Scaling properties of three-dimensional Bose-Einstein condensed gases

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Facolt`a di Scienze matematiche fisiche e naturali

Corso di Dottorato in Fisica Anno Accademico 2016/2017

Tesi di Dottorato

Scaling properties of three-dimensional

Bose-Einstein condensed gases

Candidato Relatore

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Introduction

The behavior of three-dimensional bosonic gases at low temperatures is char-acterized by the formation of a Bose-Einstein condensate (BEC) [1]. The Bose-Einstein condensation is a phase transition of a bosonic gas, occurring usually when the temperature is lowered below a certain critical value Tc,

which is of the order of a few hundred of nK [2, 3, 4]. The gas shifts from the normal phase to a new one where the lowest-energy state becomes macro-scopically occupied and we say that a condensate has been formed. Thus a Bose-Einstein condensate is a macroscopic number of particles in the same quantum sate and the wave function associated to such a state is called the condensate wave function.

Bosonic gases at low temperature have been extensively studied both theoretically and experimentally since the first appearance of a BEC in lab-oratory in 1995. In this work we will focus on their coherence properties and their behavior in the presence of a trapping potential. In particular we will show how coherence properties vary when the system shape is changed re-sulting in a change of the effective dimensionality of the system, for example when it shifts from a cube to a very long parallelepiped thus becoming an effective one-dimensional system or to a very thin slab which is instead an effective two-dimensional system. Coherence properties are also strongly af-fected by inhomogeneous conditions induced by a trapping potential, which is necessary in experimental setups. Moreover if the trapping potential is smooth enough we can assist to the peculiar presence of different phases in different spatial regions, separated by a spatial critical surface.

The phase coherence properties of the BEC phase have been observed by several experiments, see for example references [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. In three-dimensional systems the phase of the condensate wave function is usually coherent all over the condensate size. Several theoretical and experimental studies have also investigated the critical properties at the BEC transition, when the condensate begins forming, see for example refer-ences [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. Both the phase-coherence properties of the BEC phase and the critical behavior at the BEC transition turn out to be particularly sensitive to the inhomogeneous conditions arising from spatially-dependent confining potentials and to the geometry of the atomic-gas system.

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Inho-mogeneous conditions due to space-dependent trapping potentials give rise to a universal distortion of the homogeneous critical behavior, which can be cast in terms of a universal trap-size scaling [18, 30] controlled by the same universality class of the three-dimensional BEC transition. In the case of homogeneous traps, such as those experimentally realized in refer-ences [32, 35, 36, 38], the geometry of the trap may lead to quite different phase-coherence properties, when passing from 3D, to quasi-2D, or quasi-1D systems. For example atomic gases in elongated homogeneous boxes [39] and harmonic traps [40, 9, 11, 41, 12, 42, 43, 44] show a dimensional crossover from a 3D behavior to a quasi-1D behavior, which gives rise to a substan-tial phase decoherence along the longer axial direction. This crossover is characterized by a peculiar anisotropic finite-size scaling (AFSS) and can be described by an effective 3D spin-wave theory. The phase-correlation properties of the spin-wave theory are expected to be universal, i.e. they are expected to apply to any condensed particle system, at any temperature below the BEC transition.

The behavior of homogeneous gases in slab geometries is more complex. It can be described in terms of a dimensional crossover, from 3D behaviors for T & Tcto a quasi-2D critical behavior for T . TBKT, where TBKT is the

Berezinskii-Kosterlitz-Thouless (BKT) critical temperature. Indeed in two dimensions we do not have condensates but bosonic gases undergo another kind of transition to a quasi-long range ordered phase, where correlations decrease with the distance with a power which depends on the temperature. This transition is known as the BKT transition [45, 46, 47, 48, 49], which occurs in 2D statistical systems with a global U(1) symmetry. Experimental evidences of BKT transitions have also been reported for quasi-2D trapped atomic gases [50, 51, 52, 53, 54, 55, 56]. The interplay of the BEC and BKT critical modes gives rise to a quite complex behavior. In the limit of large thickness Z, the quasi-2D BKT transition temperature approaches that of the 3D BEC transition. The dimensional-crossover scenario can be described by a transverse finite-size scaling (TFSS) limit for systems in slab geometries [57, 58, 59]. An analogous description apply to 4He systems in film geometries [60], and to 3D XY spin models defined in lattices with slab geometries [61, 62, 63, 64].

All our theoretical predictions are supported by numerical analyses of the 3D hard-core Bose-Hubbard model [65], which models gases of bosonic atoms in optical lattices [66, 14]. We perform quantum Monte Carlo simu-lations of the Bose-Hubbard model at several values of the temperature and with different geometries and external potentials, measuring in particular its phase-coherence properties. Numerical results confirm finite-size scaling ansatzes, providing evidence of the dimensional-crossover scenarios.

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This work is organized as follows:

• In chapter 1 we introduce the Bose-Hubbard model, which describes gases of bosonic particles in optical lattices. We then review some phenomenological properties of Bose-Einstein condensed systems.

• In chapter 2 we summarize the main elements of the theory of phase transitions. We describe the universal properties of physical systems near a critical point, introducing the Renormalization Group theory. We also show how to deal with finite systems through the finite-size scaling theory.

• In chapter 3 we present the 3D spin-wave theory that provides an ef-fective description of the long-range phase correlations within the BEC phase, and we show the emergence of a nontrivial AFSS behavior for elongated systems. In the crossover from 3D cubic-like geometries to elongated effectively 1D systems the one-particle correlation function develops an exponential decay along the axial direction, with a coher-ence length proportional to the transverse area. QMC simulations of the 3D hard-core Bose-Hubbard model within its BEC phase confirm the universality of the AFSS behavior obtained within the 3D spin-wave theory.

• In chapter 4 we consider bosonic particle systems confined within slab geometries. We investigate their behavior at the BEC transition temperature Tc and at lower temperatures. Their low-temperature

behavior (T < Tc) is further characterized by the possibility of

un-dergoing a finite-temperature transition to a quasi-long range ordered (QLRO) phase, the BKT transition. We describe the interplay of the BEC and BKT critical modes with a transverse finite-size scaling (TFSS) limit.

• In chapter 5 we consider 3D bosonic gases trapped by an external harmonic potential, when the temperature is sufficiently low to show a BEC-phase region around the center of the trap. We study the behav-ior of the correlation functions around the center of the trap and at the boundary of the BEC. We point out that the quantum many-body system develops a peculiar critical behavior at the boundary of the BEC region, with a nontrivial scaling behavior controlled by the uni-versality class of the homogenous BEC transition. We provide some numerical evidence of this phenomenon in lattice atomic gases

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mod-eled by the Bose-Hubbard Hamiltonian.

• In appendix A we describe the details of the quantum Monte Carlo algorithm employed to simulate the hard-core Bose-Hubbard model. We also enumerate the observables measured in the simulations.

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Chapter 1

Bose-Einstein condensation

In three dimensions gases of bosonic particles undergo a phase transition when the temperature is lowered below a certain critical value Tc. The

low-temperature phase is characterized by the formation of a Bose-Einstein con-densate (BEC), that is a macroscopic occupation of the lowest-energy state. In the following sections we will review some phenomenological properties concerning the phenomenon of BEC (see e.g. [67]). We also derive the Bose-Hubbard Hamiltonian starting from the microscopic Hamiltonian of a gas of bosonic particles confined by a trapping potential and in the presence of an optical lattice.

1.1 The Bose-Hubbard model

The experiments with cold atoms that realize Bose-Einstein condensates need first of all a way to confine the atoms into a certain spatial region. Trap-ping techniques rely on the magnetic and optical properties of the atoms. In particular optical confinement is based on optical dipole forces in laser fields [68]. Optical confinement methods let us realize periodic potentials by superimposing two counterpropagating laser beams with the same fre-quency. Periodic potentials, usually referred to as optical lattices, let us obtain systems analogous to typical systems of solid-state physics, but with much more control over the system parameters and easier ways to measure correlations within the system. Moreover lower dimensional systems (two or one dimensions) are a lot easier to realize with respect to solid-state physics. Systems of cold bosonic atoms in optical lattices, whose behavior at very low temperature is characterized by the BEC phase, are well described by the Bose-Hubbard model, which constitutes our starting point in the numerical simulations of these systems. We now proceed with the derivation of the Bose-Hubbard model starting from the microscopic Hamiltonian of a gas of bosonic atoms confined by a trapping potential Vtrap and in the presence of

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Hamiltonian describing such system should look like H =X i p2 2m+ X i Vlat(ri) + X i Vtrap(ri) + X i<j Vint(ri− rj), (1.1)

where the sums run over the particles. We are interested in the low-temperature regime therefore in the low-energy properties of the system. In order to sin-gle them out it is useful to change formalism and introduce the bosonic field operators ψ(x), which destroys an atom at position x, and ψ†(x), which creates an atom at position x. They obey the usual commutation relations for bosonic fields _h

ψ(x), ψ†(y) i

= δ(3)(x − y). (1.2)

In this formalism the Hamiltonian is an operator with the following expres-sion in terms of the fields

H = Z ψ†(r) −~ 2 2m∇ 2_{+ V} lat(r) + Vtrap(r) ψ(r)d3r + 1 2 Z ψ†(r)ψ†(r0)Vint(r − r0)ψ(r0)ψ(r)d3rd3r0. (1.3)

The field operator can always be expanded in a complete orthonormal basis of single particle wavefunctions. A periodic Hamiltonian has wavefunctions of the form ϕnq(x) = unq(x)eiqx, known as Bloch functions, where unq(x)

has the same periodicity of the Hamiltonian, n is the band index and q is the quasi-momentum. The Bloch functions are plane waves modulated by a periodic function. We can obtain localized wavefunctions by taking the lattice Fourier transform of the Bloch functions

wnz(x) = 1 √ N X q e−iqzϕnq(x), (1.4)

where N is the number of lattice sites. The functions wnz(x) are known

as Wannier functions, they depend only on the relative distance x − z and satisfy the orthonormalization condition

Z

w∗_mx(r)wny(r)d3r = δmnδ(3)(x − y). (1.5)

In the lowest band the Wannier functions are exponentially damped, with a decay length of the order of the lattice spacing. We can expand the field operator in the Wannier basis

ψ(x) =X

nz

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where the operators bnz are defined by the former equation and their

mean-ing is that of destruction operators for an atom at position z in the band n. Commutation relations of the bnz operators follow from equations (1.2, 1.5)

h

bnx, b†my

i

= δnmδxy. (1.7)

If the average energy of an atom is much smaller than the energy separation between the first two energy bands we can retain in the field expansion (1.6) only the lowest band terms

ψ(x) =X

z

wz(x)bz. (1.8)

Substituting the truncated expansion (1.8) of the bosonic field into the Hamiltonian (1.3) we obtain the following new expression

H =X ij tijb†ibj+ X ij ijb†ibj+ 1 2 X ij Uijb†ib † jbibj, (1.9)

where the indices (i, j) run over the lattice sites and the coupling parameters are given by tij = Z w∗_r_i(r) −~ 2 2m∇ 2_{+ V} lat(r) wrj(r)d 3_r, _(1.10) ij = Z w∗_r_i(r)Vtrap(r)wrj(r)d 3_r, _(1.11) Uij = Z |wri(r)| 2_V int(r − r0)|wrj(r 0_)|2_d3_rd3_r0_. _(1.12)

Further simplifications can be made. Since Wannier functions in the lowest band are exponentially dumped, only the diagonal terms and the nearest-neighbors terms in (1.10) will be significantly different from zero. Diagonal terms tii are site independent because of lattice translation invariance of

the Wannier functions, so that we can set tii = −µ. For the same reason

also nearest-neighbors terms are all equal and we can set t<ij> = −t, with

< ij > denoting a nearest-neighbor pair. If the external trapping potential is smooth, as it is usually the case, for the same reasons we can neglect all off-diagonal terms in (1.11) and set

ii=

Z

Vtrap(r)|wri(r)|

2_d3_{r ≈ V}

trap(ri). (1.13)

Moreover if we make the assumption of very short range interactions, in (1.12) only the diagonal terms Uii = U are different from zero. Therefore

with the assumptions of a smooth weak trapping potential and of short-range interactions the final form of the Hamiltonian is

H = −t X <ij> b†_ibj+ b†_jbi +U 2 X i ni(ni− 1) + X i Vtrap(ri)ni− µ X i ni, (1.14)

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where we have introduced the particle number operator ni ≡ b†ibi. The

Hamiltonian (1.14) is known as the Bose-Hubbard model and it appeared for the first time in 1989 [65] within the study of the zero temperature superfluid-insulator transition.

The connection between the parameters in the model (1.14) and the ex-perimental parameters can be made more explicit. In a cubic optical lattice the characteristic parameters are the intensity V0 and the wavelength λ of

the lasers constituting the lattice. The band structure of an optical lat-tice can be obtained from the solutions to the one-dimensional Schr¨oedinger equation with a sinusoidal potential, which is also known as the Mathieu equation. Taking the recoil energy

Er= ~ 2_k2

2m , k =

2π

λ , (1.15)

as a reference energy scale (it is the energy gained by an atom which emits a photon of frequency ν = c/λ), we can distinguish the weak and the strong potential limits. In the weak potential V0 Er limit a particle is well

approximated by a free particle, while in the deep potential limit V0 Er

the eigenvalues are only weakly dependent on the quasi-momentum. The hopping parameter t is related to the width of the lowest energy band

t = max[E(q)] − min[E(q)]

4 (1.16)

and can be also given analytically as

t = √8 πEr V0 Er 4/3 exp " −2 V0 Er 1/2# . (1.17)

We can also find an analytic expression for the interaction term U if we assume the following simplified form for the interaction potential

Vint(r) = 4π~ 2_a

m δ(r), (1.18)

where a is the scattering length of the real interaction potential. In the deep potential limit we can approximate the exact Wannier function of equation (1.12) with the Gaussian ground state and thus we obtain

U = r 8 πkaEr V0 Er 4/3 . (1.19)

We want to stress again that the expressions (1.17, 1.19) are only valid in the deep potential limit V0 Er. In the general case a numerical solution

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1.2 One-particle correlation function

Let us consider a gas of bosonic particles within the formalism of quantum field theory. The bosonic field operator will be denoted by ψ(x) (ψ†(x)), which annihilates (creates) a particle at position x. The field operators satisfy the commutation relations

h ψ(x), ψ†(y) i = δ(x − y), (1.20) h ψ†(x), ψ†(y)i = [ψ(x), ψ(y)] = 0. (1.21) A crucial quantity for the characterization of the different phases of the system is the one-particle correlation function

G(x, y) = hψ†(x)ψ(y)i, (1.22)

where the average h·i is understood in a quantum-statistical sense. The function (1.22) contains informations on the particle density ρ(x), which is recovered by setting y = x, and also determines the momentum distribution

n(p) = hψ†(p)ψ(p)i, (1.23)

where ψ(p) = (2π~)−3/2R dx eipx/~_{ψ(x) is the field operator in momentum}

representation. In particular n(p) = 1

(2π~)3

Z

dRdr eipr/~ G(R − r/2, R + r/2), (1.24) where R = (x + y)/2 and r = y − x. For a uniform system in the thermo-dynamic limit (or in a finite volume with periodic boundary conditions), relation (1.24) reduces to

n(p) = V (2π~)3

Z

dr eipr/~ G(r), (1.25) where G(r) ≡ G(0, r), since in this case the one-particle correlation function depends only on the relative distance r = y − x.

The behavior of the correlation function (1.22) and the related momen-tum distribution (1.24) can be experimentally investigated by looking at the interference patterns of absorption images after a time-of-flight period in the large-time ballistic regime [14].

For simplicity, let us first consider a uniform system in the thermo-dynamic limit. In the normal phase (i.e. without BEC), the momentum distribution has a smooth behavior at small momenta and consequently the one-particle correlation function vanishes when r → ∞. In the BEC phase the momentum distribution instead can have the following singular behavior,

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due to a macroscopic occupation of the single-particle state with momentum p = 0:

n(p) = N0δ(p) + ˜n(p), (1.26)

where N0 is the number of particles in the condensate, which is of the same

order of magnitude of the total number of particles, while ˜n(p) is smooth for small momenta. By inverting relation (1.25), from a momentum distribution with the form (1.26) it follows that the one-particle correlation function does not vanish at large distances, but it approaches a finite value:

G(r) →

r→∞n0, (1.27)

where n0 = N0/V is the condensate density. The long range order displayed

by the one-particle correlation function (1.27) is a signature of the Bose-Einstein condensation.

In the most general case of a nonuniform system we can still identify an orthonormal basis of single-particle states by considering the eigenfunctions of the one-particle correlation function:

Z

d3y G(x, y)φi(y) = niφi(x). (1.28)

It can be checked that for uniform systems the solutions to (1.28) are plane waves. We can rewrite the one-particle correlation function in the diagonal-ized form

G(x, y) =X

i

niφi(x)φ∗i(y). (1.29)

In the presence of BEC one eigenstate (let us say φ0) will be macroscopically

occupied (n0 ∼ ρ, the total density) and its extension will determine the

large-distances behavior of G(x, y).

1.3 Order parameter and superfluidity

The field operator can be expanded in the basis {φi}, obtained from (1.28):

ψ(x) = φ0(x)a0+

X

i

φi(x)ai, (1.30)

where the operator ai destroys a particle in the state φi. If in the BEC phase

the state φ0 is macroscopically occupied then the following average value of

the field operator is nonvanishing:

Ψ (x) = hψ(x)i =√n0φ0(x) = |Ψ (x)|eiθ(x), (1.31)

where in the average it is understood that the state to the right has one more particle in the condensate than the state to the left. This definition

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of the average (1.31) is justified by the fact that two states differing by one particle in the condensate are physically equivalent in the thermodynamic limit.

The emergence of a new macroscopic quantity in the low temperature phase which reduces the symmetry of the system is typical of continuous phase transitions. A quantity of this kind is generally know as order pa-rameter. The order parameter (1.31) is proportional to the wave function of the condensate φ0(x) and its modulus determines the condensate

den-sity n0(x) = |Ψ (x)|2. A wave function is always defined up to a constant

phase factor eiα, therefore making an explicit choice for the value of the or-der parameter corresponds to a spontaneous symmetry-breaking of the U (1) symmetry of the system.

Interacting BEC gases display a superfluid behavior, that is they can flow without friction for low velocities and they have a non-classical re-sponse to an infinitesimal rotation. Superfluidity was originally observed in liquid 4He [69, 70] and it was soon suggested its relation to BEC [71]. The phenomenon was then explained by Landau [72], who showed that, at zero temperature, if the velocity of the fluid is less than a critical value, determined by the excitation spectrum, the flow occurs without dissipation. At finite temperatures the system can be described by a two fluid model: a superfluid component with density ρs, which has no viscosity, and a normal

viscous component with density ρn = ρ − ρs (ρ being the total density of

the fluid). The ratio ρs/ρn depends on the temperature, at T = 0 we have

ρs = ρ, while at T = Tc, ρn= ρ and ρs = 0. The velocity of the superfluid

component vs is related to the phase of the order parameter (1.31):

vs = ~

m∇θ, (1.32)

where m is the mass of the constituent particles. In particular it follows from equation (1.32) that the superfluid velocity is irrotational. It is important to stress that the superfluid density ρs does not coincide with the condensate

density mn0. The superfluid density can be experimentally probed through

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Chapter 2

Critical phenomena

The BEC transition is a continuous phase transition characterized by a spontaneous symmetry breaking of the U (1) symmetry of a bosonic gas. It therefore belongs to the XY universality class which determines the critical behavior of the gas near the transition point. In this chapter we outline phe-nomenological characteristics of critical phenomena with specific attention to the XY universality class. We also introduce the technique we employ to study critical behavior in finite systems, the finite-size scaling (FSS) theory.

2.1 Continuous phase transitions

Observable properties of a macroscopic system are determined by the values of some thermodynamic parameters such as the temperature T , the chemi-cal potential µ and the magnetic field h. For some particular values of these parameters the system may be found in two or more coexisting phases, well separated from each other. Each of them has different macroscopic proper-ties and when the parameters are changed slightly the whole system assumes the characteristics of one of these phases. Transitions of this kind are called discontinuous or first-order transitions, because some thermodynamic quan-tities vary in a discontinuous manner. A simple example is the liquid-gas transition in which the density of the system abruptly changes. A system may also undergo another kind of transitions, namely continuous or second-order transitions. The point in the thermodynamic parameters space where a continuous transition happens is called critical point. At the critical point macroscopic properties are the same for the whole system, they change from a phase to the other in a continuous way. As examples we can consider the liquid-gas critical point and a ferromagnetic-paramagnetic transition at the Curie temperature, where, respectively, the difference between liquid den-sity and gas denden-sity and the magnetization smoothly goes to zero. The BEC transition is also a continuous transition for the condensate density increases smoothly as the temperature is decreased. The term critical

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phe-nomena refers to the thermodynamic properties of systems near a critical point.

Let us now make an important observation. The system in different phases will have different symmetry properties, however at the critical point it must have the symmetry properties of both phases. Because the sym-metry at the critical point is equal to the symsym-metry of one of the phases [74], it is necessary that the symmetry of a phase is larger than that of the other (this means that the symmetry group of a phase is a subgroup of that of the other). Usually the more symmetrical phase (also disordered phase) corresponds to a greater temperature. To take into account this reduction of symmetry we introduce a quantity called order parameter, which is equal to zero in the disordered phase and assumes a non-zero value in the ordered phase (the less symmetrical phase). Order parameters may have many com-ponents, their nature depends upon the symmetry groups involved in the transition. Considering again the paramagnetic-ferromagnetic transition, we distinguish the two phases looking at the magnetization, which is the natural order parameter. If the system was initially isotropic we have a reduction of the O(3) symmetry (spatial rotations) to an O(2) symmetry (plane rotations), because the magnetization selects a preferred direction in space and the system remains invariant only under rotations around this direction. In the case of a bosonic gas the freedom of choice of the global phase of the condensate wavefunction (U (1) symmetry) is lost when the condensate wavefunction actually emerges as a macroscopic property of the system because only one possible value for the phase is then selected.

2.2 Universal critical behavior

The critical point is a singular point for thermodynamic functions. Even though the free energy and its first derivatives are continuous at the critical point, its second or successive derivatives present discontinuities or diver-gences (from this property stems the older nomenclature of second order phase transitions, now disused in favor of the more general continuous phase transitions). Non-analytic behaviors are due to the divergence at the critical point of the length scale ξ characterizing the decay of physical correlations

lim

T →Tc±

lim

L→∞ξ = ∞, (2.1)

where L denotes the size of the system in every direction, i.e. the limit in (2.1) corresponds to the thermodynamic limit. For the rest of this section we will always consider systems at the thermodynamic limit. At the critical point the correlation function decays algebraically with the distance

G(x, y; T = Tc) ∼

1

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where d is the space dimensionality of the system and η is a critical exponent. Around the critical point instead the behavior of ξ and G(x, y) is given by

ξ ∼ |T − T_c|−ν, (2.3)

G(x, y; T ) ≈ ξ−1−ηG(|x − y|/ξ), (2.4) where the long distance behavior of G is characterized by an exponential decay controlled by the correlation length ξ, while ν is another critical ex-ponent.

Critical exponents are particularly important in the characterization of the critical behavior because they are universal quantities. As experiments have shown, different physical systems with the same global symmetry, space dimensionality and symmetry breaking pattern have also the same critical exponents. It is said that such systems belong to the same universality class. Systems with a U (1) symmetry which breaks into a residual Z2 symmetry,

such as BECs, planar ferromagnets and superconductors, belong to the so called XY universality class. A theoretical explanation of this phenomenon has been achieved through the Renormalization Group theory [75, 76, 77].

The values of the critical exponents ν and η within the 3D XY univer-sality class are [78, 79, 80, 81, 82, 29, 30, 31, 33, 83, 84]

ν = 0.6717(1), η = 0.0381(2). (2.5)

2.3 Renormalization Group theory

The intuition at the base of the Renormalization Group (RG) theory is that microscopic details of physical systems are irrelevant for their critical be-havior. Thus we only need to study their long distance behavior when we are interested in their critical properties. The approach of RG consists in the application of repeated coarse-graining transformations to the degrees of freedom of the theory. After one such RG transformation the Hamiltonian is changed in such a way that its original long distance behavior is preserved, while information on the microscopic details is lost. This procedure gener-ates a flow in the Hamiltonian space. The Hamiltonian space is an infinite dimensional space parametrized by the coupling constants that are asso-ciated to all possible operator terms that can appear in the Hamiltonian, compatibly with the symmetries of the system.

The critical behavior is associated with a fixed point of the RG flow. Hamiltonians of different systems belonging to the same universality class, each one at its respective critical point, all flow to the same fixed point. If instead a system is not exactly at the critical point but only in its neighbor-hood, there will be at least one parameter in the Hamiltonian space that will lead the flow away from the fixed point. Near a fixed point we can identify the RG eigenoperators, whose associated parameters ui are called nonlinear

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scaling fields. The scaling fields ui are analytic functions of the Hamiltonian

parameters like for example t ≡ |T /Tc− 1| or h. The fixed point correspond

to ui = 0, ∀i. Under a RG transformation ui → byiui, where b is a positive

number defining the scale of the coarse-graining procedure and yi are the RG

eigenvalues, also called RG dimensions of the scaling fields. We distinguish between irrelevant fields with yi < 0, which are an infinite set, and relevant

fields with yi > 0. In the case of the XY fixed point there are two relevant

fields, say u1 and u2. For t, h → 0 we have u1 ∼ t and u2 ∼ |h|. Indeed the

number of relevant fields must match the number of physical parameters we need to fine-tune in order to reach the critical point of the system.

By comparison of the expressions of the free energy before and after a RG transformation we observe that it obeys a scaling law [85, 86, 87, 80]

Fsing(u1, u2, . . . , uk, . . . ) = b−dFsing(by1u1, by2u2, . . . , bykuk, . . . ), (2.6)

where Fsingdenotes the free energy apart from a regular additive part which

is analytic at the critical point. If we fix b by requiring by1_|u

1| = 1, we obtain

Fsing(u1, u2, ..., uk, ...) = |u1|

d

y1_F_sing_{(sign u}₁_{, u}₂_|u₁_|−y2y1 _{, ..., u}_k_|u₁_|−yky1 _{, ...),}

(2.7) The exponents yk/y1 are negative for k ≥ 3 so that uk|u1|

−yk

y1 → 0 for t → 0. Thus, provided that Fsing is finite and nonvanishing in this limit, we obtain

Fsing(u1, u2, ..., uk, ...) ≈ |t|

d

y1_F_sing_{(sign t, |h||t|}−y2y1 _{, 0, ...).} _(2.8)

The function on the right side of (2.8) is a universal scaling function of |h||t|−y2y1 _{. From the expression (2.8) of the free energy it is possible to}

estab-lish a connection between the critical exponents ν, η and the positive RG eigenvalues y1, y2: y1 = 1 ν, (2.9) y2 = d + 2 − η 2 , (2.10)

where d is the space dimensionality. Corrections to the expression (2.8) are of two types: analytic corrections due to the analytic dependence of u1 and u2 on t and |h|, and nonanalytic ones due to the irrelevant fields.

The leading nonanalytic correction is of order |u1|

−y3

y1 ∼ t∆ _{(∆ ≡ −y} 3/y1),

corresponding to the largest yi that for simplicity we have identified with

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2.4 Finite systems

A finite system of overall size L does not show divergences at the critical point because the correlation length is limited by the system size ξ ∼ L. We may think of the finite size L as another relevant field that drives the RG flow away from the fixed point. For a finite system eq. (2.6) generalizes to [57, 58, 86, 80]

Fsing(ui, L) = b−dFsing(byiui, L/b). (2.11)

From eq. (2.11) we see that L−1 can be considered as a relevant field with eigenvalue yL= 1. We can fix b by choosing b = L, obtaining

Fsing(ui, L) = L−dFsing(Lyiui). (2.12)

From the expression (2.12) of the free energy for a finite system we can derive the finite-size scaling (FSS) behavior for any thermodynamic quantity by performing the appropriate derivatives with respect to t and h. Neglecting contributions from the irrelevant fields, a generic quantity S that for h = 0 and L−1= 0 behaves as t−σ for t → 0, has a FSS behavior given by

S(t, L) = Lσν f_S(ξ_∞/L) + O(L−ω, ξ_∞−ω) , (2.13)

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Chapter 3

Phase-coherence properties:

Elongated geometries

In this chapter we show how the long-range phase correlations of BEC gases can be described by an effective spin-wave (SW) theory in three dimensions. We focus then on their shape dependence, showing how they vary during a dimensional crossover from three to one dimension, that is when the effective dimensionality of the system changes due to the fact that it becomes much longer along one direction with respect to the other two.

3.1 Spin-wave theory

Within the BEC phase, the long-range correlations in homogeneous systems can be described by using the following macroscopic representation of the particle-field operator [40]:

ψ(x) =√n0eiˆθ(x). (3.1)

In writing (3.1) we have assumed that, at long distances, fluctuations of the density are negligible. This is true in almost the whole BEC phase excluding only the relatively small critical region close to Tc. Then, the long-distance

modes of the phase correlations are expected to be described by an effective 3D SW theory for a real phase field θ(x), which is invariant under a global shift θ(x) → θ(x) + ϕ. The simplest SW action reads

Ssw =

Z

ddx α 2(∂µθ)

2 _(3.2)

where α is proportional to the superfluid density (α = (~/m)2ρs/T ). In the

framework of the effective field theories [88], the action (3.2) represents the first non-trivial term of a derivative expansion of the fundamental field θ(x). Contributions of higher-order derivatives are expected to be suppressed in the long-range correlations.

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Within the SW approximation, the one-particle correlation function (1.22) assumes therefore the form

Gsw(y − x) = n0he−iθ(x)eiθ(y)i, (3.3)

where the average is computed with the SW action (3.2).

3.2 Exact SW results

We consider the 3D SW model in a finite box of generic shape L1× L2× L3

and periodic boundary conditions (PBCs). In order to better compare SW predictions with the simulations results, the action (3.2) is rewritten on a corresponding L1× L2× L3 lattice and the partition function looks

Z = X {nµ} Z D[θ] e−α2 P x,µ(θx−θx+ ˆµ−2πnµδxµLµ)2 = X {nµ} W (n1, n2, n3) Z D[θ] e−α2 P x,µ(θx−θx+ ˆµ)2 = X {nµ} W (n1, n2, n3) Z0, (3.4)

where the shifts of 2πnµ (nµ ∈ Z, µ = 1, 2, 3) at the boundaries take into

account winding configurations on lattices with PBC. The weights W (nµ)

are given by ln W = −2π2α L2L3 L1 n2₁+L3L1 L2 n2₂+L1L2 L3 n2₃ , (3.5)

and Z0 is the plain partition function without shifts at the boundaries.

Analogous formulations of the SW theory have been considered to describe the quasi-long-range order of two-dimensional U(1)-symmetric systems [89, 90, 91, 92].

The above formulas allow us to compute the helicity modulus along the three spatial directions (A.27). The helicity modulus is a measure of the response of the system to a phase-twisting field along one of the lattice directions [93]. We obtain Y1 ≡ − L1 L2L3 ∂2log Z(φ) ∂φ2 φ=0 = (3.6) = α − 4π2α2L2L3 L1 P∞ n=−∞n2e −2π2_n2_αL 2L3/L1 P∞ n=−∞e−2π 2_n2_αL 2L3/L1

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The correlation function (3.3) can be written as Gsw(y − x) = n0he−i(θx−θy)i0 (3.7) × P {nµ}W (nµ) cos h 2πP3 µ=1nµ(xµ− yµ)/Lµ i P {nµ}W (nµ) ,

where h.i0 denotes the expectation value in a Gaussian system without

boundary shift (with PBC). Then we use the relation he−i(θx−θy)_i 0 = exp[G0(x, y)] (3.8) where G0(x, y) ≡ h(θx− θy)2i0 (3.9) = 1 αL1L2L3 X p6=0 cos[p · (x − y)] − 1 2P µ(1 − cos pµ)

with pµ = 2π{0, . . . , Lµ− 1}/Lµ. Eqs. (3.7-3.9) allow us to compute the

Gsw(x) for any lattice shape and size. The continuum limit is formally

equivalent to the finite-size scaling (FSS) limit, i.e. Lµ→ ∞ keeping

appro-priate ratios of the sizes fixed.

3.3 Anisotropic finite-size scaling

When we consider large-volume limits keeping the ratios of the sizes Lµ

finite, eq. (3.6), and the analogous equations for the other directions, give equal helicity modulus. We obtain Yµ = α for any µ = 1, 2, 3. However,

nontrivial results are obtained when considering elongated geometries, in particular when one size scales as the product of the sizes of the other two directions, giving rise to an anisotropic FSS (AFSS).

Let us now consider the particular case of anisotropic geometries: we fix L1 = L2 = L and L3 = La. More precisely, we consider the AFSS limit

obtained by the large-L limit at fixed ratio

λ ≡ La/L2. (3.10)

From eq. (3.6) we obtain

Yt= Y1 = Y2= α, (3.11) Ya= Y3= α − 4π2α2 λ P∞ n=−∞n2e−2π 2_n2_α/λ P∞ n=−∞e−2π 2_n2_α/λ . (3.12)

Therefore, the transverse helicity modulus Υt≡ T Ytcan be again identified

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0.0 0.1 0.2 0.3 0.4 0.5 0.6

Y

t

/ λ

0.0 0.5 1.0

R

Y L = 8 T = 1.5 L = 12 L = 16 L = 8 T = 1.75 L = 12 L = 16 SW theory

Figure 3.1: The ratio RY ≡ Ya/Ytfor anisotropic L2×Lalattices with PBC,

versus Yt/λ where λ ≡ La/L2. We show the curve (3.14) obtained by the

3D SW theory (full line), and QMC data for the 3D hard-core BH model, for µ = 0 and two temperature values, i.e. T = 1.5 and T = 1.75 (we use the QMC estimates Yt≈ 0.280 and Yt≈ 0.176 respectively). The error bars

of the QMC data are smaller than the size of the symbols. The QMC data clearly approach the SW AFSS curve with increasing L (the differences get rapidly suppressed, apparently as L−3). [39]

increasing λ, from Ya = α for λ → 0 to Ya → 0 for λ → ∞. Its AFSS can

be written as

RY ≡ Ya/Yt= fΥ(ζ), ζ ≡ Yt/λ, (3.13)

where

fΥ(ζ) = 1 + 2ζ ∂ζln ϑ3(0|i2πζ), (3.14)

and ϑ3(z|τ ) is the third elliptic theta function [94]. In particular, fΥ(0) = 0

and fΥ(∞) = 1. A plot of fΥ(ζ) is shown in fig. 3.1.

In the infinite-axial-size La → ∞ limit, the phase correlation function

Gsw(x) along the axial direction is essentially determined by the first factor

of the r.h.s. of eq. (3.7), since axial boundary terms become irrelevant, as indicated by the vanishing axial helicity modulus. In this limit Gsw(x) turns

out to decay exponentially along the axial direction:

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(apart from a power-law prefactor), where

ξa= 2αL2. (3.16)

This is obtained from the large-distance behavior of the Gaussian correlation function (3.9):

G0(0, 0, z 1) ≈ −

z

2αL2. (3.17)

In order to study the AFSS of the axial coherence length, we consider the axial second-moment correlation length

ξ2_a≡ 1 4 sin2(π/La) e Gsw(0) e Gsw(pa) − 1 ! , (3.18)

where eGsw(p) is the Fourier transform of Gsw(x) and pa = (0, 0, 2π/La).

Although the second-moment correlation length generally differs from that obtained from the large-distance exponential decay, they are expected to be equal for Gaussian-like theories, as supported by the direct computations at fixed L and La → ∞. We numerically compute it using eqs. (3.7-3.9), for

increasing values of L keeping λ = La/L2 fixed, up to values of L where the

results become stable with great accuracy (lattice sizes L & 10 turn out to be sufficient to get a satisfactory O(10−5) accuracy for the large-L limit). The results show that its AFSS can be written as

ξa(λ) ≈ 2 YtL2fξ(ζ) (3.19)

with ζ defined as in eq. (3.13). In particular fξ(0) = 1. The scaling function

fξ is shown in fig. 3.2.

We also report the two-point function Gsw(x, y) in the case of open

boundary conditions (OBCs), where winding effects do not arise, but trans-lation invariance is violated by the boundaries. Assuming that the site coordinates are xµ= [−(Lµ− 1)/2, ..., (Lµ− 1)/2], Gsw(x, y)obc = exp 2Mxy− Mxx− Myy 2α , (3.20) Mxy = X p6=0 2np Q µcos[pµ(xµ+ Lµ/2)] cos[pµ(yµ+ Lµ/2)] 2L1L2L3 Pµ(1 − cos pµ)

with pµ = π{0, . . . , Lµ− 1}/Lµ and np is the number of momentum

com-ponents different from zero. Analogously to the PBC case, for elongated L2× La lattices with La → ∞, the phase correlation function

asymptoti-cally behaves as Gsw(0, 0, z 1)obc ∼ e−z/ξa with ξa given by eq. (3.16).

Finally, assuming the universality of the above asymptotic AFSS for any BEC system, we would like to get information on the size of the corrections when approaching this universal limit. Corrections to the asymptotic scaling

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0.0 0.1 0.2 0.3 0.4 0.5

Y

t

/

λ

0.7 0.8 0.9 1.0

ξ

a

/ (2 Y

t

L

2

)

L = 8 T = 1.5 L = 12 L = 16 L = 8 T = 1.75 L = 12 L = 16 SW theory

Figure 3.2: The ratio ξa/(2YtL2), where ξa is the axial second-moment

cor-relation length defined in eq. (3.18), versus Yt/λ with λ ≡ La/L2. We

show results from the SW theory, and QMC data for the 3D hard-core BH model for µ = 0, on L2× L_a lattices with PBC (we use the QMC estimates Yt≈ 0.280 and Yt≈ 0.176 respectively for T = 1.5 and T = 1.75). The error

bars of the QMC data are smaller than the size of the symbols. The QMC data clearly approach the spin-wave AFSS with increasing L (the differences get rapidly suppressed, apparently as L−3). [39]

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behavior are generally expected in generic BEC systems due to the fact that the corresponding effective SW theories generally require further higher-order derivative terms. Since the SW action (3.2) is quadratic, the power law of their suppression can be inferred by a straightforward dimensional analysis of the couplings of the further irrelevant higher-order derivative terms which are consistent with the global symmetries. Therefore, in the FSS or AFSS limit, their contributions are generally suppressed by O(L−2), due to the fact that the next-to-leading terms has two further derivatives, and therefore the corresponding couplings have a square length dimension. Therefore, we expect that BEC systems rapidly approach the universal AFSS described by the SW quadratic theory. Notice that, in the presence of nontrivial boundaries, O(L−1) corrections may arise from boundary contributions.

3.4 Numerical results

The universality of the AFSS predicted by the SW theory for any BEC system is tested by means of Quantum Monte Carlo (QMC) simulations of the 3D hard-core Bose-Hubbard (BH) model (see appendix A).

We consider the 3D BH model (A.1) in the hard-core U → ∞ limit and at zero chemical potential µ = 0, on elongated L2 × L_a lattices with PBC. We present numerical results, obtained by QMC simulations, for a few values of the temperature below the BEC phase transition occurring at [31] Tc ≈ 2.0160, in particular T = 1.5 and T = 1.75 (which are both smaller

than, but not far from, Tc), and for various values of L (generally up to

L = 16) and La.

3.4.1 Particle density

At µ = 0 the particle density ρ (A.23) of the hard-core BH model is exactly one half, independently of T due to the particle-hole symmetry. Moreover, the on-site density fluctuation is exactly given by hn2_xi − hn_xi2 _{= 1/4. The}

connected density-density correlation function Gn(x, y) (A.24) turns out to

differ significantly from zero only at a distance of one lattice spacing, where it has a negative value, while it is strongly suppressed at larger distances, independently of the lattice size and shape. The corresponding values of the compressibility κ, κ ≡ ∂ρ ∂µ = 1 V X x,y Gn(x, y) = X x Gn(x), (3.21)

are κ = 0.1172(1) and κ = 0.1407(1) respectively at T = 1.5 and T = 1.75. Therefore, as expected, the observables related to the particle density do not show any relevant finite-size dependence within the BEC phase.

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3.4.2 Helicity modulus

In the case of bosonic systems and for homogeneous cubic-like systems, the helicity modulus is related to the superfluid density [93]: Υ(T ) ∝ ρs(T ).

However, for anisotropic L2× La systems we must distinguish two helicity

modulus along the transverse and axial directions. In our QMC with PBC they can be estimated from the linear winding number Wt and Wa along

the trasverse and axial directions respectively, see eq.(A.28). Therefore, we define transverse and axial helicity modulus:

Υt≡ 1 La ∂2F (φt) ∂φ2_t φt=0 = T Yt, (3.22) Υa≡ La L2 ∂2F (φa) ∂φ2 a φa=0 = T Ya, (3.23) Yt≡ 1 La hW_t2i, Ya≡ La L2hW 2 ai, (3.24)

where F = −T ln Z is the total free energy, φtand φaare twist angles along

one of the transverse directions and along the axial direction respectively. Their behaviors appear analogous to those obtained for the 3D SW the-ory in elongated geometries. Indeed, the QMC data at both T = 1.5 and T = 1.75 show that Yt is stable with respect to variations of λ = La/L2,

while Ya significantly decreases when increasing λ. Straightforward large-L

extrapolations of the finite-L data lead to the estimates Yt = 0.280(1) at

T = 1.5 and Yt= 0.176(1) at T = 1.75.

The QMC data for the ratio RY ≡ Ya/Yt behave consistently with the

AFSS behavior (3.13) of the 3D SW model. They are shown in fig. 3.1. With increasing L, the data plotted versus Yt/λ rapidly approach the SW curve

for both temperatures T = 1.5 and T = 1.75, supporting the universality of the SW results. Note that the temperature dependence enters only through the temperature dependence of the transverse helicity modulus Yt.

The convergence of the QMC data to the asymptotic universal curve is apparently characterized by an O(L−3) approach, which is a higher power then the typical O(L−2) corrections expected in SW effective theories, as we have discussed at the end of sec. 3.3. This may be explained by the fact that we are considering observables related to the axial size that scales as L2 in the AFSS limit. Thus, in the derivative expansion of the corresponding SW effective theory, a further derivative with respect to the axial space coordinate formally leads to a further power L−2, instead of L−1.

3.4.3 Phase coherence length

The decay of phase coherence along the axial direction can be quantified by the one-particle correlation function (A.30). The corresponding length scale ξa may be extracted from its exponential decay or its second moment along

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the axial direction. We consider the axial second-moment correlation length ξa, defined in eq. (3.18). Again, its large-L behavior is consistent with the

AFSS obtained within the SW theory, cf. eq. (3.19), as clearly demonstrated by the QMC data shown in fig. 3.2.

Finally, we consider the space dependence of the correlation function along the axial direction, and, in particular, of the axial wall-wall correlation function Gw(z), Gw(z) ≡ 1 L2 X x1,x2 G(x1, x2, z). (3.25)

We expect that its AFSS reads Gw(z) =

χ L2_ξ

a

fw(z/ξa, Yt/λ) (3.26)

where χ is the spatial integral of G

χ =X

x

G(x). (3.27)

In fig. 3.3 we show QMC data in the infinite axial-size limit, i.e. λ → ∞ (obtained by increasing Laat fixed L, up to the point the data become stable

within errors). Again the QMC data nicely agree with the corresponding space dependence of the SW correlation function, obtained using formulas (3.7)-(3.9). In particular, they confirm the large-distance exponential decay with axial correlation length ξa= 2YtL2, in agreement with eqs. (3.16) and

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0 2 4 6 8 10

z /

ξ

_a 10−4 10−3 10−2 10−1

G

w

/ (

χ

/

L

2

ξ

a

)

L = 8 T = 1.5 L = 8 T = 1.75 SW theory

Figure 3.3: The wall-wall phase correlation Gw(z), cf. eq. (3.25), in the

infinite axial-size limit λ → ∞. Again, the QMC data for the hard-core BH model and the computations using the 3D SW theory perfectly agree (actually their differences are hardly visible). Clearly, Gw(z) decays

expo-nentially, as e−z/ξa _{with ξ}

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Chapter 4

Phase-coherence properties:

Slab geometries

In this chapter we consider bosonic particle systems confined within slab geometries, i.e. within boxes of size L2× Z with Z L. We investigate their behavior at the BEC transition temperature Tc and at lower

tempera-tures. Their low-temperature behavior (T < Tc) is further characterized by

the possibility of undergoing a finite-temperature transition to a quasi-long range order (QLRO) phase, with long-range planar correlations which decay algebraically. This is the well-known Berezinskii-Kosterlitz-Thouless (BKT) transition [45, 46, 47, 48, 49], which occurs in 2D statistical systems with a global U(1) symmetry. The interplay of the BEC and BKT critical modes gives rise to a quite complex behavior, which can be described by considering an appropriate dimensional crossover limit.

4.1 Bose-Hubbard phase diagram for a finite

thick-ness Z

For concreteness let us consider bosonic particle systems described by the Bose-Hubbard model (A.1) and confined within boxes of size L2× Z with Z L. The phase diagram of the 3D Bose-Hubbard model sketched in fig. A.1 substantially changes if we consider a quasi-2D thermodynamic limit, i.e. L → ∞ keeping Z fixed. Indeed the length scale ξ (2.3) of the correla-tions of the critical modes remains finite at the BEC transition point when Z is kept fixed. Of course the full 3D critical behavior must be somehow recovered when Z → ∞, for which one expects ξ(Z) ∼ Z. More precisely, defining

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standard FSS arguments (see sec. 2.4) predict that at the 3D critical point Tc

RZ(Tc) = R∗Z+ O(Z−ω) (4.2)

where R∗_Zis a universal constant and ω = 0.785(20) is the scaling-correction exponent associated with the leading irrelevant perturbation at the XY fixed point [79, 80, 81]. Note that the universal constant R∗_Z depends on the boundary conditions along the transverse direction (the boundary conditions along the planar directions are irrelevant since we assume L Z and ξ ∼ Z). However, we should also take into account that 2D or quasi-2D systems with a global U(1) symmetry may undergo a finite-temperature transition described by the BKT theory [46, 47, 48, 49]. The BKT transition separates a high-temperature normal phase and a low-temperature phase characterized by QLRO, where correlations decay algebraically at large distances, without the emergence of a nonvanishing order parameter [95, 96]. When approach-ing the BKT transition point TBKTfrom the high-temperature normal phase,

these systems develop an exponentially divergent correlation length: ξ ∼ e c √ τ_, _{τ ≡} T TBKT − 1, (4.3)

where c is a nonuniversal constant. The susceptibility diverges as χ ∼ ξ7/4, corresponding to the critical exponent η = 1/4.

Consistently with the above picture, 2D BH systems (corresponding to the Hamiltonian (A.1) with Z = 1) undergo a BKT transition. Fig. 4.1 shows a sketch of the phase diagram of 2D BH systems in the hard-core U → ∞ limit. The finite-temperature BKT transition of BH models has been numerically investigated by several studies, see for example refs. [97, 98, 99, 29, 100]. In particular, TBKT = 0.6877(2) in the hard-core U → ∞

limit and for µ = 0 [100]. Note that the 2D BH systems do not show a real BEC below the critical temperature TBKT, but QLRO where the

phase-coherence correlations decay algebraically.

The phase diagram of quasi-2D systems with finite thickness Z > 1 is expected to be analogous to that of 2D BH systems, with a BKT transition at TBKTdepending on the thickness Z. Analogously to 2D systems, they are

expected to show a QLRO phase below TBKT, where correlation functions

show power-law decays along the planar directions, as described by the 2D spin-wave theory.

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Figure 4.1: Sketch of the phase diagram of the 2D BH model in the hard-core U → ∞ limit. The normal and superfluid QLRO phases are sepa-rated by a finite-temperature BKT transition line, which satisfies TBKT(µ) =

TBKT(−µ) due to a particle-hole symmetry. Its maximum occurs at µ = 0,

where [100] TBKT(µ = 0) = 0.6877(2). The superfluid QLRO phase is

re-stricted to a finite region between µ = −4 and µ = 4, which is narrower than that of the 3D phase diagram, see fig. A.1. [59]

4.2 The QLRO phase

The general universal features of the QLRO phase of quasi-2D systems with a U(1) symmetry are described by the Gaussian spin-wave theory

Hsw =

β 2

Z

d2x (∇θ)2. (4.4)

For β ≥ 2/π, corresponding to 0 ≤ η ≤ 1/4, this spin-wave theory describes the QLRO phase. The values β = 2/π and η = 1/4 correspond to the BKT transition [101].

The spin-wave correlation function

Gsw(x1− x2) = he−iθ(x1)eiθ(x2)i (4.5)

is expected to provide the asymptotic large-L behavior of the two-point function of 2D interacting bosonic gases within the QLRO phase. For |x1−

x2| L,

Gsw(x1, x2) ∼

1

|x₁− x₂|η, (4.6)

where the exponent η is related to the coupling β by η = 1

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The general size dependence of Gsw on a square L2 box with PBC is also known: [101, 102, 89, 100] Gsw(x, L) = C(x, L)η × E(x, L), (4.8) C(x, L) = e πy2 2θ0 1(0, e−π) |θ1[π(y1+ iy2), e−π]| , E(x, L) = P∞ n1,n2=−∞W (n1, n2) cos[2π(n1x1+ n2x2)] P∞ n1,n2=−∞W (n1, n2) , W (n1, n2) = exp[−π(n21+ n22)/η],

where x ≡ (x1, x2), yi≡ xi/L, θ1(u, q) and θ10(u, q) are θ functions [103].

Using eq. (4.8), one can easily compute the universal function RL(η),

where RL≡ ξ/L and ξ is the second-moment correlation length defined as

ξ2 = L 2 4π2 χ χ1 − 1 , (4.9) χ = Z d2x Gsw(x), χ1= Z d2x cos 2πx1 L Gsw(x).

Analogous results are obtained for the helicity modulus [89] Y (η) = 1 2πη − P∞ n=−∞n2exp(−πn2/η) η2P∞ n=−∞exp(−πn2/η) . (4.10)

The above asymptotic large-L behaviors (at fixed T or η) are approached with power-law corrections, indeed

RL(L, η) ≡ ξ/L = RL(η) + aL−ε, (4.11)

Y (L, η) = Y (η) + aL−ζ, (4.12)

respectively, where ε and ζ are the exponents associated with the expected leading corrections:[90, 104]

ε = Min[2 − η, κ], ζ = Min[2, κ], (4.13)

κ = 1/η − 4 + O[(1/η − 4)2]. (4.14)

With increasing T within the QLRO phase, the critical exponent η of the two-point function, cf. eq. (4.6), increases up to η = 1/4 corresponding to the BKT transition. Therefore, close to the BKT transition, i.e. for T . TBKT, we may expand the universal curves RL(η) and Y (η) around

η = 1/4, obtaining RL(η) = 0.7506912222 + 1.699451 1 4 − η + . . . , (4.15) Y (η) = 0.6365081782 + 2.551196 1 4 − η + . . . (4.16)

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4.3 Finite-size behavior at the BKT transition

The BKT transition is characterized by logarithmic corrections to the asymp-totic behavior, due to the presence of marginal renormalization-group (RG) perturbations at the BKT fixed point [105, 106, 107, 108, 89, 104].

The asymptotic behaviors at the BKT transition for RL and Y can be

obtained by replacing [89, 104] 1/4 − η ≈ 1 8w, w = ln L Λ + 1 2ln ln L Λ, (4.17)

into eqs. (4.15) and (4.16). The nonuniversal details that characterize the model (such as the thickness Z of the quasi-2D BH models) are encoded in the model-dependent scale Λ. Thus one obtains the asymptotic large-L behavior

R(L, TBKT) = R∗+ CRw−1+ O(w−2). (4.18)

for both R = Y, RL, with

Y∗= 0.6365081789, CY = 0.31889945, (4.19)

R∗_L= 0.7506912222, CRL = 0.21243137, (4.20)

for PBC.

In numerical analyses, eq. (4.18) may be used to locate the BKT tran-sition point, i.e. by requiring that the finite-size dependence of the data matches it. However we note that this straightforward approach is subject to systematic errors which get suppressed only logarithmically with increasing L. This makes the accuracy of the numerical or experimental determination of the critical parameters quite problematic. This problem can be overcome by the so-called matching method [106, 107, 89, 109, 64, 110, 100], which allows us to control the whole pattern of the logarithmic corrections, leaving only power-law corrections.

The matching method exploits the fact that the finite-size behavior of RG invariant quantities R, such as RL and Y , of different models at their BKT

transition shares the same logarithmic corrections apart from a nonuniversal normalization of the scale. Indeed, the L-dependence of two models at their BKT transition is related by the asymptotic relation

R(1)(L1, T_BKT(1) ) ≈ R(2)(L2= λL1, T_BKT(2) ), (4.21)

apart from power-law corrections, which are O(L−2) for the helicity modulus Y and O(L−7/4) for the ratio RL. The matching parameter λ is the only

free parameter, but it does not depend on the particular choice of the RG invariant quantity. The matching method consists in finding the optimal value of T matching the finite-size behavior of Y and RLof the 2D XY model

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whose value of TBKT is known with high accuracy [107, 89]. The complete

expression of RLand Y of the 2D XY model have been numerically obtained

by high-precision numerical studies [89, 109] and by extrapolations using RG results for the asymptotic behavior. For example, the L-dependence of the helicity modulus Y at the BKT transition of the 2D XY model is accurately reconstructed by the following expression [33]

e YXY(L) ≡ YXY(TBKT, L) = (4.22) = 0.6365081782 + 0.318899454 w−1 + 2.0319176 w−2− 40.492461 w−3 + 325.66533 w−4− 874.77113 w−5 + 8.43794 L−2+ 79.1227 L−4− 210.217 L−6, where w is given in eq. (4.17) with Λ = ΛXY = 0.31.

The matching method has been already applied [100] to the 2D BH models, obtaining the accurate estimate TBKT= 0.6877(2) in the hard-core

U → ∞ limit and at half filling (µ = 0).

4.4 Dimensional crossover limit

The scenario outlined in sec. 4.1 can be interpreted as a dimensional crossover from a 3D behavior when T & Tc, and ξ is finite (in particular the anisotropy

of the system is not locally relevant when ξ Z), to an effective 2D critical behavior at T . TBKT(Z) where the planar correlation length ξ diverges.

Such a dimensional crossover can be described by an appropriate trans-verse finite-size scaling (TFSS) limit, defined as δ ≡ 1 − T /Tc → 0 and

Z → ∞, keeping δZ1/ν fixed. In this TFSS limit [57, 58]

RZ ≡ ξ/Z ≈ R(X), X = Z1/νδ, (4.23)

where R(X) is a universal function (apart from a trivial normalization of the argument X), but depending on the boundary conditions along the Z di-rection. Scaling corrections are suppressed as Z−ω, analogously to eq. (4.2). In this TFSS framework the BKT transition appears as an essential singularity of the scaling function R(X):

R(X) ∼ exp b √ XBKT− X for X → X_BKT− , (4.24) where XBKTis the value of the scaling variable X corresponding to the BKT

transition point

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where

δBKT(Z) ≡

Tc− TBKT(Z)

Tc

. (4.26)

The constant b in eq. (4.24) is a nonuniversal constant depending on the normalization of the scaling variable X. R(X) is not defined for X ≥ XBKT.

Note that the above scaling equations predict that [111, 112]

δBKT(Z) ∼ Z−1/ν (4.27)

in the large-Z limit.

The TFSS of the planar two-point function (4.30) is given by

g(x, Z) ≈ Z−(1+η)G(x/Z, X), (4.28) where η = 0.0381(2) is the critical exponent of the 3D XY universality class [81], associated with the power-law decay of the two-point function at Tc. Eq. (4.28) also implies that the planar susceptibility defined as in

eq. (4.31) behaves as

χ ≈ Z1−ηfχ(X). (4.29)

It is important to note that the above features are shared with any quasi-2D statistical system with a global U(1) symmetry, and in particular standard O(2)-symmetric spin models. Numerical analyses of dimensional crossover issues for the XY model are reported in refs. [61, 62, 63, 64].

4.5 Numerical results

In order to check the dimensional crossover scenario discussed in the previous section, we present a numerical study of the equilibrium properties of the BH model (A.1) in the hard-core U → ∞ limit and at zero chemical potential µ = 0, corresponding to half filling, i.e. hnxi = 1/2 for any T . In the

hard-core limit and for µ = 0, the 3D BEC transition occurs at Tc= 2.01599(5)

and the 2D BKT transition at TBKT= 0.6877(2).

We consider slab geometries, i.e. L2 × Z lattices with Z L. We set open boundary conditions (OBC) along the transverse Z-direction; we label the corresponding coordinate as −(Z − 1)/2 ≤ z ≤ (Z − 1)/2, so that the innermost plane is the z = 0 plane. This choice is motivated by the fact that OBC correspond to gas systems trapped by hard walls, such as the experimental systems of refs. [32, 35, 36, 38]. Since the thickness Z of the slab is generally considered as much smaller than the size L of the planar directions, and in most cases we consider the 2D thermodynamic L → ∞ limit keeping Z fixed, the boundary conditions along the planar directions are generally irrelevant for our study around Tc. However, they

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diverges. We consider the most convenient periodic boundary conditions (PBC) along the large planar dimensions; the corresponding site coordinates are x = (x1, x2) with x1,2= 1, ..., L.

We want to understand how the phase diagram and critical behavior change when varying the thickness Z. We present numerical results for some values of Z, in particular Z = 5, 9, 13, various planar sizes up to L ≈ 100, and several values of the temperature T . Tc. The maximum

size Z of our numerical study is limited by the fact that the computational effort of QMC rapidly increases, because they also require larger values of the planar sizes.

We focus on the behavior of the correlation function (A.30) along the planar directions. In particular, for simplicity reasons, we study the corre-lation function between points belonging to the central z = 0 plane, i.e.

g(x1− x2) ≡ G[(x1, 0), (x2, 0)], (4.30)

where we have taken into account the invariance of the system for trans-lations along the ˆ1 and ˆ2 directions. In particular, we consider the planar susceptibility

χ =X

x

g(x) (4.31)

and the planar second-moment correlation length ξ ξ2 = 1

4χ X

x

x2g(x). (4.32)

More precisely, since we consider PBC along the planar directions, we use the equivalent definition

ξ2 ≡ 1 4 sin2(π/L) ˜ g(0) − ˜g(p) ˜ g(p) , (4.33)

where ˜g(p) is the Fourier transform of g(x), and p = (2π/L, 0).

We consider also the helicity modulus (A.27) along the planar directions ˆ 1 and ˆ2 Υa≡ 1 Z ∂2F (φa) ∂φ2 a φa=0 ≡ T Y_a, (4.34)

where F = −T ln Z is the free energy and φa are twist angles along one of

the planar directions. Note that Y1 = Y2 ≡ Y by symmetry for L2 × Z

systems.

Figure 4.2 shows data for the planar second-moment correlation length ξ defined in eq. (4.33), for Z = 5, 9, 13 and T . Tc. We observe that ξ

is small for T > Tc, and apparently L- and Z-independent (for sufficiently

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1.7

1.8

1.9

2.0 T

10

100 ξ

L=40 Z=5 L=60 L=100 L=40 Z=9 L=60 L=100 L=40 Z=13 L=60 L=100

T

c

Figure 4.2: QMC data of the planar correlation length ξ (4.33) for the hard-core U → ∞ BH model at zero chemical potential, for various values of Z, L and T . The dashed vertical line indicates the BEC transition temperature Tc, the dotted vertical lines indicate our estimates of the BKT transition

temperature for Z = 5, 9, 13 (from the left to right). The statistical errors of the data are so small to be hardly visible. [59]

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limit. Around Tc the data of ξ appear to converge to a finite value when

increasing L at fixed Z; however, they show that ξ increases with increasing Z, approximately as ξ ∼ Z. Then, for sufficiently small values of T , the data begin showing a significant dependence on L. At low temperature we observe ξ ∼ L at fixed T , suggesting that ξ diverges with increasing L even when keeping Z fixed. In the following we show that this apparently complicated behavior can be explained by the dimensional crossover scenario put forward in the previous sections.

To begin with, we investigate the nature of the low-temperature behavior where the planar correlation length ξ appears to diverge with increasing L. According to the arguments of the previous sections, at low temperature BH systems for any thickness Z should show a quasi-2D QLRO phase, whose behavior is essentially described by the 2D spin-wave theory. As discussed in sec. 4.2, this implies universal relations among the ratio RL ≡ ξ/L, the

quasi-2D helicity modulus Y and the exponent η characterizing the planar two-point correlation function. In fig. 4.3 we plot data of RL versus those

of Y , comparing them with the universal curve RL(Y ) which can be easily

obtained from the spin-wave results reported in sec. 4.2. This curve ends at the BKT point (Y∗, R∗_L) = (0.6365..., 0.7506...). For sufficiently small T , depending on the value of Z, the data approach the universal spin-wave curve RL(Y ) with increasing L. Extrapolations using the expected

power-law corrections, cf. eqs. (4.11) and (4.12), turn out to be consistent with the exact spin-wave results. Therefore, the numerical results nicely support the existence of a QLRO phase for any Z, with the expected universal spin-wave behaviors.

We also note that above a given temperature, depending on the thickness Z, the data do not approach the spin-wave curve RL(Y ) anymore, as it is

expected to occur for T > TBKT where both RL and Y vanish in the

large-L limit. Therefore, the data of fig. 4.3 allow us to approximately locate TBKT between the temperature values of the data closest to the BKT point

(Y∗, R∗_L) which respectively approach the spin-wave curve and deviate from it. We already note that TBKT increases with increasing Z. This can be

also inferred by the data of the helicity modulus Y versus the temperature, see fig. 4.4. They are generally decreasing, and for sufficiently large T they appear to cross the value Y = Y∗ ≈ 0.6365 corresponding to the BKT transition, indicating that those values of T are larger than TBKT.

More accurate estimates of TBKT can be obtained by looking for the

optimal values of T achieving the matching of the available data of Y and RL

with the finite-size dependence of the 2D XY model at its BKT transition, see sec. 4.3. In particular, TBKT(Z) is given by the value of T providing the

optimal matching of the data of Y (Z, L, T ) with the finite-size dependence of the helicity modulus of the 2D XY model, i.e.

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Y

0.0 0.2 0.4 0.6 0.8 1.0

R

L L=30 L=40 L=50 L=60 L=80 L=100 spin wave BKT

Z = 9

0.0 0.5 1.0 1.5 2.0

Y

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

R

L L=20 L=30 L=40 L=50 L=60 L=80 L=100 spin wave BKT

Z = 5

Figure 4.3: RL ≡ ξ/L versus Y for Z = 5 (bottom) and Z = 9 (top),

and for several values of L and T . The full line shows the spin-wave curve RL(Y ) which is expected to be asymptotically approached for L → ∞ within

the QLRO phase; its end point corresponds to the BKT transition. In particular, for Z = 5 the values of T of the data shown in the bottom figure are (from right to left) T = 1.1858, 1.3518, 1.5179, 1.6, 1.64, 1.65, 1.67, 1.6839, 1.85. The behavior of the data close to the BKT point suggests TBKT(Z = 5) ≈ 1.65. Analogously for Z = 9 the data are for T =1.8, 1.81,

1.82, 1.83, 1.84,1.85, 1.9, 2; they suggest that TBKT(Z = 9) ≈ 1.83. The

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1.88 1.89 1.90 1.91 1.92

T

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Y

L=60 L=70 L=80 L=90 L=100 BKT value

Z=13

1.82 1.83 1.84

T

0.6 0.7 0.8

Y

L=60 L=70 L=80 L=90 L=100 BKT value

Z=9

Figure 4.4: Data of Y for Z = 9 (bottom) and Z = 13 (top) around the corresponding BKT temperatures. Analogous results have been obtained for Z = 5. The dashed horizontal line indicates the BKT value Y∗ = 0.6365.... The dotted vertical lines indicate the interval corresponding to our best estimates of TBKT, i.e. TBKT= 1.829(1) for Z = 9 and TBKT= 1.899(1) for

Scaling properties of three-dimensional Bose-Einstein condensed gases

Scaling properties of three-dimensional

Bose-Einstein condensed gases

Contents

Introduction

Chapter 1

Bose-Einstein condensation

1.1

The Bose-Hubbard model

1.2

One-particle correlation function

1.3

Order parameter and superfluidity

Chapter 2

Critical phenomena

2.1

Continuous phase transitions

2.2

Universal critical behavior

2.3

Renormalization Group theory

2.4

Finite systems

Chapter 3

Phase-coherence properties:

Elongated geometries

3.1

Spin-wave theory

3.2

Exact SW results

3.3

Anisotropic finite-size scaling

Y

/ λ

R

Y

/

λ

ξ

/ (2 Y

L

)

3.4

Numerical results

z /

ξ

G

/ (

χ

/

L

ξ

)

Chapter 4

Phase-coherence properties:

Slab geometries

4.1

Bose-Hubbard phase diagram for a finite

thick-ness Z

4.2

The QLRO phase

4.3

Finite-size behavior at the BKT transition

4.4

Dimensional crossover limit

4.5

Numerical results

1.7

1.8

1.9

2.0

T

10

100

ξ

T

Y

R

Z = 9

Y