Phase diagram of a fully frustrated Bose-Hubbard diamond chain

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Introduction

Cold atomic systems have always attracted a lot of interest because of the rich variety of phenomena they display. One of the most interesting features is the observation of Bose-Einstein condensation (BEC) which appears when a gas of weakly interact-ing bosons is cooled below a critical temperature: at the onset of condensation a macroscopic fraction of the particles occupies the same single-particle state, finally leading to a macroscopic many-body wave function. Although the existence of this state of matter was predicted nearly one century ago, its experimental observation was achieved only recently and it contributed to improve the interest towards this phenomenon, which is nowadays a largely studied topic.

Thanks to the development of advanced experimental techniques, it has been pos-sible to realize cold atomic gases featuring properties and phases even more exotic than BECs; for example, the possibility to confine neutral atoms in artificial lattice potentials, named optical lattices, allowed to move from weakly interacting atomic gases (like BECs) to systems where particles interact strongly, the so-called ultracold strongly correlated gases.

Optical lattices exploit the dipole interaction occurring between neutral atoms and light, described by a potential proportional to the light intensity; letting two counter-propagating lasers interfere, the light intensity attains a spatial periodic pat-tern which leads to a periodic potential, i.e. a lattice potential. Optical lattices allow to implement different models and to finely tune their parameters acting on the laser configuration and intensity.

An ultracold bosonic gas in an optical lattice can be described effectively by a Bose-Hubbard (BH) model whose Hamiltonian is composed by two competing terms: a kinetic and an on-site interaction one. The former describes particles hop-ping between adjacent sites and tends to delocalize them over the lattice , the latter favors a state where particles are well localized on each site.

An interesting class of BH models is that describing frustrated lattices, that is systems where, due to a subtle interplay between both the geometry of the lattice and the form of the Hamiltonian, all the kinetic terms cannot be minimized simultane-ously. This constraint is due to the presence of suitable complex phase factors in the

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2

hopping terms which are physically induced by the action of a magnetic field on a system of charged particles. In this case indeed, the gauge invariance of the Hamilto-nian requires the presence of ad hoc hopping phases which finally lead to frustration (if the magnetic field satisfies few constraints). Actually, when dealing with optical lattices, since they are loaded with neutral particles, the effects of a magnetic field on charged particles and the presence of a gauge field in general (not only for the frustrated case) have to be artificially simulated, inducing the hopping phases either optically by means of a laser or loading the gas on a rotating lattice. Since the action of a magnetic field on charged particles is a very general and largely studied topic, it looks natural to refer to it when dealing with frustrated BH models, although one would better speak of pseudo magnetic fields since, experimentally, in optical lattices their action is “only” effectively simulated. Thus, keeping in mind that in experiments one deals with pseudo magnetic fields only, in the remainder of this thesis frustration is discussed referring to charged particles embedded in a magnetic field.

In this thesis we focus on fully frustrated lattices, i.e. systems where, as long as no interactions take place, the magnetic field is set to a value such that the energy spectrum has completely flat single-particle bands corresponding to single-particle eigenstates with a strongly limited extension over the lattice. The mechanism which leads to this strong localization can be explained in terms of the Aharonov-Bohm (AB) effect and these states are named AB cages: because of the presence of the magnetic field, each path that a particle can move along is weighted with a phase depending on the circuitation of the potential vector along that path; in a fully frus-trated system the particular geometry of the lattice leads to a destructive interference between paths linking different sites, finally letting the amplitude of probability for a particle to move over the lattice be zero outside a limited portion of it.

Contrary to what happens for an unfrustrated BH model, in the fully frustrated case interactions tend to delocalize particles, i.e. destroy cage states; furthermore since the AB cage states are no more the exact single-particle eigenstates, the descrip-tion of the physics displayed by the model becomes a difficult task to tackle. Many fully frustrated lattices with different geometries have been studied adopting various approaches and remarkable results were found like, for example, the appearance of a Bosonic quasi condensate of pairs of Cooper pairs (rather than Cooper pairs). Fully frustrated BH models are then a playground where a rich variety of phenomena may appear and, thanks to their strict connection with optical lattices, offer the concrete possibility to be experimentally realized.

This thesis is devoted to the study of a BH model describing a system of charged spinless particles at zero temperature in a fully frustrated quasi-one-dimensional dia-mond chain which, despite of its simple geometry, allows the realization of AB cage

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3

states. Rather than the BH model describing this system, we study a simpler e ffec-tive model obtained applying the so-called lowest band approximation: we assume that interactions are weak enough to let all the particles be in the single-particle low-est band state, i.e. the lowlow-est band AB cage state, finally allowing us to project the original Hamiltonian over the subspace generated by them.

The first chapter is a review about the main properties of ultracold gases in op-tical lattices and BH models as well, focusing in particular on the strict connection between them: the BH Hamiltonian is explicitly derived from the Hamiltonian de-scribing a cold gas in an optical lattice and the phases it describes are discussed; finally, the results of a prominent experiment where the BH physics was studied are presented.

In the second chapter fully frustrated systems are introduced showing the exis-tence of the AB cages in two lattices, namely the two-dimensional T3lattice and the quasi-one-dimensional diamond chain, when no interactions take place. The effects of their onset is then described recalling some of the results already known in the literature. Among the possible strategies adopted to study these systems, we discuss the lowest band approximation, focusing on its physical meaning and the effective models it leads to.

In the third chapter we present the original part of my thesis which deals with the study of the fully frustrated diamond chain in the lowest band approximation. The effective Hamiltonian for such a system is derived and it is made of a pair hopping term, a nearest neighbour and an on-site interaction term. Their magnitude can be tuned varying their couplings and we are going to study two instances, namely when on-site interactions dominate and when they are comparable to nearest neighbour interactions.

Our analysis is numerical and it is performed applying both the Exact Diagonal-ization (ED) and the Density-Matrix RenormalDiagonal-ization Group (DMRG) algorithm: the former is an exact method but it can investigate small-size chains only, while the lat-ter is an approximated approach able to simulate larger sizes. Comparing the results obtained from both methods we are finally able to characterize the phases of the effec-tive model. We discover that at semi-integer fillings ν = 1/2, 1, 3/2 incompressible phases are displayed (i.e. the filling is constant when varying the chemical potential in a finite range of values), whereas at 0 < ν < 1/2 the phases have infinite compress-ibility (i.e. a slight variation of the chemical potential lets the filling increase from 0 to 1/2). We focus on ν = 1/2, 1, discovering that the properties featured by our system are prominently affected by the competition between nearest neighbour and on-site interactions. Indeed, at ν = 1/2 occupied (by one particle) and empty sites alternate, i.e. a charge-density wave (CDW) phase minimizing both the interactions

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4

appears. At ν= 1 particles are distributed uniformly along the chain forming a Mott phase if on-site interactions dominate, whereas a CDW of pairs (i.e. doubly occupied and empty sites alternate) realizes when their strength is comparable to that of nearest neighbour interactions. The case ν = 3/2 is investigated with ED only and we can only infer it is incompressible.

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Chapter 1

Bose-Hubbard model

This chapter is an introduction to the physics of the BH model, focusing on its con-nections with ultracold strongly correlated gases.

The chapter is organized as follows: in section 1.1 we present some basic theoret-ical aspects of ultracold gases in opttheoret-ical lattices and we explicitly derive an effective Hamiltonian (BH Hamiltonian) able to describe them. In section 1.2 we first discuss the BH phase diagram by means of a mean-field approach and then we discuss briefly some problematic features of the one-dimensional case. Finally, the results of a sem-inal experiment which detected the quantum phase transition proper of the BH model are summarized in section 1.3.

1.1 Ultracold strongly correlated atomic gases

In the past years the improvements of experimental techniques to cool and trap atoms were at the base of the intense investigation of many-body quantum physics with (ultra)cold gases. In particular, optical lattices allowed to study cold bosonic and fermionic systems with very diverse properties and to establish important connections to other research fields like condensed matter, statistical mechanics and quantum information [1–4].

1.1.1 Cold gases and optical lattices

Cold gases at low temperature develop peculiar phenomena related to the quantum nature of their elementary constituents, which may be both bosons and fermions. In the former case, one of the most spectacular effects is the BEC, characterized by the presence of a macroscopic number of bosons that below a critical temperature occupy the same single-particle state [5]. This feature makes the BEC an unusual state of matter: considering the limiting case of a noninteracting gas at zero temperature, all particles are described by the same single-particle wave function and the many-body

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6 1. Bose-Hubbard model

wave function is just the product of them (in this sense correlations can be neglected) and the gas “behaves” as a system described by a unique macroscopic wave func-tion. A distinctive consequence of this property is the spatial correlation of the phase of the bosonic system, which in many experiments leads to detectable interference patterns after the release of the gas from the trap. For the case of a noninteracting three-dimensional gas in a box potential, BEC sets in when ρ1/3λth> 2.612 [6], with ρ the particle density and λth=

q 2π~2

MkBT the thermal wavelength (M is the boson mass,

kBis the Boltzmann constant and T the temperature). This inequality shows that con-densation realizes whenever the thermal wavelength becomes comparable or greater than ρ−1/3, i.e. the average interatomic distance. Thus both a low temperature and a sufficiently high density are needed, two requirements hardly achievable at the same time since increasing the density, collisions are enhanced, the temperature rises up and loss effects may take place. BEC of an atomic gas was experimentally realized1 for the first time in 1995 [8, 9] by the groups of Wieman, Cornell and Ketterle (later awarded with the Nobel prize in 2001).

At low temperature the number of physically relevant quantities involved in the description of a gas reduces. For example, as long as the particles have a sufficiently low kinetic energy (estimable as kBT), the presence of a centrifugal barrier for non-zero values of the angular momentum freezes out high angular momentum collisions in two-body scattering events, allowing to restrict the analysis of the scattering pro-cesses to s-wave collisions only. The length scale relative to interactions is then the scattering length aS of the s-wave channel and it is the most relevant quantity related to interactions. When this workframe applies (this usually happens in the sub-millikelvin regime) a gas is named ultracold [2]. For these systems two energy scales come into play: the interaction and the kinetic one. Assuming that, in addition to aS, the interaction energy Eintdepends on the density ρ, one can estimate with di-mensional argument that Eint ≈ ρ~

2_a S

M (a rigorous derivation of this result is reported in Ref. [2]); the kinetic energy is instead Ekin≈ ~

2

2Mρ

2/3_{. Eventually the ratio between} the two energy scales is thus Eint

Ekin ≈ρ

1/3_a S.

In BECs usually produced in laboratories, particles interact weakly, that is Eint

Ekin

is small or, equivalently, aS ρ−1/3. Nevertheless, increasing ρ or aS (or both of them) the system exits the weakly interacting regime (and the BEC phase as well) and enters a strongly interacting one. Unfortunately, although both the possibilities can be experimentally performed, increasing ρ the gas may become a liquid while increasing aS may shorten strongly the lifetime of the condensate2, two effects which

1_{A schematic review with various references about the main attempts historically performed to}

ob-tain BEC is reported in the introduction of Ref. [7]

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1.1 Ultracold strongly correlated atomic gases 7

reduce the stability of the system.

A more natural way to reach the strongly interacting regime is to employ optical lattice potentials. They are generated exploiting the interaction between the light-induced dipole moment of neutral atoms and the laser light [10]: atoms exposed to an electric field E(ωL) attain an (oscillating) electric dipole-moment p(ωL). The frequency of the laser ωL is set to values far from atomic resonance (far-detuned light) to avoid absorption phenomena that may cause dissipative and recoil effects (e.g. spontaneous emission). The interaction is then conservative. In the complex notation E(r, t)= êE(r)e−iωLt+ c.c. and p(r, t) = êp(r)e−iωLt+ c.c. where ê is the unit

polarization vector and the amplitude p(r) of the dipole moment is simply related to the field amplitude E(r) by

p(r)= α(ωL)E(r), (1.1)

where α(ωL) is the complex atomic polarizability. A potential Udipis then generated

Udip(r)= − 1 2hp · Ei= − 1 20c Re(α)I(r) (1.2)

where the brackets indicate a time averaging over one period of the laser light and the field intensity is I(r)= 20c|E(r)|2(0is the vacuum permittivity). The above formula (1.2) shows the main concept upon which optical dipole trapping relies: a neutral atom irradiated with (laser) light feels a potential which follows the spatial modula-tion of the laser field intensity I(r). Thus a lattice3can be formed by overlapping two counter-propagating laser beams tuned at the same frequency which create a standing wave whose intensity I(r) has a periodic pattern with period λL/2 (λL= 2π_ωc

L is the

wavelength of the laser) and its maxima coincide with the minima of Udip(r) if the frequency ωLis smaller than the atomic resonance frequency (red-detuning), with the maxima if it is larger (blue-detuning).

Optical lattices are very versatile: by interfering several laser beams, one-, two-or three-dimensional lattices potentials can be obtained; their geometrical configura-tion is under the control of the experimentalist upon acting on the angle between the interfering laser beams; finally, the depth of the potential wells can be varied acting on the intensity of the laser. Thus, the ratio between the kinetic and the interaction energy can be modified : increasing the laser intensity the wells are deepened, tunnel-ing between sites is inhibited and particles occupytunnel-ing the same site interact4strongly; therefore, the kinetic energy is reduced while the interaction energy is enhanced (the opposite situation realizes if the intensity of the laser is reduced). The opportunity to vary the ratio between these two energy scales just by acting on the optical potentials

3_{Further details required for a realistic description of dipole traps are presented in Ref. [10].} 4_{Dealing with neutral atoms no Coulomb long-range interaction can take place.}

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allows then to enter the strongly interacting regime. Within this framework correla-tions play a prominent role (oppositely to BEC, the many-body wave function is no more the product of single particle wave functions) and characterize the phases. This is the reason why these systems are referred to as ultracold strongly correlated gases.

1.1.2 Effective description of a bosonic cold gas in an optical lattice: Bose-Hubbard model

In 1998 Jaksch et al. [11] showed that the physics of bosonic optical lattices is cap-tured by the BH model. The starting point is the following Hamiltonian for bosonic atoms in a lattice potential5:

H= Z d3rψ†(r) − ~ 2 2M∇ 2_{+ V} 0(r) ! ψ(r) + 1 2 4π~2as M Z d3rψ†(r)ψ†(r)ψ(r)ψ(r), (1.3) where M is the mass of a boson, asis the s-wave scattering length, ψ†(r) (ψ(r)) is a bosonic quantum field in second quantization which creates (annihilates) a boson at position r satisfying the following commutation rules:

• hψ(r), ψ†_(r0₎_{i = δ(r − r}0_), • hψ†_{(r), ψ}†_(r0₎_{i = 0,} • ψ(r), ψ(r0₎= 0.

The lattice potential V0(r) with depth V0 at position r = (r1, r2, r3), wave vector k₀= 2π_λ and a lattice period l= λ/2 is

V0(r)= V0 3 X

i=1

sin2(k0ri). (1.4)

The natural energy scale of this system is derived from the lattice potential periodicity contained in k0 and it is the recoil energy: Er = (~2k2₀)/(2M). All the quantities of interest can be expressed as a function of V0and Er(in particular of their ratio). The last term of Eq. (1.3) is derived from the general form of the two-particle interaction term in second quantization formalism

1 2

Z

d3r d3r0ψ†(r) ψ†(r0) v(r0, r) ψ(r0) ψ(r) (1.5)

for the case of a contact-like pseudo-potential

v(r0, r) = 4π~ 2_a s 2M δ(r 0 − r). (1.6)

5_{For the sake of simplicity, we will omit any additional trapping potential, e.g. the harmonic}

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1.1 Ultracold strongly correlated atomic gases 9

This choice is valid only if two assumptions are verified: • scattering events involve only two-body processes;

• the kinetic energy EKis smaller than the centrifugal barrier, letting thus s-wave scattering length be the main parameter related to interactions [2], see section 1.1.1.

In presence of a periodic potential like (1.4) and with periodic boundary con-ditions, the single-particle Schrödinger equation leads to an energy spectrum with a band structure En(k) and corresponding Bloch eigenfunctions Ψn,k(r) where k is the quasi-momentum within the first Brillouin zone and n is an index which labels the energy bands [12]. Nevertheless the single-particle problem can be tackled us-ing an orthonormal set of states named Wannier functions [13], related to the Bloch eigenstates via Fourier transformation:

wn,Ri(r)=

X

k

e−ik·RiΨ

n,k(r), (1.7)

where Riis the position vector pointing the i-th site of the lattice. Unlike Bloch states, Wannier functions are not eigenstates of the single-particle Hamiltonian. However, they fit particularly well to the purpose of describing atoms in a lattice because of their exponentially localized profile centered over the lattice site at position R. In order to exploit this feature, we expand the bosonic operator ψ†(r) appearing in Eq. (1.3) over a set of Wannier functions. Let us consider now the case in which only the lowest band of the Bloch spectrum is populated by atoms, being the energy gap with other bands large enough (in comparison with the energy scale of interactions and temperature) to prevent atoms to populate excited levels. Thus the expansion of ψ†_{(r) includes only Wannier functions relative to the lowest band (for this reason we} will omit the band index n). In short, after defining canonical bosonic annihilation and creation operators bi and b†_i which destroy and create respectively a particle on site i and obey [bi, b

†

j]= δi j, the expansion follows immediately ψ(r) =X

i

biwi(r − Ri). (1.8)

Adopting Eq. (1.8), the starting Hamiltonian (1.3) reduces to the effective one

H= −X i, j Ji, jb†_ibj+ 1 2 X i Uni(ni− 1), (1.9)

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where ni = b†_ibiand the coupling constants are explicitly

Ji, j= − Z d3r w∗_i(r − Ri) −~ 2 2m∇ 2_{+ V} 0(r) ! wj(r − Rj), (1.10) and U = (4π~ 2_a s) M Z |w(r)|4d3r. (1.11)

Since Wannier functions are localized, Eq. (1.10) gives a non negligible Ji, j ≡ J only for nearest neighbouring sites and Eq. (1.9) eventually becomes the BH model describing nearest neighbour hopping and on-site interactions with J and U the ki-netic and interaction coupling constants, respectively. Looking at (1.9),(1.10) and (1.11), we can infer that “the dynamics of an ultracold dilute gas of bosonic atoms in an optical lattice can be described by a BH model where the system parameters are controlled by laser light” [11]. In particular, the stronger the optical potential is, the lower is J, the bigger is U; indeed, qualitatively, Wannier functions get more and more localized, the overlap between functions centered in adjacent sites decreases, eventually lowering the probability of tunneling from one site to another whereas enhancing on-site interactions. Thus, being J/U the ratio of the kinetic and the in-teraction energy, the system can be moved from the weak to the strongly correlated regime acting on laser intensities.

1.2 Properties of the Bose-Hubbard model

The BH model was first introduced by Fisher et al. [14] as a model able to describe “the behaviour of bosons with short-repulsive interaction moving in an external peri-odic potential”[14]. In order to study the system using the grancanonical ensemble, the chemical potential µ is introduced by means of an additional term −µP

ini. Thus we have H= −J X <i, j> b†_ibj+ U 2 X i ni(ni− 1) − µ X i ni, (1.12)

where < i, j > denotes that the summation is performed over nearest neighbouring sites. In what follows we name the kinetic and the interaction term Hhop and Hint, respectively, and we consider the case U > 0, i.e. a repulsive interaction. The BH Hamiltonian is invariant under a global U(1) phase transformation

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1.2 Properties of the Bose-Hubbard model 11

and the conserved charge is the total number of particles operator Ntot = Piniwhich satisfies

[H, Ntot]= 0. (1.14)

Hhop and Hint are two competing terms, the former favoring a state where par-ticles are delocalized all over the lattice, the latter privileging a configuration with a well defined number of particles in each site, i.e. a state where particles are well localized. This is readily understood looking at the ground state of the system in the two limits J = 0 or U = 0.

In the former case the Hamiltonian becomes diagonal over a basis made up by the product of local Fock states with miparticles in the i-th site, that is

|m1, . . . mLi= L Y i=1        b†mi i √ mi!       |0i, (1.15)

where |0i is the vacuum state, miruns over all non-negative integer and L is the total number of sites. Moreover, being H(J = 0) the sum of equal single-site terms, the total energy E is the summation over all sites of the on-site energy

(m) = −µm +U

2m(m − 1), (1.16)

where we dropped the site index i because our system is translationally invariant and homogeneous. Once the ratio µ/U is fixed, to minimize the total energy it is sufficient to minimize (m). Since for m − 1 < µ/U < m this minimum occurs when exactly m bosons occupy each site, the ground state6is

|ΨN=m·Li(J = 0) = L Y i=1        b†m_i √ m!       |0i, (1.17)

where we considered a system having N = L · m particles. Thus the filling is com-mensurate(i.e. integer valued)

hnii ≡ hni= m, (1.18)

fluctuations are suppressed (hn2_ii ≡ hn2i= hni2) and

hb†_ii ≡ 0. (1.19)

Let_Uµ be within the interval (m−1, m) and let us parametrize it as µ(α)= (m+α)U with α ∈ (−1, 0). The on-site energy is then (m, α) = −µ(α)m + U₂m(m − 1). This

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means that the energy cost to add7 or remove a particle on a site is finite, being δEp(J = 0) = (m + 1, α) − (m, α) = −αU the cost for the former process and δEh(J= 0) = (m − 1, α) − (m, α) = (α + 1)U for the latter; in particular, involving a pair particle-hole creation with an energy cost δEph = δEp+ δEh, hopping has a finite cost as well and, like the addition or removal of a particle, is inhibited.

In the opposite limit U = 0, H reduces to Hhop which does not commute with ni, implying that the local occupation number is not integer valued and it fluctuates strongly. Nevertheless Hhopis diagonal over a basis made up by the product of local Fock states in the momentum space (assuming periodic boundary conditions):

− J X <i, j> (b†_ibj+ h.c.) = − X k (k)b† kbk (1.20)

where (k)= 2J Pd_i₌₁cos(kil) , d is the dimension of the system, k = (k1, . . . , kd) is the quasi-momentum which lies in the first Brillouin zone, b†_kand bkare the Fourier transformed operators of b†_i and bi respectively and l is the lattice spacing. The ground state is such that each boson is in the zero-momentum k = 0 single-particle state [2]: |ΨNi(U = 0) = 1 √ N! b†_k₌₀N|0i= √1 N!        1 √ L X i b†_i        N |0i, (1.21)

implying that the one-particle density matrix displays off-diagonal long-range order8 [5]. In fact, this is an ideal BEC (at least for d-dimensional systems with d > 1) and the many-body ground state (1.21) explicitly displays the delocalizing nature of this phase.

1.2.1 Phase diagram of the BH model

Mean-field approach

It is known from the work of Fisher et al. [14] that, in the thermodynamic limit, the transition between the two limiting phases at U = 0 and J = 0 described previously occurs by means of a second-order phase transition related to the spontaneous sym-metry breaking of the U(1) symsym-metry of the model. This can be seen by means of a mean-field approach of which we give a brief description focusing on the case of a d-dimensional homogeneous ipercubic lattice with L sites at T = 0 . For the moment we consider d > 1. When dealing with the BH model, mean-field method relies on the existence of an order parameter which, when the ratios J/U and/or µ/U change,

7_{In (m}_{+1, α) the value of µ(α) is still (m+α)U even after the addition of one particle to the system.} 8_{For one-dimensional systems it is more correct to say o}_{ff-diagonal quasi long-range order.}

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varies smoothly between a zero value when the system is in a symmetric phase (like the one at J = 0) and a non-zero value when in a spontaneously broken one, like the ideal BEC described by the ground state (1.21); in fact, the order parameter is hb†_ii (the averaging is meant to be performed over the ground state). Let us perform on Hamiltonian (1.12) the following substitutions:

bi → hbi+ ∆b, b†_i → hb†i+ ∆b†,

(1.22)

where∆b = bi− hbi,∆b†= b†_i − hb†i and the site index i is dropped since our system is homogeneous. Keeping only the terms up to the first power in ∆b and ∆b† and neglecting the second order terms, we obtain the mean-field Hamiltonian

HMF = L X i=1 −zJ(hbib†_i + hb†ibi)+ U 2ni(ni− 1) − µni , (1.23)

where z = 2d is the coordination number of our ipercubic lattice. We observe that the mean-field hopping term breaks explicitly the symmetry of the original model “allowing” the appearance of broken symmetry phases. By means of a self-consistent equation for hbii it is possible to compute the values of the parameter (U, J, µ) (or (J/U, µ/U)) such that hb†_ii approaches to zero, i.e. the boundaries of each phase. The self-consistent equation is hbi= T rhe−βHMF_b i i T re−βHMF (1.24)

where β = 1/kBT and assuming we are in proximity of the phase transition, hbii will take small values, thus allowing a power expansion of the right-hand term in Eq. (1.24). At the quantum phase transition point then Eq. (1.24) gives us the boundaries of the two phases and the resulting phase diagram is shown in Fig. 1.1.

The Mott-insulator phase

The most evident feature of the phase diagram is the presence of the Mott lobes [14] which enclose the so-called Mott-insulator phase. At J = 0, indeed, a Mott-insulator phase realizes and the properties we deduced in that limiting case extend to the region of the lobes where J > 0 with slight modifications: the exact ground state is no more the one of Eq. (1.17) and small (in comparison to hnii) fluctuations of the on-site number particle appears (hn2_ii , hnii2). Nevertheless, since the U(1) symmetry of the BH model is exactly preserved, within each lobe separately the total

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0

1

2

3 J

_U

Μ

U

SF

MI

_Ν=3

MI

_Ν=2

MI

_Ν=1

Figure 1.1: BH phase diagram obtained with a mean field approach. Mott lobes are colored in blu and, starting from the bottom, lobes at filling ν = 1, 2, 3 are shown. The remaining part of the phase diagram is occupied by the superfluid phase.

number of particles is conserved9and in particular the filling ν= N/L has a constant value independent of µ. This last feature lets the Mott insulator be an incompressible phase, being the compressibility K defined as

K = ∂ν

∂µ. (1.25)

Let us observe that since the filling is commensurate at J= 0, it is commensurate all over the lobe and, being the system homogeneous, hnii= ν. Physically, the features of the Mott-insulating phase follow from the fact that on-site interactions are so strong that the most energetically convenient configuration is the one which minimizes them and, consequently, the local occupation of every site. For this reason, in the ground state, sites are homogeneously occupied and hopping is inhibited since otherwise a configuration energetically more expensive, i.e. an “excited” state, would realize. The analysis we made in section (1.2) in the J = 0 case confirm quantitatively this analysis and approximately apply even to the region of the lobes where 0 < J U: the interaction energy cost relative to the addition of a particle or a hole is still approximately δEp(J= 0) or δEh(J= 0), respectively, and as long as it is larger than

9_{The Mott phase ground states are eigenstates of the operator N; to ensure that the ground state at}

J > 0 and at J = 0 have the same eigenvalue, a non-zero energy gap between states with different eigenvalues is needed in order to let the ground state “move adiabatically” when J increases without “crossing” with eigenstates carrying a different eigenvalue; the Mott phase is gapped and this argument works.

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the gain in kinetic energy, roughly J, these processes are disadvantaged and hopping is suppressed, although not completely inhibited. However, whenever the gain in kinetic energy overcomes the interaction energy cost, the phase is no more a Mott-insulator. This is actually the reason why the Mott phase has a lobe-like structure in the phase diagram: starting at a point inside one of the Mott lobes and increasing µ keeping fixed J, one will eventually reach an upper value µ(+)_Mott(J) such that the kinetic energy gained adding an extra-particle will balance the interaction energy cost. An analogous situation realizes for an extra-hole at a lower value µ(−)_Mott(J). Finally, since the kinetic energy increases with J, the interval (µ(−)_Mott(J), µ(_Mott+) (J)) which delimits the width in µ of the Mott phase shrinks as J increases; at a critical value Jcwe eventually have µ(−)_Mott(Jc)= µ(+)_Mott(Jc) and the lobe closes. Being µ(+)_Mott(J) and µ(−)_Mott(J) the values at which the energetic balance is in favor of the addition or removal of a particle from the system respectively, the energy gap δEph(J) is then δEph(J)= µ(_Mott+) (J) − µ(−)_Mott(J).

Finally, we mention that in the Mott phase the correlation function hb†_ibji decays exponentially:

hb†_ibji ∝ e−|i− j|/ξ, (1.26)

where ξ is the correlation length; as we are going to explain in section 1.3, this behaviour is exploited to distinguish experimentally the Mott-insulator from the su-perfluid phase.

The superfluid phase

Outside the lobes, a superfluid phase (hbi , 0) with off-diagonal long-range order is displayed, that is

hb†_ibji

|i− j|→∞

−−−−−−→Λ₀ (1.27)

where the constantΛ0is different from zero being the U(1) symmetry of the system broken.

The distinctive feature of the superfluid phase is that it is gapless, i.e. the en-ergy cost to create a pair particle-hole in a hopping process is infinitesimal in the thermodynamic limit; thus the hopping is the dominant process and, unlike for the Mott-insulator, particles delocalize all over the lattice, eventually letting hnii fluctuate significantly.

Looking at the phase diagram displayed in Fig. 1.1, superfluidity occurs at J U and, if µ is in a specific and limited range of values (which shrinks as J lowers), at arbitrary small J. In other words, recalling that the chemical potential is related to the average total number of particles by means of hNtoti= −∂E0/∂µ where E0= E0(J, µ) is the ground energy of the system, this means that as long as the kinetic energy

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Figure 1.2: BH phase diagram obtained with a mean field approach for a cubic lattice. The gray dotted lines are the constant-filling lines referring to commensurate fillings ν = 1, 2, 3 and incommensurate filling ν = 1 + . In this figure, rather than with ν, commensurate and incommensurate fillings are labeled with ¯n and n, respectively. Lowering J/U, the lines at integer fillings in the superfluid phase hit at a critical value (J/U)c(depending on the filling) the corresponding Mott-insulator phases at the tips of the lobes. The line at incommensurate filling 1+ instead stays outside of the Mott lobe at every J/U since the corresponding phase is always superfluid. Reproduction from Ref. [2]

gained thanks to hopping overcomes the interactions cost (J U) superfluid occurs at every filling; on the other hand, as J lowers, below a critical value (depending on µ or equivalently on the filling), the superfluidity lasts only within a finite interval of values of the filling. Indeed, an investigation of the BH phase diagram performed fixing the average filling (see Fig. 1.2) leads to the following results [2, 14]:

• commensurate fillings: when J/U 1, particles are delocalized and, although the probability to find more than one particle in the same site is not negli-gible, interactions are too weak to prevent them from hopping. Nevertheless, diminishing J/U such delocalized configuration becomes more and more ener-getically expensive because of the interactions and a quantum phase transition toward the Mott phase occurs at a critical value (J/U)c (depending on the fill-ing); this value is the same at which the lobe closes and the phase transition is in the universality class of the (d+ 1) XY model;

• incommensurate (i.e. non-integer) filling: only a superfluid phase can realize and thus, even at arbitrary small J, no transition towards a Mott-insulator

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re-1.2 Properties of the Bose-Hubbard model 17

alizes. Physically, this is because in a system with an incommensurate filling, when increasing U, a commensurate number of particles tends to establish a Mott-like phase while the extra-particles are always “free” to hop since all con-figurations which differ only for their location have the same energy. Indeed on the phase diagram, for fillings slightly greater (less) than a commensurate one, as J decreases the constant-filling contours lie slightly above (below) the Mott phase relative to the “closest” commensurate filling and at J = 0 reaches the point where µ/U assumes an integer value, say m; indeed, recalling Eq. (1.16), at J = 0 and µ/U = m the on-site energy for a local occupation number equal to m or m − 1 is the same; then no energy “barrier” prevents the addition of extra-particles, finally leading to superfluidity. Actually, the existence of con-figurations which differ only for the total number of particles but have the same energy is a general property of the superfluid phase related to the spontaneous breaking of the U(1) symmetry of the model.

1.2.2 One-dimensional case

Some properties of the BH model are affected by dimensionality. In particular, the quantity hbi which in high dimensional systems plays the role of order parameter, is identically zero in both phases in the one-dimensional case. Here the mean-field approach fails completely.

A widely used approach able to study the one-dimensional BH model is the Density-Matrix Renormalization Group (DMRG) [15, 16]. DMRG is a numerical method able to describe successfully strongly correlated systems at d = 1 requir-ing rather modest computational efforts [17]. Its basic principles are described in Appendix B.

To study the BH phase diagram by means of the DMRG [18–21], one typically considers the canonical ensemble where the total number of particles N is fixed. Therefore, the lower and upper boundary values µ(−)_Mott(N, J) and µ(_Mott+) (N, J) respec-tively of the chemical potential enclosing the Mott phase at (integer) filling ν= N/L (L is the lattice size) can be computed by means of the following relations:

• µ(−)

Mott(N, J)= E0(N, J) − E0(N − 1, J); • µ(+)

Mott(N, J)= E0(N+ 1, J) − E0(N, J)

where E0(N, J) is the ground energy for a system with N particles at a certain J, a quantity which can be computed with high precision by the DMRG. Scanning di ffer-ent values of J then the Mott lobe boundaries in µ are obtained.

The one-dimensional BH phase diagram obtained with the DMRG is displayed in Fig. 1.3. In comparison to the higher dimensional case, Mott lobes are still present

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Figure 1.3: One-dimensional BH phase diagram obtained using DMRG. Notice that in this plot the hopping coupling constant is named t instead of J and the filling ρ rather than ν. Mott lobes have a sharper shape than the one obtained for higher dimensional systems with mean-field. Moreover now lobes close with a tip while according to mean-field theory they close smoothly. Reproduction from Ref. [18].

although having a sharper shape and a pinned tip. The inner phase is still a Mott-insulator while the outer one is no more superfluid since hbi ≡ 0 and the two point correlation function of Eq. (1.27) does not display off-diagonal long-range order, as Λ0= 0. The characterizing feature of the outer phase is indeed the power-law decay of hb†_ibji with the distance |i − j| (instead in the Mott phase they decay exponentially) between the i-th and the j-th site and one speaks of off-diagonal quasi long-range order and of quasi condensate.

Other methods of investigation of the phase diagram of the Bose-Hubbard model Other approaches (both numerical and analytical) rather than the mean field and the DMRG have been performed in literature to investigate the BH model:

• Gutzwiller ansatz: the system is studied using an ansatz for its many-body wave function made up of a product-over-sites form [22, 23]. Conceptually is analogous to the mean-field approach, that is it assumes a separable Hamilto-nian (i.e. made up of the summation of single-site terms like the mean-field Hamiltonian) simpler than the original one can describe effectively the system. It fails when dealing with one-dimensional systems;

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1.3 Experimental detection of the Superfluid-Mott transition 19

interactions, that is when they are non-neglegible but a macroscopic portion of particles still forms a BEC ;

• strong coupling expansion: it studies the BH model in the limit of strong inter-actions when the system is in a Mott-insulator phase [24, 25]. It considers the effects of the hopping term by means of a perturbative expansion in J/U of the BH Hamiltonian. It works properly in any dimension;

• Exact Diagonalization approach: it aims to compute numerically all the quan-tities of interest (ground states, correlation functions) from the explicit form of the BH Hamiltonian of the considered system. It suffers greatly the large dimension typically attained by the Hilbert space of the system;

• Quantum Monte Carlo: it is a largely used numerical method able to study the BH model (and other models) in large lattices. It works properly in any dimension.

For a list of additional references about the above topics see Ref. [1].

1.3 Experimental detection of the Superfluid-Mott

transi-tion

We conclude this chapter briefly describing the experiment performed by Greiner et al. [26] in 2002 which paved the way towards the experimental investigation of the BH model and other strongly correlated systems as well.

Taking inspiration from the work of Jaksch et al. [11], Greiner et al. [26] used an ultracold atomic gas in a three-dimensional optical lattice to find an experimental signature of BH physics: the appearance of a Mott insulator or a superfluid phase and the phase transition between them when varying the ratio U/J. Their experiment was performed using a 87RbBEC created in a magnetic trap. In order to form a three-dimensional lattice potential they ramped up three standing waves orthogonal to each other with their crossing point positioned at the center of the BEC. The optical setup was such that the lattice had a simple cubic geometry. The lattice potential was raised up slowly in order to ensure that the condensate remained always in the ground state and atoms populated the lowest Bloch band only: in this way all the assumptions made by Jaksch et al. in their work [11] were satisfied10 and the realized system was described by the BH model. Increasing or lowering the lattice potential it was possible to increase or lower the ratio U/J inducing a Mott or a superfluid phase, respectively.

10_{Having populated the Bloch lowest band only allowed us in section (1.1.2) to expand the bosonic}

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Figure 1.4: Interference pattern of an ultracold gas loaded in a three-dimensional optical lattices observed by Greiner et al. [26] for different values of U/J. From panel (a) to panel (h) U/J assumes increasing values. In terms of the potential depth V0and the recoil energy Erthe images reported correspond to the following values of V0: (a) 0Er, (b) 3Er, (c) 7Er, (d) 10Er, (e) 13Er, (g) 16Erand (h) 20Er. In panel (a) an interference pattern due to the presence of off-diagonal long-range order is displayed. As U/J is increased the interference tends to decrease (panels (e), (f)) since it vanishes completely (panels (g), (h)). Reproduction from [26].

To distinguish one phase from another, they exploited the correlation properties of the many-body wave function of the bosons, using an experimental technique named time-of-flight method: the optical lattice is turned off suddenly and the density dis-tribution of the bosons is measured after they expand freely (i.e. interactions during the expansion are negligible and the system maintains its in-trap properties) for a cer-tain amount of time; as explained in Ref. [2], since the measured signal is related to the Fourier transform of hb†_ibji, it carries information about the in-trap momen-tum distribution of the bosons. Due to the different decaying behaviour of hb†_ibji in the Mott and in the superfluid phase (see Eq. (1.26) Eq. (1.27), respectively), this allows to distinguish between them. For example for a superfluid phase, because of off-diagonal long-range order, the Fourier transform of hb†

ibji is sharply peaked at the zero momentum, corresponding to a peak in the momentum (and density) dis-tribution; actually when considering a superfluid phase in a periodic lattice, the mo-mentum distribution is sharply peaked at the reciprocal-lattice vectors and thus it has multiple peaks.

Applying the time-of-flight method for a different set of values of U/J Greiner et al.[26] obtained the interference patterns shown in Fig. 1.4 (we recall that all the physical parameters of a lattice potential, U/J too, are function of the ratio V0/Er). The sharp interference maxima displayed from panel (a) to panel (d) are a direct proof of the presence of off-diagonal long-range order. In panel (e) an incoherent back-ground appears and increases until the interference peaks vanish completely, panel (g). The quantum phase transition from a superfluid to a Mott phase was therefore detected. To check they were really facing a superfluid-Mott quantum phase

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transi-1.3 Experimental detection of the Superfluid-Mott transition 21

tion they studied the opposite transition too (from Mott to a superfluid) lowering the optical potential at values where the superfluid phase could inset and observing in-deed that coherence was rapidly restored. A further check to ensure that the observed incoherent state was a Mott insulator was performed: they created an incoherent state applying a magnetic field gradient over a superfluid phase thanks to which a dephas-ing of the condensate realized implementdephas-ing random phases between neighbourdephas-ing lattice sites. This phase displayed no interference patterns as well as the Mott one but coherence could not be restored anymore lowering the lattice potential. Thus no phase transition toward a superfluid phase occurred, another evidence that restoration of coherence was possible only if the initial state was a Mott one. They probed also the Mott energy gap between the ground and the first excited state (that is the one obtained after the creation of a particle-hole pair) obtaining another confirmation the incoherent state they observed was a Mott-insulator.

Finally they checked if the experimental critical value of U/J at which the phase transition was observed agreed with the theoretical predictions. For the lobe at ν= 1 the theoretical critical value is (U/J)(th)c ≈ 5.8z where z is the coordination number of the lattice. For the cubic three-dimensional lattice employed in the experiment z= 6 and (U/J)(th)c ≈ 34.8 which corresponds to a critical lattice depth V_0c(th) ≈ 11.9Er [2]. Combining the data about the vanishing interference pattern and those about the appearance of a gapped spectrum, Greiner et al. [26] concluded that the experimental critical value V_0c(exp)lies within the interval 12Er−13Er, in reasonable agreement with the theoretical expectations.

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Chapter 2

Aharonov-Bohm cages

The BH Hamiltonian can be straightforwardly extended in order to describe a large variety of systems more complex than those we considered in chapter 1. For example, particular geometries of the lattice or additional interaction (in addition to nearest-neighbour) terms can be taken into account leading to rich phase diagrams. A widely studied case is that of systems embedded in a magnetic field. In general, not only for cold atoms in optical lattices, the presence of a magnetic field has demonstrated to give rise to a large variety of phenomena like Landau levels, Quantum Hall effect or fractal properties of the energy spectrum (Hofstadter’s butterfly).

Using a BH Hamiltonian with few suitable modifications, in this chapter we an-alyze a system of charged bosons first in a two-dimensional lattice, namely the T3 lattice and then in a diamond chain in presence of a transversal magnetic field. The field is uniform and it is tuned on a constant ad hoc value such that the system is said to be fully frustrated. In this particular configuration, for the case of zero on-site interactions, a phenomenon of destructive interference leads to fully localized single-particle states and flat degenerate energy bands. These states are the so-called Aharonov-Bohm(AB) cages [27]. The existence of flat bands in presence of a mag-netic field is a known phenomenon but localization, in the context of cold atoms in lattices, has been shown only in the presence of disorder [28, 29]. Since AB cages do not need disorder, this novel localization mechanism deserves to be investigated and may lead to new phenomena.

If an on-site interaction term is turned on in the fully frustrated model, delocal-ization effects can take place. This result is rather bizarre, as one usually expects interactions to favor well localized states. Even more surprising is the exotic phase which can appear in these conditions, that is a condensate of pairs of particles1, being the single particles alone not affected by this interaction-induced delocalization.

Since optical lattices work with neutral atoms only, the action of the magnetic

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24 2. Aharonov-Bohm cages 00 00 00 11 11 11 00 00 00 11 11 11 00 00 00 11 11 11 −J ? −J −J

(a) Frustrated triangular lattice −J −J −J −J (b) Square lattice −J −J ? −J J

(c) Frustrated square lat-tice

Figure 2.1: The same Hamiltonian H(AF) gives rise to a frustrated triangular (panel (a)) and a non-frustrated squared lattice (panel (b)). Nevertheless changing the cou-pling of the upper bond in the square lattice the system becomes frustrated (panel (c)). We recall that J < 0.

field on a system of charged particles is experimentally simulated loading the neutral bosons on suitable lattices named gauge field lattices [30, 31].

The chapter is organized as follows: section 2.1 is devoted to a brief description about the meaning of frustration in the context of BH models. In section 2.2 and 2.3 we discuss the AB cages forming in a T3lattice and in a diamond chain, respectively. The role of interactions is described in section 2.4. Finally section 2.5 deals with the results obtained by means of an effective description relying on the so-called lowest band approximation.

2.1 Frustration of the Bose-Hubbard model

In the context of spin chains, frustration indicates that the simultaneous minimization of all the interactions between different spins is precluded. Both the geometry of the lattice and the form of the Hamiltonian play a role in the realization of frustration. A simple example is that of a classical antiferromagnetic model on a triangular lattice in absence of a magnetic field:

H(AF)= −J X <i, j>

σiσj (2.1)

where J < 0, < i, j > denotes that the sum is performed over nearest neighbour and σi, σjare classical variables which assume only the discrete values ±1/2. Each interaction term is minimized when the relative nearest neighbouring spins are an-tiparallel but, as seen in Fig. 2.1a, such configuration cannot realize simultaneously for each couple of spins. Conversely, a square lattice allows it (see Fig. 2.1b),

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al-2.1 Frustration of the Bose-Hubbard model 25

though it can be frustrated too if instead of a purely antiferromagnetic model one considers a “mixed” model with some ad hoc ferromagnetic terms which are mini-mized by aligned spins (see Fig. 2.1c). In this examples a typical feature of frustra-tion emerges and it is the degeneracy of the ground state2which lets the problem of finding the ground state of H(AF)at T = 0 non trivial.

In the context of BH models frustration can be introduced by changing appro-priately the hopping constants. In order to clarify how this modification leads to frustration, let us consider a system of bosons with charge e in a square lattice em-bedded in a uniform and constant transversal magnetic field B described by a suitable potential vector A. In what follows we choose a reference frame such that the lattice is within the x-y plane and the magnetic field is along the z axis, B = (0, 0, B). As known from the AB effect [32], the presence of a magnetic field lets a particle with charge q moving along a pathΓ acquire a phase

exp iq ~c I Γ dx · A ! . (2.2)

For this reason in a BH model the hopping constants Ji, j relative to (nearest neigh-bour) sites i and j become

Ji, j= JeiAi j, _(2.3) where Ai j = 2π Φ0 Z j i dx · A(x), (2.4)

and we have introduced the flux quantum Φ0=

hc

q. (2.5)

We notice that Ai j(= −Aji) is gauge dependent thus it cannot be a physically relevant parameter. The quantity of interest is indeed the fluxΦ of B through each plaquette of the lattice which is

Φ =I plaquette dx · A= Φ0 2π X i, j∈plaquette Ai j (2.6)

where the line integral and the sum are performed along the anticlock closed path which contours a squared plaquette. In literature, rather than the flux, it is usually taken into account the reduced flux f = Φ/Φ0[27]. Since Ai j is defined modulo 2π, f is defined modulo 1 so the interesting interval of values to study can be restricted for example to −1/2 < f ≤ 1/2; actually, since f and − f are equivalent (changing

2_{Referring to Fig. 2.1, in the frustrated systems the configuration having spin up or down on sites}

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26 2. Aharonov-Bohm cages

φ

12

23

34 φ

φ

₁₂

₂₃

1 1

+A

1 3 1

+A

2

−2JS

2

−2JS

2

−2JS

2

−2JS

2 23 12

cos( )

A +A +A +A₃₄ ₄₁ 1 4

Figure 2.2: For a squared plaquette it is possible to fix the phase at site 1 at a value φ1and then to choose the phases φ2, φ3, and φ4(written in blue) in order to set three bond terms of Hamiltonian (2.9) to the minimum value −2JS2, as displayed in figure. WheneverΦ , 0 (and then f , 0) the remaining bond cannot be minimized to that value being cos(A12+ A23+ A34+ A41)= cos(2π f ). Reproduction from Ref. [31]

the sign of f means the flux reversed its sign, i.e. the magnetic field reversed), one can restrict to 0 ≤ f ≤ 1/2. In presence of a magnetic field the BH Hamiltonian is then H= −J X <i, j> eiAi j_b† ibj+ e iAji_b† jbi + U 2 X i ni(ni− 1) (2.7)

where we adopted the same notation of chapter 1 (in particular J > 0).

In order to show explicitly that the complex hopping phases in Eq. 2.7 can lead to frustration, in what follows we consider the hard-core limit of H, U/J → ∞, by means of which the Hamiltonian is mapped in a model describing a quantum spin 1/2 chain. Indeed by means of strong interactions double occupancy is excluded and on-site occupations equal to zero or one are analogous to spin up or down states. Making the mapping sz_i = ni− 1/2, s+_i = b†_i, s_i− = bi (where sz_i, s+_i , s−_i are the z-, lowering and raising spin-1/2 operators, respectively) and considering that in this limit the interaction term is identically zero, we have the following hard-core Hamiltonian

Hh−c= −J X <i, j> eiAi j_s+ i s − j + e −iAi j_s+ js − i (2.8)

which is a quantum spin-1/2 ferromagnetic model (J > 0) with a nearest neighbour interaction. In such a model, although not rigorously, the presence of frustration can be straightforwardly pointed out considering the spin vectors like classical rather than quantum variables (i.e. in a mean-field like approximation). Thus we replace the vector of quantum spins si = (s_ix, sy_i, sz_i) with the classical vector of constant modulus S, which is, using spherical coordinates, Si= S (sin θicos φi, sin θisin φi, cos φi). One

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2.2 The fully frustrated T3lattice 27

Figure 2.3: On the left panel an extended portion of the T3 lattice is shown and a conventional cell is highlighted. A zoomed view of it is displayed in the right panel. Within it the primitive vectors t1and t2and the three sites A, B and C of the basis are shown. Reproduction from Ref. [33].

finally obtains a classical frustrated Hamiltonian H( f )_clas= −2JS2 X

<i, j>

sin θisin θjcos(φi−φj+ Ai j) (2.9)

where the φi cannot be chosen in order to maximally minimize the energy of each bond of a squared plaquette, see Fig. 2.2. In particular, at f = 1/2 the coupling relative to the bond linking site 4 and 1 takes the highest value as possible, thus fully incompatible with the minimization constraint; because of that, in this case the model is named fully frustrated.

2.2 The fully frustrated T

3

lattice

In this section we exploit the subtle interplay between the geometrical features of a T3lattice and the presence of a magnetic field which frustrates the system.

The T3 lattice is a two-dimensional periodic structure whose tiling is made of identical rhombic plaquettes oriented with the particular configuration displayed on the left panel of Fig. 2.3. Because of its appearance, T3lattice is also known as dice lattice. Its main peculiarity is that it contains two different kinds of sites: sixfold-coordinated sites named hub sites (site A in Fig. 2.3) and threefold-sixfold-coordinated sites named rim sites (sites B or C in Fig. 2.3). Hub sites are connected to rim sites only and viceversa. Rim sites form an hexagonal sublattice while hub sites form a triangular one, i.e. the T3is bipartite.

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a conventional cell with a basis made of three lattice sites (sites A, B, C in the right panel of Fig. 2.3). Following Ref. [33], the primitive vectors are

t₁= (3/2; −√3/2)l, t₂= (3/2; +√3/2)l, (2.10)

and the vector basis are

dA = (0; 0)l, dB = (0; 1)l, dC= (0; 2)l, (2.11)

where l is the rhombus edge length.

We consider now a system of bosons in a dice lattice embedded in a uniform and static magnetic field with magnitude B orthogonal to the lattice plane and inducing a flux over each rhombus tile equal toΦ = Bl2√3/2. We focus for the moment on the case where no interactions (U = 0) take place, thus the Hamiltonian of the system is made up only of hopping terms:

Hhop= −J X <i, j> eiAi j_b† ibj+ e −iAi j_b† jbi (2.12)

where the hopping constants are modified by the presence of the magnetic field as we described in section 2.1. Since in this case −J is just a global multiplicative factor, the physics is not affected by its value and we drop it from Hhop. In what follows, we describe the AB cage states displayed by the system at full frustration, focusing in particular on their strongly localized nature. Since we will explicitly show that a particle initially localized on a site can delocalize only on a very limited portion of the lattice, it is convenient to perform our description in terms of on-site localized orbital states; thus, following the analysis performed by Vidal et al.3 [34], rather than the second quantized formalism used in Eq. (2.12), we adopt a tight binding Hamiltonian:

Hhop= X

<i, j>

eiAi j_{|iih j|} _(2.13)

where |ii is a localized orbital on site i. From the equation for the eigenvalues of Hhop in (2.13), Vidal et al. [34] were able to obtain the spectrum of the dice lattice as a function of the reduced flux f and their result is displayed in Fig. 2.4. “The most spectacular and unusual feature is that, for f = 1/2, the spectrum collapses into three eigenvalues0 = 0 and ± = ±

√

6” [34]. The flatness of these bands for f = 1/2 (the so-called fully frustrated case) leads to “energy eigenstates where the probability of finding an electron is non vanishing only in a finite size cluster,[...] an

3_{Although the article of Vidal et al. deals with electrons, their description applies to a bosonic}

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Figure 2.4: The spectrum of the T3lattice vs the reduced flux f . Reproduction from Ref. [34].

Aharonov-Bohm cage”[34].

We now investigate quantitatively the localization mechanism which leads to the existence of AB cages. The first step is performed using a method introduced by Haydock-Heine-Kelly [35] which, starting from the set of orbitals {| ji} over which the Hamiltonian has the representation of Eq. (2.13), allows to build explicitly a new set of orthogonal states {|ϕni} (n ≥ 0 is an integer valued index) over which Hhophas a tridiagonal representation HT D. The “new” states {|ϕni} are a linear combination of the {| ji} and are found exploiting the following recursive relation:

cn+1|ϕn+1i= (Hhop− dn)|ϕni − cn|ϕn−1i, (2.14)

where |ϕni ≡ 0 if n < 0 and dn, cn, cn+1are the non-zero elements4of HT Dwhich are automatically defined from the above relation being

hϕm|Hhop|ϕni= (HT D)m,n =                            dn m= n, cn m= n − 1, cn+1 m= n + 1, 0 otherwise. (2.15)

By means of Eq. (2.14), once an “initial” state |ϕ0i is chosen as a combination of the “old” orbital states {| ji}, all the other states |ϕni and the matrix elements of HT Dare derived. The representation of HT Dis formally identical to the one of an Hamiltonian

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Figure 2.5: In the upper panel an AB cage centered on a hub site (white site) and extending up to black sites is displayed. At the bottom the portion of the effective one-dimensional chain equivalent to an AB cage built up by means of the Haydock-Heine-Kelly method is represented. Reproduction from Ref. [34].

with on-site and nearest-neighbouring terms (embodied respectively by dn and cn) describing a semi-infinite one-dimensional chain using as basis a set of orbitals |ϕni centered on the n-th site. Thus Eq. (2.14) maps our problem from the T3lattice to an effective semi-infinite one-dimensional chain whose initial site corresponds to |ϕ0i.

The Hamiltonian in Eq. (2.13) is purely off-diagonal and the non-zero matrix elements hi|Hhop| ji connect hub to rim sites only and viceversa. Therefore if |ϕ0i co-incides with an orbital | ji corresponding to a hub or a rim site of the dice lattice, then the |ϕni obtained using Eq. (2.14) are a combination of states |ii relative exclusively to rim or hub sites only. In particular if |ϕni is a combination of hub sites then |ϕn+1i is a combination of rim sites only. According to the properties of the {|ϕni} we stated above, thus we deduce that dn = hϕn|Hhop|ϕni ≡ 0.

To start our computations, we choose a gauge A = H(−y/2, x/2, 0). Then we identify the initial orbital |ϕ0i with a hub state that is, referring to Fig. 2.5, the 0-th site state |0i. From Eq. (2.14) we now have

Hhop|ϕ0i= 6 X i=1 |ii ⇒ c1= √ 6, |ϕ1i= 1 √ 6 6 X i=1 |ii. (2.16)

The orbital |ϕ1i of the effective chain represents thus the first shell of sites centered on the 0-th site and made of rim sites numbered from 1 to 6. Applying again the

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Figure 2.6: Absence of AB cages in the squared lattice since particles can move along the straight paths indicated by the arrows without any destructive interference occurring: a particle wave packet initially localized on the white site cannot reach black, but only grey sites.

If f = 1₂ then c2 = 0 and a wave packet which was initially localized in the hub site is disconnected from the rest of the lattice, remaining in this sense “caged” in-side the black sites of Fig. 2.5. “This effect can be simply understood in terms of the Aharonov-Bohm effect. Indeed, the amplitude of probability A_0→2→9to go, for example, in “two steps”, from (site) 0 to (site) 9 via 2, is exactly the opposite of A0→3→9, so that the resulting amplitude is zero” [34]. An AB cage state realizes as well if a rim site is chosen as the “initial” orbital |ϕ0i, although in this case the particle is trapped within a larger portion of the lattice than previously. Thanks to the superposition principle, once the AB cages around a rim and hub site are known, we have a complete description of the phenomenon of caging over the entire lattice.

We remark that the localization feature of AB cages is the result of a quantum interference process. The topology of the lattice plays a fundamental role: in the dice lattice, considering a plaquette, two hub sites are linked by two possible equivalent (equivalent in the sense that they link the initial and final site with the same number of steps) paths, one having the opposite amplitude of the other; thus destructive inter-ference realizes and particles are caged. For a square lattice such phenomenon does not occur. Indeed, looking at Fig 2.6, although a particle initially localized over the white site is prevented from reaching the black sites (because of the same destructive interference process which occurs in the dice lattice), nevertheless it can reach all the gray sites. Indeed, only one path linking directly the white and grey sites exists

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A

m

C

m

l

B

m

B

Φ

Figure 2.7: The diamond chain embedded in a uniform and static magnetic field B orthogonal to the lattice with a fluxΦ through each plaquette. The m-th conventional cell composed by a hub and two rim sites (Am, Bmand Cmrespectively) is enclosed in the dotted box. Adopting the local U(1) transformation of Eq.(2.21), the phase factor due to the presence of a magnetic field affects only the links at the bottom of each plaquette, over which an arrow is drawn.

and no alternative paths can do the same with the same number of steps. Finally no destructive interference occurs and no caging realizes.

We conclude this section, mentioning that the existence of AB cage states in the fully frustrated dice lattice has been experimentally confirmed, see Ref. [36] and [37].

2.3 The fully frustrated diamond chain

We study now the simplest geometry displaying AB cages, i.e. the quasi-one-dimensional diamond chain. In fact, this kind of lattice is interesting since it allows to study the phenomenon of caging in a framework where it is natural to expect the physics of AB cages to emerge with almost no complications related to the geometry of the lattice. The diamond chain is the repetition along one direction of a conventional cell with a basis composed by three sites, as shown in Fig. 2.7: A is the fourfolded-coordinated site (named hub even in this case) while B and C are twofolded-coordinated sites (named rim). Formally this lattice is defined by the primitive vector

t= (1; 0)l (2.18)

and the three basis vectors

dA = (0; 0)l, dB= 1 2; 1 2 ! l, dC = 1 2; − 1 2 ! l, (2.19)

(35)

2.3 The fully frustrated diamond chain 33

where l is the lattice spacing. Analogously to the T3lattice, hub sites are linked only to rim sites and viceversa, a circumstance which, in presence of a suitable magnetic field, enables the formation of AB cages via the same mechanism described in the previous section. As for the dice lattice, this happens when the no interactions take place and the system is embedded in a uniform and static magnetic field orthogonal to the lattice, inducing a flux through each plaquette such that f = 1₂, i.e. fully frustrating the system. The simple geometrical configuration of the chain allows a straightforward diagonalization of Hhop. We consider in our computation a lattice with L fundamental cells and periodic boundary conditions. According to what we said in section 2.1, the hopping constant gains a complex phase eiAi j_{, see Eq. (2.4),}

which, choosing the gauge A= (0, Bx, 0), leads to the following hopping term:

Hhop= −J L−1 X m=0 (e−iπ/4(b†_A mbBm+ b † BmbAm+1)+ e iπ/4_(b† AmbCm+ b † CmbAm+1)+ h.c.), (2.20) where b†_S

m and bSm with S = A, B, C are respectively the bosonic operators which

create and annihilate a spinless boson on the S -like site contained in the m-th con-ventional cell. According to periodic boundary conditions the L-th site coincides with the first one (we label the sites starting from 0). By means of a local U(1) transformation bAm → e iπ₂m_b Am, bBm → e iπ₂(m+1/2)_b Bm, bCm → e iπ₂(m−1/2)_b Cm, (2.21)

we obtain the following Hamiltonian:

Hhop = −J L−1 X m=0 (b†_A mbBm+ b † AmbCm + b † BmbAm+1+ e iπ_b† CmbAm+1 + h.c.). (2.22)

Notice that now only the hopping term between the Cmand Am+1sites has a phase factor. Passing to Fourier transform by means of

bSm = 1 √ L X k eiklmbS,k, b†_S m = 1 √ L X k e−iklmb†_S,k, (2.23)

Phase diagram of a fully frustrated Bose-Hubbard diamond chain

Contents

Introduction

Chapter 1

Bose-Hubbard model

1.1

Ultracold strongly correlated atomic gases

1.2

Properties of the Bose-Hubbard model

0

1

2

3

J

U

Μ



U

SF

MI

Ν=3

MI

Ν=2

MI

Ν=1

1.3

Experimental detection of the Superfluid-Mott

transi-tion

Chapter 2

Aharonov-Bohm cages

2.1

Frustration of the Bose-Hubbard model

φ

φ

12

12

23

34

φ

φ

12

23

+A

+A

+A

+A

+A

+A

−2JS

−2JS

−2JS

−2JS

cos( )

2.2

The fully frustrated T

lattice

A

C

l

B

B

Φ

Φ

Φ

2.3

The fully frustrated diamond chain

_U

_Ν=3

_Ν=2

_Ν=1

₁₂

₂₃