat functions of the energy with the possible exception of the extremes of the spectrum. If the initial state is peaked around a given energy, the Eigenstate Thermalization

Hypothesis implies the equivalence of the diagonal and the microcanonical

en-semble. If this happens ergodicity and thermalization are verified for every

initial condition.

To sum up briefly, we can state that the absence of integrability preserves both ergodicity and thermalization.

1.3 Integrability and The Generalized Gibbs

Ensemble

After having introduced the concepts of the ergodicity and the thermaliza-tion for non-integrable systems, in this paragraph we extend those concepts to integrable systems. In classical statistical mechanics, integrable systems have not an ergodic behaviour because they possess several integrals of motion and, therefore, they cannot explore the full hypersurface of constant energy. The problem of thermalization in integrable quantum systems has been stud-ied by Kinoshita et al. [54]. As we will see in paragraph 1.4, this experiment shows that integrable two-dimensional and three-dimensional systems thermal-ize while one-dimensional systems relax to a stationary state which cannot be described by the Gibbs ensemble. This is due to the fact that these systems are characterized by an infinite set of conserved charges which play a key role in the description of the stationary state. For this reason, we introduce the

Generalized Gibbs ensemble[75–78] that takes into account the extra integrals

of motion Iα i.e. : ˆρGGE = 1 Ze −P αλαIα (1.17)

where λα are Lagrange multipliers fixed by the initial state |ψ0i through the

condition:

hIαi= hψ0|Iα|ψ0i= T r[ˆρGGEIα] (1.18)

where {Iα} is a complete set of local conserved charges in evolution. In this

respect, it is worth mentioning the work of Rigol et al. [78], which deals with hard-core bosons (HCB) in one-dimensional lattice described by the following Hamiltonian: ˆ H = −J L X i=1 (ˆb† iˆbi+1+ h.c) (1.19) where ˆbiand ˆb †

i are the annihilation and creation operators and J is the hopping

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1.3. INTEGRABILITY AND THE GENERALIZED GIBBS ENSEMBLE13

Figure 1.2: Figure taken from [78] in which A is the graphical representation of the time evolution of the momentum distribution while B represents the quasi-momentum distribution after relaxation of N = 30 hard-core bosons undergoing a free expansion from an initial zero-temperature superlattice with period four of half-depth A = 8J and bound by a hard-wall box of size L = 600.

trasformation: ˆb† i = ˆc † i i−j Y j=1 e−ıπˆc†jcˆj _, ˆb i = i−j Y j=1 eıπˆc†jcˆjˆc i. (1.20)

If we rewrite the Hamiltonian (1.19) using (1.20) we find: ˆ H = −J L X i=0 (ˆc† iˆci+1+ h.c.) (1.21) where ˆci e ˆc †

i are fermionic annihilation and creation operators satisfying the

canonical anticommutation relation i.e. {ci, c

† j}= δij, {ci, cj}= 0 and {c † i, c † j}=

0. This system possesses as many conserved quantities as there are in lattice sites. They are the (quasi-)momentum distribution operators:

ˆIk = 1 L L X i=1 L X j=1 σi−j(N)e− ı L2πk(i−j)ˆc† jˆci (1.22)

where σ∆(N) = 1 for N odd and σ∆(N) = e−

ı

Lπ∆i for N even. The density

matrix reads:

ˆρGGE =

1

ZGGE

e−PkλkI(k)ˆ (1.23)

where ZGGE = T r[exp(−PkλkˆI(k)] = Qk(1 + e−λk). The values of

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Figure 1.3: Figure taken from Kinoshita et al. [54] Left: Schematic representation of the atomic bundles in the 1D harmonic trap. Right: Absorption images of an ensemble of 1D Bose gases.

Figure 1.4: Figure taken from Kinoshita et al. [54] showing momentum distribution for t > 1910τ , with τ oscillation period, and γ = 3.2 (on the left) and momentum distribution for t > 390τ , with τ oscillation period, and γ = 18 (on the right). The red curves are real data taken in a certain t_obs , the blue and green curves are simulations from different models that take into account the losses during the measure.

momentum distribution predicted by (1.23) be the same as the quasi-momentum distribution of fermions in the actual initial state of the system. This constraint leads to:

λk = ln1 − f(k)

f(k) , (1.24)

where f(k) = hψ0| ˆI(k)|ψ0i. The density matrix in (1.23), with condition in

(1.23), is assumed to predict correctly the values of local observables after a complete relaxation from an initial state with the fermionic (quasi-)momentum distribution f(k). This has been verified by numerical experiments on re-laxation. The system is prepared in the ground state of a spatially-periodic background-potential: Vext= A X i cos2πı L ˆb † iˆbi. (1.25)

As it can be seen in Figure 1.2, there is a perfect matching between the GGE predictions and the results from dynamical evolution, indeed the lines are com-pletely overlapped. Furthermore, the results show that even after a very long propagation time, the four characteristic peaks in the (quasi-)momentum

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dis-1.4. EXPERIMENTS 15

Figure 1.5: Taken from [54]. Momentum distribution in a three-dimensional Bose gas with t = 0, 2τ, 3τ, 4τ, 5τ where τ is the oscillation period. We can notice that thermalization occurs after a few periods.

tribution remain well resolved, although their shape is modified in the course of the propagation. The interpretation of the numerical results is the following: if the initial (quasi-)momentum distribution consists of several well-separated peaks, the memory of the initial distribution that is stored in the ensemble (1.23) prevents the peaks from overlapping, no matter how long the propaga-tion time is.

Other studies have dealt with integrable systems and have proved that after the quench they reach a stationary state which is described by the GGE. The most remarkable analytical works are reported in Refs [15–17, 37, 39, 40, 69, 87] . However, the vast majority of these studies refer to physical models that are either interacting or interacting but exactly equivalent to non-interacting ones, both before and after the quench. If we consider non-interacting integrable models, the derivation of the GGE predictions and the study of the time evolution are technically difficult problems that have been accomplished only in a small number of cases. For this reason, new methods have been introduced: Quench Action Method (GTBA) and Quantum Transfer Matrix Approach. The most important works on these methods, that we will not discuss here, can be found in references [5,11,17,21,38,73,86].

1.4 Experiments

In this paragraph we will present some relevant experimental evidences at the basis of the theoretical studies explained above. It is worth stressing that we will not deal with such experiments from a quantitative point of view but a more general outline will be, instead, provided. More specifically, we will briefly look at the following experiments:

• Quantum Newton’s cradle by Kinoshita et al. [54]

• Experiment on relaxation in an isolated strongly correlated one-dimensional Bose gas by Trotzky et al. [91]

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Figure 1.6: Figure taken from Trotzky et. al. [91] representing the three phases of the experiment: after having prepared the density wave |ψ(0)i (1), the lattice depth was rapidly reduced to enable tunnelling (2). Finally, the properties of the evolved state were read out after all tunnelling was again suppressed (3).

In their article A quantum Newton’s cradle, Kinoshita et al. studied the time evolution of non-equilibrium Bose gases in one dimension in order to demon-strate that they might not thermalize. The experiment starts with a Bose-Einstein condensate of Rb atoms in a bidimensional lattice. Then an array of one-dimensional lattices is built by using a red-detuned crossed dipole trap. The dynamics within each tube is strictly 1D because the lowest transverse ex-citation ~ωr exceeds all other energy scales and tunneling is negligible. Using

two shortly separated pulses the atoms in the tube are put in superposition of momentum states with p = ±2~k. Finally, the system is left evolving before removing the trap (Figure 1.3). In this way the system expands for 27ms. It is thus possible to determine the momentum distribution. After several peri-ods of oscillations the distributions are far from being gaussian meaning that there is no thermalization (see Figure 1.4). These distributions, in a certain sense, carry much information about the initial conditions. The same experi-ment has also been performed for 2D and 3D Bose systems and, in these cases, thermalization occurs after a few periods (Figure 1.5). Therefore, the spatial dimensionality plays a crucial role in the thermalization of isolated quantum systems.

The second experiment was carried out by Trotzky et al.. It focuses on the observation of relaxation dynamics in an interacting many-body system using ultracold atoms in an optical lattice. The experiment starts by loading a Bose-Einstein condensate of about 45×103_Rbatoms into a three-dimensional optical

lattice formed by retro-reflected laser beams of wavelength λxl = 1530nm

along one direction (long lattice) and λy,z = 844nm along the other two.

(The trap is set in such a way that in every chain there are about 43 atoms). Finally, they add to the long lattice another optical lattice with wavelength

λxs = 765nm = λxl/2 (short lattice) with the relative phase between the

two adjusted in order to load bosons every two short-lattice sites. Completely removing the long lattice gives an array of practically isolated one-dimensional state |ψi = |..., 0, 1, 0, 1, 0...i. Thereafter the dynamics is described by the Bose-Hubbard Hamiltonian: ˆ H =X i [−J(ˆb† iˆbi+1+ ˆb † i+1ˆbi) + U 2ni(ni−1) + K 2 nii2] (1.26)

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1.4. EXPERIMENTS 17

Figure 1.7: Figure taken from Trotzky et al. [91] showing the evolution of the average population of the odd sites after the quench. The dots are the experimental data, the blue lines are results from numerical simulations.

in which the parameter K = mω2_d2 (m is the particle mass, d the lattice

spacing) describes an external harmonic trap, present in the experiment, with trapping frequency ω = 2π × 61Hz. Moreover, everything is set in such a way that the tunneling probability is very low. The potential is suddenly changed and the tunneling between adjacent sites is activated, the system is left evolving for a certain time t. Thence the potential is restored and the tunneling is blocked; the average population on odd sites is then measured. This process is repeated for several values of t and for four values of U

J. Furthermore, in

Fig. (1.6) t-DMRG simulations (solid lines) are plotted. The Bose-Hubbard parameters used in these simulations were obtained from the respective sets of experimental control parameters. For times accessible to simulations, these averages differ only slightly from the traces obtained for a single chain with the maximum particle number Nmax = 43 of the ensemble. For interaction

strengths U

J <6, we find a good agreement of the experimental data with the

simulations. In this regime, only small systematic deviations can be observed and these are strongest for the smallest value of U

J corresponding to the smallest

lattice depth. This experiment was also verified through numerical simulations (for further details see Ref. [42]).

Other important experiments have been carried out to study the non-equilibrium dynamics of isolated systems which can be found in Refs. [23, 49,51,79].

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Chapter

2

The TEBD Algorithm

2.1 Introduction

The description of the physical properties of low-dimensional strongly cor-related quantum lattice systems has been one of the major tasks in theoretical condensed matter physics. The majority of these systems cannot be analyti-cally treated and, for this reason, several numerical methods have been devel-oped. Currently, one of the most powerful numerical methods in the study of one-dimensional quantum lattice models is the density-matrix renormalization group (DMRG) which was introduced in 1992 by Steven R. White [98]. It was originally formulated as a ground state method allowing to simulate the static properties of one-dimensional quantum lattice models. It consists of an iterative decimation procedure of the Hilbert space of a growing quantum sys-tem (a chain with increasing length). Therefore, we assume that there exists a reduced state space which can describe the relevant physics and that we can develop an algorithm to identify it.

How does this algorithm work? In a first step, we form left and right blocks (in a chain) A and B consisting of one spin each. Longer chains are now built iteratively from the left and right ends by inserting pairs of spins between the blocks such that the chain grows to length 4; at each step, previous spins are absorbed into the left and the right blocks such that block sizes grow as 1,2,3, and so on, leading to the exponential growth of the dimension of the full block state space as dl _{where l is the current block size and d is the Hilbert-space}

dimension of the single site. Our chains then have a block-site-site-block A◦◦B structure.

Let us assume that our numerical resources are sufficient to deal with a reduced block state space of dimension D, and that the blocks A and B can be described in D-dimensional reduced Hilbert space. The quantum state of the superblock A ◦ ◦B is:

|ψi=X

ij

Ψij|iiA◦|jiB◦. (2.1)

The basis dimension of the blocks A◦ and ◦B is dD. To avoid the exponen-tial growth of the dimension we truncate this block bases to D states. More

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Figure 2.1: Figure taken from [81] which represents schematically the DMRG pro-cedure for an infinite chain. New blocks are formed by integrating a site into a block, with a state space truncation according to the density-matrix prescription of DMRG. This growth happens on both sides of the chain, leading to chain growth.

specifically, we consider the reduced density matrices:

b

ρA◦ = T r◦B|ψihψ| ρb◦B = T rA◦|ψihψ| (2.2)

such that ρbA|ω

A

αi = λ2α|ωαAi and ρbB|ω

B

αi = λ2α|ωαBi and we choose as

reduced bases the eigenstates with the largest associated eigenvalues. Now it is possible to rewrite all desired operators on A◦ and ◦B into new bases. By numerical approximation we can find the |ψi that minimizes the energy:

E = hψ|HA◦◦B|ψi

hψ|ψi (2.3)

An important extension of the DMRG is the time dependent DMRG (or t-DMRG) which allows to simulate the time-evolution of strongly interacting many-body quantum systems [81, 82, 99]. However, in this thesis we will make use of another algorithm for the simulation of the time-evolution of one-dimensional quantum systems i.e. the so-called Time Evolving Block

Decima-tion(TEBD) which is based on the matrix product state (MPS) representation

of a many-body quantum system on a lattice [28,94,95]. Before analyzing the TEBD algorithm, in the next section we will explain some basic concepts, i.e. the Schmidt decomposition and the MPS.

2.2 Schmidt Decomposition and Sigular Value

Decomposition (SVD)

Throughout this chapter we will make use of one of the most important tools of the linear algebra i.e. the Singular Value Decomposition (SVD) which is at the basis of the Schmidt Decomposition. Let us briefly recall what they are. The Singular Value Decomposition is defined by the following theorem:

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2.2. SCHMIDT DECOMPOSITION AND SIGULAR VALUE DECOMPOSITION (SVD)21

Theorem. Let M be an m × n matrix, there exist a m × m unitary matrix U

a n × n unitary matrix V and a m × n positive diagonal matrix S such that:

M = USV†. (2.4)

This is the singular value decomposition of M, and S is the singular value of M. Note that the form of the decomposition implies that M has at least one and at most min(m, n) distinct singular values.

The most important property of the SVD is that the matrix Ml ₌Pl

k=1UikSkkV

†

kj

is the closest rank-l matrix to M, meaning that Ml _{minimizes the Frobenius}

norm P

ij|Mij− Mijl|2 (optimal approximation) [81].

The Schmidt Decomposition [70], instead, is defined by the following theo-rem:

Theorem. Let |ψi be a pure state in a Hilbert space HA ⊗ HB with total

dimension NANB. There exist vectors |aiA and |aiB and the scalar χ such

that: |ψi= χ X a=1 sa|aiA|aiB (2.5)

with 1 ≤ χ ≤ min(NA, NB) and s1 ≥ s2 ≥ . . . ≥ sχ ≥0 and Pαs2α = 1.

We shall provide a proof of this theorem by using the SVD in order to derive the Schmidt Decomposition of a generic quantum state. Any pure state |ψi on HA⊗ HB can be written:

|ψi=X

ij

Ψij|iiA|jiB (2.6)

where |iiA and |jiB are orthonormal bases of HA and HB with dimension

NA and NB respectively. If we carry out a SVD of matrix Ψ in the previous

where |aiA = PiUia|iiA and |aiB = PiVja?|jiB. Due to the orthonormality

properties of U and V†_{, the sets |ai}

A and |aiB are orthonormal and they can

be extended to be orthonormal bases of A and B. If we restrict the sum to run only over the χ ≤ min(NA, NB) positive non zero singular values, we obtain

the Schmidt Decomposition: |ψi=

χ

X

a=1

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If the total Hilbert space dimension is large, we could approximate |ψi by some | ˜ψi spanned over state spaces of A and B that have dimension χ0 only.

Therefore, the Schmidt Decomposition of the approximate state reads: | ˜ψi=

χ0

X

a=1

sa|aiA|aiB. (2.9)

This approximation introduces an error called Schmidt error [81,82,95]:

τ = 1 −

χ

X

a

s2_a. (2.10)

The Schmidt Decomposition allows to read off the reduced density operators for A and B. Carrying out the partial traces one finds:

b ρA = χ X a=1 s2_a|aiAAha| ρbB = χ X a=1 s2_a|aiBBha|. (2.11)

The eigenvalues of the reduced density matrix are the ωa= s2a. Therefore, we

can easly write the von Neumann entanglement entropy:

SA|B = −T rρbAlog2ρbA= −

r

X

a=1

s2_alog2s2a. (2.12)

2.3 Matrix Product States (MPS)

In this section we will deal with the decomposition of an arbitrary quantum state through the matrix product states [81, 82]. Let us consider a quantum chain with L lattice sites each of them with an associated local Hilbert space of dimension d. The most general pure quantum state on the lattice reads:

|ψi= X

σ1...σL

Cσ1...σL|σ1...σLi (2.13)

with |σji spanning the local Hilbert space at site j . First of all, we write the

components of the state |ψi into a matrix of dimension d × dL−1

Ψσ1,(σ2....σL) = Cσ1...σL. (2.14)

Then from a SVD of Ψ we obtain:

Cσ1...σL = Ψσ1,(σ2....σL) =Xr1 a1 Uσ1a1Sa1a1V † a1,(σ2...σL) =Xr1 a1 Uσ1a1Ca1σ2...σL (2.15)

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2.3. MATRIX PRODUCT STATES (MPS) 23 where the rank r1 ≤ d. We now decompose the matrix U into a collection of

d-row vectors Aσ1

a1 = Uσ1,a1. At the same time, we reshape Ca1σ2....σL into a

matrix Ψ(a1σ2),(σ3...σL) of dimension r1d × d L−2 _{to obtain:} Cσ1...σL = r1 X a1 Aσ1 a1Ψ(a1σ2),(σ3...σL). (2.16)

Through the SVD we find that:

Cσ1...σL = r1 X a1 r2 X a2 Aσ1 a1U(a1σ2),a2Sa2a2V † a2,(σ3...σL) =Xr1 a1 r2 X a2 Aσ1 a1A σ2 a1a2Ψ(a2σ3),(σ4...σL) (2.17)

where we have replaced U by a set of d-dimension matrices Aσ2 of dimension

r1× r2 with entries Aσa21a2 = U(a1σ2),a2 and multiplied S and V

† _{to be reshaped}

into a matrix Ψ of dimension r2d × dL−3 where r1 ≤ r1d ≤ d2. Finally, we can

write the components Cσ1...σ2 in the following way:

Cσ1...σL = X a1a2....aL−1 Aσ1 a1A σ2 a1a1...A σL−1 aL−2aL−1A σL aL−1. (2.18)

Therefore, the form of matrix product states of a quantum state is:

|ψi= X

σ1....σL

Aσ1_Aσ2_...AσL−1_AσL_|σ

1....σLi, (2.19)

and since U†_U _{= 1 we obtain that:}

X

σl

Aσl†_Aσl_. (2.20)

The dimensions of Aσj may maximally be (1×d), (d×d2)...(dL₂−1×dL₂)....(d2×

d), (d × 1) going from the first to the last site. This shows that, in practical

calculations, it will usually be impossible to carry out this exact decomposition explicitly, as the matrix dimensions blow up exponentially. This procedure is named Left-canonical matrix product state [81], since the decomposition starts from the left. Similarly, if we start from the right, we obtain that:

Cσ1...σL = Ψ(σ1...σL−1),σL = X aL−1 U(σ1...σL−1),aL−1SaL−1,aL−1V † (aL−1),σL = X aL−1 Ψ(σ1....σL−2),(σL−1aL−1)B σL aL−1 = Bσ1 a1B σ2 a1a2...B σL−1 aL−2aL−1B σL aL−1 (2.21)

and we obtain a MPS of the form:

|Ψi = X

σ1...σL

Bσ1_Bσ1_...BσL−1_BσL_|σ

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Figure 2.2: Graphical representation of an exact MPS starting from the left and right.

Since V†_V _{= 1 we obtain that:}

X

σl

Bσl_Bσl†= 1 (2.23)

This method is named Right-canonical matrix product state [81].

Both in the Left-canonical matrix product state and in the Right-canonical

matrix product state, if we split the lattice into parts A and B we find that:

|ψi=X

al

|aliA|aliB (2.24)

where for the Left-canonical matrix product state we have: |aliA= X σ1....σl (Aσ1_Aσ2_...Aσl−1_Aσl) 1,al|σ1....σli |aliB = X σl+1....σL (Aσl+1_Aσl+2_...AσL−1_AσL) al,1|σl+1....σLi (2.25) while for the Right-canonical matrix product state we obtain:

|aliA= X σ1....σl (Bσ1_Bσ2_...Bσl−1_Bσl) 1,al|σ1....σli |aliB = X σl+1....σL (Bσl+1_Bσl+2_...BσL−1_BσL) al,1|σl+1....σLi. (2.26) It is worth stressing that the state |ψi is not written in Schmidt Decomposition. The reason lies in the fact that for left MPS |aliA form an orthonormal set

while the |aliB is not orthonormal. In right MPS the situation is the opposite.

In order to find the Schmidt Decomposition (2.8) we divide the chain into two parts and we mix the decomposition of the state from the left and from the right where A runs from sites 1 to l and B from sites l + 1 to L (see Figure 2.2). Therefore, the Schmidt Decomposition can now be read off immediately:

|aliA= X σ1....σl (Aσ1_Aσ2_...Aσl−1_Aσl) 1,al|σ1....σli |aliB= X σl+1....σL (Bσl+1_Bσl+2_...BσL−1_BσL) al,1|σl+1....σLi sa= Sa,a. (2.27)

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2.4. MPS EXAMPLES 25

Figure 2.3: Blocks A and B put toghether at bond l

Then the state takes the form (2.8).

Furthermore, putting together a chain from a left block A and a right block B (see Figure 2.3) we can form a general superposition:

|ψi= X

al,a0l

Ψal,a0_l|aliA|a

0

liB. (2.28)

Inserting the state esplicitly we find:

|ψi=X

σ

Aσ1_...AσlΨBσl+1_....BσL_|σi. (2.29)

Finally, if we multiply the Ψ-matrix into one of the adjacent A or B matrices we obtain the exact MPS:

|ψi=X

σ

Mσ1_...MσL_|σi. (2.30)

2.4 MPS Examples

Here we give some examples of many-body quantum states which can be studied through an exact MPS [44,81].

1) AKLT (Affleck-Kennedy-Lieb-Tasaki) state [1,2]: the father of all matrix product state is the ground state of AKLT-Hamiltonian:

ˆ H =X i Si· Si+1+ 1 3(Si· Si+1)2 (2.31)

where S is the vector of spin-1 operators. The ground state of this Hamiltonian can be constructed by replacing each individual 1 through a pair of spin-1/2 which are completely symmetrized, i.e.

|+i = | ↑↑i

|0i = | ↑↓i√+ | ↓↑i 2 |−i= | ↓↓i.

On neighbouring sites, adjacent pairs of spin-1/2 are linked in a singlet state: | ↑↓i − | ↓↑i

√

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Therefore, to write the MPS of the ground state we must represent both the singlet bound and the triplet states on each site. The state with singlets on every bound reads:

|ψΣi=

X

ab

Σb1a2Σb2a3. . .ΣbL−1aLΣbLa1|abi (2.32)

with |ai = |a1. . . aLi and |bi = |b1. . . bLi representing the first and second

spin-1/2 on each site and

Σ = 0 1 √ 2 −_√1 2 0 ! . (2.33)

We now encode the symmetrized state of the two auxiliary spins-1/2. To this end, we introduce Mσ

ab|σihab|with |abi and |σi representing the auxiliary spins

and the physical spin on site i. Writing Mσ

ab as three 2 × 2 matrices, one for

each value of σi, with rows and column indices standing for the values of |ai and |bi, we find:

M+ = 1 0 0 0 ! M0 = 0 1 √ 2 1 √ 2 0 ! M− = 0 0 0 1 ! . (2.34)

The mapping on the spin-1 chain Hilbert space {|σi} then reads:

X σ X ab Mσ1 a1b1M σ2 a2b2. . . M σL aLbL. (2.35)

Substituting in the (2.32) we obtain the state which describes our chain:

|ψi=X σ X ab Mσ1 a1b1Σb1a2M σ1 a2b2Σb2a3. . .Σb_L−1aLM σL aLbLΣbLa1|σi. (2.36)

We introduce Aσ _{= M}σ_{Σ such that:}

A+= 0 1 √ 2 0 0 ! A0 = − 1 2 0 0 +1 2 ! A− = 0 0 −_√1 2 0 ! . (2.37)

The AKLT state now takes the form:

|ψi=X

σ

T r(Aσ1_Aσ2_{. . . A}σL)|σi. (2.38)

To find a MPS for AKLT state we have to renormalize matrices Aσ _because

P

σ(Aσ)†Aσ = 34I which implies that we rescale by 2 √

3.

2) Majumdar-Ghosh [64, 65] : the Majumdar-Ghosh model is an exten-sion of one-dimenexten-sional quantum Heisenberg spin model in which an extra interaction is added coupling first and second neighbour spins. The Hamilto-nian of this model is:

ˆ

H =X

i

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2.5. VIDAL’S DECOMPOSITION 27 The ground state is a superposition of two periodic states given by products of singlets on neighbouring sites. The equal weight superposition of these states is translational invariant and has a MPS representation:

A0 =    0 1 0 0 0 −1 0 0 0    A 1 ₌    0 0 0 1 0 0 0 1 0   . (2.40)

3) GHZ (Greenberger-Horne-Zeilinger) state [47]: this state is a superposition of {|0i} and {|1i} describing a system formed by N particles.

|ψi= | + + + . . . + ++i + | − − − . . . − −−i. (2.41) This state can be written through a MPS representation where the matrices

A are defined as follows:

A+ = 1 + σz A−= 1 − σz. (2.42)

4) Cluster state [10,44]: the Cluster states are the unique ground states of the three-body interactions:

ˆ

HI =

X

i

σ_izσx_i+1σ_i+2z . (2.43)

They can be represented by:

A0 = 0 0 1 1 ! A1 = 1 −1 0 0 ! . (2.44)

2.5 Vidal’s Decomposition

In this section we will introduce Vidal’s notation [71,81,94,95] that allows us to write the state |ψi as a sum over products of local tensors Γ[l] _{and local}

vectors Λ[l]:

|ψi= X

σ1....σL

Γ[σ1]Λ[1]Γ[σ2]Λ[2]Γ[σ3]Λ[3]_....Γ[σL−1]Λ[L−1]Γ[σL]_|σ

1....σLi. (2.45)

We introduce on each site l a set of d matrices Γ[σl] and on each bond l one

diagonal matrix Λ[l]_{. Such matrices are defined so that they fulfill the Schmidt} Decomposition, where the Schmidt coefficients are the diagonal of Λ[l]_{, s}

al = Λ

[l]

and the states A and B are given as: |aliA = X σ1....σl Γ[σ1]Λ[1]Γ[σ2]_....Λ[l−1]Γ[σl]_|σ 1....σli |aliB = X σl+1....σL Γ[σl+1]Λ[l+1]Γ[σl+2]_....Λ[L−1]Γ[σL]_|σ l+1....σLi. (2.46)

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Figure 2.4: Representation of a MPS in Vidal’s notation.

Figure 2.5: This figure represents the overlap hφ|ψi formed by contraction of physical indices σk. More specifically, the triangles represent the MPS matrices Mσk.

The graphic representation of a MPS in Vidal’s notation is provided in Fig. (2.4).

Vidal’s decomposition is a more explicit version of the A and B matrix notation with the advantage of keeping explicit reference to the singular values. Reduced density operators ˆρA and ˆρB in eigenbasis representation read: ρ[l]A =

ρ[l]_B = Λ[l]2. Now let us show that any quantum state can be brought into that

form. Starting from coefficients Cσ1...σL we reshape to Ψσ1,(σ2....σl) through the

SVD: Cσ1....σL = Ψσ1,(σ2...σL) = X a1a2a3 Γ[σ1]Λ[1]Γ[σ2]Λ[2]Γ[σ3]Ψ (a3σ4),(σ5....σL). (2.47)

The relation between the decomposition into left-renormalization A matrices and the Vidal’s decomposition is:

Aσl al−1,al = Λ [l−1] al−1,al−1Γ σl al−1,al (2.48)

which generates a set of orthonormal states on the part of the lattice ranging from site 1 to l. Similarly, starting the decomposition from the right by us-ing the right normalization of B matrices, the same state is obtained with a grouping: Bσl al−1,al = Γ σl al−1,alΛ [l] al,al (2.49)

we can convert Vidal’s notation in left and right MPS notation. If one intro-duces an additional scalar Λ[0] as a matrix to the very left |ψi we can use the

equations (2.48) (2.49):

(Λ[0]_Γ[σ1])(Λ[1]Γ[σ2])(Λ[2]Γ[σ3]).... =⇒ Aσ1_Aσ2_Aσ3_... (2.50)

(Γ[σ1]Λ[1])(Γ[σ2]Λ[2])(Γ[σ3]Λ[3]).... =⇒ Bσ1_Bσ2_Bσ3_... (2.51)

If we consider a mixed conversion we obtain that:

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2.6. EXPECTATION VALUES 29

Figure 2.6: This Figure shows the MPS representation of density matrix ˆρ = |ψihψ|

of the system with PBC.

Finally, if we divide the chain into two parts we will have left-normalized A matrices on the left and right-normalized B matrices on the right. Then if we multiply them we obtain the orthonormal block bases for A and B, hence a

Schmidt Decomposition.

2.6 Expectation Values

Following Refs [81, 93], to study correlated systems we need to compute the overlap hφ|ψi between different states and n-point correlations having the following form:

hψ| Ô1⊗ Ô1⊗ Ô2⊗ Ô3· · · ˆOL|ψi. (2.53)

Here we will demonstrate that all these quantities can be efficiently computed whenever |ψi and |φi are MPS. For convenience the description given for the rest of this section uses MPS with PBC (periodic boundary condition). The modification to OBC (open boundary condition) involves replacing the T r(·) operation with an appropriate scalar-product to boundary states. As regards the overlap hφ|ψi, we consider |φi and |ψi described by matrices M and ˜M:

|φi=X σ ˜ Mσ1_{· · · ˜}_MσL_|σi _|ψi=X σ Mσ1_{· · · M}σL_|σi. (2.54)

The overlap assumes the following form:

hφ|ψi=X

σ

˜

MσL†_{· · · ˜}_Mσ1†_Mσ1_{· · · M}σL_. (2.55)

We have contractions over the matrix implicit indices in the matrix multipli-cations and over the physical indices. Grouping the sum we find that:

hφ|ψi=X σL ˜ MσL†(. . . (X σ2 ˜ Mσ2†(X σ1 ˜ Mσ1†_Mσ1)Mσ2) · · · )MσL_. (2.56)

Carrying out the contraction the complexity is polynomial with total operation count O(Lχ3_d).

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Figure 2.7: The graphic represents the trace on all sites except for the site k. It is equivalent to contracting all physical indices. The uncontracted indices σ_k and σ_k0 are then the rows and columns of the density matrix ˆρk.

Now we consider the correlation expressed in the (2.53). The full density operator ˆρ = |ψihψ| of the system (see Fig. (2.6)) is given by:

ˆρ =X σ X σ0 T r[ L Y l=1 Mσl]T r[ L Y l=1 Mσl?]|σihσ0_|. (2.57)

We combine the two traces over the separate matrix products as a single trace over the Kronecker tensor product of these matrix products:

T r(X)(Y ) = T r(X ⊗ Y ) = T r(Y ⊗ X), (2.58) so we obtain that: ˆρ =X σ X σ0 T r[ L Y l=1 Eσlσ_l0]|σihσ0_| (2.59)

where Eσlσ0l = Mσl ⊗ Mσ0l. The normalization of the state is obtained by

tracing out all the physical sites:

T r( ˆρ)) = hψ|ψi = T r[ L Y l=1 Il] = T r[ L Y l=1 d X σl=1 Eσlσ_l0]. (2.60)

The matrix Eσlσ0l is called transfer matrix [81, 93]. By leaving out one site k

from this trace we obtain the reduced density operator: ˆρk = X σk X σ0_k T r{[ k−1 Y l=1 Il]Eσkσ0k[ L Y l0_=k+1 Il0]}|σkihσ0k| (2.61)

which is graphically represented in Fig. (2.7). It is possible to calculate the reduced density operators of larger numbers of sites which will be a trace of products of sets of Im _{and E}σlσ0l.

Now we can calculate the expectation value (2.53): hφ| Ô1⊗ Ô2⊗ Ô3· · · Ô1|ψi= T r[

L

Y

l=1

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2.6. EXPECTATION VALUES 31

Figure 2.8: This figure represents the correlation h ˆOlOˆl+ki formed by the contraction

of indices.

where the observables ˆOl is defined as:

ˆ Ol = d X σ0 l=1 d X σl=1 hσ_l0|Ol|σliEl σlσ 0 l. (2.63)

The calculations above consist of additionally performing L multiplications of these matrices requiring O(χ6_L_{) operations. These calculations have only}

polynomial scalings and, therefore, they remain efficient with increasing L . In fact, the naive approach outlined can be improved to give O(dχ5_L_{) and} O(dχ3_L_{) performance for PBC and OBC respectively.}

We can write these objects in the Vidal’s notation. In fact, if we find the expectation value h ˆOki, we decompose the state to isolate the single site |σli:

|ψi= X σlαl−1αl Λ[l] αl−1Γ σl αl−1αlΛ [l+1] αl |φ [1...l+1] αl−1 i|σli|φ [l+1....L] αl i. (2.64)

Tracing ˆρ = |ψihψ| on all sites except k we obtain: (ˆρl)σl,σ0 = X αl−1αl λ[l]_α l−1(Γ σl αl−1αl) ?_λ[l+1] αl λ [l] αl−1(Γ σl αl−1αl)λ [l+1] αl . (2.65)

As regards the expectation value of a two-site observable ˆB = ˆOk⊗ ˆOl at sites

k and l, we need to calculate ˆρkl which is the reduced density matrix obtained

by tracing over all sites but k and l. To calculate the two-site reduced density matrix we first assume, without loss of generality, that l < k so that we can write our Vidal decomposed state as:

|ψi= X σl...σk X αl−1...αk λ[l]_αl−1Γσl αl−1αlλ [l+1] αl ....λ [k] αk−1Γ σk αk−1αkλ [k+1] αk |φ[1...(l−1)]_α_l−1 i|σl...σki|φ[(k+1)...L]αk i. (2.66)

If we substitute the (2.66) in ˆρ = |ψihψ| we find:

|ψihψ|= X σl...σkσl0...σ 0 k X αl...αkα0l...α 0 k (λ[l] αl−1Γ σl αl−1αlλ [l+1] αl . . . λ [k] αk−1Γ σk αk−1αkλ [l+1] αl ) . . . (λ[l] α0_l−1(Γ σ0_l α0_l−1α0_l) ?_λ[l+1] α0_l . . . λ σ_k0 αk−1(Γ σ_k0 α0_k−1αk) ?_λ[l+1] α0_l )

|φ[1...(1−l)]_α_l−1 i|σl. . . σki|φ[(k+1)...L]αk ihφ

[1...(l−1)] α0 l−1 |hσ 0 l. . . σ 0 k|hφ [1...(l−1)] α0 l−1 | (2.67)

tracing over all sites except l k and using the orthonormality of the Schmidt vectors we obtain ˆρlk.

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Figure 2.9: Representation of an MPS in Vidal’s notation.

2.7 Time Evolution

Following Ref. [95], we consider an infinite array of sites in one dimension where each site l is described by a complex space of finite dimension d. Let vector |ψi denote a pure state of the lattice and operator ˆHa Hamiltonian with

nearest-neighbor interactions where we assume that they are invariant under shifts by one lattice site. Given an initial state |ψ0i, our goal is to simulate a

time-evolution according to H :

|ψti= e−ı ˆHt|ψ0i. (2.68)

In the TEBD algorithm |ψi is represented through Vidal’s decomposition as in (2.45) (see Fig. (2.9)) and the evolution operator is expanded through a Suzuki-Trotter decomposition [89, 90]. More specifically, we assume that the Hamiltonian can be written as a direct sum of "local Hamiltonians":

ˆ

H = ˆHodd+ ˆHeven (2.69)

where ˆHeven = Peven lHˆl and ˆHodd = Podd lHˆl. Our time evolution operator

which evolves the state |ψi in a time interval δt thus becomes: ˆ

U(δt) = e−ıδt(Podd lHˆl+

P

even lHˆl). (2.70)

We write this as a sequence of two-sites operators bearing in mind that ˆHl and

ˆ

Hl+1 do not commute and, consequently, the time evolution operator does not

factorize into two-site evolution. Therefore, we approximate ˆU(δt) through the

Suzuki-Trotter expansion: e−ı ˆHδt = e−ı2Hˆoddδte−ı ˆHevenδte− ı 2Hˆoddδt (2.71) where: e−2ıHˆoddδt= Y odd l e−2ıHˆlδt= ˆ_UAB e−ı ˆHevenδt= Y even l e−ı ˆHlδt= ˆ_UBA_. (2.72)

We perform a half time-step propagation over odd sites and a full time step propagation over even sites, and then another half time-step propagation over odd sites to complete a full time-step over the whole system. The time evolu-tion operator is thus described as a sequence of two-site operaevolu-tions. Moreover, since we had initially assumed that |ψi is shift invariant, it could be repre-sented with a MPS where Γ[l] and λ[l] are independent of l. However, we will

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2.7. TIME EVOLUTION 33 only partially break translational symmetry to simulate the action of gate ˆUAB

and ˆUBA on |ψi and we choose a MPS of the form:

Γ[2l] _{= Γ}A_, _Γ[2l+1] _{= Γ}B_, _Λ[2l] _{= Λ}A_, _Λ[2l+1]_{= Λ}B_. _(2.73)

The simulation of the time evolution is achieved by updating the MPS so as to account for the repeated application of gates ˆUAB _{and ˆ}_UAB _{on |ψi. The action}

of the gates preserves the shift invariance then ΓA_, _ΓB_, _ΛA_, _ΛB _{need to}

be updated.

Now we will analyse how to update the MPS in detail. To this end we consider the action of the gates U on each pair of sites [2l ,2l+1] (see Fig. (2.10)). ˆ U = X σlσl+1σ0lσ 0 l+1 Uσlσl+1 σ0_lσ0_l+1|σlσl+1ihσ 0 lσ 0 l+1| (2.74)

we write the initial state as a bipartite splitting between sites l and l+1.

|ψi= X αl−1αl+1;σlσl+1 Θσlσl+1 αl−1αl+1|φ [1....l−1] αl−1 i|σlσl+1i|φ l+2....L αl+1 i (2.75)

where the operator Θ is defined as: Θσlσl+1 αl−1αl+1 = X αl Λ[l] αl−1Γ [σl] αl−1αlΛ [l+1] αl Γ [σl+1] αlαl+1Λ [l+2] αl+1 (2.76)

with this definition we can apply the gate U

U |ψi= X αl−1αl+1;σlσl+1 ˜Θσlσl+1 αl−1αl+1|φ [1....l−1] αl−1 i|σlσl+1i|φ l+2....L αl+1 i (2.77)

where ˜Θ is defined as: ˜Θσlσl+1 αl−1αl+1 = X σ0 lσ 0 l+1 Uσlσl+1 σ_l0σ0_l+1Θ σ0 lσ 0 l+1 αl−1αl+1 = X αl Λ[l] αl−1˜Γ [σl] αl−1αl˜Λ [l+1] αl ˜Γ [σl+1] αlαl+1Λ [l+2] αl+1. (2.78)

By normalizing ˜Θ and performing a SVD of ˜Θ we can keep only the χ largest singular values and compute the updated tensor Γ:

˜Λ[l+1] αl = Sαl q P aSα2 (2.79) ˜Γ[σl] αl−1αl = U(σl−1)χ+αl−1,αl Λ[l] αl−1 ˜Γ[σl+1] αlαl+1 = V_{(σl+1−1)χ+αl,αl} Λ[l+1]_αl+1 . (2.80)

Therefore, we have found a new MPS for the state UAB_|ψi_{. Finally, if we}

apply UAB _{to |ψi and then apply again U}BA_{, through the procedure explained}

above, we can find the evolution of |ψi (see Fig. (2.10)). It is worth noticing that discarding χ(d − 1) of the dχ eigenvalues of ˜Θ leads to a truncation error which is called Schmidt error [51,95]:

τ = 1 − χ X α (Λ[l+1] α )2. (2.81)

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Figure 2.10: This Figure represents the updating of the MPS after the application of gate U. We first contract the tensor network 1) into a single tensor Θ 2). 3) We then compute the SVD and we obtain the matrix X and Y and the vector ˜λA. 4) We introduce again λ_B in the network. 5) Attaching to X and Y the inverse of the Schmidt decomposition λB _{we find the new tensor network ˜}_ΓA _{and ˜}_ΓB_.

Note that such an error is not the only one but we have to take also into account the error caused by the truncation of the Suzuki-Trotter decomposition [45,94]. In particular, performing a time evolution of interval δt we commit an error:

εδt∼(δt)p+1 (2.82)

where p is the order of expansion.

Through the TEBD algorithm it is also possible to simulate the imaginary time propagation: |ψτi= e−τ ˆH|ψ0i ke−τ ˆH_|ψ 0ik . (2.83)

This allows us to find the ground state of the system. This operation, unlike the real time evolution, takes states to non canonical form [71,83]. Imaginary time propagation involves non-unitary operations on the tensor network, and, for this reason causes non orthogonality in Γ bases. Now we are going to show how to write in canonical form a state which is not in canonical form:

|ψi=X

α

|φA_αi|φB_αi (2.84)

to restore it to a canonical form we first form the matrix MA_{, which has the}

following elements: M_α,αA 0 = hφA_α|φA_α0i= X γ (ΓσlA γα ) ?_λ2_ΓσlA γα . (2.85)

We find an orthogonal matrix X and a vector of values S from performing an SVD on this matrix MA _−→SV D _{U SV} _{with X}

αβ =

q

SαUαβ? . The new set of

vectors: |eτi= X α X_{τ α}† |φA αi (2.86)

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2.7. TIME EVOLUTION 35 form an orthonormal set. A similar procedure for B leads to a matrix Y defined as MB _−→SV D _{with Y}

αβ = λαUαβ

q

Sβ and the orthonormal set:

|fηi=

X

α

Yηα|φBαi. (2.87)

The state may now be written as:

|ψi=X

τ η

(XT

Y)τ η|eτi|fηi (2.88)

which may be brought to canonical form using the SVD in the usual way: (XT_Y_{) =}X k ˜ΓA ik˜λk˜ΓBkj =⇒ |ψi = X k ˜λk| ˜φAki| ˜φ B ki (2.89) where: | ˜φAi=X k ˜ΓA ik|iAi, | ˜φBki= X k ˜ΓB jk|jBi. (2.90)

This procedure allows to preserve the canonical form of the state |ψi when carrying out a non-unitary operation on it.

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Relaxation and (pre)thermalization in the one dimensional Bose-Hubbard model

Università di Pisa

Dipartimento di Fisica “Enrico Fermi”

Corso di Laurea Magistrale in Fisica

Relaxation and (pre)thermalization in the

one dimensional Bose-Hubbard model

Tesi di Laurea Magistrale

Candidato

Emanuele Tirrito

Relatori

Prof. Pasquale Calabrese

Dott. Mario Collura

Contents

Preface

Chapter

1

Theoretical Framework

1.1

Quantum Statistical Mechanics

1.2

Ergodicity

1.3

Integrability and The Generalized Gibbs

Ensemble

1.4

Experiments

Chapter

2

The TEBD Algorithm

2.1

Introduction

2.2

Schmidt Decomposition and Sigular Value

Decomposition (SVD)

2.3

Matrix Product States (MPS)

2.4

MPS Examples

2.5

Vidal’s Decomposition

2.6

Expectation Values

2.7

Time Evolution