Quantum simulator of link models using spinor dipolar ultracold atoms Pierpaolo Fontana SISSA and INFN Sezione di Trieste Via Bonomea 265 I-34136 Trieste Italy
2025-04-29
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Quantum simulator of link models using spinor dipolar ultracold atoms
Pierpaolo Fontana∗
SISSA and INFN, Sezione di Trieste, Via Bonomea 265, I-34136 Trieste, Italy
Joao C. Pinto Barros
Institut f¨ur Theoretische Physik, ETH Z¨urich, Wolfgang-Pauli-Str. 27, 8093 Z¨urich, Switzerland
Andrea Trombettoni
Department of Physics, University of Trieste, Strada Costiera 11, I-34151 Trieste, Italy and
SISSA and INFN, Sezione di Trieste, Via Bonomea 265, I-34136 Trieste, Italy
(Dated: March 29, 2023)
We propose a scheme for the quantum simulation of quantum link models in two-dimensional lat-
tices. Our approach considers spinor dipolar gases on a suitably shaped lattice, where the dynamics
of particles in the different hyperfine levels of the gas takes place in one-dimensional chains coupled
by the dipolar interactions. We show that at least four levels are needed. The present scheme does
not require any particular fine-tuning of the parameters. We perform the derivation of the parame-
ters of the quantum link models by means of two different approaches, a non-perturbative one tied
to angular momentum conservation, and a perturbative one. A comparison with other schemes for
(2 + 1)-dimensional quantum link models present in literature is discussed. Finally, the extension
to three-dimensional lattices is presented, and its subtleties are pointed out.
I. INTRODUCTION
Quantum simulators are of fundamental importance in the realm of quantum and science technologies: they are
quantum systems having properties that can be controlled and used to simulate some target system, whose study
is currently hindered by lack of proper classical computational, experimental or analytical tools [1, 2]. In the last
decades, there has been a formidable development in the fields of quantum optics and atomic physics, allowing for
the realization of highly precise and controllable platforms by means of trapped ions [3], superconducting circuits [4],
Rydberg atoms [5] and ultracold atoms in optical lattices [6]. For these reasons, quantum simulators play a key role
in various areas, including quantum chemistry, condensed matter and high energy physics [7–15]. Various many-body
quantum systems have been analyzed as quantum simulators [16–24], a typical example being provided by atomic
systems loaded in optical lattices, which are described at low energies by (extended) Hubbard models [25].
Over the past years, the idea and use of quantum simulators to study gauge theories has gained increased relevance.
These theories are at the basis of the Standard Model in the field of particle physics, and describe the electroweak
and strong interactions through a non-Abelian gauge theory [26–28]. In condensed matter and statistical physics,
often gauge theories arise as low energies effective descriptions of strongly correlated phenomena, such as quantum
spin liquids, quantum Hall effect and frustrated magnets [29]. The standard approaches to study gauge theories may
present various drawbacks, depending on the regimes and properties of interest. A major example is the analysis
of quantum chromodynamics through Monte Carlo simulations: due to the well-known sign problem, this numerical
method can not reliably approach the analysis of the theory with finite chemical potential, for example [30, 31].
Quantum simulators based on ultracold atomic platforms emerge as a promising alternative to investigate such
phenomena for lattice gauge theories (LGTs), by circumventing some limitations of classical simulators [7–9]. A first
point to be addressed is related to the implementation of the Hilbert space of dynamical gauge fields in a quantum
simulator, since it is infinite-dimensional for a single link in the Wilson formulation of LGTs [32]. To overcome this
difficulty, one could replace the continuum gauge groups with discrete ones that approximate the latter in the proper
limit [33, 34], or replace the link variables with discrete degrees of freedom, discretizing the Hilbert space considering
the so-called quantum link formulation of gauge theories. Even if they possess a finite number of states, quantum
link models (QLMs) preserve the gauge symmetry of the original model, paying the price of introducing non-unitary
operators on the links of the lattice [35–37]. Due to the finiteness of the Hilbert space and the preservation of the
local symmetry, they are suitable to be implemented and analysed as quantum simulators. While it is possible to
recover the full, non-truncated, Wilson formulation from QLMs [38–41], they provide an enriched playground where
new phases are expected to appear, making them interesting also from this perspective [42–47].
∗pfontana@sissa.it
arXiv:2210.14836v2 [cond-mat.quant-gas] 28 Mar 2023
2
With respect to the quantum simulation of usual many-body quantum systems, there are additional features to
be considered in the case of theories with gauge fields. The point is that the designed quantum platform should
be consistent with the local symmetry, i.e. the gauge invariance, of the theory. While in d= 1 efficient ways to
deal with the problem have been developed [8, 16, 20, 22], in d > 1 it seems that one necessarily needs to involve
rather complicated many-body interaction terms to simulate the gauge fields dynamics. We mention that interesting
physical phenomena can emerge in d > 1 even in the absence of magnetic terms [48–52]. As a general consideration,
one would like to have quantum simulation schemes which do not crucially depend on fine-tuning of the parameters
of the systems, possibly not intrinsically perturbative, and extendable to higher dimensions.
For the d > 1 case, both in the Wilson and quantum link formulations, different proposals have been put forward
involving ultracold atoms in optical lattice and Rydberg atoms [11, 12]. Concerning the first platforms, in Ref.
[53] the gauge invariance of the theory is obtained through angular momentum conservation for the gauge-matter
interaction, while the dynamics of the gauge field emerges effectively in perturbation theory, employing the so-called
“loop method” in d= 2 for the compact quantum electrodynamics (QED), realizing the plaquette term in terms of
bosons. At variance, using the dual formulation [54] of the U(1) spin−1/2 model in d= 2, plaquette interactions
are mapped into single constrained hopping processes on the dual lattice. Ref. [55] proposes to simulate this model
through Rydberg configurable arrays, in which the physical states have a blockade character. While in [53] the
plaquette terms are emerging at fourth-order of a strong coupling (cold atomic) expansion, in this proposal they are
implemented directly, without the use of any perturbative expansion. At the same time, the approach of Ref. [55]
relies on the two-dimensional nature of the system, and does not seem to be easily generalizable to higher dimensions.
In this paper we propose a quantum simulator for the U(1) spin−1/2 pure Abelian QLM using spinor dipolar Bose-
Einstein condensates (BECs) loaded in a spin-dependent optical lattice. With respect to Ref. [53], we use only bosonic
atoms of spin-2, so that we have access to five internal states that, through angular momentum conservation in the
various scattering channels, give rise to gauge invariance. As in Ref. [53], the robustness of gauge invariance is tied to
the one of angular momentum conservation, and in the present paper it is used to generate the plaquette term. The
same principle can be achieved without conservation of angular momentum, provided the ultracold atom parameters
are properly tuned in the strong coupling regime. The resulting effective Hamiltonian describes the dynamics of the
gauge field at third-order in perturbation theory for a square lattice, or at second-order for a triangular lattice.
The paper is organized as follows. In Sec. II we give a brief reminder about Abelian LGTs and then introduce
the U(1) spin-1/2 models in d= 2, discussing both the bosonic and fermionic formulations. In Sec. III we briefly
review the theory about spinor dipolar BECs. In Sec. IV we present our proposal using ultracold atomic platforms:
we show how to construct our optical lattice, and derive the plaquette interctions using two approaches, the first one,
non-perturbative (Subsec. IV A), based entirely on angular momentum conservation, and the second one based on a
perturbative expansion (Subsec. IV B). In Subsec. IV C we present the connection with the target gauge theory. In
Sec. V we discuss possible extensions and generalizations of our proposal. In Sec. VI we summarize our results and
present our conclusions.
II. U(1) LATTICE GAUGE THEORIES IN TWO DIMENSIONS
FIG. 1. The gauge fields live on the links of the two-dimensional lattice, and are highlighted in blue in the elementary plaquette.
The plaquette element Uµν is the ordered product of the links, as presented in the right-hand side.
The purpose of this Section is to set the notation regarding the target theory, and provide a brief review review of
the Hamiltonian formulation for U(1) gauge theories. We will focus on the two dimensional case, later generalizing to
higher dimensions. The matter fields live on the vertices of the lattice, here denoted by n= (n1, n2), while the gauge
degrees of freedom live on the links, denoted by the site nand the direction towards which they point ˆµ={ˆ
1,ˆ
2}.
3
The electric field operator acting on the link, connecting the site nto the site n+ ˆµ, is represented by Eµ(n) and
commutes non-trivially with the Wilson operator Uµ(n) on the same link
[Uµ(n), Eν(n0)] = −δµ,ν δn,n0Uµ(n),[U†
µ(n), Eν(n0)] = δµ,ν δn,n0U†
µ(n),(1)
with all remaining commutations set to zero. The Wilson and plaquette operators can be related to the gauge field
Aµand field strength Fµν ≡Aν(n+ ˆµ)−Aν(n)−Aµ(n+ ˆν) + Aµ(n) by Uµ(n) = eieaAµ(n), Uµν (n) = eiea2Fµν (n).
The Kogut-Susskind (KS) Hamiltonian is H=Hg+Hm, where
Hg=e2
2X
n,µ
E2
µ(n)−1
4a2e2X
P
(Uµν +U†
µν ) (2)
is the pure gauge field contribution, while Hmis the matter contribution, and depends on the employed discretization
scheme for the fermions on the lattice. When matter is absent, Eq. (2) represent the KS Hamiltonian of a pure
Abelian U(1) LGTs in 2D [56, 57]. The KS Hamiltonian is gauge invariant, i.e. it commutes with the set of local
operators
G(n) = X
µ
[Eµ(n)−Eµ(n−ˆµ)],[H, G(n)] = 0.(3)
Possible gauge invariant extensions can be added to the Hamiltonian, in the sense that the above symmetries are
preserved. More about this point will be discussed below. In the absence of static charges, which is the case considered
in the present work, the physical states |ψiof the system are the ones satisfying the Gauss’ law G(n)|ψi= 0,∀n.
A. Bosonic quantum link models
Quantum link models [35–37] realize the commutation relations in Eq. (1) using quantum spin operators as
Uµ(n) = S+
µ(n), U†
µ(n) = S−
µ(n), Eµ(n) = Sz
µ(n).(4)
In this framework, the operators Uµand U†
µare no longer unitary nor commuting, but rather satisfy
[Uµ(n), U†
ν(n0)] = 2Eµ(n)δµ,ν δn,n0,(5)
making the Hilbert space at each link finite. This difference gives rise to interesting physical phenomena [43, 45, 46],
while still providing a route to recover the Wilson discretization as one takes the spin representation Sto be large.
In the particular case of S= 1/2 there are only two states per link, associated with the values Eµ(n) = ±1/2 of
the electric field. The Hamiltonian gets simplified because (Sz
µ)2= 1/4: the electric part is trivial and we are left
with magnetic interactions only. The physics described by the Hamiltonian (2) can be enriched by introducing the
Rokhsar-Kivelson (RK) term, with coupling λ, giving rise to the Hamiltonian[58]
HRK =Hg+λX
P
(Uµν +U†
µν )2,(6)
which remains gauge invariant. Among all the possible 24states of the four links joining a vertex n, only six satisfy
the Gauss’ law in 2D. Despite the apparent simplicity of the model, its physics is very rich [7], being closely related
to the quantum dimer model [59].
An alternative way to view this model is provided by mapping spins to hardcore bosons. There + or −signs of
Eµ(n) label, respectively, the presence or absence of an hardcore boson in the link n→n+ ˆµ[47]. In terms of bosons,
the gauge operators are written as
Uµ(n) = b†
µ(n), U†
µ(n) = bµ(n), Eµ(n) = nµ(n)−1
2.(7)
The plaquette term becomes
Uµν (n) = bµ(n)bν(n+ ˆν)b†
µ(n+ ˆν)b†
ν(n) (8)
and can can be interpreted as a correlated hopping of two bosons. As will be discussed below, we will interpret the
above terms as a particle at the link (µ, n) hopping to the link (ν, n) and one at (ν, n+ ˆnu) hopping to (µ, n+ ˆnu).
4
The RK term can be written in this language as a sum of two-, three- and four-particles interactions. While this is
simple to write, it does not arise as easily in an ultracold atomic setting. For this reason, we will focus only on the
generation of the plaquette term in the present work. In this language, the generators take the form
G(n) = X
µ
[nµ(n)−nµ(n−ˆµ)].(9)
and commute with the Hamiltonian by construction.
B. Fermionic quantum link models
The particle representation opens the door to the construction of an alternative gauge theory: a gauge theory
constructed with fermionic links [44, 47], which is achieved by replacing the bosonic operators by fermionic ones. This
is still gauge theory (there is still a set of local symmetries) but possibly hosting different physics due to the different
commutation relations among the gauge field operators Uµ,U†
µand Eµ. It turns out that for 2D the theories are
equivalent, while for 3D they represent truly different models [44, 47].
For concreteness, in the fermionic case, we can choose as a basis for the two-dimensional Hilbert space, the states
|0iand |1i=c†
µ(n)|0i, and identify the Wilson and electric field through
Uµ(n) = c†
µ(n), U†
µ(n) = cµ(n), Eµ(n) = nµ(n)−1
2,(10)
where nµ(n)≡c†
µ(n)cµ(n) is the number operator. It is straightforward to verify that Eq.s (1) , (5) are satisfied with
these definitions. As anticipated, the Wilson operators anticommute.
Fermionic QLMs have been subjected to much less intense research when compared to their bosonic counterparts.
Their analysis can lead, in principle, to the characterization of new phases of matter for LGTs. At the same time,
quantum simulators of 2D LGTs with ultracold atoms may profit from the fermionic interpretation of the plaquette
interactions, as they provide an alternative equivalent way of realizing the same physics.
III. SPINOR DIPOLAR BOSE-EINSTEIN CONDENSATES
Before addressing the details of our proposal, we give a brief reminder about spinor dipolar BECs, which are the
basic tool that we need to build up the quantum simulator. Spinor BECs are degenerate Bose gases with spin internal
degrees of freedom. With respect to usual (scalar) BECs, they present multicomponent order parameters and display
richer physical phenomena, due to the interplay between superfluidity and magnetic effects. As a consequence, they
provide a useful platform for the study of different physical aspects, such as the role of symmetry breaking and
long-range order in quantum-ordered materials, quantum phase transitions and non-equilibrium quantum dynamics
[25, 60, 61].
The general atomic Hamiltonian of spinor BECs can be written on the basis of symmetry arguments, and, apart from
the usual single-particle terms, it includes quantum number dependent interaction terms. For a spin-fBEC we denote
with φm(r) the bosonic field operators, satisfying the canonical commutation relations [φm(r), φ†
m0(r0)] = δm,m0δr,r0,
where m=−f, −f+ 1, . . . , f is the magnetic quantum number and fis the hyperfine spin of the given atomic species.
The microscopic Hamiltonian is
H=H0+H(f)
int , H0=ZdrX
m
φ†
m(r)−~2∇2
2M+Utrap(r)φm(r),(11)
H(f)
int =1
2ZdrX
m1,m2,m0
1,m0
2
Cm1m2
m0
1m0
2φ†
m1(r)φ†
m2(r)φm0
1(r)φm0
2(r).(12)
The single-particle term, H0, includes the possibility of having a trapping potential Utrap(r). H(f)
int is the most general
on-site interaction term for hyperfine spin f. For our purposes, it is enough to consider the f= 2 case
H(2)
int =1
2Zdr[c0:n2(r):+c1:F2(r):+c2A†
00(r)A00(r)],(13)
5
where : O: represents the normal order for the operator O,c0, c1and c2are numerical coefficients related to the
scattering lengths aFin the various channels, and
n(r) =
2
X
m=−2
φ†
mφm, A00(r) = 2φ2φ−2−2φ1φ−1+φ2
0
√5, Fi(r) =
2
X
m,m0=−2
φ†
m(fi)mm0φm0.(14)
The dependence on r, on the right hand side, was ommitted for simplicity. The above defined quantities are n(r)
the total density operator, A00 the amplitude of the spin singlet pair, and Fithe spin density operators, with fi
representing the spin-2 rotation matrices. Without further interactions, the spinor BECs in spin-independent optical
lattices can be described by the Bose-Hubbard (BH) model [62, 63]. Expanding the field operators in terms of Wannier
functions, and introducing the associated annihilation and creation operators bim, b†
im, the BH Hamiltonian can be
written as
HBH =−tX
hi,ji,m
(b†
imbjm+ h.c.) + U0
2X
i
ni(ni−1) + U1X
i
(A†
00)i(A00)i+U2
2X
i
F2
i−µX
i
ni,(15)
with ni=Pmb†
imbim,Fiα=Pm,m0b†
im(fα)mm0bim0and A00 is the spin singlet amplitude written in terms of bim, b†
im.
The single site interactions are not enough to generate the desired plaquette terms within our proposal. However, this
can be accomplished by including magnetic dipole-dipole interaction (MDDI) terms, and considering spinor dipolar
BECs. The MDDI couples the spin degrees of freedom with the orbital ones, conserving the total angular momentum.
For spinor BECs, the MDDI can be relevant, as it is spin dependent and long-ranged. Its Hamiltonian in second
quantization is given by
Vdd =cdd
2Zdrdr0X
ν,ν0
:Fν(r)Qνν0(r−r0)Fν0(r0) :, Qνν0(r) = δνν0−3ˆrνˆrν0
r3,(16)
with the coefficient cdd ∝d2related to the electric dipole moment. In the optical lattice Hamiltonian this generates
a series of long-range terms
Hdd =1
2X
i6=j
Uij
ddninj, Uij
dd ≡cdd Zdrdr0|w(r−ri)|21−3 cos2θ
|r−r0|3|w(r0−rj)|2,(17)
where θis the angle between the dipole moment and the vector r−ri. The full Hamiltonian HEBH =HBH +Hdd
falls in the class of the so-called extended Bose-Hubbard models. Depending on the values of t, U0, U1and Udd, that
can be tuned independently, the extended BH has different quantum transitions and phases, including Mott insulator,
superfluid and even supersolid phases, provided that more than nearest neighbors interaction terms are considered in
the extended Hamiltonian [64–66].
In principle, the dipolar interaction is dominant in gases of polar molecules when the application of a strong electric
field is considered, due to their strong electric dipole moments. In this case, these are called spin-polarized dipolar
BECs. The dipole-dipole interaction can be properly tuned through a rotating field [67], allowing for the control of
the interaction strength cdd, that can be positive or negative according to the relative orientation of the dipoles. On
the other hand, the MDDI can be neglected in several ultracold atomic systems, such as scalar alkali atoms, while
they play an important role for other species, e.g. Cr and Dy [68, 69]. We refer to the reviews [60, 61], and the
references therein, for more details on the various physical properties of spinor (dipolar) BECs.
IV. PLAQUETTE TERMS FROM ANGULAR MOMENTUM CONSERVATION
In this Section we describe how the plaquette interactions in the 2D Abelian spin-1/2 QLMs can be interpreted as
a correlated hopping obtained through angular momentum conservation. The use of angular momentum conservation
in scattering processes to ensure local gauge invariance was introduced, for the first time, in Ref. [53]. In that case,
it guarantees that the gauge-matter interaction satisfies gauge invariance. By other side, plaquette terms are still
obtained perturbatively. In contrast, our target model does not include matter and uses the conservation of angular
momentum as a mean to obtain robust plaquette terms of the pure gauge theory.
In our proposal, we consider a spin-2 dipolar BEC loaded in a square optical lattice, whose structure is showed
in the left panel of Fig. 2. This figure has to be read as follows: the bosons are located on the different vertices of
the lattice, and the lattice itself has a spin-dependent structure, so an atom can sit at a generic site nif it has the
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QuantumsimulatoroflinkmodelsusingspinordipolarultracoldatomsPierpaoloFontanaSISSAandINFN,SezionediTrieste,ViaBonomea265,I-34136Trieste,ItalyJoaoC.PintoBarrosInstitutfurTheoretischePhysik,ETHZurich,Wolfgang-Pauli-Str.27,8093Zurich,SwitzerlandAndreaTrombettoniDepartmentofPhysics,UniversityofTriest...
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