Scale Invariance in the Lowest Landau Level

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Comptes Rendus

Physique

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Gold Medal Jean Dalibard / Médaille d’or Jean Dalibard

Scale Invariance in the Lowest Landau Level

Invariance d’échelle dans le niveau de Landau

fondamental

Johannes Hofmann∗,aand Wilhelm Zwerger∗,b

aDepartment of Physics, Gothenburg University, 41296 Gothenburg, Sweden

bTechnische Universität München, Physik Department, James-Franck-Strasse, 85748

Garching, Germany

E-mails: johannes.hofmann@physics.gu.se (J. Hofmann), zwerger@tum.de

(W. Zwerger)

Abstract. We show that the discrete set of pair amplitudes Amintroduced by Haldane are an angular-

momentum resolved generalization of the Tan two-body contact, which parametrizes universal short-range

correlations in atomic quantum gases. The pair amplitudes provide a complete description of translation-

invariant and rotation-invariant states in the lowest Landau level (LLL), both compressible and incompress-

ible. To leading nontrivial order beyond the non-interacting high-temperature limit, they are determined an-

alytically in terms of the Haldane pseudopotential parameters Vm, which provides a qualitative description

of the crossover towards incompressible ground states for different ﬁlling factors. Moreover, we show that for

contact interactions ∼g2δ(2)(x), which are scale invariant at the classical level, the non-commutativity of the

guiding center coordinates gives rise to a quantum anomaly in the commutator i[b

HLLL,b

DR]=(2+ℓ∂ℓ)b

HLLL

with the dilatation operator b

DRin the LLL, which replaces the trace anomaly in the absence of a magnetic

ﬁeld. The interaction-induced breaking of scale invariance gives rise to a ﬁnite frequency shift of the breath-

ing mode in a harmonic trap, which describes transitions between different Landau levels, the strength of

which is estimated in terms of the relevant dimensionless coupling constant e

g2.

Résumé. Nous montrons que l’ensemble discret des amplitudes de paires Amintroduit par Haldane est une

généralisation résolue en moment cinétique du coefﬁcient de contact à deux corps de Tan, qui paramétrise

les corrélations universelles à courte distance dans les gaz quantiques atomiques. Les amplitudes de paires

fournissent une description complète des états invariants par translation et par rotation dans le niveau de

Landau fondamental (LLL), qu’ils soient compressibles ou incompressibles. Au premier ordre non nul au-

delà de la limite de haute température sans interaction, elles sont déterminées analytiquement en fonction

des paramètres Vmdu pseudopotentiel de Haldane, ce qui fournit une description qualitative du passage

vers des états fondamentaux incompressibles pour différents taux de remplissage. De plus, nous montrons

que pour les interactions de contact g2δ(2)(x), qui sont invariantes d’échelle au niveau classique, la non-

commutation des coordonnées du centre de giration donne naissance à une anomalie quantique dans le

commutateur i[b

HLLL,b

DR]=(2+ℓ∂ℓ)b

HLLL de l’hamiltonien avec le générateur des dilatations b

DRdans le LLL,

qui remplace l’anomalie de Weyl sur la trace en l’absence de champ magnétique. La brisure de l’invariance

d’échelle induite par l’interaction conduit à un déplacement de fréquence du mode de respiration dans un

∗Corresponding authors.

ISSN (electronic) : 1878-1535 https://comptes-rendus.academie-sciences.fr/physique/

arXiv:2210.10086v3 [cond-mat.quant-gas] 16 Jun 2023

2Johannes Hofmann and Wilhelm Zwerger

piège harmonique, qui reﬂète des transitions entre différents niveaux de Landau et dont nous estimons la

valeur en termes de la constante de couplage sans dimension pertinente e

g2.

Keywords. Bose–Einstein condensates under rapid rotation, Lowest Landau Level, Virial Expansion, Scale

invariance and quantum scale anomaly, Universal/exact (Tan) relations.

Mots-clés. Condensats de Bose–Einstien en rotation rapide, niveau de Landau fondamental, développement

du viriel, invariance d’échelle et anomalie d’échelle quantique, relations universelles/exactes de Tan.

Funding. This work is supported by Vetenskapsrådet (grant number 2020-04239).

Manuscript received 16th October 2022, revised 13th December 2022, accepted 15th December 2022.

1. Introduction

The observation by Tsui, Störmer and Gossard [1] of a Hall conductance σHthat is quantized

at fractional values of the fundamental unit σ0=e2/hin a high-mobility two-dimensional (2D)

electron gas subject to a strong magnetic ﬁeld Bhas launched an immense amount of theoretical

work dealing with interacting fermions in the lowest Landau level (LLL). In physical terms,

the LLL is reached in the limit ℓB/aeff

0→0 where the magnetic length ℓB=pħ/eB is much

smaller than the effective Bohr radius aeff

0= ħ2/m∗e2, where eis the electron charge and m∗

its effective mass. Based on Laughlin’s realization that for dominant short-range repulsion the

ground state at ﬁlling factors ν=1/3,1/5.. . (or their particle-hole conjugates at ﬁlling 1−ν) forms

an incompressible ﬂuid with a ﬁnite excitation gap of order ∆≃e2/ℓB[2], an understanding

of the quantization of σHthat does not depend on any microscopic details is provided by an

effective Chern-Simons ﬁeld theory [3,4]. This description, which is based only on the underlying

symmetries, has been extended by Son and coworkers [5–7], which allows to properly account for

effects like the Hall viscosity [5] or the important issue of particle-hole symmetry at half ﬁlling

ν=1/2 [7].

While completely general, the effective-ﬁeld-theory description of interacting particles in the

LLL is restricted to incompressible ground states and it is a priori not clear whether further

universal results can be derived beyond this special class of topologically ordered states. It is the

aim of our present work to provide some steps in this direction, mostly restricting ourselves to

the case of bosons in the context of ultracold quantum gases. In the special limit of vanishing

interactions in the LLL, this problem can be mapped to a Gaussian ﬁeld ψ(x)=Pmamφm(x)

whose expansion coefﬁcients amdeﬁne a random polynomial. Its roots determine the location

of vortices, which exhibit an antibunching property [8]. Here, we are instead concerned with the

problem in the presence of interactions, which for Bose gases in 2D are usually described by

an effective pseudopotential V(x)=g2(Λ)δ(x) [9]. Since a delta function interaction does not give

rise to a proper two-body scattering problem in two dimensions, the relevant strength g2(Λ) must

be regularized by a logarithmic running with a cutoff Λ. As discussed below, for states within the

LLL, this renormalization turns out to be absent and g2(Λ)→(ħ2/m∗)e

g2can be replaced by a

dimensionless constant e

g2=p8πa/ℓz, which is determined by the 3D scattering length aand

the transverse conﬁnement length ℓz[9].

An effective magnetic ﬁeld for neutral particles may be induced by rotating the gas with a ﬁnite

angular frequency Ω[9]. In the stationary rotating frame, the Hamiltonian b

Hof the nonrotating

system is changed to

H→b

H−Ωb

Lz, (1)

where b

Lzis the angular momentum operator. This gives rise to a uniform effective magnetic

ﬁeld (eB)eff =2m∗Ωdirected along the rotation axis z, reﬂecting the mathematical similarity

between the Coriolis and Lorentz force (here, we keep m∗as the atomic mass to avoid confusion

C. R. Physique — Draft, 19th June 2023

Johannes Hofmann and Wilhelm Zwerger 3

with the angular momentum quantum number mintroduced below). Note that, in contrast to

the electronic problem, the effective magnetic length ℓB=pħ/(eB )eff =pħ/2m∗Ωnow scales

inversely with the mass m∗, while the effective cyclotron frequency ωc=2Ωis independent of

it. In the presence of an additional harmonic conﬁnement with frequency ω, the single-particle

levels in the x,y-plane are then of the form [9]

E(n)

j=ħ(ω−Ω)·j+ħ(ω+Ω)·n, (2)

where we drop a constant zero-point energy ħω. In the limit Ω→ω−and for given n=0,1, . ..,

these energies group into a series of degenerate levels labelled by their angular momentum

m=j−n= −n,−n+1.. ., which constitute an analog of the nth Landau level. In particular, the

LLL corresponds to n=0 with a degenerate set of single-particle levels labelled by the angular

momentum quantum number m=j≥0. In the following, we use as a characteristic length scale

the harmonic oscillator length

ℓ=sħ

m∗ω, (3)

which differs by a factor of p2 from the effective magnetic length ℓBin the LLL limit. Quite

generally, the restriction to the LLL requires that |Vm|/ħω→0 in order to suppress transitions

between different Landau levels, where |Vm|is the characteristic magnitude of the Haldane

pseudopotentials formally introduced in Eq. (8) below. For bosons, the dominant interaction is

V0= ħω·e

g2/2π[10], which arises from the standard zero-range pseudopotential with scattering

length adiscussed above. Exceptions arise, however, if the short-range scattering length is tuned

to zero or in the presence of dipolar interactions, where—in a minimal model—both V0and V2

need to included [11]. Note that since the dimensionless parameter e

g2parametrizes an s-wave

interaction, it takes the same form for any rotation frequency and is thus independent of the

magnetic length ℓB, cf. App. A. The condition V0/ħω∼e

g2≪1 for staying in the LLL corresponds

to the perturbative limit with respect to the harmonic oscillator spacing, for which the running

of the interaction strength due to the renormalization of the coupling is negligible. Since e

g2≲0.1

generically, the LLL condition is well obeyed in practice1.

For bosons, the ﬁlling factor ν=n2(πℓ2) in the LLL (n2is the 2D particle density) may take

arbitrary large values. In particular, for a gas with Nparticles in a rotating harmonic trap [9],

ν≃¡N(1 −Ω/ω)/ e

g2¢1/2 (4)

scales like pNat a ﬁxed value of 1 −Ω/ω≪1. In the regime of large ﬁlling factors ν≫1, which

is the one that has been explored mostly so far, bosons can be described in terms of a coherent

state picture. In particular, the ground state is a regular array of vortices, which is predicted to

melt into a strongly correlated vortex liquid at around ν≃10 [12] (for a review, see Ref. [10]). From

Eq. (4), the condition for reaching ﬁlling factors of order unity requires 1 −Ω/ω≲e

g2/N, which

places the gas very close to an unstable conﬁguration where the harmonic trap can no longer

overcome the centrifugal force. As a result, a number of alternative routes have been proposed

to realize Bose gases in the LLL in a regime where mean-ﬁeld theory no longer applies, such as

optical ﬂux lattices [13,14] or periodically driven systems, where a nonvanishing gauge ﬁeld arises

in the effective Floquet Hamiltonian [15, 16]. This idea has been implemented successfully in a

recent experiment by Léonard et al [17], where a strongly correlated state with ν=1/2 has been

observed with N=2 bosons in an optical lattice. Earlier, the physics of strongly interacting bosons

in the LLL has also been realized with photons in a twisted optical cavity, where the repulsion

results from an effective interaction mediated by Rydberg atoms [18]. Moreover, a quite different

1Note that the relevant dimensionless interaction strength within the LLL is set by V0/ħ(ω−Ω) and thus becomes

strong for Ω→ω−.

C. R. Physique — Draft, 19th June 2023

4Johannes Hofmann and Wilhelm Zwerger

approach to study interacting bosons in the LLL has recently been explored with 23Na atoms that

are geometrically squeezed into the LLL using a rotating saddle trap [19,20].

This paper is structured as follows: In Sec. 2, we show that the pair amplitudes Amorigi-

nally introduced by Haldane [21] via the number of particle pairs with relative angular momen-

tum mmay be seen as a generalization of the Tan contact parameter for non-rotating quan-

tum gases with zero-range interactions. The pair amplitudes fully describe ﬂuid states within

the LLL at arbitrary temperature and ﬁlling. An explicit result for the Amas a function of the Hal-

dane pseudopotential parameters Vmis derived within a virial expansion. It turns out that, even

at leading order, this allows to describe in qualitative terms the smooth crossover towards in-

compressible ground states, with a non-monotonic behavior of the compressibility as a function

of temperature. Moreover, data for the pair distribution function obtained in the recent exper-

iment by Léonard et al. [17] allows to extract the two lowest Haldane pair amplitudes A0and

A2in the strongly correlated state at ν=1/2. In Sec. 3, we consider the fate of scale invari-

ance and its violation by quantum ﬂuctuations of 2D Bose gases in the presence of rotation. It

is shown that the interaction-induced quantum scale anomaly of the nonrotating system is sup-

pressed in the LLL limit due to the absence of a running coupling constant. Nevertheless, the fre-

quency of the breathing mode, which here describes transitions between different Landau lev-

els, is shifted away from the symmetry-dictated value. In addition, we show that despite the ab-

sence of a running coupling constant, a quantum anomaly is still present in the LLL due to the

non-commutative nature of the guiding center coordinates. The paper ends with a conclusion in

Sec. 4 and contains two appendices that summarize the two-body problem in a rotating trap and

provide details on the virial expansion in the LLL, respectively.

2. Haldane pair amplitudes and Tan-like relations in the LLL

In order to describe interactions in the LLL, we start from the general interaction Hamiltonian

Hint =1

2Zdz1Zdz2V(z1−z2)b

ψ†(z2)b

ψ†(z1)b

ψ(z1)b

ψ(z2) (5)

with a translation invariant two-body interaction V(z) in the absence of rotation. Here, z=x+i y

is a complex coordinate and b

ψ†(z) is the creation operator for a particle at position z. To

incorporate the ﬁnite rotation Ω→ω−, where the kinetic energy is completely quenched, as

well as the restriction to the LLL, it is convenient to consider a disk geometry in the circular

gauge. Expanding the ﬁeld operator b

ψLLL(z)=Pmφm(z)b

amrestricted to the LLL in terms of the

associated single-particle eigenstates φm(z)=(z/ℓ)me−¯

zz/2ℓ2/pπℓ2m!, the projected interaction

Hamiltonian is given by

HLLL

int =X

′VmX

ξ†

mM b

ξmM =X

′Vmb

Pm. (6)

Here, the projection operator b

Pm=PMb

ξ†

mM b

ξmM is deﬁned in terms of the operator

ξmM =1

p2X

m1,m2〈mM |m1,m2〉b

am1b

am2, (7)

which annihilates a two-particle state |mM〉described by the center-of-mass quantum number

Mand a relative angular momentum m. The prime in Eq. (6) restricts the m-summation to odd

or even non-negative integers in the case of fermions or bosons, respectively. In the presence of

both translation and rotation invariance, there is no dependence on the center-of-mass quantum

number M, and the two-body interaction V(z) within the LLL reduces to a discrete set of Haldane

pseudopotential parameters [22]

Vm=mM ¯¯b

V¯¯mM®−−−−→

m≫1V³z=ℓp2m´. (8)

C. R. Physique — Draft, 19th June 2023

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ComptesRendusPhysiqueDraftGoldMedalJeanDalibard/Médailled’orJeanDalibardScaleInvarianceintheLowestLandauLevelInvarianced’échelledansleniveaudeLandaufondamentalJohannesHofmann∗,aandWilhelmZwerger∗,baDepartmentofPhysics,GothenburgUniversity,41296Gothenburg,SwedenbTechnischeUniversitätMünchen,PhysikDep...

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